Relativistic kinetics and hydrodynamics of hot collisional plasmas
In the paper, relativistic equations of local hydrodynamics for the laboratory fusion plasmas are obtained. Relativistic effects in the physics of electron transport appear primarily because of macroscopic features of relativistic thermodynamic equilibrium given by the Maxwell-Jüttner distribution f...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1958842023-12-08T12:30:22Z Relativistic kinetics and hydrodynamics of hot collisional plasmas Marushchenko, I. Azarenkov, N.A. Basic plasma physics In the paper, relativistic equations of local hydrodynamics for the laboratory fusion plasmas are obtained. Relativistic effects in the physics of electron transport appear primarily because of macroscopic features of relativistic thermodynamic equilibrium given by the Maxwell-Jüttner distribution function, and the characteristic velocity of plasma flow is significantly small: V << νₜₑ < c. We propose an approach in which the plasma electrons are treated as fully relativistic and the hydrodynamic flow is treated in the weakly relativistic approximation. For convenience, the obtained relativistic effects are divided between “quasi-relativistic” terms, which in the nonrelativistic limit coincide with well-known expressions, and fully relativistic terms, which disappear at c → ∞. The considered mixed approach can be useful for construction of transport models for numerical studies of both astrophysical objects and hot fusion plasma. Отримано релятивістські рівняння локальної гідродинаміки для плазми лабораторного термоядерного синтезу. Релятивістські ефекти у фізиці транспорту електронів проявляються насамперед через макроскопічні особливості релятивістської термодинамічної рівноваги, яка задається функцією розподілу Максвелла-Ютнера, а характерна швидкість течії плазми є суттєво малою: V << νₜₑ < c. Запропоновано підхід, у якому електрони плазми вважаються повністю релятивістськими, а гідродинамічна течія розглядається у слабкорелятивістському наближенні. Для зручності отримані релятивістські ефекти розділено між “квазірелятивістськими” членами, які в нерелятивістській межі збігаються з відомими виразами, та повністю релятивістськими членами, які зникають при c → ∞. Розглянутий змішаний підхід може бути корисним при побудові транспортних моделей для чисельних досліджень як астрофізичних об’єктів, так і плазми гарячого термоядерного синтезу. 2022 Article Relativistic kinetics and hydrodynamics of hot collisional plasmas / I. Marushchenko, N.A. Azarenkov // Problems of Atomic Science and Technology. — 2022. — № 6. — С. 44-48. — Бібліогр.: 25 назв. — англ. 1562-6016 PACS: 52.55.Dy, 52.25.Fi, 52.27.Ny DOI: https://doi.org/10.46813/2022-142-044 http://dspace.nbuv.gov.ua/handle/123456789/195884 en Problems of Atomic Science and Technology Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Basic plasma physics Basic plasma physics |
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Basic plasma physics Basic plasma physics Marushchenko, I. Azarenkov, N.A. Relativistic kinetics and hydrodynamics of hot collisional plasmas Problems of Atomic Science and Technology |
| description |
In the paper, relativistic equations of local hydrodynamics for the laboratory fusion plasmas are obtained. Relativistic effects in the physics of electron transport appear primarily because of macroscopic features of relativistic thermodynamic equilibrium given by the Maxwell-Jüttner distribution function, and the characteristic velocity of plasma flow is significantly small: V << νₜₑ < c. We propose an approach in which the plasma electrons are treated as fully relativistic and the hydrodynamic flow is treated in the weakly relativistic approximation. For convenience, the obtained relativistic effects are divided between “quasi-relativistic” terms, which in the nonrelativistic limit coincide with well-known expressions, and fully relativistic terms, which disappear at c → ∞. The considered mixed approach can be useful for construction of transport models for numerical studies of both astrophysical objects and hot fusion plasma. |
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Article |
| author |
Marushchenko, I. Azarenkov, N.A. |
| author_facet |
Marushchenko, I. Azarenkov, N.A. |
| author_sort |
Marushchenko, I. |
| title |
Relativistic kinetics and hydrodynamics of hot collisional plasmas |
| title_short |
Relativistic kinetics and hydrodynamics of hot collisional plasmas |
| title_full |
Relativistic kinetics and hydrodynamics of hot collisional plasmas |
| title_fullStr |
Relativistic kinetics and hydrodynamics of hot collisional plasmas |
| title_full_unstemmed |
Relativistic kinetics and hydrodynamics of hot collisional plasmas |
| title_sort |
relativistic kinetics and hydrodynamics of hot collisional plasmas |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2022 |
| topic_facet |
Basic plasma physics |
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http://dspace.nbuv.gov.ua/handle/123456789/195884 |
| citation_txt |
Relativistic kinetics and hydrodynamics of hot collisional plasmas / I. Marushchenko, N.A. Azarenkov // Problems of Atomic Science and Technology. — 2022. — № 6. — С. 44-48. — Бібліогр.: 25 назв. — англ. |
| series |
Problems of Atomic Science and Technology |
| work_keys_str_mv |
AT marushchenkoi relativistickineticsandhydrodynamicsofhotcollisionalplasmas AT azarenkovna relativistickineticsandhydrodynamicsofhotcollisionalplasmas |
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2025-07-17T00:09:05Z |
| last_indexed |
2025-07-17T00:09:05Z |
| _version_ |
1837850633180282880 |
| fulltext |
44 ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142).
