IC/66/94

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

SELF-CONSISTENT DESCRIPTIONOF A WARM STATIONARY PLASMA IN AUNIFORMLY SHEARED MAGNETIC FIELD

A. SESTERO

1966PIAZZA OBERDAN

TRIESTE

IC/66/94

International Atomic Energy'Agenoy

IUTERFATIOKAL CENTRE FOE THEORETICAL PHYSICSi

SSLF-COlfSISTEM1 DESCRIPTION OP A HARM STATIONARY PLASMA

111 A TJKIPOHMLT SHEARED MAGHETIC "

A . SESTERO*

TRIESTE

July 1966

T To "be subiaitted to "Physics of Fluids'1

*On leave of absence from " Laboratorio Gas Ionizzati", SURATQM-CIEE!!!,Fxaacati, Italy.

ABSTRACT

An exact, self-consistent solution of the system of Vlasov's

plus Maxwell's equations is derived, describing a warm stationary

plasma in a uniformly sheared magnetic field. It is found that there

is no upper limit for the rate of shear, that is, the distance over

which the direction of the magnetic field rotates of an angle Ztr can

be as small as desired, in particular smaller than the ion or even

the electron Larmor radius. Within the assumed model, for any value

of plasma density, magnetic field strength and rate of shear a priori

assigned, there exists a one-parameter family of solutions, differing

from each other essentially for the fraction of the total ourrent that

goes into ion or electron current respectively. The two currents can

also flow in opposite directions, subtracting thus from each other in

producing the net total current; in this case, however, they cannot

separately exceed certain limiting values, set by the requirement of

integrability for the distribution functions.

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S3LF-C0NSIST3M1 DESCRIPTION OF A WARM STATIONARY -PLASMA I1T A UMIFORKLY

SHEARED MAGNETIC FIELD

1. INTRODUCTION

The cont inuing i n t e r e s t in fusion research fo r plasma-magnetic

conf igura t ions with sheared magnetic f lux l i n e s has prompted the w r i t -

ing of the present no te , in which the simple s i t u a t i o n of a warm plasma

in a uniformly sheared magnetic f i e l d i s i n v e s t i g a t e d . By uniformly

sheared magnetic f i e l d we mean here a configurat ion in which the magnetic

f i e l d vec to r , while remaining of constant magnitude, r o t a t e s with chang-

ing i in such a way as t o descr ibe a he l ix of constant .pitch with i t s

t i p ( i t i s assumed, furthermore, t h a t there i s no x component of the

magnetio f i e l d , and t h a t the system i s uniform in the y and z d i r e c t i o n s ) .

In the fol lowing, f i r s t the general procedure i s descr ibed, which

i s appropr ia te fo r dea l ing with one-dimensional plasma-magnetic e q u i l i b r i a

in presence of shear (Seo .2) ; the equat ions employed are two Vlasov equa-

t i o n s fo r the two species of p a r t i c l e s , p lus Maxwell's equa t ions . Then,

in Sec* -3 > the spec ia l (force f r e e ) s i t u a t i o n of the uniform shear i s

considered, and an e x p l i c i t , exact so lu t ion of the equations i s derived

for this case.

2 . GENERAL FORMULATION OF THE PROBLEM

For a s t e a d y - s t a t e , one-dimensional oase, Maxwell's equat ions

read ( i n r a t i o n a l i z e d u n i t s , with yU and K as oonstants of the vacuum):

_ ^ . q • ( i )

with

Bx *• c o n s t . , Ey => c o n s t . , . E.» = oons t .

We choose here 8 ^ 5 0 , Ey ~ E-* * 0 . I n terms of the usual

vec to r and e l e c t r i c p o t e n t i a l s , the above equat ions become:

- 2 -

(2)

where

(3)

The charge and current densities in the above equations can be explicit-

ly computed from the solutions of two VIasov equations, respectively for

electrons and (singly charged) ions:

VA± [{ dx oix dx/Dii d* Dv dx

These are linear homogeneous partial differential equations of first order,

and oan "be integrated with the usual procedure. Any function of the so

called constants of the motion.

