Download Stability of Plasma in Static Equilibrium

P/365
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Stability of Plasma in Static Equilibrium
By M. D. Krusiial and С R. Oberman *
Our purpose is t ^ derive from the Boltzmann
equation in the small mje limit x, criteria useful in the
discussion of stability of plasmas in static equilibrium. At first we ignore collisions but later show
their effects may be taken into account. Our approach
yields a generalization of the usual energy principles z> 4>5 for investigating the stability of hydromagnetic systems to situations where the effect of heat flow
along magnetic lines is not negligible, and hence to
situations where the strictly hydrodynamic approach
is inapplicable.
In the first two sections we characterize our general
method of approach and delineate the properties of
the small mje limit which we use to determine the
constants of the motion and the condition for static
equilibrium. In the next two sections we calculate
the first and second variations of the energy and
conclude with a statement of the general stability
criterion. In the final three sections we state several
theorems which relate our stability criterion to those
of ordinary hydromagnetic theory,5 we show how
to take into account the effect of collisions, and
briefly discuss the remaining problem of incorporating
the charge neutrality condition into the present
stability theory.
the displacement of magnetic lines of force away
from their equilibrium positions.) We rid ourselves of
the dependence on / by minimizing the energy with
respect to it, subject to the constraint that all constants of the forementioned type have their equilibrium values. We then have a sufficient condition
for stability involving \ alone.
GENERAL METHOD
There exists another constant (the initial phase) of
more complicated behavior and involving the time
explicitly. We do not employ this latter constant.
The first order variation of the energy vanishes
since in static equilibrium x(t) vanishes. The second
order variation leads to the form
Generally, the constants of the motion of the type
we employ do not specify the motion completely so
that there exist many motions evolving from the same
equilibrium (at t = —сю). By restricting the constants of motion to their equilibrium values, the only
possible motions other than the equilibrium behavior
are instabilities (the pure modes of which have an
exponential time behavior and hence vanish at
t= —oo).
To illustrate the method we consider the simplest of
examples, the equation of motion
x = Àx.
This system has one time-independent constant,
the energy
Our method consists of writing down the energy
of the system to second order in the perturbation
fields of / and B, where / is the distribution function
in x,v space and В is the magnetic field intensity.
We eliminate the terms involving the second order
perturbation / by employing certain constraints,
namely, that certain constants of the motion have their
equilibrium values. The constants of the motion we
employ are time-independent and are functionals of /
and В which are regular at, and permit expansion about,
their equilibrium values.
The resulting expression for the energy is a quadratic
form in / and Ç jointly whose positive-definiteness
provides a sufficient condition for stability. (More
about Ç later, let it suffice for now to say % describes
* Project Matterhorn,
New Jersey.
(1)
Princeton University, Princeton,
137
for the perturbation. Clearly if Я > 0 then the
energy is a positive-definite form and the system is
stable, for there exist no motions away from equilibrium. The stable oscillatory motions must necessarily
increase the energy from its equilibrium value and
hence are disregarded. If Я < 0, however, the form
is indefinite, (3) can be satisfied nontrivially, and there
exist (exponential) motions away from equilibrium.
This method was suggested by a technique used by
W. Newcomb 2 to show stability in the much simpler
case of a plasma with a Maxwellian equilibrium distribution.
138
SESSION A-5
P/365
M. D. KRUSKAL and С R. OBERMAN
SMALL m/e LIMIT
where P is the stress dyadic, I is the unit dyadic,
In the present investigation we obtain these criteria
by examining the second order variation of the energy
2
with p± and pn given by
2
S = f dH\{B + E )
/J/
dvded*x[mf№ + W + vB)]
(4)
from its equilibrium value. Here E and В are the
electromagnetic field intensities, / is the distribution
function in x, v space of a particular species of charged
particles, the summation is over all species, and
v = a + v± + ?n,
B(x, t) = В (х, t)n(3Ct)
(5)
a=ExB/B!,
(7)
q = V»n,
(8)
v = v^\1B,
(9)
2
s = a /2 + vB.
