Basis for Classical , Neoclassical and Anomalous Transports in Torus Plasmas

Heiji Sanuki

中国科学院等離子体物理研究所外籍特聘研究員（２００９）and

Visiting Professor of ASIPP and SWIP

Lectures in ASIPP, 2011　 and 2012, May~June

　 　 Basis for Classical, Neoclassical and Anomalous Transports in Torus Plasmas　　　　　　　 Heiji sanukiPart 1:

Basis of transport theory, Classical DiffusionPart2-1:Neoclassical Transport in Tokamaks and Helical Systems Part2-2:Fluctuation Loss, Bohm, Gyro-Bohm Diffusion and Convective Loss etc.

Understanding of physics in transport process is one of the key issues associated with confinement improvement and/orDevice performance in fusion plasmas.In this lecture, some fundamental equations and physics involved in transport process are briefly reviewed.

　 　 Fluctuation Loss, Bohm, Gyro-Bohm Diffusion and Zonal Flow and GAM

PartII-1 and II-2

References1) Kenro Miyamoto: NIFS-Proc. 80, Chapt.10, pp157~pp1872) Kenro Miyamoto: NIFS-Proc. 88, Chapt.7, pp76~pp81 App. E, pp328~341(in Japanese)3) J. Q.Dong ,H.Sanuki and K.Itoh,PoP(2001),NF(2002,2003) 4) Zhe Gao , H.Sanuki, K.Itoh and J.Q.Dong, PoP(2003,2004,2005)5) Zhe Gao, K.Itoh, H.Sanuki and J.Q.Dong,PoP(2006)6) H.Sanuki and Jan Weiland(1979)7) Jan Weiland ,H.Sanuki and C.S.Liu(1980)8) Lots of references involved in this lectures

Topics (PartII-1)(a)Diffusion caused by Ion Temperature Gradient(ITG) modes ,Two cases, (1) mode overlapping and (2) no overlapping (localized)(b) Simulation Models for Turbulent Plasmas(c) Gyro-Kinetic Particle Simulation Model for trapped electron drift modes(d) Gyro-Kinetic Particle Simulation Model for ITG Modes(e) Full Orbit Particle Model(f) Experimental Observations of Heat Diffusivity(g) Brief View of Gyro-Kinetic and Gyro-Fluid Simulation of ITG and ETG Modes(h) Parameter Dependence of Critical Temperature Gradient associated with Heat Conductivity

Topics(PartII-2)(a)Topics on Zonal Flow and Geodesic Acoustic Mode(GAM)(b) Hasegawa-Mima equation in turbulent plasmas(c) Electromagnetic Drift wave Turbulence and Convective Cell Formation( Weiland-Sanuki-Liu Model)(d)Characteristics of Hasegawa-Mima Equation(e) Zonal Flow Generation Mechanism(f) Self-Regulation and Dynamics for Zonal Flow in Toroidal Systems(g) Overview of Recent Progress in Zonal Flow and GAM Studies(h) Eigenmode Behavior of GAM(i) Experimental Observations in EAST and other Tokamaks

PartII-1

From K.Miyamoto

Mixing Length Theory

　 Bohm, Gyro-Bohm Diffusion 　　　　

Bohm Diffusion

2

2

0

2

0

~~

n

k

k

ke

k

k

kn

ky

n

n

eB

T

n

nkD

In saturation level of turbulence, 1ck

,16

1

)(~ 2

2

2

02

eB

TD

x

kn

nD

eB

cx

kk

n

k

Bohm Diffusion Coefficient

　 　 Example: Diffusion caused by ITG modes

References:1) S. Hamaguchi and W. Horton, PF B4,319(1992)2) W .Horton et al., PF 21, 1366(1978)3) F. Romanelli and F. Zonca, PF B5,4081(1993)4) Y. Kishimoto et al., 6th IAEA vol.2 581(1997)

