One-dimensional Vlasov–Maxwell equilibria

Onedimensional Vlasov–Maxwell equilibriaJohn M. Greene Citation: Phys. Fluids B 5, 1715 (1993); doi: 10.1063/1.860970 View online: http://dx.doi.org/10.1063/1.860970 View Table of Contents: http://pop.aip.org/resource/1/PFBPEI/v5/i6 Published by the AIP Publishing LLC. Additional information on Phys. Fluids BJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

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One-dimensional Vlasov-Maxwell equilibria John M. Greene General Atomics, San Diego, California 921869784

(Received 1 September 1992; accepted 19 January 1993)

The purpose of this paper is to show that the Vlasov equilibrium of a plasma of charged particles in an electromagnetic field is closely related to a fluid equilibrium, where only a few moments of the velocity distribution of the plasma are considered. In this fluid equilibrium the electric field should be calculated from Ohm’s law, rather than the Poisson equation. In practice, only one-dimensional equilibria are treated, because the symmetry makes this case tractable. The emphasis here is on gaining a better understanding of the subject, but an alternate way of doing the calculations is suggested. It is shown that particle distributions can be found that are consistent with any reasonable electromagnetic field profile.

1. INTRODUCTION

The purpose of this paper is to describe a somewhat different way of considering the detailed, local magnetopause structure. The ideas presented here have more general validity, but have been selected with the magnetopause in mind.

On the scale of its thickness, the magnetopause is nearly planar. Thus, a local slab approximation is appro- priate. A further approximation is adopted that a signifi- cant portion of the magnetopause is not a slow shock, but rather can be modeled as a tangential discontinuity with a vanishing normal component of the magnetic field. As a result, the potentials of the electric and magnetic fields depend only on x, a coordinate normal to the plane of the magnetopause. Thus, the Hamiltonian of the particle tra- jectories depends only on x, and the system is integrable. This permits a considerable mathematical development of the theory.

In the kinetic treatment of this problem, the particle distributions in phase space are given in terms of the electromagnetic field, through Vlasov equations, and the fields are given in terms of averages over the particle distribution, through Maxwell’s equations. In the usual way of proceeding, the Vlasov equations are solved first, so that the particle distributions are given implicitly in terms of the electromagnetic potentials. Thus, the sources for Maxwell’s equations can be given in terms of the potentials, and Max- well’s equations can be integrated to yield the spatial profiles of the potentials. This approach was pioneered by Harris,’ and a more recent formulation has been given by Lee and Kan.”

There are several related problems with this approach. The primary difficulty is that the Vlasov equation does not place severe restrictions on the solutions, so that there remains a very large freedom to choose distribution functions. It is natural to make these choices so that the mathematical manipulations can be carried out in a tractable form. However, some quite reasonable distribution functions cannot be expressed in a simple way.

Here another approach is explored. First, reasonable profiles of the electromagnetic fields are given. Then, moments of the Vlasov equation are used to determine condi-

tions on the profiles of various plasma moments. The latter are the mass densities, the momentum densities, and the the stresses for each of the various particle species. They are the variables of a fluid theory, and they must be consistent with the plasma moment equations that define the fluid theory. With the electromagnetic fields known, the particle orbits can be calculated. These orbits are parame- trized by the values of the constants of the motion of the individual particles. Occupation numbers of the orbits yield particle distribution functions in terms of constants of the motion. They can be chosen so that moments of these distribution functions are consistent with the moment equations. One way to accomplish this is to develop the solution incrementally in x. Assume the distribution functions are consistent with the moment equations for x <x0 . The incremental freedom in the particle distribution functions in the interval between x0 and x0 + Sx is provided by particles that are reflected in this interval. This incremental freedom can be used to satisfy the incremental constraints resulting from the moment equations evaluated in this interval. It is shown here that the incremental freedom vastly exceeds the incremental constraints. As a result, finding consistent distribution functions does not provide any physically relevant information beyond that contained in the moment equations.

