# Presentation: 858514

## 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:
- MHD model with resonant (jxB) and non-resonant (NTV) plasma braking.
- Example for 18 picture frame coils.

## -resonant (jxB) braking [Y.Kikuchi et al PPCF 48(2006)169],[E. Lazzaro et al PoP9(2002)3906] 

- -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)

- 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=-0. 558  =12.620° ; c-c=20.°    

- 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.

## Toroidal harmonic n=-4  of poloidal plux perturbation.

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

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

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

- 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.

- 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.

- 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

## r

- 0.7

- 1.

- 

- 

- 

- Vt=0; t=8000A

- Vt=0.5610-2; t=8000A,
- 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=8000A)

- 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?

## 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)

- 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)

- 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=8000A

- jxB:Vt=0.0056; t=8000A

- jxB+NTV:
- Vt=0.0056; t=80000A

## 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 this 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...