  Hybrid-kinetic simulations describe ion-scale kinetic phenomena in space
plasmas by considering ions kinetically, i.e. as particles, while electrons are
modelled as a fluid. Most of the existing hybrid-kinetic codes neglect the
electron mass (see chapter 3) for a simplified calculation of the
electromagnetic fields. There are, however, situations in which delay in the
electrons response due to the electron inertia matters. This chapter
concentrates on hybrid-kinetic simulation models which take the finite mass of
the electron fluid into account. First a review is given of the history of
including the finite electron mass in hybrid-kinetic models. Then the equations
are discussed which additionally have to be solved compared to the mass-less
hybrid-kinetic models. For definiteness their numerical implementation without
additional approximations is illustrated by describing a hybrid-kinetic code,
CHIEF. The importance of the consideration of the finite electron mass are
discussed for typical applications (magnetic reconnection, plasma turbulence,
collisionless shocks and global magnetospheric simulations). In particular the
problem of guide field magnetic reconnection is addressed in some detail.
Possible next steps towards further improvements of hybrid-kinetic simulations
with finite electron mass are suggested.
