PHYSICS OF PLASMAS
VOLUME 8, NUMBER 11
NOVEMBER 2001
Relativistic bounce-averaged quasilinear diffusion equation
for low-frequency electromagnetic fluctuations
Alain J. Brizard
Department of Chemistry and Physics, Saint Michael’s College, Colchester, Vermont 05439
Anthony A. Chan
Department of Physics and Astronomy, Rice University, Houston, Texas 77005
共Received 14 June 2001; accepted 13 August 2001兲
A relativistic bounce-averaged quasilinear diffusion equation is derived to describe stochastic
particle transport associated with low-frequency electromagnetic fluctuations in a nonuniform
magnetized plasma. Expressions for the relativistic quasilinear diffusion coefficients are calculated
explicitly for magnetically-trapped particle distributions in axisymmetric magnetic geometry in
terms of drift-bounce resonant contributions associated with low-frequency fluctuations which
conserve the first adiabatic invariant. © 2001 American Institute of Physics.
关DOI: 10.1063/1.1408623兴
is equal to harmonics of the bounce frequency ␻ b ), and the
drift-resonance 共in which the wave frequency ␻ is equal to
harmonics of the magnetic-drift frequency ␻ d ).
The derivation of a relativistic quasilinear Vlasov equation describing collisionless diffusion in action space begins
with the standard relativistic Vlasov equation written in
Hamiltonian form as
I. INTRODUCTION
One of the outstanding problems in magnetospheric
physics is the nature of the mechanism共s兲 responsible for the
transport and energization of trapped 共radiation-belt兲 particles during magnetic storms. Relativistic radiation-belt
electrons are of special interest because of their damaging
effects on spacecraft electronics. During magnetic storms,
MeV electron fluxes in Earth’s magnetosphere typically drop
by a few orders of magnitude early in the storm and then rise
to values one to two orders of magnitude above pre-storm
values over a period of a day or two.1 Recent observational
studies have shown strong correlations between these flux
variations and the occurrence of hydromagnetic fluctuations
in the 2 to 10 mHz ULF frequency range.2,3 In related work,
simulation results of Elkington, Hudson, and Chan4 suggest
that the ULF fluctuations may play an important role in the
radial transport and energization of the MeV electrons
through magnetic-drift wave-particle resonances. More generally, it has long been established that trapped magnetospheric particles can exhibit stochastic behavior in the presence of multiple wave-particle resonances5–7 leading to
dissipationless transport.
⳵F
⫹ 兵 F,H 其 ⫽0,
⳵t
where F is the Vlasov distribution function, H is the singleparticle relativistic Hamiltonian, and 兵 , 其 is the Poisson
bracket. The Vlasov equation 共1兲 can be written in terms of
the action-angle coordinates (J i , ␪ i ) 共with i⫽1,2,3), with
which the Poisson bracket has the form
兵 F,H 其 ⬅
⳵F ⳵H ⳵F ⳵H
⫺
•
•
.
⳵␪ ⳵J ⳵J ⳵␪
Next, we expand the Vlasov distribution F and the Hamiltonian H in powers of a small dimensionless parameter ⑀
associated with wave-perturbation amplitude:
F⫽F 0 共 J; ␶ 兲 ⫹ ⑀ ␦ F 共 J, ␪,t 兲 ,
A. General quasilinear theory
H⫽H 0 共 J兲 ⫹ ⑀ ␦ H 共 J, ␪,t 兲 ,
Stochastic transport of charged particles interacting with
low-frequency electromagnetic waves can faithfully be modeled by a quasilinear diffusion equation in which waveparticle resonances are explicitly taken into account. Quasilinear theory was originally developed to describe the
collisionless evolution of an unstable velocity-space particle
distribution in a uniform plasma.8,9 The derivation of a quasilinear kinetic equation for a nonuniform magnetized
plasma in the full phase space using action-angle coordinates
was first performed by Kaufman.10,11 This formulation, as is
shown below, explicitly displays the following three waveparticle resonances: the gyroresonance 共in which the wave
frequency ␻ is equal to harmonics of the cyclotron frequency
␻ c ), the bounce-resonance 共in which the wave frequency ␻
1070-664X/2001/8(11)/4762/10/$18.00
共1兲
共2兲
where F 0 and H 0 are functions of the action variables J, such
that 兵 F 0 ,H 0 其 ⬅0, and F 0 is also a slow function of time ( ␶
⬅ ⑀ 2 t), i.e., ⳵ F 0 / ⳵ t⬅O( ⑀ 2 ). The first-order perturbation
terms ␦ F and ␦ H are associated with the presence of waves
perturbing the background medium whose typical wave time
scales are short compared to the quasilinear evolution time
scale of the background medium.
Upon expanding the Vlasov equation 共1兲 in powers of ⑀
and averaging over the fast-wave time scales, we obtain the
following equations for F 0 and ␦ F:
⳵F0
⫽⫺ 兵 ␦ F, ␦ H 其 ,
⳵␶
4762
共3兲
© 2001 American Institute of Physics
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Phys. Plasmas, Vol. 8, No. 11, November 2001
Relativistic bounce-averaged quasilinear diffusion . . .
⳵␦F
⫹ 兵 ␦ F,H 0 其 ⫽⫺ 兵 F 0 , ␦ H 其 ,
⳵t
共4兲
where (•••) denotes averaging with respect to the fast-angle
variables and time, and we have omitted the nonlinear term
兵 ␦ F, ␦ H 其 in the evolution equation for ␦ F 共as required by
the quasilinear analysis兲. We now consider the Fourier decomposition
冉 冊
␦F
␦H
⫽
兺␬ 兺m
冉
␦ F̃ m共 J; ␻ ␬ 兲
␦ H̃ m共 J; ␻ ␬ 兲
冊
expi 共 m j ␪ j ⫺ ␻ ␬ t 兲 , 共5兲
where m⬅(m 1 ,m 2 ,m 3 ) labels Fourier components 共each index m i takes integer values from ⫺⬁ to ⫹⬁), the sum 兺 ␬
denotes a sum over normal modes with eigenfrequencies
␻ ␬ , and ␦ F̃ ⫺m(⫺ ␻ ␬ )⬅ 关 ␦ F̃ m( ␻ ␬ ) 兴 * as follows from the
reality condition on the perturbation fields. Substituting this
decomposition into the right side of Eq. 共3兲 yields12
⫺ 兵 ␦ F, ␦ H 其 ⫽
⳵
⫽ •
⳵J
⬅
冉 冊
冉兺
冋兺
⳵
⳵␦H
" ␦F
⳵J
⳵␪
⳵
•
⳵J
m, ␬
m, ␬
* 共 J; ␻ ␬ 兲 ␦ H̃ m共 J; ␻ ␬ 兲
im␦ F̃ m
冊
册
Substituting the Fourier decomposition Eq. 共5兲 into Eq. 共4兲,
on the other hand, yields the formal solution for the Fourier
component ␦ F̃ m :
⳵F0
␦ H̃ m共 J; ␻ ␬ 兲 , 共7兲
⳵J
where ⍀ i (J)⬅ ␪˙ i ⫽ ⳵ H 0 / ⳵ J i is the unperturbed orbital frequency associated with the fast orbital angle ␪ i . Finally, substituting this solution into Eq. 共6兲, the evolution equation 共3兲
for F 0 becomes the quasilinear diffusion equation
冉
冊
⳵F0 ⳵
⳵F0
⬅ • DQL•
,
⳵␶
⳵J
⳵J
共8兲
where the components of the quasilinear diffusion tensor are
ij
⬅
D QL
兺 m i m j 关 ␲ ␦ 共 ␻ ␬ ⫺m " ⍀兲 兩 ␦ H̃ m共 J; ␻ ␬ 兲 兩 2 兴 ,
m, ␬
modeled by replacing the small denominator ( ␻ ␬
⫺m"⍀) ⫺1 with ( ␻ ␬ ⫺m"⍀⫹i ␯ ) ⫺1 , where the autocorrelation time ␶ ac⬅1/␯ represents the time scale over which a
resonant particle loses correlation with the waves.13 Hence,
using the prescription Im( ␻ ␬ ⫺m " ⍀⫹i ␯ ) ⫺1 ⬅⫺ ␶ ac for
resonant particles 共i.e., ␻ ␬ ⫽m " ⍀), and the quasilinear diffusion tensor Eq. 共9兲 is replaced with the renormalized quasilinear diffusion 共RQL兲 tensor
ij
D RQL
⬅
冉兺
m, ␬
冊
(res) 2
m i m j 兩 ␦ H̃ m
兩 ␶ ac ,
共10兲
(res)
where ␦ H̃ m
⬅ ␦ H̃ m(J; ␻ ␬ ⫽m " ⍀). Such nonlinear effects
are, however, beyond the scope of the present work and we
postpone further discussion of nonlinear resonancebroadening effects to future work on applications of relativistic bounce-averaged quasilinear diffusion in magnetospheric physics.
