Relativistic mono-energetic transport coefficients in hot plasmas
Relativistic mono-energetic drift kinetic equation for hot toroidal plasmas is analyzed. For compatibility with non-relativistic description, non-canonical thermodynamic forces with the additional temperature-dependent term in the first thermodynamic force were introduced. The transport coefficients...
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Marushchenko, I.N. Azarenkov, N.A. 2017-01-15T16:58:04Z 2017-01-15T16:58:04Z 2013 Relativistic mono-energetic transport coefficients in hot plasmas / I.N. Marushchenko, N.A. Azarenkov // Вопросы атомной науки и техники. — 2013. — № 4. — С. 112-114. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 52.27.Ny, 52.25.Fi https://nasplib.isofts.kiev.ua/handle/123456789/111939 Relativistic mono-energetic drift kinetic equation for hot toroidal plasmas is analyzed. For compatibility with non-relativistic description, non-canonical thermodynamic forces with the additional temperature-dependent term in the first thermodynamic force were introduced. The transport coefficients were defined as a convolution of monoenergetic transport coefficients with Maxwell-Jüttner distribution function and corresponding weight function. Проаналізовано релятивістське моноенергетичне дрейфове кінетичне рівняння для гарячої тороїдальної плазми. Для сумісності з нерелятивістським формалізмом були введені неканонічні термодинамічні сили, що містять додатковий температурно-залежний член у першій термодинамічній силі. Коефіцієнти переносу отримані у вигляді згортка моноенергетичних коефіцієнтів переносу з функцією розподілу Максвелла- Юттнера, що описує термодинамічну рівновагу в релятивістському газі, та з відповідною ваговою функцією. Проанализировано релятивистское моноэнергетическое дрейфовое кинетическое уравнение для горячей тороидальной плазмы. Для совместимости с нерелятивистским формализмом были введены неканонические термодинамические силы, содержащие дополнительный температурно-зависимый член в первой термодинамической силе. Коэффициенты переноса получены в виде свёртки моноэнергетических коэффициентов переноса с функцией распределения Максвелла-Юттнера, описывающей термодинамическое равновесие в релятивистском газе, и соответствующей весовой функцией. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Термоядерный синтез, коллективные процессы Relativistic mono-energetic transport coefficients in hot plasmas Релятивістські моноенергетичні коефіцієнти переносу в гарячій плазмі Релятивистские моноэнергетические коэффициенты переноса в горячей плазме Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Relativistic mono-energetic transport coefficients in hot plasmas |
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Relativistic mono-energetic transport coefficients in hot plasmas Marushchenko, I.N. Azarenkov, N.A. Термоядерный синтез, коллективные процессы |
| title_short |
Relativistic mono-energetic transport coefficients in hot plasmas |
| title_full |
Relativistic mono-energetic transport coefficients in hot plasmas |
| title_fullStr |
Relativistic mono-energetic transport coefficients in hot plasmas |
| title_full_unstemmed |
Relativistic mono-energetic transport coefficients in hot plasmas |
| title_sort |
relativistic mono-energetic transport coefficients in hot plasmas |
| author |
Marushchenko, I.N. Azarenkov, N.A. |
| author_facet |
Marushchenko, I.N. Azarenkov, N.A. |
| topic |
Термоядерный синтез, коллективные процессы |
| topic_facet |
Термоядерный синтез, коллективные процессы |
| publishDate |
2013 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
| title_alt |
Релятивістські моноенергетичні коефіцієнти переносу в гарячій плазмі Релятивистские моноэнергетические коэффициенты переноса в горячей плазме |
| description |
Relativistic mono-energetic drift kinetic equation for hot toroidal plasmas is analyzed. For compatibility with non-relativistic description, non-canonical thermodynamic forces with the additional temperature-dependent term in the first thermodynamic force were introduced. The transport coefficients were defined as a convolution of monoenergetic transport coefficients with Maxwell-Jüttner distribution function and corresponding weight function.