Mode fluctuation potential

p

mt

ii

mn

Lk

qR

rrkRq

s

rq

mn

RR

n

B

B

r

m

B

Bbi

r

mk

Rinzimrzr

11

)()(

1k

1~krate,growth maximumat ,8.0~7.0 ,

)/exp()(),,(

//

//

qR: connection length

　 Example: Diffusion caused by ITG modes (continued)

skr

nn

m

n

mrqrq

sL

qRr

sOkksRqrrr

mmm

kr

k

pi

im

1~--,

11)()(rq

modeITG for ,

)/(~)/)(/(

1m

2/12/1

//

Case1: mode overlapping

Case2: no overlapping(localized)

　 Example: Diffusion caused by ITG modes (continued)

Case1: mode overlapping(ITG)

seB

T

s

LrD

rks

Lr

kpi

kg

grpi

g

~~)(

/1~ ,

2

2/1

(Bohm DiffusionType)

Case2: no overlapping ITG(weak shear configuration )

p

i

pp

i

ppik

pi

LeB

T

sL

qR

LeB

T

eBL

Tk

sL

qRrD

sL

qRr

~~)(~ 22

2/1

(Gyro-Bohm DiffusionType)

Note: Diffusion is small around minimum q in negative shear configurations

Simulation Models for Turbulent Plasmas

１） Gyro-Kinetic Particle Simulation Model ( W.W.Lee: PF 26,556 (1983))Example1: “Trapped Electron Drift Mode ( R.D. Sydora: PF B2, 1455(1990))For given parameters:

1,1.0/,1836/

,4/,4/ ,)8006464( 0

eieei

ieizyx

mmm

TTaRLLL

#Time evolution of Intensity of (m,n)=(5,3) mode# Relation between spectrum and frequency

Saturation intensity

Growth rate and mode frequency are in good agreement with linear mode analysis

035.0/ eTe

Simulation Models for Turbulent Plasmas(continued)

１） Gyro-Kinetic Particle Simulation Model ( W.W.Lee: PF 26,556 (1983))Example2: “ITG mode ( S.E. Parker et al.: PRL 71,2042(1993)) δf/f method is applied, adiabaticity is assumed for electronsElectrostatic potential in poloidal cross section at both linear and nonlinear saturation levels

Linear Nonlinear

Experimental Observations of Heat Diffusivity (continued)

Gyrokinetic and gyrofluid simulations of ITG and ETG models

#A.M.Dimits et al. Phys. Plasmas 7(2000)969From the gyrofluid code using 1994, an improved 1998 gyrofluid closer, the 1994 IFS-PPPL model, the LLNL and U.Colorad flux-tube and UCLA (Sydora) global gyrokinetic codes, and the MMM model (Weiland QL-ITG) for the DIII-D Base case.

Good fit scaling to LLNL gyrokinetic results

iLn /(i2vti )15.4[1.0 6.0(LT / R)]

Q (R / LT R / LTeff .)Offset linear dependence

Parameter Dependence of Critical Temperature Gradient

Algebrac formula for critical gradient are proposed associated with electron anomalous transport;

Jenko et al.,PoP(2001), Ryter,PRL(2001), Dong et al.,NF(2003), many other papers

Critical gradients of the SWITG modes in wide parameter regions are evaluated to discuss the possibility of the SWITG instability as a possible candidate to explain anomalous transport( Zhe Gao, H. Sanuki, K. Itoh and J. Q. Dong)(see, SWITG and SWETG PoP(2005) )

The scaling of the critical gradient with respect to temperature ratio, toroidicity, magnetic shear and safety factor are obtianed ( 19th ICNSP and 7th APPTC conference in Nara(July, 2005))

Parameter Dependence of Critical Temperature Gradient(continued)

(J.Q.Dong, G.Jian, A.K.Wang, H.Sanuki and K.Itoh, NF43(2003)1183)

max 1

0.0173 i2 1.95 i 1.18

ckmaxTe

eBR

R

LTe

R

LTe

ct

ETG instability

e DB

0.0173 i2 1.95 i 1.18

c

pe

1

LTe

1

LTe

ct

F(n ,ˆ s ,q)

R

LTe

ct

3.51.07 i 0.5 i

2 , 0.5 i 1.5

2.35 2.59 i , 1.5 i 5.0

i Te Ti

eBcTD eB

Increasing temperature ratio,

i Te Ti

is in favor of suppression of modes due to raising TG criticaland dropping coefficient between maximam growth rate andderivation of TG from critical TG

(up to about 3)

Important comment from view point of control of confinement improvement

Experimental Observations of heat diffusivity

# Ion thermal transport could be reduced to neoclassical level with Internal Transport Barrier(ITB’s) in DIII-D# Electron ITB’s are observed in JET tokamak ECH dominant discharges with

Te Ti 3 　　　　　　　　　　　　　　　　　　　　　 # ETG driven insta.