Series: Plasma Physics (28), p. 44-48.
https://doi.org/10.46813/2022-142-044
RELATIVISTIC KINETICS AND HYDRODYNAMICS OF HOT
COLLISIONAL PLASMAS
I. Marushchenko1, N.A. Azarenkov1,2
1V.N. Karazin Kharkiv National University, Kharkiv, Ukraine;
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkiv, Ukraine
In the paper, relativistic equations of local hydrodynamics for the laboratory fusion plasmas are obtained.
Relativistic effects in the physics of electron transport appear primarily because of macroscopic features of relativistic
thermodynamic equilibrium given by the Maxwell-Jüttner distribution function, and the characteristic velocity of
plasma flow is significantly small: 𝑉 ≪ 𝑣𝑡𝑒 < 𝑐. We propose an approach in which the plasma electrons are treated
as fully relativistic and the hydrodynamic flow is treated in the weakly relativistic approximation. For convenience,
the obtained relativistic effects are divided between “quasi-relativistic” terms, which in the nonrelativistic limit
coincide with well-known expressions, and fully relativistic terms, which disappear at𝑐 → ∞. The considered mixed
approach can be useful for construction of transport models for numerical studies of both astrophysical objects and
hot fusion plasma.
PACS: 52.55.Dy, 52.25.Fi, 52.27.Ny
INTRODUCTION
Relativistic effects in astrophysical objects and fusion
plasmas do not necessarily require extremely high
temperatures and energies. They appear to be non-
negligible even for electronic temperatures 𝑇𝑒 of the
order of tens keV, i.e. when 𝑇𝑒 ≪ 𝑚𝑒𝑐2. Relativistic
effects in kinetics, hydrodynamics and transport physics
in collisional plasmas appear due to a macroscopic
features of relativistic thermodynamic equilibrium given
by the Maxwell-Jüttner distribution function (or
relativistic Maxwellian) [1]. In fusion devices such as
ITER [2, 3] and DEMO [4], where electron temperatures
must reach several tens of keV, relativistic effects for
electron transport become noticeable. The same is true
for aneutronic fusion reactors, where the expected
electron temperature should be about 50...70 keV and
above [5-9].
It has recently been shown [10, 11] that relativistic
effects can modify electron transport, making the fluxes
noticeably different from those calculated in the
nonrelativistic limit for both tokamaks and stellarators.
At the same time, virtually all transport codes developed
to date for modeling fusion reactor scenarios are based
on a nonrelativistic approach.
Usually, in the literature devoted to relativistic
kinetics and MHD of plasmas the covariant formalism
with the 4-vectors is applied [12, 13]. This is the most
general and straightforward way to obtain the transport
and MHD equations with conservation of Lorentz
invariance [14, 15]. Usually, this formalism is applied to
describe astrophysical objects. However, for the
problems, where the Lorentz invariance is of low
importance, the kinetics is considered in the same way as
in the non-relativistic limit [10, 11, 16-20].