A further difficulty with the classical approach lies in the choice of equations to be satisfied. The usual approach is to evaluate the electric field from Poisson’s equation, using the charge density as the source. However, in a plasma the net charge density is a small difference between the electron and ion densities that cannot be evaluated directly with the necessary accuracy.

The consequences for the dynamics are as follows. Un- like the charge-poor media with which we are familiar, adequate charges are easily accumulated in a dense plasma. Rather, the limitation on the charge and electric field arises from the necessity of providing the energy and momentum required to establish the plasma drift velocity, EXB/B’. As an example, when an electric field that is perpendicular to the ambient magnetic field is applied across a dense plasma, the plasma particles are set into a dielectric drift that is proportional to3 aE/&. The sum of all these drifts is the polarization current. This current simultaneously pro-

1715 Phys. Fluids B 5 (6), June 1993 0899-8221/93/061715-08$06.00 @ 1993 American Institute of Physics 1715


duces a jXB force that accelerates the plasma toward its EXB/B2 drift velocity, and it tends to short out the source of the applied electric field. Thus, energy in the electric field is transformed into plasma kinetic energy. The electron dielectric drift continues until the plasma flow is consistent with the remaining electric field. If the density is sufficiently large, this balance is reached with a nonrelativistic plasma velocity and most of the energy in the plasma.

The conclusion is that the plasma velocity should be calculated from the forces on the plasma. With the velocity known, the electric field can be evaluated from Ohm’s law, and the charge density from Poisson’s equation.

The relation of plasma moment equations to kinetic equations is outlined in Sec. II, and particular emphasis is given to the invariance of the equations in frames that are in relative motion. This is applied to the one-dimensional case in Sec. III. Section IV contains a discussion of the method of deriving distribution functions that are consistent with given electromagnetic fields. Conclusions are presented in Sec. V.

II. GENERAL CONSIDERATIONS

Vlasov-Maxwell equilibria are described in terms of the electromagnetic field, E and B, and distribution functions fj(X,y,Z,Vx ,a,, ,v,) in phase space for each of several species of particle, where the species are denoted by the index i. It is convenient to normalize the distribution functions so that fj dx dy dz dv, dv,dv, is the mass of the jth species in the given volume element. Then, numerous fac- tors of the particle masses are eliminated from the subse- quent expressions.

Several moments of these distribution functions are particularly important in the theory. The mass density in configuration space for each species is given by

(1)

the momentum density for each species is

Pj= s vf j dvs (2)

and the corresponding stresses are

nj= WfjdV* s

(3)

It is common to separate the stress tensor into two parts, by writing the velocity v as the sum of a portion associated with the gross plasma mass motion together with a remain- der that is the particular velocity. The stress associated with the particular motion is called the pressure. Such a separation does not serve a purpose in this paper, so that here nj includes the entire plasma stress.

In terms of these quantities, the total mass, momentum, and stress densities are

P”CPjy

M= CELj, i

n= CIIj.

(4)

Then, a position-dependent mean velocity of the plasma, V, can be defined,

V=M/p. (5)

The charge density, a, and electric current, j, are

(T= C Kjpj s j

j= CKjpj.

(6)

Here, the quantities Kj are the charge-to-mass ratio for each species, Kj =Zje/mj, with Zj= - 1 for electrons,

Two sets of equations form the foundation of the discussion in this paper. The first set is the time-independent Vlasov equation for each species,

V~Vfj+Kj(E+VXB)'V~j=O, (7)

where V, is the gradient in velocity space. The second fun- damental set of equations is the time-independent Max- well’s equations,

V*E=a/ee, VXE=O, (81

V*B=O, VXB=p$.

Here, y. and e. are the magnetic permeability and dielectric constant of free space. They are related by

(9)

This will be used to eliminate eo. These two sets of equations, Vlasov equations and Maxwell’s equations, are cou- pled together by the electromagnetic forces in the Vlasov equation, and by the expression for the sources in Max- well’s equations.

