### 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.

Original language | English (US) |
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Pages (from-to) | 6-20 |

Number of pages | 15 |

Journal | Contributions to Plasma Physics |

Volume | 58 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2018 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Condensed Matter Physics

### Keywords

- ELMs
- MHD
- non-linear ballooning model

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## Cite this

*Contributions to Plasma Physics*,

*58*(1), 6-20. https://doi.org/10.1002/ctpp.201700047