Download Stabilization of Plasma by Nonuniform Magnetic Fields

P/2212
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Stabilization of Plasma by Nonuniform Magnetic Fields
By B. B. Kadomtsev and S. I. Braginsky:|
wards from the boundary. This result is confirmed by
a rigorous hydrodynamic treatment.
The equations of magnetohydrodynamics for small
plasma oscillations can be written as a single selfconjugate differential equation of the second order,
for a displacement £ from an equilibrium position.
This equation can be obtained from a variational
principle with a real functional. If the plasma pressure
po and its density /><) are constant and the currents
flow only along its surface, one obtains for the proper
oscillation frequencies from the variational principle:
STABILIZATION OF PLASMA BY MEANS
OF GUARDING CONDUCTORS
The main problem in controlled thermonuclear
reactions is the confining of high-temperature plasma,
i.e., the obtaining of a stable plasma configuration
isolated from the walls. One of the most direct methods
for creating a plasma separated from the walls consists in the application of the well-known pinch effect
stabilized by a longitudinal magnetic field and a
metal casing (this problem is considered in detail by
S. I. Braginsky and V. D. Shafranov). It is of interest to study other possibilities of obtaining a stable
plasma configuration. The present paper deals with
the problem of stabilizing plasma by means of a nonuniform magnetic field of a special kind. The systems
considered here can, obviously, be used both separately and in combination with the pinch effect.
Let us consider a case when the magnetic field
within the plasma is absent. We shall assume that the
conductivity of the plasma is infinite, and all the
currentsflowingalong the plasma are concentrated on
its surface.
The criterion of stability of a plasma such as described above can be defined by. simple qualitative
considerations. Indeed, the instability of the plasma
cylinder pinched by the magnetic field of its own
current is closely connected with the decrease of the
magnetic field, outwafds from the boundary of the
plasma. This becomes particularly clear if we consider local perturbations of the boundary. If there is
no magneticfieldwithin the plasma, the plasma pressure in equilibrium is balanced by the external magnetic field P = Я2/8тг, where Я is the value of the
field at the boundary. Let us imagine that the plasma
ejects a flat "tongue" oriented parallel to the magnetic Unes of force. Such a "tongue" slightly perturbs
the magnetic field—as if it were pushing through the
lines of force and slightly drawing them apart. If the
field decreases outward from the plasma boundary,
the tip of the "tongue" touches the region of lower
magnetic pressure and will stretch out further with an
acceleration.
f -Л2
Jv{ yPo
where Vi is the volume occupied by the plasma, Ve is
the external volume, So is the plasma boundary, Hoe
and Яо1 are the unperturbed fields outside and inside the plasma respectively; p and H are the pressure
and field of small perturbations, f» is the normal component of the displacement on the boundary So and
д/дп is a derivative along the external normal to SoThe above equation shows that the observance of
the condition 8[{Н0в)2-{Н0Щдп > 0 at all points of
the plasma boundary is sufficient to ensure plasma
stability (to2 > 0).
If Ho1 = 0, an increasingfieldaway from the plasma
boundary is a sufficient and, as is seen from the previous qualitative consideration, also a necessary condition for its stability. (Similar results were obtained
in Ref. 1.) Hence, we see the way to find stable
systems. For instance, the infinite flat plasma layer
(Fig. l(a)), outside of which the field is constant,
shows no instability.2 However, a flat conductor of
infinite broadness cannot be made and for any finite
broadness instability appears on account of the
edge effect. As is well known the edge effect in a
flat condenser can be eliminated with the help of
guard rings. An analogous method can be used in
this case too. The system given in Fig. 1(6), in which
part of the plasma is substituted by metal conductors,
is stable. With increase in the magneticfieldthe plasma boundary is driven inwards, and in this case the
magnetic field increases outward from the plasma
boundary (Fig. l(c)). Such a system is all the more
stable.
Therefore, it seems natural that for stability of such
a plasma, it is necessary for the field to increase outOriginal language: Russian.
* Academy of Sciences of the USSR, Moscow.
Г
H%
J r{+ve 4тг
M a[(ffOe)2-.(ffol)2]
8n
dn
233
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SESSION A-9
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B. B. KADOMTSEV and S. I. BRAGINSKY
(b)
2212.1
Figure 1. Stable plasma configurations
One significant fact is that the guard conductors
prevent the lines of forcefrbm freely moving with the
plasma. The lines of force tighten on the conductors
like elastic strings and because of this no sausage-type
instability and distortions of the cylindrical plasma
column appear. This ejfect can be further intensified
if the plasma is surrounded by a lattice of metal rods
placed perpendicular to the Unes of force (Fig. l{d)).
