Download Heating of a Confined Plasma by Oscillating Electromagnetic Fields

P/357 USA
Heating of a Confined Plasma
by Oscillating Electromagnetic Fields
By J, M. Berger,* W. A. Newcomb,f J. M. Dawson, * E. A. Fneman,*
R. M. Kulsrud * and A. Lenard
We wish to consider the heating of a fully ionized,
infinitely long, cylindrical plasma confined by an axial
magnetic field. The method of heating originally
suggested by Lyman Spitzer, Jr. (1953) involves the
application of an external oscillating electromagnetic
field over a region of extent L in the axial direction.
The oscillating electric field vector in the plasma is
perpendicular to the lines of force of the main confining field. Since this oscillating electric field is
produced by changing the main magnetic field in
time, this method of heating is often denoted by the
term " magnetic pumping ". Since we are interested
in heating the ions to thermonuclear temperatures in
a controlled fusion reactor we shall not consider
electron heating in the following but it should be
noted that some of the methods considered can also
be utilized to heat the electrons directly.
It is found that simple analytical results can be
obtain^, in four limiting cases. These cases are
classified in terms of the sizes of four characteristic
times. These times are: (1) rCoii = 1/v, the time for
a 90° deflection of an ion due to many small angle
encounters with electrons and ions, (2) n — 2n/со,
the period of the oscillating applied field, (3) Ttr, the
time of transit of a typical ion through the length L
and (4) rCi = 2п/а)си the cyclotron period of an ion.
The four cases which have been analyzed are:
(1)
(2)
(3)
(4)
collisional heating, rCi <C TCOII ~ Tf <C Ttr
transit-time heating, TCI <C Ttr ~ Tf <C TC0H
acoustic heating,
rCi <C TCOII <C Tf ~ n r
ion Cyclotron
Tci ~ Tf <C Ttr <C Tcolb
resonance heating,
In the following four sections we treat the four
cases outlined above. In cases (2) and (4), where the
collision time is very long compared with the other
characteristic times, the question arises as to the time
necessary for actual heating in the sense of achieving
randomization of the ordered energy delivered to
the system. Finally we discuss a mechanism which
can act to thermalize the ordered energy by phase or
" fine-scale " mixing.
A further question which may be raised is, why is
the plasma not cooled by the oscillating field? In
fact, in transit time heating, for certain initial velocities, particles lose energy in traversing the oscillating
field region even after an average over the initial
phases of the field relative to the particles has been
performed. Although it is intuitively obvious that
heating will occur, a simple thermodynamic argument
is given here which shows that after averaging over
a Maxwellian velocity distribution the energy transfer
cannot be negative.
Assume that the initial state г is in kinetic equilibrium; i.e., that the velocity distribution is Maxwellian and the spatial distribution is uniform.
Denote by / the state of the gas at some future time
after the oscillating field has been applied. The
transition from state i to state / is adiabatic: i.e.,
no heat has been added to the plasma (dQ = 0). Now
let collisions act to bring the plasma to the equilibrium
state /'. Clearly the internal energy of the plasma
is the same for states / and /'. Since the entire transition from state г to state /' is adiabatic, by the second
law of thermodynamics the entropy of state /', S(f),
is greater than S(i). Since the energy of an equilibrium state is a monotonically increasing function
of entropy for fixed confining volume, the energy of
state /', E(f), is greater than E(i). Therefore,
energy has been added to the plasma by the oscillating
field. This argument assumes perfect confinement
of the plasma, since it applies only when i, /, and /'
contain the same number of particles.
COLLISIONAL HEATING i-e
It is known that the magnetic moment ¡x = mw±2/2B
is an adiabatic invariant for a particle in a magnetic
field which is sufficiently slowly varying in space and
time. 4 Here m is the particle mass and w± the
magnitude of the particle velocity perpendicular to
the magnetic field B. Thus, in the absence of collisions, a time-varying field will only cause a propor-
* Now at IBM Research Laboratory, Yorktown, New York.
t Now at University of California Radiation Laboratory,
Li ver more, California.
î Project Matterhorn, Princeton University, Princeton, New
Jersey.
112
HEATING BY OSCILLATING FIELDS
tionate variation in the perpendicular energy E±.