ECH experiments in typical Tokamaks(ASDEX Upgrade,RTP,FTU, TCV,etc.)

All four machines clearly show a strong increase of electron transport above a threshold in

Offset Linear Dependence

R LTe

Experimental Observations in DIII-DExperiment employed off-axis ECH heating to change local value of Heat Pulse Diffusivity Model is proposed

ee TT /

kcriteeeeeeHP

e HkTTTfTTH ])/(2)[(/)( .0

# No clear evidence of an inverse critical scale length on heat diffusivity was observed in DIII-D experiments# More improved theory and experimental fine data are needed

JET ECHExperiment

Simulation Models for Turbulent Plasmas

2） Full Orbit Particle Model:References: 1)R.W.Hockney and J.W.Eastwood,” Coumputer Simulation using Particles”, MacGraw-Hill, New York(1981)This model is characterized by introducing “ Super-particle” and “Shaping Factor” and study wave phenomena with long wavelength

2)H. Naitou: J.Plasma Fusion Res. 74.470(1998)

Some limitations:# Short wavelength modes are hardly studied# CPU time is huge, for 1 Ai t

Simulation Models for Turbulent Plasmas(continued)

2） Full Orbit Particle Model:“Toroidal Particle Code(TPC)”# Study electrostatic turbulence, excluding electromagnetic effect# Using Poisson equation instead of Maxwell equationsReferences: 1) M.J.LeBrun et al., PF B5,752(1993)2)Y. Kishimoto et al. Plasma Phys. Contr. Fusion 40, A&&#(1998)“ ITG mode in tokamak”, electrons are adiabatic fluid ( assumed)

Electrostatic Potential plotin negative shear tokamak

“Discontinuity around minimum -q surface in qusi-stationary state”

--minqq

　 　 Ｔ opics on Zonal Flow(Convective cell)

Hasegawa-Mima equation in turbulent Plasmas

References1) A. Hasegawa and K. Mima, PF 21,87(1978)2) A.Hasegawa, C.G. Maclennan and Y. Kodama, PF22,2122(1979)

Assumpitons involved in Model:1)Continuity eq. for ions, 2)ion inertia effect parallel to B is neglected, 3) ions: cold, 4) electron : Boltzmann distribution,4) ordering:

1~,~,~1

ni

Ldt

d

　 　 Ｔ opics on Zonal Flow(Convective cell) (continued1)

Derivation of Hasegawa-Mima eq. in turbulent Plasmas

) (,~

,

ions) (,)ˆ(1

,1

ˆ1

)ions (,0)()(

00 electronsfor

T

e

n

nnnn

forzBtdt

d

dt

d

Bz

Bv

forvnnvt

nvn

t

n

e

i

E×B drift Polarization drift

　 　 Ｔ opics on Zonal Flow(Convective cell) (continued2)

Derivation of Hasegawa-Mima eq. in turbulent Plasmas

0)1(

,0)1(

22422

22322

xyyxt

yv

xyyxc

t

iss

dsss

(Hasegawa-Mima-Charney equation)

Density gradient is negligible small(Hasegawa-Mima equation)　長谷川―三間的式

Note: This equation has two conservations“Energy conservation” and “vorticity conservation”

Electromagnetic Drift Wave turbulence and Convective Cell Formation(1)