If the lattice is sufficiently dense, it rigidly determines
the plasma boundary, provided the magnetic field
pressure outside the,lattice is larger than the plasma
pressure. Indeed, the magneticfieldcannot sag deeply
through the dense lattice. Therefore, if the plasma
(a)
Figure 2. Isolation of plasma from rods
boundary moves away from the lattice inwards
appreciably, the magnetic field on it decreases drastically. If the plasma moves outwards, the current
from the rods passes on to the plasma which will take
up the whole magnetic pressure, greater than the
plasma pressure. As a result a certain equilibrium and
stable state of the plasma boundary will be established
which, roughly speaking, coincides with the lattice
surface. In this case part of the magnetic pressure is
held by the rods and stability is reached at the cost of
some increase in the value of the magnetic field
increase balancing the plasma.
If the lattice is flat, it is possible to consider in
detail the plasma boundary near the rods. Since outside the plasma and rods, curl H = 0 and div H = 0,
and Я 2 = 8TT/> = const on the plasma boundary, the
problem of the plasma boundary is analogous to the
hydrodynamic problem of potential flow of an incompressible fluid with free boundaries. By known
methods of conformai transformations, it is possible to
find the magnetic field and the shape of the plasma
boundary. If the magnetic field pressure does not
exceed the plasma pressure too much, the plasma
boundary sags between the rods, as is shown in
Fig. 2(Й). In this case the magnetic field increases in
the outward direction from the plasma boundary, so
that it is stable with respect to small perturbations.
If there is some magnetic flux between the rods
and the plasma, the plasma boundary takes the
position illustrated in Fig. 2(6). In this case the
plasma never touches the rods, and there is no direct
heat exchange between the plasma and the rods.
Such a configuration is also stable with respect to
small perturbations.
Thus, the guard lattice gives the plasma boundary
stability with respect to small perturbations and
also leads to a constant stabilizing force for displacements which are large in comparison with the distance
between the rods.
The guard conductor systems can differ widely.
The simplest system is the plasma cylinder; the lattice
can be made either of straight rods arranged along the
generatrix of the cylinder or of rings perpendicular
to the generatrix. In the first case (Fig. 1(¿)) the
current in the plasma and in the rods flows in the
direction of the cylinder axis. This current produces
an azimuthal magnetic field. In the ring system the
longitudinal magnetic field is produced by an additional external winding. The azimuthal current flows
along the rings and the plasma in a direction reverse
to the direction of the current in the winding. Each
of these systems can be either transformed into a
torus or restricted at the ends. In the rod system, for
instance, electrodes can be placed at the ends, and the
ring system can be restricted by intensifying the magnetic field near the ends ("magnetic plug") with a
simultaneous decrease in the diameter of the extreme
rings (Fig. 3). In the last system the presence of a
longitudinal magnetic field within the plasma is very
undesirable since it facilitates the release of the
plasma from the system.
PLASMA STABILIZATION
The guard conductor systems are, generally speaking, not stationary. Their time of action is restricted
by the finite value of electrical conductivity of the
plasma, as well as of the conductors if the- latter
are passive, i.e., if they are not fed by energy frorri'the
external circuit. Indeed, the entire plasma will move
beyond the lattice, where it will be released because
of the loss of stability, during the time tB — 4iro-a2/c2,
where a is the transverse plasma dimension and a is
its conductivity. In reality the lifetime of the plasma
may be significantly less. For instance, in the system
illustrated in Fig. 3, with a sufficiently large frequency
of collisions, the plasma in the skin layer will escape
along the lines of force with a thermal ion velocity
VT- If the length of the system equals L, the time of
escape of particles along the lines of force is ir = L/VT.
As a result of the plasma loss from the skin layer a
certain thickness of the skin layer independent of the
time
is established, f and the plasma lifetime t will become,
obviously,
L\i
With sufficiently high plasma conductivity t ^ ta.
A similar situation is found in other systems, viz., as
a result of diffusion of the magneticfield,the plasma
moves beyond the rod to the line ABC in the solution
presented in Fig. 2(a) or to the dotted line of force in
the isolated solution presented in Fig. 2(b), so that
around each rod there appears a plasma "encasement". But each rod must somehow be suspended
from the walls of the chamber; the supports would
pass through this plasma "encasement". Consequently, in the system there should occur losses
on the supports. If the supports have a diameter b,
and the distance between the supports of a given
rod is /, the probability for the particle to escape
when passing through one "encasement" equals b/l.
