Physica 133A (1985) 22&246
North-Holland,
Amsterdam
SOLUTIONS OF VLASOV-MAXWELL
MAGNETICALLY
IWO BIALYNICKI-BIRULA*
Institute
EQUATIONS
CONFINED RELATIVISTIC
and ZOFIA
BIALYNICKA-BIRULA’
for ‘Theoretical Physics, Universiry of Innsbruck,
Received
18 January
FOR A
COLD PLASMA
A-6020
Innsbruck.
Austria
1985
Special representation
of the distribution
function is employed
to obtain new solutions of the
coupled Vlasov-Maxwell
equations.
This approach
combines
two modes of description
used in
plasma physics: magnetohydrodynamics
and the theory of orbits. Using this method, we described
plasma configurations
confined
in one and two directions
in space, with plane and cylindrical
symmetry,
respectively.
1. Introduction
The aim of this paper is to describe some new self-consistent
solutions of the
relativistic
Vlasov-Maxwell
equations.
These solutions
describe
stationary
streams of electrons
moving against the neutralizing
ionic background.
The
motion of the electrons is confined by their own static magnetic held.
Several exact solutions of the Vlasov-Maxwell
equations
for neutralized
cold
plasma beams have been reported in the pastle5). Our solutions, in the simplest
case of plane symmetry,
are closely related to those obtained recently by Peter,
Ron
and
general
different
Rostoker5).
method
These
which
solutions
perhaps
are obtained
with
may be also applied
from the ones treated
the
use of a fairly
to plasma
configurations
in this paper.
2. General outline of the method
We will seek solutions
the distribution
functions
* Permanent
address:
32146, Warsaw, Poland.
’ Permanent
address:
Poland.
Institute
Institute
of the coupled
f,(r,
set of Vlasov-Maxwell
p, t) describing
for Theoretical
of Physics,
Physics,
Polish Academy
various
Polish
plasma
Academy
of Sciences,
0378-4371/85/$03.30
@ Elsevier Science Publishers
(North-Holland
Physics Publishing
Division)
B.V.
equations
components
of Sciences,
Lotnikow
for
and
Lotnikow
32/46, Warsaw,
SOLUTIONS
the
electric
and
magnetic
OF VLASOV-MAXWELL
fields
EQUATIONS
self-consistently
modified
229
by
the
plasma
particles,
a,E=-VxB+4rrj,
a,B=
The
VxE,
sources
distribution
V.E=477p,
(lb)
V.B=O.
(lc)
of the electromagnetic
field, p and j, are determined
by the
functions
For relativistic
particles the relation
momentum
p has the form
between
the velocity
u and the kinetic
(3)
We will restrict
of each plasma
ourselves
component
to special
at every
solutions
point
of eqs. (1) for which the motion
decomposes
into a discrete
number
of streams,
Different
streams are labelled by the superscript
A, whereas the subscript (Y
always refers to different plasma components.
The number of different streams
does not have to be specified at this point, but it will be set equal to two or four
for the electron component
of the plasma in our specific examples.
As was to be expected, upon substituting
our Ansatz (4) into Vlasov equation
(la), we obtain for each stream the equations of relativistic hydrodynamics
of a
charged fluid without the pressure term:
230
I. BIALYNICKI-BIRULA
AND
Z. BIALYNICKA-BIRULA
(Sa)
where
v; =
db-, t)
\'m2+(7rA(r
a
a
’
(6)
t))2’
The charge and current densities,
which enter the Maxwell
given as sums over all streams and over all plasma components:
equations,
are
In the simplest
case, when there is only one stream
for each plasma
component,
our Ansatz reduces to the known hydrodynamic
representation
of
the distribution
function for cold plasma6.‘). Our generalization
of this known
Ansatz to two (or more) &functions
for one plasma component
enables us to
find new types of solutions. This approach combines two modes of description
used in plasma physics: the magnetohydrodynamics
and the theory of orbits.
While locally the dynamics of the system is completely
determined
by partial
differential
equations,
equations
global
of many-fluid
properties
hydrodynamics
of solutions
coupled
will be inferred
to the
from
the
Maxwell
particle
picture.
We will visualize
the streams of electrons
at each space point as
resulting
from the continuous
motion
of electrons
bending
in their own
magnetic field. In this way we can obtain solutions confined in a finite region of
space.