(5)

is a solution. In evaluating the charge and current integrals, it is

convenient to transform from the variables t) , V , w to the variables

p. » p » € • Correspondingly one obtainst

f£i

-3-

V * *

w i t h •

e"! s ± n

The factor "2" in the r . h . s * of Eqjs. (6) takes care of the double mapping,due to the appearance of u , but not u , in relat ions (5)« Thefunctions C (p• *p4?€ J in Eqe.(6) are in principle arbi t rary ,

d i t iand i t i s

, (7)

The distribution functions / (p J |*» > O a r e actually only defin-

ed in a oertain domain, namely, the domain of integration of the r.h.s.'s

of Eqs.(6), which in general will change from one value of x to another

(as the limits of integration e't are x-dependent). This entails the

possibility that in some region of the ( -|> , j , £ )-spaoe the func-

tions f may be assigned different values for different values of x

(possible "multivaluednsss" of the distribution functions in the space

of the constants of the motion).

This point has been thoroughly discussed in Ref. 1, Sec.II,

(actually, only for the case of two-dimensional distribution funotions,

i.e., independent of *pz $ but the appropriate extension to the present

three-dimensional case is easily inferred). Here, however, we shall

restric ourselves to considering only proper, single valued distribution

functions, which we assume to be defined by their analytic form in the

whole of the ( p , t> , £ )-space, irrespective of the boundaries

of the actual (x-dependent) domain of integration of the charge and

current integrals (6).

Also, we shall further restrict ourselves to the class of

solutions that can be obtained from distribution functions of the form

-4-

-£) *» • < * ) » . . ( * ) , ( 8 )

where s3 t ( ? v ) , S - ( t> ) » s*+ (t* )» sa-(?x ) a r e ^ principlearbitrary {dimensionless) functions of their arguments. The form (8)of the distribution functions includes of course the case of the thenaodynamic equilibrium, which corresponds to the simple choice s3± (-p ) 5 1

I t is convenient at this point to reduce a l l the equations tonon-dimensional form. Paralleling olosely the procedure of Ref.1, wedefine

and introduoe x = e^ f -j together with

6/KT ,

In terms of these non-dimensional quantities Eqs.(2) take the following form (dropping?f6r simplioity,all the asterisks from now on):

(9)

= Jo , a)

whereby the right-hand sides of the equations are given by the follow-

ing expressions;

( l 0 )

with

Eqs.(9)» together with (10) and (11), can be looked upon from

two different points of view. On one hand, assigned the sys (f •ji )

and s,t ( f i j . ) , the above equations constitute a system of three

(coupled, non-linear) ordinary differential equations for Ay , Ax

and 6 as functions of x ; this is the viewpoint that is usuallyl)2^taken, in analogous problems n ' . Alternatively, one may think of

assigning A« , A t and (j> as functions of x , thus obtaining a

set of integral equations for the s^^ (iN* ^ s*t Tit ^ a s u n k n o w n

functions. In-the next section, an example wil l be given of th i s

second procedure, in which, moreover, the equations will turn out to

be simple enough to allow an expl ic i t solut ion.

3 . UNIFORM SHEAR

As anticipated in the Introduction, we shal l endeavour in th i s

section to obtain the form of the ion and electron distr ibutions that

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corresponds to the following prescriptions for the macroscopic quantities

{non-dimensionalized as in the previous section):

with £ x - 0 f <f> = 0 > a n d

Ay (x) = A. w (fix), / * ( * ) * 4 Q s m ( * * ) ; (13)

or also

In other words, as i changes, the tip of the magnetic field vector

describes a helix of constant pitch %-w/^ , whose two possible

orientations correspond respectively to (13) or (131). The magnitude

&A0 of the magnetic field is constant, so is the (common) density

of ions and electrons, and there is no electric field. This is perhaps

the simplest situation involving a sheared magnetic field that can be

envisaged.

Within the framework of the ;pseoeding section, the conditions

(12), because of (10), are equivalent to

while, inserting (13) or (13") into the first two of Eqs.(9)» one obtains,

taking also into aooount (14):

-7 -

The above equations are certainly verified if the following is true:

with oC+ and «*. such t h a i r '

of+ + *_ = 1 . (16)

Eqs.(l5) are all of the form (dropping all subscripts, for simplicity);

or, because of ( l l ) :

(17)

This is a homogeneous Fredholm integral equation of the first type

for the unknown S (f) * We need not study it in detail here; we

shall content ourselves with obtaining a single, specific solution,

which can be shown to exist (and be explicitly derived) through the

following procedure .

By simple inspection of Eq..(l7)» we observe, first of all,

that a possible way of satisfying it would be to find a function 5(f)

such that the product s(?) exf> £-("p - yA) J were a function

symmetric with respect to the point f a yA .*• <* R A , and this

identically for varying values of A (though the function $M must

not itself depend on A, by its own meaning; see formula (8)).

