Multi-Dimensional Processes In Stellar Physics: Evry Schatzman School 2018 9782759824373

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Mul-Dimensional Processes In Stellar Physics Evry Schatzman School 2018

Michel Rieutord, Isabelle Baraffe and Yveline Lebreton, Eds

Printed in France ISBN(print): 978-2-7598-2416-8 – ISBN(ebook): 978-2-7598-2437-3 All rights relative to translation, adaptation and reproduction by any means whatsoever are reserved, worldwide. In accordance with the terms of paragraphs 2 and 3 of Article 41 of the French Act dated March 11, 1957, “copies or reproductions reserved strictly for private use and not intended for collective use” and, on the other hand, analyses and short quotations for example or illustrative purposes, are allowed. Otherwise, “any representation or reproduction – whether in full or in part – without the consent of the author or of his successors or assigns, is unlawful” (Article 40, paragraph 1). Any representation or reproduction, by any means whatsoever, will therefore be deemed an infringement of copyright punishable under Articles 425 and following of the French Penal Code. © EDP Sciences, 2020

Multi-Dimensional Processes In Stellar Physics

Sponsors Centre National de la Recherche Scientifique (CNRS, Formation Permanente) Programme National de Physique Stellaire (PNPS) of CNRS/INSU (Institut National des Sciences de l’Univers du CNRS) Centre National d’Etudes Spatiales (CNES) Centre de Recherche Astrophysique de Lyon (CRAL) Agence Nationale de la Recherche (Projet ESRR - grant ANR-16-CE31-0007-01)

Scientific Organizing Committee Isabelle Baraffe - co-Chair (University of Exeter) Boris Dintrans (Institut de Recherche en Astrophysique et Planétologie, IRAP-UMR CNRS 5277, Université Toulouse III) Yveline Lebreton (LESIA-UMR CNRS 8109, Observatoire de Paris et Institut de Physique de l’Université de Rennes 1) Julien Morin (LUPM-UMR CNRS 5299, Université de Montpellier) Michel Rieutord - co-Chair (Institut de Recherche en Astrophysique et Planétologie, IRAPUMR CNRS 5277, Université Toulouse III)

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Evry Schatzman School 2018

A few participants closing the school and waiting for the bus. From left to right: Jérôme Guilet, Sutirtha Sengupta, François Lignières, Pascale Garaud, Laurène Jouve, Bertrand Putigny, Julien Morin, Bastien Gouhier et Damien Gagnier.

List of Participants - Amard Louis, [email protected], University of Exeter - Baraffe Isabelle, i.baraff[email protected], University of Exeter et CRAL, ENS Lyon - Belkacem Kevin, [email protected], LESIA - Observatoire de Paris - Bouchaud Kevin, [email protected], obs. de Nice, obs. de Paris Meudon - Breimann Angela, [email protected], University of Exeter - Browning Matthew, [email protected], University of Exeter - Brun Allan Sacha, [email protected], DAp-AIM, CEA Paris-Saclay - Buldgen Gaël, [email protected] Observatoire de Genève, Université de Genève - Chabrier Gilles, [email protected], CRAL, ENS Lyon - Charlet Arthur, [email protected], ENS Lyon - Christophe Steven, [email protected], LESIA, Observatoire de Paris - Currie Laura, [email protected], University of Exeter - Debras Florian, fl[email protected], CRAL, ENS Lyon - Dintrans Boris, [email protected], CINES, Montpellier - Dumont Thibaut, [email protected], Observatoire de Genève / LUPM - Farnir Martin, [email protected], Université de Liège - Finley Adam, [email protected], University of Exeter - Gagnier Damien, [email protected], IRAP, Toulouse - Garaud Pascale, [email protected], University of California in Santa-Cruz - Gouhier Bastien, [email protected], IRAP, Toulouse - Guilet Jérôme, [email protected], CEA Saclay, Département d’astrophysique - Jouve Laurène, [email protected], IRAP, Toulouse - Khan Saniya, [email protected], University of Birmingham - Kupka Friedrich, [email protected], University of Göttingen - Lebreton Yveline, [email protected], LESIA, Observatoire de Paris - Lebreuilly Ugo, [email protected], ENS Lyon - Lignières François, [email protected], IRAP, Toulouse - Manchon Louis, [email protected], Université Paris-Saclay - Menu Melissa, [email protected], Université Paris-Saclay - Mirouh Giovanni, [email protected], SISSA Trieste - Morin Julien, [email protected], LUPM, Montpellier - Padioleau Thomas, [email protected], CEA Saclay - Petitdemange Ludovic, [email protected], LRA, ENS Paris - Philidet Jordan, [email protected], Observatoire de Paris - Pinçon Charly, [email protected], IAS, Université Paris XI

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Evry Schatzman School 2018

- Putigny Bertrand [email protected], IRAP, Toulouse - Raynaud Raphaël, [email protected], CEA Saclay - Reese Daniel, [email protected], LESIA, Observatoire de Paris - Rieutord Michel, [email protected], IRAP, Toulouse - Samadi Réza, [email protected], LESIA, Observatoire de Paris - Sengupta Sutirtha, [email protected], University of California in Santa-Cruz - Tremblin Pascal, [email protected], Maison de la Simulation, CEA ParisSaclay

T able of contents

List of Participants

5

Preface

11

Double-diffusive processes in stellar astrophysics Pascale Garaud

13

1 Introduction ........................................................................................................................ 2 Linear instability of double-diffusive systems ...........................................................

13 15

3 Mixing by fingering convection .....................................................................................

25

4 Oscillatory double-diffusive convection and layered convection ..........................

38

5 What next? ..........................................................................................................................

50

References ..............................................................................................................................

57

Thermo-compositional adiabatic and diabatic convection Pascal Tremblin

61

1 Introduction ........................................................................................................................ 2 Linear stability analysis of adiabatic and diabatic convection ..............................

61 61

3 Non-linear regime and temperature-gradient reduction ......................................... References ..............................................................................................................................

62 63

2.1 Model setup ............................................................................................................................... 2.2 Linear stability properties of homogeneous double-diffusive systems ..................... 2.3 Where is fingering convection likely to occur in stars? ................................................. 2.4 Where is ODDC/semiconvection likely to occur is stars? .............................................. 2.5 Beyond linear theory ............................................................................................................... 3.1 Traditional models of "thermohaline" mixing ................................................................... 3.2 Numerical simulations of small-scale fingering convection ............................................ 3.3 The Brown et al. 2013 model for small-scale fingering convection ........................... 3.4 Large-scale instabilities? ........................................................................................................ 3.5 Conclusions for now ................................................................................................................ 3.6 Applications to stellar astrophysics .................................................................................... 4.1 Traditional models of mixing by ODDC .............................................................................. 4.2 Numerical simulations of ODDC ........................................................................................... 4.3 Transport in non-layered ODDC ........................................................................................... 4.4 Criterion for layer formation .................................................................................................. 4.5 Mixing in layered convection ................................................................................................ 4.6 Conclusions for now ................................................................................................................ 4.7 Applications to stellar astrophysics .................................................................................... 5.1 Model limitations ...................................................................................................................... 5.2 More realism ..............................................................................................................................

2.1 Adiabatic case ........................................................................................................................... 2.2 Diabatic case .............................................................................................................................

15 17 22 22 24 25 26 29 31 35 36

38 41 42 44 47 48 49

51 52

61 62

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Fully compressible time implicit hydrodynamics simulations for stellar interiors Isabelle Baraffe

65

1 The MUSIC code ................................................................................................................ 2 An example of application: convective penetration ................................................. References ..............................................................................................................................

65 66 67

Thermal Convection in Stars and in Their Atmosphere Friedrich Kupka

69

1 Introduction ........................................................................................................................ 2 Convection in astrophysics and the basic physics of convection .........................

69 70

3 Modelling of convection ..................................................................................................

77

4 Challenges and pitfalls in numerical modelling ........................................................

88

5 Applications: overshooting .............................................................................................

96

2.1 The physics of convection ...................................................................................................... 2.2 Examples from astrophysics and geophysics ................................................................... 2.3 Astrophysical implications ..................................................................................................... 2.4 Local stability analysis for the case of convection .......................................................... 2.5 Conservation laws .................................................................................................................... 2.6 The challenge of scales ........................................................................................................... 3.1 MLT and phenomenological models ................................................................................... 3.2 Ensemble and volume averages ........................................................................................... 3.3 Reynolds stress approach ...................................................................................................... 3.4 Two-scale mass flux models .................................................................................................. 3.5 Comparisons ..............................................................................................................................

4.1 General remarks ....................................................................................................................... 4.2 Uniqueness of numerical solutions ...................................................................................... 4.3 Initial conditions and relaxation ........................................................................................... 4.4 Boundary conditions ............................................................................................................... 4.5 Criteria for modelling replacing the quest for parameter freeness ............................. 4.6 Computability ............................................................................................................................ 4.7 A comparison ............................................................................................................................ 5.1 Overshooting ............................................................................................................................. 5.2 Modelling non-locality ............................................................................................................ 5.3 The DA white dwarfs ............................................................................................................... 5.4 3D simulations of overshooting ........................................................................................... 5.5 The role of the Péclet number .............................................................................................. 5.6 Previous research and a deep DA WD simulation ........................................................... 5.7 Conclusions ................................................................................................................................

70 71 72 73 74 76

77 80 81 84 87

88 89 90 91 92 93 95

96 98 99 99 101 103 104

6 Summary ............................................................................................................................. 105 References .............................................................................................................................. 105

Turbulence in stably stratified radiative zone François Lignières

111

1 Introduction ........................................................................................................................ 111 2 Stability of parallel shear flows in radiative atmosphere ........................................ 112 2.1 Unstratified shear flows .......................................................................................................... 2.2 Stably stratified shear flows .................................................................................................. 2.3 Stably stratified shear flows in highly diffusive atmosphere ........................................

113 115 118

3 Turbulent transport .......................................................................................................... 122

Multi-Dimensional Processes In Stellar Physics

3.1 The eddy diffusion hypothesis .............................................................................................. 3.2 Vertical eddy diffusion in stably stratified turbulence at Pr ~ 1 ................................. 3.3 Vertical eddy diffusion due to radial shear in radiative zone ...................................... 3.4 Vertical eddy diffusion in strongly stably stratified turbulence with high thermal diffusivity ....................................................................................................................

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123 125 130 136

4 Conclusion .......................................................................................................................... 137 References .............................................................................................................................. 138

An extremely short course on stellar rotation and magnetism Matthew K. Browning

141

Preface ..................................................................................................................................... 141 Lecture 1: An introduction to stellar rotation and magnetism .................................. 142 A breezy tour of observational techniques ............................................................................... Some observational punchlines and motivating puzzles ...................................................... Some order-of-magnitude estimates about stars .................................................................... Summary of the first lecture ..........................................................................................................

143 144 146 149

Lecture 2: Descriptive MHD, and some limiting cases in fluid mechanics ............. 149 The only equations we will solve ................................................................................................. Dimensionless numbers and the momentum equation ......................................................... Dimensionless numbers and the induction equation ............................................................. Equations for a thin layer, and geostrophic balance .............................................................. Descriptive MHD .............................................................................................................................. Example: shaking of coronal loops ............................................................................................. Example: magnetospheres and accretion onto a magnetized star .....................................

149 151 153 154 155 156 157

Lecture 3: an introduction to dynamo theory ............................................................... 159 The minimum Rm for growth ........................................................................................................ Stretching and amplification ......................................................................................................... Is stretching enough? A brief survey of anti-dynamo theorems ......................................... Qualitative building blocks of dynamo action .......................................................................... Mean-field dynamo theory and the link to helicity .................................................................

159 160 161 162 164

Closing remarks: simulations, open issues, and a biased summary ........................ 166 References .............................................................................................................................. 168

Stellar magnetism: bridging dynamos and winds Allan Sacha Brun and Antoine Strugarek

171

1 Introduction ........................................................................................................................ 2 A brief summary of what is a star ................................................................................ 3 Mean-field model of stellar dynamo and nonlinear aspects .................................. 4 Stellar wind: extensions of the Parker model ............................................................ 5 Coupling dynamo and stellar wind on dynamical and secular timescales: a perspective ...................................................................................................................... 6 Conclusion .......................................................................................................................... References .............................................................................................................................. 7 Solutions of the exercises ...............................................................................................

171 172 172 177 183 186 187 189

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Multi-dimensional asteroseismology Daniel Roy Reese

195

1 Introduction ........................................................................................................................ 195 2 Impact of rotation on stellar structure and evolution .............................................. 196 2.1 Structural changes ................................................................................................................... 2.2 Baroclinic effects ......................................................................................................................

196 197

3.1 The non-rotating case ............................................................................................................. 3.2 The rotating case .....................................................................................................................

198 199

4.1 Gravito-inertial modes ............................................................................................................ 4.2 Acoustic modes ........................................................................................................................

201 202

5.1 Frequency patterns .................................................................................................................. 5.2 Mode identification techniques ............................................................................................ 5.3 Observed stars ..........................................................................................................................

204 205 206

3 Calculating pulsations ...................................................................................................... 198

4 The effects of rotation on pulsations ........................................................................... 200 5 Asteroseismology ............................................................................................................. 204

6 Conclusion .......................................................................................................................... 207 References .............................................................................................................................. 208

Multi-dimensional physics of core-collapse supernovae Jérôme Guilet

213

1 Introduction ........................................................................................................................ 213 2 Hydrodynamic instabilities ............................................................................................. 214 2.1 Ledoux convection inside the protoneutron star ............................................................ 2.2 Double diffusive convection? ................................................................................................ 2.3 Neutrino-driven convection in the post-shock layer ....................................................... 2.4 Standing accretion shock instability (SASI) ....................................................................... 2.5 Corotation instability (aka low-T/W instability) ................................................................ 2.6 g-mode excitation ....................................................................................................................

214 215 216 217 219 220

3 Magnetic field origin and impact .................................................................................. 220 3.1 Slow-rotating standard supernovae .................................................................................... 3.2 Fast-rotating (extreme?) supernovae ..................................................................................

221 221

4 Conclusion .......................................................................................................................... 224 References .............................................................................................................................. 224

Preface With the development of powerful tools in computing sciences the resolution of partial differential equations, which always govern continuous media, is now far easier than a few decades ago. As a consequence, the multi-dimensional processes, which are quite numerous in stars, can be studied and modelled with some realism. This is most welcome since new instruments, like spectropolarimeters, interferometers or space missions, deliver data that are either instrinsically multi-dimensional or directly the result of multi-dimensional processes. However, as alluded above, the modelling of such processes require learning both new methods in various area of computing sciences, and being familiar with many aspects of stellar fluid dynamics. The 2018 Evry Schatzman School, dedicated to the multi-dimensional processes in stellar physics, proposed some practical work on the computing sciences side to the participants together with a series of lectures on the associated physics. The conferences have addressed several subjects that are all at the forefront of research and therefore not present in textbooks. We hope that the present lecture notes will help the newcomers overcome the difficulties that are inherent to this field and thus be more easily acquainted with this fascinating subject.

The Editors, Michel Rieutord, Isabelle Baraffe and Yveline Lebreton

Double-diffusive processes in stellar astrophysics Pascale Garaud1,∗ 1

Department of Applied Mathematics, Baskin School of Engineering, University of California Santa Cruz, 1156 High St, Santa Cruz CA 95064 Abstract. The past 20 years have witnessed a renewal of interest in the subject

of double-diffusive processes in astrophysics, and their impact on stellar evolution. This lecture aims to summarize the state of the field as of early 2019, although the reader should bear in mind that it is rapidly evolving. An Annual Review of Fluid Mechanics article entitled Double-diffusive convection at low Prandtl number [1] contains a reasonably comprehensive review of the topic, up to the summer of 2017. I focus here on presenting what I hope are clear derivations of some of the most important results with an astrophysical audience in mind, and discuss their implications for stellar evolution, both in an observational context, and in relation to previous work on the subject.

1 Introduction Double-diffusive instabilities were first discovered in the context of physical oceanography by a group of scientists from the Woods Hole Oceanographic Institution [2, 3]. Stern in particular realized that a region of the ocean with a stable density stratification can nevertheless undergo fluid instabilities because the water density depends on both temperature and salinity, which diffuse at different rates. To see why that may be the case, consider first a scenario in which temperature is stably stratified (with temperature increasing upward) and salinity is unstably stratified (with salinity increasing upward), as shown in Figure 1a. This could correspond for instance to the near-surface stratification of the tropical ocean. Overall, the density decreases upward because the stabilizing temperature stratification is stronger than the destabilizing salt stratification. As such, this fluid is stable to standard convection: a parcel of fluid forcibly moved downward is less dense than its surroundings, and in the absence of any diffusion, would experience a buoyancy force that pushes it back upwards. However, if the displaced parcel is small enough, its temperature rapidly adjusts to the surroundings, but its salinity does not (because salt diffuses 100 times slower than temperature). As a result, the small parcel becomes denser than its surroundings (because it has the same temperature but is saltier) and continues to sink. A similar argument can be made for small fluid parcels initially displaced upward, that continue to rise. This instability, called the salt fingering instability because of its tendency to produce long thin fingers of salty or fresh water, causes a net vertical transport of salt and heat downward. It is one of a few different kinds of thermohaline (or more generally thermocompositional) double-diffusive instabilities that have been discovered since the late 1950s. These include the oscillatory double-diffusive instability (ODDC), which occurs when the signs of the gradients ∗ e-mail: [email protected]

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Evry Schatzman School 2018

Figure 1. Pictorial description of the two main double-diffusive instability mechanisms. The color represents potential temperature (or entropy) and the symbol density represents the concentration of a denser species. Left: the fingering instability. A parcel of fluid displaced downward rapidly equilibrates in temperature, but retains its high concentration. As a result it is denser than the surroundings and continues to sink. Right: the ODDC instability. In the absence of any background thermal stratification, a displaced parcel would just oscillate up and down (dashed line). Thermal diffusion at extreme phases of oscillation cools down or warms up the parcel, which amplifies the oscillation (solid line).

of temperature and salinity are both reversed [3, 4], and the intrusive instability, which occurs when horizontal gradients of temperature and salinity are also present [5]. The idea of double-diffusive instabilities and double-diffusive mixing soon spread to the astrophysical community, thanks in part to the role of the Geophysical Fluid Dynamics summer program at the Woods Hole Oceanographic Institution, that was attended in the 1960s by astrophysicists such as Edward Spiegel, Shoji Kato, Jean-Paul Zahn, Douglas Gough, and others. In stellar interiors, the role of salt is replaced by any chemical species that has a higher mean molecular weight than its environment, as for instance He in comparison with H, or any heavier species (especially C, N, O, Si, Ni, Fe, etc. ) within a H-He mixture. In addition, whether the fluid is thermally stably stratified or not depends on the sign of the gradient of potential temperature (or equivalently, the sign of the entropy gradient) rather than on the sign of the temperature gradient alone. This accounts for adiabatic cooling or heating as the fluid parcel expands or shrinks to adapt to the local hydrostatic pressure. Fingering instabilities can take place in regions that are thermally stably stratified (i.e. radiative zones) with unstable compositional gradients. The process is usually referred to as thermohaline convection in stars. ODDC takes place in thermally unstable regions (as established by the Schwarzschild criterion) that have a sufficiently large compositional gradient to be Ledoux-stable. These regions are commonly called semiconvective, and their existence was first discussed in the late 1950s [6]. The relationship between semiconvection and ODDC was only later clarified by Kato [7] and Spiegel [8]. The physics of the ODDC instability, which takes place in these semiconvective regions, are summarized in Figure 1b. If, as a thought-experiment, we ignore the unstable entropy stratification and only consider adiabatic perturbations, the fluid is stably stratified by the compositional gradient and supports internal gravity waves. As such, a displaced parcel would just bob up and down without change of amplitude owing to its own restoring buoyancy force. The temperature stratification can destabilize this oscillation, however. Thermal diffusion causes a sufficiently small parcel to adjust to the local background temperature, which then warms it up (lowering its density) when the parcel is low, and cools it down (increasing its density) when it is high. This effect enhances the buoyancy force, and causes a gradual amplification of the oscillation, thus driving the instability. Both fingering convection and ODDC/semiconvection were popularized in stellar astrophysics in the 1970s and 1980s. Compositional transport models were proposed for both

Multi-Dimensional Processes In Stellar Physics

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instabilities [9–13] and many of them are still widely used in stellar evolution codes today. Until this last decade, however, none of these transport models had ever been tested, because laboratory experiments and numerical experiments at relevant parameters are prohibitively difficult to perform. As such, I would argue that any result ever obtained in the field of stellar astrophysics that strongly relies on double-diffusive mixing (including in particular fingering convection and semiconvection) should be taken with a very healthy dose of skepticism, and ought to be revisited in the light of the recent numerical and theoretical developments I will now proceed to review. In what follows, I will first lay out for completeness and pedagogical purposes the simplest possible model setup in which to study double-diffusive instabilities, and will review its stability properties (Section 2). I will then describe in more depth recent results on mixing by the fingering instability (Section 3) and the ODDC instability (Section 4), and discuss their implications. I conclude with a list of model caveats, and a summary of the more recent results obtained including the effects of shear, rotation, and magnetic fields in Section 5.

2 Linear instability of double-diffusive systems 2.1 Model setup

In Section 1, I briefly described how the fingering and ODDC instabilities proceed, and argued that both cases involved the need for small parcels of fluids so thermal diffusion can equalize the temperature within the parcel with that of the background. The fact that doublediffusive instabilities have to be small-scale can therefore be seen as a defining property of double-diffusive convection. The lengthscale of basic double-diffusive fluid motions in both fingering and ODDC situations can usually be estimated as being of order 10d [3, 14], where ⎛ ⎞  1/4  1/4 ⎛⎜ |N 2 | ⎞⎟−1/4 ⎜⎜⎜ κT ν ⎟⎟⎟1/4 κT ν ⎜⎜⎜ T ⎟ 3 ⎜ ⎟ , d = ⎝ 2 ⎠  10 cm ⎝ −4 2 ⎟⎟⎠ 7 cm2 s−1 2 s−1 10 10cm 10 s |NT |

(1)

where κT and ν are the thermal diffusivity and kinematic viscosity of the fluid, and NT2 = −δg(∇T − ∇ad )/H p is the square of the Brunt-Väisälä frequency associated with the entropy stratification only (i.e. ignoring compositional stratification). All the quantities used here are defined with standard notations in stellar astrophysics [15, 16]. Note that NT2 can be negative if the system is unstably stratified (∇T − ∇ad > 0). While κT , ν and NT2 can vary a lot from star to star, and between the center and surface of a given star, the 1/4 power implies that the double-diffusive scale 10d itself does not vary much, taking values around 100 meters in non-degenerate regions of stars, down to about 1 centimeter in degenerate regions (assuming the fingering region extends there). We see that in all limits, d is much smaller than the stellar radius. The fact that these are small-scale diffusively driven instabilities is both a blessing and a curse from a numerical standpoint. It implies that all diffusive lengthscales must be fully resolved at all times, and that any attempt to resolve them in global hydrodynamical models of stars is futile. Double-diffusive instabilities should also never be studied with Large-Eddy Simulations or any other numerical technique involving subgrid scale parametrizations, because they rely on microscopic diffusivities to exist. On the plus side, this also implies that these instabilities are best modeled in small computational domains that can ignore complex stellar physics. In particular, the effects of curvature, compressibility and the nonlinearity of the equation of state can almost always be neglected for such small-scale instabilities. Instead, they can adequately be studied with the Boussinesq approximation for weakly compressible gases [17].

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Evry Schatzman School 2018

ez

ey

ex

rm

Figure 2. Illustration of the model setup. A local Cartesian domain is modeled, near radius rm , with the vertical defining the z-axis. In the absence of rotation the orientation of x and y are arbitrary. If the system is rotating, x is aligned in the azimuthal direction.

To do so, let’s consider from now on a small region of a star, located around a radius rm , with mean temperature, density and pressure T m , ρm and pm . This region is modeled in a local Cartesian reference frame with coordinates (x, y, z) (see Figure 2), where the local gravity defines the vertical axis: g = −gez . As such, we can define z = r − rm , for instance. Following the Boussinesq assumptions, the computational domain size must be much smaller than any of the local pressure, density, and temperature scaleheights. This guarantees that the background temperature profile T 0 (z) can be linearized close to rm , so that T 0 (z) = T m + zT 0z where T 0z is the locally constant temperature gradient1 . Similarly, we assume that the background compositional field profile C0 (z) (which could for instance taken to be the mean molecular weight, or the mass fraction of a particular chemical species), can be written as C0 (z) = Cm + zC0z where Cm is the mean value in the domain, and C0z is the local gradient. Fluid motions, written as u = (u, v, w), as well as fluctuations of temperature T , density ρ and composition C, evolve around that background state following the Spiegel-VeronisBoussinesq equations [17], namely 

∂u + u · ∇u = −∇p − ρgez + ρm ν∇2 u, (2) ρm ∂t ∂T (3) + u · ∇T + w(T 0z − T ad,z ) = κT ∇2 T, ∂t ∂C + u · ∇C + wC0z = κC ∇2C, (4) ∂t ρ = −αT + βC and ∇ · u = 0, (5) ρm where κC is the compositional diffusivity, and the coefficients of the linearized equation of −1 state are α = −ρ−1 m (∂ρ/∂T )Cm ,pm and β = ρm (∂ρ/∂C)T m ,pm taken at rm . We see that the temperature gradient does not appear alone, but is instead combined with the adiabatic temperature gradient T ad,z = −g/c p (where c p is the specific heat at constant pressure) to form the potential temperature gradient (T 0z − T ad,z ), which is the relevant one in stellar interiors as discussed earlier. With this model setup, we can conveniently require all perturbations u, T , p and C to be triply periodic in the computational domain, which enables us to study the dynamics of double-diffusive instabilities far from any solid boundaries (that would be unphysical in a star). The set of equations (2)–(5) is commonly used to study double-diffusive con1 Note that all the gradients here (except for ∇ , ∇ and ∇ ) are taken with respect to position, not with respect T μ ad to mass or pressure as it is commonly done in stellar evolution.

Multi-Dimensional Processes In Stellar Physics

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vection with given background temperature and composition gradients far from boundaries [4, 18, 19]. As an illustrative side-note, these equations can very easily be used as is to look into the stability of standard (non-diffusive) multicomponent convection, and recover the well-known Ledoux criterion for instability. This is a useful warm-up exercise for the more complicated double-diffusive case. Let’s first linearize the equations, substitute the equation of state into the momentum equation, and ignore all diffusive terms. We obtain (in addition to the incompressibility condition) ∂u = −ρ−1 (6) m ∇p + (αgT − βgC)ez , ∂t ∂T ∂C (7) + w(T 0z − T ad,z ) = 0 and + wC0z = 0. ∂t ∂t Assuming an ansatz of the form q(x, y, z, t) = q˜ exp(ilx + imy + ikz + λt) for each of the variables, yields ˜ λ˜u = −iρ−1 ˜ , λ˜v = −iρ−1 ˜ , λw˜ = −iρ−1 ˜ + (αgT˜ − βgC) m lp m mp m kp ˜ ˜ λT + w(T ˜ 0z − T ad,z ) = 0 , λC + wC ˜ 0z = 0 and l˜u + m˜v + kw˜ = 0.

(8) (9)

Assuming without loss of generality that l  0 (otherwise switch the x and y variables), we can eliminate pressure from the first two equations to get v˜ = (m/l)˜u, and so from incompressibility u˜ = −lkw/(l ˜ 2 + m2 ). We can then eliminate pressure between the u˜ and w˜ equation, and substitute u˜ , T˜ and C˜ in terms of w˜ in the result. Once completed, the process results in an equation for λ only, because (as required from linear theory) the amplitude w˜ cancels out of the equation. We then have λ2 = −

l 2 + m2 l 2 + m2 N2 − T ) − βgC αg(T = − 0z ad,z 0z l 2 + m2 + k 2 l 2 + m2 + k 2

(10)

where N 2 = NT2 + NC2 and NC2 = −βgC0z is the contribution to the Brunt-Väisälä frequency due to the compositional stratification. With the form of the ansatz selected above, instability occurs when solutions of this equation exist for which the real part of λ is strictly positive. This is the case when αg(T 0z − T ad,z ) − βgC0z = NT2 + NC2 < 0, which is equivalent to the Ledoux criterion as required. 2.2 Linear stability properties of homogeneous double-diffusive systems

When all the diffusion terms are taken into account to model double-diffusive instabilities, it is significantly more convenient to non-dimensionalize the equations first. Following standard practices in the field (see [19] for instance), the unit lengthscale from here on is d, the unit timescale is the thermal diffusion timescale across d, namely d2 /κT , the unit velocity is κT /d, the unit temperature is d|T 0z − T ad,z | and the unit composition is αd|T 0z − T ad,z |/β. Note that we use |T 0z − T ad,z | instead of T 0z − T ad,z to ensure that all units are positive. Letting, e.g. ˆ t = (d2 /κT )tˆ, (u, v, w) = (κT /d)(ˆu, vˆ, w), ∇ = d−1 ∇, ˆ T = d|T 0z − T ad,z |Tˆ , and so forth, we obtain the non-dimensional equations

 1 ∂uˆ ˆ z + ∇ˆ 2 uˆ , ∇ˆ · uˆ = 0 (11) + uˆ · ∇ˆ uˆ = −∇ˆ pˆ + (Tˆ − C)e Pr ∂tˆ ∂Tˆ + uˆ · ∇ˆ Tˆ ± wˆ = ∇ˆ 2 Tˆ , (12) ∂tˆ ∂Cˆ ˆ ˆ = τ∇ˆ 2C, + uˆ · ∇ˆ Cˆ ± R−1 (13) 0 w ∂tˆ

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where the + sign applies in (12) and (13) for fingering convection, and the − sign applies for ODDC. It is rather remarkable to see that the only mathematical difference between these two instabilities is the sign difference in the wˆ term in these equations. Since the notation is somewhat heavy as is, in what follows the hats on the non-dimensional independent variables x, y, z, t and on the operator ∇ will be dropped, but those on the non-dimensional dependent variables will be kept for clarity. Three nondimensional parameters have appeared in these equations, namely the Prandtl number Pr = ν/κT , the diffusivity ratio τ = κC /κT , and the density ratio α|T 0z − T ad,z | |NT2 | δ(∇T − ∇ad ) R0 = = 2 = (14) β|C0z | φ∇μ |NC | where the last expression has been written assuming C is the mean molecular μ, and using standard definitions in stellar astrophysics [15, 16] for δ, φ and ∇T − ∇ad and ∇μ . The Prandtl number and diffusivity ratio depend principally on the nature of the fluid considered, and their dependence on thermodynamical quantities such as temperature, pressure, composition, etc. is neglected as part of the Boussinesq approximation. In salt water for instance Pr = O(10) and τ = O(0.01). In degenerate regions of White Dwarf (WD) stars Pr = O(0.01) and τ = O(0.001). In non-degenerate regions of Main Sequence (MS) and Red Giant Branch (RGB) stars Pr = O(10−6 ) and τ = O(10−7 ). Note that these are just indicative order of magnitudes, and the user should compute these numbers more accurately for the application of their choice. Examples of such computations are given in [20] for WDs, MS stars and RGB stars. The density ratio on the other hand measures the relative strengths of the potential temperature and compositional stratifications. With this definition, a Ledoux-neutral stratification has R0 = 1. As we will now demonstrate, the density ratio is the most important parameter controlling the dynamics of both fingering convection and ODDC. The linear stability of a doubly-stratified system satisfying equations (11)–(13) can be computed more-or-less as we did before for the non-diffusive case. We first assume a similar ˆ + imy ansatz (bearing in mind everything is now non-dimensional), q(x, ˆ y, z, t) = q˜ exp(ilx ˆ + ˆ + λt). ˆ Substituting that ansatz into the linearized equations, and successively eliminating ikz p, ˜ v˜, u˜ , T˜ , C˜ and finally w, ˜ results in the following cubic for the growth rate λˆ [4]: ⎡ ⎤ ⎢ ⎥⎥⎥ kˆ 2 ˆ 2 λˆ 2 + ⎢⎢⎢⎢⎣(Pr + τ + Prτ)|k| ˆ 4 ± Pr h (1 − R−1 ˆλ3 + (Pr + τ + 1)|k| ⎥⎦ λˆ 0 )⎥ 2 ˆ |k| ˆ 6 ± Prkˆ 2 (τ − R−1 ) = 0, (15) + Prτ|k| h



0

ˆ is the total wavevector, kˆ h = lˆ2 + m ˆ 2 is the horizontal wavenumber, and where kˆ = (l,ˆ m, ˆ k) where the + sign applies for fingering convection, and the − sign applies for ODDC. As in the non-diffusive case, the existence of solutions of this cubic with positive real part demonstrates instability. Despite the strong similarity of the cubics obtained in the fingering and ODDC cases, respectively, the solutions can be quite different. Using standard properties of cubics, it can be shown that fastest-growing modes in the fingering regime (with R0 > 1) are real while they are complex in the ODDC regime (with R0 < 1). For this reason, we now discuss the two cases separately. 2.2.1 Linear stability properties of the fingering regime

To study the fingering regime, we consider (15) with the + sign. Solutions are real so the condition for marginal stability can simply be written as λˆ = 0, which implies that the critical

Multi-Dimensional Processes In Stellar Physics

19

density ratio for instability for a given mode with wavenumber kˆ is ⎡ ⎤ ˆ 6 ⎥⎥⎥−1 ⎢⎢⎢ τ|k| ˆ ⎢ R0,c (k) = ⎢⎣τ + 2 ⎥⎥⎦ . kˆ h

(16)

ˆ is obtained when kˆ = 0, and is equal to R f c = τ−1 . We have The largest possible R0,c (k) therefore demonstrated that the fingering instability exists when 1 ≤ R0 ≤ τ−1 . The system is unstable to standard overturning according to the Ledoux criterion when R0 < 1, and stable when R0 > τ−1 . These findings are not too surprising: in this regime, R0 can be interpreted as the ratio of the stabilizing stratification to the destabilizing one, so the larger R0 is, the more stable the system is. We also see that the range of instability depends only on the value of the diffusivity ratio τ = κC /κT . If temperature and composition diffuse at the same rate, then fingering is not possible. On the other hand, the smaller τ is, the larger the range of density ratios for which instability can exist. Because τ is usually asymptotically small in stars, this implies that even a very small inverse μ gradient can destabilize a radiative zone. With a little work, it can be shown [19] that the modes with kˆ = 0 are always the most rapidly-growing ones, and are often called elevator modes for obvious reasons: the flow within these elevator modes is strictly vertical, and all the quantities of interest are invariant with z. In the limit of low Prandtl number and low diffusivity ratio appropriate for stellar ˆ are small), we can interiors, and close to marginal stability (so R0 → τ−1 and both λˆ and |k| ˆ roughly estimate λ for these elevator modes by neglecting the cubic and quadratic terms in (15) (see [9]): R−1 − τ (17) λˆ  0 −1 kˆ h2 . 1 − R0 In reality, however, the horizontal wavenumber kˆ h of the fastest growing elevator mode also varies with R0 close to marginal stability, so this expression does not directly tell us what λˆ is for that mode. To do so, one needs to perform a formal asymptotic expansion of (15) in the limit of low Pr and τ. This was done by Brown et al. [21], who derived a number of approximate analytical solutions for the growth rate λˆ of the fastest-growing modes as a function of R0 , Pr and τ, in the stellar limit (Pr, τ  1). The reader is referred to their Appendix B for details. Among other results, they find that for R0 not too close to 1 nor too close to marginal stability (i.e. 1  R0  τ−1 , which seems to be a reasonable limit for stars),   Pr ˆλ  Pr 1 − τ  , (18) R0 − 1 R0 while the horizontal wavenumber of the fastest-growing mode is always of order unity, with  kˆ h2



Pr . Pr + τ

(19)

Dimensionally, this implies that the wavelength of these fingers is of order 2πd (corresponding to one up-flowing and one down-flowing finger), where d was estimated in (1), and that their growth rate is  λ

1/2  R0 −1/2 |NT |2 Pr κT NT −4 −1   10 s , √ R0 d2 10−4 s−2 104 R0

(20)

which is usually substantially smaller than the buoyancy frequency unless R0 is close to 1.

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2.2.2 Linear stability properties of the ODDC regime

To determine the stability properties of ODDC, we study the solutions of (15) with the − sign. −1 In that case, it is common to use the inverse density ratio R−1 0 as the relevant parameter; R0 is now the ratio of the stabilizing compositional stratification to the destabilizing potential temperature stratification, with R−1 < 1 being Ledoux-unstable (hence convective in the 0 −1 standard sense), while R−1 0 > 1 is Ledoux-stable. Increasing R0 correspond to increasingly stable systems. The criterion for marginal stability to ODDC is a little more difficult to derive than in the fingering case, because λˆ is not real. So we let λˆ = λˆ R + iλˆ I and first rewrite (15) in terms of λˆ R only. The same argument as in the fingering case can be used to show that the elevator modes are still the fastest-growing modes, so focussing on these only, we let kˆ = 0. With a little work (see the Appendix A.1 in [22]), we obtain     8λˆ 3R + 8kˆ h2 λˆ 2R (τ + Pr +1) + 2λˆ R kˆ h4 τ + Pr τ + Pr + (τ + Pr +1)2 + Pr(R−1 0 − 1) +kˆ h6 (τ + Pr)(τ + 1)(Pr +1) + kˆ h2 Pr(R−1 0 (τ + Pr) − (Pr +1)) = 0 .

(21)

Setting λˆ R = 0, we then find that the critical inverse density ratio for instability is achieved for kˆ h → 0 (as in the fingering case). This shows that ODDC can only occur for 1 ≤ R−1 0 ≤

Pr + 1 . Pr + τ

(22)

Since Pr τ in geophysical environments, (Pr + 1)/(Pr + τ)  1. As a result, the range of R−1 0 for which a doubly stratified system on Earth may be linearly unstable to ODDC is very small and almost never naturally realized. By contrast, both Pr and τ are very small in stars, so the range of instability to ODDC can be very substantial. Applying the same arguments as in the fingering case, we can roughly estimate λˆ R close ˆ ˆ to marginal stability (i.e. when R−1 0 → (Pr + 1)/(Pr + τ) where both λR and kh are small) by neglecting the quadratic and cubic terms in (21). In the stellar limit where Pr, τ  1, we then get [12], kˆ h2 1 (Pr +1) − R−1 0 (τ + Pr) (23)  λˆ R  kˆ h2 2 R−1 2R−1 0 −1 0 for sufficiently large R−1 0 (but not too close to marginal stability). Again, this should be viewed as a rough estimate, and does not provide information on the wavenumber kˆ h for the fastest-growing modes. A more formal asymptotic analysis needs to be carried out to get this information. Details can be found in Appendix A.2 of Mirouh et al. [22]. The solutions have an analytical but non-trivial dependence on R−1 0 , whose expression is not particularly ˆ R is proportional2 to Pr, so from a dimensional point illuminating. At best, one can state that λ √ of view, we find that λR ∝ PrNT . Finally, it is also relatively easy to show that for Pr, τ  1, the imaginary part of λˆ is well-approximated by the local buoyancy frequency including the compositional stratification, i.e.   NC2 + NT2 . (24) λˆ I  Pr(R−1 0 − 1) → λI  2.2.3 Summary for linear stability of double-diffusive systems

An illustrative summary of the findings from linear stability analysis is presented in Figure 3. In the absence of compositional stratification, the threshold between stability and instability 2 This

−1 is indeed consistent with (23) since that equation is only valid in the limit where R−1 0 → (Pr + τ) .

Multi-Dimensional Processes In Stellar Physics

21

to overturning convection is simply given by the Schwarzschild criterion, namely R0 = 0 or equivalently ∇T = ∇ad . With compositional stratification, instability to overturning convection is set by the Ledoux criterion, R0 = 1 or equivalently δ(∇T − ∇ad ) = φ∇μ . We see that double-diffusive effects destabilize regions that are Ledoux-stable. In the presence of a destabilizing compositional gradient, fingering instabilities are excited in a wide region of parameter space that would normally be purely radiative. In the presence of a stabilizing compositional gradient, ODDC is excited almost everywhere in the region of parameter space bound by the Schwarzschild criterion on one side, and the Ledoux criterion on the other. In all cases, the fastest-growing modes of instability are elevator modes, i.e. vertically-invariant structures, whose horizontal size is O(10d).
T =
ad   
  



 




= 0 T
 ad

   
  



 < 0

  
  



 > 0


 



R0 =

   
  


1


  
  


T
 ad

R0 = 1 
T
 ad =








 




R0
1 =

Pr+1 Pr+


  
  


R0
1 = 1 
T
 ad =




T
 ad

Figure 3. Illustration of the various regimes of instability as a function of R0 (or equivalently, ∇T − ∇ad and ∇μ ), for fingering convection and ODDC.

While this analysis has been performed for an idealized system without any added physics, we are of course also interested in knowing whether rotation, shear, magnetic fields, and other dynamics, could affect these results. Interestingly, neither rotation, magnetic fields, nor shear seem to affect the range of instability of fingering convection [23–25], because there always appears to be a way for the elevator modes to develop. Horizontal gradients, however, can extend the unstable fingering range significantly [5, 26]. Gravitational settling may induce a similar effect if the settling velocity of individual species approaches the value κT /d [27, 28], though this has not yet been studied in the astrophysical context. To my knowledge, there is no evidence for subcritical instabilities (i.e. instabilities that only develop given specific finite-amplitude initial conditions instead of from infinitesimal perturbations) in fingering-prone stratified fluids. The impact of added dynamics on ODDC has not yet been studied as exhaustively as in the fingering case. Rotation does not affect the unstable range [29], but shear does (at least in geophysics), through the newly discovered thermo-shear instability [30]. Whether that instability is relevant at astrophysical parameters has not yet been established. The effect of magnetic fields on the linear stability of ODDC was investigated by Stevenson [31], who found that it does not affect the overall unstable range, but can change the nature of the

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unstable modes that are present. Finally, by contrast with the fingering case, ODDC is known to have subcritical branches of instability, at least in the geophysically-relevant region of parameter space [32]. This remains to be studied more extensively at low Prandtl number [33]. Finally, note that all of these results have been stated in the context of the model setup described in Section 2.1. The presence of physical boundaries can strongly affect the instability range, especially when the size of the domain is not very large compared with the natural double-diffusive scale 10d. For bounded domains the linear stability problem needs to be investigated on a case-by-case basis, and depends on the domain size, shape, applied boundary conditions, and any added physics. 2.3 Where is fingering convection likely to occur in stars?

The linear stability analysis presented above clearly shows that fingering instabilities can take place in radiative stellar regions which are Ledoux-stable, but subject to an unstable composition gradient, even if the latter is very weak. Such gradients can arise in stellar interiors in a number of scenarios, all of which have important observational implications. A common mechanism to drive fingering convection involves the accretion of high μ material at the surface of a star, either from infalling planets or planetary debris, or from material transferred from a more evolved companion star. Planetary infall for instance can increase the apparent metallicity of the host star; this has been invoked in turn as a possible explanation to the planet-metallicity correlation [34], and to the existence of metal-rich WDs [35, 36]. It was rapidly realized however that this inverse μ-gradient would create a region below the surface of the star that is unstable to fingering convection [37], which would then significantly shorten the residence time of high-μ chemical species near the surface, with implications for derived accretion rates and/or surface abundances of light elements in both MS stars [38, 39] and WDs [40–42]. Similar arguments have been put forward in the context of binary mass transfer, with possible implications for carbon-enhanced metal poor stars, for instance [43–46] and nova outbursts [47]. Fingering regions can also appear deep within a star from off-center nuclear burning in RGB and AGB stars. In RGB stars, 3 He burning at the outskirts of the H-burning layer can produce lower-μ material, causing that region and the regions above it to become fingeringunstable [9, 48]. Instabilities associated with this inverse μ gradient have been proposed as a possible solution to the peculiar surface abundances of RGB stars beyond the luminosity bump [49, 50]. Fingering is also thought to occur below the carbon-burning shell of superAGB stars [51], and to affect the carbon flame propagation by mixing the available fuel. It may also impact the properties of hybrid C/O/Ne WDs, whose electron-to-baryon fraction becomes unstable to convection and fingering convection as the star cools [52, 53]. Finally, the microscopic segregation of various chemical species by gravitational settling and radiative levitation can produce chemically peculiar layers below the surface of intermediate-mass stars, that are prone to fingering instabilities [54–56]. Not accounting for mixing by fingering, the accumulation of such elements can be so strong that it leads to the formation of intermediate convection zones. Fingering instabilities, however, may develop much earlier and prevent the accumulation from reaching such extreme levels [55]. 2.4 Where is ODDC/semiconvection likely to occur is stars?

We also learned using linear theory that ODDC develops in regions that are traditionally called semiconvective, i.e. regions which are unstable according to the Schwarzschild criterion (∇T − ∇ad > 0), but stable according to the Ledoux criterion (φ∇μ > δ(∇T − ∇ad )).

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Semiconvective regions are often found adjacent to convective cores and are caused by the development of stabilizing μ gradients. There are two stellar mass ranges in which this occurs, and the reason for the presence of the μ gradient differs in the two cases. For reviews on the topic, see for instance [8, 12, 13, 57, 58]. In intermediate mass MS stars, with masses above 1M and below 3M , nuclear reactions can extend far beyond the radius of the convective core, owing to the relatively weak dependence of the pp-chain reaction rates on temperature. This implies that He is slowly generated outside the core, at a rate that decreases with radius. This creates a gentle μ gradient, that partially inhibits convection and causes the convective core to be smaller than what it would be according to the Schwarzschild criterion. A semiconvective region develops between this so-called Schwarzschild core radius and the Ledoux core radius, see Figure 4a. In the core of higher mass stars (with masses greater than about 5M ) where the CNO cycle dominates, the temperature dependence of the nuclear reaction rates is so strong that the latter only takes place deep within the convective core. As such, the mechanism described above for creating μ-gradient near the core of intermediate mass stars does not work. However, the convective core shrinks with time as H is converted into He, because the opacity of the material in that region is dominated by electron scattering and is simply proportional to 1 + X, where X is the hydrogen mass fraction. With a reduced opacity, more of the energy can be transported radiatively, and the convection zone retreats. This leaves behind concentric shells of increasingly high μ material, and therefore gradually builds up a substantial μ gradient outside this core (see Figure 4b). That region can be unstable to ODDC3 . Finally, semiconvection zones detached from nearby convective zones are sometimes found in models of high mass stars (15M or higher), see for instance [59]. However, whether they exist in stars or not remains to be established, because their presence or absence in models is quite sensitive to the semiconvective mixing prescription used. 








 




   
 


 



   











 





 












Figure 4. Illustration of the two mechanisms for forming semiconvective regions at the edge of a convective core. The red line represents the nuclear energy generation rate, and the other colored lines represent the μ(r) profile, with blue at early times, then green, then orange at latest time. The vertical dashed line marks the edge of the convective core according to the Ledoux criterion. Left: In intermediate mass stars, nuclear reactions extend outside the core, and their rate depends on temperature. A μ gradient thus forms in situ. Right: In high-mass stars, nuclear generation only takes place well within the core. However the core shrinks with time (see text for detail) and leaves behind material of increasing μ. A μ-gradient gradually forms as a result. In both figures rL marks the edge of the core according to the Ledoux criterion. The ODDC can exist just outside of it. 3 Note that the dependence of the opacity on X

presents an additional modeling challenge in high mass stars, which is not accounted for in the simple ODDC model presented in this review. As such whether the results presented here are applicable to semiconvection in these stars or not remains to be determined.

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2.5 Beyond linear theory

Having established the conditions under which fingering convection and ODDC may develop in stars, we can now ask the more important quantitative question of how much mixing they cause, and how this affects stellar evolution. To do so requires connecting fluid dynamics, which governs the double-diffusive instabilities on the small scales / short timescales, and stellar modeling, whose equations govern the structure and evolution of stars on the largescales / long timescales. In stellar evolution codes, turbulent transport by double-diffusive convection is traditionally modeled as a turbulent diffusion process. Assuming the star is spherically-symmetric, the conservation law for the evolution of the concentration C of a particular chemical species in a given mass shell is

 1 D(ρC) 1 DC = − ∇ · (ρFC ) + , (25) Dt ρ ρ Dt nuc where the time-derivative is a Lagrangian derivative following the shell, ρ is its density, ρC is the total mass of the particular chemical element considered, ρFC is the mass flux, and (D(ρC)/Dt)nuc is the rate of change due to nuclear (or chemical, if relevant) reactions. The assumption that turbulent transport takes a diffusive4 form implies that the compositional mass flux FC should be downgradient, with FC = −(κC + DC )∇C,

(26)

where κC is the microscopic diffusivity and DC is the turbulent diffusivity, both of which have units of cm2 /s in cgs. The assumption of spherical symmetry then implies that



 1 ∂ 2 1 D(ρC) ∂C DC = 2 r ρ(κC + DC ) + , (27) Dt ∂r ρ Dt nuc ρr ∂r which can be expressed in mass coordinates m as



 ∂ 1 D(ρC) ∂C DC = (4πr2 ρ)2 (κC + DC ) + . Dt ∂m ∂m ρ Dt nuc

(28)

This is the formula typically implemented in stellar evolution codes for compositional transport, with a turbulent mixing coefficient DC that depends on the instability driving the turbulence. It is worth remembering, however, that turbulent transport does not always take a diffusive form, so one should ideally first question whether that assumption is correct, before attempting to model DC . The case of turbulent heat transport by double-diffusive instabilities could in principle be treated in a similar way, but in practice this is rarely ever done, for two reasons. The first is that very few stellar evolution codes actually evolve the temperature profile in the star, preferring instead to solve for it at each timestep knowing the stellar luminosity. The second is that heat transport in fingering convection and semiconvection is usually negligible (with some notable exceptions discussed later), because temperature has to diffuse for the instability to exist in the first place. The rest of this lecture will be therefore dedicated to presenting models for DC for doublediffusive instabilities, with the fingering case discussed in Section 3, and the case of ODDC in 4 In

this lecture I will always use the mathematical interpretation of a diffusive process which is associated with a downgradient flux, rather than the astrophysical interpretation of element diffusion due to e.g. gravitational settling or radiative levitation.

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Section 4. The following Table summarizes the important non-dimensional parameters that characterize the properties of double-diffusive convection. All will be extensively used in this lecture. Input parameters such as Pr, τ and R0 have already been discussed in this Section. Output non-dimensional parameters based on DC (and other turbulent mixing processes) will be introduced and used in Sections 3 and 4, to help quantify turbulent transport of heat and composition in the system, both in comparison with the respective diffusive transport rates, or in comparison with one another. Table 1. Non-dimensional parameters governing double-diffusive mixing, written as the ratio of dimensional quantities. The dimensional turbulent fluxes of temperature and composition are FT = wT and FC = wC , where the notation · denotes a volume average. Assuming turbulent transport is diffusive implies that FC = −DC C0z in this notation.

Name Input Prandtl number params. Diffusivity ratio Density ratio

Symbol and Formula Pr = ν/κT

Interpretation Ratio of diffusivities

τ = κC /κT

Ratio of diffusivities

R0 =

α|T 0z −T ad,z | β|C0z |

=

|NT2 | |NC2 |

Output Nusselt number for T params.

NuT =

−κT (T 0z −T ad,z )+ wT −κT (T 0z −T ad,z )

Nusselt number for C

NuC =

−κC C0z + wC −κC C0z

Turbulent flux ratio

γturb =

α wT β wC

Total flux ratio

γtot =

=

DC +κC κC

α −κT (T 0z −T ad,z )+ wT β −κC C0z + wC

Ratio of stratifications Ratio of total to diffusive potential temperature flux Ratio of total to diffusive compositional flux Ratio of turbulent fluxes Ratio of total fluxes

3 Mixing by fingering convection 3.1 Traditional models of "thermohaline" mixing

The first turbulent diffusion model proposed for mixing by fingering convection in astrophysics was put forward by Ulrich [9], and is based on a very simple dimensional analysis. Noting, as it is common to do so in astrophysics, that the turbulent diffusion coefficient DC has the units of a velocity times a length, or equivalently, of a length squared divided by time, it is reasonable to assume that the one appropriate for fingering convection should be expressed as ˆ T, (29) DC = Dfing ∝ λd2 ∝ λκ where λ is the dimensional growth rate of the fastest-growing fingers, which is related to the ˆ 2 . Using the estimate from non-dimensional growth rate discussed in Section 2 via λ = κT λ/d (17) for the growth rate of fingers close to marginal stability, we get Dfing ∝

R−1 0 −τ 1 − R−1 0

kˆ h2 κT .

(30)

Assuming that kˆ h2 remains close to unity, then we can write Dfing = Cfing

1 − R0 τ κT . R0 − 1

(31)

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In the limit where R0 is not too close to marginal stability (which is somewhat inconsistent with the assumption made above, but let’s ignore this for now) then 1 − R0 τ  1 and we recover the formula proposed by Ulrich [9], with a proportionality constant he argues should be of order 700: 1 Dfing = CU κT . (32) R0 − 1 A very similar expression was later obtained from rather different arguments by Kippenhahn et al. [11], who argued that CKRT Dfing = κT . (33) R0 We see by comparison with (31) that this expression should only be valid in the limit where 1  R0  τ−1 . The coefficient CKRT in this model is argued to be much smaller than CU , taking a proposed value of 12. We can already see, however, that both models as they are presented fail to account for the stabilization of the system to fingering instabilities beyond the threshold R0 = τ−1 . As such, we expect that they should largely overestimate the true mixing coefficient as R0 approaches and/or exceeds that threshold. A better model, that would at least be consistent with linear stability theory, is given in (31) and was (to my knowledge) first derived by Denissenkov [60] (see his equation 15). Written in terms of the more standard astrophysical notations, the model becomes φ∇μ − δ(∇T − ∇ad )τ Dfing = CD κT , (34) δ(∇T − ∇ad ) − φ∇μ assuming the compositional field C is simply the mean molecular weight. That model, however, is ill-posed (with Dfing → ∞) as R0 tends to unity, or in other words, as we approach the Ledoux criterion for convective instability. This is obviously not physically plausible, so the model should not be used when R0 → 1. This ill-posedness is not surprising since the model was derived in the first place assuming that R0 is close to marginal stability, i.e. very large; better models (see below) have since been proposed to address the problem. 3.2 Numerical simulations of small-scale fingering convection

Until recently, no robust evidence for or against the adequacy of the various expressions for Dfing listed above existed. As mentioned in the introduction, laboratory experiments at low Prandtl numbers are almost impossible, because so few low Prandtl number fluids exist on Earth, and those that do are very expensive and notoriously difficult to manipulate (e.g. liquid mercury, liquid lithium, liquid potassium). Numerical experiments are also challenging. Indeed, the computational domain needs to contain at least 5-10 fingers in each direction for good statistics. Furthermore, we saw that the scale of the fingers is related to the thermal diffusion scale, while the scale of boundary layers between the fingers is dictated by the viscous and compositional diffusion scales, which are asymptotically small under stellar conditions (since Pr = ν/κT  1 and τ = κC /κT  1). All of these scales need to be resolved to adequately model fingering convection, which poses a hard computational challenge. Finally, it has recently been demonstrated that fingering convection at low Prandtl number cannot be modeled in 2D, as it develops unphysical pathological behavior [61]. In 2D simulations, artificial horizontal shear layers appear spontaneously from the fingering instability, and in turn affect the fingering structures. These shear layers are not present in 3D domains5 , as long as the third dimension is wide enough (i.e at least 2 finger widths), see Figure 5. 5 at

least not as ubiquitously and with such strong amplitude.


 


Multi-Dimensional Processes In Stellar Physics

 


 



 

 


  
  


Figure 5. Comparison between 2D simulations and thin-domain 3D simulations at the same parameter values (Pr = τ = 0.01, and R0 = 5). In the 2D case, horizontal shear layers spontaneously develop and tilt the fingers, which in turn reduces the turbulent fluxes. This is artificial, however, and these shear layers are not present in 3D as long as the domain is thick enough. Figure adapted from [61].

Within the last decade, however, numerical simulations with a physical grid resolution of ∼ 3003 have become routine, and state-of-the-art ones are now exceeding ∼ 30003 or more. With the smaller-sized runs, it is possible to systematically explore parameter space (i.e. cover the entire range of density ratios) for fingering convection at diffusivity ratios Pr and τ down to ∼ 0.01. The higher resolution simulations can be used to test models down to, e.g. Pr, τ ∼ 0.001, but only for a few values of the density ratio. A first series of 3D DNSs of fingering convection at low Prandtl number was presented by Traxler et al. [62], using the PADDI code developed by S. Stellmach. This code is a pseudo-spectral code that solves equations (11)–(13) in a triply periodic domain, ensuring incompressibility is maintained using a standard projection method [63]. Traxler et al. limited their study to moderate Pr, τ, from 1/3 down to 1/30. Figure 6 shows representative snapshots of the compositional field in their simulations, for two density ratios (one low, close to the Ledoux threshold, and one large, close to the marginal stability threshold). Simulations with lower Pr and τ down to 1/300 made with the PADDI code are now also available [1, 21] and show qualitatively similar features.

 


 


Figure 6. Volume renderings of the compositional field in homogeneous fingering convection at Pr = τ = 0.1, for R0 = 1.45 (left), which is strongly unstable, and R0 = 9.1 (right), which is close to marginal stability. Compositionally rich (red) fingers are flowing downward, and compositionally poor (blue) fingers are flowing upward. Figure from [62].

Each simulation can easily be used to measure a turbulent compositional diffusivity. Indeed, in a triply-periodic domain, the horizontal average (marked with an overbar) of the

27

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dimensional composition evolution equation (4) is ∂C¯ ∂2C¯ ∂ + wC = κC 2 , ∂t ∂z ∂z

(35)

which reveals wC to be the vertical turbulent compositional flux (i.e. the transport of C due to advection by vertical fluid motions). While this quantity would generally depend on both z and t, in a statistically steady state and assuming that C¯ remains small (which we can verify is true in these simulations), wC is approximately constant. We can then obtain a good estimate of the vertical turbulent flux by taking both a time-average and a volume average of wC in the simulation, and define this as FC = wC . The same can be done for the temperature. Incidentally, it is easy to show [64] that FC and FT must be negative for statistically stationary fingering convection. Indeed, multiplying equation (4) by C, integrating over the volume, and using incompressibility and the periodicity of the boundary conditions to eliminate the nonlinear advection terms and the boundary terms, we get 1 ∂ C 2 + wC C0z = κC C∇2C = −κC |∇C|2 , 2 ∂t

(36)

using one of Green’s identities. Since C0z > 0 in fingering convection, and since |∇C|2 > 0, then wC must be negative if the turbulence is statistically stationary (i.e. if we can neglect the time-derivative). The same is true for wT , showing that both T and C are transported downward (which is not surprising, given the physical mechanism described in Section 1). The volume-averaged compositional flux can finally be used to derive Dfing from each simulation as wC ˆ Dfing = − = −R0 κT wˆ C , (37) C0z where the hats denote non-dimensional quantities (see Section 2.2). The numerical results can be used to test the traditional models of Ulrich and Kippenhahn et al. [9, 11] described in Section 3.1. Writing both models as Dfing = CU,KRT Rκ0 T−1 , with only the constant differing between them, the theory would predict that Dfing (R0 − 1)/κT should be constant. The results are shown in Figure 7, for a wide range of simulations [1, 21, 62]. For ease of visualization the results are shown, not as a function of R0 , but as a function of the reduced density ratio [62] R0 − 1 , (38) r = −1 τ −1 which maps the entire fingering range into the interval [0, 1] regardless of τ, with r = 0 corresponding to the Ledoux criterion, and r = 1 corresponding to marginal stability. We can immediately make several important conclusions: that Dfing (R0 − 1)/κT is not constant across the entire range, that it seems to depend on the ratio ν/κC = Pr/τ (called the Schmidt number) and finally that the numerical results do not support the large value of CU  700 [9] but are more consistent (within the list of caveats listed) with the smaller value CKRT  12 [11]. As discussed earlier, the fact that Dfing (R0 −1)/κT is not constant across the fingering range is not surprising, since Dfing ∝ Rκ0 T−1 is both unphysically singular as R0 → 1, and does not account for the stabilization of the system as R0 → τ−1 (which is clearly visible in the data). The fact that the results depend principally on the Schmidt number is also expected, both on physical and mathematical grounds. Physically speaking [62], when κT  (κC , ν), the role of the temperature fluctuations essentially becomes negligible, and the instability is driven by the compositional field. It is therefore not surprising to find that the dynamics solely depend on ν/κC , rather than on any parameter that depends on κT . From a mathematical point of view,

Multi-Dimensional Processes In Stellar Physics

29

it can be shown [65, 66] that in the limit of κT → 0 (and for sufficiently weak fingering) the non-dimensional governing equations (11)–(13) reduce to a new set of asymptotic equations that only depend on two parameters instead of the usual three, namely the Schmidt number, and the so-called Rayleigh ratio R = (R0 τ)−1 . As such, the data dependence on the Schmidt number, rather than Pr and τ, is expected. Based on their first set of numerical experiments, Traxler et al. [62] proposed a simple empirical formula for mixing by fingering convection, namely √ Dfing = 101 κC ν exp(−3.6r)(1 − r)1.1 , (39) where the numbers 101, 3.6 and 1.1 were fitted to the data. This model is easy to use, and fits the results adequately except for very low r [21], where the system is only weakly stratified and close to the Ledoux threshold for overturning convection. In that limit (39) dramatically underestimates the mixing coefficient [21] . 100

10

Figure 7. Non dimensional ratio Dfing (R0 − 1)/κT , against the reduced density ratio r, for a wide range of simulations. This quantity should be constant according to the traditional models of fingering convection [9, 11] but is not in reality.

D fing (R0
1)


T

1

0.1

0.01

Pr=1/3, tau=1/3 Pr=1/10, tau=1/10 Pr = 1/3, tau=1/10 Pr=1/10, tau=1/30 Pr=1/10, tau=1/100 Pr=1/100, tau=1/100 Pr=1/300, tau=1/300 0.01

r=

R0
1

0.1


1



1

3.3 The Brown et al. 2013 model for small-scale fingering convection

In order to address the shortfalls of the Traxler et al. model for fingering convection, Brown et al. [21] revisited the problem and proposed a new theory for the mixing coefficient. Starting with the definition of Dfing given in (37), we see that the key to creating a model for this coefficient is to estimate both the typical non-dimensional vertical velocity of the fingers, wˆ f , and their typical compositional perturbation, Cˆ f , so ˆ = −R0 κT KB wˆ f Cˆ f Dfing = −R0 κT wˆ C

(40)

where KB is a constant that depends on the geometry of the fingers, and the typical correlation ˆ Assuming that transport is controlled by the most rapidly growing fingers between wˆ and C. (which are elevator modes) and that these fingers are controlled by the linearized equations (at least until they nonlinearly saturate) we can relate wˆ f and Cˆ f using the linearized version ˆ and horizontal derivatives are of (13), in which ∂/∂t is replaced by the finger growth rate λ, replaced by the horizontal wavenumber kˆ h , so ˆ f = −τkˆ h2Cˆ f → Cˆ f = − λˆ Cˆ f + R−1 0 w

R−1 ˆf 0 w . λˆ + τkˆ 2

(41)

h

Hence Dfing = KB

wˆ 2f λˆ + τkˆ h2

κT ,

(42)

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Evry Schatzman School 2018

and the only problem remaining is to estimate wˆ f . To do so requires specifying the mechanism by which the linear fingering instability saturates. Following Radko & Smith [67], Brown et al. [21] assumed that saturation occurs by parasitic shear instabilities that develop between up-flowing and down-flowing fingers, and that the fingers stop growing when the parasitic ˆ By dimensional analysis (or by solving the problem instability growth rate σ ˆ approaches λ. exactly, see Appendix A of Brown et al.), it can be shown that σ ˆ = Kσ wˆ f kˆ h where Kσ is a universal constant of order unity (whose value is irrelevant for reasons explained below) so the saturation condition reads Kσ wˆ f kˆ h = Kw λˆ → wˆ f =

Kw λˆ , Kσ kˆ h

(43)

where Kw is another universal constant of order unity. We see that these two constants end up folded into a single one, which is the reason why their individual values are irrelevant. ˆ kˆ h was recently The relationship between wˆ f at saturation of the fingering instability and λ/ verified [23] against a re-analysis of the existing numerical data, and revealed that Kw /Kσ  2π. With this, we finally have the prediction that

Dfing = KB

Kw Kσ

2

λˆ 2 λˆ kˆ h2 + τkˆ h4

κT .

(44)

This can easily be tested against available data, and the combination of constants KB (Kw /Kσ )2 is found to be around 49 (implying that KB  1.24). The comparison between the model and the data is very good (though not perfect) for all cases where ν/κC ≤ 1, which is indeed the case in stellar interiors, for all r, Pr and τ tested. For decreasing Pr and τ the good fit deteriorates somewhat, though remains reasonable (within a factor of order unity), see Figure 8. In summary, to compute Dfing one should first find the growth rate and wavenumber of the fastest-growing modes of the fingering instability at the desired parameter values (R0 , Pr, τ), which can be done numerically using a cubic root-finding algorithm. Once λˆ and kˆ h are known, they are used in expression (44) to predict Dfing . This method has already been implemented for instance in MESA. Alternatively, analytical approximations for λˆ and kˆ h (found in Appendix B of Brown et al. [21]) can be used instead to speed up the process. Routines providing such estimates are also available in MESA. 10000

Pr=1/3, tau=1/3 Pr=1/10, tau=1/10 Pr = 1/3, tau=1/10 Pr=1/10, tau=1/30 Pr=1/10, tau=1/100 Pr=1/100, tau=1/100 Pr=1/300, tau=1/300

1000

Dfing


C

Figure 8. Comparison between data (symbols) and theory (equation 44) (lines) for the Nusselt number Dfing /κC + 1, against the reduced density ratio r, for a wide range of parameters. Figure from [1].

+1 100

10

1 0.001

0.01

0.1 r

1

Multi-Dimensional Processes In Stellar Physics

3.4 Large-scale instabilities?

While the process of small-scale fingering convection in stellar environment is now arguably well-understood, this may not be the full story. Indeed, fingering-unstable regions of the ocean on Earth are often associated with thermohaline staircases, which are horizontallyinvariant stepped structures in the vertical profiles of temperature and salinity (see [19] for a review). These staircases are formed of stacked layers and interfaces. Within a layer, the density is very slightly unstably stratified and subject to large-scale convective overturning. Both temperature and salinity are almost constant within each layer as a result of the strong mixing. In between the layers are stably stratified interfaces undergoing fingering convection. While the layers can be tens or even hundreds of meters deep, the interfaces are much shallower (tens of centimeters), so the temperature and salinity gradients across the interface are very large (hence the characteristic appearance of the staircase). When thermohaline staircases are present, vertical mixing can be enhanced by two orders of magnitude [68] compared to a similar overall stratification without staircases. It is therefore crucial to understand why and how such staircases form in the ocean, and whether similar processes may be taking place in stars. Thankfully, significant progress in modeling the formation of oceanic thermohaline staircases (and the emergence of other large-scale dynamics) from fingering convection has been made in the past 20 years. Given the wide separation between the finger scale and the scale of the staircases, mean field hydrodynamics turns out to be a fruitful approach to the problem. In this type of theory, the large scales (such as the layer scale) are modeled exactly, while the effect of small scales (i.e. the basic fingering convection) is parameterized using some form of turbulence closure. As we saw earlier, for a given fluid (i.e. given Pr and τ), the turbulent fluxes in fingering convection only appear to depend on the local density ratio. With this in mind, it is easy to see how large-scale instabilities might develop (see Figure 9). Indeed, any large-scale perturbation in the temperature and compositional fields causes largescale modulations in the local density ratio (which depends on their gradients). This in turn modulates the temperature and compositional fluxes due to fingering, and the convergence or divergence of these fluxes can in some cases enhance the original perturbation, and in some cases suppress it. Positive feedback loops, when they exist, thus drive the growth of largescale instabilities. As we have discovered [20, 63], several distinct types of positive feedback loops can in fact exist, each leading to the amplification of different kinds of perturbations.

