On relativistic effects on electron transport in the banana regime in tokamaks
In the present work, the neoclassical transport theory in tokamaks is re-considered with the relativistic effects for electrons taken into account. Since such effects are important only in high-temperature plasmas, only the low collisional banana regime has been considered. The obtained formulations...
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irk-123456789-1945852023-11-27T17:22:59Z On relativistic effects on electron transport in the banana regime in tokamaks Marushchenko, I. Azarenkov, N.A. ITER and fusion reactor aspects In the present work, the neoclassical transport theory in tokamaks is re-considered with the relativistic effects for electrons taken into account. Since such effects are important only in high-temperature plasmas, only the low collisional banana regime has been considered. The obtained formulations give a possibility to calculate the electron neoclassical fluxes in very broad range of temperatures. Hеокласична теорія транспорту в токамаках розглядається з урахуванням релятивістських ефектів для електронів. Оскільки такі ефекти важливі лише у високотемпературній плазмі, розглянуто лише слабозіткневий банановий режим. Отримані формулювання дають можливість обчислювати електронні неокласичні потоки в дуже широкому діапазоні температур. Hеоклассическая теория переноса в токамаке пересматривается с учетом релятивистских эффектов для электронов. Поскольку такие эффекты важны только в высокотемпературной плазме, был рассмотрен только слабостолкновительный банановый режим. Полученные формулировки дают возможность рассчитать электронные неоклассические потоки в очень широком диапазоне температур. 2019 Article On relativistic effects on electron transport in the banana regime in tokamaks / I. Marushchenko, N.A. Azarenkov // Problems of atomic science and technology. — 2019. — № 1. — С. 45-48. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 52.55.Dy, 52.25.Fi, 52.27.Ny http://dspace.nbuv.gov.ua/handle/123456789/194585 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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ITER and fusion reactor aspects ITER and fusion reactor aspects |
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ITER and fusion reactor aspects ITER and fusion reactor aspects Marushchenko, I. Azarenkov, N.A. On relativistic effects on electron transport in the banana regime in tokamaks Вопросы атомной науки и техники |
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In the present work, the neoclassical transport theory in tokamaks is re-considered with the relativistic effects for electrons taken into account. Since such effects are important only in high-temperature plasmas, only the low collisional banana regime has been considered. The obtained formulations give a possibility to calculate the electron neoclassical fluxes in very broad range of temperatures. |
| format |
Article |
| author |
Marushchenko, I. Azarenkov, N.A. |
| author_facet |
Marushchenko, I. Azarenkov, N.A. |
| author_sort |
Marushchenko, I. |
| title |
On relativistic effects on electron transport in the banana regime in tokamaks |
| title_short |
On relativistic effects on electron transport in the banana regime in tokamaks |
| title_full |
On relativistic effects on electron transport in the banana regime in tokamaks |
| title_fullStr |
On relativistic effects on electron transport in the banana regime in tokamaks |
| title_full_unstemmed |
On relativistic effects on electron transport in the banana regime in tokamaks |
| title_sort |
on relativistic effects on electron transport in the banana regime in tokamaks |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2019 |
| topic_facet |
ITER and fusion reactor aspects |
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http://dspace.nbuv.gov.ua/handle/123456789/194585 |
| citation_txt |
On relativistic effects on electron transport in the banana regime in tokamaks / I. Marushchenko, N.A. Azarenkov // Problems of atomic science and technology. — 2019. — № 1. — С. 45-48. — Бібліогр.: 11 назв. — англ. |
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Вопросы атомной науки и техники |
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AT marushchenkoi onrelativisticeffectsonelectrontransportinthebananaregimeintokamaks AT azarenkovna onrelativisticeffectsonelectrontransportinthebananaregimeintokamaks |
| first_indexed |
2025-07-16T21:57:55Z |
| last_indexed |
2025-07-16T21:57:55Z |
| _version_ |
1837842383983607808 |
| fulltext |
ISSN 1562-6016. ВАНТ. 2019. №1(119)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2019, № 1. Series: Plasma Physics (25), p. 45-48. 45
ON RELATIVISTIC EFFECTS ON ELECTRON TRANSPORT
IN THE BANANA REGIME IN TOKAMAKS
I. Marushchenko, N.A. Azarenkov
V.N. Karazin Kharkiv National University, Kharkiv, Ukraine
In the present work, the neoclassical transport theory in tokamaks is re-considered with the relativistic effects for
electrons taken into account. Since such effects are important only in high-temperature plasmas, only the low
collisional banana regime has been considered. The obtained formulations give a possibility to calculate the electron
neoclassical fluxes in very broad range of temperatures.
