Dust-Gas Instabilities in Protoplanetary Disks: Toward Understanding Planetesimal Formation (Springer Theses) 9811917647, 9789811917646

Table of contents :
Supervisor’s Foreword
Acknowledgements
Contents
1 Introduction
1.1 From Molecular Cloud Cores to Protoplanetary Disks
1.2 Dust Dynamics in Protoplanetary Disks
1.2.1 Radial Drift and Vertical Sedimentation
1.2.2 The Effects of Turbulence on Dust Grains
1.2.3 Gravitational Instability of a Dust Layer
1.2.4 Secular Gravitational Instability
1.3 ALMA Observations of Annular Substructures
1.4 Purposes of This Thesis
References
2 Revision of Macroscopic Equations for Dust Diffusion
2.1 Short Introduction: Unphysical Momentum Transport Due to the Diffusion Term
2.2 Reformulation of Basic Equations from the Mean-Field Approximation
2.3 Linear Analyses
2.3.1 Basic Equations
2.3.2 Linearized Equations
2.3.3 Results
2.4 Discussion: Effects of Disk Thickness
2.5 Summary
References
3 Numerical Simulations of Secular Instabilities
3.1 Short Introduction
3.2 Numerical Methods
3.2.1 Lagrangian-Cell Method
3.2.2 The Piecewise Exact Solution for Friction
3.2.3 Methods for Turbulent Diffusion and Viscosity
3.3 Simulations of Radially Extended Disks
3.3.1 Formation of Transient Low-Contrast Dust Rings
3.4 Discussion
3.4.1 Linear Analyses with Dust Drift
3.4.2 Condition for the Thin Dense Ring Formation
3.4.3 Fate of Dense Dusty Rings
3.4.4 Observational Justification
3.4.5 Effects on Dust-to-Gas Ratio Dependence on the Dust Coefficient
3.5 Summary
References
4 Coagulation Instability: Self-induced Dust Concentration
4.1 Short Introduction
4.2 Basic Equations for Linear Analyses with Dust Growth
4.3 Linear Analyses and Results
4.4 Discussion
4.4.1 Impact of Dust Diffusion
4.4.2 Effects of Other Collision Velocities
4.4.3 Coevolution with Other Dust-Gas Instabilities: Bridging a Gap Between First Dust Growth and Hydrodynamical Clumping Toward Planetesimal Formation
4.5 Summary
References
5 Summary and Future Prospects
5.1 Summary of This Thesis
5.2 Future Prospects
References

Citation preview

Springer Theses Recognizing Outstanding Ph.D. Research

Ryosuke Tominaga

Dust-Gas Instabilities in Protoplanetary Disks Toward Understanding Planetesimal Formation

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

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More information about this series at https://link.springer.com/bookseries/8790

Ryosuke Tominaga

Dust-Gas Instabilities in Protoplanetary Disks Toward Understanding Planetesimal Formation Doctoral Thesis accepted by Nagoya University, Nagoya, Japan

Author Dr. Ryosuke Tominaga Cluster for Pioneering Research RIKEN Wako, Saitama, Japan

Supervisor Prof. Shu-ichiro Inutsuka Department of Physics Nagoya University Nagoya, Japan

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-19-1764-6 ISBN 978-981-19-1765-3 (eBook) https://doi.org/10.1007/978-981-19-1765-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Supervisor’s Foreword

Understanding the formation of rocky planets is one of the natural interests of human being because we are living on one of such astronomical objects. The most standard theory for the formation of rocky planets assumes that the formation of a myriad of smaller size rocky objects called planetesimals in a gaseous circumstellar disk called the protoplanetary disk, prior to the formation of astronomical rocky planets. The sizes of planetesimals are supposed to be on the order of 1–100 km and they are the building blocks of planets. Various developments were made in investigating the formation of planetesimals but all of them still remain controversial. Since it is virtually impossible to study the formation of such a big object in a laboratory, our theoretical study should rely on purely analytical investigations or numerical simulations. Unfortunately, none of the numerical simulations has convincingly shown the formation of planetesimals from micron-size dust grains in a protoplanetary disk that is supposed to be a reasonable initial condition for the planet formation. In this thesis, the author first formulates the basic equation for the dynamics of dust grains in gaseous protoplanetary disks. Turbulent motions in a gaseous disk affect the motion of dust grains, which result in the spatial diffusion of dust grains. Since the previous formulations have violated the momentum conservation in describing the diffusion in the rotating two-component turbulent system, the author proposes new basic equations that keeps all the conservation property. The author then studies the non-linear development of secular gravitational instability, which is supposed to operate in protoplanetary disks under a certain condition and expected to result in the formation of planetesimals. Since the condition for the instability seem to be consistent with the recent astronomical observations, this mechanism is one of the mechanisms to explain the observed multiple ring-like structures in protoplanetary disks. To perform the numerical simulations, the author has developed a new numerical algorithm based on action principle that enables very accurate long-term calculations. Finally, the author investigates the growth of dust grains in protoplanetary disks and found a new instability that is supposed to accelerate the grain growth and dust concentration. This instability is expected to bridge the realistic common initial condition of protoplanetary disks and the advent of the planetesimal forming v

vi

Supervisor’s Foreword

processes such as secular gravitational instability. Since the discovery of the first exoplanets in 1995, the scientific community has made a lot of progress in the research on planet formation processes. However, our understanding of the formation of planetesimals is limited. I hope this thesis provides a useful step toward understanding the precursors of rocky planets. Nagoya, Japan October 2021

Shu-ichiro Inutsuka

Parts of this thesis have been published in the following journal articles: 1.

“Non-linear development of secular gravitational instability in protoplanetary disks”, R. T. Tominaga, S. i. Inutsuka, and S. Z. Takahashi 2018, Publications of the Astronomical Society of Japan, Volume 70, Issue 1, 3 (1–15), Advanced Access Publication Date: 2018 January 5, DOI: 10.1093/pasj/psx143.

2.

“Revised Description of Dust Diffusion and a New Instability Creating Multiple Rings in Protoplanetary Disks”, R. T. Tominaga, S. Z. Takahashi, and S. i. Inutsuka 2019, The Astrophysical Journal, Volume 881, Number 1, 53 (17pp), Publication Date: 2019 August 12, DOI: 10.3847/1538-4357/ab25ea.

3.

“Secular Gravitational Instability of Drifting Dust in Protoplanetary Disks: Formation of Dusty Rings without Significant Gas Substructures”, R. T. Tominaga, S. Z. Takahashi, and S. i. Inutsuka 2020, The Astrophysical Journal, Volume 900, Number 2, 182 (17pp), Publication Date: 2020 September 15, DOI: 10.3847/1538-4357/abad36.

4.

“Coagulation Instability in Protoplanetary Disks: A Novel Mechanism Connecting Collisional Growth and Hydrodynamical Clumping of Dust Particles”, R. T. Tominaga, S. i. Inutsuka, and H. Kobayashi 2021, The Astrophysical Journal, Volume 923, Number 1, 34 (24pp), Publication Date: 2021 December 8, DOI: 10.3847/1538-4357/ac173a.

Figures, Tables, and some texts are reproduced in this thesis with permission from American Astronomical Society (© AAS) and Oxford University Press.

vii

Acknowledgements

First, I would like to thank my supervisor, Prof. Shu-ichiro Inutsuka, for his support and continuous encouragement. His lectures based on his comprehensive knowledge of various topics in physics were very interesting, expanded my curiosity, and enhanced the motivation to study physics. He also taught me how a theorist should be, the importance of enjoying addressing issues, and delight in deeply understanding physical processes. What I learned from him certainly helps me to get through a tough time and encourages me to move forward in my life. I would also thank collaborators of the works in this thesis, Hiroshi Kobayashi and Sanemichi Z. Takahashi. They kindly spent their time on discussion, teaching, and revising manuscripts. I could not accomplish the works in this thesis without their kind support. The studies in this thesis were supported by JSPS KAKENHI Grant Number JP18J20360. Figures, Tables, and some texts are reproduced in this thesis with permission from American Astronomical Society (© AAS) and Oxford University Press. I am also grateful to Tsuyoshi Inoue, Yuri I. Fujii, Kenji Kurosaki, Jiro Shimoda, Gabriel Rigon, Kensuke Kakiuchi, and all the students in the astrophysics group at Nagoya university. We had fruitful discussions on various topics. Interaction with them has enriched my daily life. I would also like to appreciate the kind support by the former and present secretaries in the astrophysics group, Machiko Yoshida, Kyoko Yamazaki, Ritsuko Watanabe, Yasuko Iwata, Natsu Kato and Ayako Imai. I would like to thank Kiyotomo Ichiki and Shohei Saga. Taking their lectures when I was an undergraduate student, I found it fun to study physics and astrophysics. That motivated me to start the career in astrophysics. I am grateful to Hidekazu Tanaka and Takeru K. Suzuki for fruitful discussion, and also thank former members of the astrophysics groups at Nagoya university: Masanobu Kunitomo, Shinsuke Takasao, Doris Arzoumanian, Yuki A. Tanaka, Masato I. N. Kobayashi, Torsten Stamer, Lucas Fery, Jamie Townsend, Shoji Mori, Kotaro Maeda, Keisuke Sugiura, Tomoya Miyake, Yuki Ohno, Yutaro Sato, Kenta Nakashima, Shun Arai, Naoya Tokiwa, Kaori Kawamura, Kensuke Yokosawa, Kouki Matsumoto, Chikako Nagao, Takumi Sumida, Taichi Takeuchi, Toshiyuki Tanaka, Mutsumi Minoguchi, and Teppei Minoda. I am also grateful to other former members with whom I had a great time. ix

x

Acknowledgements

Special thanks go to Akane Ochiai and my best friends, Takashi Yoshioka and Ryo Iida. We have been sharing thoughts on many things and getting closer, which is memorable and invaluable to me. I could not have enjoyed the Nagoya life over the years without them. Last but not least, I would like to thank my family, and especially I am sincerely grateful to my parents, Masasuke and Yoshiko. They gave me many opportunities to know and to experience various things, through which I found the fun of science. Their support and encouragement helped me to get through tough periods in my life. Their educational and thoughtful opinions expanded my perspective. I could not have devoted myself to studying physics at Nagoya university without their support. I sincerely appreciate their kind and dedicated support.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 From Molecular Cloud Cores to Protoplanetary Disks . . . . . . . . . . . 1.2 Dust Dynamics in Protoplanetary Disks . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Radial Drift and Vertical Sedimentation . . . . . . . . . . . . . . . . . 1.2.2 The Effects of Turbulence on Dust Grains . . . . . . . . . . . . . . . 1.2.3 Gravitational Instability of a Dust Layer . . . . . . . . . . . . . . . . . 1.2.4 Secular Gravitational Instability . . . . . . . . . . . . . . . . . . . . . . . . 1.3 ALMA Observations of Annular Substructures . . . . . . . . . . . . . . . . . 1.4 Purposes of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5 6 8 11 14 16 18

2 Revision of Macroscopic Equations for Dust Diffusion . . . . . . . . . . . . . 2.1 Short Introduction: Unphysical Momentum Transport Due to the Diffusion Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Reformulation of Basic Equations from the Mean-Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Linear Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Linearized Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Discussion: Effects of Disk Thickness . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

28 31 31 33 33 43 46 47

3 Numerical Simulations of Secular Instabilities . . . . . . . . . . . . . . . . . . . . 3.1 Short Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Lagrangian-Cell Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The Piecewise Exact Solution for Friction . . . . . . . . . . . . . . . 3.2.3 Methods for Turbulent Diffusion and Viscosity . . . . . . . . . . .

49 49 50 51 59 61

25

xi

xii

Contents

3.3 Simulations of Radially Extended Disks . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Formation of Transient Low-Contrast Dust Rings . . . . . . . . . 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Linear Analyses with Dust Drift . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Condition for the Thin Dense Ring Formation . . . . . . . . . . . . 3.4.3 Fate of Dense Dusty Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Observational Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Effects on Dust-to-Gas Ratio Dependence on the Dust Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64 74 75 75 81 83 84

4 Coagulation Instability: Self-induced Dust Concentration . . . . . . . . . . 4.1 Short Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Basic Equations for Linear Analyses with Dust Growth . . . . . . . . . . 4.3 Linear Analyses and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Impact of Dust Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Effects of Other Collision Velocities . . . . . . . . . . . . . . . . . . . . 4.4.3 Coevolution with Other Dust-Gas Instabilities: Bridging a Gap Between First Dust Growth and Hydrodynamical Clumping Toward Planetesimal Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 94 97 101 101 105

106 106 107

5 Summary and Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Summary of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 113 115

85 86 87

Chapter 1

Introduction

1.1 From Molecular Cloud Cores to Protoplanetary Disks A protoplanetary disk is a by-product of a star-forming process (Fig. 1.1). Star formation starts from self-gravitational collapse of a molecular cloud core. A molecular cloud core is a gas clump with a size of ∼0.1 pc, number density of ∼104 cm−3 (e.g., see [101]). Gas temperature of molecular cloud cores is about 10 K, which is mainly determined by thermal equilibrium with radiative cooling due to C II fine-structure line and radiative heating due to photoelectric emission from dust grains and PAHs [57, 131]. Those isothermal gas clumps collapse in self-similar manner once their self-gravity dominates thermal pressure gradient force [60]. This first isothermal collapse proceeds until inner collapsing gas becomes optically thick and behaves as adiabatic gas. Effective specific heat ratio γeff of the resulting dense region is initially 5/3 although the dense gas consists of H2 . This behavior originates from high temperature that is required for the lowest rotational transition of H2 to be excited (510 K). When compressional heating increases temperature enough, the rotational transition is excited and γeff becomes 7/5. These adiabatic gas cores can decelerate self-gravitational collapse. The condition for the expansion/contraction can be described in terms of the effective specific heat ratio as reviewed in Inutsuka [45]. For a polytropic gas, thermal pressure P and gas mass density ρ are related through P = Kρ γeff , where K is constant. Using this relation, one can roughly estimate pressure gradient force, −ρ −1 ∂ P/∂r , by −Kρ γeff −1 /r , where r is a radial dimension of the core. If the self-gravitational collapse is a spherically symmetric process, one can further simplify the pressure gradient force as −Kρ γeff −1 /r ∝ r 2−3γeff . This rough estimation demonstrates that the pressure gradient force increases as collapse proceeds if 2 − 3γeff < 0. On the other hand, self-gravity on the sphere of the core is −G M/r 2 ∝ rρ ∝ r −2 , where G and M are the gravitational constant and a core mass, respectively. Therefore, the self-gravity increases as collapse proceeds but the radial dependence is different from that of the pressure gradient force. More specifically, when the ratio of the specific heats is larger than 4/3, the pressure gradient force will dominate the self-gravity as the collapse decreases the core radius r , resulting in a hydrostatic core called a “first core”. Larson [60] performed numerical simulations of the core collapse based one © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 R. Tominaga, Dust-Gas Instabilities in Protoplanetary Disks, Springer Theses, https://doi.org/10.1007/978-981-19-1765-3_1

1

2

1 Introduction

Molecular cloud & core

Protoplanetary disk

Planetary system

~0.1 pc ~1-10 pc

~100 au

Fig. 1.1 Schematic picture showing time evolution from a molecular cloud core to a planetary system. The sizes of a molecular cloud and a cloud core are on the order of 1–10 pc and 0.1 pc, respectively (e.g., see [101]). A dust-gas disk called a protoplanetary disk with the size of ∼100 au forms around the central star. Planets form as the disk evolves and dust grains grow: a planetary system is “leftover”

dimensional hydrodynamic equations, and showed that the first core initially has a mass of ∼0.01 M and a radius of ∼4 au. The resulting first core grows in mass through material infall from a surrounding envelope, and gradually shrinks with help of radiative energy loses from its outer layer. Once the internal temperature reaches ∼2000 K, molecular gas turns into monoatomic gas through dissociation of H2 whose binding energy is 4.47 eV corresponding to >104 K [32, 132]. The H2 dissociation plays a role in cooling the gas and triggers the second collapse of the core. Because the energy of collapsing gas is used to dissociate hydrogen molecule, the gas temperature increases insignificantly and the process is closely isothermal (γeff < 4/3). The second collapse is quenched once all hydrogen molecules are dissociated, resulting again in a hydrostatic core called a second core or a “protostar”. The mass of the protostar is about 10−3 M , and it radius is ∼10−2 au corresponding to about 1 solar radius (see, [60, 70]). The protostar further evolves through mass accretion and radial expansion/contraction along with internal structure evolution, and finally goes to pre-main- and mainsequence stars (e.g., [39, 60, 102]). In reality, initial cloud cores have non-zero angular momentum. Gravitational collapse of such a rotating cloud core forms a gaseous disk around a resultant protostar. The resultant disk is called a protostellar disk, and evolves toward a protoplanetary disk that is more radially extended than a protostellar disk. Three-dimensional hydrodynamic simulations of the core collapse showed that the first core formed before the second collapse eventually turns into a protostellar disk (e.g., [15, 67, 71, 92]). A protostellar disk is thus initially a few au in size and more massive (∼ 0.01M ) than a protostar (∼10−3 M ). Along with the protostellar evolution through mass accretion from the protostellar disk, the disk also grows in size and mass through magnetic interaction with the surrounding envelope and mass infall (e.g., [68, 111]). The core collapse and disk formation and evolution has also been extensively studied based on (non-)ideal magnetohydrodynamical simulations (e.g., [2, 12–14, 68, 69, 116, 117]). When inner collapsing gas has a low density and Ohmic dissipation is ineffective, angular momentum of the gas is transferred outward by threaded magnetic fields. This process is called magnetic braking (e.g., [72, 76]). Magnetic braking on inner collapsing gas is effective when the outer gas is more massive, which is expected in the early phase of the runaway collapse. Magnetically driven

1.1 From Molecular Cloud Cores to Protoplanetary Disks

3

wind and outflow also contributes to extract angular momentum from the very inner region (e.g., [17, 116, 117]). The inner gas density increases as the collapse proceeds. Once the gas number density reaches ∼1011−12 cm−3 , Ohmic dissipation operates and dissipates magnetic field [66, 79]. Dissipation of magnetic field leads to inefficient angular momentum transfer, and a rotationally supported disk forms around a protostar. An overview of the disk formation and evolution is as follows (e.g., see Fig. 23 in Machida et al. [69]). The first core, which is a precursor of a protostellar disk, is thermally supported and a few au in size at its formation. The rotationally supported disk where Ohmic dissipation operates (a “magnetically inactive region”) is initially (sub-)au scale. Surrounding gas inside of the first core continues to accrete on the inner disk, increasing the gas density of the disk. In addition, accreting gas brings angular momentum to the disk. As a result of the increase in gas density and angular momentum, the inner magnetically inactive region expands radially outward. In other words, the disk radius increases. Machida et al. [69] showed that a disk radius exceeds 10 au a few thousands year after formation of a protostar. The disk extends further toward 100 au after the infalling gas outside the first core becomes less massive than the inner disk [68]. Because of the nature of runaway collapse, the protostellar mass is smaller than the first protostellar disk (e.g., see Fig. 2 in Inutsuka et al. [46] for schematic summary of mass evolution of a protostar and a disk). Such a massive disk evolves not only via magnetic interaction but also via its self-gravity, which triggers gravitational instability. Gravitational instability creates spiral structures and drives angular momentum transport and mass accretion in magnetically inactive region (e.g., [15, 68, 110]). The protostar grows in mass via resultant mass accretion from the disk, evolving toward a protoplanetary disk that is less massive than a central star.

1.2 Dust Dynamics in Protoplanetary Disks Dust grains in a molecular cloud core also accrete with gas onto a resultant disk. Dust abundance relative to gas was observationally derived for interstellar medium, and dust-to-gas mass ratio is about ∼0.01 [18]. This small amount of dust grains and their dynamics are important for planetesimal and planet formation in protoplanetary disks. In this section, we briefly review basic but key processes of dust grains in a gas disk and previous studies on secular GI. Dust grains aerodynamically couple with a gas disk. Hereafter, we refer to this dust-gas coupling as “friction”. Characteristic timescale of friction on a dust grain with a size of a and mass of m is called stopping time tstop (e.g., [1, 130]). Stopping time is determined by, for example, surrounding gas density ρg , sound velocity cs , a mean free path of gas λmfp , and internal dust mass density ρint as follows:

tstop

⎧ ⎨ π8 =  ⎩ π

ρint a ρg cs

4ρint a 2 8 9ρg cs λmfp

a < 49 λmfp a > 49 λmfp

,

(1.1)

4

1 Introduction

where we show only subsonic gas cases. The former is called the Epstein law while the latter is called the Stokes law. For dust in supersonic gas flow, one has to apply the Newtonian law (e.g., [1, 128, 130]). How strongly dust grains are coupled to gas is often represented by dimensionless stopping time τs ≡ tstop . Dust with τs  1 is tightly coupled while it is decoupled for τs  1. Thus, we can us τs as a measure of dust sizes in a protoplanetary disk. In the following subsections, we summarize dynamical processes mostly based on stopping time. If we neglect self-gravity of a disk, equations of motion of a dust grain orbiting around a star with mass M∗ in cylindrical coordinate (r, φ, z) are vφ2 dvr vr − u r G M∗ r − , = − 2 2 3/2 dt r (r + z ) tstop

(1.2)

dvφ vr vφ vφ − u φ , =− − dt r tstop

(1.3)

dvz G M∗ z vz − u z =− 2 − , dt (r + z 2 )3/2 tstop

(1.4)

where vr , vφ , and vz are radial, azimuthal, and vertical velocity of a dust grain. Gas velocities are denoted by u r , u φ , and u z . Gas equations are u 2φ du r ρd u r − vr 1 ∂P G M∗ r − , = − − 2 2 3/2 dt r ρg ∂r (r + z ) ρg tstop

(1.5)

du φ ur u φ 1 ∂P ρd u φ − vφ =− − − , dt r ρgr ∂φ ρd tstop

(1.6)

du z 1 ∂P G M∗ z ρd u z − vz =− − 2 − , 2 3/2 dt ρg ∂z (r + z ) ρg tstop

(1.7)

where we neglect turbulent viscosity, magnetic fields, and self-gravity. Gas pressure P is given by P = cs2 ρg , and dust density is denoted by ρd . In the absence of frictional backreaction to gas, we can derive some gas disk properties used as a zeroth-order background field. For simplicity, we consider axisymmetric disk. We then obtain a steady vertical gas density profile from Eq. (1.7): ln ρg = C +

cs2

G M∗ G M∗ − 2 , √ 2 2 cs r r +z

(1.8)

where C is an integral constant, and we consider a vertically isothermal structure. The assumption of the vertically isothermal structure is valid in interior of stellarirradiated disks (e.g., see [22, 25]). For z  r , one obtains the following Gaussian density distribution

1.2 Dust Dynamics in Protoplanetary Disks

5

  z2 , ρg = √ exp − 2H 2 2π H

g

(1.9)

where the integral constant is chosen as gas surface density. The vertical thickness  H is called a gas scale height and given by H ≡ cs /  where  ≡ G M∗ /r 3 is the Keplerian angular velocity at the disk midplane. We can also derive azimuthal gas velocity for a steady axisymmetric disk and in the absence of the frictional backreaction. Equation (1.5) shows that radial force balance leads to sub-Keplerian gas velocity: u φ < vKep ≡ r . At the disk midplane (z = 0), one obtains 

c2 ∂ ln P u φ = 1 + 2s vKep ∂ ln r 1 η≡− 2



1/2

cs vKep

vKep  (1 − η)vKep , 2

∂ ln P . ∂ ln r

(1.10)

(1.11)

The factor η is usually used as a measure of gas pressure gradient force (e.g., [1, 78]). If gas is barotoropic P = P(ρg ) and friction force on gas is neglected, Eqs. (1.5) and (1.7) give a steady gas solution whose angular velocity is uniform in the vertical direction. Based on the above equations and the derived “zeroth-order” gas profie, we will describe basic processes of dust in a gas disk.

1.2.1 Radial Drift and Vertical Sedimentation One important consequence of frictional interaction with gas is radial drift of dust grain (e.g., [1, 78, 128, 130]). Since dust grains move with Keplerian velocity in the absence of gas, they counter a head-wind in a sub-Keplerian gas disk. Friction force causes angular momentum transfer from dust grains to gas, and thus dust grains fall toward a central star. In a steady sub-Keplerian gas disk (u r = u z = 0 and u φ = (1 − η)vKep ), terminal velocity of dust is vr = −

2τs ηvKep , 1 + τs2

(1.12)

ηvKep . 1 + τs2

(1.13)

vφ = vKep −

When we include the backreaction to the gas, we obtains vr = −

2τs ηvKep , (1 + )2 + τs2

(1.14)

6

1 Introduction

vφ = vKep −

(1 + ) ηvKep , (1 + )2 + τs2

(1.15)

where ≡ ρd /ρg is dust-to-gas ratio. Equation (1.12) shows that dust grains with τs = 1 has the fastest drift speed. Once pressure profile is specified, one can estimate the drift rate of dust grains in a gas disk. A model often used in the literature is the “minimum mass solar nebula” (MMSN) model where the disk mass is minimal for the solar system planets to form [40, 58]. Based on power-law disk models including the MMSN model, Weidenschilling [128] showed that drift velocity of meter-sized dust can reach 104 cm s−1 ∼ 10−2 au yr −1 . Thus, dust grains located at r = 1 au fall onto a central star within just 100 yr, which is shorter by an order of magnitude than one orbital period of protoplanetary disks ∼100 au in size. In other words, this fast drift results in dust depletion and introduces a “barrier” against dust coagulation beyond meter sizes (e.g., [128]) while Brauer et al. [19] found that the issue is sensitive to the initial dust abundance in disks. Another important process is vertical sedimentation. Equation (1.4) shows that dust grains settle toward the midplane at the following terminal velocity vz − u z = −tstop

G M∗ z . (r 2 + z 2 )3/2

(1.16)

For the thin disk limit (z  r ), one obtains vz − u z  −τs z.

(1.17)

In a hydrostatic gas disk, dust grains settle at the velocity vz = −τs z. Small grains with τs  1 take longer time to settle than larger grains with τs ∼ 1. Note that when too large grains τs  1 show vertical oscillation with a frequency  rather than settling with the above terminal velocity. The following extrapolated velocity formula to a maximum velocity of large grains is often used [19]: vz = −

τs  z. 1 + τs

(1.18)

1.2.2 The Effects of Turbulence on Dust Grains Gas in a protoplanetary disk is thought to be turbulent to some extent, leading to angular momentum transfer and mass accretion via “turbulent viscosity”. Strength of turbulence is often scaled by the dimensionless parameter α [96], and turbulent viscosity coefficient is given by ν = αcs H . The value α depends on a driving mechanism of gas turbulence. Origins of gas turbulence in protoplanetary disks can be various hydrodynamical or magnetohydrodynamical instabilities including convective overstability (e.g., [53, 62]) and vertical shear instability (e.g., [82, 122, 123]),

1.2 Dust Dynamics in Protoplanetary Disks

7

and magnetorotational instability (e.g., [10, 11]). Some dust-gas instabilities are also known to induce turbulence in a dusty layer around the disk midplane, e.g., KelvinHelmholtz instability (e.g., [21, 24, 35, 73, 129]), streaming instability (e.g., [49, 135, 139]) and vertically-shearing streaming instability [48, 63]. Dust grains in turbulent gas suffer diffusion, which prevents vertical settling introduced in the previous subsection. An equilibrium profile of dust density can be derived from the continuity equation with mass diffusion term: ∂ρd vz ∂ ∂ρd + = ∂t ∂z ∂z



∂ρd Dz ∂z

 ,

(1.19)

where Dz is a vertical diffusion coefficient, and we ignore radial and azimuthal gradient for simplicity. For steady density profile, ρd satisfies ρd vz = Dz

∂ρd . ∂z

(1.20)

Adopting vz = −τs z for dust tightly coupled to turbulence, we obtain the following Gaussian profile (e.g., [20, 24, 26]):  

d z2 exp − ρd = √ 2Hd2 2π Hd

(1.21)

Hd ≡

Dz , τs 

(1.22)

where we use a dust surface density d for an integral constant, and Hd called dust scale height represents a vertical thickness of a dust disk. The diffusion coefficient has been estimated analytically (e.g., [24, 140]). For tightly coupled dust grains, Dz is equal to a gas diffusion coefficient Dg = αcs H . Using the Langevin equation that includes orbital motion of dust grains, Youdin and Lithwick [140] derived diffusion coefficients and dust scale height that are also applicable for large dust grains (τs  1). For uniform turbulence, the dust scale height is

  Stτe2 −1 α 1+ Hd = H , (1.23) τs 1 + St where τe ≡ teddy  is dimensionless turnover time of the largest eddies, and St ≡ tstop /teddy is so-called Stokes number. For eddies in protoplanetary disks, we often assume τeddy = 1, and thus St = τs . They also modify Equation (1.23) derived for constant ρg in order to make the formula applicable for stratified disks:

Hd = H

α α + τs

  Stτe2 −1 1+ , 1 + St

(1.24)

8

1 Introduction

For τeddy = 1 and St = τs , the above equation is reduced to the following

 1 + 2τs −1 , 1 + τs   τs 1 + 2τs −1/2 .  H 1+ α 1 + τs

Hd = H

α α + τs



(1.25) (1.26)

Dust diffusion also occurs in the radial direction. Youdin and Lithwick [140] derived the radial diffusion coefficient given as follows D=

1 + τs + 4τs2 αcs H, (1 + τs2 )2

(1.27)

where we assumed isotropic turbulence (see also [138]). For small dust grains with τs  1, diffusion coefficient becomes D  αcs H . The right hand side corresponds to a rate of gas diffusion via its eddy motion. Thus, dust grains are mixed by gas eddies when they are well coupled to gas. Relatively large grains do not follow gas eddy’s motion, and their trajectories are determined by a combination of the Coriolis force and stochastic kicks by gas eddies. The Coriolis force dominate the kicks for τs  1, and the diffusion coefficient monotonically decreases. Along with the diffusion, gas turbulence generates non-zero velocity dispersion cd . The velocity dispersion is also derived in Youdin and Lithwick [140]. For isotropic turbulence, cd2 is given by cd2 =

1 + 2τs2 + (5/4)τs3 2 αcs . (1 + τs2 )

(1.28)

The non-zero velocity dispersion acts as Reynolds stress and affects evolution of mean dust flow. The process is often modeled by effective pressure gradient force proportional to cd2 (e.g., [97, 136, 138]).

