PFC/RR-93-4
Self-Similar Variables and the Problem
of Nonlocal Electron Heat Conductivity
S. I. Krasheninnikov, 0. G. Bakunin
MIT Plasma Fusion Center
Cambridge, Massachusetts 02139 USA
October 1993
This work was supported by the US Department of Energy under contract DE-FG02-91ER-54109.
Reproduction, translation, publication, use, and disposal, in whole or in part, by or for the US
Government is permitted.
Self-similar variables and the problem
of nonlocal electron heat conductivity
S. I. Krasheninnikovl, O.G. Bakunin2
MIT Plasma Fusion Center, Cambridge, USA
Self-similar solutions of the collisional electron kinetic equation are
obtained for the plasmas with one (1D) and three (3D) dimensional plasma
parameter inhomogeneities and arbitrary Zeff. For the plasma parameter
profiles characterized by the ratio of the mean free path of thermal
electrons with respect to electron-electron collisions, XT, to the scale length
of electron temperature variation, L, one obtains a criterion for
determining the effect that tail particles with motion of the non-diffusive
type have on the electron heat conductivity. For these conditions it is
shown that the use of a "symmetrized" kinetic equation for the
investigation of the strong nonlocal effect of suprathermal electrons on the
electron heat conductivity is only possible at sufficiently high Zeff (Zeff ;
(L/AT)1/ 2 ). In the case of 3D inhomogeneous plasma (spherical symmetry),
the effect of the tail electrons on the heat transport is less pronounced
since they are spread across the radius r.
1Permanent
2Kurchatov
address: Kurchatov Institute of Atomic Energy,
Moscow, Russia.
Institute of Atomic Energy, Moscow, Russia.
2
I. INTRODUCTION
It
is
electron
well
heat
known
conductivity
for comparatively
path
with
that
the
along
small ratios
respect
to
Spitzer-Harm
a magnetic
field
of the thermal
electron-electron
theory
is
for
only valid
electron mean
collisions
This
is
due
to
the
fact
that
plasma heat conductivity
is played
mean
suprathermals,
free path
of these
suprathermal particle energy; T
the
main
by suprathermal
A~
free
to
AT
characteristic length of the plasma parameter variation, L:
10
the
role
the
7=A /L
in
the
particles.
The
XT(Ck/Te)2
(C*
is the
is the electron temperature;
c.~
(4+9)Te) is much greater than AT.
At
the
same
time,
in
the
tokamaks
scrape-off
produced
on
plasmas
plasmas
plasmas,
with matter,
etc.
Spitzer-Harm's
Such
So
theory
attempts
values
it
7
a
10-2
are
layer
interaction
seems
rather
rather
(SOL
of high
have
been
made
by
many
), 7,8
energy
attractive
for relatively high values
typical
of
space
fluxes
to
generalize
of z:
z a 10-2.
authors.
Various
approximation methods for solving kinetic equations were used. For
example,
in Ref.
account
in
the
the terms proportional
i
expressions
for
the
to z2 were taken into
heat
flux
q;
modifications of the method of momenta were used in Refs.
Let
us
consider
the
studies
of Refs.
17,18
various
12-16.
in detail,
the
results of which can be represented in rather compact form. These
are often used for various applications.
In
the
Ref.
heat
17
flux
the
integral
q(x)
from
expression
a
previously
heuristic
obtained
consideration19
for
was
3
theoretically
verified and compared with the results of numerical
calculations:
q
q(x)
=
(x')
dx'
f
exp
dl /
T(l)
X
where qSH(x) is Spitzer-Harm's heat flux.
In
Ref.
18
a
procedure
for
approximately
solving
the
electron kinetic equation is proposed in the high effective plasma
charge
Z
limit
function
f (,x)
where
,
the
electron
can be represented
as
velocity
the
distribution
sum of the
symmetric
function f0(v,x) and the small asymmetric part gf I (v,x). In this
case
deviation
of
f0(v,x)
suprathermal region
deviation
from
the
Maxwellian
function
(responsible for the heat transfer)
from Spitzer-Harm's
theory.
The main
in
the
is causes
idea of the paper
Ref. 18 is to convert the differential equation for f (v,x),
8 f
1
+
ac
into
an
-
8
integral
using
approximation
the
integral
- 0,
0)
equation
equation which
is then
the
distribution
Maxwellian
density).
was
This
0
solved
realized
conversion
by means
for the operator 82G/8C2
T (a'
f(v,
(2)
8v
=
-
G(C,C',v,
Gv'
The approach developed
+
by
the
function
(here v is the electron energy;
electron
G(C,(',v,v')
8f
+
av
v
technique,
is
f
C
of
as
a
fne(x)dx;
Eq.(2)
of the Green's
zero
n (x)
into
an
function
V3 8G/8v:
a2
(3
(3)
8f'2
in Ref.
iteration
18 is
generalized
for medium
4
Z
in
first
20.
Ref.
sight)
However,
for
this
technique
approximately
actually gives an incorrect
solving
result.
(which
the
Indeed,
seems natural
kinetic
equation
at
(2)
by direct calculation
it can readily be shown that the distribution function obtained in
Ref.