The present work is focused on description of
transport processes in a hot collisional plasmas with
relativistic electrons and macroscopic flows with
characteristic velocities 𝑉 ≪ 𝑣𝑡𝑒. The main goal is to
derive the equations of local hydrodynamics in the
weakly relativistic approach with respect to the mean
flow, i.e. neglecting the terms of the order 𝑉3/(𝑐2𝑢𝑡𝑒),
𝑉4/(𝑐2𝑢𝑡𝑒
2 ) and above, while the thermal effects
involving plasma electrons are described as fully
relativistic. The final equations are mathematically
similar to the non-relativistic ones and have a transparent
physical interpretation.
FIRST MOMENTS IN THE REST FRAME
First, it is convenient to write a relativistic kinetic
equation for the electron distribution function𝑓𝑒 in
divergent form and without 4-vectors,
𝜕𝑓𝑒
𝜕𝑡
+
𝜕
𝜕𝑥𝑘
(𝑣𝑘𝑓𝑒) +
𝜕
𝜕𝑢𝑘
(�̇�𝑘𝑓𝑒) = 𝐶𝑒(𝑓𝑒) , (1)
where 𝑥�̇� = 𝑣𝑘 is the velocity with 𝑘 = 1,2,3, 𝑢𝑘 = 𝑣𝑘𝛾
is the momentum per unit mass with 𝛾 = √1 + 𝑢2/𝑐2 as
the relativistic factor, and 𝑚�̇�𝑘 = 𝑒𝐸𝑘 +
𝑒
𝑐
[𝒗 × 𝑩]𝑘 is
the force with electric field 𝑬 and magnetic field 𝑩,
respectively. Here and below, the standard rule of
summation over the repetitive indexes is supposed. The
operator 𝐶𝑒(𝑓𝑒) describes the collisions of electrons with
themselves and ions, i.e. 𝐶𝑒(𝑓𝑒) = 𝐶𝑒𝑒(𝑓𝑒) + 𝐶𝑒𝑖(𝑓𝑒),
where ions are considered non-relativistic.
In order to derive the equations for such values as the
mean flow velocity, density and temperature of plasma
electrons, it is natural to assume that plasma is very close
to the thermodynamical equilibrium given by the
“drifting” Maxwell-Jüttner distribution function,
𝑓𝑒0 = 𝐶𝑀𝐽
𝑛𝑒
𝜋3/2𝑢𝑡𝑒
3 exp (−𝜇𝛾0 [𝛾 −
1
𝛾0
−
𝑉𝑘𝑢𝑘
𝑐2 ]), (2)
where 𝑛𝑒 is the density of electrons measured in the rest
frame which moves with mean flow velocity 𝑽, 𝛾0 =
1/√1 − 𝑉2/𝑐2 is the relativistic dilation factor, 𝑢𝑡𝑒 ≡
𝑝𝑡𝑒/𝑚𝑒 = √2𝑇𝑒/𝑚𝑒 is the thermal momentum per unit
ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142) 45
mass (formally, 𝑢𝑡𝑒 coincides with the thermal velocity
in non-relativistic limit, but is not limited by speed of
light), 𝑇𝑒 is the electron temperature and 𝜇 =
𝑚𝑒𝑐2
𝑇𝑒
> 1
(typically, μ > 10 for fusion plasmas). The normalizing
coefficient equals
𝐶𝑀𝐽 = √
𝜋
2𝜇
𝑒−𝜇
𝐾2(𝜇)
= 1 −
15
8𝜇
+
345
128𝜇2 +, (3)
with 𝐾𝑛(𝜇) as the modified Bessel function of second
kind of the n-th order.
While 𝑇𝑒 is assumed here to be arbitrary high (with only
natural limitation 𝑇𝑒 < 𝑚𝑒𝑐2, just to exclude a generation of
the electron-positrons pairs), the mean velocity satisfies the
conditions 𝑉/𝑢𝑡𝑒 ≪ 1 and 𝑉2/𝑐2 ≪ 1. The last condition
makes possible to apply the weakly relativistic approach
with respect to flow,
𝛾0 = 1/√1 − 𝑉2/𝑐2 ≃ 1 + 𝑉2/2𝑐2, (4)
and reduce 𝑓𝑒0 to
𝑓𝑒0 ≃ 𝐶𝑀𝐽
𝑛𝑒
𝜋
3
2𝑢𝑡𝑒
3
exp [−𝜇 (𝛾 − 1 −
𝑉𝑘𝑢𝑘
𝑐2 ) −
𝑚𝑒𝑉2
2𝑇𝑒
]. (5)
The form of representations of 𝑓𝑒0 in Eqs. (2) and (5) with
coefficient given by Eq. (3) is chosen in such a way that
the limit of 𝑓𝑒0 (which is the classical drifting
Maxwellian) when 𝑐 → ∞would be the most obvious.
