Non-linear MHD modelling of RMPs with toroidal rotation and resonant and non-resonant plasma

braking.

M.Becoulet G. Huysmans, E. NardonAssociation Euratom-CEA, CEA Cadarache, F-13108, St. Paul-lez-

Durance, France.Thanks to all USBPO RMPs team and especially to M. Schaffer and S. Sabbagh

Outline:1. MHD model with resonant (jxB) and non-resonant

(NTV) plasma braking.2. Example for 18 picture frame coils.

code RMHD: reduced non-linear MHD [A.Y Aydemir, Phys.Fluids B4(11)1992,3469] in cylindrical geometry, but with some new physics included:-Doppler shift due to the toroidal rotation;

-resonant (jxB) braking [Y.Kikuchi et al PPCF 48(2006)169],[E. Lazzaro et al

PoP9(2002)3906] -non-resonant braking [K. Shaing, PoP 10(2003)1443], [W.Zhu et al PRL96,225002(2006)], due to the Neoclassical Toroidal Viscosity (NTV).

2( ) , ,//

W Wv uW j p u Wt z

(1 )//

v u p jt z

2, ( 2 )//

p pv u p v j k p Spt z

1( ) (1 ) ,//2

2// (0,0) (0,0)

v vv p u p vt z

S v F Fv j B NTV

Vorticity:

Pressure(~0 here):

Poloidal flux:

Parallel velocity:(Source is adapted: )

00, 02

// tS vv

Calculation of Neoclassical Toroidal Viscosity (NTV) in collisionless regime.(W.Zhu PRL2006)

;

2 ;

1,1 12 3/2

Vf

R

piiF e B R Intv t B Rt i

3 24

13.708;1

1/2 3/2 2 3/215.18 ln 1.8210 ;19 3 19 3,(1/ ) ( ) ( )(10 ) (10 )

ln 17; 2; 1;

, ,(1/ )

iA n T n T

i s i keV i keVm mA Z

i Shaing i s

-used here

2 ( );

1 1( , ) (( ), )) ;0 0 0 0

0 01 ( , )

00

0 0 0( ) ;2 2 20 0 0

;

cos( ) sin( );0,

b real number

real

B B B B b B B bB B

b b BB

R R Z Zb B b B b Bn n nin inb b e en n B B Bn

in imb emnn m

b b m n b m nnmc nmsn m

b b bnmc mn nm

0

2 ( );

3/2 2 2 2( ) ; /2 , 0

11 2 2 2 2( ) (1 ) ( ) ( ( ) ( ) );0

2arcsin( ) 2 2 1/22 ( sin ( /2)) cos(( ) );0

2arcsin( ) 2 2 1/2( ) 2 ( sin ( /2)) sin(0

im bs mn

I n b b W r Rnmc nms nmm n

W dk E k k K k F k F knm nmc nms

kF d k m nqnmckF k d knms

( ) );m nq

Complete eliptic integrals of first (K) and second (E) kind.

R1= 8.608m; Z1=1.798; R2= 8.664; Z2=-0. 558 =12.620° ; c-c=20.° PF-coil currents (A) (nmax=4): 19140. 110000. 19140. -103400. -55000. 84260. 84260. -55000. -103400. 19140. 110000. 19140. -103400. -55000. 84260. 84260. -55000. -103400.

Example of the spectrum from 18 picture-frame coils around ports in-vessel.

Chirikov parameter and normalized radial magnetic field in cylindrical approximation for H-mode, Hybrid and ITB q-profiles. For peak current 110kAt,n=-4 edge (>0.9) is ergodized.

Islands size in cylindrical approximation for H-mode, Hybrid and ITB q-profiles.

, , , ,

;

;R Z R Zn

n

A

A

inA en

Rn

Toroidal harmonic n=-4 of poloidal plux perturbation.

Input for RMHD code: normalized bnd at r=a. n=-4

Equilibrium components needed for calculations of NTV. ITER H-mode.

