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PHYSICS OF PLASMAS

VOLUME 6, NUMBER 12

DECEMBER 1999

Nonlinear relativistic gyrokinetic Vlasov-Maxwell equations Alain J. Brizard Lawrence Berkeley National Laboratory and Physics Department, University of California, Berkeley, California 94720

Anthony A. Chan Department of Space Physics and Astronomy, Rice University, Houston, Texas 77005-1892

共Received 12 July 1999; accepted 20 August 1999兲 A set of self-consistent nonlinear gyrokinetic equations is derived for relativistic charged particles in a general nonuniform magnetized plasma. Full electromagnetic-field fluctuations are considered with spatial and temporal scales given by the low-frequency gyrokinetic ordering. Self-consistency is obtained by combining the nonlinear relativistic gyrokinetic Vlasov equation with the low-frequency Maxwell equations in which charge densities and current densities are expressed in terms of moments of the gyrokinetic Vlasov distribution. For these self-consistent gyrokinetic equations, a low-frequency energy conservation law is also derived. © 1999 American Institute of Physics. 关S1070-664X共99兲01012-5兴

I. INTRODUCTION

terms of guiding-center and gyrocenter dynamics wherein the equations of motion are independent of the fast gyromotion degree of freedom. In previous work on radiation-belt transport, various Fokker-Planck transport equations for the average phasespace density as a function of the three adiabatic invariants are used.5,17 These radiation-belt transport equations are essentially quasilinear transport equations in the space of the three adiabatic invariants. It is important to note that these equations are not derived from first principles; rather, the form of the equation is assumed and the transport coefficients for diffusion and drag are given semiempirical forms and then adjusted by attempting to reproduce measured particle fluxes. Recently, nonrelativistic quasilinear transport equations have been derived to describe the interaction of nonrelativistic particles with magnetospheric hydromagnetic waves.18 In these equations it is shown that the wave-particle interactions result in a drag-like term which can be comparable to the diffusion term. We expect that there will be a similar drag-like contribution in the relativistic quasilinear transport equation; in the past this contribution has been ignored because the drag term has been assumed to arise only from Coulomb scattering. Chen’s derivation18 began with the nonrelativistic nonlinear gyrokinetic Vlasov equation derived by Brizard.19,20 In this paper a nonlinear relativistic gyrokinetic Vlasov equation is derived, partly as a starting point for a later derivation of relativistic quasilinear transport equations for space physics applications, and partly because it is interesting in its own right as a relativistic generalization of the earlier work of Brizard.20 Other applications might include the gyrokinetic particle simulation of relativistic runaway electron transport in tokamak plasmas. The nonlinear relativistic gyrokinetic equations derived here also extend the earlier work by Littlejohn21 and by Tsai, Van Dam and Chen,22 who derived linear relativistic gyrokinetic equations.

Relativistic particles, particularly electrons, are found in many plasmas, including fusion plasmas,1–4 space plasmas5 and astrophysical plasmas.6 In addition to the general question of how these particles are transported and accelerated to relativistic energies, the particles are of interest because they can cause significant damage. In fusion plasmas, for example, relativistic runaway electrons can cause serious damage to the containment vessel.3,4 In space plasmas, on the other hand, relativistic electrons trapped in the Earth’s radiation belts are hazardous to spacecraft hardware7 and they are a radiation hazard to astronauts.8 In high-temperature tokamak plasmas, multi-MeV runaway electrons are typically generated during the start-up phase or following plasma disruptions.3 After a disruption, a large fraction of the stored magnetic energy in the tokamak plasma can be transferred to the runaway-electron population. The primary mechanism for the generation of relativistic runaway electrons involves an accelerating electric field strong enough to overcome the frictional drag produced by Coulomb collisions.4,9,10 The confinement properties of relativistic test-particles in tokamak plasmas have been investigated in Refs. 11 and 12; these studies reveal that the guiding-center approximation is appropriate for relativistic runaway-electron dynamics in axisymmetric and nonaxisymmetric tokamak plasmas. In space plasmas there is observational evidence that relativistic electrons may be produced during magnetic storms by a drift-resonant exchange of energy with hydromagnetic waves in the Earth’s magnetosphere.13,14 The driftresonant interaction conserves the first adiabatic invariant and transports particles to regions of higher magnetic field magnitude, thereby increasing their energy.15,16 The timescale separation between the short gyroperiod and the long drift-period motivates the asymptotic removal of the gyromotion time scale from the exact dynamical description. The resulting reduced dynamical description is expressed in 1070-664X/99/6(12)/4548/11/$15.00

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© 1999 American Institute of Physics

Phys. Plasmas, Vol. 6, No. 12, December 1999

Nonlinear relativistic gyrokinetic Vlasov-Maxwell equations

Finite gyroradius effects are usually small for electrons, but they are not necessarily negligible, especially for highly relativistic electrons such as those found in fusion and space plasmas.23 For generality we retain the finite gyroradius effects in this paper; in a future paper we will obtain the driftkinetic equations from the appropriate limit of the finite gyroradius equations. The remainder of the paper is organized as follows. In Sec. II, relativistic Vlasov-Maxwell equations are presented as a foundation for work done in future sections. In Sec. III, an outline of the derivation of relativistic Hamiltonian guiding-center theory based on the phase-space Lagrangian Lie-transform perturbation method is presented. Sections IV and V, where relativistic Hamiltonian gyrocenter theory and nonlinear relativistic gyrokinetic Vlasov-Maxwell equations, respectively, are derived, contain the new results in nonlinear relativistic gyrokinetic theory. Lastly, Sec. VI presents a summary of our work and discusses its applications.

II. RELATIVISTIC VLASOV-MAXWELL EQUATIONS

In this section, we present two Hamiltonian formulations of the relativistic Vlasov-Maxwell equations. Each formulation is defined in terms of a Hamiltonian function H and a Poisson bracket 兵 , 其 derived from a phase-space Lagrangian. The first formulation is based on a covariant description of relativistic charged particle dynamics in an arbitrary electromagnetic field in terms of the phase-space coordinates (x 0 ⫽ct,x; p 0 ⫽E/c,p), where (x 0 ,x) denotes the space-time position of a particle and (p 0 ,p) denotes its momentumenergy coordinates. The covariant formulation treats space and time on an equal footing. The second formulation, on the other hand, treats time and space separately and makes use of the extended phase-space coordinates (x,p;t,w), where the energy coordinate w is canonically conjugate to time t. For each Hamiltonian formulation, a relativistic Vlasov equation 兵 f ,H 其 ⫽0 for the Vlasov distribution function f is written in terms of the Hamiltonian function H and the Poisson bracket 兵 , 其 . Next, self-consistent relativistic Maxwell equations are presented in which the charge and current densities are expressed in terms of moments of the relativistic Vlasov distribution function f. Lastly, an exact energymomentum conservation law for the self-consistent relativistic Vlasov-Maxwell equations is presented. A. Relativistic charged particle dynamics

The relativistic motion of a particle of rest-mass m and charge q is described in eight-dimensional phase space in terms of the space-time coordinates x ␣ ⫽(x 0 ⫽ct,x) and the four-momentum p ␣ ⬅mu ␣ ⫽(p 0 ,p), where the four-velocity is dx ␣ ⬅ 共 u 0 ⫽ ␥ c, u⫽ ␥ v兲 , u ␣⬅ d␶

共1兲

with ␥ ⬅(1⫺ 兩 v兩 2 /c 2 ) ⫺1/2 the relativistic factor and d/d ␶ ⫽ ␥ d/dt the derivative with respect to proper time. The equation of motion for the four-momentum p ␣ is

