1. Simulating an IBW propagating transversal to a tokamak confining field: non-linear kinetic effects and possibility for turbulence suppression Chiara Marchetto Istituto di Fisica del Plasma, C.N.R., EURATOM-ENEA-CNR Ass.; Milan, Italy INFM – Iniziativa Trasversale Calcolo Parallelo, Italy In collaboration with: M. Lontano 1, F. Califano 1,2 1 Istituto di Fisica del Plasma, C.N.R., EURATOM-ENEA-CNR Ass.; Milan, Italy 2 Dipartimento di Fisica, Universita’ di Pisa, Pisa, Italy

2. OUTLINE 1. Tokamak, turbulence suppression and IBW 2. The model, the code and the HTC 3. The macroscopic quantities 4. The kinetic quantities 5. Summary

3. A way to affect turbulence: Shear Flow (flux of particles with average velocity in poloidal direction changing of sign and module along radial direction) reducing correlation length and of the amplitude of turbulence BDT turbulence suppression criterion on the shear flow, the turbulent decorrelation frequency  k, the corelation length along the shear  y and the Fourier wavenumber of the correlation length along the flow k x 3 TOKAMAK, TURBULENCE SUPPRESSION, IBW Tokamak: toroidal confinement device for nuclear-fusion purposes Turbulence: cause of poor confinement, main problem for tokamaks Ion Bernstein Waves: electrostatic waves propagating a right angles to confining magnetic field at harmonics of the cyclotron frequency.

4. y z x k,E wave B B - Aim: to model in a simplified way the interaction of an Ion Bernstein Wave with a tokamak plasma - k  B 0 ; E || k; B 0 uniform - monochromatic wave, modelled by: -   4  ci ;   li  k  li >1 - slab geometry: 3d (1x2v) - parameters from IBW-FTU : n e = 5 × 10 13 cm -3 T e = 1 keV, B 0 = 7.8 T (  e = -m i /m e,  i = 1) THE MODEL 4

5. - We start from Mangeney, Califano, Cavazzoni, Travnicek 2002, Journ. Comp. Phys., 179, 1, accurate up to second order in time, position, velocity; high resolution in velocity, requiring a large numerical effort. - Periodic spatial boundary conditions, x  [0,3 ] ( = pump wavelength). At t=0 the electron and ion distribution functions are Maxwellian and no electromagnetic field is present. - Pump applied to the system throughout the interaction time. - Upgrade: Electrons following E  B drift, System response electrostatic  Eliminated Vlasov eq for electrons, set E y  0 and B z  B 0, eliminated all Maxwell equations except Poisson (for E x ) [ Marchetto, et al., 2002]. - High Throughput Computing (large amounts of fault-tolerant computational power over prolonged periods of time) via Condor Scheduler (specialised job and resource management system, enabled for Opportunistic Computing i.e. ability to use resources whenever they are available, without requiring 100% availability). THE CODE and the HTC 5

6. THE MACROSCOPIC QUANTITIES: - high wave amplitude, - peculiar relations between wave frequency and system frequencies u iy and u ix shown at t=144 for  0 =0.35 6 - spatial response can be non-linear, as ex: - oscillations in space ( = 0 ) and time (  =  0,  =  ci ) - constant and uniform mean value - constant amplitude

7. - Plasma response to a great extent electrostatic (B unperturbed, E y negligible) - - E x of the order of the wave amplitude - Energy content of the system increasing during the interaction with the wave. 7 (the ion energy content averaged over space) plotted versus time for a=0.0001 and  0 =0.35 E x (x) as measured during the simulation

8. - The ion space-averaged fluid velocity in y direction oscillates at  ci with mean value always negative and 2 to 4 orders of magnitude smaller than the ion thermal velocity: the flow! 8 Ion space averaged fluid velocity versus time, for  0 =0.35 and a=0.0001

9. - The wave-vector spectra show the typical behaviour of a cascade toward the small scales. 9 - The frequency spectra present a maximum at the wave frequency and peaks at the ion cyclotron frequency and at its harmonics |E k | plotted versus k at t=630 (sx) and |E  | plotted versus  at x= 0.026 (dx), for  0 =0.35 and a=0.0001. All in dimensionless units.

