Non-Markovian renormalization of kinetic coefficients for drift wave turbulence
Statistical theory of non-Markovian effects in turbulent transport is applied to the description of plasma turbulence in a magnetoactive plasma in the drift-kinetic approximation. Renormalized transition probability is calculated with regard to non-Markovian effects and equations for the relevant...
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| Zitieren: | Non-Markovian renormalization of kinetic coefficients for drift wave turbulence / A. Zagorodny, J. Weiland // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 603-609. — Бібліогр.: 6 назв. — англ. |
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Zagorodny, A. Weiland, J. 2017-06-12T09:20:23Z 2017-06-12T09:20:23Z 2001 Non-Markovian renormalization of kinetic coefficients for drift wave turbulence / A. Zagorodny, J. Weiland // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 603-609. — Бібліогр.: 6 назв. — англ. 1607-324X PACS: 05.40.+j, 52.35.Ra, 25.65.Ff DOI:10.5488/CMP.4.4.603 https://nasplib.isofts.kiev.ua/handle/123456789/120518 Statistical theory of non-Markovian effects in turbulent transport is applied to the description of plasma turbulence in a magnetoactive plasma in the drift-kinetic approximation. Renormalized transition probability is calculated with regard to non-Markovian effects and equations for the relevant kinetic coefficients are derived. It is shown that memory effects can be important for the description of transport under saturated turbulence. Статистична теорія немарковських ефектів у процесах турбулентного перенесення застосована до опису насичення плазмової турбулентності плазми в зовнішньому магнітному полі. Розраховано перенормовану ймовірність переходу з урахуванням немарковських ефектів і сформульовано рівняння для перенормованих кінетичних коефіцієнтів. Показано, що ефекти пам’яті можуть бути суттєвими при описі насичення турбулентності. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Non-Markovian renormalization of kinetic coefficients for drift wave turbulence Немарковське перенормування кінетичних коефіцієнтів для дрейфової турбулентності Article published earlier |
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| title |
Non-Markovian renormalization of kinetic coefficients for drift wave turbulence |
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Non-Markovian renormalization of kinetic coefficients for drift wave turbulence Zagorodny, A. Weiland, J. |
| title_short |
Non-Markovian renormalization of kinetic coefficients for drift wave turbulence |
| title_full |
Non-Markovian renormalization of kinetic coefficients for drift wave turbulence |
| title_fullStr |
Non-Markovian renormalization of kinetic coefficients for drift wave turbulence |
| title_full_unstemmed |
Non-Markovian renormalization of kinetic coefficients for drift wave turbulence |
| title_sort |
non-markovian renormalization of kinetic coefficients for drift wave turbulence |
| author |
Zagorodny, A. Weiland, J. |
| author_facet |
Zagorodny, A. Weiland, J. |
| publishDate |
2001 |
| language |
English |
| container_title |
Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Немарковське перенормування кінетичних коефіцієнтів для дрейфової турбулентності |
| description |
Statistical theory of non-Markovian effects in turbulent transport is applied
to the description of plasma turbulence in a magnetoactive plasma in the
drift-kinetic approximation. Renormalized transition probability is calculated
with regard to non-Markovian effects and equations for the relevant kinetic
coefficients are derived. It is shown that memory effects can be important
for the description of transport under saturated turbulence.
Статистична теорія немарковських ефектів у процесах турбулентного перенесення застосована до опису насичення плазмової турбулентності плазми в зовнішньому магнітному полі. Розраховано перенормовану ймовірність переходу з урахуванням немарковських
ефектів і сформульовано рівняння для перенормованих кінетичних
коефіцієнтів. Показано, що ефекти пам’яті можуть бути суттєвими
при описі насичення турбулентності.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120518 |
| citation_txt |
Non-Markovian renormalization of kinetic coefficients for drift wave turbulence / A. Zagorodny, J. Weiland // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 603-609. — Бібліогр.: 6 назв. — англ. |
| work_keys_str_mv |
AT zagorodnya nonmarkovianrenormalizationofkineticcoefficientsfordriftwaveturbulence AT weilandj nonmarkovianrenormalizationofkineticcoefficientsfordriftwaveturbulence AT zagorodnya nemarkovsʹkeperenormuvannâkínetičnihkoefícíêntívdlâdreifovoíturbulentností AT weilandj nemarkovsʹkeperenormuvannâkínetičnihkoefícíêntívdlâdreifovoíturbulentností |
| first_indexed |
2025-11-26T00:10:46Z |
| last_indexed |
2025-11-26T00:10:46Z |
| _version_ |
1850595952463708160 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 4(28), pp. 603–609
Non-Markovian renormalization of
kinetic coefficients for drift wave
turbulence
A.Zagorodny 1 , J.Weiland 2
1 Bogolyubov Institute for Theoretical Physics, 03143 Kiev 143, Ukraine
2 Institute for Electromagnetics, Chalmers University of Technology,
S-41296 Göteborg, Sweden
Received August 29, 2001
Statistical theory of non-Markovian effects in turbulent transport is applied
to the description of plasma turbulence in a magnetoactive plasma in the
drift-kinetic approximation. Renormalized transition probability is calculated
with regard to non-Markovian effects and equations for the relevant kinetic
coefficients are derived. It is shown that memory effects can be important
for the description of transport under saturated turbulence.
