A&A 428, 1-19 (2004)
A. Beklemishev 1 - M. Tessarotto 2
1 - Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
2 - Department of Mathematical Sciences, University of Trieste, 34127 Trieste, Italy
Received 18 August 2003 / Accepted 20 July 2004
A fundamental aspect of many plasma-related astrophysical problems is the kinetic description of magnetized relativistic plasmas in intense gravitational fields, such as in accretion disks around compact gravitating bodies. The goal of this paper is to formulate a gyrokinetic description for a Vlasov-Maxwell plasma within the framework of general relativity. A closed set of relativistic gyrokinetic equations, consisting of the collisionless gyrokinetic equation and corresponding expressions for the four-current density, is derived for an arbitrary four-dimensional coordinate system. General relativity effects are taken into account via the tetrad formalism. The guiding-center dynamics of charged particles and the gyrokinetic transformation are obtained accurate to the second order of the ratio of the Larmor radius to the nonuniformity scale length. The wave terms with arbitrary wavelength are described in the second-order (nonlinear) approximation with respect to the amplitude of the wave. The same approximations are used in the derivation of the gyrophase-averaged Maxwell equations. The derivation is based on the perturbative Lagrangian approach with a fully relativistic, four-dimensional covariant formulation. Its results improve on existing limitations of the gyrokinetic theory.
Key words: plasmas - relativity - gravitation
The gyrokinetic theory is an adequate, yet simplified, description of collisionless magnetized plasmas. It strives to utilize the conservation of the magnetic moment to reduce the number of effective degrees of freedom of charged particles. In this respect it is similar to the drift-kinetic approximation. However, it goes further than the drift approximation, since it allows short-wavelength (comparable to the Larmor radius, ) perturbations of the background fields.
From the mathematical viewpoint, the dynamics of Hamiltonian systems can be expressed, in principle, in arbitrary hybrid (i.e., non-canonical) state variables. The search of phase-space transformations yielding "simpler'' equations of motion has long motivated theoretical research in physics and mathematical physics. Among such transformations, a particular case is provided by the gyrokinetic transformation, yielding hybrid variables defined in such a way that one of them, the so-called gyrophase angle, is ignorable. This transformation can only be introduced when particles move in a sufficiently strong magnetic field.
For a certain class of problems in plasma physics and astrophysics, the existing limitations of the standard gyrokinetic theory (Hahm et al. 1988; Brizard & Chan 1999; Cooper 1997; Littlejohn 1979,1984; Boozer 1996) make its use difficult or impossible. In particular, this involves the description of experiments in which the electric field may become comparable in strength to the magnetic field (so that the drift velocity becomes relativistic), and the study of relativistic plasma flows in gravitational fields, which are observed or assumed to exist in accretion disks and related plasma jets around neutron stars, black holes, and active galactic nuclei (Frank et al. 1992). The finite Larmor radius effects and the influence of short wavelength electromagnetic perturbations are also expected to play a fundamental role in particle dynamics. The present paper aims at extending the applicability range of gyrokinetic theory to include these features, and at providing an effective tool for use in plasma-physics-related problems in astrophysics.
The conventional non-relativistic gyrokinetic theory (Hahm et al. 1988; Littlejohn 1979) has been generalized to include only some relativistic effects, which are essential for the description of the confinement of fusion products in thermonuclear reactors (Brizard & Chan 1999; Cooper 1997; Littlejohn 1984; Boozer 1996), namely, the particle itself is considered relativistic, while its drift velocity is not. This deficiency has been pointed out by Pozzo & Tessarotto (1998), who attempted to write the theory in the reference frame moving with the fluid velocity, where all drifts should be first order in the expansion parameter, and the question about relativistic drifts should disappear. However, it proved to be more rewarding to create the covariant theory (Beklemishev & Tessarotto 1999), as it makes use of the term "relativistic'', i.e., with relation to any reference frame, in its full sense. This approach is inherently more general and symmetric than the non-relativistic treatment and its "relativistic'' generalizations. As a result, even the non-relativistic limit of the theory is found to have a broader applicability range than the standard derivation. In particular, covariant formulation allows self-consistent inclusion of the gravitational field, which makes the theory suitable for astrophysical applications.
