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Size and amplitude scaling of ELM-wall interaction on JET and ITER

Size and amplitude scaling of ELM-wall interaction on JET and ITER. W.Fundamenski and O.E.Garcia Euratom/UKAEA Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, UK.

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Size and amplitude scaling of ELM-wall interaction on JET and ITER

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  1. Size and amplitude scaling of ELM-wall interaction on JET and ITER W.Fundamenski and O.E.Garcia Euratom/UKAEA Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, UK This work was funded jointly by the UK Engineering and Physical Sciences Research Council and by the European Communities under the contract of Association between EURATOM and UKAEA. The view and opinions expressed herein do not necessarily reflect those of the European Commission.

  2. Open question: Size & amplitude scaling Why do bigger, more intense ELMs deposit a larger fraction of their energy on the wall ? or, in the context of a filament model, why do larger ELM filaments travel faster ? A.Loarte et al., Phys. Plasma, 11 (2004) 2668 T. Eich et al., J. Nucl. Mater., 337-339 (2005) 669

  3. Origin of plasma motion in non-uniform B-field Momentum conservation equation Denoting LHS byFand taking a curl yields a general plasma vorticityequation, where k is the magnetic curvature, eg. in MHD, the two terms on the RHS correspond to flux tube bending (kink) and interchange (ballooning). Term by term, this equation can be recast as a charge conservation or current continuity equation, which expresses the balance of polarisation, parallel and diamagnetic currents R.D.Hazeltine and J.D.Meiss, Phys. Rep., 121 (1985) 1, or Plasma Confinement (1992) Addison-Wesley, New York

  4. Inertia + curvature = interchange motion Note that the divergence of the diamagnetic current, is equal to the divergence of the current due to guiding centre, magnetic drift In toroidal geometry, grad-B points towards the major axis, which leads to a vertical polarisation of charge and an outward radial ExB drift, as shown for a plasma filament (blob), see left. This is the origin of the interchange instability and ballooning transport in tokamaks, i.e. turbulent motions in regions of unfavourable magnetic curvature (pointing along the pressure gradient).

  5. Two-field interchange model Invoking the thin layer approximation, one obtains the reduced vorticity equation where x, y, z are the radial, poloidal and parallel co-ordinates, h is the kinematic viscosity and is the electric drift vorticity. We complement this with an advection-diffusion equation for a generic thermodynamic variable q, where c is its collisional diffusivity. We normalise by a characteristic cross-field blob size l, the ideal interchange growth rate 1/g, and the characteristic variation Dq, N.Bian et al, Phys. Plasmas., Phys. Plasmas 10 (2003) 671

  6. Dimensional values for typical ELM filaments Let us set the model parameters at typical large tokamak values And choose the cross-field filament size and amplitude as This yields the following values of gyro-radius, sound speed and interchange rate, such that the ideal velocity, , is equal to 25 km/s. O.E.Garcia, N.H. Bian, W.Fundamenski., submitted to Phys. Plasmas

  7. Non-dimensional model equations This gives the non-dimensional model equations Where k and m are the non-dimensional diffusivity and viscosity. Their product and ration define the Rayleigh and Prandtl numbers, Ra is the ratio of boyancy and collisional dissipation, while Pr is the ratio of viscosity and diffusion. N.Bian et al, Phys. Plasmas., Phys. Plasmas 10 (2003) 671; O.E.Garcia et al, Phys. Plasmas, 12 (2005) 090701

  8. Ideal (non-dissipative, collisionless) limit For larger Rayleigh numbers, the viscous term is negligible Which gives the only dimensionally allowable scaling of the transverse velocity, Hence, in the ideal (collisionless) limit, the radial Mach number increases as the square root of the cross-field filament size and the relative thermodynamic amplitude, i.e. the perturbation compared to some background value. In other words, provided dissipation forces are small, we expect larger and more intense perturbations to travel faster (aside from the obvious scaling with the sound speed), in broad agreement with ELM measurements on JET, see below. O.E.Garcia et al, Phys. Plasmas, 12 (2005) 090701

  9. Initial conditions and moments Consider a gaussian filament, initially at rest, Define the centre-of-mass position, velocity and effective diffusivity as where the dispersion tensor is given by O.E.Garcia, N.H. Bian and W.Fundamenski., submitted to Phys. Plasmas

  10. Numerical simulations of filament motion density, pressure vorticity Radial distance Radial distance O.E.Garcia, N.H. Bian and W.Fundamenski., subm. to Phys. Plasmas

  11. Position, velocity and diffusivites vs. time Filament velocity, in units of , increases from 0 to < 1, then decays gradually with time. O.E.Garcia, N.H. Bian and W.Fundamenski., submi. to Phys. Plasmas

  12. Range of Rayleigh and Prandtl numbers

  13. Maximum radial velocity vs. Ra and Pr Maximum velocity, in units of , is only weakly dependent on collisional dissipative effects.