1.2.3 Gravitational Instability of a Dust Layer In the classical scenario of planet formation, planetesimals are thought to form via gravitational instability (GI) and fragmentation of a dust layer around the midplane (e.g., [35, 91]). Here, we review basic properties of GI based on one-dimensional linear analyses (e.g., see [100]). We first review GI in infinitesimally thin disks based on vertically integrated continuity equation and equations of motion of dust in local shearing sheet coordinates [34]: ∂ d vx ∂ d + = 0, (1.29) ∂t ∂x

1.2 Dust Dynamics in Protoplanetary Disks

9

∂ ∂vx ∂vx c2 ∂ d + vx = 32 x + 2v y − d − , ∂t ∂x

d ∂ x ∂x

(1.30)

∂v y ∂v y + vx = −2vx , ∂t ∂x

(1.31)

∂ 2 = −4π G d δ(z), ∂x2

(1.32)

where (x, y) = (r − R, R(φ − t))  is the co-orbital coordinate with a reference radius R and angular velocity  = G M∗ /R 3 . Self-gravitational potential of a dust disk is denoted by . We also assumed that a disk is axisymmetric and dust grains are free from diffusion and friction, for simplicity. Taking a uniform density profile

d = d,0 and Keplerian velocity field (vx , v y ) = (0, −3x/2) as an unperturbed state, we introduce linear perturbations and linearize the continuity equation and the equations of motion: ∂δvx ∂δ d + d,0 = 0, (1.33) ∂t ∂x ∂δvx ∂δ c2 ∂δ d = 2δv y − d − , ∂t

d,0 ∂ x ∂x

(1.34)

∂δv y  = − δvx , ∂t 2

(1.35)

where the unperturbed value is represented by subscripts “0”, and linear perturbations are denoted with δ. We assume perturbations proportional to exp(ikx + nt) and perform Fourier transformation of the above equations: nδ d + ik d,0 δvx = 0, nδvx = 2δv y − ikcd2 nδv y = −

δ d − ikδ,

d,0

 δvx . 2

(1.36) (1.37)

(1.38)

Perturbed self-gravitational potential is given by δ = −2π Gδ d /k, which satisfies a boundary condition that δ diminishes outside the disk (e.g., see [100]). Equations (1.36)–(1.38) and δ = −2π Gδ d /k have nontrivial solutions only when n = 0 for arbitrary wavenumber k or the following dispersion relation is satisfied: − n 2 = 2 − 2π G d,0 k + cd2 k.

(1.39)

The solution of n = 0 is called a static mode or a neutral mode, and corresponds to a steady solution of the linearized equations. Equation (1.39) is a dispersion relation of GI mode. GI can grow if n has positive real parts. The criterion is given by

10

1 Introduction

Qd ≡

cd  < 1, π G d,0

(1.40)

where Q d is the Toomre’s Q value for a dust disk [118]. Explicitly including vertical structures reduces growth rates and makes the critical mass larger. Goldreich and Lynden-Bell [33] showed that GI of uniformly rotating gas disks grows when π Gρmid /42 > 0.73 is satisfied, where ρmid is midplane gas density. Applying this criterion for Keplerian dust disks with the midplane dust density ρd,mid , one obtains Q 3D ≡

2  (0.73)−1  1.3, π Gρd,mid

(1.41)

(see also [94]). Assuming the vertical Gaussian profile for√dust density distribution, the midplane density is related to d,0 by ρd,mid = d,0 / 2π Hd . In this diffusionless arguments, the vertical thickness of a dust disk is determined by balance of effective pressure gradient force and vertical gravity as in the case of a gas disk shown in the beginning of this section. Thus, the vertical thickness Hd in this case is given by Hd = cd /. We then have Q 3D Qd = √  0.55. 2π

(1.42)

This criterion shows that twice larger disk-mass is required for GI to operate in vertically stratified disks. Since dust grains settle toward the midplane as a result of frictional interaction with gas, the midplane dust density monotonically increases in the absence of gas turbulence. However, dust settling itself can trigger Kelvin-Helmholtz instability, which stirs dust grains up. Dust sedimentation is more difficult in the presence of vertical dust diffusion, which is not considered in the above derivation of the critical Q d . Kelvin-Helmholtz instability in this context is self-regulated because the instability is powered by vertical shear resultant from dust sedimentation. If dust grains are diffused too much, Kelvin-Helmholtz instability and resultant turbulence become weak. This implies an equilibrium dust density profile. Sekiya [95] calculated the equilibrium profiles under the influence of Kelvin-Helmholtz instability. They showed that vertical diffusion driven by Kelvin-Helmholtz instability makes the midplane dust density much lower than the critical value required of GI. Sekiya [95] also showed that a high dust-to-gas ratio weakens Kelvin-Helmholtz instability, and the midplane dust layer can be GI-unstable (see also [141]). For example, for GI to operate at r = 1 au, a dust-to-gas ratio should be about 0.07–0.08 (see Fig. 2 and Table 1 therein). Nevertheless, turbulence powered by other instabilities will prevent dust sedimentation. Thus, the direct formation of planetesimals via GI still seems difficult unless other processes locally concentrate dust grains.

1.2 Dust Dynamics in Protoplanetary Disks

11

1.2.4 Secular Gravitational Instability Pure gravitational instability requires Toomre’s Q value less than unity. Toomre’s Q value represents the magnitude of Coriolis force and pressure gradient force relative to self-gravity. If some processes weaken either Coriolis force or pressure gradient force, GI will grow even in a disk with Q > 1. For example, radiative cooling weakens pressure gradient force and augments GI (e.g., [31, 64]). Dust-gas friction modifying dust and gas mean flows is another process that can make dust GI more unstable. Dust grains frictionally interacting with gas can not freely move with epicyclic frequency but tend to follow gas flow around their positions. In other words, friction weakens rotational support due to Coriolis force, which is one of the restoring forces exerted on dust. This process augments dust GI and makes its growth faster. In addition to the augmentation of dust GI, the friction triggers another instability called “dissipative GI” or “secular GI”. Secular GI is a process that most of my thesis focuses on. Here we review properties of secular GI in detail based on previous studies (e.g., [97, 107, 126, 136, 138]). The idea of secular GI was pointed out by Ward [125, 126]. Coradini et al. [23] also showed an unstable mode corresponding to secular GI, which is referred to as type I mode in the paper, although they mainly focused on two-fluid GI. Youdin [136, 137] analyzed stability of a self-gravitating dust disk embedded in a static gas disk. Thus, their analyses are based on one-fluid equations. Secular GI originates from a static mode that is present in friction-free self-gravitating disks (see the previous subsection). The static mode represents an equilibrium state that holds radial force balance of self-gravity, Coriolis force, and effective pressure gradient force. At shorter wavelength (kcd /  1), self-gravity and effective pressure gradient force are dominant, and the radial force balance is mainly achieved by these two. Perturbations with such short wavelengths are insignificantly affected by friction because stabilizing effects due to Coriolis force are small. On the other hand, the static mode at longer wavelengths shows the force balance mainly determined by Coriolis force and self-gravity, and thus significantly affected by friction. Youdin [136] showed that long-wavelength perturbations are unconditionally unstable, which also can be shown in the following. Another feature of secular GI is its slow growth in contrast to pure GI that grows at timescale of ∼ −1 . Here, we derived growth rate based on terminal velocity approximation. Considering that velocity perturbations damp at a timescale tstop , we use the following linearized equation of motion: nδvx = 2δv y − ikcd2 nδv y = −

δ d δvx − ikδ − ,

d,0 tstop

δv y  δvx − . 2 tstop

(1.43)

(1.44)

12

1 Introduction

Terminal approximation and δ = −2π Gδ d /k give 2δv y − ikcd2

δ d δvx + i2π Gδ d − = 0,

d,0 tstop

(1.45)

τs δv y = − δvx . 2

(1.46)

From Eqs. (1.36), (1.45), and (1.46), we obtain the following approximated dispersion relation: 1 2π G d,0 k − cd2 k 2 . (1.47) n= −2 tstop 2 + tstop This shows that perturbations with k < 2π G d,0 /cd2 are unconditionally unstable. For small dust with τs = tstop   1 and sufficiently long wavelengths so that the numerator can be approximated as 2π G d,0 k, the growth rate is reduced to  n  tstop 2π G d,0 k = 2π

λ tstop × 2π G d,0

−1

,

(1.48)

where λ = 2π/k is a wavelength of perturbations. The last equality of Eq. (1.48) shows that the growth rate is roughly given by a timescale for dust grains to transverse one wavelength with terminal velocity tstop 2π G d,0 (see also Sect. 1.2 of Youdin [138]). Growth rate relative to  is n/ ∼ τs × (kcd −1 )/Q d , which is less than unity because of the factor τs . Therefore, secular GI grows much slower than dust GI. Regardless of its slow growth, secular GI was proposed as a possible mechanism of planetesimal formation because “one-fluid” secular GI can grow without thresholds in contrast to GI (see also [95]). In the presence of radial diffusion of dust grains, secular GI is found to be significantly stabilized. Youdin [138] and Shariff and Cuzzi [97] performed linear analyses using hydrodynamic equations of motions for dust as in the above description and continuity equation with diffusion term: ∂ d vx ∂ 2 d ∂ d . + =D ∂t ∂x ∂x2

(1.49)

Michikoshi et al. [73] analyzed secular GI with diffusion in a more rigorous manner using Langevin equations, and showed consistent results. In both cases, shortwavelength perturbations are significantly stabilized because the diffusion operates efficiently at smaller spatial scales. Adopting the MMSN model [40] for calculating disk properties, Youdin [138] derived growth timescale, unstable wavelengths, and masses of dust grains accumulated via the instability into one ring (see Fig. 2 therein). For significantly weak turbulence with α = 10−10 , secular GI is found to grow in wide radial region from 0.1 au to 100 au (n −1 ∼ 10 − 104 yr for a = 1 mm). On the other hand, turbulence with α = 10−6 significantly stabilizes secular GI especially

1.2 Dust Dynamics in Protoplanetary Disks

13

in the inner region (n −1 > 106 yr for a = 1 mm). They concluded that secular GI creates dust rings with a mass of ∼ 0.1M⊕ and resultant rings will fragment into planetesimals if the instability operates. Youdin [138] and Shariff and Cuzzi [97] found that the growth timescale is shorter in outer radii. Therefore, secular GI can be a mechanism for creating planetesimals at the outer region (r ∼ 101−2 au) rather than the inner region (r ∼ 1 au), and weakly turbulent disks are preferable. We should emphasize that their findings are based on the MMSN model where disk masses are set to be minimal to reproduce solar system planets. Disk masses significantly affect growth timescale of secular GI through Toomre’s Q value. According to disk formation and evolution reviewed in the previous section, disk masses decrease from its formation time. In other words, increasing disk masses adopted corresponds to considering early-phase disks. Therefore, early massive disks will be one site where secular GI grows. Two-fluid analyses The previous studies reviewed above are based on one-fluid equations for dust. Including gas equations and frictional backreaction to gas introduces another property of secular GI. Takahashi and Inutsuka [107] performed two-fluid analyses of secular GI, and showed that long-wavelength perturbations are stabilized as a result of backreaction, which is in contrast to the previous studies showing unconditionally unstable secular GI (e.g., see Fig. 1.1 therein). They showed that the stabilization of long-wavelength perturbations are attributed to Coriolis force exerted on dust. In contrast to one-fluid analyses, gas slightly moves and follows radial concentration of dust grains as a result of backreaction. Radially concentrating flows in positive x−direction induce Coriolis force in the negative y−direction, and decelerate the azimuthal velocity of gas. Dust grains are also decelerated because of the azimuthal friction. This deceleration tends to reduce the radially concentrating flows, and quenches at long wavelengths (see also Latter and Rosca [61]). In this way, backreaction renders secular GI operational only at intermediate wavelengths comparable to gas scale height. Takahashi and Inutsuka [107] also derived the following growth condition for secular GI in the absence of velocity dispersion cd : D2 < ε(1 + ε)τs , (π G g,0 )2

(1.50)

where ε ≡ d,0 / g,0 is dust-to-gas surface density ratio (see Eq. (1.21) therein). They showed that “two-fluid” secular GI is operational for α/τs  10−2.5 when dustto-gas ratio is 0.1 and Toomre’s Q for gas is 3 (see Fig. 3 therein). Although their linear analysis does not include dust-gas drift motion as in the previous studies, they estimated condition for secular GI to grow in the presence of dust drift comparing the growth rate and the drift timescale tdrift ≡ r/|vr |. They found that secular GI can grow when the following condition is satisfied (see Eq. (1.39) therein):

14

1 Introduction



α 4 × 10−5



ε −2 0.1



Q 10

2 

η  1. 0.01

(1.51)

Even for high dust-to-gas ratio, turbulence should be weak (α ∼ 10−5 ). When Toomre’s Q is ∼5 during the disk evolution from its formation, disks with turbulence of α ∼ 10−4 marginally host secular GI although high dust-to-gas ratio is still required.

1.3 ALMA Observations of Annular Substructures Classically, the presence of protoplanetary disks was observationally confirmed based on infrared excesses in spectral energy distributions (SEDs). Based on SEDs at μm-wavelengths, Lada [59] proposed three classes that would divide the evolutionary timeline of stars and disks (see Fig. 2 therein). Class I refers to embedded objects that show wider SED relative to stellar black body radiations because of a surrounding envelope. Class II objects show infrared excess relative to stellar black body radiation. The infrared excess is responsible for emissions from dusty-gas disks, i.e., protoplanetary disks. Class III objects correspond to a star with a fairly dispersed disk. In addition to the three classes, Andre et al. [5] introduced Class 0 as a group of very young objects embedded in dense envelopes (see also Andre and Montmerle [4]). Recent observational developments enable us to directly see disks and their structures of ∼10-au scales. For example, optical and near-infrared observations with Subaru telescope and Very Large Telescope (VLT) with SPHERE1 have detected disks with rings, spirals, and shadows casted on disk surfaces (e.g., [8, 16, 38, 75, 77, 104, 124]). Those observations see scattered light from small dust grains floating in an upper layer of a disk. Atacama Large Millimeter/submillimeter Array (ALMA) has also been showing observational results on detailed disk structures. ALMA observations at (sub-)mm wavelengths trace dust grains around the midplane. High resolution disk observations with ALMA revealed that most of the observed and resolved disks have annular substructures, i.e., rings and gaps (e.g., [3, 6, 7, 29, 47, 65, 119]). Those rings and gaps have been observed not only in relatively old disks (∼10 Myr; e.g., TW Hya, [7, 119]; HD169142, [29, 84]) but also in very young disks (1 Myr; e.g., HL Tau, [3]; WL 17, [98]). It is reported by a very recent study that Class 0/I object also hosts a dust ring (e.g., [80, 99]). The Disk Substructures at High Angular Resolution Project (DSHARP) is a ClassII-disk survey for statistical studies of dust substructures (e.g., [6]). They observed 20 disks at 5-au resolutions. The observed disks show various substructures including rings/gaps and spirals (see Fig. 3 in Andrews et al. [6]. Eighteen disks show annular structures, and thus rings and gaps seem common in their samples. Although their observations are targeted to large bright disks, the other disk observations also 1

Spectro-Polarimetric High-contrast Exoplanet Research.

1.3 ALMA Observations of Annular Substructures

15

indicate the ubiquitousness of ring-gap structures in protoplanetary disks (e.g., [3, 7, 29, 47, 65, 119]). Huang et al. [42] studied properties of rings and gaps of the DSHARP disks. The intensity variation of adjacent rings and gaps are found typically less than 20% (see Sect. 3.2 of their paper). The observed rings and gaps are marginally resolved. They found that the widths of most of their targets are smaller than 10 au (see Fig. 4 therein). Assuming that disk temperature is determined by stellar irradiation, Dullemond et al. [27] showed that the observed ring-widths are comparable to or less than the gas scale height H . The analyses by Huang et al. [42] also showed that the observed rings and gaps are widely distributed from ∼10 au to 160 au (see Fig. 4 therein). They also find that there is no clear trend with stellar mass, mass accretion rate, and stellar age (see Fig. 10 therein). Possible origins of the observed substructures Many studies have proposed mechanisms to explain the origins of the observed rings and gaps, and the origins are still in debate. One possible mechanism is “planetbased” and that (sub-)Jupiter mass planets already exist in the observed disks (e.g., [36, 50, 143]). A planet embedded in a disk gravitationally interacts with gas and dust, leading to angular momentum transport. Gas and dust around the planetary orbit are then cleared out. This planet-driven clearing makes a gap in the disk. In addition, such a planet excites waves called density waves. Propagating density waves eventually steepen into shocks at radii away from the planetary orbit. Bae et al. [9] shows that even low-mass planets induce multiple waves (spiral arms) and shocks due to the steepening create multiple gaps. Although recent works reported kinematic signatures of Jupiter-mass planets at the observed gaps in some disks [85–88, 109], it is still unknown whether such planets also exist in the other disks. If planets actually exist in younger disks hosting annular substructures, this gives strong time-constraints on planet formation (e.g., within ∼1 Myr, see [99]). It is not still understood how to form planets within only 1 Myr at larger radii where gaps are observed. Such a fast planet formation seems difficult at least in the core accretion model ([74, 89]). For example, collisional fragmentation of planetesimals delays formation of planetary cores necessary to accrete gas at radii 10 au ([55, 56]). Planetary objects (clumps) can form via GI and disk fragmentation in very young disks, but those clumps suffer fast migration and fall onto the central star (e.g., see review on planet migration [54]). Besides, those clumps are quite massive. The mass can be 10MJ within 103 yr after its formation (see Fig. 7 in Tsukamoto et al. [120]), meaning that some down-sizing mechanism is required (e.g., [43, 81]). Mechanisms without assuming planets have also been proposed. Dullemond et al. [27] analyzed the DHSARP data using analytical models, and showed that the observed rings can be explained by dust-trapping due to “pressure bumps” [130]. A pressure bump has a positive and negative pressure gradient at inner and outer radii. According to the formula of the drift velocity (Eq. (1.14)), dust grains tend to accumulate at pressure maxima, which creates dust rings. The boundary of magnetically dead and active zones is one possible location where a pressure bump exists [30]. The magnetic activity itself is also found to form ring-like substructures through

16

1 Introduction

reconnection of the toroidal magnetic fields [105, 106]. Based on local shearing-box simulations and linear analyses, Riols and Lesur [90] shows that a disk subject to disk-wind mass loss can be unstable and host rings and gaps as a result of such a “wind-driven instability”. In the above processes, dust grains just follow the background gas structures and accumulate into rings (e.g., pressure bumps). On the other hand, dust itself can drive processes leading to ring formation. Such active processes include secular GI [107, 108], which is reviewed in the previous section. Takahashi and Inutsuka [108] performed linear analyses using disk models consistent with observations of HL Tau. They found that widths of outer rings (r 100 au) are consistent with the most unstable wavelengths of secular GI. The presence of dust leads to another type of instability called viscous ring instability [28]. Instead of the self-gravity, this instability requires the turbulent viscosity dependent on dust abundance (e.g., [44, 93]). Zhang et al. [142] discussed ring formation by rapid dust growth near snow lines where dust grains evaporate. They showed that the process explains the three prominent gaps in the HL Tau disk [3]. Sintering of dust aggregates is another important process that changes dust sizes near the snow lines. Okuzumi et al. [83] shows that sintering dust aggregates fragment around the snow lines and pile up there. The piling-up regions are observed as bright dust rings. Those dust size variation change the ionization degree and mass accretion rate across snow lines, which augments the ring-gap formation [41]. This process is similar to the instability discussed in Dullemond and Penzlin [28]. However, Huang et al. [42] showed that expected locations of snow lines of CO, N2 are not correlated with radii of the DSHARP rings/gaps although some of the substructures may result from the processes with snow lines. We should note that recent works [121, 133] show that the midplane temperature profile in irradiated disks (so-called passive disks) can be significantly disturbed because of thermal wave instability (e.g., [127]). Such a dynamical process will affect snow-line locations and should be taken into account when we compare the theoretical models and observed structures.

1.4 Purposes of This Thesis As shown in the previous sections, the recent high-resolution observations have been providing detailed information on disk structures, especially on spatial distributions of dust grains. The revealed annular substructures are some clues to reveal planetesimal formation, and planet formation. For example, if the observed substructures result from radial dust concentration without any planets, those observations indicate ongoing planetesimal formation. The ubiquitousness of dust rings may indicate that ring formation is a common process before planetesimal formation. Therefore, it is important to study ring-forming mechanisms without planets and their connection to planetesimal formation. According to Zhu et al. [144], protoplanetary disks having observed at mmwavelengths can be optically thick because of scattering of thermal radiation, and they

1.4 Purposes of This Thesis

17

indicate that disks are more massive than expected previously. Such self-scattering of dust thermal emissions is indicated from polarization observation by ALMA (e.g., [51, 52, 103, 134]). These observational results motivate us to study secular GI, which is one ring-forming process operational in massive disks. As mentioned in the previous section, there is another possibility that unseen planets create the observed substructures. If this is the case, the observations indicate that formation of planet(esimal)s have occurred at early disk-evolutionary stages. Since the early phase disks are more massive according to the disk formation theories (Sect. 1.1), understanding physics in relatively massive dusty disks seems to be the key to figure out early planet formation. Secular GI is one possible explanation of early planet formation. Therefore, studies on secular GI is also important even when we discuss formation of planets that finally carve gaps in Class II disks. Based on the above motivations, we investigate the disk evolution via secular GI in this thesis. The previous studies focused on linear growth of secular GI, and they found that turbulent diffusion is the most efficient process to stabilize secular GI. As reviewed in this chapter, turbulent diffusion is usually modeled by mass diffusion term introduced in the continuity equation. Although this modeling is widely used, just introducing the diffusion term violates the conservation law of momentum of a disk, which was indicated from Sect. 2.3 of Goodman and Pindor [37] (see an equation below Eq. (1.23) therein). Since linear/angular momentum is one fundamental physical property, the violation of its conservation law will affect not only stabilities of dusty-gas disks but also radial transport and mixing of dust grains. In Chap. 2, we revisit dust dynamics in turbulent disks and reformulate macroscopic equations that guarantee the angular momentum conservation and describe turbulent dust diffusion. Our formulation is based on mean-field approximation usually used to model turbulent fluid. Our results show that introducing turbulent effects not only in the continuity equation but also in the momentum equations holds the conservation law. We also study the stability of protoplanetary disks using the reformulated equations and revisit properties of secular GI. The linear analyses also show another instability that is unphysically stabilized because of the diffusion modeling in previous works. In Chap. 3, we investigate nonlinear growth of secular GI using the reformulated equations. In contrast to the previous works, we perform numerical simulations of secular GI and discuss how secular GI grow under the influence of radial dust drift and to what extent the instability accumulates dust grains into rings. To simplify the problem and separately consider multiple processes, we assume dust sizes limited by radial drift in the simulations presented in Chap. 3 rather than explicitly include dust growth. Our simulations show that perturbations growing via secular GI move inward with the so-called drift velocity. The drifting properties can be understood from linear analyses including dust drift in an unperturbed state. Our numerical results also show two types of growth of secular GI: formation of thin dense rings, and formation of transient rings. In the former case, secular GI grows into the nonlinear phase and accumulates over 50% of dust grains into multiple rings. In the latter case, a growth timescale of secular GI is too long for the instability to show nonlinearity. Once amplified density perturbations enter a stable region, those start to decay, that is,

18

1 Introduction

rings become transient. Thus, planetesimal formation via secular GI requires that perturbations grow into the nonlinear phase before they enter a stable region. Secular GI requires higher dust-to-gas ratio than interstellar values (0.01) although the required dust-to-gas ratio depends on other parameters including strength of turbulence. Dust-to-gas ratio in protoplanetary disks is not constrained observationally. Nevertheless, theoretical studies showed that if a disk is isolated dust coagulation and radial drift decrease dust surface density because of its inside-out nature (e.g., [19]). Such an isolated disk may correspond to a disk of a very late stage at which we can not expect mass infall from an envelope. Even when dust and gas accrete onto a disk from the envelope, dust coagulation may proceed from the inner region and inner large dust grains fall onto a central star. Thus, dust would tend to be depleted to some extent regardless of the mass infall. Since secular GI also requires large dust (e.g., τs ∼ 0.1), the dust depletion resulting from coagulation is problematic. In Chap. 4, we propose another instability triggered by coagulation as a re-accumulation process of large dust grains. We call the instability “coagulation instability”. Based on linear analyses with a single-sized coagulation equation, we show that coagulation instability can grow at tens orbital periods even when dust-togas ratio decreases down to 10−3 . We also investigate effects of diffusion and find that coagulation instability overcomes the diffusion and concentrates dust grains at a spatial scale ∼H that is comparable to the most unstable wavelength of secular GI. Therefore, coagulation instability will set up dust-rich circumstances and assist the growth of secular GI and its further development toward planetesimal formation. In Chap. 5, we summarize the present thesis and discuss issues to be addressed in the future work. Most contents in Chaps. 2, 3 and 4 are based on our published papers [112–115]. Figures, Tables, and some texts are reproduced in this thesis with permission from c American Astronomical Society ( AAS) and Oxford University Press.