4549

d p ␣ q ␣␤ ⫽ F u␤ , d␶ c

共2兲

where summation over repeated indices is assumed and F ␣␤ ⬅ ⳵ ␣ A ␤ ⫺ ⳵ ␤ A ␣

共3兲

24

is the Faraday tensor. Here, the space-time contravariant derivative is ⳵ ␣ ⬅g ␣␤ ⳵ ␤ ⫽(⫺ ⳵ / ⳵ x 0 , ⵜ), where g ␣␤ ⫽diag (⫺1,⫹1,⫹1,⫹1) is the space-like metric tensor, and A ␣ ⫽ 共 ⌽,A兲

共4兲

is the four-dimensional electromagnetic potential. B. Noncanonical Hamiltonian structures

1. Covariant noncanonical Hamiltonian structure

We now show that the equations 共1兲 and 共2兲 possess a covariant 共cov兲 Hamiltonian structure, i.e., they can be written as dZ a /d ␶ ⫽ 兵 Z a ,H其 cov , where Z a ⫽(x 0 ,x; p 0 ,p) are eight-dimensional noncanonical phase-space coordinates, H is the covariant relativistic particle Hamiltonian and 兵 , 其 cov is the covariant relativistic noncanonical Poisson bracket on eight-dimensional phase space. To derive the covariant relativistic Poisson bracket , 兵 其 cov , we proceed along the same lines as with the nonrelativistic case 共see Ref. 25 for details兲. We begin with the covariant relativistic phase-space Lagrangian,

冉

冊

q ⌫ cov⫽ p ␣ ⫹ A ␣ dx ␣ ⬅⌫ a dZ a . c

共5兲

From the fundamental Lagrange brackets 关 Z a ,Z b 兴 ⬅⍀ ab , where26 ⍀ ab ⬅

⳵⌫b ⳵Z

a

⫺

⳵⌫a ⳵Z b

共6a兲

,

we define the Lagrange (8⫻8) matrix with components ⍀ ab ,

冉

共 q/c 兲 F ␣␤

⫺ g ␣␤

g ␣␤

0

冊

共6b兲

,

where each component in 共6b兲 is a 4⫻4 matrix. The inverse of the Lagrange matrix 共6b兲 defines the Poisson matrix,

冉

0

g ␣␤

⫺ g ␣␤

共 q/c 兲 F ␣␤

冊

共7兲

,

whose elements in turn define the fundamental Poisson brackets 兵 Z a ,Z b 其 , q c

兵 x ␣ ,x ␤ 其 cov⬅0,

兵 x ␣ ,p ␤ 其 cov⬅g ␣␤ , and 兵 p ␣ ,p ␤ 其 cov⬅ F ␣␤ .

The Poisson bracket 兵 A,B其 cov of two functions A and B on eight-dimensional phase space is thus constructed from these fundamental Poisson brackets to yield

兵 A,B其 cov⬅g ␣␤

冉

⳵A ⳵B ␣

⳵x ⳵p

␤

⫺

⳵A ⳵B ␤

⳵p ⳵x

␣

冊

⳵A ⳵B q ⫹ F ␣␤ ␣ . c ⳵p ⳵p␤

共8兲

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Phys. Plasmas, Vol. 6, No. 12, December 1999

A. J. Brizard and A. A. Chan

We note that the phase-space coordinates Z a are noncanonical because of the second term in the Poisson bracket 共8兲. Furthermore, the Poisson bracket 共8兲 is guaranteed to satisfy the Jacobi identity as well as other Poisson-bracket properties since it was derived from a phase-space Lagrangian. We now determine the relativistic Hamiltonian function H by requiring that dZ a /d ␶ ⫽ 兵 Z a ,H其 cov . Substituting A ⫽x ␣ or p ␣ and B⫽H in 共8兲, we find ␣

兵 x , H其 cov⫽g

␣␤

and its associated noncanonical Poisson bracket is derived by the same procedure used in going from 共5兲 to 共8兲 above; for two arbitrary functions F and G on extended phase space, it is given as

兵 F,G 其 ⬅

冉

⫹

⳵H

冊冉

⳵F ⳵G ⳵F ⳵G ⳵G ⳵F ⫺ ⫺ •ⵜG ⫹ ⵜF• ⳵w ⳵t ⳵t ⳵w ⳵p ⳵p

冋

冉

q ⳵F ⳵G ⳵A ⳵F ⳵G ⳵F ⳵G B– ⫻ ⫹ – ⫺ c ⳵p ⳵p ⳵t ⳵p ⳵w ⳵w ⳵p

冊 冊册

.

共12b兲

⳵p␤

Using 共11兲 and 共12b兲, the relativistic noncanonical Hamilton equations for charged particle motion are now

and

兵 p ␣ ,H其 cov⫽⫺ g ␣␤

⳵H

⳵H q ⫹ F ␣␤ ␤ . c ⳵x ⳵p

⳵H dt ⬅⫺ ⫽1, dt ⳵w

␤

We recover 共1兲 and 共2兲 from these equations if the Hamiltonian function H is H⬅

1 m g ␣␤ p ␣ p ␤ ⫽ g ␣␤ u ␣ u ␤ . 2m 2

共9兲

The covariant dynamical Eqs. 共1兲 and 共2兲 can thus be written in Hamiltonian form, ␣

␣

␣

dx /d ␶ ⫽ 兵 x ,H其 cov⫽p /m⬅u

␣

d p ␣ /d ␶ ⫽ 兵 p ␣ ,H其 cov⫽ 共 q/mc 兲 F ␣␤ p ␤ ⬅ 共 q/c 兲 F ␣␤ u ␤

冎

. 共10兲

Note that covariant relativistic charged-particle motion takes place on the surface H⬅⫺mc 2 /2, where the right side is obtained by substituting 共1兲 into 共9兲. Hence the Hamiltonian H is a Lorentz scalar.24 More importantly, however, the Hamiltonian function 共9兲 has no classical analog 共i.e., the limit c→⬁ is not defined兲. This property of the covariant Hamiltonian 共9兲 forces us to seek an alternative Hamiltonian formulation for relativistic charged-particle motion with a well-defined classical limit.

H⬅ ␥ mc ⫹q ⌽⫺w, 2

where ␥ ⬅ 冑1⫹ 兩 p/mc 兩 and, since the relativistic chargedparticle motion now takes place on the surface H⫽0, w⫽ ␥ mc 2 ⫹q ⌽ represents the total energy of a charged particle 共including its rest energy兲. In contrast to the covariant Hamiltonian 共9兲, the Hamiltonian 共11兲 has a well-defined classical limit and is an energy-like quantity. The phasespace Lagrangian, on the other hand, is

冉

冊

q ⌫⬅ p⫹ A •dx⫺w dt, c

共12a兲

冊

共13b兲

p dx ⳵ H ⬅ ⫽ ⫽v, dt ⳵ p m ␥

共13c兲

冉

冊

共13d兲 This set of Hamilton equations presents an alternative description to the Hamilton equations 共10兲 derived within the covariant formulation. In the present formulation, we note from 共13a兲 that the time coordinate t can be identified with the Hamiltonian orbit parameter. C. Relativistic Vlasov equations

The relativistic Vlasov equation describes the fact that the Vlasov distribution function is conserved along a relativistic Hamiltonian orbit in phase space. First, using the covariant formalism discussed above, this statement is mathematically expressed as dF ⬅ 兵 F,H其 cov⫽0, d␶

共14兲

where F(x,p) is the covariant Vlasov distribution function on eight-dimensional phase space. It can be explicitly written as 0⫽u ␣

