On the problem of negative dissipation in multi-ion component plasmas in numerical calculations at the fundamental ICR
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irk-123456789-785072015-03-30T20:22:07Z On the problem of negative dissipation in multi-ion component plasmas in numerical calculations at the fundamental ICR Pavlov, S.S. Castejon, F. Magnetic confinement 2000 Article On the problem of negative dissipation in multi-ion component plasmas in numerical calculations at the fundamental ICR / S.S. Pavlov, F. Castejon // Вопросы атомной науки и техники. — 2000. — № 6. — С. 73-75. — Бібліогр.: 10 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/78507 533.9 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Magnetic confinement Magnetic confinement |
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Magnetic confinement Magnetic confinement Pavlov, S.S. Castejon, F. On the problem of negative dissipation in multi-ion component plasmas in numerical calculations at the fundamental ICR Вопросы атомной науки и техники |
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Pavlov, S.S. Castejon, F. |
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Pavlov, S.S. Castejon, F. |
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Pavlov, S.S. |
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On the problem of negative dissipation in multi-ion component plasmas in numerical calculations at the fundamental ICR |
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On the problem of negative dissipation in multi-ion component plasmas in numerical calculations at the fundamental ICR |
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On the problem of negative dissipation in multi-ion component plasmas in numerical calculations at the fundamental ICR |
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On the problem of negative dissipation in multi-ion component plasmas in numerical calculations at the fundamental ICR |
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On the problem of negative dissipation in multi-ion component plasmas in numerical calculations at the fundamental ICR |
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on the problem of negative dissipation in multi-ion component plasmas in numerical calculations at the fundamental icr |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2000 |
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Magnetic confinement |
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On the problem of negative dissipation in multi-ion component plasmas in numerical calculations at the fundamental ICR / S.S. Pavlov, F. Castejon // Вопросы атомной науки и техники. — 2000. — № 6. — С. 73-75. — Бібліогр.: 10 назв. — англ. |
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Вопросы атомной науки и техники |
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AT pavlovss ontheproblemofnegativedissipationinmultiioncomponentplasmasinnumericalcalculationsatthefundamentalicr AT castejonf ontheproblemofnegativedissipationinmultiioncomponentplasmasinnumericalcalculationsatthefundamentalicr |
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2025-07-06T02:34:59Z |
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UDC 533.9
Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 73-75 73
ON THE PROBLEM OF NEGATIVE DISSIPATION IN MULTI-ION
COMPONENT PLASMAS IN NUMERICAL CALCULATIONS AT THE
FUNDAMENTAL ICR
S. S. Pavlov* and F. Castejon**
*Institute of Plasma Physics, National Science Center
“Kharkov Institue of Physics and Techology”, 61108 Kharkov, Ukraine,
**Asociacion Euratom-Ciemat para Fusion, Ciemat, 28040 Madrid, Spain
1 Introduction
It is known that finite Larmor radius approximation
up to the second order (FLR2) of Fast Wave (FW)
propagation and absorption may give wrong results in
numerical simulations of multi-component plasmas in
the range of ICR. This is true not only for short
wavelength modes (Slow Wave and ion Bernstain wave)
but also for long wavelength FW. The main unphysical
effect that appears is the negative absorption of high
frequency energy of FW. This fact does not depend on a
simulation character and takes place in both, wave
equation and ray tracing approaches in numerical
calculations. This problem is closely connected with the
reliability of many past and present high frequency
heating calculations, since minority heating, one of the
most promising heating methods, has deal with multi-
component plasmas and uses absorption at the
fundamental ICR for one of the ion sorts. Probably due
to that fact, the problem of negative absorption
intensively discussed in works since 80s’ years [1-10].
In previous works [1-9] several assumptions on the
influence of short wavelength modes description on FW
absorption calculations were made. The finiteness of
k⊥ ρi parameter (k⊥ is transverse wave number, ρi is ion
Larmor radius), for short wavelength modes, provokes a
bad convergence property of the series and makes the
FLR2-approximation fails for them. The main efforts
were devoted to weaken that negative influence
considering that the wave properties are correctly
described for FW due to the smallness of k⊥ ρi parameter
for FW and, consequently, to the good convergence of
expansions for FW.
The works [6,7] are totally devoted to study this
problem and, in particular, they deal with the plasma
conditions for which negative absorption arises. The
possibilities of partially and totally avoiding this
problem are also explored. It was found in these works
that the main parameter that governs the apparition of
negative absorption was the concentration of resonant
ions: the negative absorption appears when the
concentration becomes high enough. Several studies
have been performed in the context of doing calculations
avoiding the problem of negative absorption. Those
studies play with different variants of the “hot” terms of
FLR2-expansion, that are connected with power
absorption at the fundamental ICR frequency. Firstly,
the fully omission of these terms gave the so called
“SCK”-form. This name comes from the names of their
authors: Swanson [8], Colestock and Kashuba [9].
