Geodesic mode spectrum modified by the energetic particles in tokamak plasmas

A. G. Elfimov and R. M. O. Galvão Institute of Physics, University of São Paulo, Brazil

Outline• Results of a kinetic treatment of Geodesic Acoustic Modes (GAM) that

takes into account adiabatic electrons and dynamics of hot and cold ions . • Found GAM instability is driven by electron current and hot particle drift in

combination with main ion drift.• Formation of ion-sound eigenmode by geodesic effect is demonstrated due

to or second poloidal harmonic effect (finite-orbit-width (FOW) parameter ).• Applications to experiment will be discussed.

7th IAEA Technical Meeting on Plasma Instabilities, Frascati, March 4-6, 2015

2. Introduction: Geodesic modes in tokamak plasmas

• Geodesic Acoustic Modes (GAM) are linear eigen-modes driven by anisotropic

perturbations of the ion and electron pressure in tokamak geometry.

• The standard theory presents three cases of eigenmodes in limit =r/R<<0:

high frequency geodesic mode with frequency

ion-sound branches

and the low frequency zonal flow (over-damped mode) of the equilibrium rotation (R0 is

major radius, q is safety factor, mi is mass and Ti is temperature of plasma species).

iei

eieieiGAM mR

TTqTTTTTT 2

02

22

)47()4(42322/7

...3 ,2 ,1 ,// 02 lqRmTl ieis

• Experimentally, series of geodesic oscillations have been detected in a relatively

wide range of frequencies in different tokamaks, JET, JT-60, AUG, D-III, Textor, T-

10, FT-Ioffe et al. These modes may strongly affect the drift-wave turbulence and

plasma transport that has been observed in experiments (specially in L-H transition,

Conway et al (2008) PP & CF 055009) and in numerical simulations (Hallatschek et al PRL (2001) 1223).

To find the GAM frequency for the toroidal mode number N=0, we will use standard drift kinetic equation for electrons and ions with Fe,i -Maxwell distribution in the form

To calculate GAM dispersion, gyro kinetic equation in potential approximation may be used that gives

3. Drift kinetic equation and standard GAMs

sin

2)u(2w

23)-u(w2)(

sinii1

22

20

22

20

3

0

0 ER

Ewdvkwvk

EwwmeFf

wkkV

wff

crcTT

rr

sin2

)u(2w)w(

i1

22

220

ER

Evw

vkmFef

cis

TiTii

eiiei

,4

i 12

2

Ec

cjA

p

dwfVduduvej ieireTe

ieie

r ,,0

3

,,

2~

)(isin 20, uJqF

TefwqgVk drr

tdrkgTFef

r

r i-iexp

Zonca F, Chen L (2008) EPL; Nguyen et al (2008) Phys. Plas.

Zonca et al PPCF (1996);NF(2009);

Sugama & Watanabe (2006) Phys. Plas ; Watari et al (2006) Phys. Plas.

;2)(2 ,,22

, RvuwV ieiTeire ;/1;1 where 000 qRkvk T cTv

2

23222222

874748114016

21326

431 eee

eeirGAMttt

ttk

and obtain standard the GAM continuum equation

iei

eieieiGAM mR

TTqTTTTTT 2

02

22

)47()4(42322/7

Then, we calculate the balance of the radial drift and polarization currents in div(jp+je+ji)=0 where

Ignoring Vr drift in the equation and using solution

E2,3–field components are found from qusi-neutrality

To study hot particle effect on GAM continuum, we use the drift kinetic equation via integrals of motion ( ) and

,

sin

2)cos(2-2

2)3(u2i 1

22

2

23 E

vRuE

dvvwEF

mqReffw

TcrcTT

BvBvvvu TT2

02222 ,/ 1;cos1|| ssuw

cos12,

1

1

0

cos1

10

23

sr

s

Tr

fVdddduuvej

;

cos1sin ,

1

1

0

cos1

1

cos

0

23)(,

s

sTcs

fdddduuvn

Next, the variable is changed to the new -variable for the untrapped and trapped particles . The untrapped equation is rewritten via the cn, sn, dn Jacobi functions 2)1(ˆ 2