共11兲

2

冉

⳵⌽ v ⳵A dw ⳵ H q ⳵ A ⳵ H , ⬅ ⫺ • ⫽q ⫺ • dt ⳵t c ⳵t ⳵p ⳵t c ⳵t

q ⳵A ⳵H q ⳵H v dp ⬅⫺ ⵜH⫹ ⫹ ⫻B⫽q E⫹ ⫻B . dt c ⳵t ⳵w c ⳵p c

2. Extended phase-space Hamiltonian structure

The relativistic Hamilton Eqs. 共1兲 and 共2兲 can alternately be written in terms of the reference-frame time t as dz a /dt ⬅ 兵 z a ,H 其 , where z a ⬅(x,p;t,w) are extended phase-space coordinates and the Hamiltonian structure is given in terms of the extended relativistic Hamiltonian 共for positive-energy particles兲,

共13a兲

⳵F ⳵x

or 0⫽

⳵ ⳵Z a

␣

⫹

q ⳵ F ␣␤ F u␤ , c ⳵p␣

冉 冊

dZ a F , d␶

共15a兲

共15b兲

since Hamilton’s equations are incompressible: ⳵ (dZ a /d ␶ )/ ⳵ Z a ⬅0. The covariant relativistic Vlasov equation 共14兲 has one additional degree of freedom compared to its non-relativistic version. This is due to the fact that the energy coordinate p 0 is not really independent of the other coordinates because the

Phys. Plasmas, Vol. 6, No. 12, December 1999

Nonlinear relativistic gyrokinetic Vlasov-Maxwell equations

Hamiltonian 共9兲 is constrained to be equal to ⫺ mc 2 /2 共i.e., p ␣ p ␣ ⫽⫺ m 2 c 2 ). Hence the covariant Vlasov distribution function F(x,p) must vanish for p 0 ⫽ ␥ mc 共for positiveenergy particles兲. This thus implies that the covariant Vlasov distribution F(x,p) should be written as27

densities are evaluated. The second expression on the right side of 共19兲 is obtained by substituting 共16兲 into the first expression and performing the p 0 -integration. The relativistic Vlasov-Maxwell equations 共14兲 or 共18兲 and 共19兲 satisfy an energy-momentum conservation law

冉

F共 x,p 兲 ⬅c ␦ H⫹

冊

mc 2 f 共 x,p,t 兲 2

⫽ ␦ 共 p 0 ⫺ ␥ mc 兲

⳵ ⳵x␣

f 共 x,p,t 兲 , ␥

共16兲

where f (x,p,t) is the Vlasov distribution on the sixdimensional phase space with coordinates (x,p) and ␥ ⬅(1 ⫹ 兩 p/mc 兩 2 ) 1/2 is the relativistic factor expressed in terms of the relativistic kinetic momentum p; note that f (x,p,t) is explicitly independent of the energy coordinate w. When 共16兲 is substituted into 共14兲, one obtains

冉

0⫽c ␦ H⫹

mc 2

2

冊

⫻

冋

冉

冊 册

⳵f u ⳵f u ⫹ •ⵜ f ⫹q E⫹ ⫻B • . ⳵t ␥ ␥c ⳵p

共17a兲

After performing a p 0 -integration on 共17a兲, we obtain the relativistic Vlasov equation on 6⫹1 phase space,

冉

冊

⳵f ⳵f v 0⫽ ⫹v•ⵜ f ⫹q E⫹ ⫻B • , ⳵t c ⳵p

共17b兲

which looks exactly like the nonrelativistic Vlasov equation except that p is the relativistic kinetic momentum p⫽m ␥ v. 27 The relativistic Vlasov equation 共17b兲 can also be written as df ⬅ 兵 f ,H 其 ⫽0, dt

共18兲

where the relativistic Hamiltonian is given by 共11兲 and the Poisson bracket is given by 共12b兲. Note that this new Hamiltonian formulation has separated the components of the electromagnetic four-potential (⌽,A): the electrostatic potential ⌽ now appears in the Hamiltonian 共11兲 while the magnetic vector potential A and its derivatives remain in the phasespace Lagrangian 共12a兲 and the Poisson bracket 共12b兲. D. Self-consistent relativistic Vlasov-Maxwell equations

To obtain a self-consistent relativistic Vlasov-Maxwell theory one combines the covariant relativistic Vlasov equation, 共14兲 or 共18兲, with the Maxwell equations

⳵␤ F

␣␤

共 r 兲 ⫽4 ␲

⫽4 ␲

兺 q冕

p␣ F共 x, p 兲 d Z␦ 共 x⫺r 兲 mc 8

共20a兲

␣␤ where the energy-momentum stress tensor T ␣␤ ⬅T ␣␤ M ⫹T V is divided into the Maxwell 共field兲 part,

T ␣␤ M ⬅

1 1 ␣␮ F g ␮␯F ␯␤, 共 F F ␯ ␮ 兲 g ␣␤ ⫺ 16␲ ␮ ␯ 4␲

and the Vlasov 共particle兲 part, T V␣␤ ⬅

兺

冕

d 4 pmu ␣ u ␤ F⫽

兺

冕

d3p

共20b兲

mu ␣ u ␤ f . 共20c兲 ␥

Of particular importance in the development of selfconsistent Hamiltonian gyrokinetic particle simulation techniques is the energy conservation law

兵 f ,H其 cov

⬅ ␦ 共 p 0 ⫺ ␥ mc 兲

␣␤ 共 T ␣␤ M ⫹T V 兲 ⫽0,

4551

4

⳵E ⫹ⵜ•S⫽0, ⳵t

共21a兲

where the energy density is E⫽

1 共 兩 E兩 2 ⫹ 兩 B兩 2 兲 ⫹ 8␲

兺

冕

d 3 p ␥ mc 2 f ,

共21b兲

and the energy-density flux is S⫽

c E⫻B⫹ 4␲

兺

冕

d 3 pmc 2 uf .

共21c兲

We return to this conservation law in Sec. V C, where an approximate energy conservation law is derived for the nonlinear relativistic gyrokinetic Vlasov-Maxwell equations. In summary, two Hamiltonian formulations for relativistic charged-particle motion are presented in this section: the covariant formulation based on the Hamiltonian 共9兲 and the Poisson bracket 共8兲; and the extended phase-space formulation based on the Hamiltonian 共11兲 and the Poisson bracket 共12b兲. For each formulation, self-consistent relativistic Vlasov-Maxwell equations 关共14兲 or 共18兲 and 共19兲兴 are given in addition to the exact energy-momentum conservation law 共20a兲. In the remainder of this paper, we adopt the extended phase-space formulation since its Hamiltonian function has a well-defined classical limit and the separation of the time coordinate from the other phase-space coordinates allows us to transform the latter coordinates without transforming the time coordinate itself.