Key words: plasma turbulence, drift-kinetic approximation,
non-Markovian effects
PACS: 05.40.+j, 52.35.Ra, 25.65.Ff
1. Introduction
In [1,2] we proposed statistical theory of turbulent transport giving the non-
Markovian version of the Dupree-Weinstock renormalization [3,4]. In the present
paper the transition probability approach is used to introduce non-Markovian gener-
alization of drift kinetic equations with regard to turbulent collisions. Derived equa-
tions include collision terms describing the transverse diffusion in the real space and
longitudinal diffusion in the velocity space. The main feature of the non-Markovian
description is the time-nonlocality of the collision terms, which leads to a frequen-
cy dependence of transport kinetic coefficients the real part of which describes the
modification of the growthrate while the imaginary part is responsible for coherent
wave interaction, in particular, for nonlinear frequency shift.
The proposed approach makes it possible to calculate the renormalized transi-
tion probability taking into account the influence of turbulent fields on the particle
trajectories. It is shown that non-Markovian effects could be important for turbu-
lence saturation. In the Markovian limit the Dupree-Tetreault renormalization [5] is
reproduced.
c© A.Zagorodny, J.Weiland 603
A.Zagorodny, J.Weiland
2. Drift kinetic equation with turbulent collision term
We start from the equation for the microscopic phase density of guiding centers
N(X, t) =
N
∑
i=1
δ(X −Xi(t)), X = (r, v‖) (1)
which in the drift approximation is given by
{
∂
∂t
+ (vg + vE)
∂
∂r⊥
+ v‖
∂
∂r‖
+
e
m
E‖
∂
∂v‖
}
N(X, t) = 0. (2)
Here vE = ec
B0
[E⊥ · e‖], E = −gradΦ is the microscopic electric field, vg is the
gravitation drift velocity, the other notation is conventional.
The kinetic equation for the distribution function f(X, t) ≡ 〈N(X, t)〉 can be
obtained by statistical averaging of equation (2)
L̂0f(X, t) ≡
{
∂
∂t
+ (vg + 〈vE〉)
∂
∂r⊥
+ v‖
∂
∂r‖
+
e
m
〈E‖〉
∂
∂v‖
}
f(X, t) = I, (3)
where
I = I⊥ + I‖ ,
I‖ = −
e
m
∂
∂v‖
〈δE‖δN〉, I⊥ = −
∂
∂r⊥
〈δvEδN〉. (4)
The equation for fluctuations reduces then to
{
L̂0 +
e
m
δE‖(r, t)
∂
∂v‖
}
δN(X, t) = −δvE(r, t)
∂f(X, t)
∂r⊥
−
e
m
δE‖(r, t)
∂f(X, t)
∂v‖
. (5)
The formal solution of equation (5) is given by
δN(X, T ) = δN (0)(X, t)
−
t
∫
0
dt′
∫
dX ′Wm(X,X ′; t, t′)
δvE(r
′, t′)
∂f(X ′, t′)
∂r′⊥
+
e
m
δE‖(r
′, t′)
∂f(X ′, t′)
∂v′‖
,
(6)
where δN (0)(X, t) is the fluctuation of the microscopic phase density with no self-
consistent interaction generated by the general solution of homogeneous equation,
i.e.