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114. Tominaga RT, Takahashi SZ, Inutsuka SI (2019) ApJ 881(1):53. https://doi.org/10.3847/15384357/ab25ea 115. Tominaga RT, Takahashi SZ, Inutsuka ST (2020) ApJ 900(2):182. https://doi.org/10.3847/ 1538-4357/abad36 116. Tomisaka K (1998) ApJL 502(2):L163–L167. https://doi.org/10.1086/311504 117. Tomisaka K (2002) ApJ 575(1):306–326. https://doi.org/10.1086/341133 118. Toomre A (1964) ApJ 139:1217–1238. https://doi.org/10.1086/147861 119. Tsukagoshi T, Nomura H, Muto T, Kawabe R, Ishimoto D, Kanagawa KD, Okuzumi S, Ida S, Walsh C, Millar TJ (2016) ApJL 829:L35. https://doi.org/10.3847/2041-8205/829/2/L35 120. Tsukamoto Y, Machida MN, Inutsuka SI (2013) MNRAS 436(2):1667–1673. https://doi.org/ 10.1093/mnras/stt1684 121. Ueda T, Flock M, Birnstiel T (2021) ApJL 914(2):L38. https://doi.org/10.3847/2041-8213/ ac0631 122. Urpin V (2003) A&A 404:397–403. https://doi.org/10.1051/0004-6361:20030513 123. Urpin V, Brandenburg A (1998) MNRAS 294:399. https://doi.org/10.1046/j.1365-8711.1998. 01118.x 124. van Boekel R, Henning T, Menu J, de Boer J, Langlois M, Müller A, Avenhaus H, Boccaletti A, Schmid HM, Thalmann C, Benisty M, Dominik C, Ginski C, Girard JH, Gisler D, Lobo Gomes A, Menard F, Min M, Pavlov A, Pohl A, Quanz SP, Rabou P, Roelfsema R, Sauvage JF, Teague R, Wildi F, Zurlo A (2017) ApJ 837(2):132. https://doi.org/10.3847/1538-4357/ aa5d68 125. Ward WR (1976) In: Avrett EH (ed) Frontiers of astrophysics, pp 1–40 126. Ward WR (2000) On planetesimal formation: the role of collective particle behavior, pp 75–84 127. Watanabe SI, Lin DNC (2008) ApJ 672(2):1183–1195. https://doi.org/10.1086/523347 128. Weidenschilling SJ (1977) MNRAS 180:57–70. https://doi.org/10.1093/mnras/180.1.57 129. Weidenschilling SJ (1980) Icar 44:172–189. https://doi.org/10.1016/0019-1035(80)90064-0 130. Whipple FL (1972) In: Elvius A (ed) From plasma to planet, p 211 131. Wolfire MG, Hollenbach D, McKee CF, Tielens AGGM, Bakes ELO (1995) ApJ 443:152. https://doi.org/10.1086/175510 132. Wolniewicz L (1995) J Chem Phys 103(5):1792–1799 133. Wu Y, Lithwick Y (2021) arXiv:2105.02680 134. Yang H, Li ZY, Looney L, Stephens I (2016) MNRAS 456(3):2794–2805. https://doi.org/10. 1093/mnras/stv2633 135. Youdin A, Johansen A (2007) ApJ 662:613–626. https://doi.org/10.1086/516729 136. Youdin AN (2005) ArXiv Astrophysics e-prints, astro-ph/0508659 137. Youdin AN (2005) ArXiv Astrophysics e-prints, astro-ph/0508662 138. Youdin AN (2011) ApJ 731:99. https://doi.org/10.1088/0004-637X/731/2/99 139. Youdin AN, Goodman J (2005) ApJ 620:459–469. https://doi.org/10.1086/426895 140. Youdin AN, Lithwick Y (2007) Icar 192:588–604. https://doi.org/10.1016/j.icarus.2007.07. 012 141. Youdin AN, Shu FH (2002) ApJ 580(1):494–505. https://doi.org/10.1086/343109 142. Zhang K, Blake GA, Bergin EA (2015) ApJL 806(1):L7. https://doi.org/10.1088/2041-8205/ 806/1/L7 143. Zhang S, Zhu Z, Huang J, Guzmán VV, Andrews SM, Birnstiel T, Dullemond CP, Carpenter JM, Isella A, Pérez LM, Benisty M, Wilner DJ, Baruteau C, Bai XN, Ricci L (2018) ApJL 869(2):L47. https://doi.org/10.3847/2041-8213/aaf744 144. Zhu Z, Zhang S, Jiang YF, Kataoka A, Birnstiel T, Dullemond CP, Andrews SM, Huang J, Pérez LM, Carpenter JM, Bai XN, Wilner DJ, Ricci L (2019) ApJL 877(2):L18. https://doi. org/10.3847/2041-8213/ab1f8c

Chapter 2

Revision of Macroscopic Equations for Dust Diffusion

2.1 Short Introduction: Unphysical Momentum Transport Due to the Diffusion Term In Chap. 2, we revisit macroscopic description of dust disk evolution. The contents of this chapter are based on our paper published: Tominaga et al. [15]. Most of studies use the advection-diffusion equation for dust density to describe dust evolution in a turbulent gas disk. This widely-used equation for dust does not conserve the total angular momentum. We first analytically show the violation of the angular momentum conservation due to the diffusion and a possible solution to recover the conservation law. The following statement is independent from spatial dimensions one adopts. In this chapter, we use equations of a two-dimensional disk (a razor thin disk) because we also consider such a disk in Chaps. 3 and 4. The continuity equation and the azimuthal equation of motion for the dust in the cylindrical coordinates are as follows:   ∂d rD , ∂r

(2.1)

vφ vr vφ − u φ − d , r tstop

(2.2)

1 ∂ (r d vr ) 1 ∂ ∂d + = ∂t r ∂r r ∂r  d

∂vφ ∂vφ + vr ∂t ∂r

 = −d

where we assume axisymmetric disks. This assumption does not change the following statement. As introduced in the previous chapter, the right hand side of Eq. (2.1) models turbulent dust diffusion (e.g., [18]). Using the above equations, we obtain an equation for angular momentum of dust d jd ≡ d r vφ : ∂ (d jd ) 1 ∂ vφ − u φ 1 ∂ + + jd (r vr d jd ) = −r d ∂t r ∂r tstop r ∂r

  ∂d rD . ∂r

(2.3)

The first term on the right hand side represents a torque due to dust-gas friction. If we take into account the frictional backreaction to gas, the frictional torque term does

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 R. Tominaga, Dust-Gas Instabilities in Protoplanetary Disks, Springer Theses, https://doi.org/10.1007/978-981-19-1765-3_2

25

26

2 Revision of Macroscopic Equations for Dust Diffusion

not change total angular momentum of dust and gas as follows. We use the following equations for gas   ∂g 1 ∂ r g u r + = 0, (2.4) ∂t r ∂r   ∂u φ ∂u φ u φ ur vφ − u φ . (2.5) g + ur = −g + d ∂t ∂r r tstop The equation for angular momentum of gas is    ∂ g jg 1 ∂  vφ − u φ + r u r g jg = r d , ∂t r ∂r tstop

(2.6)

where g jg ≡ gr u φ . We thus obtain an evolutionary equation of total angular momentum summing Eqs. (2.3) and (2.6):  1 ∂   1 ∂ ∂  g jg + d jd + r u r g jg + r vr d jd = jd ∂t r ∂r r ∂r

  ∂d rD . ∂r

(2.7)

Because of the term on the right hand side, the volume integral of this term has nonzero values in general, meaning that the set of the above equations violates the total angular momentum conservation. One can immediately see that this “unphysical” nonconservation originates from the mass diffusion term in Eq. (2.1) because the right hand side is proportional to the diffusion coefficient. The diffusion term directly changes the dust angular momentum and affects orbital evolution of dust. Gas motion is also affected by the unphysical angular momentum changes because gas and dust always exchange their angular momentums through friction. The effect on gas motion is, however, smaller than that on dust motion by a factor of dust-to-gas mass ratio d /g because the angular momentum transport via friction is proportional to the dust surface density (Eq. (2.6)). Thus, the violation of the momentum conservation mainly affects on dust evolution. We can see how the angular momentum nonconservation affects dust motion through the following rearrangement of Eq. (2.3):     vφ − u φ D ∂d ∂d ∂ jd ∂ (d jd ) 1 ∂ + r vr − d jd = −r d . −D ∂t r ∂r d ∂r tstop ∂r ∂r (2.8) The second term on the left hand side vanishes when one integrate the equation over all space. One the other hand, the second term on the right hand side remains and changes the angular momentum. We here consider a Keplerian disk for further discussion. For such a disk, the second term on the right hand side represents a negative (positive) torque when the dust surface density gradient is positive (negative).

2.1 Short Introduction: Unphysical Momentum Transport Due to the Diffusion Term

27

In a dust-piling-up region, an inner dust (∂d /∂r > 0) loses its angular momentum and goes inward, and vice versa. Thus, the unphysical torque prevents dust accumulation, and the previous studies underestimated it. To discuss dust accumulating process precisely, it is necessary to revise the often-used equations with the dust diffusion term. We then propose a possible solution to recover the angular momentum conservation. When dust grains are so small that their stopping time satisfies τs = tstop   1, dust diffusion is mainly driven by radial kicks by turbulent gas, which appears in the equations as radial drag force [19]. In that case, specific angular momentum of dust grains remains constant during the radial displacements. They will exchange their angular momentum at a place to which they are going. By the way, Eq. (2.1) is rearranged to     1 ∂ D ∂d ∂d (2.9) + r vr − d = 0, ∂t r ∂r d ∂r which shows that the dust advection velocity due to diffusion is −Dd−1 ∂d /∂r . Considering the advection velocity and the fact that small dust grains are displaced with their angular momentum being constant, we experimentally consider the following equation  d

   ∂ jd vφ − u φ D ∂d ∂ jd + vr − = −r d , ∂t d ∂r ∂r tstop

(2.10)

where we add the advection velocity −Dd−1 ∂d /∂r to the usual advection term. From this equation, we obtain an equation for dust angular momentum     vφ − u φ D ∂d ∂ (d jd ) 1 ∂ + r vr − d jd = −r d . ∂t r ∂r d ∂r tstop

(2.11)

From this equation, one can see that dust angular momentum changes only through friction. Using Eqs. (2.6) and (2.11), we can show that the total angular momentum is conserved      1 ∂ D ∂d ∂  g jg + d jd + r u r g jg + r vr − d jd = 0. (2.12) ∂t r ∂r d ∂r The above experimental discussion represents that considering momentum advection associated with mass diffusion is the key to recover total angular momentum conservation. In the next section, we derive such an advection term based on the mean-field approximation [15].

28

2 Revision of Macroscopic Equations for Dust Diffusion

2.2 Reformulation of Basic Equations from the Mean-Field Approximation The mean-field approximation is one formalism used to derive macroscopic equations governing evolution of mean-fields. This formalism was invoked in previous works (e.g., [1, 10]), and they derive advection-diffusion equation for dust in protoplanetary disks. Following these works, we use the Reynolds averaging, which is one technique to analyze turbulent fluid based on averaging physical properties over a timescale longer than a typical turnover time of turbulent eddies. We first revisit the derivation of the advection-diffusion equation. We assume that a physical variable A is expressed as a sum of a time-averaged term  A and a short-term fluctuation originating from turbulence A ≡ A −  A, where A = 0. We average the following equations for dust: 1 ∂ (r d vr ) ∂d + = 0, ∂t r ∂r vφ2  ∂ ∂ (d vr ) 1 ∂  + r d vr2 = d − d ∂t r ∂r r ∂r

(2.13)

  G M∗ vr − u r − − d , (2.14) r tstop

   ∂ d vφ 1 ∂  vφ vr vφ − u φ + r d vφ vr = −d − d . ∂t r ∂r r tstop

(2.15)

Substituting d = d  + d and vr = vr  + vr to Eq. (2.13) and averaging both side of the equation in time, we obtain an equation for the mean density: 1 ∂ (r d vr ) ∂ d  1 ∂ (r d  vr ) + =− . ∂t r ∂r r ∂r

(2.16)

We model the term d vi  based on the “gradient diffusion hypothesis” (e.g., [1]): d vr  = −D

∂ d  , ∂r



D ∂ d  d vφ = − = 0. r ∂φ

(2.17) (2.18)

In the last equality of Eq. (2.18), we make use of the assumption of axisymmetric disks. Finally, we obtain the Reynolds-averaged continuity equation ∂ d  1 ∂ (r d  vr ) 1 ∂ + = ∂t r ∂r r ∂r This equation is equivalent to Eq. (2.1).

  ∂ d  rD . ∂r

(2.19)

2.2 Reformulation of Basic Equations from the Mean-Field Approximation

29

In the same way, we obtain equations for mean velocity fields averaging Eqs. (2.14) and (2.15)     D ∂ d  ∂ (d  vr ) 1 ∂ vr  + r d  vr  − d  ∂r ∂t r ∂r  2  

 vφ vr  − u r  G M∗ ∂ ∂  − − d − d  − d  = d  r ∂r r ∂r tstop 1 ∂ (r σrr ) σφφ + − r ∂r r     d (vr − u r ) ∂ d  ∂ d  1 ∂ ∂ D + r vr  D − , + ∂t ∂r r ∂r ∂r tstop

(2.20)    ∂ d  vφ t ∂t

where

    D ∂ d   1 ∂ r d  vr  − vφ d  ∂r r ∂r      d  vφ vφ − u φ D ∂ d  vr  − − d  =− d  ∂r r tstop      v − u ∂ r σ  σr φ 1 rφ d φ φ + − + . (2.21) r ∂r r tstop +

  σrr ≡ − d  vr2 − d vr2 ,

(2.22)

  σr φ ≡ − d  vr vφ − d vr vφ ,

(2.23)

  σφφ ≡ − d  vφ2 − d vφ2 ,

(2.24)

  represent the so-called Reynolds stress. Using a closure relation vr2 = vφ2 = cd2 adopted in Shariff and Cuzzi [10], one obtains the effective pressure gradient force as follows:      σφφ ∂ cd2 d  1 ∂ r σrr 1 ∂ (r σrr ) σφφ − =− + − , (2.25) r ∂r r ∂r r ∂r r  σrr ≡ − d vr2 , (2.26)   σφφ ≡ − d vφ2 .

(2.27)

 for simplicity since a closure relation In this thesis, we neglect the terms σrr , σr φ , σφφ on these terms is uncertain. Moreover, we only consider cases that dust grains are so small that we can assume vr = u r , vφ = u φ . Adopting these assumptions, we obtain

30

2 Revision of Macroscopic Equations for Dust Diffusion

    D ∂ d  ∂ (d  vr ) 1 ∂ vr  + r d  vr  − d  ∂r ∂t r ∂r   2  2  

 vφ ∂ cd d  ∂ G M∗ ∂  − − − d  − d = d  r ∂r ∂r r ∂r     vr  − u r  ∂ d  ∂ d  1 ∂ ∂ D + r vr  D , − d  + tstop ∂t ∂r r ∂r ∂r (2.28)    ∂ d  vφ t ∂t

    D ∂ d   1 ∂ r d  vr  − vφ d  ∂r r ∂r      d  vφ vφ − u φ D ∂ d  vr  − − d  . =− d  ∂r r tstop

+

(2.29)

The fourth term on the right hand side of Eq. (2.28) represents self-gravitational force whose source is density fluctuations due to turbulence. In this thesis, we assume that volume-integrated density fluctuations are so small that we can neglect the term. Following Cuzzi et al. [1], we also neglect the sixth term on the right hand side assuming the term is smaller than the time derivative of d  vr . The seventh term represents the advection of momentum d vr  along the mean flow vr . This term is the same order of the advection of d  vr  along diffusion flow, and thus we keep the term in the equations. Rearrangement of Eqs. (2.28) and (2.29) with those assumptions gives the following equations for the mean velocities:    2    ∂ cd2 d  vφ ∂ vr  D ∂ d  ∂ vr  d  + vr  − = d  − d  ∂r ∂t ∂r r ∂r   G M∗ ∂  − − d  ∂r r vr  − u r  − d  tstop   ∂ d  1 ∂ r vr  D , (2.30) + r ∂r ∂r 

       d  vφ ∂ vφ D ∂ d  ∂ vφ D ∂ d  vr  − d  + vr  − =− d  ∂r d  ∂r ∂t ∂r r   vφ − u φ − d  (2.31) tstop

2.2 Reformulation of Basic Equations from the Mean-Field Approximation

31

Equation (2.31) yields           ∂ r vφ vφ − u φ D ∂ d  ∂ r vφ d  + vr  − , = − d  r d  ∂r ∂t ∂r tstop (2.32)  which is equivalent to Eq. (2.10) since the mean specific angular momentum is r vφ . One can also obtain an equation equivalent to Eq. (2.11) using Eqs. (2.19) and (2.32):           ∂ d  r vφ vφ − u φ 1 ∂ D ∂ d  d  r vφ = −r d  + . r vr  − d  ∂r ∂t r ∂r tstop

(2.33) Using the gas equation (Eq. (2.6)), one can derive an equation equivalent to Eq. (2.12), showing total angular momentum conservation. It is also possible to model diffusion with the following closure relation 

∂ d vr  = −D g ∂r



 d   , g

(2.34)

which is another often-used diffusion model (e.g., [2]). Even in this case, one can derive similar equations that guarantee total just replacing  momentum conservation   D∂ d  /∂r in the above equations by D g ∂ d  / g /∂r . In subsequent parts of this thesis, we use the reformulated equations for dust but omit the brackets representing the averaged value for convenience.

2.3 Linear Analyses In this section, utilizing the reformulated equations, we perform linear analyses of secular GI. We then compare results with those of the previous studies.

2.3.1 Basic Equations We summarize a set of basic equations including the newly formulated dust equations. We use the following equations for gas and the Poisson equation, which were also used in previous studies [12, 13]:   ∂g 1 ∂ r g u r + = 0, ∂t r ∂r  g

∂u i ∂u i + uj ∂t ∂x j

 = −cs2

∂g ∂ − g ∂ xi ∂ xi

  G M∗ − r

(2.4)

32

2 Revision of Macroscopic Equations for Dust Diffusion

+

   ∂u j vi − u i ∂ ∂u i 2 ∂u k g ν + d + − δi j , ∂x j ∂x j ∂ xi 3 ∂ xk tstop (2.35)   ∇ 2  = 4π G g + d δ(z),

(2.36)

where u i , vi are the i-th component of gas and dust velocities, and  is the gravitational potential of the dust-gas disk, respectively. The third term on the right hand size of Eq. (2.35) is turbulent viscosity with viscosity coefficient ν = αcs H [9]. The reformulated dust equations with dust diffusion are summarized as follows: 1 ∂ (r d vr ) 1 ∂ ∂d + = ∂t r ∂r r ∂r  d

  ∂d rD , ∂r

(2.1)

     vφ2 ∂vr G M∗ ∂d ∂ D ∂d ∂vr + vr − = d − cd2 − d − ∂t d ∂r ∂r r ∂r ∂r r   vr − u r 1 ∂ ∂d r vr D , (2.37) − d + tstop r ∂r ∂r



     ∂vφ vφ vφ − u φ D ∂d ∂vφ D ∂d + vr − = −d vr − − d d , ∂t d ∂r ∂r r d ∂r tstop (2.38) Note again that we omit the brackets representing the averaged value for convenience. We investigate mode properties in the local shearing sheet (x, y). In the local frame, the above equations yield ∂g u x ∂g + = 0, ∂t ∂x

(2.39)

∂u x ∂u x ∂ 1 ∂ c2 ∂g + ux = 32 x + 2u y − s − + ∂t ∂x g ∂ x ∂x g ∂ x ∂u y ∂u y ∂ + ux = −2u x + ∂t ∂x ∂x



∂u y g ν ∂x

∂d ∂d vx ∂ + = ∂t ∂x ∂x

 D



∂d ∂x

 g ν

4 ∂u x 3 ∂x

 +

d v y − u y + , g tstop

d v x − u x , g tstop

(2.40) (2.41)

 ,

(2.42)

    c2 ∂d ∂vx ∂ vx − u x D ∂d ∂vx ∂ ∂d + vx − = 32 x + 2v y − d − − vx D , + ∂t d ∂ x ∂x d ∂ x ∂x tstop ∂x ∂x (2.43)     vy − u y ∂v y D ∂d ∂v y D ∂d + vx − = −2 vx − − . (2.44) ∂t d ∂ x ∂x d ∂ x tstop

We solve the eigenvalue problem with those equations.

2.3 Linear Analyses

33

2.3.2 Linearized Equations We adopt an unperturbed state with uniform surface densities and radial velocities of u x,0 = vx,0 = 0, where subscripts “0” represent unperturbed state values. The azimuthal velocities are u y,0 = v y,0 = −3x/2, which satisfy the steady condition with the above basic equations. We consider axisymmetric perturbations: δ, δd , δu x , δu y , δvx , δv y , δ proportional to exp[nt + ikx]. Linearing Eqs. (2.36) and (2.39)–(2.44), we obtain nδg + ikg,0 δu x = 0, nδu x = 2δu y − nδu y = −

(2.45)

cs2 4 δvx − δu x ikδg − ikδ − νk 2 δu x + ε , g,0 3 tstop

(2.46)

δv y − δu y  3ν δ + ε , δu x − νk 2 δu y − ik 2 2g,0 tstop

(2.47)

nδd + ikd,0 δvx = −Dk 2 δd , nδvx = 2δv y − nδv y = −

cd2 δvx − δu x ikδd − ikδ − , d,0 tstop

  δv y − δu y  ik D δvx − δd − , 2 d,0 tstop

δ = −

 2π G  δg + δd , k

(2.48) (2.49)

(2.50)

(2.51)

where ε ≡ d,0 /g,0 is the dust-to-gas mass ratio.

2.3.3 Results In this section, we show results of linear analyses with and without turbulent viscosity separately. In both cases, there are six modes in the system considered because the basic equations include six time-derivatives. In the absence of the dust-gas friction, the dust diffusion and the turbulent gas viscosity, there are two density waves for dust and gas disks respectively, and two static modes. A static mode is a steady solution of linearized equations. Figure 2.1 shows how the six modes are changed by adding three physical processes step by step and which mode becomes unstable. As shown in Fig. 2.1, the two static modes become unstable, which we will explain in more detail.

34

2 Revision of Macroscopic Equations for Dust Diffusion

Fig. 2.1 Mode classification based on the present linear analysis with the newly formed equations. There are four density waves and two unstable modes each of which originates from a static mode. The top line shows modes obtained in the absence of friction, dust diffusion, and turbulent gas viscosity. Dust-gas friction couples the dust and gas density waves, resulting in modified density waves (DWs, the mode A’ on the second line). The modified density wave is unstable if self-gravity is strong enough, i.e., classical GI. Secular GI originates from one static mode that is destabilized by dust-gas friction (the mode B). Dust diffusion reduces growth rates of the modes on the second line while those modes remains qualitatively the same (the third line). Turbulent gas viscosity destabilizes the remaining static mode (the mode D on the bottom line). We name this destabilized static mode “two-component viscous gravitational instability” (TVGI). The modified DWs become viscousc overstable modes. This figure is originally shown in Tominaga et al. [15] ( AAS). Reproduced with permission

Without turbulent viscosity In the absence of turbulent viscosity, we find one static mode (n = 0) and one unstable mode (the modes B’ and C in Fig. 2.1). The obtained static mode is a perturbed state where dust and gas have the same azimuthal velocity and the radial force balance holds. The latter mode corresponds to secular GI. Terminal velocity approximation (tstop  n −1 ) with tstop  −1  n −1 reduces the dispersion relation as follows: A1 n + A0 = 0,     1+ε 2 ε Dk 2 2 2 2 2 + c k , A1 ≡  + ωgd tstop tstop s    ωg2 1+ε 2 εc2 k 2 1 + ε A0 ≡ Dk 2 + cd2 k 2 + ωd2 s , 1+ε tstop tstop tstop where

(2.52) (2.53) (2.54)

2.3 Linear Analyses 0.0008 0.0006

(a)

35 0.004

Previous work This work Approximate dispersion relation

0.003

Previous work This work

(b)

0.002 Im[n]/Ω

Re[n]/Ω

0.0004 0.0002 0

0.001 0 -0.001 -0.002

-0.0002 -0.0004 0

-0.003 1

2

3

4

5

6

7

-0.004 0

1

2

3

4

5

6

7

kH

kH

Fig. 2.2 Dispersion relation of secular GI for D = 10−4 cs2 −1 , cd = 0, ε = 0.1, τs = 0.01, Q = 3. The left and right panels show growth rates and frequency as a function of dimensionless wavenumber k H , respectively. The vertical axis of the both panels is normalized by the Keplerian frequency . The red line is the dispersion relation we obtain in this work while the black line is one obtained in the previous work [4, 12]. The blue cross mark on the left panel is approximated growth rates (Eq. (2.52)). In contrast to the previous works, the imaginary part Im[n] is zero for the c secular GI in this work. This figure is originally shown in Tominaga et al. [15] ( AAS). Reproduced with permission

cs2 + εcd2 2 k − 2π G (1 + ε) g,0 k, 1+ε ωg2 ≡ 2 + cs2 k 2 − 2π G (1 + ε) g,0 k,

(2.56)

ωd2

(2.57)

2 ωgd ≡ 2 +

≡ + 2

cd2 k 2

− 2π G (1 + ε) g,0 k.

(2.55)

The above dispersion relation gives a growth rate of secular GI Figure 2.2 compares the dispersion relations of secular GI obtained in the present analyses and in the previous work [12] for D = 10−4 cs2 −1 , cd = 0, ε = 0.1, τs ≡ tstop  = 0.01, and Toomre’s Q value for gas Q≡

cs  π Gg,0

(2.58)

is set to be 3. Secular GI is stabilized at long wavelengths by the Coriolis force exerted on dust as shown by the previous study [12]. Short-wavelength perturbations are stabilized by turbulent diffusion. At intermediate wavelengths where secular GI is unstable, the gas pressure gradient force dominates the Coriolis force exerted on gas. As a result, an azimuthal zonal flow form [4], and dust accumulates by the selfgravity of itself. The growth rate of secular GI we obtain is several times larger than that obtained in the previous work. The most remarkable difference is that secular GI is a monotonically growing mode while it is an overstable mode in the previous studies [4, 12], that is, secular GI is unphysically overstabilized in the previous works because of the torque exerted on dust (Eq. (2.3)). We obtain the following condition for secular GI from the approximated dispersion relation (Eq. (2.52)):

36

2 Revision of Macroscopic Equations for Dust Diffusion

Fig. 2.3 Schematic picture of mode exchange between secular GI and dust GI. The left figure shows the dispersion relations of the static mode and dust GI in the absence of friction. The right figure shows the dispersion relations obtained with friction. The gray dashed line represents a growth rate of the dust GI mode, and the blue solid line is that of the static mode or secular GI. The labels (A), (A’) and (B) shown in the legends correspond to the labels shown in Fig. 2.1. We find mode exchange at wavelengths where the eigenvalue and eigenfunction degenerate in the absence of friction. This c figure is originally shown in Tominaga et al. [15] ( AAS). Reproduced with permission

  Q 2 tstop cd2 + D    < 1.  (1 + ε) tstop εcs2 + cd2 + D (1 + ε)

(2.59)

Equation (2.59) is equivalent to the condition in Latter and Rosca [4] in the case of D = 0 (see Eq. (2.43) in their paper). If we assume cd = 0 and (1 + ε)D  εcs2 tstop , Eq. (2.59) yields  ε (1 + ε) tstop cs2 . (2.60) Q< D Equation (2.60) is equivalent to Eq. 2.51 in Latter and Rosca [4] for (1 + ε)D  εcs2 tstop . Thus, our formulation does not change the condition for secular GI if D is so small that (1 + ε)D  εcs2 tstop is satisfied. We find mode exchange between secular GI and dust GI at k = kc,− , kc,+ where the growth rate of dust GI becomes zero (Fig. 2.3). Mode exchange is reconnection of dispersion curves of two different modes. Youdin [18] also found mode exchange even in the one-fluid linear analyses (see Fig. 2.1 therein). To distinguish the different modes on the same branch of the dispersion relation, we refer to the growing mode at wavenumbers where dust GI is unstable for tstop  → ∞ as “dust GI” while we refer to the mode at wavenumbers where dust GI is stable for tstop  → ∞ as “secular GI”. Figure 2.4 shows the maximum growth rate of secular GI and dust GI as a function of τs and α for ε = 0.1, Q = 3. As in the previous chapter, we use the following equations to calculate the diffusion coefficient D and the velocity dispersion cd :

2.3 Linear Analyses

37 -2

1

10

10

Approximate formula

0

10 -3

-1

10 α

-2

10 -4

n/Ω

10

-3

10

10

-4

10 -5

10

-1

-2

10 τs

10

0

10

-5

10

Fig. 2.4 Maximum growth rate of secular GI and dust GI for ε = 0.1 and Q = 3 as a function of dimensionless stopping time τs and strength of turbulence α. The color represents the growth rate normalized by the Keplerian angular velocity . The short-dashed line shows maximum α for the growth of secular GI (Eq. (2.59)). Dust GI grows faster than secular GI for parameters below the long-dashed line while secular GI is faster above the long-dashed line. This figure is originally c shown in Tominaga et al. [15] ( AAS). Reproduced with permission

1 + τs + 4τs2 αcs H, (1 + τs2 )2

(2.61)

1 + 2τs2 + (5/4)τs3 2 αcs . (1 + τs2 )

(2.62)

D=

cd2 =

The short-dashed line in Fig. 2.4 represents the approximated maximum value of α for the growth of secular GI, which is obtained from Eq. (2.59). The exact upper limit of α is well represented by the short-dashed line. Secular GI grows faster than dust GI in the colored region above the long-dashed line. On the other hand, dust GI is the fastest below the long-dashed line. The most unstable wavelengths are kc,− along the long-dashed line. The reason why dust GI grows faster than secular GI for smaller α is that stabilization due to dust diffusion and velocity dispersion is weaker. With turbulent viscosity Next, we show results with turbulent viscosity on the gas (Step 3 in Fig. 2.1). We find yet another instability that is new and is distinct from secular GI. We name this new instability “two-component viscous gravitational instability (TVGI)”. TVGI stems from the destabilization of one static mode existing in the absence of turbulent viscosity (the mode labeled (C) in Fig. 2.1). The static mode is a steady solution where gas and dust have the same azimuthal velocity (δu y = δv y ). The radial force balance is established mainly by self-gravity and the Coriolis force, that is, 2δu y − ikδ = 2δv y − ikδ 0. This radial force balance is not realized once turbulent viscosity is in action. The viscosity smooths out the azimuthal velocity gradient and leads to a decrease in δu y , which generates the relative motion between dust and gas in the azimuthal direction. Since dust interacts with gas through aerodynamical friction, the azimuthal velocity perturbation δv y also decreases. The decrease in

38

2 Revision of Macroscopic Equations for Dust Diffusion

both azimuthal velocity perturbations causes the break-down of the radial force balances, and thus both dust and gas accumulate by the self-gravity. This is the physical interpretation of TVGI. It is important for TVGI that a combination of friction and turbulent viscosity decreases the Coriolis force on both dust and gas. In the absence of friction, this mode becomes a static mode that satisfies the radial force balance for both components and δu x = 0, − νk 2 δu y − ik δvx −

3ν δg = 0, 2g,0

ik D δd = 0. d,0

(2.63) (2.64)

(2.65)

This proves that TVGI is distinct from the one-fluid viscous instability discussed in the previous studies (e.g., [3, 5–7]). The terminal velocity approximation and the assumption of tstop  −1  n −1 allow us to reduce dispersion relation into a quadratic function of n: B2 n 2 + B1 n + B0 = 0, 

  1+ε 2 ε Dk 2 2 2 2 + c k B2 ≡  + ωgd tstop tstop s    νk 2 (1 + ε)2 2 + εωgd + ε Dk 2 3 (1 + ε) 2 + εcs2 k 2 2 tstop (1 + ε)  3  2 1+ε  2 (1 + ε) 2 2 Dk + cd + εcs k + 2 tstop tstop   ε2 4νk 2 1 + ε Dk 2 + cd2 k 2 − 2π Gεg,0 k , + + 3 tstop tstop 1+ε

(2.66)



2

(2.67)

   1+ε 2 1+ε 2 2 Dk 2 2 2 B1 ≡ Dk + c k + ενk 2π Gg,0 k + 1+ε tstop tstop d tstop   2   εωd 1 + ε 2 2 + cs k + νk 2 32 + cs2 k 2 − 2π Gg,0 k 1 + ε tstop      1+ε  2 ενk 2 cd − cs2 k 2 + Dk 2 cs2 k 2 − 2 + (1 + ε) tstop tstop      1+ε 2  2 νk 2 2 3 + cs2 k 2 − 2π G (1 + ε) g,0 k , (2.68)  + + 1+ε tstop ωg2



2.3 Linear Analyses

39

0.03 0.02

TVGI Secular GI Approximate dispersion relation

n/Ω

0.01 0 -0.01 -0.02 -0.03 0

2

4

6

8

10

kH Fig. 2.5 Growth rates Re[n]/ of TVGI and secular GI for α = 10−3 , ε = 0.1, τs = 0.3 and Q = 5 as a function of dimensionless wavenumbers k H . The solid and dashed lines represent the dispersion relation of TVGI and secular GI, respectively. The blue cross mark represents the approximated growth rates obtained from Eq. (2.66). In this case, the secular GI is stable. This c figure is originally shown in Tominaga et al. [15] ( AAS). Reproduced with permission

   1+ε 2 1+ε 2 2  2 2 Dk + cd k 3 + cs2 k 2 − 2π Gg,0 k tstop tstop  2 2  εc k Dk 2 2π Gg,0 k + . (2.69) − νk 2 s tstop tstop

νk 2 B0 ≡ 1+ε



The second and higher order terms of νk 2 are neglected in the derivation of the above relation based on another assumption of weak turbulence (α  1). Equation (2.66) yields growth rates of two modes: secular GI and TVGI. The solutions of Eq. (2.66) indeed gives the static mode (n = 0) and the growth rate of secular GI (Eq. (2.52)) in the absence of turbulent viscosity (ν = 0). Figure 2.5 shows the dispersion relations of TVGI and secular GI for α = 10−3 , ε = 0.1, τs = 0.3 and Q = 5. Secular GI is stable for those parameters, and only TVGI grows. Note that n is real for both TVGI and secular GI, and thus they are not oscillating modes. The growth of TVGI is inefficient at long wavelengths since the angular momentum transport due to turbulent viscosity is inefficient. We derive the condition for TVGI using the approximated dispersion relation (Eq. (2.66)). We consider a case where secular GI is stable and also assume that a 2 > 0, that is, disk is self-gravitationally stable and ωgd 

(1 + ε)3/2 1 + ε (cd /cs )2

< Q.