B. Orbital time scales for relativistic electrons
* 共 J; ␻ ␬ 兲 ␦ F̃ m共 J; ␻ ␬ 兲兲 . 共6兲
mIm共 ␦ H̃ m
␦ F̃ m共 J; ␻ ␬ 兲 ⬅⫺ 共 ␻ ␬ ⫺m"⍀兲 ⫺1 m "
4763
共9兲
and we have used Im( ␻ ␬ ⫺m•⍀) ⫺1 ⬅⫺ ␲ ␦ ( ␻ ␬ ⫺m•⍀),
which indicates that quasilinear diffusion is produced by
resonant wave-particle interactions. The action-angle formulation of quasilinear transport theory therefore establishes the
following paradigm: each wave-particle resonance introduces an explicit violation of the adiabatic invariance of an
action variable which results in stochastic transport 共e.g.,
bounce-resonant fluctuations destroy violate adiabatic
bounce invariance and set up stochastic transport in bounceaction space兲.
We note that the singular character of the quasilinear
diffusion tensor Eq. 共9兲 disappears when nonlinear effects
such as resonance broadening are taken into account.11,13 In
Eq. 共9兲, for example, resonance broadening effects may be
In the present work, we focus our attention on axisymmetric magnetic geometry represented by the magnetic coordinates ( ␺ , ␸ ,s), where the azimuthal angle ␸ is an ignorable
coordinate whereas the magnetic flux ␺ is a radial-like coordinate and s is the parallel coordinate along a magnetic-field
line labeled by ( ␺ , ␸ ). The background Vlasov distribution
F 0 (J g ,␧, ␺ ) is a function of the relativistic gyro-action J g
⬅ 兩 p⬜ 兩 2 /(2M ␻ g) 共an adiabatic invariant associated with the
asymptotic elimination of the gyroangle ␨ 兲,14 the particle energy ␧ 关instead of the bounce action J b ⬅(1/2␲ ) 养 p 储 ds], and
the radial-like drift action J d ⬅q ␺ /c 共an exact invariant associated with axisymmetry in ␸ ).
The type of hydromagnetic fluctuations considered in
this work are ultra-low-frequency 共ULF兲 fluctuations in the
frequency range 2–10 mHz. For a 1-MeV test electron at
geosynchronous orbit,15 the cyclotron frequency is in the
range 1 – 3 kHz, the bounce frequency is in the range 1 – 3
Hz, and the drift frequency for equatorially-trapped electrons
is in the range of a few mHz. Hence, a 1-MeV trapped electron can easily encounter magnetic-drift resonances with a
ULF wave; hence, ULF waves can lead to radial transport as
a result of the destruction of the third adiabatic invariant.
Because of the time-scale ordering ␻ Ⰶ ␻ c and ␻ c Ⰷ ␻ b
Ⰷ ␻ d , the ULF fluctuations conserve the first adiabatic invariant, while the second and third invariants may be broken
or not.
In low-frequency gyrokinetics,14 the fluctuation time
scale (2 ␲ / ␻ ) is assumed much longer than the fast cyclotron
time scale (2 ␲ / ␻ c ) and could be comparable to the intermediate bounce time scale (2 ␲ / ␻ b ) or the slow drift time scale
(2 ␲ / ␻ d ). Because ␻ Ⰶ ␻ g ⬅ ␥ ⫺1 ␻ c , an asymptotic expression for the relativistic gyroaction J g can be constructed
which is conserved in the presence of low-frequency electromagnetic fluctuations. The condition ␻ ⬃ ␻ b or ␻ d , on the
other hand, implies that the construction of the bounce and
drift invariants beyond their simplest expressions is unnecessary.
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4764
Phys. Plasmas, Vol. 8, No. 11, November 2001
A. J. Brizard and A. A. Chan
C. Relativistic drift-kinetic Vlasov equation
We now review the relativistic gyrokinetic Vlasov equation for low-frequency electromagnetic fluctuations in a nonuniform magnetized plasma with general magnetic geometry
recently derived by Brizard and Chan.14 Because electrons
have negligible Larmor radii compared to the background
scale lengths, we focus our attention on the drift-kinetic
limit. For a particle species with mass M and charge q 共particle species here is arbitrary兲, the relativistic drift-kinetic
Vlasov equation is written in terms of the gyrocenter Vlasov
distribution F(X,p 储 ,t;J g ) and the relativistic gyrocenter
Hamiltonian H(X,p 储 ,t;J g ) as
冉
冊
⳵F
B* ⳵ H
⳵F
cb̂
B*
⫹
⫹
à ⵜH " ⵜF⫺
" ⵜH
⫽0,
⳵t
⳵ p 储 qB *
⳵p储
B*
B*
储
储
共11兲
where B* ⬅ⵜ Ã 关 A⫹(c p 储 /q) b̂兴 , b̂⬅B/B, and B *
储 ⬅b̂ " B*
are quantities derived from the unperturbed background
magnetic field only. In the drift-kinetic limit, the linearized
gyrocenter Hamiltonian H is expressed as
冋冉
H⫽ 共 ␥ ⫺1 兲 M c 2 ⫹ ⑀ q ␦ ␾ ⫺
冊
␦B储
p储
␦ A 储 ⫹J g ␻ g
␥Mc
B
⬅H 0 ⫹ ⑀ ␦ H,
册
共12兲
where ␻ g ⬅ 兩 q 兩 B/M c is the rest-mass gyrofrequency and the
relativistic factor is
␥ 共 X,p 储 ,J g 兲 ⬅
冑
1⫹
2J g ␻ g
M c2
冉 冊
p储
⫹
Mc
⫽
B* ⳵ H 0
cb̂
⫹
à ⵜH 0
⳵ p 储 qB *
B*
储
储
冉
冉
冊 冋
⳵
cb̂
" ⵜF Ã ⵜ ␦ H
⫹Ẋ0 " ⵜ F⫽ ⑀
⳵t
qB *
储
冉
⫹Ẋ0 " ⵜ ␦ H
⳵F
⳵␦H
⫺ⵜF
⳵␧
⳵␧
冊册
.
共14兲
The physical interpretation of F in Eq. 共14兲 is that it represents the phase-space number density of gyrocenters; its relationship to the particle phase-space density f is that
F(X,␧;J g , ␨ ;t)⬅ f (x,p;t) to all orders in ⑀ .
Lastly, the Jacobian for the transformation from the particle phase-space coordinates (x,p) to the reduced gyrocenter
phase-space coordinates (X,␧;J g ) is (2 ␲ 兩 q 兩 /c)B *
储 / 兩 v 储兩 ,
where the factor 2 ␲ comes from the elimination of the gyroangle variable ␨ . One can easily show from Eq. 共13兲 that
D⬅B *
储 / 兩 v 储 兩 satisfies the identities
ⵜ " 共 Ẋ0 D兲 ⬅0 and ⵜ Ã
冉 冊
cDb̂
qB *
储
⬅
⳵
共 Ẋ D兲 .
⳵␧ 0
共15兲
For the axisymmetric magnetic geometry considered in this
work, we find B *
储 ⬅B and we henceforth write D⬅B/ 兩 v 储 兩
wherever it appears.
2
.
D. Organization
In Eqs. 共11兲 and 共12兲, the gyrocenter phase-space coordinates
(X,p 储 ) are the gyrocenter position X and the relativistic gyrocenter parallel momentum p 储 ⬅ ␥ M v 储 , the relativistic gyroaction J g is an invariant of gyrocenter Hamiltonian dynamics, and the background electric field is assumed to be
negligible. In Eq. 共12兲, the electromagnetic-field fluctuations
are represented by the following scalar fields: ␦ ␾ denotes the
perturbed electrostatic scalar potential, while ␦ A 储 ⬅b̂ " ␦ A
and ␦ B 储 ⬅b̂ " ␦ B denote the parallel components of the perturbed magnetic vector potential and the perturbed magnetic
field, respectively.
Before proceeding to the derivation of the relativistic
quasilinear drift-kinetic equations, we first perform a change
of gyrocenter phase-space coordinates wherein we replace
the relativistic gyrocenter parallel momentum p 储 with the
unperturbed relativistic gyrocenter particle energy ␧⬅H 0 .