The set of eqs. (.5)-(7), together with the Maxwell equations,
will be solved in
sections 3 and 4 under the assumptions
of special symmetry.
For symmetric
plasma configurations
our method of solution bears a strong resemblance
to the
a-function
method introduced
by Longmire’)
and Mjo1sness2) and used subsequently
by various authorsS5).
Two cases of symmetry
will be treated in our paper: plane symmetry
and
cylindrical
symmetry.
In both cases we shall discuss only time-independent
solutions,
although by Lorentz transforming
our solutions to a moving frame,
one can obtain also time-dependent
solutions.
SOLUTIONS
OF VLASOV-MAXWELL
EQUATIONS
231
3. Solutions with plane symmetry
Solutions considered in this section are characterized by the magnetic field
pointing in the z direction and depending on the x variable only,
B = (0, 0, B(x)).
(8)
For simplicity, we shall assume that the ions are at rest and distributed in
such a way that at each point their charge neutralizes the charge of electrons.
For the electron distribution function we shall assume at first the sum of two
S-functions (two streams) in eq. (4). The coupled hydrodynamic-Maxwell
equations reduce in this case to the following set (we omit the index cy and
A = 1,2):
V.(nAuA)=O,
(vA4qmA= e(uAxB),
(9b)
vx B = 47re(nV+
(9c)
n’v’).
If it were not for the presence of two streams (two terms on the r.h.s. of eq.
(SC)), we would not obtain any new, interesting solutions.
Under the assumption of plane symmetry, the density fields nA and the
velocity fields vA depend only on the variable x and eqs. (9) reduce to
2
nAv;) = 0,
(104
(lob)
A_
Ad
vxz
TY
-
-e(B)
WC)
dB
z = -47re(n’vb + n2v3,
We>
fl’v’ x + n2v2
=0
x
Wf)
’
232
I. BIALYNICKI-BIRULA
AND
2. BIALYNICKA-BIRULA
n’v; + n*v2,= 0 .
General
solutions
vk)
of these
equations
do not satisfy the global
constraints
of
“stream continuity”
discussed in section 2. However, under a special choice of
the integration
constants,
the two streams of electrons
form a matching pair,
leading
to global
solutions.
In order
to see how this comes
about,
we shall
rewrite the set of equations
(10) in terms of a different set of variables. To this
end, we observe first that the densities of both streams can be eliminated
by
integrating
eq. (lOa) and using the constraint
equation
(10f):
n’=,,
K
n*=-_ K
(11)
v;’
V’,
where K is an integration
constant.
For definiteness,
we shall assume that the
first stream is the one which has a nonnegative
component
of its velocity in the x
direction;
therefore
K > 0. In order to satisfy eq. (10d) and eq. (log) with two
streams only, we must set vi = 0 = vt. Eqs. (lob) and (10~) guarantee
conservation
of kinetic energy in the magnetic
field. Hence,
the velocity components
in the following way:
we can parametrize
(G(x),u:(x)>= v,(cos tiAtx(x),
sin (I/,(x>),
where
The
coupled
the v,_,,‘sdo not depend
two
angles
(LA and
(12)
on x.
the
magnetic
field
B obey
the
following
set of
equations:
d
zsin(I,,=--,
eB
PA
dB
= -4reK(tan
dx
$r - tan I+/J*),
where
(14)
Since the first and the second stream
of each other, the moduli of momenta
Pl= P2= P.
have been assumed
must be equal,
to be a continuation
(15)
SOLUTIONS
OF VLASOV-MAXWELL
EQUATIONS
233
However, this does not imply that the angles are equal. On the contrary, in
order to obtain a nontrivial solution we must keep them different and the
global continuity requirement gives
(16)
*2= *,+ n-a
This finally leads to the set of two equations:
d sin $
-=_p,
d5
dP
-=
d5
(174
(17b)
-tan *,
where $ = $i and we have introduced
the dimensionless variables
z=x$E5,
(ISa)
P
B
-.