In our previous paper (Beklemishev & Tessarotto 1999) the development of the covariant theory (derivation of the gyrokinetic transformation) was carried out through first order in the expansion parameter, and without the wave fields. It has been shown that the fundamental adiabatic invariant of a particle, i.e., its magnetic moment, as well as the ignorable gyrophase, can be defined as Lorentz invariant, independent of the reference frame. Also, the derivation itself becomes algebraically simple due to the internal symmetry of the electromagnetic field in the four-dimensional formulation. Covariant formulation allows the derived equations to be easily rendered for any coordinate system in four-dimensional Riemann space-time. It is important for astrophysics applications, as well as for problems where description in curvilinear magnetic coordinates is convenient. Although we do not use canonical variables, any coordinate system compatible with canonical variables can be imposed on the final result. Here it is necessary to note that the so-called relativistic drift Hamiltonian (Boozer 1996), and the gyrokinetic theories based on it (Cooper 1997), are limited by the applicability range of the drift equations given by Northrop (1963), i.e., assume non-relativistic drifts. The covariant drift equations require no such assumptions.
Covariant descriptions of guiding center dynamics have been published earlier, in the most complete form by Fradkin (1978). He first used the decomposition of motion of a charged particle into two quasi-independent motions in orthogonal invariant planes of the electromagnetic field tensor. Equations of motion were derived in the first order in inhomogeneity, without the wave fields. However, explicit use of the proper time for parameterization of the trajectory makes his expressions difficult to adapt to the gyrokinetic description, where parameterization should be applicable to the guiding center trajectory. Later, Beklemishev & Tessarotto (1999) showed that without such parameterization the choice of the decomposition planes becomes a matter of convenience rather than necessity.
The goal of the present work is two-fold. First, adopting the covariant gyrokinetic approach developed earlier (Beklemishev & Tessarotto 1999) we intend to describe particle dynamics in the presence of non-uniform and strong electromagnetic (EM) and gravitational fields, accurate to second order in the Larmor radius, and treating consistently the effect of short-wavelength EM perturbations. The basic features of the approach, allowing the relativistic drifts typical of many astrophysical plasmas, are the assumption of a non-vanishing parallel electric field and the description of particle dynamics in the field-related basis tetrad, which allows a simpler formulation of the relativistic equations of motion. Second, the perturbative theory is employed to derive the relativistic Vlasov-Maxwell equations expressed in gyrokinetic variables accurate through second order in (which is the ratio of the gyro-radius/gyrotime to the equilibrium gradient length/time) while the wave fields are taken into account in the second order in the amplitude. The gyrophase-averaged expression of the four-current density appearing in Maxwell equations is evaluated with the same accuracy.
In contrast to non-covariant formulations, since the space and time variables are treated in a parallel fashion, the applicability limits of gyrokinetic theory are extended - the wave fields can be fast oscillating in time ( ), as well as in space ( ), while the customary assumption is not required. As a result, we find a contribution to the magnetic moment which is proportional to the usually neglected parallel component of the wave-field. An important feature of this work is its relative simplicity, which allows us to present results with the gyrophase-averaging performed explicitly. (The usual way of doing this is by leaving the averaging along the particle trajectory to the reader's computer; see, for example, Brizard & Chan 1999.) However, currently there is also a limitation linked to our approach, namely, in the case of a vanishing background parallel electric field, only certain polarizations of the wave field are allowed. Namely, the wave field should be describable by just two components of the vector potential, both orthogonal to the magnetic field (in the reference frame where the electric field vanishes), or have a large wavelength, to be described as part of the main field. We intend to eliminate this restriction in the future.
Another work dealing with the covariant representation of the gyrokinetic theory (for a flat space-time) is the Ph.D. Thesis of Boghosian (1987). It extends the covariant description of charged particle guiding center motion developed by Fradkin (1978). In line with non-covariant treatments Boghosian assumes a vanishing parallel component of the electric field and the usual gyrokinetic ordering for admissible wave forms, and thus has to modify Maxwell equations to maintain the assumed properties. His derivation is also conducted only to first-order both in nonuniformity and in the wave-amplitude.