  
 
 
Tz ,Cz 


 


FT ,C = f (R;Pr,
)








1+
T R =
1 z R0
+ C z

Figure 9. Illustration of the possible positive feedback loop between ¯ large-scale perturbations in T¯ and C, and turbulent fluxes. See text for detail. Figure adapted from [1]. The background image is a snapshot from a simulation exhibiting the spontaneous emergence of large-scale gravity waves from [61] (see also [20]).

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To model this idea mathematically and determine whether thermocompositional layers may form in stars, we essentially follow the theory of Radko [69], and expand it to account for the diffusive contribution to the fluxes [62] (which, as we shall demonstrate, is a crucial element of the stellar problem). Since layers are horizontally invariant, we begin by taking the horizontal average of the non-dimensional equations for temperature and composition, namely ∂ ∂ ∂T¯ ∂2 T¯ ∂C¯ ∂2C¯ + wT = 2 , + wC = τ 2 , (45) ∂t ∂z ∂t ∂z ∂z ∂z where we expect w¯ = 0 otherwise mass is not conserved. Note that the hats have been dropped from the horizontally averaged quantities to avoid crowding the notations, but everything in this section is implicitly non-dimensional. If we then define the total fluxes



  ∂T¯ ∂C¯ Fˆ T,tot = wT − 1 + and Fˆ C,tot = wC − τ R−1 , (46) + 0 ∂z ∂z as the sum of the turbulent flux plus the diffusive flux of each quantity, then these equations simply become the conservation laws ∂C¯ ∂Fˆ C,tot ∂T¯ ∂Fˆ T,tot + = 0 and + = 0. ∂t ∂z ∂t ∂z

(47)

We also define two non-dimensional quantities: the Nusselt number NuT , and the flux ratio γtot , as Fˆ T,tot Fˆ T,tot NuT = − , (48) and γtot = 1 + ∂T¯ /∂z Fˆ C,tot which are similar to those defined in Table 1, but expressed as the ratio of non-dimensional ¯ The Nusselt number is the ratio of quantities, and including large-scale perturbations T¯ and C. the total temperature flux to the diffusive (potential) temperature flux, the latter being simply equal to −(1 + ∂T¯ /∂z) in these units. The key assumption made by the Radko [69] is that NuT and γtot can only depend on other local non-dimensional properties of the fluid. Aside from Pr and τ, which are fixed once the fluid is specified, the only other relevant non-dimensional quantity is the local density ratio. The latter is given by R=

1 + ∂T¯ /∂z , ¯ R−1 0 + ∂C/∂z

(49)

since 1 is the non-dimensional background potential temperature gradient, and R−1 0 is the nondimensional background compositional gradient. To determine the evolution of large-scale horizontally-invariant perturbations, we therefore simply evolve the equations in (47) together with equation (48), where the functions NuT (R; Pr, τ) and γtot (R; Pr, τ) are assumed to be known (they can be derived for instance from the Brown et al. model or from experimental data), and R given in (49). A trivial solution of these equations exists: when T¯ ≡ C¯ ≡ 0, then R = R0 , NuT = NuT (R0 ; Pr, τ) and γtot = γtot (R0 ; Pr, τ) are all constant, so the fluxes Fˆ T,tot and Fˆ C,tot are also constant. This defines a turbulent state that is spatially homogeneous and statistically steady – this is the basic state of small-scale fingering convection. When T¯ and C¯ vary with z, solutions cannot be found analytically in general because the functional form of NuT and γtot can be quite complex, and (49) is nonlinear. Instead, we proceed by linearizing the large-scale equations around the homogeneous fingering solution

Multi-Dimensional Processes In Stellar Physics

33

described above to study its stability to the development of large-scale perturbations. Let us ¯ therefore assume that T¯ and C¯ are small, so that ∂T¯ /∂z  1 and ∂C/∂z  R−1 0 . Then

 ∂C¯ ∂C¯ ∂T¯ ∂T¯ R  R0 1 + − R0 − R20 . (50) ≡ R0 + R → R = R0 ∂z ∂z ∂z ∂z We continue by linearizing NuT (R) (at fixed Pr and τ) in the vicinity of R0 , as  ∂NuT   NuT (R) = NuT (R0 ) + R . ∂R R=R0

(51)

Using this with (47) and (48), we get after successive simplifications (from linearization)  



 ∂T¯ ∂ ∂T¯  ∂NuT  = 1+ NuT (R0 ) + R ∂t ∂z ∂z ∂R R=R0  2¯ ∂T ∂NuT  ∂R  = 2 NuT (R0 ) + ∂R R=R0 ∂z ∂z 

2 ∂ T¯ ∂2 T¯ ∂2C¯ = 2 NuT (R0 ) + ANu , (52) − R 0 ∂z ∂z2 ∂z2  T   where ANu ≡ R0 ∂Nu ∂R R=R . Similarly, it is easy to show that 0

∂γ−1 ∂C¯ ∂  −1 ˆ  ∂T¯ −1 =− − Fˆ T,tot tot γtot FT,tot = γtot (R0 ) ∂t ∂z ∂t ∂z −1  ¯ ∂T ∂γtot ∂R ˆ −1 − = γtot (R0 ) FT,tot (R0 ) ∂t ∂R ∂z

2 

2   2 ∂ T¯ ∂ T¯ ∂2C¯ ∂2C¯ ∂ T¯ −1 = γtot Nu (R ) + A − R − R (53) + A NuT (R0 ), (R0 ) T 0 Nu 0 0 γ ∂z2 ∂z2 ∂z2 ∂z2 ∂z2  ∂γ−1  , and where we used the fact that in the absence of any large-scale where Aγ ≡ R0 ∂Rtot  R=R0 perturbations, Fˆ T,tot (R0 ) = −NuT (R0 ). ˆ and similarly for C, ˆ + Λt) ¯ we can substitute these Finally, assuming that T¯ ∝ exp(iKz solutions into (52) and (53), and with a little algebra, obtain a quadratic equation for the ˆ of these horizontally-invariant perturbations: non-dimensional growth rate Λ   ˆ2 +Λ ˆ Kˆ 2 ANu (1 − R0 γ0−1 ) + Nu0 (1 − Aγ R0 ) − Kˆ 4 Aγ Nu20 R0 = 0, Λ (54) where for simplicity of notation I have used Nu0 = NuT (R0 ) and γ0 = γtot (R0 ). Solutions for ˆ that have a positive real part denote instability. When unstable, perturbations of the form Λ ˆ or sin(Kz) ˆ (and similarly for C) ¯ grow exponentially with time, until the density T¯ ∝ cos(Kz) profile itself has regions that are dynamically unstable (i.e. where density increases upwards). As soon as this happens, overturning convection sets in, and stacked convective layers appear separated by sharp interfaces (see Figure 10). From the form of (54), we can immediately deduce a few important properties of the ˆ it is sufficient solutions. First, for unstable modes to exist (i.e. solutions with positive real Λ), to require that Aγ > 0, or equivalently, that γtot be a decreasing function of R. This is called the γ-instability criterion, first derived by Radko [69]. ˆ Instead, it is easy to Second, we see that there is no stabilization of the system for large K. 2 ˆ ˆ show that Λ is always proportional to K , so the smallest scale modes grow the most rapidly, which is not particularly physical. At the root of this so-called ultraviolet catastrophe, with

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Evry Schatzman School 2018

z


0 (z) +
(z)

Figure 10. Eigenmodes of the layering instability are sinusoidal perturbations in the density profile ρ(z). ¯ They grow exponentially, and eventually cause the total density profile ρ0 (z) + ρ(z) ¯ to increase with height at specific positions in the domain. Being unstable to overturning convection these regions are then fully mixed and become the layers of the staircase. Credit: S. Stellmach.

ˆ ∝ Kˆ 2 , is the anti-diffusive nature of the fingering fluxes when γtot decreases with R. To see Λ this, let us construct the evolution equation for the horizontally-averaged density perturbation (using as before a linearization of the equations around R = R0 )  ∂T¯ ∂C¯ ∂ ˆ ∂ρ¯ −1 =− + = ) FT,tot (1 − γtot ∂t ∂t ∂t ∂z ˆ ∂γ−1 ∂R ∂ F T,tot −1 = (1 − γtot − tot ) Fˆ T,tot ∂z ∂R ∂z

2  ˆ ∂ T¯ ∂2C¯ ˆ −1 ∂ F T,tot − Aγ − R0 2 FT,tot = (1 − γtot ) ∂z ∂z2 ∂z 2 ∂Fˆ T,tot ∂ ρ¯ ∂2C¯ − Aγ (R0 − 1) 2 Nu0 , = −Aγ Nu0 2 + (1 − γ0−1 ) ∂z ∂z ∂z

(55)

where I have linearized R to go from the second to the third line, and again used Fˆ T,tot (R0 ) = −Nu0 in the last line. Since Nu0 is always positive in fingering systems, we see that the first term behaves anti-diffusively if Aγ > 0 and diffusively if Aγ < 0, which confirms the statement made above. Note that in reality the mean-field equations stop being valid on scales approaching the finger scale, and should not be used in that limit. As such, this ultraviolet catastrophe is only a feature of the mean field equations but would not occur in a real system. The γ-instability model has been quantitatively validated in the geophysical context for fingering convection at high Prandtl number in both 2D [69] and 3D [70], confirming its ability to predict not only when layers should form, but also at what rate (as long as the wavenumber of the layering mode is not too large). As a result, we can use it to determine whether layers are expected to form in stars or not. To do so, we simply need to compute the function γtot (R) and apply the γ-instability criterion. Nondimensionally, in a homogeneous fingering simulation at density ratio R0 , γtot (R0 ) =

Fˆ T,tot wˆ Tˆ − 1 = , ˆ − τR−1 Fˆ C,tot wˆ C 0

(56)

ˆ from e.g. Traxler et al. so we can use the numerically-determined fluxes wˆ Tˆ and wˆ C and Brown et al. [21, 62] to compute γtot (R0 ) at moderate values of Pr and τ. The results are shown in Figure 11. In all cases, we see that γtot is a strictly increasing function of the density ˆ ratio. The same can be shown to be true at stellar values of Pr and τ, with estimates for wˆ C ˆ and wˆ T obtained using the Brown et al. model. To understand why this is the case, note that the turbulent fluxes decrease significantly when Pr and τ do, and become small compared with the diffusive fluxes [21]. As a result, for sufficiently low Pr and τ, we have γtot  R0 τ−1

Multi-Dimensional Processes In Stellar Physics

which clearly increases with density ratio. This effectively demonstrates that layering cannot be produced from the γ−instability in the stellar regime. 100

Pr = 0.333, τ = 0.333 Pr = 0.1, τ = 0.1 Pr = 0.1, τ = 0.333 Pr = 0.333, τ = 0.1 Pr = 0.1, τ = 0.033 Pr = 0.033, τ = 0.1 Pr = 0.1, τ = 0.01 Pr = 0.01, τ = 0.01


tot 10

Figure 11. The variation of γtot with R0 for a range of fingering simulations at varying Pr and τ smaller than one, showing that this function is always increasing. Figure from [21].

1

1

10

R0 Curiously, Brown et al. [21] did report that one of their simulations at very low density ratio spontaneously developed layers. However, these layers were not formed by the γ−instability, but instead, by the nonlinear development of large-scale internal gravity waves, that were themselves excited by the small-scale fingering. An example of such gravity waves can be seen for instance in the background snapshot of Figure 9. The spontaneous emergence of internal waves from fingering convection is another well-known large-scale instability, sometimes called the collective instability, that can also be modeled using mean field hydrodynamics [20, 63, 71]. The algebra associated with that calculation is however much more involved than the one outlined above to model the layering instability, so I will not derive it here. The reader is referred to the work of Traxler et al. [63] for details of the calculation (see also [20]). Using the turbulent flux laws from Brown et al. [21] to close the mean field equations, we were able to determine that fingering convection is expected to excite internal gravity waves down to Pr, τ ∼ 10−3 but not lower [1, 20]. Furthermore, they can only develop when the density ratio is close to one (i.e. close to the Ledoux limit). This finding rules out the excitation of gravity waves from basic fingering convection in non-degenerate regions of stellar interiors (where Pr, τ ∼ 10−6 ), but not in degenerate regions of e.g. WDs and evolved stars. Whether a fingering region would extend all the way into the degenerate region remains to be determined, however. Furthermore, even if gravity waves are indeed excited by this mechanism, they would remain relatively small scale, so would probably not be observable anyway [20]. As such, this effect seems to be more of an interesting curiosity rather than something that is likely to impact stellar evolution and observations6 . 3.5 Conclusions for now

Having established that fingering convection is not likely to spontaneously excite larger-scale dynamics (layers or internal gravity waves) at low Prandtl number and diffusivity ratio, we can conclude that in the absence of any other dynamical process (rotation, magnetic fields, shear, etc.), fingering-induced mixing in stars remains small scale, and is well-described by the model of Brown et al. [21] (see Section 3.3). The role of rotation, magnetic fields, 6 Of

course I would love to be proved wrong.

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Evry Schatzman School 2018

and shear on fingering-induced mixing is the subject of ongoing work and will be briefly discussed in Section 5. 3.6 Applications to stellar astrophysics

In Section 2.3, we saw a number of examples where fingering convection may occur in stars. Armed with a better quantitative understanding of the process, we are now equipped to answer the question of how much mixing it causes and what its observable effect on stellar evolution may be. Here, it is almost impossible to give a comprehensive review of the topic. Instead, I will focus on a few examples where the newly established transport laws have enabled us to make somewhat definitive statements about the role of fingering in explaining (or not explaining, in some cases) observations. Understanding the surface metallicity of stars undergoing accretion of high-μ material from infalling planets is one such example. It is thought that, through the combined effect of type I migration in a protostellar disk (which can bring terrestrial planets close to the central star, [72]) and tidal interactions (which causes orbital decay of sufficiently close-in planets even in the absence of disk), planets could regularly fall into their host star even after the disk has disappeared [73]. The event is not expected to have a noticeable effect on the surface metallicity if the star has a deep outer convective zone, but could be important for stars with sufficiently shallow outer convection zones (or none at all). Planetary infall has thus been proposed as a possible mechanism that could explain the observed planet-metallicity correlation [34, 74]. Vauclair [37] (see also [39]) correctly argued that this scenario would not work if mixing induced by fingering convection drains the metals into the interior on a short timescale compared with the time since the last infall event (for a given star). Using the mixing coefficient proposed by Traxler et al. [62] (see equation 39), which is valid in the limit where the density ratio is not too small (which is the case in these systems), I confirmed Vauclair’s idea and demonstrated that any evidence for an infall event would disappear on a timescale of around 100Myr (see Figure 12). Since this is relatively short compared with the typical age of planet-bearing stars, we are left to conclude that the planet-metallicity correlation must be of primordial origin [38]. Fingering convection similarly affects the surface metallicity of WDs undergoing accretion from a planetary debris disk, and this effect should be taken into account if one wishes to use the observed metallicities to derive the debris accretion rates [40–42]. Meanwhile, RGB stars are a good example of objects where a better understanding of fingering convection has made it harder to explain observations. Detailed spectroscopy of metal-poor stars by Gratton et al. [75] has revealed that the surface abundances of lithium and of elements participating in the CNO cycle changes noticeably along the RGB. A first sudden change occurs as expected during the first dredge-up event (i.e. when the outer convection zone penetrates most deeply into the star), see [76] for early work on the topic. A second much more unexpected change occurs once the convection zone has retreated, around the socalled luminosity bump that corresponds to the time where the hydrogen-burning shell enters the region that was previously mixed in the first dredge up, see Figure 13. Eggleton et al. [49] noted that 3 He burning (3 He + 3 He → 4 He + p + p) is the dominant reaction in the cooler outer edge of the hydrogen burning shell, and locally lowers the mean molecular weight slightly. This can cause an inversion of the mean molecular weight gradient, but only once the shell has moved into a region that was previously homogenized by the dredge-up (see Figure 14). In the 3D simulations of Eggleton et al. [49], this inversion caused the development of a Rayleigh-Taylor instability, which mixed material between the hydrogen burning shell and the convection zone above. Charbonnel & Zahn [50] however pointed out that this inverse μ-gradient would first become unstable to fingering convection, rather

Multi-Dimensional Processes In Stellar Physics

37

Figure 12. Evolution of the metallicity profile and compositional mixing coefficient Dμ (calculated using equation 39), in a 1.4M model impacted by a 1 Jupiter mass planet. Left: Evolution of the surface metallicity of the star μ s with and without taking fingering into account. The early-time drop in μ s from the initial value μ0 of the polluted material to the intermediate value μ1 is caused by mixing within the surface convection zone, while the later, slower decay in μ s corresponds to the effect of mixing below the convection zone (the decay in the case without fingering is due to microscopic molecular diffusion). Right: Evolution of the subsurface metallicity profile (top) and compositional mixing coefficient (bottom) after 105 yrs, 106 yrs, 107 yrs, 108 yrs, and 7×108 years. Figure adapted from [38].



 
 






 
 





















Figure 13. Left: Surface abundances of metal poor field stars as a function of luminosity, adapted from Gratton et al. [75]. The red line marks the approximate location of the first dredge up, and the green line marks the approximate location of the luminosity bump. The dashed horizontal green line marks the abundances a star would have in the absence of post-dredge-up mixing. Right: Kippenhahn diagram for a 1.3M , Z = 0.2 star, illustrating how the luminosity bump occurs when the hydrogen burning shell reaches the radius of lowest descent of the convection zone during prior dredge-up. Figure produced by C. Cadiou using MESA.

than the Rayleigh-Taylor instability. By including the effects of fingering convection in their stellar evolution code, modeled using the traditional formula where Dfing = C f κT /(R0 −1) (see equation, e.g. 32), they were able to explain the Gratton et al. data provided C f = O(1000), but not if C f = O(10). At the time, this was viewed as evidence in favor of the Ulrich prescription for fingering convection [9], but we now know that such a high constant C f is

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Evry Schatzman School 2018

not supported by the numerical experiments presented in this lecture (see Figure 7 and also [60, 77]). 












 

  


    


m/M  





  


m/M  





  


Figure 14. Left: Illustrative sketch of the mean molecular weight profile before the luminosity bump. Right: Same but after the luminosity bump. In that case, an inverse μ gradient forms that drives fingering convection between the hydrogen burning shell (HBS) and the convection zone (CZ).

This is a very frustrating state of affairs, since in all other respects the Charbonnel & Zahn model [50] provides a simple and elegant explanation for the data: fingering convection is triggered at exactly the right time and in the right place to cause the extra mixing needed – but its efficiency is too low. With this in mind, we dedicated the last 5 years (since 2013) trying to determine if another process, in combination with fingering convection, could increase the induced turbulent transport. This was at the heart of our attempts to establish whether thermo-compositional staircases could form through fingering in stars, but it really seems that they cannot (see Section 3.4 and [20]), at least spontaneously. It was also what prompted us to study added physics, such as rotation [23], magnetic fields [24] and shear [25], all of which are present in RGB stars. Of all these processes, magnetic fields appear to be the most promising. Preliminary results for fingering convection in a medium permeated by a vertical background magnetic field of amplitude B0 obtained by Harrington and Garaud [24] suggest that  2.5B20 NT2 + NC2 Dfing,mag (B0 )  , (57) ρ m μ0 −NT2 NC2 for sufficiently large magnetic field strengths, where μ0 is the permeability of the vacuum, and all the other quantities were defined earlier. This coefficient can be two orders of magnitude larger than one appropriate for non-magnetic fingering for B0 as low as a few hundred Gauss, which are plausibly present in these stars [24]. More work is needed, however, to establish whether this result holds for arbitrarily aligned magnetic fields.

4 Oscillatory double-diffusive convection and layered convection 4.1 Traditional models of mixing by ODDC

As discussed in Section 1, the existence of regions that are both Schwarzschild-unstable but Ledoux-stable predates the discovery of ODDC by several years, and ODDC did not really become widely known in the astrophysical literature until much later. As such, there are many different prescriptions for mixing in semiconvective regions that have very little to do with the process of ODDC. These prescriptions are usually quite simplistic, either assuming that the region is adiabatically stratified, or purely radiative, or some interpolation between the two regimes, with some associated mixing model for chemical species.

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It was only later that the first attempts to create models of semiconvective mixing based on the physics of ODDC were put forward [10, 12, 13, 31, 78]. Owing to its simplicity, the model of Langer et al. [12] is probably the most commonly-used one in stellar evolution codes today. It can be derived using the same arguments as the ones I presented in Section 3.1 to recover the Ulrich [9] and Kippenhahn et al. [11] models for fingering convection. If we simply assume that semiconvection acts as a turbulent diffusivity with coefficient DC = Dsemi , then by dimensional arguments ˆ T (58) Dsemi ∝ λd2 = λκ ˆ ˆ ˆ 2 −1 As discussed in Section 2.2.2, for sufficiently large R−1 0 , λ ∝ kh /(R0 − 1), where kh is close to one. As a result, we can estimate Dsemi as Dsemi = CL

κT , R−1 0 −1

(59)

where CL is a constant factor that can only be determined by comparison with experimental data. This expression is very reminiscent of the traditional fingering coefficient Dfing (see equation 32), with R0 replaced by R−1 0 . Using astrophysical notations that might be more familiar to this audience, this becomes Dsemi = CL

κT δ(∇T − ∇ad ) , φ∇μ − δ(∇T − ∇ad )

(60)

which recovers the prescription of Langer et al. [12]. The coefficient CL is usually argued to be of order unity. Note that heat transport in this model is assumed to be negligible, so the background temperature gradient is the radiative one. Less well known perhaps is the model of Stevenson [31], who argued that the saturation of the unstable oscillatory modes of ODDC arises from a parametric subharmonic instability, in which smaller scale modes rapidly grow once the parent mode has reached a certain amplitude. He argues, using analogies with the geophysical literature, that the mixing coefficient should take the form7 Dsemi = CS R20 κT , (61) which, for R−1 0 1, only differs from the Langer et al. proposal in the exponent applied to the inverse density ratio R−1 0 . Stevenson and his colleagues [31, 78] were also the first to clearly argue for the existence of a distinct regime where layered double-diffusive convection takes place (though Spiegel [8] hinted at its possibility) instead of the small-scale turbulence considered so far, and to discuss where in parameter space it could take place. Indeed, as discussed in Section 2.2.2, the range of linear instability for ODDC in geophysics (and in laboratory experiments) is very small, so ODDC almost never derives from it. Instead, it is excited from a subcritical branch of instability. As a result, none of the existing laboratory experiments on ODDC show the presence of unstable gravity waves, but instead, all take the form of layered double-diffusive convection (layered convection, for simplicity) [19, 79–81]. Layered convection is also ubiquitously found in nature on Earth in the combined presence of unstable temperature gradients and stable salt gradients, such as in volcanic lakes [82], in the arctic ocean [83], and under the ice shelf in the antarctic [84]. The temperature and salinity profiles associated with layered convection take the form of a thermohaline staircase similar to fingering staircases discussed in Section 3.4, although the overall gradients now have the opposite signs. An entirely different class of astrophysical semiconvection models therefore exists that assumes the presence of layered convection [13, 78, 85–87]. Despite their differences, these 7 See

equation 31 of [31].

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models generally start from the same assumption, namely that the thermocompositional staircase is in equilibrium. This implies that the fluxes of T and C through a layer must be equal to the fluxes of T and C through the adjacent interfaces on either side, otherwise the interfaces and/or the layers would have to evolve on a rapid timescale8 . As a result of this assumption, one can easily build a transport model from two ingredients only: the first is a prescription for the temperature flux within a convective layer (which is not negligible in layered convection), and the second is a prescription for the ratio of temperature and compositional fluxes across an interface. Once these two quantities are known, the fluxes of heat and composition through the entire staircase can easily be computed. The turbulent temperature flux through individual layers can be put forward on dimensional grounds, to be  T FT ≡ wT = −(NuT − 1)κT T 0z − T ad,z = κT (NuT − 1)δ(∇T − ∇ad ), Hp

(62)

where NuT is a potential temperature Nusselt number, defined in Table 1; going from the second to the third expression above uses that definition. Traditional geophysical models of overturning convection between solid boundaries separated by a distance H usually argue [64, 88–91] that for very turbulent flows the Nusselt number should be equal to some power of the so-called Rayleigh number Ra = |NT2 |H 4 /κT ν, which is the ratio of the buoyancy force driving convection, to the viscous force that damps fluid motions. In layered convection in stars, however, the viscous force is thought to be negligible, and the relevant Rayleigh number is |N 2 |H 4 (63) Ra ≡ T 2 L = RaPr. κT instead, where HL is the layer height. As such, models of layered convection in stars traditionally have (64) NuT − 1 = Csemi Raa , where the pre-factor Csemi , and the power a, vary between models (for instance a = 1/4 in [13] and a = 1/3 in [85, 86]). We see that FT depends rather sensitively on the layer height, which must also be specified by the model. Without further justification, HL is usually taken to be some fraction of the pressure scaleheight, to be specified by the user. Note that unless HL is very small, NuT is much larger than one, so the diffusive contribution to the heat transport is in this case negligible. The second ingredient of these models is a parametrization for the non-dimensional ratio −1 (see Table 1). In this case, all traditional of the temperature flux to the compositional flux, γtot astrophysical models of layered convection [13, 78, 85, 86] agree with one another and are based on laboratory experiments [79, 92] and theory [80] of layered convection in salt water. In this theory, the interface is assumed to be entirely diffusive, so the ratio of the interfacial fluxes is equal to the ratio of the diffusive fluxes, which in turn depends on the respective temperature and salinity gradients across the interface. With further assumptions, Linden & Shirtcliffe [80] arrive at the conclusion that  κC −1 = = τ1/2 , (65) γtot κT (with no proportionality constant between the two expressions). This prediction is consistent with many of results obtained in laboratory experiments, and is generally considered to be correct in the geophysical literature. 8 Evolution

on the slower stellar evolution timescale is of course allowed.

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This flux ratio can then be used to compute the compositional flux across the staircase given the temperature flux, as in  βFC α = τ1/2 → FC = − τ1/2 (NuT − 1)κT T 0z − T ad,z , αFT β

(66)

which in turn implies that the compositional mixing coefficient for layered semiconvection can be written as FC κT −1 (67) Dsemi = − = γtot (NuT − 1) −1 , C0z R0 −1 = τ1/2 . Again, unless the layer height is where the Nusselt number is given in (64) and γtot very small, this quantity is usually much larger than the microscopic diffusion coefficient κC , which can be neglected.

4.2 Numerical simulations of ODDC

As in the fingering case, none of these models had ever been tested in the low Prandtl number regime more appropriate of stellar astrophysics until recently. In particular, whether ODDC takes the form of small-scale wave-like turbulence or layered convection remained unknown. Early simulations of ODDC/semiconvection in a two-dimensional vertically bounded domain were presented by Merryfield [57] (see also [93]), but the results were limited in scope by the resolution affordable at the time. The first three-dimensional direct numerical simulations of ODDC at low Prandtl number were presented by Rosenblum et al. [94], using the PADDI code [63] with the model setup described in Section 2.1, and solving equations (11)– (13) with the minus sign in the temperature and composition equations. In this paper, we demonstrated that both layered and non-layered outcomes are possible, depending on the local thermocompositional stratification (as measured by the inverse density ratio R−1 0 ). This was later confirmed by Mirouh et al. [22], who performed a more comprehensive exploration of parameter space. These two possible outcomes are illustrated in Figure 15. For high inverse density ratios, i.e. for systems that are more strongly stratified, we have found that ODDC is excited as expected from the linear theory described in Section 2.2.2, and saturates into a state of weak wave turbulence. The fastest-growing modes are, as expected, elevator modes. However, once they reach a certain amplitude, nonlinear interactions cause a transfer of energy to modes that have a higher vertical wavenumber (see Figure 15b). This is at least qualitatively consistent with the idea put forward by Stevenson [31]. The turbulent fluxes in that regime are fairly weak, and decrease with increasing R−1 0 , and decreasing Pr and τ (see below). For low inverse density ratios, on the other hand, the initial state of wave-like turbulence always transitions to a layered state. The initial height of the convective layers is fairly small, of the order of a few tens of d, but the layers then always merge until a single one is left in the computational domain (the mergers cannot proceed beyond that owing to the periodicity of the boundary conditions). The initial formation of the layers, and each subsequent merger, is accompanied by a substantial increase in the turbulent fluxes, suggesting that the latter indeed depend on the layer height. In view of these results, it is clear that any model of ODDC/semiconvection in stars should include a way to determine which regime is expected (layered or non-layered), a prescription for transport in the layered regime and a prescription for transport in the non-layered regime [22].

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3 2.5




2




1.5 1 0.5 0 -0.5 -1 0

1000

2000

3000

4000

5000

6000

t

Figure 15. Snapshots of the compositional perturbations in the two forms of ODDC, extracted from simulations at Pr = τ = 0.03, see text for detail. Left: Layering at low inverse density ratio (R−1 0 = 1.5) causes substantial increase in the turbulent temperature (solid line) and compositional (dotted line) fluxes. Right: No layering at high inverse density ratio (R−1 0 = 5). The turbulent fluxes remain weak. Figure from [1], with data from [22].

4.3 Transport in non-layered ODDC

Starting with the latter, we can use the simulations of Mirouh et al. [22] to extract turbulent fluxes in non-layered convection9 , and compare them to the predictions of Stevenson [31] and Langer et al. [12] (which are for the non-layered regime). The turbulent diffusion coefficient ˆ As Dsemi is extracted from the simulations as usual by computing − wC /C0z = R0 κT wˆ C . a side note, it is easy to show using arguments very similar to those put forward in the finˆ and wˆ Tˆ have to be positive in a statistically gering regime (see Section 3.2), that both wˆ C stationary state. As such, Dsemi is indeed positive. The results are shown in Figure 16a. In the model of Stevenson [31], Dsemi /κT ∝ R20 −1 while in Langer et al. [12] Dsemi /κT ∝ (R−1 0 − 1) . The data clearly favors a model where − 1 and the slope is roughly equal to −1.5, which is Dsemi /κT is a power law function of R−1 0 somewhere in between these existing models, at least for large enough R−1 0 . We also see that these models fail to account for the stabilization of the system (with Dsemi → 0) that occurs when R−1 0 approaches the marginal stability threshold (Pr + 1)/(Pr + τ). In an attempt to create a new model for mixing by ODDC in the non-layered regime, we (i.e. me and my students) have tried various approaches, including an extension of the Brown et al. model to the ODDC case, and various attempts at weakly nonlinear theory to capture the nonlinear evolution of the unstable gravity waves. None of these approaches were able to model the variation of Dsemi with all input parameters (Pr, τ, and R−1 0 ) satisfactorily. This does not mean that such a theory cannot be developed – only that we have been unable to do so to date. For lack of a more theoretically-grounded model, Mirouh et al. [22] put forward an empirical parametrization of the existing data that seems to capture adequately the variation of both compositional and temperature fluxes across parameter space. In this parametrization, the non-dimensional turbulent temperature flux Fˆ T = wˆ Tˆ is proposed to be

1/4 Pr 1−τ (68) (1 − ro ), Fˆ T = 0.75 τ R−1 0 −1 9 For

simulations that become layered, we extract the fluxes prior to layer formation

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10

Dsemi
T

1

Pr = 0.3, tau=0.3 Pr = 0.1, tau=0.1 Pr = 0.03, tau=0.03 Pr = 0.01, tau=0.01

10

wˆ Tˆ

0.1

43

1

0.01 Pr = 0.3, tau=0.3 Pr = 0.1, tau=0.1 Pr = 0.03, tau=0.03 Pr = 0.01, tau=0.01

0.001 0.0001 0.1

0.1

1

10

0.01

-1 R0 -1

0

0.2

0.4

0.6

0.8

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ro

Figure 16. Left: Variation of the ratio Dsemi /κT with R−1 0 − 1, in the data, and comparison with models −2 −1 where Dsemi /κT ∝ (R−1 and (R−1 respectively (grey lines). A better fit to the data suggests 0 − 1) 0 − 1) −1.5 that Dsemi /κT ∝ (R−1 (black line), except when R−1 0 − 1) 0 approaches marginal stability. Right: Nondimensional turbulent temperature flux. The symbols are from the data [22], and the lines are from the model given in equation (68).

where R−1 oc = (Pr + 1)/(Pr + τ) is the marginal stability threshold for ODDC, and ro =

R−1 0 −1 , R−1 oc − 1

(69)

is the reduced inverse density ratio associated with ODDC. This parameter, as in the fingering −1 case, maps the instability range R−1 0 ∈ [1, Roc ] to the interval ro ∈ [0, 1]. A comparison of (68) with data from [22] is shown in Figure 16b, suggesting that this models is adequate. We also note that Fˆ T rapidly decreases with Pr and τ, to the extent that for any reasonable stellar value Fˆ T is either smaller than or negligible compared to the diffusive flux (which is one in these units). We therefore confirm that the turbulent heat flux is negligible in comparison with the diffusive flux for non-layered ODDC in stars. To model the compositional flux, we could use a similar empirical parametrization, starting for instance from the scaling with R−1 0 − 1 discovered in Figure 16a; this may be an interesting idea to pursue in the future. Generally speaking, the data suggests that Dsemi rapidly decreases with increasing stratification. Moll et al. [95] in fact suggested that compositional mixing in the non-layered regime (which takes place at large R−1 0 ) is negligible compared with pure diffusive transport at stellar parameters. This remains to be confirmed. As an alternative approach, in Appendix A.3. of Mirouh et al. [22] we extended the work of Schmitt [14, 96] (also used by Brown et al. [21]), who noted that it is possible to estimate the ratio of the turbulent temperature flux to the turbulent compositional flux, named γturb (see Table 1) in double-diffusive systems simply from linear theory, even though nonlinear arguments are needed to estimate each of them individually. Indeed, using non-dimensional ˆ Assuming that w, fluxes, we can write γturb = wˆ Tˆ / wˆ C . ˆ Tˆ and Cˆ are dominated by the fastest-growing linearly unstable modes, we can replace them by their proposed ansatz from linear theory (see Section 2), to get γturb

    ˆ Re T˜ exp(ikˆ h x + λt) ˆ Re w˜ exp(ikˆ h x + λt)     , = ˆ Re C˜ exp(ikˆ h x + λt) ˆ Re w˜ exp(ikˆ h x + λt)

(70)

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In this expression λˆ and kˆ h are the growth rate and wavenumber of the fastest-growing elevator modes for the desired set of parameters Pr, τ and R−1 0 , computed from linear theory (see Section 2). From the linearized versions of equations (11)–(13), we have the following system of equations relating the amplitudes w, ˜ T˜ and C˜ of each mode. λˆ T˜ ± w˜ = −kˆ h2 T˜ ,

˜ ˜ = −τkˆ h2C, λˆ C˜ ± R−1 0 w

(71)

˜ where as usual + refers to the fingering regime and − refers to ODDC. Solving for T˜ and C, we get R−1 w˜ w˜ ˜=∓ 0 , C . (72) T˜ = ∓ λˆ + kˆ 2 λˆ + τkˆ 2 h

h

If we finally assume, without loss of generality, that the amplitude w˜ is real, then we combine equations (70) and (72) to get the following prediction for the turbulent flux ratio γturb [22]: γturb = R0

(λˆ R + τkˆ h2 )2 + λˆ 2I λˆ R + kˆ h2 , (λˆ R + kˆ 2 )2 + λˆ 2 λˆ R + τkˆ 2 h

I

(73)

h

where we have written λˆ = λˆ T + iλˆ I . Conveniently, the ± signs cancel out, so this expression can be used for both fingering convection and ODDC. Even more conveniently, the unknown amplitude of the mode w˜ cancels out entirely from that expression, so γturb does not contain any unknown parameter. A comparison of this theory with the simulation data (in terms of γtot , rather than γturb ) is presented in the next section, and shows reasonably good (but not perfect) agreement between the two. This expression can finally be used to compute the dimensional mixing coefficient −1 ˆ = R0 κT γturb wˆ Tˆ , Dsemi = R0 κT wˆ C

(74)

with wˆ Tˆ given by (68). One may rightfully argue that this method is much more complicated than proposing a simple empirical formula for Dsemi based on the data shown in Figure 16, which would be more practical for stellar evolution purposes. I do not disagree, especially since we find that in most instances Dsemi  κC at stellar parameters, making the additional effort of computing γturb (which involves solving a cubic equation) somewhat pointless. It does however have the advantage (in my opinion) of being based on solid theoretical arguments for why the turbulent flux ratio γturb should be of a particular form, and can in addition be used to establish a criterion for layer formation, as shown below. 4.4 Criterion for layer formation

As discovered by Rosenblum et al. [94] and Mirouh et al. [22], simulations of ODDC at a low inverse density ratio always spontaneously evolve into a state of layered convection, suggesting that the mechanism leading to layer formation must be very robust. In Section 3.4, we learned about the γ-instability as a mechanism for layer formation in oceanographic fingering convection, and found that it should occur whenever the total flux ratio γtot is a decreasing function of the density ratio R0 . It is therefore natural to wonder whether a similar mechanism may be at play here. As it turns out, the mechanism is not just similar, it is indeed exactly the same. To see this, simply note that the horizontally-averaged temperature and composition equations which were used as the starting point for the development of the γ−instability theory in Section 3.4 (see equation 45) are exactly the same in fingering convection and in ODDC, since the

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horizontal average of the vertical advection terms containing ±w are identically zero (for mass conservation). As a result, Rosenblum et al. [94] demonstrated that the criterion for layer formation in ODDC is the same as in fingering convection: layers are expected to form −1 is a decreasing function of when γtot is a decreasing function of R0 , or equivalently, when γtot . R−1 0 This theory can be tested from the extended dataset of Mirouh et al. [22], and some of the results are presented in Figure 17a. All symbols in this figure show the measured values −1 extracted from the data, by computing of γtot −1 γtot =

ˆ τR−1 ˆ C 0 + w 1 + wˆ Tˆ

(75)

in the non-layered phase (this expression is the non-dimensional version of the one given in Table 1). Large symbols in this figure denote simulations that eventually transitioned into −1 is a decreasing function layered convection. We see that this indeed happens whenever γtot −1 of R0 , confirming the role of the γ-instability as suspected. A more quantitative way of testing the theory is to compare the predicted and observed growth rates of the layering modes in simulations. Let’s take for instance a simulation presented by [94], with Pr = τ = 0.3 and R−1 0 = 1.2. In this simulation, 4 layers were first observed to form, though by inspection of the spectral power in the horizontally-averaged density field (see Figure 17b) we see that the 3-layer mode is also growing. In order to ˆ of the layering modes, we first need to estimate the compute the theoretical growth rate Λ parameters of the quadratic dispersion relation (54). We can easily measure the potential temperature Nusselt number Nu0 and the total flux ratio γ0 in the non-layered phase preceeding layer formation in the simulation. We can also, by running simulations at slightly  −1  ∂γtot ∂NuT    . We find that different R−1 0 , compute ANu = R0 ∂R R=R and Aγ = R0 ∂R  0

R=R0

Nu0  3.4, γ0−1  0.55, ANu  12.9 and Aγ  0.45,

(76)

for Pr = τ = 0.3 and R−1 0 = 1.2. Using these parameters and coefficients in the quadratic equation (54), we then predict that ˆ K) ˆ  0.47Kˆ 2 Λ(

(77)

for this simulation. We can compare the temporal evolution of the spectral power in density for each growing mode with this prediction, knowing their wavenumber is Kˆ = kn , where kn is the wavenumber with n layers in the domain. We see in Figure 17b that the growth rate of the 4-layer mode is over-predicted by the model, but that of the 3-layer mode is adequately captured. As discussed in Section 3.4, the fact that the mean-field theory fails when the mode wavenumber is large is not surprising given the ultraviolet catastrophe inherent to the mean field model, but it is very encouraging to see that it works well for the 3-layer mode. Similar comparisons were later made by Mirouh et al. [22], with equally satisfactory success. This completes the validation of the γ−instability theory for layer formation in ODDC. If one believes the Mirouh et al. [22] prescriptions for the turbulent heat flux and turbulent flux ratio described above, it is easy to predict when layers are expected to form in stellar ˆ we have ODDC. Indeed, from (75) and the definition of γturb = wˆ Tˆ / wˆ C −1 = γtot

−1 ˆ Tˆ τR−1 0 + γturb w . 1 + wˆ Tˆ

(78)

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0.9

102

0.8 0.7

Spectral power in density

Pr = 0.3, tau=0.3 Pr = 0.1, tau=0.1 Pr = 0.03, tau=0.03 Pr = 0.01, tau=0.01

0.6





1 0.5 tot 0.4

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100 10-2

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(0,0,k1) (0,0,k2) (0,0,k3) (0,0,k4)

0.1 0 0

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1000 t

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−1 Figure 17. Left: The symbols show the total inverse flux ratio γtot measured from simulation data in non-layered ODDC for various Pr and τ (see [22] for a more extended dataset). Larger symbols denote −1 simulations that ultimately become layered. The solid lines are predictions for γtot from the model of Mirouh et al. [22]. Figure adapted from [1]. Right: The solid lines show the growth of the spectral power in the horizontally-average density field for various layering modes with wavenumber kn (where n is the number of layers in the domain) for a simulation with Pr = τ = 0.3 and R−1 0 = 1.2 [94]. The 3- and 4-layered modes grow until they reach an amplitude that causes inversions in the density profile (black line), at which point layered convection begins. The 1- and 2-layered modes do not initially grow but emerge out of subsequent layer mergers. The dotted lines show the corresponding prediction from the γ−instability theory for the growth of the 3-layered and 4-layered mode.

−1 We can then use (68) and (73) to predict γtot as a function of R−1 0 for any parameter set (Pr, τ). A comparison of this prediction with existing data is shown in Figure 17a, and reveals very good agreement (see [22] for a comparison with a more extended dataset), giving us some −1 is a decreasing confidence in the model. Using this method, we can determine when γtot −1 function of R0 for any Pr and τ in the stellar parameter regime. The results were first presented by Mirouh et al. [22] and are shown in Figure 18, which is a contour plot of the critical value of R−1 0 below which layering can occur, as a function of both Pr and τ. Generally speaking, we see that when Pr > τ1/2 the layering threshold is more-or-less independent of τ and is roughly equal to Pr−1 . This is not a particularly relevant situation in stellar astrophysics, however. On the other hand, when Pr < τ1/2 , then the layering threshold is roughly equal to τ−1/2 (with a weak dependence on Pr). For stellar interiors with Pr and τ of order 10−6 , we therefore expect layers to spontaneously form whenever the inverse 3 density ratio R−1 0 < O(10 ), implying that all cases of semiconvective zones near a convective core should lie in the layered regime. For planetary interiors or degenerate regions of stellar interiors with Pr and τ of order 10−2 , we expect layers to spontaneously form whenever the inverse density ratio R−1 0 < O(10) [22].

3 10 30 100 300 1000

Figure 18. Contour lines of value of the inverse density ratio R−1 0 below which layered convection is expected, as a function of Pr and τ, according to the model of Mirouh et al. [22]. The dashed line is the Pr = τ1/2 line. Figure adapted from [22].

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4.5 Mixing in layered convection

Having established when layered convection is naturally expected, we now turn to the question of modeling turbulent transport induced by the process. Figure 15a revealed that the total heat and compositional flux through layered convection depends on the layer height, and increases every time a merger occurs. To study this more quantitatively, Wood et al. [97] (see also earlier preliminary work by [94]) re-analyzed the simulations of Mirouh et al. [22] that became layered, and also presented new simulations that were produced using the same model setup and code, but in much larger domains. They measured the average turbulent temperature and compositional fluxes in the domain as a function the average layer height. With this information, the Rayleigh number Ra , the potential temperature Nusselt number −1 NuT and the total flux ratio γtot can be computed and compared with the models introduced in equation (63) for layered convection. The results for existing simulations with varying Pr, τ and R−1 0 are shown in Figure 19. Figure 19a reveals that NuT is indeed a power law function of Ra , with: 1/3 NuT  1 + Csemi (τ, R−1 0 )Ra ,

(79)

with Csemi ∼ O(0.1) when fitted to the existing data. The exponent of that power law therefore rules out [13] but is consistent with [85, 86]. The pre-factor Csemi seems to have an additional weak dependence on both R−1 0 and τ, which could not be determined without much more demanding numerical simulations [97]. Indeed, we just saw that layered convection only −1/2 ). spontaneously emerges when R−1 0 is smaller than the layering threshold (which is ∼ τ −1 As such, running simulations of layered convection for a wider range of R0 requires using much smaller values of τ, which has so far been computationally prohibitive. Figure 19b tests the second assumption of traditional models [13, 78, 85, 86], namely −1 −1 that γtot = τ1/2 [80]. We see that this prescription does not hold: γtot is generally larger 1/2 that τ (see also the results of Moll et al. [33]), having instead a complicated dependence −1 on Pr, τ and R−1 0 , and perhaps even on the layer height (though there are hints that γtot may become independent of HL for sufficiently tall layers). The discrepancy between these numerical results and the Linden and Shirtcliffe [80] model for the interfacial flux ratio is easy to understand. The interfaces in their model (and more generally in the high Prandtl number laboratory experiments of layered convection for which the model was created) are quiescent so the interfacial transport is purely diffusive. However, the interfaces in our low Prandtl number simulations are quite turbulent (see Figure 15). This example nicely illustrates the risk of applying geophysically-derived parametrizations to stellar astrophysics without verifying them first with low Prandtl number numerical experiments. Based on their limited data, Wood et al. [97] proposed the following empirical parametrization for the compositional mixing coefficient: Dsemi = (NuC − 1)κC  0.03Pr−0.12 Ra 0.37 κT .

(80)

Note, however, that combining this with (79) would imply that −1 γtot = τR−1 0

NuC  0.3R−1 0 , NuT

(81)

(neglecting very weak powers of Pr and Ra and assuming that NuC , NuT 1). This is not −1 may be decreasing instead of really consistent with Figure 19b, where it appears that γtot −1 increasing with R0 . This discrepancy most probably comes from the fact that neither (79) nor (80) correctly account for the weak dependence of NuT and NuC (and in turn Dsemi ) on

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106

107

Ra


108

0.4

0.2

109

0 103

104

105

106 R P

107

Ra


108

109

Figure 19. Left: Nusselt number as a function of Rayleigh number Ra for simulations at various Pr, τ 1/3 (see symbols legend) and R−1 0 (see color legend). The solid line shows the relationship NuT = 0.2Ra . −1 Right: Total inverse flux ratio γtot as a function of Ra for the same simulations. Figure from [1] based on data from [97].

−1 R−1 0 . This problem can only be resolved by running simulations for a wider range of R0 , which is computationally very challenging as discussed above. Ultimately, however, the main uncertainty in modeling both heat and compositional fluxes in layered convection is their dependence on the layer height (i.e. with FT,C ∼ HL4/3 ). In all the simulations performed to date, the layers were ultimately seen to merge until a single one remains in the domain. This poses two problems. First, it has prevented us from proposing a model for what the layer height may be in a star, since it is not clear what mechanism (if any) eventually stops the merger process. Secondly, these results seem to challenge the original assumption of an equilibrium staircase, suggesting instead that layered convection may be a short temporary phase in the evolution of a star, whose duration depends on the layer merger rate.

4.6 Conclusions for now

Our numerical results and theoretical investigations have confirmed that there are two regimes of ODDC / semiconvection: one that is layered and one that is not layered. The layered form arises spontaneously from the γ−instability [22, 69, 94] at sufficiently low inverse density ratio. The threshold for the layering instability to occur can be computed in the manner −1/2 described in Section 4.4 [22], and is approximately given by R−1 at stellar (and plan0 τ −1 etary) parameters. Note that layered convection can also exist for R0 beyond that threshold, but only if layers are seeded, i.e. if they are present in the initial conditions applied to the problem (see [57], [93], [33]). Whether this regime is relevant for stars or not remains to be determined, since one would have to specify why these layers should be present in the first place. Both turbulent heat and compositional transport in layered convection are very efficient, and can be computed as follows [97]: Fheat,semi = ρc p FT =

ρc p T κT (NuT − 1)δ(∇T − ∇ad ), Hp

(82)

with NuT − 1  Csemi Ra1/3  ,

(83)

Multi-Dimensional Processes In Stellar Physics

where Ra depends on the layer height as HL4 (see equation 63), and Csemi is of the order of 0.1, with some possible weak dependence on τ and R−1 0 that remains to be determined. What the layer height would be in a star, or whether the concept of an equilibrium staircase even applies in the first place, remains to be determined. Meanwhile, the turbulent mixing coefficient DC = Dsemi is κT −1 Dsemi = γtot (84) (NuT − 1) −1 , R0 −1 where γtot is of order one (but smaller than one), with some possible weak dependence on Pr, −1 τ and R0 that also remains to be determined. In the non-layered regime, by contrast, we have demonstrated that the turbulent heat flux is negligible, and the turbulent compositional flux also appears to be negligible, at least for non-degenerate regions of stellar interiors where Pr and τ are asymptotically small. At more moderate Pr and τ, the turbulent compositional flux could be significant, and can be determined from the model of Mirouh et al. [22] presented in Section 4.3, or by fitting their data empirically (which remains to be done).

4.7 Applications to stellar astrophysics

The impact of these newly developed theories of mixing by ODDC (layered or non-layered) on stellar evolution have not really been studied in detail yet, so this section will be very limited in scope. As discussed in Section 2.4, semiconvective regions are most commonly found just outside the convective cores of MS stars, in two mass ranges: intermediate-mass stars with M between about 1M and 3M , and higher-mass stars with M > 10M . In both cases, the typ−1/2 . The turbulent ical inverse density ratios are well within the layered regime, with R−1 0 τ transport of heat and composition can be substantial and depends on the unknown heights of the layers (which need not be constant in space nor time). With all this uncertainty, the best one can do (for now) is to fix the layer height to some fraction of a pressure scaleheight, and qualitatively ascertain what the effects of layered convection on the star’s evolution are as a function of the assumed layer height . It has long been known that one of the main effects of semiconvection in these stars is to prolong their lifetime on the MS by increasing the amount of hydrogen available for fusion reactions. Consequently, it also increases the size of the helium core at the end of the MS. Moore and Garaud [98] attempted to quantify the effect, to determine more specifically how the helium core size of intermediate-mass stars varies with the assumed semiconvective layer height. To do so, Moore implemented both heat and compositional transport prescription by layered ODDC in MESA. The compositional transport is easy to implement because it merely requires computing the mixing coefficient Dsemi with equation (67). Including a model for the turbulent heat transport on the other hand is much more complicated, since MESA computes the temperature profile within the star given the heat flux, instead of computing the heat flux given the temperature gradient (which is the way in which equation 82 is written). As such, including heat transport by layered semiconvection involves the numerical solution of a fourth-order polynomial in the quantity ∇. Details of the implementation are given in Section 3.1.2 of [98]. The conclusions of our investigation on intermediate-mass stars are remarkably simple. We found that unless the height of the layers was assumed to be unphysically small, compositional mixing by layered convection was so efficient that it would erase any existing compositional gradient on a short timescale compared with the stellar evolution timescale. As a result, the semiconvective region rapidly becomes fully convective, and the star’s core grows

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to be the size one would have predicted much more simply using the Schwarzschild criterion and ignoring semiconvection altogether. This is illustrated in Figure 20, which shows the predicted (fully) convective core size of a 1.3M star, under four different scenarios: ignoring semiconvection and with either the Schwarzschild or Ledoux criteria to compute the core sizes, or, including layered convection and assuming that the layer height HL is either 10−6 or 10−14 times the pressure scaleheight H p . We see that with HL = 10−6 H p the model already behaves almost as if the Schwarzschild criterion had been used instead, and it is only for vanishingly small layer heights (HL = 10−14 H p ) that a substantial semiconvective region survives above the core throughout the star’s evolution.

Figure 20. Kippenhahn diagram for the evolution of a 1.3M star under different scenarios. Left column: Models that ignore semiconvection, and use either the Schwarzschild (top) or Ledoux (bottom) criterion to compute the convective core size. Right column: Models that include layered semiconvection, with varying layer height, as shown in the figure. The dark blue regions are fully convective, while the light blue regions are semiconvective. Figure from [98].

These results are consistent with recent asteroseismic determinations of the convective core sizes in MS stars [99, 100], which reveal them to be even larger than what stellar evolution predicts using the Schwarzschild criterion. This implies not only that semiconvection is irrelevant (as we predicted), but also that substantial overshoot must be present. We therefore conclude that ignoring semiconvection and using the Schwarzschild criterion in these stars is a good approximation for stellar evolution purposes (for intermediate mass stars), which is particularly convenient given how technically difficult the implementation of layered convection is (for heat transport) in standard stellar evolution codes. Finally, I would like to add that MESA, to my knowledge, is the only stellar evolution code to date that includes the complete Wood et al. [97] prescription for transport by layered convection for both heat and composition in semiconvective zones. I encourage readers to use it to see what other effects layered convection may have, in particular for more massive stars (M > 10M ), or for stars in more evolved stages of evolution.

5 What next? In the previous Sections, I have attempted to summarize what is known so far about doublediffusive instabilities and double-diffusive convection at low Prandtl number, which is the relevant limit for astrophysical fluids. However, there are a number of limitations of the model we have used to arrive at these results, which I would be remiss not to mention. Much also remains to be done in terms of modeling double-diffusive convection more realistically, including for instance the effects of rotation, magnetic fields, shear, and other relevant processes. I now discuss both of these in turn.

Multi-Dimensional Processes In Stellar Physics

5.1 Model limitations

The numerical model used in the simulations presented in this lecture uses the SpiegelVeronis-Boussinesq [17] approximation, and triply-periodic boundary conditions, so it is important to determine when the limitations of this setup may begin to invalidate the results obtained. The Spiegel-Veronis-Boussinesq approximation is valid a long as two conditions are satisfied: (1) that the typical fluid velocities are much smaller than the sound speed c s and (2) that the region modeled is much smaller than a pressure scaleheight H p . In Section 2.2, we learned that the typical size of basic double-diffusive structures (prior to the formation of layers, if they form) is of order 10d = 10(κT ν/|NT2 |)1/4 and provided some estimates for that quantity. We also saw that the structures remain small even in the nonlinear regime (again, prior to layer formation), so 10d is a good estimate for the typical eddy size in small-scale homogeneous double-diffusive convection (of both fingering and ODDC type). In terms of the typical velocity of fingering convection and ODDC, we saw that it can be estimated using λd, where the growth rate λ is at most equal to NT , but usually much smaller (depending on the regime considered). It is clear that well within the interior of a star, we always have 10d  H p and NT d  c s . Closer to the stellar surface, however, especially for intermediate and high-mass stars, κT increases rapidly, and so will d. Meanwhile, the pressure scaleheight H p and the sound speed c s both decrease, so it is conceivable that fingering convection or ODDC in regions sufficiently close to the surface of these stars may need to be modeled more realistically including the effects of compressibility. Once layers form in ODDC, the typical size of turbulent structures increases to be the entire layer height, and the turbulent velocities also increase to take typical convective (rather than double-diffusive) values. If the layers grow to a substantial fraction of a pressure scaleheight, the effects of compressibility should therefore also be taken into account. Finally, note that the Boussinesq approximation also assumes that the diffusivities are constant within the domain. This could be a limitation of the model when ν, κT or κC depend sensitively on temperature or composition. Another set of caveats of the results comes more specifically from the use of the triplyperiodic boundary conditions for the perturbations in our DNSs. The first is relatively well understood, and pertains to the relative size of the double-diffusive structures modeled compared with the computational domain size. If the structures are much smaller than the domain size (in all directions), then the latter has no effect on the double-diffusive dynamics. This is usually the case in the homogeneous regime of fingering convection and ODDC. On the other hand when the double-diffusive structures approach the domain size, it is quite likely that the results become dependent on the boundary conditions. This is the case for instance in layered convection, when there is only a single layer left. This is also the case of rapidly rotating double-diffusive convection and magnetized fingering in the very strong field regime, where the dynamics become invariant along the rotation axis and the magnetic field, respectively. When this happens, the measured fluxes can be artificially enhanced [23, 24, 29]. To control this problem, it is important to run additional DNSs in increasingly large domains until a point where the results become independent of the domain size. In all the simulations presented in these lectures, this was always verified. A second problem associated with triply-periodic boundary conditions is more subtle, and requires care in interpreting the turbulent flux data. In our computational model setup, the background potential temperature and composition gradients are fixed, and the developing instabilities drive turbulent heat and composition fluxes. Crucially, there is nothing in the system that limits these fluxes beyond what drives them. However, in a real star the situation is somewhat different because the total heat flux that needs to be transported through

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a given shell of material is fixed by the nuclear reaction rates taking place within or below. In that case, we anticipate that (on long timescales) the double-diffusive dynamics may adjust themselves to carry the required flux, rather than the flux being solely controlled by the double-diffusive dynamics. The problem is not significant when the double-diffusive dynamics transport a negligible amount of heat (which is the case of homogenous fingering convection and ODDC), but needs to be born in mind for layered convection and magnetized fingering convection with strong fields, where heat transport can be substantial (see below). These types of problems are then perhaps best studied in a model setup that can impose a fixed heat flux through the domain. 5.2 More realism

In the past few years, our group has begun to investigate the effects of other physical processes on the dynamics of double-diffusive convection, including those of lateral gradients of potential temperature and composition [26], rotation [23, 29] magnetic fields [24], and shear [25]. Given the dramatic increase in the dimensionality of parameter space that needs to be explored, our results are in most cases preliminary, and much remains to be done. This section briefly summarizes the state of the field as of May 2019. 5.2.1 Horizontal gradients

The effects of horizontal gradients have been investigated in the context of fingering-like stratifications only [26] (the ODDC-like case has never been looked at, to my knowledge), and affect the results described in Section 3 only in the limit of very strong stratification, i.e. for density ratios approaching marginal stability or exceeding it. Horizontal gradients greatly increase the region of parameter space that is unstable [5, 26], to the extent that almost any inverse μ gradient can cause vertical mixing. However that instability is slowly growing, and does not appear to cause very strong enhancements in the turbulent mixing coefficient above the purely diffusive value. This conclusion was obtained from simulations at very moderate Prandtl number and diffusivity ratio, and it remains to be seen whether it holds for more stellar-like parameter values. 5.2.2 Rotation

The effects of rotation have been investigated in preliminary work by Moll and Garaud [29] for ODDC, and Sengupta and Garaud [23] for fingering convection. In both cases, rotation does not affect the overall instability range, nor the growth rate of the fastest-growing modes of instability. Whether rotation has a weak or strong effect on the nonlinear saturation of double-diffusive instabilities can be determined by computing the turbulent Rossby number, Ro = urms /Ωd where urms is the typical velocity of turbulent eddies, and Ω is the rotation rate. Rotation is important for Rossby numbers of order one or much smaller than one, but is negligible when Ro 1. At moderate Pr and τ (appropriate for giant planets, or degenerate regions of stellar interiors), the turbulent velocities can be estimated roughly using κT /d, so the Rossby number is Ro = κT /Ωd2 = Pr−1/2 |NT |/Ω [29]. With that assumption and the fact that Pr ∼ 0.01 in these objects, we see that the effects of rotation will only be negligible if Ω  10|NT |, roughly speaking. In stellar interiors, the turbulent velocities associated with the basic instability are typically much smaller, of the order of (κT /d)(Pr/R0 )1/2 for fingering convection and (κT /d)(PrR0 )1/2 for non-layered ODDC. As a result, the Rossby number is of −1/2 |NT |/Ω for fingering convection and Ro = (R−1 |NT |/Ω for nonthe order of Ro = R−1/2 0 ) 0 layered ODDC. In other words, at very low Pr and τ and for sufficiently strongly stratified

Multi-Dimensional Processes In Stellar Physics

53

systems (i.e. large R0 for fingering, and large R−1 0 for ODDC), the Rossby number can be of order one or even smaller and rotation could be important. For layered ODDC, the situation is a little different since the estimate for the turbulent velocities urms must be replaced by an estimate for convective velocities, that depends on the assumed layer height. Most of our numerical simulations of rotating double-diffusive convection to date have been performed at the poles (where the rotation axis is aligned with gravity), with a few cases only looking at mid-latitudes. We have found that weak rotation (with Ro 1) tends to reduce the vertical compositional flux slightly compared with non-rotating simulations, in both fingering and ODDC. By contrast, when the Rossby number is of order unity, we found that rotation causes the formation of large-scale vortices that span the entire domain in both vertical and horizontal directions10 . These vortices have a tendency to cause a segregation of the compositional field in their core, either expelling it or concentrating it (see Figure 21) depending on the initial conditions selected. Denser cores drive stronger downward velocities, while lighter cores drive stronger upward velocities. This increase in the correlation between w and C causes a substantial increase in the vertical compositional flux compared with a non-rotating simulation at otherwise similar parameters. The effect is very substantial, and could potentially be invoked as a solution to the missing mixing problem in RGB stars (see Section 3.6) [23]. However, why these vortices form, and whether they would also form in stars (rather than in idealized simulations), remains to be determined. In particular, based on our results, and similar results obtained for rotating Rayleigh-Bénard convection [101, 102], it is not clear that they would form at lower latitudes [29], or in computational domains that have different horizontal dimensions (i.e. L x  Ly ) [103].




Figure 21. Left: Illustrative snapshot of a large-scale vortex formed in a fingering simulation at Pr = τ = 0.1, and R0 = 1.45, with Ta∗ = 4Ω2 d2 /κT = 10. This is the compositional field perturbation, which shows that the vortex core is low C, and flows upwards. Figure adapted from [23]. Right: Illustrative snapshot of a large-scale vortex formed in an ODDC simulation at Pr = τ = 0.1, and R−1 0 = 1.25, with Ta∗ = 10. This time the vortex core is high C, and flows downwards, but has a more complex radial structure that remains to be explained. Figure adapted from [29],

Finally, for rotationally dominated systems (with Ro  1), large-scale vortices do not form. Instead, the dynamics of both fingering and ODDC are dominated by narrow tubular structures that are invariant along the axis of rotation. In that case, the periodicity of the domain becomes a limitation, and it is not clear that the numerical results obtained are reliable 10 Note

that they have only been seen in polar cases so far

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(see previous section). More remains to be done, using perhaps a different formalism (or much taller periodic domains) to characterize ODDC and fingering convection in that limit. 5.2.3 Magnetic fields

As discussed earlier, Charbonnel and Zahn [104] and Stevenson [31] investigated the linear stability of magnetized fingering convection and magnetized ODDC, respectively, but very little is known so far about the effects of magnetic fields on double-diffusive convection in the nonlinear regime. Beyond linear theory, the first numerical study of magnetized fingering convection has just been completed by Harrington and Garaud [24], in a model that assumes the presence of a vertical11 large-scale background field of amplitude B0 . The impact of the field can be characterized by a new non-dimensional number, namely HB =

B20 d2 ρm μ0 κT2

,

(85)

which is the square of the ratio of the Alfvén velocity associated with that field, to the selected unit velocity κT /d (see Section 2.2). Note however that since κT /d is not always a good estimate for the actual fingering velocities (see Section 3), especially at very low Prandtl numbers, HB  1 does not necessarily imply that the magnetic field is negligible (see below for more on this topic). We have run a number of DNSs of fingering convection that include a vertical magnetic field, varying the quantity HB , with otherwise fixed parameter values. We have found that vertical transport can be increased by orders of magnitude compared with non-magnetic simulations, because the field stabilizes the fingers partially against the parasitic shear instabilities that normally destroy them (see Section 3). As a result, larger vertical velocities can be achieved before the shear instability causes saturation (see Figure 22), which in turn enhances the vertical fluxes.

Figure 22. Snapshots of the vertical velocity field in simulations of magnetized fingering convection with a vertical magnetic field, at Pr = τ = 0.1 and R0 = 1.45. The vertical coherence of the fingers increases with the magnetic field strength (as measured by HB ), and so does the vertical velocity. Figure from [24].

Our results can be quantitatively explained by a simple extension of the Brown et al. [21] model, that contains no additional free parameter. Recall that the mixing coefficient Dfing can 11 The

study of arbitrarily inclined fields is in progress.

Multi-Dimensional Processes In Stellar Physics

55

be estimated using equation (42), where KB  1.24 and where wˆ f is the anticipated vertical velocity within the finger at saturation obtained by matching the finger growth rate λˆ to the parasitic shear instability growth rate σ. ˆ The latter decreases in the presence of a magnetic field [105], because the field rigidifies the fingers. This effect can be captured using the approximate expression (see [24] for detail): 2/3 . σ ˆ  0.42wˆ f kˆ h (0.5 − HB wˆ −2 f )

(86)

This equation recovers the hydrodynamic limit studied by Brown et al. [21] when HB = 0, as required. It also shows that when HB  0 the fingering elevator modes are unaffected by √ parasitic shear instabilities until their vertical velocities reach a threshold value of wˆ f = 2HB (which corresponds to energy equipartition). Beyond that threshold, the finger velocities conˆ where C H  1.66 is not a free parameter, but instead, is fixed tinue to grow until σ ˆ  C H λ, by the requirement that our theory recovers the hydrodynamic limit when HB = 0 (see [24] for detail). 2/3 All that remains is to solve the algebraic equation C H λˆ = 0.42wˆ f kˆ h (0.5 − HB wˆ −2 for f ) (42) to compute the mixing coefficient wˆ f numerically, then substitute the result in equation √ Dfing . Note that for large HB , we find that wˆ f  2HB so using this expression in (42) yields Dfing 

2KB HB κT , λˆ + τkˆ 2

(87)

h

which recovers the aforementioned equation (57) when KB  1.24, τ  1, and if we use the √ approximation λˆ  Pr/(R0 − 1) (see equation 18). Figure 23 compares the new model for magnetized fingering convection with results of our DNSs including the vertical field, at moderate parameter values (Pr = τ = 0.1, R0 = 1.45), and confirms that it correctly predicts the variation of Dfing with magnetic field strength as HB increases. Applying the model to more stellar-like conditions (where the fingering growth rate is much smaller), we predict that magnetic fields become important for HB ∼ 10−4 in WDs, and 10−8 in RGB stars. With a very moderate magnetic field of the order of 300G, Dfing increases by two orders of magnitude compared with the non-magnetic case for both types of stars (see Figure 23). This strongly suggests that magnetic fields should be taken into account when modeling fingering convection in stars, and could be the answer the RGB star abundance conundrum described in Section 3.6.

Dfing


C

Figure 23. Dfing /κC in simulations (symbols) and models (lines). The simulations were run at Pr = τ = 0.1 and R0 = 1.45, and the model corresponding to these parameters is the black line. The red and blue lines are predictions for RGB and WD stars respectively, see [24] for detail. The shaded areas are values of HB corresponding to B0 ∼ 100G. Figure adapted from [24].

Of course, this conclusion remains to be tested in the more general case where the magnetic field is inclined with respect to gravity. Preliminary results suggest that transport is also increased in that case, but new effects also appear (such as a dynamo, see [24] and Harrington’s MS thesis), whose impact needs to be better understood.

