Filamentary plasma eruptions: Results using the non‐linear ballooning model

Abstract

This paper provides an overview of recent results on two distinct studies exploiting the non-linear model for ideal ballooning modes with potential applications to edge-localized modes (ELMs). The non-linear model for tokamak geometries was developed by Wilson and Cowley in 2004 and consists of two differential equations that characterize the temporal and spatial evolution of the plasma displacement. The variation of the radial displacement along the magnetic field line is described by the first equation, which is identical to the linear ballooning equation. The second differential equation is a two-dimensional non-linear ballooning-like equation, which is often second order in time but can involve a fractional time derivative depending on the geometry. In the first study, the interaction of multiple filamentary eruptions is addressed in a magnetized plasma in a slab geometry. Equally sized filaments evolve independently in both the linear and non-linear regimes. However, if filaments are initiated with slightly different heights from the reference flux surface, they interact with each other in the non-linear regime: lower filaments are slowed down and are eventually completely suppressed, while the higher filaments grow faster because of the non-linear interaction. In the second study, this model of non-linear ballooning modes is examined quantitatively against experimental observations of ELMs in Mega Amp Spherical Tokamak (MAST) geometries. The results suggest that experimentally relevant results can only be obtained using modified equilibria.

Nonlinear ballooning modes in tokamaks: stability and saturation

“…with n ≫ 1) -thus in this case δF ⊥ ∼ n −1/2 sin θF ⊥ ≪ F ⊥ . In the weakly nonlinear theory [7,8,12,13] the linear eigenfunction evolves into an even narrower elliptical flux tube. The weakly nonlinear theory includes the external perturbations and the interaction of filaments because the displacement is ordered to be small ∆r ∼ δ 1 ∼ R 0 n and the system is assumed to be close to marginal stability so that δF ⊥ ∼ F ⊥ .…”

“…) giving r = r r , sat 0 q ( )-solutions of equations ( 17), ( 6) and (12). For small displacements about this lowest order solutions we can write r r r r r , , 0 s a t 0…”

“…Equation 15with B and B ⊥ given by equation (13) determines the force given the shape of the field line, r(θ, ψ 0 ) for each r 0 . Note that by definition B in • ∇r 0 = (B B 0 + B ⊥ e ⊥ ) • ∇r 0 = 0 and therefore B in • ∇ ≡ B in • ∇θ ∂ ∂θ r 0 .…”

“…The weakly nonlinear theory includes the external perturbations and the interaction of filaments because the displacement is ordered to be small r R n 1 0 d D ~~and the system is assumed to be close to marginal stability so that F F d ~^. However as the filaments evolve in the weakly nonlinear theory they evolve into the isolated tubes considered here [10,12].…”

“…In a series of papers we have shown that the early nonlinear stage of the ballooning mode generates explosively unstable elliptical flux tubes-'filaments' [8][9][10]. The interaction between filaments (flux tubes) tends to suppress the weaker filaments leading to isolated filaments [11,12]. Thus we have conjectured that the fully nonlinear state of the ballooning type modes is isolated displaced elliptical flux tubes [10].…”