DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E 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.

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Open question: Size & amplitude scaling A.Loarte et al., Phys. Plasma, 11 (2004) 2668 T. Eich et al., J. Nucl. Mater., (2005) 669 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 ?

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Origin of plasma motion in non-uniform B-field Momentum conservation equation Denoting LHS by F and taking a curl yields a general plasma vorticity equation, where  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

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Note that the divergence of the diamagnetic current, is equal to the divergence of the current due to guiding centre, magnetic drift Inertia + curvature = interchange motion 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).

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E 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,  is the kinematic viscosity and is the electric drift vorticity. We complement this with an advection-diffusion equation for a generic thermodynamic variable , where  is its collisional diffusivity. We normalise by a characteristic cross-field blob size l, the ideal interchange growth rate 1/  and the characteristic variation  N.Bian et al, Phys. Plasmas., Phys. Plasmas 10 (2003) 671

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Dimensional values for typical ELM filaments O.E.Garcia, N.H. Bian, W.Fundamenski., submitted to Phys. Plasmas 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.

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Non-dimensional model equations N.Bian et al, Phys. Plasmas., Phys. Plasmas 10 (2003) 671; O.E.Garcia et al, Phys. Plasmas, 12 (2005) This gives the non-dimensional model equations Where  and  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.

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E 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)

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E 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

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Numerical simulations of filament motion Radial distance density, pressure vorticity O.E.Garcia, N.H. Bian and W.Fundamenski., subm. to Phys. Plasmas

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Position, velocity and diffusivites vs. time O.E.Garcia, N.H. Bian and W.Fundamenski., submi. to Phys. Plasmas Filament velocity, in units of, increases from 0 to < 1, then decays gradually with time.

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Range of Rayleigh and Prandtl numbers

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Maximum radial velocity vs. Ra and Pr Maximum velocity, in units of, is only weakly dependent on collisional dissipative effects.

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Sheath dissipation: earlier theories S.I.Krasheninnikov, Phys.Lett. A, 283 (2001) 368; D.A.D’Ippolito et al, Phys. Plasmas 9 (2002) 222 In earlier theories, the interchange term was assumed to be non-linear in  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,

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Sheath dissipation: improved model O.E.Garcia, N.H. Bian and W.Fundamenski., subm. to Phys. Plasmas In line with the earlier derivations, we also introduce the sheath-dissipative term, but not the strange,  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.

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Effect of sheath dissipation (Ra = 10 4, Pr = 1)

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E O.E.Garcia, N.H. Bian and W.Fundamenski., subm. to Phys. Plasmas Maximum radial velocity vs. Ra and  (Pr =1) Maximum velocity, in units of, decreases substantially due to sheath dissipative effects Earlier, ideal sheath- dissipative solution

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Expression for ELM energy to wall on JET Interchange driven amplitude scaling with convective ion losses combined with moderate-ELM (  W/W = 5%,  W/W ped =12%) e-folding length, yields so that fraction of ELM energy to wall can be approximated as where  ped is the pedestal width and  SOL is the separatrix-wall gap. eg. when  W/W reduced by a third, then (W wall /W 0 ) = 10 % for 3 cm gap, see below. W.Fundamenski et al, PSI 2006; subm. to J.Nucl.Mater

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Comparison with JET data presented earlier Model prediction plotted for a range of  SOL = 1 – 5 cm, assuming  ped = 3 cm Comparison with JET data reveals fair agreement, considering scatter in data and approximate nature of the model

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E ELM-wall & limiter interaction on ITER W.Fundamenski et al., Plasma Phys. Control..Fusion, 48 (2006) 109 Same prescription as used to match JET data (Type-I ELMs,  W/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 2 nd separatrix movable limiter Normalised ELM filament quantities Normalised time since start of parallel losses R.Aymar et al., PPCF 44 (2002) 519

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Prediction for ITER: ELM amplitude scaling Approximate amplitude scaling (in fact, the e-folding length increases with distance) Nominal Type-I ELM size on ITER ELMs size required from material limits 8 5 cm cm

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E 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,  W/W ped < 3%, less than 1% of ELM energy deposited on the wall (near 2 nd separatrix at upper baffle); contact with main wall is negligible.

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E W.Fundamenski et al., Plasma Phys. Control..Fusion, 48 (2006) 109 Radial distance from mid-pedestal location Peak ion impact energy Ion impact energies on JET and ITER Predicted peak ELM filament quantities on JET and ITER (moderate Type-I ELMs) JET: T i,max (r lim ) ~ 185 eV (ion impact energy ~ 0.6 keV) at 4 cm ITER: T i,max (r lim ) ~ 350 eV (ion impact energy > 1 keV) at 5 cm; ~ 100 eV at 15 cm Lower bound estimates for moderate (  W/W ~ 5 %) Type-I ELMs JETITER n max (m  3 ) 8.25   T i,max (eV) T e,max (eV)74140 n,max (mm) Ti,max (mm) Te,max (mm)

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E 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. Energy equation for interchange motions

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E 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

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Parallel loss model of ELM filament evolution W.Fundamenski et al., Plasma Phys. Control..Fusion, 48 (2006) 109 Key elements of parallel loss model Temporal evolution of n, T e and T i 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

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Filament evolution for nominal JET conditions Normalised ELM filament quantities Ion-to-electron temperature ratio W.Fundamenski et al., Plasma Phys. Control..Fusion, 48 (2006) 109 Normalised time since start of parallel losses n’ Ti’Ti’ Te’Te’ As expected T e decays faster than T i, which decays faster than n As the initial density is increased, e-i equipartition becomes more effective and the two temperatures converge more quickly

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Good agreement with all dedicated JET data ExperimentParallel loss model Limiter probes + TC: n e ELM (r lim ) ~ 2.4  m -3 T e ELM (r lim ) ~ eV n,max ELM ~ 50 mm, Te,max ELM ~ 30 mm Nearly all power found on the divertor n e ELM (r lim ) ~ 2.2  m -3 T e ELM (r lim ) ~ 30 eV n,max ELM ~ 47 mm, Te,max ELM ~ 32 mm Fraction of ELM energy to wall ~ 5 % Outer gap scan + IR & TC: W ELM ~ mm, W,max ELM ~ mm W ELM ~ 36 mm, W,max ELM ~ 22 mm RFA measurements of ion energies: J sat, I coll, T e (reproduced by model) T i,max ELM (r lim ) ~ 100 eV, T e,max ELM (r lim ) ~ 40 eV, n e ELM (r lim ) ~ 4.3  m -3 n,max ELM ~ 48 mm, Ti,max ELM ~ 52 mm Te,max ELM ~ 30 mm, W ELM ~ 32 mm Fraction of ELM energy to wall ~ 15 % ELM energy deficit based on IR: ~ 30 % for ~ 3 cm gap and  W/W ~ 5 %~ 28 % for 3 cm based on W ELM ~ 35 mm