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This paper investigates hybrid kinetic-MHD models, where a hot plasma(governed by a kinetic theory) interacts with a fluid bulk (governed by MHD).Different nonlinear coupling schemes are reviewed, including thepressure-coupling scheme (PCS) used in modern hybrid simulations. This latterscheme suffers from being non-Hamiltonian and to not exactly conserve totalenergy. Upon adopting the Vlasov description for the hot component, thenon-Hamiltonian PCS and a Hamiltonian variant are compared. Special emphasis isgiven to the linear stability of Alfv\'en waves, for which it is shown that aspurious instability appears at high frequency in the non-Hamiltonian version.This instability is removed in the Hamiltonian version.

Hybrid Vlasov-MHD models: Hamiltonian vs. non-Hamiltonian