III. RELATIVISTIC HAMILTONIAN GUIDING-CENTER THEORY

兺 q 冕 d 6 z ␦ 3共 x⫺r兲 mc ␥ f 共 x,p;t 兲 , p␣

A. Preliminary coordinate transformation

共19兲 where the sum is over particle species and r ␣ ⫽(ct,r) denotes the space-time location where the charge and current

Relativistic charged-particle motion in a strong magnetic field is characterized by three disparate time scales: the fast gyromotion time scale associated with gyration about a single magnetic field line, the intermediate parallel 共or

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bounce兲 time scale associated with motion along a field line, and the slow drift time scale associated with motion across magnetic-field lines.28 In the following analysis 共only results are presented here兲, we work in a preferred reference frame in which the background magnetic field B0 ⬅ⵜ⫻A0 is time independent ( ⳵ A0 / ⳵ t⬅0) and the background electrostatic field ⌽ 0 is zero. We note that the choice ⌽ 0 ⬅0 is not crucial to the present analysis and that nonrelativistic gyrokinetic theory in the presence of background nonuniform electrostatic fields has been systematically derived elsewhere using phase-space Lagrangian methods.19,29 In the present paper, any quasistatic electrostatic potential is thus treated as a perturbation. We further note that the covariant formulation of relativistic guiding-center Hamiltonian theory has been developed in Ref. 30 in which the background electromagnetic field is expressed in terms of ⌽ 0 and A0 . The analysis begins with the introduction of the following local momentum coordinates (p 储 0 , ␮ 0 , ␪ 0 ), ˆ ⫺ 冑2m ␮ 0 B 0 共 aˆ cos ␪ 0 ⫹cˆ sin ␪ 0 兲 , p⬅p 储 0 b

共22兲

where p 储 0 ⬅m ␥ v•bˆ is the component of the relativistic momentum parallel to the local background magnetic field (B0 •bˆ⬅B 0 ), ␮ 0 ⬅m ␥ 2 兩 v⬜ 兩 2 /2B 0 is the relativistic magnetic moment, while ␪ 0 ⬅tan⫺1 关 (⫺p•cˆ)/(⫺p•aˆ) 兴 is the local gyrorangle, and (aˆ,bˆ,cˆ) form a right-handed orthogonal unitvector set. The Jacobian for this preliminary coordinate transformation is mB 0 . Inserting the local momentum coordinates into 共12a兲, the unperturbed phase-space Lagrangian is ⌫ 0⬅

冉

冊

q ⳵␳ ˆ ⫹m ␻ B 0 •dx⫺wdt, A ⫹p 储 0 b c 0 ⳵␪0

冑

2 ␮ 0B 0 共 cˆ cos ␪ 0 ⫺aˆ sin ␪ 0 兲 , m

共24兲

共25兲

After this preparatory transformation, we now proceed with the asymptotic elimination of the fast gyromotion time scale from the relativistic Hamiltonian system represented by 共23兲 and 共25兲.

B. Relativistic guiding-center Hamiltonian dynamics

Using the phase-space Lagrangian Lie-perturbation method on the unperturbed phase-space Lagrangian 共23兲 and the unperturbed Hamiltonian 共25兲, we obtain 共to zeroth-order in magnetic-field nonuniformity兲 the unperturbed relativistic guiding-center (gc) phase-space Lagrangian,

冉

冊

q ˆ •dR⫹ ⑀ 共 mc/q 兲 ␮ d ␪ ⫺W dt, A ⫹p储 b ⌫ 0gc ⫽ ⑀c 0

⬅mc 2 冑1⫹ 共 2 ␮ B 0 /mc 2 兲 ⫹ 共 p 储 /mc 兲 2 ⫺W.

共27兲

In 共26兲, R denotes the guiding-center position, (p 储 , ␮ , ␪ ) are the guiding-center momentum coordinates, and (W,t) are the guiding-center energy coordinate and time coordinate 共we henceforth omit the subscript 0 to denote the background magnetic field兲. In addition, the dimensionless parameter ⑀ Ⰶ1 denotes terms of order of the ratio of the gyroradius 兩 ␳兩 over the magnetic-field nonuniformity length scale L B 共see Ref. 28 for details兲. The relation between the guiding-center phase-space coordinates Z a ⫽(t,W,R,p 储 , ␮ , ␪ ) and the local phase-space coordinates z a ⫽(t,w,x,p 储 0 , ␮ 0 , ␪ 0 ) is given to lowest order in magnetic-field nonuniformity as w⫽W x⫽R⫹ ⑀ ␳ p 储0⫽ p 储

␮0 ⫽␮ ␪0 ⫽␪

冧

共28兲

,

where ␳( ␮ , ␪ )⬅ ␳0 ( ␮ 0 , ␪ 0 ) denotes the gyroradius vector expressed in guiding-center coordinates. The Poisson bracket corresponding to the relativistic guiding-center phase-space Lagrangian 共26兲 is defined in terms of two arbitrary functions F and G on the guidingcenter eight-dimensional phase space as

兵 F,G 其 gc ⬅

冉

冊

冉 冉

⳵F ⳵G ⳵F ⳵G ⳵G ⳵F B* ⫺ ⫹ • ⵜF ⫺ ⵜG ⳵W ⳵t ⳵t ⳵W ⳵ p储 ⳵p储 * B储

⫺

B m ␻ BB * 储

•ⵜF⫻ⵜG⫹

冊

冊

q ⳵F ⳵G ⳵F ⳵G ⫺ , mc ⳵ ␪ ⳵ ␮ ⳵ ␮ ⳵ ␪ 共29a兲

while the unperturbed Hamiltonian function is H 0 ⫽mc 2 冑1⫹ 共 2 ␮ 0 B 0 /mc 2 兲 ⫹ 共 p 储 0 /mc 兲 2 ⫺w.

H 0gc ⫽ ␥ mc 2 ⫺W

共23兲

where ␻ B ⬅qB 0 /mc is the signed 共rest-mass兲 gyrofrequency and the lowest-order relativistic gyroradius vector ␳0 is bˆ⫻u 1 ␳0 ⬅ ⫽ ␻B ␻B

and the unperturbed relativistic guiding-center Hamiltonian function

where the small parameter ⑀ was omitted for simplicity and B* ⬅B⫹ 共 cp 储 /q 兲 ⵜ⫻bˆ ˆ ˆ ˆ B* 储 ⬅b•B* ⫽B 关 1⫹ 共 p 储 /m ␻ B 兲 b•ⵜ⫻b 兴

.

共29b兲

We note that the Jacobian for the guiding-center transformation is B * 储 /B and thus the Jacobian for the total transformation (t,w,x,p)→(t,W,R,p 储 , ␮ , ␪ ) is mB * 储 . Using the relativistic guiding-center Hamiltonian 共27兲 and the relativistic guiding-center Poisson bracket 关共29a兲 and 共29b兲兴, we obtain the relativistic guiding-center equations of motion: dZ a /dt⫽ 兵 Z a ,H 0gc 其 gc . The relativistic guidingcenter equations for the canonically conjugate coordinates (t,W) are dt/dt⫽⫺ ⳵ H 0gc / ⳵ W⫽1 dW/dt⫽ ⳵ H 0gc / ⳵ t⫽0

共26兲

冎

冎

.

The relativistic guiding-center velocity

共30a兲

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Nonlinear relativistic gyrokinetic Vlasov-Maxwell equations

dR ⳵ H 0gc B* cbˆ ⫽ ⫹ ⫻ⵜH 0gc dt ⳵p储 B* qB * 储 储 ⬅

q q ⌫ 1gc ⫽ A1 共 R⫹ ␳,t 兲 • 共 dR⫹d ␳兲 ⬅ A1gc • 共 dR⫹d ␳兲 , c c 共32a兲

p 储 B* cbˆ ⫹ ⫻ ␮ ⵜB ␥m B* q␥B* 储 储

共30b兲

includes parallel motion (bˆ•dR/dt⬅p 储 / ␥ m) as well as the gradient-B and curvature drifts 共which characterize drift motion兲. The relativistic guiding-center force equation for the parallel momentum dp 储 ␮ B* B* ⫽⫺ •ⵜH 0gc ⬅⫺ •ⵜB, dt B* ␥B* 储 储

共30c兲

shows the mirror force associated with magnetic-field nonuniformity along the magnetic field lines 共which characterizes the bounce motion兲. We point out that 共30b兲 and 共30c兲 agree with the standard relativistic guiding-center equations found in Northrop28 if we neglect phase-space volumepreserving terms 共such as the difference between B * 储 and B); see 共30e兲 below. Lastly, the relativistic guiding-center equations for the canonically conjugate coordinates ( ␮ , ␪ ) are d ␮ /dt⫽⫺ 共 q/mc 兲 ⳵ H 0gc / ⳵ ␪ ⫽0 d ␪ /dt⫽ 共 q/mc 兲 ⳵ H 0gc / ⳵ ␮ ⬅ ␻ B ␥ ⫺1

冎

⳵Za

冉

B* 储

冊

dZ a ⬅0, dt

共30d兲

共30e兲

i.e., the relativistic guiding-center Hamiltonian flow is incompressible; note that the appearance of the Jacobian B * 储 in 共30e兲 ensures the property of phase-space volume conservation.

