ULF Waves’ Interaction with Cold and Thermal Particles in the Inner Magnetosphere 9789813293786

Table of contents :
Table of contents (7 chapters)

Background and Motivation

Pages 1-33

Ren, Jie
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Data Sources

Pages 35-37

Ren, Jie
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Interactions Between ULF Waves and Cold Plasmaspheric Particles

Pages 39-59

Ren, Jie
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Simultaneous Interaction of ULF Waves with Different Ring Current Ions

Pages 61-73

Ren, Jie
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Phase Relationship Between ULF Waves and Drift-Bounce Resonant Particles

Pages 75-93

Ren, Jie
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Substorm-Related ULF Waves and Their Interaction with Ions from Different Sources

Pages 95-104

Ren, Jie
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Summary

Pages 105-106

Ren, Jie
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Springer Theses Recognizing Outstanding Ph.D. Research

Jie Ren

ULF Waves’ Interaction with Cold and Thermal Particles in the Inner Magnetosphere

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 http://www.springer.com/series/8790

Jie Ren

ULF Waves’ Interaction with Cold and Thermal Particles in the Inner Magnetosphere Doctoral Thesis accepted by Peking University, Beijing, China

123

Author Dr. Jie Ren School of Earth and Space Sciences Peking University Beijing, China

Supervisor Prof. Qiugang Zong School of Earth and Space Sciences Peking University Beijing, China

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-32-9377-9 ISBN 978-981-32-9378-6 (eBook) https://doi.org/10.1007/978-981-32-9378-6 © Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved 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

Shortly after this Chinese New Year, a movie called The Wandering Earth was released and quickly became very popular with the masses. The background of this movie is that our Earth is faced with the fate of being devoured by the constantly expanding sun that has got out of the main sequence. In reality, scientists speculate that the lifespan of the sun will continue for about several billion years, when it will collapse into a white dwarf. However, the space around Earth is always impacted by the solar wind and interplanetary magnetic fields, and in the extremes, the effect of solar activity can cause satellite damages, radiation hazards to airline passengers and astronauts, navigation and telecommunication problems, outages of power and electronic systems, etc. Energy from the Sun can be transferred into Earth’s magnetosphere in a variety of ways, part of which is mediated by ultra-low-frequency (ULF) waves. ULF waves can propagate electromagnetic energy over the magnetosphere and into the ionosphere, and play an important role in the acceleration and transportation of charged particles trapped in the magnetosphere. In Fall 2012, Jie joined the Institute of Space Physics and Applied Technology (ISPAT) in Peking University, as a Ph.D. candidate, without examination. Under my supervision, he started his research work on the interactions of ULF waves with ring current ions and plasmaspheric particles. Besides, he also systematically studied the properties of field-aligned currents in the high-latitude energetic electron boundary, which is not included in this doctoral thesis. Dr. Ren’s Ph.D. thesis focuses on exploiting the combined observations from spacecraft and ground-based magnetometers to investigate the excitations of ULF waves during storms and substorms, the behaviors of ring current oxygen ions and cold plasmaspheric particles in different harmonic mode ULF waves, and the possible scenarios for the parallel structures of standing wave electric fields. He reported the accelerating effect of ULF waves on both cold plasmaspheric electrons and ions and the modification of main plasmaspheric populations due to the adiabatic acceleration, and proved the existence of localized ULF waves using multi-spacecraft observations. Next, he studied the behaviors of different ions simultaneously interacting with ULF waves and provided a potential explanation for the formation of O+-dominated ring current. Furthermore, Jie found a way to v

vi

Supervisor’s Foreword

study the phase relationship between ULF waves and drift-bounce resonant particles, and proposed that their phase relationship can be used to diagnose the parallel structures of standing wave electric fields and energy transfer directions between waves and particles. I believe that the research results of this thesis will lead to great progress in our understanding of the role of ULF waves in the dynamics of the inner magnetosphere. Beijing, China June 2019

Prof. Qiugang Zong

Publications Parts of this thesis have been published in the following journal articles: 1. Ren Jie, Q. G. Zong, Y. F. Zhu, X. X. Zhou and S. J. Gu, Field-aligned structures of the poloidal-mode ULF wave electric field: phase relationship implications. Journal of Geophysical Research: Space Physics, 124(5), 3410–3420, DOI: 10.1029/2019JA026653, 2019 2. Ren Jie, Q. G. Zong, Y. Miyoshi, R. Rankin, H. E. Spence, H. O. Funsten, J. R. Wygant and C. A. Kletzing, A comparative study of ULF waves’ role in the dynamics of charged particles in the plasmasphere: Van Allen Probes observation. Journal of Geophysical Research: Space Physics, 123(7), 5334–5343, DOI: 10.1029/2018JA025255, 2018 3. Ren Jie, Q. G. Zong, Y. Miyoshi, X. Z. Zhou, Y. F. Wang, R. Rankin, C. Yue, H. E. Spence, H. O. Funsten, J. R. Wygant and C. A. Kletzing, Low-energy (1 MeV) can easily damage spacecraft in Earth’s orbit and threaten the human security in space [3, 4].

1.1 The Inner Magnetosphere

3

Fig. 1.2 Radiation belt structures including inner and outer belts, and a slot region

Fig. 1.3 SAMPEX satellite observations of energetic electrons in the radiation belt from 1 January 1994 to 31 December 1995. (From Li et al. [54])

1.1.2 Ring Current Magnetic storms are mainly associated with the interplanetary structures with longterm and intense southward magnetic fields that continuously interconnect with Earth’s magnetic field and transport solar energy into the magnetosphere [35]. The

4

1 Background and Motivation

strong decrease of magnetic fields observed by ground-based magnetometers is thought to be generated by the intensification of ring current. The main carriers of ring current are trapped energetic particles (e.g. H+ , O+ ) in the energy range from tens to hundreds of keV [103–105]. The symmetric ring current is formed by ions in closed drift orbits, while the asymmetry one is formed by ions in open orbits. There are three main stages during a storm: storm onset, main phase and recovery phase [35]. Storm onset occurs in coincidence with the sharp southward turning of the magnetic field in the interplanetary medium; the following is the rapid development of the main phase; the recovery phase in the final is initiated by a gradual turning of the interplanetary fields to a northward direction, which will decrease rapidly in the first step due to the rapid loss of ring current ions and slowly in the next step. Spacecraft observations have demonstrated that H+ ions are the dominant species in quiet times, while there is strong energization and enhancement of ring current O+ ions during geomagnetic storms [12, 30, 118]. Figure 1.4 summarizes five possible scenarios that can contribute to energization and supply of the ring current O+ ions. The first scenario is that the O+ population is directly supplied by the topside ionosphere, and accelerated up to several tens of keV in the upgoing process [13, 33, 86]. The second is that cold O+ ions are transported into the magnetotail and further experience adiabatic accelerations by the convection electric field in the earthward motion [27, 58, 69]. The third is that O+ ion population is supplied in the same way as the second scenario, but energized due to the non-adiabatic acceleration associated with near-Earth dipolarization [17, 66, 73]. The forth is that there is an oxygen torus region in the outer plasmasphere (L = 2–5) [29, 41], and these preexisting thermal O+ ions are accelerated by the dipolarization in the deep inner magnetosphere [72]. The last one is that O+ ions are accelerated and transported by ULF waves [82, 108, 120].

