On the Poisson brackets of hybrid plasma models with kinetic ions and massless electrons

We investigate the conditions under which the Jacobi identity holds for a class of recently introduced anti-symmetric brackets for the hybrid plasma models with kinetic ions and massless electrons. In particular, we establish the precise conditions under which the brackets for the vector-potential-based formulations satisfy the Jacobi identity, and demonstrate that these conditions are fulfilled by all physically relevant functionals. Moreover, for the magnetic-field-based formulation, we show that the corresponding anti-symmetric bracket constitutes a Poisson bracket under the divergence-free condition of the magnetic field, and we provide a direct proof of the Jacobi identity. These results are further extended to models incorporating electron entropy as well as more general hybrid kinetic-fluid models.