Hence, the lifetime of particles in the skin layer is
tT = L/VTJ/Ь, where L ~ a is the distance between
the rods. By repeating the above calculations we
obtain
,
.....
'№)*
/4яО«» L I \*
(
j
which, at b/l ~ 10~2, is one order greater than the
lifetime of the system presented in Fig. 3. The supports submerged in hot plasma are subject to a
powerful plasma ion bombardment which results in
their evaporation and in contamination of the system.
The evaporation of the supports can be considerably
decreased if they are supplied with a larger current,
so that the magnetic Unes of force near the supports
close. Then the ions can reach the supports only by
t The mean skin layer thickness
Larmor radius of electrons if 8 < />e.
should be taken as a
235
Winding
2212.3
Figure 3. Ring and magnetic plug geometry
diffusion, and the density of the energy loss will be
of the order of
Tnp2
-г-—
О
T
where n is the plasma density, T is its temperature, p is
the mean Larmor ion radius, т is the ion-ion collision
time and w = eH/Mc is the ion cyclotron frequency.
If p/b ~ 1, the energy loss decreases от times in
comparison with the energy release nTv^ on the
currentless support. In this case there appears along
the supports at some distance from it a region of
very weak magnetic field which will play thé part
of a "canal" for the plasma leakage. But if the width of
the canal is of the order of b\ this leakage, in any case,
will not exceed the direct loss to the supports.
It follows from the estimates given that though
the plasma lifetime is less than the skin-time, it can
reach several hundredths of a second at a ~ 102 and
T of the order of several kev. The estimates show
that for a D-T mixture the thermonuclear reaction
energy released in this time can reach the value of the
thermal energy. It may be expected that the most
promising is the use of the guarding conductors in
combination with the pinch effect. As is known from
Ref. 3, the plasma pinch in the toroidal system possessing an azimuthal symmetry can be in equilibrium only
in the presence of a current along the pinch. Therefore, alternation of the current is not permissible in
such a system: at the moment of zero current equilibrium is lost. If a torus with a strong magnetic
field is supplemented by guard rings, the plasma at
the moment of zero current will be retained by the
rings,' and this will make it possible to extend the
discharge over many periods of the axial current.
On the other hand the self field of the current compressing the pinch apparently leads to a decrease in
loss on the suspenders. These problems require
experimental investigation.
MAGNETIC TRAPS
The problem of plasma stabilization may be
approached by a different, in some sense opposite,
method. The condition for stability obviously becomes
more and more severe with, an increase in plasma
pressure. Therefore, if we consider the case of a plasma
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B. B. KADOMTSEV and S. I. BRAGINSKY
Both in closed-line traps and in adiabatic traps the
particles travel quasi-periodically; they move mainly
along the Unes of force, gradually drifting from one
Une to another. If the magnetic field is strong enough
so that the total drift during one passage of the Une
of force is small, then during such a movement the
longitudinal invariant / = \udl, as can be shown, is
conserved (u is the longitudinal velocity component,
<п is the element of length of the Une of force; the drift
velocity itself is proportional to the gradient J). If the
electricfieldis absent, then v = const, and the equation
for the surface along which the particle drifts can be
written in the form
Figure 4
with a pressure much lower than that of the magnetic
field, it will enable us to determine the minimum
requirements for the field which is capable of confining the plasma. Under such conditions it is a
property of the field itself, since the low-pressure
plasma plays only the part of a "test body"—
practically, it does not distort the field under investigation. A field capable of confining a quasi-neutral
plasma with an infinitesimal pressure will be called
a magnetic plasma trap. Consideration of magnetic
traps makes it possible to obtain a simple plasma
confinement criterion which provides some orientation in more complicated cases.
Magnetic Particle Traps
Since plasma confinement implies the confinement
of separate plasma particles and since the investigations of the behaviour of charged particles can
often predict the behaviour of plasma, it would be
of interest to investigate the simpler problem of magnetic particle traps, i.e., magnetic fields that can
hold charged particles in a limited space for a long
time. | We shall start our consideration with these
particle traps.