PACS: 52.55.Dy, 52.25.Fi, 52.27.Ny
INTRODUCTION
A general trend for fusion researches is already
oriented to development of the commercial reactor. The
focus of researches is set on physics of the processes
related to burning plasmas, where the self-heating by
fusion reactions is balanced predominantly by transport
processes with minimized losses. Quantitative
requirements for steady state in burning plasmas are
known as Lawson criterion [1], ,
and, as a more modern criterion, the triple product [2],
(here, is
plasmas temperature, is the density, is
the energy confinement time with as total plasma
energy confined and as total input
(heating) power balanced by losses from plasmas,
; numbers are given for the D-T plasma).
Actually, these criteria can be reached already in ITER
(under construction in Cadarache, France) with plasma
density about and temperatures about
25…30 keV. At the same time, the fusion reactors based
on aneutronic schemes seem more attractive for future
due to minimization of the neutron flux which produces
a destructive influence on the reactor itself. Being based
on other reactions (the best one seems to be a hot D-
3
He
plasmas, where D-D parasitic branch can be minimized
by relative concentration), this kind of reactors requires,
however, a higher plasma temperatures, in the range of
50…70 keV.
As follows from a simplest estimations, relativistic
effects become important for electrons practically in all
kinds of fusion plasmas. These effects have been
studied in details for radiactive losses from hot plasmas
as well for plasma heating by high-frequency waves [3].
And while the neoclassical transport theory is well
established [4], the theory of relativistic transport
processes for electrons in hot plasmas is still far from
completeness. In particular, only several aspects were
investigated, such as specific feature of electron-ion
energy interchange [5] and influence of relativistic
effects on neoclassical radial fluxes in 1/-regime for
stellarators [6].
In the present work, we continue the same line as in
our previous papers [6-8] devoted to stellarators and re-
consider now the neoclassical transport theory for
electrons in hot tokamak plasmas. Only the low-
collisional banana regime is considered.
RELATIVISTIC DRIFT KINETIC EQUATION
As follows from the formulations given in [9, 10],
the relativistic drift equations of motion in weakly
inhomogeneous magnetic field , where an adiabatic
invariance of
(with ) is
satisfactory, can be represented as
̇
(
)
̇
Here all the values correspond to the gyrocenter, is
the drift trajectory, is the parallel momentum
per unit mass with √ and
| | is the electron gyrofrequency. Only the
longitudinal electric field, , is accounted
while the radial electric field leads to a plasma rotation
in poloidal direction without any contribution to
transport. The terms proportional to , which are
related to the poloidal drift perpendicular to the
magnetic field line, do not contribute in radial and
longitudinal fluxes and are omitted from further
consideration. It can be shown also that the second term
in ̇ (so-called the mirror-force in (2)) does not
perform any total work (no contribution on current) and
can be omitted as well.
Magnetic field in tokamaks, i.e. in toroidal traps
with an axial symmetry, is usually represented [4] as
Here are local cylindrical coordinates in torus,
is the poloidal magnetic flux that corresponds to the
given magnetic surface, , and is
the flux-function.
Generally, the linearized relativistic drift kinetic
equation (rDKE) for electrons can be written as
following:
̇
where is the local deviation from
thermal equilibrium induced by and neoclassical
effects; the thermal equilibrium of relativistic electrons
is given by the Maxwell-Jüttner distribution function,
46 ISSN 1562-6016. ВАНТ. 2019. №1(119)
√
with √ and
. The
collisions are described by the linearized Coulomb
operator, where is taken as
linearized one and handles only the pitch-scattering
(Lorentz term). The value , given by the second term
in (1), describes the radial drift and is responsible for
the radial flux in (4).
Using the definitions introduced above, rDKE can be
represented in the following form:
(
)
This equation is quite general and can be applied for any
collisional regime in axisymmetrical traps (tokamaks).
Below, we consider only the high temperature case,
when collisionality is low. For tokamaks, this case is
traditionally called the "banana" regime.