IV. RELATIVISTIC HAMILTONIAN GYROCENTER THEORY

When small-amplitude electromagnetic-field fluctuations ⌽ 1 (x,t) and A1 (x,t) are introduced into the relativistic guiding-center Hamiltonian theory, the guiding-center phasespace Lagrangian 共26兲 and the guiding-center Hamiltonian 共27兲 are perturbed, ⌫ gc ⫽⌫ 0gc ⫹ ⑀ ␦ ⌫ 1gc H gc ⫽H 0gc ⫹ ⑀ ␦ H 1gc

冎

,

and the first-order guiding-center Hamiltonian is H 1gc ⫽q ⌽ 1 共 R⫹ ␳,t 兲 ⬅q ⌽ 1gc ,

共32b兲

where the perturbation fields ⌽ 1 and A1 are evaluated at the particle position x⬅R⫹ ␳ at time t and we define ⌽ 1gc ⬅⌽ 1 (R⫹ ␳,t) and A1gc ⬅A1 (R⫹ ␳,t). The subscript gc is used here to denote the fact that the fields ⌽ 1 (R 储 ,R⬜ ⫹ ␳,t) and A1 (R 储 ,R⬜ ⫹ ␳,t) have acquired gyroangle-dependence through the gyroradius vector ␳. The perturbations in 关共32a兲 and 共32b兲兴 reintroduce gyroangle dependence into the guiding-center Hamiltonian dynamics 共31兲. To remove this fast time-scale dependence, we again use the phase-space Lagrangian Lie-perturbation method. The outcome of this analysis yields the gyrocenter Hamiltonian dynamics, i.e., perturbed guiding-center Hamiltonian dynamics from which the fast gyroangle dependence has been asymptotically eliminated. In what follows, we shall treat the nonuniformity in the background field only to lowest order, i.e., d ␳⬅( ⳵␳/ ⳵ ␮ )d ␮ ⫹( ⳵␳/ ⳵ ␪ )d ␪ in 共32a兲. A. Low-frequency gyrokinetic ordering

.

These equations indicate that, through the asymptotic elimination of the fast gyromotion time scale, an adiabatic invariant, the relativistic magnetic moment ␮ , has been constructed. Hence, within relativistic guiding-center Hamiltonian theory, the relativistic magnetic moment ␮ is a constant of the motion. We also note that the guiding-center Hamilton equations 关共30a兲–共30d兲兴 satisfy the identity

⳵

4553

共31兲

where ⑀ ␦ is an ordering parameter associated with the amplitude of the electromagnetic field perturbations. The first-order guiding-center phase-space Lagrangian is20

The electromagnetic-field perturbations in 关共32a兲 and 共32b兲兴 are assumed to satisfy the following space-time scale ordering,31

␻ / ␻ c ⫽O共 ⑀ ␦ 兲 k 储 ␳ ⫽O共 ⑀ ␦ 兲 兩 k⬜ 兩 ␳ ⫽O共 1 兲

冎

,

共33兲

where ␳ and ␻ c ⬅ ␻ B / ␥ are a typical gyroradius and gyrofrequency for a particle of mass m and charge q, respectively, while ( ␻ ,k 储 ,k⬜ ) are the characteristic frequency, parallel and perpendicular wave numbers of the electromagnetic-field fluctuations of interest. This is the usual low-frequency gyrokinetic ordering31 in which the characteristic fluctuation time scales are long compared to the gyroperiod 2 ␲ / ␻ c , the characteristic wavelengths parallel to the magnetic field 2 ␲ /k 储 are long compared to the typical gyroradius ␳ , while the characteristic wavelengths perpendicular to the magnetic field 2 ␲ / 兩 k⬜ 兩 are comparable to ␳ . The space-time scale ordering used for the background field, on the other hand, considers a characteristic time scale ordered at O( ⑀ ␦3 ) compared to the gyroperiod 2 ␲ / ␻ c and a characteristic spatial scale ordered at ⑀ ⬅O( ⑀ ␦ ) compared to the typical gyroradius ␳ . Since the present nonlinear analysis retains only terms up to second order in ⑀ ␦ in the Hamiltonian, the background fields are thus considered as timeindependent nonuniform fields with weak spatial gradients ( ␳ 兩 ⵜ ln B兩Ⰶ1). B. Phase-space Lagrangian Lie-perturbation method

To remove the gyroangle dependence reintroduced by the electromagnetic field perturbations in 共31兲, we use the phase-space Lagrangian Lie-transform perturbation method20,32 wherein a new 共gyrocenter兲 phase-space La-

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A. J. Brizard and A. A. Chan

grangian ⌫ gy and a new 共gyrocenter兲 Hamiltonian H gy are constructed. These new gyrocenter expressions are given as asymptotic expansions in powers of ⑀ ␦ , ⌫ gy ⫽⌫ 0gy ⫹ ⑀ ␦ ⌫ 1gy ⫹ ⑀ ␦2 ⌫ 2gy ⫹••• H gy ⫽H 0gy ⫹ ⑀ ␦ H 1gy ⫹ ⑀ 2␦ H 2gy ⫹•••

冎

共34兲

,

where the lowest-order terms ⌫ 0gy ⬅⌫ 0gc and H 0gy ⬅H 0gc are given by the unperturbed guiding-center expressions 共26兲 and 共27兲, respectively. The first- and second-order terms for the phase-space Lagrangian are20,32 ⌫ 1gy ⫽⌫ 1gc ⫺i 1 •⍀ 0gc ⫹dS 1

⌫ 2gy ⫽⫺ i 2 •⍀ 0gc ⫺i 1 •⍀ 1gc ⫹i 1 •d 共 i 1 •⍀ 0gc 兲 /2⫹dS 2

冎

, 共35a兲

and the first- and second-order terms for the Hamiltonian are H 1gy ⫽H 1gc ⫺g 1 •dH 0gc

H 2gy ⫽⫺ g 2 •dH 0gc ⫺g 1 •dH 1gc ⫹g 1 •d 共 g 1 •dH 0gc 兲 /2

冎

,共35b兲

where g n and S n are the nth-order generating vector and the phase-space gauge function, respectively. In 共35a兲, where the elements of the two-form ⍀ gc are defined as in 共6a兲, we have i n •⍀ gc ⫽g an (⍀ gc ) ab dZ b , while in 共35b兲 we have g n •dH gc ⫽g an ⳵ H gc / ⳵ Z a . We note that, within the phase-space Lagrangian formalism, the Hamilton equations are independent of the choice of phase-space gauge functions S n .