1.1.3 Plasmasphere Plasmasphere was discovered independently by spacecraft measurements and ground-based whistler wave observations in 1960s [53]. Plasmasphere as the upward extension of Earth’s ionosphere is filled with dense (102 −104 cm−3 ), cold (∼eV) electrons and ions that mainly comprise H+ (∼80%), He+ (∼10 − 20%) and O+ (∼5 − 10%) [88]. Plasmasphere is dynamically changing during geomagnetic activities. According to previous knowledge, the dynamics of plasmasphere is dominantly controlled by the superposition of corotation and convection electric fields. During geomagnetic activities, erosion or inward motion of the plasmasphere occurs in the time scale of a few hours in response to the enhanced convection electric field (e.g. [32, 34, 93]); in the recovery phase of a geomagnetic storm, the erose part of the plasmasphere will be refilled along with the decrease of convection electric field over a few days (e.g. [80, 84]). The outer boundary of the plasmasphere not only exhibits a plasmapause with a sharp gradient density, but has complicated density structures like plumes [68]. In Fig. 1.5, plasmasphere has various density structures

1.1 The Inner Magnetosphere

5

Fig. 1.4 Primary acceleration mechanisms of ring current O+ ions during geomagnetic storms: 1. O+ ions are directed supplied by the topside ionosphere and accelerated into the ring current region; 2. Ionospheric O+ ions are transported through the lobe region into the plasma sheet and then transported into the ring current region after adiabatic acceleration by convection electric fields; 3. Ionospheric O+ ions are transported in the same way as mechanism 2 but experience non-adiabatic acceleration by the dipolarization in the plasma sheet; 4. Non-adiabatic acceleration of the local cold O+ ions in the oxygen torus; 5. Acceleration and transportation by ULF waves. (From Keika et al. [45])

including large-scale (plumes and notches), medium-scale (fingers, shoulders, creaulations and channels), and small-scale (irregularities) features, which unfold in the time scale from minutes to hours [16]. The plume as a large-scale structure can extend beyond the main plasmapause to outer magnetosphere, and can affect the magnetic reconnections at the magnetopause [100, 101].

1.2 ULF Waves and Their Interaction with Particles 1.2.1 Introduction to ULF Waves Magnetohydrodynamic Waves In 1942, Hannes Alfvén first proposed the existence of transverse “electrohydrodynamic” waves in the magnetized plasma [1], and then won the Nobel Prizing for his excellent work on magnetohydrodynamics. The magnetohydrodynamic

6

1 Background and Motivation

Fig. 1.5 An example EUV image of the plasmasphere at 0734 UT on 24 May 2000, where various density structures are visible. (From Sandel et al. [85])

(MHD) theory was first applied to explain the observed geomagnetic pulsations in Earth’s magnetosphere by Dungey [19]. Nowadays, Dungey’s theory has been greatly confirmed and extended by observations from ground-based magnetometers, aurora radars as well as satellites. MHD theory is a combination of theory of fluid dynamics and electromagnetic theory. In the ideal MHD theory, there are some fundamental assumptions including: 1. The electrical conductivity satisfies σ → ∞, which is the frozen-in condition; 2. The displacement current in the Maxwell equations is neglected. Under the first assumption, (1.1) E⊥ + V × B  0 E  0

(1.2)

We first consider a simplest case that the background magnetic field is uniform and constant, plasmas are cold (in other words, the kinetic pressure can be neglected), and there is no boundary conditions. Assume that all the perturbations are small enough to use the linearized equations. Then the Maxwell equations can be simplified into

1.2 ULF Waves and Their Interaction with Particles

∇×E=−

7

∂b ∂t

∇ × b = μ0 j

(1.3) (1.4)

where μ0 is the space permeability; E, b and j are small perturbations. The motion equation can be simplified into ρ0

∂v = j × B0 ∂t

(1.5)

where ρ0 and B0 are the plasma density and magnetic field in equilibrium, respectively; v is a small perturbation. Hence, from Eqs. (1.1), (1.3), (1.4), (1.5), we can get ∂2 E + V A2 ∇ × (∇ × E) = 0 ∂t 2

(1.6)

where V A = √μB00ρ0 , V A is the Alfvén velocity. In a cartesian coordinate, adopting an uniform background magnetic field B0 = B0 zˆ , Eq. (1.6) can be rewritten as (V A−2

∂2 ∂2 ∂2 ∂2 − − )E + Ey = 0 x ∂t 2 ∂ y2 ∂z 2 ∂x∂ y

(1.7)

(V A−2

∂2 ∂2 ∂2 ∂2 Ex = 0 − 2 − 2 )E y + 2 ∂t ∂x ∂z ∂x∂ y

(1.8)

Assume that all the perturbations are in a plane-wave form, ei(k y y+kz z−ωt) . Since the plasma is uniform, there is no difference between x and y directions. Adopt that wave vector is in the yz plane, k 2 = k 2y + k z2 . So ∂/∂x ≡ 0, and Eqs. (1.7)–(1.8) will reduce to (1.9) ω 2 /k z2 = V A2 ω 2 /k 2 = V A2

(1.10)

Equation (1.9) is the dispersion relation for the shear Alfvén wave. Its phase velocity = ±V A zˆ . It is a kind of transverse wave, driven is kωz = V A , and group velocity is dω dk by magnetic tension, and its energy flow is only along the magnetic field line. Equation (1.10) is the dispersion relation for the compressional wave. Its phase velocity is V A , and ground velocity is ±V A ek . It is a kind of longitudinal wave, driven by magnetic pressure. Since the energy flow direction has nothing to do with B0 , the compressional wave can propagate across the magnetic field line. If the plasma is not cold, the thermal pressure can not be neglected. Equation (1.10) will be divided into the dispersion relations for fast and slow compressional sound waves,

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1 Background and Motivation

ω 2 /k 2 =

1 2 4c2 V 2 cos 2 θ (cs + V A2 )[1 ± (1 − s2 A 2 2 )1/2 ] 2 (cs + V A )

(1.11)

where cs = (γ P0 /ρ0 )1/2 , cs is the sound velocity, θ is the angle between k and B0 . For the fast magnetosonic mode, the magnetic and thermal pressures are in-phase; for the slow magnetosonic mode, they are anti-phase. Drift-bounce Resonance Theory There is large Pedersen conductivity in the ionosphere E layer. So in Earth’s magnetosphere, the MHD waves can be reflected by the ionosphere, and the reflection coefficient [90] is given by Er 1 − μ0  P V A = Ei 1 + μ0  P V A

(1.12)

where Er and E i are the reflected and incident wave electric fields, respectively;  P is the Pedersen conductivity. When  P → ∞, Er and E i are anti-phase. Therefore, when the Alfvén wave frequency agrees with the eigenfrequency of the magnetic field line, it will form standing waves with its node of electric field oscillations located in the outer boundary of ionosphere. 1. Standing Alfvén Waves in the Uniform Background Magnetic Field We initially look at the simplest magnetospheric model that the background magnetic field is uniform and constant, plasmas are cold, the length of the straight field line is L, and E y (z, t) = 0 at the end of lines (z = ± L2 ). Based on Eq. (1.7), the Alfvén wave equation is ∂2 ∂2 (1.13) ( 2 − V A2 2 )E y = 0 ∂t ∂z Under the boundary condition, the general solutions are ∞ (Cn sin E x (z, t) = n=1

nπ nπV A nπV A t + Dn cos t)cos z(n = 1, 3, 5...) L L L

(1.14)

∞ E y (z, t) = n=1 (Cn sin

nπ nπV A nπV A t + Dn cos t)sin z(n = 2, 4, 6...) L L L

(1.15)

Equation (1.14) is the solution for odd harmonic mode waves with electric field symmetrical about z = 0 (magnetic equator); Eq. (1.15) is the solution for even harmonic mode waves with electric field asymmetrical about z = 0. The eigenfrequencies of A the standing waves can be described as f n = nV (n = 1, 2, 3...). 2L 2. Standing Alfvén Waves in a Dipole Field Due to the curvilinear magnetic field line, the electric field oscillations are different in the radial and azimuthal directions, expressed as E = E ν eν + E  e . E ν and E  are the toroidal and poloidal mode electric fields, respectively. In Fig. 1.7, the unit vector eν is in the direction normal to the magnetic field line, e is in the