Particle tiaps naturally fall into two groups: traps
with closed Unes of force, and adiabatic traps based
on the conservation of the transverse invariant
I = œ>2/ff, where w is the velocity component perpendicular to the magnetic field. Only those particles
which experience a reflection from the "plugs", i.e.,
from the strong magneticfieldregions, are confined in
the adiabatic traps suggested by G. I. Budker.
Particles with a small w/v ratio pass out through the
"plugs" and one would like by some means to return
them. However, an ordinary closing of the lines of
force will not bring us to our aim: the particles moving
in a closed line will gradually drift to the walls. An
analogous drift takes place in all closed-line traps,
and the problem of confining the particles leads to the
investigation of this drift.
$ Magnetic particle traps are also of interest from the point of
view of confinement of the charged products of the thermonuclear reactions.
Thus, in order to keep a particle with a given v and
J within a certain volume, the ф surface must be
closed and remain entirely inside this volume.
The condition is simplest for particles with ujv = 1,
i.e., J = 0. For these particles ф simply denotes the
length of the Une of force, and consequently, they will
be confined if the magnetic field is "corrugated", i.e.,
if the peripheral Unes of force are longer than the
internal Unes.
As an example, we shall consider two corrugated
magnetic field traps. Each of them consists of two
corrugated traps with plugs coupled to one another
with a "cross-link" A. The presence of plugs with
field Hm will lead to the confinement of a great
number of particles in the corrugated parts and only
particles with w%\v% < Ho¡Hm ^ 1 will pass over the
cross-link, travelling along the closed Unes of force.
The first system (Fig. 4(a)), which can be called a
closed magnetic plug trap, operates to a certain extent
like the ordinary adiabatic trap. If L > R, the length
of the Une of force is determined by the corrugated
part and the share of the cross-Unk is insignificant.
Therefore the particles that have passed over the
cross-link will be stabilized: corrugation leads to a
compulsory circular drift of particles near the axis of
the system which compensates the drift in the crosslinks, providing ДХ/ТГЙ > 1, where AL is the elongation of the peripheral Une in comparison with the
central Une. Therefore, the main process of particle
loss in such a system is due to particle-particle colUsions in the cross-links A which wiU lead to their
drifting out of the cone of wz/v2 < Ho/Hm and
to a subsequent drifting out to the cross-Unk wall.
But since the density in the cross-Unk is Ho/Hm times
less than the density in the straight parts and their
volume is TTR/L times less than the volume in the
straight part, the leakage will decrease HmLIHOirR > 1
times.
Magnetic Plasma Traps
The second system (Fig. 4(6)) is some modification of
the torus. Here the bulk of particles in the cross-links
is also stabilized. However, the particles with small
u/v ~ p/a in the cross-links are able to settle on the
wall during the time of theirflightover the cross-links,
237
PLASMA STABILIZATION
which leads to an additional energy leakage exceeding
the leakage caused by thermal conductivity if A > L,
where A is the mean free ion path.
'
Besides, in both traps there occurs an increase in
the thermal conductivity on account of the radial
displacement caused by the drift in the cross-links
("mixing"). Therefore, if the plasma touches the wall,
the linear part must be of a very great length to
fulfil the condition LLJTTR > 1.
Corrugated Traps
Corrugated magnetic traps can confine individual
particles, or at least most of them. But this does not
mean that they will confine aquasi-neutral plasma, since
there occur in the plasma proper electric fields which
can essentially change the character of the individual
particle motion. The next problem consists in finding
conditions of confinement for a plasma with an infinitesimal pressure. In the hydrodynamic approximation
and with the assumption of an infinite conductivity these
conditions have a simple form and can be deduced from
simple qualitative considerations. For the low-pressure
plasma only such motions can exist that shift separate
tubes of force almost without distorting the magnetic
field. For such motions the tube volume of a tube of
plasma changes proportionally to the integral / (1/H)dl
along the line of force. Plasma trying to expand will
induce each tube to tend towards an increase of this
integral.
As a result, it appears5 that the behaviour of lowpressure plasma is analogous to that of a nonuniformly heated gas in gravitational field; the plasma
pressure corresponds to the gas density and the
function U = —f(l/H)dl is the potential energy. If
the plasma at the initial moment has an arbitrary
distribution, convection sets in: the tubes of force in
which the plasma pressure is higher start moving
towards U, ousting the tubes with lower pressure.