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5.2.4 Shear

The final physical process whose impact on stellar double-diffusive convection had not yet been investigated yet is shear, though numerical and experimental studies of sheared doublediffusive convection in geophysics exist, such as for instance [106–109] for the fingering case, and [30, 110] for ODDC. Although sheared ODDC in stars has not been studied yet, the effect of a moderate shear on fingering convection is now relatively well understood [25]. While shear tends to suppress the development of fingering structures that vary in the streamwise direction, it has no effect on streamwise invariant perturbations. As a result, the initial instability takes the form of sheets aligned with the flow, rather than fingers [106]. In the nonlinear regime, however, these sheets rapidly break up into 3D fingers, that are tilted in the direction of the shear. That tilt has two effects: to reduce the horizontal lengthscale of the fingers (by squeezing them together), and to add a horizontal component to the flow within the fingers. The Brown et al. [21] model for mixing by fingering convection can easily be extended to take both effects into account [25], and ultimately predicts that the mixing coefficient Dfing must be corrected to account for shear in the following way:

 S 2 −2 , Dfing (S ) = Dfing (S = 0) 1 + χ2 λ

(88)

where Dfing (S = 0) is the value of Dfing in the absence of shear, S is the local shearing rate, λ is the growth rate of fastest-growing unsheared fingers, and χ is a number of order unity12 . This prescription holds only for moderate shearing rates, i.e. S < 3λ or so. Beyond that, the shear itself begins to contribute to the vertical mixing as expected, but no quantitative model of the process exists (so far).

1.2

Z

1

Dfing (S)

0.8

Dfing (S = 0) 0.6

0.2 0 0.01

R0 = 1.5 R0 = 1.75 R0 = 2 R0 = 2.25 R0 = 2.5 0.1

Z

0.4

X

1 S/h

10

100

X

Figure 24. Right: Illustrative snapshots of the compositional perturbations in sheared fingering convection, in two simulations where the background shear is sinusoidal. Parameters are Pr = τ = 0.3, and R0 = 2.5 for both of them. The shearing rate is about 5 times larger in the bottom figure than in the top figure. Left: Comparison of the model prediction from equation (88), (black line) and the data, for Dfing (S )/Dfing (S = 0), as a function of S /λ where S is the maximum shearing rate in the domain. Figure adapted from [25]. 12 In the simulations we have performed, which have a sinusoidal shear flow, χ = 1/3, though this value could be different for linear shear flows.

Multi-Dimensional Processes In Stellar Physics

Finally, sheared fingering convection can also contribute to substantial momentum transport. We have found empirically that νturb  0.25Dfing in the limit where S < 3λ, growing to νturb  Dfing for larger shearing rates. This can be used to inform models of angular momentum transport due to fingering convection in stars, which may (or may not) be useful in modeling the observed evolution of their interior rotation rates, as in [111] for example. Acknowledgements: The majority of this work was funded by the National Science Foundation through various grants over the past 10 year, including NSF CBET-0933759, NSF AST-1211394, NSF AST-1412951 and NSF CBET-1437275. These lectures were written up while on sabbatical at the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, supported by a visiting by-fellowship from Churchill College. I thank both institutions for their welcoming support during my visit. I would like to thank all my students, postdocs and collaborators on these projects, including Justin Brown, Nic Brummell, Jonathan Fortney, Peter Harrington, Anuj Kumar, Chris Mankovich, Michael Medrano, Giovanni Mirouh, Ryan Moll, Kevin Moore, Timour Radko, Erica Rosenblum, Sutirtha Sengupta, Josiah Schwab, Stephan Stellmach, Adrienne Traxler, Sylvie Vauclair, Toby Wood, and Varvara Zemskova. It has been fun and inspiring to work with them. I also thank Christopher Tout for his invaluable feedback on this manuscript and the organizers of the meeting, Michel Rieutord and Isabelle Baraffe for asking me to write these notes, which I hope will be useful to others. Finally, my special thanks go to Timour Radko, who inspired me to start working on this topic in the first place, Nic Brummell, with whom I always enjoy bouncing ideas, and to Stephan Stellmach, who developed the PADDI code that is at the heart of all this research.

References P. Garaud, Annual Review of Fluid Mechanics 50, 275 (2018) H. Stommel, A.B. Arons, D. Blanchard, Deep Sea Research 3, 152 (1956) M. Stern, Tellus 12, 172 (1960) P. Baines, A. Gill, J. Fluid Mech. 37, 289 (1969) J.Y. Holyer, J. Fluid Mech. 137, 347 (1983) M. Schwarzschild, R. Härm, Astrophys. J. 128, 348 (1958) S. Kato, Proc. Astro. Soc. Japan 18, 374 (1966) E.A. Spiegel, Comments on Astrophysics and Space Physics 1, 57 (1969) R.K. Ulrich, Astrophys. J. 172, 165 (1972) D.J. Stevenson, Proceedings of the Astronomical Society of Australia 3 (1977) R. Kippenhahn, G. Ruschenplatt, H. Thomas, Astron. Astrophys. 91, 175 (1980) N. Langer, K.J. Fricke, D. Sugimoto, Astron. Astrophys. 126, 207 (1983) H.C. Spruit, Astron. Astrophys. 253, 131 (1992) R.W. Schmitt, Phys. Fluids 26, 2373 (1983) J.P. Cox, R.T. Giuli, Principles of stellar structure (Gordon and Breach, 1968) R. Kippenhahn, A. Weigert, A. Weiss, Stellar structure and evolution, Vol. 192 (Springer, 1990) [17] E.A. Spiegel, G. Veronis, Astrophys. J. 131, 442 (1960) [18] C.Y. Shen, Phys. Fluids 7, 706 (1995) [19] T. Radko, Double-diffusive convection (Cambridge University Press, 2013) [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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[92] T.G.L. Shirtcliffe, Journal of Fluid Mechanics 57, 27 (1973) [93] F. Zaussinger, H.C. Spruit, Astron. Astrophys. 554, A119 (2013), 1303.4522 [94] E. Rosenblum, P. Garaud, A. Traxler, S. Stellmach, Astrophys. J. 731, 66 (2011), 1012.0617 [95] R. Moll, P. Garaud, S. Stellmach, Astrophys. J. 823, 33 (2016), 1506.07900 [96] R. Schmitt, Deep Sea Res. 26A, 23 (1979) [97] T.S. Wood, P. Garaud, S. Stellmach, Astrophys. J. 768, 157 (2013), 1212.1218 [98] K. Moore, P. Garaud, Astrophys. J. 817, 54 (2016), 1506.01034 [99] V. Silva Aguirre, S. Basu, I.M. Brandão, J. Christensen-Dalsgaard, S. Deheuvels, G. Do˘gan, T.S. Metcalfe, A.M. Serenelli, J. Ballot, W.J. Chaplin et al., Astrophys. J. 769, 141 (2013), 1304.2772 [100] S. Deheuvels, I. Brandão, V. Silva Aguirre, J. Ballot, E. Michel, M.S. Cunha, Y. Lebreton, T. Appourchaux, Astron. Astrophys. 589, A93 (2016), 1603.02332 [101] C. Guervilly, D.W. Hughes, C.A. Jones, Journal of Fluid Mechanics 758, 407 (2014), 1403.7442 [102] B. Favier, L.J. Silvers, M.R.E. Proctor, Physics of Fluids 26, 096605 (2014), 1408.6483 [103] K. Julien, E. Knobloch, M. Plumley, Journal of Fluid Mechanics 837, R4 (2018), 1711.01685 [104] C. Charbonnel, J.P. Zahn, Astron. Astrophys. 476, L29 (2007), 0711.3395 [105] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability (1961) [106] P.F. Linden, Geophysical and Astrophysical Fluid Dynamics 6, 1 (1974) [107] S. Kimura, W. Smyth, Geophys. Res. Lett. 34 (2007) [108] W.D. Smyth, S. Kimura, J. Phys. Oceano. 41, 1364 (2011) [109] T. Radko, J. Ball, J. Colosi, J. Flanagan, Journal of Physical Oceanography 45, 3155 (2015) [110] J.M. Brown, T. Radko, Journal of Fluid Mechanics 858, 588 (2019) [111] J.P. Marques, M.J. Goupil, Y. Lebreton, S. Talon, A. Palacios, K. Belkacem, R.M. Ouazzani, B. Mosser, A. Moya, P. Morel et al., Astron. & Astrophys. 549, A74 (2013), 1211.1271

Thermo-compositional adiabatic and diabatic convection Pascal Tremblin1,∗ 1

Maison de la Simulation, CEA, CNRS, Univ. Paris-Sud, UVSQ, Université Paris-Saclay, 91191 Gifsur-Yvette, France Abstract. We propose a general theory of convection that is able to account

for the presence of different types of thermal and compositional source terms (diabatic processes). The linear stability analysis of the equations of stratified hydrodynamics with these source terms shows that, thermohaline convection in Earth’s oceans, fingering convection in stellar interiors or atmospheres, and moist convection in Earth’s atmosphere are derived from the same general diabatic instability. Generalized to the CO/CH4 chemistry and radiative transfer in the atmospheres of brown dwarfs and extrasolar giant exoplanets, this process could explain the spectral reddening observed in the spectrum of L dwarfs and also explain its disappearance at the L/T transition.

1 Introduction Field brown dwarfs of effective temperatures ≥∼ 1200K (spectral type L) have a reddening in their spectra usually associated with the presence of clouds that would disappear for T dwarfs with effective temperatures ≤∼ 1200K. Since this transition seems to exactly correlate with the carbon chemistry in the atmosphere (L dwarfs are CO-dominated and T dwarfs are CH4 dominated), in [1] we have proposed that the reddening is caused by a reduction of the temperature gradient in the atmosphere, a process that could be linked to convection being impacted by the carbon chemistry. In order to further explore this idea, we have proposed in [2] a general framework for the study of convection in the presence of thermal and compositional source terms and show that this framework can encompass many known convective systems in Earth climate physics. We present here the important points of this analysis and our understanding of the temperature-gradient reduction that is observed in many convective systems.

2 Linear stability analysis of adiabatic and diabatic convection 2.1 Adiabatic case

We consider an ideal gas with pressure P, temperature T , a mean molecular weight μ(X) that depends on a compositional mass fraction X, and adiabatic index γ. The equation of state is given by P = ρkb T/(μ(X)mH ) We define the potential temperature θ = T (Pref /P)(γ−1)/γ . The gas is assumed to be initialy at hydrostatic equilibrium in the z direction with a scale ∗ e-mail: [email protected]

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height defined by h p = −1/(∂ log P/∂z). Assuming c p is constant, the equations of evolution of potential temperature and composition are given by ∂ log θ  log θ = 0 + u · ∇ ∂t ∂X  (X) = 0 + u · ∇ ∂t

(1)

The linear stability analysis of the complete system with mass and momentum balance leads to the Ledoux criterion for instability ∇θ − ∇μ = ∇T − ∇ad − ∇μ > 0

(2)

with ∇T = −h p ∂(log T )/∂z, ∇ad = (γ − 1)/γ, and ∇μ = −h p ∂(log μ)/∂z. This is the standard instability criterion for thermo-compositional “adiabatic” convection that reduces to Schwarzschild criterion in the absence of mean-molecular-weight gradient. 2.2 Diabatic case

With the addition of a thermal source term H(X, T ) and compositional source term R(X, T ) the system becomes ∂ log θ  log θ = H(X, T ) + u · ∇ ∂t T ∂X  (X) = R(X, T ) + u · ∇ ∂t

(3)

In addition to Ledoux criterion, a second criterion for instability is found in the analysis (∇T − ∇ad )ωX − ∇μ ωT < 0 ωX = RX + RT (T ∂(log μ)/∂X) ωT = HT + HX /(T ∂(log μ)/∂X)

(4)

with HT,X and RT,X the partial derivatives of the source terms with respect to temperature and composition. This is the criterion for thermo-compositional “diabatic” convection since the source terms play a role in the criterion itself. Replacing the source terms by thermal and compostional diffusion with diffusion coefficients κT and κμ , we get in Fourier space ωX = −k2 κμ and ωT = −k2 κT . The criterion then reduces to the standard form for the thermohaline or fingering instability (∇T −∇ad )κμ −∇μ κT > 0 (see [3]). Replacing the source terms by evaporation/condensation of liquid/steam water and the associated pumping/release of latent heat L for the thermal source term and assuming that water-vapour concentration is at saturation X = Xeq (P, T ), we get the standard criterion for moist convection ∇T − ∇ad (1 + Xeq L/Rd T )/(1 + Xeq L2 /c p Rv T 2 ) > 0 with Rv and Rd the vapour and dry air gas constant (see [4]). Replacing the compositional source term by the carbon chemistry for the conversion of CO to CH4 and the thermal source term by radiative transfer in the atmospheres of brown dwarfs and extrasolar giant exoplanets, we get a system that is equivalent to thermohaline convection in Earth’s oceans and moist convection in Earth’s atmosphere.

3 Non-linear regime and temperature-gradient reduction Knowing that thermohaline convection and moist convection are observed to produce temperature gradient reductions and temperature inversions in Earth’s atmosphere and oceans

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(see [5] and [6]), we can expect that CO/CH4 radiative convection will lead to such a temperature gradient reduction in the atmospheres of brown dwarfs and extrasolar giant planets in the non-linear regime. This regime can be studied through numerical simulations (e.g. [7]) or with a mean field theory such as the mixing-length theory. For adiabatic convection, the energy flux from the mixing-length theory is given by Fad = ρc p T vconv (∇T − ∇ad )

(5)

with vconv = ωlconv , lconv the mixing-length parameter, and ω the linear growth rate of the instability. In the diabatic case, the convective flux can be approximated by   Fd = ρc p T vconv ∇T − ∇ad − ∇μ ωT /ωX (6) The extra-term in the flux is linked to a definition of a new potential temperature that is conserved in the linear regime log θ = log θ − X

∂ log μ ωT ∂X ωX

(7)

which reduces in the case of moist convection to log θ = log θ − XL/(c p T ) and is known in Earth’s climate physics as the moist potential temperature. This new potential temperature is linked to the fact that an energy transport is hidden in the small-scale fluctuations in the composition: composition can be transported in the convective process and release energy at small-scale through the thermal source (e.g. moist convection) term or through the contraction/expansion work associated with the equation of state and the compositional source term (e.g. thermohaline/fingering convection). For a given value of a convective flux, the extra term ∇μ ωT /ωX in the diabatic flux allows the temperature gradient ∇T to be reduced compared to the adiabatic flux. This process can therefore explain the temperature-gradient reduction in Earth atmosphere associated with moist convection and the temperature inversions in the oceanic staircases associated with thermohaline convection. Hence the diabatic instability for CO/CH4 radiative convection can also lead to a temperature-gradient reduction explaining the reddening in L-dwarf spectra. When the whole atmosphere is CH4 dominated, there is no more chemical conversion associated to the carbon chemistry and the diabatic instability dissipates. This leads to a warming up of the deep atmosphere associated to the transition between the diabatic convective transport Fd and the adiabatic convective transport Fad . This transition can be seen as a cooling crisis and could be responsible for the L/T transition in the atmospheres of brown dwarfs and extrasolar giant exoplanets. Further studies with 3D numerical simulations will help to understand better these processes.

References [1] [2] [3] [4] [5] [6] [7]

Tremblin P. et al., ApJL 817, L19 (2016) Tremblin P. et al., ApJ in press (2019) Stern M. E., Tellus 12, 172 (1960) Stevens B., Annu. Rev. Earth Planet. Sci. 33, 605 (2005) Manabe S. & Strickler R. F., J. Atmos. Sci. 21, 361 (1964) Gregg M., Elvesier Ocealography Series 46, 453 (1988) Padioleau T. et al., ApJ 875, 128 (2019)

Fully compressible time implicit hydrodynamics simulations for stellar interiors Isabelle Baraffe1,2,∗ 1 2

Physics and Astronomy, University of Exeter, Exeter EX44QL UK Univ Lyon, ENS de Lyon, Univ Lyon 1, CNRS, CRAL, UMR5574, F-69007, Lyon, France Abstract. I present the recent development of a multi-dimensional, fully com-

pressible hydrodynamical time implicit code MUSIC devoted to stellar and planetary interiors. I discuss the main challenges in the computation of multidimensional stellar structures and some of the advantages of our approach with MUSIC for the study of stellar fluid dynamics problems. MUSIC is an implicit large eddy simulation code that uses implicit time integration. It is interfaced with a stellar evolution code which provides initial 1D stellar structures for any type of star and allows exploration of a wide range of stellar phases and masses. A major motivation for these studies is to derive new prescriptions to be implemented in stellar evolution codes and to be tested against various observational constraints.

1 The MUSIC code The Music code is a three dimensional hydrodynamic code optimised for the description of stellar/planetary interiors. It includes a performant and successfully benchmarked timeimplicit integration method. Its main specificities are summarised below ([1-3]): • Use of cartesian (for tests) and spherical coordinates (more natural for stellar interiors) • Finite volume method on a staggered grid • Solves the fully compressible hydrodynamic equations • Includes radiative transfer within the diffusion approximation (valid for optically thick stellar interiors) • Includes a performant, low storage time implicit solver based on a Jacobian-free Newton Krylov method • Accurate for a range of Mach numbers ∼ 10−6 − 1 • Uses realistic stellar input physics (opacities and equations of state EOS) • Centrifugal and Coriolis forces are implemented • Transport of passive and active scalars. ∗ e-mail:

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Figure 1. Angular structure of the penetration layer in a 2D simulation of a young Sun (see [5]).

The MUSIC code is particularly suited to study turbulent convection and convective penetration into stably stratified regions. The global geometry of the code allows to capture the dynamics of the convective flows in the entire convective zone and their penetration in the stably stratified layers, with all these processes shaped self-consistently by rotation, if the effects of rotation are included. Because it solves the fully compressible Euler equation, we can include strongly stable stratifications that are difficult or impossible to model self-consistently within the anelastic approximation commonly used to study stellar hydrodynamics; unlike most other fully compressible codes, however, the numerical timestep is not limited by the Courant-Friedrich-Levy condition on sound waves.

2 An example of application: convective penetration One of the major uncertainty of stellar evolution models is the treatment of mixing taking place at convective boundaries. Convective motions do not abruptly stop at the classical Schwarzschild boundary, but extend beyond it. The complex dynamics resulting from turbulent convection penetration in stable layers is a major process in stars that drives the transport of chemical species and heat, strongly affecting the structure and the evolution of many types of stars. This process is usually called overshooting or convective penetration. In the study of a pre-Main Sequence solar-like star, we show the existence of extreme (rare) events of convective down-flows (or plumes) that penetrate much deeper in the stable region below the convective envelope than the average plume (Fig. 1). These extreme events could contribute substantially to mixing on the long term evolution of the star [5]. Applying a statistical method (extreme value theory) to calculate the probability of rare events, a quantitative description of the mixing driven by convective plumes in the penetration layer can be provided [5]. The success of a statistical approach to derive diffusion coefficients was demonstrated by implementing preliminary derived statistical diffusion coefficients in a stellar evolution code [6], showing very promising results to explain lithium depletion in solar

Multi-Dimensional Processes In Stellar Physics

like stars. The study of [6] also shows how direct links can be established between numerical simulations and observations, and thus the way to progress. This work is supported by the ERC grants 320478-TOFU and 787361-COBOM

References 1. 2. 3. 4. 5. 6.

M. Viallet, I. Baraffe, R. Walder, A&A, 531 (2011) M. Viallet, T. Goffrey, I. Baraffe et al., A&A, 586 (2016) T. Goffrey, J. Pratt, M. Viallet et al., A&A, 600 (2017) J. Pratt, I. Baraffe, T. Goffrey et al., A&A, 593 (2016) J. Pratt, I. Baraffe, T. Goffrey et al., A&A, 604 (2017) I. Baraffe, J. Pratt, T. Goffrey et al., ApJL, 845 (2017)

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Thermal Convection in Stars and in Their Atmosphere Friedrich Kupka1,2,∗ 1

formerly at: Institute for Astrophysics, Georg-August-University Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany 2 MPI for Solar System Research, Justus-von-Liebig-Weg 3, D-37077 Göttingen, Germany Abstract. Thermal convection is one of the main mechanisms of heat transport and mixing in stars in general and also in the photospheric layers which emit the radiation that we observe with astronomical instruments. The present lecture notes first introduce the role of convection in astrophysics and explain the basic physics of convection. This is followed by an overview on the modelling of convection. Challenges and pitfalls in numerical simulation based modelling are discussed subsequently. Finally, a particular application for the previously introduced concepts is described in more detail: the study of convective overshooting into stably stratified layers around convection zones in stars.

1 Introduction Convection is an important mechanism for energy transport and mixing in stars. Both its analytical and its numerical modelling are particularly challenging and have remained key topics in stellar physics ever since it had been realized that stars can have convectively unstable regions at their surface and in their interior ([1–5]). This review article provides an introduction into the subject, but has to neglect some important varieties of convection: magneto-convection, double-diffusive convection in multicomponent fluids, and, mostly, also convection in (rapidly) rotating objects. Those subjects are covered by further articles in this book. Including them would have been prohibitive for keeping this exposition within reasonable limits. For completeness though some references on these subjects are also provided just below. Concepts from fluid mechanics are essential in the study of convection. Readers interested in an introduction to fluid dynamics as a supplement to this review are referred to [6, 7]. Those introductions omit the subject of magnetohydrodynamics (MHD) which is covered, for instance, in [8] or also in [9], who provides a modern exposition of MHD. All these books are useful for finding further introductory texts on their main topics, too. Concerning the physics of turbulent flows we refer in particular to [10] for an introduction to statistical concepts, one-point closures, and large eddy simulations, as well as to [11] for an introduction to two-point closures and other techniques not covered by the previous reference, such as 2D turbulence and turbulence in geophysics. These topics are also of interest to readers specializing in astrophysics. A critical review of the different concepts used in studying the physics of turbulence can be found in [12]. ∗ e-mail: ,

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There is a vast mathematical literature about the foundations of numerical methods for solving the partial differential equations of fluid dynamics which are at the core of theoretical studies of convection as well as on strategies concerning their implementation in a numerical simulation code. For beginners the introduction by [13] is particularly useful. It focusses on finite volume methods and on incompressible flows but it also provides a discussion of more general methods and the case of compressible flows. The reader interested in the basics and implementations of finite difference methods may find both modern introductions ([14] and later re-editions) as well as classical books such as [15] useful. As a standard reference for spectral methods, popular in numerical models of stellar convection, the books of [16, 17] may be consulted. Finally, a more mathematical account that includes the case of finite element methods but also the basics of modern shock resolving methods is given in [18]. These books should help in finding further literature on this still rapidly evolving subject. As is discussed below 2D and 3D numerical approximations to the basic equations of fluid dynamics are too expensive for a direct application to stellar evolution calculations. Classical stellar structure and evolution models are thus still rooted in computationally much less demanding, semi-analytical models. In astrophysics the most frequently used among them is mixing length theory ([3, 19]). A detailed account of this approach is given in classical texts on the theory of stellar structure and evolution such as [20]. For specialized subtopics concerning convection in stars the comprehensive and freely accessible reviews of the Living Reviews series on Solar Physics and on Computational Astrophysics are a good starting point to find further information. Subjects dealt with by these series include the large scale dynamics of the solar convection zone and tachocline [21], an account of solar surface convection for the case where magnetic fields are neglected [22], the problem of interaction between convection and pulsation [23], and the modelling of stellar convection with both semi-analytical and numerical methods [24]. For advanced subjects such as supergranulation [25] specialized reviews are available as well. This also holds for extensions concerning magnetic fields and the solar cycle, for example, dynamo models of the solar cycle [26], solar surface magneto-convection [27], or modelling of magnetism and dynamo action for both the solar and the stellar case ([28]). This introduction hence limits itself to the following topics: Sect. 2 gives an overview on the role of convection in astrophysics and on the basic physics of convection. Sect. 3 deals with the modelling of convection, with a focus on semi-analytical models. Sect. 4 provides a discussion of some challenges and pitfalls in numerical simulations of convection. Sect. 5 discusses the problem of calculating the extent of overshooting of convection into neighbouring, locally stably stratified regions, as a challenging example for the different modelling techniques introduced in the previous sections. Sect. 6 provides a summary of this material.

2 Convection in astrophysics and the basic physics of convection 2.1 The physics of convection

Convection is caused by a hydrodynamical instability which can occur in a fluid layer stratified due to gravitational force and subject to a temperature difference between the “top” and the “bottom” of that layer. Gravity specifies this distinct “vertical” direction as the one aligned with its action. In a sufficiently slowly rotating and hence spherically symmetric star this direction of course coincides with the radial one. Thus, the top of such a layer is located towards the surface of a star, its bottom towards the centre. These definitions are also evident for laboratory models such as convection occurring between two horizontally

Multi-Dimensional Processes In Stellar Physics

mounted plates.1 Consider a fluid stratified such that it is initially in hydrostatic equilibrium and ρtop < ρbottom . Linear stability analysis demonstrates that this configuration can become unstable depending on the distribution of temperature along the vertical direction. As shown in Sect. 2.4, if we consider adiabatic expansion (without viscous friction) of a fluid volume that is perturbed (considered moved away) vertically from its initial position such that through this expansion it attains the mean pressure of its new environment again, it is then crucial whether this displaced fluid has a lower density than its new environment. If that is the case a net buoyancy force prevails and the fluid is unstable to convection. In practice, at least in a stellar context small perturbations are always present and they will initiate such a convective instability. This leads to velocity fields building up with time. The evolution of this process can be very accurately predicted by the conservation laws of hydrodynamics: the Navier-Stokes equation (NSE) which ensures conservation of momentum, and its associated conservation laws for mass (continuity equation), and energy. Eventually, this causes heat transport in the fluid and mixing of the fluid. The convectively driven velocity field also couples to pulsation and shear flows in stars induced by pulsational instabilities and rotation. Since the fluid in stars is actually a plasma, it is generally magnetic and its velocity field can hence either cause or react to magnetic fields as well. The key criterion for convective instability in stars was originally suggested by [1] and is also derived in Sect. 2.4. The Schwarzschild criterion assures that a given stratification is unstable to convection, if the local (horizontally averaged) temperature gradient is steeper than the adiabatic one. With the definition of the dimensionless temperature gradient ∇ := (d ln T/d ln P) this can be written as ∇ > ∇ad . In turn, the gradient of radiative diffusion for a spherically symmetric star in hydrostatic equilibrium ([20]) can be written as ∇rad = (3κross P Lr )/(16πacGT 4 Mr ). Here, P is the pressure, Lr the luminosity at radial coordinate r, Mr is the mass inside a radius of r, T is the temperature, and κross is the Rosseland opacity. For a first analysis it suffices to compare ∇rad and ∇ad with each other to find the main sources of convective instability. The first one was identified to be partial ionization (by [2] for the zone of partial ionization of hydrogen in the Sun). Basically, in cool stars of spectral type A to M the quantity ∇ad drops from ∼ 0.4 down to 0.05 . . . 0.1 in this zone whence ∇ad < ∇rad (see also [20]). An even more important source is triggered simultaneously in this zone: high opacity due to partial ionization of hydrogen, but in general also of helium or “iron peak” elements. The former ones are important for A to M type stars. Iron peak opacity is important for hot stars (O and B type, but also A type stars when radiative diffusion is accounted for, see [29, 30]). Again, ∇ad < ∇rad in this case, but this time primarily due to a large ∇rad instead of a small ∇ad . The third main reason for convection in stars is high luminosity by efficient energy generation from nuclear fusion. There, εc = dLr /dMr ∼ Lr /Mr for small Mr which is large for energy production dominated by the CNO cycle and all later burning stages. Thus, along the main sequence convective cores occur from O down to F type stars. There, ∇ad < ∇rad because of a large ∇rad . A steep temperature gradient hence triggers the convective instability and the first model for this process in astrophysics was proposed by [3]. 2.2 Examples from astrophysics and geophysics

On Earth convection can occur both in the ocean (or lakes, etc.) and in the atmosphere and in a different physical parameter range also in its interior. In the atmosphere of the Earth convection may be caused by the specific moisture content of air, by the Sun heating the surface during the day, or by wind moving cold air, for instance from above the arctic sea ice, 1 This setup can be generalized by including other forces or by letting an electrical force take over the role of gravitation, but we focus on the standard case here and in the following.

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over warmer, open ocean water (see [31] for images of this scenario). The latter two scenarios are known under the name of the convective planetary boundary layer (CBL/PBL). This is a local phenomenon: the lowermost part of the atmosphere of the Earth can be convectively unstable in some area, but stable elsewhere and this even changes as function of time for each area. That is different from the solar or stellar case, where the convective instability usually occurs at all locations at a given radius (apart from special local conditions caused by magnetic features such as sun spots and the like). In astrophysics solar granulation is the most well-known observational feature caused by convection (see [32] and [33] for some early photographic images, more recent images can be found in the reviews mentioned in the introduction). Its characteristic structure is caused by hot upflows which are separated from each other by regions of cold downflows. This is quite different from the hot and narrow upwards flowing plumes observed in the convective planetary boundary layer of the atmosphere of the Earth. Clearly, the boundary conditions of each system are important, i.e., whether the system becomes convective due to a heating source at the bottom (PBL), a cooling layer on top (Sun, convection in the ocean), or a mixture of both (other PBL scenarios), determines the flow topology. Extensive studies on this subject have been performed in the atmospheric sciences ([34–38]), but see also [39] for the astrophysical case. The astrophysical and geophysical scenarios for which convection occurs are nevertheless sufficiently closely related to each other such that studies of results on this subject in the neighbouring field are highly worthwhile for either side. 2.3 Astrophysical implications

On a phenomenological level one can characterize the effects of convection on the physical properties of stars as follows. Through its action on temperature gradients and through the inhomogeneity of physical quantities such as temperature it causes (for instance, at stellar surfaces through granulation), convective heat transfer modifies the emitted radiation of stellar atmospheres and thus it changes the photometric colours of stars (see also [40]), the profiles of absorption lines generated in their photosphere (cf. [41], or [22] for a summary), and the chromospheric activity of stars. From a more general perspective this leads to uncertainties in secondary distance indicators when deriving stellar parameters from analyzing the atmosphere of a star to determine its absolute magnitude and hence its distance. Stellar structure and stellar evolution are influenced by convection especially along the pre-main sequence and during post-main sequence evolution. In typical calibrations convection contributes to the uncertainties in determining the effective temperature for pre-main sequence models up to T eff ± 175 K (e.g. [42]). Along the main sequence it influences stellar radii through the size of the convective envelope of F to M type stars and particularly the stellar luminosity at the end of the main sequence. For the lower red giant branch (RGB) for stars with 1M evolutionary tracks are subject to an additional uncertainty of T eff ±100 K (e.g. [42]). Different convection models and calibrations based on 3D hydrodynamical simulations have been proposed to solve this problem (e.g. [43, 44]). In the end convection influences the mass determination of stars and the interpretation of Hertzsprung-Russell diagram (HRD). Through its capability of overshooting into stable layers convective mixing modifies concentration gradients and thus eventually the evolution of convective cores and hence stellar lifetimes, stellar chemical composition, and stellar (element) yield rates at the end of stellar evolution (cf. the discussion in the introduction of [45]). The effective depth of a convection zone caused by mixing can determine the destruction of trace elements such as 7 Li (T b ∼ 2.5 · 106 K, see also [46]) and likewise also the structure and composition of progenitors of core collapse supernovae (cf. [45]), and the production of Li and other elements in late stages of stellar evolution.

Multi-Dimensional Processes In Stellar Physics

Convection also couples to mean velocity fields and the magnetic fields of stars. It can hence also excite and damp pulsation. Examples include the computation of frequencies of pressure modes in the Sun and solar-like stars ([47, 48]) and the excitation and driving of pulsation ([49–51])). Convection is thus also investigated with the tools provided by helioand asteroseismology. Its role in angular momentum transport, the generation of magnetic dynamos, and the origin of long-term cycles of solar and stellar variability are now increasingly studied not only by observational means, but also by global numerical simulations of convection, which resolve the largest scales of the convective zones (e.g., [21, 28, 52]). 2.4 Local stability analysis for the case of convection

The local stability analysis for convection is usually based on the following assumptions: 1. Consider a fluid element of size , 2. with  less than the lengths along which the stellar structure substantially changes:  < H p = −P/(dP/dr) and  < R∗ , etc., 3. and assume instantaneous pressure equilibrium of fluid elements with their environment (thus: subsonic flow), 4. furthermore that there is no occurrence of “acoustic phenomena” such as stellar pulsation due to sound waves (pressure modes), shock fronts, etc., 5. with temperature and density fluctuations small against the mean temperature and density at a given height. Assumptions 1 to 5 are essentially the Boussinesq approximation. 6. For stellar applications also assume the viscosity to be very small 7. and consider that we can neglect non-local effects, shear flow and rotation, concentration gradients and magnetic fields. Apart from neglecting non-local effects, assumptions 6 to 7 can also be relaxed as part of more general linear stability analyses. We now discuss a simple variant of this kind of analysis. Consider a fluid without compositional gradient in an external, gravitational field which points towards the negative z direction. A fluid element of specific volume V(P, S ) were to move adiabatically, i.e., without heat exchange with its environment, against the direction of gravity from a height z to a height z + ξ. There, a pressure P holds and thus after the displacement the specific volume of the mass element has changed to V(P , S ) while keeping its entropy S . The stratification is stable, i.e., the fluid element is moved back to its original position z, if the fluid element is heavier than the fluid otherwise located at z + ξ, which has a specific volume V(P , S  ). The pressure is P at z and P at z + ξ, both inside and outside the fluid element. Thus, pressure equilibrium follows for a low Mach number flow with no acoustic phenomena (assumptions 3+4). The assumption of adiabaticity requires small temperature and density fluctuations to hold (that is assumption 5). Generalizations beyond assumptions 6+7 would have to be introduced at this point, too. We consider them to hold and hence stability requires V(P , S  ) − V(P , S ) > 0 (the displaced fluid element has a lower volume per mass than other fluid in the environment at z + ξ). Using S  − S ≡ ξ(dS /dz) and the relation (∂V/∂S )P = (T/C P )(∂V/∂T )P > 0 we obtain the stability condition dS > 0, (1) dz

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whence entropy has to increase with height for a convectively stable stratification. For the temperature gradient this implies



 dS ∂S dT ∂S dP = + > 0. (2) dz ∂T P dz ∂P T dz To obtain Eq. (2) we have used assumption 2. It is at this point where inconsistencies come in once   H p . The key criterion for stability in this analysis is the entropy gradient. Through the specific volume of the fluid element it can be linked to buoyancy. Using standard thermodynamical relations (see Chap. 9 of [20]) we obtain the stability criterion for the entropy gradient expressed in terms of the pressure and temperature gradient. The corresponding criterion for chemically homogeneous stars reads: stars are convectively stable where their entropy increases as a function of radius, i.e., if dS > 0. dr

(3)

For a perfect gas, S ∝ ln[PT γ/(γ−1) ] whence (∂S /∂T )P = [γ/(γ − 1)]T −1 and (∂S /∂P)T = P−1 . With Eq. (3) we thus find γ d ln T d ln P dS = + > 0. dr γ − 1 dr dr

(4)

Using dP/dr < 0 (hydrostatic equilibrium), the chain rule of calculus, and the thermodynamic relation ∇ad ≡ (γ − 1)/γ the famous Schwarzschild criterion for convective stability follows: d ln T < ∇ad . d ln P

(5)

The spatial temperature gradient of a star expressed by ∇ := (d ln T/d ln P) hence has to be smaller than the temperature change of a fluid element due to adiabatic compression to ensure stable stratification with respect to convection (see [20] for further details). Eq. (5) is the standard form of the convective stability criterion used in astrophysics: for unstable stratification, ∇ > ∇ad . In meteorology, the gradient of potential temperature is commonly used instead: ∇Θ > 0. Physically, these criteria are all equivalent. 2.5 Conservation laws

For convenience let us consider the case of radiation hydrodynamics for a one-component fluid in the non-relativistic limit. Further extensions to this setup are the subject of other lectures described in this volume. The equations of hydrodynamics describe the time evolution of mass density ρ (continuity equation), ∂t ρ + div (ρu) = 0,

(6)

and momentum density ρu (compressible Navier-Stokes equation), ∂t (ρu) + div (ρ(u ⊗ u)) = −div Π − ρ grad Φ,

(7)

where ∂ f /∂t is abbreviated by ∂t f and the fluid velocity is denoted by u. We first complete this set of equations before the meaning of all variables is explained just below. The conservation law for the total energy density ρ E, where E = (|u|/2) + ε, and thus ∂t (ρE) + div ((ρE + p)u) = qsource + div(πu) − ρu · grad Φ,

(8)

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is used to close this system. The pressure tensor Π in Eq. (7) is defined as Π = pI − π,

(9)

where I is the unit tensor with its components given by the Kronecker symbol δik . The components of the tensor viscosity π in turn are given by

 2 (10) πik = η ∂ xk ui + ∂ xi uk − δik div u + ζδik div u , 3 where ∂ f /∂x j ≡ ∂ x j f . From Eq. (10) the standard form of the Navier-Stokes equation, ∂t (ρu) + div (ρ(u ⊗ u)) = −grad p + div π + ρ g ,

(11)

is obtained and identifying the local sources sources and sinks of energy for the fluid, ∂t (ρE) + div ((ρE + p) u) = div(πu) − div f rad − div h + ρu · g + qnuc .

(12)

We now summarize the meaning of all variables not yet introduced. The dynamical variables (ρ, ρu, ρ E) are vector fields as function of time t and one, two, or three space coordinates x. Moreover, e = e(x, t) is total energy density (sum of internal and kinetic one), while E = e/ρ = E(x, t) is specific total energy or total energy per unit of mass, and ε = E − 12 |u|2 = ε(x, t) is the specific internal energy. The latter is related to the temperature T = T (ρ, ε, c˜ ) through an equation of state and we use c˜ to denote the different quantities specifying chemical composition. For idealized microphysics T = T (x, t) is frequently used. The gas pressure p = p(ρ, T, c˜ ) is also obtained from an equation of state and in Boussinesq approximation, p = p(x, t). For zero bulk viscosity ζ, the viscous stress tensor π is usually denoted as σ and depends only on one microphysical quantity, the shear or dynamical viscosity η, which follows from the kinematic viscosity: ν = η/ρ. Each of these are in general functions of (ρ, T, c˜ ). The radiative transport properties are characterized by the radiative conductivity, Krad = Krad (ρ, T, c˜ ), which for idealized model systems is often a function of location, Krad = Krad (x). It is related to the radiative diffusivity χT = K/(c p ρ) = χ(ρ, T, c˜ ) (knowing the specific heat at constant pressure) and requires the computation of opacity, κ = κ(ρ, T, c˜ ). The equivalent of Krad for the case of heat conduction is the conductivity Kh . Finally, g = −grad Φ is the solution of the Poisson equation div grad Φ = 4πGρ, where G is the gravitational constant, g is the gravitational acceleration in vertical (or radial) direction, and g is the vector of gravitational acceleration, g = (g, 0, 0), if the vertical component is given first. In practice, the gravitational force is often computed from (approximate) analytical solutions or held constant. Since in all cases of interest here qnuc is a function of local thermodynamic parameters (ρ, T , chemical composition, cf. [53]), we find that Eqs. (6), (11) and (12) together with equations for the radiative and conductive heat flux, f rad and div h, as well as (10) form a closed system of equations provided the material functions for κ, Kh , ν, ζ, and the equation of state are known, too. Usually, they are used in stellar modelling in pretabulated form. At high densities, heat conduction contributes: qcond = −divFcond = Kh ∇T . Otherwise, the non-convective energy flux is mostly due to radiation, qrad = −divFrad , for which inside stars the diffusion approximation Frad = −Krad ∇T holds. In the photosphere, the radiative transfer equation is solved, along several rays, usually assuming also LTE, and grey or binned opacities. Its stationary limit reads (see [54] who also provides validity ranges for this form) r · ∇Iν = ρ κν (S ν − Iν ),

(13)

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where r denotes position, Iν intensity at frequency ν, κν opacity for given ν, and S ν is the source function which in LTE equals the Planck function Bν . Angular and frequency integration of intensity then yield qrad and Frad . In numerical simulation of stellar convection the hydrodynamical equations are solved for a finite set of grid cells, with special geometrical assumptions on the simulation box: for whole stars or parts thereof, spheres and spherical shells are common which implies using a polar geometry and an appropriate coordinate system. Box-in-a-star (“local simulations” of solar granulation and the like) and “star-in-a-box” concepts are also popular, since they can rely on Cartesian geometry. In more recent simulation codes, mapped grids, Ying-Yang grids, and grid interpolation in principle allow for an arbitrary or “mixed geometry” (transition from a Cartesian grid near the centre to a polar one near the surface of a star). Boundary conditions for those simulations are either idealized (such as setting the vertical velocity zero for some layer), or are based on realistic physical models, or lateral periodicity is assumed (for global, whole sphere simulations this is trivial, for local simulations in Cartesian geometry it is often a good approximation). 2.6 The challenge of scales

The huge range of temporal and spatial scales is one of the main challenges in modelling turbulent convection in astro- and geophysics. Such flows are characterized by very large scales L on which stratification, heating, and cooling act. Viscous dissipation in turn occurs on very small scales ld . The very high temperatures of astrophysical fluids makes them distinct in comparison with geophysical flows due to the very large mean free path for photons. As a result, in astrophysics the Prandtl number is computed from the ratio of kinematic molecular viscosity to the radiative diffusivity instead of molecular heat diffusivity and consequently, Pr  1 instead of Pr ∼ O(1). For the Sun, L may be taken as large as 180000 km compared to ld which for the interior is in the range of 1 cm to 10 cm. As a result, the Reynolds number Re = U(L) L/ν for the convective zone is of order 1010 to 1014 for typical velocities U(L) depending on the specific length scales L considered whereas Pr is in the range of 10−6 to 10−10 . In comparison, the convective boundary of the atmosphere of the Earth has L ∼ 1 km with ld ∼ 1 mm and Re ∼ 108 at Pr ∼ 0.7. For oceans on Earth L may range form a few km to a few 103 km with ld ∼ 1 mm and Re in the range of 109 to 1012 whereas Pr is around 6 to 7 (varying by a factor of 2 according to specific local conditions). Hence, the scale range for these systems is very similar. Differences among these systems originate from the Prandtl number, the boundary conditions, the specific microphysics with phase transitions, and, in the solar or stellar case, of course the possibility of magnetic fields interacting with the flow. For simulations of solar convection one can make more specific estimates on Reynolds and Prandtl numbers, Péclet numbers, and required grid sizes. A detailed calculation can be found, for instance, in Sect. 2.2 of [24]. This issue is addressed again in Sect. 4 and 5 below.

Multi-Dimensional Processes In Stellar Physics

3 Modelling of convection Most convection models which are currently used in astrophysical calculations assume local isotropy and horizontal homogeneity of the flow as well as that non-local transport can be described as a diffusion process. Geometrical properties of the flow are generally neglected. Some consider a range of bubbles or “eddy-sizes” and dispersion lengths which all depend on local quantities such as temperature or pressure. None of these assumptions strictly hold for convection in astrophysics! Where does this preference for such simplifications come from and what are its consequences?

Figure 1. Numerical simulation of convection at the solar surface, presented originally in [55]. The isosurface where T = 6000 K is shown by a dark yellow shade. Blue tubes mark where a certain value of the magnitude of vorticity is reached. The domain shown is a region with double grid refinement, 1.2 Mm wide horizontally and 1 Mm deep, at a resolution of 3.7 km horizontally and 2.5 km vertically. It is embedded inside a domain 2.5 Mm to 2.8 Mm wide and 1.9 Mm deep at a grid spacing of 5.1 km vertically and 7.4 km horizontally. The latter is the resolution at which the structures visible at this instant of time have formed. This outer box resides inside one which is 3.7 Mm deep and 6 Mm wide and which has half of its vertical and one third of its horizontal resolution (figure by courtesy of H.J. Muthsam).

3.1 MLT and phenomenological models

If we consider a numerical simulation of convection at the surface of a star such as our Sun (e.g., as in [56–59] or in the comparison of [60]), it is immediately evident that the flow field is non-isotropic and inhomogeneous (there are structures of up- and downflows), a whole range of flow structures co-exist, and it is not at all evident why such a flow could be successfully described by a diffusion process (see Fig. 1). This holds especially since the background quantity driving the flow, the entropy gradient, varies substantially on the scale of the flow structures visible at the surface while it also leads to an inhomogeneous pattern in emitted intensity (solar granulation) that coincides well with observations (cf. [56]).

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Figure 2. Numerical simulation of convection at the solar surface, presented originally in [55]. This is the same simulation as shown in Fig. 1, but after further evolution of about 143 s. In this time shearing stresses accounted for on the fine computational grid used for this simulation have led to a highly turbulent flow (figure by courtesy of H.J. Muthsam).

3.1.1 The concept of turbulent viscosity

Even for present supercomputers it is challenging to fully reveal the turbulent nature of stellar convection (see Fig. 2). The ideas traditionally used in convection models of course are much older. In his “Essai sur la théorie des eaux courantes” J. Boussinesq introduced the concept of turbulent viscosity [61]. Contrary to molecular viscosity, its computation is model dependent, since it is not a property of a fluid, but a property of the flow. Its underlying idea is to assume that the main effect of turbulence is a boost of molecular viscosity. Thus, we express velocity fluctuations relative to this mean as ui = ui − < ui >= ui − Ui for a velocity field ui (written in index notation) that has an average of Ui . The Reynolds stress is defined to be < ui uj > and is assumed to be proportional to the mean strain rate S i j = (Ui, j + U j,i )/2 where f, j = ∂ f /∂x j . The proportionality constant in this relation is the turbulent viscosity, νturb , which has to be computed from a model and allows the introduction of an effective or dynamical viscosity, νeff = ν + νturb , which is used to replace ν in expressions for the (mean) strain rate. In the end this model assumes that in a turbulent flow the main effect of small (and in computations possibly unresolved) scales on large scales is that of an “extra viscosity”. The first and most simple model for νturb had been to take it constant. But this assumption already fails for turbulent boundary layers where νturb must vary along the cross-flow direction to obtain sensible results (cf. [10]). To solve this problem L. Prandtl proposed in 1925 his mixing length theory [62]. It assumes νturb ∼ urms , with urms ∼ S and S = (2 S i j S i j )1/2 . The mixing length  is taken from the geometry of the system or other constraints. In particular for pipe flow or boundary layer flow,  is taken as the distance to the (nearest) boundary. For turbulent boundary layer flow this model can explain some basic data ([10]) and is clearly an improvement in comparison to the model where νturb = constant.

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3.1.2 Turbulent diffusivity and the MLT model of stellar convection

Similar to turbulent viscosity the concept of a turbulent diffusivity “generalizes” the law of gradient diffusion which had been proposed to model the transport of heat and concentration on molecular scales. Originally introduced in 1915 by G.I. Taylor [63] and further developed in 1929 by D. Brunt [64], the idea behind turbulent diffusivity is to consider a driving gradient which would lead to a transport of some quantity at the molecular level (heat conduction or, alternatively, radiation). A gradient diffusivity χ can hence be associated with a (diffusive) flux −χ∇ < φ >. The idea now is to replace χ by χeff = χ + χturb and approximate the (advective) transport of φ through a turbulent flow with a gradient diffusion hypothesis: < ui φ > ∼ −χturb ∇ < φ >. If χ is the heat diffusivity, related to the kinematic viscosity ν by the Prandtl number Pr = ν/χ, we obtain an approximation for the turbulent heat flux. Together with the concept of turbulent viscosity this provides the basis for stellar mixing length theory (MLT) as introduced in 1932 by L. Biermann [3]. Just as in linear stability analysis the driving gradient can either be given by entropy (dS /dr), potential temperature (∇Θ ), or superadiabaticity (∇ − ∇ad ), the latter being most popular in astrophysics. In the end stellar MLT is a (down-) gradient diffusion model for the heat flux with a particular description for deriving the turbulent heat conductivity. Various derivations of it yield equivalent results (cf. [19, 20, 65]). The catch of this approach is that it is completely unclear whether it is applicable beyond the linear regime considered during its derivation. In [65] a particularly compact variant of the MLT is discussed which is repeated here for convenience. The convective flux is computed from ∗ Fconv = Kturb β = Krad T H −1 p (∇ − ∇ad )Φ(∇ − ∇ad , S )

(14)

where ∇ > ∇ad with ∇ = ∂ ln T/∂ ln P and ∇ad = (∂ ln T/∂ ln P)ad . Here, Krad = c p ρχ is the radiative conductivity (in the diffusion approximation of radiative transfer), c p is the specific heat at constant pressure, ρ is the density, T is the temperature, P is the pressure (usually the sum of gas and isotropic radiative pressure), and H p = −P/(∂P/∂r) is the local pressure scale height along the radius coordinate r. The turbulent (heat) conductivity Kturb has to be computed from a model (the function Φ) while the superadiabatic temperature gradient is calculated from basic quantities of stellar structure:

 

dT dT (15) = T H −1 − β=− p (∇ − ∇ad ) . dr dr ad The convective efficiency is expressed by the squared ratio of the (radiative) diffusion time scale to the buoyancy time scale, which we call S ∗ here to avoid its confusion with entropy. It can be computed from gαv βl4 ν · , (16) S ∗ = Ra · Pr = νχ χ where αv is the volume expansion coefficient, g the (radial component of) gravitational acceleration, and l is the mixing length. This allows a compact expression for the turbulent heat conductivity of MLT: Φ(S ∗ ) is given by Φ(S ∗ ) = ΦMLT with 3

729 ∗ −1 2 (S ) (1 + S ∗ )1/2 − 1 , ΦMLT = (17) 16 81 and where S ∗ is computed from 81 S = Σ, 2 ∗

Σ = 4A (∇ − ∇ad ), 2

Q1/2 c p ρ2 κl2 A= 12acT 3



g , 2H p

(18)

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and Q = T V −1 (∂V/∂T ) p = 1 − (∂ ln μ/∂ ln T ) p is the variable, average molecular weight. Finally, the mixing length l is usually parametrized as l = αH p ,

(19)

where α is calibrated by comparison with some data or another, integral constraint. 3.1.3 Countergradient flux and non-locality, shortcomings of popular models

Already in 1947, C.H.B. Priestley and W.C. Swinbank realized [66] that the downgradient diffusion model is insufficient to explain observed convective turbulence in the planetary boundary layer of the atmosphere of the Earth (note that this was more than 10 years before publication of the variant of MLT which is commonly used now in astrophysics, [19]). The problem they discovered is that the Schwarzschild criterion predicts a convective instability to exist only where ∇Θ > 0 (or ∇ > ∇ad ), an assumption intrinsic to the MLT prescription for the computation of the convective heat flux. However, turbulence is observed in the atmosphere of the Earth next to a convectively unstable zone also where ∇Θ < 0. This points towards a non-local nature of the flow in the planetary boundary layer. As is discussed below (Sect. 5.1), regions for which Fconv  0 while ∇ < ∇ad are called overshooting zones in astrophysics. A first modification of the downgradient model was proposed in [66]. Deardorff [67, 68] then demonstrated the essential role of non-local fluxes both from the viewpoint of observational data and from theoretical considerations and introduced the concept of countergradient flux into models of the convective planetary boundary layer of the Earth. In his model, Fconv > 0 is compatible with ∇Θ < 0 (∇ < ∇ad ). We return to this question in Sect. 5.1. The main shortcomings of the most popular models of convection used in astrophysics can be summarized as follows: • the commonly used models are of pure diffusion type with a locally specified diffusivity; • they ignore non-local production and thus cannot explain the mixing next to but outside a convectively unstable region; • they ignore apparent geometrical properties such as spatial inhomogeneity of a flow; • for use in astrophysics they require a calibration with solar data, but global quantities such as luminosity are used for this purpose despite the quantity calibrated (primarily the mixing length) describes effects at a more local scale and has been demonstrated to vary over the Hertzsprung-Russell diagram (HRD) and as a function of radius. Alternatives to these models have resulted in the demand for numerical simulations needed for stars all over the HRD. But what about less expensive non-local models of convection? 3.2 Ensemble and volume averages

Recalling the huge range of length (and time) scales (see Sect. 2.6) on which convection in stars operates some kind of averaging has to be applied to the dynamical equations introduced in Sect. 2.5. Two different types of averages are useful in this context. The first one is a volume averaged interpretation of all functions f (t, x, y, z) that appear in the dynamical equations. The goal here is to explicitly account for the dynamically most important length scales in the modelling. In the Large Eddy Simulation (LES) approach these are the length scales where most of the kinetic energy is concentrated, such as the up- and downflow patterns of stellar granulation. This allows for numerical simulations with realistic microphysics. The second approach is based on an ensemble average interpretation of f (t, x, y, z) and allows the

Multi-Dimensional Processes In Stellar Physics

computation of statistical properties of the flow as expressed by the average < f (t, x, y, z) >. This is the basis of (semi-) analytical convection models and typical quantities computed with this approach are Fconv (the convective or enthalpy flux), Pturb (the turbulent pressure), or vrms , the root mean squared (flow) velocity. Let us return to the first type of average. In a grid based numerical approach to solve the hydrodynamical equations the volume average interpretation of ρ(t, x), ρu(t, x), and ρe(t, x) occurs quite naturally: either the function values given at the grid points are considered to be interpolation nodes for a numerical approximation which has to ensure to conserve the basic variables within a grid cell as in a conservative finite difference scheme. Or the function values on the grid are considered as numerical approximations to the averages of the basic variables over the grid cell (finite volume approach, conservative by construction). Both concepts to solve the hydrodynamical equations thus are quite naturally compatible with the idea of a volume average. The LES approach in a strict sense is closely related to the finite volume approach, but not identical with it. As indicated the idea is to choose a grid such that scales carrying most of the kinetic energy are explicitly resolved by the grid. For simulations of stellar granulation this includes those spatial scales on which radiative cooling takes place at the stellar surface. Smaller scales are accounted for through a “subgrid scale model” or simply, in the sense of the concept of turbulent viscosity discussed in Sect. 3.1.1, by some form of extra (artificial, etc.) viscosity. The influence of scales larger than the domain size has to be taken care of by boundary conditions. As starting point for such a numerical simulation one assumes “typical” initial conditions and first simulates the (kinetic and thermal) relaxation of the system before a statistical interpretation of the results based on a quasi-ergodic hypothesis is done (cf. [24]). Continued time integration is used to generate ensembles and “typical cases” where the latter may be inspected by some visualization software. This approach is based on the assumption that a sufficiently long time integration provides statistical realizations of the physical system with the same probability distribution as if they were generated directly from a whole set of simulations with (slightly) different, randomly generated initial conditions. The computation of averages over long time intervals allows a comparison of the simulation with astronomical observations and direct ensemble averages. But this approach is of course computationally quite expensive (see again [24]). The goal of ensemble averaged models on the other hand is to directly compute functions such as < ρ(t, x, y, z) > where the ensemble consists of different realizations of ρ (obtained from “sensible” initial conditions which differ only by a small amount from each other). Of course, these have to be compatible with the boundary conditions. There are clear differences for such models in comparison with the strict derivation of the Navier-Stokes equations (NSE). The macro-states considered for a turbulent flow are much more complex than the “fluid-elements” invoked in the classical, phenomenological models such as MLT (Sect. 3.1.2). There is no generally valid and fully self-consistent procedure known to derive such models from first principles, i.e., the NSE, only. Thus, convection models, just as any other models of turbulent flows, need additional hypotheses, the closure approximations. These might be simply inspired from analyses of laboratory systems, but the non-linearity of this problem and its sensitivity to boundary conditions make an approach based just on data from laboratory scenarios highly incomplete and likely to fail in applications to astrophysical flows. Hence, a more general strategy is needed. 3.3 Reynolds stress approach

An ensemble average can be computed straight from a numerical simulation assuming that the quasi-ergodic hypothesis holds for it. When deriving (semi-) analytical models it is common to first perform a splitting of the fields of the dynamical variables into an often slowly

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varying background and any fluctuations around it. The background state is simply taken to be the ensemble average of the basic variable, say ρ. This Reynolds decomposition or Reynolds splitting was first suggested in 1894 by [69]. Formally, an equation for A(t, x, y, z) is additively split by introducing A = A + A . The dynamical equation for A is averaged to obtain an equation for A. Since the NSE are non-linear, the equation for A actually depends on averages over products, A A , of the fluctuations around the mean state. This gives rise to the idea of a moment expansion of the hydrodynamic equations, which was first suggested in 1925 by Keller and Friedmann, [70]. This procedure requires to subtract the dynamical equation for A from the complete equation for A to obtain a dynamical equation for A . A dynamical equation for A A is obtained from computing the product of the latter, averaging it, and using basic calculus and algebra. But due to the non-linearity of the underlying equations an infinite hierarchy of moments is obtained this way, the basic closure problem of hydrodynamics. To proceed from there requires assumptions in addition to the basic equations themselves. This is its main disadvantage over the volume average. The additional assumptions for closing ensemble averaged moment equations directly involve the physics of the large, energy carrying scales and those are very difficult to model analytically. One-point closure turbulence models are the most widely used class of such models. Their main idea is to perform the ensemble averaging directly in physical space (x, y, z). At lowest order this generates equations for the mean thermal structure, i.e., temperature T and pressure P, as well as for a non-zero mean flow. Fluctuations around these mean values describe the “turbulent component” of the flow: w = W − W, θ = T − T . The dynamical equations for those are used to derive dynamical equations for the second order moments (SOMs) of the basic variables: the quantities w2 and θ2 quantify the kinetic and thermal (potential) energy contained in the turbulent component of the dynamical fields and their cross correlation wθ is associated with the contribution of the turbulent component to the mean heat flux in the system. The next higher order statistics, the third order moments (TOMs) describe non-local transport by the turbulent component (flux of turbulent kinetic energy) and basic asymmetries of the flow field (skewness). Usually though not necessarily, the ensemble averaged equations are constructed for the horizontal average of the fields. Various isotropy assumptions are common as well. The most popular closure assumptions are based on expressing all higher order moments in terms of SOMs and sometimes also TOMs. For instance, the fourth order moments are frequently assumed to follow a Gaussian distribution. If no further restrictions are put on the TOMs in this case, this approach is known as quasi-normal (QN) approximation. “Local models” on the other hand assume that the TOMs are zero contrary to “non-local models” which have non-zero TOMs. Instead of the plain QN approximation the so-called damped QN approximation is often preferred which limits the size of the TOMs (cf. [71, 72]) by explicit clipping or by damping through boosting some of the closure terms. An attractive property of this approach is that it is straightforward to understand the physical meaning of at least the lower order moments. The mean structure of the object modelled is given by the averages of T and P. The second order moments describe the effects of convection on the mean structure: the enthalpy flux Fconv = c p ρwθ can modify the thermal structure in comparison with pure radiative or conductive heat transfer. The turbulent pressure Pturb = ρw2 can change the hydrostatic equilibrium structure. In addition, if the mean structure is perturbed by a large scale velocity field (global oscillations), the turbulent fields provide feedback and interact (see [23] for further details). Third order moments describe non-local transport through advection and are thus related to quantities such as the filling factor (fraction of horizontal area covered by upflows, for example) and, as already indicated, the skewness of the velocity and temperature field. Standard local models of convection such as

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mixing-length theory consider horizontal averages of the mean structure and the SOMs and ignore non-local transport: the TOMs are set to zero. The SOMs are treated using algebraic relations and concepts from the diffusion approximation are used Non-local convection models thus have to first and foremost suggest a non-trivial approximation for the TOMs. An advanced model of this type was derived in [73, 74] and [75]. It consists of 5 differential equations which describe the time evolution of turbulent kinetic energy q2 = u2 +v2 +w2 (i.e. the sum over squared fluctuations of horizontal (u, v) and vertical (w) velocity components), of temperature fluctuations θ2 , of the (convective) temperature flux wθ, and of the vertical component of turbulent kinetic energy, w2 . An additional differential equation provides the kinetic energy dissipation rate . Heat loss effects on temperature fluctuations are accounted for by algebraic closures derived from turbulence modelling (see also [76] for further details). Compressibility effects and anisotropy effects are included in this model following [74], see also [77]. Various models for the TOMs have been suggested in this framework, [73–75, 78]. These have been applied, among others, to models of the convective planetary boundary layer of the Earth (see [78, 79]). That work included comparisons to numerical simulations and to the Deardorff and Willis laboratory experiment [80]. To compare the complexity of this approach with MLT a variant of the Reynolds stress convection model of [73–75] as used in [77] is given in the following. Using the notation of [75] and abbreviating ∂t ≡ ∂/∂t and ∂z ≡ ∂/∂z this convection model reads as follows: 1 ∂t K + Df (K) = gαv J −  + Cii + ∂z (ν∂z K), 2 1 1 1 1 2 2 ∂t ( θ2 ) + Df ( θ2 ) = βJ − τ−1 θ θ + ∂z (χ∂z θ ) + C θ , 2 2 2 2 2 1 1 ∂t J + Df (J) = βw2 + gαv θ2 − τ−1 pθ J + ∂z (χ∂z J) + C 3 + ∂z (ν∂z J), 3 2 2 1 2 1 2 2 1 1 2 1 2 2 ∂t ( w ) + Df ( w ) = −τ−1 pv (w − K) + gαv J −  + C 33 + ∂z (ν∂z w ), 2 2 3 3 3 2 2  ∂t  + Df () = c1 K −1 gαv J − c2  2 K −1 + c3  N˜ + ∂z (ν∂z ), N˜ ≡ gαv |β|,

1 Df () ≡ ∂z (w) = − ∂z (νt + χt )∂z  . 2 To calculate the mean stratification one has to solve these equations alongside

(20) (21) (22) (23) (24) (25)

∂z (P + pt ) = −gρ,

(26)

cv ρ∂t T + ρ∂t K = −∂z (Fr + Fc + Fk ),

(27)

which are the equations of hydrostatic equilibrium and of flux conservation. They have to be extended by an equation for mass conservation and for radiative transfer. Non-locality is represented by the terms Df (K) = ∂z (q2 w/2), Df ( 12 θ2 ) = ∂z (w θ2 /2), Df (J) = ∂z (w2 θ), and Df ( 12 w2 ) = ∂z (w3 /2), which require closure approximations for the third order moments q2 w, w θ2 , w2 θ, and w3 . An example for the latter is the downgradient model Eq. (25) for w. Instead of closing with only lower order moments, also other moments of the same order can be used. Often though not necessarily these provide better performance: w ≈ 0.6 q2 w/τ with τ = q2 / has been found in [81] to be a much more accurate closure than Eq. (25). The other variables introduced in Eq. (20)–(27) are the following ones: K = q2 /2 is the turbulent kinetic energy (per mass unit). θ2 /2 are the squared fluctuations of the temperature

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field around its mean value and J = w θ is the cross-correlation between vertical velocity and temperature, both introduced further above. Moreover, αv is the volume compressibility (for a perfect gas this is = 1/T ) and β is the superadiabatic gradient. The radiative (thermal) diffusivity χ = Krad /(c p ρ) and the kinematic viscosity are related to each other through ν = χPr. The variables denoted with the letter τ are all time scales. Equations (25a), (27b), and (28b) of [75] can be used to relate these to the dissipation time scale τ = 2K/. Compressibility effects are included through the additive terms Cii , Cθ , C3 , C33 taken from Equations (42)–(48) of [74]. Depending on their actual size some of their contributions can also be neglected, see [77]. In Eq. (25) the turbulent viscosity νt = Cμ K 2 / is introduced where Cμ is a constant given by Eq. (24d) of [75], for which we take the Kolmogorov constant to be Ko = 1.70. The turbulent heat diffusivity χt is given by the low viscosity limit of Eq. (11f) of [75]. Depending on the specific problem molecular dissipation can be included through restoring the largest second order moment terms containing ν, i.e., ∂z (ν∂z K), etc. They are important when Pr is of order unity rather than zero, [77], as is often the case when comparing solutions of convection models to numerical simulations for idealized microphysics. Therefore, such a contribution can also be included in (24). For the latter, c1 = 1.44, c2 = 1.92, and c3 = 0.3 where β < 0 while c3 = 0 elsewhere, as suggested in [75]. At this level of complexity the model has been used in [76, 77, 82]. When compared to 3D hydrodynamical simulations it allows accurate predictions for cases with large radiative losses, as in sufficiently hot A type and DA type stars ([83]), but not for solar-like convection. 3.4 Two-scale mass flux models

For deep convection zones with low radiative heat losses inside the convection zone, such as in our Sun, a comparison with numerical simulations for idealized microphysics reveals [83, 84] that the models for the TOMs suggested in [74, 78] cannot any more reproduce higher (third) order moments. Those cases of numerical simulations of compressible, vertically stratified convection for which they had been found to work, such as discussed in [77], had been characterized by small values of skewness of the vertical velocity and temperature fields. A large skewness is related to flow topology and results from suitable boundary conditions. Non-local transport then leads to inhomogeneity of the flow, the formation of an asymmetric distribution of up- and downdrafts. This has been analyzed in meteorology already about 30 years ago. The physical mechanism behind the asymmetry between top-down and bottom-up “diffusion” due to a turbulent flow was discussed in [34] while the nature of the vertical-velocity skewness for the planetary boundary layer was discussed in [36]. Narrow convection zones with strong radiative losses reduce the influence of the boundary layers which leads to a smaller skewness of the vertical velocity and temperature field. For the atmosphere of the Earth the most detailed in situ measurements of these and related quantities has been made during the aircraft campaign ARTIST which took place from 4–9 April in 1998 (see [85]). To explain these data a new model was derived in [86] while a competing model was suggested by [87]. In the following an overview on the model by [86] is given. From an analysis of the PBL (planetary boundary layer) aircraft data [86] suggested that coherent structures contributed most to the higher order moments of the vertical velocity and temperature field and thus also to skewness. They hence considered averages over up-/downflow areas and separately averages over hot and cold areas. The result is a two-scale mass flux average which replaces the previously suggested mass flux average (also known as “elevator model”) to compute higher order moments in the ballistic limit of large skewness. In that case either the up- or the downdrafts (or either the hot or the cold regions) are so localized within each horizontal plane that normal to the latter the fluid moves at high velocity (or large temperature difference) with

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just limited interaction with its environment. This extreme, asymptotic limit for very large skewness gives rise to the idea of considering n-δ probability distribution functions for the velocity and temperature distributions of turbulent flow fields. The level of averaging required to arrive at this model was visualized through a data record obtained from a flight through the convective planetary boundary layer at a certain altitude [86]. Horizontally and in time the unaveraged data vary highly. A top-hat average then segments this function into constant functions between sign changes for each variable. Here, the sign is obtained from subtracting the horizontal (and temporal) mean from each quantity measured at a particular vertical depth and instant of time. Downflow regions have a negative velocity, upflow regions a positive one, and likewise for regions of the flow where the temperature is lower or higher than the specified average. The two-scale mass flux average is then obtained by determining the average over each of the upflow segments, the downflow segments, the hot flow regions, and the cold flow regions. This is more general than the one-scale mass flux average introduced by [88, 89] where an average value is computed for any variable over all upflow regions joined by a second average which is performed over all downflow regions. An alternative representation of the two-scale mass flux are 4-δ probability distribution functions which for each horizontal layer describe the distribution as averages over all four combinations of signs for vertical velocity and temperature. For the convective planetary boundary layer, 35% to 45% of the total horizontal area are covered by cold downflows between which an area of 25% to 30% is covered by hot, upwards rising plumes originating from the heating of the flow at the surface. But between 25% and 40% are covered by flow characterized through the other two possible combinations of signs. With the role of signs reversed (hot upflows dominating in area over cold downflows) this can also be found for solar granulation (see [39]), although the detailed distribution varies depending on whether averages are taken over horizontal planes or planes of constant optical depth. Using correlations computed with the 4-δ probability distribution function to approximate ensemble averages in the limit of large skewness, the influence of turbulent fluctuations caused by shear between the up- and downflows is accounted for in a second step. [86, 90] suggested these to be obtained from linear interpolation between the quasi-normal, Gaussian limit for the case of zero skewness and the ballistic two-scale mass flux limit in the other case. This allows specifying closed expressions for fourth order moments of the flow and for crosscorrelation third order moments. The skewness for both vertical velocity and temperature has to be specified from some other source and the same holds also for second order moments (estimates for w θ from the two-scale mass flux turned out to be not useful for improved closure relations, as it is much less correlated to up- and downflow regions than its higher order counterparts, see [86]). In this form the model was tested with both aircraft data and simulation data for the convective planetary boundary layer. Root mean square errors between model and data were found to be drastically smaller than for the quasi-normal approximation. The top-hat average two-scale mass flux relations of [86] read as follows: < vz >mf < T >mf < w2 θ >mf < wθ2 >mf < w >mf 4

< θ4 >mf < w θ >mf 3

< wθ3 >mf

= a wu + (1 − a) wd , = b θh + (1 − b) θc . = S w σw < wθ >mf ,

(28) (29) (30)

= S θ σθ < wθ >mf ,   = 1 + S w2 σ4w ,   = 1 + S θ2 σ4θ ,   = 1 + S w2 σ2w < wθ >mf ,   = 1 + S θ2 σ2θ < wθ >mf .