1.2 ULF Waves and Their Interaction with Particles

9

azimuthal direction, and eμ is parallel to the magnetic field line. The coordinates are: ν = (sin 2 θ)/r , which is constant along a field line; , which is the azimuthal spherical polar coordinate; μ = (cosθ)/r 2 , which is constant in the radial direction. In this coordinate system, Eq. (1.6) can be written as [

h 2 ∂ 2 ∂2 ∂ 1 ∂ ∂ 2  ( )]

− − = − ν ∂2 ∂μ h 2ν ∂μ ∂ν∂ V A2 ∂t 2

(1.16)

[

h 2ν ∂ 2 ∂2 ∂ 1 ∂ ∂ 2  ( 2 )]  = − − 2− 2 ∂t 2 ∂ν ∂μ h  ∂μ ∂ν∂ν VA

(1.17)

where h ν , h μ and h  are metrical coefficients; h ν = [ν(1 + 3cos 2 θ)]−1/2 r 3/2 , h  = ν 1/2 r 3/2 h μ = h ν h  , ν = h ν E ν and  = h  E  . When ∂/∂ ≡ 0, the azimuthal wave number m ≡ 0. Then Eqs. (1.16) and (1.17) are decoupled and degenerated into ∂ 1 ∂ h2 ∂2 ( )] ν = 0 (1.18) [ 2 2 − ∂μ h 2ν ∂μ V A ∂t [

h 2ν ∂ 2 ∂2 ∂ 1 ∂ ( )]  = 0 − − 2 ∂t 2 2 ∂ν ∂μ h 2 ∂μ VA

(1.19)

Equations (1.18) and (1.19) are the wave equations for toroidal and poloidal modes, respectively. According to the frozen-in condition (Eq. 1.1), the magnetic field oscillations of the toroidal mode are in the azimuthal direction, which are the oscillations of the whole magnetic field shells; the magnetic field oscillations of the poloidal mode is in the radial direction, which resembles the “breath” of the field line in the meridian planes, which are shown in Fig. 1.6. Another extreme is that m → ∞ and the poloidal mode is dominant with  ν [18, 77]. Equation (1.16) will be in balance if ∂ ν ∂   ∂ν ∂

(1.20)

h 2ν ∂ 2 ∂ 1 ∂ ( 2 )]  = 0 − 2 ∂t 2 ∂μ VA h  ∂μ

(1.21)

Hence, Eq. (1.17) is rewritten as [

Then Eqs. (1.18) and (1.21) become [

ω ∂ 1 ∂ ( ) + h 2 ( )2 ] ν = 0 ∂μ h 2ν ∂μ VA

(1.22)

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1 Background and Motivation

Fig. 1.6 Schematics of standing poloidal and toroidal mode Alfvén waves: a Fundamental mode; b Second harmonic mode

[

∂ 1 ∂ ω ( 2 ) + h 2ν ( )2 ]  = 0 ∂μ h  ∂μ VA

(1.23)

Once the plasma density model is given, we can obtain the eigenfrequencies of the magnetic field lines by solving these equations [11]. Due to the difference between h ν and h μ , the toroidal and poloidal modes in the same harmonic mode have different eigenfrequencies. Rankin et al. [78] proposed a general formalism to describe the standing Alfvén waves in a general magnetospheric model, which makes it possible to calculate the eigenfrequencies of the standing Alfvén waves in arbitrary magnetic field topology. ULF waves can be decomposed into different modes including compressional, poloidal and toroidal modes. The features of these modes are listed in Table 1.1. In order to study the properties of these modes with spacecraft observations, the magnetic and electric field measurements need to be projected into a local meanfield-aligned (MFA) coordinate system [82, 94, 116], which is shown in Fig. 1.8. The unit vector e p is in the parallel direction of the background magnetic field, which is obtained through a running average in several wave cycles; the unit vector in the azimuthal direction is e = e p × rs , where rs is the radius vector of the spacecraft; the unit vector in the radial direction is er = e × e p . 3. The phase relationship between electric and magnetic field components of standing Alfvén waves After Eqs. (1.14) and (1.15) are put into Eqs. (1.1) and (1.3), we can get the relation formula of magnetic field perturbations and plasma velocity or velocity of the oscillating field line. The plasma velocity and electric field perturbations are in-phase,

1.2 ULF Waves and Their Interaction with Particles

11

Table 1.1 Properties of different modes of ULF waves Mode Magnetic field Electric field Compressional Poloidal Toroidal

Bp Br B

E E Er

Azimuthal wave number (m) large∼10 s large∼10 s small∼1 s

Fig. 1.7 An illustration of the orthogonal dipole coordinate system. (From Cummings et al. [11]) Fig. 1.8 An illustration of the mean-field-aligned coordinate system

er

ep

rs

ea

B and there is ±90◦ phase difference between magnetic and electric field perturbations. The left part in Fig. 1.9 shows the largest displacement of the field line at t = 4/T in the er -e p plane; the right part shows the time series of magnetic and electric field oscillations in the northern hemisphere (Mlat > 0◦ ). For the fundamental mode, E lags Br by 90◦ ; for the second harmonic mode, E leads Br by 90◦ . In the southern

12

1 Background and Motivation Field Dsiplacement at t=T/4 Ionosphere

Magnetic and Electric Field Perturbation Phase Relations at Satellite

ep

Br

ea Satellite Equator

er

T/4

Ea T

T/2 3T/4

Ionosphere

Fundamental Mode Ionosphere

ep ea Equator

Br

Ea

Satellite

er

T/4

T

T/2 3T/4

Ionosphere

Second Harmonic Mode Fig. 1.9 Phase differences between magnetic and electric field components for fundamental and second harmonic modes. (From Singer et al. [87]) Table 1.2 Phase difference between magnetic and electric fields for different harmonic mode, + and − signs represent that E leads and lags Br by 90◦ , respectively Northern Southern hemisphere(Mlat>0◦ ) hemisphere(Mlat xr to x < xr , oscillating fast magnetosonic waves turn into decay. But, if k y = 0, Eqs. (1.27) and (1.29) are still coupling, and can be simplified into d2 Ey d Ey + (K (x)2 − k 2y − k z2 )E y = 0 −C dx2 dx

(1.30)

k 2y d K (x)2 . In this equation, there are two singularities (K (x)2 −k z2 )(K (x)2 −k 2y −k z2 ) d x (xr and xc ), which satisfy K (xr )2 − k 2y − k z2 = 0 and K (xc )2 − k z2 = 0, respectively.

where C =

In the damping region, there is a special point at x = xc , named as coupling point. At this point, ω 2 = V A2 k z2 and the frequency of shear Alfvén waves is equal to the frequency of fast magnetosonic waves, indicating the possible resonance of these two modes. This is the so-called field line resonance [97]. Around x = xc , the amplitude of poloidal mode (Ex ) is given by Ex =

−E 0 [ + i(x − xc )] k y [(x − xc )2 + 2 ]

(1.31)

1.2 ULF Waves and Their Interaction with Particles

17

Fig. 1.12 Modeled spin modulation related to finite larmour effect using Eq. (1.26) for different ρL λ⊥ . (From Min et al. [65])

where E 0 depends on E y and depends k z (see more details in Fenrich and Samson [26]). Figure 1.14 shows the amplitude and phase shift of E x around x = xc ; there is the largest oscillation of the shear Alfvén wave electric field at the coupling point.

