Effects of Flow on Radial Electric Fields

Shaojie WangDepartment of Physics, Fudan University

Institute of Plasma Physics, Chinese Academy of Sciences

Outline

• Introduction

• Basic equations

• Zonal flows in rotating systems

• Summary

1. Introduction

• The dynamics of zonal flows (ZFs) is very important in tokamak fusion plasma physics researches, because the flow shear can suppress the drift-type turbulence that degrades the confinement performance.

• ZFs are electrostatic perturbations with the spatial structure of toroidal symmetry and poloidal symmetry.

• Two branches of ZFs: the low-frequency branch (ω~0), and the high-frequency branch (ω~c_s/R). that is also known as the Geodesic Acoustic Mode (GAM).

• In a non-rotating system, ZFs are linearly stable and the GAMs are standing waves.

• There exists an Equilibrium Toroidal Rotation Flow in a tokamak plasma.

• Clearly, it is of great interest to investigate the effects of ETRF on ZFs and GAMs.

2. Basic Equations

governing equations

0 ut

Bjpuut

0 put

0 BuE

The equilibrium solution with an ETRF

20 Ru T

T

RmmN Tii 2exp

22

0

imTp /2 00

ddT 0

Two components of the momentum equation

0 puuB t

02

puuB

B

dRtp

p

Linearized Equations

00011 vut

01000001 puvvuuuB t

01000001

2

puvvuuuB

B

dRtP

P

0/11 02

12

1 vccp stst

3. Zonal Flows in Rotating Systems

• Perturbation form

• Large-aspect-ratio tokamak • Ordering ansatz

~/ sE cv~/|| scv ~/ 01

~/ 01 pp ~/ tscr

RBRdrdvE // 01

]exp[]cossin[ tfff cs

Eigenmode Equation

012

||,0

02

20

2

0

0,1

cE

s

Ts v

qRv

c

R

Ri

0||,0

0,1 sc vqR

i

012 002

,12

,1 ETsss vRcipi

0,12

,1 csc cipi

01

,100

||, cETs pqR

vvi

01

,100

||, sc pqR

vi

02

1,1

0

02

,100

||, sT

ssTE

Rp

Rvvi

Dispersion relation

,01

22

112

1 42

242

42

2

TT M

qM

qq

scR /0

sTT cRM /0

Solution to the dispersion relation

22 /1 qSW

2/1

1200

2 FFFGAM

2/1

1200

22 FFFZFZF

420 22

11, TT M

qMqF

421

1, TT M

qMqF

• When the speed of ETRF approaches the sound speed, the GAM frequency is significantly reduced and becomes sensitive to the safety factor.

• The low-frequency branch of zonal flows is linearly unstable in a rotating system, while it is linearly stable in a non-rotating system.

• When the speed of ETRF approaches the sound speed, the linear growth-rate of the ZFs in a rotating system can exceed the SW frequency, which is comparable to the collisional damping rate of ZFs.

Eigenfunction

TT

SWs

c

MMi

1

2

1

,1

,1

2

2

22,1

,1

11

2/

GAM

TT

GAMT

GAMs

c

M

qM

qiM

2

2

22,1

,1

11

2/

ZF

TT

ZFT

ZFs

c

M

qM

qM

• SWs, GAMs and ZFs are poloidal standing waves in a non-rotating system.

• In a rotating system SWs and GAMs can propagate in the poloidal direction.

)exp()cossin( ,1,11 tics

4. Summary

• The low-frequency branch of zonal flows, which is linearly stable in the non-rotating system [2], becomes linearly unstable in a rotating system due to the centrifugal force and the induced poloidal asymmetry of the equilibrium plasma pressure distribution.

• This new result may be applied to analyze the physics of transport barrier control by tangential neutral beam injection.

• GAM frequency in a rotating system is lower than in a non-rotating system; in the regime of sonic toroidal rotation, the GAM frequency is significantly reduced and becomes sensitive to the safety factor.

• This new result may be applied to resolve the GAM frequency scaling issue raised by recent experimental observations.

• SWs and GAMs, which are poloidal standing waves in a non-rotating system, can propagate in the poloidal direction in a rotating system.