Now we will adapt to our notations the definitions
given by other authors; see [12, 13, 16]. In order to obtain
the equations for density, momentum and energy, one
needs to integrate kinetic equation Eq. (1) with the
corresponding weight functions: 1, 𝑚𝑒𝑢𝑘 and 𝑚𝑒𝑐2(𝛾 −
1), respectively. For that, following to the algorithm of
Braginskii [21], the Lorentz transformation from the
local coordinate system to the rest frame is required,
where𝑽 = 0 and 𝛾0 = 1. The variables that correspond
to the rest frame are labeled by prime. For compactness,
let us introduce the notations: 〈𝐹〉 = (1/𝑛𝑒) ∫ 𝐹𝑓𝑒𝑑3 𝑢
and 〈𝐹′〉 = (1/𝑛𝑒) ∫ 𝐹′𝑓𝑒
′𝑑3 𝑢′. Evidently, that in the rest
frame 〈1〉 = 1and〈𝑣𝑘
′ 〉 = 0.
For Maxwell-Jüttner distribution function, the
relation between the total relativistic energy and
temperature is well known [12],
ℇ𝑡𝑜𝑡𝑎𝑙 = 𝑛𝑒𝑚𝑒𝑐2〈𝛾′〉 = 𝑛𝑒 (𝑚𝑒𝑐2 𝐾3(𝜇)
𝐾2(𝜇)
− 𝑇𝑒). (6)
Alternatively, the internal thermal energy Eq. (6) can be
represented in different form [10],
𝑊 ≡ 𝑛𝑒𝑚𝑒𝑐2〈𝛾′ − 1〉 = (
3
2
+ ℛ) 𝑛𝑒𝑇𝑒 , (7)
which reminds the classical expression, where ℛ is the
relativistic correction term,
ℛ = 𝜇 (
𝐾3(𝜇)
𝐾2(𝜇)
− 1) −
5
2
=
15
8𝜇
−
15
8𝜇2 +
135
128𝜇3 + (8)
Here, Eqs. (7) and (8) give a quasi-classical form for
energy. Similarly, also the heat flux can be defined,
which, however, is equal in the rest frame to the energy
flux,
𝑞𝑘 = 𝑛𝑒𝑚𝑒𝑐2〈(𝛾′ − 1)𝑣𝑘
′ 〉, (9)
which is also related to the averaged momentum as
follows,
𝑛𝑒𝑚𝑒〈𝑢𝑘
′ 〉 =
1
𝑐2 𝑞𝑘. (10)
It is useful to mention that the moment in Eq. (10)
represents a purely relativistic effect and is equal to zero in
the classical limit, while the heat flux Eq. (9) is “quasi-
classical” in the above sense. Indeed, for
𝑐 → ∞𝑚𝑒𝑐2(𝛾 − 1) → 𝑣2/𝑣𝑡𝑒
2 , and the values 𝑢𝑘
′ and
𝑣𝑘
′ become indistinguishable, while 〈𝑣𝑘
′ 〉 =0.