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1 Background and Motivation

Fig. 1.13 A schematic of field line resonance. a Alfvén velocity as a function of x; b Wave vector as a function of x. xr indicates the demarcation point where the oscillating fast magnetosonic wave turns into decay; xc is a point where the fast magnetosonic wave is coupled to the shear Alfvén wave

Fig. 1.14 The amplitude (solid line) and phase (dashed line) around the field line resonant point (x = xc ). (From Fenrich and Samson [26])

All the aforementioned contents depend on the ideal MHD theory. When considering the two-fluid MHD theory, E y can be expressed as [106] A4

d4 Ey d3 Ey d2 Ey d Ey =0 + A + A + A1 3 2 dx4 dx3 dx2 dx

(1.32)

Bellan [7] suggested that the coefficient A4 is not zero at x = xc and xc is not a singularity, which means that field line resonance can not occur at this point. But

1.2 ULF Waves and Their Interaction with Particles

19

[106] pointed out that although Eq. (1.32) is true in math, the two-fluid MHD theory is meaningless to ULF waves in reality. Because two-fluid effects become important after 2000 cycles, but ULF waves typically live for several to tens of cycles. Therefore, the coefficients A3 and A4 in Eq. (1.32) is zero, and the field line resonance truly exists.

1.2.2 Wave-Particle Interactions The General Theory

 In Hamiltonian dynamics, the action of pdq is the integral over a period of motion in q, where p and q are conjugate canonical variables. If the particle trajectory changes slowly in the time scale of the periodic motion, the integral will be almost constant, which is known as adiabatic invariant [6]. Unlike total energy or momentum, invariants are not absolutely constant, but change in space and time. The number of adiabatic invariants depends on the periodicities that characterize the motions. There are tree kinds of motions for the charged particles trapped in the magnetosphere: gyration around the field line, bounce motion along the field line, and drift motion across the field lines in the azimuthal direction. Corresponding to these particle motions, there are three adiabatic invariants: 1. the magnetic moment, μ = WB⊥ , where W⊥ and B are particle’s energy in the perpendicular direction and total magnetic field, respectively; 2. the longitudinal invariant, J = mv ds, where v and ds are the parallel particle velocity and  an line element of the guiding center path, respectively;  where d S is the surface element. 3. the drift invariant,  = B · d S, The essence of energy transfer between waves and particles is to break particle’s adiabatic invariants. For ULF wave-particle interactions, it requires that the characteristic periods of particle’s motions are comparable with the ULF wave period. Figure 1.15 presents an overview of ULF wave-particle interaction regions for electrons and ions. The overlapped regions reveals the possibility of drift resonance with energetic electrons and both drift and drift-bounce resonance with energetic ions. In addition, the bounce period of ∼eV electrons is also comparable with the ULF wave period, which is shown in Table 1.3. Therefore, besides radiation belt electrons and ring current ions, cold plasmaspheric electrons also can interact with ULF waves through drift-bounce resonance. The energy variations of a particle trapped in the magnetosphere can be expressed as [71] ∂ B dW = q E⊥ · vd + q E v + μ (1.33) dt ∂t where E⊥ is the perpendicular electric field, vd is particle’s drift velocity, E is the parallel electric field, v is particle’s parallel velocity, and B is the compressional magnetic field. The first term on the right hand indicates the interactions between ULF wave electric fields and particles, where the drift motion consists of magnetic field curvature and gradient drift, E × B drift and polarization drift; the second

20

1 Background and Motivation

Fig. 1.15 An overview of the ULF wave-particle interaction regions for energetic electrons (left panel) and ions (right panel). This figure shows the power flux density of various magnetospheric signals in different frequency ranges (left Y-axis), and the gyration, bounce and drift frequencies at different L shells (right Y-axis). Their overlapped regions indicate that drift/drift-bounce resonance is possible. (From Zong et al. [119]) Table 1.3 Time scales of gyration, bounce and drift motions for different particles in the inner magnetosphere. L = 5 and αeq = 60◦ , where αeq is the equatorial pitch angle e− 100

keV eV H+ 10 keV O+ 10 keV e− 10

Gyration

Bounce

Drift (mHz)

6.9 kHz 6.9 kHz 3.7 Hz 0.2 Hz

1.6 Hz 18.0 mHz 13.5 mHz 3.4 mHz

1.8 ×10−1 1.8 ×10−5 6.4 ×10−3 6.4 ×10−3

term is the effect related to parallel electric field, which is usually neglected in a collisionless plasma; the third term is the compressional effect of the magnetic field. Next we will focus on the first term; the effect of the third term can refer to Kivelson and Southwood [51, 52], Liu et al. [55]. Interactions between Standing Alfvén Waves and Particles If there is only interaction between standing Alfvén waves and particles, Eq. (1.33) can be simplified into dW A = q E⊥ · v d (1.34) dt where subscript A indicates the averaged energy change over many gyrations. The accelerating process represented by Eq. (1.34) is much slower than the gyration acceleration. Therefore, the amount of acceleration is independent of gyrophase [91].

1.2 ULF Waves and Their Interaction with Particles

21

1. Interactions between Poloidal Mode ULF Waves and Particles In the equatorial plane, the poloidal mode electric field can be expressed as E = E  ei(m−ωt) e

(1.35)

In a dipole field, particle’s drift velocity is given by vd = L R E ωd e

(1.36)

L P(α) , P(α) = 0.35 + 0.15sinα [37]. For 90◦ particles, Eq. (1.36) where ωd = − 6W q B E R 2E can be rewritten as 3L 2 W vd = − e (1.37) q BE R E

With Eqs. (1.35) and (1.37), the averaged energy change rate is 3L 2 W dW A =− E  ei(m−ωt) dt BE R E

(1.38)

The unperturbed orbit of particles is given by  = ωd t + 0 , where 0 is the initial longitude. We can obtain δW A by integrating back in time to t → −∞. In reality, the interacting time is finite; therefore, a growth index (γ) can be introduced into the electric field amplitude. When t → −∞, E  eγt → 0. So the integral of Eq. (1.38) is 3L 2 W E Phi ei(m−ωt) (1.39) δW A = −i B E R E ω − mωd When particle’s drift velocity equals to the wave velocity in the azimuthal direction, drift resonance will occur, described as ω − mωd = 0

(1.40)

In Fig. 1.16b and d, the odd modes (e.g. fundamental mode, third harmonic mode, etc) have a electric field symmetry around the equator and an anti-node at the equator. Therefore, drift resonance only occurs between odd modes and particles. Next we proceed to the case particle’s equatorial pitch angle is not 90◦ . The  sthat ωb ds bounce phase is given by δ = eq v . E  is a function of position (s) along the field line. Then Eq. (1.34) becomes dW A = qvd (s)E  (s)ei(m−ωt) dt This equation can be expressed in a form of fourier integral

(1.41)

22

1 Background and Motivation

(a)

Electron drift

(c)

(b)

Ion drift

(d)

Fig. 1.16 A schematic of behaviours of drift/drift-bounce resonant particles in the wave frame of different harmonic mode electric fields. a, b Fundamental mode; c, d Second harmonic mode. Plus and minus signs correspond to westward and eastward electric fields of the poloidal mode, respectively. The red dashed lines indicate the guiding center trajectories of drift/drift-bounce resonant particles; the blue line indicates that particles can not satisfy any resonant condition. (Adapted from Zong et al. [121])

∞  dW A  = W˙ N ei N θ ei(m d −ωt) dt N =−∞

(1.42)

d is the averaged drift frequency and W˙ N are the fourier coefficients. where  For odd mode, Eq. (1.42) can be rewritten as ∞

 dW A ωd −ω)t = W˙2l cos(2l)θei(m dt l=0

(1.43)

Integrating along the unperturbed trajectory backward to t → −∞, it becomes δW A =

∞  l=0

ωd )cos2lωb t − 2lωb sin2lωb t ωd −ω)t i(ω − m · W˙2l ei(m (ω − m ω )2 − (2lωb )2

(1.44)

1.2 ULF Waves and Their Interaction with Particles

23

For even mode, Eq. (1.42) can be rewritten as ∞

 dW A ωd −ω)t = W˙ 2l+1 sin((2l + 1)θ)ei(m dt l=0

(1.45)

