Renormalized non–modal theory of turbulence of plasma shear flows
In our report, we present the results of the non-linear investigations of the temporal evolution and saturation of drift turbulence in shear flows, which have the non-modal approach as their foundation. Подано результати нелінійних досліджень часової еволюції та насичення дрейфової турбулентності у...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2011
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| Цитувати: | Renormalized non–modal theory of turbulence of plasma shear flows / V.S. Mikhailenko, V.V. Mikhailenko, K.N. Stepanov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2011. — № 1. — С. 41-43. — Бібліогр.: 5 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859472484686888960 |
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| author | Mikhailenko, V.S. Mikhailenko, V.V. Stepanov, K.N. Azarenkov, N.A. |
| author_facet | Mikhailenko, V.S. Mikhailenko, V.V. Stepanov, K.N. Azarenkov, N.A. |
| citation_txt | Renormalized non–modal theory of turbulence of plasma shear flows / V.S. Mikhailenko, V.V. Mikhailenko, K.N. Stepanov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2011. — № 1. — С. 41-43. — Бібліогр.: 5 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | In our report, we present the results of the non-linear investigations of the temporal evolution and saturation of drift turbulence in shear flows, which have the non-modal approach as their foundation.
Подано результати нелінійних досліджень часової еволюції та насичення дрейфової турбулентності у зсувних течіях плазми, які базуються на немодальному підході.
Представлены результаты нелинейных исследований временной эволюции и насыщения дрейфовой турбулентности в сдвиговых течениях, основанных на немодальном подходе.
|
| first_indexed | 2025-11-24T10:53:33Z |
| format | Article |
| fulltext |
BASIC PLASMA PHYSICS
RENORMALIZED NON–MODAL THEORY OF TURBULENCE
OF PLASMA SHEAR FLOWS
V.S. Mikhailenko1, V.V. Mikhailenko3, K.N. Stepanov1,2 , N.A. Azarenkov1
1V.N. Karazin Kharkov National University, Kharkov, Ukraine;
2National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
3University of Madeira, Largo do Municipio, 9000 Funchal, Portugal
E-mail: vmikhailenko@kipt.kharkov.ua
In our report, we present the results of the non-linear investigations of the temporal evolution and saturation of drift
turbulence in shear flows, which have the non-modal approach as their foundation.
PACS: 94.05.Lk, 94.20.wf
1. INTRODUCTION
The discovery[1] of the L–H transition phenomena is
one of the greatest successes in the investigation of the
magnetized fusion related plasmas. That transition
characterized by a sudden suppression of edge density-
and magnetic turbulence, which follows by rapid drop of
turbulent transport at the plasma edge that resulted in the
development of steep edge gradients indicating the set up
of a transport barrier in the outermost few cm of the
confinement region. The experiments revealed that
suppression of the drift turbulence conditioned by shear
flows, which developed prior to the transition in the
boundary layers of plasma. The discovery of the
connection of the observed turbulence suppression with
shear flows determined the development of the turbulence
theory of shear flows as the one of the most important
task in the theory of the controlled fusion and of the
plasma theory in whole.
The contemporary theory of plasma shear flows
turbulence meets with great obstacles in its development.
That theory grounds on two approaches. The first is called
as the normal mode or modal approach, in which
perturbations of the fields and density, temperature, ets.
are considered as spatially inhomogeneous in the
direction of the flow shear and the application of the
spectral transform in time is assumed. The solution
obtained on this way in linear approximation has as a rule
the singularities at the critical level, where phase velocity
of the perturbations is equal to the local magnitude of the
flow velocity. Because of that singularity plasma
turbulence grounded on the modal approach is still absent.
Even the simplest turbulence theory grounded on the
weak interaction approximation is not developed yet
because of the divergence of the power series expansions
used in this approach. The phenomenological shear flow
turbulence theory (in which the problem of the solutions
secularity even not notice) was presented in Refs.[2, 3].