We choose the generating vectors g n in 关共35a兲 and 共35b兲兴 so that the gyrocenter phase-space Lagrangian 共31兲 retains the guiding-center form given by 共26兲

冉

冊

q ˆ •dR ¯储 b ¯ ⫹ 共 mc/q 兲 ␮ ¯ d¯␪ ⫺W ¯ d¯t , A⫹p c

共36兲

i.e., (⌫ gy ) n ⬅0 for n⭓1 and the gyrocenter phase-space ¯ ,R ¯ , ¯p 储 , ␮ ¯ ,¯␪ ) are transformation is canonical. Here ¯Z a ⫽(¯t ,W the new gyrocenter phase-space coordinates 关see 共45a兲–共45f兲 below兴. Solving 共35a兲 for ⌫ 1gy ⬅0 and ⌫ 2gy ⬅0 共by inverting the Lagrange ⍀-matrices兲 yields the following expressions for the first- and second-order generating vectors, q ¯ a 其 gc ⫹ A1gc • 兵 R ¯ ⫹ ¯␳,Z ¯ a 其 gc g a1 ⫽ 兵 S 1 ,Z c

The first- and second-order phase-space gauge functions S 1 and S 2 in 关共37a兲 and 共37b兲兴 are chosen by demanding that the first- and second-order gyrocenter Hamiltonians H 1gy and H 2gy be gyroangle-independent, respectively. From 共32b兲, 共35b兲 and 共37a兲, we find the expression for the firstorder gyrocenter Hamiltonian

冉

H 1gy ⫽⫺ 兵 S 1 ,H 0gy 其 gc ⫹q ⌽ 1gc ⫺ ⬅⫺ 兵 S 1 ,H 0gy 其 gc ⫹K 1gc .

q ¯ ⫹ ¯␳,Z ¯ a 其 gc , G ⫻B1gc • 兵 R 2c 1

冊

共38兲

The phase-space gauge functions S n are chosen to have the following properties 共for all n): ¯ ⬅0, 具 S n 典 ⬅0 and ⳵ S n / ⳵ W

共39a兲

where 具典 denotes averaging with respect to ¯␪ 共with the other gyrocenter phase-space coordinates held constant兲. The first choice in 共39a兲 simplifies the analysis while the second choice ensures that the time coordinate remains unaffected by the gyrocenter extended phase-space transformation. Since we want H 1gy to be gyroangle-independent, noting that the gyroangle-averaging operation commutes with the guiding-center Poisson bracket, it can be shown that the firstorder gyrocenter Hamiltonian can be written as H 1gy ⬅ 具 K 1gc 典 .

共39b兲

共37a兲

˜ 1gc , 兵 S 1 ,H 0gy 其 gc ⫽K 1gc ⫺ 具 K 1gc 典 ⬅K

共37b兲

¯ ⫹ ¯␳)⬅ 兵 S 1 ,R ¯ ⫹ ¯␳其 gc and B1gc ⬅ⵜ ¯ ⫻A1gc where G1 ⫽g 1 •d(R is the perturbed magnetic field expressed in guiding-center phase-space coordinates. At this point, the gauge functions S 1 and S 2 are still arbitrary; they will be chosen next by demanding that the gyrocenter Hamiltonian H gy be independent of the gyrocenter gyroangle ¯␪ .

共40a兲

which is obtained by subtracting 共39b兲 from 共38兲. This equa˜ 1gc ⬅ ␥ ⫺1 L ␶ S 1 , where the option can also be written as K ¯ 储 /m) bˆ–ⵜ ¯ ⫹ ␻ B ⳵ / ⳵¯␪ denotes the total erator L ␶ ⬅ ¯␥ ⳵ / ⳵ t⫹(p proper-time derivative along unperturbed particle orbits. To lowest order in the low-frequency gyrokinetic ordering 共33兲, the operator L ␶ becomes ␻ B ⳵ / ⳵¯␪ , and the solution for S 1 is written explicitly as ˜ ¯ S 1 ⫽ ¯␥ L ⫺1 ␶ K 1gc ⬅ 共 ␥ / ␻ B 兲

冕

˜ 1gc d¯␪ . K

共40b兲

We easily verify that 共39a兲 is satisfied for n⫽1. We note that for high-frequency gyrokinetics,22 the inverse operator L ␶⫺1 involves an integration along unperturbed particle orbits. Next, from 共35b兲 and 关共37a兲–共37b兲兴, we find the expression for the second-order gyrocenter Hamiltonian

and ¯ a 其 gc ⫹ g a2 ⫽ 兵 S 2 ,Z

¯u •A ¯␥ c 1gc

The phase-space gauge function S 1 is determined from the equation

C. Gyrocenter phase-space Lagrangian

⌫ gy ⬅

D. Nonlinear gyrocenter Hamiltonian

H 2gy ⫽

q2 2c 2

冓 再 冎 冔 ¯ ⫹ ¯␳, A1gc • R

¯u ¯␥

•A1gc

gc

1 ˜ 1gc ,K ˜ 1gc 其 gc 典 , ⫺ 具 兵 ¯␥ L ␶⫺1 K 2

共41兲

where the second term is evaluated only to lowest order in the low-frequency gyrokinetic ordering 共33兲. This expression shows how ponderomotive effects enter into the lowfrequency gyrokinetic formalism 共the derivation of a second-

Phys. Plasmas, Vol. 6, No. 12, December 1999

Nonlinear relativistic gyrokinetic Vlasov-Maxwell equations

order ponderomotive Hamiltonian for arbitrary frequencies is discussed in Ref. 33兲. The two terms in 共41兲 contain the following relativistic corrections:

再

¯u ¯ ⫹ ¯␳, R ¯␥

冎

冉

¯u¯u 1 ⫽ I⫺ 2 2 ¯␥ c m ¯␥ gc

冊

共42a兲

and

˜ 1gc ,K ˜ 1gc 其 gc 典 ⫺ ⫽ ¯␥ 具 兵 L ␶⫺1 K

1 ¯␥ mc

˜ 1gc 兲 2 典 , 具共 K 2

共42b兲

where the first terms in 关共42a兲 and 共42b兲兴 correspond to relativistic generalizations of the classical terms (c→⬁) while the second terms are relativistic corrections. The nonlinear gyrocenter Hamiltonian function H gy ⫽H 0gy ⫹ ⑀ ␦ H 1gy ⫹••• can now be written as ¯, H gy ⬅ ¯␥ mc 2 ⫹⌿ gy ⫺W

共43a兲

where the nonlinear perturbation gyrocenter Hamiltonian is

⫹ ⑀ 2␦

q

2

2c

2

⑀ 2␦ ˜ 1gc ,K ˜ 1gc 其 gc 典 具 兵 ¯␥ L ␶⫺1 K 2

冓 再 冎 冔 ¯ ⫹ ¯␳, A1gc • R

¯u ¯␥

•A1gc .

共43b兲

We note that all perturbation effects appear exclusively in the gyrocenter Hamiltonian 共43a兲, while the gyrocenter Poisson bracket is identical to the unperturbed guiding-center Poisson bracket 共29a兲. In the classical limit, we recover the previous nonrelativistic nonlinear gyrokinetic results20 from 共43b兲.