This means writing for S^) the functional equation

to be satisfied identically in A (though A itself must not enter in

the expression for • ( ? ) ) . Taking the logarithm of both numbers,

-8-

and introducing r(y) - -£<*j s^?' , one obtains

Deriving alternatively with reepeot to £ and with respect to A, one

has

with the prime denoting derivation of r^O with respect to its argument

Solving the latter expressions for the derivatives gives

which ohviouBly define the same function y1

This is actually independent of A, as desired. Prom it one obtains

(where C is an arbitrary, positive constant), and thus

Upon re-introduotion of the appropriate subscripts, the latter actual-

ly gives origin to the four relations

Cy.

(18)

•where the parameters /nj, are defined in terms of the oi^ by

Inserting the above expressions into the first and third of Eqs.(ll) one

obtains

and since it is (from (13) or (131))

Ay + A\ = /4O ,

one sees that Eqs»(l4) oan also be satisfied,-"by simply taking the

constants of integrations C ^ , C^_, Ci¥) Cr_ such as to verify the two

conditions

= fi-TJ «P[->i,(i-i.jr1 jr* A* 1 • (20)

TTotioe that only the products Cy+. CZi. , Cy_ -z._ have meaning

(and not the four constants of integration separately), since only

such products appear in the final expression for the distribution functions

(see (21) below). • . '

For the convenience of the reader* we shall give here the final

result in terms of the original dimensional variables. For the K (fy }?*J e)

one obtains the expression:

(21)_ v v i ^ T A ^/^y i fj -_L*: ' * M ~ i /

-10-

(with the products C^t C 2 l given "by (20)), which corresponds to

the following form of the more familiar (

where the Ay(*) and Ax(x) are to be taken as given by (13) or (131)

It may be of interest to evaluate also the (total) kinetic

energy densities of the two species of partioles, which are given ~by

the triple integrals:

• & 0 0 - f olA dA AW l ^ /M ' tA * 1 ) F* ( M / , ^ , *) ••Lo» Lf> -L<w ^

Using (22), one obtains a result which is independent of x (as it was

clearly to be expected):

6. -

For -R r 0 (hence , in - 0 ) , this gives the appropriate result

for a Maxwellian plasma, [£t = •£ NKTJ whereas in the

opposite limit, that is>for -ft »e (/u/V/mi) , by making use of (19), one

obtains the following aaymptotio expression for the energy;

(24)

where Bo • A<> is the (constant) magnitude of the magnetic field

vector. Notice that this differs from the corresponding oold plasma

result - easily obtainable - just because of the presence of the second

term in the square brackets, which then represent the "disordered" or "~

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thermal component of the kinetic energy,

Inoonolusion, we can say that the following results have

"been established. For any given value of the plasma density A/ , of

the magnetic- field stregth Bo , and of the helix pitch. "2-tfjk ,

a one-parameter family of solutions has "been obtained. The free para-

meter in the solutions is related to the free choice cfoCvand °C_ ,

sub jet I: to the condition y oc . + oc_ « d . This partition of

the unity corresponds in fact to the partition of the total current into

ion current and electron current. Notice that the oi+ are also per-

mitted negative values, that is,the ion and electron currents nay also

subtract from each other, rather than add. However, the condition of

integrability of the distribution functions restricts the range of values

that are permitted for the <*+ . The integrability condition requires

in faot^' ^2 ** » from which one obtains

and correspondingly

We shall also point out that there appears to be no upper limit for

the values of M that can be prescribed; in particular -fc may be

larger than the reciprocal ion Larmor radius or even electron Larmor

radius (though the energy densities (24) become very large in these

cases). This is to be contrasted with other solutions, obtained else-l) 2)where n , connecting two asymptotically constant states, where the

electron Larmor radiua or ion Larmor radius were found to give a lower

bound for the width of the transition regions.

ACOOWLED (MEETS

The author is grateful to the IAEA and Professor Abdus Salamfor the hospitality extended to him at the International Centre forTheoretical Physics in Trieste. He also wishes to thank Dr. R.Sagdeevand Professor P .Budini, who personally invited the author to visit thePlasma Division of the Trieste Centre.

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s AITO POOTITOTSS

1) A.S3STERO, Phys. Fluids X » 44 (1964).

2) A.S3ST3RO, "A Vlasov Equation Study of Plasma Motion Across

Magnetic Fields", to be published in Phys. Fluids.

3) The condition of integrabillty for the distribution functions

poses further conditions, aotually, for the parameters <x+ and

ct_ to be admissible. This point will be commented upon further

on in this section.

4) To be sure, the conditions 71+ < d also imply the existence of

al l the moments of the distribution functions.

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