Non-linear MHD modelling of RMPs with toroidal rotation and resonant and non-resonant plasma braking. M.Becoulet G. Huysmans, E. Nardon Association 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). Vorticity: Pressure (~0 here): Poloidal flux: Parallel velocity: (Source is adapted: )

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

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

R1= 8.608m; Z1=1.798; R2= 8.664; Z2=  =12.620° ;  c-c= 20.° PF-coil currents (A) (n max =4): 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.

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.

-magnetic field strength along non-perturbed line.

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

Integral I  in 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 force Damping time

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

Central islands are more screened, but edge ergodisation persist : smaller rotation, larger resistivily=>less screening at the edge r    V t =0; t=8000  A V t = ; t=8000  A, only resonant braking

More external harmonics are less screened by rotation.  mn q=-m/n with (V t =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=8000  A )

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

Initial rotation profile corresponds to f t (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 = It’s a bit higher (to see more rapid braking in modelling) compared to our estimations ~ on p.13) It is not stationary profile yet! Braking continues.

Here similar weak screening for m=10 with jxb resonant braking and both jxB and NTV. V t =0; t=8000  A jxB:V t =0.0056; t=8000  A jxB+NTV: V t =0.0056; t=80000  A

More external harmonics are less screened by rotation=> edge erdodisation.  mn q=-m/n with (V t =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 y mn on 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...