The structure of the paper is as follows: The hybrid Hamilton variational principle is formulated in Sect. 2. In Sect. 3 we present the definition of the basis tetrad and its link to the Faraday tensor. In Sect. 4 the relativistic gyrokinetic transformation is derived to second order in , and in the wave amplitude. The Euler equations for the gyrocenter trajectories are formulated in Sect. 5. Maxwell equations with the current and charge densities, expressed in terms of the distribution function as a function of the gyrokinetic variables, and integrated in the gyrophase, are presented in Sect. 6. In the Conclusion the results of the paper are summarized.
Construction of the gyrokinetic transformation for the one-particle Hamiltonian system requires a description of the particle dynamics in phase-space. It is possible to redefine the Hamilton variational principle in such a way that it becomes suitable for the phase-space description of motion in non-canonical variables (Cary & Littlejohn 1983). We refer to this as "the hybrid Hamilton variational principle''. In the framework of general relativity the four-dimensional formulation of the Hamilton variational principle is well known (Landau & Lifshits 1975). It makes it possible to describe the particle trajectory (or its world-line) as a four-dimensional extremal curve. In this section a hybrid form of this variational principle, suitable for the description of particle dynamics in the seven-dimensional phase-space (Beklemishev & Tessarotto 1999), is recalled.
The relevant tensor notations are standard. Thus,
metric tensor components, characterizing the coordinate system (and the
underlying space-time structure). It also provides the connection between
the co- and countervariant components of four-vectors (henceforth referred
to as 4-vectors)
The signature of
is assumed to be
while the invariant interval
is then defined in terms of its countervariant
components according to
Finally, we note that the above variational principle, as any Lagrangian
variational principle, is gauge invariant, i.e., any change of the
generating differential 1-form of the type
The use of a tetrad of four-vectors is a well-known type of description of curved space-time (Landau & Lifshits 1975), which makes it possible to replace complex derivatives of the metric tensor by derivatives of the tetrad components. We adopt this approach below, while also finding it convenient to link our tetrad to properties of the electromagnetic field, or the Faraday tensor,
We first introduce an orthogonal basis of unit 4-vectors
so that the last three
4-vectors are space-like, and
A special choice of orientation links the basis to the electromagnetic field tensor, . We shall refer to this choice of basis as field-related and use it henceforth. With this choice the -plane coincides with the space-like invariant plane of the antisymmetric tensor In some sense this corresponds to the magnetic coordinates of non-relativistic treatments. Fradkin (1978) used these invariant planes of the electromagnetic field tensor for the decomposition of motion of a charged particle.
Any non-degenerate antisymmetric tensor of rank 4,
particular, has two orthogonal invariant planes. In our case one of them is
space-like, i.e., any vector belonging to it has negative length, while the
other contains both time-like and space-like vectors. Each of the invariant
planes has an associated eigenvalue. This means that if
is the first invariant plane, then
other, and if H and E are the eigenvalues of
Due to these properties of the basis, the Faraday tensor can be fully
expressed via its components as
The object of gyrokinetics is to introduce a new set of phase-space variables (called the gyrokinetic variables) such that the variable describing the rotation angle along the Larmor orbit (i.e., the gyrophase ) becomes ignorable. This happens, by definition, when the Lagrangian (or, more generally, the functional) is independent of . Once an ignorable variable is found, the number of corresponding Euler equations is reduced by one, and the new variables allow simplified numerical calculations, as the motion is effectively integrated over the fast Larmor rotation. The one-to-one transformation from the original set of phase-space variables to the gyrokinetic variables is called the gyrokinetic transformation. In what follows to find these variables we use the Lagrangian perturbative approach, which is equivalent (in broad terms) to the Lie-transform method, though more direct.
