Effects of Three-Dimensional Geometry and Radial

Electric Field on ITG Turbulence and Zonal Flows

GROKINETICS IN LABORATORY AND ASTROPHYSICAL PLASMASWorkshop “Kinetic-scale turbulence in laboratory and space plasmas: emprical constraints,

fundamental concepts and unsolved problems” Issac Newton Institute for Mathematical Sciences, Cambridge, July 23, 2010

H. Sugama

National Institute for Fusion Science,Graduate University of Advanced Studies

Toki 509-5292, Japan

in collaboration with T.-H. Watanabe M. Nunami

OUTLINE

Introduction

Linear ITG Mode Analysis for High-Ti LHD plasmas

Zonal Flows and ITG Turbulence in Helical Systems

Effects of Equilibrium Electric Field Er on Zonal Flows in Helical Systems

Momentum Balance and Radial Electric Field in Quasisymmetric Systems with Stellarator Symmetry

Summary

Introduction

Tokamak

Helical System

B = B0 [ 1 − εt cos θ − εh cos (Lθ −Mζ) ]

B

Classification ofparticle orbits

passing

passing

trapped

trapped

!

vdr b

B = B0 (1 − εt cos θ )

vdr radial drift of helically-trapped particles

toroidally-trapped particles

Watanabe et al. NF2007

Eigenfunction of linear ITGmode electrostatic potential

Sugama & Watanabe PoP2006

Zonal-flow response(GAM, residual ZF)

Helical geometry influences ITG mode and zonal flow.

Gyrokinetic Equations (for ITG Turbulence)

!

"

"t+ v

||ˆ b # $ + vd # $ %µ ˆ b # $&( ) "

"v||

'

( )

*

+ , -f +

c

B0

.,-f{ } = v/ % vd % v||ˆ b ( ) # e$.

TiFM + C -f( )

!

v" = #cT

i

eLnB

0

1+$i

mv2

2Ti

#3

2

%

& '

(

) *

+

, -

.

/ 0 ̂ y , µ =

v12

22

!

J0

k"v" #( )$f d3v% & 1&'

0k"

2( )[ ]e(

Ti

=e

Te

( & (( ) , k"2 = kx + ˆ s zky( )

2

+ ky

2

Diamagnetic drift

Quasineutrality condition & Adiabatic electron assumption

Ion gyrokinetic equation for

Ion polarization

!

k"#i $1, k"#e <<1

δ f(x, v||, µ, t)

!

vd" #Gyrocenter drift

Mirror force

!

"µ b # $%( ) & /&v||

Effects of magnetic geometry

Linear ITG Mode Analysis forHigh-Ti LHD plasmas

Results from Linear ITGMode Analyses by GKV-X

(See Poster by M. Nunami)

Zonal Flows and ITG Turbulencein Helical Systems

For low collisionality, better confinement is observed in theinward-shifted magnetic configurations, where lower neoclassicalripple transport but more unfavorable magnetic curvature drivingpressure-gradient instabilities are anticipated.

H. Yamada et al. (PPCF2001)

Scenario:Neoclassical optimization contributesto reduction of anomalous transportby enhancing the zonal-flow level.

10-1

100

10-1 100 101

Rax=3.6m

Rax=3.75m

Rax=3.9m

!Eexp/!E

ISS95

Reactor

condition

"*b

Anomalous transport is alsoimproved in the inwardshifted configuration.

Results from LHD experiments

Inward-shifted

Standard

Outward-shifted

!

KL (t) =1" (2 /# )1/ 2 (2$H )

1/ 2{1" gi1(t,%}

1+G + E(t) n0k&2'ti

2( )

Collisionless Time Evolution of Zonal Flows in Helical Systems

Response of the zonal-flow potential to a given initial potential

Response function = GAM component + Residual component

GAM response functionLong-time response function

[Sugama & Watanabe, PRL (2005), Phys.Plasmas (2006)]

k ρi < 1

E(t) represents effects of shielding of potential due to helical-ripple-trapped particles.

!

"k(t) = K(t) "

k(0)

!

K(t) = KGAM

(t)[1"KL(0)]+ K

L(t)

!

K(t)"KL(t), K

GAM(t)" 0 as t" +#

!

K(t = 0) =1

!

KGAM

(t) = cos("G)exp(# | $ | t)

(L=2, M=10)

B = B0 [ 1 − εt cos θ − εh cos (Lθ −Mζ) ]

!

E(t) =2

"n0(2#H )

1/ 2{1$ gi1(t,%)} $

3

2k&2'ti

2(2#H )

1/ 2{1$ gi1(t,%)}

(

) *

+Ti

Te(2#H )

1/ 2{1$ ge1(t,%)}

+

, -

Linear time evolution of zonal-flow potential

Smaller χi and larger zonal flows arefound in the saturated turbulent state forthe inward-shifted configuration than forthe standard one !