Under the same rule as above, the integration yields

δW A =

∞ 

i(ω − m ωd )sin(2l + 1)ωb t − (2l + 1)ωb sin2lωb t ˙ ex pi(m W2l+1 ωd − ω)t · (ω − m ω )2 − (2l + 1)2 ωb2 l=0

(1.46)

The denominators in Eq. (1.44) and (1.46) show the drift-bounce resonance, expressed as ω − mωd = N ωb (1.47) where m is the azimuthal wave number, N is an integer representing the number of wavelengths through which particles pass in one bounce period. For odd mode, N = 0, ±2, ...; for even mode, N = ±1, ±3, .... Figure 1.16a and c illustrate the behaviors of drift-bounce resonant particles in the fundamental mode under the resonant condition of N=2 and second harmonic mode under the resonant condition of N=1, respectively. 2. Interactions between Toroidal Mode and Particles The toroidal mode electric field oscillations are in the radial direction, but there is no radial motion of a particle in the dipole field. So the toroidal mode has no effect on these particles. However, under the impact of solar wind, geomagnetic field will show a noon-midnight asymmetry in Fig. 1.17, given by B(L , ) =

B0 + b1 (1 + b2 cos) L3

(1.48)

where the first term on the right hand represents the strength (B0 ) of the dipole field; the second term represents the compressional effect, and the coefficients b1 and b2 are determined from observations;  is the azimuthal angle, which is zero at the local noon and increasing in a counter-clockwise sense. For 90◦ particles, the drift . For toroidal mode, the drift velocity in the radial direction is given by vr = b1qbB2 sinW 2LR E resonance is ω − (m ± 1)ωd = 0; the drift-bounce resonance is ω − (m ± 1)ωd = N ωd [22, 23]. 3. Extensions of Drift Resonant Theory Equation (1.39) is the relation formula between particle’s energy variations (δW A ) and the poloidal mode electric fields (E ). Their phase relationship can be expressed as [114] ωr − mωd (1.49)  = ar g(δW A /E  ) = 180◦ + ar ctan ωi

24

1 Background and Motivation

Fig. 1.17 A sketch of the drift trajectory of an electron in a compressed field. The azimuthal wave number of the toroidal mode electric field is 2. (From Elkington et al. [23])

where ωr and ωi are the real and imaginary parts of wave frequency, respectively. In drift resonance, ωr = mωd and  = 180◦ . In Fig. 1.18a, ωr ωi ; when particle’s d is in the range from +∞ to −∞, so  energy is changing from 0 to +∞, ωr −mω ωi ◦ ∼ 180 , which is consistent with the conventional theory proposed by Southwood and Kivelson [91]. In Fig. 1.18b, ωr ∼ ωi ; when particle’s energy is changing from d is in the range from 1 to −∞, so  < 180◦ . Overall, the total 0 to +∞, ωr −mω ωi d . Since the phase shift across energies from 0 to +∞ equals 180◦ + ar ctan ωr −mω ωi ◦ energy range of an instrument is finite,  should be smaller than 180 . There is a basic assumption in the conventional drift theory that the real and imaginary parts of wave frequency should be constant. But in reality, the imaginary frequency will decrease from positive in the growth stage to negative in the decay stage. So it is important to introduce a time-dependent imaginary part to accommodate the growth and decay of ULF waves. For simplicity, the imaginary part is assumed to be linearly decreasing with time [113]. Then the electric field in Eq. (1.35) can be expressed as E = Ee−t

2

/τ 2 i(m−ωr t)

e

e

(1.50)

The wave amplitude grows until t = 0; after that, it starts to decay. Integrate the particle’s drift trajectory backward in time to obtain its energy variation, given by √ δW A = −

π 3L 2 W E  k(τ )g(t, τ )ei(m−ωt) 2 BE R E

(1.51)

where k(τ ) = τ e−(mωd −ωr ) τ /4 and g(t, τ ) = er f ( τt − i mωd τ2−ωr τ ) + 1. The resonant condition is still ωr = mωd . At the resonant energy, k(τ ) reaches its maximum and the 2 2

1.2 ULF Waves and Their Interaction with Particles

25

Fig. 1.18 The amplitude and phase relationship between δW A and E in a complex plane when a ωr ωi (quasi-steady waves), b ωr ∼ ωi (rapidly damping waves). (From Zong et al. [121])

complex function g(t, τ ) degenerates into a real function er f (t/τ ) + 1. Figure 1.19 shows the behaviours of electrons interacting with ULF waves in the growth and decay stages, which includes theoretical calculations (Fig. 1.19a) and Van Allen Probe B observations (Fig. 1.19b). The resonant energy is about 250 keV marked by the horizontal line. In the growth stage, the amplitudes of E  and δW A gradually increase and their phase difference across the resonant energy increases from a small value to 180◦ at t = 0 when waves stop growing. In the decay stage, the amplitude of δW A and the phase shift still increase until phase mixing attenuates the phase space density perturbations. Energy Transfer between ULF Waves and Particles The above has illustrated the resonant theory between ULF waves and a single particle. But in reality, it is impossible for a spacecraft to detect the behaviors of a sing particle. But it can detect the particle count, which can be further transferred into

26

1 Background and Motivation

Fig. 1.19 The response of energetic electrons to ULF waves in the growth and decay stages. a Theoretical predictions, b Van Allen Probe B observations. (From Zong et al. [113])

phase space density, flux, energy flux, etc. The phase space density (f) is a function of magnetic moment (μ), spatial position (L) and particle’s energy (W). In the driftbounce resonance, μ is constant, and particles diffuse from regions with high density along the curves to regions with low density. If particles diffuse towards higher energies, they will be accelerated; but if particles diffuse towards lower energies, they df [92]. will be decelerated. So energy gain or loss depend on the sign of dW

1.2 ULF Waves and Their Interaction with Particles

∂f df dL ∂ f = |μ,L + |μ,W dW ∂W dW ∂ L

27

(1.52)

df The sign of dW can be determined from spacecraft observations [65, 102]. When df df > 0, energy will be transferred from particles to waves; when dW < 0, energy dW flow will be in the opposite.

1.3 Research Contents More and more spacecraft observations have demonstrated that ULF waves play an important role not only in the dynamics of radiation belt electrons including acceleration [28, 60, 116, 117], loss [64, 70, 98], and transportation [21, 25, 75], but also in the dynamics of ring current O+ ions via drift-bounce resonance [108, 120]. However, there are still lots of significant issues to address, which include: 1. How could ULF waves affect the dynamics of plasmasphere? Can the spatial distribution of ULF waves be affected by plasmaspheric density structures like plume? 2. What are the similarities and differences of different ring current ions simultaneously interacting with ULF waves? 3. What is the phase relationship between ULF waves and drift-bounce resonant particles? And what are the physical meanings of their phase relationship? 4. What are the features of substorm-related ULF waves and their interaction with particles from different origins in the magnetosphere? The objective of this work is to study and address these issues using observations mainly from Van Allen Probes and Cluster, and to shed new light on the understanding of ULF waves’ role in the solar wind-magnetosphere coupling processes.