RELATIVISTIC ELECTRON TRANSPORT IN
BANANA REGIME
In the banana regime, the effective collision
frequency for electrons is much lower than the bounce-
frequency. The collisions and a presence of gradients
(thermodynamic forces) lead to a generation of a
diffusive fluxes in both radial and longitudinal
directions (the latter is called the bootstrap current).
While the radial fluxes in tokamaks are intrinsically
ambipolar, the electron component of the bootstrap
current can be quite different from the ion component.
Apart from this, due to a permanent existence in
tokamaks of the inductive electric field, , this current
is also needs to be estimated correctly, i.e. with account
of the relativistic effects.
Two driving terms are present in (7): the term
, which describes the radial forces due to the
gradients, and the term associated with induced electric
field . The latter can be accounted by the Spitzer
function , defined by the equation
Regarding now the Spitzer function as a known
function, rDKE (7) can be represented now as:
where
[
(
)
]
with and
Note that all definitions obtained so far a very similar to
non-relativistic ones, apart from instead of and the
additional term in (12); see also [6-8].
The electric field no longer appears explicitly,
while direct calculation of relativistic Spitzer function
can be regarded as the separated problem.
Following the logic used in [4], let's further expand
the distribution function in a smallness of the
parameter with as a collisional frequency and
as a bounce frequency:
. Then rDKE appears as a pair of equations:
(
)
(
)
The first equation in this pair, (14), can be solved by
ansatz
, where is the constant at the
given magnetic surface labeled by , i.e. and
with as the electron energy,
magnetic momentum and the sign of parallel velocity,
respectively.
Performing an averaging over the tokamak magnetic
surfaces,
〈 〉 ∫
∫
the contributions from trapped electrons are annihilated.
As result, the equation for which contains only the
contribution from passing electrons is obtained:
〈
〉
In tokamaks, where the toroidal rotation of both
electrons and ions due to the collisional drag of plasma
components can be a significant factor and cannot be
ignored in transport physics, an accuracy of the models
related to the parallel momentum conservation is one of
the most important points. Since the linearized collision
operator for electrons contains both the "field" part and
the integral part, there is no way to solve (17)
analytically without some sort of simplification.
Fortunately, it can be simplified without significant loss
of the physical content if one describes the
collisions using a "drifting Maxwellian" model which
conserves both the number of particles and the
longitudinal momentum:
[
]
Here is the parallel drift velocity for the test
particles and
is the deflection
frequency due to pitch scattering on the field particles
; for definition of
see [11].
ISSN 1562-6016. ВАНТ. 2019. №1(119) 47
Applying (15) for and collisions, the model
operator for electrons in this approach can be written s
following:
(
)
(
)
with
for deflections; and are the parallel drift
velocities of electrons and ions respectively. Here
is the Lorentz operator with as a pitch. In the
following, however, it is more convenient to use instead
of the normalized magnetic moment
with , where is the maximum of B at
the given magnetic surface.
Since and hence , our kinetic
problem (16) can now be rewritten as following:
〈 〉
〈
〉 〈 ̂ 〉
with ̂ . Note that in steady state tokamak
plasmas the toroidal rotation velocities of electrons and
ions are usually approximately equal to each other,
.
The kinetic problem is now simplified to an ordinary
differential equation due to small Larmor radius and the
transport ordering. Since vanishes for the trapped
particles, we need to consider only the passing particles
with (this condition follows directly from the
definition of ). Hence, , where
is the Heaviside step function. In the passing
region , while in the trapped region,
, . For convenience, the
value is introduced,
∫
〈
〉
In a large-aspect-ratio torus, , the magnetic
field strenght is almost constant, and the trapped-
passing boundary is located at with
. In this case the quantity is
approximately equal to the parallel momentum of
electron, , in most of velocity space, i.e. if
, then .
Combining all pieces together, we thus obtain the
electron distribution function: Combining all pieces
together, we thus obtain the electron distribution
function,
(
)
〈
̂ 〉
which is required for calculation of the neoclassical
fluxes in tokamaks with relativistic effects accounted.