That theory bases on the suggestion, that observed
suppression of the drift turbulence is the result of the
enhanced decorrelation of the plasma displacements,
which follows from the coupled action of the turbulent
scattering and convection by shear flow. The experiments,
however display the results, which are opposite to the
prediction of that theory: the correlation times grow in
plasma shear flow. In this report, we present the results of
the development of the hydrodynamic and kinetic drift
turbulence theory of the plasma shear flows. This theory
is grounded on Kelvin’s method of shearing modes or a
so-called non-modal approach. The non-modal approach
appears very effective in the development of the linear
and weak nonlinear theories of plasma waves and
instabilities in shear flow. This theory gives simple, exact,
and uniformly bounded for all times, solutions, which are
free from the problem of the singularities, which is
inherent to modal approach. Particularly, the solution of
the initial value problem, obtained in this approach,
reveals that a drift wave in the shear flow gradually
transformed into a convective cell and normal-mode
solution is not the steady-state limit for the initial value
problem considered.
2. RENORMALIZED HYDRODYNAMIC THEORY
OF DRIFT TURBULENCE OF SHEAR FLOWS
We investigate the temporal evolution of drift modes
in time-dependent shear flow using the Hasegawa–
Wakatani equations for the dimensionless density
and potential = / en n n% = / ee Tφ ϕ perturbations ( is
the electron background density, is the electron
temperature),
en
eT
( )
( )
2 2
0
2
2
,
= ,
s
cV x t
t y B y x x y
a n
z
φ φρ φ
φ
⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂
+ − −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠
∂
−
∂
∇
( )
( )
0
2
2
,
= ,
de
cV x t n v
t y B y x x y
a n
z
y
φ φ φ
φ
⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂
+ − − +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠
∂
−
∂
∂
where ( )0 ,V x t is the velocity of the sheared flow. We
transform these equations to new spatial variables ,ξ η ,
0= , = , = ' , = .t t x y V xt z z−ξ η (1)
In these coordinates the linear convection terms are absent
in above system:
( )
2
2 2
2= ,s
c a n
t B z
⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂
− − ∇⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠
φ φ
−ρ φ φ
η ξ ξ η
(
2
2= .de
c n v a n
t B z
⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂
− − +⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠
φ φ φ )−φ
η ξ ξ η η
(2)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2011. № 1. 41
Series: Plasma Physics (17), p. 41-43.
It is interesting to note that transformation (1)
conserves the E B× convective nonlinear derivative in
in the form similar to one in a plasma without any flows.
With new variables 1ξ , 1η determined by the nonlinear
relations
1 1 1
0
= , =
t
t t
c dt dt
B B
∂
+ −
∂ ∂∫ ∫
φξ ξ η η
η ξ
42
1
0
,
tc ∂φ (3)
the convective nonlinearity in Eqs.(2) becomes of the
higher order with respect to the potential φ . Omitting
such nonlinearity, as well as small nonlinearity of the
second order in the laplacian, resulted from the
transformation to nonlinearly determined variables 1ξ ,
1η , we come to linear equation with solution
( ) ( ) ( ) 1 1, , = , , 0 , , ,ik ilt d k d l k l g k l t e ξ ηφ ξ η φ +⊥
⊥ ⊥ ⊥∫ ∫ (4)
where wave numbers , are conjugate there to
coordinates
k⊥ l
1ξ , 1η respectively. With variables ξ and η
this solution has a form
( ) ( )
( ) % ( ) % ( )(1 1
, , = , ,0
, , exp ,
t dk dl k l
)1g k l t ik il ik t il t
⊥ ⊥
⊥ ⊥ ⊥× + − −
∫ ∫φ ξ η φ
ξ η ξ η
(5)
where
% ( ) ( ) % ( ) ( )1
1
0 0
= , =
t t
t t
tc ct dt t
B B
∂
−
∂∫ ∫
φ
ξ η
η
1
1.
t
dt
∂
∂
φ
ξ
(6)
Eq.(5) is in fact a nonlinear integral equation for
potential φ , in which the effect of the total fourier
spectrum on any separate fourier harmonic is accounted
for. The functions % ( )tξ and in the exponential of
Eq.(5) involve through eq.(6) integrals of
% ( )tη
φ , which in
turn, involve in their exponentials the integrals (5) and so
on. This form of solution, however, appears very useful
for the analysis of the correlation properties of the
nonlinear solutions to Hasegawa-Wakatani system and for
the development of the approximate renormalized
solutions to Hasegawa-Wakatani system, which
accounted for the effect of the turbulent motions of
plasma on the saturation of the drift-resistive instability.