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94. Takahashi K, McEntire R, Lui A, Potemra T (1990) Ion flux oscillations associated with a radially polarized transverse Pc 5 magnetic pulsation. J Geophys Res 95(A4):3717–3731. https://doi.org/10.1029/JA095iA04p03717 95. Takahashi K, Ohtani S, Anderson BJ (1995) Statistical analysis of Pi 2 pulsations observed by the AMPTE CCE spacecraft in the inner magnetosphere. J Geophys Res 100(A11):21,929C21,941. https://doi.org/10.1029/95JA01849 96. Thorne RM, Horne RB (1997) Modulation of electromagnetic ion cyclotron instability due to interaction with ring current O+ during magnetic storms. J Geophys Res 102(A7):14155– 14163. https://doi.org/10.1029/96JA04019 97. Tsutomu T (1965) Transmission and coupling resonance of hydromagnetic disturbances in the non-uniform Earth’s magnetosphere. Sci Rep Tohoku Univ Ser 5 Geophys 17(2):43–72 98. Turner DL, Shprits Y, Hartinger M, Angelopoulos V (2012) Explaining sudden losses of outer radiation belt electrons during geomagnetic storms. Nat Phys 8(3):208–212. https://doi.org/ 10.1038/nphys2185 99. Van Allen JA, Ludwig GH, Ray EC, McIlwain C (1958) Observation of high intensity radiation by satellites 1958 Alpha and Gamma. Department of Physics, State University of Iowa 100. Walsh B, Foster J, Erickson P, Sibeck D (2014) Simultaneous ground-and space-based observations of the plasmaspheric plume and reconnection. Science 343(6175):1122–1125 101. Walsh B, Phan T, Sibeck D, Souza V (2014) The plasmaspheric plume and magnetopause reconnection. Geophys Res Lett 41(2):223–228. https://doi.org/10.1002/2013GL058802 102. Wei C, Dai L, Duan S, Wang C, Wang Y (2019) Multiple satellites observation evidence: High-m Poloidal ULF waves with time-varying polarization states. Earth Planet Phys 3:1–14. https://doi.org/10.26464/epp2019021 103. Williams D (1981) Ring current composition and sources: an update. Planet Space Sci 29(11):1195–1203 104. Williams D (1983) The Earth’s ring current: causes, generation, and decay. Space Sci Rev 34(3):223–234 105. Williams D (1985) Dynamics of the Earth’s ring current: theory and observation. Space Sci Rev 42(3–4):375–396. https://doi.org/10.1007/BF00214994 106. Wright AN, Allan W (1996) Are two-fluid effects relevant to ULF pulsations? J Geophys Res 101(A11):24991–24996. https://doi.org/10.1029/96JA01947 107. Wygant J et al (2000) Polar spacecraft based comparisons of intense electric fields and Poynting flux near and within the plasma sheet-tail lobe boundary to UVI images: an energy source for the aurora. J Geophys Res 105(A8):18675–18692. https://doi.org/10.1029/1999JA900500 108. Yang B, Zong Q-G, Fu SY, Li X, Korth A, Fu HS, Yue C, Reme H (2011) The role of ULF waves interacting with oxygen ions at the outer ring current during storm times. J Geophys Res 116. https://doi.org/10.1029/2010JA015683 109. Yang B, Zong Q-G, Wang Y, Fu S, Song P, Fu H, Korth A, Tian T, Reme H (2010) Cluster observations of simultaneous resonant interactions of ULF waves with energetic electrons and thermal ion species in the inner magnetosphere. J Geophys Res 115(A2). https://doi.org/ 10.1029/2009JA014542 110. Yeoman TK, Wright D, Baddeley L (2006) Ionospheric signatures of ULF waves: active radar techniques. In: Synthesis and new directions, magnetospheric ULF waves, pp 273–288 111. Yumoto K, Saito T, Tsurutani BT, Smith EJ, Akasofu S-I (1984) Relationship between the IMF magnitude and Pc 3 magnetic pulsations in the magnetosphere. J Geophys Res 89(A11):9731– 9740. https://doi.org/10.1029/JA089iA11p09731 112. Zhao L, Zhang H, Zong Q (2017) Global ULF waves generated by a hot flow anomaly. Geophys Res Lett 44(11):5283C5291. https://doi.org/10.1002/2017GL073249 113. Zhou X-Z, Wang Z-H, Zong Q-G, Rankin R, Kivelson MG, Chen X-R, Blake JB, Wygant JR, Kletzing CA (2016) Charged particle behavior in the growth and damping stages of ultralow frequency waves: theory and Van Allen Probes observations. J Geophys Res. https://doi.org/ 10.1002/2016JA022447 114. Zhou X-Z et al (2015) Imprints of impulse-excited hydromagnetic waves on electrons in the Van Allen radiation belts. Geophys Res Lett 42(15):6199–6204. https://doi.org/10.1002/ 2015GL064988

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115. Zhu X, Kivelson MG (1989) Global mode ULF pulsations in a magnetosphere with a nonmonotonic Alfvén velocity profile. J Geophys Res 94(A2):1479–1485 116. Zong Q-G et al (2007) Ultralow frequency modulation of energetic particles in the dayside magnetosphere. Geophys Res Lett 34(12):105–+. https://doi.org/10.1029/2007GL029915 117. Zong Q-G et al (2009) Energetic electrons response to ULF waves induced by interplanetary shocks in the outer radiation belt. J Geophys Res 114(A10):204. https://doi.org/10.1029/ 2009JA014,393 118. Zong QG, Wilken B, Fu SY, Fritz T, Pu ZY, Hasebe N, Williams DJ (2001) Ring current oxygen ions in the magnetosheath caused by magnetic storm. J Geophys Res 106:25–541. https://doi.org/10.1029/2000JA000127 119. Zong Q-G, Wang YF, Yang B, Fu SY, Pu ZY, Xie L, Fritz TA (2008) Recent progress on ULF wave and its interactions with energetic particles in the inner magnetosphere. Sci China Ser E Technol Sci 51(10):1620–1625. https://doi.org/10.1007/s11431-008-0253-z 120. Zong Q-G, Wang YF, Zhang H, Fu SY, Zhang H, Wang CR, Yuan CJ, Vogiatzis I (2012) Fast acceleration of inner magnetospheric hydrogen and oxygen ions by shock induced ULF waves. J Geophys Res 117(A11):206. https://doi.org/10.1029/2012JA018,024 121. Zong Q, Rankin R, Zhou X (2017) The interaction of ultra-low-frequency pc3-5 waves with charged particles in Earths magnetosphere. Rev Mod Plasma Phys 1(1):10

Chapter 2

Data Sources

Since 1980s, several important science programs were proposed to address the fundamental physical processes of the sun, its interaction with Earth and the solar system, which include International Solar-Terrestrial Physics (ISTP) Program, Living With a Star (LWS) Program, and Solar Terrestrial Probes (STP) Program. ISTP is an international cooperation program led by NASA, ESA as well as ISAS, and its primary missions include ACE, Wind, SOHO, Geotail, Cluster, GOES, LANL, etc. LWS and STP are two strategic programs of NASA in the new century. The primary missions in the LWS Program are SDO, Van Allen Probes, BARREL, Solar Probe Plus, Solar Orbiter, etc. And the primary missions in the STP Program are TIMED, MMS, Solar-B, STEREO, etc. Two main data sources of this thesis are from Cluster and Van Allen Probes. The Cluster mission with four identically instrumented spacecraft was originally placed in high inclination orbits with a perigee at ∼4R E , an apogee at ∼19.6 R E , and a period of about 57 h [1], which is shown in Fig. 2.1. Each satellite carries a set of 11 instruments including FGM, EFW, CIS, RAPID, PEACE, ASPOC, WHISPER, EDI, DWP, STAFF and WBD. CIS instrument includes a Hot Ion Analyzer (HIA) and a time-of-flight ion COmposition and DIstribution Function analyzer (CODIF) [2]. HIA instrument provides the three-dimensional distributions of ions in the energy range of 5 eV/e–32 keV/e without distinguishing ion species. CODIF instrument is capable of measuring the three-dimensional distributions of major ions (H+ , O+ , He+ and He++ ) in the energy range of 25 eV/e–40 keV/e. RAPID provides the measurements of energetic electrons in the range of 20–400 keV. The magnetic and electric fields are measured by the Fluxgate Magnetometer (FGM) [3] and the Electric Field andWave experiment (EFW) [4], respectively. The Van Allen Probes mission consists of two spacecraft (Probe A and B) in low inclination orbits with a perigee at ∼1.1 R E , an apogee at ∼5.8 R E , and a orbital period of about 9 h [6], which is shown in Fig. 2.2. The spectral measurements of Helium, Oxygen, Proton and Electron (HOPE) mass spectrometer cover an energy © Springer Nature Singapore Pte Ltd. 2019 J. Ren, ULF Waves’ Interaction with Cold and Thermal Particles in the Inner Magnetosphere, Springer Theses, https://doi.org/10.1007/978-981-32-9378-6_2