Thus, the plasma tends toward a stable state in which
its pressure declines with increasing U. Besides this,
other stable equilibrium states exist. Just as a stable
equilibrium is possible in the atmosphere of the earth
when the air temperature does not drop too fast with
height, in a plasma there is no loss of stability if its
pressure rises not very quickly with increase in U.
Indeed, let the plasma be in equilibrium and let a
certain tube shift towards an increase in U by some
infinitesimal quantity, moving the other tubes apart.
If the process occurs adiabatically, the plasma pressure in this case will change by a quantity
Thus, the problem of plasma confinement by a
given magnetic field requires an investigation of the
potential U for this field. For a magnetic trap the
potential U should be minimum within a bounded
volume and increase towards the walls. It appears
that the field of the guard conductor systems satisfies
this condition completely. Let us consider, for example,
a guard ring system the magneticfieldof which has the
form presented in Fig. 5. Since the system in Fig. 5 is
periodic, U may be regarded as an integral over one
period, — a/2 < z < a/2. The potential U is a function
of the line of force; therefore it is convenient to consider it in the plane z = 0, crossing all the lines of force.
In this plane U has two "wells" at the points A and
С located on the dotted line of force that passes
through A, where the field vanishes. At A, U logarithmically approaches minus infinity. Near the rod U has
a maximum and is close to zero. At v = 0, U = — u/Hi,
\
Winding
Rings
Figure 5. Corrugated trap geometry
where Hi is a certain mean value of the field on the z
axis; and at point D, U £ —a/Ho > — a/Hi.
Hence, we see that any plasma configuration
whose pressure decreases in the direction away from
the dotted line of force is certainly stable. Moreover, an
arbitrary plasma configuration in such a system
changes by itself, so that the pressure decreases in the
direction away from the dotted line, i.e., the plasma
itself will tend to a stable equilibrium.
Besides states in which there are plasma encasewhere у is the adiabatic index. If the pressure in the
ments near the rings, we can find in the ring system
tube appears to be lower than the surrounding plasma
a stable plasma state whose pressure differs from zero
pressure, which is equal to p + dp, the tube win move
only within the dotted line of force, and vanishes on
with acceleration towards further increase in U; if
this Une; for stability it is sufficient that the plasma
8p < dp, the tube win be forced back and the plasma
pressure decreases with U slower than | U\-v.
will be stable. Thus, to obtain stability it is necessary
However, the actual attainment of such a state is a
and sufficient that dp/dU < yp/\U\, i.e., the plasma more subtle matter; it requires a special preparation of
pressure should drop with U not faster than \U\-v.
the plasma. Moreover, this state actually corresponds
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SESSION A-9
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B. B. KADOMTSEV and S. I. BRAGINSKY
to the plasma touching the wall, since the state assumes
that there should be constantly a pressure equal to
zero on the dotted line, otherwise plasma encasements will appear.
Being stable, the system presented in Fig. 5 can be
transformed into a torus: for a very small toroidality
there will be no loss of stability.
An analogous consideration shows that there are
stable low-pressure plasma states in other guard
conductor systems. This means that guard conductor systems are also magnetic plasma traps. This
circumstance, although not well proven, can serve as
an important argument in favour of a possible plasma
stabilization in guard conductor systems in a broad
pressure region.
The stability conditions for magnetic traps with a
corrugated field follow also from the above considerations. Indeed, the magnetic field in the system
presented in Fig. 5 is a field of a corrugated type, and
the fact that the plasma in such a field tends to take
a position along the dotted line shows that, at least
for the case of plasma states separated from the wall,
m'agnetic traps with corrugated fields should be
practically realized as guard conductor systems.
ACKNOWLEDGEMENT
The authors' best thanks are due to Prof. M. A.
Leontovich for his participation in the discussion of
the work and for his valuable suggestions in the process of its fulfilment.
REFERENCES
1. I. Bernstein, E. Frieman, M. Kruskal and R. Kulsrud,
Proc. Roy. Soc, A., 224, 17 (1958).
2. M. Kruskal and M. Schwarzschild, Proc. Roy. Soc, A.,
223, 348 (1954).
3. V. D. Shafranov, Zhur. Eksp. i Teoret. Fiz., 33, 710 (1957).
4. M. Rosenbluth and С Longmire, Ann. Phys., 1, 120 (1957).
5. R. Z. Sagdeev, B. B. Kadomtsev, L. I. Rudakov and A. A.
Vedenov, Dynamics of Rarified Plasma in a Magnetic
Field, P/2214, Vol. 31, these Proceedings.