We have obtained[4] the renormalized form of the
potential (5), in which the average effect of the random
convection is accounted for,
( ) ( )
( )
0
, , = , ,0
exp , , .
t
d
t dk dl k l
i t t d tC k l t ik il
φ ξ η φ
ω γ ξ η
⊥ ⊥
⊥ ⊥
⎛ ⎞
× + − + +⎜ ⎟
⎝ ⎠
∫ ∫
∫ $ $
(7)
The saturation of the instability occurs when
, i.e. when ( )( )2
, , / = 0t t∂ φ ξ η ∂
( ) ( ) ( )
[ ] ( )
( )
2
2
1 1 1 12
2 1 1
1 2
1 1
, = , , = , ,
, ,
.
,d
ck l C k l t dk dl k l t
B
C k l t
k l
⊥ ⊥ ⊥ ⊥
⊥
⊥ ⊥
⊥
× ×
∫ ∫
k k
γ
ω
φ
(8)
From the double Eq.(8) we obtain the equation, which
determines the level of the instability saturation
( ) ( ) [ ] ( )
( )
2
22 1
1 1 1 1 12 2 .
k1
= , ,
d
kck dk dl k l t
B ⊥ ⊥ ⊥ ⊥×∫ ∫ k k
γ
γ φ
ω
(9)
The sought-for value is a time satt at which the
balance of the linear growth and nonlinear damping
occurs for given initial disturbance and
dispersion. With obtained
( 1 1, ,0k l⊥φ )
satt the saturation level will be
equal to
( ) ( )2 2
1 1 1 1, ,0satt dk dl k lφ φ⊥ ⊥∫ ∫; ( )( )1 1exp 2 , satk l tγ ⊥× .
Also, the well known order of value estimate for the
po-tential φ , in the saturation state is obtained easily
from Eq.(9), Obtained results show
that the nonlinearity of the Hasegawa-Wakatani system of
equations in variables
( ) 1/ e ne T k L −
⊥φ : .
ξ and η , with which frequency
and growth rate are determined without spatially
inhomogeneous Doppler shift and wave number is time
independent, does not display any effects of the enhanced
decorrelations provided by flow shear.
3. RENORMALIZED KINETIC THEORY
OF DRIFT TURBULENCE OF SHEAR FLOWS
It was obtained in Ref.[5], that application the
transformation (1) to Vlasov equation jointly with trans-
formation of the velocity to convective set of reference
resulted in Vlasov equation, in which inhomogeneities
conditioned by shear flow are absent. With leading center
coordinates, determined by relations
= cos , = sinx yv v v v⊥ ⊥α αφ η φ ,
( )
( ) ( )
1
1 0 1
= sin
= cos sin
c
c
c c
c c
vX x t
v vY y t V t t
α α α
α
α α α α
α α
φ ηω
ηω
φ ηω φ ηω
ηω ηω
⊥
⊥ ⊥
+ −
′− − − −
Vlasov equation has a form
1 = 0,
c
n n n n n
c z
F e F F
t m v v v
e F F e F
m X Y Y X m z v
αα α α α
α α
α α α α
α α α α α α α α α
ηω ϕ φ
φ φ
ϕ φ ϕ
ηω
⊥ ⊥
⎛ ⎞∂ ∂ ∂∂ ∂
+ −⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠
⎛ ⎞∂ ∂ ∂∂ ∂ ∂
+ − −⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠
α
(10)
where perturbed electrostatic potential is determined as
( ) ( )
( )
( ) ( )( )1
, = , , , ,
exp ( )
exp sin .
x y z x y z
x y z
c
c
t dk dk dk t k k k
ik X ik Y ik z i t
ik t v
t t
αϕ ϕ
δ
φ ηω θ
ηω
⊥ ⊥
× + + +
⎡ ⎤
× − − −⎢ ⎥
⎢ ⎥⎣ ⎦
∫r
k r (11)
In (11) ( )i tk rδ term,
( )
1( ) ( ) ( ) ( )
( )sin( ( )) cos( ( )) ( ) ,
x y ct k X t k Y t k t
v t t v t t
δ δ δ ω
δ φ θ φ θ δφ
−
⊥
⊥ ⊥
= + −
× − + −
k r
(12)
determines the nonlinear phase shift of the potential (11)
due to turbulent scattering of ions in electrostatic
turbulence. We find, that for the times ( ) 1
0<t V −′ the main
effect, which determines the nonlinear scattering of ions
by long wavelength drift turbulence with < 1ik ρ⊥ is the
scattering of the leading center coordinates, Xδ and Yδ .