’ =lhn-Kp
(18b)
Eqs. (17) can be reduced to a quadrature,
ip2+cos$=
a,
because they possess one integral,
(19)
where a is a constant. This integral has a simple physical interpretation. It is
the XX component of the stress tensor and its independence of x expresses the
vanishing of the divergence of the stress tensor. Since by our convention cos IJ
is always positive, so is the constant a.
With the use of (19), the solution of eqs. (17) can be written as
p = kd2(a
- cos f)),
(2Ob)
where & is the (trivial) integration constant, which only fixes the origin of the .$
coordinate. Both choices of the sign in eqs. (20) give a solution.
I. BIALYNICKI-BIRULA
234
Depending
on whether
has a different character.
varies
continuously
AND
Z. BIALYNICKA-BIRULA
the constant a is greater or less than one, the solution
In the first case, the strong field regime, the angle I/J
over the whole
range,
-%-/2Sl)SlrJ2,
(21)
whereas in the second case,
either of the two conditions
-7Tl2S
These
* S -arccos
a
the weak
its range
arccos a S +!Id 42 .
or
two types of solutions
held regime,
is restricted
by
(22)
are shown in figs. 1 and 2. As seen in these figures,
all orbits lie inside a strip of width d. Thus, our solutions
describe plasma
configurations
confined
to a slab of the width d. The absolute
value of the
:--=-_.
=._
,/_.
-.‘.‘-
----r
__--
‘t
/’
‘S
‘4
I
1,
I
k.
i
i
,,’
-._-.._
\t
=-_____
--^
___----
__z-
./-
.-
:
Fig. 1. Plane symmetry, strong field regime (& = -1.0040). (a) Magnetic field p and velocity angle II,
as functions of coordinate .$ (b) Individual particle trajectory.
SOLUTIONS
OF VLASOV-MAXWELL
aa,’
EQUATIONS
(4
235
Si
/
----
/I
---_
-.
‘.,
‘%
:’
I
5
Fig. 2. Plane symmetry,
weak field regime (&, = -0.9899).
This value gives the same confinement
as in fig. 1. (a) Magnetic field /3 and velocity angle Cc,as functions of coordinate
5. (b) Individual
particle trajectory.
outside magnetic field B, is always the same on both sides, but for ~1< 1, the
field changes its sign. Since at the boundary cos I,!J= 0, the constant a is related
to B0 by the formula
a=
(B,Z
Bcr>’
(23)
where
B, = Xkh~Kp
.
(24)
236
I. BIALYNICKI-BIRULA
From
AND
(20a) one can determine
Z. BIALYNICKA-BIRULA
of d on B,:
the dependence
42
d+ cos 4
d=L
I
V/z
1
~(B,IB,,)~-
~0~
(254
I41 > B,, 1
II,
- d2
(25b)
of d on 5 is shown
The dependence
The integral
elliptic
integrals.
~0s
in fig. 3.
(20a) can also be explicitly
Upon
evaluated
and expressed
in terms
of
the substitutions
4, = sin($/2),
lB,I
> B,, ,
cos2 42 = 1 - k2 cos2(+/2),
@a)
JB,I < B,, ,
(26b )
where
k2 =
2
I + (&l&J2
(27)
’
this integral reduces to the canonical,
In this way, we obtain
?t=
(k2-k)F(d,jk)-EE(d,jk)+const.,
15 = F(42jkm1)where
Legendre
2E(&
km’) + const.
F and E are the elliptic
integrals
forms of the elliptic
+ 2 cos($/2)
+ const
(28a)
lB,I > B,, 3
,
lBol < B,, ,
(2W
of the first and the second
In the limiting case, when JB,I = B,,, both expressions
elliptic integrals reduce to the elementary
functions
‘5 = In(tan($/4))
integrals.
kind*).