(6)
(10)
The quantity (Б/д) dvde represents the volume element
in velocity space. We assume for simplicity that any
boundaries present are such as to present no complications, e.g., rigid and perfectly conducting walls with
В entirely tangential. The properties of the small
tn/e limit we employ are:
(a) v is constant following a particle motion.
(b) f is rotationally symmetric in velocity space
about a line parallel to В and passing through the
point a.
(c) a is the common drift velocity of all particles.
This last fact, as is well known,1' 3> 4> 5 permits the
introduction into the formalism of a displacement
vector Ç(x, t) which governs not only the development
of the field quantities but also the transverse motion
of the particles. We defer consideration of the
additional property of charge neutrality until later,
but we remark at this time that in contradistinction to the Chew-Goldberger-Lowx (CGL) theory,
where one particle species is taken to have a much
smaller mass than another in order to satisfy the
condition that E*n vanishes, we treat all particle
species on an equal footing, regard E and В as participants in the mje expansion and find that E»n is
indeed zero to lowest order in m¡et which is all that
is necessary for the evolution of the expansion.
The equilibrium distribution function we denote
by g(v, s, L) where L labels the line of force passing
through a point in space. We take g to be monotonic in s with
ge<0
(11)
for reasons we shall see later. The equilibrium condition is
0 = - V-P+ (VxB)xB
= - V.(/>+l + £_nn)+ VB 2 - B.VB
and
pL - mjj(B/q)dvd8 vBg.
FIRST ORDER VARIATION OF ENERGY
The first order change in
g?=
B.B
given by
)- шíJ
dvdecPx
) + Bq(f+ efe)].
X [{g
(16)
Here we take for convenience the change following
the displacement Ç rather than at a fixed point, and
we now take equilibrium quantities without designation and denote perturbation quantities with
circumflexes. (We shall frequently perform an integration by parts with respect to e, as we have in
arriving at (16), by making use of the fact that \\q
equals qe, which follows immediately from (7). We
do this in order to avoid the appearance of nonintegrable integrands like l/<73.) The perturbed
magnetic field intensity is given up to second order in
(17)
= В + [B.V Ç + i {В [(V-Ç)2 +
- 2(y.Ç)B-VÇ},
and the volume element at the displaced point is
given by
Ç) = d4{\ + V-Ç
(18)
A.
However, we find S vanishes trivially when we make
use of the general constraint condition that all constants of the motion have their equilibrium values.
Indeed, to first order, we have
| Cj C(B¡q)dvded*x G(/, v, L)|A = 0,
(19)
where G(f,v,L) is an arbitrary function of the distribution function / (remember / = 0), v, and L where
again L labels a line of force passing through a point
in space. That magnetic lines of force maintain their
identity during a displacement 6 is a consequence of
the fact that E«n is zero (to lowest order in m/e) ;
that particles stick to magnetic lines of force is a
consequence of (c). We may write condition (19) as
0 - - J fjdvd8d4Gf(g{v, e, L), v, L)
X
(12)
(15)
-
(B/qffl
(20)
139
PLASMA STABILITY IN STATIC EQUILIBRIUM
and now regard Gf as an arbitrary function of e, v, L
because, by (11), g is monotonie in e. Accordingly
we may strip (20) to the basic constraint condition
We now make the particular choice
Gf(g(v,e, L), v, L) = — me
(23)
for Gf in (20), add the resulting expression to (16) in
order to eliminate /, and obtain
X ( - V.Ç + nn:VÇ)(<?2 - vB)ge - Bf/q],
(21)
where the integration is over a thin tube of force T.
(We may, in general, transform integrals over thin
tubes of force of flux dtp to integrals along lines of force
according to the prescription
JT<Px BA (x) - dy)fLdl A (x),
+ B.B)
f
- mf f fdvded*xg [BqV-% + {Bq)"].
(24)
The right-hand side of (24) now vanishes identically as
stated when use is made of (10), (17), (12), (14), and
(15).
(22)
SECOND ORDER VARIATION OF ENERGY
for arbitrary A(x).
The second order change in energy is given by
B.B
mfffdvded*x
Щ3))*
(Чее + 2ge)
(eL + 2fe)+Bq*(efee + 2fe)}.