8 begins
energies of v
means
of
actually
5
:
(see
Coulomb
it
6
and
value
term
follows
,
v
at
f0(v,x)
large
the
theory
already
between
that
one
at
particle
2
occurs
sufficiently
from the the Maxwellian
f3/7 . Meanwhile, from the solution of Eq.(2) by
perturbation
difference
fact
to differ
2
1
the
and
the
This
this
at
Maxwellian
below).
in
,
that
v
difference
5
::.
92/
function
is
explained
kinetic
equation
2
the
becomes
by
the
for
the
distribution functions close to the Maxwellian one is in fact the
small
difference
heating
of
Incorrect
between
electrons
two
(the
large
second
operators:
and
separation of these operators
third
deceleration
terms
results
in
in an
and
Eq.(2)).
artificial
overestimation of the role of Coulomb electron-electron collisions
and in a reduction of the deviation of f 0(v,x) from the Maxwellian
function.
Unfortunately,
approximate
solutions
values
not
are
criteria
of
the
applicability
of the electron kinetic
available
and
it
can
be
equation
tested
by
of
the
at high 7
means
of
numerical simulations in only a rather limited range of parameters
because numerical solution of the kinetic equation is cumbersome.
Moreover, all solutions of the electron kinetic equation mentioned
above
were
found
for
the
case
of
one
dimentional
(1D)
plasma
parameter inhomogeneity. It is still not clear what role nonlocal
5
effects
two
play
(2D)
in the
and
electron heat conductivity
three
(3D)
dimentional
for
the
cases
plasma
of
parameter
inhomogeneity.
In Ref.
21
a
class of
solutions
equation which allow representation
found.
The
transition
comparatively
easy
to
to
of the
in self-similar
self-similar
find
collisional
exact
variables was
variables
solutions
kinetic
of
makes
the
it
kinetic
equation. Analytical and numerical studies of these solutions21,22
showed,
in particular,
of a strongly
that Eqs.(1),(3)
anisotropic electron
do not describe the effect
distribution
function
tail
on
the heat flux q, which under certain conditions can turn out to be
decisive.
In
the
present
paper
solutions
of
the
electron
kinetic
equation are found for the case of 1D and 3D (spherical symmetry)
plasma parameter
of self-similar
inhomogeneity
variables.
for arbitrary
Criteria
of the
plasma Zeff
by means
strong effect
of tail
electrons on the electron heat transport are obtained.
Sections
plasma.
II-IV are devoted to investigating
In Sec.
II
the main equations
analyzed
1D inhomogeneous
in the
paper
are
given and the limits of their applicability are specified. In Sec.
III solutions of the kinetic equation for moderate values of
Zeff'
(1+Ze
>
7-1/2)
)2
< z-1/2,
limit
inhomogeneous
are obtained.
In Sec.
is
investigated.
In
plasma
is considered.
The
Sec.
IV the high Z
V
results
the
case
(3
of
3D
are discussed
in
Sec. VI and the main conclusions are summarized in Sec. VII.
6
II.
SELF-SIMILAR VARIABLES FOR THE 1D KINETIC EQUATION
Let us assume ions to be at rest and consider the stationary
kinetic
equation
for
an
electron
distribution
function
that
is
inhomogeneous along the x-axis:
af
V9
e E
--
f
ay!
-M
a
12
+
g
_
(4)
Z
2ne A
2
St(
+
,f)
n (x)8
8f
(1_9 )
e3e
where
=-
St(Vf
U
a13 =(u
af (V)
a)r
a13
-u
u
aa3c
a
f
f
v
U adv',
V/
)/u3V
a
o is the angle between the particle velocity vector V and the axis
x; g = cosa; f ,(V,g,x)
m, n (x)
ion
are
its
charge;
A
collisional
is
charge,
is
the electron
mass
the
and
density;
Coulomb
function;
St(V,f)
the
ambipolar
allows
solutions
is
e,
is the effective
Z ef
logarithm;
E(x)
operator;
Coulomb
distribution
is
the
electric
field.
As
shown
in
Ref.
21
Eq.(4)
in
the
self-similar variables
f (V,x)= N F(0)/[Te (x)]',
where
N
adjustable
is
the
normalization
parameter;
characteristic
average
the
0'=
j*(m/2Te(x))l/2
factor;
function
electron
JF(v)dv
Te(x)
energy.
plays
In this
(5)
=
1;
the
a
role
is
of
an
a
case the kinetic
7
equation (4) is converted in the following way:
1-g 2 8F
8F
-Ev+
v
~
Z
8F
TvyaF
-2
(6)
1
-
Z
St(, F)
2 aF
fa
+
8g
-
2
)
where the parameter z E = eET /(2ne An ) is found from the particle
flux
ambipolarity
T (X)(
2
a)J
=
condition,
const,
automatically
found
from
assumed that
where
J
=
satisfied at a = 2,
TE can be arbitrary
is
which
the
j
in this case,
relation
J =
reduces
to
the
SF(v)vudv.
and,
condition
as a result,
while
0.
This
j
relation
is
the value of
at a # 2 the quantity
However,
it
C
will
always
7E
be
= 0.
Here it is useful to note that Eq.(6) is similar in structure
to
the
equation
electric
field,
representing
the
electron
runaway
effect
which allows one to use the approaches
in
a
developed
in Ref. 23-26 for it solution.
