TH/6 - 1Rb Non - linear MHD modelling of Edge Localized Modes and their interaction with Resonant Magnetic Perturbations in rotating plasmas. M.Bécoulet 1 , F.Orain 1 , J. Morales 1, X. Garbet 1 , G. Dif - Pradalier 1 , C.Passeron 1 , G. Latu 1 , E. Nardon 1 , A. Fil 1 , V. Grandgirard 1 , G.T.A.Huijsmans 2 , S. Pamela 3 , A. Kirk 3 , P. Cahyna 4 , M.Hoelzl 5 , E. Franck 5 , E. Sonnendrücker 5 , B. Nkonga 6 1 CEA, IRFM, 13108 Saint - Paul - Lez - Durance, France 2 ITER Organization, Route de Vinon sur Verdon, 13067 Saint - Paul - Lez - Durance, France 3 CCFE, Culham Science Centre, Oxon, OX14 3DB, UK 4 Institute of Plasma Physics ASCR, Prague, Czech Republic 5 Max Planck Institute for Plasma Physics, Boltzmannstr. 2, 85748 Garching, Germany 6 Nice University, INRIA, Sophia Antipolis, Equipe CASTOR, France marina.becoulet@cea.fr 1. Introduction. The intensive experimental and theoretical study of the Edge Localized Modes (ELMs) and methods for their control has a great importance for ITER [1]. The application of small external Resonant Magnetic Perturbations (RMPs) has been demonstrated to be efficient in ELM suppression/mitigation in present day tokamaks [2]. RMPs are foreseen as one of the promising methods of ELM control in ITER [3]. However in order to make reliable predictions for ITER, a significant progress is still required in order to understand the ELM dynamics and the interaction of RMPs with ELMs. In the present work the dynamics of a full ELM cycle including both the linear and non-linear phase of the crash and the possible explanation of the mechanism of the ELM mitigation/suppression by RMPs are presented based on the results of the multi-harmonic non-linear resistive reduced MHD modeling using the JOREK code [4]. These simulations are performed in realistic tokamak geometry including X-point and Scrape-Off-Layer (SOL) with relevant plasma flows: toroidal rotation, bi-fluid diamagnetic effects, and neoclassical poloidal friction, which have recently been included in the model [5], so that both the plasma rotation and the radial electric field are self- consistently described during MHD activity. The introduction of flows in the modelling demonstrated a large number of new features in the physics of the ELMs and their interaction with RMPs compared to previous results without flows [4]. JET and ITER parameters were used for modelling. 2. ELM modelling with flows. The detailed description of JOREK model with flows and neoclassical effects can be found in [5], here we just recall that the main flows used in modelling. The normalized fluid velocity (for ions) in JOREK units [5] is taken in the 2 R 2 following form: V ( ) v . Here the first term represents R u p B IC || V V E B || * V i the E convection, the second term is the ion diamagnetic drift and the last one is the B motion parallel to the magnetic field. Here u is the electrostatic potential , - is the mass density normalized to the central value , ( ) is the normalized scalar total p T T T i e pressure, T e,i are the electron/ion temperatures, φ – is the toroidal angle and R- the major [4], where – is radius. The magnetic field is represented in the form: B F 0 the poloidal magnetic flux, and , B being the toroidal field on the magnetic axis. F B R 0 ,0 0 ,0 T , but the model is bi-fluid, since the electron diamagnetic terms For simplicity here / 1 T e i are kept in Ohm’s law [5]. The normalized diamagnetic parame ter can be written as: 3 3 / (2 ) For JET and ITER cases the typical value is ~ 4.10 4.510 m e F 0 0 0 IC IC i respectively. Both resistivity and viscosity are temperature dependent: 4 and ~ 5.610 IC
S 3/2 8 3.310 , ~ ( T T / ) . The Lundquist number in the center was taken for ITER ||, max S 7 simulations and 5.510 - for JET, which is for numerical reasons about two orders of magnitude smaller than the realistic values. The parallel conduction has a Spitzer-like 5/2 temperature dependence: . Typically the ratio to the perpendicular K ~ K ( T T / ) || ||,0 max 9 conductivity in the pedestal was K || / K ~10 . The normalized neoclassical coefficients [6] 5 were taken constant for simplicity as in [5]: 10 ; k i = -1. A toroidal rotation source i neo , was introduced in the equation for parallel velocity to maintain the rotation profile at the initial value compensating losses due to the parallel viscosity ( ). Bohm S V || ||, 0 V t boundary conditions are set for the parallel velocity on the divertor plates: the parallel velocity is equal to the value of ion sound speed near the wall in the sheath entrance: [4,5]. Realistic JET parameters corresponding to the pulse #77329, similar to those V C || s used in [5], were used: R 0 =2.9m, a=0.89m, B tor =1.8T, q 95 =3.8. The initial density and temperature values at the plasma center and in the pedestal are: T e (0)=6keV, T e,ped =1.8keV n e (0)=5.10 19 m - 3 , n e,ped =3.310 19 m - 3 . The toroidal rotation profile is taken parabolic with central frequency 0)=38krad/s . The magnetic energy of modes n=2,4,6,8 taken for ELM modelling with and without flows are presented in Fig.1. Note that after the linear growth, all modes are unstable during the non-linear phase (ELM crash), however n=8 remains the most unstable mode in both cases. Note also the stabilizing effect of the plasma flows which decreases the growth rates of the modes and the ELM size. The ELM power deposited on the inner and outer divertor targets are smaller and almost symmetric with diamagnetic drifts (Fig.2-3). This is closer to the experimental observations [7], compared to simulations without flows where the outer divertor received almost all ELM power because of the Low Field Side (LFS) location of the ballooning instability [4]. With the E and diamagnetic advection B taken into account, more density reaches the inner divertor than the outer. Fig.1 Magnetic energy of the modes Fig.2. Density near the X-point Fig.3. Maximum power flux on the n=2,4,6,8 and kinetic energy of n=0 and maximum power flux in the inner and outer divertor targets in the ELM modelling without flows divertor due to an ELM without due to an ELM in modelling with (in bold) and with flows (in dashed). (a) and with flows (b). and without flows. 3. ELM precursors and filament dynamics. The rotation of the ELMs and their associated filaments during the ELM crashes was observed in many machines (KSTAR [8], ASDEX Upgrade [9], MAST[10]). The general observation is that an ELM precursor is observed in a time scale of about ~0.2ms before the ELM crash rotating poloidaly mainly in the electron diamagnetic (which is the same as the E ) direction with poloidal velocity of about 5- B 10km/s, which is in the range of the values of the E velocity in the pedestal. Approaching B the ELM crash, this rotation usually decreases and sometimes is reversed for ELM filaments in SOL [10]. In the present modelling of ELMs with flows, we could reproduce these generic features of the dynamics of the ELM precursors and filaments. Without diamagnetic rotation, the instabilities at the onset of the ELM have a static growth (Fig.4-top), whereas the
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