Lorentz-covariant uid description of relativistic

magnetized plasma

R. D. Hazeltine and S. M. Mahajan

Institute for Fusion Studies

The University of Texas at Austin

MHD

What is MHD? It is a uid description of a strongly magnetized plasma.

� \Fluid" description requires that the distribution function be smooth

(no resonant singularities, for example). In fact it is taken to be

Maxwellian|an assumption that is explicit in the relativistic case|

but distribution details beyond smoothness are not crucial.

� \Magnetized" limit means that the gyroradius is allowed to vanish.

MHD is intended to describe what happens in a zero{gyroradius plasma

uid.

If one seeks a uid description of a plasma in the limit of vanishing

gyroradius, does one �nd MHD as the general, inevitable result?

Other uid descriptions

Magnetized uid descriptions beyond MHD include

� Braginskii uid equations require short mean-free path, not applicable

to hot plasmas of frequent interest.

� Double-adiabatic (CGL) theory does not allow heat ow along the mag-

netic �eld, but this ow can be strong in hot plasmas.

� More recent uid models, keeping various e�ects and neglecting others,

often disagree with each other: if often clever, they are not inevitable.

We seek a description that follows systematically from the words \ uid"

and \magnetized." Requiring Lorentz covariance helps �x the form of the

equations and covers both astrophysical and laboratory plasmas|while

saving chalk.

Magnetized plasma

A plasma system is magnetized if

1. The magnetic �eld dominates:

W � B2 � E2 > 0;

2. The gyroradius is small:

�=L � Æ � 1

3. The parallel electric �eld is small:

� � E �BW

� 1

Recall that � and W are scalars. We use the maximal ordering � � Æ.

Magnetized closure

Recall the energy-momentum conservation law:

@T ��

@x�� F ��J� = 0

Here T �� is the energy-momentum tensor, or stress tensor, F �� is the Fara-

day (or \�eld-strength") tensor and J� is the 4-vector current density. The

product F ��J� includes the Lorentz force.

All uid closures for magnetized plasma use this relation to compute the

perpendicular current. Thus every uid closure for a magnetized plasma is

de�ned by its choice for the stress tensor T ��.

Projection operators

A covariant meaning is given to \perpendicular" and \parallel" by the

operators

e �� � �F��F ��=W

b �� � � �� � e ��

These tensors act as projection operators in the magnetized limit, � ! 0.

In the rest-frame they act on an arbitrary 4-vector C = (C0;C) to give

(b �� C�)R = (C0;Ck)

(e �� C�)R = (0;C?)

These are approximate projectors, e.g., e � e = e+O(Æ2). But they �t our

purposes and are much simpler than Fradkin's exact projectors.

Closing Maxwell

Perpendicular components of energy-momentum conservation law,

e��J� = �F�

�W

@T ��

@x�:

combined with charge conservation and quasi-neutrality,

@J�@x�= 0; J�U� = 0

where U� = (�1;V) is the 4-velocity, determine J�.

The 4-current density is determined, closing Maxwell's equations, once

the energy-momentum tensor is known. This is not new; it is the closure

procedure of MHD.

Single{species moments

We compute the stress tensor from �rst three moments of the (relativistic)

kinetic equation

@��@x�= 0

@T ��

@x�� eF ���� = 0

@M���

@x�

� e(F ��T �� + F ��T �� ) = 0

Here � = nRU , where nR is the rest{frame density, is the particle ux, T

the single{species stress tensor andM is the third{rank stress{ ow tensor.

In each case, the gradient{term is smaller than the e{term by a factor of

Æ, so the magnetized limit corresponds to e!1.

Simple `closure engine'

1. Construct T �� from e!1 limits of the conservation laws

@T ��

@x�� eF ���� = 0

@M���

@x�

� e(F ��T �� + F ��T �� ) = 0

2. Add to compute the total stress tensor

T �� =

XspeciesT ��

and thus close Maxwell's equations.

The same recipe is sometimes said to describe MHD closure.

Magnetized ow

We test the engine by applying it to the second moment,

@T ��

@x�� eF ���� = 0;

which becomes

F ���� = 0

For any 4{vector C�,F ��C� = (E �C;�EC0 +C�B)

Since �� / U�, we can infer E +V �B = 0, or

V = Vk +VE:

Hence in this case our closure engine reproduces MHD.

Magnetized stress tensor

In the same way we infer from

@M���

@x�

� e(F ��T �� + F ��T �� ) = 0

that the magnetized stress tensor must satisfy

F ��T �� + F ��T �� = 0 (�)

This relation is the basis for magnetized closure. It requires the stress

tensor, like the MHD ow, to produce vanishing electromagnetic force.

Notice that the electromagnetic force alone determines the form of T ; col-

lisions and thermodynamics don't enter.

Form of the magnetized stress tensor

We use indicial symmetries and properties of the projection operators to

�nd the exact, general solution to (*):

T �� = b��pk + e��p? + hU�U � + q�U � + U�q�

where pk, p? and h are Lorentz scalars corresponding respectively to par-

allel pressure, perpendicular pressure and enthalpy density, and where the

four-vector q� is constrained bye��q� = 0

U�q� = 0

Thus there is only one independent component in q�; this represents parallel

heat ow in the rest frame and is denoted by qk.

Comparisons

� MHD does not derive the form of the stress tensor. Instead MHD uses

the thermodynamic equilibrium form

T ��MHD = ���p + hU�U �

MHD uses the electromagnetic force F to �nd the form of �, but ignores

F in its expression for T .