In their full generality, Maxwell’s equations and the Vlasov equation sit together rather uncomfortably. Max- well’s equations are naturally fully relativistic, while the Vlasov equation is invariant under Galilean transforma- tions, Thus, there are minor discrepancies when comparing results that are derived in frames that are in relative motion. IIowever, in a plasma medium where the AlfvCn speed is much less than the speed of light there is a COA- sistent approximation to Maxwell’s equations that is Gal- ilean invariant. In this approximation the ratio of electric to magnetic field is small, in the sense that the dimension- less quantity E/c3 is of order v/c. This is in contrast to the electromagnetic field produced by an isolated charged ob- ject, near which E/c-B is of order c/u.

One of the consequences of the smallness of the electric field is that the charge density, IT of Eq. (6), can consis- tently be taken to vanish. This does not mean that V l E= 0

1716 Phys. Fluids B, Vol. 5, No. 6, June 1993 John M. Greene 1716


in Maxwell’s equation, it means that V l E(K,~&,c’, where K,P, is the charge density associated with the electrons. Then, insignificant perturbations of the electron density have a large effect on the electric field, and it is quite im- practical to use Maxwell’s equation to determine the electric field.

Another consequence of the smallness of the ambient electric field is that its value in a given frame is comparable to the electric field that is induced by nonrelativistic motion of the magnetic field in the given frame. The motion of magnetic field lines is a somewhat slippery concept, and there is considerable virtue in taking the relativistic point of view that the induced electric field is a property of space- time that is expressed in the rule for Galilean transforma- tions. Thus, we say that if the electric field is E in one frame, and E’ in another, then

E’=E-UXB, (10)

when the relative velocity between the frames is U. This is the leading order of the small U/c limit of the relativistic transformation of an electric field.4 Then, for example, the force on a particle moving with velocity v given in Eq. (7), E+vXB, is Galilean invariant. It is also the electric field in a frame in which the particle is stationary.

It follows from Bqs. (8) and (10) that the charge density differs significantly between frames. That is,

o’=u+U- j/C2, (11)

where (T’ and u are the charge densities in two frames with relative velocity U. In a fully relativistic treatment, charge density is the fourth component of a four-vector, and Eq. (11) is the leading term in the relativistic transformation between frames that are in relative motion. The fact that this small term must be evaluated to obtain consistency between frames is another indication of the difficulty of determining the electric field from the charge distribution in a plasma. There is a cancellation of leading-order terms, so that small corrections to the charge density are signifi- cant.

To complete the discussion of the Galilean invariance, note that the magnetic field B, the current j, and the distribution functions f / are invariant. That is, the relativistic corrections are sufficiently small that they can be ignored in the Galilean approximation. It is then straightforward to show that the Vlasov equations, Eq. (7), are Galilean invariant.

The sources of the electromagnetic fields in Eq. (8) are moments of the distribution functions. A number of rela- tions between the various moments can be derived by taking moments of the the Vlasov equations, Eq. (7). They can be used to express the information in Maxwell’s equations in different forms.

The first of these moment equations is momentum balance. It can be obtained by multiplying the Vlasov equation by v, averaging over velocity space, and summing over particle species. The result, in terms of quantities defined in Eq. (4), is

V-II-irE-jXB=O. (12)

With the aid of Maxwell’s equations, Bq. ( 12) can be re- written in the form

V*[H+--$(+EB)+;(fBZI--BB)]=O, (13)

where I is the unit dyad. This is the basic equation of fluid equilibrium force balance. Thus, a Vlasov equilibrium must be consistent with the fluid equations for force balance. The electromagnetic forces and stresses have been written out in their full form in Eqs. (12) and (13) for illustrative purposes. However, consistent with the nonrelativistic approximation that E/cB is small that was discussed above, the electric terms will be dropped from these equations. Note that when the electric field is large enough to contribute significantly to Bq. ( 13), the EXB/B2 drift velocity is comparable to the speed of light.

