1. G.HuysmansETFP2006, Krakow11-13/9/2006 Edge Localised Modes: Theory/Simulation Guido Huysmans Association Euratom-CEA Cadarache, France ETFP2006, Krakow 11-13/9/2006 Acknowledgement: M. Becoulet, D. Brennan, S. Medvedev, E. Nardon, S. Saarelma, P. Snyder, P. Tamain, H. Wilson

2. G.HuysmansETFP2006, Krakow11-13/9/2006 Outline Introduction –ELMs at the ideal MHD stability limit Linear MHD questions/extensions –(diamagnetic) flow –Separatrix –Resistivity ELM models/simulations –Simple 0D model –Non-linear evolution Analytic theory Non-linear MHD Simulations Conclusion

3. G.HuysmansETFP2006, Krakow11-13/9/2006 Introduction Good agreement between edge pedestal pressure gradients and linear MHD stability limits due to ballooning and peeling (external kink) modes. –Improvement with plasma current and triangularity –ELMs caused/triggered by ideal MHD instability: ballooning and/or peeling mode P. Snyder, NF2004 (DIII-D, GA and UKAEA Culham) ELITE

4. G.HuysmansETFP2006, Krakow11-13/9/2006 Ballooning Modes Ballooning mode results from a competition between the local destabilising pressure gradient (in ‘bad’ curvature region, low-field side) and the stabilising field line bending –n=∞ ballooning mode most unstable limit increases with shear –n=∞ ‘second stable regime’ at large edge current density at high q, high poloidal beta and large plasma shaping –Finite-n ballooning modes remain unstable in second stable regime mainly localised in edge transport barrier, tail into plasma stability improves with plasma shaping radius n=32

5. G.HuysmansETFP2006, Krakow11-13/9/2006 External Kink (Peeling) Modes m=5 (  ) destabilised by edge current (J edge /, I 90 ) stabilised by pressure gradient (due to favourable average curvature) –up to ballooning limit, then destabilised low-n most unstable –peeling mode is an external kink mode with one main resonant poloidal harmonic at the edge of the plasma requires a rational q just outside the plasma more localised at boundary then ballooning modes

6. G.HuysmansETFP2006, Krakow11-13/9/2006 Peeling-Ballooning Modes driven by edge pressure gradient and edge current density –connects stability limits of peeling and ballooning modes –most relevant in second stable regime radius n=20 peeling-ballooning mode

7. G.HuysmansETFP2006, Krakow11-13/9/2006 Edge MHD Stability diagram Ideal MHD instabilities driven by: –pressure gradient : ballooning modes –parallel current density : external kink modes (peeling modes) pressure gradient MISHKA-1 Edge stability diagram (JET low-triangularity):

8. G.HuysmansETFP2006, Krakow11-13/9/2006 P. Snyder (ELITE)

9. G.HuysmansETFP2006, Krakow11-13/9/2006 Open Questions Extensions to the ideal MHD model: –Diamagnetic flow Stabilisation : Radial variation of  * : –reduced stabilisation –remaining small growth rate –Interaction with Alfvén continuum –Toroidal flow –Resistivity, Viscosity, Diffusivity, FLR many unstable modes, interpretation? Separatrix geometry –Peeling mode requires rational surface just outside plasma but all rational q are inside the plasma with separatrix  (  *)    ideal MISHKA-D A/cA/c

10. G.HuysmansETFP2006, Krakow11-13/9/2006 Toroidal Flow Toroidal is stabilising for high-n ballooning modes –Less important for relevant medium-n ballooning modes –Stabilisation stronger at small aspect ratio (spherical tokamaks, Saarelma 2006) –Combination with diamagnetic flow yields co/counter difference (Chapman, MISHKA-D-flow, PoP2006) P.Snyder EPS2005 DIII-D Experimental level  0.2 C s Saarelma (MAST) 2006

11. G.HuysmansETFP2006, Krakow11-13/9/2006 D. Brennan (NIMROD)

12. G.HuysmansETFP2006, Krakow11-13/9/2006 Peeling modes stabilised by Separatrix KINX code shows ideal MHD peeling modes are strongly stabilised by the separatrix (Medvedev, PPCF2006) Ideal kink stability limit at : J // ~ Stabilisation confirmed by CASTOR & JOREK results (Huysmans, PPCF2005) Ideal MHD kink limit not relevant for ELMs?  p/  p c J // / ballooning kink Medvedev, PPCF2006 displacement

13. G.HuysmansETFP2006, Krakow11-13/9/2006 Peeling-tearing Mode Resistive peeling modes are also stabilised by separatrix Resistive peeling-tearing mode remains unstable –Mode structure like kink/peeling mode but with phase change at X_point –Does not sensitively depend on q close to boundary –Stability limit J // / similar to limit of peeling modes without separatrix –Small growth rates (diamagnetic stabilisation likely?) peeling modepeeling-tearing  =0.99  =0.998 X-point Growth rate(n=1)

14. G.HuysmansETFP2006, Krakow11-13/9/2006 ELM cycle Linear MHD (ideal MHD, rotation, diamagnetic, resistivity) –provides the relevant MHD instabilities (ballooning/kink) –the MHD stability limits on the pressure gradient/current density –Information on the width of the instability Size of ELM : energy loss per ELM –How far can one cross the MHD stability boundary? What is the noise level for the ideal MHD mode in stable plasma? –What determines the final state after the ELM? Is there a correlation with the width of the linear eigenmode? –What is the saturation mechanism? Why not a saturated instability but a discrete event? ELM time scales