(2.70)

2 One finds B2 > 0 for ωgd > 0 and weak turbulence that satisfies ενk 2 /tstop  2 . In such a case, the unstable condition is that Eq. (2.66) has one negative solution and

40

2 Revision of Macroscopic Equations for Dust Diffusion

one positive solution for a certain wavenumber k. This is equivalent to a condition that wavenumbers satisfying B0 < 0 exist. From Eq. (2.69), we obtain the following quadratic equation for k > 0:       B0 tstop 1+ε D 1+ε =3 D + cd2 2 − 2π Gg,0 D + cd2 + εcs2 k + + cd2 cs2 k 2 . 4 νk tstop tstop tstop

(2.71) Thus, such wavelengths exist if the discriminant of the right hand side of Eq. (2.71) is positive. We then find the following condition for the instability:    3Q 2 tstop cd2 + D tstop cd2 + (1 + ε) D < 1.    2 tstop cd2 + εcs2 + (1 + ε) D

(2.72)

The left hand side of Eq. (2.72) is independent from ν. This is partly because we assume weak turbulence. Another reason is that infinitesimally small viscosity is enough for TVGI. Leaving the leading term in Eq. (2.72) under the assumptions of α  τs  1 and cd2 /cs2 ∼ D/cs2 ∼ α gives  3 (1 + ε) or

QD εtstop cs2

2  1,

−1    D H −2 −1 −1 . tstop π Gg,0 H √ 3 (1 + ε)

(2.73)

(2.74)

The left hand side of Eq. (2.74) gives a timescale at which dust grains travel across a length of H = cs −1 with the terminal velocity. The right hand side gives a diffusion timescale of dust surface density perturbation with the length scale ∼H . Thus, Eq. (2.72) means that TVGI grows if dust grains can overcome turbulent diffusion and concentrate at the terminal velocity. This physical picture is similar to that of the one-component secular GI discussed in Youdin [18]. One can also estimate the most unstable wavelength if the higher order terms of νk 2 are negligibly small. Since the growth rate of TVGI is determined by the efficiency of viscous angular momentum transport, the growth rate is small when νk 2 is small. The growth rate is then approximately given by −B0 /B1 (see Eq. (2.66)). Neglecting the higher order terms of νk 2 , one obtains     n − νk 2 32 D (1 + ε) − 2π Gg,0 tstop εcs2 + D (1 + ε) k + Dcs2 k 2      × tstop εcs2 + D (1 + ε) 2 − 2π G (1 + ε) g,0 tstop εcs2 + D (1 + ε) k + D (1 +

ε) cs2 k 2

−1 (2.75)

2.3 Linear Analyses

41

0.008 0.006

Exact dispersion relation Approximate dispersion relation

0.004 n/Ω

0.002 0 -0.002 -0.004 -0.006 -0.008 0

5

10 kH

15

20

Fig. 2.6 Growth rates Re[n]/ of TVGI and secular GI for α = 10−4 , ε = 0.1, τs = 0.03 and Q = 4 as a function of dimensionless wavenumbers k H . The solid lines show the exact dispersion relations. The blue cross mark is the approximated growth rates (Eq. (2.66)). In this case, both c TVGI and the secular GI grow. This figure is originally shown in Tominaga et al. [15] ( AAS). Reproduced with permission

The term of cd2 k 2 is neglected in the above derivation since this term has insignificant effect to stabilize TVGI compared to dust diffusion. The most unstable wavenumber kmax is of the order of a wavenumber at which n/νk 2 has the local maximum, which is kmax

  π Gg,0 tstop εcs2 + D (1 + ε) ∼ Dcs2 1 + ε −1 ετs  H −1 . H + = Q Dcs−2 Q

(2.76)

The right hand side of the Eq. (2.76) is about 4.5H −1 for α = 10−3 , ε = 0.1, τs = 0.3 and Q = 5, which is consistent with the most unstable wavenumber seen in Fig. 2.5. Note that the unstable condition depends on ν when the higher order terms of νk 2 are not negligible. In the case that both TVGI and secular GI are unstable, those appear on one branch of the dispersion relation. Figure 2.6 shows the dispersion relations for α = 10−4 , ε = 0.1, τs = 0.03 and Q = 3. Those parameters satisfy the unstable conditions of both TVGI and secular GI (Eqs. (2.59) and (2.72)). However, in Fig. 2.6, there is only one unstable branch. We find that this apparent “single” unstable mode is due to mode exchange between TVGI and secular GI (Fig. 2.7). In the absence of turbulent viscosity, the static mode and secular GI share growthrates at wavenumbers where the growth rate of the secular GI is zero. In the presence of small but finite turbulent viscosity, curves of the dispersion relations of the destabilized static mode (i.e., TVGI) and secular GI reconnect at their crossing points, which results in single

42

2 Revision of Macroscopic Equations for Dust Diffusion

Fig. 2.7 Schematic picture of the mode exchange between TVGI and secular GI. The mode exchange occurs because of turbulent viscosity. Left figure shows the dispersion relations of secular GI and the statice mode in the absence of turbulent viscosity, and right figure shows those obtained in the presence of the turbulent viscosity. The blue line shows a branch of secular GI, and the red line shows the static mode or TVGI. The labels (B’), (B”), (C) and (D) shown in the legends correspond to the labels in Fig. 2.1. The mode exchange occurs at wavelengths where eigenvalues and eigenfunctions degenerate in the absence of turbulent viscosity. This figure is originally shown c in Tominaga et al. [15] ( AAS). Reproduced with permission 1

-2

10

10

Approximate formula (TVGI)

0

10

-1

-3

α

-2

10 -4

n/Ω

10

10

-3

10

10

-4

10 -5

10

-2

10

-1

10 τs

0

10

-5

10

Fig. 2.8 Maximum growth rates of TVGI and secular GI as a function of τs and α for  = 0.05 and Q = 10. The short-dashed line shows the maximum α for which TVGI can grow (Eq. (2.72)). TVGI is the fastest growing mode in the colored region above the dotted line. In the region between the dotted and long-dashed lines, secular GI grows the fastest. Dust GI is the most unstable mode below the long-dashed line as in Fig. 2.4. This figure is originally shown in Tominaga et al. [15] c ( AAS). Reproduced with permission

unstable branch. As shown in Fig. 2.7, we designate the growing mode as secular GI at wavenumbers where secular GI is unstable for ν = 0. On the other hand, we call the mode TVGI at wavenumbers where secular GI is stable for ν = 0. Figure 2.8 shows the maximum growth rate of TVGI and secular GI for ε = 0.05 and Q = 10 as a function of τs and α. TVGI grows faster than secular GI in the colored region above the dotted line. The dotted line is almost equivalent to a line

2.3 Linear Analyses

43

of the maximum α for which secular GI can grow (Eq. (2.59)). On the other hand, secular GI grows faster in the region enclosed by the dotted and long-dashed lines. Figure 2.8 also shows that TVGI can grow for larger α and smaller τs that stabilize secular GI. We thus expect that TVGI grows earlier than secular GI since the stopping time becomes larger as dust grains grow in protoplanetary disks. As described above, the dust grains concentrate via TVGI. Therefore, TVGI has the potential to explain planetesimal forming process. We should note that the above analyses do not include the radial drift of dust grains at the unperturbed state. In the presence of the significant radial drift, secular GI is more important than TVGI. Linear analyses with the radial drift are shown in the next Chapter.

2.4 Discussion: Effects of Disk Thickness In the previous section, we neglect the vertical thickness of a disk and regard it as an infinitesimally thin disk. The vertical thickness however stabilizes unstable modes to some extent because the thickness reduces self-gravity estimated for an infinitesimally thin disk. In this section, we investigate such effects on growth rates of secular GI and TVGI. According to [11, 16], the self-gravitational potential δ reduced by the disk thickness is approximately given by δ = −

2π G k



δg δd + 1 + kH 1 + k Hd

 ,

(2.77)

where Hd is the dust scale height:   τs 1 + 2τs −1/2 Hd = H 1 + . α 1 + τs

(2.78)

Using this equation rather than Eq. (2.51) gives the reduced growth rate. Figure 2.9 demonstrates the effect of the disk thickness on the dispersion relations of secular GI and TVGI for  = 0.05 and Q = 10. The most unstable mode is secular GI and TVGI on the left and right panels, respectively. In both cases, the disk thickness decreases the maximum growth rates by a factor of a few. Figure 2.10 shows the maximum growth rate as a function of τs and α for the same parameters as in Fig. 2.8. Although the maximum growth rates are smaller than those in Fig. 2.8, the extent of the unstable region is almost the same, and the maximum α for which the instabilities can grow does not change more than a factor of two. Equation (2.78) shows that a dust disk is generally thinner than a gas disk. In reality, gas above a dust disk hardly interact with dust grains via friction because of less dust abundance at the upper layer. In the above analysis, however, the back reaction is averaged in the full vertical extent of a gas disk. Thus, we may need

44

2 Revision of Macroscopic Equations for Dust Diffusion 0.008

0.02

w/ thickness w/o thickness

(a)

0.006

0.01 n/Ω

n/Ω

0.004 0.002

0.005

0

0

-0.002

-0.005

-0.004 0

w/ thickness w/o thickness

(b)

0.015

15 kH

10

5

20

25

30

-0.01 0

10 kH

5

15

20

Fig. 2.9 We compare dispersion relations of secular GI (left panel) and TVGI (right panel) for ε = 0.05 and Q = 10 in the case with the disk thickness (solid lines) and in the case of an infinitesimally thin disk (dashed lines). The dimensionless stopping time and strength of turbulence (τs , α) are set to be (0.1, 2.5 × 10−5 ) for the left panel and (1, 2.5 × 10−4 ) for the right panel (see also, Fig. 2.10). The turbulent viscosity is neglected on the left panel so that we allow only the secular GI to grow. We can see that the reduced self-gravity due to the disk thickness decreases both growth rates and c unstable wavenumbers decrease. This figure is originally shown in Tominaga et al. [15] ( AAS). Reproduced with permission 1

-2

10

10

Approximate formula (TVGI)

0

10

-1

-3

α

-2

10 -4

n/Ω

10

10

-3

10

10

-4

10 -5

10

-2

10

-1

10 τs

0

10

-5

10

Fig. 2.10 Maximum-growth-rate map modified by the disk thickness as a function of α and τs . Toomre’s Q for gas and dust-to-gas ratio ε are set to be the same value as those of Fig. 2.8. The shortdashed line is the same as that shown in Fig. 2.8. TVGI grows the fastest in the colored region above the dotted line. In the region below the dotted line, secular GI is the fastest growing mode. Both instabilities are stable in the white region. This figure is originally shown in Tominaga et al. [15] c ( AAS). Reproduced with permission

to exclude gas located above a dust disk from our analysis (e.g., see also [4]). We should also note that, even though gas above a dust disk is frictionally decoupled from dust, upper gas and the midplane dust interact with each other through their selfgravity. Thus, it is unclear to what vertical extent dust and gas should be considered. Although multidimensional analyses will give solutions to this problem, we focus on one-dimensional analyses in this work and discuss with some simplifications, which is described below. Assuming the following Gaussian functions for the unperturbed density structures, we investigate stability in a dust disk:

2.4 Discussion: Effects of Disk Thickness

45

ρg,0

  g,0 z2 , ≡√ exp − 2H 2 2π H

(2.79)

ρd,0

  d,0 z2 , ≡√ exp − 2Hd2 2π Hd

(2.80)

where ρg,0 and ρd,0 are the mass density of gas and dust, respectively. Vertically integrating these density profiles gives surface densities within a dust disk although there might be some uncertainty in the appropriate range of the vertical integration. Here, we integrate these densities in −3Hd ≤ z ≤ 3Hd . Although the dust density at z = 3Hd is smaller by orders of magnitudes than that at the midplane, the dustto-gas mass ratio is large because dust concentrates around the midplane and gas has much larger vertical scale height. Besides, friction force per unit mass exerted on dust grains does not depend on dust density, indicating that dust grains at a low dust-density region might take part in instabilities. We then obtains gas and dust surface densities within the dust sublayer:  g,0



3Hd ≡

ρg,0 dz = g,0 erf −3Hd

 d,0



3Hd ≡

ρd,0 dz = d,0 erf −3Hd

3 √ 2

3Hd √ 2H

 ,

(2.81)

 0.997d,0 ,

(2.82)

where erf(x) is the error function. The relation between the mid-plane dust-to-gas   /g,0 is mass ratio and d,0  d,0 H ρd,0 (z = 0) =  erf ρg,0 (z = 0) g,0 Hd



3Hd √ 2H

  −1 3 erf √ . 2

(2.83)

We perform linear analyses using these surface densities and the dust-to-gas mass   /g,0 . We introduce the modified Toomre’s Q parameter for ratio in the sublayer d,0  a gas disk Q˜ ≡ cs /π Gg,0 . We also modify the self-gravitational potential using Hd for both gas and dust: δ = −

2π G δg + δd . k 1 + k Hd

(2.84)

Figure 2.11 shows the maximum growth rates as a function of total dust-to-gas mass ratio d,0 /g,0 and Q = cs /π Gg,0 for τs = 0.1 and α = 10−4 . In this case, ˜ is about 14, meaning that Q˜ is 14 times larger Hd /H is about 0.03, and, thus, Q/Q than Q in the whole parameter space of Fig. 2.11. The dust-to-gas mass ratio in the

46

2 Revision of Macroscopic Equations for Dust Diffusion 0

1

10

Q for the whole gas disk

10

-1

10

-2

-3

10

n/Ω

10

-4

10

-5

10 0

10 -3 10

-2

10 Total dust-to-gas mass ratio

-1

10

-6

10

Fig. 2.11 Maximum growth rates of secular GI, TVGI, and dust GI as a function of total dust-togas mass ratio d,0 /g,0 and Q = cs /π G0 for τs = 0.1 and α = 10−4 . The color shows the growth rates normalized by . TVGI is the fastest growing mode in the colored region above the dotted line. Secular GI grows the fastest in the region enclosed by the dotted and dashed lines. Dust GI is the most unstable mode below the dashed line. This figure is originally shown in Tominaga c et al. [15] ( AAS). Reproduced with permission   dust disk d,0 /g,0 is also about 10 times larger than d,0 /g,0 . Secular GI grows the fastest in the parameter space bounded by the dotted and dashed lines. This is because Q˜ is too large and dust GI grows faster than secular GI. We also find that TVGI can grow in a larger parameter space. TVGI grows even for large Q˜ since the self-gravity of dust is important for its growth. Thus, at least in the absence of the radial drift, TVGI can grows even when we consider the vertical structures. We again note that gas above the dust disk will affect the motion in the dust disk via gravitational interaction. Thus, we may underestimate the self-gravity in this analysis. Increasing the vertical extent of the integration results in smaller Q˜ and a larger parameter space where secular GI is the most unstable. To examine the effects of the upper gas on the instabilities in the dust disk, we need to perform multidimensional analyses, which are beyond the scope of this thesis. Moreover, dust is less diffusive in dust rich regions [8, 14, 17]. The smaller radial diffusivity makes the disk more unstable to secular GI than in Fig. 2.11.

2.5 Summary In Chap. 2, we revisit dust dynamics in turbulent gas and reformulate the macroscopic equations of dust based on the mean-field approximation (the Reynolds-averaging). The reformulated equations conserve total angular momentum, which is in contrast to the previous studies that simply use the advection-diffusion equation for dust density. The difference comes from the fact that we introduce the advection terms originating from the diffusive flow in momentum equations. The reformulated equations can be used for various studies because of its simplicity.

2.5 Summary

47

Using the reformulated equations, we perform linear stability analyses on secular GI. We find that secular GI grows monotonically with a few times larger growth rate than expected in the previous studies. The property of the monotonic growth is in contrast to the previous studies that found overstabilized secular GI. Our linear analyses show that the overstability in the previous studies is resultant from the violation of angular momentum conservation. The present linear analyses show that turbulent viscosity introduces a new instability that we refer to as TVGI. TVGI grows for larger α and smaller τs for which secular GI can not grow. Because τs increases as dust grains grow, we may expect that TVGI becomes operational earlier than secular GI at least in the absence of significant radial drift (cf. Chap. 3). The vertical structure is a possible source stabilizing the instabilities. Simply assuming the Gaussian density profiles, we investigate its effect on the maximum growth rates of secular GI and TVGI. Although it is unclear to what vertical extent dust and gas should be considered, we consider the extent of −3Hd ≤ z ≤ 3Hd and perform linear analyses. Results show that secular GI grows the fastest in a region bounded by the dotted and dashed lines in Fig. 2.11. This is because gas mass in −3Hd ≤ z ≤ 3Hd is about 10 times smaller than the overall gas mass. On the other hand, TVGI can grow even larger Q and smaller dust-to-gas mass ratio. We should note that increasing the vertical extent in consideration expands unstable regions of those instabilities. Because upper gas interacts with the midplane dust through gravity, we may underestimate the growth rate and extents of parameter space unstable to secular GI. To explore more precisely, we need to perform linear analyses where vertical structures are explicitly considered. Those multidimensional analyses are our future work.

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14. Takeuchi T, Muto T, Okuzumi S, Ishitsu N, Ida S (2012) ApJ 744(2):101. https://doi.org/10. 1088/0004-637X/744/2/101 15. Tominaga RT, Takahashi SZ, Inutsuka SI (2019) ApJ 881(1):53. https://doi.org/10.3847/15384357/ab25ea 16. Vandervoort PO (1970) ApJ 161:87. https://doi.org/10.1086/150514 17. Xu Z, Bai XN (2021) arXiv:2108.10486 (2021) 18. Youdin AN (2011) ApJ 731:99. https://doi.org/10.1088/0004-637X/731/2/99 19. Youdin AN, Lithwick Y (2007) Icar 192:588–604. https://doi.org/10.1016/j.icarus.2007.07. 012

Chapter 3

Numerical Simulations of Secular Instabilities

3.1 Short Introduction The previous studies on secular GI focused on linear growth of the instability (e.g., [16, 27, 32, 36, 38, 39]). The linear analyses showed the possibility that secular GI creates some observed ring-like structures [27]. Moreover, resultant dust concentration is expected to lead to planetesimal formation (e.g., [28, 39]). However, the process from ring formation to planetesimal formation is a nonlinear process, which has not been studied. Studies on nonlinear secular GI are necessary toward understanding planetesimal formation. Although the linear growth at the local frame is well studied (e.g., [27, 32, 39] and Chap. 2), it is also necessary to investigate the growth of secular GI toward planetesimal formation as a global problem. This is because dust grains drift inward throughout a disk where the growth efficiency of secular GI varies radially. Numerical simulations are useful to study such a nonlinear global problem. We thus perform numerical simulations of secular GI and explore how the instability grows in a radially extended disk and to what extent the instability concentrates dust grains into rings. For numerical simulations of secular GI, it seems necessary to take notice especially of numerical errors and diffusion. A growth timescale of secular GI is typically about 100 times longer than one Keplerian period. Such a slow process will suffer numerical diffusion during long-term integrations. In addition, numerical diffusion due to spatial discretization and advection prevents growth of secular GI because dust grains drift inward in a disk and the associated advection numerically smoothes out seed perturbations of secular GI. Motivated by these numerical issues, we develop a Lagrangian-cell method that utilizes a symplectic integrator for time integrations [31]. Using Lagrangian cells, we avoid numerical diffusion due to the spatial advection. A symplectic integrator is one method used in N-body simulations for orbital evolutions because the method preserves an error in total energy throughout a calculation for Hamiltonian systems. Since secular GI requires frictional dissipation and the system is not exactly Hamiltonian, we adopt the operator splitting and use a symplectic integrator only for time integration with Hamiltonian part. We solve momentum evolution due to friction using the piecewise exact solution [12] that is free from time-step requirement due © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 R. Tominaga, Dust-Gas Instabilities in Protoplanetary Disks, Springer Theses, https://doi.org/10.1007/978-981-19-1765-3_3

49

50

3 Numerical Simulations of Secular Instabilities

to small tstop . Adopting these methods, we perform radially one-dimensional simulations secular GI [33]. Most of the contents in this chapter are based on our papers published: Tominaga et al. [31, 33]. In Sect. 3.2, we describe numerical methods and show some test calculations presented in Tominaga et al. [31]. We then show results of simulations of secular GI in Sect. 3.3. Discussions and summary are given in Sects. 3.4 and 3.5.

3.2 Numerical Methods As in the previous chapter, we consider the following gas and dust equations:   ∂g 1 ∂ r g u r + = 0, ∂t r ∂r  g

∂u i ∂u i + uj ∂t ∂x j



  G M∗ ∂ = − − g ∂ xi r    ∂u j vi − u i ∂ ∂u i 2 ∂u k g ν + d + + − δi j , ∂x j ∂x j ∂ xi 3 ∂ xk tstop (3.2) ∂g −cs2 ∂ xi

1 ∂ (r d vr ) 1 ∂ ∂d + = ∂t r ∂r r ∂r  d

 d

(3.1)

  ∂d rD , ∂r

(3.3)

     vφ2 G M∗ ∂vr ∂ D ∂d ∂vr 2 ∂d = d − cd − + vr − − d ∂t d ∂r ∂r r ∂r ∂r r   vr − u r 1 ∂ ∂d r vr D , (3.4) − d + tstop r ∂r ∂r

     ∂vφ vφ vφ − u φ D ∂d ∂vφ D ∂d + vr − = −d vr − − d , ∂t d ∂r ∂r r d ∂r tstop (3.5)   (3.6) ∇ 2  = 4π G g + d δ(z).

As shown in the previous chapter, the total angular momentum is conserved and the time evolution of the dust angular momentum is governed by        ∂ r vφ D ∂d ∂ r vφ vφ − u φ + vr − . d = −d r ∂t d ∂r ∂r tstop

(3.7)

3.2 Numerical Methods

51

We solve these equations utilizing the operator splitting method. In the following, we describe the methods in detail.

3.2.1 Lagrangian-Cell Method First, we formulate the Lagrangian-cell method, which we will utilize for nondissipation parts of hydrodynamic equations. To demonstrate the formulation clearly, we fist treat one-dimensional gas with the Cartesian coordinates and test the scheme against the traveling of sound wave to demonstrate that the scheme is free from the numerical diffusion due to the advection. Next, we formulate the method in the cylindrical coordinates. Our formulation of the scheme is based on the action principle. We consider only pressure as a source of momentum transfer. The Lagrangian for such 1D gas is given by    2

x˙ −u , (3.8) L = dx ρ 2 where x denotes a position of a fluid parcel, ρ is mass density, and u is specific internal energy. The dot mark represents time derivative, e.g. x˙ represents gas velocity. For a barotropic gas, i.e. P = P(ρ), one obtains

u=

P dρ. ρ2

(3.9)

Regarding the 1D gas as a set of N gas parcels (cells), we rewrite the above Lagrangian by discretizing the integral: L=

N −1 

m i+1/2

i=1

m i+1/2 =

2 x˙i+1/2

2

−

N 

mi ui ,

(3.10)

i=1

m i+1 + m i , 2

(3.11)

Physical properties defined at the ith cell are represented with the subscript i. xi+1/2 denotes a boundary position between the ith and (i + 1)th cells. Our Lagrangianbased formulation assumes that mass enclosed in the ith cell, m i , is conserved. We then calculate the density ρi at each time step as follows ρi =

mi . xi+1/2 − xi−1/2

The Euler-Lagrange equation in this discretized system is

(3.12)

52

3 Numerical Simulations of Secular Instabilities

d dt



∂L ∂ x˙i+1/2

 −

∂L = 0. ∂ xi+1/2

(3.13)

Substituting Eq. (3.10) into the Euler-Lagrange equation, we obtain m i+1/2 x¨i+1/2 = −

∂

N 

∂ xi+1/2

j=1

m ju j.

(3.14)

Equation (3.9) and the right hand side of Eq. (3.14) give −

∂

N 

∂ xi+1/2

j=1

m j u j = −Pi+1 + Pi .

(3.15)

The equation of motion of cell boundaries is finally given as follows m i+1/2 x¨i+1/2 = −(Pi+1 − Pi ).

(3.16)

We conduct time integration of Eq. (3.16) with a symplectic integrator. The leap-frog integrator, which is one second-order symplectic integrator, is adopted in this work. Our scheme (Eq. (3.16)) is second-order accurate in xi . To demonstrate it, we assume small cell widths (xi  xi ) below. Conducting the Taylor-series expansion on the right hand side of Eq. (3.16) around xi+1/2 , we obtain m i+1/2 x¨i+1/2 = −

xi + xi+1 d P + O(xi3 ), 2 dx

(3.17)

  where xi = xi+1/2 + xi−1/2 /2 is the position of the cell center and xi+1 = xi + O(xi2 ). The gas mass in the ith cell boundary m i+1/2 is then 1 (m i + m i+1 ) 2 ρi+1/2 = (xi + xi+1 ) + O(xi3 ). 2

m i+1/2 =

(3.18)

Using the above equation and Eq. (3.16), we obtain x¨i+1/2 = −

1 dP + O(xi2 ). ρi+1/2 d x

(3.19)

Thus, our method is second-order accurate in xi . Next, we conduct test simulations of 1D plane wave propagation. A numerical domain has the length L. We equally space the domain by N =128 cells. We assume the isothermal equation of state. A time-step is t = 0.5L/N cs . We adopt code units where the unperturbed gas density is 2 and L = cs = 1. We consider a sinusoidal

3.2 Numerical Methods

53

Fig. 3.1 Results of the test simulation on a one-fluid plane wave propagation. The time variation of the error in the total energy is on the left panel. The amplitude of errors is constant for a longerterm calculation. The right panel shows density perturbations at three time steps and compares the simulation results (black circles) with the exact solution (green line). The perturbation amplitudes are constant over 100 periods although there are dispersive errors. This figure is originally shown in Tominaga et al. [31] (Fig. 16 therein), and is reproduced with permission from Oxford University Press

initial perturbation and displace cells with an amplitude of ξ = 1.0 × 10−6 and a wavelength of λ = 0.5. The time evolution of errors in the total energy is shown in the left panel of Fig. 3.1. On the right panel, we show the time evolution of the density perturbation δρ. The oscillating behavior in the energy errors stems from the fact that the method utilizes the symplectic integrator. Although we found dispersive errors in the density evolution (the right panel of Fig. 3.1), the amplitude of the density perturbations is constant for over 100 periods, meaning that the method is free from numerical diffusion. We also check convergence and spatial accuracy performing other test simulations. We use the same setups as in the above wave-propagation tests, and conduct two test simulations: (1) wave propagation tests with the periodic boundary condition, and (2) standing wave tests with the fixed boundary condition. We let the fluid evolve until t = 3 periods adopting the time step of t = 0.1L/512cs for both simulations. In Fig. 3.2, we show L 2 norm error in the density as a function of N . The figure shows that the error is proportional to N −2 , which demonstrates the second-order spatial accuracy of our scheme. We finally adopt the above methods in the cylindrical coordinates (r, φ). We treat an infinitesimally thin axisymmetric disk and its radially 1D evolution. The discretized Lagrangian and surface density at each cell are given as follows L=

N −1  i=1

m i+1/2

2 r˙ i+1/2

2

+ (ri+1/2 ) −

2 2 2 2 r˙ i+1/2 φ˙ i+1/2 ≡ r˙i+1/2 + ri+1/2 ,

N 

mi ui ,

(3.20)

i=1

(3.21)

54

3 Numerical Simulations of Secular Instabilities

Fig. 3.2 N -dependence of L 2 norm error from the convergence tests. Squares and triangles represent the results of the wave propagation and standing wave tests, respectively. This figure is originally shown in Tominaga et al. [31] (Fig. 17 therein), and is reproduced with permission from Oxford University Press

i =

mi ,  2 2 π ri+1/2 − ri−1/2

(3.22)

where  denotes potential energy. Equation (3.20) and the Euler-Lagrange equation yield m i+1/2 r¨i+1/2 =

2 Ji+1/2 3 m i+1/2 ri+1/2

− 2πri+1/2 (Pi+1 − Pi ) − m i+1/2 d Ji+1/2 = 0, dt Ji+1/2 ≡

∂L , ˙ ∂ φi+1/2

∂(ri+1/2 ) , (3.23) ∂ri+1/2 (3.24) (3.25)

where Ji+1/2 is angular momentum defined at each cell boundary r = ri+1/2 . In Fig. 3.3, we schematically show the radial locations at which we assign each physical variables. For two-fluid calculations, we also use the same cell-structures for dust and gas. Our scheme is a symplectic scheme in the absence of aerodynamical friction between dust and gas. Therefore, the scheme is free from numerical diffusion and enables accurate long-term calculations. Self-gravity solver To calculate disk’s self-gravity, we superimpose gravitational forces from infinitesimally thin rings that are (1) coaxial with the center at (r, φ) = (0, 0), (2) located in z = 0 plane, and (3) having uniform line densities. Gravitational potential ring (r, z; Rring ) associated with a ring with a mass of Mring and a radius of Rring is given by ring (r, z; a) = Mring U (r, z; Rring ),

(3.26)

3.2 Numerical Methods

55

Fig. 3.3 Schematic picture to show radial locations at which each physical variable is defined. Each cell boundary has radial velocity and angular momentum that we update. R(i) is a radius that equally divides a mass in a ith cell. This figure is originally shown in Tominaga et al. [31] (Fig. 1 therein), and is reproduced with permission from Oxford University Press

U (r, z; Rring ) ≡ − where p and n are p≡



2G K (n) , πp

(r + Rring )2 + z 2 ,

n≡

(3.27)

(3.28)

4Rring r . p2

(3.29)

K (n) is the complete elliptic integral of the first kind

π/2 

K (n) ≡ 0

dφ 1 − n sin2 φ

.