If we imagine a square-wave pulse field superimposed
on the main field and further assume that a collision
occurs during the time the pulsed field is non-zero,
energy will be transferred from E± to E „. After the
field has returned to zero we see that a net energy
transfer to the gas has taken place.
Taking into account the two assumptions that the
magnetic moment is a constant and the departure
from equipartition during a collision time is small, we
obtain an approximate set of equations governing this
effect :
dt
В dt
dE»
= -?(£,-*£,.),
dt
(1)
where v is the collision frequency. Note that because
of our assumption that rt r ^> TCOIL changes in v or В
with time resulting from motion in the axial direction
can be neglected. If we assume that В is given by
В — B0(l + ecos со/),
(2)
and treat e as small compared to unity we can solve
Eq. (1) by a perturbation treatment. A simple application of the Floquet theory for differential equations
with periodic coefficients leads to the increase in
energy AE, in a time 2я;/со,
113
Because of the constancy of t h e magnetic moment ¡u,
it is clear t h a t in this mode of heating all t h e energy
changes will be due t o t h e parallel velocity changes.
T h a t the parallel velocity will change is easily seen
if E q . (8) is interpreted as representing a mechanical
system in which a particle passes through a timevarying potential well. I t is not so clear, however,
what t h e net effect of t h e oscillating field on a distribution of particles will be.
Equation (8) can be easily solved b y perturbation
theory assuming BJB0 to be small. The energy change
of a particle AE = ^m(w\\f2 — w¡\i2), where m¿n a n d
w\\f are the initial a n d final parallel velocities respectively, can then be obtained a n d averaged over values
of t h e phase oc. I n this calculation i t is necessary t o
2
compute w^fto second order in Вг/В0.
The result is
d
(AE\ = -
dco
[co^j
X
dz
], (9)
where w {l is the zeroth-order parallel velocity. To
enable us to analyze this further we have chosen the
specific field shape f(z) = exp — (z/a)2. With this
choice (9) becomes
coa \ 2 - 2
w 2/
TI
Í
~m\
]. (10)
(О*
(3)
The difference Eq. (3) can be replaced by the differential equation
dE
= [is2co2v/(9v2/4
+ co2)]E = ocE.
(4)
In investigating the rate of heating from (4), note that
the energy dependence of v is
v ~ E-l\
(5)
Equation (10) clearly shows that particles whose transit time is somewhat greater than the field period are
heated, while those for which the converse is true
are, in fact, cooled.
A quantity of greater interest is the flux of energy
of heated particles passing out of the field section.
We obtain this by multiplying <(АЕУа by w „ n, where
n is the particle density, and then averaging over both
the perpendicular and parallel velocities. Thus, we
find that the flux F is
TRANSIT-TIME HEATING *
=
In the limiting case of transit-time heating, the
assumption is made that the particles pass through
the heating section sufficiently fast so that they suffer
no collisions during the transit time. It is also
assumed that the transit is slow enough for the
adiabatic approximation to hold. The equation of
motion of an ion moving in the z (axial) direction is
then
m'z = — ¡xdB/dz.
(6)
(7)
where Bo and B1 are constants, f{z) describes the field
shape, and a i s a phase angle specifying the time the
ion enters into the field region. Equation (6) then
becomes
z = - {¡LiB1lfn){df{z)/dz) cos {cot — a ) .
(8)
l
nkT (kT/m)iG(x)
(11)
C, (x) + Кг> {%)]
(12)
where
and
x — aco (tn/kT)?
/
/1 О\
G(x) is a function with a single maximum at x = 1.55
where G(1.55) — 0.24. The limiting forms are given
We assume for this treatment that В is given by
В = Bo + BJ(z) cos (cot - a),
(я/2)*
G(x) =
x\ 1
— In —
2
x
2"
m
(14)
e
When this calculation was originally made a different
average from that given by (11) was computed and
therefore the results given here do not agree with
those of Ref. 2.
