Transcript スライド 1 - Zhejiang University

Dynamics of ITG driven turbulence in
the presence of a large spatial scale vortex flow
Zheng-Xiong Wang,1J. Q. Li,1 J. Q. Dong,2 and Y. Kishimoto1
1 Graduate School of Energy Science, Kyoto University, Japan
2 Southwestern Institute of Physics and Zhejiang University, China
outline
Introduction: Background
Model and equations
Simulation results:




Linear stabilization of ITG mode by vortex flows (VFs)
Nonlinear dynamics of ITG turbulence w/o zonal flows
Characteristics of Zonal flows in the presence of VFs
Transport structure
Summary
Effects of magnetic island on ITG evolution
Background Interactions in multiple T-S scales
Interaction between a large spatial scale vortex flow
and ion-scale micro-turbulence
Zonal structures
Turbulent transport due to
ITG instability is crucially
important for improved
operation!
Background Internal transport barrier (ITB)
A inner region of reduced anomalous transport
Radial profile of conductivity
Radial profile of pressure
A key factor for ITBs
 EXB shear flows often lead to ITB formation
Background Mean flows (MFs)
MFs
 toroidally and poloidally symmetric
 suppress turbulence by radial flow shear effects
 lead to internal transport barriers
 driven by external momentum source
or by neoclassic effect
y
uy ( x)  f ( x)
y
uy ( x)  f ( x)
eddies
eddy
 Flow shearing
 Mode coupling
 Spectral scattering
(a)
x
(b)
x
Background Vortex flows (large scale VFs)
 Varying in both radial and poloidal directions
VFs
 Often observed in natural and laboratory plasmas
 Inevitable due to the inherent poloidal asymmetry
of equilibrium pressure or magnetic field anisotropy
 Induced by numerous plasma instabilities, such as
Kelvin–Helmholtz (KH) or tearing mode,
KH induced vortex in simulations
Observation in experiments
Moderate scale VFs Large scale VFs
Pegoraro et al., JoP, conference, 2008
M.G. Shats et al., PRL, 2007
Background purpose
In comparison with MFs
What are the roles of VFs in ITG evolution?
 Linear stabilization of ITG mode by VFs
 Direct interaction between VFs and ITG turbulence
 Multiple interactions among turbulence, zonal flows, and VFs
Understanding of these fundamental processes is
crucially desirable but seldom
Model and equations
A stationary large scale VF may be represented
T  TX ( x)sin(kT y)  m f ( x)sin(kT y)
Streamer-like flows(SF)
kT
---poloidal wave number
m
---flow strength
The VFs may then be understood as a
combination of MF and SF structures.
Model and equations: ITG with imposed VFs
k
k
k
2
2
 y
 y
 y  
ky
ky
ky
4 ky
(1      )
 (1  K   )
  v  [ ,   ]    
t
y
y x
2
 vol k y  ks vol k y ks
i s 0
i
k y  ks
k y  ks

[i(k y  ks )vol
 i(k y  ks )vol
]  (iks )s 0 ( x) 

2 x
2

x
x

v
ky
t

  (
ky
 p y )  [ y , v y ]    2 v
k
k
i s 0 ( x)
k k
[i (k y  k s )v y s
2 x
p



ky
k
Multiplied
ky
k y  ks
k y  ks


v

v
i
k k
 i (k y  k s )v y s ]  (ik s )s 0 ( x) 


2
x
x





 y
8
k
k
k
k
k
k
 K
  v y  (  1)
k ( p y   y )  [ y , p y ]    2 p y
t
y

Global
k
k y  ks
k y  ks


p

p


(
x
)
i s0
i
k k
k k

[i (k y  k s ) p y s  i (k y  k s ) p y s ]  (ik s )s 0 ( x) 


2 x
2
x
 x
MF effect
Effect!!!
 x , SF effect ikS (T )
ITG !!!




Satisfying wave matching condition
Model and equations: ITG with imposed VFs
In equations,   1for ZF component and   0 for ITG fluctuations.
Ly  20i
An initial value code with simulation box Lx  100and
i
Finite difference method in x direction with periodic boundary condition
Fourier decomposition in y direction.
Parameters
Normalization
i  2.2
 x y z vti 
( x, y, z, t )   , , , 
 i i Ln Ln 
      0.1
kT  0.1
Ln  e v p 
( , v , p)   , ,