(3.30)

The ring gravity per unit mass Fr (r, z; Rring ) is thus given by ˜ z; Rring ) Fr (r, z; Rring ) ≡ Mring F(r, ˜ z; Rring ) = − G F(r, πp



 K (n) E(n) + A(r, z; Rring ) 2 , r q

(3.31) (3.32)

56

3 Numerical Simulations of Secular Instabilities

A(r, z; Rring ) ≡ q≡



2 r 2 − z 2 − Rring

r

,

(r − Rring )2 + z 2 ,

(3.33) (3.34)

where E(n) is the complete elliptic integral of the second kind

E(n) ≡

π/2

1 − n sin2 φdφ.

(3.35)

0

In this work, we use approximated functions presented in Hastings et al. (1955) [10] to evaluate the elliptic integrals. We found that using the following “perturbed ring mass” δm i+1/2 to calculate the self-gravitational force well reproduces the linear evolution of GI and secular GI, δm i+1 + δm i , 2   2 2 δm i ≡ π ( − 0 ) ri+1/2 , − ri−1/2 δm i+1/2 ≡



δm j+1/2 F˜r (ri+1/2 , 0; r j+1/2 ),

(3.36) (3.37) (3.38)

i= j

where 0 is an initial (unperturbed) surface density. We can regard this treatment based on δm as solving the Poisson equation for gravitational source of δ ≡  − 0 . For a problem with dust and gas, we calculate the self-gravity using this approximation for both dust and gas based on their surface densities d , g , unperturbed densities d,0 , g,0 , and their cell masses m d,i , m g,i . We also introduce a softening length h to avoid divergence of ring gravity at ring positions: 

δm j+1/2 F˜r (ri+1/2 , h; r j+1/2 ).

(3.39)

i= j

We find that the error in the above self-gravity solver is reduced if we include ring’s gravity exerting on itself. To describe this prescription, we consider self-gravity on a ring at r = ri+1/2 as an example. In addition to the formula (Eq. 3.38), we introduce gravity from “sub-rings” at r = ri , ri+1 . To calculate gravity from the sub-ring at r = ri , we divide the mass m i into two parts: m −,a and m −,b (see Fig. 3.4):   2 − ri2 i , m −,a ≡ π ri+1/2   2 m −,b ≡ π ri2 − ri−1/2 i .

(3.40) (3.41)

3.2 Numerical Methods

57

Fig. 3.4 This figure shows how to distribute a mass in the ith cell when we introduce the correction terms for self-gravity. The mass assigned atr = ri is m − = m −,a + m −,b /2, where 

2 2 i . We replace the mass at r = ri−1/2 , m −,a ≡ π ri+1/2 − ri2 i and m −,b ≡ π ri2 − ri−1/2   i.e. m i−1/2 , with m i−1 + m −,b /2. This figure is originally shown in Tominaga et al. [31] (Fig. 18 therein), and is reproduced with permission from Oxford University Press

Using these masses, we change masses defined at r = ri (m = m − ), ri−1/2 (m = m i−1/2 ) as follow m −,b , 2 m i−1 + m −,b . = 2

m − = m −,a + m i−1/2

(3.42) (3.43)

We also calculate masses m + and m i+3/2 of sub-rings at r = ri+1 , ri+3/2 respectively using m +,a and m +,b as follows:   2 2 i+1 , − ri+1/2 m +,a ≡ π ri+1   2 2 m +,b ≡ π ri+3/2 − ri+1 i+1 , m +,b , m + = m +,a + 2 m i+2 + m +,b m i+3/2 = . 2

(3.44) (3.45) (3.46) (3.47)

We calculate gravity due to the sub-rings at r = ri−1/2 , ri , ri+1 , and ri+3/2 in the same way as Eq. (3.38). For dust-gas systems, we make use of the above correction for both dust and gas based on their cell masses m d,i , m g,i . We perform a test simulation of one-fluid GI. We consider a Keplerian disk around a solar mass star. A local domain is set around r = 80 au with a radial width L of twice the most unstable wavelength. Only in this test simulation, we use a piecewise polytropic relation (cf., [18])

58

3 Numerical Simulations of Secular Instabilities

 P=

2 cs,0 0

 + 0



 0

5/3 ,

(3.48)

where cs,0 is an isothermal sound speed, and 0 is set to be the unperturbed surface density. The most unstable wavelength is scaled by cs,0 /. To compare results with local linear analyses, we assume small cs,0 (cs,0  0.18 m s−1 ) and make the most wavelength ∼ 10−4 times shorter than the orbital radius. The assumed sound speed is a thousand times smaller than a sound speed with mean molecular weight 2.37 and temperature 10 K. We set the unperturbed surface density 0 assuming Toomre’s Q of 0.999 at r = 80 au. We initially put sinusoidal perturbation in the density with the amplitude of δ/0 ∼ 10−5 . Perturbations in the other variables are set according to the eigenfunction of the most unstable mode. Although we adopt the fixed boundary condition, we let a surface density outside the domain evolve at the linear growth rate of GI. The external density perturbations on both sides are resolved by 128 cells. We space the domain using 512 cells. Time interval t is set to be t = L/512cs . We also use the softening length h = L/4N to avoid divergence of ring gravity. Figure 3.5 shows the time evolution of the density peak at the center of the domain. The growth rate is about 4.5 × 10−2  for Q = 0.999. The numerical results reproduce analytically derived growth rate. In this way, our scheme can accurately solve such a very slow evolution with our symplectic method.

Fig. 3.5 Time evolution of the surface density peak via GI. The horizontal axis is time in the unit of −1 . The black circles are simulation data, and the solid green line is the linear growth expected from local linear analyses. The numerical results show good agreement with the linear growth rate, meaning that our scheme can accurately solve such a long-term evolution. This figure is originally shown in Tominaga et al. [31] (Fig. 2 therein), and is reproduced with permission from Oxford University Press

3.2 Numerical Methods

59

3.2.2 The Piecewise Exact Solution for Friction We adopt the piecewise exact solution to calculate frictional momentum transport [12]. The stopping time is much smaller than one Keplerian period for small dust grains considered in this study. In such a case, time stepping for stable simulations is restricted by the small stopping time, which makes long-term simulations difficult. The piecewise exact solution enables us to get rid of such a restriction. The method is based on an operator splitting method. In our cases, we split the equations into time evolutions due to (1) friction, (2) dust diffusion, (3) turbulent viscosity, and (4) the other hydrodynamical force that are solved based on the Lagrangian-cell method (e.g., pressure gradient, self-gravity). In the part (1), the momentum equations can be solve analytically. The piecewise exact solution utilizes the analytical solution and let momentums evolve. This method is not only free from the above restriction but also unconditionally stable. In the following, we describe the scheme in more detail. The part (3) is described afterward. When solving the frictional interaction, we only update linear and angular momentums of dust and gas whose positions are unchanged. Because our method is based on the Lagrangian description, positions of dust cells do not necessarily coincide with gas cells’ positions. Thus, we first interpolate physical variables in ri ≤ r < ri+1 using the√physical values √ at r = ri+1/2 . We use linear functions in r for radial velocities, jg / r and jd / r , where jd and jg are specific angular momentum of dust and gas. Coefficients of r 1 and r 0 in the linear functions are determined so that spatial integrations of the interpolation functions give the radial momentum and angular momentum of the cell boundary at r = ri+1/2 . In this way, we avoid numerical diffusion due to the interpolation. We integrate the interpolation functions and calculate masses, radial momentums, and angular momentums in a region where jth dust cell and ith gas cell overlap with each other. Those values are denoted by m kg , m kd , Pgk , Pdk , Jgk , and Jdk for gas and dust at kth overlap region (see Fig. 3.6). These momentums are updated according to the following equations: d Pgk

Uk − V k , k tstop

(3.49)

d Pdk V k − Uk = −m kd , k dt tstop

(3.50)

dt

d Jgk

= −m kd

= −m kd

jgk − jdk

,

(3.51)

jdk − jgk d Jdk = −m kd k , dt tstop

(3.52)

dt

k tstop

60

3 Numerical Simulations of Secular Instabilities

Fig. 3.6 This figure shows sub-regions in cells (colored with dark and light gray) used when we calculate dust-gas momentum transport with the piecewise exact solution. Especially, the figure highlights regions used for momentum changes at the gas cell boundary at r = rg,i+1/2 .Their boundaries are either the cell boundary or the cell center of gas and dust. The number k characterizes each region. Cells enclosed by thick lines represent gas cells while those enclosed by thin lines represent dust cells. This figure is originally shown in Tominaga et al. [31] (Fig. 3 therein), and is reproduced with permission from Oxford University Press

where U k ≡ Pgk /m kg and V k ≡ Pdk /m kd denote radial velocity, and jgk ≡ Jgk /m kg and jdk ≡ Jdk /m kd are specific angular momentum. Assuming m kg and m kd to be constant, one can analytically solve the above differential equations as follows:   Pgk (t + t) = Pgk (t) − U k (t) − V k (t) f k (t),

(3.53)

  Pdk (t + t) = Pdk (t) + U k (t) − V k (t) f k (t),

(3.54)

  Jgk (t + t) = Jgk (t) − jgk (t) − jdk (t) f k (t),

(3.55)

  Jdk (t + t) = Jdk (t) + jgk (t) − jdk (t) f k (t),

(3.56)

f (t) ≡ k

m kg m kd m kd + m kg





m kd + m kg t 1 − exp k m kg tstop

,

(3.57)

where we assume the interval of time integration to be t. Using these analytical solutions, we update linear and angular momentums at each region where a dust cell overlaps with a gas cell. Summing up updated values Pgk , Pdk , Jgk , and Jdk in each k k cell gives radial linear momentums, Pg,i+1/2 and Pd,i+1/2 , and angular momentums, k k Jg,i+1/2 and Jd,i+1/2 , of cell boundaries:

3.2 Numerical Methods

61

Pg,i+1/2 (t + t) =



Pgk (t + t),

(3.58)

Pdk (t + t),

(3.59)

Jgk (t + t),

(3.60)

Jdk (t + t),

(3.61)

k

Pd,i+1/2 (t + t) =

 k

Jg,i+1/2 (t + t) =

 k

Jd,i+1/2 (t + t) =

 k

This method exactly conserves total linear momentum and total angular momentum.

3.2.3 Methods for Turbulent Diffusion and Viscosity We use the second-order Runge-Kutta integrator for the time integration with dust diffusion and viscosity parts. To calculate the mass flux de to dust diffusion, we interpolate dust surface density using a quadratic function. The mass flux D∂d /∂r at a dust cell boundary is given by the r -derivative of the interpolation function. We then displace the dust cell boundaries using velocity −d−1 D∂d /∂r . We do not update angular momentum of the dust cell boundaries when displacing them. This is because Eq. (3.7) states that angular momentum is carried along the diffusion flow (see the Lagrange derivative). The radial linear momentum of ith dust cell boundary, m d,i+1/2 vr,i+1/2 , is updated in the diffusion part based on the last term on the right hand side of Eq. (3.4). We evaluate F(r ) ≡ r vr D∂d /∂r at r = rd,i+1 , rd,i , and update m d,i+1/2 vr,i+1/2 with using F(rd,i+1 ) − F(rd,i ), where we omit brackets representing the averaged value for the simplicity. Dust diffusion tends to limit the time step as dust grains concentrate in small radial regions. To relax the limitation due to dust diffusion, we adopt the super-time-stepping (STS) [1] if t is limited by dust diffusion. The number of substeps and the stability parameter in the STS scheme are fixed to be 4 and 0.1, respectively. Because the total time step in the STS should be shorter than a timescale “physically” required by dust diffusion, we just moderately accelerate the time stepping. In the viscous evolution part, we interpolate u r and jg , calculating radial and angular momentum fluxes at r = rg,i+1 , rg,i . In the cylindrical coordinates, the momentum changes due to turbulent viscosity are written as follows 

∂g u r ∂t





≡ vis

∂g jg ∂t

2u r ∂r 2 ν 1 ∂ Fvis,r − 3 , r ∂r 3r ∂r  ≡ vis

1 ∂ Fvis,φ , r ∂r

(3.62)

(3.63)

62

3 Numerical Simulations of Secular Instabilities

∂u r 4 , Fvis,r ≡ r ν 3 ∂r Fvis,φ ≡ r 3 ν

(3.64)

∂  uφ  . ∂r r

(3.65)

Spatial integration of the above equations give a net change in momentums of ith gas cell boundary, m g,i+1/2 u r,i+1/2 and m g,i+1/2 jg,i+1/2 :  r ∂m g,i+1/2 u r,i+1/2 4π = 2π Fvis,r rg,i+1 − g,i ∂t 3

r g,i+1

rg,i

u r ∂r 2 ν dr, r 2 ∂r

 r ∂m g,i+1/2 jg,i+1/2 = 2π Fvis,φ rg,i+1 , g,i ∂t

(3.66)

(3.67)

where [A(r )]rr21 ≡ A(r2 ) − A(r1 ). We update radial and angular momentums of gas cell boundaries according to Eqs. (3.66) and (3.67). We evaluate the last term on the   rg,i+1 2 2 right hand side of Eq. (3.66) by − 4π u r,i+1/2 /3rg,i+1/2 r ν rg,i . Test simulations on secular GI and TVGI To validate our numerical methods, we perform test simulations of secular GI and TVGI. As in the test simulations of GI, we use the piecewise polytropic equation of state for gas pressure, and initially set flat surface density profiles for dust (d,0 ) and gas (g,0 ). The Toomre’s Q of gas is set to be 5, and strength of turbulence α is 2 × 10−4 . We also assume that tstop  is uniform in the domain and set tstop  = 0.1. We consider two values of the dust-to-gas ratio d,0 /g,0 : 0.08 for a simulation of secular GI and 0.04 for a simulation of TVGI. Although diffusion coefficient D and those are small for tstop  < 1 and we velocity dispersion cd have tstop -dependences, √ simply assume D = αcs H and cd = αcs . To guarantee the locality, as in the test simulation of one-fluid GI, we place the center of the domain at r = 80 au and set 100 time smaller gas scale height H than that for a disk around solar mass star. The domain width is 4 times the most unstable wavelength. We adopt periodic boundary condition for dust and gas surface density profiles, which we utilize to calculate self-gravity. The number of cells is 512 for both dust and gas, and the domain is equally spaced. The softening length is initially set to be one fourth of the cell width. To follow nonlinear evolution, we change the softening length in time at a region of positive surface density perturbations and reset the length to be one fourth of the cell width at each time. The time step is set based on the following:   t = min tg , td , tdiff , tvis ,

(3.68)

where tg and td are the time steps determined by the Courant-Friedrich-Levy (CFL) condition for gas and dust equations without the dust diffusion or the viscosity, tdiff and tvis are time steps required for the stability by the diffusion term and the

3.2 Numerical Methods

Fig. 3.8 Time evolution of surface density peaks of dust (red line) and gas (blue line) via secular GI. The black segments represent the linear growth rate derived from the linear analyses. Numerical simulations reproduce the linear growth of secular GI

Surface density [Σg,0]

1

tΩ=0 tΩ=456 tΩ=657

10

0

10-1

10-2

-1

102

-0.5

0 x [H]

0.5

1

Dust Gas

1

10 δΣ/Σg,0, δΣd/Σd,0

Fig. 3.7 Surface density profile normalized by the unperturbed gas surface density and its evolution via secular GI. The horizontal axis is normalized by the assumed gas scale height. Red and blue lines represent surface density of dust and gas, respectively. The short-dashed, long-dashed, and solid lines correspond to the profile at t = 0, 456, 657

63

100

e

th rat

r grow

Linea

10-1 10-2 -3

10

10-4 10-5

0

100

200

400

300

500

600

700

tΩ

viscosity term: tdiff = 0.125 × min((rd,i+1/2 − rd,i−1/2 )2 /D), and tvis = 0.125 × min((rg,i+1/2 − rg,i−1/2 )2 /ν). The CFL number is set to be 0.5. Figure 3.7 shows dust and gas surface density evolution via secular GI. The amplitude δd /d,0 is larger by an order of magnitude than δg /g,0 . Secular GI concentrates dust into narrow regions with a width of H while its linear growth proceeds at a wavelength ∼H . The resultant concentration increases dust-to-gas ratio above unity in the dust rings. Figure 3.8 shows the time evolution of the surface density peak. The black segment represents the linear growth rate obtained from the linear analyses. The numerical results at the long-term linear growth until t  500 are in good agreement with the linear analyses, which validates the efficiency of our scheme for exploring long-term evolution even in the presence of the friction. The simulation shows divergent behavior at the nonlinear growth phase of secular GI. To understand the physics of the nonlinear growth, we fit the dust surface density evolution in 500 ≤ t ≤ 670 using the power-law function:

3 Numerical Simulations of Secular Instabilities

δΣd/Σd,0

64 10

2

10

1

10

0

(tc-t)-1

Simulation Fitting function

10

-1

100

101

102

103

(tc-t)Ω

Fig. 3.9 Dust surface density evolution at the nonlinear growth phase. The horizontal axis is the looking-back time from the collapse time tc obtained by fitting the data. The divergent nature with proportional to (tc − t)−1 is consistent with the freefall collapse of a self-gravitating ring

δd a = , d,0 (tc  − t)b

(3.69)

where a, b, tc are parameters. The time tc represents collapse time. The fitting gives a = 103 ± 5, b = 1.05 ± 0.01, and tc  = 663.0 ± 0.1. Figure 3.9 shows the resultant fitting function and the original data. The density evolution is almost proportional to (tc − t)−1 . This behavior originates from gravitational collapse of a ring as explained below. A timescale of self-gravitational collapse √ √ is given by the freefall time: tff ∼ 1/ Gρd . Assuming d ∼ ρd λJ where λJ ∼ cd / Gρd is the Jeans length, one obtains d ∼ cd /G × tff , Thus, dust surface density increases with being proportional to (tc − t)−1 . Figures 3.10 and 3.11 show results of the test simulation of TVGI. TVGI also concentrates dust into thin rings through its nonlinear growth. Although the growth timescale of TVGI is much longer than secular GI, our scheme can trace such a long-term linear growth until t  2250 and the result shows in good agreement with the linear analyses. This validates the efficiency of our scheme.

3.3 Simulations of Radially Extended Disks We investigate evolution of radially extended disks via secular GI using Eqs. (3.1), (3.2), (3.3), (3.4), (3.5), and (3.6), which are summarized at the beginning of this chapter. A dust layer around the midplane seems to be the most important region for secular GI since the instability is triggered by the dust-gas friction. This is also discussed in the linear analyses presented in Chap. 2. Gas above the dust layer will also contribute to the growth of secular GI via the gravitational interaction although the frictional

3.3 Simulations of Radially Extended Disks

65

Surface density [Σg,0]

101

tΩ=0 tΩ=1756 tΩ=2599

100

10

-1

-2

10 -2

-1

0 x [H]

1

2

Fig. 3.10 Surface density profile normalized by the unperturbed gas surface density and its evolution via secular GI. The horizontal axis is normalized by the assumed gas scale height. Red and blue lines represent surface density of dust and gas, respectively. The short-dashed, long-dashed, and solid lines correspond to the profile at t = 0, 1756, 2599 2

10

1

10 δΣ/Σg,0, δΣd/Σd,0

Fig. 3.11 Time evolution of surface density peaks of dust (red line) and gas (blue line) via TVGI. The black segments represent the linear growth rate derived from the linear analyses. Numerical simulations reproduce the slow linear growth of TVGI

Dust Gas

0

10 10

-1

10

-2

10

-3

te

rowth ra

Linear g

10-4 -5

10 0

500

1000

1500 tΩ

2000

2500

3000

interaction is weak. In the present study, we thus adopt Eqs. (3.1), (3.2), (3.3), (3.4), (3.5), and (3.6) for a “lower layer” that includes those dust and gas driving secular GI. As also mentioned in the previous chapter, the vertical extent of such a lower layer is unknown unless one perform multidimensional analyses. Hereafter, we do not concern the vertical extent since it is beyond the scope of this study. Setups The initial inner boundaries for dust and gas are set at 10 au. We use 1024 cells for both dust and gas and space the radial domain so that each cell has the same mass. Such a spacing gives the outer boundaries located at r  300 au. We fixed the gas and dust outer boundaries in simulations. We let the gas inner boundary move so that the innermost gas surface density becomes constant in time. On the other hand, we allow dust cells to move to the inner region of r < rg,1 where rg,1 denotes the radius of the fist gas cell center. The radial velocities of those inner dust cells are fixed to the steady drift velocity (e.g., [19]) estimated for the initial density profiles

66

3 Numerical Simulations of Secular Instabilities

Table 3.1 Summary of parameters and results. This table is originally shown in Tominaga et al. c [33] (AAS). Reproduced with permission. The layout is modified Label Q 100 a α Results tfin b Q4a10

4

1 × 10−3

Q4a20

4

2 × 10−3

Q5a5

5

5 × 10−4

Q5a8

5

8 × 10−4

Q5a8Lc

5

8 × 10−4

Q6a3

6

3 × 10−4

Q6a5

6

5 × 10−4

Thin dense dust rings Transient low-contrast rings Thin dense dust rings Transient low-contrast dust rings Thin dense dust rings Thin dense dust rings Transient low-contrast dust rings

2.1 × 104 yr 5.9 × 104 yr 1.9 × 104 yr 5.6 × 104 yr

2.5 × 104 yr 2.7 × 104 yr 6.2 × 104 yr

Q value for the lower layer of a gas disk at r =100 au that simulations last for c The letter “L” means a run with six times larger perturbations a Toomre’s b Time

at r = rg,1 . We weakens gravity from each cell adopting a softening length of half cell’s width (∼0.1 au). We rescale the softening length when the cell width becomes larger in time. We consider a disk around a star whose mass is 1M . The following power law functions are adopted as initial gas and dust surface density profiles:  r −q r  , exp − 100 au 100 au  r −q  r  d (r ) = d,100 , exp − 100 au 100 au g (r ) = g,100



(3.70) (3.71)

where g,100 and d,100 are constants. In this study, we use the Toomre’s Q value of gas (Q = cs /π Gg ) to show how massive the lower layer is. One obtains 100 from the assumed Q value at r = 100 au (Table 3.1). The dust surface density at r = 100 au is determined by the assumed initial dust-to-gas ratio in the lower layer d /g . We note that d /g is different from “total” dust-gas ratio d,tot /g,tot . d,tot and g,tot denote total surface densities of dust and gas disks: a mass in both upper and lower layers is included. In very weakly turbulent disks, d /g can be higher by an order of magnitude than d,tot /g,tot . In this work, we consider dust rich disks with d,tot /g,tot = 0.05, and assume d /g = 0.1. We adopt q = 0.5 in this study. This choice is motivated by Kitamura et al. [15], which observationally

3.3 Simulations of Radially Extended Disks

67

found that disks around T Tauri stars show density profiles with the power-law index between 0–1 in most cases. We assume the following temperature profile T (r ) is T (r ) = 10 K



r −1/2 . 100 au

(3.72)

The gas scale height is then H  6.3 au (r/100 au)5/4 . The initial azimuthal velocity is given by the radial force balance without friction, diffusion or viscosity. The dimensionless stopping time tstop  is one important parameter determining the efficiency of secular GI. Although it is important to explicitly implement dust growth in our code and explore the coevolution of dust grains and secular GI, we focus only on secular GI in radially global disks and assume a uniform tstop  profile to make the problem as simple as possible. Because a timescale of the dust coagulation is expected to be shorter than that of secular GI when dust grains are small [27], dust sizes will be well approximated by the drift-limited value (e.g., [20]). The driftlimited value is a maximum value expected in the dust coagulation, especially at outer radii where dust fragmentation is insignificant (see also [6, 7]). We thus adopt the drift-limited tstop , which is about 0.6 for d,tot /g,tot = 0.05 (see Appendix A of [33]). The diffusion coefficient√D and the velocity dispersion cd are well approximated by D  αcs2 −1 and cd  αcs for tstop  < 1 although they have weak dependence on tstop  and α [41]. We use those simplified relations in the present simulations for simplicity. In this study, we perform numerical simulations with different Toomre’s Q values and strength of turbulence α. We summarize the parameters in Table 3.1. Disks considered in our simulations are massive, and our choice of the parameters is optimistic. Previous observations found a disk (Elias 2–27, e.g. [21]) which is a Class II disk but very massive (∼0.04 − 0.1M , [4, 13, 23]) compered to other Class II disks. Thus, the present results might correspond to time evolution of such massive disks. We should also note that disk masses are not well-constrained by observations because of some underlying assumptions: for example, dust-gas ratio, dust opacity and low optical thickness. Ansdell et al. [5] shows variation in dust-gas ratio across disks that they observed using ALMA. Regarding the assumption of optical thickness, some observational studies show that mm-continuum-flux from disks is proportional to the area of disks (the square of a disk size), indicating that disks might be optically thick with some filling factor (e.g., [2, 3, 17, 30, 34]). Thus, optically thin approximation to derive disk masses, which is widely adopted, might be invalid to some extent. In addition, recent work shows that neglecting scattering effects of dust thermal emissions underestimates dust masses [43], i.e., disks might be actually massive. Therefore, in this study, we assume such massive disks where secular GI operates and conduct first investigation of its radially global evolution. As an initial perturbation, we randomly displace the cell boundaries and also add random fluctuations in their velocity. Amplitudes of the initial perturbations are five percent of the cell widths and cd , respectively.