The nonrelativistic form of Bq. (13) is equivalent to (yuj -VXB) XB =O. Thus, if the distribution function satisfies the Vlasov equation, Eq. (7)) and stress II satisfies Eq. ( 13)) the particle drifts necessarily3 yield the perpendicular current that is the source of the field of Eq. ( 13). This result is quite general and does not depend on particular approximations to the particle orbits, or on the one- dimensional symmetry used in this paper. Since there are no subtractions in the calculation of the total particle stress, it is preferable to evaluate II rather than the particle drifts required in the direct evaluation of j.

Just as the perpendicular current is given by the condition of force balance, the parallel current is given by the twisting and shearing of the magnetic field. Thus, the momentum equation, Eq. (13), can be supplemented by a relation for the parallel current derived from Eq. (8))

B*VXB=pL$*B. (14)

In the small Alfvtn velocity regime that we are considering, we have seen that the Poisson equation is a very difficult way of determining the electric field. The charge density is a small difference of large quantities. Its place is taken by Ohm’s law, obtained by multiplying each Vlasov equation by Kj, and then taking averages as above.5 Then the momentum equation, Eq. ( 13)) and Ohm’s law together are equivalent to the separate equations of momentum balance for ions and electrons, when there is only one ion species. Amongst other terms, Ohm’s law contains a Hall term, jXB, that can be eliminated using the momentum equation, Bq. ( 12). When time dependence is retained this leads to an awkward inertial term dM/&, but in the equilibrium problem this is irrelevant. A straightforward way to eliminate the Hall term is to divide each Vlasov equation by Kj before summing over species. The result is

pE+MXB=V* z k II, > I

(15)

where p and M are the total plasma mass and momentum densities, defined in Eq. (4). Note that M/p is V, the plasma velocity, from Eq. (5). Thus, the right side is the electric field in a frame where the mean plasma velocity vanishes. This version of Ohm’s law is unique, in that it



depends on no approximations other than those used in deriving the Vlasov equation, and it has no Hall term.

The statement that Ohm’s law is an equation for the electric field carries the implication that the other quantities can be determined independently. In support of this, note that the velocity V is determined by the forces applied to the plasma. Further, the stresses in Eq. (15) are determined by the past history of the plasma. Thus, it is the electric field in Ohm’s law that is otherwise unknown. One consequence is that any reasonable velocity profile, V(X), can be accommodated in this formalism.

The appearance of the electric field in Ohm’s law, but not in the momentum equation, shows that the electric field mediates the exchange of momentum between ions and electrons, but does not contribute to the exchange of momentum between the material plasma and the electromagnetic field.

III. ONE-DIMENSIONAL EQUILIBRIA

Here we specialize to the case of one-dimensional equilibria. We return to the basic Vlasov and Maxwell equations, and rederive the momentum balance and Ohm’s law for this system. This permits comparison to some previous work, and shows how it fits into a more general scheme. The distribution functions fj are taken to be independent of the coordinates y and z, and only the x component of the electric field, and y and z components of the magnetic field, are nonvanishing. As discussed below, we take the distribution function to be even in v,

We introduce scalar and vector potentials, that depend only on x, for the electric and magnetic fields. Thus

a4 Ex= --ax,

Bz=Z. The equations for these potentials, in terms of their sources, are from Eq. (8),

824 Q= ---cLOC2~>

a2A, &T= -Po.iy~ (17)

2

t&g= -&j, .

The solution of the Vlasov equation can be written in terms of these potentials,

fj(X*ux ,Vy ,Vz) =Pj(Wy ,Pz), (18)

where Fj is an arbitrary function of the particle constants of the motion,

h=fV2+Kj&X),

Pu=Vy+KjAy(X)t

p,=V,+KjA,(X).

(19)

These quantities are normalized to be the energy and momentum per unit mass, to eliminate many unnecessary fac- tors of particle mass. In Eq. ( 18), Fj includes particles going in both the positive and negative x directions, which leads to the factor of 2.