0 0 0 020 0

0 0 020

0

( );

0;

1 1( , ) (1 ( , )) (1 );

1; R R Z Zn n n

n

b b B b B b B

here b

B B B B b B B bB B

inb b en n Bn

-magnetic field strength along non-perturbed line.

0,2 ( ); 2 ( )

cos( ) sin( );, n m

real b im bmn nm

im inb b e b m n b m nnm nmc nmsn m

b bnmc nms

Poloidal harmonics for magnetic field strength. Non-resonant m=0 is the largest=> typical for one-row coils.

; / ;3/2 2 2 2( )2 0,

r RI n b b Wnmc nms nmn m

Integral Iin the expression for NTV. Here only n=4 is taken into account. Possibly n=14 will be important. Non-resonant harmonics are more important. Also they are not screened by rotation, so one can take vacuum amplitudes for these harmonics.

Plasma parameters (H-mode) for estimations of NTV from 18pf coils (n=4).

NTV force and damping time (~0.4s on r=0.4) for ITER H-mode parameters with 18 picture-frame coils at I=110kA. Here only n=4 is taken into account. Possibly with n=14 damping time will be a bit shorter.

NTV forceDamping time

Non-linear MHD modelling with rotation and only resonant braking. n=-4, m=10:14; bnd=(2.5m=10;2m=11;1.5m=12; 1.25m=13;1m=14; )10-4; =10-8 (here plasma resistivility is higher compared to real one 10-9-10-10)

Resistivity profile q-profile

Central islands are more screened, but edge ergodisation persist : smaller rotation, larger resistivily=>less screening at the edge.

0.7 1.r

Vt=0; t=8000A

Vt=0.5610-2; t=8000A,

only resonant braking

More external harmonics are less screened by rotation.mn q=-m/n with (Vt=0.0056) and without rotation.

Resonant braking near q=-m/n surfaces

6

How the most central (most screened) harmonics n=-4,m=10 looks like: (t=8000A)

Convective cells are formed in the ergodic zone (seen also in JOREK code E. Nardon PoP 2007)=>density transport?

Initial rotation profile corresponds to ft(0)~1kHz (ITER-like).Resonant (jxB) braking is localized near q=-m/n surfaces. With NTV global braking is observed. Here ‘normal’ toroidal viscosity ://=10-6, NTV has a

calculated form (p.13) with maximum NTV,max=10-6 . It’s a bit higher (to see more rapid braking in modelling) compared to our estimations ~5.10-7 on p.13)

;1 2( ) (1 ) ,// //2

;

://

v vv p u p v S v F Fv j B NTVt zF vNTV NTV

Here v v

It is not stationary profile yet!Braking continues.

Here similar weak screening for m=10 with jxb resonant braking and both jxB and NTV.

Vt=0; t=8000A

jxB:Vt=0.0056; t=8000A

jxB+NTV:Vt=0.0056; t=80000A

More external harmonics are less screened by rotation=> edge erdodisation.mn q=-m/n with (Vt=0.0056) and without rotation.

Total braking near q=-m/n surfaces

Conclusions (from previous presentation): -Penetration time increases like ~1/resistivity. For ITER~to the top of the pedestal~1s!-Larger amplitudes are less screened by rotation.-Edge islands are much less screened than ymnon q=-m/n.-Edge is ergodised even with strong rotation( DIII-D like).-ITER rotation screens central (m<8) non-resonant ,edge is ergodised.-Non-resonant harmonics are not screened by rotation.

Conclusions (from this presentation): -one row design (here 18 picture-frame coils, but it’s typical for one row designs) give large amplitudes of non-resonant harmonics in the plasma centre, notice also that they are not screened by plasma rotation.-The NTV calculated according to K. Shaing in collisionless regime gives damping time ~0.4s at r~0.4 (ITER H-mode,18pf coils, n=4);-Edge ergodisation here is weakly influenced by plasma braking, since the initial rotation was already weak. More external islands (here m>10) are less screened by rotation, since resistivity is larger and rotation is slower. However , here we are still two orders of magnitude larger resistivity on the top of the pedestal, so screening is expected to be larger. To be continued...