The next required moment is the momentum flux,
𝑛𝑒𝑚𝑒〈𝑣𝑘
′ 𝑢𝑗
′〉 = 𝑝𝑒𝛿𝑘𝑗 + 𝜋𝑘𝑗 , (11)
which, similarly to the non-relativistic representation,
decomposes into hydrostatic scalar pressure𝑝𝑒,
𝑝𝑒 =
1
3
𝑛𝑒𝑚𝑒 〈
𝑢′2
𝛾′
〉 = 𝑛𝑒𝑇𝑒 , (12)
and (traceless) viscous stress tensor𝜋𝑘𝑗,
𝜋𝑘𝑗 = 𝑛𝑒𝑚𝑒〈𝑣𝑘
′ 𝑢𝑗
′〉 − 𝑝𝑒𝛿𝑘𝑗 . (13)
The moments related to the collisional operator are
also required. Since the conservation laws of momentum
and energy in Coulomb collisions of electrons with
themselves are satisfied automatically, only the
contribution from electron-ion collisions survives in
integration. Then, by definition, electron-ion collisional
friction force is the following:
𝑅𝑘
𝑒𝑖 = ∫ 𝑚𝑒𝑢𝑘
′ 𝐶𝑒𝑖(𝑓𝑒
′) 𝑑3𝑢′ (14)
Similarly, the stress tensor generated by the electron-ion
collisions can be defined as
𝐹𝑘𝑗
𝑒𝑖 = ∫ 𝑚𝑒𝑣𝑘
′ 𝑢𝑗
′𝐶𝑒𝑖(𝑓𝑒
′)𝑑3𝑢′. (15)
The collisional rate of the heat-flux generation is,
respectively,
𝐺𝑘
𝑒𝑖 = ∫ 𝑚𝑒𝑐2(𝛾′ − 1)𝑣𝑘
′ 𝐶𝑒𝑖(𝑓𝑒
′)𝑑3𝑢′. (16)
The rate of collisional energy exchange between
relativistic electrons and classical ions is:
𝑃𝑒𝑖 = ∫ 𝑚𝑒𝑐2(𝛾′ − 1)𝐶𝑒𝑖(𝑓𝑒
′)𝑑3𝑢′. (17)
In this case, only the dominant part of the energy
exchange is taken into account for the calculation, i.e. in
Eq. (17) both electrons and ions distribution functions are
assumed to be equilibrium (Maxwellian for ions and
Maxwell-Jüttner for electrons, but with their own
46 ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142)
temperatures). Then, the result of integration can be
presented as follows [22],
𝑃𝑒𝑖 = 𝑃(𝑐𝑙)
𝑒𝑖 𝐶𝑀𝐽 (1 +
2
𝜇
+
2
𝜇2), (18)
where 𝑃(𝑐𝑙)
𝑒𝑖 is the classical (non-relativistic) electron-ion
energy exchange rate [23],
𝑃(𝑐𝑙)
𝑒𝑖 = −
4
√𝜋
𝜈𝑒0
𝑚𝑒
𝑚𝑖
𝑛𝑖𝑍𝑖
2(𝑇𝑒 − 𝑇𝑖) ∝ −
𝑇𝑒−𝑇𝑖
𝑇𝑒
3
2
. (19)
In somewhat different form Eq. (18) was obtained also in
[14, 16].
HYDRODYNAMIC EQUATIONS IN
LABORATORY FRAME
For integration in the local coordinate system, a
Lorentz invariance of 4-momentum volume has to be
taken into account, that can be written in our notations as
𝑑3𝑢/𝛾 = 𝑑3𝑢′/𝛾′. The Lorentz transformation of the
momentum and energy from the local coordinate system
into the rest frame [24] can be reformulated in our
notations as following,
𝑢𝑘 = 𝛾0𝛾′𝑉𝑘 + 𝑢𝑘
′ + (𝛾0 − 1)
𝑉𝑘𝑉𝑗
𝑉2
𝑢𝑗
′,
𝛾 = 𝛾0 (𝛾′ +
𝑉𝑗𝑢𝑗
′
𝑐2 ). (20)
Relations in Eq. (20) are precise. However, below we
will apply a weakly relativistic approach with respect to
𝑉; see Eq. (4). In the local frame, where𝑉 ≠ 0, the
moments get an additional contributions related to the
mean flow, which are accounted in the weakly relativistic
approach, neglecting the terms of order 𝑉3/(𝑐2𝑢𝑡𝑒),
𝑉4/(𝑐2𝑢𝑡𝑒
2 ) and above.
It is convenient to represent all moments as a sum of
two parts: “quasi-classical” contribution and the term of
purely relativistic correction that completely disappear in
a non-relativistic limit 𝑐 → ∞. Thus, it was found more
appropriate to group the relativistic correction terms with
the formal factor 1/𝜇.
