Plasma turbulence in localized sheared flows
Plasma configurations with strong velocity shear are considered taking into account magnetic shear. Ion temperature gradient driven drift instability and Kelvin-Helmholtz type instability are analyzed for such sheared plasma flows. Possible instability saturation mechanisms and estimation of particl...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2007
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| Cite this: | Plasma turbulence in localized sheared flows / A.Yu. Chirkov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 58-60. — Бібліогр.: 6 назв. — англ. |
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Chirkov, A.Yu 2017-01-04T08:05:20Z 2017-01-04T08:05:20Z 2007 Plasma turbulence in localized sheared flows / A.Yu. Chirkov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 58-60. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 52.35.Ra; 52.35.Qz; 52.35.Mw https://nasplib.isofts.kiev.ua/handle/123456789/110383 Plasma configurations with strong velocity shear are considered taking into account magnetic shear. Ion temperature gradient driven drift instability and Kelvin-Helmholtz type instability are analyzed for such sheared plasma flows. Possible instability saturation mechanisms and estimation of particle diffusivity are discussed. Розглядаються плазмові конфігурації із сильним широм швидкостей, а також з обліком магнітного шира. Для таких течій плазми аналізуються іонно-температурно-градієнтна нестійкість і нестійкість типу Кельвіна-Гельмгольца. Обговорюються механізми насичення нестійкостей і оцінки коефіцієнта дифузії. Рассматриваются плазменные конфигурации с сильным широм скоростей, а также с учетом магнитного шира. Для таких течений плазмы анализируются ионно-температурно-градиентная неустойчивость и неустойчивость типа Кельвина-Гельмгольца. Обсуждаются механизмы насыщения неустойчивостей и оценки коэффициента диффузии. Work was supported by Russian President grant MK-3755.2005.8. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Plasma turbulence in localized sheared flows Турбулентність плазми в локалізованих ширових течіях Турбулентность плазмы в локализованных шировых течениях Article published earlier |
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Plasma turbulence in localized sheared flows |
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Plasma turbulence in localized sheared flows Chirkov, A.Yu Basic plasma physics |
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Plasma turbulence in localized sheared flows |
| title_full |
Plasma turbulence in localized sheared flows |
| title_fullStr |
Plasma turbulence in localized sheared flows |
| title_full_unstemmed |
Plasma turbulence in localized sheared flows |
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plasma turbulence in localized sheared flows |
| author |
Chirkov, A.Yu |
| author_facet |
Chirkov, A.Yu |
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Basic plasma physics |
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Basic plasma physics |
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2007 |
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English |
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Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Турбулентність плазми в локалізованих ширових течіях Турбулентность плазмы в локализованных шировых течениях |
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Plasma configurations with strong velocity shear are considered taking into account magnetic shear. Ion temperature gradient driven drift instability and Kelvin-Helmholtz type instability are analyzed for such sheared plasma flows. Possible instability saturation mechanisms and estimation of particle diffusivity are discussed.
Розглядаються плазмові конфігурації із сильним широм швидкостей, а також з обліком магнітного шира. Для таких течій плазми аналізуються іонно-температурно-градієнтна нестійкість і нестійкість типу Кельвіна-Гельмгольца. Обговорюються механізми насичення нестійкостей і оцінки коефіцієнта дифузії.
Рассматриваются плазменные конфигурации с сильным широм скоростей, а также с учетом магнитного шира. Для таких течений плазмы анализируются ионно-температурно-градиентная неустойчивость и неустойчивость типа Кельвина-Гельмгольца. Обсуждаются механизмы насыщения неустойчивостей и оценки коэффициента диффузии.
|
| issn |
1562-6016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/110383 |
| citation_txt |
Plasma turbulence in localized sheared flows / A.Yu. Chirkov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 58-60. — Бібліогр.: 6 назв. — англ. |
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AT chirkovayu plasmaturbulenceinlocalizedshearedflows AT chirkovayu turbulentnístʹplazmivlokalízovanihširovihtečíâh AT chirkovayu turbulentnostʹplazmyvlokalizovannyhširovyhtečeniâh |
| first_indexed |
2025-11-24T16:02:15Z |
| last_indexed |
2025-11-24T16:02:15Z |
| _version_ |
1850850513900273664 |
| fulltext |
58 Problems of Atomic Science and Technology. 2007, 1. Series: Plasma Physics (13), p. 58-60
PLASMA TURBULENCE IN LOCALIZED SHEARED FLOWS
A.Yu. Chirkov
Bauman Moscow State Technical University,
2nd Baumanskaya Str. 5, 105005, Moscow, Russia, e-mail: chirkov@power.bmstu.ru
Plasma configurations with strong velocity shear are considered taking into account magnetic shear. Ion temperature
gradient driven drift instability and Kelvin-Helmholtz type instability are analyzed for such sheared plasma flows.