The mutual intersections of surfaces of constant h, P,,, and P, in the four-dimensional space (x,u, ,I+ ,v,) are curves. In fact, they are the orbits of individual particles in the prescribed electric and magnetic fields. The particles maintain constant values of h, P,,, and P, while streaming along these orbits. The solution, Eq. ( 18), says that the particle density must be constant along these orbits, so that the configuration remains steady while particles A ow along. The particle velocity along a given orbit is given by

Vx= f [2h-2Kj~i-(P,-KjA,)2-(P,-~KiAZ)2]1’2,

U,,=Py-Kji’i,,,

Vz=Pz-Kjfd,.

(20)

Here, h, P,,, and P, are constant, but #(x), A,(x), and A,(x) vary along an orbit. The velocity u, is real, and the orbit is actually present, only for those values of x for which

(211

so that the argument of the square root in Eq. (20) is positive. Typically, the components of the vector potential, A, and A, approach infinity for large values of [X f , so that hmin forms a well. Thus, the orbits are restricted in their excursion in the x direction. This is magnetic gyration, modified by the electrostatic potential. The value of x at the bottom of the well is the center of gyration, and is determined for each orbit by the value of P,, and Pr When all particles gyrate at equilibrium, there are equal numbers flowing back and forth in X. This justifies the assumption that the distribution functions are even in v,.

Since the particles gyrate in the magnetic field, there is oscillation but not growth of the values of x and of the three components of velocity of each particle. However, the time average of v,, and v, in Eq. (20) do not vanish, so that the particles drift in the y and z directions. This separation between gyration and drifts can be used in finding approximate expressions for these orbits. In a strong magnetic field, so that the gyration radius is smaller than other length scales, the orbits can be separated into three components: a gyration, a streaming along field lines, and a drift perpendicular to the field lines. Even in a weaker field, this separation can be accomplished to a reasonable approximation. There are a variety of approximations to the orbits that can be used that are known as fluid, FIX, etc. They should all yield the nearly same results3



The next step is to evaluate the moments of the distributions using Eqs. (l)-(3) and (18). It is convenient to change the integration variables of Eqs. (l)-(3) from (0, ,ud, ,v,) to (h,P, ,P,), using Eq. (20). The Jacobian of this transformation is

dv, dv,, dv,= 2dh dP, - 21 (22)

where the factor of 2 comes from combining particles with opposite signs of u,. The full velocity space is covered by

- co <Pr, Pz< co,

h>hmin(xJ’y ,Pz), where hmin is defined in Eq. (21). Then

(pj)p= j- dPz [ dPy j- dh (py;;j$)F’,

(pji),= IdP,ldPyldh (pz;~;)F’,

(nj)~=ldP,~dP,Sd’IV~lF,~

(24)

where v, stands for the function defined in Eq. (20). Note that lIxrr is the only component of the stress tensor that contributes to V l lI in the symmetric configuration adopted in this section. Equation (24) says that the moments of the particle distribution at each value of x are given by the sum of the contributions from each orbit, where the orbits are identified by the values of h, P,,, and P,. The weighting of the density along each orbit is proportional to l/l uXj.

A useful identity can be derived by taking the x derivative of the stress in Eq. (24). The x dependency of the integral occurs in the lower limit of the integration over h, and in the funct.ion u,(x,h,P, ,P,>. The former does not contribute to the derivative since the integrand, 1 v, 1 Fj, vanishes when h = hmin . The result is

g (“,),=KjJ‘d~ld’Yldh(-~ f (Py-K/y) f$+ (P~--K#z) 2 5

x

--KjPI~+Kj(~j)V~+Kj~~j)=~’ (25)

This is the momentum moment of the Vlasov equation, Eq. (7). Since Fj of Eq. (18) is a solution of the Vlasov equation, Eq. (7)) moments of the Vlasov equation are automatically satisfied.

We now proceed to derive the one-dimensional form of the momentum equation, Eq. ( 13), from Eq. ( 17). This reproduces the derivation in Sec. II, but it is instructive. Summing Eq. (25) over species, and using Eqs. (4) and (17), we find

g JL= 1 a$ a2$J 1 &4, a%, 1 a&%4,

----y--~-------~. ,U~C axax p. ax ax j.io ax ax (26)

It can be seen directly that Eq. (26) is Eq. ( 13), simplified to the one-dimensional equilibria that are being considered in this section. Thus, if the distribution functions depend only on the constants of the motion, and yield the stresses that are consistent with the given electromagnetic field, the configuration is in force balance.