(31) (32) (33) (34) (35)

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The quantities wu and wd are the horizontal averages over all up- and downflows relative to the mean upwards velocity, < vz >mf . In the same sense, θh and θc are the horizontal averages over all areas with a temperature higher and lower than the mean temperature < T >mf , respectively. For the two-scale mass flux averages the external parameters a, b, wu − wd , and θh − θc can be replaced self-consistently by the quantities σw , σθ , < wθ >mf , S w , and S θ . This has already been used here for Eq. (30)–(35). From this starting point [86, 90] have suggested that the ensemble averages   w2 θ = S w σw wθ = w3 /w2 wθ, (36)   wθ2 = S θ σθ wθ = θ3 /θ2 wθ, (37)

 2 1 w4 = 3 1 + S w2 w2 , (38) 3

 2 1 θ4 = 3 1 + S θ2 θ2 , (39) 3

 1 w3 θ = 3 1 + S w2 w2 wθ, (40) 3

 1 wθ3 = 3 1 + S θ2 θ2 wθ, (41) 3 w2 θ2

=

2

w2 θ2 + 2wθ + S w S θ wθ σw σθ ,

(42)

hold in the limit of both large and small skewness. Here, the cross-correlation for Eq. (42) has been added which has been derived in [90] to complete the closure relations of [86]. The relations for Eq. (36)–(37) had already been suggested in earlier work (see [91, 92]). Note that for zero skewness this new model coincides with the Millionshchikov 1941 (quasinormal) approximation [93]. Alternatively, it was suggested in [86] to consider optimized parameters ai and di instead of 3 and 1/3. The improvement gained from such a procedure is much smaller though than that one already gained by Eq. (36)–(42) relative to the QN, [93]. The realizability of the closure approximations Eq. (36)-(42) was demonstrated in [90] for the planetary boundary layer using aircraft data and numerical simulations as well as in [94] for granulation in the Sun and in a K-dwarf using hydrodynamical simulations. The quasi-normal approximation yields non-realizable results for the temperature field of the atmosphere and for both the velocity field and the temperature field for both types of stars. Studies which show the substantial improvement of either some or all of Eq. (36)-(42) when compared to previous models include a resolved numerical simulation of free oceanic convection, [95], the numerical simulation of deep, compressible convection with idealized microphysics, [81, 83, 84, 96], the numerical simulation of solar granulation, [24, 94, 97], and granulation in a K-dwarf, [94], as well as in a DA white dwarf, [24], and again simulations of compressible convection with idealized microphysics, [98]. Given these credits, what are the known limitations of this model apart from still existing (though smaller) discrepancies between direct computations of the ensemble averages and their approximation through Eq. (36)-(42) ? Clearly, the up- and downflows trigger shear induced turbulent fluctuations at their boundaries (see [56] and also [58]). In the model of [86, 90] a quasi-normal distribution is assumed to hold for the case of zero skewness of a field of fluctuations of velocity or temperature. But solar granulation provides a counter example: at the bottom of the solar superadiabatic peak the skewness S w = w3 /(w2 )1.5 drops from ≈ −0.2 to −1 whereas the kurtosis Kw = w4 /(w2 )2 ≈ 2.5 instead of 3 in that region. Likewise, S θ = θ3 /(θ2 )1.5 ≈ 0 there, whereas the kurtosis Kθ = θ4 /(θ2 )2 ≈ 1.3 instead of 3.

Multi-Dimensional Processes In Stellar Physics

Values of Kθ ≈ 2 at the bottom of the surface superadiabatic peak are also observed for numerical simulations of DA white dwarfs, [99], or K dwarfs, [94], and independently of vertical boundary conditions or the particular simulation code used. At the same time the large values of the kurtosis deep inside thick convection zones are still somewhat underestimated by the model of [86, 90] (see again [94]). Both the modelling of statistical properties of the large scale coherent structures and the local turbulence created by the flow still require further improvement, although the discrepancies are already much lower, typically by factors of 2 to 3, than those obtained with the quasi-normal approximation. Plume models for the downdrafts may provide some of these improvements (see [100]). Recalling the Reynolds stress models it might seem attractive to combine the closures suggested in [86, 90] to a more general model (see [86]). However, the naive combination of “favourite closures” including those by [86, 90] easily triggers instabilities as has been shown in [83] and confirmed by [98]. 3.5 Comparisons

For systems with efficient convection the comparison with numerical simulations implies that the non-local convection models based on the downgradient approximation for third order moments or the quasi-normal approximation of fourth order moments (such as those used in [76, 77, 82], but also [101, 102] or [103]) cannot reproduce third order moments and thus non-local transport effects (cf. [81, 83, 84, 96]) despite such models claim to account for these effects. The cases of interest characterized by efficient convection all feature large absolute values of skewness which have to be a consequence of boundary conditions and non-locality (see [34] and [36], e.g.). This leads to a strong spatial inhomogeneity of the flow with strong asymmetric up- and downflow areas, as has also been found for geophysical cases such as convection in the atmosphere of the Earth or in the ocean. But what about cases in which such models have been shown to work, as in [76, 77, 82] or in [78]? Can they be distinguished from the other cases? One important difference is that these “good cases” all have small absolute values of skewness. Hence, it is less important that coherent structures and flow topology have not been taken into account in much detail in these convection models. In turn, convection models that explicitly account for flow structure are clearly favourable over those which do not, as these features are an essential part of the physics of convection.

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4 Challenges and pitfalls in numerical modelling Hydrodynamical simulations are now a key tool of stellar astrophysics. A summary on codes and simulation strategies for modelling convection has been given elsewhere, [24]. The topics highlighted in the following are usually paid less attention for in discussions of numerical modelling of solar and stellar convection. Their closer consideration should help avoiding some of the main pitfalls and allow a better understanding of the challenges of this approach. 4.1 General remarks

Due to finite computational resources any hydrodynamical simulation of stellar convection for the case of realistic microphysics and realistic radiative fluxes has to consider a volume averaged approximation of ρ(t, x), ρu(t, x), ρe(t, x). In the language of engineering sciences these are hence large eddy simulation (LES). As has already been explained in Sect. 3.2 in that case the numerical simulation directly resolves the scales carrying most of the kinetic energy (e.g., scales on which radiative cooling takes place at the stellar surface) on the computational grid. Smaller scales are taken care of by some subgrid scale model. Often this may just be some numerical or artificial viscosity. Scales larger than the computational grid have to be handled by boundary conditions. In practice the simulation is started from “typical” initial conditions and relaxed towards a statistical equilibrium state (see Sect. 3.2). A statistical interpretation of the results of a numerical simulation of convection is thus part of the computational procedure. The time integration of the simulation is used to generate both ensembles and the typical cases (shown as snapshots such as Fig. 1). The computation of averaged values over sufficiently long time intervals allows a comparison with observations, for instance of spectral absorption lines, and the computation of direct ensemble averages. Other tools used for the interpretation of hydrodynamical simulations include also graphical visualization or interpretation of the simulation data in Fourier space. How well does this concept deal with turbulent flows in practice? To answer this question some terminology has to be introduced first. For the phenomenology of turbulent flows there is no commonly accepted definition. In a strict sense ([12]) it refers to anything but direct experimental results, direct numerical simulations, and a few results obtainable from first principles. From this viewpoint any, necessarily large eddy type, simulation of stellar convection with realistic microphysics would be considered phenomenology. However, in astrophysics the usage of the word phenomenology is restricted to models that introduce a concept such as rising and falling bubbles which cannot be derived from first principles nor confirmed by experiment or numerical simulation, but is used in deriving mathematical expressions of the model. Stellar mixing length theory is a classical example for that. Turbulent flows are distinguished by being capable to give rise to large scale, coherent structures in spite of the long-term unpredictability of the flow itself. The claim that statistics and structure contrapose each other is among the common misconceptions about turbulent flows, as explained in [12]. Indeed, the conduction and evaluation of numerical simulations of (turbulent) flows are based on assuming statistical stationarity and quasi-ergodicity from the beginning. Thus, any numerical simulation of this kind is based on a statistical approach to the physical problem. Of key importance in this context is the quasi-ergodic hypothesis. It is frequently worded as follows: “The time average of a single realization, which is given by one initial condition, is equal to an average over many different realizations (obtained through different initial conditions) at any time t in the limit of averaging over a large time interval and a large ensemble.” (cf. [12]). For an introduction into statistical descriptions and ensemble averages of turbulent flows see Chap. 3 of [10].

Multi-Dimensional Processes In Stellar Physics

This hypothesis is fundamental to any numerical simulation of stellar convection and one consequence thereof is that the detailed initial conditions on the velocity fields are “forgotten” by the flow after a relaxation time trel . This cannot be proven to hold for all flows: there are known counterexamples, but for some flows such as statistically stationary (time independent), homogeneous (location independent) turbulent flows it can be corroborated even directly from numerical simulations (Chap. 3.7 of [12]). 4.2 Uniqueness of numerical solutions

The Navier-Stokes equations depend on non-linear fluxes which are purely algebraic combinations of the basic, dependent variables and of pressure. In Eq. (6)–(8) they appear on the left-hand side as objects on which the divergence operator acts. Together with the pressure gradient term in Eq. (7) contained in the first term on the right-hand side (separated in Eq. (11) from contributions containing viscosity) they provide the hyperbolic part of the hydrodynamical equations. They are also contained in the Euler equations of hydrodynamics which are distinguished from the Navier-Stokes equations by assuming zero viscosity (η = 0, ζ = 0), whence from Eq. (10) we obtain π = 0 for this case. One may wonder why one should not ignore viscosity altogether from the very beginning. The reason for this idea is that in stars the kinematic viscosity ν is “small”. This is suggestive for neglecting terms containing it and thus to just solve Euler’s equations instead of the full Navier-Stokes equations (NSE). Indeed, if one numerically approximates the corresponding terms in the NSE, the resulting contributions are extremely small. If one decides to just not care about viscosity and solve the Euler equations without any viscosity, one has to face an unpleasant problem: the resulting solutions can violate the fundamental laws of thermodynamics and there can be even an infinite number of solutions to a given initial condition. A unique, physically consistent solution is needed instead. At the root of this problem is the fact that hyperbolic conservation laws can have discontinuous solutions developing in finite time for smooth initial conditions. Strong solutions (as obtained for the partial differential equation form of a conservation law) are unavailable in that case while weak solutions (of the integral form of a conservation law) can still be found. Already for the special case of the Riemann problem one can find out about and understand the consequences of these properties. In the Riemann problem initial conditions with a jump between two constant states and different initial velocities in those regions are considered. If those initial conditions have characteristics (special invariants of the solution) going into discontinuity, a unique solution exists for that problem. If the characteristics are instead going out of it, even an infinite number of solutions is possible. Many of these solutions will be unphysical. But a unique physical solution exists for fairly general cases (see [18]), if one requires the solution to be that one of a more general, second order partial differential equation for which viscosity has not been neglected yet and takes the limit ν → 0. This solution is known as the “entropy solution”: it is consistent with thermodynamics and it exists for the Euler equations if they are interpreted as limit ν → 0 of the Navier-Stokes equations. For the Euler equations they can be obtained directly from imposing the RankineHugoniot (jump) conditions at discontinuities of the solution. This ensures the uniqueness of the solution and as a result the conservation laws also hold across solution discontinuities. Finally, consistency of this solution with the laws of thermodynamics is ensured. Consequently, all numerical simulation codes (and in particular any LES of stellar convection) must use some prescription of viscosity. Examples include numerical viscosity, artificial viscosity, and others. Its contribution is often difficult to extract from simulations which can become inconvenient if one has to compute the dissipation rate of (turbulent) kinetic energy or related

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quantities. It is important to note that this form of viscosity is required in addition to others which are introduced to account for unresolved scales as in any LES of a turbulent flow (such as hyperviscosity or subgrid-scale viscosity, ...). The inevitability of having to introduce viscosity makes any claims of parameter freeness of numerical simulations of convection for systems with small viscosity pointless. This has been discussed in detail in Chap. 6 of [24] and is briefly addressed below in Sect. 4.5. 4.3 Initial conditions and relaxation

Numerical simulations of convection have to start from some initial state of the system. A “statistically characteristic” initial condition for the given problem, say a simulation of solar granulation, cannot be constructed analytically from first principles. Rather, some more idealized state will have to be considered instead, such as the horizontal average given by a one-dimensional stellar structure model. In the first part of the numerical simulation the system is evolved until it forgets this peculiar initial state. Data for detailed analysis are collected only for relaxed states of the simulation. For the case of numerical simulations of the solar surface which are based on small boxes with Cartesian geometry (box-in-a-star simulations) the relaxation process was studied in more detail by [104]. Three solar structure models based on two different stellar evolution codes with two different convection models being used provided initial conditions for three numerical simulations distinguished from each other only by their initial state. Except for a very small difference caused by slightly different entropies of each initial condition the simulation averages after one hour of solar time turned out to converge to the same solution. This was tested by comparing entropy and the superadiabatic gradient as a function of depth and of pressure (see Fig. 9 and 10 in [104]). As the initial states differ by more than 1% only for the superadiabatic peak and for the photosphere, this is not surprising: the thermal time scale for those layers is less than one solar hour. Once these layers are statistically relaxed and since the region underneath them is quasi-adiabatic with just a small difference in entropy between each of the three solar structure models, the simulations are thermally relaxed and thus have the same mean thermal structure. This is discussed in further detail in [24]. In [99] relaxation is studied for a DA white dwarf with a shallow surface convection zone. The good agreement for solar granulation simulations made with different numerical simulation codes (cf. [60, 97]) is thus not a complete surprise: as long as resolution, microphysics, chemical composition, and the entropy in the quasi-adiabatic layers are comparable, relaxation proceeds to a similar physical state after just one solar hour. This is the thermal relaxation time for the superadiabatic zone and also the convective turnover time scale which guides relaxation of kinetic energy, as it is tied to the entire energy being transported by the flow, i.e., the convective motions. The question of how long do we have to relax a simulation of convection before proceeding with its statistical analysis can thus be answered as follows [24]: relaxation has to ensure that the mean stratification is in thermal equilibrium. Thus, trel ∼ ttherm at least. Since this can become computationally very expensive, the main trick for fast relaxation when doing twoand three dimensional (2D and 3D) numerical simulations of stellar convection is to guess a thermally relaxed stratification from a proper one-dimensional model. For simulations of the top layers of a deep, quasi-adiabatic convection zone such as our Sun this is quite easy. There, trel ∼ tKH (r), the local, depth dependent Kelvin-Helmholtz time scale (see both [104] and [24]). This is the time it takes to exchange the thermal energy contained in the layers above r with the environment for a fixed luminosity or input flux. Only if the dominating relaxation process is radiative diffusion, then trel ∼ tdiff (r) instead. Here, the timescales are based on integrals ranging from the top of the simulation to the layer r below which the mean

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stratification remains roughly constant with time due to the layers underneath being relaxed from the beginning. Since no good initial condition can be guessed for the velocity field, the relaxation of kinetic energy requires trel  tconv (rbottom ). Generally, trel ∼ max(tconv (rbottom ), tKH (rrel ))

(43)

with an initial condition chosen which hopefully permits rrel > rbottom . In [104] it was also shown that the relaxation of kinetic energy may take much longer for a 2D simulation of stratified (compressible) convection than for a 3D one. The 2D case may show a plateau in kinetic energy for some time (see their Fig. 12), but then this quantity can start to increase or decrease again. This is likely connected to the fact that in 2D large scale (vertically oriented) vortices form with a long life time and their merging and disappearing and reappearing may cause major changes in the amount of kinetic energy in the flow. As discussed in [12], 2D flows are very often non-ergodic even when their 3D counterparts are quasi-ergodic. Hence, their relaxation and statistical evaluation may set requirements quite different from those of 3D simulations for the same scenario. Successful relaxation then sets the stage for further analysis of the results of numerical simulations of convection. How long do we have to average and evaluate a simulation? For time independent, quasi-stationary quantities, this is determined by the number of realizations required to obtain statistical stationarity and tstat , the time scale over which we want to do averages to compute statistical properties, may hence vary from a few snapshots to many hundreds of sound-crossing times. In fact tstat can also depend on the width of the simulation box or, equivalently, on the number of granules contained in it. If the quantity of interest is truly time dependent, tstat depends on the time scales of the underlying physical process such as stellar oscillations, mode growth rates, or mode damping time scales. Provided sound waves and radiative transfer are also properly taken care of, Ntot = (trel + tstat )/(δtadv ) ∼ 106 . . . 108 time steps can be afforded for a 3D simulation of stellar convection with about 4003 grid points. As an order of magnitude estimate this separates computable from unaffordable problems on top department clusters (on top supercomputer clusters calculations which are some two orders of magnitudes more expensive are typically still affordable, see Sect. 4.6). If this is to be circumvented, radiative transfer has to be accelerated, local resolution has to be reduced, or other measures have to be taken to proceed with a computing project in acceptable time. 4.4 Boundary conditions

Vertical boundary conditions in simulations of stellar convection are parametrized. The reason for this is that the inflow of mass and energy at the vertical boundaries has to be modelled. Even without explicit numerical constants involved, this inevitably requires some form of parametrization. The most simple assumption is that of a closed vertical boundary with zero vertical velocity. Since that keeps the fluid completely inside the computational box, any inflow and outflow of energy has to be due to radiation (or artificial conduction). If inflow and outflow of fluid are permitted, the parametrization becomes more delicate. Due to the extreme stratification of the fluid and a comparatively moderate flux of energy through stellar convection zones, a feature illustrated by the ratio of kinetic and potential energy in simulations of solar granulation, [104], the mean thermal structure of stellar convection zones is quite robust. The mean temperature structure in the superadiabatic layer is even independent of whether closed (impenetrable and reflecting) boundaries or open boundaries have been used for the vertical velocity field in published simulations (see [59, 97]). However, this is not the case for many higher order correlations. Especially, quantities depending on horizontal velocities are sensitive (cf. [59]), but also higher order correlations

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which depend on the temperature field and vertical velocity field (see [104]). A safety margin of between 1 and 2 local pressure scale heights as measured at each boundary is needed even in case of the best vertical boundary condition models currently available (cf. [94, 104]). Evidence for this comes from dedicated 3D numerical simulations of solar granulation made with the ANTARES code which differ only in the boundary conditions chosen (see [104]). There, the mean temperature was found to be invariant to variations of the vertical boundary conditions. However, the flux of kinetic energy turned out to be sensitive to those changes in the entire lower half of the simulation domain (of about 5 Mm), despite the vertical boundary conditions considered all permitted free outflow and an inflow, as specified by the particular boundary condition model. Closer inspection demonstrated that also the probability distribution of vertical velocities is equally sensitive to details of the boundary conditions. In the superadiabatic layer the velocity distribution was found insensitive to the boundary conditions applied several pressure scale heights away from it. However, half-way to the bottom of the simulation box, some 2 Mm below the optical surface, systematic difference are apparent which close to the bottom become striking [104]. The recommendation drawn from this was to opt for the boundary condition causing the smallest change in the shape of the velocity distribution as function of depth. This choice also minimizes spurious features in the fluxes of kinetic energy and enthalpy and other quantities such as skewness and kurtosis. 4.5 Criteria for modelling replacing the quest for parameter freeness

The task of choosing the “best” vertical boundary condition motivates the question which criteria have to be applied when choosing parameters in any approach to model convection. The parameters to be worried about do not include those directly linked to the physical system studied (mass of a star, radius, etc.), but those introduced by physical and numerical approximations. Against widespread opinion numerical simulations have this problem, too. The quest for a parameter free modelling of convection has a long tradition in astrophysics. In the end the only way to completely avoid “free parameters” is to consider a flow field so idealized as to become unrealistic. Examples include assuming the flow field is 1. irrotational and allows the Boussinesq approximation. Then, a closed model can be derived, [105], without the need to specify closure constants. This approach completely suppresses turbulence of the flow and can hold only for small contrasts in density. 2. A two-point Dirac distribution function describes the vertical velocity field. This is the idea behind the mass flux average, [88, 89], which averages quantities separately over up- and downstreams. This yields exact closures for higher order correlations but ignores small scale turbulence, pressure fluctuations, among others. Such assumptions can at best give hints to new closure approximations, but for applications to real world flows those again have to be parametrized. Less commonly known are the unavoidable parameters introduced in 2D and 3D numerical simulations of convection, [24]. These include 1. mathematical ones (time integration, spatial discretization); 2. the viscosity model (subgrid-scale models, hyperviscosity, artificial diffusion, numerical viscosity); 3. and the boundary conditions.

Multi-Dimensional Processes In Stellar Physics

The key point to be remembered is that the mere existence of model parameters is not crucial. What actually matters is the nature of the entire model (completeness of physics included) and how the parameters are adjusted. How is this problem solved in the case of numerical simulations of convection? 1. For the mathematical ones, optimization procedures are performed by mathematicians based on generic criteria (stability, etc.). Any retuning later on should be avoided. 2. The viscosity model is calibrated with idealized test problems (shock tubes, etc.), but not with astrophysical ones and should not be retuned later on. 3. The boundary conditions are calibrated with single or a few astrophysical test cases, but once fixed should not be altered any more. This motivates an alternative approach to the problem of free model parameters, [24]. The idea is to define properties that both numerical simulation based models and any local or non-local model of convection should have and use this as a set of criteria replacing the (unrealistic and not very helpful) quest for parameter freeness. Those criteria were suggested and discussed in detail in [24]. The model or approximations it contains should 1. have the correct physical dimension; 2. ensure invariance of tensor properties and proper behaviour with respect to standard transformations (coordinate systems); 3. respect sign- and other symmetries of basic variables; 4. allow physical and mathematical realizability; 5. ensure robustness, i.e., predictions should be robust with respect to reasonable changes of the parameters or replacement of the model component containing them by one which is robust in this sense, too; 6. allow for universality, i.e., it should be unnecessary to recalibrate the internal parameters for different types of astrophysical objects (e.g., Sun, DA type white dwarfs, etc.); 7. permit computability, i.e., the formalism must be affordable on present computing means; 8. allow physical verifiability. This requires the model to be falsifiable with observational data or direct numerical simulation or an approach having passed such tests; 9. ensure independence of its internal parameters from the object being modelled. 4.6 Computability

Due to the enormous increase in computer power during the last five decades hydrodynamical simulations of astrophysical flows are now a standard tool of research in all fields of astrophysics. The success in solving some problems including those involving stellar convection has spread hope to apply this tool ever more widely. It has hence become a common expectation in astrophysics in general and in stellar astrophysics in particular that “we just need to wait until computers are powerful enough to perform a 3D simulation” of a given problem. Thus, also studies of stellar convection, stellar pulsation, and stellar evolution are confronted with this demand. Sometimes this takes place during discussions at scientific conferences where it certainly is raised as a legitimate and interesting question. However, the

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same question or demand can change into a serious issue, if it is voiced by peer-reviewers of research publications or even grant applications. It is thus important to become aware that some problems can or will be solved this way. However, others cannot, at least not with semiconductor based technology. In [24] a detailed analysis of several examples from stellar convection studies has been given in which some problems had been concluded to be “computable” in this sense while others were not. It is important to be aware that this computability analysis is not about “exact” predictions, since details vary depending on software and coding efficiency, numerical methods, hardware platforms, and the like. However, for the main conclusions this has limited impact: presently doable problems and their requirements are known. They can be distinguished from problems for which this is not the case yet and from problems for which this will not be the case for unforeseeably long times to come. The detailed estimates have already been given in [24]. For convenience and to avoid misunderstandings the underlying assumptions are summarized here along with some additional remarks. First and foremost, a hydrodynamical simulation of stellar convection is time step limited by τ ∼ Δtadv . This is the time scale on which fluid is advected between neighbouring grid cells. That restriction cannot be circumvented by implicit methods which is possible for constraints such as those caused by sound waves or rapid radiative cooling: those may operate rapidly but they do not have to be followed in detail on time scales on which they do not cause a large relative change of the solution (see [24]). Since a spatial resolution h  12 km is needed to obtain a properly resolved solar superadiabatic layer, there τ ∼ Δtadv ∼ 1 sec. Typical moderately high resolution 3D simulations done today have ∼ 400 grid points per direction. To relax and average the simulation at such a time step requires Nt = (trel + tstat )/(Δtadv ) ∼ 106 time steps with N ∼ 108 grid points. This is readily solvable on present supercomputers and has been assigned a reference computational complexity of C := 1. The estimate is also based on a thermally relaxed initial solution for the adiabatic interior ([24]). A similar estimate can be obtained for simulations excluding the stellar surface, accounting for cases of implicit integration of radiative diffusion or sound waves (or filtering), or solving the radiative transfer equation using binned opacities. The variation of C caused by these changes is less than about a factor of 10. In 2D simulations one can trade grid points for time steps to arrive at similar values of C. Grid refinement methods can gain another factor of ∼ 10. Optimized parallelization (including GPUs) may gain factors anywhere between 10 and 100 (this is naturally the least certain part of the estimate). At the bottom line though this allows sufficiently robust estimates to distinguish doable from undoable problems. Problems doable or becoming doable today have C  103 and include well-resolved simulations of solar granules in a 6 Mm wide (and about 4 Mm deep) box, a fully turbulent simulation of the same problem (h  3 km), low resolution simulations of very large and deep boxes (200 Mm wide and deep), subsurface simulations of global solar convection, and 2D simulations of short period Cepheids that include their first interior node (42% of the stellar radius from the top) for both narrow wedges and full azimuth covering simulations. With the upcoming generation of exascale computing systems problems with a complexity of C in the range of 104 to 106 and perhaps even 107 might become doable. This includes 3D simulations of the entire solar surface or 3D simulations of Cepheids (both have C  105 ). The limiting factor for the next step, a low resolution simulation of the entire solar convection zone, is then to find a compromise between how low the resolution of the surface layers can still become to yield acceptable results and the question to what extent one can accept simulations based on kinetic relaxation only. The example given in [24] refers to a problem with a vertical resolution of 12 to 28 km for the surface layers and a kinetic relaxation over one year resulting in C > 107 which seems unlikely to be achievable shortly. It is of course possible to downscale this computation with respect to resolution and to push the problem into the realm

Multi-Dimensional Processes In Stellar Physics

of simulations which will become affordable with new codes and new hardware expected to be developed and released during the upcoming decade. However, a 3D simulation of the entire solar convection zone that would be usable to study damping of p-modes would be much more demanding, since downscaling of relaxation is not an option there while simulated periods have to cover one or, rather, several decades. A technology suitable to cope with C > 1010 appears unforeseeable as of now. One might of course speculate about the capability of quantum computers in the future, but this cannot be linked to a technological development path with predictable production cycles. This restriction also includes low resolution 3D simulations of stellar evolution for any foreseeable time (C > 1017 ). The only option in that field is to rather use numerical simulations to calibrate advanced 1D models. 4.7 A comparison

The complex challenges of calibrating advanced 1D models of convection with numerical simulations can be elaborated when considering the case of Cepheids [106]. The standard 1D non-local models of convection used to quantify the convective flux in radial simulations of Cepheid pulsation require the specification of closure parameters which change as a function of time, pulsation phase (since the mean stratification varies substantially contrary to the case of solar-like oscillations), and location inside the convection zone or in the overshooting region. In the case of a short-period Cepheid (P∗ ∼ 4 d, M∗ ∼ 5 M , R∗ ∼ 38.5 R , L ∼ 913 L ) one of the parameters of either convection model studied ([103, 107]) was found to have to be varied by at least a factor of 2 as a function of each of the dependent quantities considered (i.e., pulsation phase, location inside the convection zone, etc.). Evidently, reliably calibrating such models is a highly non-trivial issue. With respect to the criteria listed in Sect. 4.5 one might thus compare the different approaches to modelling convection as follows. The requirements of dimension, invariance, and symmetries are generally fulfilled by the models in active use and definitely for all well-tested numerical simulations (some closure approximations may violate sign symmetries though). Realizability is mostly an issue for some of the more advanced 1D models of convection. However, the other five criteria clearly divide the different modelling concepts. Classical mixing length theory benefits from being the cheapest computationally and can be (and indeed has been) falsified in astrophysical applications. It performs poorly though with respect to robustness of its results, universality, and the demand of object independence. This is actually the real problem behind adjusting the mixing length in astrophysical applications, especially, if this is done based on astronomical observations. At increased computational costs the non-local models of convection and in particular the Reynolds stress models aim at improving on those short-comings. The extent to which this is possible remains a part of active research and depends largely on the class of objects and the physical process chosen to be modelled with the desired level of accuracy. Numerical simulations on the other hand perform well for each of the criteria except for computability, as can also be concluded from the discussion in Sect. 4.6. For that reason in stellar astrophysics and in related fields in planetary science one has to cope with a larger variety of concepts to model convection for the foreseeable future.

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5 Applications: overshooting One important problem in the modelling of thermal convection in stars is to quantify the amount of mixing next to an existing convection zone. Some results on this process can be obtained in a fairly general form through considering balanced budgets of ensemble averages, even on the level of Reynolds stress models. After introducing such more general principles the problem of overshooting in white dwarfs and the role of the Péclet number in characterizing overshooting are highlighted. 5.1 Overshooting

In the following the convective planetary boundary layer (PBL) of the atmosphere of the Earth is considered as a model system. For this case one can explicitly show that using an analysis just based on Reynolds stress modelling it is possible to demonstrate that convective overshooting, the penetration of fluid from a zone convectively unstable according to the Schwarzschild criterion into neighbouring layers which are stably stratified from that point of view, is an inherently non-local phenomenon. Advective fluxes play a key role in establishing energy transport in an overshooting zone and determine its basic physical properties. We start from the Reynolds stress equations for the second order moments introduced in Sect. 3.3 (cf. [73], [75], [77]): turbulent kinetic energy K, root mean square fluctuations of temperature θ2 , and their cross relation wθ (essentially, the convective flux). They read 

1 2 ∂t K + ∂z q w + pw = gαv wθ − ε 2 1 + ∂z (ν∂z K) + Cii , (44) 2

∂t



 1 2 1 2 θ + ∂z wθ2 = βwθ − τ−1 θ θ 2 2  1 1  + ∂z χ∂z θ2 + Cθ , 2 2

    ∂t wθ + ∂z w2 θ = βw2 + (1 − γ1 )gαv θ2 − τ−1 pθ wθ     1 1 + ∂z χ∂z wθ + C3 + ∂z ν∂z wθ . 2 2

(45)

(46)

Here, K = q2 /2 = (u2 + v2 + w2 ). We ignore contributions from compressibility (Cii = 0, Cθ = 0, C3 = 0), assume a low Prandtl number (drop ∂z (ν∂z K) and ∂z (ν∂z wθ)) with convection still efficient enough to neglect ∂z (χ∂z θ2 ) and ∂z (χ∂z wθ). In the stationary limit we then obtain 

1 2 ∂z q w + pw = gαv wθ − ε, (47) 2

∂z

 1 2 2 wθ = βwθ − τ−1 θ θ , 2

  ∂z w2 θ = βw2 + (1 − γ1 )gαv θ2 − τ−1 pθ wθ.

(48)

(49)

In this form the equations also apply to the convective planetary boundary layer after taking moist convection into account via β, the superadiabatic or potential temperature gradient.

Multi-Dimensional Processes In Stellar Physics

For these three equations, the non-local transport (flux) terms on the left-hand side are exact within the Boussinesq approximation, as also holds for the terms gαv wθ, βwθ, βw2 , and gαv θ2 . 2 The term τ−1 θ θ results from a closure of the dissipation rate of (potential) temperature fluctuations εθ which must be positive definite. Thus, for further analysis it suffices to consider its closure form. Likewise, ε is the dissipation rate of turbulent kinetic energy, which is positive definite, too.

This leaves the closure of −Πθ3 = −θ∂z p = −γ1 gαv θ2 −τ−1 pθ wθ as the only non-exact relation one has to pay attention for here. From the Canuto-Dubovikov turbulence model [108, 109] one obtains γ1 = 1/3 (values between 1/3 and 1/5 are suggested in the literature) whereas the time scale τpθ , just as τθ , is necessarily positive (see also [75]). Hence, this closure takes into account that Πθ3 in principle changes sign like wθ although not necessarily at the same location (the exact shift being subject to closure uncertainty). As is shown below this is not relevant to recover the countergradient flux nor is it the key element to obtain a negative buoyancy (convective, enthalpy) flux, even though it is tempting to conclude just that from a purely local point of view. The latter clearly leads to wrong conclusions on overshooting. Consider Eq. (47) and abbreviate J ≡ wθ. For the non-local pressure flux pw we note that it is often closed by assuming it to be a fraction of the flux of kinetic energy and thus being proportional to q2 w. It is in any case a non-local term and hence does not change any of the following conclusions where for simplicity we assume it to be absorbed into the term ∝ q2 w. Assume J > 0. Then, non-zero velocities require ε > 0 and we can find a local solution, 1 2 q w = 0, inside a convectively unstable zone. This is essential for MLT to work! 2 Assume J < 0, then no non-trivial local solution exists. Thus, we must have ∂z ( 12 q2 w)  0 for any solution with non-zero velocities. This realization is at the heart of all “1-equation models” of overshooting which consider a dynamical equation for turbulent kinetic energy or its flux, whether as a differential equation, in an integral form (I.W. Roxburgh’s integral constraint, [110] and [111]), or in a formalism based on that ingredient, among others (such as the plume model for convective penetration by J.-P. Zahn, [112]). However, this structure of a convective overshooting zone is at variance with observations from meteorology. Following their publication [66] in 1947, the countergradient transport of heat in atmospheric turbulent flows became known as the Priestley-Swinbank effect. Its first convincing explanation was given in 1966 by J.W. Deardorff [67] who made it clear why there is a region with a potential temperature gradient opposite to the direction of the turbulent heat flux by analyzing a variant of Eq. (48) with data from meteorology and laboratory experiments (work with G.E. Willis 1966, see [67] — an earlier suggestion by Deardorff [113] had still lacked the observational data to confirm the idea further elaborated in [67]). Consider β > 0 and J > 0 in Eq. (48). This permits local solutions, where 12 wθ2 = 0. Now if β < 0 while J > 0 there can be no non-trivial local solution. Instead, ∂z ( 12 wθ2 )  0. For this reason, wθ2  const. is the key ingredient for the existence of the countergradient region in overshooting. If both β < 0 and J < 0, Eq. (48) would permit a local, non-trivial solution, but this is not the case for Eq. (47) for which already J < 0 excludes a non-zero local solution. Hence, we need to model both ∂z ( 12 q2 w) and ∂z ( 12 wθ2 ) by dynamical equations or closure conditions. Deardorff [67] also realized the connection between wθ2 and S θ = θ3 /(θ2 )3/2 . He noticed that skewness and the location of convective driving are related to each other (heating from below or both heating from below and cooling from above as in the water tank experiment with Willis). This was corroborated by work reported, among others, in [34–38, 114, 115],

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more than 20 years after Deardorff’s pioneering work, for then numerical simulations could be used to support Reynolds stress budgets and plume model based arguments. In the absence of a non-local flux w2 θ, the specific form of Eq. (49) is compatible with a solution where β > 0 and J > 0. This is also the case for β < 0 and J > 0 and even for β < 0 and J < 0. Hence, Eq. (49) appears to permit local solutions for convectively stable and unstable stratifications. However, whether the correct signs of positive definite quantities are realizable is only a necessary condition for local solutions to exist, not a sufficient one. In this sense, only a detailed analysis of Πθ3 itself can demonstrate the necessity for a non-zero ∂z (w2 θ). Modern LES and observational data of the PBL show that this is indeed the case ([79], [116], [86]): also this term is clearly non-zero in overshooting regions. This chain of arguments also holds for the compressible case since using the density weighted Favre average [117] as in [118] leads to structurally similar equations and the smallness of additional terms can be established from detailed budget equations (unclosed Reynolds stress equations with contributions computed from 3D numerical simulations, as in [116] for the Boussinesq case). A 3-equation model (for K, J, and θ2 ) such as that one by R. Kuhfuß, [103], appears to be needed to capture the behaviour of β and J in overshooting zones. To also self-consistently account for how the ratio between vertical and total kinetic energy changes with depth an equation for w2 has to be added, too, as in the Reynolds stress models of D.R. Xiong and V.M. Canuto ([101, 102],[119]). Computationally, such non-local models are much more costly than the “recipies” used for convective overshooting in stellar evolution calculations. Finally, overshooting is a fundamentally non-local phenomenon, a result demonstrated in meteorology more than 50 years ago without requiring any numerical simulations even though the latter are useful to further corroborate this finding. 5.2 Modelling non-locality

One important tool to test convection models in general and closure assumptions in particular are budget equations derived from unclosed ensemble averages. Assuming again a quasiergodicity hypothesis one can calculate these budgets from numerical simulations. Examples for the case of the planetary boundary layer of the Earth can be found, for instance, in [116] who also computed these statistics for two different rotation rates in addition to the nonrotating case. Indeed, the flux of temperature fluctuations, wθ2 , is shown to be essential for a balanced, quasi-stationary stratification for both cases of rotation and for the non-rotating scenario. The same kind of simulation can also be used to probe closure expressions ([78, 91]). For a most convincing test the simulation data can also be combined with observational data, as has been done by [86, 87] for the tests of their closure models. Claiming universality of a closure suggested, say, for wθ2 or w2 θ, however, requires a larger region of applicability. For instance, in [24] it was demonstrated that the two-scale mass flux closure Eq. (37) yields excellent results both in comparison with a numerical simulation of solar granulation and with a numerical simulation of convection in a moderately hot (T eff ∼ 11800 K) DA type white dwarf. It was also shown that compressibility (as accounted for by using Favre averages) has only very little impact on this result. The traditional onescale mass flux closure of this quantity, which ties it to the skewness of the vertical velocity instead of the temperature field, fails in the overshooting regions as it violates a sign symmetry. Thus, in spite of its complexity, the non-local flux of temperature fluctuations can be modelled fairly well over a very large parameter space by a formally rather simple algebraic expression. This is not the case for all closure relations, but it does hold for quite a few (see [83] for some further examples).

Multi-Dimensional Processes In Stellar Physics

5.3 The DA white dwarfs

White dwarfs turn out to be particularly useful objects for the study of convective overshooting. At effective temperatures around 12000 K, DA white dwarfs are characterized by a thin atmosphere which enshrouds a zone of notable temperature change which in turn contains an isothermal, electron degenerate core. Around T eff ∼ 12000 K the surface convection zone, which is caused by (partial) ionization of hydrogen, is thin enough so as to be barely deeper than the atmosphere itself. Thus, a box-in-a-star-type simulation in Cartesian geometry is possible: for a box with a depth of about 8 km and a width around 15 km, the curvature of a star with a radius of 6000 km is negligible near its surface. The DA white dwarfs have originally been characterized by their hydrogen lines in the optical spectrum and the lack of helium and metal lines in the same wavelength region. Extensive spectroscopic observations have demonstrated that about 25% of these objects in fact do have metal lines (DAZ stars, [120–122]). The standard model on how white dwarfs could assemble a metal polluted atmosphere is steady-state accretion, as discussed in [123, 124], who provide both the theoretical models and further data needed for such calculations. Possible accretion sources are planetary systems and remnants of them (some authors have previously doubted this type of source, but it has now become the standard explanation). However, diffusion inside the stars spreads the accreted material into the star. Counteractions of gravitational setting and radiative diffusion on a microscopic scale vary element by element. In addition, a convection zone at the stellar surface provides mixing on very short time scales, practically instantaneously. This defines the size of the reservoir into which accretion occurs and is counteracted by diffusion. Overshooting extends the convectively mixed region. But by how much and how reliable are models of this process? Clearly, numerical simulations are needed to quantify this more accurately. 5.4 3D simulations of overshooting

In numerical simulations of overshooting the following length and time scales have to be considered either directly or indirectly (see [24, 99]). They are first of all based on the underlying physical processes. Among the length scales of interest are: • The length scale of the maximum of kinetic energy transport which is a function of radius, H(r). At the stellar surface this is roughly the horizontal extent of a granule: H(R) ∼ Hgran . • The length scale of viscous dissipation ld , also known as the Kolmogorov scale. In numerical simulations of stellar convection with realistic microphysics this scale is of course always unresolved and a much larger turbulent viscosity is taking over its role. This shifts dissipation of kinetic energy to much larger length scales. • The size of the radiative (or thermal) boundary layer, Lt . Once more a function of radius it is hence different for the upper and for the lower boundary of a convection zone. An estimate ([24]) is given by Lt  δ ∼ Pr−1/2 ld . Here, δ is the thermal boundary due to (radiative) diffusion. The Prandtl number compares momentum to (radiative) heat diffusion, Pr = ν/χ. The estimate can be improved, [112], which does not change the conclusions drawn below. The most important time scales are: • The thermal relaxation time scale ttherm , which for non-degenerate stars can be approximated by ttherm (x) ∼ tKH (x), as discussed in detail in [24] (see [53]). Both time scales are

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again defined in a more general form as functions of location and the latter is given by

! Ms (rb )  −1 tKH = −3 p ρ dM s /L. (50) M s (ra )

Here, M s (r) is the mass contained from the surface of a star till a radius r and rb > ra , while L is the luminosity (at the local radius or constant). p and ρ are pressure and density. • The convective turn over time tconv , tconv =

!

rb

ra

u−1 x (r) dr.

(51)

Here, u−1 x (r) is the ensemble average of the root mean square difference between the local vertical velocity and its horizontal average. • This allows the definition of the relaxation time trel (see Sect. 4.3 and [24]) as follows: trel ≈ max(tconv (rbottom ), ttherm (rrel )).

(52)

Thus, relaxation has to occur both with respect to the kinetic energy of the flow (tconv ) and the potential (thermal) energy of the stratification (tKH ) and hopefully, rrel > rbottom . These scales are linked to crucial dimensionless ratios: the Reynolds number Re = (u(l)l)/ν, 2 . the Péclet number Pe = Re Pr = (u(l)l)/χ, and the Rayleigh number Ra = (tvisc trad )/tbouy Evidently, these are scale dependent numbers and the length scale they are based on is crucial for their interpretation. Thus, u(l) refers to a vertical (or total) velocity which can relate either to a length scale on which most kinetic energy is transported, u(H(r)), or to a scale l on which only some small scale fluctuations occur. Note that tvisc = l2 /ν and trad = l2 /χ (radiative −1 , the inverse of the Brunt-Väisälä frequency. diffusion) and tbouy = NBV The scales have to be accounted for in 2D and 3D numerical simulations of overshooting. To this end simulation based (effective) dimensionless parameters can be introduced (cf. [24, 125]). For the 3D case one can estimate whether it is possible to resolve turbulence generated by shear between resolved upflows and downflows by estimating Reeff ∼ (H/h)4/3 . Here, h refers to the grid spacing used and H to the scale to which the effective Reynolds number refers to. For judging whether the shear between granules and downflows can generate turbulence, H is set to Hgran . Vortex tubes may be found at rather moderate values of a few 100 or less, but their character changes once Reeff significantly exceeds values of 2000. To check this statement it is instructive to compare some simulations published in the literature: large vortex tubes are clearly visible in downflow lanes of Fig. 13 and Fig. 14 of [56] who at h ≈ 23.7 km horizontal resolution and assuming Hgran ≈ 1300 km achieve Reeff ≈ 208. Somewhat more small turbulence and a highly upwards reaching “tornado” are visible in Fig. 15 and 16 of [58], where h ≈ 9.8 km, whence Reeff ≈ 677. A change only occurs at even higher Reeff and was first shown in [55] who demonstrated the transition to a highly turbulent flow once the resolution is increased from 7.4 km (and thus Reeff ≈ 984) to 3.7 km (with Reeff ≈ 2479) and can be inspected by comparing Fig. 1 with Fig. 2 in Sect. 3 further above. This intense level of turbulence is also observed for a more quiet state of granule evolution that is shown in Fig. 13 of [24] (where h ≈ 2.5 km and thus Reeff ≈ 4182). Thus, only once Reeff clearly exceeds 1000, small scale turbulence can be observed on the computational grid itself. Many simulations of astrophysics in general and of stellar convection in particular have Reeff  1000. They feature rather thick vortex tubes and do not credibly show a Kolmogorov inertial range: at lower resolution the kinetic energy carrying scales and the energy dissipating grid scale are still close and thus tightly coupled to each other and the hypotheses underlying the

Multi-Dimensional Processes In Stellar Physics

Kolmogorov theory cannot hold (cf. [10, 12]). Note that the box Reynolds number, Rebox = (L/h)4/3 , is not the best guide to judge on the level of turbulence: L may include stable layers, get boosted by horizontal box extent, or even L Hgran . A second important parameter is the effective Prandtl number, Preff . It compares turbulent or grid viscosity with radiative or microscopic heat diffusivity. In numerical simulations of convection in astrophysics one usually aims at achieving 1  Preff  0.1. Another important parameter compares convective to conductive heat transport on the grid: the effective Péclet number, Peeff = Reeff Preff . In astrophysical applications that fulfil Preff  1 it is possible to achieve Peeff  1000. It may be tempting to invoke 2D simulations to boost Peeff , since they follow a different scaling (see [11]): Reeff ∼ (H/h)2 , whence Peeff  105 . However, this scaling is bought at the price of an inversely operating energy cascade and 2D flows are much more often non-ergodic, [12]. Estimates of overshooting based on 2D simulations thus have to be done with great care. What happens if some of those scales are ignored in a numerical simulation of overshooting? If trel  max(tconv (rbottom ), ttherm (rrel )), then the temperature gradient dT/dr and the extent of overshooting may be unrealistic, since the relaxed structure may not have been known in advance. If Peeff  Pe, we have to expect the amount of overshooting to become wrong, too, since the thermal structure and the amount of overshooting are highly sensitive to Pe as was shown in [126]. The role of Pe and the case Preff > 1 are considered further in Sect. 5.5. If, finally, Lt or δ remain unresolved, one might have to deal with flow artifacts, as are known from numerical simulations of Rayleigh-Bénard convection in similar situations. 5.5 The role of the Péclet number

How do numerical simulations of the solar surface, of the solar tachocline, and of shallow convection zones in DA white dwarfs differ from each other? For the Sun, at the surface H ∼ Hgran ∼ 1300 km, ld ∼ 1 m, Lt ∼ 30 km, Pr ∼ 10−9 and thus Pe ∼ 10 (further details on how to estimate those numbers are given in [24]). The requirements Peeff  1000 and 1  Preff  0.1 can easily be fulfilled and highly realistic simulations are possible for those layers. Indeed, for a simulation of solar granulation within a box that is 4 Mm deep and convective for all layers more than 0.7 Mm away from the top and which is quasi-adiabatic for layers that are more than 1.5 Mm away from the top, at h = 11 km, one finds trel = max(tKH (1.5 Mm), tconv (4 Mm)) = max(45 min, 57 min) < 1 hr. At a time step Δt = 0.25 s this requires Nt ∼ 1.4 × 104 time steps. This is easily affordable and in fact is a problem with a computational complexity C < 1 (see Sect. 4.6). The problem of simulating overshooting underneath the solar convection zone is of a very different kind. Taking again the values estimated in [24], one finds ld ∼ 1 cm, Lt ∼ 1 km as a more realistic estimate in comparison to δ ∼ 30 m, Pr ∼ 10−7 , and Pe ∼ 106 for the bottom of the solar convection zone. If we want to keep 1  Preff  0.1, then necessarily Peeff  1000 and the resulting overshooting calculation cannot be expected to be realistic. What happens, if one gives up on 1  Preff  0.1 to achieve a higher value of Peeff ? To answer this question it is helpful to recall some discussion originally presented in Sect. 4.1 of [125] and elaborate it with some additional comments. A stable numerical simulation requires that the grid cell Reynolds number Regrid = U(h)h/νeff = O(1). In this context νeff is the viscosity the numerical scheme has to provide to achieve Regrid = O(1). For the linear advection equation and basic finite difference schemes one can prove that Regrid  2 (see Chap. 6.4 in [14]). For that case U(l) = a = const. at all l and thus Reeff = U(l) l/νeff  Re = U(l) l/ν if νeff ν. One can rewrite this into Reeff = U(l) (l/h) h/νeff = (l/h) (U(l) h/νeff ) ∼ l/h, since for a stable simulation U(l) h/νeff = U(h) h/νeff ∼ O(1) and U(l) = U(h) here.

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For a turbulent convective flow this is too pessimistic because U in the definition of Regrid refers to U(h) at the grid cell level. Contrary to linear advection, however, U(h) is smaller in magnitude than U(l) at the energy carrying scales. Thus, U(l) l/νeff ∼ O(1) would require an unnecessarily viscous simulation whereas U(l) l/νeff 1 is both desirable and possible. Thus, one can insert U(h)/U(h) in the previous expression for Reeff and rewrite it into Reeff = (l/h) (U(l)/U(h)) (U(h) h/νeff ) from which we see that it is sufficient to demand (U(h) h/νeff ) = Regrid = O(1) for a stable simulation. Violating the latter constraint though would cause unwanted oscillations in the numerical solution. It is now left to explain the estimate (l/h) (U(l)/U(h)) ∼ (l/h)4/3 to justify the definition Reeff ∼ (H/h)4/3 as the effective Reynolds number of a stable, 3D numerical simulation of a turbulent flow. To this end the first and second hypotheses of Kolmogorov have to be invoked (cf. [10] for further details about them). The first hypothesis assumes that the velocity difference of two fluid elements separated by a distance λ depends only on the flux of kinetic energy between different scales around λ and on the viscosity ν. Clearly, l > λ. In practice, also l λ can be necessary. The second hypothesis assumes that there is a restricted range called inertial range where the velocity difference is also independent of ν. Clearly, λ > ld , where ld is the Kolmogorov scale for which U(ld ) ld /ν ∼ O(1). For the inertial range the energy flux is proportional to u(λ)3 /λ and independent of λ (see [10]). Consequently, U(l)/U(h) ∼ (l/h)1/3 and (l/h) (U(l)/U(h)) ∼ (l/h)4/3 which completes the derivation. Note that it has been important to select the “correct” value of U from a scaling relation that had to be derived separately (in this case from Kolmogorov’s theory). For this reason, in 3D one finds Reeff ∼ (H/h)4/3 whereas in 2D, Reeff ∼ (H/h)2 . Although for low numerical resolution the Kolmogorov inertial range scaling cannot hold, the kinetic energy of such a simulation varies slowly and smoothly around the energy carrying scales and decreases quite steeply close to the grid scale (this is also a result of the construction principles behind numerical and artificial viscosities). This allows an O(1) estimate from Reeff ∼ (H/h)4/3 for those cases. Having clarified the physical meaning of Reeff it is now possible to clarify the role of Preff and Peeff . Their definitions are straightforward following their molecular counterparts, [125]: Preff = νeff /χ and Peeff = Reeff Preff . There is a pitfall ahead. At face value, νeff cancels in this definition and Peeff = Pe even if the simulation has νeff ν. Following again [14] it turns out that also for the grid Péclet number Pegrid = U h/χeff = O(1). More precisely, Pegrid  2 for linear heat diffusion with advection, in addition to Regrid  2. But that condition also has to be fulfilled for the hydrodynamical equations in the sense of Pegrid = O(1) alongside Regrid = O(1), otherwise the positivity of diffusion is violated and fluctuations of energy pile up at the grid scale (this is revealed by a more detailed analysis of Eq. (12) which describes how the advection of internal energy and radiative diffusion contribute to energy transport). Assume now that νeff < χ or Preff < 1. Then it is sufficient to set χeff = χ and both conditions on Regrid and Pegrid can be fulfilled at the same time since νeff was assumed to be large enough to make Regrid = O(1) in first place. It thus makes perfect sense to identify Pe = Peeff . This is the case for simulations of convection at the solar surface and for the DA white dwarf we consider below. But in other cases νeff χ, i.e. Preff 1, may be inevitable when assuring that Regrid = O(1). In this case U h/χ O(1) by the definition of Regrid . Thus, the simulation would be unstable unless there is a sufficiently large numerical heat diffusion χeff χ which permits Pegrid = U h/χeff = O(1). In subgrid scale models of turbulent flows this is expressed by setting a turbulent Prandtl number, Prturb = νeff /χeff which is typically somewhat less than 1. Then again the stability conditions are fulfilled. However, such a change also requires to redefine Peeff = U(l) l/χeff = (U(l) l/νeff )(νeff /χeff ) = Reeff Prturb . For a numerical simulation where 1  Preff  0.1 one has Prturb ∼ Preff and one achieves Pe = Peeff . If instead the resolution of the simulation is so low that Preff > 1 or even Preff 1,

Multi-Dimensional Processes In Stellar Physics

then the simulation can only be stable, if either explicitly through a turbulent heat diffusivity or by some other form of numerical viscosity associated with the spatial discretization, or implicitly, through a damping included in the time integration scheme, sufficient numerical heat diffusion is introduced, i.e., χeff > χ or even χeff χ. In this case one has to use the second definition of Peeff , Peeff = Reeff Prturb < Reeff . Thus, Peeff  Pe or even Peeff  Pe. In conclusion, if the grid resolution is too coarse to resolve (radiative) heat diffusion, then the Péclet number which can be achieved by a numerical simulation using such a grid is limited basically by the effective Reynolds number. This is just what happens when performing a numerical simulation for overshooting below the solar convection zone: the Péclet number which can be achieved by the simulation is several orders of magnitudes smaller than the actual Péclet number of the problem. Thus, systematic differences between overshooting predicted from such simulations and the actual overshoot have to be expected. Since Pe ∼ 106 at the bottom of the solar convection zone, realistic Péclet numbers cannot be reached for currently affordable spatial resolutions in simulations of global solar convection. Their relaxation is also troublesome. With trel = max(tKH (180 Mm), tconv (180 Mm)) = max(∼ 106 yrs, 1 month) we obtain due to the problem of long thermal relaxation and considering time steps from 0.25 sec to a few hours that Nt ∼ 2 × 109 . . . 1014 . This problem has already been addressed in Sect. 4.6. Overshooting below stellar convection zones is more easily studied for realistic microphysics when remaining closer to the surface of a star. For DA white dwarfs it is possible to consider cases with strong convection that are readily affordable with current computational resources. The object studied in [99] is a DA white dwarf with T eff = 11800 K, a surface gravity log(g) = 8, and pure hydrogen composition. The simulation box was designed for a layer about 7.5 km deep for which the thermal structure deviates from a radiative one for about the top 4 km. In this case, H ∼ Hgran ≈ 0.45 km . . . 1.35 km. Likewise, ld ∼ 25 cm δ ∼ 3.4 km, Pr ∼ 5.4 × 10−9 . Consequently, Pe ∼ 10 . . . 30. With h ≈ 30 m it is possible to achieve Preff  0.2 while Reeff ∼ 40 . . . 170. Consequently, Peeff ∼ 10 . . . 30 and thus Peeff ∼ Pe. This permits realistic 3D hydrodynamical simulations of overshooting and indeed the entire range of Pe  1 to Pe  a few100 is accessible for T eff in the range of 14000 K to 11600 K at the same values of surface gravity and chemical composition. The realistic upper boundary condition of this case also prevents artifacts and can hence yield reliable benchmarks for non-local convection models. Thus, trel = max(tKH (4 km), tconv (∼ 7 km)) = max(25 s, 30 s). With a relaxation time of 30 s at a time step Δt = 22 ms we have Nt ∼ 1.4 × 105 . Hence, a relaxed simulation resolving the relevant scales in space and time has been possible. 5.6 Previous research and a deep DA WD simulation

DA type white dwarfs have raised interest as goals of numerical studies of convection due to a number of specific astrophysical questions related to pulsational stability, spectroscopic determinations of stellar parameters, and the already introduced problem of mixing of accreted material (Sect. 5.3). Another, practical advantage are the readily fulfillable demands on resolution and on statistical relaxation (Sect. 5.5). DA white dwarfs have hence been studied with numerical simulations for quite a while. The earliest work considered 2D simulations, [127], which have demonstrated the existence of strong overshooting below the actually convectively unstable region in hotter objects (T eff of 12200 K, for example) and indicated an exponential decay of the velocity field in that region. While those findings appeared convincing for higher values of effective temperature (T eff ≈ 13400 K), a need for deeper simulation boxes can also be concluded from the velocity profiles shown for lower T eff in their work. This turned out to be at variance with velocity profiles derived from Reynolds stress models, [82]. There, in the overshooting region proper where notable negative convective fluxes exist

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(plume-dominated region) and in the lowest part of the countergradient region (as described by Deardorff), the velocity field was found to decay linearly as a function of distance from the convectively unstable zone. Similar was found in the equivalent region of DB white dwarfs studied in [82] and for A-type main sequence stars analyzed in [76]. Eventually, in [128–130] 3D numerical simulations of surface convection in DA white dwarfs have been presented and analyzed in [130] with respect to overshooting. Their results appeared to confirm the previous findings of [127]. However, closer inspection of their Fig. 4 and Fig. 5 revealed differences in the profiles of convective flux and vertical root mean square velocity between hotter (T eff  13000 K) and colder (T eff  12500 K) models when compared as a function of distance from the convectively unstable region. Since indications for the influence of the lower vertical simulation boundary could be argued for (their Fig. 12) in case of the cooler models due to their larger region of convective instability and overshooting, a detailed simulation for this parameter region for a larger volume appeared necessary. This has been presented in [99]. There, the authors identified the presence of self-excited waves dominating the lower part of the simulation box and thus also the overshooting region at greater distance from the convectively unstable zone. Closer to the latter, agreement was found with the Reynolds stress model discussed in [82]: a linear decay of the velocity field provides clearly a better fit for the plume-dominated region of overshooting than an exponential model. The latter only begins to work when overshooting finally fades away, in the transition layer to the wave-dominated region and within the latter. A detailed analysis of the statistical properties of the different regions of the overshooting zone was given and a lower estimate for the convectively mixed region was suggested. 5.7 Conclusions

The modelling and the quantitative description of overshooting due to stellar convection zones allows some interesting conclusions drawn when considering a case for which physically reliable numerical simulations can be made. This is possible for the case of DA white dwarfs. Lessons learned from that work include the following. • Concerning simulation extent and resolution a sufficiently wide (high aspect ratio) and well resolved numerical simulation is necessary to obtain accurate statistics. For the case considered in [99] it must extend to more than 4 pressure scale heights below Schwarzschild unstable zone to obtain a wave-dominated region which is not strongly influenced by the lower boundary condition (this also depends on the stellar parameters considered). • With respect to the properties of the overshooting zone it is interesting to note that there is a linear decay of root mean squared velocities in the plume-dominated region just as in the Reynolds stress models whereas an exponential decay can be argued for only in the wave-dominated region and in the transition region to the latter. This exponential decay appears to be even faster for horizontal velocities. • A conservative estimate for accretion concludes a 1.6 to 2.5 times higher accretion rate is needed due to the larger amount of mass mixed by convective overshooting compared to the convectively unstable zone alone. This value is obtained if only the plume-dominated region is considered. It increases to a factor of 3.2 to 6.3, if the linear extrapolation suggested in [99] is used. • The main limitations of this approach are the strong modes excited in the simulation which are enhanced by the small mode mass of the simulation box. The mixing caused by waves requires a proper scaling of amplitudes since otherwise very strong overestimations of its efficiency can occur.

Multi-Dimensional Processes In Stellar Physics

6 Summary Given the necessarily limited extent of the original lectures the associated notes presented in this review have focussed on some core topics of the physics of thermal convection and its study in astrophysics. A guide to further literature is hence given in the introductory Sect. 1 where readers can also find reviews on subjects other than those covered here. The physics of convection is introduced in Sect. 2 first from a heuristic point of view and explained further by examples from astrophysics and geophysics. Some of the main implications of this process for astrophysics are then presented. The section concludes by first repeating the classical linear stability analysis which yields the Schwarzschild criterion of convective instability, followed by introducing the basic hydrodynamical conservation laws, on which this analysis is actually based on. A first hint on problems ahead is then given by introducing the challenge posed by the huge spread of scales caused by convection in stellar scenarios. The different modelling approaches to convection are introduced in Sect. 3. Emphasis is given on modern approaches including non-local models of different types, advanced closure models, and by a comparison of these methods to those used in hydrodynamical simulations. The latter are then discussed in Sect. 4. Since there are several recent reviews on this subject, the focus of this part is on important basic questions which nevertheless find less attention: the relation of simulations to statistical physics, the uniqueness of numerical solutions which entails the inevitable introduction of some form of diffusivity and viscosity, the proper relaxation of a simulation to compute reliable statistics, the role of boundary conditions, and a list of criteria to judge on the amount of uncertainty introduced by the different modelling approaches to convection. Affordability issues related to numerical simulations are summarized before a specific topic, convective overshooting, is selected as a process which in Sect. 5 serves as an example for the application of the different modelling principles introduced in the preceding sections. The fundamental non-locality of overshooting is demonstrated using a semi-analytical approach, the Reynolds stress formalism. Its extension by moment budgets computed from numerical simulations is discussed before the DA white dwarfs are introduced as a highly convenient class of stellar objects for the study of overshooting. The necessary preconditions for a solid study of this process are then explained including the constraints put by length and time scales and by several dimensionless quantities, in particular the Péclet number. Finally, the main results on studying overshooting in DA white dwarfs as published when writing this introduction are presented. The concepts presented in this text, while developed here also for rather specific applications, are nevertheless of general interest in studies of convection in both astrophysics and geophysics. They are hoped to be useful also for approaching more advanced subjects related to convection in further detail.

Support of the author by the Austrian Science Fund (FWF), project P29172-N27, is gratefully acknowledged.

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Turbulence in stably stratified radiative zone François Lignières1,∗ 1

Institut de Recherche en Astrophysique et Planétologie, Université de Toulouse, CNRS, CNES, UPS, France Abstract. The topic of turbulent transport in a stellar radiative zone is vast and

poorly understood. Many physical processes can potentially drive turbulence in stellar radiative zone but the limited observational constraints and the uncertainties in modelling turbulent transport make it difficult to identify the most relevant one. Here, we focus on the effect of stable stratification on the radial turbulent transport of chemicals and more particularly on the case where the turbulence is driven by a radial shear. Results of numerical simulations designed to test phenomenological models of turbulent transport will be presented. While this may appears little ambitious, stable stratification will influence the radial turbulent transport whatever the mechanism that drives the turbulent motions and the present considerations should be useful for these other mechanism too.

1 Introduction Most constraints on the transport of angular momentum and chemicals in radiative zones are indirect. Maybe, the most direct one is the observation of abundance anomalies at the surface of intermediate-mass stars, called chemically peculiar stars. These anomalies are due to the gravitational settling and radiative levitation of chemical elements, both being slow processes. This can only happen if the stably stratified subphotospheric layers are nearly quiescent showing that the macroscopic hydrodynamic transport is very inefficient there [1]. In other stars surface abundances are the signature of deep mixing instead. This is the case in massive stars where the observed surface abundances of elements involved in the CNO cycle require mixing down to the stellar core (e.g. [2]) or in the Sun where the surface Lithium depletion indirectly shows that a radial transport occurs in the radiative zone below the convective envelope (e.g. [3]). Then, helio and asteroseismology provide detailed informations on the interior of certain stars, in particular the sound speed, the Brunt-Väisälä frequency and the rotation rate, the latter being the best data available to understand the dynamics [4–7]. All these constraints are compared with results of 1D stellar evolution codes that include the transport of chemical elements and angular momentum using radial diffusion models (e.g. [8]). This comparison provides estimates of the transport required to reproduce the surface and/or seismic data. Contrary to planetary atmospheres or stellar convective envelopes, we have no direct constraint on the length scale and the velocity that characterize the flow in radiative zones. We know however one important thing, that is the radial angular transport, whatever its cause, is weak in the sense that rotation velocities ∼ 2 − 100 × 105 cms−1 and lengthscales ∼ 1011 cm ∗ e-mail: [email protected]

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only produce an effective radial transport with diffusion coefficient Deff of the order 103 − 104 cm2 s−1 in solar type stars [9–11] and up to 108 cm2 s−1 in the external layers of massive star models [8]. In a radiative zone, the heat transport is ensured by radiation and the associated thermal diffusivity, κ, of the order of 107 cm2 s−1 in the solar radiative zone, is much higher than Deff . Thus the motions that are responsible for the effective transport of chemicals or angular momentum should generate a heat transport that is negligible with respect to the radiative heat transport. This means that, contrary to typical planetary atmospheres, the mean radial thermal stratification is not expected to be modified by hydrodynamical transport. This assumption is implicit in stellar evolution codes. The most obvious reason for the small radial transport in radiative zones is the strength of the stable stratification relative to the dynamics. The Brunt-Väisälä frequency N, that measures the time scale of the restoring buoyancy force 1/N, is of the order of a fraction of the star dynamical frequency (GM/R3 )1/2 . Except may be for the most rapid rotators, it will dominate motions taking place on a rotation time scale 1/Ω or those induced by differential rotation if the gradient scale if of the order of the radius r dΩ dr ∼ Ω. The N/Ω ratio is ∼ 360 in the solar radiative zone and is still ∼ 15 in a main-sequence intermediate-mass star with a rotation period of 1 day. While radial motions are strongly limited by the buoyancy force, horizontal motions are not and may drive an efficient transport in these directions. Hydrodynamical models of chemical and angular momentum transport in stellar radiative zones have been developed under the assumption that the horizontal transport is efficient enough to strongly limit the differential rotation in latitude. Distinguishing a large scale axisymmetric meridional circulation from 3D turbulent motions, Zahn [12] derived radial equations for the transport of angular momentum and chemical elements where the circulation velocities and the turbulent diffusion coefficients are expressed in terms of the prognostic variables Ω(r), C(r)... When compared to observations, this model has been quite successful for massive stars where a dominant transport process is the radial transport induced by radial differential rotation [13, 14]. It is this process that we shall consider in detail in this lecture. However, for solar-type stars, the model predicts larger rotation rates than observed in the interior of the Sun [15] and of the sub-giants [11]. This led to consider gravity waves generated by convective motions at the radiative/convective interface as a possible alternative for transport in solar-type stars [16]. Instabilities involving the magnetic fields are also considered as potential candidates to increase the angular momentum transport [17]. We shall not consider these processes in the following, although as long as they involve turbulent motions they will be also affected by stable stratification. In the following, we first consider the shear instability of parallel flow in stellar radiative zones (Sect. 2) and then discuss models of the vertical turbulent transport in stably stratified turbulence (Sect. 3). Recent numerical simulations allowed us to test models for the vertical transport of chemicals driven by a vertical shear flow in stellar conditions and we shall present their results.

2 Stability of parallel shear flows in radiative atmosphere

The stability of parallel shear flows is first considered in a fluid of constant density (Sect. 2.1), then in the presence of a vertical stable stratification (Sect. 2.2) and finally adding the effect of a high thermal diffusivity as in stellar interiors (Sect. 2.3).

