Relativistic effects in electron neoclassical transport
The parallel momentum correction technique is generalized for relativistic approach. It is required for proper calculation of the parallel neoclassical flows and, in particular, for the bootstrap current at fusion temperatures. It is shown that the obtained system of linear algebraic equations for p...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Relativistic effects in electron neoclassical transport / I. Marushchenko, N.A. Azarenkov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 37-40. — Бібліогр.: 15 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860132683503370240 |
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| author | Marushchenko, I. Azarenkov, N.A. |
| author_facet | Marushchenko, I. Azarenkov, N.A. |
| citation_txt | Relativistic effects in electron neoclassical transport / I. Marushchenko, N.A. Azarenkov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 37-40. — Бібліогр.: 15 назв. — англ. |
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| description | The parallel momentum correction technique is generalized for relativistic approach. It is required for proper calculation of the parallel neoclassical flows and, in particular, for the bootstrap current at fusion temperatures. It is shown that the obtained system of linear algebraic equations for parallel fluxes can be solved directly without calculation of the distribution function if the relativistic monoenergetic transport coefficients are already known, while the latter can be calculated by any non-relativistic solver.
Предложено обобщение метода сохранения параллельных импульсов на случай релятивистского приближения. Это необходимо для корректного вычисления параллельных неоклассических потоков и, в частности, бутстреп-тока при температурах, которые требуются для осуществления термоядерного синтеза. Показано, что полученная система линейных алгебраических уравнений может быть решена непосредственно, без вычисления функции распределения, если известны релятивистские моноэнергетические коэффициенты, причем последние могут быть вычислены любым нерелятивистским солвером путём переобозначения величин, входящих в дрейфово-кинетическое уравнение.
Запропоновано узагальнення методу збереження паралельних імпульсів на випадок релятивістського наближення. Це необхідно для коректного обчислення паралельних неокласичних потоків і, зокрема, бутстреп-струму за температур, які потрібні для здійснення термоядерного синтезу. Показано, що здобуту систему лінійних алгебраїчних рівнянь можна розв’язати безпосередньо, без обчислення функції розподілу, якщо відомі релятивістські моноенергетичні коефіцієнти, причому останні можуть бути обчислені будь-яким нерелятивістським солвером шляхом перепозначення величин, що входять до дрейфово-кінетичного рівняння.
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| first_indexed | 2025-12-07T17:45:07Z |
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ISSN 1562-6016. ВАНТ. 2015. №1(95)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2015, № 1. Series: Plasma Physics (21), p. 37-40. 37
RELATIVISTIC EFFECTS IN ELECTRON
NEOCLASSICAL TRANSPORT
I. Marushchenko, N.A. Azarenkov
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
The parallel momentum correction technique is generalized for relativistic approach. It is required for proper
calculation of the parallel neoclassical flows and, in particular, for the bootstrap current at fusion temperatures. It is
shown that the obtained system of linear algebraic equations for parallel fluxes can be solved directly without
calculation of the distribution function if the relativistic monoenergetic transport coefficients are already known,
while the latter can be calculated by any non-relativistic solver.
PACS: 52.55.Dy, 52.25.Fi, 52.27.Ny
INTRODUCTION
Since the fusion reactor scenarios require high
temperatures, the role of relativistic effects for electrons
in toroidal devices also becomes an actual question. In
particular, the relativistic effects in the transport
processes need to be examined for the next generation
of such devices as ITER and DEMO [1, 2], where the
expected electron temperatures are sufficiently high,
> 25 keV. Apart from this, the aneutronic fusion
reactors with , and reactions
require actually relativistic temperatures of up to
~ 100 keV [3, 4]. It was already shown [5, 6] that
relativistic effects in electron transport can be non-
negligible in the range of electron temperatures typical
for fusion. In the previous publications [6, 11, 12] the
relativistic approach has been developed mainly for
radial neoclassical transport. However, the account of
relativistic effects is much more important for parallel
neoclassical fluxes since there is no anomalous transport
along the magnetic surfaces and neoclassical fluxes are
in good agreement with experiment.
