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...