We also introduce the unperturbed relativistic guiding-center
velocity
Ẋ0 ⬅
represent grad-B and curvature magnetic drifts across field
lines, respectively. After making the substitution p 储 →␧ and
using Eq. 共13兲, the relativistic drift-kinetic Vlasov equation
共11兲 becomes
冊
p 2储
p储
cb̂
à J g ⵜ ␻ g ⫹ b̂ " ⵜb̂ ,
b̂⫹
␥M
M
␥ qB *
共13兲
where the first term in Eq. 共13兲 represents parallel motion
along a magnetic-field line, while the second and third terms
The remainder of the paper is organized as follows. In
Sec. II, relativistic quasilinear drift-kinetic equations are derived in axisymmetric magnetic geometry in which the background fields are independent of the azimuthal angle ␸ . For
this purpose, magnetic coordinates ( ␺ , ␸ ,s) are introduced,
where the magnetic flux ␺ is an exact invariant of the unperturbed guiding-center motion and the parallel spatial coordinate s denotes a location along a magnetic field line labeled
by ( ␺ , ␸ ). The perturbed Vlasov distribution ␦ F is then decomposed into the adiabatic part ( ␦ H ⳵ ␧ F 0 ) and the nonadiabatic part 共denoted ␦ G). Quasilinear drift-kinetic equations
are subsequently written for the slow-time evolution of F 0 in
terms of adiabatic and nonadiabatic contributions and the
fast-time evolution of the nonadiabatic part of the perturbed
Vlasov distribution ␦ G. In Sec. III, the quasilinear evolution
drift-kinetic equation for F 0 is bounce averaged and we
show that only nonadiabatic contributions remain, i.e., under
general assumptions, the adiabatic contributions to quasilinear diffusion vanish. In Sec. IV, we solve explicitly for the
components of the relativistic bounce-averaged quasilinear
diffusion tensor associated with stochastic transport in ( ␺ ,␧)
space. We summarize our work in Sec. V and discuss possible applications in magnetospheric physics. Lastly, Appendices A and B present additional details associated with general axisymmetric magnetic geometry 共Appendix A兲 and the
form of the perturbed electric and magnetic fields 共Appendix
B兲.
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Phys. Plasmas, Vol. 8, No. 11, November 2001
Relativistic bounce-averaged quasilinear diffusion . . .
where Eq. 共18兲 was used. We can rewrite Eq. 共20兲 by decomposing the perturbed gyrocenter Vlasov distribution ␦ F in
terms of its adiabatic and nonadiabatic components,16 respectively:
II. RELATIVISTIC QUASILINEAR DRIFT-KINETIC
EQUATIONS IN AXISYMMETRIC MAGNETIC
GEOMETRY
A. General relativistic quasilinear drift-kinetic
equations
Relativistic quasilinear drift-kinetic equations can now
be derived from Eq. 共14兲 by first introducing the following
decomposition on the gyrocenter Vlasov distribution:
F⬅F 0 ⫹ ⑀ ␦ F,
共16兲
where the gyrocenter Vlasov distribution F 0 is defined as
having a slow 共transport兲 time scale dependence while the
gyrocenter Vlasov distribution ␦ F has a fast 共wave兲 time
scale dependence.
Next, we introduce the fast time scale and azimuthalangle average with the property that F 0 ⬅F 0 and ␦ F⬅0; the
notation (•••) is henceforth used to mean fast time scale and
azimuthal-angle averaging only and appropriate notation for
bounce averaging will be introduced later. Upon averaging
Eq. 共14兲, we obtain
冉
冋 冉
冊
⳵
⳵␦F
⳵␦H
⫺ⵜ ␦ F
⑀
⫹Ẋ0 " ⵜ F 0 ⫽ ⑀ 2 Ẋ0 " ⵜ ␦ H
⳵␶
⳵␧
⳵␧
2
⫹
cb̂
qB *
储
册
" 共ⵜ␦F Ã ⵜ␦H兲 ,
冊
共17兲
where F 0 ⫽F 0 ( . . . , ⑀ 2 t⬅ ␶ ), i.e., the slow time scale dependence of F 0 enters at second order in ⑀ . At order ⑀ 0 , Eq. 共17兲
yields
Ẋ0 " ⵜF 0 ⬅0,
共18兲
i.e., F 0 is a constant along an unperturbed guiding-center
orbit. At order ⑀ 2 , on the other hand, Eq. 共17兲 yields
冋
⳵F0
cb̂
⫽
" 共ⵜ␦F Ã ⵜ␦H兲
⳵␶
qB *
储
冉
⫹Ẋ0 " ⵜ ␦ H
⳵␦F
⳵␦H
⫺ⵜ ␦ F
⳵␧
⳵␧
冊册
共19兲
This quasilinear drift-kinetic equation states that the slow
time scale relativistic gyrocenter Vlasov distribution F 0
evolves as a result of quadratic wave nonlinearities associated with the perturbed relativistic gyrocenter Vlasov distribution ␦ F and the perturbed relativistic gyrocenter Hamiltonian ␦ H. To derive a relativistic quasilinear drift-kinetic
diffusion equation from Eq. 共19兲, we must now establish a
relation between ␦ F and ␦ H.
For this purpose, we consider the linearized relativistic
drift-kinetic Vlasov equation. On the fast time scale 共order
⑀ ), Eq. 共14兲 yields
冉
冊 冉
␦ F⬅ ␦ H
⳵F0
⫹ ␦ G,
⳵␧
共21兲
where ␦ G represents the nonadiabatic part of the perturbed
gyrocenter Vlasov distribution. The linear relativistic driftkinetic Vlasov equation for the nonadiabatic part ␦ G is then
obtained by substituting Eq. 共21兲 into Eq. 共20兲:
冉
冊 冉
冊
⳵
cb̂
⳵F0 ⳵
⫹Ẋ0 " ⵜ ␦ G⫽
" ⵜF 0 Ã ⵜ⫺
␦ H.
⳵t
⳵␧ ⳵t
qB *
储
冊
⳵
cb̂
⳵F0
⫹Ẋ0 " ⵜ ␦ F⫽
" ⵜF 0 Ã ⵜ ␦ H⫹Ẋ0 " ⵜ ␦ H
,
⳵t
⳵␧
qB *
储
共20兲
共22兲
This relativistic drift-kinetic equation generalizes the earlier
nonrelativistic works of Antonsen and Lane,17 Catto, Baldwin, and Tang,18 and Chen.19
B. Axisymmetric magnetic geometry
In the present paper, we are interested in deriving quasilinear drift-kinetic equations 共19兲 and 共22兲 for an axisymmetric magnetic field geometry 共however, note that deviations
from axisymmetry can be incorporated as magnetic perturbations ␦ B with zero frequency; see Appendices A and B for
details兲. The axisymmetric magnetic field geometry is represented by the magnetic coordinates ( ␺ , ␸ ,s), where
B⬅ⵜ ␺ Ã ⵜ ␸ ⬅B 共 ␺ ,s 兲
⳵X
.
⳵s
共23兲
Here, the magnetic flux ␺ represents a radial-like coordinate,
␸ denotes the azimuthal angle, and s denotes the position of
a point along a magnetic field line labeled by ( ␺ , ␸ ). Note
that the background magnetized plasma is assumed to be
independent of the azimuthal angle ␸ . By definition, the parallel coordinate s is such that
b̂⬅ⵜs⫹a ⵜ ␺ ,
.
4765
共24兲
with a( ␺ ,s)⬅b̂ " ⳵ ␺ X, i.e., the magnetic coordinates ( ␺ , ␸ ,s)
are nonorthogonal coordinates. Moreover, from Eq. 共24兲, we
find ⵜ Ã b̂⫽b̂ Ã ( ⳵ s a ⵜ ␺ ), and thus b̂ " ⵜ Ã b̂⫽0 so that
B*
储 ⬅B. We note that in axisymmetric magnetic-dipole geometry, we have B⫽ⵜ ␹ ( ␺ ,s), with ⳵ s ␹ ⬅B and ⳵ ␺ ␹ ⬅a B.
C. Perturbed magnetic and electric fields
The perturbed vector potential can be written in purely
covariant form as
␦ A⬅ⵜ ␦ ␣ ⫹ ␦ ␺ ⵜ ␸ ⫺ ␦ ␹ ⵜ ␺ ,
共25兲
where ␦ ␣ , ␦ ␺ , and ␦ ␹ are perturbation scalar potentials 共see
Appendix B for details兲. Note that the parallel component of
the perturbed vector potential is ␦ A 储 ⬅b̂ " ␦ A⫽ ⳵ s ␦ ␣ . Hence,
using Eq. 共25兲, the perturbed magnetic field ␦ B⬅ⵜ Ã ␦ A
can be expressed in explicit divergenceless form as
␦ B⬅ⵜ ␦ ␺ Ã ⵜ ␸ ⫹ⵜ ␺ Ã ⵜ ␦ ␹ .