Multi-Dimensional Processes In Stellar Physics

Figure 1. Instability of a sheet of vorticity submitted to an harmonic perturbation. The arrows show the velocities induced by the vorticity sheet on itself, leading to the reinforcement of the perturbation [19].

2.1 Unstratified shear flows

Let us first consider the kinetic energy potentially available in shear flows. If we consider an inviscid parallel flow U = U(z)ex between two horizontal plates where the vertical velocity "H vanishes, the initial horizontal momentum integrated across the layer 0 Udz is conserved. The uniform flow Um ex that has the same total horizontal momentum, determined by Um H = "H Udz has a lower kinetic energy than the initial flow. Shear instabilities can then be viewed 0 as a mechanism to extract the kinetic energy stored in the shear. This is realized for example when two parallel streams with different velocity are superposed and create a turbulent mixing layer [18]. The instability mechanism has been described physically by [19] by considering a sinusoidal perturbation of the vorticity sheet formed by the superimposed streams. The sheet being a material element, a perturbation ∝ ei(kx x−ωt) with k x the horizontal wave number and ω = ωr + iσ, will disturb the vorticity sheet as in Fig. 1. To predict the evolution of the perturbation, the self-advection produced by this vorticity distribution is analyzed. Expanding the vorticity sheet into elementary elements and considering the velocities induced by each of these elements at point A (or C), we see that by symmetry the added velocities at point A vanish. On the other hand, the velocities induced at the point B do not be compensate and will cause a horizontal displacement (to the left as shown on the figure) of the corresponding vorticity element. This process reinforces the local vorticity in A rather than in C. The asymmetric contribution from A and C on B will consequently displace the B element further up, leading to the growth of the perturbation. The rigorous mathematical treatment consists in applying continuity conditions at the interface and retaining solutions that vanish at large vertical distances. It shows that all disturbances are unstable with a growth rate σ = k x ΔU/2 that depends on the horizontal wavelength of the perturbation (Note that the dependence on k x ΔU could have been anticipated from dimensional analysis). The absence of conditions on the lengthscale of the perturbation is due to the velocity jump between the streams. We now consider the case of a continuous piecewise linear velocity profile presented in Fig. 2 (left panel) where the shear has a proper lengthscale H that will constrain the wavelength of the unstable modes. This configuration also allows us to present another physical interpretation of the shear instability. Again using continuity conditions at the lower and upper fluid interfaces z = ±H, the dispersion relation of the perturbations

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 S2  reads ω2 = 40 (1 − 2k x H)2 − e−4kx H , where S 0 = ΔU/(2H) = U0 /H is the shear rate. Accordingly, the horizontal wavelength of the perturbation, λ = 2π/k x , must be sufficiently large λ ≥ 9.81H to grow. The maximum growth rate is σmax ∼ 0.2S 0 and it is reached for kmax H ∼ 0.40. In the stable k x H ≥ 0.64 regime, the solutions of the dispersion relation are two waves propagating horizontally in opposite directions. In the large k x H limit, their wave speeds take the simple form cr = k x ωr = ±(U0 −S 0 /2k) and their amplitudes are concentrated around the upper and lower interface, respectively. The solutions of the dispersion relation are displayed on Fig. 2 (right panel).

Figure 2. Piecewise linear shear (left). Growth rate as a function of the horizontal wave number of the perturbation (blue), oscillation frequency of propagative waves in the stable regime (green).

The long waves of the stable regime are known as vorticity waves (also called shearRayleigh or Rossby waves) that are in general present at vorticity jumps. For example, at the interface between a layer of uniform velocity and a linearly sheared layer, U(z) = {U0 for z > 0 ; S 0 z + U0 for z < 0}, a vorticity wave propagates upstream at a phase velocity c = U0 − S 0 /(2k) and its amplitude decreases as e−kx |z| away from the vorticity jump. Baines & Mitsudera [20] proposed an interpretation of the shear instability based on the interaction between the two vorticity waves produced at the upper and lower vorticity jumps of the piecewise shear flow. First, the requirement that the velocity field induced by one vorticity wave at the level of the other interface is not negligible imposes a condition on the horizontal wavenumber, i.e. k x H  1 since the wave amplitude decreases as e−kx |z| . Then, a phaselocking condition between the two waves should ensure a sustained interaction between them. Imposing the same phase velocity for the two waves reads U0 − S 0 /(2k) = −U0 + S 0 /(2k) leading to k x H = 1/2. These two physical conditions are reasonably close to the exact con-

Multi-Dimensional Processes In Stellar Physics

ditions for instability and maximum growth rate given above. See [20] for more details on how the waves reinforce each other. Other instabilities like the baroclinic [21], the Holmboe [20] or the Taylor-Caufield [22] instabilities can be also interpreted as due to the resonant interaction between waves supported by the system. For general profiles U = U(z)ex of inviscid parallel flows, the modal linear stability analysis provides necessary conditions for instability. The Rayleigh criterion says that U(z) must have an inflexion point inside the fluid domain, while the stronger Fjørtoft criterion specifies, that, given a monotonic profile U(z), d2 U/dz2 [U(z) − U(zi )] must be negative somewhere in the domain, zi being the location of the inflexion point. In terms of the vorticity distribution, these criteria say there must be an extremum of vorticity in the domain (Rayleigh) and this extremum must be a maximum of the (unsigned) vorticity (Fjørtoft) [23]. As known for a long time, however, experimental results on various shear flows show a transition to turbulence even when modal linear analysis finds stability. This holds for the plane Couette flow which is linearly stable for all Reynolds number and yet experimentally unstable for Reynolds numbers higher that ∼ 350. The plane Poiseuille flows is linearly unstable above a critical Reynolds number of 5772 but experiments show transition to turbulence at a much lower Reynolds number ∼ 1000. To understand these discrepancies, a first step is to recognize that modal linear stability does not guarantee that perturbations monotonically decrease over time. In general, the evolution of perturbations can be put in the form ∂ ˆ = L(ˆu) where L is a linear operator acting on the spatial part of the perturbation uˆ . The ∂t u modes are the eigenvectors of the linear operator and in the modal analysis the system is linearly stable if the imaginary part of all the eigenvalues ω is negative. When L does not commute with its adjoint operator, the linear stability operator is said to be nonnormal and for shear flows, it is generically the case. Then, even if modal analysis says the system is linearly stable, the kinetic energy of small initial perturbations can grow during a transitory phase before decreasing exponentially over larger time (in the linear approximation). The transient growth is a consequence of the non-orthogonality of the set of eigenmodes of nonnormal operators [24–26]. On the other hand, the linear stability operators of the centrifugal instability or of the Rayleigh-Bénard convection are normal and in these cases, the critical parameters derived from the modal analysis correspond to the experimental ones. The maximum energy gain of the perturbations during the transient growth has been investigated for different shear flows and it was found that is can be very large [27]. Thus, even if perturbations initially behaves linearly, the transient growth will induce non-linear interactions which can lead to instability. A well documented mechanism of non-modal energy growth in shear flows is the lift-up process whereby counterrotating streamwise vortices generate growing streamwise velocity streaks. It is part of a generic self-sustained mechanism for shear turbulence proposed in [28, 29]. Accordingly, the streaks induced by the lift-up mechanism are then unstable to three-dimensional instabilities and non-linearly regenerate the streamwise vortices. The transition to turbulence in shear flows involving nonnormal amplification and non-linear interactions is an active field of research in fluid dynamics [30]. In astrophysics this type of transition has been considered to investigate magnetorotational dynamos in accretion disks [31]. Before the origin of these non-modal and non-linear instabilities were elucidated, J.P. Zahn [32] invoked the experimental results on shear flows as an evidence that all shear flows are unstable above some critical Reynolds number Rec , of the order of 1000. 2.2 Stably stratified shear flows

In an atmosphere stably stratified in the vertical direction, vertical motions necessarily come with a work of the buoyancy force. If the fluid elements are initially at their equilibrium level

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Figure 3. Potential energy required to exchange two fluid parcels in a stably stratified atmosphere

in the atmosphere, this work is negative and transforms kinetic energy into potential energy. As a parallel shear flow instability induces vertical motions, its development will be hindered by the stable stratification. Following Chandrasekhar [33] (see also [34]), the energy required to interchange two fluid elements can be compared to the energy available in the shear in order to derive stability conditions. Considering that the two fluid elements are initially at rest in the atmosphere and are respectively located at z and z + Δz, the difference in potential energy between the initial and the final states is ρ0 N 2 (Δz)2 , where N 2 = −g/ρ0 dρ/dz is the Brunt-Väisälä frequency characterizing the background stratification. As shown on Fig. 3, E p2 − E p1 = gρ(z + Δz) +

gz(ρ + Δρ) − gρz + g(ρ + Δρ)(z + Δz) = −gΔρΔz where Δρ < 0 if Δz > 0 and N 2 ∼ −g/ρ0 (Δρ/Δz). The energy that can be drawn from a shear flow is estimated considering that the initial horizontal velocities of the fluid elements at z and z + Δz are respectively U(z) and U(z) + ΔU, and assuming a perturbation preserving horizontal momentum that brings both velocities to U(z) + ΔU/2. The kinetic energy available is thus given by Ek2 − Ek1 = −ρ0 (ΔU)2 /4. The transformation will not be allowed if the total, kinetic plus potential, energy of the system increases, that is if Ek2 − Ek1 + E p2 − E p1 > 0 which implies Ri = N 2 /(ΔU/Δz)2 > 1/4. The stability of a parallel shear flow in a stratified atmosphere thus appears to be controlled by a nondimensional number, called the Richardson number, that compares the shear time scale tS = 1/(dU/dz) to the buoyancy time scale tB = 1/N. For an inviscid ν = 0 and diffusionless fluid κ = 0, it is possible to obtain a rigorous derivation of this criterion. Here we summarize the main steps of the calculation, the details can be found in [23, 35]. The basic state to be perturbed is again a general horizontal shear

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flow U = U(z)ex in a vertically stratified Boussinesq fluid with Brunt-Väisälä frequency N(z) given by the thermal stratification N 2 = βgdT/dz where g is the gravity and β the thermal expansion coefficient. In the Boussinesq approximation [36], linear perturbations to this equilibrium are governed by the following equations : ∂u ∂w + ∂x ∂z ∂u dU ∂u +U +w ∂t ∂x dz ∂w ∂w +U ∂t ∂x ∂Θ ∂Θ +U ∂t ∂x

=

0

1 ∂p + νΔu ρ0 ∂x 1 ∂p = − + βgΘ + νΔw ρ0 ∂z dT = −w + κΔΘ dz = −

(1) (2) (3) (4)

where u and w are the velocities in the streamwise and vertical directions respectively, p and Θ are the pressure and temperature perturbations. Alternatively, the buoyancy perturbation field b = βgΘ can be used in place of Θ, in which case the Brunt-Väisälä frequency appears explicitly in the heat equation. Neglecting viscosity and thermal diffusion, and looking for normal mode solutions i(k x x−ωt) w(x, z) = w(z)e ˆ , this set of equation can be reduced to one equation, called the TaylorGoldstein equation :   ˆ d2 Ψ N2 U  2 ˆ − kx Ψ = 0 + − (5) dz2 (U − c)2 (U − c) ˆ ˆ where Ψ(z) is the amplitude of the streamfunction and c = ω/k x . Introducing Φ,  √ complex ˆ ˆ U − c Φ, multiplying the Taylor-Goldstein equation by the complex conthrough Ψ = ∗ ˆ jugate Φ , integrating across the fluid layer bounded by two rigid plates where the vertical velocities vanish, and taking the imaginary part of the expression obtained, leads to : ⎧!  ⎫  ! top

top  ⎪  ˆ 2 ⎪ ⎪ ⎪ U 2  Φ ⎨ 2 2 ˆ 2 2  dz⎬ ˆ σ⎪ N − |Φ | + k |Φ| dz + =0 (6) ⎪  ⎪ ⎪  ⎩ bot ⎭ 4 U − c bot As σ  0 is a necessary condition for instability, the integrand must be zero which imposes 2 that N 2 − U4 < 0 somewhere in the fluid layer. In other words, a necessary condition for modal linear instability is : Ri = 

1 N2 2 < 4 dU

somewhere in the fluid

(7)

dz

It is the so-called Miles-Howard criterion. For the simple case of an hyperbolic-tangent parallel shear flow U(z) = ΔU tanh(z/H) in a linearly stratified atmosphere T (z) = T 0 + ΔT z/H, this condition is found to be sufficient. Indeed, according to [37], all perturbations with horizontal wavevector verifying k x H < 1 and Ri < (k x H)2 [1 − (k x H)√2 ] are unstables. Thus, if the Richardson number is less than 1/4, perturbations with k x H = 2/2 will be unstable. Before evaluating Richardson numbers in stars, we should mention that different types of instability mechanism have been identified for sheared stratified atmosphere. In addition to the Kelvin-Helmholtz type which involves two interacting vorticity waves, two other instabilities involving resonant interaction with gravity waves, the Holmboe and the Taylor-Caufield

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instabilities, have been identified. The Holmboe instability is relevant when the stable stratification varies on a length scale that is sufficiently smaller than the lengthscale of the velocity gradient [38, 39]. At first sight, this situation should not be generic in stellar radiative zones as angular momentum diffuses more slowly than heat. For example, in [40], when considering an hyperbolic-tangent shear in a polytropic atmosphere, the Kelvin-Helmholtz instability was found to dominate over the Holmboe instability. It might nevertheless be relevant at the edge of evolved convective core where sharp compositional gradients are present. Richardson numbers in stars can be estimated from the typical Brunt-Väisälä frequencies found in stellar evolution models and from our knowledge of surface or internal rotation rates. For the solar tachocline, helioseismolgy provides us with both the differential rotation across the layer ΔΩ = 0.15Ω and an estimate of the layer thickness H = 4 × 10−2 R . Together with a typical value of the Brunt-Väisälä frequency N = 10−3 s−1 , we get a very high Richardson 2 2 H 4 number Ri = r2N(ΔΩ) 2 = 2 × 10 . In the other cases where stellar seismology revealed differential rotation in radiative zones, in subgiants [6] or in red giants [5], the scaleheight of the rotation gradient is not known. It is nevertheless reasonable to assume that the gradients take place over a distance of the order of the stellar radius. With this assumption, we can write d ln Ω N2 dU/dz ∼ r dΩ dr = −αΩ with α = − d ln r ∼ 1. Thus the Richardson number reads Ri ∼ Ω2 and we can estimate its value for the Sun and for a typical early-type star. For the Sun, using Ω = 2.76 × 10−6 rad/s and the previous value of N, we obtain Ri = 1.3 × 105 . For a main-sequence intermediate-mass star with a period of rotation of 1 day and a Brunt-Väisälä frequency N = 10−3 s−1 , taken from stellar structure models of [41], we get Ri = 2 × 102 . These estimates show that Richardson numbers in stellar radiative zones are typically very high. The Richardson criterion for instability is thus very unlikely to be verified unless for some reason the rotation rate varies significantly over much smaller radial distance. 2.3 Stably stratified shear flows in highly diffusive atmosphere

In the previous section, we considered a parallel shear flow in a stably stratified atmosphere and found that, neglecting viscosity and thermal diffusion, stable stratification should hinder the shear instability in typical stellar radiative zones. However, we left aside many physical ingredients that may play a role in stars, including rotation, sphericity, viscosity, thermal diffusion, density stratification, compressibility, magnetic field. Among them, thermal diffusion can significantly alleviate the stabilizing effect of the stratification. Indeed, the buoyancy force gρ /ρ is equal to gβΘ in the Boussinesq approximation. By damping temperature deviations, thermal diffusion thus also reduces the amplitude of the buoyancy force and this will favor vertical motions. At the same time, thermal diffusion is known to damp gravity waves. Thus it does not always favors vertical motions in stably stratified fluid. The efficiency of these dynamical effects will depend on the motion lengthscale  since the thermal diffusion time scale is 2 /κ while the restoring buoyancy force acts on a time scale 1/N. Interestingly, in the presence of a strong thermal diffusion, it is not a trivial matter to decide whether a fluid element thrown vertically from its equilibrium position will end up above or below the maximum level of its adiabatic oscillations. We can anticipate that the dynamical effect of thermal diffusion will depend on the time scale Thus, before we study how thermal diffusion affects the stability of stably stratified parallel shear flows, let us first clarify this point by considering the simple case of small perturbations in an otherwise quiescent atmosphere in hydrostatic equilibrium. The Brunt-Väisälä frequency is taken uniform so that modal perturbations ∝ ei(kx x+kz z−ωt) are also harmonic in the vertical direction, where as before ω = ωr + iσ. The linear Boussinesq equations Eqs. (1-4), with U=0 and ν=0, lead to the dispersion relation : ω2 + iτω − ω2g = 0 where τ = κk2 is the thermal damping rate with k2 = k2x + kz2 and ωg = ± kkx N is the frequency of adiabatic

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Figure 4. Inflectional parallel shear flow (left) in a linearly stably stratified atmosphere (right)

gravity waves. The nature of the two solutions of this dispersion relation changes as thermal diffusivity is increased. For low thermal diffusivity, they are damped gravity waves and, as long as the damping rate τ is significantly smaller than ωg , their frequencies ωr ∼ ±ωg are practically not modified by thermal diffusivity. Then, when thermal diffusivity exceeds some values, more precisely when 2|ωg | ≤ τ, the damping is too strong to allow wave solutions.  The perturbations are damped without propagation at a rate σ± = 12 (−τ ± τ2 − 4ω2g ). What is more interesting for our current purpose is the regime of still larger thermal diffusivity when the two solutions σ± show very different behaviours. In the limit 2|ωg |  τ, σ− = −τ and σ+ = −ω2g /τ, the first mode corresponds to a fast thermal damping that does not depend on the stratification while the second mode is also decreasing in amplitude but with a damping rate that decreases for larger thermal diffusivity and for smaller lengthscales. This latter mode characterizes the effect of the buoyancy force when it is strongly affected by the thermal diffusivity. For isotropic perturbations, i.e. k x ∼ kz , the damping time scale of this modified buoyancy reads tBM = κ/(N 2 2 ). It can be expressed as a function of the adiabatic buoyancy time scale tB = 1/N and the diffusion time scale tκ , as tBM = t2B /tκ , that is, the adiabatic buoyancy time increased by the factor tB /tκ which is necessarily larger than one in the regime considered, 2|ωg |  τ. We now consider the effect of thermal diffusion on the modal linear stability of a stably stratified Kelvin-Helmholtz shear layer characterized by an hyperbolic tangent profile U(z) = ΔU tanh(z/H) in a uniform Brunt-Väisälä frequency background. For perturbations i(k x −ωt) ∝ w(z)e ˆ , the linear equations (1-4) reduce to a vertical 1D boundary value problem

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Figure 5. Effect of the thermal diffusivity on a Kelvin-Helmholtz instability in a linearly stratified medium. Neutral stability curves k x H = f (Ri), relating the horizontal wavenumber of the perturbation k x scaled by the shear lengthscale H to the Richardson number, delimit stable vs unstable domains for different values of the Peclet number.

that is solved numerically. This problem was studied by Lignières et al. [42] but also with minor variations by Dudis [43] and Jones [44]. Witzke et al. [40] considered the effects of compressibility using a polytropic background. Figure 5 displays, in the inviscid case ν = 0, the neutral stability curves delimiting the stable and unstable domains as a function of the Richardson number Ri = N 2 H 2 /(ΔU)2 , the Peclet number Pe = ΔUH/κ, and the horizontal wavenumber of the perturbation (k x on the figure corresponds to k x H in the present notation). One observes that the instability occurs at large Richardson numbers if the Peclet number is small enough. This can be interpreted by comparing the three time scales involved, i.e. the buoyancy time scale tB = 1/N, the thermal diffusion time scale tκ = H 2 /κ and the shear time scale tS = H/(ΔU). According to the piecewise shear studied in Sect. 2.1, the maximum growth rate of shear instability is ∼ 1/tS and this occurs for a perturbation such that k x H ∼ 0.5. Thermal diffusion will play a role on the dynamics only when buoyancy has an effect on the dynamics. This last condition is verified when tB < tS or Ri > 1. Moreover, we have seen that thermal diffusivity significantly reduces the stabilizing effect of buoyancy when tκ < tB . Thus, thermal diffusivity will affect the dynamics if tκ ≤ tB ≤ tS . It is possible to go further by using the concept of the modified buoyancy force introduced previously. Assuming that in the regime tκ < tB , the combined effect of buoyancy and diffusion acts on a tBM time scale, the stability threshold can be determined by comparing tBM and tS . Applying the condition tS ≤ tBM , we obtain the criterion RiPe ≤ 1 for instability. This is essentially the correct result as shown on Fig. 6, where the results presented in the previous

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Figure 6. See previous figure. The stability curves are now displayed as function of the product RiPe.

figure are now plotted in terms of the product RiPe. Modes with k x H ∼ 0.5 are unstable for RiPe  1 and their growth rates are of the order of 1/tS . However, this figure also shows that disturbances with very large horizontal scales are unstable in the domain RiPe > 1, the condition on the horizontal wavenumber being k x H < 0.117/(RiPe) [42]. This low k x H tongue actually corresponds to a different type of modes that benefit from the fact that mostly horizontal motions are practically not affected by the modified buoyancy. However, this set of modes must be considered with caution for astrophysical applications. First, their growth rate is much smaller than the dynamical one as σmax = 0.005tS /(RiPe). In addition, their horizontal lengthscale is so large that it may exceed the star radius. Indeed, taking into account viscosity, [42] shows that RiPe must be smaller than 1.67 × 10−3 Re for instability and that the most unstable modes have a ratio of the horizontal to the vertical lengthscale equal to ∼ Re/20. Applying this constraint to the solar tachocline values leads to a horizontal lengthscale of the unstable mode much larger that the solar radius, showing that the unstable modes in the RiPe > 1 regime are not relevant for the tachocline. We note also that the low k x H tongue is not always present as for example in [43] where the temperature background is an hyperbolic tangent profile instead of the linear one considered here. I take this opportunity to make a side remark on the use of the term "secular" to describe instabilities in the stellar physics literature. This terms in general refers to long time variations as compared to shorter periods present in the system (e.g. in celestial mechanics) and for

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shear instabilities it is appropriate if the time scale of the instability is much larger than the dynamical time scale tS . This holds to describe the mostly horizontal modes of the k x H  1 tongue with growth rates σmax  1/tS . The Goldreich–Schubert–Fricke instability [45] where mostly horizontal modes strongly affected by thermal diffusion are also present is another example where the term secular is appropriate. But the growth rate of the k x H ∼ 1 modes being of the order of 1/tS , the present shear instability should not be called secular, as it is usually done. In a stellar context, the difference between an adiabatic shear instability at Ri < 1/4 and and a diabatic shear instability at Ri > 1 is that the vertical length scale of the latter has to be small enough to benefit from the destabilization effect of thermal diffusivity, but their time scales are of the same order. Modal linear stability shows that a Kelvin-Helmholtz shear layer is dynamically unstable whenever PeRi  1. In an attempt to generalize these linear results (at the time due to [46] and [43]) to the non-linear instability of general shear layers, Zahn [32] added the constraint that the Reynolds number should exceed some critical value Re > Rec (see Sect. 2.1). As by definition Pe = RePr, these two conditions lead to RiPr < 1/Rec as the condition for instability. For stellar applications, Zahn proposed to take Rec ∼ 1000 from the transitional Reynolds observed in laboratory experiments (Couette, Poiseuille flows), that is RiPr < 10−3 for instability. Numerical simulations of shear-driven turbulent flows using different setups, homogeneous shear turbulence in Prat et al. [47], Couette flow in Garaud et al. [48], both found that RiPr must be lower than 0.007 for the turbulence to be sustained. They confirm the existence of a critical RiPrc though somewhat larger than Zahn’s estimate. These simulations also study the turbulent transport induced by unstable shear flows. Their results will be described with more details in Sect. 3.3.

3 Turbulent transport As commented in the introduction, the turbulent transport of chemical elements and angular momentum in stars is modelled through turbulent diffusivities associated with each identified instabilities. In the following we first recall what are the basis of the turbulent diffusion model and most importantly the quite restrictive conditions for its validity Sect. 3.1. These conditions can nevertheless be met and we present two examples in Sect. 3.2.1 that concerns the vertical transport of tracers in stably stratified turbulence at Pr ∼ 1. How the associated eddy diffusivity relates to the flow properties is discussed in Sect. 3.2.2. As compared to the Earth atmosphere or the oceans, a specificity of the stellar fluid is its very low Prandtl number meaning that at microscopic scales heat diffusion is much more efficient than momentum diffusion. This is taken into account to study the vertical transport driven by shear turbulence in stellar conditions (Sect. 3.3) and to comment on the case where a mostly horizontal turbulence is forced on large horizontal scales (Sect. 3.4). In Tab. 1, in addition to Prandtl numbers, some dynamical parameters of the stellar radiative zones and the Earth atmosphere and ocean are compared. Except in the convective boundary layer, motions in the atmosphere and in the oceans are typically strongly affected by the stable stratification. The time scale of the observed large scale horizontal motions can be compared with the buoyancy time scale through the horizontal Froude number Frh = uh /(Nh ), where uh and lh are the horizontal velocity and horizontal length scale of the large scale motions. As displayed in Tab.1, these horizontal Froude numbers are very low in the oceans and in the atmosphere which means that the turbulence is strongly affected by the stable stratification. We can not estimate Froude numbers from observations in stars. On the other hand, we know from the ratio N/Ω, also displayed in Tab. 1, that in stars also the effect of stable stratification on vertical motions is dominant as compared to that of the Coriolis force.

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Table 1. Relevant parameters of different stably stratified and rotating atmospheres.

N/Ω 150 150 360 14

Earth atmosphere Earth ocean Sun Intermediate-mass star (3M )

lh /uh (days) Fr = uh /(lh N) 1 10−3 10 10−4 ? ? ? ?

Pr 0.7 ∼ 10 10−6 − 10−5 10−7 − 10−6

3.1 The eddy diffusion hypothesis

A basis for the eddy diffusion hypothesis is the work of G.I. Taylor [49], who considered the dispersion of tracers in a turbulent flow under the assumption that the Lagrangian velocities are statistically homogeneous. The main steps of his model are recalled below as this is useful to understand the limitations of the eddy diffusion concept. It consists in relating the dispersion of fluid particles to the correlation of the velocity field. Here we focus on [z(t) − z0 ]2 , the mean square vertical displacement of fluid particles from their intitial position " t z0 = z(t = 0). From dz/dt = W, the individual displacement is given by z(t) − z0 = 0 W(t , z0 )dt where W(t, z0 ) is the Lagrangian vertical velocity. The mean square displacement then reads : ! t! t [z(t) − z0 ]2 = W 2 R(t , t”)dt dt” (8) 0

0

where R(t , t , z0 )

=

W(t , z0 )W(t”, z0 ) W 2

(9)

is the normalized autocorrelation of the Lagrangian vertical velocities. Assuming statistical homogeneity of W(t) allows us to limit the dependence of the autocorrelation to the time delay τ = t − t”. The double integral can then be simplified (see details in [21]) to get : ! t [z(t) − z0 ]2 = 2 W 2 (t − τ)R(τ)dτ (10) 0

A distinctive property of turbulence being its finite correlation time, one should be able to define a Lagrangian correlation time T L as : ! +∞ R(τ)dτ (11) TL = 0

Depending on the time over which the dispersion is considered, two distinct regimes emerge from the expression (10). A ballistic regime at short time, t  T L , where the velocities are still well correlated and the mean dispersion is linear in time [z(t) − z0 ]2 = W 2 t2 . Then, over longer times t T L , the memory is lost and the dispersion increases as if fluid parcels experienced a random walk : [z(t) − z0 ]2 = 2 W 2 T L t.

(12)

By analogy, a diffusivity Dt = W 2 T L is defined and is called turbulent (or eddy) diffusivity.

(13)

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The t T L condition imposes that a tracer field must vary on a sufficiently large length scale to experience eddy diffusion. Indeed, if LC denotes this length scale, the tracer distribution will evolve on a LC2 /Dt time scale and this has to be much larger than T L . This is equivalent to LC LT , where LT = WT L is the Lagrangian turbulent lengthscale associated with T L . On the other hand, if a tracer patch has a size smaller than LT , its dispersion is described by considering pair dispersion [50, 51]. Back to the condition on the turbulent velocity field, we note that Lagrangian statistical homogeneity is never strictly met in real flows. In terms of Eulerian statistics, a flow that would be statistically stationary and homogeneous would satisfy Lagrangian statistical homogeneity. But some level of Eulerian inhomogeneity is always present in real conditions for example near the boundaries of the system. Nevertheless, if the system has clear scale separation between the turbulent length scale LT and the lengthscale that characterizes the variation of the mean flow LS , turbulent diffusion should still be a good approximation to model the evolution of the tracer at the intermediate lengthscales. These conditions really need to be checked for the particular flow considered. For example, the hypothesis of scale separation is far from being guaranteed. In free shear flows like jets, wakes or mixing layer, the mean shear has a well-defined lengthscale which is also the lengthscale of the largest eddies. An eddy diffusion approximation can also be derived from the Eulerian equations using the mixing length theory. The evolution of a conserved quantity C in an incompressible turbulent flow with no mean velocity, u = 0, is governed by : ∂ C = −∇ · C  u + Dc Δ C (14) ∂t where C  are the fluctuation to the mean C  = C − C and Dc is the molecular diffusivity. This equation simplifies to : 2   ∂ C(z) = − ∂ C∂zw + Dc ∂ C(z) (15) ∂z2 ∂t when the mean distribution C only depends on z and the turbulence is homogeneous in the horizontal directions. The basic assumption of the mixing length theory is that a fluid parcel carries its conserved quantity over a distance mix before its mixes with the surroundings. Consider a fluid parcel that moves from z −  to z in the presence of a mean vertical gradient d C /dz. As C is conserved, its deviation from the mean will be C  = C (z −  ) − C (z) = − d C /dz + .. at z before it mixes. If  is smaller than the scale over which the mean vertical gradient changes, the first order of the Taylor expansion is dominant. The turbulent flux can then be estimated by : C  w = − w  d C /dz = −Dt d C /dz, where Dt = w  is a turbulent diffusivity. For generalized distributions C (x, y, z), C  ∼ − z d C /dz −  y d C /dy −  x d C /dx so that the three components of the turbulent flux C  u are written uiC  = −Di j ∂ j C in terms of the components of a diffusivity tensor : Di j = ui  j . We shall not consider here the implications of the tensor nature of the eddy diffusion as we investigate turbulent transport in the radial (or vertical) direction in stars and we assume that turbulence homogenize the mean flow in the horizontal directions. However, this aspect should be taken into account in 2D stellar models, where mean quantities depends on two spatial coordinates. We note that in this case the anti-symmetric part of the tensor acts as an additional meridional flow [21]. In the mixing length theory, the two important hypothesis are the scale separation between the length of variation of the mean gradient of C and the mixing length, and the near conservation of the transported quantity, that is conservation except for the effect of a small molecular diffusion. Irreversible mixing with the surrounding material is also important because the memory of C  must be lost to ensure a significant mean correlation between the motion and the fluctuations of C.

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Even if the diffusion hypothesis is correct, the value of the eddy diffusivity still needs to be related to the flow properties. The basic idea is to express it as Dt = wt mix , where mix is the mixing length and wt a r.m.s vertical velocity. But, to close the equations, the difficult part is to express both quantities as a function of the mean flow properties. The mixing length theory has been first proposed by L. Prandtl [52] to describe the transport of horizontal momentum, a non conserved quantity, in shear flows. The eddy viscosity νt = αut mix is said to be proportional to the product of the mixing length mix and a velocity scale ut estimated by mix |d U /dz|. In free shear flows like jets, wakes or mixing layers, the characteristic length scale of the mean shear flows is a natural choice for mix . It turns out that this mixing length model provides reasonably accurate profiles of the mean velocity for calibrated value of the non-dimensional parameter α, which varies between 0.07 and 0.18 for the three mentioned free shear flows [53]. Considering that both the assumptions of materially conserved quantity and scale separation are not verified in these cases, the success of this mixing length model is somewhat surprising. Tennekes & Lumley [53] attribute it to dimensional analysis as these flows are characterized by one length scale LS and one time-scale 1/(d U /dz). In the general case however, and in particular when additional physical effects come into play, using dimensional analysis is not sufficient to prescribe how the eddy diffusivity depends on the mean flow properties. This remark holds in particular when stable stratification adds a new time scale 1/N in a turbulent shear flow. To summarize this part, previous works indicate that the eddy diffusion hypothesis can be a valid approximation to describe turbulent transport under certain circumstances [21]. Indeed, for a materially conserved quantity whose initial distribution C0 (z) varies over length scales larger than the turbulent lengthscale, a turbulent flow with homogeneous Lagrangian statistics and finite correlation time T L , will act as a diffusion process on its mean distribution C (z, t). Whether these conditions are met depends on the particular flow considered. Even so, there is no obvious choice in general to express the eddy diffusivity in terms of the mean flow property. 3.2 Vertical eddy diffusion in stably stratified turbulence at Pr ∼ 1

In this section, we consider the problem of vertical turbulent transport in a stably stratified atmosphere. The turbulence is feeded by some forcing mechanism, the stable stratification is specified by the Brunt-Väisälä profile N(z), and the fluid properties by a molecular viscosity and a thermal diffusivity. Except for an example of turbulent transport in the ocean, we focus on results of numerical simulations. Indeed in numerical simulations, the forcing mechanism can be chosen to generate statistically stationary and homogeneous turbulent flows which is a major advantage to study turbulent transport. As implied by Eulerian homogeneity in space and time, the Lagrangian statistics is homogeneous and the transport of a tracer if it occurs on time scale larger than the Lagrangian time scale T L should be described by an eddy diffusion. In the following we illustrate with two examples that eddy diffusion can indeed be a good model for passive scalar vertical transport in stably stratified turbulence (Sect. 3.2.1). Then, we comment on the physical mechanism behind this transport in terrestial conditions where Pr ∼ 1 (Sect. 3.2.2). 3.2.1 Examples of vertical eddy diffusion

While estimates of the vertical (or diapycnal to take into account the fact that isodensity surfaces are not strictly horizontal) eddy diffusion in the oceans often rely on indirect measurements [54], the release and following of tracers allow direct in-situ transport measurements and constitute a nice illustration for our purpose. In an experiment reported in [55], inert

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Figure 7. Vertical profiles of the concentration (scaled) of a tracer released at a depth of about 300m in the ocean and followed during several months [55]

chemicals (139kg of dissolved SF6) were released in the ocean at a depth of about 300m and followed over seven months. The initial patch is dispersed mostly horizontally (or parallel to the density surfaces) but a slow vertical dispersion also takes place as shown in Fig. 7 where the tracer vertical distribution is displayed at different epochs. The increase of the width of a Gaussian-like profile resembles what is expected for a diffusion process. The width is measured by the second moment of the concentration profile M2 and its (almost linear) rate of growth provides a value for the effective vertical diffusivity using 2Deff = (1/2)dM2 /dt. A value of 0.11 cm2 s−1 is found in this experiment. Much higher eddy diffusivity, up to 10 2 −1 cm s , have also been measured by the same method over rough topography in the abyssal ocean [56]. We now turn to numerical simulations of homogeneous stably stratified turbulence where a random forcing term injects energy into large scale vortical motions (the forcing is concentrated around an horizontal length scale equal to L f = Lh /4, where Lh is the horizontal size of the numerical domain). The energy is transferred to small scale through a turbulent cascade where it is dissipated and a statistically stationary state is reached. A uniform Brunt-

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Figure 8. Vertical profiles of the mean concentration field of a tracer released in the midplane of a numerical simulation of randomly forced stationary and homogeneous stably stratified turbulence. The initial vertical profile is Gaussian and the mean concentration appears to be diffused vertically. The solid curves are analytical solutions of the 1D diffusion equation with Dt = P /N 2 [57].

Väisälä frequency and periodic boundary conditions in the three spatial directions insure that the turbulence is statistically homogeneous [57]. The advection-diffusion equation of a passive scalar field initially concentrated in a layer located at the mid-plane of the numerical domain is then solved numerically. The initial field is homogeneous horizontally C0 (z) and its vertical distribution is a Gaussian profile of width h0 . Figure 8 displays the initial profile together with the horizontal average of the scalar field at two later times. The first observation is that again the vertical transport of the tracer apparently behaves like a diffusion as the width of the Gaussian increases in time. Indeed, solving a 1D vertical diffusion equation with the initial condition C0 (z) and using the diffusion coefficient Dt = P /N 2 , where P = κ ∇b·∇b is the mean dissipation rate of potential energy, closely models the evolution of the mean scalar field in the numerical simulation (as shown by the superposition of the model and numerical results on Fig. 8). We recall that b = βgΘ is the buoyancy fluctuation field. Such a good agreement requires that the initial layer √ width h0 is much larger than the vertical length scale of the turbulence defined by v = E P /N, where E P = 1/2 b2 is the mean turbulent potential energy. This ratio is equal to 18.5 for the simulations shown on figure 8. When h0 /v = 4.8, there is a clear disagreement between the model and the simulation, which can not be improved significantly by tuning the eddy diffusivity (see details in [57]).

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This example shows that a gradient diffusion model for the vertical turbulent transport can be excellent in stably stratified turbulent flows. Stable stratification helps as it strongly constrains the vertical length of the turbulence and thus opens a range of scales that are at the same time larger than the vertical turbulent lengthscale v and smaller than the vertical lengthscale of the mean flow. As mentioned above, such a scale separation is not often achieved in turbulent flows like unstratified shear layer or thermal convection. In these simulations instead, the ratio between v and the vertical size Lz of the numerical domain can be controlled by tuning Lz and the rate of energy injection by the random forcing. This example also provides an expression for the eddy diffusivity that contains no free parameter. Its physical ground are exposed in the next section. 3.2.2 Modelling the vertical eddy diffusion

An important specificity of the vertical turbulent transport in stable atmosphere is to require some irreversible mixing to take place. The vertical displacement of fluid parcels from their equilibrium position indeed produces buoyancy fluctuations that increase the mean potential energy E P . Therefore, if the process is adiabatic, a monotonically increasing vertical dispersion [z(t) − z0 ]2 comes with an ever increasing mean potential energy. This would require a corresponding increase of the turbulent kinetic energy that is not possible if the turbulence is stationary. A finite vertical dispersion of particle is indeed observed at intermediate times in stationary turbulence [58]. This might seem at odds with Taylor diffusion model but numerical simulations of slowly decaying homogeneous stably stratified turbulence [59] show that the Lagrangian autocorrelation of the vertical velocity oscillates as a function of the time lag τ which leads to a vanishing Lagrangian correlation time T L [59]. Note that these strongly stratified flows have been modeled as a statistical superposition of gravity waves. Thus, a monotonic vertical dispersion in a stationary turbulence must involve interchanges of density between fluid elements and in a turbulent flow this will occur through stirring and diffusive irreversible mixing at small length scales. Building on [60], Lindborg & Brethouwer [61] proposed a model for the vertical dispersion in stationary turbulence that include these effects : [z(t) − z0 ]2 =

[b(t) − b0 ]2 P + 2 2t 2 N N

(16)

where the first term increases with time up to a finite limit 4E P /N 2 , that corresponds to the maximum vertical dispersion in adiabatic conditions. The second term dominates the first one for times larger than an eddy turnover time and corresponds to a vertical diffusion with eddy diffusion Dt = P /N 2 . According to the authors, the main assumption in their derivation is that the interaction time scale for exchange of density between fluid elements is equal to the Kolmogorov time scale (ν/)1/2 , where  is the mean dissipation rate of kinetic energy. This model has been tested in numerical simulations of homogeneous stably stratified turbulence sustained by large scale random forcing, like the ones mentioned in the previous section [62]. The two main parameters of these simulations are the horizontal Froude number Frh = uh /(Nlh ), that controls the buoyancy effects on the large horizontal scales, and the buoyancy Reynolds number R = /(N 2 ν), that controls the scale separation between the scales not affected by the stratification and the Kolmogorov scale (ν3 /)1/4 . Snapshots of the buoyancy fluctuations in a vertical plane are shown in Fig. 9 for different values of these two parameters. They show that the flow is more anisotropic as the Froude number decreases. At at low Frh , strong vertical shears develop between the horizontal layers and, if the Reynolds number is sufficient, they break down into 3D turbulence as seen in the R = 9.5 and R = 38

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F r = 0.01, R = 0.9

F r = 0.04, R = 9.5

F r = 0.07, R = 38

F r = 0.8, R = 5900

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Figure 9. Snapshots of the buoyancy fluctuations in a vertical plane, for different simulations of randomly forced stably stratified turbulence [62].

simulations. In this case, the vertical length scale adapts to v ∼ uh /N, which explains the difference in the layer thickness between the Frh = 0.04 and Frh = 0.07 simulations. We shall come back to this point in Sect. 3.4. The vertical dispersion computed in these different flows shows a good agreement with the model of Eq. (16), especially for the vertical diffusion term and when the buoyancy Reynolds number is large enough (see details in [62]). Actually this expression of the vertical eddy diffusivity is known from Osborn & Cox [63, 64] who derived it by assuming a balance between production and dissipation in the equation of conservation of the buoyancy fluctuations. In the context of the Kolmogorov energy cascade, P is independent of the molecular diffusion κ and is instead related to the dynamics. This led [64] to express the eddy diffusivity as a function of  : Dt = Γ

 N2

(17)

where Γ = P / is referred to as the "mixing efficiency". The value of Γ was assumed to be 0.2 by [64] although it has since been shown to vary with the type of flow considered and with the Froude number [39, 65, 66]. Nevertheless, according to [65], it approaches a constant value 0.33 at low Froude numbers.

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3.3 Vertical eddy diffusion due to radial shear in radiative zone

Here we consider the special case of shear driven turbulence in a stably stratified atmosphere with high thermal diffusivity. We have seen in Sect. 2.3 that, in stars, for radial differential rotation with shear rate of the order of Ω, stable stratification inhibits shear instability unless thermal diffusion comes into play. Linear stability then showed that a strongly stratified shear layer (Ri 1) can still be unstable on a dynamical time scale if the thermal diffusion time scale tκ = v2 /κ is smaller than the buoyancy time scale tB = 1/N. This condition puts√a strong constraint on the vertical length scale of the motions involved namely v < c = κ/N. A typical value for c in the solar radiative zone is 1km, which means we are considering very small length scales as compared to the star radius. By the way, this tends to justify the use of the plane parallel geometry and of the Boussinesq approximation. Zahn [12] proposed that the vertical turbulent transport driven by a vertical shear can be modeled by a turbulent diffusion of the form : Dt ∼ (Ric /3)κRi−1

(18)

with Ric = 1/4. The starting point of Zahn’s model was the linear stability analysis of Dudis [43] stating, as above, that instability requires RiPe < (RiPe)c . He added that in a turbulent regime it is the turbulent Peclet number Pe = u/κ that controls whether an eddy of size  and velocity u behaves adiabatically or not in the stably stratified atmosphere. Then, the largest eddies authorized by the stability constraint are such that RiPe ∼ (RiPe)c . The prescription Eq. (18) follows from the usual relation between the turbulent diffusivity and the eddy scale and velocity Dt ∼ u/3 (and using Ric instead of (RiPe)c ). In the following we review numerical simulations designed to test Zahn’s prescription. We can distinguish the scaling law Dt ∝ κRi−1 that contains the main physical assumptions of the model from the scale factor, Ric /3, that is not supposed to be better than an order of magnitude estimate. In principle, numerical simulations or laboratory experiments of turbulent flows are best suited to constrain such a factor. However, the usual stellar conditions being Ri 1, the consequence of Zahn’s prescription is that turbulent flows with very low Peclet numbers must be simulated. This in turn implies that the Prandtl numbers must be very low, which is indeed a specificity of the stellar fluid. Very low Prandtl numbers put strong constraints on laboratory experiments, because of the lack of low Prandtl fluids, but also on numerical simulations. Typically, the smallest time scale that needs to be resolved in numerical simulations of turbulent flows is the viscous dissipation at lengthscales close to the spatial resolution. The numerical time step must then be smaller than ∼ (Δx)2 /ν, with Δx the spatial resolution. But at low Prandtl number, the thermal diffusion time scale at this lengthscale is much smaller. This means the numerical time step must be decreased by a factor 1/Pr, which is huge for typical stellar Prandtl numbers. 3.3.1 The small-Peclet-number approximation

This difficulty can be avoided by considering the Boussinesq equations in the limit of small Peclet numbers. The Boussinesq equations are written around an hydrostatic solution, for which the thermal background is a solution of the heat equation. In a plan parallel geometry, these equations read : ∇·v

= 0

∂v + (v · ∇)v = ∂t

∂Θ + v · ∇Θ + w ∂t

=

−∇p + RiΘez + 1 ΔΘ Pe

(19) 1 Δv Re

(20) (21)

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where the dimensionless numbers Ri = (N 2 L2 /U 2 ), Pe = (UL)/κ, Re = (UL)/ν are expressed in terms of units of length L, velocity U and the Brunt-Väisälä frequency N 2 = βgΔT/L associated with the temperature difference ΔT between the upper and lower horizontal limits of the layer. All the fields are expanded in ascending powers of the Peclet number (w = w0 + Pew1 + ..., Θ = Θ0 + PeΘ1 + ...). At zero order, the solution of the heat equation is Θ0 = 0 which means that the thermal background remains unchanged. At the next order, the Lagrangian time derivative of the temperature is negligible except for the term of vertical advection against the thermal background that cannot disappear. Replacing the fluctuations of temperature Θ by a new variable Ψ = Θ/Pe then leads to the small-Peclet-number equations [67] : ∇·v = 0 1 ∂v + (v · ∇)v = −∇p + RiPeΨez + Δv ∂t Re w = ΔΨ

(22) (23) (24)

Without time derivative in the heat equation, there is no more constraint on the numerical time step associated with thermal diffusion. But one should keep in mind that formal asymptotic expansions can be singular, a well-known example being the singular behaviours of the viscous dissipation in the large Reynolds number limit. This question has been investigated by mathematicians who confirmed the validity of the small-Peclet-number approximation [68]. Physically, the linear evolution of harmonic perturbations in a quiescent atmosphere that we considered in Sect. 2.3 provides some clues. We have seen that for large enough thermal diffusivity, pure thermal diffusion modes separate from the dynamics. The small Peclet number expansion is a way to filter them out. This is reminiscent of the Mach number expansion that enables to filter out acoustic waves, leading to the anelastic approximation [69]. The next interrogation is how small the Peclet number has to be for the approximation to be relevant. For the Kelvin-Helmholtz instability described above, it was shown that the small-Peclet-number approximation is already very close to the Boussinesq calculations at Pe = 0.1 [42]. Similar results have been obtained in the non-linear regime by [70, 71]. The small-Peclet-number approximation also has the advantage of simplifying the physical interpretation of the effects of the stable stratification and of the thermal diffusion. Both effects indeed combine in a single process, a buoyancy modified by thermal diffusion, with a time scale tBM = t2B /tκ . In Eqs. (22-24), this translates into the combinaison of two independent non-dimensional numbers Ri and Pe, into a single one RiPe that controls the relative importance of the modified buoyancy with respect to the dynamics. Moreover, the energy conservation equation shows that the work of the buoyancy force always extract kinetic energy [67] (as already observed above with the damped linear modes in an otherwise quiescent atmosphere). Thus, the concept of available potential energy, that is the energy stored into buoyancy fluctuations that can be transformed back to kinetic energy, is no longer relevant in this limit. This property incidently clarifies a physical interrogation related to the fact that Zahn’s eddy diffusion is proportional to κ. On the one hand, a higher thermal diffusion indeed favors the vertical transport by reducing the temperature deviations thus also the amplitude of the buoyancy force. But, the opposite and concomitant effect, namely that the kinetic energy extracted by the buoyancy will be more efficiently dissipated by thermal diffusion, does not seem to be taken into account in Zahn’s criterion. This issue is solved in the smallPeclet-number regime because, as we just mentioned, all the kinetic energy extracted by the buoyancy force is immediately and irreversibly lost thus, obsviously, increasing the thermal diffusion cannot increase the dissipation of kinetic energy. In this regime, increasing thermal diffusion just favors the dynamics which is in agreement with Zahn’s eddy diffusion.

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Figure 10. Snapshot of the concentration field when a mean vertical gradient d C /dz is imposed in a homogeneous and stationary stably stratified shear turbulence [70].

3.3.2 Numerical simulations of shear driven turbulence in a stably stratified atmosphere with high thermal diffusivity

We now present numerical simulations performed in an attempt to test Zahn’s prescription for the radial turbulent transport in differentially rotating radiative zones. They have been performed in plan parallel geometry and with no effect of the Coriolis force. Prat & Lignières [70, 72] considered a flow configuration where a constant mean shear d U /dz and a constant Brunt-Väisälä frequency are maintained by body forces. The shear feeds a turbulent flow initialized by a statistically isotropic velocity field of given power spectral density. The Richardson number can be tuned to reach a statistically stationary state. The Reynolds number of these simulations is high enough to get a ∝ k−5/3 power spectral density while the aspect ratio of the numerical domain ensures that between four and eight large structures in each horizontal direction and at least six in the vertical direction are present in the flow. The generic self-sustained mechanism of shear turbulence described by [29] (see Sect. 2) is also observed in these simulations. These stationary and homogeneous turbulent shear flows are obtained on one hand using the Boussinesq equations with decreasing values of the Prandtl number or equivalently of the Peclet number. In [70, 72] the Prandtl number is decreased from 1 to 10−3 while the Reynolds number remains fixed. This allowed us to reach a turbulent Peclet number of 0.34 for a Reynolds number of u/ν = 340. On the other hand, one small-Peclet-number simulation, obtained for a stationary RiPe, allows us to consider the asymptotic regime Ri 1, Pe  1 relevant for stars.

Multi-Dimensional Processes In Stellar Physics

Figure 11. Dt /(κRi−1 ) as a function of the turbulent Peclet number. Dots correspond to Boussinesq simulations. The solid line represents the value obtained with the small-Peclet-number simulation.

The vertical turbulent transport of a passive scalar is then determined using either the vertical dispersion of Lagrangian tracers, the evolution of the width of a Gaussian layer of passive scalar or the mean turbulent flux C  w in the presence of a forced gradient d C /dz. For illustration, Fig. 10 shows a snapshot of the concentration field in this last case. The two first methods show that the eddy diffusion model is approximatively valid in these simulations. The eddy diffusions obtained with the three methods are consistent. The measured turbulent eddy diffusion scaled by the Zahn model is displayed in Fig. 11 for the different numerical simulations characterized by their turbulent Peclet numbers. They are shown for turbulent Peclet numbers around 1 and for the asymptotic small-Pecletapproximation. It appears that the Zahn scaling becomes valid for turbulent Peclet numbers of the order or smaller than one, and that the asymptotic value of Dt /(κRi−1 ) obtained with the small-Peclet-number equations is close to the value found at the smallest turbulent Peclet number reached with the Boussinesq equations, Pe = 0.34. As expected, the Zahn scaling is not valid at higher Peclet number. In that regime, as shown in Fig. 12, the eddy diffusivity Dt = P /N 2 fits the data much better. We have seen in Sect. 3.2.1 that this eddy diffusivity reproduces the vertical transport in the homogeneous stably stratified turbulence generated by a random forcing at Pr ∼ 1. In this regime, thermal diffusivity allows vertical transport by dissipating the buoyancy content of fluid parcels but this process takes place at the dissipative length scales of turbulence in such a way that P , and thus Dt , does not depend on κ. This is no longer the case in the regime of small Peclet numbers where thermal diffusion plays a role in the large scale dynamics, and the vertical transport linearly increases with κ.

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Figure 12. Dt /(P N −2 ) as a function of the turbulent Peclet number. Dots correspond to full Boussinesq simulations. The solid line represents the value obtained with the small-Peclet-number simulation.

These results validate Zahn’s scaling in the regime it was made for, that is when the transport is enabled by thermal diffusivity in strongly stratified shear layer Ri > 1. They also validate the small-Peclet-number approximation as a tool to explore the asymptotic regime Ri 1, Pe  1. A series of papers [47, 48, 71] have investigated the robustness of these results mostly by varying the Reynolds number and the way the mean shear is forced in the simulations. Most of them, except [71] who also performed Boussinesq simulations, used the small-Pecletnumber approximation for the parametric study. In Prat et al. [47], it was found that Dt /(κRi−1 ) = 0.0356 at Re = 7080, the largest Reynolds number considered in their study, to be compared with Dt /(κRi−1 ) = 0.0558 at Re = 336 displayed in Fig. 11. For these simulations, they used the shearing-box configuration as a different way to force statistically constant shear and stratification, and found no difference with the restoring forcing used in Prat & Lignières [70]. Garaud et al. [48] considered a stratified plane Couette flow, where the shear is forced through the boundaries. The shear remains nevertheless quasi-uniform away from relatively small boundary layers. Their results are fully compatible with those obtained by Prat et al. [47]. By decreasing the Reynolds number, both Prat et al. [47] and Garaud et al. [48] found that Dt /(κRi−1 ) increases slightly before it goes sharply to zero at RiPr ∼ 0.007. In Sect. 2.3, we already mentioned these results, as they provide the first determinations of the critical RiPrc for non-linear stability. The dependence of Dt /(κRi−1 ) at low Reynolds numbers, or equivalently at RiPr smaller but close to RiPrc , has been quantified through an empirical law in both studies. Garaud et al. [48] in addition considered the regime of low RiPe where the

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dynamics becomes unaffected by buoyancy. Non-uniform shear profiles were also considered in two other numerical studies [71] and [73]. In the latter, turbulence is sustained in localized shear layers within the numerical domain which allows us to investigate the transport at the edge of the turbulent layer. In short, for the regime Ri 1 relevant for stars, the Zahn model for transport and for stability has been confirmed in various numerical simulations that in turn provide in principle more reliable values for Dt /(κR−1 ) and for RiPrc . Despite some evidences given in [47], the assumption that Dt /(κR−1 ) will not vary by increasing the Reynolds number needs to be further confirmed. One must bear in mind that effective Reynolds numbers in stellar radiative zones are not extremely high (Deff /ν ∼ 20 for the Lithium transport in the Sun [9] and about 1000 − 10000 to account for the angular momentum transport in the core of subgiants [11]). Using the buoyancy modified time scale, Zahn’s scaling can be derived as a mixing length model for shear flow, that is Dt ∼ m2 d U /dz where m is specified by the condition that the shear time scale 1/(d U /dz) equals the buoyancy modified time scale tBM . This is a slightly more direct, although similar, derivation as compared to Zahn’s that introduces √ a marginal stability argument. The length scale that arises using this argument is Z = (RiPe)c m and it has been named Zahn’s scale in [48]. The fact that the dynamically relevant lengthscale of the flow, m or Z , can be directly related to the mean flow characteristics may be the reason why such a simple parameterization of the turbulent transport exists. This is not the case in turbulent stably stratified shear flows at Pr ∼ 1 where numerical simulations of homogeneous shear turbulence do not find an univocal relation between the eddy diffusivity and the Richardson number [74]. During stellar evolution, a stable gradient of chemical composition develops at the outer boundary of nuclear-burning convective cores. This introduces an additional term in the buoyancy force that is not sensitive to thermal diffusion but rather to the much lower molecular diffusivity. The consequence is a potentially very efficient barrier for the radial transport, with important effects on the abundance of the CNO cycle elements at the surface of massive stars [75]. Various prescriptions for the turbulent transport induced by a shear across the chemical composition gradient have been proposed in the literature [76–78]. They basically adapt the Zahn model to the presence of this additional stabilizing effect. From their numerical simulations, Prat & Lignières [72] derived the following prescription : Dt

= ακRi−1 (Ricr − Riμ ) with α = 0.45 and Ricr = 0.12

(25)

where Riμ = Nμ2 /(d U /dz)2 is the Richardson number defined from the Brunt-Väisälä frequency Nμ2 = −β gd μ /dz associated with the gradient of the mean molecular weight μ, β being the coefficient of compositional contraction of the fluid. The model of [76] agrees with the numerical results while the model proposed in [77] is not compatible with them. However, according to [75], the latter best fits the observations. Thus, if we trust the results of numerical simulations, this comparison indicates the need for extra mixing at the edge of the massive star convective cores. In [78], the effect of an horizontal turbulence on diminishing the buoyancy content was proposed as a possibility to increase the radial transport. The horizontal turbulence present in the homogeneous and stationary shear simulations does not seem to have an effect but, in the context of [78], this turbulence is not attributed to the radial shear but rather to an additional horizontal shear. This point remains to be studied using numerical simulations. Finally Prat et al. [47] and Garaud et al. [48] also computed the turbulent flow of horizontal momentum u w in their simulations. Deriving a turbulent viscosity defined by νt = u w /d U /dz, [47] found νt ∼ 0.8Dt when RiPr < 3 × 10−3 and in [48] νt /Dt was comprised between 0.8 and 1. Note that determining a value of νt in this way provides an esti-

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mate of the horizontal momentum transport, but it does not tell whether the gradient-diffusion approximation is correct or not. 3.4 Vertical eddy diffusion in strongly stably stratified turbulence with high thermal diffusivity

In the previous section we considered the case of a dynamical radial shear instability that √ produces approximatively isotropic motions at vertical length scales below c = κ/N. But other processes, other instabilities, may sustain turbulent motions of different types and we are obviously interested in their transport properties. In Zahn (1992) [12], the horizontal turbulence that is invoked to strongly limit the latitudinal differential rotation is assumed to be generated by barotropic instabilities of the latitudinal differential rotation itself. Horizontal motions being authorized by stable stratification, this turbulence is believed to be predominantly horizontal w/uh ∼ v /h  1. This could also be the case for other sources of turbulence whose driving mechanism is not constrained by the condition  ≤ c as it is the case for the dynamical instability of the vertical shear. In this category one can think of the turbulent motions enforced by convective overshooting in the radiative zone or the turbulence driven by baroclinic instabilities [79]. As already mentioned the atmospheric and oceanic largescale motions are characterized by very low Froude numbers Frh = uh /(Nh ) (see Tab. 1). This fact has motivated studies where a turbulent flow is forced at low horizontal Froude number, this situation being referred to as strongly stratified turbulence. A theoretical approach of strongly stratified turbulence is to consider the asymptotic limit of the Boussinesq equation in the Fr → 0 limit. In the scaling analysis of Riley et al. [80] and Lilly [81], the vertical Froude number defined by Frv = uh /(Nv ) is assumed to vanish also when Frh → 0, leading to asymptotic equations where the horizontal flow is governed by purely two-dimensional equations. Billant & Chomaz [82] reconsidered this scaling without a priori assumption on the vertical length v . They built on the self-similar properties of the invisicd Frh  1 equations with respect to zN/uh to show that the vertical length scale adapts to the system through v ∼ uh /N, that is Frv ∼ 1. The resulting asymptotic equations are not two-dimensional, in particular ruling out the possibility of an inverse cascade. They instead predict a direct energy cascade with an horizontal energy spectrum Eh (kh ) ∼  2/3 kh−5/3 . Various numerical simulations have confirmed the relevance of this scaling analysis in the Frh  1 regime, as long as the buoyancy Reynolds number is high enough (R = N2 ν  10) [65, 83–86]. Thus, in the double limit Frh  1, R 1, we may assume that turbulence is strongly anisotropic with v /h ∼ w/uh ∼ Frh . Horizontal motions show a direct energy cascade in between h and the Ozmidov scale O = ( N3 )1/2 that separates the lengthscales affected by buoyancy from the isotropic inertial range. Below O , the return to isotropy is possible and a Kolmogorov turbulent cascade develops between O and the Kolmogrov scale η = (ν3 /)1/4 . The condition R 1 is crucial for the existence of the two inertial ranges below and above O [83]. An extrapolation of these results to stellar conditions can be attempted as follows. Two regimes are to be distinguished depending on the rate of energy input into the turbulence u3h /h ∼ . If it is high enough so that return to isotropy takes place at lengthscales which are not affected by thermal diffusion, the previous scaling holds. On the other hand, if c = √ κ/N > O , the buoyancy effects will be strongly diminished at scales larger than O . In this  3/8 case, we can define a modified Ozmidov scale OM = κ 1/3 /N 2 ) for return to isotropy as the scale for which the eddy turnover time /u, where u = ()1/3 , is equal to the modified buoyancy time κ/(N 2 2 ).

Multi-Dimensional Processes In Stellar Physics

In Sect. (3.2.2), we have seen that when Pr ∼ 1 the vertical eddy diffusivity in strongly stratified turbulence reads Dt = Γ/N 2 with Γ ∼ 0.33 in the Frh → 0 limit. This turbulent diffusivity can be equivalently expressed using the Ozmidov scale as Dt = ΓuO O with uO = (O )1/3 . Using the same expression but with the modified Ozimdov scale yields an estimate of the vertical transport in the c > O regime. Finally, for strongly stratified turbulence in radiative zones, two regimes of vertical turbulent transport are expected depending on the characteristics of the horizontal turbulence, namely :  → Dt = Γ 2 (26) if O > c N  κ 1/2 → Dt = Γ 2 (27) if c > O N where Γ is a yet unknown constant that plays the same role as Γ . This derivation of a vertical eddy diffusion in a strongly stratified turbulence Frh  1 affected by thermal diffusion has not been published elsewhere. Nevertheless, I found the same expression in [12] as a model for the vertical turbulence driven by horizontal shears. According to [12], it has been derived in [87] following Townsend’s arguments [46] to account for thermal diffusion effects. It turns out that a quite general statement can be deduced by combining these estimates and the observational constraints showing that the effective transport of chemicals is in general much smaller than the thermal diffusivity, Deff  κ (see Sect. 1). Indeed, a consequence of the previous scalings is that Dt > κ when O > c , whereas Dt < κ when c > O . Thus we conclude that if turbulent motions are indeed responsible for the observed transport, the fact that Deff < κ implies that the eddies involved in the vertical √ transport are affected by thermal diffusion, that is their length scales are smaller than c = κ/N. This same conclusion was reached by [88] using similar arguments regarding thermal diffusion effects, but without reference to the strongly stratified turbulence scalings of [83].

4 Conclusion We have reviewed physical processes involved in the vertical turbulent transport produced by a plan parallel shear flow in a vertically stably stratified atmosphere with high thermal diffusivity. This led us to discuss shear instabilities, eddy diffusion, and stratified turbulence in conditions encountered in the Earth fluid envelope (Pr ∼ 1) up to stellar radiative zones (Pr  1). We showed that numerical simulations have been successful in testing Zahn’s model for the radial turbulent transport induced by radial differential rotation. Regarding the broader issue of modeling the transport of chemical elements in differentially rotating radiative zones, many more questions can be approached through dedicated numerical simulations. Local simulations such as the ones presented here are well suited to investigate turbulent transport processes occurring at small length scales. But they do not provide informations on the largescale flow. This requires global simulations in spherical geometry. As a first step, this type of simulation should tell us more about the differential rotation and the laminar largescale flows driven when torques are applied to radiative zones. These torques might be due to stellar winds, or to Reynolds stresses at the interface with a differentially convective zone or to structural changes during expansion or contraction phases. A better knowledge of the resulting large scale configurations should help us identify the instabilities than can power a transition to turbulence, and provide us with some constraints on the scale of the dominant eddies. For example the main instabilities triggered in a differentially rotating star embedded in a poloidal magnetic field are being studied thanks to a combination of axisymmetric and 3D simulations [89, 90]. In non-magnetic radiative zones, global numerical simulations should be designed to test the hypothesis of weak latitudinal differential rotation which is at the base of the current models of rotationally induced transport.

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An extremely short course on stellar rotation and magnetism Matthew K. Browning1 1 Dept

of Physics and Astronomy, University of Exeter, UK These notes cover most of the material conveyed during a series of lectures I gave at the 2018 Evry Schatzman School (on “multi-dimensional processes in stellar physics”) in Roscoff, France. These lectures dealt with stellar rotation and magnetism in a way meant to be accessible to (post)graduate students without much previous background in fluid dynamics or stellar astronomy. They try, to the extent possible in a few hours (or pages), to explore how a star’s life is shaped by magnetism or rotation – which, in practice, means exploring a few topics from stellar astronomy and introducing some basic elements of magnetohydrodynamics (MHD). The first “lecture” here introduces some basic observational facts about stars, and some essential stellar physics, to motivate the rest of our discussion. The second lecture introduces the equations of (rotating, magnetised) fluids, describes some important limiting cases of these equations, and uses them to solve a couple of example problems. The third focuses specifically on the problem of how magnetic fields in stars are amplified and maintained by dynamo action.

Abstract.

Preface The central question addressed in these lectures is, “How is a star’s life shaped by rotation or magnetism?” It has to be admitted at the outset that I won’t really answer this question, and indeed that there is not (in my view anyway) a complete answer available – but we do have many constraints, and the lectures will try to lay these out, to the extent possible in four hours or so. To do so, we need to understand a bit about both stellar astronomy and about magnetohydrodynamics: for example, we need to know how rapidly stars rotate; how magnetic they are; how this depends on gross properties like luminosity; how these fields in turn react back on the star’s structure and evolution; and so forth. So at a practical level, the lectures are a sort of amalgam of “stars 101” and “MHD 101,” aimed most squarely at postgraduate students who know a bit about stars but not so much about magnetohydrodynamics, or possibly at students who are familiar with MHD but less acquainted with stellar astronomy. (Students who already knew about both topics would have been well-advised to spend their four hours elsewhere!) The first lecture introduces some basic observational facts about stars – namely, that they all rotate and are all magnetic at some basic level – and uses these to motivate the rest of our discussion. Much of this material is adapted from a recent review I co-wrote with Sacha Brun for Living Reviews in Solar Physics [1]. It also develops some order-of-magnitude estimates of the conditions that prevail in stellar interiors, and near their outer surfaces, so that we can have some idea of whether magnetism or rotation ought to matter at some detectable level. The second might be called “descriptive MHD with rotation,” and derives (or states)

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some of the most important results from this broad subject; here, I have drawn heavily on introductory MHD lectures taught to me long ago (at the University of Colorado-Boulder) by Ellen Zweibel. The third focuses on dynamo action in particular – that is, the growth of an initially small magnetic field by motion, and its subsequent maintenance against Ohmic decay. (Portions of this are also dealt with in more detail both in our Living Review, and in the excellent Les Houches lecture notes by [2].) The fourth lecture in Roscoff introduced some topics in current research, and was effectively a summary of some outstanding issues, with less derivation and more recourse to numerical simulation. Because this material will so quickly become out-of-date, and portions of it are treated in our recent Living Review [1] in any case, I have chosen not to duplicate most of it here and instead provide only a very quick summary. The actual lectures as delivered in Roscoff were a mix of Keynote slides and blackboard work; not all of this lends itself particularly well to reproduction here, so I have picked and chosen which topics to cover in these notes. This is also an extremely informal writeup, written long after the lectures themselves, so it inevitably differs from them in various other ways.