114
SESSION A-5
P/357
Simple inequalities for the validity of the second
order theory are easy to obtain from the integrated
form of Eq. (6). These are:
w.
wn
< 1 if
w,
Bn
«1
if
aco
w
\\
aco
w„
The usual equations of motion for small amplitude
sound waves are :
' = - Ъфг
И = — др/dz
2
P = C Q,
(17)
2
(15)
One efíect which has ostensibly been neglected in the
above treatment is the reflection of some of the particles before a complete transit of the oscillating field
region has been made. Necessary conditions for
nonreflection can be written down which are satisfied
if Eq. (15) is satisfied. Note, however, that in the
course of integrating the particle flux over the Maxwell
distribution conditions (15) may be violated. Therefore, Eq. (11) should properly be regarded only as an
estimate of the flux.
There is one further effect which a more sophisticated treatment of transit-time heating should take
into account ; namely, that in the analogous situation
of mechanical particles passing through a timevarying potential well there will be bunching some
distance from the well. In this case, where the particles are charged, it is possible that positive ion waves
will be set up in the plasma regions outside the
oscillating field section. Under such a circumstance,
however, it seems quite likely that either collisions
between positive ions of different e/m or the fine-scale
mixing phenomena discussed in section (6) would
eventually thermalize the energy in the positive
ion waves.
ACOUSTIC HEATING2
If the effective mean-free path is small, the oscillating field will produce density variations which will
result in the propagation of sound waves in the plasma.
The small effective mean-free path may be due to
high densities or to cooperative phenomena which are
at present little understood. We consider here the
production of energy in the form of sound waves
but do not treat the problem of their absorption by
the plasma. The following assumptions are made
in this calculation: (1) the plasma is treated as an
ideal gas satisfying an adiabatic equation of state,
(2) the particles do not diffuse appreciably during the
period of oscillation, and (3) there is no appreciable
mass motion out of the ends of the field region.
Further, only those waves are treated which propagate
in the axial direction. Under these approximations,
the effect of the oscillating field is to produce a radial
mass velocity. Let us assume that the plasma has
uniform density within the radius r, and zero density
outside. Since the material follows the lines of force,
the total magnetic flux through the cross section
uzr2 is constant. Thus
vr = dr/dt £Ё (r/2B0) dBJdt,
J. M. BERGER et al.
(16)
where we have again assumed that Вг/В0 = s is small
compared to unity.
where p, Q, and C = ykT/m are the pressure, density
and square of sound velocity, respectively. We
further assume that the field is given by
В = BQ {1 + s exp - [(z/a)2 + icot]} .
(18)
Noting that we are only interested in the forced
oscillations, we obtain the solution to Eq. (17) in the
form
P = — i ÍQ0CCOS exp + ioot
x
{ J l oo e x P [— (*'A*)2 — Щ* ~ *')¥*'
+ J/ 0 exp [ - (z'/a)2 + ik (z - z')] dzj
(19)
where k = со/С. However, since it is sufficient for
our purposes to consider the traveling wave solutions
outside of the forcing region, we let z-> + oo in the
above expression. The result is
p = p0ks{7ta2/4)% sin (kz — cot) exp — (ak/2)2.
(20)
The flux of energy is pvz and, therefore, we easily
obtain t h e average flux
F =
X {BJB0)2
nkT
x2 exp (— %2/2y), (21)
where x is given by (13).
Upon comparison of this expression for the flux
with Eq. (11) for the flux obtained for transit-time
heating, it is seen that there is good agreement
although these two cases represent extremes as far
as the approximations are concerned.
ION CYCLOTRON HEATING 5 6
In this case the external field is set up in the same
way as in the other cases but its frequency is near the
ion cyclotron frequency and thus the magnetic moment
of the ions need not be constant but increases rapidly.
The transit time of the ions across the heating section
is long compared to the ion cyclotron period so that
the ions undergo many gyrations in their passage
through the heating section when collisions are neglected entirely. The heating of ions by this method
is very efficient. The ions gain large energies during
a single passage through the heating section since the
ion particle velocities are always in phase with the
electric field and the ions are continually accelerated.
Further, the motions of all the separate ions are in
phase, being correlated by the external field, so that
large currents are developed in the plasma. Therefore, we must consider thefieldproduced by the plasma
currents in calculating the heating of the ions.
The ions will, in general, have a radial motion
relative to the magnetic lines of force due to their
correlated motion and their larger Larmor radii.
HEATING BY OSCILLATING FIELDS
Since the electrons are tied to these lines, a space
charge will be set up producing a radial electric field.