Solutions of type
(5)
correspond
mean free path of electrons
to a constant
ratio of the
with an energy of about
T (x) to the
characteristic scale length of T (x):
S=
- eT2
Taking
relation
(7)
(dnT/dx)/(2Te4 An ) = const.
account
results
> 0
of
the
distribution
in
the
following
(7)
.
function
temperature
form
and
(5),
density
profiles:
T (a- / 2 )(dT /dx) = const,
ne
T (3/2-a)
(8)
8
Considering
the
expression
for
the
energy
flux
density,
corresponding to Eq.(5),
q(x) = QTe(x)(3 -a)N(2/m)2 ,
Q =
F(v)v 3 dv,
(9)
it is easy to see that the dependence on x in Eq.(9) vanishes at a
=
3.
Note
that
conservation
conduction
Eq.(8)).
either
the
case
other
temporal
=
classical 1
for the
coefficient
In
a
terms
corresponds
dependence
on temperature
Kc
cases
3
of the
K
sources
i.e.
or
energy
the
sinks
flux
electron
T 5/2 as
a
one has dq/dx * 0,
or energy
to
well
heat
(see
presence
not taken
of
into
account in Eqs.(4),(6). However, when 7 << 1, the effect of T (X)
inhomogeneity on the distribution function in the range of thermal
velocities
~
v
(2T /m) /2
is
small
and
the
function
in
f
this
range is close to the local Maxwellian one. Therefore for the case
a * 3 and z << 1 it can therefore be assumed that Eq.(6) describes
the effect of suprathermal particles
on the heat flux q(x)
in the
presence of the sinks or sources in the thermal velocity range.
In the limit of interest, v >> 1, Eq.(6) transforms to
2
-
F
+
+ _y
(_
1~ -a8(F
where C = v 2 ;
Further
related
to
(+Zeff
)/2.
simplification
fast
function at high g,
(10)+a(,,)a
of
the
symmetrization
where F(v)
of
kinetic
the
equation
electron
can be represented
(10)
is
distribution
in the form of
the sum of the symmetric part F 0 (v) and the small asymmetric part
9
gF1 (v).
In
this
approximation
the equations
for F0 (v)
and
F (v)
have the forms:
(1-)d
E d
({F )
yt
3d
dF
({F ) =
~
(11)
F , - =y
F2
d
{
pd
- ( C F O)
Substituting
F (v),
72
the
+
E
expression
2 dF0
d
for
'
F1 (v)
in the
equation
for
one obtains
(1-a)d(
a[
( 3 -a)d
a
-(C F,,) +
3
6 d(F
a
)d
[(3-C
2d
jC (F
FO) +
)]
6C d~~F)
)
(F 0(2
3
(12)
dF
d
C dC ~o
where
6
=
T7.
Essentially
doJ
are
Eqs.(11),(12)
a
self-similar
analog of the equation used in Ref. 18.
Analyzing
Eq.(12),
one
can
see
that
distortion
of
Maxwellian distribution function starts at the energies C6
The local
the
192
C5
of the electrons with the energies
"temperature"
j3/-2 exceeds the temperature of the bulk electrons by a factor of
two.
in order for the condition F
On the other hand,
true,
it
energy
is necessary,
be
rather
as
small,
seen
C3
from Eq.(11),
j9/7.
,
Hence,
it
that
>> F
to be
the electron
follows
that
the
applicability of Eqs.(11),(12) and of Eq.(2) derived in Ref. 18 is
restricted by the relation 0 : 7-1/2
At
energies
C4
>
12,
i.e.
where
electrons
have
time
to
10
diffuse
path)
and
at
a
distance
distortion
f (,x)
is
anisotropy
in
conditions
of
of the
about
Maxwellian
determined
fe(V,x)
(11),(12)
-d1 (measured
by
the
and
function
integral
violation
can be expected
the
becomes
T,(x)
of
at
in
mean
free
very
strong
profile.
Strong
the
applicability
2 2 p/z, where the mean
free path becomes greater than the characteristic scale of Te(X)In
the
Maxwellian
opposite
distribution
anisotropization
and
(
case
z -1/2
function
one
should
is
distortion
accompanied
consider
the
by
complete
of
the
strong
equation
(10).
III. SOLUTION OF THE SELF-SIMILAR 1D EQUATION (
Let us
find
solutions
of
Eq.(1O)
ranges of the dimensionless energy
A.
Low energy range,
<
in various
1/2
characteristic
with their further matching.
C2(#/T)1/3
In this energy range the distribution function F(v)
close
to
Maxwellian
and
can
be
found
by
expansion
in
is quite
Legendre
polynomials P1 (w):
Co
F(v) = (
F4()P(g)
(13)
i=0
Leaving the mean expansion terms only and substituting F.(i)
in the form
F
= f-3/2 e-
(2k3
k=0
(2k+i)a
,
(14)
11
where
a0=1,
one
0
finds
from
1-1
i+1
-a
Eq.(10)
the
following
recurrent
relation:
a
21-1
a
=
k
It
(
can
+
2i+3
al
k-
(15)
1(1+1) + 6(2k+i)/
be
seen
that
F0(()
deviates
noticeably
from
the
Maxwellian function at energy C a (g/2 ) / 6 , while the role of the
asymmetric term F 1 (C) only becomes essential for higher energies (
1/3
7-1/2
B. Energy range (g/1)1/3
In
this
energy
range
it
is
convenient
to
introduce
the
variable z = C1/2 and to represent the function F(V) in the form
F(v)
=
(16)
-/exp(-p(z,g)).