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Fig. 2.2 Van Allen Probes orbit. (From https://www.nasa.gov/)

range of ∼15 eV–50 keV for electrons and ∼eV–50 keV for ions (H+ , He+ and O+ ) [7]. Magnetic and electric fields are obtained from the Electric andMagnetic Field Instrument and Integrated Science (EMFISIS) [8] and the Electric Field and Waves (EFW) [9], respectively. In addition, a list of satellites provide enormous supplementary observations, which include LANL, GOES, MMS, etc. The groundbased magnetic field measurements are provided by OMNI data (ftp://spdf.gsfc.nasa. gov/pub/data/omni/omni_cdaweb/).

References 1. Escoubet CP, Fehringer M, Goldstein M (2001) Introduction the cluster mission. Ann Geophys 19(10/12):1197–1200 2. Rème H et al (2001) First multispacecraft ion measurements in and near the Earth’s magnetosphere with the identical Cluster ion spectrometry (CIS) experiment. Ann Geophys 19:1303– 1354 3. Balogh A et al (1997) The cluster magnetic field investigation. Space Sci Rev 79:65–91 4. Gustafsson G et al (2001) First results of electric field and density observations by Cluster EFW based on initial months of operation. Ann Geophys 19(10/12):1219–1240

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5. Ren J, Zong Q, Zhou X, Zhang H, Fu S, Wang Y, Liu YC-M (2016b) Statistics of the fieldaligned currents at the high-latitude energetic electron boundaries in the nightside: Cluster observation. J Geophys Res 121(3):1979–1989. https://doi.org/10.1002/2015JA021754 6. Mauk B, Fox NJ, Kanekal S, Kessel R, Sibeck D, Ukhorskiy A (2013) Science objectives and rationale for the Radiation Belt Storm Probes Mission. Space Sci Rev 179(1–4):3–27. https:// doi.org/10.1007/s11214-012-9908-y 7. Funsten H et al (2013) Helium, Oxygen, Proton, and Electron (HOPE) mass spectrometer for the radiation belt storm probes mission. Space Sci Rev 179(1–4):423–484. https://doi.org/10. 1007/s11,214-013-9968-7 8. Kletzing C et al (2013) The electric and magnetic field instrument suite and integrated science (EMFISIS) on RBSP. Space Sci Rev 179:127–181. https://doi.org/10.1007/s11,214013-9993-6 9. Wygant J et al (2000) Polar spacecraft based comparisons of intense electric fields and Poynting flux near and within the plasma sheet-tail lobe boundary to UVI images: an energy source for the aurora. J Geophys Res 105(A8):18675–18692. https://doi.org/10.1029/1999JA900500

Chapter 3

Interactions Between ULF Waves and Cold Plasmaspheric Particles

3.1 Introduction In terms of particle energy and species, the inner magnetosphere can be divided into three regions including radiation belts, ring current, and plasmasphere, which are overlapped in space. In theory, radiation belt energetic electrons, ring current ions and cold plasmaspheric electrons can be accelerated by ULF waves via drift/driftbounce resonance, which is shown in Fig. 3.1. A mass of spacecraft observations have demonstrated that ULF waves can cause the rapid generation of “killer” electrons that can severely threaten the satellites, and the acceleration of ring current ions (e.g. [10, 19, 21, 24, 31, 34]). But it has not been extensively investigated what is the role of ULF waves in the dynamics of cold plasmaspheric populations. The plasmasphere as the upward extension of Earth’s ionosphere is filled with dense populations of cold ions and electrons, and highly dynamics during geomagnetic activities. According to previous knowledge, the dynamics of plasmasphere is dominantly controlled by the superposition of corotation and convection electric fields. The outer plasmasphere can be diminished and refilled along the increase and decrease of the convection electric field, respectively. Adrian et al. [1] first proposed that the standing ULF waves probably account for the bifurcated density features of He+ ions in the outer plasmasphere. There are many density structures in the plasmasphere such as plume, shoulder, finger, etc. The time scales of these features are varying from minutes to hours [8]. If cold plasmaspheric particles can be affected by ULF waves, it will provide one of the potential explanations for the formation and evolution of the density structures in the plasmasphere. In addition, the spatial and temporal dynamics of cold plasmaspheric particles will influence the excitation criteria for various waves such as EMIC waves, chorus and hiss (e.g. [2, 11, 18]). The objectives of this chapter are: 1. To investigate the acceleration and modification of cold plasmaspheric particles by ULF waves, 2. To explore whether the spatial distribution of ULF waves can be controlled by plasmaspheric density structures (e.g. the drainage plume).

© Springer Nature Singapore Pte Ltd. 2019 J. Ren, ULF Waves’ Interaction with Cold and Thermal Particles in the Inner Magnetosphere, Springer Theses, https://doi.org/10.1007/978-981-32-9378-6_3

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Fig. 3.1 Drift-bounce resonant conditions for a electrons, b H+ and c O+ ions in the inner magnetosphere. (From Zong et al. [37])

3.2 Fast Acceleration of Cold Plasmaspheric Electrons Figure 3.2 shows an overview of Van Allen Probe B observations from 2000 UT on 9 September 2015 to 1600 UT on 10 September 2015. In Fig. 3.2a and b, the magnetic field measurement (Bx ) from EMFISIS instrument presents that there are ULF wave oscillations in the wave period of about 1 min during two time intervals, which are 0100-0400 UT, labelled as Part I and 0900-1300 UT, labelled as Part II. These ULF waves are also observed in the electric field measurement (E y ) from EFW instrument in Fig. 3.2c and d. Besides, the wavelet spectrum of electric fields in Fig. 3.2c also

3.2 Fast Acceleration of Cold Plasmaspheric Electrons

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shows that there are 2-min period ULF waves in Part II, which is not observed in the wavelet spectrum of magnetic fields in Fig. 3.2a. Since Probe B is near the magnetic equator (Mlat∼2-3◦ ) where is the node of the fundamental mode magnetic field oscillations, 2-min and 1-min period ULF waves are probably fundamental and second harmonic modes, respectively, which will be further verified in other ways. Figure 3.2e–g present the HOPE instrument observations of electron energy spectrum and pitch angle distributions for >200 and 200 eV still show the same pancake-like distributions as there are no ULF waves in the two orbits of Probe B. Figure 3.2h–k show the proton energy spectrum and pitch angle distributions in some representative energy channels, respectively. The bi-directional pitch angle distributions appear in the energy channel of 11.2 keV in Part I and II. The SYMH index in Fig. 3.2l indicates that the wave-particle interactions observed by Probe B occurred during the late recovery phase of a geomagnetic storm. There are substorm activities ahead of the appearance of ULF waves in both Part I and Part II, which is indicated by the AL index in Fig. 3.2m. Therefore, the excitation of 1-min period ULF waves should be associated with substorm activities.