Turbulent thermal diffusivityand squared zonal-flow potential

Larger residual zonal flow is foundfor the inward-shifted case.

!"#$

"

"#$

"#%

"#&

"#'

(

" (" $" )" %" *" &" +" '"

Standard ( kr ! = 0.25)

Inward ( kr ! = 0.28)

,"-./0123,"

-."12

/000.Ln034

/0i001

inward-shiftedinward-shifted configurationconfiguration standard configurationstandard configuration

Results from GKV simulation ( flux tube, Er = 0 )

Watanabe, Sugama & Ferrando, PRL(2008)Sugama, Watanabe & Ferrando, PFR(2009)

The GKV turbulence simulations were carriedout by the Earth Simulator (JAMSTEC).

ITGturbulence

Potential contours obtained from six copies of flux tube

Effects of Equilibrium Electric Field Er

on Zonal Flows in Helical Systems

In helical systemsEr is given from ambipolar condition of radial particle fluxes.Er reduces neoclassical ripple transport.

How does Er influence zonal flows and anomalous transport?

Effects of Er on gyrokinetic equation and zonal flows

Gyrokinetic equation for

!

k" = kr#r

!

"

" t+ v||

ˆ b # $ + ik r # vd %µ ˆ b # $&( )"

"v||

+'E

"

"(

)

* +

,

- . /f = %ik r # vd

e 0(x + 1)

TiFM

!

"E

= #c E

r

r0B0

angular velocity due to ExB drift

!

" = # $% /q

gyrophase average ofzonal-flow potential

field line label

In helical systems, α -dependence appears in and .

!

µ ˆ b " #$( )

!

kr" v

d

Therefore, even if the zonal-flow potential φ is independent of α , δ f comes to depend on α .

Thus, ωE influences δ f and accordingly φ through quasineutrality condition.

Classification of particle orbits in the presence of Er

θ

κ2 trapping parameter

π−π 0

κ2=1

passinghelicallytrapped(closed)

ExB drift

ΔE

helicallytrapped(unclosed)

toroidallytrapped

ExB drift transition

pointsΔE

radial displacement ofhelically-trapped particle

!

"E~ r

0

vdr

vE#B

!

" 2 =1# $B

0[1#%

T(&) #%

H(&)]

2$B0%H(&)

trapping parameter

Cary et al., PF (1988)Wakatani (1998)

!

" =1

2mv

2 µ

passing

helically trapped (closed)

helically trapped(unclosed)

toroidally trapped

Solution of gyrokinetic equation to describe long-timeevolution of zonal flows [Sugama & Watanabe, PoP(2009)]

!

" fk (t) = #e

T$k (t)FM 1# e

# i kr % r e#i kr & r e

i kr & r J0(kr%r )orbit

[ ]

+ e#i kr % r e# i kr & r ei kr & r " fk

(g )(0) + FM Sk (t)dt

0

t

'[ ]orbit

Perturbed particle distribution function

Polarization(classical & neoclassical)

Initial condition &Turbulence source

!

Lorbit

= Ldl

(dl /dt)"

dl

(dl /dt)"

Average along the orbit

For particles which show transitions

!

dl

(dl /dt)" = (1# P

t)

dl

v||$ 2>1v ||>0

" +dl

v||$ 2>1v ||<0

" + Pt

d%

&E$ 2<1

"

toroidallytrapped

helicallytrapped

Pt transition probability

1-Pt

helicallytrapped

toroidallytrapped

ExB drift transition

points

!

"rL

#rL

gyro motiondrift motion

1-Pt

Pt 1

!

Pt "4 2

#

c Er

v B0

$

% &

'

( ) R0q

r0

$

% &

'

( ) *H( )

+1/ 2 ,*H ,-

, *H +*T( ) ,-

.

/ 0

1

2 3 P+

Zonal-flow generation can be enhanced whenGht and Gh decreases with neoclassical optimization (which reduces radial drift velocity vdr )and when poloidal Mach number increases

with increasing Er and using heavier ions.

Long-time zonal-flow response to theinitial condition and turbulence source

!

e

Ti"k (t) =

n0#1

d3v 1+ i kr $ i r # $ i r

orbit( )[ ] % f i k(g )(0) + FM Sik (t)dt0

t

&[ ]&

kr'ti( )21+Gp +Gt + Mp

#2(Gh t +Gh )(1+ Te /Ti)[ ]

For radial wavenumbers krρti < 1 (ITG turbulence) and krΔE < 1, the zonal-flowpotential is derived from the quasineutrality condition as [Sugama & Watanabe, PoP(2009)]

!

Mp " (cEr B0) (rvt i Rq)

Geometrical factors G’s represents shielding effects of neoclassical polarization due toparticles motions in different orbits.

Gp : passingGt : toroidally-trapped

Ght : helicallly-trapped (unclosed orbit)

Gh : toroidally-trapped (closed orbit)

!

G " (population)

# $r%( )

2

No transitions occur.

Response to the initial condition

!