Here Q is the mass density. If we write the constraint
condition (19) to second order,
(G'f> +
0 =
(25)
+ (VxB).(ÇxQ)
G'fjy
- [-
Bq*)ee + UG>f)ee(Bq*)
I) 2 - V? :
G'fV
X
(26)
and make the same choice (23)for Gf, we can eliminate
-s
Ъ
/ in the same way / was eliminated in first order, and
S then becomes a quadratic form in Ç and / jointly.
(We have not explicitly introduced the next order
correction to the displacement \ since its contribution
to S vanishes in second order, as the contribution of
\ to S vanished in the first order.) We now have
ÔW
(27)
with ÔW defined by
ÔW =
nn:VÇ) 2 ].
(30)
We are now prepared to state our stability criterion :
ÔW is a quadratic form in \ and / jointly, otherwise
depending only on equilibrium quantities. If this
form is positive definite (i. e. positive for all nontrivial
permissible \ and /), then our system is stable. Indeed,
the only \ and / for which ê can vanish (as it must
since S is a constant of the motion) are the trivial ones
and hence no instability can develop. We hope to
show this condition is necessary as well as sufficient
for stability.
Let us now minimize this expression with respect to
/ (find the worst / from the point of view of stability)
subject to the general constraint condition (21).
To do this we multiply (21 ) by the Lagrange multiplier
À(e, v, L), integrate over v and s and then integrate
(sum) over tubes of force to obtain
0 = J J fdvded*xl(v, e, L)[BqgeV-%
+
+ (Bf) * },
- vB) ~ Bf/q).
(28)
(31)
and
(29)
We find after using (10), (17), (14), and (15) that (28)
becomes
We now add this expression to (30) and then vary
with respect to /, obtaining the Euler equation
- f/g6 + Я = 0.
(32)
140
SESSION A-5
P/365
M. D. KRUSKAL and С R. OBERMAN
present particle theory implies stability under the
CGL. fluid theory. For by means of Schwarz' inequality
If we now use this to eliminate / i n (16), we find
X = Г dH(B¡q)[q^-% + ( - V.Ç + nn:VÇ)
d4{Blq).
X («f-
(33)
2vB
These give the minimizing / in terms of Ç, so equation
(30) now represents a quadratic form in Ç alone, and
otherwise involving only equilibrium quantities. In
the hydromagnetic г> 5 fluid theory, where the pressure
develops according to the adiabatic laws
d {p „B*lQ*)jdt = 0
d {pJqB)jdt
(38)
If we now insert this inequality into (37) we find
(34)
= 0
(39)
the corresponding expression SWD is
When the right-hand side of (39) is expressed in terms
of p- and p+, it becomes precisely the last integral
on the right-hand side of (36). Hence,
n = i¡<Px{Q* + (VxB).ÇxQ + \p
(Ç)
( № ) C
- 2
:VÇ + 3 (
4(nn:VÇ)2
(40)
ÔW
(35)
+ n.V|.(Ç.Vn)]}.
We can now write (30) as
We can obtain an important inequality in the
opposite direction when the equilibrium distribution
function is isotropic (gv = 0). In this case
g(et I),
ÔW = ÔWD + I /35)
and we may write
where
(42)
/ = _ i mffJ{B/q)dvde<Pxge{iï - v* В*
-уВ)~Щ1
X =
(37)
(This expression for aW can be shown to be independent of the component of Ç parallel to В as it
should be on physical grounds.)
COMPARISON THEOREMS
If our condition is necessary as well as sufficient for
stability, it is easy to prove that stability under the
- h
= (15/4)
Я
\¡Bm
dsxdy
Bp
(
~0Г=ГуЩг{
- |yB)nn:VÇ
+ \yB V. %]/ f BdH{\
-yB)-*,
where
у - v\z.
