Upload
builiem
View
214
Download
1
Embed Size (px)
Citation preview
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.
- 1 -
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
- 6 -
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 "~
-11-
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.
-12-
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.
-13-