15. G.HuysmansETFP2006, Krakow11-13/9/2006 Simple 0D ELM cycle Linear MHD model –Exponential growth ( )  of linear instability for  >  crit –Small damping (D) when mode is stable –low noise level A 0 for stable mode Pressure gradient Mode amplitude  min cc AmAm 1 2 34  max noise

16. G.HuysmansETFP2006, Krakow11-13/9/2006 Simple 0D ELM cycle 1)Initial Growth of MHD Instability (passing through marginal stability): timescale : Analytic solution: 2)Maximum gradient: 3)Relaxation to minimum : 4)Final level MHD exactly identical to initial noise level: A min, A max do not depend on details of growth rate and damping dependence E. Nardon, P. Tamain

17. G.HuysmansETFP2006, Krakow11-13/9/2006 Simple 0D ELM cycle  A

18. G.HuysmansETFP2006, Krakow11-13/9/2006 Non-linear Ballooning Modes (theory) Wilson/Cowley : Analytic theory for high- n modes: – a tour de force of algebra, based on a number of approximations: – an ordering of spatial scales derived from linear theory – close proximity to marginal stability – periodicity boundary conditions assumed not to be important Predictions include: – Hot filaments (flux tubes) of plasma are formed – They are ejected explosively from the pedestal – The filaments are extended along field lines,but localised toroidally helps explain why ELM event is fast, even close to marginal stability

19. G.HuysmansETFP2006, Krakow11-13/9/2006 ELM models Filament model (Wilson et al.) –‘Leaky filament’ Filament remains connected at inboard (large reservoir) Diffusive losses from filament into SOL due to large pressure gradients Secondary instabilities? –‘squirting hosepipe’ Filaments reconnects at X-point Direct connection from main plasma to wall –return to L-mode Filament crossing flux surfaces stops shear flow : removes turbulence stabilisation

20. G.HuysmansETFP2006, Krakow11-13/9/2006 JOREK (1.0) JOREK Non-linear MHD code (under development) –Reduced MHD model in toroidal geometry Evolving poloidal flux, vorticity, density and temperature –Finite element discretisation in poloidal plane Fourier or FE in toroidal direction Finite elements aligned on equilibrium flux surfaces Includes X-point and open field lines Mesh refinement in 2D, variable number of harmonics –Fully implicit time evolution Linearised Crank-Nicholson Allows large time steps –Parallelised using MPI –Parallel direct sparse matrix solvers (MUMPS, WSMP, PASTIX*) *P. Ramet et al., LaBri, University of Bordeaux 1

21. G.HuysmansETFP2006, Krakow11-13/9/2006 Non-Linear Peeling-Tearing Mode n=1 Peeling-tearing mode in separatrix geometry saturates into steady state (JOREK) –Peeling-tearing mode causing large deformation of density profile at X-point –Consistent with long-lived saturated JET ELM precursors identified as external kink modes (Outer Modes) –Relevant for Edge Harmonic Oscillation in QH modes? kinetic magnetic time  (Alfvén) energy Huysmans PPCF2005 density profile

22. G.HuysmansETFP2006, Krakow11-13/9/2006 Non-linear Ballooning Mode n=6 ballooning mode: –  (0) = (0) = 5x10 -6 ; D  =   =10 -5 ;  // =0.2 –Pedestal :  >0.9 : D  =   =10 -6 FE grid, equilibrium flow

23. G.HuysmansETFP2006, Krakow11-13/9/2006 Non-linear Ballooning Mode Density ‘Blobs’ are sheared of by large n=0 induced (zonal) flow –number of blobs (in time) depends on linear growth rate –blobs are cold –no obvious magnetic structure (outside separatrix) densityflow

24. G.HuysmansETFP2006, Krakow11-13/9/2006 n=0 (Zonal) Flow n=0 flow driven by [  n,J n ] non-linearity Zonal flow limits amplitude of ballooning mode –n=0 flow larger than ballooning mode flow (but same order) totaln=0 projection Kinetic energy balance:

25. G.HuysmansETFP2006, Krakow11-13/9/2006 Density & Temperature Profiles Mid-plane density profile shows ‘blobs’ (density filaments) –life time depends on particle diffusivity in SOL Temperature profiles remain monotonic –due to large parallel transport (T ~constant on perturbed flux surfaces) Losses from plasma into SOL : 3-4% for density and pressure

26. G.HuysmansETFP2006, Krakow11-13/9/2006 D. Brennan (NIMROD)

27. G.HuysmansETFP2006, Krakow11-13/9/2006 D. Brennan (NIMROD)

28. G.HuysmansETFP2006, Krakow11-13/9/2006 P. Snyder (BOUT)

29. G.HuysmansETFP2006, Krakow11-13/9/2006 Conclusion Ideal MHD external kink/peeling modes not likely relevant to ELM crash –Strong stabilisation from separatrix moves limit up to J // / ~1 Resistive peeling-tearing mode remains unstable with in presence of separatrix –Mostly kink-like mode except at X-point –Small growth rates, no strong q-edge dependence –Saturates non-linearly : consistent with Outer Modes, QH mode EHO. ELM crash : non-linear evolution of medium-n ‘ideal’ MHD ballooning modes –Explosive expulsion of filament(s) (analytic theory/ BOUT simulations) Energy loss mechanism not clear (reconnection, locking of edge rotation, …) –Or : Out-flowing density following ballooning mode flow pattern, sheared off by induced n=0 flow, leading to ‘blobs’ (density filaments) (JOREK simulations) Discrete losses in one or more bursts Saturation of burst due to induced n=0 flow