共26兲
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4766
Phys. Plasmas, Vol. 8, No. 11, November 2001
A. J. Brizard and A. A. Chan
Using the covariant representation Eq. 共25兲 for the perturbed
vector potential ␦ A, the perturbed electric field ␦ E
⬅⫺ ⵜ ␦ ␾ ⫺c ⫺1 ⳵ t ␦ A, on the other hand, is written as
冉
D Ai␧ ⬅
冊
while the nonadiabatic coefficients (R i ,R ␧ ) are defined
1 ⳵␦␣
.
c ⳵t
共28兲
冋 冉
D. Quasilinear equations in axisymmetric magnetic
geometry
In axisymmetric magnetic geometry, the unperturbed
guiding-center operator Ẋ0 " ⵜ is thus written in terms of
magnetic coordinates ( ␺ , ␸ ,s) as
⳵
⳵
,
⫹␻d
⳵s
⳵␸
共29兲
where v 储 ⬅p 储 / ␥ M is the parallel component of the relativistic guiding-center velocity and the magnetic-drift frequency
is defined as
共30兲
We note that since Ẋ0 " ⵜ ␺ ⫽0, the drift-action J d⬅q ␺ /c is
an invariant for unperturbed particle motion in axisymmetric
magnetic geometry. We also find from Eq. 共18兲 that F 0 (I, ␶ )
depends on the three invariants I⬅(␧, ␺ ,J g ) of unperturbed
guiding-center motion in axisymmetric magnetic geometry.
Using the magnetic coordinates ( ␺ , ␸ ,s)⬅ ␺ i and the decomposition Eq. 共21兲, the slow-time relativistic quasilinear
drift-kinetic Vlasov equation 共19兲 can be written in divergence form20 as
冊
⳵
⳵␦H
cDb̂
à 共 ␦ Fⵜ ␦ H 兲
⫹
共 DF 0 兲 ⫽⫺ⵜ " DẊ0 ␦ F
⳵␶
⳵␧
qB
⫹
⳵
关 DẊ0 " 共 ␦ Fⵜ ␦ H 兲兴
⳵␧
冋冉
冋冉
⳵ ␦ H cb̂
⫹
à ⵜ␦H
⳵␧
qB
R i ⬅ⵜ ␺ i " ␦ G Ẋ0
冊册
共33兲
and
R ␧ ⬅Ẋ0 " 共 ␦ Gⵜ ␦ H 兲 .
Hence, the parallel component of the perturbed electric field
is ␦ E 储 ⬅b̂ " ␦ E⫽⫺ ⳵ s ␦ ⌽.
冋 冉
共32兲
D A␧␧ ⬅ 21 Ẋ0 " ⵜ ␦ H 2 ,
共27兲
where the scalar potentials ␦ ␾ and ␦ ␣ are combined to define ␦ ⌽ as follows:
Ẋ0 " ⵜ⫽ v 储
冊
and
⳵␦␹
1 ⳵␦␺
ⵜ␸⫺
ⵜ␺ ,
␦ E⫽⫺ ⵜ ␦ ⌽⫺
c ⳵t
⳵t
␦ ⌽⬅ ␦ ␾ ⫹
冉
1 ⳵␦H2
cb̂
Ẋ0 ⫹
à ⵜ␦H2 " ⵜ␺i
2 ⳵␧
qB
D i
⳵F0
⬅⫺B i
R ⫹D Ai␧
⳵␧
⳵␺ B
冊册
⳵
⳵F0
⫹
D R ␧ ⫹D A␧␧
⳵␧
⳵␧
,
⳵
冊册
册
The linear relativistic drift-kinetic Vlasov equation 共22兲 for
␦ G can be written in terms of the magnetic coordinates
( ␺ , ␸ ,s) as
冉
冊 冉
冊
⳵
⳵
⳵
c ⳵F0 ⳵
⳵F0 ⳵
␦ G⫽
⫺
␦ H,
⫹v储 ⫹␻d
⳵t
⳵s
⳵␸
q ⳵ ␺ ⳵␸
⳵␧ ⳵t
共34兲
where the total time derivative appearing on the left side
contains bounce and magnetic-drift contributions. Although
the adiabatic coefficients Eq. 共32兲 are shown in the next section to vanish under bounce averaging, we keep them here
for completeness.
III. RELATIVISTIC BOUNCE-AVERAGED QUASILINEAR
DRIFT-KINETIC DIFFUSION EQUATION
Since the time-independent background medium is axisymmetric in ␸ , we decompose the perturbation fields in
Fourier components in time t and azimuthal angle ␸ :
冉 冊
␦G
␦H
⬁
⬅
兺␬ m⫽⫺⬁
兺
冉
␦ G̃ m 共 s;I, ␻ ␬ 兲
␦ H̃ m 共 s;I, ␻ ␬ 兲
冊
e (im ␸ ⫺ ␻ ␬ t) .
共35兲
In what follows, we shall use the notation f̃ to indicate explicit s-dependence in a function f and the notation f̂ to indicate s-independence 共i.e., ⳵ s f̂ ⬅0). For two arbitrary functions f (X,t) and g(X,t), the fast time scale average ( f g) is
now explicitly written as
共 f g 兲⬅
兺
m, ␬
* 共 ␺ ,s; ␻ ␬ 兲 g̃ m 共 ␺ ,s; ␻ ␬ 兲 .
f̃ m
共36兲
Thus we find from Eqs. 共32兲 and 共35兲 that D A␺ ␧ ⬅0 and D A␧␧
⬅ 12 v 储 ⳵ s ␦ H 2 .
Next, since F 0 is independent of the parallel coordinate
s, we introduce a bounce-average operation defined as
具 f 典 共 I兲 ⬅
共31兲
where the adiabatic 共A兲 coefficients (D Ai␧ ,D A␧␧ ) are defined
1
␶b
兺 冕
␴ ⫽⫾1 s
sU
L
ds
f 共 s, ␴ ;I兲 ,
兩 v 储兩
共37兲
where the sum is over the two possible signs of v 储 共note that
D/B⬅1/兩 v 储 兩 ), and the bounce period is defined as
␶ b共 I兲 ⬅
兺 冕
␴ ⫽⫾1 s
sU D
L
B
ds⫽2
冕
sU
sL
ds
⳵Jb
,
⬅2 ␲
兩 v 储兩
⳵␧
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Phys. Plasmas, Vol. 8, No. 11, November 2001
Relativistic bounce-averaged quasilinear diffusion . . .
with s L (I) and s U (I) denoting the lower (L) and upper (U)
turning points. We note that since F 0 is independent of the
parallel spatial coordinate s, we have 具 F 0 典 ⬅F 0 .
We now perform the bounce-average operation on Eq.
共31兲 to obtain the relativistic quasilinear bounce-averaged
drift-kinetic diffusion equation:21
冋冓
⳵F0 1 ⳵
cb̂
à ⵜ ␺ " 共 ␦ Gⵜ ␦ H 兲
⬅
␶b
⳵␶
␶b ⳵␺
qB
⫹
冔册
1 ⳵
关 ␶ 具 Ẋ " 共 ␦ G ⵜ ␦ H 兲 典 兴 ,
␶b ⳵␧ b 0
冓
共38兲
冔
Hence, only nonadiabatic effects contribute to quasilinear
diffusion.
Using the Fourier decomposition Eq. 共35兲 in the linear
relativistic drift-kinetic Vlasov equation 共34兲 for the nonadiabatic part of the perturbed Vlasov distribution, we obtain
共39兲
where F(I, ␻ ␬ )⬅ ␻ ␬ ⳵ ␧F 0 ⫹(mc/q) ⳵ ␺ F 0 and we henceforth
omit displaying the dependence of the perturbation fields on
I and ␻ ␬ . In Eq. 共39兲, the differential operator L is defined
as
冋
L␦ G̃ m 共 ␴ ,s 兲 ⬅ ␴ 兩 v 储 兩
册
⳵
⫺i 共 ␻ ␬ ⫺m ␻ d兲 ␦ G̃ m 共 ␴ ,s 兲 ,
⳵s
共40兲
and the perturbed Hamiltonian defined in Eq. 共12兲 becomes
冉
˜ m共 s 兲 ⫺ ␴
␦ H̃ m 共 ␴ ,s 兲 ⫽q ␦ ␾
冊
兩 v 储兩
␦ B̃ 储 m 共 s 兲
␦ Ã 储 m 共 s 兲 ⫹J g ␻ g
.
c
B
共41兲
The operator L defined in Eq. 共40兲 has the following property associated with bounce-averaging operation defined in
Eq. 共37兲:
具 f * 共 Lg 兲 典 ⫽⫺ 具 共 Lf 兲 * g 典
共42兲
for two arbitrary functions f and g. Using Eq. 共35兲 in Eq.