Проаналізовано релятивістське моноенергетичне дрейфове кінетичне рівняння для гарячої тороїдальної плазми. Для сумісності з нерелятивістським формалізмом були введені неканонічні термодинамічні сили, що містять додатковий температурно-залежний член у першій термодинамічній силі. Коефіцієнти переносу отримані у вигляді згортка моноенергетичних коефіцієнтів переносу з функцією розподілу Максвелла- Юттнера, що описує термодинамічну рівновагу в релятивістському газі, та з відповідною ваговою функцією.
Проанализировано релятивистское моноэнергетическое дрейфовое кинетическое уравнение для горячей тороидальной плазмы. Для совместимости с нерелятивистским формализмом были введены неканонические термодинамические силы, содержащие дополнительный температурно-зависимый член в первой термодинамической силе. Коэффициенты переноса получены в виде свёртки моноэнергетических коэффициентов переноса с функцией распределения Максвелла-Юттнера, описывающей термодинамическое равновесие в релятивистском газе, и соответствующей весовой функцией.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111939 |
| citation_txt |
Relativistic mono-energetic transport coefficients in hot plasmas / I.N. Marushchenko, N.A. Azarenkov // Вопросы атомной науки и техники. — 2013. — № 4. — С. 112-114. — Бібліогр.: 10 назв. — англ. |
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2025-11-25T15:37:51Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2013. №4(86) 112
RELATIVISTIC MONO-ENERGETIC TRANSPORT COEFFICIENTS
IN HOT PLASMAS
I.N. Marushchenko, N.A. Azarenkov
V.N. Karazin Kharkоv National University, Kharkov, Ukraine
E-mail: i.marushchenko@gmail.com
Relativistic mono-energetic drift kinetic equation for hot toroidal plasmas is analyzed. For compatibility with
non-relativistic description, non-canonical thermodynamic forces with the additional temperature-dependent term in
the first thermodynamic force were introduced. The transport coefficients were defined as a convolution of mono-
energetic transport coefficients with Maxwell-Jüttner distribution function and corresponding weight function.
PACS: 52.27.Ny, 52.25.Fi
INTRODUCTION
Non-relativistic neoclassical theory for toroidal de-
vices is commonly considered to be a matured field of
research given the extensive body of scientific literature
dealing with the topic [1, 2]. One of the most convenient
computational methods developed inside the theory for
determining neoclassical contributions to the transport
of plasma observables such as density, temperature and
current is the mono-energetic approach, which leads to
the so-called mono-energetic transport coefficients (see,
for example, [3] and the references therein).
In contrast to the widespread opinion [1, 2] that rela-
tivistic effects in fusion plasmas can be important only
for high-energetic groups of electrons with energies
close to me0c2 or exceeding it (say, runaway electrons),
relativistic effects was found to produce the non-
negligible contribution in collisional transport properties
of plasmas for the temperatures about few tens keV due
to the features of relativistic thermodynamic equilibrium
of electrons [4 - 7]. In order to take these effects into
account, one needs to reformulate the mono-energetic
drift kinetic equation and the definition of transport co-
efficients.
In this paper, we derive the neoclassical mono-
energetic transport coefficients for relativistic electrons
in hot plasmas. Rigorous relativistic description gener-
ally requires a covariant formulation [8]. However,
while Lorentz invariance is not important for neoclassi-
cal transport because the latter is limited to relatively
small characteristic velocities, non-covariant formula-
tion is better suited for the task since our goal is to keep
the form of equations as close as possible to the formu-
lation generally accepted in non-relativistic neoclassical
theory. This approach allows us to calculate the relativ-
istic mono-energetic transport coefficients using already
existed mono-energetic non-relativistic transport codes
like DKES [9]. Apart from this, it may be considered as
the tool for estimation of the applicability range of non-
relativistic neoclassical approach.
In section 2, relativistic mono-energetic drift kinetic
equation is formulated and a set of non-canonical ther-
modynamic forces is introduced. It is shown that for
compatibility with non-relativistic description it is nec-
essary to include the explicit temperature dependence in
the first thermodynamic force. In section 3, the transport
coefficients are derived as a convolution of mono-
energetic coefficients with the relativistic Maxwell-
Jüttner distribution function and corresponding relativis-
tic weight function.