Expanding V(z,g) in a power series in
P(zg)
and
= PO/71/2 + P/1/4
substituting
this
+
expansion
+ .
in
-
(17)
,
Eq.(10)
after
appropriate
calculations, one finds
PO(Z)
= z -
z /3
8,)
(18)
12
))1/
2
Z3
2
1/2
1+g
(-)
(19)
Z
+ g1/2f( (1z' 2
1/2dz'
(1-Z,2y
0
2(3
1
2
+6-10z 2
~+610z ln2[1+
C
where z*
=
+y 1/2 -1
2
(a'-3/4)1n(1-z 2)
-
-
C1
+ C1
/2
9 11/3 1/6 and C
+
-
Oz2
+
22
+ 4
(1-y)
-
4
ln(2/(l+y))
qz2
distribution functions (13),
- 1
(20)
2
+ C,
Z1-21
+ C1 31n(z.) + C 4
1
1/2
12
(21)
are the matching constants of the
(17),
realized at energies z = z,.
C. Energy range z = 1
Let us introduce the variable y = (z 2 _1/7 11 6
the function V as a power series in -d
= V01/6
o(Y,)
+ P,
+ ...
+ C2'
and represent
-
(22)
Expanding the function V1 in Taylor series in the vicinity of
13
00
V
(y, W )
(
= I
(1
k=O
)k
(23)
k!
and substituting Eqs.(22),(23) in Eq.(10), one finds
(P 0= 0
Rp/2
_ p24
4/4
(24)
(4
P
(25)
,
(
1
11+21p3
(6 3)'
--
1
-
n(p) + 11n+
1 + (2/3)z-1/2 + (2T)3
C2 =
13p 3
+
3
1 12
12
-1
2
2
[(1/4)]
p3 '
3
--
(26)
2
(27)
(a-(6+jS)/4)1n(g/2) -
-
where f(x)
the
is the gamma
solutions
(17),
(22)
12
function;
in the
1nT + 6 -
C2
1 -
1n
,
is the matching constant of
energy
range
s v-1/2
z
where
both expansions are applicable, and the function p(y) = P(Y)//
where Y = y/g1/3,
is determined by positive roots of the equation
P3 + YP - 1
0.
(28)
They are
P=
1 3 cos
2(-Y/3)
1
(
2 13 YKj 3
3
12
for
2 2 13
< -1
13
+1
=1]
1
21/3[
(29)
3/
/
3
23
1/
arccos[2
(30)
22/3311/2
[(3l+1]
111/3
-
22/3 Y
,
for
3
At the limit of applicability of the expansion (22)
(y
y4 =
14
),
C
3
=C
= 1)
the function V(y,
+
2
-+
6
Applicability
= C 3 turns out to be equal:
+
2
12
of the
C-
expansions
4)
(17),
1nz + 1n.
(22)
of
(31)
(31)
the
function
can be verified by direct comparison between the magnitudes
q(gi)
of successive terms
in the series
(17),
(22).
It
is easy to show
that the results of a given section are true at
71/2
13
(32)
This characteristic
g-values.
the
energy
The distribution
spherically
obtain
(1/7)1/2
Energy range (2/1)1/2
D.
it.
Let
only exists
function F(4)
symmetric
us
range
one,
neglect
the
and
one
is
high
found to be close to
should
ambipolar
at rather
use
electric
Eq.(12)
field
to
effect
which is not essential here (the terms proportional to 6) Equation
(12) is then transformed in the following way:
Id2
(1-U)d(C )
The
solution
(1/72)1/4,
taking
3
(C F.)
of
Eq.(33)
account
of
d
= -
(F 0
the
in
+
dF0
range
joining with
its
.
(33)
(2/7)1/2
the
<
C
solutions
in
lower energy ranges, has the form
0
(-3 :
F= 7-3/2 exp-
~
C
+ 3((2/z)
At higher energies
-1)
(9/z71/4
Eq.(33) has the power law dependence
(34)
71/2 the solution
of
15
3 12
F = Tr
The
exp -C
power
represents
a
practically
spectrum
1/4 a.
3
law
dependence
(35)
(V2a
see Ref.
F (v)
"collectivization"
of
high-energy
insensitive to Coulomb collisions.
of
such
electrons
is
determined
by
electrons
In this case,
just
21)
the
parameters of electron temperature and density profiles
the
global
(a in the
case under consideration). A similar dependence F(v) a v-2a occurs
at
energies
anisotropic
C
,
p
and the
where
applicability
F(4)
again
becomes
strongly
condition of Eqs.(11),(12)
is
violated.
E. Energy range { a ((/7)1/2
We seek the solution of Eq.(10)
2
(v) = it (z,i) + 7
F
c
(Z' g)
+
in this range in the form
...
(36)
.
Substituting Eq.(36) in Eq.(1O), one obtains the equation for
it (z,g):
2
It
z
(1-a) 8
easy
is
parabolic
type
(Z
Co)
+ ,
=
2
-jy
.