共38兲, we therefore find
具 Ẋ0 " 共 ␦ Gⵜ ␦ H 兲 典 ⬅ 兺 ␻ ␬ Im具 ␦ G̃ m ␦ H̃ m* 典 ,
m, ␬
and
冓
冔
b̂
*典 .
m Im具 ␦ G̃ m ␦ H̃ m
à ⵜ ␺ " 共 ␦ Gⵜ ␦ H 兲 ⬅
B
m, ␬
兺
共43兲
共44兲
By introducing the quasilinear perturbation potential
⌫ˆ m 共 I; ␻ ␬ 兲 ⬅F
⫺1
*典 ,
Im具 ␦ G̃ m ␦ H̃ m
冋冉
冋冉
⳵F0 1 ⳵
⳵F0
⳵F0
␧␧
␧␺
⫹D QL
⬅
␶ b D QL
⳵␶
␶b ⳵␧
⳵␧
⳵␺
⫹
␧␧
D QL
共 I兲 ⬅
␺␧
D QL
共 I兲 ⬅
␺␺
D QL
共 I兲 ⬅
1
1
⳵␦H2
2
⬅0.
具 Ẋ " ⵜ ␦ H 典 ⫽ v 储
2 0
2
⳵s
L␦ G̃ m 共 ␴ ,s 兲 ⬅i F␦ H̃ m 共 ␴ ,s 兲 ,
the relativistic quasilinear bounce-averaged drift-kinetic diffusion equation 共38兲 now becomes
冊册
1 ⳵
⳵F0
⳵F0
␺␺
␶ D ␺␧
⫹D QL
␶ b ⳵ ␺ b QL ⳵ ␧
⳵␺
冊册
,
共46兲
where the relativistic quasilinear bounce-averaged diffusion
coefficients are defined as
where the s-term in Eq. 共31兲 is averaged out while the
␸ -term vanishes as a result of axisymmetry. Here, we used
Ẋ0 " ⵜ ␺ ⫽0 in the first term in Eq. 共38兲 while the adiabatic
coefficient D A␧␧ vanishes upon bounce-averaging since
具 D A␧␧ 典 ⫽
4767
兺 ␻ ␬2 ⌫ˆ m共 I; ␻ ␬ 兲 ,
m, ␬
␧␺
共 I兲 ,
兺 共 mc/q 兲 ␻ ␬ ⌫ˆ m共 I; ␻ ␬ 兲 ⬅D QL
m, ␬
兺 共 mc/q 兲 2 ⌫ˆ m共 I; ␻ ␬ 兲 .
m, ␬
This quasilinear diffusion tensor describes stochastic transport in ( ␺ ,␧)-space induced by low-frequency electromagnetic fluctuations which conserve the first 共adiabatic兲 invariant J g . We now proceed to obtain an explicit expression for
⌫ˆ m in terms of the perturbation Hamiltonian ␦ H̃ m .
IV. SOLUTION OF THE NONADIABATIC DRIFT-KINETIC
EQUATION FOR TRAPPED PARTICLES
To obtain an explicit expression for the quasilinear perturbation potential ⌫ˆ m , we need to solve the linearized relativistic drift-kinetic equation 共39兲 for the nonadiabatic part
␦ G̃ m ( ␴ ,s) of the perturbed gyrocenter Vlasov distribution in
terms of the perturbed relativistic gyrocenter Hamiltonian
␦ H̃ m ( ␴ ,s).
The fact that the perturbed relativistic gyrocenter Hamiltonian Eq. 共41兲 depends on ␴ introduces some difficulties in
solving Eq. 共39兲. The solution of Eq. 共39兲 can, however, be
simplified as follows. First, using ␦ A 储 ⬅b̂ " ␦ A⫽ ⳵ ␦ ␣ / ⳵ s
from Eq. 共25兲, we can write ␦ H̃ m as
q
c
␦ H̃ m 共 ␴ ,s 兲 ⬅ ␦ K̃ m 共 s 兲 ⫺ L␦ ˜␣ m ,
共48兲
with
q
c
˜ m ⫹J g ␻ g 共 ␦ B̃ 储 m /B 兲 ⫺i 共 ␻ ␬ ⫺m ␻ d 兲 ␦ ˜␣ m
␦ K̃ m 共 s 兲 ⬅q ␦ ␾
⬅q ␦ ⌽̃ m ⫹J g ␻ g 共 ␦ B̃ 储 m /B 兲 ⫹
imq
␻ d ␦ ˜␣ m ,
c
共49兲
˜ m ⫺i( ␻ ␬ /c) ␦ ˜␣ m 关see Eq. 共28兲兴; the reader
where ␦ ⌽̃ m ⬅ ␦ ␾
is referred to Appendix B for a representation of ␦ K̂ m in
terms of components of the perturbed electric and magnetic
fields. When the decomposition Eq. 共48兲 is inserted into Eq.
共39兲, we obtain the new equation
⬘ 共 ␴ ,s 兲 ⬅iF␦ K̃ m 共 s 兲 ,
L␦ G̃ m
共45兲
共47兲
共50兲
⬘ ( ␴ ,s) is defined as
where the new nonadiabatic part ␦ G̃ m
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4768
Phys. Plasmas, Vol. 8, No. 11, November 2001
q
c
␦ G̃ m ⬅ ␦ G̃ m⬘ ⫺i F ␦ ˜␣ m .
A. J. Brizard and A. A. Chan
共51兲
Using properties Eq. 共42兲 of L and the decompositions Eqs.
共48兲 and 共51兲, we can show that the quasilinear perturbation
potential Eq. 共45兲 can also be written as
⬘ ␦ K̃ m* 典 ,
⌫ˆ m ⫽F ⫺1 Im具 ␦ G̃ m
共52兲
e i⌰ ␦ ĝ m ⫹i
⬘ „␴,s…
A. General solution for ␦ G̃ m
⳵ ⫺i ␴ ␪ (s)
␦ G̃ m⬘ 共 ␴ ,s 兲兴 ,
关e
⳵s
ds ⬘
共 ␻ ␬ ⫺m ␻ d 兲 .
sL 兩 v 储兩
s
共53兲
共54兲
⬘ ( ␴ ,s) 共for
which implies that its general solution ␦ G̃ m
trapped particles with turning points at s⫽s L and s⫽s U ) can
be written as
␦ G̃ m⬘ 共 ␴ ,s 兲 ⬅F e i ␴ ␪ (s) 关 ␦ ĝ m 共 ␴ 兲 ⫹ ␦ g̃ m 共 ␴ ,s 兲兴 .
共55兲
The homogeneous term ␦ ĝ m ( ␴ ) is determined from two
⬘ ( ␴ ,s) for ␴ ⫽⫾1 at s⫽s L and
matching conditions for ␦ G̃ m
s⫽s U 共see below兲.
册
␶ b i␪
具 e ␦ K̃ m 典 ,
2
共58兲
␶b
␲
共 ␻ ␬ ⫺m 具 ␻ d 典 兲 ⬅
共 ␻ ⫺m 具 ␻ d 典 兲 .
2
␻b ␬
共59兲
The matching condition Eq. 共58兲 can now be used to solve
for the s-independent function ␦ ĝ m :
Hence, Eq. 共50兲 can be written as
⳵
⬘ 共 ␴ ,s 兲兴 ⫽iF␦ K̃ m 共 s 兲 ,
e i ␴ ␪ (s) v 储 关 e ⫺i ␴ ␪ (s) ␦ G̃ m
⳵s
册
where Eq. 共57兲 was used and the phase factor ⌰
⬅ ␪ (s L ,s U ) is defined from Eq. 共53兲 as
⌰⫽
where ␪ (s) is defined as
冕
冋
␶ b ⫺i ␪
具 e ␦ K̃ m 典
2
⫽e ⫺i⌰ ␦ ĝ m ⫺i
To proceed with the solution of the linearized driftkinetic equation 共50兲 we first write the operator L 关defined in
Eq. 共39兲兴 in terms of the integrating factor ␪ :
␪ 共 s L ,s 兲 ⬅
We now solve for the homogeneous term ␦ ĝ m ( ␴ ). The
⬘ (⫹1,s L )⫽ ␦ G̃ m⬘ (⫺1,s L ) at the turnmatching condition ␦ G̃ m
ing point s⫽s L yields ␦ ĝ m (⫹1)⫽ ␦ ĝ m (⫺1)⬅ ␦ ĝ m , while
⬘ (⫹1,s U )⫽ ␦ G̃ m⬘ (⫺1,s U ) at the
the matching condition ␦ G̃ m
turning point s⫽s U yields
冋
⬘ and ␦ K̃ m .
in terms of ␦ G̃ m
⬘ 共 ␴ ,s 兲 ⬅e i ␴ ␪ (s) v 储
L␦ G̃ m
2. Homogeneous term
␦ ĝ m ⬅⫺
␶b
共 cot ⌰ 具 ␦ K̃ m cos ␪ 典 ⫹ 具 ␦ K̃ m sin ␪ 典 兲 .