(43)
If we now take у and e as variables in velocity space
rather than v and s wefindwe may write I in terms of the
moment p after an integration by parts in e and
obtain
-УВ)-Щ\
-
¡TBd4(\
(44)
where Bm\n is the minimum value of В along a line of
force. For this isotropic case we now have
ÔW = \
+ (VxB).ÇxQ
This result has been also obtained independently by
M. Rosenbluth7 using another method. Since the
integrand in 1г is positive, we may take Schwarz'
inequality in the opposite direction, perform the
у integration, and obtain
ÔW > ÔWH =
(46)
where
We may conclude that if for a y = 5/3 hydromagnetic
fluid we can show stability (ÔWH < 0) then we may
conclude the system will indeed be stable under our
more refined particle picture.
PLASMA STABILITY IN STATIC EQUILIBRIUM
COLLISIONS
141
and
In case collisions are not negligible, the situation is
somewhat altered in that we lose most of the constants
of the motion. Those of the type in (19), for which G
is independent of / and v, remain. However the fact,
that the equilibrium distribution function is now
locally Maxwellian (as it must be for static equilibrium
with collisions) enables us to proceed with the argument. We do not lose the property that particles
stick to magnetic lines of force, however, since the
size of the step away from a magnetic line of force
after a collision goes to zero with m/e.
The Boltzmann Ж function
T dHdvde (B/?)fea
+ (-V-Ç + nniV
(56)
- vBgt) - / j .
This leads to
(57)
for the value of the integral involving / 2 . (The
meaning of <y • Ç> is the same as in (46).) It follows at
once that
d*x{B/q)dvdeO{L)f{e, L) In /
Ж = (jj
Жо= + Ж +Ж +..
(58)
(48)
has the well-known property
Ж < 0.
CHARGE NEUTRALITY
(49)
We now assume that all regular, time-independent,
phase functions have their equilibrium values at
t = — oo and in particular obtain
= - oo) =
i.e., stability is not destroyed by the occurrence of
collisions.
We conclude our presentation with a brief discussion of the charge neutrality condition which is also
a consequence of the small m\e limit. This condition
is
(50)
0 = S nl el
Now
i
§ = 0
(51)
Ж < 0.
(52)
= — S e* Íf/Bqdvds.
and, therefore,
But аЖ/dt is linear in / and hence a reversal in the
sign of / leads to аЖ\&1 > О. We conclude, therefore,
Ж = 0
We must now minimize (25) with respect to / subject
to the present constraint as well as (16). This leads
to a coupled set of linear integral equations for
the multipliers with which the constraints are introduced. We have not solved these equations and defer
further discussion to future work.
(53)
and finally obtain
ACKNOWLEDGEMENT
(54)
Now S is still a constant of the motion and we may
use the expression for Ж to eliminate / from (25),
obtaining expression (28) for this Maxwellian case
with the additional positive term —Ж on the righthand side. We could have used the theorem that 8 S
+ Ж is a minimum for the Maxwellian distribution
with modulus в, to arrive at this result. We minimize
this expression with respect to / now with the constraints that the Ж function for each tube of force is
constant to first order (see (53)) and the number of
particles in each tube is constant. That is, using
(16) and (20) we may minimize subject to the constraints
0 = J f fT dfixdvde{B¡q)e[q*V*Cge
(59)
(55)
+ ( - V.Ç + nil VÇ) (q2 - vB)ge - /]
We are indebted to M. Rosenbluth for a valuable
discussion.
REFERENCES
1.
G. F. Chew, M. L. Goldberger and F. E. Low, The Boltzmann Equation and the One-Fluid Hydromagnetic Equations in the Absence of Particle Collisions. Proc. Roy.
Soc. A, 236, 112 (1956).
2. I. B. Bernstein, Plasma Oscillations in a Magnetic Field,
Phys. Rev. 109, 10 (1958).
3. S. Lundquist, Magneto-Hydro static Instability, Phys. Rev.
83, 307 (1951).
4. S. Lundquist, Studies in Magneto-Hydrodynamics, Ark.
Mat. Ast. Fys., 5, 297 (1952).
5. I. B. Bernstein, E. A. Frieman, M. D. Kruskal and R. M.
Kulsrud, An Energy Principle for Hydromagnetic Stability
Problems, Proc. Roy. Soc. A, 244, 17 (1958).