Lecture 1: An introduction to stellar rotation and magnetism In this first lecture, we seek partly to motivate the other three, and to provide some preliminary estimates about when and where rotation or magnetism might plausibly matter. So the central question today is, why should you – and by you, I really mean “someone who doesn’t already work on stellar MHD specifically” – care about stellar rotation or magnetism? From the stellar point of view, are there really any aspects of a star’s life that are noticeably affected by these phenomena? Or from the MHD point of view, are there any insights we can gain from the study of stars specifically (as opposed to laboratory experiments or basic theory)? We are motivated partly by a few key observations, which I’ll show a few times during these lectures. Among these are the finding that some stars turn out to have rather strong magnetic fields – meaning, for example, fields strong enough to influence the visible appearance of the star, its observable high-energy emission, and so forth – whereas others do not. We’d like to know why. Among the stars that do show magnetic fields, some of these exhibit a remarkable degree of spatial and/or temporal order: they show cycles, for example, or a large-scale dipole field; others don’t. Again, it would be nice to know why this is, and whether/how it depends on basic parameters like mass and rotation rate. Another motivation is that we live next to a star with some magnetism – the Sun has observable magnetic fields that range from the smallest scales we can probe up to the largest scales present in the system (i.e., the overall stellar radius). The most famous visible manifestations of the magnetism are sunspots: dark blemishes on the photosphere, which occur in regions where the emergent magnetic field is strong enough to impede convective heat transport (and so make the region a little cooler and dimmer). Peering more closely, there is a great deal of small-scale magnetic “chaff” that comes and goes, and the Sun certainly looks “turbulent”: any of the classic measures of fluid mechanics, measuring something like inertia (or magnetic induction, or buoyancy driving) relative to viscous or thermal dissipation, are absurdly big. But the Sun also shows a stunning degree of organisation in time and in space. One example is the “butterfly diagram”: spots tend to appear first at mid-latitudes in the Sun, then progressively nearer the equator over the course of roughly an 11-year cycle, with the number of spots also waxing and waning over the course of the cycle. So we’d like to understand not just how you build fields, but also how you build ordered fields (either in space or in time).

Multi-Dimensional Processes In Stellar Physics

A breezy tour of observational techniques

To understand how any of this comes about, though, let’s talk for a minute about how you actually measure anything about a star’s magnetism or rotation. If you have managed to reach this School without literally any exposure to observational astronomy, an extremely quick summary is that all we get are photons (with apologies to the nascent field of gravitational wave astronomy). You can measure the incoming flux of photons from an object as a function of wavelength, as a function of time, or as a function of position on the sky; if you’re really lucky you’ll have polarization information, too. That’s it. But it’s enough to learn a great deal! We can measure magnetism (and/or rotation) provided we have enough precision along any of these observational dimensions. Many other resources go into all this in much more detail, so I will just illustrate a few examples. Let’s talk about information in “wavelength space” first. If you have sufficient spectral resolution, you can measure the broadening of spectral lines due to rotation and/or magnetism; if you add polarization information you can also deduce some aspects of the magnetic field geometry. (See, for example, early measurements of magnetic broadening in a magnetic M-dwarf by [3].) Conceptually, there are two classes of effects here: one that changes the actual energy levels of the atoms or molecules you’re observing, and another that changes something about how the photons propagate to you. You would expect that the former tell you something about the total energy of a magnetic field, or some equivalent quantity (like the “unsigned flux”) – the atoms don’t know which direction you’re looking at them from, so for example the overall Zeeman broadening of a magnetically-sensitive line doesn’t depend on the field orientation. In contrast, the latter type of effect might in principle tell you something about the orientation of the field (since the photons do propagate in some well-defined direction relative to that field). You need rather high spectral resolution to learn much about the rotation or magnetism in this way, and you also need the effects of these processes to be “large enough” compared to other effects. Consider, for example, the Doppler broadening due to rotation: this just comes from the fact that portions of the star are moving towards you while others are moving away, which will broaden some spectral line occuring at λ0 by an amount of order Δλ/λ0 ∼ v/c, with v the rotation velocity. Given this, how big does rotation have to be before you could measure it? Well, a very good spectrograph might be quoted as having “R =60,000,” meaning that it can give you λ/Δλ ≈ 60000 at some fiducial wavelength – and some playing around with numbers here will convince you that something of order the Sun’s rotational velocity (about 2 km/s, though this varies from pole to equator) is not likely to be detectable. Basically, you need 1/R ∼ v/c, so to detect v ∼ 3 km/s, you’d really like to have an R = 100, 000 spectrograph (which is hard to come by). These estimates are a little pessimistic, but not wrong to order of magnitude. To complicate matters further, the spectral lines are also thermally broadened: that is, we might assume the atoms we’re  observing have some Maxwellian distribution, with a typically thermal velocity of order (kT /m) (with m the mean mass). Plugging in numbers, H atoms at the photosphere might be expected to have thermal broadening of order a few km/s. The punchline here is just that stars have to be rotating pretty rapidly in order to measure their rotational broadening in wavelength space. (We haven’t done the equivalent calculation for magnetic fields here, but similar considerations apply.) Alternatively, you can examine a star’s variation in time (i.e., its variability) to learn something about rotation and/or magnetism. See, for example, some stunning examples of this in the data from Kepler or CoRoT [e.g., 4, 5]. The simplest model you might imagine is of a single dark spot (caused by an emergent magnetic field) that is steady in the star’s rotating frame of reference: in this case you’d see a periodic modulation of the light curve, and by measuring its period and depth you’d learn something both about the rotation rate and

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the brightness of the spot (and hence maybe something about the field). In reality, the relation between magnetism and irradiance is considerably more complicated: for example, the Sun is actually brighter at solar maximum (when there are lots of spots), because the small dimming provided by the spots is more than offset by a brightening induced by numerous faculae. See, for example, [6]. Whether the spots or the faculae “win” in setting the irradiance is likely to depend on the star’s overall level of magnetism; see, e.g., recent analysis by [7]. Similarly (and remarkably), it is now becoming possible to learn something about stellar magnetism and rotation by measuring the brightness of a star as a function of position: that is, by making an actual image of the star. See for example [8], who employed optical interferometry to image the spotted surface of the nearby star Zeta Andromedae. More generally, precision along each of these observational “dimensions” can be combined to yield even more information – so, for example, the technique of spectropolarimetry combines a series of high-precision spectra to infer information about the change in surface magnetism over time. See, for example, reviews in [9], [10], [11]. For a recent example of analysis combining information from interferometry, spectropolarimetry, and Doppler imaging, see [12]. There are a number of more exotic observational techniques that we won’t have time to talk about here – but I will mention one of them just because I think it’s really neat. If a star hosts a transiting exoplanet, that planet masks out different portions of the stellar photosphere as a function of time (and, if the spin and orbit are misaligned, even different latitudes on the star’s surface). If you have enough transits, it is thus possible to use the exoplanet as a probe of the stellar surface – see [13] for an example. After all this, though, it must be admitted that much of our knowledge about stellar magnetism comes from measurements of various “proxies” – that is, observable things that are linked to the magnetism in some direct way. The most well-known of these are chromospheric and coronal emission. It is evident from any side-by-side comparison of a Solar magnetogram and Solar high-energy emission that the two phenomena are linked, even if the precise details behind how the magnetism heats the upper atmosphere are still imperfectly understood. This link can, in the case of the Sun at least, be made quantitative [14], lending us some confidence that observations of stellar chromospheres and coronae really are telling us something about magnetic field strengths. See, for example, [15] for a review. Some observational punchlines and motivating puzzles

So when you go out and observe stars using all of these techniques, what do you actually find? Our Living Review [1] has a much more detailed review and link to (many) other references, so I will just mention a few “take-away” points, before discussing a few open questions that motivate some of our discussion in the following lectures. One key qualitative point is just that all stars are probably magnetic at some level, and the surface magnetism appears to be linked to convection. If you go looking for the signs of surface magnetic activity (e.g., via proxies like chromospheric or coronal heating), you tend to find these things are common in stars that have near-surface convection and rare otherwise. So that’s a pretty big clue, independent of any theoretical model for how the fields are built, that convection probably has something to do with it (either directly, or in principle indirectly, for example by establishing differential rotation that in turn builds the magnetism). Similarly, all stars presumably rotate at some level, and in practice we observe a wide range of rotation rates at different ages and masses (e.g., [16]). Strikingly, the rotation and the magnetic activity are linked: various measures of surface activity increase with rotation up to some point, beyond which the activity seems to plateau (often called “saturation”). You see some version of this link both in “proxies” (see, e.g., [17–20]) and in more “direct” Zeeman

Multi-Dimensional Processes In Stellar Physics

Doppler Imaging (ZDI) measurements (e.g., [21]). This “rotation-activity” correlation is also a pretty powerful clue that rotation probably has something to do with the field generation process. Another basic observational fact is that some stars exhibit magnetic cycles, and others don’t appear to. The situation here is a bit complex, with different authors coming to different conclusions about the relation between cycle period and (for example) rotation rate; see [15], [22], [23], [24], or [25] for discussions. Stars also lose mass all the time. In the Sun this mass loss is “low”: the Sun has probably lost only about 0.01 percent of its mass, with a current mass-loss rate of the same order as the mass loss due to its radiation alone, L/c2 , though of course in terrestrial terms the result is still pretty big (about 1.3 million tons/second). But even this paltry outpouring can still have an effect on us: for example, the solar wind’s interaction with the Earth’s magnetosphere is ultimately responsible for all sorts of interesting phenomena observable here on Earth, ranging from aurorae to geomagnetic “storms” that can knock out power grids and disrupt satellite communications. Solar storms could in principle have a devastating effect on unshielded satellites or (for example) on astronauts on long-term missions outside the protection of Earth’s magnetosphere. (As a motivating aside, if you hang out with Solar people for very long, you will inevitably here about the “Carrington event,” the largest solar storm in recent history. This occurred in 1859, and accounts from that time – see quotes compiled in [26]– are breathtaking, with aurorae reported at essentially all latitudes, telegraph wires sparking, etc. Recovery from a similar storm today might cost of order a trillion dollars within the first year, according to a 2008 National Research Council study.) As another motivation, consider that interaction with the solar wind is probably part of how Mars lost its atmosphere – see recent results from the MAVEN mission, which showed that atmospheric erosion (of the current, very tenuous Martian atmosphere) increases during solar storms. The outpouring matter also carries away angular momentum, and this inexorably spins down the star. The star’s magnetic field effectively sets the “lever arm” for the wind: both its strength and morphology matter here (see, e.g., [27]), but qualitatively stronger, organised fields lead to more rapid spindown. So the relation between magnetism and rotation is a two-way street: rotation probably affects the field generation process (see our discussion of theory below, or just the rotation-activity correlation), but magnetism feeds back on the star’s rotational evolution, too. Various models of stellar spindown try to incorporate parameterizations of all this in order to understand how the stars spin down over time (as a function of mass, metallicity, etc) – see, e.g., [28]. Observations of stars today have a precision that astronomers a generation ago could only dream of, and cover large enough samples that meaningful statistical inferences can be made. The recent GAIA DR2 data release is a compelling example: positions and brightnesses for 1.7 billion stars! Parallax, proper motion, and colour for more than 1.3 billion! Other powerful new(ish) sources of data include the Kepler satellite, CoRoT, and now TESS, which have measured the light curves of hundreds of thousands stars to very high precision. These new data inevitably lead to new puzzles. For example, a particularly interesting puzzle comes from the asteroseismic study of red giants, whose core regions appear to rotate quickly, but not as quickly as simple theoretical models would predict (see [29], [30]). This finding suggests a need for additional angular momentum transport, with magnetic fields a possible culprit (alongside internal gravity waves, shear turbulence, and other mechanisms) – see discussion in [31]. Other authors have pointed to the possible impact of magnetism (and/or rotation) on stellar evolution and end states – see, e.g.,[32], [33], [34]. One particular puzzle that I’ve found interesting, and will concentrate on a bit just to focus our discussion, involves the radii of low-mass stars. You can measure these quite precisely in eclipsing binary systems, and the finding over the past decade or so has been

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that some objects are consistently “inflated” (by perhaps 3-15 percent) over the predictions of standard 1D models – e.g., [35]. (This also translates to saying that the effective temperatures in these objects are lower than the model predictions, since the luminosities are roughly in accord with the models.) Where does this come from? Several authors have suggested that convective inefficiency of some sort – perhaps caused by magnetism – or magnetic “spots” directly might be to blame (e.g., [36], [37], [38] and subsequent papers). Should you be surprised by this argument? Is it plausible that magnetic fields could “inflate” a star directly or indirectly? The field in a strong sunspot is maybe 1500 Gauss (a kitchen magnet is 10-100; the Earth’s field at its surface is around 0.5) – is this strong enough to have any significant effect, for example? Some order-of-magnitude estimates about stars

To understand whether fields of that strength – or magnetism/rotation in general – have any chance of modifying the internal structure or evolution of a star, we first need to understand roughly what conditions prevail in stellar interiors. This will let us figure out whether the pressures associated with the presence of a magnetic field are “large” in the stellar context, or not. First, let’s try estimating the deep interior pressures and temperatures to order-ofmagnitude. Stable stars are in hydrostatic equilibrium, so we have dP = −gρ dr (assuming g points radially inward). Approximating dP/dr to be ∼ Pc /R, with Pc the central pressure, we then have GM 2 Pc ∼ gR ¯ ρ¯ ∼ 4 R with g, ¯ ρ¯ average values of gravitational acceleration and density. (You could have derived the same thing by dimensional analysis.) You can plug in numbers for your favourite star to find that pressures are “big,” in the sense that only extraordinarily strong magnetic fields (many MG) would have Pmag ∼ Pgas . To estimate the temperature, we could just assume the interior is an ideal gas, and use our estimates of pressure above, but it’s also a convenient excuse to introduce the virial theorem. You are probably familiar with this already (e.g., from classical mechanics) but if not, you can think of it in this context as essentially a global version of hydrostatic equilibrium: ultimately, it relates the gravitational binding energy of the object to its internal energy (which is in turn related to pressure). (For a more extensive discussion, see any standard stellar text – e.g., [39]) To be specific, consider a γ-law equation of state of the form P = (γ − 1)ρum with um the energy per unit mass arising from microscopic processes. Let U=



um dm

be the total internal energy, and the gravitational potential energy (defined here as the energy needed to assemble the star, or the negative of the energy required to disperse it – meaning it’s negative for a gravitationally-bound object) as Ω=−



GMr GM 2 dMr = −q r R

Multi-Dimensional Processes In Stellar Physics

where q is a constant of order unity that depends on the mass distribution. (For a uniformdensity sphere, q = 3/5.) In equilibrium, the virial theorem tells us that U =−

1 Ω 3(γ − 1)

which reduces to U = −Ω/2 if γ = 5/3. The relation expressed here has a number of consequences that can seem odd at first glance. Consider, for example, the behaviour of a star that is contracting on the pre-mainsequence: it’s radiating away energy, but isn’t yet undergoing nuclear fusion in its interior. Notice that the total energy W = U + Ω = Ω/2 (if γ = 5/3) – and again, noting that W < 0 for a bound object – so as the star loses total energy, ΔW and ΔΩ are both negative. (Note that ΔΩ ∝ (GM 2 /R2 )ΔR), so that indeed ΔR is negative too – that is, the star is contracting.) But it’s internal energy goes up (ΔU is positive)! Assuming the internal energy tracks temperature (as it does for a non-degenerate ideal gas), this means the star is losing energy but heating up. This is what people mean when they speak of self-gravitating objects having a “negative heat capacity.” We can also use the virial theorem to estimate the average temperature in the star. Assuming U ∼ NkT ∼ (M/μmh )kT ∼ (1/2)qGM 2 /R, we end up with something of order kT ∼ GMm/R (with m the average per-particle mass). That is, the thermal and gravitational energies per particle are comparable. Numerically, this implies temperatures of order T ∼ (4 × 106 μ)(M/Ms )2/3 ρ1/3 K [39], with μ the mean molecular weight and Ms the solar mass. What about conditions near the surface, or at other depths? Let’s start with the pressure again. At the photosphere, the mean free path for photon absorption is of order the scale height: 1 kT 1 lm f p = = ≈H = nσ κρ mg with κ a mean opacity of some sort, m the mean per-particle mass, σ the cross-section, and n the number density of absorbers. This implies that the density at the photosphere is of order ρ ph =

mg κkTph

where T here is the temperature at the photosphere, which we will assume is the effective temperature Te . If the gas is ideal, so that Pph =

ρkTph mg

then it follows that Pph =

g . κ

That is, the photospheric pressure is determined by the gravity and the opacity. The nearsurface opacity in stars is typically temperature and density dependent, and calculating it here would take us too far afield; but it is straightforward enough to show that the above implies photospheric pressures that are “much smaller” than the central pressures we deduced earlier. So a priori it’s more plausible that magnetic fields could play a significant dynamical role near the surface.

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We have already heard (see the lectures by Kupka, this volume) that efficient (and nonrotating, non-magnetized) convection largely tends to isotropize specific entropy. That is, the entropy per unit mass   P sm ∝ ln ργ is approximately constant throughout the convection zone. So, for example, consider a lowmass star (lower than about a third of a solar mass), which we expect to be convective throughout its interior. The constancy of specific entropy implies that Pph Pc γ ≈ γ ρc ρ ph with the subscripts c and ph denoting the values at the center of the star and at its photosphere, respectively. The point here is that the stellar properties near the surface and those in the deep interior are linked, at least over long time scales: if you change conditions near the surface (by magically changing the opacity, for example), you must change the whole star. It’s also worth noting that the central entropy is, using our results for the central pressure above and adopting a mean density of order M/R3 ,   5/3    GM 2 R3 1/3 ∝ ln GRM sm,c ∝ ln R4 M implying in turn that at fixed mass, R ∝ esm,c . That is, the radius of the star is related to its entropy. For much more discussion of this point, see for example [40]. This motivates us to understand quantitatively the entropy distribution established by convection (potentially under the influence of rotation and magnetism). So what does the entropy in a standard stellar structure model actually look like? In a fully convective star of the type we’ve considered above, and with a standard “mixing length theory” model of the convective transport, you will end up with something that is basically isentropic throughout the interior, with a steep entropy drop (that is, a stronger super-adiabatic gradient, with specific entropy decreasing outwards) near the photosphere. The exact slope of this drop – and hence the total entropy jump from photosphere to stellar center, and hence the radius of the star – depends to some extent on the model; for example, in MLT it depends directly on the value of the “mixing length” parameter α. (We will come back to this later in these notes, but you can also just see, e.g., [41] for discussion.) The discussion above implies that in order to understand whether magnetic fields (or rotation) can affect the radius of the star, we need to understand how they affect the heat transport (and hence the entropy distribution). An extreme example of this, as initially discussed in [42], would be to have fields in the interior that are so strong that the interior is actually stably stratified – that is, the magnetism effectively suppresses convection altogether – which would in turn imply larger central entropies and larger stellar radii. Fields of that strength are, however, difficult to generate and maintain – see [43] for discussion – so there is considerable interest in what fields of lower strength do to the heat transport as well. Unfortunately, there is (as yet) no universally-agreed way of incorporating the effects of magnetism (or rotation) into 1D models. Various models have tried to do so in plausible ways, for example by modeling the additional contribution to pressure arising from the magnetism or by adopting a model in which the equilibrated entropy gradient is some function of the field strength and/or geometry. See [41, 44, 45] for examples and discussions.

Multi-Dimensional Processes In Stellar Physics

One punchline from the above analyses is that if you affect the layers near the surface, the overall structure of the star must adjust. The layers near the surface are more plausibly affected by magnetism than is the deep interior – for example, as we calculated above, the pressures near the photosphere are immensely lower, so that a given fixed magnetic pressure might be expected to exert a greater influence. This local influence might conceivably change the structure of the star – but to say whether it actually does so or not, we need to be more quantitative about how strong the fields get, how they feed back on the heat transport, and so on. Summary of the first lecture

An honest “punchline” for our discussion so far is: look, to zeroth-order, the gross structure of a star is remarkably insensitive to lots of things we (i.e., people reading these notes) might care about. This includes rotation and magnetism. Indeed, as we have discussed, the observable structure of stars at any one time is even fairly insensitive to lots of physics that is arguably much more fundamental – like, say, nuclear fusion. The basic facts of a star’s existence – how bright it is, how hot, the length of time for which it shines, and so on – are set mostly by its mass, and you don’t need magnetohydrodynamics to sort them out, at least approximately. But we are living in a time when we can do much better than 0th-order models: we are flooded now with data on stars (and increasingly planets, too), at increasingly high precision, and rotation and magnetism can and do make a difference at this level. To take a concrete example: if you can only measure stellar radii to, say, an accuracy of 15 percent, you probably don’t need to care too much about magnetism; but if you are able to measure it to 1 percent (and want to understand what you’re finding), you might need to care. Because the effects of magnetism and/or rotation are bound to be fairly small perturbations on the gross structure, we really need a quantitative understanding of (for example) how strong they can be, and what they can do, rather than just a qualitative picture. We will try to develop this in the following “lectures” (represented as successive sections of these notes).

Lecture 2: Descriptive MHD, and some limiting cases in fluid mechanics Last time, we introduced some of the observations and key findings that motivate these lectures. A very short summary is that all stars rotate, all stars are convective, and all stars are magnetic – where “all” should probably have a small asterisk beside it to account for oddball cases here and there. The gross structure of a star is relatively insensitive to many of the details of these processes, but at higher precision – now available thanks to a wide variety of exquisite observations – the magnetic field and rotation rate may matter at an observable level. In this lecture, we’ll introduce some of the tools used to study these phenomena quantitatively. In particular, I’ll introduce a simple form of the equations of rotating MHD, discuss important limiting cases of these, and use them to solve a couple of simple example problems. The idea here is just to build a little bit of intuition about what magnetic fields do, and when they are likely to be important. The only equations we will solve

We seek equations for velocity, density, and in general other thermodynamic variables (temperature or entropy, say) as a function of space and time. We’ll assume these are all continuous fields – i.e., it’s a fluid, and there aren’t any weird discontinuities. In many of our

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discussions today, we’ll assume our reference frame is “locally Cartesian” – a small box on a rotating planet, say, with x eastwards, y northward, and z upward. It’s conventional to label the components of fluid velocity u = (ux , uy , uz ) such that u = (u, v, w); that is, u (no vector sign or boldface) is the component in the x direction, v the component in y, and w the component in z. When we add magnetism, we will also assume the equations of magnetohydrodynamics (MHD) apply. (In the form written here, this assumes among other things that the flow is non-relativistic; we will not always assume the fluid is a perfect conductor, but will assume that only Ohmic dissipation is important, ignoring effects like ambipolar diffusion and the Hall effect that are important in some contexts. ) What constraints do we have? Mass is conserved. Momentum is conserved. Energy might be conserved, too. Each of these will give us an equation that constrains u and/or the thermodynamic quantities. (We won’t have occasion to discuss energy conservation in this section, just momentum and mass.) Without further delay, let me just state without without proof that for a fluid rotating at constant angular velocity Ω, the Navier-Stokes equation expressing momentum conservation can (in certain circumstances that often apply in stellar interiors) be written   ∇ × B × B ∂u 1 + (u · ∇)u = − ∇P +g − 2Ω ×u + + ν∇2u (1) ∂t ρ 4πρ where the symbols take their usual meaning (e.g., ν is the kinematic viscosity, u the velocity, B the magnetic field in cgs units, ρ the density, and g the gravitational acceleration). Again, this is “just” a = F/m for a fluid: the left hand side is effectively the acceleration of a fluid parcel, and the right hand side consists of the forces acting on that parcel (here pressure gradients, gravity, the Coriolis force present in a rotating frame, the Lorentz force associated with the magnetic field, and a viscous term). In addition to momentum conservation, we also need an expression for mass conservation, which for a fully compressible fluid can be written ∂ρ + ∇ · (ρu) = 0, ∂t

(2)

which is usually called the continuity equation. If ρ is constant, this reduces to the simpler special form ∇ ·u = 0 (3) and we would say the fluid is “incompressible.” (There are many other approximations that lie somewhere in between the full, “fully compressible” continuity equation and the incompressible limit; see, e.g., [46].) For magnetism, the main result from non-relativistic MHD we will need today is the “induction equation,” which can be written as     ∂B = ∇ × v × B − ∇ × η ∇ × B , ∂t

(4)

where η is the magnetic diffusivity (units of length squared over time), η=

c2 4πσ

in cgs units, with σ the conductivity. A very good conductor has σ tending to infinity, and hence a magnetic diffusivity η tending towards zero; conversely, a poor conductor has a high

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diffusivity. If η is independent of position, we can write this in a simpler way as   ∂B = ∇ × v × B + η∇2B ∂t

(5)

which we will interpret physically below. First, though, let’s talk about some of the dynamics occurring in (rotating) fluids without magnetism. Dimensionless numbers and the momentum equation

The Navier-Stokes equation (with or without magnetism) is hard to solve, and in many realistic cases we must resort to numerical simulation. We’ll talk about that a little in the last part of these notes. But one of the big ideas in fluid dynamics is that in one circumstance or another, some terms in this equation may dominate over others; we might then be able to ignore certain terms (or at least parameterise their effects in some simple way) and end up with equations we can actually solve. The other big idea is one that runs through much of physics, namely that the “laws of physics” shouldn’t depend on what coordinate system you measure them in, or what set of units you use. One way that manifests is in dimensional analysis: e.g., when we say that an equation must be dimensionally consistent (i.e., the units have to match), we’re really insisting that there isn’t a preferred set of units or a preferred frame of reference. Another manifestation of this, and one that gets used a lot in fluids, is that many seemingly disparate problems “collapse” to the same solution when viewed in terms of certain nondimensional parameters. For example, for flow in a pipe, the individual values of velocity u, length l, and kinematic viscosity ν are less relevant than their combination as a Reynolds number, Re = ul/ν, which is dimensionless (i.e., independent of the unit system adopted). That is, different problems exhibiting different values of l, u, and ν, but having the same Reynolds number, might exhibit the same behaviour. So we’ll introduce a few of these “nondimensional” (i.e., dimensionless numbers), which can be derived by taking ratios of different terms in the momentum equation, and talk about what they mean. Then we’ll discuss some other aspects of the flow in a couple of “limiting cases,” where one term dominates over others or is balanced by a particular other term. Moving in order from the left to the right of the Navier-Stokes equation as quoted above: the first two terms together are the “advective derivative,” or the rate of change of acceleration “following the fluid.” The first piece of this (∂u/∂t) is of course just the acceleration at a specific place; the second piece, sometimes called the “inertial term” or the “nonlinear advection” term, describes how the velocity field carries around other quantities. The first term on the right-hand-side is the pressure gradient; the next is gravity, which is (importantly, for convection in stellar interiors) linked to the buoyancy of the fluid. The next term (−2Ω × u) is the Coriolis force that arises in a rotating coordinate system. The last term (ν∇2u) is the viscous term (assuming the fluid is Newtonian, incompressible, and also that its viscosity doesn’t vary from place to place). Recall that ν is the kinematic viscosity, related to the dynamic viscosity μ by ν = μ/ρ. (Frequently the viscosity isn’t independent of position, e.g. because it depends on temperature or something, so sometimes we’ll have to use a slightly more complicated version of the viscous term – but this is mostly just a mathematical annoyance.) Not all of these are important in any one problem; frequently, one or two terms dominate over the others. We can take ratios of these terms to see what’s important – and these ratios will of course be dimensionless (since the units of each term in the momentum equation must be the same for dimensional consistency). These “nondimensional” numbers can help us characterise a given fluid flow.

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Let’s suppose that U is a typical velocity scale in the flow and L is a typical length scale. (Of course in many contexts there may be a range of velocities, and also length scales. Ignore that for the moment.) Then we can form the following dimensionless variables, approximating spatial derivatives ∇n by 1/Ln : Reynolds number Re =

U·U UL inertial u · ∇u L = ∼ = U viscous ν∇2u ν ν L2

(6)

defining the Reynolds number Re; Ekman number Ek =

ν U2 ν∇2u ν viscous ∼ L = = Coriolis 2Ω ×u 2ΩU 2ΩL2

(7)

defining the Ekman number Ek; and 2

Rossby number Ro =

U inertial u · ∇u U = ∼ L = Coriolis 2Ω ×u 2ΩU 2ΩL

(8)

defining the Rossby number Ro. Qualitatively, large or small values of these numbers can tell us roughly what dynamical regime the flow is in. If the Reynolds number is high (a few hundreds or more, let’s say), then viscous forces are relatively insignificant (at least, on the scale L for which we’ve calculated the Reynolds number) and the flow is often turbulent. If Re is low, we instead have smooth “laminar” flow. Similarly, if the Rossby number is small, this implies that rotation exerts a significant influence on the dynamics (again, on this particular scale). To order of magnitude, we can understand the Rossby number as being Ro =

1 Ω L U

∼

τrot τturnover

,

(9)

i.e., the ratio of the rotation period (1/Ω) to the dynamical (or “overturning”) time of the flow (∼ L/U). On planets, the large-scale flows are often strongly influenced by rotation; that is, they have small Rossby number. Qualitatively we might say this is because slow-moving, large-scale flows have plenty of time for Coriolis forces to deflect their trajectories. This does not mean that every scale of motion is strongly affected by rotation. E.g., on Earth, hurricanes all tend to turn the same way in the northern hemisphere, and the opposite way in the southern hemisphere; but meanwhile the water in your toilet doesn’t much care which hemisphere you’re in. The hurricane (large L, and a slow overturning time) has a low Rossby number; the water swirling in your toilet bowl (much smaller L, and much faster dynamical time) doesn’t. (Put another way, on the scale of your toilet a bunch of other stuff matters more than the rotation of the Earth.) The Ekman number similarly quantifies rotation relative to viscosity; small Ekman number means that rotation has a bigger influence than viscosity. One other point here: as a really big generalisation, true viscous effects on large-scale flows are generally unimportant far from any physical boundaries. (Sometimes smaller-scale flows can “mimic” the effects of viscosity in a way, but this is conceptually distinct from the effects of real, microscopic-level viscosity.) One way of saying this is that the Reynolds number of many large-scale flows in astrophysics is really big – e.g., for the convection zone in the Sun it might be 1012 or more. But a corollary is that there is almost always some region, e.g., close to a boundary, and/or on very small spatial scales, where the Reynolds number is not large and viscous effects are important. In everyday fluid dynamics, we might separate this “boundary layer” flow, in which viscosity is extremely important, from the larger-scale

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flow with high Re (which might be treated as an essentially inviscid flow). (To pick one practical example: look at a ceiling fan sometime. Even if you use the fan a lot, you will find that the surface of the fan is dusty, unless you’ve cleaned it recently. The point here is that the velocity field around the fan transitions smoothly to a “no-slip” condition at the fan’s surface – meaning that in a thin boundary layer near the blades of the fan, the air doesn’t move much and the dust is undisturbed.) Dimensionless numbers and the induction equation

Let’s spare a moment to do the same kind of analysis for the induction equation as given above. Consider first the case where there is no flow at all (v = 0), in which case the induction equation is just the diffusion equation:   ∂B = ∇ × v × B + η∇2B = η∇2B. ∂t

(10)

That is, a field initially present will diffuse away on some characteristic timescale. To order of magnitude, if there is only one spatial scale L present, this characteristic time τ satisfies B L2 B ∼η 2 →τ∼ . τ L η

(11)

You will probably recognise this “magnetic diffusion time” as being of the same form (that is, length squared over a diffusivity) as the more familiar thermal diffusivity (which, among many other useful things, provides a nice estimate of how long it takes to cook a steak). If the conductivity σ is high (so that the diffusivity η = c2 /(4πσ) is small) then the practical implication is that fields on large length scales can survive at observable strengths for a long time, even if they are not actively maintained. Now let’s allow for there to be a non-zero velocity, and consider the ratio of the induction term (which is trying to build fields) to the diffusive one (which is trying to make them decay away). This is   UB ∇ × v × B UL ≡ Rm (12) ∼ LB ∼ 2  η η 2 η∇ B L

defining the magnetic Reynolds number Rm. Hand-wavily, if Rm is large enough, we might expect induction to be able to “win” over diffusion, and give us a growing field. (I should hasten to say, though, that it is not as simple as just making η low – only certain types of flow will lead to growing fields, as we will explore in a moment.) As a very crude rule of thumb, a person-sized thing moving at a person-sized pace (so, L ∼ 1 m and u ∼ 1 meter per second) has Rm ∼ 1 at a temperature of about 106 K (in a fully ionized plasma). (What the person is doing walking around casually in a million-degree plasma will remain unanswered today.) In astrophysics, Rm is often big mainly because the relevant length scale L is big. The Galaxy as a whole might be regarded as having Rm ∼ 1021 , the Solar convection zone of order 1010 or more, and so forth. A corollary to this, though, is that if the length scale under consideration is small, Rm can be of order unity (or less) for any system: that is, on small enough scales diffusion is always important. To put some concrete numbers on this that are relevant for the Sun, consider a portion of the solar core where η ≈ 2.3 × 102 cm2 s−1 : in one second, the field would diffuse about √ η ≈ 15 cm; the Ohmic diffusion time for a field on very large scales would be (for L = 2 × 1010 cm) of order τ ∼ (2 × 1010 )2 /η ∼ 1018 seconds, or a few times 1010 years. The practical upshot is that a relic field in the solar core might survive for a very long time (billions

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of years) even if not actively maintained. (The catch is that this calculation assumes L remains big; if, for example, flows act to structure the field on smaller scales, this will change the rate on which the magnetism diffuses.) Equations for a thin layer, and geostrophic balance

Let’s move on to a particularly useful “limiting case” of the momentum equation, relevant for stars or planets (or portions thereof) that are rotating “rapidly,” in the sense that rotation is important relative to both inertia and to viscosity. What will the resulting dynamical balance look like? Let’s work this out for the specific case of a thin, Cartesian layer on a rotating sphere – that is, something like a local patch of our atmosphere at a given (fixed) latitude. (We’ll develop the equations for this thin, rotating layer first, and then simplify them for the case where rotation dominates over inertial forces and viscosity.) Consider a reference frame that is “locally Cartesian” – a small box on a rotating planet, say, with x eastwards, y northward, and z upward, and u = (ux , uy , uz ) = (u, v, w). Let’s also assume our region of interest is thin, in the sense that the vertical scale is small in comparison to the horizontal scale. (This is really motivated by consideration of planetary atmospheres, where the approximation of a thin atmosphere is often really good; but we’ll assume it here too just to illustrate things.) The continuity equation for incompressible flow (∇ ·u = 0) then suggests that we expect vertical motions to be small as a result. Replacing derivatives with d/dz ∼ 1/H, d/dx ∼ d/dy ∼ 1/L, with H the characteristic z lengthscale and L the horizontal lengthscale, we have U W W H ∼ → ∼ 1 L H U L

(13)

which we will use to simplify things in a minute. The rotation rate Ω has components Ωx = 0, Ωy = Ω cos θ, Ωz = Ω sin θ, with θ the latitude of our Cartesian box. If we a) work out the Coriolis term 2Ω ×u, b) assume w is much less than v (as suggested by the continuity equation), c) define f = 2Ω sin θ to be the “Coriolis parameter” (or “planetary vorticity”), and d) assume that the vertical Coriolis force is much less than gravity, we are left with the following x, y, and z components of the momentum equation:  2  ∂ u ∂2 u ∂2 u Du −1 ∂p − fv = +ν + + (14) Dt ρ0 ∂x ∂x2 ∂y2 ∂z2  2  ∂ v ∂2 v ∂2 v −1 ∂p Dv + + + fu = +ν (15) Dt ρ0 ∂y ∂x2 ∂y2 ∂z2  2  ∂ w ∂2 w ∂2 w Dw −1 ∂p gρ +ν + 2 + 2 . = − (16) Dt ρ0 ∂z ρ0 ∂x2 ∂y ∂z Here I’ve allowed for the possibility of small variations around the mean density ρ0 . Note that gravity appears only in the z direction; the balance without any motion is just hydrostatic equilibrium. Now suppose that the Rossby number and Ekman number are both small, and that the flow is steady, so that we can drop the D/Dt term from each equation (Ro is small) and the viscous terms as well (Ek is small). The only things left in the horizontal direction are the Coriolis term and the pressure gradient: − fv =

−1 ∂p ρ0 ∂x

(17)

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and

−1 ∂p . (18) ρ0 ∂y (In the z direction, we are just left with hydrostatic equilibrium.) This state, in which horizontal pressure gradients are balancing Coriolis forces, is called geostrophic balance. This is often realised in planets and stars. What is the flow in this geostrophic state? You might expect naively that flow would be from regions of high pressure to low – but clearly the Coriolis force will muck with this at some level. To examine this quantitatively, consider the relation between the flow and contours of constant pressure:     −1 ∂P ∂P 1 ∂P ∂P (uex + vey ) · ∇P = ex + ey · ex + ey . (19) f ρ0 ∂y f ρ0 ∂x ∂x ∂y fu =

Substituting in from our geostrophic balance equations, and using the equality of mixed partial derivatives, gives that the above dot product is zero! The flow is along, not across, lines of constant pressure: that is, the flow is “along isobars.” The flow is counter-clockwise around a low-pressure region and clockwise around a high-pressure one. The bottom line here is that we can understand a fair bit about the overall dynamics of the flow – like the fact that it spirals a certain way around regions of low pressure in the northern hemisphere – without having to solve the full Navier-Stokes equations in all their glory. Knowing that the main dynamical players are rotation and pressure allowed us to simplify the problem enough to get a solution. Descriptive MHD

We return now to problems including magnetism. Before solving any real equations, it’s useful to have a physical picture of what magnetic fields “like to do” in a conducting fluid. To that end, let me state without proof that magnetic fields in a perfectly conducting fluid behave much like a set of infinitely long elastic strings (field lines), which obey certain rules. These strings can form loops, but can’t end; the orientation of the strings defines the orientation of the field; the density of the strings (that is, the number per unit area threading a locally perpendicular surface) is proportional to the strength of the field. The strings are carried along with the field as if they were frozen to it, but are unchanged by motions parallel to the strings. The energy density of the magnetic field is B2 /8π (cgs), and we will define the tension τ of a “bundle” of such strings (with radius a) to be aB2 /4π. Given this physical picture, what happens if we “pluck” a bundle of magnetic field lines? How does the field respond? Recall that if a mass m were attached to a string pinned between (say) −L and L, and I displaced the mass vertically, the tension in the string would resist this; if I let the mass go, and if the displacement were small, the resulting motion would be oscillatory. The equation of motion for small vertical oscillations would be d 2 (Δy) −2τ Δy (20) = dt 2 m L (using the small-angle approximation), for which the solution is simple harmonic motion with characteristic frequency 2τ ω2 = . (21) mL Here, plugging in our expression for the tension in a flux “bundle” from above, we have   2 B2 a 2 ω = . (22) mL 4π

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If we suppose the mass is distributed uniformly along the string instead of concentrated at one point, and define the average density ρ = m/(2aL), then the above implies ω2 =

 v 2 B2 a = 4πρL2 L

(23)

where we have defined the Alfven velocity B . va = √ 4πρ

(24)

This is the characteristic speed at which signals driven solely by magnetic tension – that is, the bending of fieldlines – propagate. Numerically, this turns out to be va ∼ 2 × 1011

B n1/2

(25)

in cm/s, if the number density n is in cm−3 , B is in Gauss, and the plasma is 90 percent H and 10 percent He. The fieldlines (i.e., our bundles of elastic strings) don’t just resist being “plucked” or bent; they also resist being squeezed together uniformly. This is called the “magnetic pressure,” Pm = B2 /(8π). The Alfven speed is also the characteristic speed for signals driven by magnetic pressure. (Signals which have both magnetic and gas pressure as restoring forces can  propagate at the magnetosonic speed, cms = c2s + v2a .) Just to provide some brief mathematical reassurances that all this stuff about magnetic “pressure” and “tension” is legitimate, note that the actual Lorentz force term in the momentum equation can be decomposed into “tension” and pressure components:    2 ∇ × B × B B · ∇B B = −∇ . (26) 4π 4π 8π The first term on the left-hand-side is the “tension”; the second the “pressure.” It’s really useful in constructing order-of-magnitude estimates to have these ideas in your head – but note that the analogy with tension and pressure isn’t perfect. (In particular, the full Lorentz force term is always perpendicular to B, while the tension and pressure terms in the above equation may not be; the tension and pressure components along B must ultimately cancel.) Example: shaking of coronal loops

Merely knowing the characteristic timescales and energies associated with the magnetism can often allow you to figure out whether or not magnetic fields are likely to be important in a given problem. As an example, consider the shaking of magnetic fieldlines that thread through the Solar photosphere, extend up into the corona, and come back down again. The average loop is say 109 cm in length, has a number density n ∼ 109 cm−3 , and has B ∼ 100 G. The footpoint of the loop is being dragged around by convection, which has a characteristic timescale of say 10 minutes. The question is, does this convective dragging lead to waves within the coronal loop? We can estimate the Alfven velocity in the loop to be of order 2 × 1013 /104.5 , or around 6700 km/s, implying that signals propagate along the loop in about 2/3 of a second. This is different enough from the ∼ 600 second timescale associated with the convection that we don’t expect wave-like behaviour: instead, the loop can just quickly adjust to the footpoint motion. In contrast, a flare – with timescale of 0.1 seconds or so – might drive waves in the loop, because this driving corresponds much more closely to the characteristic Alfven time.

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Example: magnetospheres and accretion onto a magnetized star

Consider a star with mass M, radius R, and an initially dipolar magnetic field, embedded in a spherically symmetric accretion flow. What happens? First, let’s do a hand-wavy analysis using our ideas about magnetic pressure, tension, and so forth, and then we’ll try to analyze the situation more formally. Intuitively, we expect the accretion will compact the magnetosphere to some degree. We’d anticipate that if the field is really strong, it will more easily resist this; if it’s weak (compared to the infalling material), it will be crushed. We can quantify this by looking at the magnetic pressure and at the pressure of the infalling material. The magnetic pressure is Pm = B2 /8π, and we’ve assumed a dipolar field Br = 2M cos θ/r3 , Bθ = M sin θ/r3 , which integrated over latitude implies Pm =

M2 . 4πr6

(27)

Meanwhile the dynamic pressure of the accretion flow is Pw = ρw v2w . If we assume the ac˙ then creting material is in free-fall, and the accretion rate (mass per unit time) is M,  vw =

2GM r

1/2 (28)

and the density in the wind is related to M˙ by ρ=

M˙ . 4πr2 vw

(29)

From looking at the form of the accretion and magnetic pressures, it’s clear that at large distances the accretion dominates (i.e., will “win” over magnetic effects); at small enough r the magnetic pressure should dominate. The two are equal at the “magnetospheric radius” Rm =

M2 1/2 ˙ M(2GM)

2/7 (30)

which gives an estimate of the size of the compacted magnetosphere of the star. Numerically, for say a neutron star with R ∼ 10 km, M a solar mass, B ∼ 109 G, accreting at 0.1 times the Eddington rate ∼ 10−9 solar masses yr−1 , we end up with Rm of about 10 times the star’s radius. You could do exactly the same sort of calculation for the interaction of Earth’s magnetosphere with the dynamic pressure of the solar wind, and so derive a decent estimate of the size of Earth’s magnetosphere. (Try it!) Now let’s try to analyze the same kind of situation a bit more rigorously. Let’s work in Cartesian geometry for simplicity, and suppose a vertical accretion flow vz and an initially horizontal magnetic field Bx . The overall accretion rate (which we’ll now define to be mass ˙ The induction equation is per unit area per unit time) is M.     ∂B = ∇ × v × B − ∇ × η ∇ × B , ∂t

(31)

and for the simple geometry assumed here we have v × B = vz Bx ey

(32)

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  ∂ ∂Bx  ∇ × (η∇ × B) = η ∂z ∂z

(33)



∂ ∂Bx ∂Bx = −vz Bx + η =0 ∂t ∂z ∂z

(34)

which together imply that

in a steady state. Integrate once to find η

∂Bx − vz Bx = K ∂z

(35)

with K some constant; this constant must be zero if B = 0 at infinity. Hence we have ∂Bx vz = Bx ∂z η which has the solution

(36)

 vz

Bx (z) = Bx (0)e

η

dz

.

(37)

To interpret these results physically, note that vz is related to the accretion rate, vz = −m/ρ, ˙ and let Σ be the surface density of matter that has already piled up, so that we can define a ˙ characteristic “accretion time” needed to accumulate Σ at rate M, tacc =

Σ . M˙

(38)

The characteristic height H of the accreted material will be defined by ρH = Σ, and the magnetic diffusion time across this height will be of order H2 . η

(39)

−Σ −M˙ −H = = ρ ρtacc tacc

(40)

tη = So we have vz = and hence

−H tη −Bx tη ∂Bx = Bx = . 2 ∂z tacc H H tacc

(41)

That is, there is a critical accretion rate that will bury the field, much as in our more intuitive estimate above. So the “punchlines” for this section are basically as follows: The interiors of stars are generally well-described by the MHD equations – but these equations are often very difficult to solve in a general setting. So we frequently seek limiting cases in which one term dominates over another, or when some effects can be ignored altogether, possibly allowing us to find analytical or semi-analytical solutions without going crazy. The example we used for hydrodynamic flow was geostrophic balance, which often applies when rotation is strong. Another example is that of “ideal MHD,” which applies when the conductivity of the medium is infinite – and in this limit the field behaves as a collection of elastic strings that are “frozen in” to the fluid. Understanding the behaviour of fields in this intuitive limit can take you a long way! We’ll explore more of this in the following section.

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Lecture 3: an introduction to dynamo theory Today, we’ll talk about how the magnetic fields we observe in stars are built. In most cases, the fields result from a dynamo: a process by which flows can amplify seed magnetic fields and maintain them against Ohmic decay, essentially by trading off kinetic energy to build magnetic energy. In essence, this process relies on the fact that a moving conductor in a magnetic field generates currents (Faraday’s law); meanwhile currents (i.e., moving electrons) lead to magnetic fields (Ampere’s law). For suitable flows, it is possible for the motion through some weak field to lead to currents that amplify that initial field. This may sound like pulling yourself up by your own bootstraps, and indeed for many years it was not clear that a real self-excited fluid dynamo could exist; we will show below that many particularly simple flows definitely won’t work. But some other flows do work, and we’ll explore today what properties those flows need to have, and what kinds of fields they build. The minimum Rm for growth

First, the good news: we’re only basically going to be working with one equation! That’s the induction equation,   ∂B (42) = ∇ × v × B + η∇2B ∂t written here in the form appropriate when the magnetic diffusivity η doesn’t vary spatially. Recall that we defined the magnetic Reynolds number Rm = UL/η by taking the ratio of the two terms on the left-hand-side of this equation; qualitatively, this measures induction relative to magnetic dissipation. We would reasonably expect we need Rm greater than unity in order for induction to “win,” and so lead to growing fields, but by how much? It’s worth noting that the problem is not just as simple as lowering η. You can sort of see why by trying to estimate the size of the diffusive term, η∇2 B ∼ ηB/L2 – but what’s the suitable L here? Presumably it is some length scale over which B is varying, but the trouble is that this gets smaller and smaller as η gets smaller: that is, the field gets structured on very fine scales in the high conductivity limit, and the diffusion term (scaling like one over this length scale, squared) can get very large. So determining whether the flow can “win” is a matter of some subtlety. (For more on the mathematical problem of dynamo action, the monograph Stretch, Twist, Fold: the Fast Dynamo, by Childress and Gilbert [47], is a good place to start. Also, note that much of what I talk about today is covered at more length in the Les Houches notes on dynamo theory by [2].) In any case, let’s try to give a more concrete estimate of the minimum Rm needed for dynamo action. We can form an evolution equation for the magnetic energy by dotting the induction equation with B; in SI units this looks like d dt



B2 = 2μ



   j · u × B dV − μη j2 dV

(43)

where η = 1/(μσ) now. The first term on the RHS is the work done; the second arises from dissipation. This can be recast as μ

dEm = −η dt



(∇ × B)2 dV +



  (∇ × B) · u × B dV

(44)

defining the magnetic energy Em . Recall that for any vectors a,b,c, a ·b ×c ≤ |a||b||c|

(45)

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so we have that 1/2  1/2    (∇ × B) · u × B ≤ umax |∇ × B|2 dV |B2 |dV .

(46)

It is also possible to prove that for divergence-free fields in a sphere of radius a, matching to a potential field outside, that 

|∇ × B|2 dV ≥

π2 a2



|B|2 dV .

(47)

Intuitively, this last equation makes a bit of sense: the current (in cgs, j ∼ c∇ × B/4π) is of order cB/l, for some length scale l; the lowest possible current associated with a given B is for the maximum possible l, here the overall radius a: j ∼ cB/a. Putting these together, we find that   2   au max ∇ × B dV (48) (∇ × B) · u × B ≤ π and hence that   2 dEm  aumax μ ≤ −η (49) ∇ × B dV . dt π Thus, a growing dynamo (positive growth rate of Em ) requires aumax ≥ π, η

(50)

which you will recognise as a sort of magnetic Reynolds number with length scale L equal to the overall radius a, and u equal to umax , the maximum velocity within the domain. For any possible dynamo, this Rm needs to be at least π (rather than unity as we first guessed). This was a lot of work for a factor of pi! Stretching and amplification

How do you amplify a field in MHD? What does this imply for our intuitive picture of fieldlines as collections of elastic strings? Imagine a single flux tube (i.e., bundle of field lines, pointed in the same way) having some initial length L0 and initial area A0 , stretched from L0 to some new length L . Let us suppose the fluid is incompressible, so mass conservation implies that ρV is constant, and hence that ρL0 A0 = ρL A , and thus that A0 L = . L0 A

(51)

That is, the cross-sectional area of the tube diminishes in proportion to the stretching. Now suppose we are in the “ideal” MHD limit of low η, so the magnetic field is “frozen in” to the fluid; in this limit the magnetic flux Φ = B0 A0 is conserved (I have not proven this, but any textbook on this subject would do so), so that B0 A0 = B A . Thus A0 B L B =  → = . B0 A B0 L0

(52)

That is, if you stretch fieldlines, you amplify the field. To take a specific example, suppose that we have an initially vertical fieldline (By ), passing through a series of anchor-points separated from each other by a vertical distance d. At each

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later instant in time, suppose that the anchor-points move horizontally in a random fashion, either to the left or to the right; this action will leave By unchanged but give the field a Bx component. (To be specific, between points j and j + 1, say, we will have Bx /By = (x j+1 − x j )/(y j+1 − y j ) = (x j+1 − x j )/d.) If the motions of the anchor-points are truly random, the fieldline will zig and zag vertically, and the displacement between points will grow as a random walk, with length proportional to the square root of the number of steps. If all √ the anchor-points move once per unit time, say, then this means the displacement grows as t. Per our discussion above, this means the field strength must grow in the same fashion, and hence that the magnetic energy B2 ∝ (t/τ)B20 at late times, with τ the time interval between steps. So that’s one way to amplify fields. One bit of trouble is that this kind of random walk would typically lead to a very tangled field, and in many astrophysical contexts this doesn’t seem to be what we see. E.g., in our own Galaxy, the current field strength is of order 10−6 G; if it started out at 10−17 G as suggested by theory, then this implies stretching of fieldlines (and a consequent degree of tangling) by a factor of 1011 ! Another example is the Solar magnetic field, which clearly exhibits large-scale order (the butterfly diagram, the 11-year cycle, etc) rather than just a mess of small-scale fields. Still, the stretching of fieldlines is an important piece of the puzzle. We saw in the first lecture/section that the Sun rotates differentially – portions of its interior rotate at different rates – so it’s natural to ask whether this is enough to lead to dynamo action. The differential rotation will certainly stretch fieldlines, and could amplify an initially weak magnetic field for a time – but this turns out not to be enough to maintain a self-sustaining dynamo, as we will see below. (It is, however, an important ingredient in theories of the Solar dynamo and in stellar dynamos more generally.) Is stretching enough? A brief survey of anti-dynamo theorems

It turns out that not just any flow will work as a proper dynamo, even if it is capable of fieldline-stretching for some finite time. A hand-wavy summary of a lot of hard work is that you “need 3D” in some fashion, either in the flow or in the field (or often both). It is possible to prove various “anti-dynamo” theorems that rule out self-sustaining dynamo action for certain simple types of flows and fields. We’ll do this for one particular case and then quote some more general results. Here’s the claim: if we consider Cartesian coordinates, no field independent of z that vanishes at infinity can be maintained by dynamo action. (That is, it is impossible to have a 2D field.) To prove this, suppose we decompose the field into horizontal and vertical components, with the former relatable to a vector potential, like so: B = B h + Bzez = Bzez + ∇ × Aez .

(53)

If the field is 2D, so is the flow (think of what the fieldlines are doing), so similarly u = uh + uzez = uzez + ∇ × φez

(54)

where φ is a streamfunction. The components of the induction equation can then be written ∂A + ( uh · ∇) A = η∇2 A ∂t

(55)

∂Bz h · ∇uz . + ( uh · ∇) Bz = η∇2 Bz + B ∂t

(56)

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Now multiply the first of these by A, and integrate over volume, to find ∂ ∂t



1 2 A dV + 2



1 ∇ · uA2 dV = −η 2



(∇A)2 dV .

(57)

So far, perhaps not terribly illuminating. But notice now that the ∇ · ... term on the LHS vanishes, because we can convert it to a surface integral that vanishes at infinity (given our statement of the problem). The term on the right-hand-side is negative-definite. Together, these imply that the integral over A2 must continually decay (i.e., d/dt of this term is less than zero). Once A is zero, Bh is also zero – so from that point on, there is no source term in the second induction equation above, so Bz must decay, too. (Note, though, that it could get large before this happens, while Bh · ∇uz is non-zero. But it must decay eventually.) This is just one example. Other versions of the same kind of argument give similar “antidynamo” results, e.g.: No dynamo can be maintained by a planar flow (ux , uy , 0). An axisymmetric magnetic field (vanishing at infinity) can’t be maintained by dynamo action (often called “Cowling’s theorem”). No purely toroidal flow can maintain a dynamo. (That last one is important for understanding why, for example, though differential rotation almost certainly plays a part in the Solar dynamo, it can’t be the only player – you need a “poloidal” flow to help close the dynamo loop, too.) Qualitative building blocks of dynamo action

So if a purely toroidal flow, or a purely planar one, or lots of other simple flows don’t actually end up working as a dynamo – what does? And what kinds of fields can these flows produce? We will turn qualitative for a minute, and describe some of the “building blocks” that have been identified (by either basic theory or through simulation) as useful or necessary for dynamo action. Much more comprehensive descriptions of dynamo theory can be found in the classic book by Moffatt [48], or in [2, 49, 50]. A very crude summary of dynamo theory is this: building magnetic energy is in some sense straightforward. A sufficiently complex flow in a sufficiently conducting medium will probably work – where I am going to be vague about what exactly I mean by “complex,” because we don’t really have time to discuss this in detail. Hand-wavily, you can imagine that a chaotic flow, with particle trajectories diverging rapidly away from some initial location, might be good candidates for providing lots of constructive field-line stretching – and this expectation is largely correct. Stars are intensely turbulent places, and thus it’s reasonable to expect that the convection alone would act as a good small-scale dynamo. (By “smallscale” we mean something on the scale of the convective eddies themselves, or smaller; we’ll contrast that with a “large-scale” or “system-scale” dynamo that generates fields on the largest size scales available, e.g. the overall stellar radius.) But to build an ordered field – ordered either in space (so that, say, there is a net nontrivial magnetic flux through some surface) or in time (with, e.g., magnetic cycles) – you need something more than just small-scale fieldline stretching. Some symmetry-breaking mechanism is essential, and in astrophysical settings this can typically be provided by rotation. This rotation can impart a handedness to the chaotic convection and at least allow for the possibility of large-scale field generation. In many models of solar/stellar/planetary dynamos, it is this convection, coupled with rotation and acting in concert with shear (i.e., differential rotation), that builds the fields. To see how this might work, it’s useful to define “toroidal” and “poloidal” fields – for our “cartoon” purposes in these notes, and as a sort of mnemonic, you can imagine toroidal fields as being ones that look like a torus wrapped around the equator (i.e., they are mostly Bφ ,

Multi-Dimensional Processes In Stellar Physics

oriented longitudinally), and poloidal fields as being ones that poke through radial surfaces around the poles. (Beware: really, the toroidal fields are generally defined as ones that don’t puncture a radial surface; poloidal fields do. So by this definition, it’s perfectly possible to have a “toroidal” field that is oriented to be Bθ rather than Bφ .) To avoid the various antidynamo theorems stated above, you really need your dynamo process to build both poloidal and toroidal fields, not just one individually; that is, you want a process that generates toroidal fields from poloidal and vice versa. With this in mind, a classic picture for the operation of a (stellar or planetary) convective dynamo (essentially that of Parker [51]) invokes the cyclonic convection as the way of building poloidal fields from toroidal, and differential rotation as the primary means of building toroidal fields from poloidal. The latter step is easier to understand: suppose I drape one of our elastic strings from pole to pole (i.e., oriented so as to have only Br and Bθ components), and suppose also that different latitudes rotate at different rates (i.e., there is differential rotation). If, say, the equator rotates faster than the poles, then the string (i.e., fieldline) will be stretched at the equator, and re-oriented to have a Bφ component as well. This is the poloidal-toroidal step – but again, by itself this is not enough. To construct poloidal fields again (from toroidal), imagine our elastic string circling the star at the equator, and picture it (as Parker did [51]) being carried up or down by a convective eddy. As that fluid parcel rises, the Coriolis force imparts a “twist” to the fieldline, so that what starts out as a purely toroidal field acquires a poloidal component as well. (This is a bit hard to describe in words, but easy to demonstrate for yourself with a belt.) Some of that new poloidal field can be stretched again by differential rotation, etc, so that we have a complete dynamo loop. The “cyclonic convection” piece of this loop is often called the α-effect, and the “stretching by shear” bit is usually called the Ω-effect, after their parameterisations in mean-field dynamo theory (described briefly below). Other “building blocks” may also figure prominently in the generation of fields. One that gets discussed a lot is the “Babcock-Leighton” effect [52], [53, 54], which basically goes like this: in the Sun, active regions emerge with a systematic equatorward tilt. They decay over some timescale, and magnetic flux is transported poleward (by meridional circulation). If the active regions arise from sub-surface toroidal fields, their systematic tilt implies a poloidal component as well – and when the active region breaks up, the transport of this poloidal flux poleward (and downwards, after that) may provide a “seed” for the next phase of the solar cycle. That is, there is again conversion from poloidal to toroidal field by shear, but now the conversion from toroidal back to poloidal is dominated by this process (the Babcock-Leighton effect) rather than cyclonic convection. There is a whole class of dynamo models, called Babcock-Leighton flux transport dynamos, in which the principal agents are a near-surface Babcock-Leighton effect, a sort of “conveyor belt” provided by meridional circulation, and shear (which is after all observed by helioseismology). How all these building blocks interact to build the observed fields in stars and planets is not entirely clear, but personally I find it plausible that they all play a role at some level. In the Sun specifically, the flux associated with the Babcock-Leighton effect is surely significant (see [55]). But equally, the observed correlation between surface magnetic activity and nearsurface convection in other stars is a pretty big clue that convection is involved at some level – though in principle its role could be somewhat indirect, e.g. through generation of shear and by providing a turbulent diffusion process to break up active regions near the surface, rather than as a direct source of induction distributed throughout the convection zone. (Turning to theory and simulation, which we will do later, it’s also worth noting that simulations that don’t include the Babcock-Leighton effect at all still manage to look fairly solar-like in some important respects: that is, if you simulate convection in a rotating spherical domain, it can quite naturally produce differential rotation, which in certain parameter regimes interacts with

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the convection to build cyclical large-scale magnetism. These simulations come with lots of caveats – in particular, they operate in parameter regimes very remote from those of real stars – but I do think it would be a remarkable coincidence if they were producing sensible results for completely incorrect reasons.) Likewise, purely from observations it’s clear that rotation is important in the dynamo process. Ultimately, to sort out the roles each of these processes plays, and to build a quantitative model of stellar magnetism, we need more than just a cartoon picture. Most work on the subject has proceeded by one of two main routes: mean-field dynamo theory or numerical simulation. We will chat briefly about both. Mean-field dynamo theory and the link to helicity

The basic idea of mean-field theory (MFT) is this: first, we expand quantities (like the field or flow) into suitable means and fluctuations around those means; next, we write down evolution equations for the mean field (which is often what we are particularly interested in) in terms of both the large-scale flows and properties of the small-scale (fluctuating) fields and flows; finally, we model the statistical properties of those small-scale fields and flows in terms of the large-scale fields and/or flows, so that the evolution equations for those large-scale quantities can be solved. In equations, this means assuming that B = B + b , with b = 0 and  denoting some suitable mean operator; the evolution equation for the mean field then looks like ∂B = ∇ × (v × B + E − η∇ × B ) (58) ∂t where E = v ×b is the “turbulent emf” (electromotive force) associated with induction by the small scale flows. We expand this emf in a linear series involving the large-scale field B and its derivatives, and hope this converges rapidly enough to be useful; the simplest version of this is of the form Ei = αi j B j , with α encapsulating the famous “alpha-effect,” which has something to do with the properties of the small-scale flows. Substitution into our large-scale induction equation then gives an eigenvalue problem of the form ∂B = ∇ × (V × B + αB − (η + β)∇ × B ) ∂t

(59)

which can be solved provided you have formulae for α and β. What do the formulae for α, β, and so on typically look like? How do they relate to basic properties of the flow? It turns out that in both MFT and in the results of numerical simulations, the kinetic helicity of the turbulent convection, defined as k = u · ∇ ×u

(60)

frequently emerges as an important quantity in the dynamo process. Before diving deeply into MFT or the simulations, let’s consider one more analytical problem that serves to illustrate why this might be so. (This example is drawn from Moffatt 1978.) Consider the induction provided by a single helical wave, acting on an initially uniform 0 (and adopting the incompressible approximation that ∇ ·u = 0). The induction mean field B equation for the small-scale field is   ∂b 0 + η∇2b (61) = ∇ × u × B ∂t or ∂b − η∇2b = B0 · ∇u. (62) ∂t

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Take u = u0 (sin (kz − ωt), cos (kz − ωt), 0) (the real part of u0 eik·x−ωt with u0 = (−i, 1, 0), k = (0, 0, k)). The solution is then b =

0 ·k iB u0 −iω + ηk2

(63)

and so the emf is

E = u ×b =

  0 ·k −ηu20 k2 B ω2 + η2 k4

(0, 0, 1).

(64)

So indeed, we have ended up with Ei = αi j B0 j with α i j = α3 =

−ηu20 k3 . ω2 + η2 k4

(65)

There are some important bits worth noting about this solution for α. • It’s not isotropic (i.e., there is a preferred direction). • It goes to zero as η → 0. Evidently diffusion induces a phase lag between u and b that is essential for generating our α-effect. • When constructing the MFT equations, you have to figure out what to do with the so-called “pain in the neck term” G = u × b − u × b : e.g. neglect it, assume some closure for it, etc. Here, this term is exactly zero. (The “first order smoothing approximation,” which you will run into if you dive into the MFT literature, is exact for motions with only one Fourier component like this one.) • Note that the wave had vorticity and helicity: here the vorticity ω = ∇ ×u = ku, and the kinetic helicity u · ∇ ×u = ku20 . Some of these results turn out to be relevant to much more complex flow fields as well. In particular, the link between the α-effect and the helicity of the flow runs deep (though in general, it is not the only thing that matters for determining the value of α). With this example under our belt, let’s return to some very brief comments on meanfield models in general. (You can find more comments on the subject in the other references noted above.) Typically, the way you’d proceed to construct a mean-field model is to make some simplifying assumptions about the nature of the turbulent flow field (e.g., “first order smoothing”), which then allow you to calculate α and higher-order terms in the expansion (which are typically a function of the kinetic helicity, current helicity, etc), and hence to solve for the evolution of the mean field. These models are extremely successful, in the specific sense that when these assumptions are made, the resulting solutions for the mean field can (in certain parameter regimes) look much like the Solar magnetic field, with (e.g.) cyclical fields that propagate towards the equator, exhibit long-term modulation, and so forth. There are, however, a few issues. Some of these are fairly fundamental, and they are part of why mean-field theory is – though undeniably a powerful tool for building intuition about the dynamo process – not (in my personal opinion) the end of the quest for a proper, predictive dynamo theory. One issue involves the relevance of MFT at high magnetic Reynolds numbers, which are usually attained in astrophysical settings: in this limit, many of the simplifying assumptions made in typical MFT models do not apply, and it is not obvious why the MFT solution should have direct relevance. A central assumption of the theory is that the small-scale field b is linked to the large-scale field B by the fluctuating velocity field; that is, the small-scale field vanishes if the large-scale field does. This need not be true at high

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Rm, when small-scale dynamo action may build plenty of b on the eddy-scale whether or not there’s a large-scale field. A central issue is whether the large-scale field generation encapsulated by MFT really dominates over the field generation by all the small-scale stuff (which, in general, may still have some non-zero spatial average that doesn’t behave much like the MFT mean field) – naively, because the turnover time on small scales is much faster than that on large scales, you might expect this small-scale dynamo to “win.” (See, e.g., discussions in [56], [57].) Another major issue involves sorting out when exactly the field will stop growing: i.e., at what field strength does the dynamo equilibrate? Our discussion above hasn’t touched on this. Physically, you might anticipate that something will change when the energy density of the growing magnetism becomes comparable to that of the flows building the fields (that is, when the flows and fields are in “equipartition”), since we might reasonably expect there to then be strong Lorentz-force feedbacks on the flows. But even if that basic idea is correct (and it’s not entirely clear that it is; see, e.g., discussion in [43]), the flow has components on many scales and of many strengths – and in particular, if the mean-field dynamo process stops when the small-scale field reaches equipartition, this would typically imply extremely small mean fields. In MFT this needs to be modeled in some fashion, e.g. by assuming the alpha effect is “quenched” at strong enough field strengths (and/or by assuming the growing fields also impact the zonal flows giving rise to the Omega effect), in order to give an equilibrated solution. Ultimately, these kinds of uncertainties are part of why many of us have turned to numerical simulations of the dynamo process, as a complement to analytical or semi-analytical theory. These simulations have many issues of their own, and it is frequently difficult to sort out just which aspects are astrophysically relevant – but they do provide some guidance about the types of magnetic fields that can be generated when convection and rotation interact in different regimes.