2
By combining the inhomogeneous solution Eq. 共57兲 and
the homogeneous solution Eq. 共60兲, the general solution of
Eq. 共50兲 is therefore written as
冋
␦ G̃ m⬘ 共 ␴ ,s 兲 ⫽F e i ␴ ␪ (s) i ␴ ␦ k̃ m 共 ␴ ,s 兲
⫺
册
␶b
共 cot ⌰ 具 ␦ K̃ m cos ␪ 典 ⫹ 具 ␦ K̃ m sin ␪ 典 兲 .
2
共61兲
1. Inhomogeneous term
We first solve for the inhomogeneous term ␦ g̃ m ( ␴ ,s).
After substituting the inhomogeneous term F␦ g̃ m exp(i␴␪)
into Eq. 共54兲, we find ␴ 兩 v 储 兩 ⳵ s ␦ g̃ m ⬅i ␦ K̃ m e ⫺i ␴ ␪ . This equation is solved, subject to the boundary condition ␦ g̃ m ( ␴ ,s
⫽s L )⬅0, as
␦ g̃ m 共 ␴ ,s 兲 ⬅i ␴
冕
ds ⬘ ⫺i ␴ ␪ (s ,s )
L ⬘ ␦ K̃ 共 s ⬘ 兲
e
m
兩
sL v 储兩
s
⬅i ␴ ␦ k̃ m 共 ␴ ,s 兲 .
共56兲
By definition, the s-dependent function ␦ k̃ m ( ␴ ,s) satisfies
the following relations: ␦ k̃ m ( ␴ ,s⫽s L )⬅0 and
␦ k̃ m 共 ␴ ,s⫽s U 兲 ⫽
⬅
共60兲
B. Quasilinear perturbation potential
The quasilinear perturbation potential Eq. 共52兲 can now
be evaluated explicitly. First, using Eq. 共61兲 and the bounceaverage definition Eq. 共37兲, we obtain
* e i ␴ ␪ 共 ␦ ĝ m ⫹i ␴ ␦ k̃ m 兲 典
⌫ˆ m ⫽Im具 ␦ K̃ m
冓
* cos ␪ 典 兲 ⫹Re ␴ 兩 v 储 兩
⫽Im共 ␦ ĝ m 具 ␦ K̃ m
冔
⳵ ␦ k̃ m*
␦ k̃ m ,
⳵s
共62兲
where, in the first term, we have used the fact that
␶b
具 ␦ K̃ m e ⫺i ␴ ␪ 典
2
␶b
共 具 ␦ K̃ m cos ␪ 典 ⫺i ␴ 具 ␦ K̃ m sin ␪ 典 兲 .
2
共57兲
and, in the second term, we have used Eq. 共56兲 to obtain
␦ K̃ m* e i ␴ ␪ ⬅ 兩 v 储 兩 ⳵ ␦ k̃ m* / ⳵ s. Next, using Eq. 共60兲, the first term
in Eq. 共62兲 becomes
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Phys. Plasmas, Vol. 8, No. 11, November 2001
* cos ␪ 典 兲 ⫽⫺
Im共 ␦ ĝ m 具 ␦ K̃ m
⫹
Relativistic bounce-averaged quasilinear diffusion . . .
⬁
␶b
兩 具 ␦ K̃ m cos ␪ 典 兩 2 Im共 cot ⌰ 兲
2
⌫ˆ m 共 I; ␻ ␬ 兲 ⬅
␶b
* sin ␪ 典 兲 .
Im共 具 ␦ K̃ m cos ␪ 典 具 ␦ K̃ m
2
共63兲
The second term in Eq. 共62兲, on the other hand, becomes
冓
Re ␴ 兩 v 储 兩
冔 冓
⳵ ␦ k̃ m*
1
⳵ 兩 ␦ k̃ m 兩 2
␦ k̃ m ⫽
␴ 兩 v 储兩
⳵s
2
⳵s
1
⫽
2␶b
⫽
兺␴
␴
冕
sU
sL
冔
Substituting Eq. 共69兲 into Eq. 共47兲, the components of
the relativistic quasilinear bounce-averaged quasilinear diffusion tensor in ( ␺ ,␧)-space are now expressed as
␧␧
⬅
D QL
1
关 兩 ␦ k̃ m 共 ␴ ⫽⫹1,s U 兲 兩 2
2␶b
␺␺
D QL
⬅
冔
which then cancels the second term in Eq. 共63兲. By combining the homogeneous and inhomogeneous terms, the quasilinear perturbation potential Eq. 共62兲 therefore becomes
␶b
兩 具 ␦ K̃ m cos ␪ 典 兩 2 Im共 ⫺ cot ⌰ 兲 .
2
共66兲
We now show that the quasilinear diffusion of magneticallytrapped particles arises from resonant magnetically-trapped
⬁
(⌰⫺n␲)⫺1
particles only. Using the identity cot ⌰⫽兺n⫽⫺⬁
共for ⌰⫽0, ⫾ ␲ , . . . ), we obtain, after substituting Eq. 共59兲,
the expression
cot ⌰⬅
1
␲
⬁
␻
b
,
兺
n⫽⫺⬁ 共 ␻ ␬ ⫺m 具 ␻ d 典 ⫺n ␻ b 兲
共67兲
where ␻ b ⬅2 ␲ / ␶ b is the bounce frequency. We note that the
denominator in Eq. 共67兲 is a function of the invariants I
⬅(␧, ␺ ,J g ) with ( ␻ ␬ ,m,n) appearing as parameters. Using
the Plemelj formula22
lim 共 x⫺a⫹i ␩ 兲 ⫺1 ⬅P关共 x⫺a 兲 ⫺1 兴 ⫺i ␲ ␦ 共 x⫺a 兲 ,
␩ →0 ⫹
we readily find Im(x⫺a) ⫺1 ⬅⫺ ␲ ␦ (x⫺a). Hence, by applying the Plemelj formula in Eq. 共67兲, we obtain the following expression for Im(⫺cot ⌰):
⬁
Im共 ⫺cot ⌰ 兲 ⬅
兺
n⫽⫺⬁
␻ b ␦ 共 ␻ ␬ ⫺m 具 ␻ d 典 ⫺n ␻ b 兲 ,
and the final expression for ⌫ˆ m is therefore
共68兲
兺
m,n, ␬
␻ ␬2 关 ␲ ␦ 共 ␻ ␬ ⫺m 具 ␻ d 典 ⫺n ␻ b 兲
⫻ 兩 具 ␦ K̃ m cos ␪ 典 兩 2 兴 ,
共64兲
⳵ ␦ k̃ m*
␶b
␦ k̃ m ⫽⫺ Im共 具 ␦ K̃ m cos ␪ 典 具 ␦ K̃ m* sin ␪ 典 兲 ,
⳵s
2
共65兲
⌫ˆ m ⬅
共69兲
This expression contains contributions from magnetic-drift
resonances ( ␻ ␬ ⬃m 具 ␻ d 典 ) and bounce resonances ( ␻ ␬
⬃n ␻ b ).