The standard technique for solving drift kinetic
equation (DKE) includes the expansion of the
distribution function in Sonine (associated Laguerre)
polynomials [7-10]. In relativistic approach,
due to the additional relativistic term in right-hand-side
(RHS) of DKE, the standard technique with Sonine
polynomials becomes inefficient, and the
generalization of this technique for arbitrary
temperatures is proposed in this paper.
1. RELATIVISTIC DRIFT KINETIC
EQUATION
Generally, relativistic consideration at fusion
temperatures only for electrons is required. However,
for convenience, below we consider all species and
distinguish between electrons and ions only when it is
necessary.
It was shown before [6, 11, 12] that the relativistic
DKE (rDKE) for neoclassical electron transport can be
written as:
(1)
where is the disturbance of the electron distribution
function driven by the thermodynamic forces , is
the relativistic Vlasov operator,
is the
relativistic linearized Coulomb operator [13] (here,
and if ), = Vdr where is
the flux-surface label, is the
normalized kinetic energy of the electrons with
, with and
is the magnetic field normalized by its
reference value. The thermodynamic forces in RHS of
Eq. (1) are defined as
(2)
where is the thermodynamic pressure, is
the inductive electric field, and prime means .
Note that in contrast to definitions used in [6, 11, 12],
the term is not included in the thermodynamic force.
Instead, in Eq. (1) only the standard definition is
applied.
The additional term in the RHS of Eq. (1) with
(3)
appears due to specific features of , and is
the modified Bessel function of the second kind of n-th
order.
The electron equilibrium is given by the Juttner-
Maxwellian distribution function [13, 14]
(4)
where is the Lorentz factor and
is the thermal momentum per unit mass
with . The Maxwellian is normalized
by density, , and
(5)
The ion distribution function, , is considered as non-
relativistic Maxwellian one.
2. MOMENT EQUATIONS FOR PARALLEL
FLUXES CALCULATION
In the moment-equation method for the parallel
neoclassical fluxes calculation [8-10], the non-
relativistic distribution function was expanded by
40 ISSN 1562-6016. ВАНТ. 2015. №1(95)
Sonine polynomials , and its zeroth and first
moments gave the parallel fluxes of particles, , and
heat, , respectively. However, the relation between
fluxes of the heat, the energy and particles in the
relativistic approach differs from the classical one [7] by
the additional relativistic term [6, 11], and
(6)
where and
are the fluxes of
particles and energy, respectively. Since the expression
for the relativistic heat flux was derived in [6, 11] only
phenomenologically, the rigorous derivation of Eq. (6)
is given in the Appendix.
From Eq. (1) with one can easily recognize
that, in contrast to the non-relativistic approach
and ), the expansion of RHS with the Sonine
polynomials cannot be represented by only
zeroth and first terms. However, if the generalized
Laguerre polynomials with are
chosen, the standard procedure can be applied. Here,
, , etc. [15]. The use of
is perfectly applicable also for the
representation of the heat flux, given by Eq. (6).
Following [10], let us introduce the parallel-flow-
velocity moments,
(7)
Then the moment with can be identified with the
parallel flow velocity or flux of particles,
, while the moment with is
related to the parallel heat flux,
.
Representing fa as the Legendre polynomials series
and taking into account that only the first Legendre
harmonic contributes in parallel fluxes, one can replace
fa with , where and
. Solution of Eq. (1) can be sought
as with the series
(8)
where is the weight function. Direct substitution
of Eq. (8) into Eq. (7) gives
(9)
( when ) and
(10)
Here it is taken into account that
(11)
Find that in the case , i.e. with and ,
the expressions (8-10) perfectly coincide with the non-
relativistic formulas given in [10].
For calculation of neoclassical parallel fluxes,
collision operator with parallel momentum conservation
can be approximated as [8]
(12)
and can also be sought as series
(13)
with the parallel collisional friction forces
. (14)
Taking Eq. (8) into account, one can see that the
integrals in Eq. (13) are well defined and can be
calculated directly through the parallel fluxes as
[8 – 10]. The coefficients can be
easily calculated and they are not shown here due to its
cumbersomeness.
Now, following [8], let us introduce the adjoint
monoenergetic rDKE,
, (15)
with the solution which gives the mono-energetic
transport coefficients. Here, is the
Lorentz operator, is the deflection
frequency and is the characteristic spatial scale.