1. MONO-ENERGETIC LINEAR DRIFT
KINETIC EQUATION FOR RELATIVISTIC
ELECTRONS
Similar to the non-relativistic consideration [1 - 3],
in order to obtain the neoclassical mono-energetic trans-
port coefficients for relativistic electrons in hot plasmas
(the ions are taken as non-relativistic) on the given
magnetic surface with label ρ, we start from the lin-
earized relativistic drift kinetic equation (rDKE) for
deviation from the equilibrium, fe1 = fe – fe0, induced by
gradients of thermodynamic quantities. Using the vari-
ables (u,ξ), where u = vγ is the momentum per unit mass
with γ = (1+u2/c2)1/2, ξ = (u·B)/(uB) is the pitch and B is
the vector of the magnetic field, the mono-energetic
rDKE can be written as
( )
1 1
0
0
( ) ( ) ( )
,
− ν =
∂ ξ ⋅
− ⋅∇ρ −
∂ρ γ
E BV
e D e
e
dr e
e
V f u L f
f u e f
T B
(1)
where V = Vdr·∇s + ξ& ∂/∂ξ is the mono-energetic Vlasov
operator,
2
2
(1 )+ ( ) ,
2s
cEu uV B
B B
ρ⎛ ⎞ξ − ξ ∂
= ∇ρ× ⋅∇ − ⋅∇⎜ ⎟γ γ ∂ξ⎝ ⎠
h h B (2)
h = B/B is the magnetic field unit vector, ∇s is a gradi-
ent within the magnetic surface, E = Eφ + Eρ∇ρ is the
electric field separated to the toroidal (inductive) field
Eφ and the radial field, Eρ = -∂Φ/∂ρ with Φ as plasma
potential; L = (1/2)∂/∂ξ((1-ξ2)∂/∂ξ) is the Lorentz opera-
tor which describes the pitch-angle scattering of elec-
trons and νD(u) is the relativistic electron deflection
frequency [10]. The radial component of the relativistic
drift velocity can be represented as
2 2
0
3
(1 )
( ) .
2
+ ξ
ρ ≡ ⋅∇ρ = ×∇ ⋅∇ρ
γ
& V Be
dr
m cu
B
e B
(3)
In order to exclude the local dependencies which do
not contribute to transport, the local equilibrium fe0 can
be represented as follows [9]:
0 2 2
1 ' ,
l
e eMJ
e
ef Bdl f
T B B
⎛ ⎞⎛ ⎞⋅⋅⎜ ⎟⎜ ⎟= + −
⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∫
E BE B (4)
where the angle brackets 〈...〉 denote the flux-surface
average and the relativistic thermodynamic equilibrium
feMJ is given by Maxwell-Jüttner distribution function,
( 1)
3/2 3 ( ) ,re
eMJ MJ r
th
nf C e
u
−μ γ−= μ
π
(5)
ISSN 1562-6016. ВАНТ. 2013. №4(86) 113
with Boltzmann factor, /
0
ee T
e en n e Φ= , included. Here,
uth = pth/me0 is the thermal momentum per unit mass
with pth = (2me0Te)1/2, μr = me0c2/Te and
2
2
151 (1/ ),
2 ( ) 8
r
MJ r
r r r
eC O
K
−μπ
= = − + μ
μ μ μ
(6)
with Kn the modified Bessel function of n-th order.
Then, the right-hand-side of Eq. (1) can be written as
2
ln ln3 ,
2
eMJ
e
e e
eMJ
e
u eRHS Bf
T B
eEn TR f
T
ρ
⋅ξ
= −
γ
⎡ ⎤∂ ∂⎛ ⎞−ρ − + − κ +⎜ ⎟⎢ ⎥∂ρ ∂ρ⎝ ⎠⎣ ⎦
E B
&
(7)
where κ = μr(γ - 1) is the relativistic kinetic energy,
normalized by temperature, and
23
2
5 151 (1/ )
2 8r r
r
KR O
K
⎛ ⎞
= μ − − = + μ⎜ ⎟ μ⎝ ⎠
(8)
is the relativistic correction term which appears due to
the specific feature of Maxwell-Jüttner distribution
function [4 - 7].