Secondly, the partial omission of them, in which only
the imaginary parts of the terms are omitted. And a third
group exists that performs the phenomenological
modification of these terms, justified by numerical
calculations.
Such simplified treatments are usually based on the
fact that the “hot” terms must be much smaller than
zero-order ones, as a consequence of the smallness of
the parameter k⊥ ρi for FW. Zero-order terms are
themselves resonant at the same frequency and therefore
they are dominant in the series. However such a
justification can be questioned in the case of
quasiperpendicular propagation of FW, when the angle
of propagation goes to π/2, equivalently, longitudinal
refractive index, N║, goes to 0.
It is shown in work [10] that the expansion of FW
dispersion equation in the parameter k⊥ ρi converges in
different ways for different propagation angles with
respect to magnetic field. In particular, in the case of
quasitransverse propagation the convergence of the
expansion is slower than in the case of other angles.
Therefore, more terms are necessary in the case of
calculation of absorption at first harmonic of ICR. This
fact implies the necessity to keep more terms of the
expansion in the dispersion equation for both, ray
tracing and wave equation calculations.
Consequently, the value of the longitudinal refractive
index, N║, is proved to be a key parameter for the
localization of negative absorption in a plasma. It must
be noted that this conclusion is equivalent to the one
presented in the works [6,7] in which the main
parameter is the concentration of resonant particles. This
is because the ion concentration, ni, and the parallel
refractive index, N║, appear in dielectric tensor through
the combination ni/ N║ and any arising of the
concentration, ni, may be replaced by decreasing of
parallel index N║ . Nevertheless, the contrary is not true,
since the upper values of the concentration are limited
by the known restrictions.
In the work [10] it was also shown that for a correct
description of FW in a single-ion component plasma it is
necessary and enough to use the FLR4-expansion in
numerical calculations.
In the present report the results of the work [10]
derived in the FLR4 frame are developed and
generalized for a multi-ion component plasma.
74
2. The reasons of the apparition of negative
absorption
To display the reasons of appearing of negative
absorption it is useful to analyze the structure of whole
FLR∞-expansion of the dispersion relation for FW in the
range of the fundamental ICR for the resonant sort of
ions. The dispersion equation for FW in the case of
neglecting the electron mass has the form:
(ε11-N║
2)N⊥
2-(ε11-N║
2)(ε22-N║
2)-ε12
2=0, (1)
where εik are components of dielectric tensor and
N⊥ =ck⊥ /ω is transverse refractive index (c is the speed
of light, ω is the angular frequency of FW). Let us
expand the components of dielectric tensor for FW in
the small parameter (k⊥ ρα)2 (from now on index α
denotes the resonant sort of ions). Then let us group the
terms proportional to the parameter Z0α=ω/(2k║VTα)
(VTα= (Tα/mα)1/2 is thermal velocity) in the antihermitian
part which is connected with absorption at he first
harmonic, leaving unchanged the hermitian terms. In this
way, the equation (1) can be written as:
[hermitian terms]+Z0α
2[O(k⊥ ρα)6]+Z0α
2[O(k⊥ ρα)4] +
Z0α[O(k⊥ ρα)2]+Z0α[O(1)]=0, (2)
The main peculiarity of this expression is the fact
that the parameter Z0α appears with first power in two
older terms and with second power in all other ones.
Since the longitudinal refractive index N║ =ck║ /ω is
contained in Z0α parameter it is easy to see that the value
N║ =0 is a singular point of the expression (2). When N║
goes to 0 the dominant term will be of order (k⊥ ρα)4.
Really, we may neglect the terms of higher order since
the parameter (k⊥ ρα)2 is small for FW. Thus, from the
expression (2) it follows that it is necessary and enough
to keep in the expression (2) the three older terms for
the exact description of absorption at the first harmonic
of ICR, i.e. to be limited by frame of FLR4-expansion.