0 the main
see
that
at
a:
equation
(37)
is
played
to
left-hand side. The solution of Sq.(37)
for the corresponding matching,
by
)(
-3 12 e
Z
- C4
T-3/exp
can therefore
a
)/2)
ID O(y)
the
at g a 0,
the form
IV~Z, g)
(37)
,(38)
be
role
in
the
terms
on
the
with allowance
represented
in
16
where
C 4 = C3
+ (3/4)g + (a/4)1ng
The function t (w)
112
S(13/7)2
analytically
represents the angular dependence F ()
Unfortunately,
defined.
(39)
It
can
this
only
be
Therefore
,
taking
F (M) at C = (p/7)/2 112
into
account
at
dependence
cannot
be
shown
J0(-141)
<<
that
owing to the absence of a particle source at v =
i(IgI)
0.
.
that
the
angular
and g s
asymmetry
approaches unity, the following estimate can
be assumed to be valid:
~
j 0(g)g dW
Thus the
(40)
1.
structure
of the
solution of Eq.(1O)
at p < 1-1/2
can be represented in the following way (Fig.1). At low energies (
:
(9/i)/,
approaches
the
the
distribution
Maxwellian
(9/)1/3
-/2,
function
one
it
is
at
F(v)
(
is
isotropic
anisotropic
and
it
significantly from the Maxwellian function; at high
(2/7)1/2
1/2,
Maxwellian one
but
it
again becomes
differs
at
2 1/6;
energies
deviates
3, at energies
considerably. from
isotropic
(at
n- 1
this
practically vanishes) and finally, at energies C a (/1)1/2,
far
this
from being Maxwellian and
so-called
spectrum
is
temperature
determined
and
is strongly anisotropic.
"collectivization"
density
by
the
profiles
of
electrons,
global
parameters,
(F(v)
consideration), takes place at energies
v-2a
in
and
a (g/72 )1/4
range
it is
Note that
where
the
the
the
their
electron
case under
17
IV. SOLUTION OF THE SELF-SIMILAR ID EQUATION
The solution of Eq.(12),
representing
1/2)
(g >
the
case of high Zef'
where 1 > 7-1/2, is readily found at a = 2 since all the parts of
the equation are complete differentials:
F
(
J
EXP
Unfortunately,
313/ 2 + (2-S)'3(, -5)
313/72 + (1--)5
' (C -5(
the
integral
J,
as
will
be
shown
later,
diverges at a s 2. Therefore we shall consider Eq.(41) as a formal
expression with which one can, nevertheless, compare the solutions
obtained in some indirect way.
Let us represent the function F0 ()
Fo()
=
Since,
-32
exp(-O(C)) .
as
follows
considerations
given
(42)
from
in
the
distortion of the Maxwellian
in the form
Eq.(41)
second
and
from
section,
distribution
qualitative
a
function
the local "temperature") occurs at energies (
introduce the variable u = C((
noticeable
(doubling of
2 1/5
let us
1/5 and represent the function
as a power series in s-1 2 , where s = (v2)1/5.
0 =
0/s
+
1/s1/2 +
2
+...
(43)
Substituting Eqs.(42),(43) in Eq.(12), one finds
U
3
(44)
u'
(u'
+3)
i
18
(45)
=0,0
2
6
5
It
is
expressions
proved
of the
Maxwellian
.
by
(46)
direct
(41),
expression
is
(44)-(46),
>> 1),
5+
53
easily
applicability
(/72 )120
(u-1)ln
8u 5 +
5 +3
u +3
and,
violated
at
at the time when
i.e.
distribution
verification
as
a
result,
range
function
already
occurs
and
it
(U
?
and
(see expression
Section II).
is similar to the solution (35) of Eq.(33).
2:
the
strong distortion of the
The solution of Eq.(12) in the energy range (P1/2
(P/7)1/2
the
of
(/z2 ) 1/4
C
"collectivization" of high-energy electrons starts
(37) and estimates in
that
1/2
In the energy
(p/7)1/2, where the applicability of Eq.(12) is violated
is necessary
to consider
Eq.(10),
we obtain a solution of
the type (36), (38):
F(v)
= n
3
2
exp(-
j
C5
g)
(47)
where
C5
3
5
(1/5)(4/5)
2 1/5 +
(
(a-l)ln
2
) 1/4
(48)
+ (a/4)1n9 +
At
the
represents
the
same
time
the
(6/5)6 -
function
angular dependence
F(v)
i
(g),
3/4
like
to
at high energies
$(g),
and the
estimate (40) is also valid for it.
Thus the structure of the solution of Eq.(10), for P > z-1/2
19
can be represented in the following way (Fig.2).
The distribution
function
F(v)
(13/z)1/2;
energies,
C
is
s(
energies (g/2
Maxwellian
isotropic
it
2 )1/6,
)1/6 <
energies
approaches
the
takes
of
place;
the
at
Maxwellian;
energies
electrons
starts
differs from the Maxwellian function and,
(13/)
s
at
low
at
the
(9/2) 1/4 considerable deviation from the
function
"collectivization"
up to
(
and
finally,
a
(g/2 1/4
F 0 (v)
greatly
at energies
/2 the function F(4) becomes strongly anisotropic.
V. SELF-SIMILAR VARIABLES FOR 3D KINETIC EQUATION
Let
shown
us
that
parameters
now consider
a
in
of
the
case
self-similar
3D
inhomogeneous
spherical
variables
plasma.
symmetry
can
be
of
It
will
the
be
plasma
introduced.