Now, let us introduce the relativistic thermodynamic
forces as follows:
1
2
3 2
ln ln3( ) ,
2
ln( ) ,
( ) .
e e
e
e
e
eEn TA R
T
TA
eA
T B
ρ∂ ∂⎛ ⎞ρ = − + +⎜ ⎟∂ρ ∂ρ⎝ ⎠
∂
ρ =
∂ρ
⋅
ρ =
E B
(9)
They are similar to the non-relativistic “canonical”
thermodynamic forces, but there is one very important
difference: in contrast to the “canonical” definition,
where only the dependencies from the local gradients
are present, A1 contains an additional relativistic factor
R and thus represents an explicit function of the electron
temperature Te.
Finally, we obtain the mono-energetic rDKE, which
can be solved by the DKES code [9]:
[ ]1 2 3
+ ( )
( ) ( ) ( ) .
ργ⎛ ⎞ γν
ξ ∇ρ× ⋅∇ − =⎜ ⎟
⎝ ⎠
γ
− ρ ρ + κ ρ − ξ ρ
h h
&
D
s e e
eMJ eMJ
Ec f L f
B u u
A A f BA f
u
(10)
Due to the lack of derivatives of fe1 with respect to u
and ρ in Eq. (10), their values can be treated as parame-
ters that leads to a considerable simplification of the
drift kinetic equation from five phase-space variables to
three (two angles at the magnetic surface and pitch).
Similar to the non-relativistic formulation [3], solution
of Eq. (10) is defined only by parameters γEρ/u and
γνD(u)/u, which are, actually, nothing else as Eρ/v and
νD/v, respectively. This approach is sufficient to cover
the main features of the neoclassical radial transport
when applied for calculations of the transport coeffi-
cients and fluxes.
2. RELATIVISTIC MONO-ENERGETIC
TRANSPORT COEFFICIENTS
Similar to [3], let us look for the solution of Eq. (10)
as
[ ]0
1 1 2 0 3
ˆ ˆ ,= + κ +d
e eMJ e eMJ e
R u
f A A f f R A f g
u
(11)
where R0 is the reference value of the torus major ra-
dius, ud = me0c2/(2eγR0B0) is characteristic of the radial
drift velocity and B0 is the reference value of magnetic
field strength. Then the original drift kinetic equation
splits into a system of two independent dimensionless
differential equations:
0 0
0 0
ˆ ˆ( ) ( ) ( ) ,
ˆ ˆ( ) ( ) ( ) ,
γ γ γ
− ν = − ρ
γ γ
− ν = −ξ
&e D e
d
e D e
R R
V f u L f
u u u
R R
V g u L g b
u u
(12)
with b = B/B0. Here, first equation describes the radial
transport due to radial gradients, contained in A1 and A2,
and second equation describes the parallel transport due
to a parallel electric field, contained in A3 (the factor γ in
the Eq. (12) for êf is kept with the only purpose to
keep the form of equations as similar as possible to the
corresponding non-relativistic equations).
Within neoclassical formalism, the relationships be-
tween the flux-surface-averaged fluxes, Ii, and the ther-
modynamic forces which drive them, Ai, can then be
expressed as
3
1
,
=
= − ∑i e i j j
j
I n L A (13)
where Lij is the matrix of transport coefficients.
As was shown in [5], the relativistic flux-surface-
averaged flow I1, which is related to the radial compo-
nent of the particle flux density, Γe , can be written in
the same form as the non-relativistic one,
3
1 1 .= ⋅∇ρ = ρ∫Γ &e eI d u f (14)
Next, I2, which is the radial component of the energy
flux density, Qe , is equal
3
2 1 .= ⋅∇ρ = κρ∫
Q
&e
e
e
I d u f
T
(15)
And the last, I3, is the parallel component of the
electron current density, Je , is equal
3
3 1
0
.
⋅
= = ξ
γ∫
J Be
e
uI d u b f
eB
(16)
Expressing the fluxes Ii through the thermodynamic
forces, the mono-energetic solutions of Eq. (12) may be
used to determine the transport coefficients by energy
convolution with the local Maxwell-Jüttner distribution
function,
32 1( ) ( ) ,
2i j MJ r i j i jL C d e D h h−κ γ +
= μ κ κ γ κ
π ∫ (17)
where h1 = h3 = 1, h2 = κ and Dij(κ) are the mono-
energetic transport coefficients, defined below. If one
compares the expression for relativistic energy convolu-
tion given by Eq. (17) to the corresponding non-
relativistic formula [3], one may find that an additional
ISSN 1562-6016. ВАНТ. 2013. №4(86) 114
relativistic factor ( 1) / 2γ γ + appears under integral,
along with expected normalization coefficient CMJ(μr),
which arise from Maxwell-Jüttner distribution function
and the use of relativistic kinetic energy, κ = μr(γ - 1),
instead of non-relativistic one, K = me0v 2/2Te.