It can be seen that this radial electric field when
combined with the externally produced azimuthal
electric field will produce a circularly polarized electric
field whose sense is just opposite to the motion of the
ions. This circularly polarized component produces
no energy increase of the ions. Thus, in some sense
we may say that the plasma has shielded itself from
the circularly polarized component of the external
field which rotates in the correct sense to increase the
energy of the ions.
The way around this difficulty was pointed out by
S t i x 6 who suggested that the space charge could
be cancelled by the electrons if the electric field did
not have a uniform phase along the axial direction.
Thus, in some regions, the electric field has an opposite
sign to the field in adjacent regions. In these regions
a positive space charge is built up while simultaneously
in adjacent sections a negative space charge is built up.
Electrons must then flow along the lines of force to
cancel these space charges. If the electrons are able
to cancel out the space charge totally, one would have
the total externally applied field acting to heat the
ions. However, the actual current of the electrons
oscillating at the cyclotron frequency produces a
back emf on the electrons. Since the total electric
field along the lines of force must be zero, the total
space charge cannot be completely cancelled and
there must be a residual space charge to produce a
field along the lines to cancel this back emf. This
space charge also produces a radial field which combines with the applied electromagnetic field so that
even in this case the circularly polarized field, in the
favorable sense, is somewhat shielded. Stix 7 has
considered this situation under slightly different
circumstances and has established that the favorably
polarized component is able to penetrate the plasma
appreciably.
In analyzing this problem, the electromagnetic
fields are not known in advance but must be solved
for in a self-consistent manner. With complex notation their behavior was assumed to be given by
s = Er + iEe = (е+еш + е-е-ш)
f{z),
(22)
with the shape factor f(z) given by
sin kz
0
-— a < z <
z\> a
(23)
with ka — 7iN, N being an integer. e_ and e+
represent the two circularly polarized components of
the electric field in the favorable and unfavorable
senses, respectively. The assumed form is then
justified by the self-consistent calculation and the
forms of e+ and £_ determined. It should be emphasized that, due to the form of /, the ions will not see
the frequency of the impressed electric field but a
Doppler shifted frequency со ± kw „, so that for œ
different from coCi some ions will always be heated.
The motion of the ions is solved in the field (22)
neglecting variations of e± (r) across the orbit, neglect-
115
ing the effect of the oscillating magnetic field, and
assuming that the axial velocity is uniform. Upon
summing over particles with all possible axial velocities, one finds that to SL good approximation the
ion current may be given by
jr + ije = {е2п/тк¥т) e_ (r) / 0 (a) sinkz в~ш,
(24)
where n is the ion number density, VT the mean axial
velocity, a = со — coCi/kVT, and / 0 (a) is a nearly
constant complex quantity of order unity. The
transverse electron current is negligible and the
axial current is chosen to make V»j = 0 since the
space charge must be small compared to V*jj_.
Since E l{ = 0 due to the mobility of electrons along
the lines of force, we may solve
V x ( V x E ) = {4n/c)d)/dt
(25)
which results Irom Maxwell's equations, to find
(26)
where
у = {4ne2n/m)
co/kVT
(27)
and £ + * is the complex conjugate to s+. For high
densities у is large and (26) says that the field produced by the plasma must be very large compared to
the favorably circularly polarized component which
is driving the ions. Thus it is clear that the s+
component must predominate over £_. On imposing
the boundary conditions, it is found that e + is nearly
that expected from the impressed field in the absence
of plasma so that e_ is shielded by the plasma. On
the other hand, if у is much less than unity, the back
reaction of the plasma may be neglected.
The total energy flux of the plasma passing out the
ends is
F =
>Nne2
2mkWT
71
1 + iy Io
(28)
This result is in agreement with that obtained by
Stix. 6
F seems to be a maximum for a — 0 but a closer
examination of the particles shows that those with
small parallel velocities are heated to large energies
while those with moderate velocities are not heated
at all. All the particles would be heated to a somewhat smaller total energy if oc ^ 1, which corresponds
to that frequency at which particles with the mean
axial velocity see the cyclotron frequency.