Firstly, let us keep in the expansion (2) only two older
terms, i.e. will limit ourself by frame of FLR2-expansion
and try to find out the details of the apparition of
negative absorption. In this case expansion (2) may be
written in the form:
2
⊥N ( )
( )
−
−+−+ − 2
//1
2
//21
12
2
2
02
2
2
11 1
N
NW
C
Zi Cp
ε
εεωρπ
ω
ω
α
α
αα
α =
( )( )
( )2
//1
2
//21
2
//
0
2
0
1
N
NN
−
−+−−
ε
εεεε , (3)
where ωpα, ωcα is plasma and cyclotron frequencies of
resonant ions, W-1α=W(Z-1α) is dispersion function of
argument Z-1α=(ω-ωcα)/ 2 k║Vtα, ε1
0, ε2
0 are the zero
order terms of the expansion of components of
dielectric tensor ε1 and ε2 (ε1=ε11, ε2=iε12) in the
parameter (k⊥ ρα)2. ε1, ε2 contain zero and first order
terms of the same expansion. To understand why the
imaginary part of perpendicular refractive index changes
and even becomes negative in the FLR2-approximation,
it is convenient to expand in the equation (3) the ratio
(ε1+ε2-N║
2)/( ε1-N║
2) up to 2nd order in the parameter
(k⊥ ρα)2:
2
//1
2
//21
N
N
−
−+
ε
εε =(2-x)+i[( ( )ξαα
βα
−wzn
zn
+ ( )12
21 2
+
++
ww
ww )(2+2x)-( ( )22 ξ
ξ
αα
ββ
−wzn
zn
+
( )12
1
+ww
)(4+2x)]
ααπ 10
2
−WZ
w , (4)
where x=(k⊥ ρα)2, W=ω/ωcα, ξ=(Zβ mα)/(Zα mβ) and Zβ,
Zα,mβ, mα,nβ,nα are charges, masses and concentrations
of nonresonant and resonant ions. Now it is easy to see
that an imaginary part of the expression (4),
multiplication times the clearly imaginary term before
square brackets in the equation (3), will be added to the
real part of the transverse refractive index and will not
present singularities. This product will not have
singularities for any value of N║ , since Z0α in the
expression (3) and imaginary part of expansion (4) will
cancel one another. The real part of the expression (4),
which after the same multiplication will add to the
imaginary part of the refractive index, consists of two
addendums: one is of zero order and the second one is
of first order in parameter (k⊥ ρα)2 . The addendum of
zero order transforms the expression in square brackets
in (3) into zero and, therefore, does not change the
imaginary part of N⊥
2. But small second addendum, of
first order, as a consequence of multiplication times the
big parameter Z0α will essentially change the imaginary
part of N⊥
2 when N║ decreases. The sign of this
addition will be positive and therefore the addition will
make the absorption at first harmonic of ICR to
decrease. And even it will make it negative. To get
convinced of that fact it is enough to solve a double
quadratic equation, which is obtained from equation (3),
after substituting the expression (4) into it. It is
necessary to note that in right part of the expression (3),
instead of (ε1+ε2-N║
2 )/( ε1-N║
2 ), it is enough to
substitute (ε1
0+ε2
0-N║
2 )/( ε1
0-N║
2), as a consequence of
the smallness of the parameter (k⊥ ρα)2 for FW. Then
we shall have
4
2
1
⊥N i π
ω
ω α
2
2
p Z0α 24
1
4
4
wC
W α
α
ωρ − +
2
⊥N -( 0
FN ⊥ )2=0, (5)
where (N0
⊥ F)2=(ε1
0-ε2
0-N║
2) (ε1
0+ε2
0-N║
2 )/( ε1
0-N║
2) is
the square of refractive index at the FLR0-
approximation. The roots of this equation describe FW
and ion Bernstein wave in the FLR2-approximation. For
FW we will have
ImN⊥ = 2
1
11 ImRe)1(
α
αα
−
−− −−
Wd
WyWx
ReN⊥ = 2
1
11 ReIm)1(
α
αα
−
−− +−
Wd
WyWx , (6)
where we have defined the quantities d=-2 2
απ AN Z0α
ω4 /C4 , NAα
2= ωpα
2/ωcα
2;
75
X=[ 2
1 ([1+d2 ( ) 2
1
220
α−WN F +2d(Im( 0
FN )2ReW-1α
+Re( 0
FN )2ImW-1α)]1/2+1+
d(Im( 0
FN )2ReW-1α+Re( 0
FN )2ImW-1α))]1/2,
Y=[ 2
1 ([1+d2 ( 0
FN )2 2 W-1α 2+2d(Im( 0
FN )2ReW-1α
+Re( 0
FN )2ImW-1α)]1\2-1-
d(Im( 0
FN )2ReW-1α +Re( 0
FN )2ImW-1α))]1/2
At the resonance ω=ωcα for not very small values of N║
(d<<1) we will have from the expression (6)
ImN⊥
2=Im( 0
FN )2--
2
1 ( 0
FN )2 2 π 2
αAN Z0αω4 4
αρ /C4
ReN⊥
2=Re( 0
FN )2 , (7)
From expressions (6),(7) it follows that in the case of
FLR2-approximation for quasiperpendicular FW
propagation the imaginary part of N⊥ may decrease and
even become negative. Therefore the frame of FLR2-
approximation is not enough for finding an exact value
of refractive index.