6. W. Newcomb (to be published).
7. M. Rosenbluth (private communication).
8. R. C. Tolman, The Principles of Statistical Mechanics,
p. 550, Oxford University Press (1948).
142
SESSION A-5
P/365
M. D. KRUSKAL and С R. OBERMAN
Mr. Kruskal presented Paper P/365, above, at the
Conference and added the following remarks :
I should like to summarize the development of the
theory of plasma stability and, in particular, to
compare the work of Rosenbluth and Rostocker, as
described in Paper P/349 presented to this Conference,
with the Kruskal-Obermann theory as expounded in
Paper P/365.
The main problem in the production of controlled
thermonuclear energy is the confinement of a completely ionized plasma of hot nuclear fuel by means of
strong magnetic fields. A major obstacle to such
confinement, however, is the possible existence of
instabilities which may quickly disrupt an otherwise
satisfactory equilibrium configuration. Accordingly,
it is a major task of theory to predict the stability of
given configurations, as well as to devise ever better
means for so doing.
In the earliest theoretical approaches the plasma
was treated as a continuous hydrodynamic fluid described at each point of space-time by a few parameters, for
which, together with the electromagnetic fields, one
could write a complete set of partial differential
equations determining the possible motions. This
type of description is appropriate when there is a
mechanism which keeps neighboring particles close
together, so that a set of neighboring particles can
form a coherent element of fluid. In many applications the frequent collisions between the particles
constitute such a mechanism.
At fiist stability was treated by a number of
investigators on the basis of the hydromagnetic
equations, using the method of normal modes. This
involves linearizing the equations for the perturbations
of the equilibrium state under investigation, and
looking for solutions of these linearized perturbation
equations which are purely exponential in their
time-dependence, with real or complex exponents.
The equilibrium is unstable if growing exponential
solutions exist.
Except for rather simple equilibrium states, however,
this normal mode method is very difficult to carry
through. Following the pioneering work of Lundquist,3 several groups of authors therefore developed
an energy principle which greatly enlarges the class of
stability problems which can be investigated practicably. This energy principle states that a static
equilibrium is stable if and only if a certain associated
homogeneous quadratic form is positive definite. The
quadratic form represents physically the second-order
variation in the potential energy of the system due to
an arbitrary virtual displacement of the fluid.
The energy principle provides a very satisfactory
general stability theory, at least for static equilibria,
when there is some mechanism (almost necessarily
collisions) that permits the plasma to be treated
hydrodynamically.
In controlled thermonuclear
energy applications, however, the effects of collisions
are negligible for processes that take place as quickly
as the growth of most instabilities. There is, instead,
a mechanism which is in a sense two-thirds effective
in keeping neighboring particles close together;
namely, a strong magnetic field forces the charged
particles to gyrate around a point which sticks to
and moves with a line of force. As a result, the particles cannot disperse in the two directions perpendicular to the magnetic lines of force, but only in a
direction parallel to the field lines.
Several groups of authors, notably Chew, Goldberger,
and Low,1 have shown how to treat the plasma in the
mathematically appropriate way when collisions are
not important. They employ the so-called collisionless Boltzmann equation with a term {q/fn)
(E + v X B) in place of the usual collision term.
The electric and magnetic fields satisfy the Maxwell
equations and the current and charge density are
expressed as sums over the different species of integrals
over velocity space, utilizing the distribution function
which satisfies the Boltzmann equation as a weighting
function in the integrand.
One now wishes to obtain the limiting form of
these equations as the radius of gyration of a particle
in the magnetic field becomes very small compared to
the characteristic length of the system under consideration, and the period of gyration becomes very small
compared to a characteristic time. This limiting
process may be formalized in various ways, the
simplest of which is to treat the charge q as being
very large. It is not trivial to carry this program
through systematically. However, it has been shown
that the resultant reduced system of equations is
easier to handle than the original system. The reduced
system, obtained in this fashion, may be used to
investigate the stability of equilibria by the normal
mode or equivalent methods. This is essentially
the approach adopted by Chandrasekhar, Kaufman,
and Watson. Unfortunately, compared to the hydromagnetic equations, it is more difficult to carry through
the normal mode method with the reduced system
when the equilibrium is not simple.