3.2.1 The Properties of the 1-Min Period ULF Waves There are three magnetic field components from EMFISIS instrument, but just two electric field components from EFW instrument. To project the electric field measurements into a local mean-field-aligned coordinate system, we can obtain the third component using E · B = 0, which requires that the angle between spin axis and magnetic field should be smaller than 15◦ [6]. This condition can be satisfied only during 0100-0330 UT, which will be used to analyze the properties of the 1-min period ULF waves. In Fig. 3.3a, E (red line) leads Br (black line) by 90◦ in the northern hemisphere (Mlat∼10◦ ), which is further confirmed by the cross wavelet analysis in Fig. 3.3d. The Poynting vector in the parallel direction oscillates around 0, indicating that both poloidal and toroidal modes are standing waves. So the phase relationship between E and Br also infers that the 1-min period ULF waves are second harmonic mode. The upper hybrid waves in Fig. 3.4a are used to calculate the plasma density, which is shown in Fig. 3.4b. During the appearance of 1-min period ULF waves in Part I and II, the plasma density is in the range of 10–30 cm−3 , indicating that Probe B is located at the plasmasphere boundary layer (PBL). The concept of PBL was first proposed by Carpenter and Lemaire [3], which refers to the transient region between dense (>102 cm−3 ) plasmasphere and tenuous (∼1 cm−3 ) plasma sheet. Based on previous studies [17, 20, 34], the region with plasma density larger than 10 cm−3 belongs to the PBL. The peak frequency of ULF waves is closer to the eigenfrequency

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of second harmonic mode calculated according the method in Ren et al. [23]. The inconsistence in Part I may originate from the plasma density model along the field line or the particle composition. Overall, the 1-min period ULF waves are identified as second harmonic mode according to: 1. There are 2-min period ULF waves in Part II, 2. E leads Br by 90◦ in the northern hemisphere, 3. Their peak frequency is consistent with the calculated eigenfrequency of the second harmonic mode.

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Fig. 3.4 a Frequency-time spectrum of electric field spectral density observed by Van Allen Probe B, the upper hybrid wave is marked by black line; b Electron number density inferred from upper hybrid waves; c Calculated eigenfrequency (black line) of second harmonic mode and the observed peak frequency (red line). (From Ren et al. [22] with kind permission from Wiley)

3.2.2 Simultaneous Drift-Bounce Resonance with Cold Electrons and Energetic Protons Figure 3.5a, b and c show Br component, cold electron and energetic proton observations, respectively. In Part I and II, there is a good correlation between the appearance of ULF waves and bi-directional pitch angle distributions for < 200 eV electrons and ∼10 keV protons, which indicates that there occurs drift-bounce resonance between ULF waves and particles. In order to further verify the cold electron acceleration by ULF waves, Fig. 3.6 shows the phase space density spectra observed by Probe B in five consecutive orbits, including two orbits ahead of ULF wave appearance (dark cyan and green lines), two orbits during ULF wave appearance (red and purple lines), and one orbit after ULF wave appearance (blue line). When there are no ULF

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5 4 3

0 180 90

5 4 3

0 180 90

5 4 3

0 180

5

90

4

0 180

3 5

90

4

0 180

3 5

90

4

0

3

Log Flux Log Flux Log Flux Log Flux Log Flux Log Flux Log Flux Log Flux Log Flux

5 4

90

PA (deg)

0 180

PA (deg)

3

9.6keV 13.1keV 17.8keV 24.1keV 32.7keV

4

PA (deg)

2 5

90

7.1keV

0 180

PA (deg)

4

90

5.2keV

2

PA (deg)

0 180

3.8keV

4

90

10000 1000 100 10 1 180

PA (deg)

E (eV)

6 4

Log Flux Log Flux Log Flux Log Flux Log Flux Log Flux Log Flux Log Flux Log Flux

PA (deg)

PA (deg) PA (deg) PA (deg) PA (deg) PA (deg) PA (deg) PA (deg)

9.6keV 13.1keV 17.8keV 24.1keV 32.7keV 7.1keV 5.2keV 3.8keV

8

10000 1000 100 10 1 180

E (eV)

(f)

(e)

Proton

09:00 10:00 11:00 12:00 13:00 14:00 15:00

Log Flux Log Flux Log Flux Log Flux Log Flux Log Flux Log Flux

6 9 8 7 6

PA (deg)

7

90 0 180

PA (deg)

90 0 180

2 1 0

10000 1000 100 10 180

66.7eV 105.5eV 166.9eV 264.2eV 418.0eV

90 0 180

RBSP-B/HOPE

16 32 64 128 256 512

(d)UT08:00

PA (deg)

PA (deg)

PA (deg) PA (deg) PA (deg) PA (deg)

66.7eV 105.5eV 166.9eV 264.2eV 418.0eV E (eV) 42.1eV

PA (deg) PA (deg)

26.6eV 16.8eV

9 8 7 6 5 7. 5 7 6. 5 6 5. 5 7. 5 7 6. 5 6 5. 5 8

42.1eV

06:00

PA (deg)

05:00

26.6eV

04:00

PA (deg)

03:00

16.8eV

02:00

PA (deg)

01:00

Log10nT2/Hz

1

0

PA (deg)

Electron

2

10000 1000 100 10 180

2015/09/10 08:00-15:00 Part II

(b) 3

Log Flux Log Flux

Br Period(s)

UT 00:00

(c)

RBSP-B/HOPE

16 32 64 128 256 512

Log Flux Log Flux Log Flux Log Flux Log Flux Log Flux Log Flux

2015/09/10 00:00-06:00 Part I

(a)

45

UT 08:00 09:00 10:00 11:00 12:00 13:00 14:00 15:00

Fig. 3.5 a, b Wavelet spectrum of Br during 0000 UT-06000 UT and 0800-1500 UT on 10 September 2015, respectively; c, d Pitch angle distributions of cold electrons at some representative energy channels; e, f Pitch angle distributions of energetic protons at some representative energy channels. (From Ren et al. [22] with kind permission from Wiley)

46

3 Interactions Between ULF Waves and Cold Plasmaspheric Particles

2015/09/09 - 09/11 RBSP-B 10

electron

1: 2015/09/09 06:20 - 2015/09/09 10:14

5

2: 2015/09/09 14:56 - 2015/09/09 19:45

10

3: 2015/09/10 00:12 - 2015/09/10 04:24

4

PSD(s3 km-6 )

4: 2015/09/10 09:22 - 2015/09/10 13:24

10

3

10

2

10

1

10

0

10

-1

10

-2

10

-3

5: 2015/09/11 07:00 - 2015/09/11 15:29

0

10

1

10

2

10

3

10

Energy (eV) Fig. 3.6 Phase space density spectra of cold electrons in five consecutive orbits of Probe B including two orbits ahead of the appearance of ULF waves, two orbits during the appearance of ULF waves, and one orbit after the appearance of ULF waves. (Adapted from Ren et al. [22])

wave-particle interactions, the phase space density of cold electrons shows a power law distribution, while it is deviated from the background and enhanced up to 5 times on average during the ULF wave-cold electron interactions. The black and red shadow regions represent the resonant energy ranges inferred from the wave bandwidth under the resonant condition of N = 1, which is consistent with HOPE instrument observations. From Probe B observations, there are bump-on-tail distributions for protons at ∼10 keV and wave-particle interactions indicated by the bi-directional pitch angle distributions. So the 1-min period ULF waves are probably excited by substorm-injected protons, which further energize the cold plasmaspheric electrons through drift-bounce resonance.

3.3 Energization of Cold Plasmaspheric Ions by ULF Waves

47

3.3 Energization of Cold Plasmaspheric Ions by ULF Waves Figure 3.7 shows Cluster observations of ULF waves and their interaction with O+ ions in different energy ranges on 7 November 2004. After the interplanetary shock arrival at about 1827 UT (marked by the vertical dashed line), both poloidal and toroidal mode ULF waves are rapidly excited and last for about half an hour, which are observed by Cluter C3 and C4 in Fig. 3.7a–d. Figure 3.7e–g show the O+ ion flux in the energy ranges of 0.3–2.6 MeV, 1.0–40.0 keV, and