  14. Sheath dissipation: earlier theories In earlier theories, the interchange term was assumed to be non-linear in q and the effect of parallel currents was included via a so-called sheath-dissipative term, This form allows an analytical solution in the ideal (collisionless) limit which gives the following transverse velocity. Note the strong, inverse size scaling, and no amplitude dependence, S.I.Krasheninnikov, Phys.Lett. A, 283 (2001) 368; D.A.D’Ippolito et al, Phys. Plasmas 9 (2002) 222

  15. Sheath dissipation: improved model In line with the earlier derivations, we also introduce the sheath-dissipative term, but not the strange, q non-linearity, Spectral decomposition (Fourier transform) of this equation reveals the major difference between viscous and sheath dissipation. The former damps small spatial scales (large k), while the latter damps large spatial scales (small k), which should affect the morphology of plasma filaments. O.E.Garcia, N.H. Bian and W.Fundamenski., subm. to Phys. Plasmas

  16. Effect of sheath dissipation (Ra = 104, Pr = 1)

  17. Maximum radial velocity vs. Ra and L (Pr =1) Maximum velocity, in units of , decreases substantially due to sheath dissipative effects Earlier, ideal sheath-dissipative solution O.E.Garcia, N.H. Bian and W.Fundamenski., subm. to Phys. Plasmas

  18. Expression for ELM energy to wall on JET Interchange drivenamplitude scaling with convective ion losses combined with moderate-ELM (DW/W = 5%, DW/Wped=12%) e-folding length, yields so that fraction of ELM energy to wall can be approximated as where Dped is the pedestal width and DSOL is the separatrix-wall gap. eg. when DW/W reduced by a third, then (Wwall/W0) = 10 % for 3 cm gap, see below. W.Fundamenski et al, PSI 2006; subm. to J.Nucl.Mater

  19. Comparison with JET data presented earlier Model prediction plotted for a range of DSOL = 1 – 5 cm, assuming Dped = 3 cm Comparison with JET data reveals fair agreement, consideringscatter in data and approximate nature of the model

  20. ELM-wall & limiter interaction on ITER Same prescription as used to match JET data (Type-I ELMs, DW/W = 5 %) ~ 8 % of ELM energy onto main wall at 5 cm (omp) ~ 1.5 % of ELM energy onto limiter at 15 cm (omp) ITER 2nd separatrix movable limiter Normalised ELM filament quantities Normalised time since start of parallel losses W.Fundamenski et al., Plasma Phys. Control..Fusion, 48 (2006) 109 R.Aymar et al., PPCF44 (2002) 519

  21. Prediction for ITER: ELM amplitude scaling Approximate amplitude scaling (in fact, the e-folding length increases with distance) ELMs size required from material limits Nominal Type-I ELM size on ITER 8 % @ 5 cm 0.6 % @ 5 cm

  22. Conclusions • JET data indicates that bigger (more intense) ELMs deposit a larger fraction of their energy on the main chamber wall, which suggests that the radial Mach number increases with ELM size • Two-field interchange model used to study size & amplitude scaling • It was found that over a wide range of conditions, the radial Mach number is expected to increase as the square root of both ELM size and amplitude, • This implies that radial e-folding length of ELM filament energy also increases • Model predictions in fair agreement with JET data • Preliminary predictions for ITER indicate the added benefit of reducing the ELM size: for small ELMs, DW/Wped < 3%, less than 1% of ELM energy deposited on the wall (near 2nd separatrix at upper baffle); contact with main wall is negligible.

  23. Ion impact energies on JET and ITER Predicted peak ELM filament quantities on JET and ITER (moderate Type-I ELMs) • JET: Ti,max(rlim)~ 185 eV (ion impact energy ~ 0.6 keV) at 4 cm • ITER: Ti,max(rlim)~ 350 eV (ion impact energy > 1 keV) at 5 cm; ~ 100 eV at 15 cm • Lower bound estimates for moderate (DW/W ~ 5 %) Type-I ELMs Peak ion impact energy Radial distance from mid-pedestal location W.Fundamenski et al., Plasma Phys. Control..Fusion, 48 (2006) 109

  24. Energy equation for interchange motions The change in kinetic energy is related to the compression of the polarisation current, Compression of diamagnetic current is related to the magnetic curvature, Hence, plasma motions are amplified when energy flows opposite to the magnetic curvature vector, i.e. in region of bad curvature.

  25. Advective-diffusive description of ELM filament • Conservation equations for mass & energy • Green’s function = advective-diffusive wave-packet (filament) • Radial velocity and diffusivity prescribed W.Fundamenski et al., Plasma Phys. Control..Fusion, 48 (2006) 109

  26. Parallel loss model of ELM filament evolution Key elements of parallel loss model • Temporal evolution of n, Te and Ti in the filament frame of reference • Above quantities represent averages over the filament volume • Time and radius related by filament radial velocity, which is not calculated by the model • Parallel loss treated by convective and conductive removal times • Acoustic loss of plasma • Electrons cooled faster than ions W.Fundamenski et al., Plasma Phys. Control..Fusion, 48 (2006) 109

  27. Filament evolution for nominal JET conditions • As expected Te decays faster than Ti, which decays faster than n • As the initial density is increased, e-i equipartition becomes more effective and the two temperatures converge more quickly n’ Ti’ Normalised ELM filament quantities Ion-to-electron temperature ratio Te’ Normalised time since start of parallel losses W.Fundamenski et al., Plasma Phys. Control..Fusion, 48 (2006) 109

  28. Good agreement with all dedicated JET data

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