West Lake International Symposium on Plasma Simulation; 18-20 April, 2012 Influence of magnetic...

West Lake International Symposium on Plasma Simulation; 18-20 April, 2012

Influence of magnetic configuration on

kinetic damping of the resistive wall mode

Influence of magnetic configuration on

kinetic damping of the resistive wall mode

Shichong Guo1,

In collaboration with

Zhirui Wang1 ,Yueqiang Liu2

1.Consorzio RFX, Euratom –ENEA, Padova Italy2.Euratom/CCFE, Culham Science Center, UK

Shichong Guo1,

In collaboration with

Zhirui Wang1 ,Yueqiang Liu2

1.Consorzio RFX, Euratom –ENEA, Padova Italy2.Euratom/CCFE, Culham Science Center, UK


Consorzio RFX (Consortium Reversed Field eXperiment)Consorzio RFX (Consortium Reversed Field eXperiment)

RFX-modRFX-mod


Resistive Wall Mode

Importance of Resistive Wall Mode (RWM)

● Understanding the physics of RWM is an important task for the successful operation of the present day and future fusion devices.

● For advanced tokamak scenarios, including those foreseen for ITER, and aiming at simultaneous maximization of the plasma pressure and a steady state operation, the RWM poses a severe beta limit.

● For Reversed Field Pinch (RFP) plasmas, the RWM appears as potentially disruptive instability, whenever the duration of the discharge is longer than the penetration time of the passive conducting structure (resistive wall).

RFX Feedback coilsRFX Feedback coils


Ideal external kink

-Idea external kink + Resistive wall

-The growth rate are comparable to the resistive diffusion time of the wall w ~ 1

-Idea external kink + Resistive wall

-The growth rate are comparable to the resistive diffusion time of the wall w ~ 1 ' '( ) ( )

( )r w r w

w wr w

b r b rr

b r

(cylindrical mode and thin shell)(cylindrical mode and thin shell)

Resistive Wall Mode

w o wr h

vPd

P vb

δWδW +τ =

δW +δW

RWM dispersion relationRWM dispersion relationVacuum energy without wallVacuum energy without wall

Vacuum energy with ideal wall at r=b

Vacuum energy with ideal wall at r=b

Magnetic penetrating time of resistive wallMagnetic penetrating time of resistive wall

Plasma potential energyPlasma potential energy

d w


RWMRWM

RFP and Circular Tokamak Configurations

RFP B ~BTokamak B<< B

B reverses near the plasma edge

F = B(a)/<B> = B(a)/<B>.

RWMs are current driven and rational surfaces outside the plasmas.

RWMs are current driven and rational surfaces outside the plasmas.

Usually Pressure driven RWMsRational surfaces inside/outside the plasmas

(m,n)(m,n)

RWM (n=1)RWM (n=1)

RFX-mod system parameters e.g. =a/R=0.2295, b/a=1.12


RWM in RFP and Circular Tokamak

● Usually only one poloidal harmonic is dominant --less importance of toroidal effects

● RFP has weaker poloidal asymmetry;

● Low q:The distance between the two neighboring rational surfaces (m ± 1)/n is larger than tokamak.

● Usually only one poloidal harmonic is dominant --less importance of toroidal effects

● RFP has weaker poloidal asymmetry;

● Low q:The distance between the two neighboring rational surfaces (m ± 1)/n is larger than tokamak.

● Strong toroidal mode coupling due to poloidal asymmetry (toroidal effects).

● Many m modes grow together and give contributions to mode growth rate.

● Strong toroidal mode coupling due to poloidal asymmetry (toroidal effects).

● Many m modes grow together and give contributions to mode growth rate.

Poloidal harmonics of the normal perturbed displacement.

TokamakTokamak

n=1n=1RFPRFPn=6n=6

MARS result in toroidal geometry


Magnetic field curvature is dominated by B, so the “bad curvature” everywhere along the poloidal angle--- almost no ballooning character.

2D plot of normal perturbed displacement

TokamakTokamak n=1n=1

Magnetic field curvature is dominated by B (bad curvature on the low field side, good curvature on the high field side) ballooning character.

RWM in RFP and Circular Tokamak

RFPRFP n=6n=6

MARS result in toroidal geometry


The linearized MHD equations including drift kinetic effects via perturbed pressure tensor (only consider the thermal particles)

-*N and *T are the diamagnetic drift frequencies-d and b are the precession and bounce frequencies- the trapped particle =0, the passing particle =1.