68

3 Numerical Simulations of Secular Instabilities

Results We first overview the radially global evolution of secular GI. The results show two regimes of the evolution: formation of “thin dense dust rings” and “transient lowcontrast dust rings” (see also Table 3.1). In the following, we show those results in detail. Formation of Thin Dense Dust Rings

Dot-dashed lines: t = 0.0 yr 4 Dashed lines: t = 1.5 x10 yr Solid lines: t = 2.1 x104yr

100

10

Gas

1

Dust

0.1 40

50

60

70 80 90 Radius [au]

100

110

120

Surface density (Σ and Σd) [g/cm2]

Surface density (Σ and Σd) [g/cm2]

Figure 3.12 shows time evolution of dust and gas surface densities from Q4a10 run. We also show the surface density evolution from a run in which we switch off selfgravity to eliminate secular GI. Note that the gas disk is self-gravitationally stable and the dust GI is also stabilized because of dust diffusion within the set of the parameters. The results show formation of multiple dust rings and gaps only when we switch on self-gravity. Thus, the ring-gap formation is associated with secular GI. Secular GI grows at wavelengths ∼cs /, resulting in the ring-gap formation in the dust disk (see the dashed line in Fig. 3.12). Nonlinear growth of secular GI makes the resultant rings much thinner as in the test simulations, and the dust surface density increases by a factor of  10. On the left panel of Fig. 3.13, we show trajectories of dust cells that compose the resultant rings and gaps. We show a trajectory of the 525th cell (red dashed line) as a reference cell in one dust ring. We note that a reduced number of cells are shown on the left panel of Fig. 3.13. Because our numerical scheme is based on the Lagrangiancell method, the motion of cells represents actual motion of dust. A region where a large number of cells exist corresponds to a high density region. The dust initially moves inward with the steady drift velocity vdri (Fig. 3.14)

Dot-dashed lines: t = 0.0 yr 4 Dashed lines: t = 1.5 x10 yr Solid lines: t = 2.1 x104yr

100

10

Gas

1

Dust

0.1 40

50

60

70 80 90 Radius [au]

100

110

120

Fig. 3.12 Surface density evolution from Q4a10 run (left panel) and from a run where we use the same parameters as in Q4a10 run but switch off self-gravity. Red and blue lines in both panels show dust and gas surface density profiles. The different line types show a density profile at different time steps. Multiple dust rings and gaps form within 104 yr in Q4a10 run while the perturbations do not grow in the right panel. We note that the classical GI is stable in this simulation because of the dust diffusion. Therefore, the multiple ring-gap formation observed in this Q4a10 run is due to c the development of secular GI. This figure is originally shown in Tominaga et al. [33] (AAS). Reproduced with permission

3.3 Simulations of Radially Extended Disks

69 100

1

80 70 60 50 0

0.5

1 1.5 Time [104yr]

0.1

2

i=520 - 524 i=526 - 530 i=525

95 Radius [au]

Radius [au]

90

Dust surface density [g/cm2]

100

90 85 80 75 70-3

-2.5

-2 -1.5 -1 -0.5 Radial velocity [10-3au/yr]

0

Fig. 3.13 (Left) Trajectories of dust cells in Q4a10 run. Color of each line shows dust surface density at each dust cell. We used a reduced number of cells to plot this figure. Red dashed line is the trajectory of the 525th dust cell. (Right) The radial velocity of some dust cell boundaries (r = rd,i+1/2 where i = 520 − 530) at each radius. The trajectory of the 525th cell boundary (red line) roughly corresponds to motion of the ring peak. Gray and black lines show the inner and outer cells, respectively. The lines spread in vr − r plane, which means that there is a collapsing motion toward the ring center. The softening of ring gravity suppresses the accelerated collapse at r  77 au. The dust drift is decelerated by the backreaction as d /g increases, which can be seen c at r  77 au. This figure is originally shown in Tominaga et al. [33] (AAS). Reproduced with permission Dust (t = 2.1 x104yr) Gas (t = 2.1 x104yr) Steady drift velocity

Radial velocity [10-3au/yr]

4.0 2.0 0.0 -2.0 -4.0 -6.0 -8.0 -10.0 -12.040

50

60

90 80 70 Radius [au]

100

110

120

Fig. 3.14 Radial velocity profile at t = 2.1 × 104 yr in Q4a10 run. Dust and gas radial velocities and the steady drift velocity [19] are shown by the red, blue, and gray-dashed lines, respectively. The dust velocity largely deviate from the steady drift velocity around the rings. We note that the gas has small positive velocity because of frictional back-reaction from the dust drift. This figure c is originally shown in Tominaga et al. [33] (AAS). Reproduced with permission

vdri ≡ −

2tstop   2 η2Dr , (1 + ε0 )2 + tstop 

(3.73)

where ε0 is the initial surface density ratio d /g and η2D

  cs2 ∂ ln cs2  . ≡− 2 2 2r  ∂ ln r

(3.74)

70

3 Numerical Simulations of Secular Instabilities

Cumulative dust mass M( 60 au in all runs. Over half of the dust masses are collected into the rings in most of the runs. Especially, 88% of the dust grains are saved in the dust rings in the case of Q6a3 run with Rd,in = 60 au and Rd,out = 200 au. The dust-gas mass ratio in the ring becomes comparable to or higher than unity as a result of nonlinear secular GI. The increase in dust surface density is saturated once the balance between diffusion and self-gravitational collapse is established. Figure 3.16 compares two velocities around one dust ring with a radius of r  73.6 au at t = 2.1 × 104 yr: (1) dust velocity with respect to the ring velocity vring = −1.5 × 10−3 au/yr that we measured from the data and (2) the velocity due to dust diffusion vdiff ≡ −Dd−1 ∂d /∂r . The sum of the two velocity components is shown in red. The red filled circles show cells’ location. One can see that the dust ring is well resolved while the adjacent gaps are not. We find that the above two velocity components are similar in magnitude, i.e. |vr − vring |  |vdiff | > |vr − vring + vdiff |. This demonstrates that further self-gravitational collapse is quenched by the diffusion. Especially, the balance is well estabilished at the inner half of the ring. The red line shows an increasing trend at the outer half of the ring, meaning that the diffusive flow is stronger than the radially converging flow. We find that the last term on the right hand side of Eq. (3.4), i.e., r −1 ∂ F(r )/∂r where F(r ) = r vr D∂d /∂r decelerates dust flow and results in the slow converging flow, which is demonstrated in

72

3 Numerical Simulations of Secular Instabilities

Table 3.3 List of the evaluated masses. This table is originally shown in Tominaga et al. [33] c (AAS). Reproduced with permission. The layout is modified Rd,in = 60 au, Rd,out = 120 au

Rd,in = 60 au, Rd,out = 200 au

Label and Time

Mring,tot a

Md a

Mring,tot /Md Mring,tot a

Q4a10 (t = 2.0 × 104 yr)

4.7×102

6.5×102

0.72

Q5a5 (t = 1.6 × 104 yr)

4.5×102

5.2×102

Q5a8L (t = 2.3 × 104 yr)

3.2×102

Q6a3 (t = 1.8 × 104 yr)

3.5×102

a masses

Md a

Mring,tot /Md

8.5×102

1.2×103

0.68

0.85

7.4×102

9.9×102

0.75

5.2×102

0.62

6.9×102

9.9×102

0.69

4.3×102

0.79

7.3×102

8.2×102

0.88

are in the unit of M⊕

Velocity/|ring velocity|

4.0 2.0

vdiff/|vring| (vr-vring)/|vring|

0.0

(vr-vring+vdiff)/|vring|

-2.0 -4.0 72.5

73

74 73.5 Radius [au]

74.5

75

Fig. 3.16 Radial velocity profile normalized by the ring velocity vring at t = 2.1 × 104 yr in Q4a10 run. Black solid and dashed lines show the dust velocity with respect to the ring velocity, vr − vring , and diffusion velocity vdiff ≡ −Dd−1 ∂d /∂r , respectively. Red line shows the sum of those velocities: vr − vring + vdiff . Filled circles show the data point. This figure is originally shown in c Tominaga et al. [33] (AAS). Reproduced with permission

Fig. 3.17. The figure compares the four forces per mass exerted on dust: self-gravity, pressure gradient force (d−1 cd2 ∂d /∂r ), the sum of the curvature term and the stellar gravity, and d−1r −1 ∂ F(r )/∂r . The force d−1 r −1 ∂ F(r )/∂r is directed outward at outer half of the ring, and its magnitude is comparable to that of the self-gravity. This means that the term strongly decelerates the inward dust motion.

3.3 Simulations of Radially Extended Disks

4

Force per mass

3

73

Self-gravity Pressure gradient force Curvature term + Stellar gravity Force due to the last term of Eq. (5)

2 1 0 -1 -2 73.3

73.4

73.6 73.5 radius [au]

73.7

73.8

Fig. 3.17 Radial forces per mass exerted on dust in the ring with radius of 73.6 au. The vertical axis is normalized by 10−6 G M /(1 au)2 . We plot four forces: the self-gravity (black dashed line), the pressure gradient force (d−1 cd2 ∂d /∂r , gray short dashed line), the sum of the curvature term and the stellar gravity (gray dot-dashed line), and the force coming from the last term on the right hand side of Eq. (3.4), i.e., d−1 r −1 ∂ F(r )/∂r where F(r ) = r vr D∂d /∂r (blue solid line). At the outer half of the ring, the fourth term becomes comparable to self-gravity in magnitude, and c decelerates the inward dust motion. This figure is originally shown in Tominaga et al. [33] (AAS). Reproduced with permission. Note that the legend of the blue line is based on the numbering of equations in Tominaga et al. [33]

Because the constant gravitational softening term weakens self-gravity, the final dust surface density might be underestimated.1 Below, we evaluate a resultant dust surface density d,f which one would obtain if the gravitational softening is neglected, and check whether or not the dust surface density is underestimated. Once the nonlinear growth is saturated, we expect that the diffusion timescale is comparable to the timescale of self-gravitational collapse: D 2 kc4 ∼ 2π Gd,f kc ,

(3.75)

where kc−1 represents the length scale of a spiky ring. This gives kc ∼ 1/3  . If we assume a ring mass to be constant during the linear and non2π Gd,f /D 2 linear growth, we can relate the final surface density and wavenumber to the unperturbed density d,0 and a wavenumber at the linear regime k0 : d,f /kc = d,0 /k0 . This relation and Eq. (3.75) yield

1

The finite thickness of the disk indeed weakens the self-gravity for short-wavelength modes [25, 35]. To some extent, the softening term is then expected to mimic this weakning. Thus, the following estimation without the softening might be regarded as a reasonable upper limit.

74

3 Numerical Simulations of Secular Instabilities

1 2π Gd,0 2 − 23 k0 D2 −1  1   1  3  d,0 /g,0 2 α Q − 2 k0 H − 2  9.3 , 0.1 1 × 10−3 4.5 8

d,f = d,0



(3.76)

where g,0 is the unperturbed gas surface density. This estimation is in good agreement with the resultant dust density of the ring at r  73.6 au in Q4a10 run (see Fig. 3.12). Simulations with the constant softening length would underestimate d,f when the adopted α is smaller.

3.3.1 Formation of Transient Low-Contrast Dust Rings

100

10

Gas

1

Dust

0.1 40

50

60

70 80 90 Radius [au]

100

110

120

-3 Radial velocity [10 au/yr]

Dot-dashed lines: t = 0.0 yr 4 Dashed lines: t = 3.2 x10 yr Solid lines: t = 5.6 x104yr

2

Surface density (Σ and Σd) [g/cm ]

Next, we show the case of formation of transient low-contrast dust rings. In Fig. 3.18, we show the surface density and radial velocity profiles obtained from Q5a8 run. We observe formation of dust rings and gaps at t = 3.2 × 104 yr. However, their amplitude decreases as they move inward in contrast to Q4a10 run (Fig. 3.12): dust rings are transient. From the right panel of Fig. 3.18, we confirm that the radial converging motion with respect to the background drift motion is insignificant in this run. In this way, dust grains drift further without significant concentration into rings. Figure 3.19 clearly show the decay of dust rings and gaps. In Fig. 3.19, we plot deviation of dust surface density from the initial profile: d − d (t = 0). The rings with spatial scale of  5 au grow during the inward drift from r  100 au. However, 1.6 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 -2.440

4

Dust (t = 3.2 x10 yr) Gas (t = 3.2 x104yr) Steady drift velocity

50

60

70 80 90 Radius [au]

100

110

120

Fig. 3.18 (Left panel) Dust and gas surface density profiles at t =0.0 yr (dot-dashed line), 3.2 × 104 yr (dashed line) and 5.6 × 104 yr (solid line) from Q5a8 run. We found that dust rings and gaps forming in this run show lower contrast than the rings and the gaps in Q4a10 run (Fig. 3.12). Besides, their amplitude decrease as they move inward. (Right panel) Radial velocity profiles of dust and gas at t = 3.2 × 104 yr. The gray line is the steady dust drift velocity [19], which is in good agreement with the obtained mean radial velocity in the simulation as in Fig. 3.14 (Q4a10 run). In contrast, the relative motion with respect to the background drift is small. This figure is originally shown in c Tominaga et al. [33] (AAS). Reproduced with permission

3.3 Simulations of Radially Extended Disks

75

Fig. 3.19 Amplitudes of substructures in the dust disk (d − d (t = 0) as a function of radius and time in Q5a8 run. The substructures (i.e., perturbations) move inward, and they become faint c at the inner region r  65 au. This figure is originally shown in Tominaga et al. [33] (AAS). Reproduced with permission

the amplitudes decrease after the rings move across r  60−70 au. We also observed such transient low-contrast rings and gaps in Q4a20 and Q6a5 runs. Those cases host small radial extent of the unstable region, and the radius where rings decay is compatible with the inner boundary of the unstable region, which is discussed in the next section.

3.4 Discussion 3.4.1 Linear Analyses with Dust Drift To understand the physics seen in the numerical results, we perform linear analyses including the radial drift of dust. We also show which mode appears as secular GI in the dust-drifting system. We first show one-fluid (dust) linear analyses to readily understand mode properties of secular GI although the simulations treat both dust and gas. Even this simplified analysis seems valid for qualitative comparison with two-component simulations because of the lack of significant evolution of the gas disk in the simulations. We present two-fluid linear analyses later. As in the previous chapter, we explore the linear stability in the local coordinates (x, y) rotating around the central star with the angular velocity 0 = (r = R). Basic equations for dust are the following: ∂ ∂ 2 d ∂d + , (d vx ) = D ∂t ∂x ∂x2

(3.77)

76

3 Numerical Simulations of Secular Instabilities

  ∂ ∂vx D ∂d ∂vx c2 ∂d + vx − =320 x + 20 v y − d − ∂t d ∂ x ∂x d ∂ x ∂x   vx 1 ∂ ∂d vx D , − + tstop d ∂ x ∂x     v y − Ug,0 ∂v y D ∂d ∂v y D ∂d + vx − = −20 vx − − , ∂t d ∂ x ∂x d ∂ x tstop

(3.78)

(3.79)

where Ug,0 ≡ −30 x/2 − η2D,0 R0 is an azimuthal velocity of a steady gas disk, and η2D,0 ≡ η2D (r = R) (see Eq. (3.74)). We add the dust diffusion term in the way that dust momentum is conserved in the absence of the drag force (see Chap. 2). The unperturbed state adopted here is the steady drift solution, vx,0 , v y,0 , with uniform surface density d = d,0 : vx,0 = −

v y,0

2tstop 0 2 η2D,0 R0 ,  1 + tstop 0

(3.80)

2  tstop 0 = Ug,0 + 2 η2D,0 R0 .  1 + tstop 0

(3.81)

After the Fourier transform, we obtain the following linearized equations: (n + ikvx,0 )δd + ikd,0 δvx = −Dk 2 δd , (n + ikvx,0 )δvx = 20 δv y − ikcd2 (n + ikvx,0 )δv y = −

(3.82)

δd δd δvx − ikδ − − vx,0 Dk 2 , d,0 tstop d,0 0 2

  δv y δd − δvx − ik D . d,0 tstop

(3.83)

(3.84)

We assume that the background gas is not perturbed in this analysis and thus the perturbed self-gravitational potential is related only to the dust surface density perturbation: δ = −2π Gδd /k. The above linearized equations give the following dispersion relation:  γ+

1 tstop



 FDW (γ , k) + γ

1 tstop

  + Dk 2 + Dk 2 −ikvx,0 +

1



tstop

FDW (γ , k) ≡ γ 2 + 20 + cd2 k 2 − 2π Gd,0 k, where we define γ ≡ n + ikvx,0 .

=

20 , tstop

(3.85)

(3.86)

3.4 Discussion

77

In the absence of the dust diffusion (D = 0), Eq. (3.85) for γ is equivalent to the dispersion relation derived in Youdin [39] (see Eq. (3.23) therein). One solutions for γ under D = 0 is the one-component secular GI (e.g., [38, 39]), which is denoted by γSGI in the following, and we obtain n = −ikvx,0 + γSGI . Note that Eq. (3.85) for D = 0 is a cubic equation of γ with the real coefficients, leading to Im[γSGI ] = 0. This is mathematically and physically expected: the drift motion can be removed by the Galilean transformation (see also [38].) The phase velocity is thus given by the steady drift velocity vx,0 . This property is consistent with the finding of our numerical simulations (see Figs. 3.14 and 3.18). Secular GI is aa destabilized static mode as shown in Chaps. 1 and 2, and thus one obtains γSGI → 0 for tstop  → ∞. Note that the growth rate n also shows n → 0 for tstop  → ∞ because the drift motion vanishes in friction-free cases. Weak dust diffusion does not qualitatively change the mode properties of secular GI of the drifting dust. It just limits the growth of short wavelengths. Strong dust diffusion leads to Im[γSGI ] = 0 at short wavelengths but secular GI is stable there. Note that, because the diffusion modeling is different, Eq. (3.85) is different from the dispersion relation derived by Youdin [39] in which one-fluid secular GI is discussed with diffusion. As mentioned in Chap. 2 (see also [32]), the diffusion modeling in Youdin [39] unphysically changes dust angular momentum while ours does not. Hence, our dispersion relation (Eq. (3.85)) describes the mode properties more precisely. For example, the previous work [39] found mode coupling between the static mode and GI mode in the presence of dust diffusion. In contrast, the dispersion relation derived here does not exhibit the mode coupling. We also conduct two-fluid analyses assuming the steady drift solution in an unperturbed state. The drift motion can not be removed in contrast to the above one-fluid analysis because dust and gas have different drift speeds. We use the following equations for gas: ∂ ∂ + (u x ) = 0, ∂t ∂x ∂ ∂u x ∂u x c2 ∂ + ux = 320 x + 20 u y + 2η2D,0 R20 − s − ∂t ∂x  ∂x ∂x   ∂u x d v x − u x 1 ∂ 4 ν + , +  ∂x 3 ∂x  tstop ∂u y ∂u y 1 ∂ + ux = −20 u x + ∂t ∂x  ∂x

  ∂u y d v y − u y ν + . ∂x  tstop

(3.87)

(3.88)

(3.89)

The external force 2η2D,0 R20 in the radial equation of motion mimics the effect of global pressure gradient force, which is the source of the dust drift (e.g., [40]). The dust equations are the almost same as Eqs. (3.77)–(3.79); we replaced Ug,0 by u y in Eq. (3.79).

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3 Numerical Simulations of Secular Instabilities

The unperturbed state is the steady drift solution with uniform surface density profile: 2tstop 0 vx,0 = − (3.90) 2 η2D,0 R0 ,  (1 + ε)2 + tstop 0 v y,0

 2  tstop 0 η2D,0 R0 3 , = − 0 x − 1 − 2  2 2 1+ε (1 + ε) + tstop 0 u x,0 =

u y,0

2tstop 0 ε 2 η2D,0 R0 ,  (1 + ε)2 + tstop 0

 2  tstop 0 ε 3 η2D,0 R0 , = − 0 x − 1 +  2 2 2 1+ε (1 + ε) + tstop 0

(3.91)

(3.92)

(3.93)

where ε = d,0 /g,0 . We use the fixed tstop for comparison with the numerical simulations.2 The linearized continuity equation for dust is the same as Eq. (3.82). The other linearized equations are (n + iku x,0 )δg + ikg,0 δu x = 0,

(3.94)

δg 4 − ikδ − νk 2 δu x g,0 3 δg vx,0 − u x,0 δd vx,0 − u x,0 δvx − δu x + −ε + , (3.95) g,0 tstop g,0 tstop tstop

(n + iku x,0 )δu x =20 δu y − ikcs2

δg 0 3 δu x − ikν − νk 2 δu y 2 2 g,0 δg v y,0 − u y,0 δv y − δu y δd v y,0 − u y,0 + −ε +ε , (3.96) g,0 tstop g,0 tstop tstop

(n + iku x,0 )δu y = −

δd δd δvx − δu x − ikδ − − vx,0 Dk 2 , d,0 tstop d,0 (3.97)   δv y − δu y 0 δd − δvx − ik D , (3.98) (n + ikvx,0 )δv y = − 2 d,0 tstop

(n + ikvx,0 )δvx = 20 δv y − ikcd2

where δ = −2π G(δg + δd )/k.

2

The g,0 -dependence of tstop does not change the mode properties much.

3.4 Discussion 0.08

0

Re[nap]/Ω0 w/o viscosity or drift Exact Re[n]/Ω0 w/o viscosity Exact Re[n]/Ω0 w/ viscosity

-0.1 -Im[n]/Ω0

0.06 Re[n]/Ω0

79

0.04 0.02 0 -0.02 0

-0.2 -0.3 -0.4 -0.5

2

4

6

8 kH

10

12

14

-0.6 0

vx,0k/Ω0 Exact -Im[n]/Ω0 w/o viscosity Exact -Im[n]Ω0 w/ viscosity 12 10 8 6 4 2 kH

14

Fig. 3.20 Growth rate (left panel) and frequency (right panel) of secular GI as a function of dimensionless wavenumber k H . The physical parameters are tstop 0 = 0.6, ε = 0.1, Q = 4.463 and η2D = 0.003014, which are taken from the initial-state values at r = 75 au of Q4a10 run. In both panels, the cross symbols and the filled circles show n calculated with and without the turbulent viscosity, respectively. The gray line on the left panel shows the approximated growth rate obtained without the turbulent viscosity or the drift motion (Eq. (26) in [32]). The black line on the right panel is vx,0 k/0 , which is in good agreement with the frequency of secular GI. This figure is c originally shown in Tominaga et al. [33] (AAS). Reproduced with permission

Figure 3.20 shows the dispersion relation of secular GI. We use the physical values from Q4a10 run for the unperturbed state. The qualitative properties are the same as in the previous studies that neglected the drift motion. The back reaction from dust to gas stabilizes long-wavelength perturbations. Short-wavelength perturbations are stabilized by the dust diffusion, and thus secular GI operates only at the intermediate wavelengths. Figure 3.20 also show the growth rates in the viscosity-free case (filled circles) and the approximated growth rates n ap obtained in the our previous study without turbulent viscosity and drift motion (gray line; Equation (26) in [32]). The approximated one is in good agreement with the growth rate in the dust-drifting system. The frequency −Im[n] is shown on the right panel of Fig. 3.20, and is well reproduced by vx,0 k (black line) as in the one-fluid analyses. The frequency deviates from vx,0 k because of the dust diffusion at short wavelengths . The growth rate of secular GI is insignificantly changed by the drift motion, which is similar to the above one-fluid analyses (see the filled circles and the gray line in Fig. 3.20). To demonstrate this, we also show growth rates of secular GI for various power-law indices q of gas surface density g in Figs. 3.21 and 3.22. We exclude turbulent viscosity when plotting Fig. 3.21 because secular GI does not require viscosity. We find that q insignificantly affects the growth rate. Thus, we conclude that the dust drift insignificantly change the growth rate of secular GI. The oscillation frequencies are compatible with vx,0 k/0 for various q at long wavelengths (see the right panel of Fig. 3.21). Thus, the power-law index only change the frequency mostly through the term vx,0 . Including turbulent viscosity only slightly change the dispersion relation (Fig. 3.15).

80

3 Numerical Simulations of Secular Instabilities 0.08

Exact Re[n]/Ω0 (q=0.5) Exact Re[n]/Ω0 (q=1) Exact Re[n]/Ω0 (q=1.5)

-Im[n]/Ω0

Re[n]/Ω0

0.06 0.04 0.02 0 -0.02 0

2

4

6

8 kH

10

12

14

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 0

vx,0k/Ω0 (q=1/2) vx,0k/Ω0 (q=1) vx,0k/Ω0 (q=1.5) Exact -Im[n]/Ω0 (q=0.5) Exact -Im[n]Ω0 (q=1) Exact -Im[n]Ω0 (q=1.5) 2 4 6 8 10 kH

12

14

Fig. 3.21 Growth rate (left panel) and frequency (right panel) of secular GI as a function of dimensionless wavenumber k H in the absence of turbulent viscosity. The parameters adopted are tstop  = 0.6, d,0 /0 = 0.1 and α = 1 × 10−3 , and Q = 4.463. Those are taken from the physical values at r = 75 au in Q4a10 run (q = 0.5). The filled circles, the red cross symbols and the blue triangles show dispersion relations for q = 0.5, 1, 1.5, respectively. The different types of the lines on the right panel show vx,0 k/0 for different q valuses: q = 0.5 in solid, q = 1 in dashed, and c q = 1.5 in short dashed lines. This figure is originally shown in Tominaga et al. [33] (AAS). Reproduced with permission 0.08

Exact Re[n]/Ω0 (q=0.5) Exact Re[n]/Ω0 (q=1) Exact Re[n]/Ω0 (q=1.5)

-Im[n]/Ω0

Re[n]/Ω0

0.06 0.04 0.02 0 -0.02 0

2

4

6

8 kH

10

12

14

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 0

vx,0k/Ω0 (q=1/2) vx,0k/Ω0 (q=1) vx,0k/Ω0 (q=1.5) Exact -Im[n]/Ω0 (q=0.5) Exact -Im[n]Ω0 (q=1) Exact -Im[n]Ω0 (q=1.5) 10 8 6 4 2 kH

12

14

Fig. 3.22 Growth rate (left panel) and frequency (right panel) of secular GI as a function of dimensionless wavenumber k H . Here we also consider the turbulent viscosity. The adopted parameters are tstop  = 0.6, d,0 /0 = 0.1 and α = 1 × 10−3 , and Q = 4.463 as in Fig. 3.21. The filled circles, the red cross symbols and the blue triangles show dispersion relations for q = 0.5, 1, 1.5, respectively. The different types of the lines on the right panel show vx,0 k/0 for different q valuses: q = 0.5 in solid, q = 1 in dashed, and q = 1.5 in short dashed lines. This figure is originally shown c in Tominaga et al. [33] (AAS). Reproduced with permission

The present analyses do not exhibit a mode corresponding to TVGI. According to Chap. 2 and Tominaga et al. [32], a static mode that becomes TVGI has dust velocity perturbations that are in phase with gas velocity perturbations. The significant drift prevent this situation from establishing. This could be the reason for the lack of TVGI.

3.4 Discussion

81

3.4.2 Condition for the Thin Dense Ring Formation Formation of thin dense rings is important for planetesimal formation via secular GI since resulting high dust density will promote dust coagulation or lead to ring fragmentation. We first discuss its condition based on the previous linear analyses. Thin dense rings form if the radial drift is decelerated enough due to the resulting enhancement of dust-to-gas mass ratio in rings. This is expected from the simulation results. In Q4a10 run, which shows the thin dense rings (Fig. 3.12), the dust-to-gas mass ratio increases up to unity. On the other hand, the maximum dust-to-gas ratio is smaller than unity in Q5a8 run, which only exhibits the transient low-contrast dust rings (Fig. 3.18). Thus, the critical dust-to-gas mass ratio sits around unity. Dust-to-gas ratio that is finally achieved depends on the radial extent of an unstable region. Figure 3.23 shows radial profiles of growth timescales of secular GI in Q4a10 run on the left panel and in Q5a8 run on the right panel. The presented three lines correspond to the timescales at different wavelengths: λ =3 au, 5 au and 10 au. The radial extent of plotted unstable regions in Q5a8 run are smaller for each wavelength than the Q4a10 run. As shown in Figs. 3.12 and 3.18, the wavelength of λ  3 − 5 au is unstable enough to form annular substructures in both cases. Figure 3.23 shows that unstable region of those perturbations is r  50 au, which is consistent with Fig. 3.12. The dust-to-gas ratio reaches unity and the drift is decelerated before rings enter the stable region. This leads to the thin dense ring formation. On the other hand, in Q5a8 case the growth timescale at λ = 5 au significantly increases as r decreases from r  60 − 70 au (see the right panel of Fig. 3.23). Since perturbations keep moving inward without significant growth, they finally move into the stable region (r  58 au for λ = 5 au) and decay as seen in Fig. 3.19. We should note that the final dust-to-gas ratio achieved in rings depends on initial amplitudes of perturbations. To demonstrate this, we conduct a simulation in which the background disk model is the same as in Q5a8 run but initial random perturbations

10 Growth timescale [10 yr]

8

3

3 Growth timescale [10 yr]

10

6 4 2 040

Q4a10, λ = 3au Q4a10, λ = 5au Q4a10, λ = 10au 90 80 70 60 50 Radius [au]

100

110

120

8 6 4 2 040

Q5a8, λ = 3au Q5a8, λ = 5au Q5a8, λ = 10au 90 80 70 60 50 Radius [au]

100

110

120

Fig. 3.23 Growth timescales of secular GI for three wavelengths: λ = 3 au, 5 au, 10 au. We assumed the background disk profiles of Q4a10 run on the left panel and of Q5a8 run on the right panel, respectively. The unstable region is located at an outer region in Q5a8 run compared to Q4a10 run. Perturbations at long wavelength can grow only at outer radii. This figure is originally c shown in Tominaga et al. [33] (AAS). Reproduced with permission

3 Numerical Simulations of Secular Instabilities 4

Dashed lines: t = 1.6 x10 yr Solid lines: t = 2.5 x104yr

2

Surface density (Σ and Σd) [g/cm ]

82

100

10

Gas

1

Dust

0.1 40

50

60

100

90 80 70 Radius [au]

120

110

Fig. 3.24 Surface density profiles of dust and gas at t = 1.6 × 104 yr (dashed lines) and 2.5 × 104 yr (solid lines) from a run in which we put the same parameters as Q5a8 run but initial perturbations with six times larger amplitudes (Q5a8L run). This figure is originally shown in Tominaga et al. c [33] (AAS). Reproduced with permission

2

Dust surface density [g/cm ]

100

Radius [au]

90

1

80 70 60 50 0

0.5

1 1.5 Time [104yr]

2

2.5

0.1

Fig. 3.25 Trajectories of dust cells in Q5a8L run. Color on each line represents the dust surface density at each dust cell. We used a reduced number of cells to plot this figure. This figure is c originally shown in Tominaga et al. [33] (AAS). Reproduced with permission

have six times larger amplitudes. An inner region of 10 au < r < 20 au is not initially perturbed to avoid sudden dust concentrations near the inner boundary due to the initial large perturbations. Figure 3.24 shows surface density profiles of dust and gas and their evolution. Spiky dust rings form before they enter the stable region, resulting in thin dense rings with d /g  1 (see also Fig. 3.25). Disks are thought to be continuously disturbed by disk turbulence or, for instance, external disturbances due to mass accretion form the envelope. Those processes will determine the perturbation amplitudes. However, those are unknown and still in debate, and we thus do not qualitatively discuss perturbation amplitudes in the present thesis. We conducted the simulations only for q = 0.5 in this work. Disks with larger q can have denser inner region, and thus the unstable region shifts inward in such a case. Since the growth timescale is scaled by the Keplerian orbital periods 2π −1 ∝ r 3/2 , it becomes shorter for the same α, tstop , Q and d /g . are the same. Note that the

3.4 Discussion

83

power-law index insignificantly affects the growth timescale according to the linear analysis presented above. Although the drift timescale r/|vr | also becomes shorter, its change is smaller than that of the growth timescale. The drift speed is faster for larger q. η2Dr  is constant throughout the disk if we neglect the exponential cutoff term, meaning that the drift timescale is proportional to r . The growth timescale has stronger r -dependence (r 3/2 ). Therefore, secular GI will grow and more easily create thin dense rings in steeper disks. Note again that this statement is justified if α, tstop , Q and d /g is the same in the inner region. For example, when one increases q with fixing the surface densities at the inner region, the inner region remains stable to the secular GI as in the present simulations. In such a case, we may not expect the growth of secular GI. We also note that unstable wavelengths become shorter at inner regions because they are scaled by the gas scale height. The widths of rings are thus smaller than we present in this thesis, which will require higher numerical resolutions.