Possible instability saturation mechanisms and estimation of particle diffusivity are discussed.
PACS: 52.35.Ra; 52.35.Qz; 52.35.Mw
1. INTRODUCTION
The objective of this work is to study instabilities and
features of turbulence in localized plasma flows.
Turbulence depends on strong velocity shear sufficiently
in the case under consideration. Magnetic shear is taken
into account, too. Ion temperature gradient driven drift
instability (ITG instability) and Kelvin-Helmholtz type
instability (KH instability) are of the interest in this work.
These instabilities are considered in framework of non-
local approach. Eigenfrequencies and eigenfunctions for
ITG modes are calculated using wave equation resulting
from gyrokinetic approach [1]. For KH instability we use
wave equation [2] in low-frequency limit (|ω|<<ωci, ωci is
ion cyclotron frequency). To estimate amplitudes of the
wave modes and particle diffusivity the model of particle
interactions with waves [3] can be used.
2. ITG MODES
For the calculation we use the slab model, where the
magnetic field is given by
B e e= +
B x
Lz
s
y , (1)
where B is the absolute value of the magnetic field, x is
the “radial” distance from considered (resonant) surface,
1)/( −= dxdBBL yzs is scale length of the magnetic
shear, ez is the unity vector directed along the magnetic
field, and ey is the transversal unity vector. For localized
modes in such a geometry the component of the wave
vector parallel to the magnetic field is sy Lxkk /|| = ,
where ky is the transversal component of the wave vector.
For the radial profile of the wave potential Φ(x) one
can use the wave equation [1]
0),(
2
2
=Φ+
Φ
ωxQ
dx
d , (2)
where
Ω
+
+ΩΩ
−
+Ω
Ω−
+−= −
2
22
22
~)~(~~
~1),( xs
K
sxsu
K
kxQ uz
sy ρω , (3)
ey
yy
Vk
uk
*
1
~~ −
=Ω
ω
, yyuk 0
~ −= ωω , ω is the complex
frequency, 01 )( uuu −= x , u(x) and u0 are the ion flow
velocity and it’s value at x=0, )/(* neBe eBLTkV = ,
)/)(/1( eiTn TTLLK += , 1)/( −−= dxdnnLn is the scale
length of density gradient, 1)/( −−= dxdTTL iiT is the
scale length of the ion temperature gradient, 1−= sxx ρ ,
eBis TkmBe 11 −−=ρ , e is the electron charge, mi is the
ion mass, kB is the Boltzmann constant, Te is the electron
temperature, Ti is the ion temperature, un LLs /= ,
1
,,0 )/( −= dxduuL zyzyu is the scale length of the ion
velocity shear.
Eq. (3) for Q function obtained under the following
condition:
1~<Tiyk ρ , Tiρδ >⊥
~ , ωυ ~
|| <Tik ,
where ρTi is the thermal ion gyroradius, δ⊥ is the radial
width of the radial profile Φ(x), υTi is the thermal velocity
of the ions.
We consider sheared flow velocity profiles
−=
2
2
,0,
2
exp
b
xuu zyzy , (4)
where b is the width of sheared flow.
To find eigenfrequencies and solve wave equation we
use WKB method. Corresponding dispersion equation is
πω
ω
ω
)2/1(),(
)(
)(
+=∫
+
−
ldxxQ
x
x
, (5)
where )(ω−x and )(ω+x are zeros of the Q function
(turning points), l is radial mode number. WKB solution
(radial eigenfunction) is ( )∫±=Φ − dxQiQx exp)( 4/1 .
In Figs. 1 and 2 results of the calculations of the
growth rate )Im(ωγ = are presented. The scale of the
growth rate in these figures is 1
*
−= ses V ρω .