)1(22

),(dn21),(sn),(cn),(sn4

),(cn),(sn)2(2is2

2c2

1

22unun

EEvm

RqFe

Evm

HquFseu

ff

sT

Tc

N

p pp

pp

Tc Kp

uQKuQH

vmEFqef

12223,

2221)(

un1, )(sin

))(1()()2(22i

Tp vpKqR 2)(0,

integral elliptic kindfirst )( K

The density and current may be obtained via the dispersion functionwhere , but rest of integration has problem:

)()exp( 2 xttdtZ 2,px

2)2/sin(1), (dn

);, (cn2/cos);,(sn2/sin

)K(

)K(-

1

02 sin2

dfd

un

Using Q-series of the Jacobi functions, we get solution 2/1)1(12)1(1 4/124/12 Q

Taking solution of the equation, we have to calculate the sinθ/cosθ density and radial current

4.Trapped-untrapped particle kinetic equation (Jacobi function technique, Elfimov (2014) Ph.Let-A, 378, 3533)

Then, taking into account the shifted electron distribution , we have for electrons eMTee FvVwF 0||1

5. Integration for hot and cold particles

Elfimov(2014) Phys. Let-A, 378,3533.

Assuming that the bounce frequency is larger than the GAM frequency, hot ions have small fraction rh<<1

, and taking into account the order in equation, we perform integration that givesRvTh 2 1O

c

5

00

3

0

1

5

002

)(un

c100)(

un

i23.01)23(06.0i27.0

25.0i4.0

;sini56.0

:particlesUntrapped

EvR

hdER

vR

EvR

vmqrnej

EET

qRrnen

Thh

s

Thh

ThTh

h

chh

hhh

chh

hhh

.2)23(8.1i8.0

21.51.2i

i6.1

givesparticelstrappedforprocedureSimiar

3

000

1

5

01

02

0

02

)(t

1200)(

t

sThh

h

h

s

ThThchh

Thhhh

Thchh

hhh

EvR

hdR

hdER

EvRE

vR

Rmqvrnej

EvmRnqren

cs

Teci

ersc

Tee

eccs

Tee

es EE

vV

mqne

jEEvV

TqRen

nEEvV

TqRen

n22

i ;2

,2

002

000000

For basic cold ions , the trapped bounce effect is compensated by untrapped one02 Rv Ti

Chavdarovski, Zonca (2009) PPCF, 115001;sini42 2

120

22

2

0

0)(tr

)(un

qEE

Rqv

qRmqnenn c

ci

Ti

i

iii

qE

Ev

Rqv

qRqR

vRmnve

jj cci

TiTh

Ti

cii

Tiiii

i2

2exp

42i

223

27i

12

20

225

022

02

2

20

20

22)(

tr)(

un

6. Hot ion mass effect and GAM instability

and using the electric field in the balance of the polarization, electron and trapped/untrapped ion radial currents equations, we get GAM frequency equation from the real part of the balance Re(jp +je+ji +jh=0)

Finally, increment/decrement values are found from the imaginary part of the radial current

Combining quasi neutrality for the sin and cos density components , we obtain equation for the electric field

12

220

22 6.11

)47()4(4232

27

i

hh

e

eee

TiGAM m

qrmtq

tttRv

12

20

25

0002

20

2

40

3

4

2exp2i2i2 E

vR

vR

vvhdRVt

vR

RvtE

TiTiTiTei

ie

Tici

Tiec

10

020

2i2- EvV

Rhdvt

RvtE

Tei

TieiTieis

Effective hot ion mass

2/12

20

22

222

2

2

0

3

0

03

4

2/12

2

20

22

22

40

45

220

20

0

6.11)23(9.125.522

6.112

exp2

32222

i

hh

G

Tiei

ih

h

Th

G

Ti

G

Th

Tih

i

hh

Ti

G

TeTi

Gei

GTei

TiieTi

mqrm

Rqvt

hddvqR

vqR

Rvvqr

mqrm

vRq

vvRqt

qRvhdvV

Rtv

(1) by electron current V0 combined with ion density gradient )( rnnd iii

(2) The other possibility is related to combinations of gradients of the basic and hot ions