Direct integration of Eq. (1) requires two lowest
moments for the local coordinate system,
𝑛𝑒〈1〉 = 𝛾0𝑛𝑒 and 𝑛𝑒〈𝑣𝑘〉 = 𝛾0𝑛𝑒V𝑘 ≡ 𝛾0Γ𝑘, which
correspond to density and particles flux, respectively.
Here we accounted that 〈𝑣𝑘
′ 〉 = 0. From that, the
continuity equation can be obtained,
𝜕
𝜕𝑡
(𝛾0𝑛𝑒) +
𝜕
𝜕𝑥𝑘
(𝛾0Γ𝑘) = 0. (21)
Note that formally this equation has exactly the same
form as in a fully relativistic approach. The weakly
relativistic expansion Eq. (4) is supposed, but not applied
directly here for compactness.
The next equation is the momentum balance that has
to be obtained integrating Eq. (1) weighed by 𝑚𝑒𝑢𝑘.
Here, both the momentum and the momentum flux are
required. It can be shown that the momentum Eq. (10) in
the rest frame can be represented as
𝑛𝑒𝑚𝑒〈𝑢𝑘〉 = 𝑛𝑒𝑚𝑒(𝑉𝑘 + 𝛿𝑈𝑘
(𝑟)
), (22)
where the additional term, which has the meaning of
relativistic correction for momentum per particle of unit
mass, is
𝛿𝑈𝑘
(𝑟)
=
1
𝜇
[(
5
2
+ ℛ) 𝑉𝑘 +
1
𝑝𝑒
(𝜋𝑘𝑗𝑉𝑗 + 𝑞𝑘)]. (23)
Here, the hydrostatic pressure 𝑝𝑒 and viscous stress
tensor 𝜋𝑖𝑗 are given by Eqs. (12) and (13), and the terms
of order 𝑉3/(𝑐2𝑢𝑡𝑒) and above are neglected.
Similarly, the flux of momentum Eq. (11) in the rest
frame can be represented as
𝑛𝑒𝑚𝑒〈𝑣𝑘𝑢𝑗〉 = Π𝑘𝑗 + 𝛿Π𝑘𝑗
(𝑟)
, (24)
with the lowest “quasi-classical” term formally
coinciding with the non-relativistic definition [23],
Π𝑘𝑗 = 𝑝𝑒𝛿𝑘𝑗 + 𝜋𝑘𝑗 + 𝑛𝑒𝑚𝑒𝑉𝑘𝑉𝑗 , (25)
while and the correction term is
𝛿Π𝑘𝑗
(𝑟)
=
1
𝑐2 [𝑞𝑘𝑉𝑗 + 𝑞𝑗𝑉𝑘 + (
5
2
+ ℛ) 𝑝𝑒𝑉𝑘𝑉𝑗 +
1
2
(𝜋𝑘𝑙𝑉𝑗 + 𝜋𝑗𝑙𝑉𝑘)𝑉𝑙]. (26)
Using Eq. (23), it may be convenient to rewrite the
relativistic correction term Eq. (26) as following,
𝛿Π𝑘𝑗
(𝑟)
=
1
𝑐2 [𝑝𝑒𝛿𝑈𝑘
(𝑟)
𝑉𝑗 + 𝑞𝑗𝑉𝑘 +
1
2
(𝜋𝑘𝑙𝑉𝑗 + 𝜋𝑗𝑙𝑉𝑘)𝑉𝑙].
(27)
The collisional friction that required for momentum
balance can also be represented in the similar form:
∫ 𝑚𝑒𝑢𝑘𝐶𝑒𝑖(𝑓𝑒) 𝑑3𝑢 = 𝑅𝑘
𝑒𝑖 + 𝛿𝑅𝑘
𝑒𝑖(𝑟)
, (28)
with zero-order term equal to that defined in Eq. (14),
while the relativistic correction is
𝛿𝑅𝑘
𝑒𝑖(𝑟)
=
3𝑉2
2𝑐2 (
𝑉𝑘𝑉𝑗
𝑉2
+
1
3
𝛿𝑘𝑗) 𝑅𝑗
𝑒𝑖 +
1
𝑐2
(𝑉𝑘𝑃𝑒𝑖 + 𝐹𝑘𝑗
𝑒𝑖). (29)
Taking into account the terms, given by Eqs. (22-29),
the momentum balance equation can be written as
follows,
𝜕
𝜕𝑡
[𝑛𝑒𝑚𝑒(𝑉𝑘 + 𝛿𝑈𝑘
(𝑟)
)] +
𝜕
𝜕𝑥𝑗
(Π𝑘𝑗 + 𝛿Π𝑘𝑗
(𝑟)
) =
= 𝑒𝑛𝑒𝐸𝑘 +
1
𝑐
[𝑱 × 𝑩]𝑘 + 𝑅𝑘
𝑒𝑖 + 𝛿𝑅𝑘
𝑒𝑖(𝑟)
. (30)
Here, 𝑱 = 𝑒𝑛𝑒𝑽 = 𝑒𝑛𝑒(𝑽𝑒 − 𝑽𝑖) is the electron electric
current that corresponds to the mean flow.
