Closure of the Strongly Magnetized Electron Fluid Equations in the Adiabatic Regime

We derive closure relations for a plasma fluid model, issued from the BGK equation for electrons in a strong magnetic field. Our scaling of the BGK equation leads in the asymptotic limit $\varepsilon \to 0$ towards the adiabatic electron regime, where $\varepsilon$ embodies the scaled Larmor radius as well as a low Mach number. In this regime the electron density adjusts instantaneously to perturbations of the electric potential via a Boltzmann relation, i.e., $n=ce^{\phi/T}$. The fluid closures are obtained in the small-$\varepsilon$ regime from a Hilbert ansatz of the distribution function; the ensuing hierarchy of kinetic equations is solved exactly up to the desired order. Different closures emerge depending on the importance of the ratio $\nu/\omega_c$ between the electron-electron collision frequency $\nu$ and the cyclotron frequency $\omega_c$. Anisotropy in the transport coefficients is found when $\nu/\omega_c \ll 1$. Moreover, bringing into play the obtained closures, we present a drift-fluid model valid for $\varepsilon \ll 1$ and identify the correct limit model as $\varepsilon \to 0$. The limit model can be used to avoid the crude approximations $c=const.$ and $T=const.$ in the electron Boltzmann relation, frequently used for plasma simulations.