As discussed in the previous section, the electric potential term on the right of Eq. (26) is very small, and should be dropped. Further, this equation can be integrated once with respect to x, yielding

(27)

where PO is a constant of integration. In the same way that the electric potential term in Eq.

(26) is dropped, there must be strong cancellation in the terms proportional to &#/ax in the sum of Eq. (25) over particle species. As a result, the particle densities must satisfy the charge quasineutrality condition

(28)

The meaning of this statement is that the net charge density is smaller than any term on the left, so that this relation is a constraint on the particle densities and not on the charge density.

Equation (14) for the parallel component of the current can be expressed as

2 F’a-=f&g-;$$-

+( FKj(,),) 2, (29)

in the one-dimensional model of this section. The last two equalities, or magnetic portion, of Eq. ( 17) can be derived from the sum over species of Eq. (25), the x derivative of Eq. (27), and Eq. (29). Thus, the magnetic part of Eq. ( 17) is automatically satisfied if the distribution functions satisfy the Vlasov equation so that Eq. (25) is satisfied, charge quasineutrality is obeyed, the particle stress satisfies Eq. (27)) and the parallel particle drifts yield the current required by Eq. (29).

The other moment equation that was identified in the previous section is Ohm’s law, Eq. ( 15). In the one- dimensional model this reduces to

(30)



This can be used to find an expression for the charge density,

i a2+ u=-zdJcl

1 a =-p&is

(31) For this relation to be consistent with charge quasineutrality, we need to show that OQK@, where KiPi is the ion charge density. On setting

a z-L-', $--A,

I

where R, is an estimate for the gyroradius, we find the estimate based on the first terms on the right of Eq. (3 1 ),

u a2+/aX2 R, ~2 --= KiPi f?'W2Ki -XC 2 '

(32)

Thus, if the Alfven velocity, B2/ppo , is much less than the velocity of light, the charge density is a small difference of large quantities, and Eqs. (28) and (31) are compatible. That is, Eq. (28) provides a good estimate for the mutual distribution of ions and electrons while Eq. (30) is a good estimate for the electric field.

IV. DETERMINING THE PARTICLE DISTRIBUTION FUNCTIONS

In Eqs. (27), (29)) and (30), the derivatives of the electromagnetic potentials have been expressed in terms of functionals of the distribution functions, Fj, and of the potentials, 4, A,,, and A,. There are now two ways of proceeding. First, the functions Fj can be specified, and the dependence on the potentials of the moment terms of Eqs. (27)-(30) can be evaluated explicitly. Then the resulting differential equations for the potentials can be solved. Note that all the useful information in IQ. (17) can be derived from Eqs. (27)-( 30) so that Eq. ( 17) is not needed. An alternate way of proceeding is to specify the potentials as functions of x, and ask for the distribution functions, Fj, that are consistent. A similar analysis was carried out by Bernstein et aL6 for the electrostatic case. This second problem is adopted here. Following either procedure, un- reasonable choices can be identified and discarded.

These two ways of treating the equilibrium problem are not mutually exclusive, in fact, they are complemen- tary. It is generally useful to be able to understand both how the orbits determine the electromagnetic field and how the fields determine the orbits.

Here we describe an instructive way of looking at the problem. We choose a magnetic field profile, and for each species of particle we choose profiles of pj, CLj l B, and (II,),. Thus, the variables of the fluid theory are specified. We demand that these profiles be consistent with Eqs. (27)-( 29). These conditions leave considerable freedom in the choice of the particle moments. Rather than specify the electric field we fix the perpendicular velocity profile VXB,

since it should be more easily estimated. Then the electric field profile is determined from Eq. (30),and the perpendicular particle velocities lujX B can be determined from Eq. (25). These automatically sum to the desired perpendicular current and velocity. Note that if there is only one species of ion the values of current and velocity determine the particle moments pj so that Eq. (25) contains no additional information.