mailto:chirkov@power.bmstu.ru
59
-2 -1 0 1 2
0
0.1
0.2
0.3
0.4 γ/ωs
u0y/V*e
2
1
4
56
3
Fig. 1. Influence of perpendicular flow shear on ITG
growth rate (no parallel flow shear):
1 – s = 0.1, K = 4, kyρs = 0.5, b = 3ρs; 2 – s = 0.1, K = 4,
kyρs = 0.5, b = 10ρs; 3 – s = 0.1, K = 4, kyρs = 0.1, b = 3ρs;
4 – s = 0.1, K = 4, kyρs = 0.8, b = 3ρs; 5 – s = 0.01, K = 4,
kyρs = 0.5, b = 3ρs; 6 – s = 0.01, K = 2, kyρs = 0.5, b = 3ρs
-3 -2 -1 0 1 2 3
0
0.1
0.2
0.3
γ/ωs
u0zLn/(υsρs)
3
2
1
4
Fig. 2. Influence of parallel flow shear on ITG growth
rate (no perpendicular flow shear):
1 – s = 0.1, K = 4, kyρs = 0.5; 2 – s = 0.1, K = 4, kyρs = 0.1;
3 – s = 0.1, K = 4, kyρs = 0.8; 4 – s =0.01, K = 4, kyρs = 0.5
––––––– – b = 10ρs, – – – – – – b = 3ρs
0 5 10 15
0
4
8
12
16
ω R1/ω 0
kyb
a
0 5 10 15
0
0.4
0.8
1.2
1.6
γ /ω 0
kyb
b
Fig. 3. Real frequencies (a) and growth rates (b) of KH modes l = 0 (–––––––), l = 1 (– – – – –) and l = 2 (– - – - – - –)
3. KH MODES
In transport barriers strong sheared radial static
electric field Er < 0 is usually observed [4, 5]. In this case
perpendicular sheared velocity 0/ >−= BEu ry leads to
sufficient dumping of ITG turbulence (see Fig. 1). But,
Kelvin-Helmholtz type instability can be induced by
strong E×B velocity shear.
Wave equation for KH eigenfunctions has a form
Eq. (2) and it can be solved using WKB theory. In low-
frequency limit Q function is
)(/
)(
),(
''
2
xVk
xV
kxQ
Ey
E
y −
+−=
ω
ω , (6)
where VE(x) is E×B drift velocity.
Here we consider the following flow velocity profile:
)/1( 22
0 bxVVV EE −∆+= inside flow ( bx <|| ), and
VE = V0 = const outside flow ( bx >|| ).
Results of the calculations of the real frequency
60
)Re( 01 Vk yR −= ωω and the growth rate )Im(ωγ = are
presented in Fig. 3. Scale the of real frequency an the
increment is bVE /0 ∆=ω . These results obtained for the
case when radial profile is centered in shear layer. Radial
structure of low-frequency KH modes was considered in [6].
For 1>>bk y turnpoints are fare from the boundaries
of the flow shear layer. They rich boundaries if bk y
decrease down to some minimal value 1~)( minbk y . For
min)( bkbk yy < there are no solutions with given flow
velocity profile.
4. PARTICLE DIFFUSIVITY
In this chapter we estimate particle diffusivity and
saturation level in framework particle-waves interaction
calculations [3].
For steady-state turbulence one can use the instability
growth and dumping rates balance:
>∆<++= ⊥
−
⊥ yyy ukDk
2
1)( 22 δγ , (7)
where D⊥ is the radial particle diffusivity,
+
∆>≈∆<
2
22
2 )(1)(1
||2
1
dr
rud
rdr
rud
r
k
ru yy
R
y
y ω
(8)
is perpendicular drift velocity averaged over particle drift
orbit, ||/6 RDr ωπ ⊥=∆ is the radial diffusion shift of
the particle during oscillation period, ωR is real frequency
(taking into account Doppler shift), r is the radial coordinate.
One can suppose for strong E×B shear case that KH
instability in dominant source of turbulence. Using
calculation results (presented in Fig. 3) and Eq. 7 we
estimate maximal collisionless diffusivity as
bVD E∆−
⊥
310~ at 1~bk y .
Work was supported by Russian President grant MK-
3755.2005.8.
REFERENCES
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temperature gradient modes in the presence of sheared
flows // Phys. Fluids. 1992, v. B4, p. 1102–1114.
2. G. Ganguli, Y.C. Lee, P.J. Palmadesso. Kinetic theory
for electrostatic waves due to transverse velocity shears
// Phys. Fluids. 1988, v. 31, p. 823–838.
3. V.I. Khvesyuk, A.Yu. Chirkov. Analysis of the
mechanisms for the scattering of plasma particles by
non-steady-state fluctuations // Tech. Phys. 2004, v. 49,
N 4, p. 396–404.
4. R.C. Wolf. Internal transport barriers in tokamak
plasmas // Plasma Phys. Control. Fusion. 2003, v. 45,
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5. J.W. Connor, T. Fukuda, X. Garbet et al. A review of
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