22i in R v p j B J Q Z v v

2i in R ξ v ξ

2i in R Q v B Q

i in p P P v ξ

j Q 1 0

,

b

min t im il ti t in

L k m mll

lm

f e X H ef

ˆ ˆp p p p I bb I bb

2 1

,

i t inL

e i

p e d Mv f 2 1

,

1

2i t in

Le i

p e d Mv f

Thermal Maxwelliandistribution

* *3 / 2k

m

N

b effl

T

d

n

n m nq l n i

- the plasma rotation frequencyThin shell is located at r=b

Resonant operator:

Mars-K formulation (MHD-Kinetic hybrid code)

Kinetic effect enter into MHD formulations though the non-perturbative way

replaced by kinetic

pressure

replaced by kinetic

pressure


● The generalized dispersion relation for RWM (Neglecting inertia term )● The generalized dispersion relation for RWM (Neglecting inertia term )

wkb

kr

WWi

WW

● The new module for analyzing the potential energy components is integrated into Mars-F/K

Magnetic compressibility

Pressure driven

F mb mc pre curW W W W W Magnetic bending

Current driven

F vW WW

b F vbW WW Bo Hu and R. Betti PRL 2004Bo Hu and R. Betti PRL 2004

21

2 P

mb

V

W Jdsd d Q

2212

2 P

mc

V

W B Jdsd d ξ ξ κ

*1

2 P

pre

V

W P Jdsd d ξ κ ξ

*1

2 P

cur

V

W J Jdsd d ξ b Q

Vacuum energy terms when ideal wall is at infinity r=∞ and r=b

Kinetic energy replaces the plasma compressibility term in fluid theory and can be complex number (mode – particle resonance) Wk=Wk

re+ i Wki

* * *1 1

2 P

K

V

W Jdsd d p Q B pB

ξ κ ξ

211

1 1

2 2 p

nv sS

V

W Jdsd d b V J d d

Q

211

1 1

2 2 p

b

bn

vb sSV

W Jdsd d b V J d d

Q

Kinetic energy:

Vacuum energy:

Plasma fluid energy:

Quadratic energy terms


Kinetic stabilization of RWM in RFP

n=6, ne0=ni0=2.5e19m-3, a/R=0.2295 ,Ti/Te =0.7, b/a=1.12 F ≈ -0.062, q0 ≈0.144. ≈1.46-1.56

n=6n=6

Full kinetic scan

■ Fluid theory predicts that the stabilization of RWM requires the plasma rotation in the range of Alfven frequency. nc ≈(k||VA)r=0 or r=a, ≈0.15A, n=6 ■ Kinetic theory predicts much lower plasma rotation =0.03-0.04Afor stabilizing n=6 RWM in the region of ion acoustic frequency for high beta RFP(p>0.15).

Uniform plasma flow is assumedUniform plasma flow is assumed


■ The mode resonance with transit frequency of passing ions plays a principle role (the ion acoustic landau damping).

Kinetic damping of RWM in RFP

n=6, ne0=ni0=2.5e19m-3, a/R=0.2295 ,Ti/Te =0.7, b/a=1.12 F ≈ -0.062, q0 ≈0.144. ≈1.46-1.56

n=6p=0.17

n=6p=0.17

Uniform plasma flow is assumedUniform plasma flow is assumed

Which kinetic mechanism leads to the stabilization in RFP? Which kinetic mechanism leads to the stabilization in RFP?


Kinetic effect on RWM in Tokamak

■ The current driven RWM (qa=1.64) can not be stabilized by the kinetic effects.

■ The pressure driven RWM (qa=3.68) can be stabilized at very slow plasma rotation (<4.810-3A) or even without rotation.

n=1, ne0=ni0=2.5e19m-3, a/R=0.2295 ,Ti/Te =0.7, b/a=1.12

Current driven

Pressure driven

Full kinetic scann=1


Kinetic effect on RWM in Tokamak

n=1, ne0=ni0=2.5e19m-3, a/R=0.2295 ,Ti/Te =0.7, b/a=1.12

q0=1.14 qa=3.68 =0.0105,C=0.66

■ The pressure driven RWM (qa=3.68) can be stabilized by the precession resonance of trapped particles (the principle candidate).

■ >4.810-3A, the mode becomes unstable. Further increasing the rotation frequency, the transit resonance will play a stabilizing role by ion acoustic damping, and the mode growth rate will decrease; however, a full stabilization is not found in this case.

Which kinetic mechanism leads to the stabilization in pressure driven tokamak? Which kinetic mechanism leads to the stabilization in pressure driven tokamak?


2D plot of d averaged over velocity space of Tokamak2D plot of d averaged over velocity space of Tokamak

In tokamak (~0)■d ~ 0(Tokamak), d~0.01A(RFP), larger ratio of */d (ml ∝ */d ) than RFP

■ Trapped particles fraction in Tokamak ~ 1/2> 1/2 in RFP

■ Many m modes can resonate with trapped particles, but RFP has only one mode that can have resonance.

Physical understanding of Kinetic effect

Why can RWM be stabilized by low rotation in tokamak but not in RFP?