"k (t) ="k (0)

1+Gp +Gt + Mp

#2(Gh t +Gh )(1+ Te /Ti)

Assume the initial distribution to have Maxwellian dependenceThen, we obtain

For the single-helicity configuration

!

B = B0[1"#t cos$ "#h cos(L$ "M% )] (#h: independent of$)

!

" fi k(g )(0) = [" nk

(g )(0) /n

0]FM

" nk(g )(0) /n

0= (kr#ti)

2e$(0) /Ti

(no turbulence source)

!

"k (t) ="k (0)

1+Gp +Gt + (15# /4)Mp

$2q2(2%h )

1/ 2(1+ Te /Ti)!

Gh t = 0, Gh = (15" /4)q2(2#h )1/ 2

This corresponds to the case considered by previous works.Mynick & Boozer, PoP(2007)Sugama, Watanabe & Ferrando, PFR(2008)

The residual zonal-flow potential as a function of kr ρtifor Mp = 0 and Mp = 0. 3

Theoretical results are derived

by assuming kr ρti << 1 .

Different kr ρti dependences

for Mp = 0 and Mp = 0. 3 aretheoretically predicted andconfirmed by simulation.

Inward-shited configuration

[To be published in CPP]

Momentum Balance and Radial Electric Field inQuasisymmetric Systems with Stellarator Symmetry

Basic Boltzmann Kinetic Equation for description ofCollisional and Turbulent Transport

Equilibrium magnetic field

Boltzmann kinetic equation

Ensemble-averaged kinetic equation

Classical, Neoclassical, and Anomalous Transportof Particles and Heat [Sugama et al. PoP1996]

The gyrophase (ξ) -average part and the oscillating part of an aribtrary

function F is defined by and respectively.

The ensemble-averaged kinetic equation is divided as

Particle flux

Heat flux

Second order part of in δ ~ ρ / L

Momentum Balance

density particle flux

pressure tensor

friction force

turbulent electromagnetic force

Momentum Balance in the direction tangential to the flux surface

!

"

" tnam

ac1ua# +

(SEM)#

c2

$

% &

'

( ) + c2 ua* +

(SEM)*

c2

$

% &

'

( )

+ , -

. / 0 a

1

= 21

V '

"

" sV ' 3s 4 P

a

a

1 2TEM

$

% &

'

( ) 4 c1

" x

"#+ c

2

" x

"5

$

% &

'

( )

6

7 8 8

9

: ; ;

+1

c2c

1<'+c

2='( ) e

anaua

s

a

1

(c1, c2 : constants)

(s, θ, ζ) : Hamada coordinates

The surface-averaged radial current

Quasisymmetry[Boozer(1983), Nuhrenberg(1988), Helander&Simakov (2008)]

quasi-axi-symmetry (c1, c2) = (0, 1) quasi-poloidal-symmetry (c1, c2) = (1, 0)

The ambipolarity

is satisfied automatically up to O(δ).

The O(δ) viscosity component in the quasisymmetrydirection vanishes :

Stellarator Symmetry

Magnetic field strength

Metric tensor components

Magnetic field components

Parity Transformation associated with Stellarator Symmetry

Expansion in η ~ δ ~ ρ / L

Parity operator is defined by

In the presence of stellarator symmetry, Boltzmann andMaxwell equations are invariant under parity transformation

!

ea"#$1

ea

(Put in Boltzmann and Maxwell eqs.)

Momentum Transport Fluxes in Stellarator Symmetric Systems

Parity of solutions

When j is even, the O(δ j ) part of radial transport fluxesof poloidal and toroidal momentum vanish.

Momentum Balance in Quasisymmetric Systemswith Stellarator Symmetry

In quasisymmetric systems with stellarator symmetry,the momentum transport fluxes vanish up to O(δ 2 ), andthe ambipolarity is automatically satisfied up to O(δ 2 ).

!

= "1

V '

#

# sV ' $s % P

a

(3)

a

& "TEM

(3)'

( )

*

+ , % c1

# x

#-+ c

2

# x

#.

'

( )

*

+ ,

/

0 1 1

2

3 4 4

The momentum balance equation determing Es is of O(δ 3 ) :

Summary Fluctuations observed in a high Ti LHD plasma are considered as ITG modes predicted from linear calculation by GKV-X.

Zonal-flow response theory and simulation show that zonal flow generation and turbulence regulation are enhanced when the radial displacements of helical-ripple-trapped particles are reduced either by neoclassical optimization of the helical geometry lowering the radial drift velocity or by strengthening the radial electric field Er to boost the poloidal rotation.

The Er effects appear through the poloidal Mach number Mp. For the same magnitude of Er, higher zonal-flow response is obtained by using ions with heavier mass (favorable deviation from gyro-Bohm scaling).

The momentum balance equation determining Er in quasisymmetric helical system with stellarator symmetry is shown to be of O(δ 3 ) by using a novel parity operator.