Indeed, we can search for the necessary transformation, representing it as a Taylor-series expansion in small parameters (inhomogeneity and wave amplitude). Then, in each order we can require the transformed functional to be independent of the gyrophase (the ignorable variable), and thus find the transformation and the new phase-independent functional order by order. For this we should be sure that 1) the solution in the necessary order exists; and 2) that the procedure converges. The existence of the solution depends on the choice of the expansion form and is proven operationally, i.e., by direct construction. Then the true dynamical system becomes replaced by the new artificial one, described by the phase-independent Lagrangian of that order. There is also the problem of convergence, i.e., how accurate the gyrokinetic description is. This problem is common to all gyrokinetic (and drift) theories and has not yet been explored in detail. From the physical viewpoint, the artificial system has no cyclotron resonances, while the real one has them, but their width decreases with the small parameters. The gyrokinetic approach cannot work within this thin but possibly interlinking Arnold web of resonances, where the adiabatic invariants are not conserved. Outside the resonances one can claim that since the difference between the real and the artificial Lagrangians is limited and tends to zero within some suitable measure, the behaviour of the gyrokinetic system is similar to that of the real systems.
We begin the search for the gyrokinetic transformation by a suitable
definition of the small parameters, i.e., we assume that the curvature
radius of the space-time and the gradient lengths of the background
electromagnetic fields are much larger than the Larmor radius characterizing
the particle path. However, we allow for the existence of wave fields with
sharp gradients [
], and rapidly varying in time [
], while such fields are assumed suitably smaller in strength than the
background field. (We stress that, unlike in conventional formulations of
gyrokinetic theory, this type of ordering is required in a covariant theory
due to the reference-frame dependence of the ordering assumptions involving
space and time scale lengths of the perturbations.) For this purpose we
introduce the ordering scheme following the notation of Littlejohn (1984):
This search consists in applying the expansion (15) to the fundamental 1-form (14), and imposing the requirement that it is independent of in each order. Since the 1-form itself depends on expansion in this parameter will also appear.
Since the difference between
small compared to the gradient lengths, the background field components
(unlike the wave terms) can be expanded in the Taylor series
Equation (15) define only four out of seven transformation rules for the gyrokinetic transformation . The rest are chosen in the following way.
First, we express the four-velocity in the new variables as
Furthermore, we assume that
and uo are independent
The validity of this assumption is justified by the existence of
the solution (at least for a non-degenerate Faraday tensor). Then, the -independent part of the 4-velocity
is not completely
arbitrary, but satisfies certain restrictions following from the requirement
for all :
Let us substitute expressions (15), (17) into (14) and
drop terms of order
Now it is straightforward to eliminate from terms oscillating in by properly defining displacements We carry out the calculations for each order in and .
Initially we drop all terms in Eq. (24), which are of order
or higher, while retaining contributions of order .
The conditions for
to be ignorable for the gauge-modified
where R is an arbitrary Gauge function look like
If the above requirements (27) and (28) are satisfied and
is ignorable, the phase-space variational principle in our
approximation can be expressed as
If the Faraday tensor is degenerate but the magnetic field is non-zero, i.e., E=0, , a solution of Eq. (29) exists for a special choice of the wave potential satisfying i.e., when the projection of is zero. Then the above expressions are valid with the replacement . For other choices of the wave potential there is no solution, meaning that we need to modify our representation of the gyrokinetic transformation, Eqs. (15) and (18). This case will be discussed elsewhere.
The evaluation of the gyrophase averages involving the wave field is, of course, the difficult part. The reason for this is the necessity of
transforming the given function of space-time coordinates
into a function of new variables, while the transformation rule (15), (31) is itself dependent on .
A solution of the equation
The solution of the problem in the limit
has already been
published by Beklemishev & Tessarotto (1999). By varying the functional (32) with respect to ,
Now it is possible to rewrite the functional (32) using Eq. (19) and
in place of w2:
Using the -independent functional (36) in the variational principle, defines the particle trajectory in terms of the new gyrokinetic variables . This set is non-canonical, but further transformations of the variables (not involving ) also lead to -independent functionals, and thus can be used to transform to canonical variables, if necessary.