The non-modal effects are negligible at this time. At times
( ) 1
0>t V −′ right the non-modal effects determine the
nonlinear evolution of drift turbulence with dominant
nonlinear phase shift due to scattering of the angle δφ in
4. CONCLUSIONS velocity space. For times ( ) 1
0 < < sV t−′ t and for times
> st t we have, respectively
43
(13)
( )
( )
3
0
2
0
/
/
i x y i
i x
k k X k V t
k k X V t
ρ δφ δ ρ
ρ δφ δ
⊥
⊥
′
′
: ?
: ?
1,
1.
We obtain for ( ) 1
0 < < sV t−′ t the renormalized
solution in the form
( ) ( ) ( )
( ) ( )
2
0 0 2
1 12
0
1, = , exp 1
3
, ,
2
i i s
t
i s
tt t i
a b t
t t C t dt
a t
⎡ ⎛ ⎞+
− −⎢ ⎜ ⎟
⎢ ⎝ ⎠⎣
⎤⎛ ⎞
+ − − ⎥⎜ ⎟
⎥⎝ ⎠ ⎦
∫
k k k
k k
τϕ ϕ ω
γ
The results presented in this report prove that any
"universal rules" or "paradigms", that thoroughly
determines the turbulence suppression by shear flow, are
absent. The suppression of turbulence by shear flows is a
mode dependent process, which includes the sequence of
different non-modal linear and non-linear processes with
different time scales for different parts of the spectrum of
the unstable waves. Presented nonlinear non-modal
analysis of the resistive drift and kinetic (universal) drift
instabilities reveals that non-modal effects lead to the
decreasing the frequency and growth rate at time
( ) 1
2 0= y st t V k
−
′≤ ρ and lead to rapid non-modal
suppression of turbulence at time ( ) 1
2 0> = y st t V k
−
′ ρ .
The time dependence of the wavenumber ( )k t⊥
becomes the key element in the proper kinetic treatment
of the long-time evolution of the perturbations in shear
flow. In such kinetic analysis the nonlinear non-modal
turbulent scattering of the phase angle of ion Larmor orbit
is the dominant effect, which determines rapid
suppression of the drift turbulence by shear flow.
t
(14)
where ( ),C k t is determined by the equation
( ) ( )
( ) ( ) ( )
62
02 2
2
4
2 1
1 1 1 2
1
, =
8
, ,
y i
y
V tcC t k
B
k
d t C t
ρ
ϕ
ω
′
×∫
k
k k k
k
.
(15)
If we omit linear non-modal terms in Eq.(14), the
condition of the balance of the linear modal growth of the
kinetic drift instability and non-linear non-modal dumping
is determined by the equation . By using
this equation in Eq.(15), we obtain the equation, which
determines the time, at which that balance occurs,
( ) ( )= ,k C k tγ
REFERENCES
( )
( )
( ) ( ) ( )
42
2 12 2
1 1 16 2 2
10
= ,
8
y
y i
kc k d t
BV t
γ
ρ ϕ γ
ω′ ∫
k
k k k
k
. (16)
1. K.H. Burrell. Effects of ExB velocity shear and
magnetic shear on turbulence and transport in
magnetic confinement devices // Phys. Plasmas (4).
1997, v. 5, p. 1499-1518.
2. H. Biglari, P.H. Diamond, and P.W. Terry. Influence of
sheared poloidal rotation on edge turbulence // Phys.
Fluids B, 1990, v. 1, p. 1-3.
The effect of the shear flow reveals in the reducing
with time as the magnitude of the growth rate in
the left part of the balance equation (16). That causes
rapid asselerated suppression of the drift turbulence. This
balance does not correspond to the steady state for drift
turbulence in shear flow. The evolution of drift turbulence
continues on times
( ) 6
0V t −′
st t≥ . It follows by strongly non-
modal way, where Markovian approximation, which is
appropriate for the solution Eq.(14), when the growth rate
and non-modal terms are small with respect to the
frequency ( )kω , ceases be valid.
3. K.C. Shaing, E.C. Crume, Jr., and W.A. Houlberg //
Phys. Fluids B, 1990, v. 6, p. 1492-1498,
4. V.S. Mikhailenko, V.V. Mikhailenko, K.N. Stepanov.
Renormalized theory of drift turbulence in plasma
shear flow // Plasma Physics and Controlled Fusion.
2010, v. 52 p. 055007.