2D plot of Wk contributed by the precession resonance2D plot of Wk contributed by the precession resonance


2D plot of |WK

m=1,n=6 b/.a=1.12 F ≈ -0.062, q0 ≈0.144 m=1,n=6 b/.a=1.12 F ≈ -0.062, q0 ≈0.144

Why transit resonance can stabilize RWM in RFP but the precession and bounce resonance can not?


■The maximum value of dominant term ·⊥in perturbed Lagrangian HLleading to max. |WK| .

p=0.1

WK(Transit)

2D plot of | · ⊥ |

0

2 1LH Q Bh B

ξ κ

m=-1,n=6 RFP p=0.17 =0.02A(marginal instability)m=-1,n=6 RFP p=0.17 =0.02A(marginal instability)

The transit resonance gives the dominant contribution to Wk, when (m-nq)p/n

■ Precession resonance gives small contribution to Wk since the fraction of trapped particles is smaller than that of passing particles in RFP.


F ≈ -0.062, q0 ≈0.144. ≈1.46-1.56F ≈ -0.062, q0 ≈0.144. ≈1.46-1.56

RFPn=6RFPn=6

Tokamakn=1

Tokamakn=1

(a) q0=1.13, q(a)=1.64,=0.02(b) q0=1.14 qa=3.68 =0.01,C=0.66

drive r rn p e cuW WW

Max |WK|Max |WK|

Marginal stability

Marginal stability

Marginal stability

Marginal stability

Max |WK|Max |WK|


RFP Tokamak

Why can kinetic effect stabilize RWM in pressure driven tokamak and high beta RFP?Why can kinetic effect stabilize RWM in pressure driven tokamak and high beta RFP?

kw

kbr

Wi

W

W

W

F vW WW F vb bWWW

Vaccum energy:

bn: the normal magnetic perturbation

Plasma potential energy:

Wv∞, Wvb ∝ bn(s=a)

mc pre umb c rF WW W WW The generalized dispersion relation for RWM (Neglecting inertia term )

■ In RFP, high effect :

1) large fraction of Wpre

2) Wmc ↑(b|| ↑), Wv∞↓,Wvb ↓(bn ↓ at plasma edge);

Common features in RFP and tokamak:A big enough fraction of Wpre is required for the kinetic stabilization, which cause big enough Wk

W∞, Wb need to be small, and comparable with Wk

Bo Hu and R. Betti PRL 2004Bo Hu and R. Betti PRL 2004



■ Wk becomes significant to determine RWM stability when W ∞ → 0 (RWM marginal instability)

or, W b → 0 (Marginal stable ideal kink with an ideal wall r=b)■ In RFP: (only Wb(→ 0) has been observed)

- ↑ Wk ↑ P ↑, ∝- High causes b|| ↑, bn ↓, which leads to Wvb ↓ - moving the resistive wall farther from plasma b/a ↑

■ In tokamak: (both Wb →0 and W∞∞ →0 can occur) - is just above N

nowall ,W ∞ ~ 0; N≈Nideal, Wb ~ 0

- A large q(a) leads to an easier stabilization. (strong toroidal magnetic field), W vb↓and Wb ↓

RWM stabilizing condition:RWM stabilizing condition:

2 2Re 0Re( ) Im( )( )K b K KbW W WW W W W

<0<0 <0 or >0<0 or >0 >0>0

■ A large enough fraction of Wpre is required for the kinetic stabilization, which causes large enough Wk

■ W∞, Wb need to be small and comparable with Wk.


■ W∞, Wb need to be small and comparable with Wk.

Common features of RFP and tokamak:Common features of RFP and tokamak:


Comparison of RWM between Tokamak and RFP

Tokamak RFP

■ Usually pressure driven RWM

■ RWMs can be either “resonant” or “non-resonant” modes

■ Strong toroidal coupling, ballooning character

■ Pressure Driven RWM can be stabilized by very low rotation frequency( < 0.5%A), even without rotation.

■ Kinetic stabilization at slow rotation is mainly due to the mode resonance with precession drifts of trapped particles

■ Current driven modes can not be stabilized by the precession resonance.

■ Current driven RWM( nowall=0).

■ All RWMs are “non-resonant” modes.

■ Weak toroidal coupling (only one poloidal dominant harmonic), almost no ballooning character.

■ In high beta RFP, the mode can be stabilized in ion acoustic frequency range (e.g. in RFX-mod, n=6, 0.03A).

■ Kinetic stabilization is mainly due to the mode resonance with transit frequency of passing ions (ion acoustic Landau damping).

■ The precession resonance of trapped particles can not stabilize the RWM.

In both configurations:


■W∞, Wb need to be small, Wk becomes significant in determine the mode stability.

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