#Convective cell formations( electrostatic ) have attracted much interest from turbulence and related anomalous transportin 1970s and 1980s#Electrostatic convective cell formation based on nonlinear driftwave model (Hasegawa-Mima eq.)(1978)

t

( ˆ ˆ )[(ˆ ˆ z ) ][ ˆ ln

n0

ci

]0

# Electromagnetic convective cell formation based on nonlinear drift Alfven wave model ( Sanuki-Weiland model(J. Plasma Phys.(1980))

me mi 1

me mi

( 2

t2 vA

2 2

z 2 VDi

t

t

)

c

B0

(1k//vA

2

2)t

(yxxy

)

Viscosity term

E B nonlinear term

　 　 Characteristics of Hasegawa-Mima Equation

Hasegawa-Mima eq. in turbulent Plasmas

2

ˆ2

ˆˆ2

ˆˆ2

ˆ

11

22322

ˆ1

ˆ)(ˆ

asgiven issolusion A

0ˆ)ˆˆ(ˆ)ˆˆ(ˆ)()ˆˆˆ(

ˆ,,ˆ,ˆ,ˆg,Introducin

,0)1(

k

k

kkttyyxx

yv

xyyxc

t

ysnk

yxxyysnt

isss

dsss

(Hasegawa-Mima-Charney equation)

Non-dimensional form

　 　 Characteristics of Hasegawa-Mima Equation (continued)