5. V.S. Mikhailenko, V.V. Mikhailenko, K.N. Stepanov.
Turbulence Evolution in Plasma Shear Flows //
Plasma and Fusion Research. 2010, v. 5, p. S2015.
Article received 29.11.10
ПЕРЕНОРМИРОВАННАЯ НЕМОДАЛЬНАЯ ТЕОРИЯ ТУРБУЛЕНТНОСТИ
СДВИГОВЫХ ТЕЧЕНИЙ ПЛАЗМЫ
В.С. Михайленко, В.В. Михайленко, К.Н. Степанов, Н.А. Азаренков
Представлены результаты нелинейных исследований временной эволюции и насыщения дрейфовой
турбулентности в сдвиговых течениях, основанных на немодальном подходе.
ПЕРЕНОРМОВАНА НЕМОДАЛЬНА ТЕОРІЯ ТУРБУЛЕНТНОСТІ
ЗСУВНИХ ТЕЧІЙ ПЛАЗМИ
В.С. Михайленко, В.В. Михайленко, К.М. Степанов, М.О. Азарєнков
Подано результати нелінійних досліджень часової еволюції та насичення дрейфової турбулентності у
зсувних течіях плазми, які базуються на немодальному підході.
V.S. Mikhailenko1, V.V. Mikhailenko3, K.N. Stepanov1,2 , N.A. Azarenkov1
ПЕРЕНОРМИРОВАННАЯ НЕМОДАЛЬНАЯ ТЕОРИЯ ТУРБУЛЕНТНОСТИ
СДВИГОВЫХ ТЕЧЕНИЙ ПЛАЗМЫ
В.С. Михайленко, В.В. Михайленко, К.Н. Степанов, Н.А. Азаренков
ПЕРЕНОРМОВАНА НЕМОДАЛЬНА ТЕОРІЯ ТУРБУЛЕНТНОСТІ
ЗСУВНИХ ТЕЧІЙ ПЛАЗМИ
В.С. Михайленко, В.В. Михайленко, К.М. Степанов, М.О. Азарєнков
|
| id | nasplib_isofts_kiev_ua-123456789-90617 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-24T10:53:33Z |
| publishDate | 2011 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Mikhailenko, V.S. Mikhailenko, V.V. Stepanov, K.N. Azarenkov, N.A. 2015-12-28T20:36:25Z 2015-12-28T20:36:25Z 2011 Renormalized non–modal theory of turbulence of plasma shear flows / V.S. Mikhailenko, V.V. Mikhailenko, K.N. Stepanov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2011. — № 1. — С. 41-43. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 94.05.Lk, 94.20.wf https://nasplib.isofts.kiev.ua/handle/123456789/90617 In our report, we present the results of the non-linear investigations of the temporal evolution and saturation of drift turbulence in shear flows, which have the non-modal approach as their foundation. Подано результати нелінійних досліджень часової еволюції та насичення дрейфової турбулентності у зсувних течіях плазми, які базуються на немодальному підході. Представлены результаты нелинейных исследований временной эволюции и насыщения дрейфовой турбулентности в сдвиговых течениях, основанных на немодальном подходе. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Фундаментальная физика плазмы Renormalized non–modal theory of turbulence of plasma shear flows Перенормована немодальна теорія турбулентності зсувних течій плазми Перенормированная немодальная теория турбулентности сдвиговых течений плазмы Article published earlier |
| spellingShingle | Renormalized non–modal theory of turbulence of plasma shear flows Mikhailenko, V.S. Mikhailenko, V.V. Stepanov, K.N. Azarenkov, N.A. Фундаментальная физика плазмы |
| title | Renormalized non–modal theory of turbulence of plasma shear flows |
| title_alt | Перенормована немодальна теорія турбулентності зсувних течій плазми Перенормированная немодальная теория турбулентности сдвиговых течений плазмы |
| title_full | Renormalized non–modal theory of turbulence of plasma shear flows |
| title_fullStr | Renormalized non–modal theory of turbulence of plasma shear flows |
| title_full_unstemmed | Renormalized non–modal theory of turbulence of plasma shear flows |
| title_short | Renormalized non–modal theory of turbulence of plasma shear flows |
| title_sort | renormalized non–modal theory of turbulence of plasma shear flows |
| topic | Фундаментальная физика плазмы |
| topic_facet | Фундаментальная физика плазмы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/90617 |
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