Applications of geometric mechanics in multi-physics hybrid models for multi-scale systems.

Energetic particle effects in magnetic confinement fusion devices are commonly studied by simulation codes utilising the equations of a hybrid kinetic-fluid model. Typically the underlying continuum equations lack the correct energy balance. This thesis studies the two main hybrid models used in fusion plasma studies (current-coupling and pressure-coupling schemes) in the light of geometric techniques such as geometric reduction, variational principles and Hamiltonian methods. New results in the study of Euler-Poincar´e reduction for semidirect product group structures are presented. Further innovations to suit the drift-kinetic approximation are also presented. Outcomes of the study and development of geometric methods include the explanation of the geometric relationship between the two coupling schemes, and variational and Poisson bracket derivations for a newly conservative (energy-conserving) hybrid model in the pressure-coupling scheme, with energetic particles undergoing guiding centre motion. Kelvin circulation theorems for the new model are presented. The bridge between variational and Poisson structures is considered. This results in a construction that yields the variational and Poisson structures of a generalised, non-canonical Maxwell-Vlasov model in both Lagrangian and Eulerian variables. Achievements are summarised and avenues of future research identified