FINE SCALE MIXING AS A MEANS FOR
RANDOMIZING ORGANIZED MOTION
In collisional and acoustic heating the possibility
exists that the energy transferred to the plasma by the
electromagnetic field goes initially into organized
motion. It is important to know the rate at which
oscillations produced by such pumping are converted
into random motion of the ions. If this rate is too
low, then the pumping mechanism is inadequate for
practical purposes. Of course, collisional damping
116
SESSION A-5
P/357
will always occur, but for the temperatures of interest
this damping mechanism is relatively slow.
However, there is another damping mechanism
which is very effective. It is similar to the damping
found by Landau 8 and Van Kampen 9 for electrostatic plasma oscillations. This mechanism depends
on phase mixing of the wave due to thermal motions
in the plasma. It is this process which will be
discussed in this section.
The physical picture for this type of damping is the
following. Originally one has an organized wave in
the plasma and because of thermal motion, ions which
are in one region of the wave move into adjoining
regions where the phase of the wave is different. Thus
the various parts of the wave become mixed and the
organized motion disappears. This process does not
lead to a Maxwell distribution in general, but it does
lead to random motions of the ions. Once this
mixing has occurred, collisions become much more
effective and probably produce a Maxwell distribution
in a short time. Even if this does not happen,
however, the random motion is equivalent to a temperature.
To illustrate this mechanism we shall consider the
case of ion cyclotron heating. A simple model of the
plasma which is used is the following: the plasma is
taken to consist of a number of cold beams of particles
traveling along the magnetic field. This is an
approximation to the thermal motion along the field
but does not take into account thermal motions
perpendicular to the field. This is done since thermal
motion perpendicular to the field has little influence
on the effect under consideration.
The linearized equations of motion for this case are
i/dt +
= [«/c](E + ViXB0 + i X В )
В
V x E == _
dt
¿•VVÍ)
v. E =
Vx В
v. В =
4тт
1
c2 dt
ж
0.
V
Що ei
(29)
The quantities appearing in the above equations
have the following meanings :
mi = mass of the particles in the ith beam
d = charge of the particles in the ith beam
V¿ = the velocity perturbation of particles in the
ith beam
"У"г = the zeroth-order velocity of the ith beam, with
E and В are the electric and magnetic fields respectively due to the disturbance, щ and щ0 are the
perturbed and zeroth-order number densities in the
ith beam respectively, and B o is the zeroth-order
magnetic field with B o = B o e 2 . The beams may be
composed of either ions or electrons.
In investigating this problem, the simplest type of
waves which show the effect were considered. These
J. M. BERGER et ai.
were transverse, circularly polarized waves traveling
along the magnetic field. It should be mentioned
that Gershman10 has treated a closely-related problem.
He solved the initial value problem for hydromagnetic
waves traveling along the magnetic field. His method
of treatment was the same as that used by Landau
for treating the electrostatic oscillation problem. The
approach presented here is closely akin to van
Kampen's treatment of electrostatic oscillations. Our
problem is also different in that we are interested in
the propagation of waves out of a pumping region
rather than in the time behavior of an initial perturbation. Our treatment covers waves with frequencies near the cyclotron frequency as well as
hydromagnetic waves.
The various quantities entering in the calculation
were taken to be of the form
VilBo, B1BO, E l B o
E = E°
В = B°
(30)
When these expressions are substituted in the equations of motion the following relations are found
coci
со —
В а
со —
CO —
kVi
(31)
(32)
where
= I ¿г
(33)
The first equation is the dispersion relation, the second
gives the perturbation velocity in terms of the magnetic field. The plus and minus sign depends on the
polarization and the charge on the particles. If the
positive sign is used for ions then the negative sign
is used for electrons. Observe that for certain values
of со and k the denominator vanishes in the expression
for ViQ. This occurs for those particles which see
their cyclotron frequency when the polarization of the
wave is appropriate. It is these particles which are
close to resonance which are most strongly perturbed.
The dispersion relation is most easily analyzed by
making a plot like that shown in Fig. 1. This figure
is obtained by fixing со and plotting the left- and righthand sides of the dispersion relation. The left-hand
side is the parabola (ck)2 and theright-handside is the
more complicated curve. Every place one of the
denominators in the dispersion relation goes to zero,
the right-hand side goes to infinity. The two curves
cross at a number of points. Each crossing gives a
possible k for the given со. Thus the plasma can
transmit many waves with the frequency со.