E. Gyrocenter phase-space coordinates

The new gyrocenter phase-space coordinates ¯Z a ¯ ,R ¯ ,p ¯ 储 ,␮ ¯ ,¯␪ ) are related to the old guiding-center ⫽(¯t ,W phase-space coordinates Z a ⫽(t,W,R, p 储 , ␮ , ␪ ) by the asymptotic expansion in powers of ⑀ ␦ ,

兲

共

共44兲

where g a1 and g a2 are defined in 共37a兲 and 共37b兲. When 共44兲 is written explicitly using 关共37a兲 and 共37b兲兴, we find first that the time coordinate is unaffected by the gyrocenter extended phase-space transformation t⫽¯t 共 to all orders in ⑀ ␦ 兲 ,

共45a兲

because of the choice 共39a兲 for the phase-space gauge functions S n . The remaining expressions are ¯ ⫹O共 ⑀ 2␦ 兲 , W⫽W ¯ ⫹⑀␦ R⫽R

共45b兲

冉

冊

⳵S1 q ¯S ⫹ A ˆ ⫹O共 ⑀ ␦2 兲 , ⫻ ⵜ b 1 1gc ⫹ ⑀ ␦ c ¯ qB * ⳵ p 储 储 共45c兲 cbˆ

冊 冊

共45e兲

¯ ␻B ⳵S1 q ⳵␳ ⫹ A1gc • ⫹O共 ⑀ ␦2 兲 , B ⳵␮ c ¯ ¯ ⳵␮

共45f兲

where the low-frequency gyrokinetic ordering 共33兲 was used. ¯ is shifted We note from 共45c兲 that the gyrocenter position R from the guiding-center position R as a result of the electromagnetic fluctuations and that this shift has both gyroangledependent and gyroangle-independent parts. Furthermore, the gyrocenter parallel momentum ¯p 储 can be interpreted from 共45d兲 as a canonical momentum, while 共45e兲 shows that the ¯ is constructed as gyrocenter relativistic magnetic moment ␮ an asymptotic expansion in powers of ⑀ ␦ which makes it into an adiabatic invariant for gyrocenter Hamiltonian dynamics 关see 共46d兲 below兴.

F. Nonlinear relativistic gyrocenter Hamiltonian dynamics

gc

¯ a ⫺ ⑀ ␦ g a1 ⫺ ⑀ 2␦ g a2 ⫺ 21 g 1 •dg a1 ⫹O共 ⑀ 3␦ 兲 , Z a ⫽Z

冉 冉

共45d兲

¯ ␻B ⳵S1 q ⳵␳ ⫹ A1gc • ⫹O共 ⑀ 2␦ 兲 , B ⳵¯␪ c ⳵¯␪

¯ ⫺⑀␦ ␮⫽␮

␪ ⫽¯␪ ⫹ ⑀ ␦

˜ 1gc ,K ˜ 1gc 其 gc 典 具 兵 ¯␥ L ␶⫺1 K

⌿ gy ⬅ ⑀ ␦ 具 K 1gc 典 ⫺

q p 储 ⫽ ¯p 储 ⫺ ⑀ ␦ A1gc •bˆ⫹O共 ⑀ 2␦ 兲 , c

4555

The nonlinear relativistic gyrocenter Hamilton equations ¯ a /dt⫽ 兵 ¯Z a ,H gy 其 gc are dZ ¯ dW ⳵ ⌿ gy ⫽ , dt ⳵¯t

共46a兲

冉

冊

冉

冊

¯ ¯p 储 ¯ ⳵ ⌿ gy B* ␮ cbˆ dR ¯ⵜ B⫹ⵜ ¯⌿ ⫹ ⫽ ⫹ ⫻ gy , dt ¯␥ m ¯ * ⳵ ¯p 储 B * qB ␥ 储 储 共46b兲

冉

冊

¯ B* ␮ d ¯p 储 ¯ⵜ B⫹ⵜ ¯⌿ ⫽⫺ • gy , dt ¯ B* ␥ 储

共46c兲

¯ d␮ ⬅0. dt

共46d兲

In 共46b兲, the term ⳵ ⌿ gy / ⳵ ¯p 储 is associated with perturbed parallel motion and perturbed curvature drift, while the term ¯ ⌿ is associated with a perpendicular perturbed E⫻B ⵜ gy ¯ ⌿ includes the effects of a flow. In 共46c兲, the term B* •ⵜ gy parallel electric field, although the usual induction term is absent from the right side of 共46c兲. This is due to the fact that ¯p 储 , as defined in 共45d兲, is the parallel component of the gyrocenter canonical momentum; this important feature provides useful computational advantages in gyrokinetic particle simulations.20,34 We also note that the nonlinear relativistic gyrocenter Hamilton equations 关共46a兲–共46c兲兴 satisfy the incompressibility condition

冉

冊

¯a ⳵ dZ *储 B ⫽0. dt ⳵ ¯Z a

共47兲

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Note that since ¯p 储 is the canonical parallel momentum 关see 共45c兲兴, the perturbed parallel induction term bˆ• ⳵ A1gc / ⳵ t does not appear in the gyrocenter Hamilton equations 关共46a兲–共46c兲兴.

We note that, by construction, the gyrocenter Vlasov distribution function f gy is independent of the gyrocenter gyroangle ¯␪ to all orders in ⑀ ␦ . B. Low-frequency gyrokinetic Maxwell equations

The low-frequency gyrokinetic Maxwell equations are

V. NONLINEAR RELATIVISTIC GYROKINETIC VLASOV-MAXWELL EQUATIONS

⫺

A. Nonlinear relativistic gyrokinetic Vlasov equation

⑀␦ 2 ␣ ⵜ A 共 r,t 兲 ⫽ 4␲ ⬜ 1

The nonlinear relativistic gyrokinetic Vlasov equation in gyrocenter phase space is simply

兵 f gy ,H gy 其 gc ⫽0,

共48a兲

or using 共36兲 and 共29a兲, 0⫽

冉

冊

⳵ f gy B* ⳵ H gy cbˆ ¯H ¯ ⫹ ⫹ ⫻ⵜ gy •ⵜ f gy ⳵t B* ⳵ ¯p 储 qB * 储 储 ⫺

B* B* 储

¯H •ⵜ gy

⳵ f gy . ⳵ ¯p 储

¯ ⫽ ¯␥ mc 2 ⫹⌿ gy , h gy ⬅H gy ⫹W

共49a兲

the nonlinear relativistic gyrokinetic Vlasov equation can also be written as

⳵ f gy ⫹ 兵 f gy ,h gy 其 gc ⫽0. ⳵t

共49b兲

From 共48b兲 we recover the linear gyrokinetic Vlasov equation previously derived by Littlejohn,21 who used the Hamiltonian Lie-perturbation method, and by Tsai, Van Dam and Chen,22 who used the standard method of gyroangleaveraging the relativistic Vlasov equation directly. The relationship between the gyrocenter Vlasov distribution function f gy , the guiding-center Vlasov distribution function f gc , and the particle Vlasov distribution function f is discussed in terms of two operators:20 the guiding-center operator Tgc and the gyrocenter operator Tgy . To lowest order in magnetic-field nonuniformity, the guiding-center operator is Tgc ⬅exp(⫺ ⑀ ␳•ⵜ) and the relation between the guiding-center Vlasov distribution function f gc and the particle Vlasov distribution function f is expressed in terms of the scalar-invariance property: f gc (R, p 储 , ␮ , ␪ ,t) ⫺1 f (R,p 储 , ␮ , ␪ ,t)⫽ f (R⫹ ␳, p 储 , ␮ , ␪ ,t). ⬅T gc For the gyrocenter transformation, the scalar-invariance property yields f gy ⬅T ⫺1 gy f gc , where Tgy ⬅exp共 ⑀ ␦ g 1 •d⫹ ⑀ 2␦ g 2 •d⫹••• 兲 ,

共50a兲

with g 1 and g 2 given in 关共37a兲 and共37b兲兴. When 共50a兲 is expanded up to second order in ⑀ ␦ , we find f gy ⬅T