The
stationary kinetic equation for the electron distribution function
f,(
,r) inhomogeneous along the radius r is
af
e E
Vg
-
af
e
1-g 2 a
2
(49)
2Te A
Z
)
=St(,f
where
the
-
is the
axis
r;
g
+
angle between
=
cosO;
.n (r) a
eff
2
e
the particle
E(r)
is
the
8f
e
velocity vector
ambipolar
electric
V and
field
directed along the r axis.
It
is
easy
to
show
that
Eq.(49)
allows
solutions
in
the
following self-similar variables:
fe (,r)
where,
= N F[Or/(vr),v]/[T (r)],
as in Eq.(5)
v = V(m/2T (r)) 1 /2 ,
N is the normalization factor;
(50)
fF(v)dv = 1; a
20
is an adjustable parameter;
of
a
characteristic
and the function Te(r) plays the role
average
electron
energy.
Equation
(49)
is
transformed to
dv g (aF + 2f
iFgE8F
+F1aF - +1-g2a8F
( 11.T
2m
T'E
j8k
- (a+1/2)(1- w22 )8F
v
v F)
(1
(51)
1
Z
St(4,F) +
=-
which
differs
derived
in
8
from
the
[21,23]
for
8F
-(1-g)
corresponding
the
case
)2
self-similar
of
1D
plasma
equations
parameter
inhomogeneity by the term proportional to I8F/8g on the left-hand
side of Eq.(51).
This term describes
the spread of the electrons
across the radius r.
As
in the case of 1D inhomogeneity
corresponds
electrons
T
to
with
a
an
constant
energy
of
ratio
of
about
Te(r)
the solution of Eq.(51)
the
mean
to
the
free
scale
path
for
length of
(r) and to the same dependencies of T (r) and n, (r) on r.
In the limit of high energies, v >> 1, Eq.(51) is transformed
to
+F
+
aF) -
(a+12)(1-g2
F +
2F)
(52)
1
For
the
high-p
a
F +
approximation
F (C) will have the forms:
+
the
2
(1- )F
equations
for
F (C)
and
21
1-a)d
2
(
2 dF
F)
F
2
2 (a+
2 d
TC(
)d
6
-
-
+
a
FC,)
E
(53)
(1-a)dF
+ SEC
i
(4-a ) d
a
C(
a)(C FO) +
3 dF0
EC
(54)
dF
3d
-d (FO +
)
We shall not demonstrate here the solutions of Eqs. (52),(54)
in
all
energy
ranges,
unlike
in
the
case
of
1D
inhomogeneous
plasma. We consider below just the tail electrons, since it can be
shown
that
functions
the
at
difference
low
between
energies
is
the
not
as
1D
and
3D
important
distributionas
for
tail
electrons with energies C a: (/j)/
The distribution function of these electrons is described by
the equation
y aF + C
-
(a+1/2)(1-
2
2 F
= 0.
)
(55)
Assuming that the angular distribution function at g s 1 and
at energies
C
((/)/
has the form
0 (g) = exp(-CM (1-u)), where
1 is the decay scale of the distribution function along the g
coordinate, the asymptotic solution of Eq.(55) for tail electrons,
S>>
112
(,/')/,
F(v )
The
<xexp
can be represented in the following way:
Tmoh e) n
normalization
ca
f r
-C 9(1 -gJ (7
constant
for
( a- 1 / 2 )
1/2
Eq.(56)
.
is
practically
( 56 )
the
22
same as in the case of 1D inhomogeneous plasma.
Thus, the main effect of 3D inhomogeneity or, in other words,
the main effect of of the electron spread across the radius r is
the beam-like distribution function of the tail
to
some
extent
leads
to
suppression
of
electrons,
the
influence
which
of
tail
electrons on the heat flux.
VI. DISCUSSION
First of all the presence of a strongly anisotropic power law
tail
in the distribution function at energies C ? (p/z)1/2 in the
above derived equations should be emphasized. This result is quite
natural if one takes into account that electrons are collectivized
at such energies and their spectrum should be determined by global
characteristics
of
the
ne(x),
by
their
not
but
distribution
the
form
condition,
temperature
local
function f e(V,x)
(5),
the
only
will be f ,
e
Formally,
and
and
values.
density
Since
profiles
we
the dimensionless
dependence
f
on
v
Te(,X),
the
consider
energy C
satisfying
in
this
1/V 2 a.21
such a power law dependence of f (V)
results
in a
divergence of the dimensionless heat flux Q in the expression (9)
at
a
s 3
(for
suprathermal
1D
inhomogeneity)
particles
for
any
owing
values
to
the
of
T.
corresponds to a rather steep dependence Te(x):
contribution
Relation
a
of
s
3
dinT /dlnx 2 2/7.
At a = 3 the divergence is a logarithmic, while at a < 3 it obeys
the power law. At a much steeper dependence of Te(x), where a s 2,
the dimensionless particle flux J diverges and it turns out to be
23
impossible
to determine
the
quantity
s,
while
at
a
s
3/2
the
quantity N turns out to be divergent. Here it should be remembered
that
the
case
a =
corresponds
3
to
the
constant
heat
flux q
according to Spitzer's electron heat conduction law.