3.4.3 Fate of Dense Dusty Rings In resulting dense dusty rings, we can expect accelerated coagulation and gravitational collapse, which lead to planetesimal formation. Because the rings are massive with mass of 10 M⊕ and self-gravitating, they will azimuthally fragment into solid bodies. We can estimate the timescale of the ring fragmentation from the freefall ˜ d is time tff . The averaged dust surface density in a dense ring  ˜d = 

Mring , 2π Rring wring

(3.99)

where Mring , Rring , wring are mass, radius and width  of a dust ring, respectively. √ ˜ d / 2π Hd , we obtain tff = 3π/32G ρ˜d . We obtain tff  55 If we assume ρ˜d =  yr for Mring = 49.7M⊕ , Rring = 73.6 au and wring = 0.3 au, which is based on one ring in Q4a10 run. This timescale is much shorter than one Keplerian period at r = 73.6 au ( 631 yr). We also note that the very recent work [22] following our work [33] conducted two-dimensional simulations and showed that the ring forming via nonlinear secular GI can fragment via Rossby wave instability triggered by the resulting spiky structure. Therefore, further multidimensional analyses and simulations are important. Because the coagulation timescale tgrow is known to be inversely proportional to the total dust-to-gas ratio d,tot /g,tot in the Epstein regime (e.g., [20]), dust grains coagulate faster in dusty rings than other regions. If we can assume the height of the lower layer to be larger enough than the dust scale height and d  d,tot , the enhancement of d,tot /g,tot becomes similar to that of d /g . This is also based on the fact that the gas surface density insignificantly changes via secular GI. In the case of Q4a10 run, if the initial total dust-to-gas ratio is 0.05 and increases as

84

3 Numerical Simulations of Secular Instabilities

well as d /g by a factor of 9.3 (see Eq. (3.76)), we have the growth timescale tgrow  199 yr at r = 73.6 au. Thus, coagulation also proceeds within one Keplerian period although the coagulation timescale is still longer than the freefall time. In Q5a8L run, d /g increases by an order of magnitude at r  80 au (Fig. 3.24). The coagulation timescale in the dust ring is about 209 yr and shorter than the radial drift timescale. This means that the dust grains in the ring will grow without significant radial migration, and finally be decoupled from gas. This combination of dust ring formation and coagulation will be the key to keep dust grains in a disk. Thus, to consider both ring fragmentation and dust growth is important for further discussion on planetesimal formation. Multidimensional simulations with the dust growth are required for more quantitative analyses, which will be the scope of our future work.

3.4.4 Observational Justification Our simulations show significant substructures only in a dust disk. This is in contrast to the case where a relatively large planet (e.g., Jupiter-mass planets) creates disk substructures: it carves a gap and induces rings in both dust and gas disks (e.g., [9, 14, 42]). Therefore, we expect that to observe gas around the midplane is important to distinguish the ring formation mechanisms: one requires gap-like profiles in a gas disk (i.e., hidden high-mass planet scenario) while another shows a relatively smooth gas profile (i.e., the secular-GI-based mechanism). It should be noted that massive planetary objects that may form after nonlinear secular GI will create some substructures in the gas disk. Thus, there can be degeneracy between those mechanisms. If the gas gap is as wide as the dust gap, it might be difficult to distinguish the secular-GI-based mechanism and the planet-based mechanism. In such cases, we should regard secular GI as one possible formation mechanism of the hypothetical planets in the gap. A lower mass planet can carve a gap only in a dust disk. Because of the difference between dust and gas scale heights, a low mass planet takes longer time to carve a significant gap in a gas disk than in a dust disk (e.g., [37]). I is possible that observations can resolve high-contrast substructures only in a dusk disk. In such a case, we may not distinguish the scenario based on secular GI from the scenario with low mass planets. Nevertheless, underlying processes in both scenarios should have dependence on the disk properties and dust sizes in different ways. Thus, accurate measurement of a gas density close to the midplane is still important. As mentioned above, we can expect formation of larger solid bodies at the locations of thin dense rings. It thus seems possible that the ring regions will be dark at millimeter wavelengths. Since the rings are close to each other, multiple spiky rings would be observed as a single dark region, i.e., a gap. The emissivity at the “apparent” gap region might be determined by small dust grains that are re-supplied by collisional fragmentation after the formation of larger bodies. The re-supplied dust grains will contribute to the growth of secular GI and subsequent fragmentation

3.4 Discussion

85

supplies small dust grains again. This recycling process indicates the existence of an equilibrium between dust-to-planetesimal conversion via secular GI and dust supply via planetesimal fragmentation. A similar process is investigated by Stammler et al. [26] although they focus on a different instability called streaming instability [40]. Stammler et al. [26] claims that planetesimal formation in dust rings via streaming instability stalls the growth of itself because dust grains are depleted as a result of planetesimal formation. This self-regulating process can limit the optical depth at sub-mm wavelengths. The DSHARP observations revealed that the optical depth in dust rings are limited to around  0.2 − 0.5 (see also, [8, 11]). To consistently examine substructures and optical depth profiles resultant from secular GI, we have to explicitly include both dust growth and fragmentation in our simulations. Because self-gravitational ring fragmentation will occur simultaneously, nonaxisymmetric analyses and simulations are important. Those will be addressed in our future studies.

3.4.5 Effects on Dust-to-Gas Ratio Dependence on the Dust Coefficient We adopted the dust diffusion coefficient D independent from dust abundance. In contrast, some studies suggest that the dust diffusivity decreases as dust-to-gas mass ratio once it becomes larger than  1. Schreiber and Klahr [24] measured the dust diffusivity under the influence of turbulence driven by streaming instability, and showed that the diffusivity is inversely proportional to ρd /ρg for ρd /ρg  1 although there is a dependence on a size of a simulation box. This dependence had been already expected in the analytical work [29]. If streaming instability sets turbulence in the rings in our simulations, the diffusion becomes weaker as secular GI develops, and dust surface density of thin dense rings will increase beyond the saturated level estimated in Eq. (3.76) with the background diffusion coefficient. It is also possible that dust diffusion becomes too inefficient to support the radial contraction of a dense ring. Those will result in dust growth and planetesimal formation at larger rate than expected above. The saturated level of d of a dense ring with the reduced diffusivity can be estimated as follows. We assume the dust diffusion coefficient to have a power-law dependence on d,f /g,0 : D = α1

cs2 



d,f g,0

−β

,

(3.100)

where α1 is turbulent strength for d,f /g,0 = 1. Adopting the same way to derive Eq. (3.76), we obtain the following equation to evaluate the saturation level:

86

3 Numerical Simulations of Secular Instabilities 1   β−1  1+2β   1   3  d,0 /g,0 2−2β α1 Q 2β−2 k0 H 2β−2 d,f = f (β) , d,0 0.1 1 × 10−3 4.5 8 (3.101) 1 f (β)  (8.7 × 101−2β ) 2−2β . (3.102)

It should be noted that there is no solution for β = 1 because f (β) diverges. In such a case, the dust diffusion becomes too weak to stop the radial concentration for β ≥ 1. There is a solution for β > 1 but it is not realized during the growth of perturbations because the diffusion timescale becomes shorter than the collapse timescale before the dust surface density reaches d,f . The velocity dispersion of dust grains cd will also decrease as the dust-to-gas ratio increases beyond unity. The Coriolis force solely repulse self-gravity once the velocity dispersion significantly decreases. The characteristic length scale λcrit in such a system is 4π 2 Gd . (3.103) λcrit ≡ 2 Thus, a dust clump whose size is smaller than λcrit will experience self-gravitational collapse. We obtain λcrit  17 au for one ring obtained in Q4a10 run (d = 9.5 g/cm2 at r = 73.6 au). This scale is much larger than the ring width (see Fig. 3.12). A circumference of a ring with r = 73.6 au is about 462 au and much longer than λcrit , indicating that ring fragmentation will operate. The resulting fragment mass will depend on the azimuthal wavenumber. If the ring mass is ∼50 M⊕ as one in Q4a10run, the fragment mass is about a few M⊕ or smaller for azimuthal wavenumber 30.

3.5 Summary In this chapter, we show numerical simulations of secular GI in radially extended disks. Because dust grains suffer radial drift toward a central star, exploring the growth of secular GI with a radially wide region is essentially important to discuss formation of dust substructures and planetesimals in a protoplanetary disk. Numerical simulations are powerful tools to investigate such a problem. However, numerical diffusion prevents the slow growth of secular GI. Motivated by this issue, we develop the Lagrangian-cell method utilizing the symplectic integrator. The Lagrangian-cell method is free from numerical diffusion due to advection, and the symplectic integrator reduces the accumulation of errors due to time integration. Combining the method with the piecewise exact solution for dust-gas friction, we performed numerical simulations of linear/nonlinear secular GI. Nonlinear growth of secular GI shows the gravitational collapse of dust rings whose timescale is characterized by the freefall time. As a result, the dust surface density increases by an order of magnitude while the gas surface density insignif-

3.5 Summary

87

icantly changes. This results in high dust-to-gas ratio in thin dense rings. The dust enrichment suppresses dust drift through the backreaction to the gas and saves dust grains in a disk. If a growth timescale of secular GI is too long, secular GI only creates low-contrast rings. Such rings eventually drift into an inner stable region and start to decay. Thus, resultant substructures are transient. Because rings smoothly decay, it seems possible that rings resultant from secular GI are observed even in the stable region. According to the above results, planetesimal formation requires dust enrichment up to d /g  1 via thin dense ring formation by secular GI. We simply estimate the coagulation timescale and the freefall timescale and show that both ring fragmentation and accelerated coagulation will proceed within one Keplerian period once secular GI develops into the nonlinear phase. Thus, secular GI can be an efficient process to cause planetesimal formation. The mass conversion from dust grains to planetesimals will make resultant rings darker at sub-mm wavelengths. This indicates that resultant multiple spiky rings will be observed as a single wide gap substructure. Planetesimal fragmentation after its formation will re-supply small dust grains that continue to accumulate via secular GI. This recycling of dust grains indicates the existence of a state at which dust depletion via planetesimal formation and dust supply via planetesimal fragmentation are in equilibrium. Such a self-regulating process might explain the observed marginally optically thin substructures as discussed in [26]. In contrast to the ring-gap formation by high-mass planets, secular GI creates prominent substructures only in a dust disk. Therefore, observations of a midplane gas density profile will provide the key to understand what process actually operates and forms substructures in the observed disks. We should note that low-mass planets carve prominent dust gaps and low-contrast gas gaps, which might observationally degenerate with the secular-GI-based scenario if observations cannot resolve the low-contrast gas gaps. For further quantitative studies, we need to perform multidimensional simulations with dust coagulation and fragmentation.

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7. Birnstiel T, Klahr H, Ercolano B (2012) A&A 539:A148. https://doi.org/10.1051/0004-6361/ 201118136 8. Dullemond CP, Birnstiel T, Huang J, Kurtovic NT, Andrews SM, Guzmán VV, Pérez LM, Isella A, Zhu Z, Benisty M, Wilner DJ, Bai XN, Carpenter JM, Zhang S, Ricci L (2018) ApJL 869:L46. https://doi.org/10.3847/2041-8213/aaf742 9. Gonzalez JF, Laibe G, Maddison ST, Pinte C, Ménard F (2015) MNRAS 454(1):L36–L40. https://doi.org/10.1093/mnrasl/slv120 10. Hastings C, Hayward JT, Wong JP (1955) Approximations for digital computers 11. Huang J, Andrews SM, Dullemond CP, Isella A, Pérez LM, Guzmán VV, Öberg KI, Zhu Z, Zhang S, Bai XN, Benisty M, Birnstiel T, Carpenter JM, Hughes AM, Ricci L, Weaver E, Wilner DJ (2018) ApJl 869:L42. https://doi.org/10.3847/2041-8213/aaf740 12. Inoue T, Inutsuka SI (2008) ApJ 687(1):303–310. https://doi.org/10.1086/590528 13. Isella A, Carpenter JM, Sargent AI (2009) ApJ 701(1):260–282. https://doi.org/10.1088/0004637X/701/1/260 14. Kanagawa KD, Muto T, Tanaka H, Tanigawa T, Takeuchi T, Tsukagoshi T, Momose M (2015) ApJL 806(1):L15. https://doi.org/10.1088/2041-8205/806/1/L15 15. Kitamura Y, Momose M, Yokogawa S, Kawabe R, Tamura M, Ida S (2002) ApJ 581(1):357– 380. https://doi.org/10.1086/344223 16. Latter HN, Rosca R (2017) MNRAS 464:1923–1935. https://doi.org/10.1093/mnras/stw2455 17. Long F, Herczeg GJ, Harsono D, Pinilla P, Tazzari M, Manara CF, Pascucci I, Cabrit S, Nisini B, Johnstone D, Edwards S, Salyk C, Menard F, Lodato G, Boehler Y, Mace GN, Liu Y, Mulders GD, Hendler N, Ragusa E, Fischer WJ, Banzatti A, Rigliaco E, van de Plas G, Dipierro G, Gully-Santiago M, Lopez-Valdivia R (2019) ApJ 882(1):49. https://doi.org/10.3847/15384357/ab2d2d 18. Machida MN, Matsumoto T, Hanawa T, Tomisaka K (2006) ApJ 645:1227–1245. https://doi. org/10.1086/504423 19. Nakagawa Y, Sekiya M, Hayashi C (1986) Icar 67(3):375–390. https://doi.org/10.1016/00191035(86)90121-1 20. Okuzumi S, Tanaka H, Kobayashi H, Wada K (2012) ApJ 752(2):106. https://doi.org/10.1088/ 0004-637X/752/2/106 21. Pérez LM, Carpenter JM, Andrews SM, Ricci L, Isella A, Linz H, Sargent AI, Wilner DJ, Henning T, Deller AT, Chandler CJ, Dullemond CP, Lazio J, Menten KM, Corder SA, Storm S, Testi L, Tazzari M, Kwon W, Calvet N, Greaves JS, Harris RJ, Mundy LG (2016) Science 353(6307):1519–1521. https://doi.org/10.1126/science.aaf8296 22. Pierens A (2021) MNRAS 504(3):4522–4532. https://doi.org/10.1093/mnras/stab183 23. Ricci L, Testi L, Natta A, Brooks KJ (2010) A&A 521:A66. https://doi.org/10.1051/00046361/201015039 24. Schreiber A, Klahr H (2018) ApJ 861:47. https://doi.org/10.3847/1538-4357/aac3d4 25. Shu FH (1984) In: Greenberg R, Brahic A (eds) IAU Colloq. 75: Planetary Rings, pp 513–561 (1984) 26. Stammler SM, Dr¸az˙ kowska J, Birnstiel T, Klahr H, Dullemond CP, Andrews SM (2019) ApJL 884(1):L5. https://doi.org/10.3847/2041-8213/ab4423 27. Takahashi SZ, Inutsuka SI (2014) ApJ 794:55. https://doi.org/10.1088/0004-637X/794/1/55 28. Takahashi SZ, Inutsuka SI (2016) AJ 152:184. https://doi.org/10.3847/0004-6256/152/6/184 29. Takeuchi T, Muto T, Okuzumi S, Ishitsu N, Ida S (2012) ApJ 744(2):101. https://doi.org/10. 1088/0004-637X/744/2/101 30. Tazzari M, Clarke CJ, Testi L, Williams JP, Facchini S, Manara CF, Natta A, Rosotti G (2021) MNRAS 506(2):2804–2823. https://doi.org/10.1093/mnras/stab1808 31. Tominaga RT, Inutsuka SI, Takahashi SZ (2018) PASJ 70:3. https://doi.org/10.1093/pasj/ psx143 32. Tominaga RT, Takahashi SZ, Inutsuka SI (2019) ApJ 881(1):53. https://doi.org/10.3847/15384357/ab25ea 33. Tominaga RT, Takahashi SZ, Inutsuka SI (2020) ApJ 900(2):182. https://doi.org/10.3847/15384357/abad36

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Chapter 4

Coagulation Instability: Self-induced Dust Concentration

4.1 Short Introduction As discussed in the previous chapters, secular GI has the potential to locally concentrate dust grains and accelerate further dust growth toward planetesimals in resultant rings. The numerical simulations in Chap. 2 show that dust surface density increases by an order of magnitude once secular GI develops into nonlinear regime. As shown in Chap. 2 and Tominaga et al. [43], the unstable condition of secular GI is given by  ε (1 + ε) tstop cs2 . (4.1) Q< D In terms of the dust-to-gas surface density ratio, disks satisfying the following condition become unstable to secular GI:  ε > 0.016

   D τs −1 Q 2 , 10−4 cs2 / 0.1 4

(4.2)

where τs = tstop  is dimensionless stopping time and we assume ε  1 to reduce Eq. (4.1). The above condition means that the dust-to-gas ratio for dust with τs = 0.1 should be higher than 0.016 for secular GI to operate. Assuming that dust grains of a size a are in the Epstein regime τs =

π ρint a , 2 g

(4.3)

one obtains

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 R. Tominaga, Dust-Gas Instabilities in Protoplanetary Disks, Springer Theses, https://doi.org/10.1007/978-981-19-1765-3_4

91

92

4 Coagulation Instability: Self-induced Dust Concentration

2 τs H M∗ (4.4) π 2 ρint Q r r 2 −1  −1      τ  Q H/r  r −2 M∗ ρint s .  0.6 cm 0.1 4 0.05 50 au 1M 1.4 g/cm3 (4.5)

a=

Thus, for secular GI to grow in a disk, Eq. (4.2) should be satisfied for mm- or cmsized dust grains although the required dust size depends on the other parameters and the radial location. Collisional coagulation will grow dust grains up to the required sizes (∼1 cm). However, according to numerical studies on coagulation in an isolated disk, dust surface density significantly decreases as dust grows into the size of ∼1 cm (τs  0.1) unless the surrounding envelope supplies dust grains to the disk (e.g., [4, 27]). In other words, pure coagulation that provides large dust tends to violate the condition on the dust density (i.e., dust-to-gas ratio, Eq. (4.2)). This is readily understood as follows (see [4, 27, 32]). The dust growth timescale for spherical grains is given by  tgrow = 3m

dm dt

−1

=

3m , ρd g v

(4.6)

where m is mass of a single dust grain. The cross section and the relative velocity of dust grains are denoted by σ and v, respectively. For compact spherical dust grains with the radius of a and the internal density of ρint , Eq. (4.6) yields tgrow =

ρint a . ρd v

(4.7)

 Assuming turbulence-driven collisions of equal-sized dust grains, v = 3tstop αcs [28], the Epstein drag regime, and vertically Gaussian profiles for dust and gas disks gives the following dust growth timescale at the midplane (z = 0):  tgrow   2

2 g . 3π d

(4.8)

Thus, the dust growth timescale in the unit of −1 is solely determined by dust-gas surface density ratio [4, 27, 32]. If the initial dust-gas ratio is uniform in a disk, dust coagulation proceeds in the inside-out manner because the timescale is proportional to the Keplerian orbital period 2π/  ∝ r 3/2 (see Eq. (4.8)). As a result of the insideout coagulation, an inner region hosts larger grains that have larger drift velocity, leading to the decrease in the dust surface density and thus in the dust-gas ratio. The previous studies [4, 27] considered an isolated disk and assumed that there is no dust supply to the disk from its outside, which corresponds to the very last stage of the disk evolution. The tendency for dust density to decrease as a result of coagulation will hold even in the early disk [18] although the density decrease may be less prominent

4.1 Short Introduction

93

Dust concentration at a pressure bump ∂ ln P 0 ∂ ln r

Diffusion Condensation Evaporation

Dust fragmentation

Radial distance vr ∝

∂ ln P ∂ ln r

vKep

T~160K for H2O ice

Fig. 4.1 Schematic pictures of two of currently proposed mechanisms for dust retention: (1) concentration at a pressure bump (e.g., [46]) and (2) combination of recondensation and traffic jam of dust grains (e.g., [12, 39]). A pressure bump concentrates dust grains at its center because the radial drift velocity is proportional to the pressure gradient ∂ ln P/∂ ln r . The second mechanism takes place near the H2 O snow line where water ice on grains evaporates. If silicate grains inside the snow line is fragile enough, inner dust grains fragment into smaller solids and pile up. Water vapor that is diffused outward across the snow line recondense onto grains, which enhances dust-gas ratio

if gas and dust infall from the envelope is significant. In this way, there is a “gap” between dust-gas ratio resulting from the inside-out coagulation and dust-gas ratio required for the onset of secular GI. The gap mentioned above is problematic not only for secular GI but also for other dust-gas instabilities expected as a promising mechanism to form planetesimals. Streaming instability is one example (e.g., [19, 21, 48, 49]). Streaming instability has the potential to cause dust clumping at much smaller scales than secular GI, and resultant clumps eventually collapse self-gravitationally once those mass densities exceed Roche density (e.g., [20, 36]). The validity of streaming instability also depends on dust-to-gas ratio ε = d /g and dimensionless stopping time (e.g., [5, 23, 47]). The previous studies [5, 47] numerically investigated conditions of dust clumping via streaming instability. They found the required dust-to-gas ratio larger than 0.02 for dust of τs = 0.1, which is similar to secular GI (Eq. (4.2)). Higher dust-to-gas ratio is required for smaller dust (τs = 10−2 − 10−3 ). In the presence of turbulent diffusion, strong clumping via streaming instability will require much higher dust abundances [9, 17, 44]. Therefore, in-advance concentration of large dust grains produced via coagulation is necessary for the dust-gas instabilities to operate and develop toward planetesimal formation. Because of the growth wavelength of the instabilities is  H , a spatial scale of concentration should be  H at least. Two mechanisms are schematically summarized in Fig. 4.1. Pressure bumps or zonal flows in a gas disk are possible sites of dust concentration (e.g., [2, 14, 22, 25, 46]). If a “bump” structure in a gas pressure profile exists, dust grains in the inner-half region of the bump move outward while those in the outer-half region fall inward because their drift velocity is proportional to ∂ ln P/∂r . Thus, dust grains pile up at the center of the bump. Streaming instability in pressure bumps has been investigated both analytically [1]

94

4 Coagulation Instability: Self-induced Dust Concentration

and numerically [6, 40]. The previous works [1, 6] showed that streaming instability develops for dust-to-gas ratio of 0.01 if a relative amplitude of a pressure bump is larger than 10–20%. However, we should note that deformation of a bump due to frictional backreaction potentially inhibits subsequent gravitational collapse [40]. Gas vortices also trap dust particles (e.g., [3, 8, 26, 30]). Some disks are considered to host a vortex [7, 15, 45] but other disks hosting annular substructures show few evidences that vortices are present. The water snow line is another possible location where dust grains are retained (e.g., [10, 11, 33, 34, 39]). The water snow line is a location where water ice on grain’s surface evaporates. Resultant bare silicate grains are relatively fragile and fragment into smaller grains. The radial speed decreases inward across the snow line, leading to traffic jam. Because vaporized water diffuses outward and recondenses onto dust grains outside the snow line and they become larger, dust grains successively pile up around the snow line (see Fig. 4.1). The process however is highly dependent on a critical velocity at which collisional fragmentation becomes efficient. It is found that dust grains efficiently pile up around the snow line if water vaporization inside the snow line provides fragile silicate grains whose critical velocity is ∼ a few m/s. However, experiments suggest that dry silicate grains are less fragile than previously considered (a critical velocity 10 m/s, [24, 38]). If this is the case in protoplanetary disks, the dust retention around the snow line may not operate (see discussion in [10]). In this chapter, we propose another mechanism for dust reconcentration. Most of the contents in this chapter are based on the published paper: Tominaga et al. [42]. We find that dust coagulation itself triggers an instability that causes dust concentration. We name the process coagulation instability. Coagulation instability grows even when dust is highly depleted and dust-to-gas ratio decreases down to ∼10−3 in a disk. In contrast to the above mechanisms that operate at a specific location in a disk (e.g., the snow line), coagulation instability can operate and concentrate dust grains throughout a disk. We first describe basic equations for linear analyses in Sect. 4.2 and show results in Sect. 4.3. Discussion and summary are present in Sects. 4.4 and 4.5.

4.2 Basic Equations for Linear Analyses with Dust Growth Dust coagulation is described by the Smoluchowski equation for a column number density N (r, m) per unit dust particle mass m (e.g., [31, 35, 37]): m ∂m N = ∂t 2

m

dm  K (r, m  , m − m  )N (r, m  )N (r, m − m  )

0

∞ − m N (r, m) 0

dm  K (r, m, m  )N (r, m  )dm  −

1 ∂ (r vr (r, m)m N (r, m)) , r ∂r (4.9)

4.2 Basic Equations for Linear Analyses with Dust Growth

95

where K (r, m 1 , m 2 ) is a collision kernel representing a vertically integrated collision rate between dust particles of masses m 1 = 4πρint a13 /3 and m 2 = 4πρint a23 /3.1 In the presence of the dust diffusion, one has to add the diffusion term on the right hand side as follows: m ∂m N = ∂t 2

m

dm  K (r, m  , m − m  )N (r, m  )N (r, m − m  )

0

∞ − m N (r, m)

dm  K (r, m, m  )N (r, m  ) −

0

1 ∂ + r ∂r

1 ∂ (r vr (r, m)m N (r, m)) r ∂r

  ∂m N (r, m) r D(r, m) , ∂r

(4.10)

where we assume mass-dependence of diffusion coefficient, D(r, m). The expression of the collision kernel is (e.g., [4]) σcoll K (r, m 1 , m 2 ) ≡ 2π Hd (m 1 )Hd (m 2 )

∞ −∞



z2 vpp exp − 2



1 1 + 2 Hd (m 1 ) Hd (m 2 )2



dz,

(4.11)

where vpp is collision velocity and σcoll is a cross section: σcoll ≡ π(a1 + a2 )2 .

(4.12)

Note that the radial velocity vr and the dust scale height Hd depend on dust particle mass. Analytical treatments of the above Smoluchowski equation are difficult in general. In this thesis, we thus utilize moment equations of the Smoluchowski equation. The moment equations are already formulated by previous studies [13, 29, 32, 41] in the absence of the diffusion term. We adopt the moment equations derived by Sato et al. [32], in which the moment values are defined with the mass-weighted averaging. The 0th and 1st moment equations derived in Sato et al. [32] are 1 ∂ ∂d + (r vr d ) = 0, ∂t r ∂r √ ∂m p 2 πa 2 vpp ∂m p dm p = + vr = d , dt ∂t ∂r Hd

(4.13)

(4.14)

(see Appendix A in Sato et al. [32] for the detailed derivation). The 0th moment corresponds to the dust surface density d . The 1st moment is called a “peak mass”, 1

In this thesis, we focus on collisional growth of compact spherical dust grains for simplicity. Collisional growth of porous dust aggregates are investigated in [27].

96

4 Coagulation Instability: Self-induced Dust Concentration

m p ≡ M1 /d [29]. The peak mass is a dust size that dominates dust masses. Sato et al. [32] conducted numerical simulations of the Smoluchowski equation and the derived moment equations, and found that the above moment approach well reproduces the full-size simulations when adopting a collision speed with a dust size ratio of 0.5 to calculate vpp . The radial dust diffusion term is ignored in the previous studies [13, 29, 32, 41]. When we use Eq. (4.10) to derive moment equations, we need assumptions to simplify the resulting equations. Here we assume the following relations (see [42]): ∞

∂m N (r, m) ∂d  D(r, m p ) , ∂r ∂r

(4.15)

∂d ∂m N (r, m)  D(r, m p )m p . ∂r ∂r

(4.16)

dm D(r, m) 0

∞ dm D(r, m)m 0

These assumptions will be justified since (1) the diffusion mass flux should be dominated by the motion of mass-dominating dust motion, i.e. dust grains of m = m p , (2) the diffusion flux should be determined by density gradient (or concentration gradient) and have nothing to do with size gradient ∂m p /∂r . Armed with these assumptions, we can derive the following moment equations with dust diffusion: 1 ∂ 1 ∂ ∂d + (r vr d ) = ∂t r ∂r r ∂r

  ∂d r D(r, m p ) ∂r

√   2 πa 2 vpp ∂m p D(r, m p ) ∂d ∂m p + vr − = d . ∂t d ∂r ∂r Hd

(4.17)

(4.18)

Note that the left hand side of the second equation is the Lagrangian derivative of m p when dust grains move at the velocity of vr − d−1 D(r, m p )∂d /∂r . Further simplification and equations at the local frame We assume the Epstein regime throughout this chapter and calculate the dimensionless stopping time τs ≡ tstop  using the size of the-peak-mass dust  1/3 a = 3m p /4πρint :  π ρint a τs = . (4.19) 8 ρg cs We also use the midplane value of τs and vr because most of the dust particles are distributed √ near the midplane. For a vertical gas density profile, ρg (z = 0) is given by g / 2π H

4.2 Basic Equations for Linear Analyses with Dust Growth

  z2 , ρg = √ exp − 2H 2 2π H g

and thus one obtains τs =

π ρint a . 2 g

97

(4.20)

(4.21)

In the absence of the diffusion, Eqs. (4.14) and (4.21) yield √    vpp ∂τs ∂τs π d τs . + vx = ∂t ∂r 4 g,0 τs Hd

(4.22)

In this chapter, we mainly focus on turbulent-induced collisions at which dust parti√ cles collide with a relative velocity vpp = Cτs αcs [28], where C is a numerical factor and C  2.3 for dust grains with a size√ratio of 0.5. We also use the simplified expression for the dust scale height: Hd = α/τs H . These simplifications reduce Eq. (4.22) as follows: d τs ∂τs ∂τs τs ∂g τs 1 ∂  + vr = r g u r − , + vr ∂t ∂r g 3t0 g r ∂r g ∂r

(4.23)

√ where t0 ≡ (4/3 Cπ )−1  0.49−1 . In the following sections, we perform linear analyses in the Cartesian local shearing sheet [16] whose radial distance from a star is R and Keplerian orbital frequency is . In the local frame, the basic equations for dust are summarized as follows: ∂d vx ∂d + = 0, ∂t ∂x

(4.24)

∂τs d τs τs ∂g ∂τs τs ∂g u x + vx = − , + vx ∂t ∂x g 3t0 g ∂ x g ∂ x

(4.25)

The analysis with dust diffusion is shown in the subsequent section.