RADIAL NEOCLASSICAL FLUXES OF
RELATIVISTIC ELECTRON IN BANANA
REGIME
For calculation of the neoclassical fluxes, we apply
the standard definition for both radial and longitudinal
fluxes. And most important from physics are the radial
heat flux, 〈 〉 and longitudinal electron flux, i.e.
the electron component of the bootstrap current,
〈 〉. Important is that, contrary to
stellarators (e.g. see [6,7]), the radial and longitudinal
fluxes in tokamaks are coupled.
Generally, the neoclassical relativistic fluxes have to
be defined from kinetics with linear plasma response
induced by thermodynamic forces, i.e. the radial
gradients and the inductive longitudinal electric field.
All necessary information has been already obtained
and the particle flux can be calculated as:
〈 〉 〈
∫
〉
Similarly, the radial energy flux is:
〈 〉 〈
∫
〉
where is the normalized kinetic energy
of electron (equal to
in a non-relativistic limit).
Now, extracting the mechanical and advective
contributions from the energy flux, the heat flux can be
found as
, where
∫
(
)
is the energy density related to the Maxwell-Jüttner
thermal equilibrium, and
is the radial flow
velocity. Finally, the relativistic heat flux is [6,7]:
〈 〉 〈 〉 (
) 〈 〉
The particle and heat neoclassical radial fluxes can be
calculated from (26) and (28) using relativistic
collisional operator. However, obtained results are
somewhat cumbersome, so it is more illustrative to look
at the limit of large Z, which corresponds to pitch-angle
scattering of electrons (Lorentz term in collisional
operator) and
. In this particular limit the
fluxes can be calculated and then expanded by the order
of
. The first term in such expansion is the non-
relativistic limit ( ), and the second term is the first-
order relativistic correction ( ).
For the radial particle flux we obtain:
〈 〉 〈
〉
〈
〉
48 ISSN 1562-6016. ВАНТ. 2019. №1(119)
〈
〉
[(
)
]
and for the radial heat flux we obtain:
〈 〉 〈
〉
〈
〉
〈
〉
[
(
)
]
Here is the "effective fraction of trapped particles",
√ , and respective non-relativistic radial
fluxes can be found elsewhere (for example, see [4]).
SUMMARY
In the present work, relativistic effects in the
electron neoclassical transport in tokamaks were
considered. Only the low collisional "banana" regime
was investigated. Starting from the general form of
guiding center equations and the drift kinetic equations
with the relativistic corrections taken into account, the
standard method for calculation of particle and heat
radial fluxes was revisited. As the main result, the
relativistic corrections for the radial neoclassical
electron fluxes were obtained. The Lorentz invariance is
lost, however the advantage of the approach used is a
possibility to include the obtained formulations in any
standard transport code based on non-relativistic
transport theory.
REFERENCES
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4. P. Helander and D.J. Sigmar. Collisional Transport in
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5. I. Marushchenko, N.A. Azarenkov, N.B. Maru-
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6. I. Marushchenko, N.A. Azarenkov, and
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8. I. Marushchenko and N.A. Azarenkov. Relativistic
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9. H.K. Wimmel. Lagrangian Formulation of a
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Article received 17.12.2018
О РЕЛЯТИВИСТСКИХ ВЛИЯНИЯХ НА ЭЛЕКТРОННЫЙ ТРАНСПОРТ В БАНАНОВОМ
РЕЖИМЕ В ТОКАМАКАХ
И. Марущенко, Н.А. Азаренков
Hеоклассическая теория переноса в токамаке пересматривается с учетом релятивистских эффектов для
электронов. Поскольку такие эффекты важны только в высокотемпературной плазме, был рассмотрен
только слабостолкновительный банановый режим. Полученные формулировки дают возможность
рассчитать электронные неоклассические потоки в очень широком диапазоне температур.
ПРО РЕЛЯТИВІСТСЬКИЙ ВПЛИВ НА ЕЛЕКТРОННИЙ ТРАНСПОРТ У БАНАНОВОМУ РЕЖИМІ
В ТОКАМАКАХ
І. Марущенко, М.О. Азарєнков
Hеокласична теорія транспорту в токамаках розглядається з урахуванням релятивістських ефектів для
електронів. Оскільки такі ефекти важливі лише у високотемпературній плазмі, розглянуто лише
слабозіткневий банановий режим. Отримані формулювання дають можливість обчислювати електронні
неокласичні потоки в дуже широкому діапазоні температур.
|