Closing remarks: simulations, open issues, and a biased summary We began these lectures with a question – how is a star or planet’s life shaped by magnetism and rotation? – that we haven’t really attempted to answer yet. From observations, we’ve seen that many stars show measurable surface magnetism, and that the fields are observably linked to convection and rotation; from theory, we have learned that the fields may be built by dynamo action, influenced by rotation, and can react back on fluid flows in interesting ways. To understand whether the fields have any measurable impact on the structure or evolution of a star or planet, we need quantitative predictions for how strong the fields are, and what they do to heat transport, mixing, and so forth. Such quantitative predictions are difficult to come by. In lectures 2 and 3 we introduced the basic equations that govern these phenomena – but also saw that in many cases these are too formidable to be solved analytically in a fully general way. To cope with this, we might choose to analyze limiting cases where the dynamics are dominated by one term or another – as explored briefly in lecture 2, and more generally via the kinds of asymptotic theories discussed in, e.g., [58]. Or we might turn to mean-field theory, which can solve for the global field evolution provided certain simplifying assumptions are made; these assumptions are in some cases reasonable and in other cases aren’t. Or we can turn to simulations of the full underlying equations. In Roscoff, I spent most of the fourth lecture giving a summary of such simulations – how they work, what they find, what their limitations are, and so forth. I’ve chosen not to do that here in detail, for a few reasons. The main one is that, unlike the basic theory in lectures 2 and 3, this material will be out of date so rapidly that inclusion in a textbook might do

Multi-Dimensional Processes In Stellar Physics

more harm than good; another is that our Living Review [1] is after all supposed to remain “living,” in the sense of being updated from time to time, so you can always probably find my thoughts on the matter there. Here, I will mention only a few quick points, summarizing the simulations and the topics of these lectures more generally. One key topic of interest involves how strong the fields get. Many proposed “scaling laws,” inspired by theory or simulation or both, have wrestled with the problem at some level. A very crude summary is that in non-rotating convection, “equipartition” of kinetic and magnetic energy densities is probably a decent rule of thumb; but it is not a law of nature (the magnetic energy can in principle exceed the kinetic by large factors), and indeed in rapidly rotating systems it is probably not what occurs. Fields much stronger than equipartition may be possible in such systems, though eventually other factors – like the “magnetic buoyancy” of strong fields, or the Ohmic dissipation associated with them (see discussion [43]) – can come into play. Another recent topic of interest involves the role of the tachocline, the shear layer at the base of the Solar convection zone. This figures prominently in “interface dynamo” models, but less so (or not at all) in others, and simulations both with and without tachoclines have yielded similarly organised fields. My own best guess is that the tachocline probably does matter in the Sun (or in any other star that has one), in the sense of shaping the fields and their temporal evolution; but it does seem to be possible to generate highly organised fields (both in space and in time) without a tachocline, too. For example, fully convective stars show a “rotation-activity correlation” just like that in stars that have tachoclines ([59, 60]). One very broad summary of simulations on both global and local scales is that, unsurprisingly, rotation matters a lot. It matters for heat transport (e.g., [61]), for angular momentum transport (e.g., [62, 63]), and for field generation (e.g., [64]). Stratification matters, too – for example, in stratified convection the viscous and Ohmic dissipation can become vastly greater than in the unstratified case (see [65]), and the interaction between the influence of stratification and rotation is fascinating and complex (e.g., [66]). Another broad summary is that the simulations can yield solutions that look remarkably akin to the observations in some respects. They can yield steady dipoles in planetary-like regimes, and cyclical, propagating magnetism in Solar-like ones; the resolved convective flows can also produce differential rotation that looks pretty reasonable, in the sense of being similar to (e.g.) constraints from helioseismology. This is good news, since the simulations typically have no (or few) truly “free” parameters – they have enormous diffusivities that are linked to the numerical resolution, but these are ideally made as low as possible rather than being “tuned” to fit observations. It isn’t yet clear exactly how the surface fields (i.e., what we observe) are linked to the deep dynamo-generated fields produced in some of these simulations (see, e.g., [67, 68]), but both global-scale processes and near-surface ones may well play roles. Those fields must impact the heat transport at some level, but we still do not (in my view) have a really convincing quantitative model of how this works. For my views on the subject, see for example [41]; for others, see, e.g., [36], [44], [38]. It is now beginning to be practical to run enough simulations, with broad enough ranges of the relevant parameters, to make quantitative predictions about a few things. For example, recently some authors have begun to use simulations to predict, e.g., relations between the periods of magnetic cycles and the overall rotation rate – see [69], [70]. It is worth noting that this particular observable is quite a stringent test: you basically need to get the convection, the shear, and the magnetism all right, since all these elements probably figure in determining the cycle period. So it wouldn’t (in my view) be terribly surprising if some of the trends identified in these simulations turn out to be correct, but if the exact place in parameter space at which they occur is wrong. Time will tell.

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Broadly, the goal of these simulations – and of dynamo theory generally – is to understand the “nonlinear outcomes” of the dynamo process. That is, we care not just about when you get a dynamo, or about the kinematic growth rate, but about how strong the fields are, what they do, and all the rest of it. Nonlinear simulations provide, in principle, answers to all this – but it is essential to understand that they operate in very different parameter regimes than real stars or planets, and it is often very difficult to sort out which aspects of the simulated flow echo those in a real astrophysical object, and which are partly artefacts of our limited computational resources. It is always striking to me, though, that such simulations do seem to get some aspects of the flows and fields “right,” despite their many limitations; the same is true in some sense of mean-field theory, despite its own obvious problems and caveats. Somehow, despite operating in a turbulent regime we cannot simulate or effectively model, the stars and planets manage to do things that are remarkably akin to what happens in these low-Rm models. At minimum, the simulations teach us about what can happen in different parameter regimes, and – together with observations and basic theory – help fuel our imaginings about what does happen in a star or planet. I will end by noting that many of the tools used to do these simulations are now publicly available. For example, the widely-used convection and/or dynamo codes Magic, Rayleigh, the Pencil code, or Dedalus can all be easily downloaded, and you can be up and running with them in a day or so. The best way of learning more about this stuff is by trying to answer questions of your own – so please dive in! We could, as I hope is clear by now, use your help. Acknowledgments

It is a pleasure to acknowledge conversations with many of my colleagues at Exeter and my long-term collaborators abroad on these topics. I am also grateful for grant support provided by the European Research Council (ERC under grant agreement number 337705 “CHASM”), and for computing allocations on various UK and EU machines. Some calculations referred to here were performed on the University of Exeter supercomputer, a DiRAC Facility jointly funded by STFC, the Large Facilities Capital Fund of BIS, and the University of Exeter; others were performed on DiRAC facilities at Edinburgh and at Leicester. I also acknowledge PRACE for awarding us access to computational resources Mare Nostrum based in Spain at the Barcelona Supercomputing Center, and Fermi and Marconi based in Italy at Cineca.

References [1] A.S. Brun, M.K. Browning, Living Reviews in Solar Physics 14, 4 (2017) [2] C. Jones, Dynamo Theory, in Dynamos: Les Houches Summer School, edited by Ph. Cardin and L. F. Cugliandolo (2008), Vol. 88 of Les Houches, pp. 45–135 [3] C.M. Johns-Krull, J.A. Valenti, Astrophys. J. Lett.459, L95 (1996) [4] G. Basri, L.M. Walkowicz, N. Batalha, R.L. Gilliland, et al., Astron. J.141, 20 (2011), 1008.1092 [5] B.T. Montet, G. Tovar, D. Foreman-Mackey, Astrophys. J.851, 116 (2017), 1705.07928 [6] C. Fröhlich, J. Lean, Astron. Astrophys. Rev.12, 273 (2004) [7] T. Reinhold, K.J. Bell, J. Kuszlewicz, S. Hekker, A.I. Shapiro, Astron. Astrophys.621, A21 (2019), 1810.11250 [8] R.M. Roettenbacher, J.D. Monnier, H. Korhonen, A.N. Aarnio, F. Baron, X. Che, R.O. Harmon, Z. K˝ovári, S. Kraus, G.H. Schaefer et al., Nature533, 217 (2016), 1709.10107 [9] J.F. Donati, J.D. Landstreet, Annu. Rev. Astron. Astrophys.47, 333 (2009), 0904.1938 [10] N. Piskunov, O. Kochukhov, Astron. Astrophys.381, 736 (2002)

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[11] S.V. Berdyugina, Living Rev. Solar Phys. 2, lrsp-2005-8 (2005) [12] R.M. Roettenbacher, J.D. Monnier, H. Korhonen, R.O. Harmon, F. Baron, T. Hackman, G.W. Henry, G.H. Schaefer, K.G. Strassmeier, M. Weber et al., Astrophys. J.849, 120 (2017), 1709.10109 [13] B.M. Morris, L. Hebb, J.R.A. Davenport, G. Rohn, S.L. Hawley, Astrophys. J.846, 99 (2017), 1708.02583 [14] A.A. Pevtsov et al., Astrophys. J.598, 1387 (2003) [15] J.C. Hall, Living Reviews in Solar Physics 5, lrsp-2008-2 (2008) [16] J. Irwin, J. Bouvier, The rotational evolution of low-mass stars, in IAU Symposium, edited by E. E. Mamajek, D. R. Soderblom, & R. F. G. Wyse (2009), Vol. 258 of IAU Symposium, pp. 363–374 [17] R.W. Noyes, L.W. Hartmann, S.L. Baliunas, D.K. Duncan, A.H. Vaughan, Astrophys. J.279, 763 (1984) [18] X. Delfosse, T. Forveille, C. Perrier, M. Mayor, Astron. Astrophys.331, 581 (1998) [19] N. Pizzolato, A. Maggio, G. Micela, S. Sciortino, P. Ventura, Astron. Astrophys.397, 147 (2003) [20] N.J. Wright, J.J. Drake, E.E. Mamajek, G.W. Henry, Astrophys. J.743, 48 (2011), 1109.4634 [21] V. See, M. Jardine, A.A. Vidotto, J.F. Donati, C.P. Folsom, S. Boro Saikia, J. Bouvier, R. Fares, S.G. Gregory, G. Hussain et al., Mon. Not. R. Astron. Soc.453, 4301 (2015), 1508.01403 [22] S.H. Saar, A. Brandenburg, Astrophys. J.524, 295 (1999) [23] E. Böhm-Vitense, Astrophys. J.657, 486 (2007) [24] R. Egeland, Ph.D. thesis, Montana State University, Bozeman, Montana, USA (2017) [25] T. Reinhold, R.H. Cameron, L. Gizon, Astron. Astrophys.603, A52 (2017), 1705.03312 [26] J.L. Green, S. Boardsen, S. Odenwald, J. Humble, K.A. Pazamickas, Advances in Space Research 38, 145 (2006) [27] A.J. Finley, S.P. Matt, Astrophys. J.854, 78 (2018), 1801.07662 [28] S.P. Matt, A.S. Brun, I. Baraffe, J. Bouvier, G. Chabrier, Astrophys. J. Lett.799, L23 (2015), 1412.4786 [29] S. Deheuvels, R.A. García, W.J. Chaplin, S.e. Basu, Astrophys. J.756, 19 (2012), 1206.3312 [30] P.G. Beck, J. Montalban, T. Kallinger, J. De Ridder, et al., Nature481, 55 (2012), 1112.2825 [31] M. Cantiello, C. Mankovich, L. Bildsten, J. Christensen-Dalsgaard, B. Paxton, Astrophys. J.788, 93 (2014), 1405.1419 [32] N. Langer, Annu. Rev. Astron. Astrophys.50, 107 (2012), 1206.5443 [33] I. Brott, S.E. de Mink, M. Cantiello, N. Langer, A. de Koter, C.J. Evans, I. Hunter, C. Trundle, J.S. Vink, Astron. Astrophys.530, A115 (2011), 1102.0530 [34] M.P.L. Suijs, N. Langer, A.J. Poelarends, S.C. Yoon, A. Heger, F. Herwig, Astron. Astrophys.481, L87 (2008), 0802.3286 [35] I. Ribas, Astrophys. and Space Sci.304, 89 (2006), astro-ph/0511431 [36] G. Chabrier, J. Gallardo, I. Baraffe, Astron. Astrophys.472, L17 (2007) [37] G.A. Feiden, B. Chaboyer, Astrophys. J.789, 53 (2014), 1405.1767 [38] J. MacDonald, D.J. Mullan, Astrophys. J.765, 126 (2013), 1302.2941

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[39] C.J. Hansen, S.D. Kawaler, Stellar Interiors. Physical Principles, Structure, and Evolution. (Springer, 1994) [40] S.W. Stahler, Publ. Astron. Soc. Pac.100, 1474 (1988) [41] L.G. Ireland, M.K. Browning, Astrophys. J.856, 132 (2018), 1803.02664 [42] D.J. Mullan, J. MacDonald, Astrophys. J.559, 353 (2001) [43] M.K. Browning, M.A. Weber, G. Chabrier, A.P. Massey, Astrophys. J.818, 189 (2016), 1512.05692 [44] G.A. Feiden, Astron. Astrophys.593, A99 (2016), 1604.08036 [45] J. MacDonald, D.J. Mullan, Astrophys. J.834, 67 (2017), 1608.02136 [46] G.M. Vasil, D. Lecoanet, B.P. Brown, T.S. Wood, E.G. Zweibel, Astrophys. J.773, 169 (2013), 1303.0005 [47] S. Childress, A.D. Gilbert, Stretch, Twist, Fold: The Fast Dynamo, Vol. 37 of Lecture Notes in Physics (Springer, Berlin; New York, 1995) [48] H.K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids, Cambridge Monographs on Mechanics and Applied Mathematics (Cambridge University Press, Cambridge; New York, 1978) [49] A. Brandenburg, K. Subramanian, Phys. Rep. 417, 1 (2005), astro-ph/0405052 [50] P. Charbonneau, Living Rev. Solar Phys. 7, lrsp-2010-3 (2010) [51] E.N. Parker, Astrophys. J.122, 293 (1955) [52] H.W. Babcock, Astrophys. J.133, 572 (1961) [53] R.B. Leighton, Astrophys. J.140, 1547 (1964) [54] R.B. Leighton, Astrophys. J.156, 1 (1969) [55] R. Cameron, M. Schüssler, Science 347, 1333 (2015), 1503.08469 [56] F. Cattaneo, D.W. Hughes, Mon. Not. R. Astron. Soc.395, L48 (2009), 0805.2138 [57] S.M. Tobias, F. Cattaneo, Nature497, 463 (2013) [58] K. Julien, E. Knobloch, A.M. Rubio, G.M. Vasil, Physical Review Letters 109, 254503 (2012) [59] N.J. Wright, E.R. Newton, P.K.G. Williams, J.J. Drake, R.K. Yadav, Mon. Not. R. Astron. Soc.479, 2351 (2018), 1807.03304 [60] E.R. Newton, J. Irwin, D. Charbonneau, P. Berlind, M.L. Calkins, J. Mink, Astrophys. J.834, 85 (2017), 1611.03509 [61] A.J. Barker, A.M. Dempsey, Y. Lithwick, Astrophys. J.791, 13 (2014), 1403.7207 [62] T. Gastine, R.K. Yadav, J. Morin, A. Reiners, J. Wicht, Mon. Not. R. Astron. Soc.438, L76 (2014), 1311.3047 [63] P.A. Gilman, J. Atmosph. Sci. 32, 1331 (1975) [64] U.R. Christensen, J. Aubert, Geophys. J. Int. 166, 97 (2006) [65] L.K. Currie, M.K. Browning, Astrophys. J. Lett.845, L17 (2017), 1707.08858 [66] L.D.V. Duarte, J. Wicht, M.K. Browning, T. Gastine, Mon. Not. R. Astron. Soc.456, 1708 (2016), 1511.05813 [67] M.A. Weber, M.K. Browning, Astrophys. J.827, 95 (2016), 1606.00380 [68] M.A. Weber, M.K. Browning, S. Boardman, J. Clarke, S. Pugsley, E. Townsend, The Suppression and Promotion of Magnetic Flux Emergence in Fully Convective Stars, in Living Around Active Stars, edited by D. Nandy, A. Valio, P. Petit (2017), Vol. 328 of IAU Symposium, pp. 85–92, 1703.04982 [69] A. Strugarek, P. Beaudoin, P. Charbonneau, A.S. Brun, J.D. do Nascimento, Science 357, 185 (2017), 1707.04335 [70] J. Warnecke, Astron. Astrophys.616, A72 (2018), 1712.01248

Stellar magnetism: bridging dynamos and winds Allan Sacha Brun1,∗ and Antoine Strugarek1,∗∗ 1

Department of Astrophysics (DAp)/AIM, CEA/IRFU, CNRS/INSU, University of Paris VII, CEA Paris-Saclay, France Abstract. In this lecture on stellar magnetism we discuss how the dynamo gen-

erated magnetic field shapes the extended hot atmosphere and how the feedback loop between rotation, convection, turbulence, dynamo action and braking by stellar wind influences the secular evolution and the rotational history of solarlike stars. We discuss each key physical mechanism such as dynamo action and wind dynamics and discuss angular momentum transport inside and outside the star. In order to illustrate these complex processes and their nonlinear interaction we use both pedagogical exercises and discuss more advanced magnetohydrodynamics numerical simulations. We propose seven problems and their solution to help getting a good first understanding of stellar magnetohydrodynamics.

1 Introduction Stars are self-gravitating turbulent rotating objects made of plasma and subject to a large range of turbulent flows in particular inside their convective surface envelope or deep core. Such turbulence is often accompanied by intense and highly time dependent magnetic fields. Understanding how stars generate and maintain their magnetic field is essential because this has a direct consequence on how the star’s dynamical state evolves and impacts its surrounding through its wind of particles and its eruptive behavior. Indeed, in solar-like stars1 for instance, the surface convective envelope drives an intense dynamo and injects energy into to the solar atmosphere that yields complex time dependent surface magnetic fields and chromospheric/coronal states. The hot corona further extends into a wind of particles (so called solar/stellar wind) that reaches supersonic and superalfvénic speeds and that impacts directly any objects surrounding the star, such as planets. In this lecture, we intend to give an introduction as how stars can generate a magnetic field –or even a cyclic activity– via dynamo action, and how the time dependent global magnetic geometry impacts the corona and wind. We will then conclude by describing the feedback loop between the dynamo and the wind, via the so-called magnetic braking effect of the star’s rotation state (e.g. Skumanich’s Law, [1]). Indeed, stars spin down as they age on the main sequence (MS), which impacts the efficiency of their dynamo. ∗ e-mail: [email protected] ∗∗ e-mail: [email protected] 1 e.g. stars with a mass ranging between 0.5 and 1.3 M , possessing a convective envelope and a stably stratified  radiative interior.

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All along this short lecture, in addition to illustrations supporting the various topics discussed, we will propose exercices (with solutions found in the appendix) to ease the understanding of the basic physical ingredients and mechanisms operating in magnetic stars and behind their dynamo and wind. The article is organized as follows: In section §2 we make a brief presentation of what is a star. In §3 we explain the concept of α − ω dynamo and how it extends to stellar dynamos, in section §4 we present the inevitability of stellar winds by discussing Parker’s wind model and more recent improvements. In section §5 we discuss how dynamo and wind work together and present the concept of rotational history of stars due to wind braking. We finally conclude in §6.

2 A brief summary of what is a star Stars in the main sequence are self-gravitating objects in mechanical and thermal equilibrium. Thermonuclear fusion reactions of hydrogen in their dense and hot core provide the necessary gas pressure to sustain the star against its own gravity. Stars are bright because they have a hot surface. The stellar luminosity can be related to an effective surface temperature 4 and to the size of the star by L = 4πR2 σT eff , with σ the Stefan-Boltzmann constant (cf Figure 1). One can further show that on the main sequence, when hydrogen nuclear burning is occurring, L ∝ M4 . The more massive the star is, the more luminous it is. Depending on their mass stars can have a convective surface (M < 2M ) or a convective core. Very low mass stars (M < 0.35M ) can even be fully convective. In this lesson, we will focus on solar-like stars possessing an outer convective layer. Exercise I: Stellar properties a) The main equilibrium in a star is hydrostatic (pressure compensates gravity). Assuming pressure is negligible, calculate the free-fall time of the Sun (the averaged density of the Sun is 1.4 g/cm3 ). b) Why are stars luminous? By supposing that the Sun radiates its potential gravitational energy (the solar mass is 1.99 1033 g and its radius 6.96 1010 cm), calculate how long it could sustain its present luminosity (the present solar luminosity is 3.83 1033 erg/s). Knowing that the Sun is 4.5 Gyr old, what do you deduce? c) The energy radiated by a star like the Sun thus has to come from another source. Calculate the energy by proton necessary to sustain the solar luminosity. Do you know any physical mechanism able to deliver such a high amount of energy by proton?

3 Mean-field model of stellar dynamo and nonlinear aspects Currently, in most low mass stars, a fluid dynamo operating inside and at the base of the convective envelope is considered to be the physical mechanism at the origin of stellar magnetic field. In some cases such as the Sun it can even lead to a cyclic magnetic activity. First let’s recall the definition of a fluid dynamo: The ability that a moving conducting fluid (plasma) has to generate (by self-induction) a magnetic field and to maintain it against Ohmic dissipation. One practical way of understanding the concept of a fluid dynamo is to think of an electromagnet. The electric current going through the copper wire induces a magnetic field. In a fluid dynamo, the motions of the conducting fluid induce the magnetic field and the source of energy is not external as it is the case for an electro-magnet that relies on a battery or

Multi-Dimensional Processes In Stellar Physics

Figure 1. Top: Hertzsprung-Russell diagram positioning stars as a function of their luminosity vs surface temperature (courtesy of Cosmic Perspective, 7th edition). The main diagonal is called the main sequence, location in the H-R diagram where stars burn their hydrogen. We can see that stars come in all sizes and colors. The Sun is found around 5800 K and a normalized luminosity of 1 by definition. Once they have depleted their inner core hydrogen, stars like the Sun leave the main sequence to become subgiant and then red supergiant (moving up-right in the H-R diagram). They fail to burn elements beyond Helium, contrary to stars having an initial mass greater than about 9 M that will burn all elements up to Iron and end their life as a supernova. Bottom: Location of stellar convection zone vs stellar mass. We note that massive stars have a convective core, whereas low mass stars have a convective envelope that thickens to become fully convective for very low mass stars.

source of power but the kinetic energy associated with the plasma flow. Fluid dynamo can be either steady (DC) or time dependent (AC). Of course in celestial bodies non linear interplays between the flow, the field and thermodynamic variables such as density and temperature are

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always at the heart of the dynamo mechanism. Historically, the mean field theory approach, which omits or simplifies the treatment of nonlinearities, has been used to study in details dynamo action in experimental devices or in cosmic objects such as stars [2, 3, 4]. The basic concept is to focus on the large scale field by splitting the magnetic and velocity fields into mean and fluctuating parts, e.g. B = B + b and V = V + v. Then from the induction equation (where we note the ohmic diffusion coefficient η) ∂B = ∇ × (V × B − η∇ × B) ∂t

(1)

one gets

∂ B = ∇ × ( V × B + v × b − η∇ × B ) (2) ∂t The term v × b is called the mean electromotive force (emf). In order to be able to solve for B only, one needs to find a closure*relation for *the+ emf, the simplest being a Taylor + expansion around B , e.g. v × b i = αi j B j + βi jk ∂ B j /∂xk , with αi j a pseudo tensor of rank 2 and βi jk a tensor of rank 3. The mean field dynamo equation is then ∂ B = ∇ × ( V × B + α B − (η + β)∇ × B ) , ∂t

(3)

where we have assumed isotropy for αi j and βi jk for the sake of simplicity. This equation is composed of 3 terms; i) large scale flows, ii) turbulence, iii) magnetic diffusion. To explain the source of magnetic field and of the 11 year modulation of the number of sunspots appearing on the solar surface, the concept of α − ω dynamo has been proposed by several authors in the late 60’s/early 70’s. Helicoidal turbulence within the convection zone of solar like stars regenerate the poloidal component of the magnetic field, while large scale flow shears the poloidal field into its toroidal component. These 2 effects are illustrated in figure 2, top row. In the mid 90’s, the concept of flux transport dynamo has been put forward, relying on both the transport of the field from mid latitude to the pole and below the convection zone, and on the existence of a source of poloidal field based on the Babcock-Leighton effect, e.g. the fact that bipolar sunspots tend to have some tilt when they emerge. The flux transport mechanism can be realized by a large scale meridional circulation, turbulent pumping or even turbulent diffusion. This scenario is nowadays the favored one and it is illustrated in figure 2, bottom row. We refer the reader to these 2 recent reviews [5, 6] that discuss in depth the advantages and disadvantages of the various scenarii to explain the solar 22 yr cycle. To get a better understanding on the dynamo mechanism we propose to derive in Exercise II a dispersion relation of a dynamo solution in the simplest case of a α − ω dynamo. Exercise II: α − ω dynamo: basic concepts We assume that the magnetic field follows the mean field induction equation of the α − ω type see eq. (3). We start with a Cartesian box, (x, y, z) for which we assume that the magnetic field takes the form B (x, t) = ∇ × (Aey ) + Bey = B pol + Btor ey . We assume that there is a large scale shear such that V = ω0 zey , with ω0 a constant, constant α-effect α0 and turbulent magnetic diffusivity ηt . Hence, the mean velocity is V = (0, ω0 z, 0) and the mean magnetic field B = (0, B, ∂A ∂x ). a) Starting from the mean field induction equation (3), derive the time dependent equations for A and B (for the sake of simplicity we will neglect the α-effect for the toroidal field B) b) Derive the dispersion relation, assuming that (A, B) = (A0 , B0 )ei(kx+σt) , with σ an imaginary number. c) Show that the cycle period and the dynamo threshold depend on ω0 , ηt and α0 . For the

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Figure 2. Cartoon view of the solar interface dynamo concept, showing ω-effect converting poloidal field into toroidal field and two possible scenarii for the regeneration of the poloidal field by the toroidal field, α-effect top row and Babcock-Leighton effect bottom row [7]

sake of simplicity we will assume that k > 0 and that the product α0 ω0 < 0 (inspired by the fact that in the Sun the kinetic helicity is negative in the northern hemisphere and dω/dr > 0 at low latitudes). Another property of solar magnetism important to explain is the intermittent state of the field, e.g. the observed solar sunspot cycle varies both in amplitude and duration. Classical mean field dynamo models such as the one illustrated in Figure 3 left panel (from [8], see also [5]), do not explain a varying cycle period without introducing fluctuations in the model ingredients as the models are usually fine tuned to provide an 11-yr cycle. In reality, historical records show that the solar cycle period varies by more or less 3 years around 11 yr, the longest cycle having lasted 14 yr and the shortest 8 yr. How can this be explained? One easy explanation is to introduce stochasticity in the α, ω, meridional circulation or BabcockLeighton effects (see for instance [9, 10]). Another explanation is to couple the induction equation to the so-called Malkus-Proctor effect, e.g. the feedback of the large scale Lorenz force on the flow [11, 12]. This leads to a non linear dynamo by adding to the large scale flow a fluctuation that depends on the field strength: ∂vϕ 1 = [(∇ × B ) × B ]ϕ + νΔvϕ , ∂t 4πρ¯

(4)

along with Ω = Ωbg + vϕ /(r sin(θ)) in the induction equation. By introducing a new time scale linked to the kinematic viscosity ν, it can be shown that low magnetic Prandtl number Pm = ν/ηt dynamos tend to be intermittent (as illustrated in Figure 3 right panels), whereas Pm = 1 dynamos are steady and regular. Since in stars Pm is very small of order 10−3 or less, it is no surprise that solar-like stars undergo variations of their main cycle period/activity. In fact, Grand minima of activity such as the Maunder minimum during the late 17th century are likely due to a strong modulation of the basic state of the 11 yr dynamo via non linear feedback and the interplay of the dipolar and quadrupolar dynamo families [13, 14]. These nonlinear dynamo models have the added advantage that one can link variations of the cycle period and dynamo waves to the feedback from the field itself. So-called torsional oscillations

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of the solar large scale differential rotation have been associated to such feedback, or to the thermal shadow that intense magnetic fields can generate inside a convective region [15, 16, 17].

Figure 3. Left panels: Typical dynamo solution of a flux transport Babcock-Leighton type [8]. Color contours of the longitudinal (top) and radial (bottom) magnetic fields. Note the 11 yr cycle and the main equatorward branch of the dynamo waves. Right panels: Nonlinear mean field dynamo with feedback from the large scale Lorentz force on the differentiual rotation via the so-called Malkus-Proctor effect. Top right panel: butterfly diagram over several millenia for a flux transport Babcock-Leighton dynamo with a Pm = 10−3 in which we notice a secular modulation of the short period 11 yr cycle. Bottom right panel: time evolution of the magnetic energy at 45 degree of latitude in the same model and over the same time interval showing the large modulation of the 11 yr cycle amplitude. (see Brun et al. 2018 in prep)

Having now explained how a turbulent plasma can generate a magnetic field by dynamo action we now turn to describe how such magnetic fields can further shape the extended atmosphere of solar-like stars. Magnetic fields at the surface of the Sun and in the Corona are often observed to have a loop-like structure. One easy way to understand these structures is to reconstruct magnetic field lines in the solar atmosphere from the observations of photospheric magnetograms. A first guess can be obtained from the Navier-Stokes equation by assuming stationarity (d/dt = 0), no motions and neglecting the gravity g and the viscosity to obtain that 0 = −∇P + 1/cJ × B, with J = c/4π∇ × B, the electric current, c the speed of light and P the gas pressure. In the corona, P is much less than the magnetic pressure (P ∼ 0.4%Pmag ), i.e. the corona is a low-β plasma, where the magnetic field controls the dynamics. Since P  Pmag , we deduce that the magnetic field is roughly speaking force-free (i.e. 1/cJ × B = 1/4π(∇ × B) × B = 0) in the solar corona, which implies that either J = 0 or J = γB. Force-free fields have interesting properties, the most commonly used are so-called potential (vacuum) magnetic field with J = 0 or γ = 0. (see Exercice III). We illustrate in Figure 4 how a potential magnetic field can be extrapolated outside of a convective spherical dynamo by assuming vacuum conditions. Exercise III: Magnetic field lines reconstruction in solar-like star’s atmosphere Assuming that the magnetic field at the surface of a solar-like star is force-free J × B = 0,

Multi-Dimensional Processes In Stellar Physics

such that J = γB, a) Derive the expression for the potential magnetic field in spherical geometry for γ = 0, such that B = −∇Φ. b) Knowing the surface magnetic field Br (r = 1, θ, φ) at the surface of the star (we will assume that the surface is at r = 1) and considering that the field is purely radial (i.e. Φ = 0) at a spherical source surface of radius r = r ss , derive the full expression for the magnetic field. c) Derive the expression for the force-free magnetic field for γ  0, assuming that γ is independent of spatial coordinates.

Figure 4. 3-D dynamo solution with potential field line extrapolation with r ss = 2.5 stellar radius (cf. Exercise III to see how potential field line extrapolation works). 3-D color rendering of the turbulent convective flow superimposed by magnetic field lines inside and outside the stellar surface. Note how the magnetic field lines form loops of all sizes at the stellar surface (from Strugarek et al. 2017).

4 Stellar wind: extensions of the Parker model The first evidence of existence of the solar wind, i.e. a continuous stream of charged particles emanating from the Sun, was first intuited from the deviation angle of comets [18] and theoretically predicted by [19]. The inevitability of the solar wind was at that time conceived using the following argument:   if the solar corona is hydrostatic, the pressure at infinity is p∞ = p exp −GM /(v2c R ) , where p is the pressure in the solar corona and vc is the local sound speed. At that time it was realized that the electronic density in the solar corona was of the order of 107 − 108 cm−3 , which corresponds to a million degree corona. This translates into a pressure ratio of p∞ /p ∼ 10−4 , whereas estimates of the interstellar pressure rather

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suggest p∞ /p ∼ 10−14 . This large discrepancy led Parker [19] to suggest that the solar corona is not static, but is actually accelerating away from the Sun up to a point where it becomes supersonic, giving a theoretical fundation to the concept of solar –and stellar– wind. The wind solutions found by Parker obey the implicit formulation F(M, x) = M 2 − 2 ln M − 4 ln x −

4 +C = 0, x

(5)

where x = r/rc is the radial distance normalized to the sonic radius, M = vr /vc is the Mach number (velocity of the wind normalized to the local sound speed vc ), and C an integration constant. The solutions to Equation 5 are shown in the left panel of Figure 5, and it was quickly realized that the only possible solution for the solar corona was the transonic solution (black line) accelerating outwards (see Exercise IV). Few years later, the Luna and Mariner programs detected hints of ionic speeds of the order of 400 to 700 km/s, confirming the existence of a stationary corona in supersonic expansion. ACE and Ulysses missions later on unambiguously showed the existence of two components in the stellar wind (the so-called slow and fast solar wind, see Figure 1 in [20]) and showed a strong modulation of the overall coronal wind along the solar magnetic cycle (see §3). The analytical development of the initial Parker wind model is proposed for the reader in exercise IV.

v/vc

1.111

1.50

7 0.66

22 0.2

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7 66 -0.

1.5 r/rc

-0.6 67 -0 .22 2

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Weber & Davis wind solutions .000-0.778 -0.556-0.333

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Figure 5. Left. Parker wind solution. We show the contours of function F(M, x) (Equation 5). The Mach number is shown as a function of the distance to the star normalized by the critical radius rc . Each contour is a mathematical solution to the isothermal spherically-symmetric solar wind. Right. Weber and Davis wind solution. We show the contours of function F A (Ma , xA ) (Equation 7), for rA = 25 R √ and rc = rA /5, and assuming that vc (rc ) = GM /rc . The alfvénic Mach number is shown as a function of the distance to the star normalized by the Alfvén radius rA . Each contour is a mathematical solution to the isothermal spherically-symmetric solar wind under the influence of rotation and magnetism. Note that on the bottom left, we recover a Parker-like wind solution (left panel).

Exercise IV: Parker’s wind solution for solar-like star We consider a spherically-symmetric environment around the star, and suppose the corona is isothermal. We further suppose that the plasma is composed of fully-ionized hydrogen, for which we have 2kB ρT = P.  a) Express the temperature T as a function of the sound speed vc = P/ρ. b) Using the stationary Euler equations, and supposing the star has a mass M , derive the

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differential equation determining the wind velocity vr . c) Deduce from this equation that any wind solution satisfying vr = vc at some distance from star must do so at the critical radius rc = GM /(2v2c (rc )). d) Integrate the differential equation to find that the wind solutions are described by F(M, x) = M 2 − 2 ln M − 4 ln x −

4 +C = 0, x

(6)

where M = vr /vc is the Mach number and x = r/rc the distance to the star normalized to the critical radius and C is a constant to be determined. e) The contours of F(M,x) are plotted in the left panel of Figure 5 (try to plot them yourself using your prefered software). Argue why only the transonic solution, with a wind accelerating outwards, is physically reasonable for the Sun. The Parker wind model is thought to apply generically to solar-type stars, i.e. stars possessing an outer convective envelope. These stars are also able to sustain strong magnetic fields at their surfaces (typically from a few Gauss to a few 104 Gauss) through dynamo action thanks to their rotation (§3) during their life on the main sequence. The corona and the astrosphere are consequently magnetized, and the wind theory of [19] was quickly extended in a full magneto-hydrodynamical regime in rotation by [21, 22, 23]. The original derivation of [21] leads to a slightly different formulation of the wind solution, and the Parker wind equation 5 becomes

2

2 vc vc GM 1 2 F A (Ma , xA ) = Ma − 2 ln(Ma ) − 4 ln(xA ) − 2 2 vA vA vA rA xA ⎛  2 ⎞  2Ma x2A − 1 1 − x2A ⎟⎟⎟⎟ Ω2 r2 ⎜⎜⎜⎜ ⎟⎟⎟ − C = 0 , (7) + 2 2A ⎜⎜⎜⎜1 + 2  ⎟⎠ xA vA ⎝ 2 M x −1 a A

where xA = r/rA is the radial distance normalized to the Alfvén radius (distance where the wind speed matches the local Alfvén speed va ), and Ma = vr /vA is a modified alfvénic Mach number where the wind speed is normalized by the wind speed at the Alfvén radius, i.e. vA = vr (rA ). The Weber & Davis wind solutions are shown in the right panel of Figure 5 with the same layout as the Parker wind solution (left panel). The solar-like solution is again the transsonic then transalfvénic solution shown by the thick black line. The derivation of the Weber & Davis wind solution is proposed to the reader in Exercise V. Due to this magnetized wind, stars lose mass and angular momentum over time, offering a physical source for the deceleration of stars over the main sequence identified by [24] (Ω ∝ t−1/2 ). It was quickly realized that the two were related, and [21] were the first to show that the angular momentum loss rate J˙ could be written as ˙  RA 2 , J˙ = MΩ

(8)

˙ is the mass loss rate and RA is the averaged Alfvén radius of the wind (see Exercise where M V). This formulation is nevertheless not very useful without a robust and simple estimate of the Alfvén radius RA . For the sake of simplicity, let us consider here that the coronal magnetic field is B(r) = B (R/R )l+2 (l depends e.g. on the topology of the field). By definition, we have vr (RA )2 =

B2 R2l+4 B2 . = 2l+4  4πρ(RA ) RA 4πρ(RA )

(9)

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˙ = R2 ρv must be constant in radius. We note that because mass is conserved, the mass loss M If we suppose that the wind radial velocity at the Alfvén radius is vr (RA ) = Kvesc (Ra /R )q √ (vesc = 2GM /R is the escape speed), we then find that

RA R

2l+2+q = KΥ ,

(10)

where Υ=

B2 R2 ˙ esc Mv

(11)

is the so-called magnetization parameter of the wind [25]. Here it must be noted that the exponents l and q are not independent, and great care is in order when supposing a priori their values. The derivation above completely neglected the azimuthal component of the wind velocity. For fast rotators, [26] proposed to generalize the formulation using a modified 2Ω2 R2 velocity v2mod = v2esc + K2  which finally gives 2

⎛ ⎜⎜⎜ ⎜⎜ RA = K1 ⎜⎜⎜⎜  R ⎝⎜

⎞m ⎟⎟⎟ ⎟⎟⎟ Υ ⎟⎟⎟ , 2⎟ 1 + ( f /K2 ) ⎠

(12)

√ where f = 2ΩR /vesc . Rather than supposing the values of q and l, the new constants K1 , K2 and m can be derived from 3D numerical simulations of stellar winds. This is the approach that was followed by [27], using an extensive set of numerical simulations and comparing various idealized magnetic topologies (dipole, quadrupole, octupole). They found that the value of the exponent m depends on the topology of the field, making a practical application of Equation 12 tedious. To overcome this aspect, they proposed a refined Alfvén " radius formulation depending on the open magnetic flux Φopen = lim s |B · dS|, rather than r→∞ the magnetic field amplitude only, with Υopen =

Φ2open . ˙ esc R2 Mv

(13)

This formulation is topology-independent, but requires to have an estimate of the open flux of the magnetic topology. For a given star, this can be achieved through 3D numerical simulations, or from a semi-empirical model of the stellar corona calibrated on numerical simulations [27]. For a series of young solar-like stars, [28] showed that this approach can be routinely used to estimate the average value of the Alfvén radius (beige surface on Figure 6) and of the magnetic torque applied to the rotating star. This torque can then in turn be used to assess the subsequent rotational evolution of the star. Exercise V: A simple Weber-Davis magnetic wind model We continue on the Parker wind model developed in Exercise IV to extend it to a rotating, magnetized wind model. We note Ω the rotation rate of the star and still assume a spherically-symmetric wind for the sake of simplicity. a) Show that Br /(ρv) is constant under the spherically symmetric assumption. b) We Suppose that the corona is in a perfect conductor state (∇ × E = 0) and that we can neglect resistivity. Using the azimuthal projection of the  momentum equation, show that the  modified angular momentum L = r vϕ − Br Bϕ /(4πρvr ) is constant.

Multi-Dimensional Processes In Stellar Physics

Figure 6. 3D views of the corona of young solar-like stars [28]. The magnetic field lines are colourcoded by polarity (red is positive, blue negative). An inset zoom shows a close-up of the stellar surface, where the surface radial field observed by Zeeman-Doppler Imaging can be seen. On the left, we see BD-16351, a 42 Myr-old solar analog. On the right, the Sun as of 1996 is shown using the same layout.

 c) We define the alfvénic Mach number as MA = 4πρvr /Br . Find the expression of vϕ as a function of Ω , r, MA and L only. d) We now suppose that an Alfvén radius rA exists in the wind where MA = 1. Show that in that case, L = rA2 Ω . What does this inspire you? e) The same exercise as in the case of the Parker-Wind Model (Exercise IV) can be played to integrate the differential equation determining the wind solution. This is a much longer (but straightforward) calculation that we leave for the most motivated readers to derive by themselves. We define another alfvénic Mach number Ma = vr /vA , where vA = vr (rA ) is the wind speed at the Alfvén radius, and normalize the radius to the Alfvén radius xA = r/rA . The wind solutions are defined in Equation 7. We recognize the same form as Equation 6, with an additional term coming from the influence of rotation and magnetism on the solution. Fix rA and rc to values of your choice (with rc < rA ), and plot the contours of F A (Ma , xA ) for a family of solutions satisfying both transsonic (F A (vc /vA , rc /rA ) = 0) and transalfvénic (F A (1, 1) = 0) solutions (an example with rA = 25 R and rc = rA /5 is given in the right panel of Figure 5). What can you conclude? We have seen that solar-like stars loose angular momentum through the torque applied by their magnetized wind. In order to characterize their rotational history we must evaluate, in addition to the wind braking efficiency, how the internal angular momentum distribution evolves as a consequence of that external stellar wind torque. An easy way to understand the key ingredients involved in angular momentum redistribution inside stars is the 2-layers model proposed first by [29]. In this model, the solar-like star is treated as a sphere divided into two zones: an inner stably stratified core and a turbulent convective envelope both independently rotating around an axis aligned with the eˆ z direction, where eˆ z is the unit vector along the z axis of a 3-D (x, y, z) cartesian system (see Fig 7 left panel). Each possess its own moment of inertia I and angular velocity Ω f ; both structural evolution and magnetized winds

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can generate differential rotation. If an external torque, such as the one coming from a stellar wind is applied to the star, the angular momentum evolution of the convective envelope and radiative interior can be expressed as [29]: dJcore dt dJenv dt

ΔJ tc ΔJ Jenv − , tc tw

= −

(14)

=

(15)

with tc the coupling time scale between the two zones due to the combined action of turbulence, waves, magnetic fields and viscous stresses and tw the wind braking timescale and −Icore Jenv (see Exercise VI). These formula further assume that the convective enΔJ = Ienv JIcore core +Ienv velope transmits instantaneously the applied surface torque to the base of the convective envelope. Such models have been used to explore the relevant coupling time scales between the convective envelope and the radiative interior in solar-like stars over the course of their evolution ([29, 30, 31, 32], and for recent developments see [33, 34, 35, 36, 37]). It is found that a time scale of tens to hundreds of Myr can explain the core-envelope coupling in young open cluster stars (see Figure 7).

Figure 7. Left: The two layers model of a star, showing the exchange of angular momentum ΔJ between a core (dashed lines in the right panel) and an envelope (plain lines in the right panel) rotating at different rates and the braking action of a stellar wind. Right: Evolutionary track for two-layers models for 1 solar mass star showing how the coupling time scale can explain slow and fast rotators (from [38]).

Exercise VI: Angular momentum redistribution in solar-like star a) Express the initial angular momentum of the radiative core Jcore and convective envelope Jenv . b) Express the final angular momentum of the radiative core Jcore and convective envelope Jenv when both zones rotate at Ω f . c) Deduce ΔJ the exchange of angular momentum between the two zones to achieve the final state where they both rotate at the same rate Ω f . Finally, the magnetized stellar wind also shapes the whole astrosphere and its structure on the long star-planet distances. It can be a priori estimated from simple Parker or Weber &

Multi-Dimensional Processes In Stellar Physics

Davis wind solutions. An interesting property of the Weber & Davis wind solution is that at sufficiently large distance from the star, e.g. outside of the Alfvén radius, the wind becomes essentially radial and converges towards a constant wind speed. Because of the stellar rotation, though, the magnetic field lines wind up more and more away from the star, giving rise to the so-called Parker spiral. We propose a simple derivation of this spiral to the reader in Exercise VII and illustrate it in Figure 10. The spiralling magnetic field lines have the important consequence that the magnetic connectivity of the Earth to the solar surface is not radial. Energetic particles accelerated during eruptive events on the Sun roughly follow the wind field lines and their spiralling pattern. Consequently, geoeffective energetic particles come from regions on the western part of the solar disk, as particles originating from the eastern part of the solar disk will be spiralling away from our Earth. The magnetic spiral of the solar wind has thus important consequences for the space weather around the Earth. Exercise VII: Parker’s spiral in solar-like star Let’s assume that we are in a reference frame rotating with the Sun (denoted with primes) and that the corona can be approximated by a perfect conductor (∇ × E = 0). In such context, it can be shown that v //B. a) Show that B makes an angle Ψ to the radial direction er such that: tan(Ψ) ∼ −Ω∗ r/vr . b) Assuming the wind radial speed vr is constant with radius and equal to vsw (this is a reasonable assumption far from the central star), calculate the parametric equation φ() of the magnetic field lines on the equatorial plane in polar coordinates (, φ). What is the shape of the field lines? [bonus] Plot them. c) Assuming a wind radial speed vr at 1 AU of 400 km/s, deduce the angle Ψ at the Earth orbit.

5 Coupling dynamo and stellar wind on dynamical and secular timescales: a perspective We have seen that solar-like stars develop intense dynamo generated magnetic fields inside their surface convective envelope and that they also possess a supersonic magnetized wind of particles at their surface extending far out until it reaches the interstellar medium, hence forming so-called helio/astro-sphere (of typical size 100 AU). This magnetized wind applies an efficient torque to the surface of the star hence spinning it down over secular ages (cf. Figure 7, right panel and Exercices VI and VII). Since dynamo action depends on the rotation rate of the star there is a fundamental feedback loop between the dynamo and the wind. To give a quantitative assessment of the coupling between the dynamo field and the wind, it is useful to list typical time scales involved in their evolution. Convection overturning time in stars are of the order of minutes to months, rotation period between half a day to half a year and magnetic cycles have been found to last between few years to few decades. Based on our knowledge of the solar wind, we can expect that the typical wind speed depends on the magnetic field amplitude, rotation rate and surface temperature of the star. In the Sun the wind speed varies from 400 km/s to 800 km/s, reaching the Earth at 1 AU in about 3 to 5 days. Faster (younger) rotating stars are likely to have much faster winds in excess of 1000 km/s. When the field topology evolves at the stellar surface, the wind adapts itself very fast (on an alfvénic time scale, of the order of few hours). Eruptive events such as coronal mass ejections can be much faster than the wind they are embedded in, and travel the Sun-Earth (1 AU) distance in about 2 days, the fastest recorded so far having reached the Earth in 14 hours. Hence we see that there is a large diversity of time and spatial scales, that makes this problem particularly difficult to solve self-consistently.

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To make the link between stellar dynamo and wind more obvious we now discuss how stellar winds can be affected by the complex and time varying geometry of dynamo generated magnetic field. From this point forward we discuss current research topics and our lecture becomes more a review listing what we believe are promising developments. Most studies of solar magnetized wind have been done with simple magnetic geometry (so called split monopole or dipole as in [24]) and only recently more complex multipolar fields have been considered [e.g. 27, 39, 40, and references therein]. The Sun exhibits such complex field geometry [13] and dynamo models have been rather successful at reproducing them both in 2.5D mean field models [5] or full 3-D MHD solutions (see for instance [41, 42]).

Figure 8. Top row and bottom left panels: Snapshots of a solar wind solution computed with realistic magnetic field geometry coming from a mean field dynamo model [8, 43]. Color contours represent the projected magnetic Mach number, field lines are in solid white and velocity arrows indicate the flow direction and amplitude. A change of background color corresponds to a reversal of the magnetic field. The dash black line represents the Alfvén surface, i.e. the zone where the wind becomes superalfvénic. Bottom right: change of the wind speed latitudinal distribution along an 11 yr magnetic activity cycle. Note the typical cross like pattern that is in qualitative good agreement with radio scintillation map [44] (see text).

In [43] we have published the first attempt to couple the output from a dynamo model with a magnetized wind solution. We display on Figure 8 three snapshots of the wind solution obtained. Starting from a minimal state of the dynamo field (mostly dipolar), a current

Multi-Dimensional Processes In Stellar Physics

Figure 9. Top left (a): Evolution (in year using log scale) of the total radius (blue dash line - right blue y-axis) and of the radius of the base of the convective envelope (solid line - black left y-axis) of one solar mass star. Top right and bottom row (b, c, d): Color contours of the longitudinal component of the magnetic field Bφ in an equatorial slice at 3 different phases along the PMS evolutionary track: fully convective phase (purple disk symbol on first panel), a radiative core extending up to 0.4 stellar radius (green square symbol) and at the ZAMS, when the core occupies more than 70% of the star (red star symbol). We note how the magnetic field in the convective envelope evolves under the joint influence of a strong structural change and a rotation rate going from 3.5 to 4.5 and 14 times solar rate respectively (From [45]).

sheet forms at mid latitude as the field topology becomes more complex by approaching the maximum of the 11-yr cycle, that occurs here between years 3 and 4. The field topology is more multipolar, with so-called helmet and/or pseudo streamers shifting from low to mid latitude. Then as the new magnetic flux tends to cancel the polarity of the existing polar cap, the magnetic field reverses sign and the field slowly returns to a minimum mostly dipolar state at year 11. One interesting way of calibrating the dynamo-wind models over the course of an 11 yr cycle is to compare the outcome of the solution to observations of the interplanetary medium. [44, 46] used scintillations radio maps to estimate the variations of the wind velocity versus the heliographic latitude. Such observations show that during maximum of activity the slow wind extends at higher latitude than during minimum, results that are qualitatively reproduced in [43] (see bottom right panel of Figure 8). One major difference is that the solar magnetic field is not symmetric with respect to the equator and the observational data is less

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regular than the numerical (symmetric) dynamo solution. [47] have recomputed new dynamo wind solution taking into account the north-south asymmetry of the solar magnetic field including the time lag of reversal of about one year between the hemispheres. They find a better agreement with the observations of [44], with the slow wind high latitude structure being shifted from one another in the southern and northern hemisphere as a direct consequence of the strong quadrupolar component of the dynamo field. In a complementary study, [48] have used magnetograms from the Wilcox observatory to reconstruct the wind solution in 3-D from 1989 to 2002 (e.g. cycle 23). They show that the cycle variations of quantities such that the Alfvén radius or mass loss do not vary as much as in the work of [43]. This is due to a very interesting property of the solar magnetic field: when the field is at maximum of intensity (at maximum of the cycle) it possesses the most complex geometry, whereas at minimum of activity it is mostly dipolar. Since the dipolar field is the field with the weakest radial decay, one finds that there is a compensation effect between intensity of the field and geometry such that this limits the overall variations of mass loss and Alfvén radius. In [43] this was not as obvious and the cycle variations were amplified beyond what is expected in the Sun. In a recent attempt to characterize the magnetic field history of a young Sun, [45] have computed from ab-initio convection and dynamo computations how the field geometry changes as the solar-like star goes from being fully convective in the pre main sequence up to having a radiative interior occupying the inner 70% of the star when it reaches the zero age main sequence (ZAMS) and starts hydrogen nuclear burning (see Figure 9). They found that as the star approaches the ZAMS, its magnetic field becomes more non axisymmetric and smaller scale, in qualitative agreement with the work of [49] and [50]. Other questions such as what happens to the primordial magnetic field once the convective zone recedes towards the surface have been addressed. It has been found that the left-over fossil magnetic field is stable to Tayler-like kink instability. This was not known before. Likewise stellar wind solutions have been computed using the field geometry coming from the 3-D simulations and a powerful wind and magnetic torque has been obtained in agreement with the work of [28] using spectropolarimetric observations. We are thus in a situation where numerical simulations of stellar dynamos and winds can be efficiently coupled to start assessing the dynamical behavior of stars on longer secular time scales, by doing precise modelling of various phases of their life.

6 Conclusion We have seen that understanding the origin of stellar magnetic fields and how stellar magnetic activity impacts the space conditions around stars is not trivial but that simple toy models can help us doing so by setting the stage to more advanced numerical simulations. Dynamo action is a key physical mechanism in stars as it is the process at the origin of their magnetic fields and of both their large scale and their variability. Likewise stellar winds of solar like stars are a powerful driver for their long term evolution, in particular of their rotational history, as they carry both mass and angular momentum away from stellar surface. There is a subtle interplay between magnetic field geometry and the wind speed and efficiency, and we attempted in this short lecture to explain how dynamo and wind are coupled. Many challenges remain, as it is still difficult to anticipate what will be for instance the geometry of magnetic fields for a given star or its cycle period, and to quantify the exact mass loss for a given coronal temperature.

Multi-Dimensional Processes In Stellar Physics

Acknowledgement We are grateful to CNES and its Solar Orbiter program and ERC Synergy grant Whole Sun 810218, INSU/PNST and INSU/PNP for their financial support. We thank Victor Réville, Rui Pinto and Sean Matt for useful discussions and Yveline Lebreton for her careful read of the paper.

References [1] A. Skumanich, Astrophys. J.171, 565 (1972) [2] H.K. Moffatt, Magnetic field generation in electrically conducting fluids (Cambridge, 1978), http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode= 1978mfge.book.....M&link_type=ABSTRACT [3] F. Krause, K.H. Raedler, Mean-field magnetohydrodynamics and dynamo theory, Oxford, Pergamon Press, Ltd., (1980) [4] P.A. Davidson, An introduction to magnetohydrodynamics, cambridge univ pr edn. (Cambridge Univ Pr, 2001), ISBN 9780521794879, http://books.google.com/ books?id=t4dg3scoE5AC [5] P. Charbonneau, Liv. Rev. in Solar Phys.7, 3 (2010) [6] A.S. Brun, M.K. Browning, Liv. Rev. in Solar Phys.14, 4 (2017) [7] S. Sanchez, A. Fournier, J. Aubert, Astrophys. J.781, 8 (2014) [8] L. Jouve, A.S. Brun, Astron. Astrophys.474, 239 (2007) [9] P. Charbonneau, M. Dikpati, Astrophys. J.543, 1027 (2000) [10] M. Ossendrijver, Astron. Astrophys. Rev.11, 287 (2003) [11] D. Moss, J. Brooke, Mon. Not. Roy. Astron. Soc.315, 521 (2000) [12] P.J. Bushby, Mon. Not. Roy. Astron. Soc.371, 772 (2006) [13] M.L. DeRosa, A.S. Brun, J.T. Hoeksema, Astrophys. J.757, 96 (2012) [14] N.O. Weiss, S.M. Tobias, Mon. Not. Roy. Astron. Soc.456, 2654 (2016) [15] H.C. Spruit, Solar Phys.213, 1 (2003) [16] E. Covas, D. Moss, R. Tavakol, Astron. Astrophys.429, 657 (2005) [17] M. Rempel, Astrophys. J.647, 662 (2006) [18] L. Biermann, Zeitschrift fuer Astrosphysik 29, 274 (1951) [19] E.N. Parker, Astrophys. J.128, 664 (1958) [20] D.J. McComas, R.W. Ebert, H.A. Elliott, B.E. Goldstein, J.T. Gosling, N.A. Schwadron, R.M. Skoug, Geophys. Res. Lett.35, 1007 (2008) [21] E.J. Weber, L.J. Davis, Astrophys. J. Supp.148, 217 (1967) [22] L. Mestel, Mon. Not. Roy. Astron. Soc.138, 359 (1968) [23] L. Mestel, Mon. Not. Roy. Astron. Soc.140, 177 (1968) [24] S.D. Kawaler, Astrophys. J.333, 236 (1988) [25] A. ud Doula, S.P. Owocki, Astrophys. J.576, 413 (2002) [26] S. Matt, R.E. Pudritz, Astrophys. J.678, 1109 (2008) [27] V. Réville, A.S. Brun, S.P. Matt, A. Strugarek, R.F. Pinto, Astrophys. J.798, 116 (2015) [28] V. Réville, C.P. Folsom, A. Strugarek, A.S. Brun, Astrophys. J.832, 145 (2016) [29] K.B. MacGregor, M. Brenner, Astrophys. J.376, 204 (1991) [30] R. Keppens, K.B. MacGregor, P. Charbonneau, Astron. Astrophys.294, 469 (1995) [31] A. Krishnamurthi, M.H. Pinsonneault, S. Barnes, S. Sofia, Astrophys. J.480, 303 (1997) [32] S. Allain, Astron. Astrophys.333, 629 (1998) [33] P.A. Denissenkov, M. Pinsonneault, D.M. Terndrup, G. Newsham, Astrophys. J. Supp.716, 1269 (2010)

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[34] J. Bouvier, Role and Mechanisms of Angular Momentum Transport During the Formation and Early Evolution of Stars 62, 143 (2013) [35] F. Gallet, J. Bouvier, Astron. Astrophys.556, 36 (2013) [36] R.L.F. Oglethorpe, P. Garaud, Astrophys. J.778, 166 (2013) [37] M. Zhang, K. Penev, Astrophys. J.787, 131 (2014) [38] M. Benbakoura, V. Réville, A.S. Brun, C. Le Poncin-Lafitte, S. Mathis, Submitted to Astron. Astrophys.(2018) [39] P. Riley, R. Lionello, J.A. Linker, E. Cliver, A. Balogh, J. Beer, P. Charbonneau, N. Crooker, M. DeRosa, M. Lockwood et al., Astrophys. J.802, 105 (2015) [40] J.A. Linker, R.M. Caplan, C. Downs, P. Riley, Z. Mikic, R. Lionello, C.J. Henney, C.N. Arge, Y. Liu, M.L. Derosa et al., Astrophys. J.848, 70 (2017), 1708.02342 [41] K. Augustson, A.S. Brun, M. Miesch, J. Toomre, Astrophys. J.809, 149 (2015) [42] A. Strugarek, P. Beaudoin, P. Charbonneau, A.S. Brun, J.D. do Nascimento, Science 357, 185 (2017) [43] R.F. Pinto, A.S. Brun, L. Jouve, R. Grappin, Astrophys. J.737, 72 (2011) [44] M. Tokumaru, M. Kojima, K. Fujiki, Journal of Geophysical Research: Space Physics (1978–2012) 115 (2010) [45] C. Emeriau-Viard, A.S. Brun, Astrophys. J.846, 8 (2017) [46] J.M. Sokół, P. Swaczyna, M. Bzowski, M. Tokumaru, Solar Phys.290, 2589 (2015) [47] B. Perri, A.S. Brun, V. Réville, A. Strugarek, Submitted to the Journal of Plasma Physics (2018) [48] V. Réville, A.S. Brun, Astrophys. J.850, 45 (2017) [49] S.G. Gregory, J.F. Donati, J. Morin, G.A.J. Hussain, N.J. Mayne, L.A. Hillenbrand, M. Jardine, Astrophys. J.755, 97 (2012) [50] C.P. Folsom, P. Petit, J. Bouvier, A. Lèbre, L. Amard, A. Palacios, J. Morin, J.F. Donati, S.V. Jeffers, S.C. Marsden et al., Mon. Not. Roy. Astron. Soc.457, 580 (2016) [51] M. Velli, Astrophys. J.432, L55 (1994)

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7 Solutions of the exercises Solutions for exercise I: Stellar properties a) If the pressure in a star is negligible, the plasma accelerates due to gravitation such that d2 r GM(r) =− 2 . 2 dt r

(16)

d2 r du 1 d(u2 ) = = dt 2 dr dt2

(17)

We first note that

where we noted u = dr/dt. As a result, we obtain that d(u2 ) = −2

GM dr . r2

(18)

By integrating over the star and defining the average density such that M  ρ¯ 43 πR3 , one finds  3π the free fall-time to be τ f f = 32G ρ¯ . In the case of the Sun, this gives a life-time of 29 min! This dramatically shows the importance of pressure for the stability of a star. GM 2 b) The gravitational energy of a star is given by UG = β R . If we suppose the star is homogeneous, β = 3/5. A star radiates its gravitational energy in τ = UG /L . For the Sun, this gives τ ∼ 20 × 106 years. As we know the Sun is 4.5 Gyr old, this means that the energy source for the solar luminosity has to come from another source. c) The Sun is composed of approximately M /m p  1057 protons. The luminosity of the Sun corresponds consequently to ∼ 0.38 Mev/proton. Known chemical reactions typically deliver 1 eV/proton, while the gravitational energy in the Sun typically gives 2 keV/proton. Only fusion nuclear reactions are known to deliver up to 1 MeV/proton, which is enough to sustain the solar luminosity. This consequently shows that fusion reactions triggered in stellar cores are necessary to explain their strong luminosity during their lifetime. Solutions for exercise II: α − ω dynamo: basic concepts a) To derive the equation for A and B, we start by injecting B = ∇ × Aey + Bey into the mean field induction equation: ∂(∇ × Aey + Bey ) ∂t

= ∇ × ( V × (∇ × Aey + Bey ) + α(∇ × Aey + Bey ) − ηt ∇ × (∇ × Aey + Bey )) ,

We notice that for the equation for A, we have a ∇× operator that can be simplified by uncurling the equation, since for instance the time derivative and the curl operator can be swapped. We can then project along ey both the uncurled induction equation and the induction equation and obtain an equation for A and B respectively. Recall that for the equation for B we neglect the α-effect and only keep the ω-effect. Also recall that V = (0, ω0 z, 0) and the mean magnetic field B depends only on x and t. With all these assumptions, we find the following set of equations for A and B: ∂A ∂t ∂B ∂t

= =

∂2 A ∂x2 ∂A ∂2 B ω0 + ηt 2 ∂x ∂x αB + ηt

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b) Introducing the solutions (A, B) = (A0 , B0 )ei(kx+σt) into the above equations, results in the following dispersion relation: (iσ + ηt k2 )2 = α0 ω0 ik c) Expanding the relation above, one gets: σ2 − 2iσηt k2 − η2t k4 + α0 ω0 ik = 0. Since σ = σr + iσi is a complex number, one furthers gets: σ2r + 2iσr σi − σ2i − 2iσr ηt k2 + 2σi ηt k2 − η2t k4 + α0 ω0 ik = 0. Separating the imaginary from the real parts we obtain the following 2 equations: σ2r 2iσr σi

− σ2i + 2σi ηt k2 − η2t k4 = 0 − 2iσr ηt k2 + α0 ω0 ik = 0

The first equation yields a quadratic equation for σi : The determinant is Δ = 4σ2r , and σi = ηt k2 ± |σr |. From the second equation one deduces that: 2σr (σi − ηt k2 ) = −α0 ω0 k. We select solutions such that α0 ω0 < 0, and after injecting the expression of σi , one finds that only σ2r = −α0 ω0 k/2 can be real (recall √ that σr is real by definition). The solution is σ√ = iηt k2 + (1 ± i) |α0 ω0 k/2|. The threshold for dynamo action√is given by −σi = −ηt k2 ∓ |α0 ω0 k/2| ≥ 0, so this leads to selecting the solution −ηt k2 + |α0 ω0 k/2|, hence the marginal state σi = 0, implies that  ηt k2 = |α0 ω0 k/2|. Hence the product α0 ω0 determines the growth rate against Ohmic dissipation √ ηt . = ± |α0 ω0 k/2|. We The frequency of oscillation of the dynamo wave is given by the σ r √ choose the negative solution σr = − |α0 ω0 k/2|, to have a wave propagating in the positive x direction, such as to get exp(σr t + kx), i.e. a direction of propagation corresponding to an equatorward dynamo branch as in the Sun’s low latitude activity branch. Positive solution is also correct, but the main dynamo branch is then poleward unlike the Sun’s. Solutions for exercise III: Magnetic field lines reconstruction in solar-like star’s atmosphere a) If J = 0, this means that the magnetic field is curl free ∇ × B = 0. The field can consequently derive from a potential Φ, such that B = −∇Φ, since ∇ × (∇Φ) = 0 is always satisfied. Given that B is always divergence-free (∇ · B = 0), then ∇ · (∇Φ) = 0 implies ΔΦ = 0, i.e. the potential Φ follows Laplace’s equation. In spherical coordinate systems, spherical harmonics Ym = am Pm (cos(θ))eimφ , with Pm associated Legendre polynomials , are solutions to the equation, such that Φ = m (Cm r + Dm r−(+1) )Ym (you can easily verify that a posteriori). , b) Let’s assume we know the surface radial field Br (r = 1, θ, φ) = m Fm Ym , with Fm known spherical harmonics coefficients. Let us also assume that the field is purely radial at r = r ss , i.e. Φ(r = r ss ) = 0, we deduce that ∂Φ/∂r|r=1 = −Br (r = 1, θ, φ) and we get: Cm  − ( + 1)Dm = −Fm . From Φ(r = r ss ) = 0, we get Cm rss + Dm r−(+1) = 0. ss So Cm = −Dm r−(2+1) and Dm = Fm /(r−(2+1) + ( + 1)). Since we know Fm from the ss ss projection of the radial magnetic field at the surface of the star (using the magnetogram) onto spherical harmonics, we are able to deduce Cm and Dm and hence Φ everywhere in the domain.

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c) If γ  0 then ∇ × B = γB. Applying the divergence one gets: ∇ · (∇ × B) = 0 = ∇ · (γB) = B · ∇γ + γ∇ · B, which implies that B · ∇γ = 0 and that γ is constant along field lines. From ∇ × (∇ × B) = ∇ × (γB) and assuming that γ is constant, then one finds that ∇ × (γB) = γ∇ × B = γ2 B. Since ∇ × (∇ × B) = −ΔB, one finds that ΔB = −γ2 B. If there is no velocity field v = 0, then from the induction equation one can write: ∂B/∂t = ηt ΔB = −ηt γ2 B and solve a simple equation for the temporal evolution of the field. When γ is not constant one cannot derive such a simplified equation. Solutions for exercise IV: Parker’s wind solution for solar-like star a) Using the ideal gaz law, we immediately find that T = m p v2c /(2kB ). We suppose in this exercise that the temperature is constant in the corona, as a corollary the sound speed is consequently also constant. b) The isothermal stationary Euler equations can be written as   ∂r ρvr r2 = 0 , (19) GM 1 vr ∂r vr = − ∂r P − 2 . ρ r

(20)

We note that P(r) = v2c ρ(r) = v2c A/(vr r2 ) using Equation 19 and where A is an unknown constant. Injecting this result in Equation 6, we find the differential equation

 v2c 2v2 GM vr − ∂r vr = c − 2 . vr r r

(21)

c) If vr is a wind solution such that vr (rc ) = vc , using Equation 21 we find that at r = rc the left hand side is zero and thus rc = GM /(2v2c ). d) We first multiply Equation 21 by rc /v2c to obtain 

2 2 1 (22) ∂x M = − 2 . M− M x x We can integrate Equation 22 and multiply it by 2 to obtain F(M, x) = M 2 − 2 ln M − 4 ln x −

4 +C = 0, x

(23)

Constant C can be determined for a solution going through the critical point M = 1, which can be realized only when x = 1. In that particular case, C = −3. e) (bonus) We plot the contours of Equation 23 in the left panel of Figure 5. 6 classes of solutions can be identified: two unique solutions are shown by the black thick lines, which separate four other classes of solutions in red and blue. The red solutions are clearly unphysical, as two wind speeds are possible at a given distance from the star which is in contradiction with the spherical symmetry assumed here. The decelerating wind (black) and upper decelerating-then-accelerating blue solutions are all unrealistic, as the wind speed increases towards infinity close to the stellar surface. The breeze solutions, where the wind speed remains sub-sonic (lower blue lines), do not overcome the pressure paradox that lead to the development of the stellar wind theory by Parker. For information, it has further been shown by [51] that the breeze solution is unstable and instead leads to accretion. As a result, the only acceptable solution is the transsonic accelerating wind (colored in black), which becomes supersonic at the sonic point rc .

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Solutions for exercise V: A simple Weber-Davis magnetic wind model a) The magnetic field must satisfy ∇ · B = 0, thus r2 Br is constant (assuming axisymmetry, only the radial component of the field matters). Consequently, Br r2 Br = ρvr ρvr r2

(24)

is constant as well. b) The perfect conductor assumption naturally gives   r vr Bϕ − vϕ Br = constant .

(25)

We project the momentum equation on the aziumthal direction and obtain   ∂r rvϕ =

  Br ∂r rBϕ . 4πρvr

We recognize the factor Br /(4πρvr ) which is constant and thus

 Br L = r vϕ − Bϕ = constant . 4πρvr

(26)

(27)

c) We inject Equation 25 into 27 to obtain vϕ = Ω r

MA2 Lr−2 Ω−1  −1 MA2 − 1

.