(28) are valid and the
(29)
This special solution
has been known for a long time’). It describes
a neutralized plasma beam (in the Rosenbluth
limit) which originates
at infinity and
is deflected by a magnetic barrier. The case IB,J 2 B,, has been recently studied
SOLUTIONS
OF VLASOV-MAXWELL
EQUATIONS
231
“I
d
2-
1 -
i
0 /
0
L
1
I
I
I
2
I
I
3
I pal
Fig. 3. The width d of the confinement
slab as a function
&,. Confinement
disappears
for & = B,&,
= 1).
of the absolute
value of the outside
field
in detail by Peter, Ron and Rostoke8) in their analysis of plasma injection into
a magnetic field. Since we are interested here in plasma configurations which
are confined in the x direction, our choice of boundary conditions is different
and, therefore, our solutions differ slightly from those of ref. 5. Moreover, our
choice of boundary conditions allows for the existence of solutions also when
l&l < %
In the simple case of two electron streams we were forced to assume that
each electron orbit lies in the plane. In order to allow for motion in the
direction of the magnetic field, we must introduce a second pair of streams. In
this case the equations (lOa)-(10d) have, of course, the same form, but now
A = 1,2,3,4, whereas eqs. (lOe)-(log) contain contributions from all four
streams:
$J = -4Te(n’vi
+ n2vz + n3vz + n4v4y) ,
(304
238
I. BIALYNICKI-BIRULA
Z. BIALYNICKA-BIRULA
n’v~ + n2vt + n3vZ + n4v: = 0,
(30b)
n’vi + n2uz+ n3vz + n4v4 = 0.
(3Oc)
We shall again
tions
AND
eliminate
(lOa). In order
integration
n’vi=
the densities
to satisfy
nA by integrating
eqs. (30b) and
the continuity
(30~) we choose
equa-
the following
constants:
K,
and the following
n’vf = -K,
n3v:=
relations
between
K,
n4vf=
-K,
(31)
the z components:
vf=vf=v,, v;=v;= -v,.
(32)
Thus, the net current has only the y component.
In view of eq. (IOd), the
velocity component
v, in the field direction must be constant. Global continuity
requirements
for electron orbits are satisfied if we choose
v~=v~=v,=v,c0s$b(x),
v;=v;=-v,,
(33)
Vi = us = vz = v: = vY = v, sin I/J(X),
where
v, is the magnitude
(34)
of the projection
of the electron
velocity
on the xy
plane. In dimensionless
units, the equations
for the velocity angle I,+ and the
magnetic
field /3 are exactly the same as for two streams.
Therefore,
the
magnetic field is unchanged.
The trajectories
now have a spiral form, but their
projections
on the xy plane are the same as before.
Under the special choice of velocities
of our four
streams
distribution
of 6-functions
function
can be reduced
to a single product
(31)-(34),
the
(35)
(36)
and
H(r)
= ti/rrZ2+ (m(r))2.
For rz = 0, this expression
function used in ref. 5.
(37)
coincided
with
the
formula
for the
distribution
SOLUTIONS
OF VLASOV-MAXWELL
EQUATIONS
239
Plane-symmetric solutions are confined only in one direction. To achieve
confinement in two directions, we will consider in the next section the case of
cylindrical symmetry.
4. Solutions with cylindrical symmetry
We shall assume the same general form (4) of the electron distribution
function as before and start from eqs. (9) for the stationary configuration.
These equations will now be solved under the assumption of cylindrical
symmetry. The components of all physical quantities in cylindrical coordinates
r, 8 and z will depend on r only and the magnetic field will be taken again in
the z direction. For simplicity, we restrict ourselves to two streams, which
excludes any motion in the direction of the magnetic field, ZIP= 0. Generalization to four streams to allow for motions in the z direction proceeds in exactly
the same way as in the case of plane symmetry discussed in the previous
section. Under these assumptions, eqs. (9) reduce to
3
rn”vt)
= 0,
(384
(38b)
1
Ad
--T$s-v:‘TT$=
“dr
r
dB
z = -47re(n’vi+
ev;B,
(38~)
n2v2,>,
n’v’,+n2v~=0.
(384
As before, we use the continuity equations (38a) and eq. (38e) to eliminate nA,
+--, r
rf-b
n 2---;;1,
r
(39)
r
where again r is a positive integration constant.
Assumptions concerning the existence of global solutions, analogous to those
240
I. BIALYNICKI-BIRULA
made
in the
preceding
dimensionless
variables:
dsin $
p=
dp
sin Cc,
---P,
P
section,
AND
lead
Z. BIALYNICKA-BIRULA
to the
following
set of equations
for
(40a)
do _-p tan *
(40b)
p’
dpwhere
p=-r
4rre’T
(4la)
’
P
p=B
47reT
@lb)
’
and the angle +(p) determines
the direction of the stream of electrons moving
outwards.