From the expression (7) it follows that negative
absorption appears when 1+2nβzβ/nαzα(1+ξ)
< π Z0α(k⊥ ρI)2 (hereξ=zβmα/zαmβ), since Im(NF
0)2=
Re(NF
0)2(1+2nβzβ/nαzα(1+ξ)/(2 πZ0α). Thus, if the
concentration of resonant ions, nα, is more or order of
the concentration of nonresonant ions, nβ, then negative
dissipation appears when Z0α(k⊥ ρI)2>1 or N//<
(k⊥ ρI)2C/VTα. This condition coincides with analogical
one for single-ion plasmas [10].
3.About avoiding negative absorption
From the discussion presented above, it follows that one
of the possibilities to avoid the negative absorption is to
search the solution in the frame of FLR4-approximation.
So, let us keep in expression (2) three older addendums
and expand the components of dielectric tensor ε1 and ε2
up to 2nd order in the parameter (k⊥ ρα)2. In this case the
expansion (2) may be reduced into next equation :
N⊥
4[i π
ω
ω α
2
2
p Z0α
4
αρ
24
4
wC
ω W-1α(-2+
8
5
)(
)(
2
//
0
1
2
//
0
2
0
1
N
N
−
−+
ε
εε + 2
1 )]+ 2
⊥N -( 0
FN ⊥ )2=0, (8)
This equation differs from equation (5) in the coefficient
of N⊥
4. In this one there appear two new terms which
change the sign of the coefficient from plus into minus,
since (ε1
0+ε2
0-N//
2 )/(ε1
0-N//
2 )=2. Now the absorption
will increase in comparison with the absorption in the
case of FLR0-approximation. So in the frame of FLR4-
approximation the problem of negative absorption
disappears due to the fact that the main term in
expression (2), which is proportional to (k⊥ ρα)4, has the
whole line of addendums and, consequently, is correctly
described.
Of course, using of the FLR4-aproximation
essentially complicates the numerical calculations,
especially those performed in 2-D and 3-D geometry,
therefore it is extremely desirable to look for any
acceptable phenomenological solution in the frame of
FLR2-approximation for any angle of FW propagation,
including the quasiperpendicular ones. However, the
strict analysis shows that by means of changing in
dielectric tensor of the "hot" terms, connected with the
absorption at the first harmonic, it is impossible to
obtain the equation (8) with the correct sign at N⊥
4. The
sign always is positive that corresponds to decreasing of
absorption in comparison with FLR0-approximation. It
follows from the conditions corresponding to receiving
of the coefficient at N⊥
2 in equation (8). As to
phenomenological modifications of the same "hot"
terms used in works [1-9], all they are reduced to
receiving the value 2 in the real part of expression (4)
instead (2-x). In this case the coefficient at N⊥
4 in the
equation (8) automatically equals 0 and, consequently
N⊥
2=(N⊥ F
0)2, i.e. it corresponds to formulae received in
the FLR0-approximation, which is not correct in the case
of the quasiperpendicular propagation. Thus, the fact
that FLR4-approximation is necessary for correct
description of FW absorption for any an angle of
propagation may be strictly proved.
References
1. Jaeger, E. F., Batchelor, D. B., and Weitzner, H.,
Nuclear Fusion 28 (1988) 53.
2 Brambilla, N., Plasma Phys. Contr. Fusion 31
(1989) 723.
3. Vaclavik, J. and Appert, K., Plasma Phys. Contr.
Fusion 29 (1987) 257.
4. Smithe, D. N., Plasma Phys. Contr. Fusion 31 (1989)
105 .
5 Smithe, D. N., "Parallel Gradient Effects on ICRH in
Tokamaks", Ph. D. thesis, University of Michigan,
1987; Appendix C.
6. Alava, M. J., Heikkinen, J. A., Hellsten T., J.
Comput. Phys.114 (1994) 85.
7 Alava ,M. J., Heikkinen, J. A., Hellsten T., Pavlov, I.,
and Shcherbinin, O. N., Physica Scripta, Vol. 50 (1994)
275.
8. Swanson, D. G., Phys. Fluids 24 (1981) 2035.
9. Colestock, P. L., Kashuba, R. J., Nuclear Fusion 23
(1983) 763.
10. Pavlov S.S. and Castejon F., 27th EPS Conf. on
Contr. Fusion and Plasma Physics, Budapest, June, 2000
References
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