␺␧
⬅
D QL
冓
␲ ␦ 共 ␻ ␬ ⫺m 具 ␻ d 典 ⫺n ␻ b 兲
⫻ 兩 具 ␦ K̃ m 共 s;I, ␻ ␬ 兲 cos ␪ 共 s;I, ␻ ␬ 兲 典 兩 2 .
where we have used the fact that ␦ k̃ m ( ␴ ,s L )⬅0 共by definition兲. Using Eq. 共57兲 and the identity 共for arbitrary complex
functions a and b) 兩 a⫾i ␴ b 兩 2 ⫽ 兩 a 兩 2 ⫹ 兩 b 兩 2 ⫾2 ␴ Im(ab * ),
the inhomogeneous term Eq. 共64兲 finally becomes
Re ␴ 兩 v 储 兩
兺
n⫽⫺⬁
C. Relativistic bounce-averaged quasilinear diffusion
tensor
ds
⳵ 兩 ␦ k̃ m 兩 2
兩 v 储兩
兩 v 储兩
⳵s
⫺ 兩 ␦ k̃ m 共 ␴ ⫽⫺1,s U 兲 兩 2 兴 ,
4769
兺
m,n, ␬
共 mc/q 兲 ␻ ␬ 关 ␲ ␦ 共 ␻ ␬ ⫺m 具 ␻ d 典 ⫺n ␻ b 兲
␧␺
⫻ 兩 具 ␦ K̃ m cos ␪ 典 兩 2 兴 ⬅D QL
,
兺
m,n, ␬
共70兲
共 mc/q 兲 2 关 ␲ ␦ 共 ␻ ␬ ⫺m 具 ␻ d 典 ⫺n ␻ b 兲
⫻ 兩 具 ␦ K̃ m cos ␪ 典 兩 2 兴 .
As expected, these expressions are very similar in form to
those obtained from general quasilinear theory 关see Eq. 共9兲兴.
In the equatorial plane 共where ␪ ⫽ ␲ ), the term 兩 ␦ K̃ m 兩 2 is
directly proportional to the energy spectral density of the
waves. For particles which mirror off the equator, the diffusion tensor elements are calculated by bounce-averaging an
s-dependent energy spectral density weighted by the factor
cos ␪, as shown in Eq. 共70兲.
V. SUMMARY AND DISCUSSION
In this paper a relativistic quasilinear bounce-averaged
drift-kinetic diffusion equation 关Eq. 共46兲兴 has been derived
and explicit expressions for the diffusion tensor elements
have been obtained 关Eq. 共70兲兴.
In the Introduction, beginning with the relativistic Vlasov equation in Poisson-bracket form, the standard two time
scale decomposition of quasilinear theory is made, actionangle variables are introduced, and the solution of the linear
Vlasov equation in action-angle form is substituted into the
F 0 evolution equation and averaged over all fast variables to
obtain the general action form of the relativistic quasilinear
diffusion equation 共8兲 and the corresponding diffusion tensor
elements 关Eq. 共9兲兴.
The derivation of Eqs. 共46兲 and 共70兲 follows steps analogous to the general action-angle derivation, but with important differences because of the use of different guiding-center
phase-space coordinates and magnetic coordinates. The starting point is the relativistic drift-kinetic equation 共11兲 obtained from the corresponding relativistic gyrokinetic equation derived recently by Brizard and Chan.14 After changing
Downloaded 19 Oct 2001 to 192.80.64.41. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp
4770
Phys. Plasmas, Vol. 8, No. 11, November 2001
from the (X,p 储 ;J g ) phase-space coordinates to (X,␧;J g ) coordinates the two time scale quasilinear decomposition is
introduced and an F 0 -evolution equation and linear driftkinetic equation are obtained 关Eqs. 共19兲 and 共22兲兴. Next,
magnetic coordinates ( ␺ , ␸ ,s) are introduced via Eq. 共23兲
and a bounce-averaged equation 共38兲 for F 0 and a linearized
relativistic drift-kinetic equation 共39兲 are obtained. After introducing a quasilinear perturbation potential ⌫ˆ m 关Eq. 共45兲兴,
the final form of the relativistic quasilinear bounce-averaged
drift-kinetic equation 共46兲 is obtained. Finally, in Sec. IV the
linearized relativistic drift-kinetic equation 共39兲 is solved and
the results are used to yield explicit expressions for the quasilinear diffusion tensor elements in Eq. 共70兲.
By Eq. 共18兲 and the discussion following Eq. 共30兲, F 0 is
independent of gyrophase, distance along the field line s, and
azimuthal angle ␸ , so to lowest order F 0 can be regarded as
being averaged over these three variables. Thus Eq. 共46兲 describes slow, diffusive changes in a gyro-, bounce-, and driftaveraged phase-space density due to breaking of the second
and third adiabatic invariants by drift-bounce wave-particle
resonances associated with ultra-low-frequency 共ULF兲 electromagnetic fluctuations 共which, however, conserve the first
adiabatic invariant兲. We note that ULF fluctuations associated with radial and energy quasilinear diffusion are not necessarily associated with strong pitch-angle scattering 共which
moves the particle mirror points along field lines5,23兲.
Lastly, we have remarked in the Introduction that the
singular character of the quasilinear diffusion tensor disappears when nonlinear effects are taken into account.11,13
Analogous expressions exist for the relativistic bounceaveraged quasilinear diffusion tensor Eq. 共70兲, and these expressions will be used in future work to calculate renormalized diffusion coefficients for the drift-resonant transport of
magnetospheric relativistic electrons in the presence of lowfrequency hydromagnetic waves.
ACKNOWLEDGMENTS
The authors dedicate this paper with gratitude and respect to the memory of Professor Thomas H. Stix, one of our
first teachers of the theory of plasma waves.
This work was supported by National Science Foundation Grant No. ATM-9901127 and NASA Grants No. NAG58136 and No. NAG5-9077.
APPENDIX A: AXISYMMETRIC MAGNETIC
GEOMETRY
Magnetic field geometry is defined in terms of magnetic
coordinates ␺ i ⬅( ␺ , ␸ ,s), where B⬅ⵜ ␺ Ã ⵜ ␸ and b̂⬅B/B
⫽ ⳵ s X. In axisymmetric magnetic-field geometry, where the
azimuthal angle ␸ is an ignorable coordinate, the magnetic
field is represented in contravariant form as B⬅B( ␺ ,s) ⳵ sX,
where s denotes the spatial coordinate along a magnetic-field
line labeled by ( ␺ , ␸ ). By definition, ⵜs " ⵜ ␺ Ã ⵜ ␸ ⬅B and
the Jacobian for the transformation X→( ␺ , ␸ ,s) is B ⫺1 , i.e.,
d 3 X⫽B ⫺1 d ␺ d ␸ ds.
A. J. Brizard and A. A. Chan
The magnetic geometry is further defined by the components of the two-contravariant metric tensor g i j ⬅ⵜ ␺ i " ⵜ ␺ j
and the two-covariant metric tensor g i j ⬅ ⳵ i X " ⳵ j X. These
tensors are used in the following relations:
⳵ i X⬅g i j ⵜ ␺ j and ⵜ ␺ i ⬅g i j ⳵ j X,
共A1兲
where summation over repeated indices is implied. We also
find det(gi j )⬅B ⫺2 and det(gi j )⬅B 2 , and we note that g ij
⬅ ⳵ i X " ⵜ ␺ j ⬅ ␦ ij . For axisymmetric magnetic-field geometry,
the components of the two-covariant metric tensor are given
explicitly as
⳵ s X⫽ⵜs⫹a ⵜ ␺ ⬅b̂,
⳵ ␺ X⫽ 共 ⵜ ␺ / 兩 ⵜ ␺ 兩 2 兲 ⫹a b̂,
共A2兲
⳵ ␸ X⫽R 2 ⵜ ␸ ,
and
a( ␺ ,s)⬅b̂ " ⳵ ␺ X⫽⫺ ⵜs
where
R⬅ 兩 ⵜ ␸ 兩 ⫺1
" ⵜ ␺ /( 兩 ⵜ ␺ 兩 2 ). The components of the two-contravariant
metric tensor g i j are obtained by inverting the matrix gi j .
APPENDIX B: PERTURBED MAGNETIC FIELD IN
AXISYMMETRIC MAGNETIC GEOMETRY
Using the magnetic coordinates ( ␺ , ␸ ,s), the perturbed
magnetic vector potential ␦ A is written 共in the gauge
␦ A " ⵜ ␸ ⬅0) as
␦ A⫽ ␦ A 储 ⵜs⫺ ␦ ␤ ⵜ ␺ ⬅ⵜ ␦ ␣ ⫹ ␦ ␺ ⵜ ␸ ⫺ ␦ ␹ ⵜ ␺ ,
共B1兲
where ␦ A 储 ⬅ ⳵ s ␦ ␣ , ␦ ␺ ⬅⫺ ⳵ ␸ ␦ ␣ , and ␦ ␹ ⬅ ␦ ␤ ⫹ ⳵ ␺ ␦ ␣ .
Hence, the perturbed magnetic field ␦ B⬅ⵜ Ã ␦ A is written
as
␦ B⫽ⵜ ␦ ␺ Ã ⵜ ␸ ⫹ⵜ ␺ Ã ⵜ ␦ ␹ .