The lastshould be the energy balance equation, which
should be obtained by integrating kinetic equation
weighted by kinetic energy 𝑚𝑒𝑐2(𝛾 − 1). Processing as
above and taking into account Eq. (7), we obtain
𝑛𝑒𝑚𝑒𝑐2〈𝛾 − 1〉 = (
3
2
+ ℛ) 𝑝𝑒 + 𝑛𝑒
𝑚𝑒𝑉2
2
+ 𝛿ℰ(𝑟), (31)
ISSN 1562-6016. Problems of Atomic Science and Technology. 2022. №6(142) 47
with, respectively,
𝛿ℰ(𝑟) =
𝑉2
𝑐2 (
5
2
+ ℛ) 𝑝𝑒 +
1
𝑐2 (𝜋𝑘𝑗𝑉𝑗 + 𝑞𝑘)𝑉𝑘. (32)
Additionally, comparing Eq. (32) and Eq. (23), one can
find a useful relation,
𝛿ℰ(𝑟) =
1
𝑐2 𝑝𝑒𝛿𝑈𝑘
(𝑟)
𝑉𝑘 . (33)
Here, the standard rule of summation over the repetitive
indexes is supposed.
In the same way, the energy flux can be obtained,
𝑛𝑒𝑚𝑒𝑐2〈(𝛾 − 1)𝑣𝑘〉 = 𝑄𝑘 + 𝛿𝑄𝑘
(𝑟)
,
where “quasi-classical” part formally coincides with the
classical definition,
𝑄𝑘 = 𝑞𝑘 + (
5
2
+ ℛ) 𝑝𝑒𝑉𝑘 + 𝜋𝑘𝑗𝑉𝑗 + 𝑛𝑒
𝑚𝑒𝑉2
2
𝑉𝑘, (34)
while the term of pure relativistic correction is
𝛿𝑄𝑘
(𝑟)
=
2𝑉2
𝑐2 [(
5
2
+ ℛ) 𝑝𝑒𝑉𝑘 +
1
2
(
𝑉𝑘𝑉𝑗
𝑉2 + 𝛿𝑘𝑗) 𝜋𝑗𝑙𝑉𝑙 +
3
2
(
𝑉𝑘𝑉𝑗
𝑉2 +
1
3
𝛿𝑘𝑗) 𝑞𝑗]. (35)
Note that the terms proportional to 𝑚𝑒𝑉2/2 in Eq.
(31) and Eq. (34), which are related to the mean flow
kinetic energy, can be excluded from the energy balance
by simple manipulation and using the continuity equation
Eq. (21). After that, the energy balance equation would
describe only the balance of internal thermal energy.
The last term to be considered is the collisional
energy exchange, which can also be written as follows,
∫ 𝑚𝑒𝑐2(𝛾 − 1)𝐶𝑒𝑖(𝑓𝑒) 𝑑3𝑢 = 𝑃𝑒𝑖 + 𝑅𝑘
𝑒𝑖𝑉𝑘 + 𝛿𝑃𝑒𝑖(𝑟), (36)
where𝑃𝑒𝑖 is given by Eq. (18) with classical part given
by Eq. (19), 𝑅𝑘
𝑒𝑖is given by Eq. (14), and the correction-
term to relativistic flow is
𝛿𝑃𝑒𝑖(𝑟) =
2𝑉2
𝑐2 [𝑃𝑒𝑖 + 𝑅𝑘
𝑒𝑖𝑉𝑘 + 𝐺𝑘
𝑒𝑖𝑉𝑘 +
𝑉𝑘𝑉𝑗
𝑉2 𝐹𝑘𝑗
𝑒𝑖]. (37)
Here, 𝐺𝑘
𝑒𝑖 and 𝐹𝑘𝑗
𝑒𝑖 are given by Eqs. (15) and (16),
correspondingly.