Then, applying the operation to
Eq. (15), where and is the averaging
over the magnetic flux surface, and using then the
adjoint properties of the Vlasov and collision operators,
one can obtain the following:
.
(16)
Here, the Onsager symmetry, , is used,
and are the relativistic mono-energetic coefficients
[12]. The operation of the energy convolution with the
relativistic Maxwellian is defined as
(17)
One can see that this system of algebraic equations
with respect to the parallel fluxes can be easily solved if
the mono-energetic transport coefficients are already
calculated. If the series in Eq. (8) and Eq. (13) are cut
off with only first two terms taken into account (i.e.
40
ISSN 1562-6016. ВАНТ. 2015. №1(95) 39
with ), then the heat and particles fluxes are
directly defined.
SUMMARY
In this paper, the moment-equation technique,
previously developed for non-relativistic plasmas
[8-10], is generalized to use in the relativistic approach.
It opens a possibility to apply the neoclassical transport
theory at arbitrary temperatures. It is shown that the
obtained system of linear algebraic equations for
parallel fluxes, and
, can be solved directly without calculation
of the distribution function if the monoenergetic
transport coefficients are already known. Note, that the
relativistic monoenergetic transport coefficients can be
calculated by any non-relativistic solver [12].
APPENDIX: DERIVATION OF THE
RELATIVISTIC HEAT FLUX IN THE
COVARIANT FORMULATION
It is shown that the definition of the relativistic heat
flux introduced in Eq. (6) and [6, 11] conforms well
with the definition accepted in the relativistic kinetics,
based on the covariant formalism [14].
In the relativistic covariant formalism, all values
required for transport equations (in particular, the
energy, as well as fluxes of particles, the energy and the
heat) are usually defined with the help of the four-vector
of the particle flow,
(A1)
and the four-tensor of the energy-momentum,
, (A2)
where and (we
do not consider the moments of higher order). Note that
naturally contains the rest energy and its flow,
while the transport equations do not require these
values. Since , the spatial flow of particles is
with , while the time-like flow is
related to the density as . The internal energy
enclosed in the electron distribution function can be
expressed from the energy-momentum tensor as
, which for the relativistic
Maxwellian is
. (A3)
Now, one can easily prove that with
given by Eq. (3).
In a similar way, the spatial energy flux, , and the
heat flux, , can be calculated [14],
, (A4a)
, (A4b)
where is the total energy flow and
is the enthalpy of electrons, that for the relativistic
Maxwellian is [15]. The heat flux,
Eq. (A4b), can be written as
(A5)
and, finally, using the definition of given by Eq. (3),
one can easily obtain Eq. (6).
REFERENCES
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4. P.E. Stott. The feasibility of using D-
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5. I. Marushchenko, N.A. Azarenkov, and
N.B. Marushchenko. On stability of collisional coupling
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N.B. Marushchenko. Relativistic neoclassical radial
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7. P. Helander and D.J. Siegmar. Collisional Transport
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9. H. Sugama and S. Nishimura. Moment-equation
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10. H. Maaßberg, C.D. Beidler, and Y. Turkin.
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11. I. Marushchenko, N.A. Azarenkov, and
N.B. Marushchenko. Relativistic neoclassical fluxes in
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Technology. Series “Plasma Physics”. 2013, v. 1, p. 67.
12. I. Marushchenko and N.A. Azarenkov. Relativistic
mono-energetic transport coefficients in hot plasmas //
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Article received 22.12.2014
40 ISSN 1562-6016. ВАНТ. 2015. №1(95)
РЕЛЯТИВИСТСКИЕ ЭФФЕКТЫ В НЕОКЛАССИЧЕСКОМ ПЕРЕНОСЕ ЭЛЕКТРОНОВ
И. Марущенко, Н.А. Азаренков
Предложено обобщение метода сохранения параллельных импульсов на случай релятивистского
приближения. Это необходимо для корректного вычисления параллельных неоклассических потоков и, в
частности, бутстреп-тока при температурах, которые требуются для осуществления термоядерного синтеза.