In the two papers which I wish to discuss here,
(P/365 and P/349) an energy principle for determining
stability is derived based on the reduced system.
The methods used differ considerably from each other.
The conclusions reached in each paper are in some
respects more general than those in the other, but
where they overlap they agree.
In the paper by Rosenbluth and Rostoker, the
equilibrium state is assumed to have isotropic distribution functions to lowest order in the gyration
radius. The first part of the paper is devoted to
obtaining the equations of a normal mode, and it is
shown that the first-order perturbations from equilibrium of all quantities are proportional to the exponential of cot. The quantity œ, which may be complex,
is the characteristic growth or frequency parameter of
the mode. In this analysis, all the first-order perturbations are expressed in terms of the usual first-order
vector \, which describes at each point the perpendicular displacement of the magnetic line of force.
One obtains finally an integro-differential equation
for \ analogous to the differential equation for Ç obtainable from hydromagnetic theory. The presence
PLASMA STABILITY IN STATIC EQUILIBRIUM
of the integrals, which are taken along equilibrium
lines of force, is of course a consequence of the spatially
non-local character of the treatment. The normal
mode equations constitute an eigenvalue problem
for со.
The next step in the Rosenbluth-Rostoker method
is to construct the second-order change in energy
due to the first-order perturbation. This can be
expressed as a quadratic functional W in Ç. For œ = 0,
the integrand of the functional contains three terms
which are the same as in the hydromagnetic case
because they represent the second-order change in the
energy of the magnetic field. The fourth term in the
integrand involves two integrals which, for each
point x and each value of the dummy variable of
integration, are line integrals taken along part of the
equilibrium line of force through x.
It turns out that the quadratic functional W
vanishes and is stationary for a vector field \ which
is not identically zero if and only if \ is a solution of
the eigen-value problem described earlier and has
the eigenvalue œ = 0. This suggests strongly that
if W is negative for some Ç, then there is an instability.
A proof is given that if W is positive-definite then the
equilibrium is stable.
In the other paper, we (Oberman and Kruskal)
have made no effort to obtain conditions for instability,
but have sought the weakest conditions we could for
stability. To do this we look for constants of the
motion of the system which do not depend explicitly
on the time and which are regular near the equilibrium
configuration. Since any purely unstable motion
has been arbitrarily close to the equilibrium state far
enough back in time, such constants of motion must
have the same values for an unstable motion as for
equilibrium. That is, the first and second-order
perturbations of these constants of motion must
143
vanish. This leads to severe restrictions on the
first-order perturbations of the physical quantities;
when they are so severe that these perturbations must
vanish, stability is assured.
We do not assume that the equilibrium distribution
functions are isotropic, but we do require of the equilibrium that the mass velocity vanish to lowest order
in the gyration radius for each species of ion. We
obtain finally a quadratic form in Ç, the positivedefiniteness of which implies stability. In the case of
isotropic equilibrium distributions it reduces exactly
to the form W obtained by Rosenbluth and Rostoker.
Our method of using constants of the motion can
be applied even when the collision terms of the
Boltzmann equation are retained. There are then
far fewer suitable constants available, but this is
compensated for by the necessary restriction to those
equilibria which are invariant during collisions to
lowest order in the gyration radius. Such equilibria
are those having Maxwellian distributions with constant
temperature along lines of force. The final result
turns out to be the same energy principle as before.
Both papers also give essentially the same comparison theorems. One of these is that W is bounded
above by the simpler result derived from the standard
hydromagnetic theory with two distinct pressures, one
parallel and one perpendicular to the magnetic field,
each governed by its own adiabatic equation of state.
The other comparison theorem is that in the isotropic
case, W is bounded below by the result of standard
hydromagnetic theory with one scalar pressure.
Because of these comparison theorems, fortunately,
we have demonstrated in these two papers that the
stability results previously obtained from standard
hydromagnetic theories still have considerable significance when viewed in terms of our more accurate
calculations.