It
for
should be noted that the Luciani-like nonlocal expression
the
heat
flux,
Eq.(1),
for
the
electron
temperature
and
density profiles under consideration does not make any difference
between
the
cases
with
a
and
>3
a
<
and
3
q
gives
=
qqSH
SH(x)/r1-T 2 (3-a) 21 .
However,
electrons
the
is
emergence
related
E
in real limited systems,
s
T
the
a
great
to their transport
(electric field effect,
to T
of
x
number
from
of
some
suprathermal
hotter
dT/dx
is inessential here).
e
where the maximal temperature
formally
emerging divergences
can be
zones
Therefore
is limited
eliminated
by cutting off the divergent integrals at the energy level mv2/2
(we
T
assume
self-similar
C
S
one
T
Cm
that
IT
the
distribution
function
in
Eq.(9),
one
tail electrons to the integral Q (a
<
finds
(3c
(3-
7ma
)
x
)
exp(C
into
account
the
electrons to the integral of Q,
of
-,
the
effect
of
tail
fact
of
0<-1/2
4)
exp(-C 5 )
Taking
Assuming that
contribution
the
a
3):
exp(C )
AQ2
tail
).
in the energy range mV /2 s T
,
fe approaches
that
(57)
2
-1/2
'
the
contribution
of
where C s (g/1/2 is of the order
electrons
on
the
heat
flux
essential when the temperature difference is rather large:
becomes
24
T ax(3-a)
>a/2
4
1/2
-1/2
>r
.
2
exp(C5(
In
formulating
constants C4
Eq.(58)
we
(58)
>1/5-/
left
the
main
term
only
in
the
~ 2/3, & 5 ~ 311 5 r(1/5)r(4/5)/5.
and C 5 ; here C 4
In the case of smooth T (x) profiles, where a > 3, the effect
of tail
electrons
on the
heat
flux
only
becomes
noticeable
at
rather high values z ~ 1.
Note,
however,
generalizations
that
made
in
the
this
solutions of the collisional
estimate
section
are
(58)
based
kinetic equation
and
on
that
other
self-similar
were obtained
for the electron temperature and density profiles characterized by
z. It is obvious that they do not spread to the zones with uniform
electron
temperature
Moreover,
a definite
exerted
by
some
in
the
effect
boundary
step-like
on the heat
effects
(e.g.
temperature
profile.
flux production
interaction
is also
between
the
plasma and tokamak divertor plates). The study of such effects is
beyond the scope of this paper.
In
the
case
of
beam-like
character
(56),
effect
the
become
of
of
3D
the
tail
plasma,
electron
tail
electrons
at much
pronounced
inhomogeneous
steeper
on
the
effective
owing
to
function
distribution
heat
flux
the
starts
temperature
to
profiles
(a < 7/4) than in the case of the 1D inhomogeneous plasma.
Let
based
of the
us
now
consider
on representation
sum of the
when
the
high-Z
of the distribution
symmetric
part
and
the
approach,
function
small
which
in
is
the form
asymmetric
one,
25
can
be
applied
for
investigating
the
suprathermal
particles
on
the
necessary
note
if
inequality
to
that
effect
electron
heat
(58)
of
nonmaxwellian
conduction.
not
is
It
fulfilled
is
the
energy range responsible for the electron heat conduction is10
< c < c 2 , where c
Let
us
shown
function
symmetric
energies
strong
temperature
electron
this
noticeably
the
in
in
that
deviates
C >
heat
the
to
profile
be
by the magnitudes of AT/L = I < l and 9 < 1-1/2.
above
anisotropic.
the
= (3+5)Te, C2 = (7+9)Te.
assume
characterized
was
1
range
(p/z)1/3
case
from
1/6
2/7)
(
the
electron
C
the distribution
function
The effect of this deviation
conductivity
when
(g/z2 1/6
when
<
is
still
1/3;
at
higher
But
becomes
becomes
2I e
<
(g/-2)1/6
c 1 /T..
(/d)
<
this
distribution
and
Maxwellian
<
it
and
effect
strongly
noticeable
for
becomes
very
can
only
be
described by high-Z approach when the distribution function in the
range
Both
be
of
interest
inequalities
fulfilled
elements
with
if
is
(
still
(g/72 1/6 S C /T
zef
this
symmetric,
is
charge
extremely
exist.
that
is
and c 2/T
high,
Therefore
c2
e<
(9/7)
< (g/7)1/3
Zff
for
a
260,
<
113
) can
but
X-12
no
the
analysis of strong deviation of the electron heat conduction from
the Spitzer-Harm theory must take into account the energy range
> (9/7)1/3 where the distribution function can not be described by
high-Z approach.
Thus, high-Z approach
can only be applied
the investigation of the nonlocal effects under the condition
-1/2
for
26
VII.
CONCLUSIONS
i. Solutions of the collisional electron kinetic equation are
found
for
1D
and
3D
(spherical
profiles
The criterion
(58)
characterized
by
thermal particles, X ,
It
determines
inhomogeneous
plasmas
means of self-similar variables. 2 1
with arbitrary Zeby
ii.
symmetry)
is obtained
the
ratio
for the plasma parameter
of the
mean
free
path
for
to the electron temperature scale length L.
the effect
that tail
particles
with motion of the
non-diffusive type have on the electron heat conductivity.
iii.
