Short wavelength ion temperature gradient driven instability in

toroidal plasmas

Zhe Gao,a) H. Sanuki,b) K. Itohb) and J. Q. Dongc)

a) Department of Engineering Physics, Tsinghua University, Beijing 100084, Chinab) National Institute for Fusion Science, Toki, Gifu 509-5292, Japan

c) Southwestern Institute of Physics, Chengdu 610041, ChinaElectronic mail: gaozhe@mail.tsinghua.edu.cn

1st Plasma Theory and Simulation (PTS) WorkshopChengdu Sep. 13-15 2004

OUTLINE

Background Model Local slab limit Parameter dependences Critical temperature gradient Conclusions

Background (I)

microinstabilitiesmicro-turbulenceanomalous transport

ion transport － reduced to NC level; electron transport － still anomalous / ETB Experimental observations: Turbulent diagnostics anomalous electron transport is

possibly attributable to short wavelength modes

Profile stiffness a threshold in temperature gradient Two kinds of well-known microinstabilities: ETG modes e.g. F. Jenko et al, PoP 8, 4096 (2000),J. Q. Dong et al, NF 43, 1 (2003)

TE modes e.g. J. Weiland et al NF 29, 1810 (1989)

Background (II)

comparisons with experimental results e.g F Ryter et al, PPCF 43, A323 (2001)

some on ETG: s/q dependence in Tore Supra

some on TEM: modulated Te evolution in ASDEX-U

other remarkable experiment results

Not all tokamaks observed “stiffness” in Te; e.g. JC DeBoo EPS conf. 2002

Clear link between long and short wavelength fluctuation;

e.g. GD Conway PRL 84,1463 (2002)

Measured high k fluctuation in the ion diamagnetic direction;

e.g. KL Wong PLA 236, 339 (1997)

electron transport might be controlled by multi-source instabilities simultaneously, or by different instability depending on different discharge condition.

Background (III)

Short wavelength ITG modes ( , ) Local mode in a shearless slab AI Smolyakov et al, PRL 89, 125005 (2002).

A new TG driven short wavelength mode or “ double-humped behavior” in Pu et al, PF 28,1722 (1985) ?

Nonlocal mode in a toroidal geometry A. Hirose et al, PoP 9, 1659 (2002).

“electron transit effect driving”, “ essentially a slab mode” , “shear is destabilizing”

0r 1iyk

Model

Gyro-kinetic equation ( ballooning representation in s-a equlibrium )

Poisson’s Equ. & Ampere’s law

Both ions and electrons are nonadiabatic: the transit effect, the finite Larmor radius effects and the curvature and magnetic gradient drift

Local slab limit:

The local short wavelength mode is attributable to the Landau damping stabilization/destabilization mechanism and the nonmonotonic behaviors of the real frequency with wavelength variation.

Some explanations in Smolyakov et al are not correct

ctirti vkvk //max// ~

Local slab limit : influences of electron kinetics and k//Ln

2.0// nLk

Toroidal modes : analytical discussion

ˆˆ 20

////

*i

Di

Ti

i

Mii

i

Miii J

vkT

Fq

T

Fqf

////* vkDii

)exp()(

12ˆˆ~0

*0 ii

ii

i

i

i

ii bbI

T

qn

T

qn

)exp()(121 0* iiii

effi

effi bbIZ

Z

2* intii bLv iii bbbI 21)exp()(0 ib

inDi *~

ion response

and it integral

(under the condition )

If electron is adiabatic

Noted: , as

Local slab limit: Landau mechanism

Toroidal: magnetic drift resonance

cvkvk //////// ~

cDiDi ~

Toroidal modes : wave spectrums

Parameters: , , , , , The SWITG mode still exists even with adiabatic electrons

assumption.

0 1 2 3 40.00

0.05

0.10

0.15

0.20

0.25

k

s/*e

ks

(a)

0 1 2 3 4

-0.6

-0.4

-0.2

0.0 l=0 mode l=1 mode l=2 mode with adiabatic electrons

rk s/

*e

ks

(b)

5.2e 5.2i 1 effi ZM8.0ˆ s 5.1q1.0n

Toroidal modes: toroidicity

more easily stabilized by toroidicity than the conventional ITG mode

condition broken

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.200.00

0.02

0.04

0.06

0.08

0.10

/*e

n(=L

n/R)

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

-0.45

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

r/*e

n(=L

n/R)

l=0 l=1 l=2

////* vkDii

Toroidal modes : eigenfunctions

-40 -20 0 20 40-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

real part imaginary part

(a)

-40 -30 -20 -10 0 10 20 30 40-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

(b)

-40 -20 0 20 40

-1.0

-0.5

0.0

0.5

1.0

1.5

(c)

-4 -3 -2 -1 0 1 2 3 4

0.0

0.4

0.8

1.2

(d)

-4 -3 -2 -1 0 1 2 3 41.2

0.8

0.4

0.0

-0.4

-0.8

-1.2

(e)

-4 -3 -2 -1 0 1 2 3 4-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

q

(f)

Broaden structures along the magnetic field line and have oscillatory tailswhen nonadiabatic electron response are considered

Toroidal modes : temperature gradients dependence

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.60.00

0.02

0.04

0.06

0.08

0.10

/*e

i

l=0 l=1 l=2

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.60.00

0.02

0.04

0.06

0.08

0.10

/*e

e

l=0 l=1 l=2

Each of the three modes has a eta_i threshold and the threshold of the fundamental mode is lower that those of higher order modes.

