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 the pressure-coupling scheme (PCS) used in modern hybrid simulations. This latter scheme suffers from being non-Hamiltonian and to not exactly conserve total energy. Upon adopting the Vlasov description for the hot component, the non-Hamiltonian PCS and a Hamiltonian variant are compared. Special emphasis is given to the linear stability of Alfven waves, for which it is shown that a spurious instability appears at high frequency in the non-Hamiltonian version. This instability is removed in the Hamiltonian version.
Plasma Physics and Controlled Fusion
Multiscale Dynamics

Tronci, C., Tassi, E., Camporeale, E., & Morrison, P. J. (2014). Hybrid Vlasov-MHD models: Hamiltonian vs. non-Hamiltonian. Plasma Physics and Controlled Fusion, 56(9), 095008:1–095008:11. doi:10.1088/0741-3335/56/9/095008