Instability may be driven:

where the reflection points are defined by respective continuum position

Converting , we have solution via respective Airy functions

)(4 and )( 2222

12

isisisis rr

To calculate geodesic wave dispersion via magnetic drift , a potential approach is used in gyro-kinetic

equation in the limits q>>1, Te/Ti>>1, , ignoring a trapped ion effect and taking into

account second harmonic effect that gives for ions

7. Geodesic Ion-Sound Eigenmode dispersion

...3 ,2 ,1 where2

ppdrkis

is

r

rr

drrVk

,2cossin 20 cs 2cossin~20 csee Tenn

qvR T /10

;247

27~

22

22

22

02

222

0

0s

rc

rr

i

i kq

kknenT

crrs

rr

ri

si kkqq

kqkknenT

2

22

222222222

022

0 2273141

421132

~

2222

22

202

22

0

2 4474

27~

c

rr

sr

i

ci kqq

kknenT

A general dispersion equation is found from this equation set

Where is normalized ion-sound frequency and

22222222 4 isisGAMr Dk 22 qTT ieis

To find the eigenmode frequencies in the interval , we use quantization rule at the reflection points

222 4 isis

rrC 2,13/1

2,12,1 Ai

)(42 222222isisee ttD

drdkr i

Radial mode numbers and frequencies are calculated for typical T-10/Textor parameters

8. Ion-Sound Eigenmode calculations

.)1(10.2 ;)1(10

),1(120,5.1 ;140 ;55.1321325.223

20

cmxneVxT

xTmRkAITB

ee

ipt

Fig.2. Comparison of theoretical GAM spectrum with experiment in Ohm discharges in T-10.Melnikov et al. PPCF, (2006) and 37th EPS, Dublin (2010)

#57406

6p7p

Recently, the geodesic modes have been observed during plasma current ramp up with counter injection of the NB heating that forms reversed shear configuration in JT-60 (G. Matsunaga et al 39th EPS

Conference, July 2012, Stockholm). The reported modes have a smaller frequency by half of the value of the core GAM frequency, and approximately coincide with the local GAM frequency at q-minimum.

8. Discussion of possible applications

We suggest that this geodesic mode is the above-discussed unstable GAM driven by the electric current and localized at the minimum of the continuum that formed due to reversed shear. Two observations are important:•the instability appears during counter-injection that means along electron current velocity;•the estimated velocity of the current at the q=4minimum is much higher then the GAM phase velocity that is necessary for instability;

•the time delay is about few electron-ion collision time to transfer the ctr-NB energy and momentum to inverse the electron distribution; •reducing frequensy masfactor

RqRqrcV ciA 20 2

.6.112

i

hh

mqrm

Conclusions• A novel method of Jacobi functions has been successfully applied to solve the drift

kinetic equation for the energetic particles with high bounce frequency.

• It is shown that the standard GAM continuum frequency is reduced by mass factor

of energetic particle,

• The calculations demonstrate that the standard geodesic mode may be unstable

when the current electron velocity of the Ohm’s current is above the wave phase

velocity V0> GRq and/or for the sharp hot ion density profiles during ICR heating.

• The dispersion related to the second harmonic effect strongly modifies the GAM

spectrum due to the finite orbit width parameter that produces the coupling the

standard GAM with second harmonic of the ion-sound mode;

• the effect manifests itself as formation geodesic ion-sound eigenmode below the

standard GAM continuum.

.6.112

i

hh

mqrm

Thank you for attention

•M.P.,Petrov, et al, Phys. Plasmas, 6, 2430 (1999).

10. Possible Applications to ICRH experiments