Показано, что полученная система линейных алгебраических уравнений может быть решена
непосредственно, без вычисления функции распределения, если известны релятивистские
моноэнергетические коэффициенты, причем последние могут быть вычислены любым нерелятивистским
солвером путём переобозначения величин, входящих в дрейфово-кинетическое уравнение.
РЕЛЯТИВІСТСЬКІ ЕФЕКТИ В НЕОКЛАСИЧНОМУ ПЕРЕНЕСЕННІ ЕЛЕКТРОНІВ
І. Марущенко, М.О. Азарєнков
Запропоновано узагальнення методу збереження паралельних імпульсів на випадок релятивістського
наближення. Це необхідно для коректного обчислення паралельних неокласичних потоків і, зокрема,
бутстреп-струму за температур, які потрібні для здійснення термоядерного синтезу. Показано, що здобуту
систему лінійних алгебраїчних рівнянь можна розв’язати безпосередньо, без обчислення функції розподілу,
якщо відомі релятивістські моноенергетичні коефіцієнти, причому останні можуть бути обчислені будь-
яким нерелятивістським солвером шляхом перепозначення величин, що входять до дрейфово-кінетичного
рівняння.
|
| id | nasplib_isofts_kiev_ua-123456789-82106 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:45:07Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Marushchenko, I. Azarenkov, N.A. 2015-05-25T09:30:13Z 2015-05-25T09:30:13Z 2015 Relativistic effects in electron neoclassical transport / I. Marushchenko, N.A. Azarenkov // Вопросы атомной науки и техники. — 2015. — № 1. — С. 37-40. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 52.55.Dy, 52.25.Fi, 52.27.Ny https://nasplib.isofts.kiev.ua/handle/123456789/82106 The parallel momentum correction technique is generalized for relativistic approach. It is required for proper calculation of the parallel neoclassical flows and, in particular, for the bootstrap current at fusion temperatures. It is shown that the obtained system of linear algebraic equations for parallel fluxes can be solved directly without calculation of the distribution function if the relativistic monoenergetic transport coefficients are already known, while the latter can be calculated by any non-relativistic solver. Предложено обобщение метода сохранения параллельных импульсов на случай релятивистского приближения. Это необходимо для корректного вычисления параллельных неоклассических потоков и, в частности, бутстреп-тока при температурах, которые требуются для осуществления термоядерного синтеза. Показано, что полученная система линейных алгебраических уравнений может быть решена непосредственно, без вычисления функции распределения, если известны релятивистские моноэнергетические коэффициенты, причем последние могут быть вычислены любым нерелятивистским солвером путём переобозначения величин, входящих в дрейфово-кинетическое уравнение. Запропоновано узагальнення методу збереження паралельних імпульсів на випадок релятивістського наближення. Це необхідно для коректного обчислення паралельних неокласичних потоків і, зокрема, бутстреп-струму за температур, які потрібні для здійснення термоядерного синтезу. Показано, що здобуту систему лінійних алгебраїчних рівнянь можна розв’язати безпосередньо, без обчислення функції розподілу, якщо відомі релятивістські моноенергетичні коефіцієнти, причому останні можуть бути обчислені будь-яким нерелятивістським солвером шляхом перепозначення величин, що входять до дрейфово-кінетичного рівняння. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Магнитное удержание Relativistic effects in electron neoclassical transport Релятивистские эффекты в неоклассическом переносе электронов Релятивістські ефекти в неокласичному перенесенні електронів Article published earlier |
| spellingShingle | Relativistic effects in electron neoclassical transport Marushchenko, I. Azarenkov, N.A. Магнитное удержание |
| title | Relativistic effects in electron neoclassical transport |
| title_alt | Релятивистские эффекты в неоклассическом переносе электронов Релятивістські ефекти в неокласичному перенесенні електронів |
| title_full | Relativistic effects in electron neoclassical transport |
| title_fullStr | Relativistic effects in electron neoclassical transport |
| title_full_unstemmed | Relativistic effects in electron neoclassical transport |
| title_short | Relativistic effects in electron neoclassical transport |
| title_sort | relativistic effects in electron neoclassical transport |
| topic | Магнитное удержание |
| topic_facet | Магнитное удержание |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82106 |
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