321

22

23322

1

3,21

032132,

111

1

321

k

case heconsider t we,discussion following In the

))(ˆ)(()1(2

1

~~~~

0process, coupling wavehreeConsider t

~~ ),exp()(

~),(

~

32

kk

kkzkkk

idt

d

kkk

xkittx

kkk

kkkkkkk

kkk

k

kkk

　 　 Characteristics of Hasegawa-Mima Equation (continued2)

312

312321

2/122

22,13

3,21

2

22,13

3,212

1

2

22,13

3,211

21

2

321

212,133

323,211

2

3122

, ,0mismatchfrequency if also

and ),( wavesintodecay cascade )( , Since

)4

14(

is rategrowth and unstable becomes mode ,04

0d

g.mismatchinfrequency is )( where

)exp(dt

dA ),exp( .,

)exp(~

~,

~~ excited,strongly is mode If

kkkwavekkk

A

AIf

AAdt

dAi

dt

A

tiAAtiAAdt

dAconstA

tiA

k

iii

　 　 Characteristics of Hasegawa-Mima Equation (continued3)

3) and modes(1 ofgain energy toleads mode(2) of lossEnergy

..,.,

~)1( ,

~)1(

as energy mode andnumber gIntroducin

213213

22

22

22

constNNconstNNconstNN

kWkk

kN

WN

ppp

rq

pp

p

pp

kWkW

W

kx- dependence ky- dependence

Linear-nonlinearterm are comparableat k-critical

Zonal Flow in DriftTurbulnce

　 　 Characteristics of Hasegawa-Mima Equation (continued4)

22/2/22

22

22

22

222

k

p

)2()exp(

~~)1(

2

1),,(

,)1(

2),( ,)1(

2),(

~)

~(

~)

~(

~)()

~~(

~: ofpart frequency low scale Small

~: ofpart frequency high scale Large

ofdensity power spectrum-k ofEvolution Time

pdxpikktxkn

kdnkk

kktxBkdn

kk

kktxA

ABA

kpkpk

kxy

kyx

yyyxxx

LyLxLxLyLysnLLt

n(k,x,t): Power density of high frequency spectrum of ~

　 　 Characteristics of Hasegawa-Mima Equation (continued5)

xycv

k

kkvvk

nkxxk

sE

s

sEdynlk

lkk

kkk

~,

~1

)(

0t

330(1992)Lett.A165, Phys. al.et Dyachenko A.I. Ref.

n:~

of spectrumfrequency high ofdensity Power

22

22

k

　 　 Zonal Flow Generation

Recent development in both theory and simulation leads physics involved in zonal flow mechanism more understandableReferences:1) Z. Lin et al., Science 281,1835(1998)2) A.I. Smolyakov et al. PRL 84,491(2000)3) P.N. Guzdar , R.G. Kleva and Liu Chen, Phys. Plasmas 87,459(2001)

Shear flow

Zonal flow

Gyro-kinetic particle simulation(Lin et al.)

(A)shearing effect by zonal flow, (B) no shearing effect

　 　 Zonal Flow Generation (continued1)

.0)~

))ˆ~

(()(

,0))ˆ~

((~

))ˆ(((

~)ˆ()ˆ

~(

~)-(1

eqs. decoupled following toreduces eq. Mima-Hasegawa

)1(~,~

),(~ ),(~~

,

:

~)( ,

as modified is responceelectron adiabatic and )0

but ,0(surface magneticon cont. is potential flow Zonal

322

3

0

022s

0

ddsszs

zddzss

dzssdssd

zdddze

de

d

ed

x

zyz

zct

zzc

zcn

nzc

t

OOOT

e

Ordering

T

e

T

e

n

n

k

kk

　 　 Zonal Flow Generation (continued2)

,~

2

)1()(k

asgiven isy instabilit almodulationfor condition Unstable

)1(

)(~

2

)1)(1(

)1(

asgiven is rategrowth

.,~

and ),exp( form in thegiven is on termperturbati

and ,~

,~~

excited,strongly is Drift wave If

and ~

,~

,~

for equations coupled 4

)(1 ).cos()(

)sin()sin()(~

)cos()sin()(~

)cos()(~~

Model Coupling Wave4consider weHere

2

0

422

2xc

2222

22

0

222222

2222

224

2

0

0

0

20

0

000d

d

dssys

ss

dsxd

ssxsy

si

yxs

d

zdcdsd

zdsdcd

s

dyzxzz

yzxds

yzxdcyd

v

ck

kc

vkkkk

k

kkc

constt

k

vktxkt

tyktxkt

tyktxkttykt

Zonal flow excitation by drift waveFor

xcx kk

Electromagnetic Drift Wave Turbulence and Convective Cell Formation(4)

Coefficients of NS Equation

s(2 kyvDi kyc0 )(km2 ky

2 ),

U ( vDi c0 )(km2 ky

2 ),

Q (1k//

2vA2

2)ky

2(3km2 ky

2 )

L( vDi c0 )

c

B0

2

,

(km2 ky

2 )

Following the theory by H. Sanuki et al.(1972)

ˆ 1(1) a(t)sech[( g

2p)1 2 a( t)( 2vt)exp( iv

p( vt) i

g

2a2(t)dt

0

t

)

p U

s, g

Q

s, at 2a

Modulational insta. condition

UQ (3km2 ky

2 )0

3

1Elongated structure of Convective cellJ.Weiland, H.Sanuki and C.S.Liu, PoF(1980)

Discovery of new truths by studying the past through scrutiny of the old(温故知新 )

Zonal Flow

Typhoon, Giant Red Spot in Jupiter (Zonal flow)、 El Nino, La Nina、 etc Review of vortics

International J. of Fusion Energy (1977-1985, particularly, 77~78, F1R26)# Hermann Helmholtz(1858)(general)# Winston H. Bostick (Vortex Ring)# D.R.Wells and P.Ziajka (Theory and Experiment) , others

What kind of dynamics determines Structure of Vortices(2D) ？　（ unsophisticated question）

lx ly or lx ly or lx ly

Vertex(Convective cell,zonal) Motions in Nature

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　　　　　　　落紅不是無情物　　　　　　化作春泥更護花　　　　　（龚自珍　己亥雑詩）

ACKNOWLEDGMENTS

I would like to acknowledge many collaborators and friends for their continuous and fruitful discussions. This visit is supported

by Prof. Li Jiangang and the Chinese Academy of Sciences 、 Visiting Professor for Senior International Scientists(2009 fiscal year) ,and also supported by Prof. Liu Yong as a guest professor of SWIP.

The present topic is also partially discussed under close collaborations with NIFS( K. Itoh, A. Fujisawa et al.),Tsinghua University (Gao Zhe et al.) and SWIP ( Dong Jiaqi ,Wang Aike

et al.) Finally I would like to acknowledge all friends and staffs, students who take care of lots of arrangements of my visiting ASIPP 　 since my first visit, 1991.