i  Ti 0 vti pi 0 
Numerical results
2-dimensional linear calculations
Linear growth rate of ITG modes versus amplitude of flows
VFs show strong stabilization effect in comparison with MFs and SFs
Numerical results
2-dimensional linear calculations
Wang, Diamond,
Rosenbluth, 1992
MFs
Mechanism
growth rate versus MF
growth rate of
Hamiltonian number L
Linear growth rate of ITG modes versus amplitude of flows
 MFs couple the modes with different radial scale
 The results of MFs agree with the previous works
Hamaguchi and Horton, PoF B, (1992)
Wang, Diamond, Rosenbluth, PoF B, (1992)
Numerical results
2-dimensional linear calculations
Linear evolution of each poloidal mode
usual ITG
With SF
SF
m  3
Linear growth rate of ITG modes versus amplitude of flows
SFs couple together all the poloidal modes
and lead to a global ITG mode with
an identical, greatly reduced growth rate
Growth rate of eachk y
Li and Kishimoto PoP (2008)
Numerical results
Multiplied effect of MF and SF
m  3
VF
Growth rate of each k y
Linear growth rate of ITG modes versus amplitude of flows
Normalized spectra in linear ITG modes
Growth rate versus VF wave number
Numerical results
Effects of magnetic shear s
Decreasing s reduces the stabilizing roles of MS, SF, and VF
Numerical results
Effects of magnetic shear s on role of MFs
W/o MFs
s=0.4
With MFs
W/o MFs
s=0.1
s=0.1
ITG structures
• Decreasing s enlarges the width of ITG linear structure
beyond the effective shear region of MFs
• MFs does not change mode width
Numerical results
Effects of magnetic shear s on role of SFs
W/o SFs
Streamer-like flows(SF)
With SFs
ITG structures
s=0.1
Ｌ ｎｕｍｂｅｒ for each poloidal mode
s=0.1
Numerical results
Effects of magnetic shear s on role of SFs
Small s
Hamaguchi
large s
Horton, 1990
Growth rate of L with magnetic shear s
Terry, et al 1988
Growth rate of L with different s
When s becomes small, the increasing of L number is a destabilizing
Effect, counteracting the stabilizing effect of poloidal mode coupling
induced by SF
Numerical results
Effects of magnetic shear s on role of VFs
VFs structure
W/o VFs
With VFs
ITG structures
s=0.1
s=0.1
Numerical results
(a)
3-dimensional nonlinear simulations w/o ZFs
Ion heat conductivity:
i ( x)[ (i / Ln )(cTi / eB) p / y /(1 i )]
(b)
Contours of electric potential
(a) w/o VF and (b) w VF m  5
Radial profile of conductivity in the quasisteady state.
(a) For m  3 and (b) for m  5
<> the average in y and z directions and in time
VFs can suppress turbulent transport more
effectively than MFs with same amplitude
Numerical results
3-dimensional nonlinear simulations w/o ZFs
Turbulent heat flux F  p / y  p 1/2  / y
1/2
cos 
Suppression by VFs is due mainly to
synergetic decreases of EPF and CPF
rather than PF
This trend is similar to the results of MFs in
toroidal simulations by Benkadda and Garbet
Profiles of electric potential fluctuations
(EPF), pressure fluctuations (PF), and
cross phase factor (CPF) for VFs
X. Garbet, et al., PoP (2002).
C.F. Figarella, et al., PRL (2003).
Numerical results
3-dimensional nonlinear simulations with ZFs
Coupling of VF and
Reynolds
ZF equation
ITG k y  0.1
stress
Viscosity
term
 t  0,0  [ ,   ]0,0  [TX ,  k y 0.1 ]0,0   4 0,0
2
2
low frequency q1
2
finite frequency q 2
q 2  TX  k
y 0.1
Three wave resonance condition
Assumption of zero frequency VF TX  0  q 2
~ ky 0.1
Numerical results
3-dimensional nonlinear simulations with ZFs
Conductivity profile
w/o ZF
Region (I) suppression by VFs
Region (III) suppression by ZFs
ITB
Time history of ZF profile
Region (II) suppression by VFs is
reduced due to finite frequency
Frequency spectra of ZF and k y  0.1
Summary
 VFs can stabilize linear ITG mode more effectively
than MFs due to multiplied effect.
 VFs can suppress turbulent transport more strongly
than MFs with same amplitude.
 An oscillatory ZF is found due to the interaction
between VFs and ITG turbulence.
 The resultant transport structure is consistent with
that of ITBs observed experimentally.
Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ
Island structure
B  B0 Z  By  Bx
k  k0 Z  k y  k x
ITG mode structure
Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ
Equations:
k
k
ky
 
2
2
 y
 y


ky
ky
ky
k
4
y
(1     )
 (1  K  )
  v  [ ,  ]   
t
y
y x
2
v
ky
t
p
  (  p )  [ , v ]   v
ky
ky
t
 K
 f 
ky
ky
ky
2 ky


8
k
k
k
k
k
k
  v y  (  1)
k ( p y   y )  [ y , p y ]    2 p y
y

ky
1
1
1
Bf  (B0 f  Bf )  (B0 f  eˆz f )
B0
B0
B0

1
1
f
f
(B0 f  Bf )  (B0 z
 B0 y
 eˆz f )
B0
B0
z
y

1
f
f 1
(B0 z
 B0 y
 [ , f ])
B0
z
y 
Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ
1 2
  x  s ( x) cos(ks y )
2
ks
Poloidal wave number of island
 f  f
[ , f ] 

x y y x
 f 1  
k y  ks
k y  ks
ik y


i
(
k

k
)
f

i
(
k

k
)
f
e
y
s
y
s

x y 2 x 
 f i

 ks
y x 2
 f k y  ks f k y  ks


x
 x
 ik y
e

k
k
ky
 
2
2
 y
 y


ky
ky
ky
k
4
y
(1     )
 (1  K  )
  v  [ ,  ]   
t
y
y x
2
Coupling mechanism: parallel compressibility or acoustic wave
Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ
w/o island
2D linear calculation
with island
Time evolution of poloidal modes
Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ
2D linear calculation
ITG growth rate versus island width
Island can stabilize
linear ITG mode
Linear spectra of ITG mode in different
island width
Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ
2D nonlinear simulation without ZFs
Time evolution of heat conductivity in the whole system
The presence of island will increase heat transport
Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ
2D nonlinear simulation without ZFs
Magnetic
field
Contour of
conductivity
Contour of
potential Fluctuation
Contour of pressure
Fluctuation
O
X
Conductivity is enhanced around X point but O point of island
Further work
 Ｅｆｆｅｃｔ ｏｆ ｉｓｌａｎｄｓ
Dynamics of ZFs in the presence of islands (2d/3d
simulation)
 Combined effects of vortex flows and islands