{
L̂(0) +
e
m
δE‖(r, t)
∂
∂v‖
}
δN0(X, t) = 0, (7)
Wm(X,X ′; t, t′) is the “microscopic” probability of particle transition from X ′ to X
during the time interval t− t′. Obviously the equation for such probability is
{
L̂(0) +
e
m
δE‖(r, t)
∂
∂v‖
}
Wm(X,X ′; t, t′) = 0 (8)
604
Non-Markovian renormalization of kinetic coefficients. . .
with the initial condition Wm(X,X ′; t, t′) = δ(X−X ′). The solution of equation (8)
is
Wm(X,X ′; t, t′) = δ (X −X ′ −∆X(X ′, t′; t)) , (9)
where ∆X(X ′, t′; t) is the guiding center displacement in the course of particle mo-
tion in the microfield
∆r‖(X
′, t′; t) = v‖(t
′)(t− t′) +
e
m
t
∫
t′
ds
s
∫
t′
ds′ E‖ (r(s
′), s′) ,
∆r⊥(X
′, t′; t) = vg(t− t′) +
t
∫
t′
dsvE (r(s), s) ,
∆v‖(X
′, t′; t) =
e
m
t
∫
t′
dsE‖ (r(s), s) .
We now assume that the distribution function slowly changes within the spatial
and velocity fluctuation scales. Then, combining equations (4), (6) and (8) it is
possible to introduce the following kinetic equation with the non-Markovian collision
term
L̂(0)f(X, t) =
t
∫
0
dt′
{
∂
∂v‖
[
v‖
(
β(t, t′, v‖) + 2Uf(t, t′, v‖)
∂
∂r⊥
)
+
∂
∂v‖
d(t, t′, v‖)
]
−
(
U
P (t, t′, v‖) +U
f(t, t′, v‖)
) ∂
∂r⊥
+Dij(t, t
′, v‖)
∂2
∂r⊥i∂r⊥j
}
f(X, t′), (10)
where
β(t, t′, v‖) = β1(t, t
′, v‖) + β2(t, t
′, v‖),
β1(t, t
′, v‖) = −i
4πe2n
m
∫
dk
(2π)3
∫
dω
2π
ki〈Wmkω(v‖)〉
k2ε(k, ω)
δ(t− t′) + c.c.,
β2(t, t
′, v‖) = −
1
v‖
∂
∂v‖
(
e
m
)2 ∫ dk
(2π)3
k2
‖〈δΦ(t)δΦ(t
′)〉k〈Wmk(v‖, t
′; t)〉,
UP
i (t, t
′v‖) = −i
4πe2
mΩ
eijz
∫
dk
(2π)3
∫
dω
2π
k⊥i〈Wmkω(v‖)〉
k2ε(k, ω)
δ(t− t′) + c.c.,
U
f
i (t, t
′, v‖) = eijz
(
e
m
)2 ∫ dk
(2π)3
k⊥jk‖
Ωv‖
〈δΦ(t)δΦ(t′)〉k〈Wmk(v‖, t
′; t)〉,
d(t, t′; v‖) =
(
e
m
)2 ∫ dk
(2π)3
k2
‖〈δΦ(t)δΦ(t
′)〉k〈Wmk(v‖, t
′; t)〉,
Dij(t, t
′; v‖) =
=
(
e
m
)2 ∫ dk
(2π)3
k2
⊥
Ω2
(
δ⊥ij −
k⊥ik⊥j
k2
⊥
)
〈δΦ(t)δΦ(t′)〉k〈Wmk(v‖, t
′; t)〉 (11)
605
A.Zagorodny, J.Weiland
and
〈Wmk(v‖, t, t
′)〉 =
∫
d∆Xe−ik∆r〈W (X +∆X,X ; t, t′)〉,
〈Wmkω(v‖)〉 =
∞
∫
0
dτeiωτ 〈Wmk
(
v‖; t+ τ, t
)
〉,
ε(k, ω) = 1−
4πe2
m
∫
dv‖〈Wmkω(v‖)〉
{
eijz
k⊥j
Ω
∂f(X, t)
∂r⊥i
+ k‖
∂f(X, t)
∂v‖
}
. (12)
The kinetic coefficents given by equation (11) describe the influence of different
physical mechanisms on averaged (kinetic) particle dynamics. Namely, β 1(t, t
′; v‖)
and β2(t, t
′; v‖) describe the polarization friction and friction due to the particle
scattering by turbulent fields, respectively, U p(t, t′; v‖) and Uf (t, t′; v‖) are associat-
ed with the polarization drift and zonal flows, d(t, t ′; v‖) and D(t, t′; v‖) characterize
the velocity diffusion along the external magnetic field and the transverse diffusion
in the real space. Thus, we see that in the case under consideration the collision
term in equation (10) includes diffusion in the velocity space in longitudinal direc-
tion, diffusion in the real space in the transverse directions, drift generated by the
polarization forces and drift due to the field fluctuations. All kinetic coefficients are
expressed in terms of fluctuation potential correlation functions and averaged transi-
tion probability which describes particle transitions taking into account fluctuation
(in particular turbulent) field influence on particle trajectories.