The fundamental SWITG mode is unstable when both eta_i and eta_e exceed thresholds. But the higher order modes persist unstable even eta_e decreases to zero

Toroidal modes: Te/Ti dependence

2ik

The SWITG mode can be stabilized by hot ions (small Te/Ti) ITG driven modes

0 1 2 3 4 50.0

0.1

0.2

0.3

(a)

-/ *i

Zeff

Te/T

i

0 1 2 3 4 5

-0.35

-0.30

-0.25

-0.20

-0.15

l=0 mode at Zeff

=1

l=1 mode at Zeff

=1

l=2 mode at Zeff

=1

l=0 mode at Te/T

i=1

l=1 mode at Te/T

i=1

l=2 mode at Te/T

i=1

(b)

Zeff

Te/T

i

-r/

*i

Toroidal modes : isotope effects

The normalized growth rate and frequency is insensitive to isotope mass

1 2 3 40.02

0.04

0.06

0.08

0.10

0.12

/*e

mi/1836m

e

l=0 mode l=1 mode l=2 mode

(a)

1 2 3 4

-0.30

-0.25

r/*e

mi/1836m

e

(b)

0.2ik Maximum growth rate

2/1.~ imconstk

Toroidal modes : magnetic shear

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.00.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

/*e

s

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.31

-0.30

-0.29

-0.28

-0.27

-0.26

-0.25

-0.24

-0.23

-0.22

-0.21

-0.20

-0.19

-0.18

-0.17

-0.16

-0.15

l=0 l=1 l=2

r/*e

s

The stabilizing effect of shear is confirmed although the shear needed to stabilize the fundamental mode is rather strong

The grow rate of the fundamental mode at negative shear is almost same as that at positive shear, while obvious low growth rate exists at negative shear for higher order modes

The high order mode may grows faster than the fundamental mode in the weak shear regime

Toroidal modes : beta effect

0.000 0.002 0.004 0.006 0.008 0.0100.00

0.04

0.08

0.12

/*e

e

(a)

0.000 0.002 0.004 0.006 0.008 0.010

-0.30

-0.25

-0.20

-0.15 l=0 mode l=1 mode l=2 mode ES model with

r/*e

e

(b)

The fundamental mode is easily stabilized by finite beta, mainly due to the coupling effect to shear Alfven waves (SAWs), not the ballooning parameter

The higher order modes are more hardly stabilized, which might be explained

by the fact that the higher order modes have lower frequencies.

Critical temperature gradient vs. Ln/R

0.10 0.15 0.20 0.25 0.30 0.35

2

4

6

8

10

12

14

16

18

20

22

24

(R/L

ti) c

n

i=3

i=1

i=0.75

0.10 0.12 0.14 0.16 0.18 0.20

10

15

20

25

30

35

40

45

50

n

(R/L

Te) c

Unstable

Stable

R/LTi: saturation in the large enough εn region

R/LTe : cannot cross the εn barrier

Critical temperature gradient Vs. Te/Ti

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.07

8

9

10

11

12

13

14

15

16

17

18

19

20

21

(R/L

ti) c

Te/T

i

n=0.2

0 1 2 3 4 5 6

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

max

/[*e

(i-

ic)]

Te/T

i

R/LTi deceases as Te/Ti increase

Critical temperature gradient Vs. kθ

“sub-steady-state” region for ITG critical electron temperature gradient varies with wavelength

0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0

-2

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

k

(R/L

te) c

i=3,

n=0.2

i=1,

n=0.1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

4

6

8

10

12

14

16

18

(R/L

ti) c

k

i=3,

n=0.2

i=1,

n=0.1

Coupling between l=0 and l=2 modes

0.0 0.5 1.0 1.5 2.0 2.5

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

e

1 2

k=1.4

0.0 0.5 1.0 1.5 2.0 2.5

-0.56

-0.54

-0.52

-0.50

-0.48

-0.46

-0.44

-0.42

r

e

1 2

-0.5 0.0 0.5 1.0 1.5 2.0 2.50.00

0.04

0.08

0.12

0.16

e

1 2

k=1.2

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

e

1 2

k=1.6

-0.5 0.0 0.5 1.0 1.5 2.0 2.5

-0.44

-0.42

-0.40

-0.38

r

e

1 2

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6-0.64

-0.60

-0.56

-0.52

-0.48

r

e

1 2

Critical temperature gradient for the l=1 mode

0.10 0.15 0.20 0.25 0.3010

11

12

13

14

15

16

17

(R/L

Ti) c

n

i=1

i=3

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.05

10

15

20

(R/L

Ti) c

i

0 1 2 3 4 50.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

/[ *e

(i-

ic)]

i

ηi- ηe diagram for l=0 and l=1 modes

electron kinetics strongly influences the l=0 mode but only slightly influences the l=1 mode

in the region where ηe is relative smaller, the l=1 mode is dominant

1 2 3 4 5

-1

0

1

2

3

4

1=

2

e

i

1=0

2=0

=3, n=0.2

0 1 2 3 4 50

1

2

3

4

5

6

1=0

0=0

e

i

A

B

C

D

E

0=

1

=1,n=0.1

Conclusions

The SWITG instability occurs due to the toroidal drift resonance mechanism in the short wavelength region and exist even if adiabatic electrons are assumed

Parameter dependences of the SWITG mode is similar as those of the conventional ITG mode

Nonadiabatic electron response can influence the fundamental SWITG mode.

The SWITG instability has a medium wavelength, so might be more difficult to be suppressed by the E*B flow shear than the ITG instability while could induce higher transport than the ETG mode.

The SWITG mode may be attributed to electron transport, however, it seems the ion kinetic is necessary and basic.

ACKNOWLEDGMENTS

Most of this work is performed during Z.G.’s visiting at the National Institute for Fusion Science, Japan.