We return to the approximation (24) with
the lower-order requirement (29). This yields
and in view of the antisymmetric properties of
Eq. (41) becomes
In the current approximation,
Using the explicit expression for
it is possible to evaluate averages, so that
As one can see, it is not necessary to actually solve for to find the transformed Lagrangian. It is only necessary to be sure that the solution exists, which is true for the "field-related'' choice of the basis.
We proceed by considering the linear approximation in the wave-field amplitude without high order curvature effects. As noted above, the main problem here is the transformation of the highly-local wave field , where is the particle position, to the guiding-center coordinates . The Taylor-expansion procedure used above for the transformation of the equilibrium field fails if the wavelength is sufficiently short. To avoid this problem, Eq. (34) is solved using Fourier analysis and expansion in powers of .
Assume that the wave field is given in terms of its Fourier components
then Eq. (34) becomes
Here the ordering of the wave-vector is such that as usual (Littlejohn 1984) , except that the time-like component of is proportional to the wave frequency, and the ordering for it is equivalent to where is the cyclotron frequency, and not as in conventional approaches; besides, the above ordering is also applicable to i.e., is assumed in place of the usual ordering . Therefore, our approach in this respect has a wider applicability range. This is inevitable, for the reference frame may move with relativistic velocity across the short-wavelength wave field, so that fast oscillations in time may result anyway. However, the particle, drifting across the wave field, will experience a quite definite oscillation frequency as well as the wavelength, which is taken into account by our definition of , introduced below.
We can calculate the required averages in Eqs. (32) and (33) by
using the expansion of Eq. (46) into a Fourier series in .
use the following identity (Abramovits & Stegun 1972)
To the order
in the expression (48) can be set
to 0, so that its zero-order solution is just
Now the averages, present in expressions (32), (33), become
The second-order nonlinear corrections are responsible, in particular, for three-wave interactions. Calculating them for a general Fourier-decomposed wave field would allow subsequent construction of a weak-turbulence theory. For other effects, such as the estimate of the Miller force of the high-frequency field on a particle, just the eikonal representation of the wave is sufficient. Below we adopt the first, more complete description.
In this approximation we can use the -independent variational principle (32) with properly defined parameters. Indeed, besides which is added to the effective vector potential, , there is one other significant contribution to the Lagrangian from the wave terms. It enters into the definition of the adiabatic invariant via Eq. (33), so that , where is given by Eq. (54).
The second order contribution to the adiabatic invariant is given by
Expanding the last exponent in Eq. (48) in powers of
Using the definition of
and taking into account the identity
(60) with l=0, we reduce the first-order correction to
to the following expression
Proceeding now to calculation of the second term in Eq. (56) we note
As one can see, the above results are not gauge invariant with respect to
gauge transformations of the wave potential .
This problem can be
resolved using gauge-invariance of the variational principle (4), (14). Indeed, any transformation of the potential
The above program of calculating the gyrokinetic transformation and the -independent description of motion can be carried out for an arbitrary
gauge function f. This means that expressions obtained in the above
subsections should also be valid for
Finally, we summarize the results of this section by presenting the
transformed variational principle valid through the second order in
and second order in
i.e., with terms of the order
yields the particle phase-space trajectory with
The single-particle distribution function can be written in general
relativity either in the eight-dimensional phase space
or in the seven-dimensional phase
where only 3 components of the 4-velocity
are independent, so that
Because of the general properties of variable transformations it is obvious
that any non-degenerate transformation of the phase-space variables
will lead to the same form of the
then the kinetic equation becomes
While the above derivation seems simple, there is a trick. Even if we prove that there is no -dependence in all orders, this does not mean that there is no such dependence at all. In fact, this is what happens close to the cyclotron resonances - the exponentially small (and thus having zero expansion coefficients) resonant factor destroys the conservation of the magnetic moment, rendering the gyrokinetic theory invalid. It should always be kept in mind that the approximation works well for particles far from the low-order cyclotron resonances.