4.3 Linear Analyses and Results We perform linear analyses considering only dust motion: gas is assumed to be static. We assume that the gas surface density is uniform at the local frame, g = g,0 , and the radial dust velocity is given by the so-call drift velocity vx = −

2τs η R. 1 + τs2

(4.26)

98

4 Coagulation Instability: Self-induced Dust Concentration

The assumption of the steady uniform gas reduces Eq. (4.25) as follows: d τs ∂τs ∂τs + vx = . ∂t ∂x g,0 3t0

(4.27)

Besides, we also neglect the dust diffusion in this section since the property of coagulation instability can be clearly understood in such a case. The impact of dust diffusion is discussed in the next section. We have to first construct an unperturbed state for the linear analysis. We assume uniform surface densities for gas and dust. There is a general problem in constructing the unperturbed state if one consider the dust coagulation. That is, dust grains grow in size in the background state because of the first term on the right hand side of Eq. (4.27), and thus there is no steady solution. Assumption of constant dust sizes is valid if background coagulation proceeds slower than the instability. As we show in the following, the growth timescale of coagulation instability is shorter than that of the background coagulation, 3t0 g,0 /d . Therefore, we can safely assume constant dust size and thus constant τs in the unperturbed state. The dust velocity in the unperturbed 2 ). state is vx,0 = −2τs,0 η R/(1 + τs,0 Adopting the above unperturbed state, we obtain the linearized equations from Eqs. 4.24), (4.26) and (4.27) (n + ikvx,0 )δd + ikd,0 δvx = 0, δvx =

2 1 − τs,0 δτs vx,0 , 2 τ 1 + τs,0 s,0

(n + ikvx,0 )δτs =

δd τs,0 d,0 δτs + , g,0 3t0 g,0 3t0

(4.28) (4.29)

(4.30)

where we assume plane-wave perturbations as in the previous chapters, i.e., δd ∝ δvx ∝ δτs ∝ exp(ikx + nt). Then, we readily obtain the following dispersion relation 

 2 ε 12t0 1 − τs,0 ikvx,0 , 1± 1− (4.31) n = −ikvx,0 + 2 6t0 ε 1 + τs,0 where ε ≡ d,0 /g,0 . One mode has a positive growth rate: there is an unstable mode. We find that at short-wavelengths the growth rate n of the unstable mode is approximately given as ε Re[n]  3t0



2 3t0 1 − τs,0 k|vx,0 |. 2 2ε 1 + τs,0

(4.32)

Equation (4.32) is always positive, meaning that a disk is unconditionally unstable. Figure 4.2 shows the obtained growth rate as a function of k H (the left panel) and

4.3 Linear Analyses and Results

99

Fig. 4.2 Growth rate of coagulation instability as a function of wavenumber for three different parameter sets. The growth rate is normalized by the background coagulation rate in both panels. The horizontal axis of the left panel is k H while that of the right panel is k L gdl . From the right panel, we can see that there is a universal form of coagulation instability. This figure is originally c shown in Tominaga et al. [42] ( AAS). Reproduced with permission

k L gdl (the right panel). One can see that the growth rate is much larger than ε/3t0 at sufficiently short wavelengths, k ε/t0 |vx,0 |. In other words, this instability develops faster than the background coagulation whose the timescale is 3t0 /ε. We find a characteristic length scale of coagulation instability: L gdl ≡ 3t0 |vx,0 |/ε. We call it a growth-drift length in the following since the length scale is determined by coagulation and radial drift. The growth-drift length is a distance over which dust moves at vx,0 within the coagulation timescale. L gdl /H is as follows:     L gdl t0   ε −1 |vx,0 |/cs = 30 . H 0.5 0.001 0.02

(4.33)

If the disk’s aspect ratio H/R is of order 10−2 , the growth-drift length can be a comparable to the disk size. This is consistent with the fact that, when dust-gas ratio ε is smaller, dust grains quickly drift inward and fall onto the central star without ˜ significant size growth. If we normalize wavenumber by L −1 gdl , i.e., k ≡ k L gdl , we obtain    1 k˜ 3t0 = + . (4.34) Re[n] ε 2 2 2 where we assume k˜ 1 and 1 ± τs,0  1. The latter is valid for dust grains that efficiently drift τs,0 ∼ 0.1. Equation (4.34) signifies that coagulation instability has the self-similarity in their dispersion relation (see the right panel of Fig. 4.2). Thus, ˜ one can derive growth rates for different dust sizes scaling wavelengths k. Coagulation instability is triggered by a combination of dust coagulation and traffic jam. The mechanism is schematically shown in Fig. 4.3. Let’s consider a case where there is a perturbation in dust surface density. Then, dust grains at denser

100

4 Coagulation Instability: Self-induced Dust Concentration

Fig. 4.3 Schematic picture to show the mechanism of coagulation instability. The present instability is a positive feedback process due to a combination of coagulation and traffic jam. This figure is c originally shown in Tominaga et al. [42] ( AAS). Reproduced with permission

regions grow faster than those at less dense regions since the coagulation timescale is is inversely proportional to d . In other words, there is a radial dependence in size-growth rate. This results in size perturbations: δd increases δτs . By the way, the dust velocity depends on the dimensionless stopping time. Thus, δτs leads to radial variation in drift velocity δvx . In such a case, dust density increases at a region where the velocity gradient is negative, i.e. traffic jam. Note that the negative velocity means inward velocity here. In this way, the size perturbations δτs enhances the surface density perturbations δd . This is a positive feedback process and further cycle augments both δd and δτs . This successive dust growth and traffic jam lead to coagulation instability. Although the size perturbations are important for the instability, they are quite small compared to the surface density perturbations. From the linearized equations, we obtain the ration of those perturbations as follows  π δτs /τs,0  exp −i δd /d,0 4



2 1 1 + τs,0 , 2 k L gdl 1 − τs,0

(4.35)

where we assume short-wavelength perturbations. The ratio decreases as k increases. For example, δτs /τs,0 is smaller than δd /d,0 by an order of magnitude for k L gdl = 100. Therefore, at least in the linear regime, coagulation instability is a process that produce more significant radial structures in the dust surface density profile than in the dust size profile.

4.3 Linear Analyses and Results

101

An effect of gas motion and profiles In Tominaga et al. [42], we also performed two-fluid linear analyses by taking gas equations into account (Sect. 4.3 therein). We found that the difference from the above one-fluid analysis is quite small. The effect of gas motion is negligible, and the gas surface density structure in the unperturbed state slightly affects growth rate. In other words, modes in gas are irrelevant to coagulation instability. The dispersion relation obtained in the two-fluid analysis is actually well reproduced by using the following equation instead of Eq. (4.27): d τs ∂τs τs ∂g ∂τs + vx = , − vx ∂t ∂x g 3t0 g ∂ x

(4.36)

where g and ∂g /∂ x are input values and time-independent. The modified onefluid analysis with Eq. (4.36) gives the following dispersion relation (see Sect. 3.3 in [42]): ε n(k, W ) ≡ −ikvx,0 + 6t0



W+

W ≡1+

W2

 2 12t0 1 − τs,0 − ikvx,0 , 2 ε 1 + τs,0

L gdl 2 . 2  /  1 + τs,0 g,0 g,0

(4.37)

(4.38)

The effect of the second term on the right hand side of Eq. (4.36) is that dimensionless stopping time decreases when dust grains move into the inner region where gas is denser. Thus, the increase rate of τs is modified. The detailed discussion and comparison with the two-fluid dispersion relation can be found in Sect. 4.3 of Tominaga et al. [42].

4.4 Discussion 4.4.1 Impact of Dust Diffusion In this subsection, we discuss how coagulation instability is affected by the dust diffusion. As shown in Sect. 4.2, the diffusion term appears not only in the continuity equation but also in the evolutionary equation of dust sizes as a part of advection velocity. The equations used here are ∂d vx  ∂ ∂d + = ∂t ∂x ∂x

 D

∂d ∂x

 (4.39)

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4 Coagulation Instability: Self-induced Dust Concentration

    d τs D ∂d ∂τs D ∂d ∂g ∂τs τs vx  − + vx  − = , (4.40) − ∂t d ∂ x ∂x g 3t0 g d ∂ x ∂x where vx  is the mean-flow component representing the mean radial drift: vx  ≡ −

2τs η R. 1 + τs2

(4.41)

Here we also take into account the effect of the background g profile in Eq. (4.40). Note that if one describe diffusion whose rate is proportional to the gradient of concentration d /g , one obtain the equations just replacing the diffusion term as Dg ∂ D ∂d → − d ∂ x d ∂ x



d g

 .

(4.42)

However, the form of the diffusion form insignificantly affects the growth rate. We refer readers to Sect. 4.1 of Tominaga et al. [42] for the detail, and here we only consider the diffusion using the gradient of d . As in the analysis without diffusion, we assume uniform dust surface density and uniform τs in the unperturbed state. Therefore, the unperturbed velocity is   R . vx,0 = vx,0 = −2τs,0 η 2 1 + τs,0

(4.43)

Then we obtain the following dispersion relation



A0 = ik vx,0



    (n + ik vx,0 )2 + A1 (n + ik vx,0 ) + A0 = 0,

(4.44)

   2 vx,0 g,0 ε A1 = − + + Dk 2 , 2  3t0 1 + τs,0 g,0

(4.45)

2 1 − τs,0 2 1 + τs,0



 g,0 ε + ik D 3t0 g,0



+ Dk 2

    2 vx,0 g,0 ε − . + 2  3t0 1 + τs,0 g,0

(4.46)

The growth rate of the unstable mode is given by 

  2   1 2 g,0  1 − τs,0 1 3t0 2  2 ˜ ˜ ˜ ˜ ˜ = ik + W − βk + n + 4i k 1 + i kβ L gdl , W + βk ε 2 2 g,0 1 + τ2

(4.47)

s,0

where we use L gdl and ε/3t0 to normalize spatial and time scales, and β = DL −2 gdl (3t0 /ε) is a dimensionless diffusion coefficient. The dimensionless diffusivity β depends on multiple physical values as follows:

4.4 Discussion

103

Fig. 4.4 Growth rate with the dust diffusion. The left panel shows growth rates calculated without  in Eq. (4.47) while the right panel shows growth rate that we derive the terms proportional to g,0 with all terms in Eq. (4.47). The dashed line shows wavenumber estimated by Eq. (4.49). This figure c is originally shown in Tominaga et al. [42] ( AAS). Reproduced with permission

 βα

H L gdl

2

 1.7 × 10−4

3t0  ε 

α 1 × 10−4



t0  0.5

−1 

ε 1 × 10−3



|vx,0 |/cs 0.02

−2

,

(4.48)

where we assume that dust particles are so small that they satisfy τs,0  1 and D  αcs2 −1 (see [50]). Note that β depends on dust sizes through the drift speed |vx |. Figure 4.4 shows growth rates as a function of k L gdl and the dimensionless diffusion coefficient β for τs = 0.1. On the left panel, we show growth rates calculated without the last term on the right-hand side of Eq. (4.40) while on the right panel we show growth rates that we derived using all terms. Short wavelength perturbations are stabilized by diffusion. Larger diffusion, i.e. larger β, stabilizes coagulation  instability more significantly. The decrease of τs originating from the term with g,0 reduces the growth rates as in the diffusion-free cases. The reduction factor is however only a few. Coagulation instability grows 2–10 times faster than background coagulation in less turbulent disks of β  10−4 . According to Eq. (4.48), such a fast growth occurs for α ×   6 × 10−8 (see Eq. (4.48)). As shown in Fig. 4.4, the most unstable wavenumber kmax appears in contrast to the diffusion-free case. We find that the following formula well reproduces the most unstable wavenumber: kmax L gdl

1  3

2 4 1 − τs,0 2 β 2 1 + τs,0

1/3 .

(4.49)

The dashed line plotted in Fig. 4.4 shows the wavenumber given by Eq. (4.49), which shows good agreement with the most unstable wavenumber. The most unstable wavelength λmax ≡ 2π/kmax is

104

4 Coagulation Instability: Self-induced Dust Concentration

2 )/27(1 + τ 2 ))1/3 (Eq. (4.49)), i.e. the maximum Fig. 4.5 Growth rate at k L gdl = (4β −2 (1 − τs,0 s,0 growth rate, as a function of dimensionless stopping time τs,0 and unperturbed dust-to-gas surface density ratio ε = d,0 /g,0 . The left and right figures show the growth rate for α = 10−4 and  in Eq. (4.47) in both panels. The α = 10−5 , respectively. We include the terms proportional to g,0 white line traces the parameter set for which the steady solution exist in the unperturbed state. This c figure is originally shown in Tominaga et al. [42] ( AAS). Reproduced with permission

 λmax  1.1H

α 1 × 10−4

2/3 

t0  0.5

1/3 

ε 1 × 10−3

−1/3 

|vx,0 |/cs 0.02

−1/3

,

(4.50) 2  1. Therefore, the efficiency of coagulation instability is where we assume 1 ± τs,0 maximized at a spatial scale of ∼H . Figure 4.5 shows the growth rate at a wavelength given by Equation (4.49). This corresponds to the maximum growth rate. We assume α = 10−4 and 10−5 on the left and right panels, respectively. In such weakly turbulent cases, coagulation instability can develop within  10 − 30 Keplerian period even when the dust-to-gas ratio is less than 0.01 if dust grains grow up to τs,0  0.1. This is in highly contrast to the previous dust-gas instabilities including secular GI: those instabilities can develop for ε  0.01. The previous dust coagulation simulations show that dust growth (an increase in τs ) results in dust depletion (a decrease in ε). In such a case, on the τs,0 −ε plane in Fig. 4.5, dust grains move toward the right bottom. Figure 4.5 shows roughly constant growth rates along a line from the left top to the right bottom. This indicates that coagulation instability does develop unless a disk is only weakly turbulent and reconcentrate dust grains that are otherwise depleted as a result of inside-out size growth and fast radial drift. Therefore, coagulation instability is a promising mechanism for retaining dust grains and setting preferable sites for secular GI or other dust-gas instabilities (e.g., streaming instability [49]) to develop and to lead to planetesimal formation.

4.4 Discussion

105

4.4.2 Effects of Other Collision Velocities We consider turbulent-induced collisions in the above sections. This is because the coagulation timescale has the very simple form 3t0 /ε and useful for the first analysis of coagulation instability. However, there are other components in the collision velocity in reality:  vpp =

2  (vt )2 + (vB )2 + (vr )2 + vφ + (vz )2 ,

(4.51)

where we consider collision velocities due to turbulence vt , Brownian motion vB , and differential drift speeds vr , vφ , vz . The actual coagulation rate is thus higher than the rate for vpp = vt . In the following, we roughly estimate the enhancement factor of the growth rate due to the multiple components of the collision velocity. We neglect vB since coagulation instability operates for drifting dust while the Brownian motion significantly contributes to the growth of very small grains that hardly drift [4]. The differential velocities are given by vr = −

vφ =

2τs (2 − τs2 ) η R, (4 + τs2 )(1 + τs2 ) η R

(4.53)

τs z, (2 + τs )(1 + τs )

(4.54)

(4 +

vz = −

3τs2 2 τs )(1

(4.52)

+ τs2 )

where we assume dust grains √ whose size ratio is 0.5 and neglect the√backreaction. If we assume v = Cατs cs , we have vr /vt ∝ vz /vt ∝ τs /α and t √ vφ /vt ∝ τs τs /α as leading-order terms. Thus, vr and vz are larger than vφ for τs < 1. Motivated by this order estimation, we approximate the collision velocity as follows  τs (4.55) vpp  vt 1 + f , α   vz 2 + , vt   2   z 2 η R 2 2 − τs2 1 4 + = C (4 + τs2 )(1 + τs2 ) cs C(2 + τs )2 (1 + τs )2 H

α f ≡ τs



vr vt

2



(4.56)

106

4 Coagulation Instability: Self-induced Dust Concentration

Equations (4.55) and (4.22) yield ∂τs ∂τs + vx = ∂t ∂x

 1+ f

τs d τs . α g,0 3t0

(4.57)

We thus  expect that growth rates of coagulation instability becomes larger by a factor of 1 + f τs,0 /α. If the dust scale height is determined by turbulent stirring  √ (z  Hd  α/τs H ), one obtains 1 + f τs,0 /α  2.5 for τs,0 = 0.1, α = 10−4 , C = 2.3, and η R = 0.11cs . The adopted value of η R is taken from the MMSN model for R = 20 au.

4.4.3 Coevolution with Other Dust-Gas Instabilities: Bridging a Gap Between First Dust Growth and Hydrodynamical Clumping Toward Planetesimal Formation The present instability is triggered by dust coagulation. In that sense, coagulation instability is distinct from any other previous dust-gas instabilities including secular GI and TVGI discussed in Chaps. 2 and 3. The growth rate converges to zero if we take the limit of t0 → ∞, meaning that coagulation is essential for the instability. The previous global dust coagulation models showed dust depletion due to the fast radial drift and inside-out size growth if a disk is isolated and there is no mass infall (e.g., [4, 27]). Such dust depletion is a crucial issue for the previous dust-gas instabilities since they can not operate for low dust-to-gas ratio. On the other hand, coagulation instability can grow even for d /g ∼ 10−3 . Coagulation instability re-concentrates dust grains into small spatial scales ∼k −1 . In the presence of dust diffusion, we expect the spatial scale of re-concentration to be ∼H (see Eq. (4.50)). If nonlinear growth increases d /g beyond 0.01, dust sizes will also significantly increase as a result of enhanced coagulation. Thus, both d /g and τs will locally increase as coagulation instability develops. We expect secular GI to operate in the resulting dust-piling-up regions. Therefore, we expect that coagulation instability is a powerful mechanism to connect the first bottom-up coagulation and planetesimal formation via secular GI. Nonlinear simulations are necessary for more qualitative discussions, which will be addressed in future works.

4.5 Summary Planetesimal formation via dust-gas instabilities has an issue in their onset conditions. Secular GI requires larger dust-to-gas ratio (>0.01) for large dust grains (τs,0  0.1; see Chaps. 2 and 3). The other dust-gas instabilities (e.g., streaming instability) also

4.5 Summary

107

require such enrichment of large dust grains. On the other hand, the first coagulation and the fast radial drift lead to dust depletion in the absence of dust supply from the infalling envelope. Thus, the previous dust-gas instabilities require some dust retention mechanisms in advance to their onset. In this chapter, we present a new instability driven by coagulation (“coagulation instability”) as a mechanism of dust re-concentration. Coagulation instability operates as a result of a positive feedback between coagulation and traffic jam: coagulation is accelerated at dust-rich regions and amplifies dust size perturbations while traffic jam due to the size perturbations locally amplifies dust density perturbations. In the absence of dust diffusion, coagulation instability grows faster at shorter wavelengths, which is because a timescale of traffic jam is shorter at shorter wavelengths. For example, the growth timescale of the instability is tens Keplerian periods for (τs,0 , ε) = (10−1 , 10−3 ), which is 20–30 times shorter than the coagulation timescale (ε/3t0 )−1 (see Fig. 4.5). In the presence of dust diffusion, short-wavelength perturbations are stabilized, and thus the dispersion relation of coagulation instability shows the most unstable wavelength at ∼H (see Eq. (4.50)). Coagulation instability still grows only within a few tens of the Keplerian periods regardless of the stabilization due to dust diffusion. Therefore, coagulation instability is a promising mechanism for re-concentrating dust grains, and bridges the gap between the first coagulation and planetesimal formation via the dust-gas instabilities.

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Chapter 5

Summary and Future Prospects

5.1 Summary of This Thesis Planetesimal formation from dust grains in a protoplanetary disk is the first step in the planet forming processes. However, the formation mechanism is under debate because there are barriers against dust growth, e.g., the radial drift and fragmentation (e.g., [2, 23, 24]). Recent ALMA observations have been showing some clues for revealing planetesimal formation. One of the most highlighted results of the observations is the discovery of ubiquitous annular substructures in dust distributions, i.e., rings and gaps. The existence of multiple dust rings in disks are in contrast to the classical theories that assumed smooth profiles and showed fast depletion of mm-sized dust because of the radial drift. Therefore, investigating the origin of multiple rings and connections to dust growth will provide the key to reveal planetesimal formation and unify the disk evolution theory and planet formation theory. In this thesis, we first focus on disk evolution via secular GI, which is one possible mechanism of ring and planetesimal formation. Secular GI is one of the dust-gas instabilities and originally proposed as a mechanism of planetesimal formation (e.g., [22, 25, 26]). Takahashi and Inutsuka [16, 17] showed that secular GI can create multiple dust rings with a width of H , which is consistent of the observed rings. However, the previous studies have some issues: 1. the previous equations with dust diffusion violate angular momentum conservation, 2. the previous studies focused on the locally linear growth while nonlinear growth in a radially extended disk is important to explore ring and planetesimal formation 3. secular GI requires high dust-to-gas ratio for mm- or cm-sized grains although the first bottom-up coagulation toward those sizes results in dust depletion. This thesis addresses these issues, and the results are already reported in our published papers [18–21].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 R. Tominaga, Dust-Gas Instabilities in Protoplanetary Disks, Springer Theses, https://doi.org/10.1007/978-981-19-1765-3_5

111

112

5 Summary and Future Prospects

In Chap. 2, we first reformulate equations that can describe diffusion without violating the momentum conservation. Our formulation is based on the Reynold averaging as done in Cuzzi et al. [3], which divides physical properties into mean-flow parts and fluctuating parts due to turbulent motion. Averaging the usual hydrodynamic equations naturally reproduce the diffusion equation (see also [3]) and at the same time introduces a new term in momentum equations: momentum advection along diffusion flow. We found that such an advection term in momentum equations is important to gurantee the momentum conservation. The new term will be valid for small dust grain of τs  1 since diffusion of such small grains is driven by the radial force due to gas turbulence [27]. Based on the reformulated equations, we perform linear analyses of secular GI. In contrast to the previous studies that showed overstability, our results show that secular GI is an exponentially growing mode without oscillation. The overstability in the previous studies was found to be due to the violation of angular momentum conservation law. We also found another unstable mode that we name two-component viscous GI (TVGI). TVGI is triggered by a combination of friction and turbulent gas viscosity. Although the linear analyses in Chap. 2 show that TVGI grows for wider parameter space than secular GI, including dust drift stabilizes TVGI as shown in Chap. 3. Thus, TVGI can be a powerful mechanism for forming planetesimals at a region where dust insignificantly drifts. In Chap. 3, we numerically investigate nonlinear evolution of secular GI. We adopt the numerical method developed in Tominaga et al. [19] and implement dust diffusion and gas viscosity [20]. Secular GI has long growth timescales, 100 orbital periods, and thus one needs long-term integrations. However, the dust drift throughout a gas disk potentially introduces significant numerical diffusion due to advection, which numerically prevents the growth of secular GI. Motivated by this issue, in Tominaga et al. [19] we develop the Lagrangian-cell method, which is free from the numerical diffusion. We also utilizing the symplectic integrator and reduces the accumulation of errors due to time integration. Test simulations with local radial domain show that combining the method with the piecewise exact solution for dust-gas friction allows simulations of linear/nonlinear secular GI. We perform numerical simulations of secular GI in radially extended disks while assuming uniform profile of dimensionless stopping time for simplicity. We found that nonlinear growth of secular GI is similar to the gravitational collapse of a dust ring whose timescale is well represented in terms of the freefall time. As a result, the dust surface density increases by an order of magnitude. On the other hand, the gas surface density insignificantly changes, leading to high dust-to-gas ratio in thin dense rings. If the dust-to-gas ratio increases enough, the dust drift is suppressed because of strong backreaction to the gas. Thus, dust grains are saved in a disk once secular GI grows into the highly nonlinear regime. If the growth of secular GI is too small to create high-contrast rings and gaps, those substructures enter the inner stable region and finally become transient. According to those results, planetesimal formation via secular GI requires dust enrichment toward around the gas density, i.e., dust-to-gas ratio 1.

5.1 Summary of This Thesis

113

Simple estimates of the coagulation timescale and the freefall timescale indicate that accelerated coagulation and ring fragmentation will result in planetesimals within one Keplerian period at the ring location. This implies that multiple rings forming through secular GI would be dark at mm wavelengths and would be observed as a single wide gap structure. Subsequent fragmentation of planetesimals will supply smaller dust grains that determine a floor intensity at the wide gap. Because secular GI creates only insignificant substructures in a gas disk, observations of gas profiles around the midplane will provide hints to understand which ring-forming process actually operates in the observed disks. In Chap. 4, we address the third issue: secular GI requires re-concentration of mm- and cm-sized dust grains. Although previous studies already proposed some dust retention mechanisms including dust-piling-up near the water snow line and the dead-zone inner boundary, those operate at a specific location. We propose a new instability as another mechanism for dust re-concentration. The instability is triggered by a combination of dust coagulation and small scale traffic jam, and thus we call it “coagulation instability”. In the absence of dust diffusion, coagulation instability shows larger growth rate at shorter wavelengths, which is because a timescale of traffic jam becomes shorter at shorter wavelengths. Even in the presence of dust diffusion and in a dust-depleted region, coagulation instability grows at a wavelength comparable to the gas scale height. Its growth timescale is about a few tens of the Keplerian periods. Therefore, coagulation instability efficiently concentrates mm- and cm-sized dust grains, which will bridge the gap between the first coagulation and the top-down planetesimal formation via secular GI investigated in Chaps. 2 and 3.

5.2 Future Prospects Numerical investigation of coagulation instability toward planetesimal formation In this thesis, we only explore the linear growth of coagulation instability. Nonlinear simulations of coagulation instability are necessary to investigate to what extent the instability re-concentrates dust grains. Besides, its growth in a radially extended disk is important. Radially global evolution inevitably leads to dust depletion in the inside-out manner, i.e., the disk loses dust grains as time proceeds. If coagulation instability develops in the dust-depleted regions, some amounts of dust grains will be trapped in rings and they are saved from the fast drift if they become large enough or dust-gas ratio becomes large enough that the backreaction stops the radial drift. The amount of saved dust grains is one important physical values for discussing planetesimal formation, i.e., it gives the upper limit of planetesimal masses. Dust concentration into rings might be important for explaining the observed spectral indices at mm-wavelengths. The previous observations show that the spectral index at 1–3 mm is about 2.0–2.5 (e.g., [13]). On the other hand, previous dust

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5 Summary and Future Prospects

coagulation model predicted that the spectral index becomes 3–3.5 after dust grains grow and they are depleted because smaller dust grains that insignificantly drift contribute to the spectral index after the depletion of large dust grains [12]. This indicates that some mechanism of dust retention is necessary. Pinilla et al. [12] demonstrated that dust traps in rings can explain the observed low spectral index. Coagulation instability can be the promising mechanism of the required dust retention. Further investigation is required. If nonlinear coagulation instability creates spicky dust rings as secular GI does, multidimensional analyses and simulations will be significantly important. The very recent work [11] shows that a spiky ring forming via secular GI becomes unstable to Rossby wave instability (RWI; e.g., [8]). This is due to a resulting extremum of potential vorticity [7]. The development of RWI will lead to axisymmetric fragmentation of resulting dust rings, which may play a key role to determine final masses of solid objects. Pierens [11] shows that azimuthal scale of fragmentation due to RWI depends on the timescale of the radial concentration. To address such a nonlinear process, we need multidimensional numerical simulations with dust coagulation, which will be our future work. Coagulation instability at the early disk-evolution stage If turbulence is not so strong, coagulation instability always grows because a disk is unconditionally unstable (see Chap. 4). On the other hand, as mentioned in Chap. 1, secular GI grows in relatively massive disks, for example, Q  6, although the required disk mass depends on the other parameters. Thus, it is worthwhile to investigate whether coagulation instability grows and sets up conditions preferable for secular GI at the early disk-evolution stage where a disk is thought to be massive. Coagulation instability in young disks is also important in the context of substructure formation. Some works reported that young disks with an age of 1 Myr already host dust ring structures (e.g., [1, 4, 9, 14, 15]). These observations may indicate that dust grains have grown up to millimeter sizes. Ohashi et al. [10] showed that the growth front, which naturally forms in the inside-out dust evolution via coagulation, can explain the observed rings in young disks. They also showed that the rings in Class II disks should be located in inner regions compared to the growth front. This might indicate that some dust concentration mechanism operates after the growth front passes and dust grains are once depleted. Coagulation instability can be such a mechanism. Because coagulation instability accelerates dust coagulation at the nonlinear growth phase, the instability potentially creates the first-generation planetesimals directly. Such an early planetesimal formation will support the hypothesis that planets already form in Class II disks and carve gaps (e.g., [5, 6, 28]). Therefore, investigating coagulation instability at the very early stage is important for both planet-based and secular-GI-based ring formation scenarios. For further discussion, we need simulations that describe disk formation and early evolution of dust grains consistently.

5.2 Future Prospects

115

Secular GI with dust growth and multidimensional analyses In Chap. 3, we showed that secular GI can create multiple thin dense rings, where one can expect dust growth or planetesimal formation via ring fragmentation within one Keplerian period. Thus, we expect that those multiple rings would be observed as a dark gap. To obtain further observational implications, we have to implement dust growth in simulations of secular GI. Collisional fragmentation is also necessary because it supplies small dust grains that we can observe at the ALMA bands. The equilibrium between dust supply due to fragmentation and dust depletion due to planetesimal formation via secular GI will determine the intensity at the wide gap. In future studies, we will explore the coevolution of secular GI and dust growth and also perform synthetic observations aiming at direct comparison with the observed intensity profiles. Multidimensional analyses including simulations are also important to understand the disk evolution via secular GI (see [11]). Introducing the azimuthal direction, we can directly treat ring fragmentation and quantify planetesimal formation rates. Numerical simulations with radial and azimuthal directions will be necessary because planetesimal formation occurs at the nonlinear stage. The inclusion of the vertical motion is also important. As mentioned in Chaps. 2 and 3, secular GI will be operational around the midplane. It is however unclear to what vertical extent we have to consider dust and gas because gas at the upper layer will interact with midplane dust through gravity. Multidimensional linear analyses will reveal the vertical extent. These are the scope of our future studies.

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