Finally, the relativistic mono-energetic transport co-
efficients Dij for electrons are defined here as follows:
2 10
11 12 21 22 1
10
13 23 1
10
31 32 1
10
33 1
ˆ ,
2
ˆ ,
2
ˆ ,
2
ˆ .
2
+
−
+
−
+
−
+
−
ρ
= = = = − ξ
ρ
= = − ξ
= = − ξξ
γ
= − ξξ
γ
∫
∫
∫
∫
&
&
d
e
d
d
e
d
d
e
d
e
u R
D D D D d f
u u
u R
D D d g
u
u R
D D d b f
u R
D d bg
(18)
Of these mono-energetic coefficients, D11 is related
for description of the radial transport, D33 of the parallel
transport, D13 is characteristic of the Ware pinch and
D31 of the bootstrap current. Only three of these coeffi-
cients are independent, however, as D13 = −D31 due to
Onsager symmetry.
CONCLUSIONS
Following the standard approach to neoclassical the-
ory, the relativistic mono-energetic drift-kinetic equa-
tion for hot electrons is considered. Due to a specific
features of the Maxwell-Jüttner distribution function,
the relativistic correction term appears in the first ther-
modynamic force. By splitting the mono-energetic
rDKE in two independent equations which correspond
to the different thermodynamic forces, the set of trans-
port coefficients is obtained. Using this scheme, relativ-
istic transport coefficients can be found by re-
interpretation of the solution from the non-relativistic
transport codes. The solution of rDKE for given values
of γEρ/u and γνD(u)/u and velocity γ/u = v, should be
interpreted as the same non-relativistic function of
pitch-angle with different velocity v and parameters
Eρ/v and νD(u)/v, such that these parameters should co-
incide numerically. Then the transport coefficients can
be calculated through the convolution of mono-
energetic transport coefficients with Maxwell-Jüttner
distribution function and specific relativistic weight
factor.
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3. C.D. Beidler et al. Benchmarking of the mono-
energetic transport coefficients // Nuclear Fusion
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p. 1355-1368.
Article received 16.05.2013
РЕЛЯТИВИСТСКИЕ МОНОЭНЕРГЕТИЧЕСКИЕ КОЭФФИЦИЕНТЫ ПЕРЕНОСА
В ГОРЯЧЕЙ ПЛАЗМЕ
И.Н. Марущенко, Н.А. Азаренков
Проанализировано релятивистское моноэнергетическое дрейфовое кинетическое уравнение для горячей
тороидальной плазмы. Для совместимости с нерелятивистским формализмом были введены неканонические
термодинамические силы, содержащие дополнительный температурно-зависимый член в первой термоди-
намической силе. Коэффициенты переноса получены в виде свёртки моноэнергетических коэффициентов
переноса с функцией распределения Максвелла-Юттнера, описывающей термодинамическое равновесие в
релятивистском газе, и соответствующей весовой функцией.
РЕЛЯТИВІСТСЬКІ МОНОЕНЕРГЕТИЧНІ КОЕФІЦІЄНТИ ПЕРЕНОСУ В ГАРЯЧІЙ ПЛАЗМІ
І.М. Марущенко, М.О. Азарєнков
Проаналізовано релятивістське моноенергетичне дрейфове кінетичне рівняння для гарячої тороїдальної
плазми. Для сумісності з нерелятивістським формалізмом були введені неканонічні термодинамічні сили,
що містять додатковий температурно-залежний член у першій термодинамічній силі. Коефіцієнти переносу
отримані у вигляді згортка моноенергетичних коефіцієнтів переносу з функцією розподілу Максвелла-
Юттнера, що описує термодинамічну рівновагу в релятивістському газі, та з відповідною ваговою функцією.
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