For
these
"symmetrized"
kinetic
investigation
of
electrons
heat
on
the
conduction
conditions
it
is
shown
equation
of
the
type
the
strong
electron
nonlocal
heat
can not be described
symmetry),
less
In
the
of
3D
the effect of tail
pronounced
radius r.
case
since the
tail
effect
by
of
when
?: ( L/? T
inhomogeneous
use
18
of
for
the
are
the
electron
theory,
is
1/2
plasma
(spherical
on the heat transport
electrons
a
suprathermal
Spitzer-Harm
(Zef
electrons
the
of Ref.
conductivity,
only possible at sufficiently high Z ef
iv.
that
spread
across
is
the
27
Harm R., Phys. Rev. 89,
1 Spitzer L.,
2 Gurevich A.V.,
3 Bell A.R.,
977 (1953).
Istomin Ya.N., JETP 77, 933
(1977) (in Russian).
Evans R.G., Nicholas D.J., Phys. Rev. Lett. 46, 243
(1981).
4 Mason R.J.,
Phys. Rev. Lett. 47, 652 (1981).
5 Matte J.P., Virmont J.,
Phys. Rev. Lett. 49, 1936 (1982).
6 Khan S.A., Rognlien T.D.,
Phys. Fluids 24, 1442 (1981).
7 Igikhanov Yu.L., Krasheninnikov S.I., Kukushkin A.S.,
Yushmanov
P.N. "Specific features of transport processes in a tokamak
edge plasma."
in: Rev. of Science and Technology, Plasma
Physics, edited by V.D.Shafranov (Moscow, USSR Inst. of
Scientific and Technological Information, 1990),
Vol.11, P.5
(in Russian).
8 Nedospasov A.V.,
Uspekhi Fiz. Nauk 152, 479 (1987)
(in Russian).
9 Galeev A.A., Natanzon A.M., Fizika Plazmy 8, 1151 (1982)
(in Russian).
10 Hares J.D., Kilkenny J.D., Key M.H., Lunney J.G.,
Phys. Rev.
Lett. 42, 1216 (1979).
11 Kishimoto Y., Mima K.,
J. Phys. Soc. of Japan 52, 3389
12 Hassam A.B., Phys. Fluids 23,
13 Shirasian M.H., Steinhauer L.,
14 Zawaideh E.,
(1982).
38 (1980).
Phys. Fluids 24, 843 (1981).
Kim N.S., Najmabadi F.,
Phys. Fluids 31,
3280
(1989).
15 Belyi V.V., Demaulin W.,
B1,
305
(1989).
Paiva-Veretennikoff I.,
Phys. Fluids
28
16
Belyi V.V., Demaulin W., Paiva-Veretennikoff I.,
Phys. Fluids
B1, 317 (1989).
17 Luciani J.F., Mora P.,
Pellat R., Phys. Fluids 28, 835 (1985).
18 Albritton J.R., Williams E.A., Bernstein I.B., Swartz K.P.,
Phys. Rev. Lett. 57, 1887 (19816).
19 Luciani J.F., Mora P., Virmont J., Phys. Rev. Lett.
51, 1664
(1983).
20 Murtazs G., Mirza M., Qaisar M.S., Phys. Lett. 141, 56 (1989);
144, 164 (1990).
21 Krasheninnikov S.I., Sov. Phys. JETP 67, 2483 (1988).
22 Krasheninnikov S.I., Dvornikova N.A., Smirnov
A.P.,
Contributions to Plasma Physics 30, 67 (1990).
23 Gurevich A.V., JETP 39, 1296 (1980) (in Russian).
24 Lebedev A.N.,
JETP 48,
1392 (1965)
(in Russian).
25 Gurevich A.V., Zhivlyuk Yu.N., JETP 49, 214 (1965)
(in Russian).
26 Connor J.W., Hastie R.J.,
Nucl. Fusion 15 (197) 415.
29
1
.
3
2
5
4
'
6
-4
2 1/6
(f3d ) 1/3
(11/d)1/2
Fig.
1.
2 1/4
(9/7) 1/2
30
1
2
3
4
4
2 1/6
2 1/4
Fig.
2.
09/7) 1/2
31
Figure captures
Fig.1
Characteristic
function
energy
change
at
moderate
Region 1 -
1/7 /.
function,
F(v)
deviation
F(v)
function,
-
4
F(4)
electrons,
F(V)
is
F(V)
symmetric;
5
symmetric;
from
6
3
F(4)
from
from
essential
-
function,
F(4)
)/2
Maxwellian
weak deviation
deviation
symmetric;
distribution
(1+Zer
Maxwellian
strong
is
is
F(4)
from
electron
Zeff:
2 -
is symmetric;
function,
of
deviation
linear
Maxwellian
asymmetric;
zones
F(v)
is
.
Maxwelli
-
"collectivization"
c:f
-
"collectivization"
of
electrons, F(4) is asymmetric.
Fig.2
Characteristic
function
change
Region 1 F(V)
is
Maxwellian
energy
zones
at moderate
Zeff:
linear deviation F(4)
symmetric;
function,
"collectivization"
of
2
-
of
j
electrons,
a
distribution
(1+Zef )/2 2:
from Maxwellian
strong
F(v)
electron
deviation
is
F(V)
i/1/2
function,
F(v)
symmetric;
is
symmetric;
"collectivization" of electrons, F(v) is asymmetric.
from
-
3
4
-