共B2兲
Since magnetic fields are most easily measured in terms of
magnetic fluxes across surfaces S 共i.e., the perturbed magnetic flux through a constant-␺ k surface is
␦ ⌿ k⬅
冕
S
␦Bk
B
⑀ i jk d ␺ i d ␺ j ,
where ⑀ i jk B ⫺1 d ␺ i d ␺ j denotes the infinitesimal surface element on the constant-␺ k surface兲, the perturbed magnetic
field Eq. 共B2兲 has the following contravariant components
␦ B i⬅ ␦ B " ⵜ ␺ i:
␦ B ␺ ⫽⫺B ⳵ s ␦ ␺ ⬅B ⳵ s 共 ⳵ ␸ ␦ ␣ 兲 ,
␦ B ␸ ⫽⫺B ⳵ s ␦ ␹ ⬅⫺B ⳵ s 共 ␦ ␤ ⫹ ⳵ ␺ ␦ ␣ 兲 ,
共B3兲
␦ B s ⫽B 共 ⳵ ␺ ␦ ␺ ⫹ ⳵ ␸ ␦ ␹ 兲 ⬅B ⳵ ␸ ␦ ␤ .
Note that the perturbed magnetic flux through a constant-␺
surface is ␦ ⌿ ␺ ⫽ ␦ ␣ and that the parallel component of the
perturbed magnetic field ␦ B 储 ⬅b̂ " ␦ B is written in terms of
␦ B 储 ⫽ ␦ B s ⫹a ␦ B ␺ ⫽B ⳵ ␸ ( ␦ ␤
these
components
as
⫹a ⳵ s ␦ ␣ ).
The perturbed electric field is written in terms of the
perturbed scalar potential ␦ ␾ and the perturbed vector potential Eq. 共B1兲 as
␦ E⫽⫺ⵜ ␦ ␾ ⫺c ⫺1 共 ⳵ t ␦ A 储 ⵜs⫺ ⳵ t ␦ ␤ ⵜ ␺ 兲 .
共B4兲
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Phys. Plasmas, Vol. 8, No. 11, November 2001
Relativistic bounce-averaged quasilinear diffusion . . .
Since electric fields are most easily measured in terms of
voltages along curves C 共i.e., the perturbed loop voltage
along loop C is 兰 C ⫺ ␦ E i d ␺ i ), the perturbed electric field
Eq. 共B4兲 has the following covariant components ␦ E i
⬅ ␦ E " ⳵ i X:
␦ E ␺ ⫽⫺ ⳵ ␺ ␦ ␾ ⫹c ⫺1 ⳵ t ␦ ␤ ,
␦ E ␸ ⫽⫺ ⳵ ␸ ␦ ␾ ,
共B5兲
␦ E 储 ⫽⫺ ⳵ s ␦ ⌽⬅⫺ ⳵ s 共 ␦ ␾ ⫹c ⫺1 ⳵ t ␦ ␣ 兲 .
Lastly, we note that the perturbation Hamiltonian equation 共49兲
␦ K̃ m ⬅q ␦ ⌽̃ m ⫹J g ␻ g
␦ B̃ 储 m
⫹
B
imq
␻ d ␦ ˜␣ m
c
␦ ˜␣ m ⬅⫺
冕␦
冕␦
␺
B̃ m
B
ds⫽
ic
c
␦ Ẽ ␸ m ⫺
m␻␬
␻␬
冕␦
Ẽ 储 m ds,
where the second expression for ␦ ˜␣ m is based on the identity
␺
⫺i ⳵ ␦ Ẽ ␸ m ␻ ␬ ␦ B̃ m
⫹
.
␦ Ẽ 储 m ⬅
m
⳵s
mc B
Thus we obtain two useful expressions for the perturbation
Hamiltonian
␦ K̃ m ⫽⫺q
⫽
冉冕
␻d
␦ Ẽ 储 m ds⫺
c
q
共 m ␻ d⫺ ␻ ␬ 兲
mc
冕␦
冕
␺
B̃ m
B
␦ B̃ m␺
B
冊
ds ⫹J g ␻ g
ds⫹J g ␻ g
␦ B̃ 储 m
B
␦ B̃ 储 m
B
⫹
共B6兲
iq
␦ Ẽ ␸ m .
m
共B7兲
Using Eq. 共B6兲, we obtain the following expression for the
perturbed parallel force:
␦ ṗ 储 m ⫽⫺
冉
⳵ ␦ K̃ m
qẊ0
⫽b̂ " q ␦ Ẽm ⫹
à ␦ B̃m
⳵s
c
⫺
X. Li and M. A. Temerin, Space Sci. Rev. 95, 569 共2001兲.
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3
R. A. Mathie and I. R. Mann, Geophys. Res. Lett. 27, 3261 共2000兲.
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5
M. Schulz and L. J. Lanzerotti, Particle Diffusion in the Radiation Belts
共Springer-Verlag, New York, 1974兲.
6
A. A. Chan, L. Chen, and R. B. White, Geophys. Res. Lett. 16, 1133
共1989兲.
7
H. P. Warren and M. E. Mauel, Phys. Plasmas 2, 4185 共1995兲.
8
A. A. Vedenov, E. P. Velikhov, and R. Z. Sagdeev, Nucl. Fusion 1, 82
共1961兲; Nucl. Fusion Suppl. 2, 465 共1962兲.
9
W. E. Drummond and D. Pines, Nucl. Fusion Suppl. 3, 1049 共1962兲; Ann.
Phys. 共N.Y.兲 28, 478 共1964兲.
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A. N. Kaufman and T. Nakayama, Phys. Fluids 13, 956 共1970兲.
11
A. N. Kaufman, Phys. Fluids 14, 387 共1971兲; 15, 1063 共1972兲.
12
Note: for two arbitrary complex-valued variables a m and b m , we find
⬁
⬁
⬁
兺 m⫽⫺⬁
ima m* b m ⫽ 兺 m⫽1
im(a m* b m ⫺a m b m* )⬅ 兺 m⫽⫺⬁
m Im(a m b m* ), where
(a ⫺m ,b ⫺m )⬅(a m* ,b m* ).
13
J. A. Krommes, Handbook of Plasma Physics, Vol. 2: Basic Plasma Physics II, edited by A. A. Galeev and R. N. Sudan 共North-Holland, Amsterdam, 1984兲, Chap. 5.5.
14
A. J. Brizard and A. A. Chan, Phys. Plasmas 6, 4548 共1999兲.
15
R. A. Wolf, in Introduction to Space Physics, edited by M. G. Kivelson
and C. T. Russell 共Cambridge University Press, Cambridge, 1995兲, Chap.
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16
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17
T. M. Antonsen and B. Lane, Phys. Fluids 23, 1205 共1980兲.
18
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19
L. Chen, J. Geophys. Res. 104, 2421 共1999兲.
20
In terms of magnetic coordinates ␺ i ⫽( ␺ , ␸ ,s), the divergence of an arbitrary vector field G is written as ⵜ " G⬅B ⳵ i (B ⫺1 G i ), where G i
⬅G " ⵜ ␺ i and ⳵ i ⬅ ⳵ / ⳵ ␺ i .
21
We note that if we were to use the bounce-action J b ⬅(2 ␲ ) ⫺1 养p 储 ds instead of the energy variable ␧, the factor ␶ b appearing in Eq. 共38兲 would
disappear since ⳵ J b / ⳵ ␧⬅ ␶ b /2␲ .
22
T. H. Stix, Waves in Plasmas 共American Institute of Physics, New York,
1992兲.
23
J. G. Roederer, Dynamics of Geomagnetically Trapped Radiation
共Springer-Verlag, New York, 1970兲.
1
Ẽ 储 m ds,
i
m
ror force and where we have used b̂ " (Ẋ0 Ã ␦ B)
⫽⫺ ␻ d ␦ B ␺ /B. Using Eq. 共B7兲, on the other hand, we find
that the first term can be written as ⫺i( ␻ ␬ ⫺m ␻ d ) ␦ J d ,
where ␦ J d ⬅(q/c) ␦ ␣ is the perturbed drift action.
Finally, although we assume the stationary magnetic
field B is axisymmetric, deviations from axisymmetry can be
incorporated by adding a zero-frequency magnetic perturbation ␦ Ba, in addition to the time-dependent 共fluctuation兲
fields. For example, the day–night asymmetry of the magnetosphere can be accounted for by adding an m⫽1 perturbation to the axisymmetric field.
2
can now be expressed in terms of the perturbed magnetic and
electric fields, with
␦ ⌽̃ m ⬅⫺
4771
冊
J g␻ g ⳵ ␦ B̃ 储 m
,
B
⳵s
in which the last term represents the standard perturbed mir-
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