Finally, taking into account the terms, given by Eqs.
(31)-(37), the equation for the balance of thermal energy
can be written in the following form:
𝜕
𝜕𝑡
[(
3
2
+ ℛ) 𝑝𝑒 + 𝛿ℰ(𝑟)] +
𝜕
𝜕𝑥𝑗
(Q𝑘 + 𝛿Q𝑘
(𝑟)
) =
= 𝐽𝑘𝐸𝑘 + 𝑃𝑒𝑖 + 𝑅𝑘
𝑒𝑖𝑉𝑘 + 𝛿𝑃𝑒𝑖(𝑟). (38)
The equations Eqs. (21), (30), and (38) describe the
collisional relativistic hydrodynamics, derived in the
mixed approach, fully relativistic for the thermal
electrons and weakly relativistic for the mean flow.
FURTHER STEPS
Since the final transport equations require, as in the
classical treatment, a knowledge of only the few first
moments and only in the rest frame, it is necessary to
make a closure of the model. Here we will draw only the
preliminary sketch of such closure, which itself is beyond
the scope of the present paper.
The first step is to formulate and solve a linearized
kinetic equation with thermodynamic forces on the right-
hand side (gradients of plasma parameters and electric
field). As it was shown in [25], where the relativistic
effects in radial neoclassical fluxes for the toroidal
systems were considered, the most adequate method for
solving the linearized relativistic kinetic equation is to
represent the solution in the form of a series of the
generalized Laguerre polynomials of order 𝛼 = 3/2 +
ℛ. For the low temperature limit, 𝑇𝑒/𝑚𝑐2 → 0, this
representation comes to the classical form with the
Sonine polynomials.
As a final step, we need to calculate the necessary
moments of the distribution function using the obtained
solution. To make the results most transparent, the
obtained moments can be expanded into a 1/𝜇 series,
retaining only the first relativistic correction term.
CONCLUSIONS
In this paper, the relativistic hydrodynamics of
collisional hot plasmas is considered. Equations for first
moments are obtained, which allow us to create a
numerical transport model for the study of astrophysical
and fusion hot plasmas.
The main point of the model is the use of “mixed”
approach, when plasma electrons are described in fully
relativistic approach and mean plasma flow is considered
in weakly relativistic approach.
The equations obtained in the paper have been written
in a form convenient enough for implementation of the
relativistic approach into the transport codes which so far
are based only on the classical approach. Such
modification is necessary for the development and
predictive investigation of the fusion reactor scenarios
with hot plasmas and relativistic electrons.
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Article received 17.10.2022
РЕЛЯТИВІСТСЬКІ РІВНЯННЯ ЛОКАЛЬНОЇ ГІДРОДИНАМІКИ З ПОВІЛЬНИМИ
ПОТОКАМИ
І. Марущенко, М.О. Азарєнков
Отримано релятивістські рівняння локальної гідродинаміки для плазми лабораторного термоядерного
синтезу. Релятивістські ефекти у фізиці транспорту електронів проявляються насамперед через макроскопічні
особливості релятивістської термодинамічної рівноваги, яка задається функцією розподілу Максвелла-
Ютнера, а характерна швидкість течії плазми є суттєво малою: 𝑉 ≪ 𝑣𝑡𝑒 < 𝑐. Запропоновано підхід, у якому
електрони плазми вважаються повністю релятивістськими, а гідродинамічна течія розглядається у
слабкорелятивістському наближенні. Для зручності отримані релятивістські ефекти розділено між
«квазірелятивістськими» членами, які в нерелятивістській межі збігаються з відомими виразами, та повністю
релятивістськими членами, які зникають при 𝑐 → ∞. Розглянутий змішаний підхід може бути корисним при
побудові транспортних моделей для чисельних досліджень як астрофізичних об'єктів, так і плазми гарячого
термоядерного синтезу.
|