Eqs. (40) do not have any analytic
integrals
and cannot
be solved in
quadratures.
We have studied
their solutions
numerically.
They depend
on two
integration
constants,
but this time none of them is trivial. Except for special
values of these constants,
all solutions
are confined
between
two concentric
cylinders whose axes lie along the magnetic field. To parametrize
the solutions,
it is convenient
to use the radius of the outer cylinder p,,, and the magnitude
of the outer magnetic field pout.
Three types of solutions are depicted
in figs. 4-6. Again,
the velocity
and the magnetic
field fi are shown, but this time as functions
variable p. The same radius pm, of the outer cylinder has been
three
cases;
they differ in the value of the outer
close correspondence
between
the plane
magnetic
(1,
of the radial
chosen in all
field p,,,.
and the cylindrical
angle
There
symmetry,
is a
which
can be noticed by comparing
figs. 1-2 and figs. 4-6. In particular,
the diagrams
la (strong field regime) and 2a (weak field regime) resemble
the diagrams 4a
and 6a, respectively.
The differences
between
the corresponding
trajectories
may be attributed
to the drift which is caused by the field gradient in the radial
direction;
field p,,, outside the larger cylinder is always different from the field
inside
the smaller
cylinder.
Diagrams
5a and 5b represent
a special
Pin
situation,
in which the inner cylinder disappeared
and the whole interior of the
larger cylinder is filled. Such a solution exists for every radius of the cylinder.
For each p,,, there is one value of pm= for which this happens.
In fig. 7 we plot the difference between the radii of the outer and the inner
SOLUTIONS
Fig. 4. Cylindrical
velocity angle +
symmetry,
as functions
OF VLASOV-MAXWELL
strong field regime
of radial coordinate
EQUATIONS
(/_JO,,= -1.65); pmar = 2. (a) Magnetic
p. (b) Individual particle trajectory.
241
field p and
cylinders, D = prnax-pmin, as a function of the outer field p,,, for different
choices of the radius p,. Each curve has two wings; the left wing corresponds
to the weak field solution (type 6a), while the right wing corresponds to the
strong field solution (type 4a). The two wings always join at a point corresponding to a special solution of the type 3b. For strong fields p,,,, the
difference of the two radii becomes independent of pOUtlong before it reaches
the zero value.
Even though our equations for 0 and (1,in the case of cylindrical symmetry
cannot be explicitly solved, the behavior of p and $ near the boundaries of
the confinement region can be described analytically (cf. the appendix). In
particular, the behavior of the magnetic field at both boundaries is governed by
the square root dependence
(42)
I. BIALYNICKI-BIRULA
242
,’
_:’
AND
Z. BIALYNICKA-BIRULA
SOLUTIONS
OF VLASOV-MAXWELL
EQUATIONS
243
3
c
2
1
C
as function of
Fig. 7. Thickness of confinement
region, D = prnal - pm,“, for cylindrical symmetry,
absolute value of the outer magnetic field. Three curves correspond
to the values of pmax equal to 1,
2 and 3.
where p, is either P,,,, or pmi.. Thus, the derivative of /!I at the boundary is
infinite, as in the case of plane symmetry.
Confined, cylindrically symmetric solutions of the Vlasov-Maxwell equations
were studied some time ago by Marx’). Although his solutions are qualitatively
similar to ours, there is an important difference. He assumed the electron
distribution function as a product of two (not three as in our case) delta
functions. From our point of view, such a distribution function describes a
statistical mixture of trajectories which differ in the values of the r and z
components of canonical momenta. The two-delta Ansatz leads to the linear
(Bessel) equation for the magnetic field, whose solutions differ quantitatively
from those described here, especially at the boundaries of the confinement
region.
244
I. BIALYNICKI-BIRULA
In addition
AND
to the confined
confined
solutions
originate
and end at infinity.
5. Concluding
(p,,
Z. BIALYNICKA-BIRULA
solutions
= m), for
discussed
which
There
the
above,
orbits
is one solution
there
of individual
are also unelectrons
of this type for each p,,,.
remarks
Our treatment
of Vlasov-Maxwell
equations
stresses the particle
orbits,
which underlie the statistical description;
the symmetry
of the solution results
from the averaging over the orbits which differ only in their positions in space,
not
in their
intrinsic
shapes.