(28)

d) To keep a finite azimuthal velocity, the numerator in Equation 28 has to go to zero when r = rA and MA = 1. As a result, we find L = rA2 Ω . The conserved angular momentum L is thus determined by the rotation of the star Ω and the Alfvén radius rA . The Alfvén radius can thus be viewed as a lever arm exerting a torque of magnetic origin effectively spinning down the star. e) The first thing to note is that the requirement that a solution both transsonic and transalfvénic exists fixes the values of vA and C for a given set of rA , rC (assuming here that √ vc = GM /rc ). We plot the contours of F A for the Sun and rA = 25R , rc = rA /5 in the right panel of Figure 5. Again, we find an accelerating solution for the wind, which crosses successively the sonic point (dashed black lines) and the alfvénic point (thin black lines), which corresponds to the solar situation following the same arguments than in Exercise IV. Solutions for exercise VI: Angular momentum redistribution in solar-like star c e a) Their initial angular momentum is respectively Jinit = Jcore = Icore Ωcore and Jinit = Jenv = Ienv Ωenv . c e b) Their final angular momentum is respectively Jfinal = Icore Ω f and Jfinal = Ienv Ω f . c) Let’s now find the exact quantity of angular momentum exchange ΔJ between the two zones required to have them rotate uniformly at the rotation rate Ω f . c e c e Applying total angular momentum conservation e.g. Jinit + Jinit = Jfinal + Jfinal , implies that c e the final state is: Jfinal = Jcore − ΔJ = Icore Ω f and Jfinal = Jenv + ΔJ = Ienv Ω f . A simple substitution yields: Ienv Jcore − Icore Jenv . (29) ΔJ = Icore + Ienv

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Solution for Exercise VII: Parker’s spiral in solar-like star a) As in Exercise V, the perfect conductor assumption, ∇ × E = 0, naturally translates into   (30) r vr Bϕ − vϕ Br = constant . In a frame at rest with the Sun (denoted with primes), v //B so vr Bϕ = vϕ Br and Equation 30 becomes (on the equatorial plane)     (31) r vr Bϕ − vϕ Br = r vr Bϕ − (vϕ + Ω r)Br = −Ω r2 Br , which we rearrange into

Bϕ vϕ − Ω r = . Br vr

(32)

Let’s note Ψ the angle between the magnetic field and the radial direction. Using Equation 32, we obtain Bϕ −Ω r + vϕ = . (33) tan (Ψ) = Br vr Far from the star, the wind in the non-corotating frame is essentially radial and thus vϕ vanishes to leave Ω r tan (Ψ) ∼ − . (34) vr b) In order to have an idea of the field lines pattern, on may try to parametrize Equation 34 with a parametric equation for (, φ). Using Equation 34, we naturally obtain on the equatorial plane  ˙ = vr , (35) rφ˙ = −Ω r which is easily solved for a starting point of the field line at (R , φ0 ) and assuming vr = vsw is constant with radius (at sufficiently large distance from the star)  = vsw t + R . (36) φ = −Ω t + φ0 Combining the equations, we find the parametric equation of the magnetic field lines to be φ() = −Ω

 − R + φ0 . usw

(37)

This form can be recognized as a standard Archimedean spiral, which was named Parker spiral in the context of the solar wind. The shape of the field lines is shown in Figure 10. c) The rotation rate of the Sun is Ω ∼ 2.6 × 10−6 s−1 , we thus find ψ ∼ 0.77 ∼ π/4 at the orbit of the Earth.

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Mercury

1 AU

Figure 10. The Parker spiral of the interplanetary magnetic field for a constant solar wind speed of usw = 400 km/s.

Multi-dimensional asteroseismology Daniel Roy Reese1,∗ 1

LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Univ. Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France Abstract. Asteroseismology enables us to probe stellar interiors using pulsa-

tion modes. Accordingly, much progress has been made in understanding stellar structure and evolution. However, various phenomena including rapid rotation, magnetic fields, and tidal effects, break spherical symmetry thus considerably complicating the use of asteroseismology to study stellar interiors. In this contribution, I describe some of the latest progress in applying asteroseismology to such stars, starting from stellar modelling and pulsation calculations, to asteroseismic interpretation.

1 Introduction Much progress has been made in stellar physics during the past century. Indeed, an improved understanding of the physical processes which occur in stars, both on a microscopic and a macroscopic scale, as well as the advent of computers has led to the development of a large variety of stellar evolution codes that provide a realistic description of stellar structure and its evolution as a result of nuclear reactions occurring in stars. This has enabled us to explain most of the main features in the Hertzsprung-Russell (HR) diagram which displays stellar luminosity as a function of temperature, as recently illustrated through Gaia’s DR2 catalogue [1]. Furthermore, asteroseismology, i.e. the study of stellar pulsations, has enabled us to probe stellar interiors using a variety of methods, thus placing increasingly stringent constraints on stellar models. For instance, helioseismology, the study of solar pulsations, played a key role in the solar neutrino problem by showing that the source of the problem was not coming from standard solar models but likely from neutrino physics, as was later confirmed. More recently, it has lead to a detailed description of the solar rotation profile, which continues to challenge current theoretical models. Asteroseismology has enabled us to probe rotation profiles within red giants thus showing the need for supplementary transport processes within stellar evolution codes. It is also the most promising way of obtaining stellar ages, a key requirement for the forthcoming PLATO mission. Nonetheless, these models and pulsation calculations rely on the assumption that the star is spherically symmetric, thereby allowing the use of 1D numerical methods. In a number of cases, this assumption is simply not true. For instance, roAp stars have very strong magnetic fields with dipolar or more complex geometries. The members of close binary systems undergo strong tidal effects which distort their geometry. A large proportion of intermediate mass and massive main sequence stars rotate rapidly, as shown for instance by spectroscopic ∗ e-mail: [email protected]

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surveys of v sin i values, i.e. equatorial velocities projected on the line-of-sight [2]. This leads to centrifugal distortion of the stellar structure as well as the Coriolis acceleration which affects all dynamical phenomena such as convection and pulsations. Some stars combine these effects thus leading to a very complex description of their physical structure. In order to take into account these effects while modelling stars and their pulsations, it is necessary to apply a multi-dimensional numerical approach. In what follows, I will focus on the effects of rapid rotation as being illustrative of the difficulties introduced by multidimensional effects. The following section will deal with the impact of rotation on stellar structure and evolution. This will then be followed by a description of how to calculate stellar pulsations with and without rotation. Section 4 will then describe the impact of rotation on stellar pulsations. This will then be followed by a description of various strategies for interpreting pulsations in rapidly rotating stars and deducing constraints on stellar structure. The last section will provide a brief conclusion.

2 Impact of rotation on stellar structure and evolution As will be described in what follows, stellar rotation has both short and long term consequences on stellar structure and evolution. The former are easy to understand and observe, whereas the latter become extremely important for stellar lifetimes and chemical yields. 2.1 Structural changes

One of the first and most obvious effects of stellar rotation is the centrifugal deformation. Indeed, gravity is counteracted by the centrifugal acceleration, especially near the equator, thus causing the equator to bulge. The second is gravity darkening. A simple explanation as proposed in [3] is that since the equator is further from the centre of the star, it is less heated by it, and therefore cooler and dimmer. Both centrifugal deformation and gravity darkening have recently been observed in various nearby rapidly rotating stars thanks to interferometry [e.g. 3–5]. A first approach to modelling rapidly rotating stars is to impose a conservative rotation profile, i.e. the centrifugal force derives from a potential. Given the orientation of the centrifugal force, a necessary and sufficient condition is that the rotation profile, Ω, only depends on s, the distance to the rotation axis. The equation of hydrostatic equilibrium, then takes on the form:   − ∇Ψ  C 0 = − ∇P − ∇Ψ (1) ρ where P is the pressure, ρ the density, Ψ the gravitational potential and ΨC the centrifugal potential. Taking the curl of this equation then leads to the relation:   0 = ∇ρ × ∇P 2 ρ

(2)

thus implying that lines of constant density and pressure coincide, i.e. the model is barotropic. Examples of models which make this assumption are rotating polytropic models [e.g. 6], and models based on the Self-Consistent Field (SCF) method [7, 8]. The advantage of such an approach is that a 2D description is only needed for the lines of constant total (gravity plus centrifugal) potential. Other quantities, such as as pressure, density, temperature etc. only require a 1D description as they only depend on the total potential (i.e. they are constant on isopotentials).

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In order to model gravity darkening, a popular approach is to assume that the effective temperature is related to the effective gravity via a power law: T eff ∝ gβeff

(3)

where the effective gravity includes the gravity and centrifugal acceleration. As initially predicted by [9], the exponent β takes on the value 0.25 for a radiative envelope. For a convective envelope, one should use β = 0.08 according to [10]. Interferometric observations favour values between these two extremes. More recently, [11] came up with a more realistic model based on two assumptions: the radiative flux lines up with the local effective gravity  · F = 0 (as F = − f (r, θ)geff , and the divergence of the radiative flux in the envelope is zero, ∇ expected in regions without nuclear reactions). This provides an alternate gravity darkening law which compares favourably with observations and 2D numerical simulations based on the ESTER code (which will be described below). A practical implication of the centrifugal deformation and gravity darkening is that the position of a star in an HR diagram is modified compared to the non-rotating case and depends on the stellar inclination. Indeed, a rapidly rotating star seen pole-on appears brighter and hotter than the same star seen equator-on. Various authors have taken this effect into account when studying clusters [12] or explaining a new class of pulsating stars [13]. 2.2 Baroclinic effects

Although barotropic models provide a good description of centrifugal deformation, they remain unrealistic with respect to the star’s thermal structure. Indeed, lines of constant temperature tend to be more spherical because they depend on energy propagation within the star whereas lines of constant pressure tend to be more oblate as a result of hydrostatic equilibrium. Accordingly, the two are not expected to coincide in stars, as opposed to what barotropic models predict, and one has to deal with a baroclinic structure instead. This implies that a full 2D approach is necessary to describe the stellar structure, in particular the various thermodynamic quantities such as pressure, temperature and density. The mismatch between isobars, isotherms, and hence isochores leads to baroclinic flows, namely meridional circulation and differential rotation. This, in turn, leads to various forms of turbulence, for instance through shear instabilities, and enhances transport of chemical species and angular momentum. As a result, stellar lifetimes and chemical yields are modified, thanks to fresh nuclear fuel being transported to the stellar core and nuclear by-products being transported out to the surface. The HR diagrams provided in [14] illustrate the impact of rotation on stellar evolution. The extensive monograph by [15] provides a detailed description of the above effects. Using the approach described in [16, 17], various groups have implemented stellar evolution codes with a perturbative description of the effects of rotation, including baroclinic effects. These include the Starevol code [18], the Geneva code [19], and CESTAM [20]. As described in [21, and references therein], including the effects of rotation has helped to improve the agreement between various observations in massive and intermediate mass stars, and theoretical predictions. Nonetheless, some discrepancies remain such as N enrichment in early B stars in the Large Magellanic Cloud [22, 23]. In order to go further, the goal of the ESTER project [24, 25] is to use a full 2D approach to model the effects of rotation on stellar structure and evolution. Currently, it fully includes baroclinic effects thus allowing it to derive a self-consistent differential rotation profile and meridional circulation. The next steps consist in including convection in envelope regions and improving the implementation of nuclear reactions in the core in order to implement stellar evolution.

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3 Calculating pulsations Once an equilibrium model of a star has been calculated, it is possible to derive pulsations modes by perturbing this model. In contrast to running waves, pulsation modes are standing waves inside of stars with specific resonance pulsation frequencies. Accordingly, their time dependence is of the form exp(iωt) where ω is the pulsation frequency. These modes are obtained by applying the fluid dynamic equations to the perturbed model. In what follows, I will first describe pulsation modes in non-rotating stars before addressing the case of rotating stars. 3.1 The non-rotating case

There are two main type of pulsation modes in non-rotating (non-magnetic) stars: acoustic modes (or p-modes) and gravity modes (or g-modes). The restoring force for the former is pressure, and these tend to have high pulsation frequencies. The restoring force for the latter is buoyancy and these tend to have low frequencies. Also, one should not confuse gravity modes with gravitational waves which are general relativistic perturbations to space-time. The pulsation modes are governed by the following set of equations: δρ   +∇·ξ ρ0 ⎛ ⎞

 

  0 ⎟⎟⎟ ⎜⎜⎜ ξ · ∇P P0  δP ∇P0 δρ δP    ⎜ ⎟⎟ = − ∇ − + − ∇Ψ + ∇ ⎜⎝ ρ0 P0 ρ0 ρ0 P0 ρ0 ⎠

0 = −ω2ξ ΔΨ δP P0

= 4πGρ δρ = Γ1 ρ0

(4) (5) (6) (7)

where quantities with a subscript “0” are equilibrium quantities, quantities preceded by a “δ” Lagrangian perturbations, and those with a prime, Eulerian perturbations. ω is the pulsation frequency, ρ the density, P the pressure, ξ the Lagrangian displacement, and Ψ the gravitational potential. The first equation is the continuity equation and represents the conservation of mass. The second is Euler’s equation and corresponds to the conservation of momentum. The third is Poisson’s equation. The fourth is the adiabatic relation, and corresponds to neglecting energy transfers during the pulsations, i.e. applying the adiabatic approximation. If instead one decides to take energy transfers into account, then the adiabatic relation should be replaced by the perturbed energy conservation equation, and a perturbed version of the energy flux equation. Furthermore, various perturbed equations of state and opacities are needed to close the system. Given the spherical symmetry of the star, pulsation modes are described by functions of the following form: Φ(r, θ, φ) = Φn (r)Ym (θ, φ) (8) where Ψ is some scalar quantity such as the pressure or density perturbation, Φn a radial function, and Ym a spherical harmonic. A vector quantity, such as the Lagrangian displacement, would take on the form: ξ(r, θ, φ) = ξn (r)R  m (θ, φ) + ηn (r)S m (θ, φ)  

(9)

 m = Y mer , and S m = ∂θ Y meθ +(∂φ Y m / sin θ)eφ . Injecting where ξn and ηn are radial functions, R      these forms into Eqs. (4) to (7) and recalling that the time dependence is of the form exp(iωt) leads to a set of ordinary differential equations in terms of the variable r, the solution of which

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provides the unknown radial functions Ψn (r), ξn (r) etc. This set of equations also turns out to be an eigenvalue problem where ω is the eigenvalue, and the radial functions the eigenvector. Three quantum numbers intervene in these solutions. The quantum numbers  and m apply to spherical harmonics and are respectively the harmonic degree and azimuthal order.  − |m| is the number of surface nodal lines parallel to the equator. m is the periodicity in the eφ direction and obeys the relation − ≤ m ≤ . In particular, m = 0 corresponds to axisymmetric modes, and m  0 to non-axisymmetric modes. The former are stationary waves whereas the latter travel around the equator. The non-axisymmetric modes can further be subdivided into sectoral modes ( = |m|), and tesseral modes (0 <  < |m|). Finally, n is the radial order and applies to the radial functions. In most cases, it can be defined as the number of nodes in the radial direction, where acoustic type nodes are counted positively and gravity type nodes negatively [26–28]. More recently, [29] derived a rigourous mathematical definition applicable in all cases. 3.2 The rotating case

In the rotating case, Euler’s equation takes on a more complicated form:      Ω2e  × ξ − ξ · ∇  × (ω + mΩ)ξ − Ω  × Ω −(ω + mΩ)2ξ = −2iΩ ⎛ ⎞

 

  0 ⎟⎟⎟ ⎜⎜⎜ ξ · ∇P ∇P0 δρ δP P0  δP    ⎜ ⎟⎟ − + − ∇Ψ + ∇ ⎜⎝ − ∇ ρ0 P0 ρ0 ρ0 P0 ρ0 ⎠      0 − ξ · ∇ρ  0  0 ∇ρ  0 ∇P ξ · ∇P + ρ20

(10)

 = Ωez the rotation vector, ez the unit vector along the where Ω is the rotation profile, Ω rotation axis,  the distance to the rotation axis, and e the unit vector in the direction of increasing . In this equation, the first term after the equality corresponds to the Coriolis acceleration. Also, the term on the last line does not vanish when the stellar structure is baroclinic, i.e. when isobars and isochores do not coincide. The continuity equation, Poisson’s equation, and the adiabatic relation keep the same form. If non-adiabatic calculations are performed, supplementary terms resulting from stellar rotation intervene in the energy conservation equation. A further complication in the rotating case is that the centrifugal force distorts the stellar structure as mentioned earlier. In order to be able to apply the boundary conditions without losing numerical accuracy, it is necessary to set up a spheroidal coordinate system which follows the shape of the star. Accordingly, the pulsation equations need to be expressed explicitly in terms of this coordinate system. For this, one can use tensors. In what follows, (ζ, θ, φ) will denote this new coordinate system, where ζ is a pseudo-radial coordinate. ζ = 1 will correspond to the stellar surface. Given that spherical symmetry is lost, pulsation modes are no longer proportional to a single spherical harmonic but are instead described as a sum of spherical harmonics. One has: ∞ . Φ(ζ, θ, φ) = Ψm (ζ)Ym (θ, φ) (11) =|m|

for scalar quantities and ξ(ζ, θ, φ) =

∞  . =|m|

 m (θ, φ) + ηm (ζ)S m (θ, φ) + τm (ζ)T m (θ, φ) ξm (ζ)R   



(12)

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for vectorial quantities, where Ψm , ξm , ηm , and τm are radial functions, and Tm = (∂φ Ym / sin θ)eθ − ∂θ Ymeφ . The summation is only carried out over  and not over m because the star is symmetric with respect to the rotation axis. Furthermore, since the star is symmetric with respect to the equatorial plane, only even  or odd ’s intervene in the above sums, except for the τm terms which have the opposite  parity. From a physical point of view, modes with even  + |m| values are symmetric with respect to the equator, and those with odd  + |m| values antisymmetric. Inserting the above expressions into the pulsation equations and projecting the equations onto the spherical harmonic basis (by calculating the dot product between the equations and the complex conjugates of the different spherical harmonics, and integrating over 4π steradians) then leads to an infinite system of ordinary differential equations in terms of the radial variable r, the solution of which provides the unknown radial functions. In practise, such a system is truncated at some maximal harmonic degree before being solved numerically. Such a system is also an eigenvalue problem where ω is the eigenvalue and the radial functions correspond to the eigenmode. Besides carrying out full 2D pulsation calculations, various other methods have been used to take into account, at least approximately, the effects of rotation on stellar pulsations. These include: • the perturbative approach: the effects of rotation are included as corrections to the nonrotating solutions. Furthermore, these corrections are described as a power series in terms of Ω, the rotation rate, which is treated as a small parameter. [30, 31] developed these corrections to first order, [32–35] to second order, and [36–38] to third order. In the third order methods, two small parameters are used: one for the effects of centrifugal distortion, and one for the effects of the Coriolis force. [39–41] came up with validity domains for this approach by comparing it with full 2D calculations.  and the centrifugal distor• the traditional approximation: the horizontal component of Ω tion are neglected. As a result, the pulsation equations become separable in r and θ. The horizontal structure of the pulsation modes is no longer described by spherical harmonics, but by Hough functions [42]. This approach was first used in the stellar context by [43] and was subsequently developed by several authors [e.g. 44–47]. • characteristics and ray dynamics: this does not provide pulsation modes directly, but rather provides insights on propagation domains, mode geometry, mode classification, and quantification. This type of approach has been used to study acoustic modes [48–51] as well as inertial and gravito-inertial modes [e.g. 52–57].

4 The effects of rotation on pulsations Having described how to calculate pulsation modes in rotating stars, we now turn our attention to the effects of rotation on stellar pulsations. A first effect which appears even at low rotation rates are rotational multiplets. Indeed, in the non-rotating case, modes with the same n and  values but different azimuthal orders have the same frequencies, i.e. these modes are degenerate. Introducing rotation removes this degeneracy, and the modes no longer have the same frequencies. Hence, a group (or multiplet) of 2 + 1 frequencies appears, one for each value of m between − and . This group is further subdivided into axisymmetric modes (i.e. with m = 0), prograde modes (i.e. non-axisymmetric modes that travel in the same direction as the star’s rotation) and retrograde modes (i.e. modes which travel in the opposite direction from the star’s rotation). Given the time dependence of the pulsation modes (∝ exp(iωt)) and their azimuthal behaviour (∝ exp(imφ)), prograde modes have negative azimuthal orders and retrograde modes positive azimuthal orders. A time dependence of the form exp(−iωt) is also

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used by many authors, and leads to the opposite result, i.e. prograde modes having positive azimuthal orders. At this point it is also useful to distinguish between pulsation frequencies in the corotating frame, ωcorot. , and those in an inertial frame, ωinert. . The former correspond to the pulsation frequencies a hypothetical observer would see if they were travelling around the star at the same speed as the rotation rate. The latter corresponds to the pulsation frequencies observed by a stationnary observer (i.e. this is what corresponds to actual observations). The two are related by the following formula (using our convention for the time dependence): ωinert. = ωcorot. − mΩ

(13)

where −mΩ could be described as a Doppler term or an advection term. We note that Eq. (10) is expressed in an inertial frame. A second effect of rotation is to increase the occurrences of avoided crossings. Avoided crossings occur when the frequencies of two coupled modes would cross each other as a function of some stellar parameter such as age or rotation rate, but cannot because the two modes are coupled. Instead of crossing each other, the two frequencies come close and then separate again, somewhat like two branches of a hyperbola, and the modes gradually exchange characteristics. Hence, at the closest encounter, the two modes have a mixed character. An example of a rotationally-induced avoided crossing is given in [58]. In rotating stars, avoided crossings are much more frequent because many more modes are coupled than in the nonrotating case. Indeed, modes with the same m and symmetry with respect to the equator are coupled. In the non-rotating case, modes would furthermore have to have the same  value. At this point it is useful to distinguish between gravito-inertial modes and acoustic modes. 4.1 Gravito-inertial modes

Like their non-rotating counterparts, gravito-inertial modes have low pulsation frequencies. The restoring force for these modes is a combination of buoyancy and the Coriolis acceleration. The effects of the Coriolis acceleration scale with 2Ω/ω, hence they become more important at low frequencies. Stars which are likely to pulsate with such modes are γ Dor stars, Slowly Pulsating B (SPB) stars, and Be stars. A subclass of these modes are the inertial modes which owe their existence to the Coriolis force. These can furthermore be subdivided into regular modes and modes which become singular when the viscosity approaches zero. An extensive literature describes these modes [e.g. 52, 53, 59–62]. In what follows, I will focus on modes that become regular g-modes in the non-rotating limit. In non-rotating stars, gravity modes with a given  value are uniformly spaced in period according to asymptotic theory [63]. When rotation is introduced, the period spacing also depends on m and 2Ω/ω. This was first shown using the traditional approximation [43] and later confirmed using full 2D calculations [64, 65]. Furthermore, strong chemical gradients introduce glitches in the period spacing in much the same way as in the non-rotating case [65]. From a geometrical point of view, these modes can be divided into the following categories: • sub-inertial modes: the co-rotating pulsation frequencies, ωcorot. , of these modes are below the Coriolis frequency 2Ω. Their geometric structure is confined to near-equatorial regions as a result of critical surfaces [40, 53, 62]. • classical super-inertial modes: their co-rotating frequencies satisfy ωcorot. > 2Ω and their geometry is very similar to their non-rotating counterparts.

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• Rosette modes: their frequencies also satisfy ωcorot. > 2Ω but their geometric structure is very different from that of their non-rotating counterparts. These modes were first discovered by [64] using full 2D calculations, and later studied by [66–68] using a perturbative approach. In particular, [66, 67] showed that these modes are caused by families of nearly degenerate modes in the non-rotating case. Figure 1 illustrates meridional cross-sections of these different types of modes.

Figure 1. Meridional cross-sections of various gravito-inertial modes (lower row) and their non-rotating counterparts (upper row) in polytropic models. The kinetic energy is plotted. Taken from [69].

4.2 Acoustic modes

At the upper end of the pulsation spectrum are acoustic modes, for which the restoring force is pressure. These modes are especially affected by the centrifugal distortion of the star. To evaluate the effects of centrifugal distortion, one can compare the amount of flattening,  = 1 − Rp /Req ∝ Ω2 , with the mode wavelength λ ∝ 1/ω, where Rp and Req are the polar and equatorial radii respectively. Hence, these effects scale with ωΩ2 . Rapidly rotating stars which pulsate with such modes are δ Scuti stars and β Cephei stars. Using ray dynamics in polytropic models, [48, 49] found different classes of modes characterised by different mode geometries and frequency distributions. [70] showed that such classes are also present in more realistic models provided the rotation profile is not too differential. These classes include: • island modes: the structure of these modes closely follows a 2-periodic ray trajectory which goes around the equatorial region. These are the rotating counterparts to modes with low  − |m| values, i.e. with few nodal lines parallel to the equator. • chaotic modes: these modes are characterised by a chaotic spatial distribution, where the node placement is uneven. The associated ray trajectories are chaotic. These are rotating counterparts to modes with medium  − |m| values.

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203

• 6-period island modes: the structure of these modes closely follows a 6-periodic ray trajectory which goes around the star. These are also rotating counterparts to modes with medium  − |m| values. • whispering gallery modes: these modes are the most similar to their non-rotating counterparts and are characterised by a geometric structure near the stellar surface with an internal caustic. These are the rotating counterparts to modes with high  − |m| values. Meridional cross sections of modes from three of these classes are shown in Fig. 2. Apart, from the chaotic modes, the pulsation spectra of the various classes of modes are characterised by asymptotic formulas [49]. The frequencies of the chaotic modes instead tend to follow a statistical distribution, although recent work by [71] has shown the presence of quasi regularities including a recurrent spacing close to the large frequency separation. Island

Chaotic

Whispering gallery

n

Figure 2. Meridional cross-sections of various acoustic modes in a 3 M ESTER model at 70 % of the break-up rotation rate. The Lagrangian pressure perturbations, normalised by the square root of the equilibrium pressure, are plotted.

Much attention has been paid to the island modes since they are the most visible of the regular (non-chaotic) modes. In particular, an alternate set of quantum numbers, appropriate for island mode geometry can be derived as illustrated in the left panel of Fig. 2. n˜ corresponds to the number of nodes along the ray orbit and ˜ to the number of nodal lines parallel to the orbit. These quantum numbers can be related to the usual spherical quantum numbers thanks to what can be described as “node conservation” [72]: n˜ = 2n + ε  − |m| − ε ˜ = 2 m ˜ = m

(14)

ε =  + m modulo 2

(17)

(15) (16)

An empirical fit to the island mode frequencies yields the following formula [70]: ω  Δn˜ n˜ + Δ˜ ˜ +

Δm˜ m2 − mΩ + α˜ n˜

(18)

where the parameters Δn˜ , Δ˜ , Δm˜ and α˜ depend on stellar structure. [48, 49] showed that Δn˜ is the inverse travel time along the periodic orbit from surface to surface, and [50, 51] derived a semi-analytical formula for Δ˜ . Also, Δn˜ corresponds to half the large separation from non-rotating stars, as can be seen thanks to Eq. (14). Furthermore, [72] showed that this parameter scales with the mean density, even at large rotation rates. A more detailed description of the behaviour of the modes as a function of m may be obtained thanks to the generalised rotational splittings. These are defined as (ω−m − ωm )/2m,

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where ω−m is the frequency of a prograde mode and ωm the frequency of its retrograde counterpart. Indeed, these splittings are approximately related to the rotation profile via the following integral formula: + Ωeff Cm + C−m ω−m − ωm Ωeff −m  m + 2m 2 2 where

" V

Ωeff = "

Ωρo ξ2 dV

V

ρo ξ2 dV

" i C= m

V

   · ξ∗ × ξ dV ρo Ω " ρ ξ2 dV V o

(19)

(20)

Ωeff quantifies how much the mode is advected by the differential rotation profile, whereas C is a first order description of the effects of the Coriolis force. These formulas are valid when the Coriolis force only plays a minor role, as is the case for high frequency acoustic modes, and when the prograde/retrograde pair of modes have a very similar geometric structure (for instance, they should not be involved in an avoided crossings as this can cause the two to differ). They are essentially the same as what is obtained in the non-rotating case, except that the integrals are carried out using the pulsation modes and stellar geometry at rapid rotation rates. Eq. (19) may provide the basis for probing the rotation profile via seismic inversions, provided a sufficient number of prograde/retrograde mode pairs are observed and identified.

5 Asteroseismology We now turn our attention to the interpretation of observed pulsations in rotating stars. We first discuss various approaches used for interpreting the observations before describing a couple of rapidly rotating stars which have been studied seismically. 5.1 Frequency patterns

One of the first and most obvious strategies for interpreting the observations is to search for theoretically predicted frequency patterns in the observations. Indeed, pulsation frequencies are the most readily available data, especially with the advent of high-precision space photometry missions such as MOST, CoRoT and Kepler. For gravito-inertial modes, a first strategy is to look for clumps of modes roughly separated by the rotation rate. Indeed, when co-rotating frequencies are substantially smaller than the rotation rate, the frequencies in the inertial frame are dominated by the advection term −mΩ. Hence, the frequencies of modes with the same azimuthal order are clumped together and the separation between these clumps can be used as a first estimate of the rotation rate. Furthermore, non-adiabatic calculations can be used to predict which clumps contain excited modes. This strategy has been used to interpret various observed stars [e.g. 73–77]. [78] has however noted some discrepancies between such seismic determinations of the rotation rate and other independent estimates. More recently, regular period spacings, which vary as a function of the period, have been detected in a number of stars, thus providing constraints on the internal rotation rate [e.g. 79–83]. [84] has devised a new approach based on the traditional approximation and the use of a stretched periodogram in order to identify such mode sequences along with their  and m values. This approach has been used by [85] to obtain near core-rotation rates as well as buoyancy radii for a number of γ Dor stars, thus revealing a larger number of slow rotators than expected from theoretical predictions.

Multi-Dimensional Processes In Stellar Physics

In the acoustic domain, recurrent frequency spacings have been found in a number of rotating δ Scuti stars using techniques such as the Fourier transform of the frequency spectra, their auto-correlation function, or sequence search methods [86–92]. [93] has taken on a complementary approach by sorting and stacking pulsation spectra from multiple δ Scuti stars, thus revealing common patterns. In a number of cases, these spacings have been interpreted as the large frequency separation, Δω. [90] was able to confirm that the scaling relation between Δω and the mean density depends little on the rotation, thus providing a way to constrain the mean density. [94] has also shown how to combine this with the parallax in order to constrain the surface gravity. Once the large separation is obtained, it is possible to construct an échelle diagram, i.e. a plot which shows the pulsation frequencies, ω, as a function of the frequencies modulo the ˜ m) values in the rotating large separation, ω mod Δω. Modes with similar (, m) values or (, case should approximately line up as vertical ridges in such diagrams. Such diagrams have been produced for a number of rotating stars [89, 95] and may help with identifying the observed modes, i.e. find their quantum numbers. Nonetheless, mode identification remains a difficult problem for such stars, and supplementary constraints are needed.

5.2 Mode identification techniques

Two types of methods have been developed in order to observationally constrain the identification of modes: those based on multi-colour amplitude ratios and phase differences, and those based on spectroscopic line profile variations. These methods have mostly been applied to slowly rotating stars and need to be adapted to rapid rotation. The first method consists in observing a pulsating star in multiple photometric bands and measuring the pulsation amplitudes and phases in each of these bands. The ratios between the amplitudes in different bands and the associated phase differences for a given pulsation mode depends only on the mode’s geometry. This can then be compared with theoretical predictions in order to constrain the mode’s identification. One of the advantages of this method is that the intrinsic mode amplitude, which is very difficult to predict theoretically, has been factored out. Accordingly, it is sufficient to calculate the ratio and phase differences of geometric disk integration factors, also called mode visibilities, in order to come up with theoretical predictions. Various authors have looked into calculating mode visibilities in rapidly rotating stars [49, 96–99]. Using such mode visibilities, [100] identified modes thanks to a χ2 minimisation in the SPB star μ Eridani. Later on, [101] described a strategy for grouping together modes with similar identifications using their amplitude ratios, provided the star has a sufficient number of modes. The second method consists in making multiple spectroscopic observations of a given star to see how the line profiles vary as a function of time. Indeed, the oscillatory movements induced by pulsation modes lead to a Doppler signature which can be seen as deformations or bumps within spectroscopic line profiles that vary over time. These line profile variations (LPVs) depend on the geometry of the mode thus providing constraints on its identification. Compared to multicolour amplitude ratios and phase differences, it provides more detailed information even if it is also more difficult to decipher. LPVs have been observed in several rapidly rotating stars [e.g. 102, 103]. Various authors have also calculated theoretical LPVs in rapidly rotating models [45, 104–106], such as the one shown in Fig. 3. Tools such as FAMIAS [107] can be used to identify modes based on their LPVs but need to be extended in order to include the effects of rapid rotation.

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Figure 3. (upper left) Line profile variations based on non-adiabatic pulsation calculations in a 9 M ESTER model rotating at 50 % of the break-up rotation rate. The other panels show: (upper right) the first and second moments, (lower left) a harmonic decomposition of the variations as a function of Doppler shift from the centre of the line, and (lower right) a meridional cross-section of the pulsation mode. Taken from [106].

5.3 Observed stars

Two stars which have recently been studied seismically using a combination of some of the above results are α Ophiuchi (also known as Rasalhague) and β Pictoris. α Ophiuchi is a δ Scuti star in a binary system which has been observed with the MOST satellite [108] and the CHARA interferometric array [109]. Accordingly, the star has a well-determined mass, as well as polar and equatorial radii, and 57 pulsation frequencies have been detected. Various authors have carried out asteroseismic investigations of this star [110–112]. The study by [112] is particularly interesting because they look at mode frequencies, visibilities, and non-adiabatic excitation to see which theoretical modes are most likely to match the observed ones. Due to current limitations in either the ESTER stellar structure code or the

Multi-Dimensional Processes In Stellar Physics

TOP pulsation code, it proved to be difficult to find excited acoustic modes to match the observations, but future improvements may address these limitations. β Pictoris is a δ Scuti star for which an exoplanet has been detected through direct imagery [113]. At the expected time of the transit of the planet’s Hill sphere, β Pictoris was observed by multiple ground-based and space-based instruments, including the BRITE constellation, thus providing photometric time series in five bands. Using a set of 1.8 M SCF models, [114] carried out an MCMC search for seismic solutions which simultaneously reproduce the pulsation spectra and multi-colour amplitudes. Figure 4 shows one such solution corresponding to a near equator-on configuration. Although large discrepancies remain between the observations and theoretical predictions, it is hoped that improvements such as the use of full non-adiabatic calculations or larger sets of theoretical modes may lead to better agreement.

Figure 4. One of the seismic solutions obtained for β Pictoris. The upper panel shows the fit to the multicolour pulsation amplitudes whereas the lower panel compares the observed and theoretical spectra. Taken from [114].

6 Conclusion In summary, various phenomena such as stellar rotation, magnetism, and tidal effects break the spherical symmetry of stars. This causes multiple effects on stellar structure and evolution and leads to many unanswered questions. This also has an important impact on stellar pulsations and seismology. Indeed, a multi-dimensional approach is needed to correctly calculate

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pulsation modes and come up with predictions (mode frequencies, visibilities, line profile variations) which can be compared with observations. As can be seen, much progress has been made in understanding these effects and interpreting seismic data. Nonetheless, more work is needed to fully exploit the data from current (MOST, CoRoT, Kepler, BRITE) and future space-missions (TESS, PLATO) as well as ground-based instruments (e.g. SONG).

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Multi-dimensional physics of core-collapse supernovae Jérôme Guilet1,2,∗ 1 2

Département d’Astrophysique, IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France Université Paris Diderot, AIM, Sorbonne Paris Cité, CEA, CNRS, F-91119 Gif-sur-Yvette, France Abstract. Multidimensional fluid motions are one of the key ingredients of core-collapse supernovae. In this chapter, we give a pedagogical introduction to the physics of the different instabilities responsible for this multidimensional dynamics. We first describe hydrodynamic instabilities such as Ledoux convection inside the protoneutron star, neutrino-driven convection between the protoneutron star surface and the shock, the standing accretion shock instability and the low-T/W instability. We also review the different mechanisms that can amplify the magnetic field, in particular the magnetorotational instability and the convective dynamo.

1 Introduction Core collapse supernovae are cataclysmic events that mark the final instants of massive star evolution. The few seconds that separate the start of the collapse and the launch of the explosion provide a crucial link between stellar evolution and compact object formation and determine many of the important consequences of core-collapse supernovae, such as their energy budget and the explosive nucleosynthesis. Several lines of observational evidence show that core-collapse supernovae are fundamentally aspherical explosions. These include the large-scale asymmetry in the distribution of heavy elements in supernova remnants [1, 2], the polarisation of the light emitted in the nebular phase [3, 4] and the large velocities that are imparted to neutron stars during their birth [5]. Theoretical and numerical studies suggest that this asymmetry can be traced back to hydrodynamic instabilities, which develop in the short time interval between the collapse and the launch of the explosion. This chapter focuses on the dynamics taking place just after the formation of a protoneutron star (PNS), i.e. when the collapse of the iron core is stopped because of the stiffening of the equation of state when nuclear densities are reached. A shock is formed at a radius around 10 km and starts to propagate outwards. The shock, however, quickly loses energy by neutrino emission and the dissociation of atomic nuclei into free neutrons and protons to such an extent that it finally stalls at a radius between 100 and 200 km. This stalled shock phase, during which material continues to accrete onto the PNS, is the subject of intense investigations aiming to understand the mechanism by which the shock can be revived to create a supernova explosion. The leading scenario for standard supernovae is the so-called neutrino-driven mechanism [see 6, for a review]. In this picture, a small fraction of the intense flux of neutrinos emitted at the PNS surface are reabsorbed by the semitransparent ∗ e-mail: [email protected]

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medium located between the PNS surface and the shock. Above a critical neutrino luminosity, this heating is sufficient to push the shock outwards and drive an explosion. advanced numerical simulations of core-collapse supernovae show that multi-dimensional dynamics is a crucial ingredient for a successful explosion in this scenario. One-dimensional simulations in spherical geometry, however, fail to provide successful explosions except for the lowest mass progenitors [7]. Two-dimensional and three-dimensional simulations show that the multidimensional character of the dynamics provides a crucial support to neutrino-driven explosions. These simulations also predict that gravitational waves and modulations of the neutrino flux should be observed in the next nearby galactic supernova, thereby providing a direct probe into this complex dynamics. A spherically symmetric collapse and explosion would not emit any gravitational wave, such that any signal detected from a future nearby supernova would be a direct probe of the multidimensional dynamics. In this chapter, we will only briefly describe examples of signals characteristic of a few of the instabilities considered. A more complete review of gravitational waves in core-collapse supernovae can be found in [8] and in [9] for the neutrino signal. In section 2, we describe successively the different hydrodynamic instabilities that can grow inside and around the protoneutron star. In section 3, we focus on the origin and the role of magnetic fields in this multidimensional dynamics.

2 Hydrodynamic instabilities Neutron stars possess a solid crust and part of the liquid interior is probably in a superfluid and/or superconducting state which is arguably very different from usual fluid dynamics. This chapter focuses on the first seconds after the formation of the neutron star, when it is extremely hot and not yet very neutronized (it is then referred to as a protoneutron star or hereafter PNS). At the high temperatures characteristic of this phase, a solid crust and superfluid/superconducting states do not occur such that the dynamics can be described with the usual equations of (magneto)hydrodynamics, albeit with the specific (and uncertain) equation of state of these ultra-high densities and temperatures. Several kinds of convection take place in core-collapse supernovae. First of all, the propagation of the shock creates a negative entropy gradient because the shock weakens with time. This negative entropy gradient drives a transient phase of so-called prompt convection, which dies out within a few tens of milliseconds after a few turnover times because it lacks a continuous driving. We describe in more details the two self-sustained kinds of convection: Ledoux convection inside the protoneutron star (section 2.1) and neutrino-driven convection in the post-shock layer (section 2.3). We also discuss the possible existence of double diffusive convection inside the PNS (section 2.2). Finally, we turn to two other hydrodynamic instabilities: the standing accretion shock instability (section 2.4) and the corotation or low-T/W instability (section 2.5). 2.1 Ledoux convection inside the protoneutron star

The diffusion of neutrinos inside the PNS and their emission from the PNS surface lead to its cooling and deleptonization. This drives negative gradients of both entropy and lepton fraction in some region of the PNS, which are sustained for a few tens of seconds. The lepton fraction Yl , defined as the lepton number per nucleon, is a conserved quantity if neutrinos are trapped and plays the role of a composition variable. The combination of these two gradients gives rise to Ledoux convection, which takes place if the square of the Brunt-Väisälä frequency is negative

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⎡  ⎤

 ∂ρ ds dYl ⎥⎥⎥ dΦ 1 ⎢⎢ ∂ρ ⎥⎦ + , N 2 = − ⎢⎢⎣ ρ ∂s Yl ,p dr ∂Yl s,p dr dr

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(1)

where s is the entropy, P the pressure, ρ the density, Φ the gravitational potential and r the radius. The sign of the thermodynamic derivatives can be non-trivial but is typically negative such that instability is driven by negative gradients of entropy and/or lepton fraction. Although the existence of protoneutron star convection is a robust result consistently found by many independent studies [e.g. 10–12], the position and duration of the convective activity is uncertain because it is sensitive to the details of the high-density equation of state [13]. A particularly important role is played by the slope of the symmetry energy, which describes the energy difference between matter with a given electron fraction and symmetric matter made of an equal number of protons and neutrons (Ye = 0.5), because it determines the destabilizing character of the composition gradients. Numerical simulations find that Ledoux convection drives convective motions with typical velocities of 1000 − 3000 km/s. The more efficient transport of energy induced by convection leads to an increase of the neutrino flux escaping the PNS by 15 − 30% in the first few 100 milliseconds [10, 11] or by as much as a factor 2 in the case of low-mass progenitors if the convective zone comes close to the PNS surface [12]. The presence and properties of PNS convection thus leave an imprint in the time evolution of the neutrino flux that may be detectable from the next galactic supernova [13]. The development of the PNS convection has also been observed to induce a hemispherical asymmetry of lepton fraction and of the neutrino emission in the so-called LESA (lepton emission self-sustained asymmetry) [14– 16].

Figure 1. Ledoux convection and neutrino-driven convection. Left panel: space-time diagram of the radial velocity in the equatorial plane as a function of the time after bounce (from [11]). The PNS convection is visible as the band between 15 and 25 km with large velocities. The green bands above and below are the regions of the PNS stable to Ledoux convection. Right panel: neutrino-driven convection in the post-shock layer (from [17]). The color bar of the 2D cuts represents entropy per nucleon. The central object is a 3D visualization of a surface of constant entropy of 17 kb per nucleon, surrounded by the supernova shock (white, transparent surface).

2.2 Double diffusive convection ?

Another possible form of convection in a PNS is double-diffusive convection (e.g. thermohaline convection or semiconvection, see chapter by P. Garaud in this volume). Under certain

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conditions on the diffusion processes, such instabilities can take place if one of the gradients would be unstable by itself but is stabilized by the other gradient according to the Ledoux criterion. Neutron finger convection, analogous to thermohaline convection, has for example been argued to be driven by the negative lepton fraction gradients in Ledoux stable zone [18]. A necessary condition is that neutrino diffusion equilibrates entropy perturbations faster than lepton fraction. Later studies concluded that this condition is generally not fulfilled therefore preventing neutron finger convection [19, 20]. They suggested that other kinds of doublediffusive instabilities may grow instead, for example lepto-entropy fingers and lepto-entropy semiconvection. These stability analysis are, however, very sensitive to the treatment of neutrino transport and the validity of the approximations used may be questioned especially in the vicinity of the neutrinosphere. So far numerical simulations including sophisticated neutrino transport have not found conclusive evidence for the development of double-diffusive convection in a PNS [e.g. 10, 11], but it cannot be excluded that it would develop at lengthscales shorter than the resolution that can be afforded by current numerical simulations. 2.3 Neutrino-driven convection in the post-shock layer

A fraction of the neutrinos emitted at the neutrinosphere are absorbed in the so-called gain layer, defined as the region where neutrino absorption overcomes emission such that a net heating is induced. In this region located below the shock, material is being accreted while being heated, which creates a negative radial gradient of entropy. In a steady state, this gradient can be expressed as L ∇S = , (2) Pvr where L is the neutrino heating rate density, P the pressure and vr is the (negative) radial component of the velocity. The gain layer is therefore unstable according to the classical Ledoux or Schwarzschild criteria. A fundamental difference with classical convection is the presence of advection, which causes material to spend only a finite amount of time in the gain layer. [21] have shown that a fast enough advection can stabilize convection if the advection time becomes comparable to the buoyancy timescale defined by the Brunt-Väisälä frequency. The competition between advection and buoyancy is controlled by the following dimensionless number, sometimes called the Foglizzo parameter: ! dr χ= N , (3) vr  where the integral is performed over the gain region. Linear analysis showed that convection is linearly unstable if χ > 3. Unstable modes are limited to a range of horizontal wavelengths centered around twice the vertical size of the gain region, which corresponds approximately to circular convective cells and to a spherical harmonics index: l=

π rsh + rgain , 2 rsh − rgain

(4)

which gives l  5 for typical values of the shock radius of rsh = 150 km and of the gain radius (i.e. the lower bound of the gain region) rgain = 70−80 km. Non-linear numerical simulations confirmed that such a critical χ can in general distinguish cases with and without neutrinodriven convection [21, 22]. Below the linear instability threshold, convective motions may still develop if sufficiently large perturbations are present in the progenitor star, moderate density perturbations of just 1% have been shown to be sufficient [23]. These convective

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motions are however not self-sustained below the linear instability threshold and would decay unless new perturbations are continuously fed into the post-shock region [24]. The value of χ depends on time and on the progenitor structure, in particular through the values of the neutrino luminosity and the accretion rate onto the shock. In a number of progenitors, for example those with rather low masses and a relatively steep decline of the density outside the iron core, the parameter χ can take values above 3 and allow the development of convective motions [e.g. 25]. These motions drive well developed turbulence [see review and references therein by 26]. They support the development of a neutrino-driven explosion by pushing the shock thanks to turbulent pressure [27, 28]. Higher mass progenitors with shallower density profiles can have a more complex behavior with time intervals where convection is stabilized with χ < 3 [e.g. 17, 29, 30], during which the dominant instability is the standing accretion shock instability described in section 2.4. 2.4 Standing accretion shock instability (SASI)

Like neutrino-driven convection, the standing accretion shock instability (SASI) takes place in the post-shock region, which can make it difficult to distinguish between the two instabilities in numerical simulations. In contrast to the latter, SASI does not need any neutrino heating to develop and, as a matter of fact, was first discovered in a simplified setup without any heating [31]. SASI modes are oscillatory with a frequency in the range 30 − 100 Hz and the most unstable modes develop at the largest angular scales (l = 1 − 2). After some debate with a competing purely acoustic mechanism [32], the linear growth of SASI has been attributed to an advective-acoustic cycle as illustrated in Figure 2 [31, 33–37]. This cycle is based on two coupling processes between an acoustic wave and an advected wave composed of perturbations of entropy and vorticity. An acoustic wave reaching the shock generates an oscillating deformation of the shock which creates perturbations of entropy and vorticity. These perturbations are advected toward the protoneutron star and, when they cross a region of steep gradients near the protoneutron star surface, they perturb the pressure equilibrium which generates an acoustic feedback that closes the loop. The duration of a radial advective-acoustic cycle reads ! rsh ! rsh 1 dr 1 taac = + dr = , (5) c−v r∇ v r∇ v(1 − M) where v is the velocity, c the sound speed, M = v/c the Mach number and the integral is performed between the coupling radius r∇ , where the deceleration is maximum, and the shock radius rsh . This duration approximately sets the oscillation period of SASI fundamental modes1 . The growth rate can be expressed as a function of the advective-acoustic timescale and of the cycle efficiency constant Q, which is the ratio of the wave amplitude at the end of the cycle and at its beginning: log |Q| . (6) ωi  taac The non-linear saturation of SASI can be caused by the growth of parasitic instabilities on the SASI mode: the Kelvin-Helmholtz instability that feeds off the vorticity and the Rayleigh-Taylor instability, which grows on the entropy gradients generated by SASI [38]. These instabilities generate turbulence and break the large-scale structure of the mode thus terminating its growth. SASI can take the form of sloshing modes, obvious for example in axisymmetric simulations as oscillations of the shock along the axis of symmetry, or of 1 The oscillation period can be somewhat shorter when the transverse structure of the mode is taken into account [37].

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Figure 2. Left panel: Schematic illustration of the advective-acoustic cycle responsible for the growth of SASI (adapted from [37]). Blue oscillating arrows represent acoustic waves, while the red circular arrows represent advected waves composed of entropy and vorticity. The global efficiency of the cycle Q = Qsh Q∇ is the product of the coupling efficiency at the shock Qsh and in the region of gradients near the PNS surface Q∇ . Right panel: 3D numerical simulation of core-collapse which displays a SASI spiral mode (from [17]). Colors and 3D surfaces have the same meaning as in the right panel of Figure 1.

Figure 3. Rotation induced by SASI spiral modes. Left panel: photograph of the SWASI experiment (shallow water analog of a shock instability) illustrating the rotation induced by a spiral mode (from [44]). A horizontal bar in the central part of the flow allows to visualize the rotation induced in the direction opposite to the mode rotation. Right panel: schematic view explaining the sign of the Reynolds stress induced by a spiral mode at the shock.

non-axisymmetric spiral modes rotating around an axis [39–41]. In the absence of rotation, these two kinds of modes have exactly the same growth rate in the linear phase (in fact a sloshing mode can be understood as a superposition of two counter-rotating spiral modes). In the absence of artificial symmetry constraints, the non-linear phase is generally dominated by spiral modes, even if the early evolution is dominated by a sloshing mode, because of a spontaneous symmetry breaking between the two counter-rotating spiral modes [42]. SASI spiral modes redistribute angular momentum by inducing a Reynolds stress resulting from the non-linear interaction of the acoustic wave and the vorticity contained in the mode [39, 43]. As a consequence, the region below the shock rotates in the same direction as the mode while the neutron star is spun up in the opposite direction as illustrated by Figure 3. First observed in numerical simulations, this effect was then confirmed with an experimental analog in a shallow water flow [44]. In this way, a neutron star may rotate up to a few dozen times per second, even if it is born from a non-rotating progenitor [42, 43, 45].

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Figure 4. Numerical simulations of the corotation instability in core-collapse supernovae. Upper panels: density (left), pressure (middle) and radial velocity (right) in a simulation from [49]. Lower panels: pression variation from the azimuthal average (left) and rotation frequency in a simulation from [45]. The region rotating faster than the mode is highlighted in dark red and the corotation radius is represented by a black line. The characteristic pattern of open spiral wave appears only when a clear corotation is present.

The presence of rotation in the progenitor has a strong impact on the linear growth of SASI by increasing the growth rate of prograde modes and stabilizing modes rotating in the opposite direction [39, 46]. This effect is larger for increasing azimuthal number m such that fast rotation favors somewhat larger m than the usually dominant l = 1, m = 1 mode [47]. The shock oscillations driven by SASI, and in particular spiral modes in the presence of rotation, foster neutrino-driven explosion by pushing the shock and allowing large shock expansion [e.g. 22, 41, 48] 2.5 Corotation instability (aka low-T/W instability)

A cold, isolated neutron star in solid body rotation is unstable to a dynamical bar-mode instability above a critical ratio of kinetic energy (T) to gravitational energy (W) of T/W  0.27. Another instability called "low-T/W" has later been found to exist in differentially rotating neutron stars for much lower values of T/W down to at least 0.01 [50, 51]. The cause of this instability is thought to be the existence of a corotation radius [52–54], where the rotation frequency matches the phase velocity of the unstable mode: Ω(rcorot ) =

ω , m

(7)

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where Ω(r) and ω are respectively the rotation and mode angular frequency, m is the mode azimuthal number (m = 1 or 2 for the low-T/W instability). The existence of a corotation radius is a source of instability in several different contexts, from accretion disks in the Papaloizou-Pringle instability [55, 56] to hydrodynamic experiments including the so-called radiative instability of a flow around a rotating cylinder [57] and the free surface flow entrained by a rotating plate [58]. The instability is driven by the coupling between waves of negative (respectively positive) energy on either side of the corotation radius: this coupling allows an over-reflection of a wave at the corotation radius [59–61] owing to the transmission of a wave of opposite energy on the other side of the corotation radius. Depending on the context, the wave may be of different nature such as an acoustic wave, a surface gravity wave or an inertial wave. The low-T/W instability of neutron stars is due to the destabilization of the f-mode [53]. A similar instability with azimuthal wave number m = 1 or 2 was discovered in numerical simulations of core-collapse supernovae with fast rotation [49, 62, 63]. The development of this instability is characterized by the appearance of an open spiral mode structure as illustrated by Figure 4. The associated emission of acoustic waves towards the shock could be important to launch the explosion by depositing energy below the shock [49]. The deformation of the PNS is furthermore responsible for an intense emission of gravitational waves in a narrow frequency band. The low-T/W instability in core-collapse supernovae is thought to be driven by a corotation radius, but it is so far unclear if the unstable mode is the f-mode like in cold neutron stars. The structure of the PNS surface is indeed very different from that of a cold neutron star and the presence of the post-shock region allows for additional acoustic and advective-acoustic modes that may also be destabilized by the presence of a corotation radius. The appearance of the characteristic open spiral structure when a corotation radius is present in the idealized numerical simulations by [45], which does not include the PNS, may be an indication in this direction. 2.6 g-mode excitation

The excitation of large-amplitude g-modes in the PNS was observed in some numerical simulations because of SASI and convective activities. These modes were argued to be responsible for a new explosion mechanism due to the emission of acoustic waves [64]. More recent simulations did not confirm such large amplitudes of the g-modes and their role in the explosion mechanism remains minor [e.g. 65, 66]. The excitation of surface g-modes of the PNS is nonetheless important for gravitational wave emission: they give rise to a robust signal whose frequency is increasing with time from 100 to 1000 Hz as the PNS contracts. The frequency of the signal can be explained by the Brünt-Väisäla frequency at the PNS surface [e.g. 67, 68].

3 Magnetic field origin and impact Most studies of core-collapse supernovae neglect the influence of the magnetic field as has been done in section 2. This is sometimes justified by the negligible influence of a magnetic field of the same magnitude as the dipolar component deduced from pulsar spin-down (1012 − 1013 G). This argument can be questioned because multipolar components of the magnetic field are not constrained and may be strong enough to influence the dynamics and because the magnetic field at the time of the explosion might have been stronger than it is at the age of observed pulsars (typically of the order of a million years). At the very least, in the case

Multi-Dimensional Processes In Stellar Physics

of magnetars, which are a class of neutron stars with much stronger dipolar magnetic fields in the range 1014 − 1015 G, magnetic effects cannot be neglected in the dynamics. A strong dynamically relevant magnetic field may be obtained by magnetic flux conservation if the progenitor’s core magnetic field is strong enough. The contraction of an iron core with a radius of riron ∼ 1.5 − 3 × 103 km to a PNS with a radius of r pns ∼ 50 km at early times down to the radius of a cold neutron star 10 − 15 km after a few seconds, leads to an 2 /r2pns ∼ 103 − 105 . In an alternative sceamplification of the magnetic field by a factor riron nario, a weak initial magnetic field may be amplified by dynamo action due to one or several of the multi-dimensional processes discussed in section 2. 3.1 Slow-rotating standard supernovae

A strong dipolar magnetic field of 1012 G in the iron core helps neutrino driven explosions by providing an extra source of pressure that pushes the shock outwards [69]. A somewhat lower magnetic field, which does not modify the pressure significantly, can still have a dynamical impact if the magnetic  energy is comparable to the kinetic energy (or equivalently if the Alfvén speed vA ≡ B/ 4πρ is comparable to the velocity) [70]. In such a situation, the propagation of vorticity through slow magnetosonic waves and Alfvén waves splits the advective-acoustic cycle into up to 5 different cycles, which interfere either constructively or destructively [71]. As a consequence, SASI growth can be either stabilized [72], slowed down or accelerated depending on magnetic field orientation and strength. Alfvén waves are generated and can furthermore accumulate and amplify near an Alfvén surface where the advection velocity equals the Alfvén speed [73] as was observed in core-collapse simulations by [69]. When the initial magnetic field is too weak to have a dynamical impact, it can be amplified by the turbulence generated by SASI and convection. [72, 74] showed that SASI can amplify the magnetic field by several orders of magnitude and generate a small-scale magnetic field of the order of 1014 G. In their simulations, the magnetic field reduces the kinetic energy of the small scales but does not affect the overall dynamics of the large scales and the shock evolution. [69] found only a moderate amplification of the magnetic field by neutrino-driven convection in their axisymmetric simulations, but the situation may be different in 3D. 3.2 Fast-rotating (extreme ?) supernovae

Magnetic effects are much more important in the presence of fast rotation because the associated kinetic energy represents an important reservoir that can be tapped by the magnetic field. For neutron star rotation periods shorter than P = 5 ms, the kinetic energy of a rotating neutron star is sufficient to be a major part of the explosion energy: E K = IΩ2 /2  1051 (5 ms/P)2 erg (for a moment of inertia I = 1.5 × 1045 g cm2 ). Numerical simulations of the collapse of a fast rotating progenitor core containing a strong dipolar magnetic field show the launch of strong magnetorotational explosions driven by jets along the rotation axis [e.g. 75–78]. The magnetic field assumed in these simulations is however artificial and probably inconsistent with the fast rotation of the progenitor, magnetic stars being slow rotators because of magnetic braking [79]. The initial magnetic field may instead be interpreted as a way to incorporate a field that should be generated by dynamo processes in the PNS, which cannot be described by current simulations of the explosion. The next two paragraphs describe our current understanding of the two main processes envisaged for this dynamo in a fast rotating PNS. Most recent results suggest that both processes can produce strong magnetic fields but that their structure is more complex than a pure dipolar magnetic field. Strong multipolar magnetic fields can also drive magnetorotational explosions but less energetic and

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1015 ideal

1014 1013 10

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1011 1

O09 viscous too slow

mix drag

2 3 radius (106 cm)

4

Figure 5. Left panel: different regimes of MRI growth in a PNS as a function of radius and magnetic field strength (adapted from [81]). Right panel: Numerical simulation of an MRI driven dynamo in a simple model of PNS (from [82]). The color represents the magnetic field intensity (in G).

with less collimated outflows than dipolar magnetic field of similar intensity [80]. This shows that numerical simulations of magnetorotational explosions need an improved description of the magnetic field to reach reliable conclusions. 3.2.1 Magnetorotational instability

The magnetorotational instability (MRI) is well studied in accretion disks because of its ability to drive vigorous magnetohydrodynamic turbulence and to transport angular momentum [e.g. 83, 84]. In its simplest formulation in the absence of dissipation and buoyancy, the instability criterion is that the angular frequency Ω decrease outwards. If the magnetic field is parallel to the rotation axis, the most unstable modes are the so-called channel modes with a wave vector pointing in the same direction. Their growth rate is independent of the magnetic field strength σ = qΩ/2,  where q ≡ d (log Ω)/d (log r), while their wave number is inversely proportional to it k = q(1 − q/4) Ω/vA . In a fast rotating neutron star, differential rotation is produced during the collapse because the density profile of a PNS differs from that of the iron core. The angular frequency decreases outwards at radii larger than 10 km with q = 1 − 1.25 [85]. Fast rotation with periods in the millisecond range should therefore lead to fast growth on millisecond timescales [86, 87]. If the magnetic field in the PNS is initially in the range of pulsars, the wavelength would be very short −1/2

−1  B

ρ Ω λ=6 m, (8) 1012 G 1013 g cm−3 1000 s−1 highlighting the difficulty of resolving the MRI in numerical simulations of core-collapse supernovae. In a PNS, these ideal estimates are modified by the presence of both buoyancy [88, 89] and the interaction of neutrinos with matter [81]. Deep inside the PNS, neutrinos are in the diffusive regime and induce a large viscosity that reduce drastically the growth rate if the magnetic field is weak [81]: ⎛ ⎞  B 1013 g cm−3 1/2 2 × 1010 cm2 s−1 1/2 Ω 1/2 ⎜⎜⎜ qEν ⎟⎟⎟1/2 σ = ⎜⎜⎝  s−1 , ⎟⎟⎠ Ω = 17 ρ ν 1012 G 1000 s−1 2(2 − q) (9)

Multi-Dimensional Processes In Stellar Physics

where Eν ≡ v2A /(νΩ) is the viscous Elsasser number (the MRI is significantly affected by viscosity for Eν < 1). At intermediate depth, the MRI can grow in a regime where the mean free path of neutrinos is larger than the MRI wavelength, such that they induce a drag force. In this regime the growth rate is reduced but, in contrast to the viscous regime, it remains independent of the magnetic field strength. Still closer to the neutrinosphere, the MRI can grow unimpeded in the ideal regime described above (see left panel of Figure 5). Because of its short wavelength, the MRI has mostly been studied using local simulations describing a small portion of the PNS [87, 90–93]. A first phase of magnetic field amplification by channel modes is stopped by parasitic instabilities of Kelvin-Helmholtz type growing on the velocity field of the mode. An amplification of the magnetic field by a factor 10 to 100 can be achieved but is probably not sufficient to reach magnetar’s strength [92, 93]. A further amplification in the following turbulent phase is therefore necessary through a MRI-driven dynamo [94, 95]. Local simulations of MRI-driven turbulence in the presence of stable stratification show that magnetic field in the magnetar range 1014 − 1015 G can be generated [91]. This magnetic field is, however, necessarily confined to the small scales described by the local model, such that only global simulations describing the full PNS can probe the magnetic field morphology on the large scales and the generation of the dipolar magnetic field of magnetars. [96] performed high resolution simulations of a quarter of the PNS and showed that the growth of the MRI led to a dynamo generating azimuthal magnetic field of alternating polarity. The generation of the dipolar component of the magnetic field could, however, not directly be studied because the simulation was started from a relatively strong dipolar magnetic field. Using a simplified global model of a PNS, [82] studied the generation of a large-scale magnetic field by a MRI dynamo starting from a small scale magnetic field of similar intensity as obtained in local models (right panel of Figure 5). They found that the MRI generates a dipolar component of the magnetic field, which is preferentially misaligned with the rotation axis. The dipolar component remains subdominant but still reaches an interesting intensity of a few 1014 G comparable to the lower end of galactic magnetars. 3.2.2 Convective dynamo

The seminal papers by [98] and [99], who first recognized that soft-gamma repeaters and anomalous X-ray pulsars were due to extremely magnetized neutron stars termed magnetars, were based on the suggestion that a convective dynamo would amplify the magnetic field in a fast rotating PNS. Using a simple assumption that the magnetic energy reaches equipartition with the kinetic energy of the PNS convection described in section 2.1 leads to an estimate of the magnetic field strength in the magnetar range 1/2  v   ρ B ∼ 4πρv  2 × 1015 G. (10) 108 cm 3 × 1013 cm [99] argued that in a slowly rotating neutron star the magnetic field would be dominated by small scales, with a much weaker dipolar component comparable to normal pulsars. A magnetar strength dipolar magnetic field may be generated only in the case of fast rotation with a period of a few milliseconds. This theoretical prediction based on rough scalings and orders of magnitudes was surprisingly not checked with dedicated numerical simulations until very recently. [97] performed the first direct numerical simulations of a convective dynamo in a PNS, using numerical methods similar to those used in stellar and planetary dynamos (Figure 6). They found the existence of two distinct branches of dynamo action depending on the rotation period: a strong branch for fast enough rotation, where the magnetic energy exceeds the turbulent kinetic energy, and a weak branch for slower rotations with magnetic energy below or

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a

b

b

y=

x

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Perfect conductor
outer b. c. Pseudo-vaccum
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Slow rotation

Fast rotation

Figure 6. Numerical simulation of a convective dynamo in a rotating PNS (from [97]). Left and middle panels: Three-dimensional rendering of the weak (a) and strong (b) field solutions. Magnetic field lines are colored by the total field strength and the inner boundary by the entropy. Blue (red) isosurfaces of the radial velocity materializes the downflows (outflows). Columnar convective flows typical of rotating convection are visible on the weak branch but are broken by the strong axisymmetric azimuthal magnetic field on the strong branch. Right panel: Ratio of magnetic to kinetic energy as a function of the inverse of the Rossby number Ro. A strong dynamo branch appears for fast enough rotations characterized by Ro < 0.2.

close to equipartition with the kinetic energy. The strong dynamo follows a scaling characteristic of magnetostrophic force balance between Coriolis and Lorentz forces: E B /E K ∝ 1/Ro, where E B is the magnetic energy, E K the kinetic energy and the Rossby number is defined as Ro ≡ U/(Ωd) with U the rms velocity and d the shell width. On the strong branch the magnetic field is dominated by its axisymmetric azimuthal component, which reaches intensities up to 1016 G. The dipolar component is weaker but still strong enough to explain magnetars with dipolar magnetic field strengths up to a few times 1015 G.

4 Conclusion Multidimensional dynamics is a key ingredient in core-collapse supernovae. We have reviewed our theoretical understanding of the physics of several hydrodynamic and magnetohydrodynamic instabilities that can drive multidimensional motions and amplify the magnetic field. This understanding is so far based on analytical and numerical studies as well as an experimental approach with a shallow water analog in the case of SASI. It can be tested indirectly by comparing the predictions of numerical simulations with the distribution of heavy elements observed in supernovae remnants such as Cas A [100], as well as the distribution of neutron stars kick velocities, rotation and magnetic field. In the future, multimessenger signals will be a much more direct probe of this multidimensional physics when the next nearby supernova will allow their detection. The time evolution of the neutrino fluxes will be detected by neutrino detectors such as Ice Cube or super/hyper-Kamiokande. Gravitational wave signals in the LIGO/VIRGO frequency band are also expected from several of the phenomena described in this chapter [8]. A future detection would give very precious constraints on this rich multidimensional physics.

References [1] B.W. Grefenstette, F.A. Harrison, S.E. Boggs, S.P. Reynolds, C.L. Fryer, K.K. Madsen, D.R. Wik, A. Zoglauer, C.I. Ellinger, D.M. Alexander et al., Nature 506, 339 (2014)

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a

b

b

y=

x

0. 5

Perfect conductor
outer b. c. Pseudo-vaccum
outer b. c.

Slow rotation

Fast rotation

Figure 6. Numerical simulation of a convective dynamo in a rotating PNS (from [97]). Left and middle panels: Three-dimensional rendering of the weak (a) and strong (b) field solutions. Magnetic field lines are colored by the total field strength and the inner boundary by the entropy. Blue (red) isosurfaces of the radial velocity materializes the downflows (outflows). Columnar convective flows typical of rotating convection are visible on the weak branch but are broken by the strong axisymmetric azimuthal magnetic field on the strong branch. Right panel: Ratio of magnetic to kinetic energy as a function of the inverse of the Rossby number Ro. A strong dynamo branch appears for fast enough rotations characterized by Ro < 0.2.

close to equipartition with the kinetic energy. The strong dynamo follows a scaling characteristic of magnetostrophic force balance between Coriolis and Lorentz forces: E B /E K ∝ 1/Ro, where E B is the magnetic energy, E K the kinetic energy and the Rossby number is defined as Ro ≡ U/(Ωd) with U the rms velocity and d the shell width. On the strong branch the magnetic field is dominated by its axisymmetric azimuthal component, which reaches intensities up to 1016 G. The dipolar component is weaker but still strong enough to explain magnetars with dipolar magnetic field strengths up to a few times 1015 G.

4 Conclusion Multidimensional dynamics is a key ingredient in core-collapse supernovae. We have reviewed our theoretical understanding of the physics of several hydrodynamic and magnetohydrodynamic instabilities that can drive multidimensional motions and amplify the magnetic field. This understanding is so far based on analytical and numerical studies as well as an experimental approach with a shallow water analog in the case of SASI. It can be tested indirectly by comparing the predictions of numerical simulations with the distribution of heavy elements observed in supernovae remnants such as Cas A [100], as well as the distribution of neutron stars kick velocities, rotation and magnetic field. In the future, multimessenger signals will be a much more direct probe of this multidimensional physics when the next nearby supernova will allow their detection. The time evolution of the neutrino fluxes will be detected by neutrino detectors such as Ice Cube or super/hyper-Kamiokande. Gravitational wave signals in the LIGO/VIRGO frequency band are also expected from several of the phenomena described in this chapter [8]. A future detection would give very precious constraints on this rich multidimensional physics.

References [1] B.W. Grefenstette, F.A. Harrison, S.E. Boggs, S.P. Reynolds, C.L. Fryer, K.K. Madsen, D.R. Wik, A. Zoglauer, C.I. Ellinger, D.M. Alexander et al., Nature 506, 339 (2014)

Multi-Dimensional Processes In Stellar Physics

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