Filamentary plasma eruptions in tokamaks

The nonlinear MHD ballooning model is exploited for two distinct studies: firstly, the interaction of multiple filamentary eruptions in magnetised plasmas in a slab geometry is investigated and secondly, this model is examined quantitatively against experimental observations of ELMs in MAST and JET-like geometries. The model consists of two differential equations which characterise the spatial and temporal evolution of the displacement: the first differential equation describes the displacement along the field line, the second differential equation is a two-dimensional nonlinear ballooning-like equation which is often second order in time, but can involve a fractional derivative in a tokamak geometry. Filaments always evolve independently in the linear regime and equally sized filaments evolve independently in the nonlinear regime. However, we find that filaments with varying heights interact with each other in the nonlinear regime: Smaller filaments are slowed down and eventually are completely suppressed by the larger filaments which grow faster due to the interaction. This mechanism is explained by the down-draft caused by the nonlinear drive of the larger filaments which pushes the smaller filaments downwards. To employ the second differential equation for a specific geometry one has to evaluate the coefficients of the equation which is non-trivial in a tokamak geometry as it involves field line averaging of slowly converging functions. The coefficients of a Type I ELMy equilibrium from MAST and a Type II ELMy JET-like equilibrium have been determined. In both cases the two coefficients of the nonlinear terms are negative which would imply imploding rather than exploding filaments. By changing the equilibrium the signs of these coefficients can be inverted. This suggests that either the nonlinear Ballooning model does not capture the behaviour of Type I and Type II ELMs, or that the calculation of the coefficients are too sensitive to a given equilibrium.