From
this point
of view
one
can
look
at the
configuration
with the cylindrical
symmetry
as a deformation
of the configuration
with plane symmetry.
After all, the hollow cylinder
can be obtained by rolling the slab. As is seen in fig. 4b, this bending,
accompanied
by
unbalanced
gradients of the magnetic field, results in a drift of electron orbits.
One may say that the rolling of the slab results in the compactification
of the
solution
in the y direction.
Thus, in order to obtain a fully confined plasma
configuration
one should perform the second compactification
by rolling the
cylinder into a torus. However,
the corresponding
solutions
would be much
more complicated.
There will be a second drift of the orbits associated with the
bending
of the cylinder and one would have to introduce
also an additional
current
flowing through
the center of the torus. This would destroy
the
cylindrical
symmetry
of the magnetic
field.
Acknowledgments
We are greatly
indebted
to Professor
F. Ehlotzky
for his kind hospitality
for getting us interested
in the subject and to other members
for Theoretical
Physics in Innsbruck
for helpful discussions.
This work
National
has been
Bank,
Project
supported
by the Jubilee
Foundation
and
of the Institute
of the Austrian
No. 1923.
Appendix
Eqs. (17) and (40) for the velocity angle $ and the magnetic
field p are
singular at the boundaries
of the confinement
region, because at the turning
points tan + is infinite. Since the singularity
is the same in both cases, from the
knowledge
of the analytic solution
for the plane symmetry
we can find the
behavior of the solutions for the cylindrical
symmetry near the boundary,
SOLUTIONS
A (p -
sin JI = +(l-
245
EQUATIONS
p,,l),
(A.la)
CIP-PP~
P-P,+
where p,( p,) is either p,,(
the weak/strong
Upon
OF VLASOV-MAXWELL
(A.lb)
p,,)
or pmin( j?,). The +/-
field types of solution,
inserting
expressions
(A.l)
sign in (A. la) corresponds
to
respectively.
into
eqs.
(40)
we can
determine
the
constants A and C.
At the outer boundary:
A =
-&+-,
(A.2a)
P max
(A.2b)
At the inner
A =
boundary:
i&++
(A.3a)
Pmin
c=
r
&-l- 2
Pmin
Pi’
(A.3b)
that A must be positive and hence the magnitude
of the outer
needed to produce the cyclotron radius pm=.
field must exceed the value l/p,,,
Formulas
(A.l) not only determine
the ranges of variation
of p,,, and pin
with respect to the corresponding
radii, but they are also needed to integrate
Let us notice
numerically
eqs. (40). They are telling us what the first step should look like,
when the integration
procedure
is started at the boundary.
Obviously, eq. (A.3a) and hence our Ansatz (A.l) break down for the special
solution, when p,+ = 0. This solution fills the whole cylinder and is regular at
the origin. One can check by a direct substitution
into eqs. (40) that in this case
near p = 0 we have
sin IL=&:,
(A.4a)
(A.4b)
I. BIALYNICKI-BIRULA
246
Thus, the special
labelled
solutions
by the values
AND Z. BIALYNICKA-BIRULA
form a one-parameter
of the magnetic
family and are conveniently
field p,, on the cylinder
axis.
References
1)
2)
3)
4)
5)
6)
7)
8)
C.L. Longmire,
Elementary
Plasma Physics (Interscience,
New York, 1963) chap. 5.
R.C. Mjolsness, Phys. Fluids 6 (1963) 1730.
K.D. Marx, Phys. Fluids 11 (1968) 357.
G. Benford and D.L. Book, in: Advances
in Plasma Physics. A. Simon and W.B. Thompson.
eds. (Interscience,
New York, 1971).
W. Peter, A. Ron and N. Rostoker.
Phys. Fluids 22 (1979) 1471.
V.P. SiIin, Introduction
into Kinetic Theory of Gases (Nauka, Moscow, 1971) (in Russian), p.
119.
R.C. Davidson, Theory of Nonneutral
Plasmas (Benjamin,
Reading,
1974). chap. I.
M. Abramovitz
and I. Stegun, Handbook
of Mathematical
Functions (Dover, New York, 1970),
p. 589.