The next step is to choose distributions functions Fj that are consistent with the particle moments. One way to construct the Fj is to consider that the problem has been solved for x<xo , where x0 is an arbitrarily chosen point. That is, the distribution functions for each species yield the particle moment profiles for pj, pj 9 B, and (flj)XX in the range XGX~, which were introduced in the previous paragraph. This implies that the distribution functions Fj are known for h>h,i,(xo), for all Py and P,. Then, consider the configuration between x0 and X,+&X, and, in particular, the restrictions on the distribution functions for values of h lying in the range between h,i,(xo) and hmi,(xo+tix), that result from the consistency requirements.

The only particles that can influence the desired moments in the interval x0 < x <x0+6x, and are not determined by the solution for x <x0, reach their minimum value of x within this interval and are reflected back toward larger values of X. Thus, they lie in a region of phase space in which h,in(x) is a decreasing function of X. The derivative of hmin is

aA $-(P~-K#~) $- (P,-Ic#~) y$ ,

(33)

so that on some parts of the (Py ,P,) plane htij, is decreasing, and on other parts it is increasing. Thus, the problem posed here is to understand the restrictions that are placed on the distribution functions for values of h in the range hmin (xc) > h > h,i”(xo+ 6x), and for values of P,, and P, in the range where dh,,&dx ~0~

First, note that if the potentials are analytic functions of x for x <x0 and the distribution functions are analytic functions of the particle invariants (h,P, ,P,) for h > hmin(xo), the analytic continuations solve the consistency problem for x >xo . Thus, the interesting case is where there is some nonanalyticity. Here, we make a simple assumption about this breaking of analyticity that can be easily generalized. In particular, the particle moments that will be evaluated have a break in a derivative at x=x().

We take the distribution functions to have the form7

F=F,, h>hmin(xo),

=F,,+~[h,i”(xo)-h]“-“2

X4Py ,Pz)~CP, ,P,), h < h,i,(xc,), (34)

where F,, is analytic, F(Py ,P,) is arbitrary, and



Ah. >o, dPJ 2,) =Q ax I mm x=x 0

d-h’ <o. =h ax m’n x=x 0

(35)

Now, consider a density moment. From Eq. (24), and noting that

1 u.xI Efi[h-hdn(X) 1 1’29 we find

r$(n+f) =pan+ r(n+ 1) s s

dPz dP,

X [htnin(xo) -htia(x) I”&,

rgyn+f>

+an+-Rn+l) - (x-xo)njdPzjdP,,

x ( A~~xGxJn&T

Similarly,

x I^dp’ldpy( -~~,-xj’

and

X [ BJPy-Kj’ly) + Bz(PzeKjAz) 1 EF

(36)

(37)

(38)

x j-d&j- dP,( -~~._,)‘“&-.

(391

Thus, the assumed form for the distribution function yields a discontinuity in the nth derivative of pi and ~j l B, and a break in the (n + 1) th derivative of (Bj),.

This series of self-consistent choices produces the spatial dependence of the magnetic and electric fields. It is now possible, in principle at least, to determine the particle orbits. Following the scheme outlined in Sec. IV, it is then possible to find distribution functions, Fjl so the particle moments have the desired dependence on x.

The x dependence of the moments in Eqs. (37)-(39) There remains the question of how the particles arrive depends on the h dependence of the distribution function in at their expected positions. Closely related is the question Eq. (34). This is expressed in the dependence on the ex- of uniqueness. Two principles controlling the plasma ponent n. The coefficients depend on Y-. Only three mo- sources are suggested here. Fist, the positioning of the ments of the latter are needed to obtain the desired coeffi- electrons is sensitive to the outcome of the dielectric drifts cients. Thus there is considerable arbitrariness in the way that are associated with the formation of the structure. 9 can be chosen. Thus, there should be no problem concerning the source of

1721 Phys. Fluids B, Vol. 5, No. 6, June 1993

It is interesting to compare this with the electrostatic case.6 When the magnetic field vanishes, h,,,(x) does not depend on Py or P, so that the x dependence in Eq. (37) can be brought outside the integral. Then the arbitrariness of the previous paragraph appears to be irrelevant. The x dependence of the potential fixes the moment of 3-. In the general case it is not clear how to make the arbitrariness of the distribution functions appear to be irrelevant.