When the plasma is pumped at the frequency со,
each of these waves will be excited to some degree.
The amplitude of each wave will depend on the details
of the pumping mechanism. In the region where the
117
HEATING BY OSCILLATING FIELDS
{kc)2/œVi2(oCiF[(œ
+
œc\)/k].
6
Stix has obtained a similar result by another method.
If the predominant k falls in a region where many
beams are near resonance, the damping is fast and
occurs in less than a wave length. If one again goes
to the limit of an infinite number of beams, one finds
that in this case no coherent wave leaves the pumping
section.
CONCLUSION
Figure 1.
Typical
dispersion current for
cyclotron waves
pumping occurs the various waves will be forced to
be in phase. As one leaves this region the various
waves will get out of phase with each other. If the
pumping mechanism is such as to excite a large
number of waves almost equally, then this phase
mixing will result in the disappearance of all macroscopic quantities. This is not true, however, for the
microscopic motion. Each beam is excited mainly
by one k; that k which is closest to resonance for
it. This can be seen from the expression for V{.
Thus, what was originally an organized motion is
converted into effectively random motion of the ions.
More detailed calculations bear out this result. It
is found that as the plasma is pumped so that a
hydromagnetic wave is produced, then little or no
damping results. This is the case if the predominant
k is far from resonance for most of the particles.
Going to the limit of an infinite number of beams so
as to represent a continuous velocity distribution
function F(V), normalized to unity, we find that the
damping distance for hydromagnetic waves is
In summary, it is of interest to examine the temperature dependence of the rates of heating of the various
types of magnetic pumping. The collisional heating
is more effective at lower temperatures, the rate of
heating dropping off as 1Г/* at higher temperatures2
since the collision cross section becomes small as 1/T .
However, the transit time heating and acoustic
heating improve at higher temperatures; the rate of
heating varying as T¡* if the frequency is optimized at
each temperature. In ion cyclotron heating the
electromagnetic shielding is given by the factor y
which decreases at higher temperatures. Thus ion
cyclotron heating increases as Ti for y greater than
one at higher temperatures and then decreases as
1/Г* for y less than one. It should be remarked,
however, that this type of heating is rather sudden;
the particles can be heated from near zero temperature
to extremely high temperature on passing through
the heating section. Finally, it should be mentioned
that the randomization of organized motion by fine
scale mixing is more effective at higher temperatures.
In conclusion the authors must express their
deepest gratitude to Lyman Spitzer Jr. for not only
initiating the basic ideas involved in this paper but
for constant encouragement and advice in the carrying
out of their development.
REFERENCES
1. L. Spitzer and L. Witten, On the Ionization and Heating
of a Plasma, Atomic Energy Comm. Report No. NYO999, PM-S-6 (1953).
2.
J. M. Berger and W. A. Newcomb, Heating of a Plasma
by Magnetic Pumping, Atomic Energy Comm. Report
No. NYO-6046, PM-S-13 (1954).
3.
A. Schluter, Der Gyro-Relaxations-Effekt, Z. Naturforsch.
72a, 822 (1957). The concept of collisional heating was
arrived at independently in this paper.
4.
M. Kruskal, The Gyration of a Charged Particle, Atomic
Energy Comm. Report No. 7903 PM-S-33 (1958). References to earlier work are given in this report.
5. A. Lenard and R. Kulsrud, Resonance Heating of Ions
in a Strong Magnetic Field, Atomic Energy Comm.
Report No. NYO-7902, PM-S-32 (1958).
6. T. Stix, Generation and Thermalization of Plasma Waves,
P/357, this Volume, these Proceedings. A similar result to
that obtained from fine-scale mixing is given in this
reference.
7. T. Stix, Oscillations of a Cylindrical Plasma, Phys. Rev.,
106, 1146 (1957).
8. L. Landau, On the Vibrations of the Electronic Plasma,
J. Phys., 10, 25 (1946).
9. N. G. van Kampen, On the Theory of Stationary Waves
in Plasmas, Physica, 21, 949 (1955).
10. Gershman, Zhur. Eksptl. i Teoret. Fiz.. 24, 453 (1953).