10. - The intensity of the wave-induced flow presents a peak for a value of the wave frequency comprised between  0 =0.34 and  0 =0.38 (4  ci =0.32) 10 (a) (b) Flow versus frequency for a=0.0001 (a) and a=0.00005 (b). All in normalised units.

11. 11 Preliminar estimates of the shear flow - Variation of the flow intensity with the pump wave frequency interpreted as a variation of flow intensity with the position, ie as shear flow - By assuming profiles for B(r) and n(r), it is possible to relate a variation in the normalised wave frequency  /  pi to a variation in the normalised coordinates x/a - Shear flow: - BDT criterion in our geometry: - For drift turbulence ( T=1keV, B=7.5  10 4 gauss, L n =a=30cm,  x k =  li /0.2 ): - Almost sufficient for turbulence suppression - Recent calculations show it is enough ( F. Califano, et al., 30 th EPS Conf. on Contr. Fus. and Plasma Phys., S.Petersbourgh, July 2003 ).

12. - Fdp: plateau and population inversion for v x   v , secondary peaks for v x =  2v   Perpendicular Ion Landau Damping (ILD) despite the presence of a strong confining magnetic field -Up to now: plasma in magnetic field considered as unmagnetised for high frequencies (  0  30  ci ) and for short times (few wave cycles). -Our simulations: ILD for  0  4  ci and several wave cycles 12 The ion distribution function f i (x,v x,v y ) is plotted versus v x, at x=0.026 and v y =0, for  0 =0.35 and a=0.0001, for t=0 (dashed line) and t=630 (solid line). THE KINETIC QUANTITIES:

13. - The contour plot (x,v x ) of Fdp shows vortexes, typical symptom of particle trapping: same structures as in simulations performed with B = 0, and as found for Electron Landau Damping in literature ( Brunetti, Califano, Pegoraro, Phys Rev E 62 4109 (2000) ) - velocity trapping region fits well with plateau 13 Contourplot (x,v x ) of the ion distribution function if B=0,  0 =1,.9305 and a=0.001, for t=13. Contour plot (x,v x ) of the ion distribution function for  0 =0.35 and a=0.0001, for t=630.

14. - Fdp presents plateau and population inversion also as a function of v y, i.e. for v y   v , with secondary peaks for v y =  2v  14 The ion distribution function f i (x,v x,v y ) is plotted versus v y at x=0.026 and v x =0, for  0 =0.35 and a=0.0001, for t=0 (dashed line) and t=630 (solid line).

15. - Even the contour plot (x,v y ) presents vortexes: the magnetic field adds a mixing effect that makes it possible to propagate the perturbation to the Fdp also in the v y direction, and the effect is stronger around v y  - v , than around v x  v  - This localisation of the perturbation of the distribution function in the negative range of v y is responsible for the negative flow in the y direction 15

16. 16 Contour plot (v x,v y ) of the ion distribution function at x=0.026, for  0 =0.35 and a=0.0001, for 4 instants of time for a=0.0001 and a=0.00005. The contour plots are seen as “ellipses”, instead of as “circles”, due to the difference of the abscissa and the ordinata scales in the plot. The finger-like structures:

17. SUMMARY - This wave-plasma interaction produces a flow in y direction, its intensity is from 2 to 4 orders of magnitude smaller than v thi Flow intensity: peak for  strictly larger than 4  ci, - Flow shear almost enough to reduce plasma turbulence (for drift- like turbulence) - Ion distribution function noticeably different from Maxwellian: plateau, population inversion, Transverse Ion Landau Damping, trapping - Magnetic field mixing effect, causing perturbation propagation to the vy direction, is supposed to be the origin of the flow 17