3. Renormalized transition probability and dielectric res ponse
function
Since the equation for “microscopic” transition probability (equation (8)) coin-
cides with the equation for microscopic phase density (equation (2)), it is easy to
derive the equation for averaged transition probability given by
L̂0〈Wm(X,X ′; t, t′)〉 =
=
t
∫
t′
dt′′
{
∂
∂v‖
[
v‖
(
β(t, t′′, v‖) + 2Uf(t, t′′, v‖) +
∂
∂r⊥
)
∂
∂v‖
d(t, t′; v‖)
]
−
(
U
P (t, t′, v‖) +U
f (t, t′′; v‖)
) ∂
∂r⊥
+Dij(t, t
′′; v‖)
∂2
∂r⊥i∂r⊥j
}
× 〈Wm(X,X ′; t′′, t′)〉. (13)
This equation, as well as the kinetic equation (10) are non-Markovian. In various
particular cases solutions of equation (13) can be found explicitly. The examples of
solutions of the equation describing non-Markovian diffusion in the velocity space
are given in [1,2]. We generalize these results to the case under consideration, i.e.
to equation (13). However, in order to illustrate the basic points of the approach,
606
Non-Markovian renormalization of kinetic coefficients. . .
we do not reproduce here the general solution but restrict ourselves to the case
when the dynamical friction, polarization and fluctuation drift, could be neglected
(β = 0, UP = U
f = 0). This approximation is valid in the case of wide k-spectrum
of turbulence and low level of turbulent fluctuations.
In this case equation (13) reduces to
L̂0〈Wm(X,X ′; t, t′)〉 =
=
t
∫
t′
dt′′
d(t, t′′, v‖)
∂2
∂v2‖
+Dij(t, t
′′; v‖)
∂2
∂r⊥i∂r⊥j
〈Wm(X,X ′; t′′, t′)〉. (14)
It follows from the solution of this equation that in the case of stationary system
〈Wmkω(v‖)〉 =
=
∞
∫
0
dτ exp
[
i
(
ω − k‖v‖ − k⊥vg + ik⊥ik⊥jDijω(v‖)
)
τ +
ik2
‖dω(v‖)
3
τ 3
]
, (15)
where
dω(v‖) =
∞
∫
0
dτeiωτd
(
t+ τ, t; v‖
)
, Dijω(v‖) =
∞
∫
0
dτeiωτDij
(
t + τ, t; v‖
)
.
In the Markovian limit these quantities should be replaced by their zero-frequen-
cy values. Equation (15) generalizes the renormalizations of the transition probabil-
ity for guiding centers due to the diffusion in the real and velocity spaces to the case
of non-Markovian diffusion.
The renormalized dielectric response function has the form
ε(k, ω) = 1− i
4πe2
m
∫
dv‖
∞
∫
0
dτ
× exp
[
i
(
ω − k‖v‖ − k⊥vg + ik⊥ik⊥jDijω(v‖)
)
τ +
k2
‖dω(v‖)
3
τ 3
]
×
{
eijz
k⊥j
Ω
∂f(X)
∂ri
+ k‖
∂f(X)
∂v‖
}
. (16)
Solutions of the equation ε(k, ω) = 0 define the renormalized spectra of eigenexci-
tations.
4. Non-Markovian transport and turbulence saturation
Substituting equation (15) into equations (11) it is possible to formulate equa-
tions for renormalized kinetic coefficients. For examples,
Dijω(v‖) =
(
e
m
)2 ∫ dk
(2π)3
∫
dω′
2π
k2
⊥
Ω2
(
δ⊥ij −
k⊥ik⊥j
k2
⊥
)
〈δΦ2〉kω′ (17)
607
A.Zagorodny, J.Weiland
×
∞
∫
0
dτ exp
[
i
(
ω − ω′ + k‖v‖ + k⊥vg + ik⊥ik⊥jDijω−ω′(v‖)
)
τ −
k2
‖dω−ω′(v‖)
3
τ 3
]
.