� The double-adiabatic theory of Chew, Goldberger and Low derives and

uses the tensor

T ��CGL = b��pk + e��p? + hU�U �

CGL accounts for the electromagnetic force but neglects parallel heat

ow. Note that such heat ow is large at small collision frequency.

Stess tensor dynamics

Since nR is determined by particle conservation, we need equations for the

dynamical variables

Vk, pk, p? and qk.

Two of the necessary equations come from energy-momentum conserva-

tion; after multiplying the conservation law by the dual Faraday tensor F ,

we �nd that

F��@T ��

@x�= eEkB��

This provides only 2 independent equations, advancing pk and Vk in time.

(The other 2 components of energy{momentum conservation are needed to

compute the current.) Hence we must consider the equation for stress{ ow,

M�� .

Stress{ ow dynamics

We �nd 2 independent annihilators of the large terms on the right-hand

side of

@M���

@x�

= e(F ��T �� + F ��T �� );

giving the 2 relations

e��@M���

@x�

= 0

(U�q� + q�U �)@M���

@x�

= �2eEkh

These equations advance p? and qk, onceM�� has been expressed in terms

of the dynamical variables.

Stress{ ow tensor

The small-Æ argument used for U� and T �� can be applied to the stress- ow

tensor M to derive its general magnetized form. The complicated result

has the schematic formM = M(Vk; qk; m1; m2; m3)

analogous to the stress tensor,

T = T (Vk; qk; pk; p?; h)

Here red ink indicates dynamical variables|quantities for which we have

evolution equations.

Truncation

To close the uid system we need to express h and the mi in terms of

the dynamical variables. To assure the self{consistency of these expres-

sions, we use an assumed form for the distribution function: a relativistic

Maxwellian, modi�ed to allow stress anisotropy and heat ow.

The distribution, like the MHD version, is parametrized by the dynamical

variables and thus evolves according to the uid equations. It yields, for

example,

h =

mnR

K2(�)2

664K3(�) +2�

�0

BB@K4(�)K2(�)�K3(�)2

K2(�)

1CCA

3775

where � � mnR=pk and the Ki are MacDonald functions. This expression

generalizes a well{known form (�! 0) used in relativistic MHD.

Closure summary I

The general form of the energy{momentum tensor in the limit of vanish-

ing gyroradius isT �� = b��pk + e��p? + hU�U � + q�U � + U�q�

This tensor represents an anisotropic plasma with heat ow along the mag-

netic �eld. It is the essential feature of our closure.

The stress tensor depends on the scalar quantities,

pk; p?; Vk; qk and h,

the �rst 4 of which are taken to be the basic dynamical variables of the

system.

Closure summary II

The evolution of the dynamical variables is determined from the 4 equations

F��@T ��

@x�= eEkB��

e��@M���

@x�

= 0

(U�q� + q�U �)@M���

@x�

= �2eEkh

There remain 4 scalar functions: h (appearing in T ��) and the 3 mi (ap-

pearing in M���). These are expressed in terms of the dynamical variables

using a representative distribution, parameterized by the dynamical vari-

ables.

Non-relativistic (NR) limit

The NR limit, V � (T=m)1=2 � 1; qk � pV , has particular interest. Here

we also include the collisional moments, and �nd the following equations

for Vk, pk, and p?:

mnb � dVdt+rkpk + (p? � pk)rk logB = enEk � �1qk ��0

eJk;

ddtlog

0BB@pkB2

n3

1CCA +6

5qk

pkrk log

0@ qk

B1=31

A = 0;

ddtlog

0@ p?

Bn1

A +2

5qk

p?rk log

0@ qk

B21

A = �2pk � p?

p?

;

CGL double{adiabatic laws are reproduced when qk = 0.

Non-relativistic heat ow

dqkdt+ qk

2649

5b � (rkV)� 7

5d log n

dt

375 +

0B@7

2pk � p?

1CAb � dV

dt� 1

2enEkV2

k

+

7T2m(p? � pk)rk logB +rk

264T

m0

B@3

2pk + p?

1CA

375

=

emEk

0B@3

2pk + p?

1CA� 3

2�3qk:

Note that the collisional limit, � � d=dt, reproduces Chapman{Enskog

transport theory. Indeed, treating qk as a dynamical uid variable is an

e�ective way to span collisional and collisionless regimes.

A benchmark: ion sound

Consider ion{acoustic waves in a plasma with Ti � Te, a well{known

hurdle for uid theories.

1. Kinetic theory: ! = kkcs (the right answer).

2. MHD: ! =r

(5=3) kkcs (but MHD can't tell that Ti is small).

3. Double{adiabatic theory: ! =p

3 kkcs (worse than MHD).

4. Present theory: ! =r

15=11 kkcs (closest to kinetic theory).

In the case Ti � Te, ion{acoustic waves are strongly Landau{damped. Our

uid theory does not capture Landau damping, which involves singularity

in the (perturbed) distribution.

Applications

Relativistic version:

� galactic jets, pulsar atmospheres, etc.

� some laser{plasma interactions

NR version:

� nonlinear uid simulation

� linear theory? Note that kinetic MHD is more accurate.

� long mfp heat ow

{ divertor and edge physics

{ magnetic island evolution: temperature equilibration over evolving

ux surfaces

Extension to �nite gyroradius

is straightforward:

� Same dynamical variables

� Same closure scheme

Only di�erence is more terms. Recall:

@M���

@x�

= e(F ��T �� + F ��T �� )

For FLR physics, retain M , which is known to suÆcient accuracy from

e!1 theory. Then solve for stress tensor as usual:

T �� = T ��(0) + T ��(1)

0B@1

e@M(0)

@x1

CA

Systematic, and deterministic: a uniquely determined set of FLR correc-

tions.