Another question is: why is it so important to have parameterization-independent equations of motion? Would it be possible to use, say, Fradkin's drift equations with the proper-time parameterization to construct the gyrokinetic equation in an alternative way? The answer follows from the transformation of the Liouville Eq. (76) with the transformation of the phase-space variables. Since its meaning is the conservation of world lines, the form is retained for canonical-like transformations (with a constant Jacobian). Otherwise, for an arbitrary set of non-canonical drift equations of motion, one has to prove that the Jacobian is at least independent of the gyrophase, in order to exclude this variable from the gyrokinetic equation. In other terms, kinetic Eq. (77) (with coefficients independent of ) cannot be transformed into the gyrokinetic equation, unless is also independent of . This requirement is not fulfilled for Fradkin's equations, since the proper time s depends on the gyrophase with periodic corrections due to changes of velocity along the orbit.
The equations of motion, or the relationships between differentials tangent
to the particle orbit, can be obtained as Euler equations of the transformed
variational principle (74). Let us find the first variation of Sassuming
Extrema of the functional are achieved for world lines where
for all variations of independent variables. This yields the Euler equations
It is convenient to rewrite the 4-vector Eq. (83) through its
projections on the orthogonal vectors of the basis
Taking scalar products with
Searching for the solution in the form
and substituting it into
Eqs. (84), (85), we get
Consider a coordinate system chosen in such a way that the dependence of all
(with some particular )
This means that
In general, the kinetic description of plasmas involves a combination of the kinetic equation and Maxwell equations, which describe the evolution of the collective (Hartree-Fock) electromagnetic fields. The same is true for the gyrokinetic theory. However, since the gyrokinetic equation is written in specific gyrokinetic variables, and thus yields the distribution function in terms of these variables, one should have a special procedure of integration in velocity space to calculate the source terms of Maxwell equations. This procedure is presented below. Fortunately, the Jacobian of the gyrokinetic transformation in the present formulation is simple enough to allow integration in the gyrophase, so that expressions for the charge and current densities via the gyrokinetic distribution function are found explicitly.
The general form of Maxwell equations in the presence of an arbitrary
gravitational field is well known (Landau & Lifshits 1975). The first pair of equations
can be written as
The gyrokinetic transformation has been found by expansion in orders of
and this expansion has to be exploited here
The background distribution function
is slowly changing
on the Larmor-radius/gyrotime-scale, and thus can be expanded into a
Taylor series around the particle position. Note that the displacements
are given by the gyrokinetic transformation as functions of
and thus should be expanded
as well. Keeping terms of the order of
and larger, we
The first order displacement of the gyrocenter is given by
Similarly, it is necessary to solve Eq. (42) for in order to calculate the second-order current density. The form of this equation (for ) is such that may have components proportional to first and second powers of trigonometric functions of namely, It is multiplied by the velocity, which has terms with the zero and first powers of the same, so that a range of first- to third-power terms are produced. However, of that range only the second-power terms can produce a non-zero contribution after integrating in , so that in the following we ignore other terms. The same argument applies to the expansion (104) as a whole.
First we solve for
The last remaining terms in Eq. (104) contribute to the current via
The parallel component, , represents the FLR corrections to the charge density and the parallel current density. It is rarely used, since it is two orders smaller than and has the same direction. The perpendicular component, , describes currents caused by the drifts in inhomogeneous fields.
First we calculate the transformation to the particle position variables
The nonlinear corrections follow while the formerly neglected last term in
the exponent of Eq. (116) is taken into account. The resulting
first-order correction to
is a prescribed function of the particle position,
but its oscillating part is not. It is
necessary to transform the zero-order solution, Eq. (50),
Now we can calculate
One can see that the nonlinear correction to the current density is due to the interaction of the perturbation of the distribution function with the perturbation of the particle motion, satisfying the three-wave interaction criteria.
A closed set of relativistic gyrokinetic equations, consisting of the collisionless gyrokinetic equation and the averaged Maxwell equations, is derived for an arbitrary four-dimensional coordinate system.
The basic features of the approach are as follows:
This work has been conducted via the cooperation program between the Trieste University, Italy, and the Budker Institute of Nuclear Physics, Novosibirsk, Russia.