⫻

⫺1 gy f gc ⫽ f gc ⫺ ⑀ ␦ g 1 •d f gc

⫺ ⑀ 2␦ 关 g 2 •d f gc ⫺ 21 g 1 •d 共 g 1 •d f gc 兲兴 ⫹O共 ⑀ ␦3 兲 . 共50b兲

冉 冊

¯u ␣ ¯ ,p ¯ 储 ,␮ ¯ ,¯␪ ;t 兲 , T f 共R ¯␥ c gy gy

共51兲

¯ ), and we use the gauge where A 1␣ ⬅(⌽ 1 ,A1 ), ¯u ␣ ⬅( ¯␥ c,u 35 Also in 共51兲, we find d 6¯Z condition ⵜ⬜ •A1 ⫽0. 3¯ ¯ ¯ ¯ 3 ¯ ¯ ⬅mB * 储 d R d p 储 d ␮ d ␪ , the term ␦ (R⫹ ␳⫺r) relates the particle source point r at which the fields are evaluated to the ¯ , and 共ignoring terms of order ⑀ 2␦ ), gyrocenter position R

共48b兲

By defining the new gyrocenter Hamiltonian

兺 q 冕 d 6¯Z ␦ 3共 R¯⫹ ¯␳⫺r兲

Tgy f gy ⬅ f gy ⫹ ⑀ ␦ 兵 S 1 , f gy 其 gc ⫹ ⑀ ␦

qA1gc ¯ ⫹ ¯␳, f gy 其 gc . •兵R c 共52兲

Here terms of order ⑀ 2␦ are necessarily omitted in 共52兲 to ensure energy conservation.20,36,37

C. Nonlinear gyrokinetic energy conservation law

The nonlinear relativistic gyrokinetic Vlasov-Maxwell equations 共48b兲 and 共51兲 possess the following energy conservation law: d 共 E ⫹E GV 兲 ⬅O共 ⑀ ␦3 兲 , dt M

共53兲

i.e., energy is conserved to the order considered in this work. Here, the field energy is EM⬅

冕

d 3x 共 兩 ⑀ ␦ ⵜ⬜ ⌽ 1 兩 2 ⫹ 兩 B⫹ ⑀ ␦ ⵜ⬜ ⫻A1 兩 2 兲 , 8␲

共54兲

and the gyrokinetic particle energy is E GV ⬅

兺

冕

d 6¯Z 具 T

⫺1 ¯ 2 gy 共 ␥ mc 兲 典 f gy

,

共55兲

where 2 ¯ 2 具 T ⫺1 gy 共 ␥ mc 兲 典 ⬅h gy ⫺ ⑀ ␦ q 具 ⌽ 1gc 典 ⫹ ⑀ ␦ q 具 兵 S 1 ,⌽ 1gc 其 gc 典 .

共56兲

We note that the energy conservation law 共53兲 is consistent with the time-scale ordering of the background fields discussed in Sec. IV A. We demonstrate the approximate energy conservation law 共53兲 as follows. After some simple manipulations, the expression for d(E M ⫹E GV )/dt can be written as

兺

冕 冋

d 6¯Z 共 ⳵ t f gy 兲 h gy ⫺

册

⑀ 2␦ ˜ 1gc 其 gc 典 f 共 具 兵 ⳵ t S 1 ,K 2 gy

˜ 1gc 其 gc 典 兲 . ⫺ 具兵S 1 , ⳵ tK

共57兲

Phys. Plasmas, Vol. 6, No. 12, December 1999

Nonlinear relativistic gyrokinetic Vlasov-Maxwell equations

The first term on the right vanishes because

ACKNOWLEDGMENTS

共 ⳵ t f gy 兲 h gy ⫽⫺ 兵 f gy ,h gy 其 gc h gy ⬅⫺ 兵 共 f gy h gy 兲 ,h gy 其 gc , 共58a兲

and the guiding-center Poisson bracket has the following property 兰 d 6¯Z 兵 a,b 其 gc ⬅0 for arbitrary functions a and b. The second and third terms satisfy the following identity: ˜ 1gc 其 gc 典 ⫽ 具 兵 S 1 , ⳵ t K ˜ 1gc 其 gc 典 具 兵 ⳵ t S 1 ,K ⫹ 具 兵 H 0gy , 兵 S 1 , ⳵ t S 1 其 gc 其 gc 典 ,

共58b兲

which is obtained by using 共40a兲 and the Jacobi identity for the gyrocenter Poisson bracket. Using the gyroangleindependence of the guiding-center Poisson bracket 共29a兲 and the low-frequency gyrokinetic ordering 共33兲, the last term in 共58b兲 yields

⑀ ␦2 具 兵 H gy , 兵 S 1 , ⳵ t S 1 其 gc 其 gc 典 ⫽ ⑀ ␦2 兵 H gy , 具 兵 S 1 , ⳵ t S 1 其 gc 典 其 gc ⬅O共 ⑀ ␦3 兲 , and thus we recover the energy conservation law 共53兲 for the nonlinear relativistic gyrokinetic Vlasov-Maxwell equations. The relativistic gyrokinetic energy conservation law 共53兲 is a simple extension of the classical gyrokinetic conservation law derived in Refs. 20, 36, and 37; note that this result requires the Maxwell equations to retain only first-order terms in ⑀ ␦ in 共51兲 and 共52兲, while the gyrocenter Hamiltonian 共43a兲 and the gyrokinetic particle energy 共55兲 retain second-order terms.

VI. DISCUSSION

We now summarize our work. In Sec. II, relativistic Vlasov-Maxwell equations were presented as a foundation for the remainder of the paper. In Sec. III, an outline of the derivation of relativistic Hamiltonian guiding-center theory based on the phase-space Lagrangian Lie-transform perturbation method was presented. Relativistic Hamiltonian gyrocenter theory 关based on 共46a兲–共46d兲兴 was constructed in Sec. IV while nonlinear relativistic gyrokinetic Vlasov-Maxwell equations 关共48b兲 and 共51兲兴 were derived in Sec. V. These two sections contain new results which extend previous nonrelativistic nonlinear gyrokinetic work20 and previous linear relativistic gyrokinetic work.21,22 A nonlinear relativistic gyrokinetic energy conservation law 关共53兲兴 was also derived in Sec. V. In future work we plan to initially obtain nonlinear relativistic drift-kinetic equations by taking the small- (k⬜ ␳ ) limit of the gyrokinetic equations, and perhaps later consider the finite gyroradius effects. From the nonlinear drift-kinetic equations, following the work of Chen,18 quasilinear transport equations can be obtained, presumably in a relativistic Fokker-Planck form. Such equations are expected to be very useful for evaluating, for example, the effectiveness of lowfrequency waves in the transport and acceleration of radiation-belt electrons.

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The authors wish to thank Professor Liu Chen for useful discussions and for his constant support and encouragement. Professor Allan N. Kaufman is gratefully acknowledged for his careful reading of the manuscript and for his suggestions toward improving it. The comments of the anonymous referee are also acknowledged concerning applications of our work to the investigation of relativistic electrons in tokamak plasmas. This work was supported by National Science Foundation Grants ATM-9613800 and ATM-9901127 and partially supported by U.S. Department of Energy Contract No. DE-AC03-76SFOO098.

1

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R. L. Liboff, Kinetic Theory: Classical, Quantum and Relativistic Descriptions 共Prentice Hall, Englewood Cliffs, New Jersey, 1990兲, Chap. 6. 28 T. G. Northrop, The Adiabatic Motion of Charged Particles 共Wiley, New York, 1963兲. 29 T. S. Hahm, Phys. Plasmas 3, 4658 共1996兲. 30 B. M. Boghosian, Ph.D. dissertation, University of California at Davis, 1987. 31 E. A. Frieman and L. Chen, Phys. Fluids 25, 502 共1982兲. 32 J. R. Cary and R. G. Littlejohn, Ann. Phys. 共N.Y.兲 151, 1 共1983兲.

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