The conclusion of this section is that it is always possible to find particle distribution functions that are consistent with any solution of the fluid equations. Thus kinetic considerations do not constrain the profiles of electric and magnetic fields.

V. CONCLUSIONS

One-dimensional Maxwell-Vlasov equilibria, Eqs. ( 7 ) and ( 8)) must be consistent with gross force balance, Eq. (27), charge quasineutrality, Eq. (28), field line shearing, Eq. (29), and Ohm’s law, Eq. (30). One way of constructing these equilibria is to start with the moment equations and then determine consistent particle distribution functions. In the body of this paper we have given the details; here they are summarized.

The choice of moment equations was determined by the assumption that the plasma is dense, in the sense that the Alfven velocity is much smaller than the speed of light. In such a plasma, phenomena that would tend to increase the electrical energy, E2/,uoc2, instead increase the plasma energy by driving an EXB/B2 drift, or they drive an electric current that twists and shears the magnetic field and thus increases its energy. Thus, the electric energy remains small. As discussed in Sec. II, when the electric field is small, it is best calculated from Ohm’s law.

The first step in constructing the equilibrium is to choose profiles of the electric and magnetic fields, together with profiles of the particle density, momentum, and stress moments, that satisfy Eqs. (25) and (27)-( 30). Note that the ratio of the stress and density moments yields an estimate for the average particle velocity. If the distribution functions were approximately Maxwellian, this ratio would be the temperature, but nothing in this paper restricts the distribution functions in this way. Even if no temperature can be defined, this ratio yields an estimate for the gyration radii of the various species, and thus provides a length scale against which to measure the length scales of the various profiles.

John M. Greene 1721


the electron density that is required to satisfy charge quasineutrality and Ohm’s law, and there is no virtue in struggling to construct an equilibrium without trapped particles.8-‘0 Second, if, as seems plausible, particles are communicated to the magnetopause through the cusps, the details of magnetopause structure can only be understood in terms of the details of the cusp structure. As another example, Whipple et al.” based their choices on a reasonable guess as to the sources of particles.

To conclude, profiles of the electric and magnetic field, plasma density, momentum density, and stress that are consistent with the moment equations, Eqs. (26)-(28), determine one-dimensional equilibria. It is straightforward, if sometimes laborious, to determine particle distribution functions that are consistent with these equilibria. However, these kinetic aspects of the equilibria do not constrain the fluid aspects. Thus all the physically useful information contained in the moment equations.

ACKNOWLEDGMENTS

I am indebted to Dr. E. G. Harris for much insight into this problem.

This is a report of work sponsored by National Aero- nautics and Space Administration Headquarters, Washing- ton, D.C. under Contract No. NASW-4393.

‘E. G. Harris, Nuovo Cimento 23, 115 (1962). 2L. C. Lee and J. R. Kan, J. Geophys. Res. 84, 6417 (1979). ‘C. L. Longmire, Elementary Plasma Physics (Wiley, New York, 1963). 4J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975). ‘J. M. Greene, Plasma Phys. 15, 29 (1973). 6L B. Bernstein, J. M. Greene, and M. D. Kruskal, Phys. Rev. 108, 546

(1957). ‘B. Abraham-Shrauner, Phys. Fluids 11, 1162 ( 1968). sV. C. A. Ferraro, J. Geophys. Res. 57, 15 (1952). ‘A. Sestero, Phys. Fluids 8, 739 (1965). “D. M. Willis, Rev. Geophys. Space Phys. 9, 953 f 1971). “E C. Whipple, J. R. Hill, and J. D. Nichols, J. Geophys. Res. 89,

GO8 (1984).



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One-dimensional Vlasov–Maxwell equilibria