Within the approximation
∞
∫
0
dτe−ατ3+iωτ ≃
i
ω + iα1/3
equation (17) reduces to
Dijω(v‖) =
(
e
m
)2 ∫ dk
(2π)3
∫
dω′
2π
k2
⊥
Ω2
(
δ⊥ij −
k⊥ik⊥j
k2
⊥
)
〈δΦ2〉kω′
×
i
ω − ω′ + k‖v‖ + ik⊥ik⊥jDijω−ω′(v‖) + i
3
√
k2
‖dω−ω′(v‖)
3
. (18)
At dω(v‖) = 0, k‖v‖ = 0 equation (18) gives the non-Markovian generalization of
the result by Dupree-Tetreault [5]
Dijω(v‖) =
(
e
m
)2 ∫ dk
(2π)3
∫
dω′
2π
k2
⊥
Ω2
(
δ⊥ij −
k⊥ik⊥j
k2
⊥
)
i〈δΦ2〉kω′
ω − ω′ + ik⊥ik⊥jDijω−ω′
.
(19)
In the case of one-mode turbulent spectrum equation (19) gives
k⊥ik⊥jDijω =
1
2
∑
β=±1
k4
⊥
Ω2
∆k
k⊥
i
(
e
m
)2
δΦ2
ω − βωk + ik⊥ik⊥jDijω−βωk
, (20)
where k⊥ and ∆k are the wave number related to the maximum growthrate and the
turbulent spectrum width, ωk is the eigenfrequency, δΦ2 is the turbulent spectrum
amplitude.
As follows from equation (20) low-frequency value of the diffusion coefficient is
given by
k⊥ik⊥jDij =
(
e
m
)2 k2
⊥
Ω2
∆k
k⊥
δΦ2γk
ω2
k + γ2
k
≃
γ3
k
ω2
k + γ2
k
, (21)
where γk is the instability growthrate. When deriving (21) we assume that in the
saturated state renormalized growthrate is equal to zero, i.e. k⊥ik⊥jDωk
= γk. The
estimate (21) is in agreement with the results of mode-coupling simulations and
improved mixing length transport theory (see, for instance [6]).
5. Summary and conclusions
The microscopic derivation of the drift-kinetic equation with a non-Markovian
collision term is done for the case of turbulent plasmas in an external magnetic
608
Non-Markovian renormalization of kinetic coefficients. . .
field. Renormalized non-Markovian kinetic coefficients are expressed in terms of
the transition probability calculated with regard to the time-nonlocal diffusion in
the real and velocity spaces. In the approximation of the time-local diffusion the
results of the quasilinear theory, as well as the Dupree-Tetreault renormalization
are reproduced. It is shown that memory effects are important for the estimates of
the diffusion coefficients for saturated turbulence.
References
1. Zagorodny A., Weiland J. // Condens. Matter Phys., 1998, vol. 1, p. 835.
2. Zagorodny A., Weiland J. // Phys. Plasmas, 1999, vol. 5, p. 2359.
3. Dupree T. H. // Phys. Fluids, 1966, vol. 9, p. 1773.
4. Weinstock J. // Phys. Fluids, 1969, vol. 12, p. 1045.
5. Dupree T. H., Tetreault D. // Phys. Fluids, 1978, vol. 21, p. 425
6. Connor J.W., Pogutse O.P. // Plasma Phys. Control. Fusion, 2001, vol. 43, p. 155.
Немарковське перенормування кінетичних
коефіцієнтів для дрейфової турбулентності
А.Загородній 1 , Я.Вейланд 2
1 Інститут теоретичної фізики ім. М.М. Боголюбова НАН України,
03143 Київ, вул. Метрологічна, 14б
2 Інститут електромагнетизму,
Чалмерський технологічний університет,
S-41296 Ґетеборґ, Швеція
Отримано 29 серпня 2001 р.
Статистична теорія немарковських ефектів у процесах турбулентно-
го перенесення застосована до опису насичення плазмової турбу-
лентності плазми в зовнішньому магнітному полі. Розраховано пе-
ренормовану ймовірність переходу з урахуванням немарковських
ефектів і сформульовано рівняння для перенормованих кінетичних
коефіцієнтів. Показано, що ефекти пам’яті можуть бути суттєвими
при описі насичення турбулентності.
Ключові слова: турбулентність плазми, дрейфове кінетичне
наближення, немарковські ефекти
PACS: 05.40.+j, 52.35.Ra, 25.65.Ff
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