First-principles Model for the Anomalous Frictional Heating of Electrons in Hall Thrusters

An analytical expression has been developed for the anomalous frictional power density of electrons, Q_ea^R, produced by their interactions with electrostatic turbulence in Hall thrusters. It accompanies another expression derived recently for the anomalous friction force density, R_eα, that led to a closed-form formula for the anomalous momentum-transfer collision frequency (MTCF). The derivations are part of a larger effort to develop first-principles closure models for hybrid numerical simulations. Our theory on which these models are based is that a two-stage evolution of unstable waves occurs in the Hall thruster discharge. First, short-wavelength (kρ_e>1), high-frequency (|ω|~ω_ce) modes that are driven by the cross-field drift E×B/B^2 grow and saturate at a level of turbulence too low to explain the observed measurements. Then the wave energy is dominated by modes of longer wavelength (kρ_e<1) and in the range of the lower-hybrid frequency ω_LH. The anomalous MTCF model we developed compared extremely well with a large set of empirical profiles derived from laser-induced fluorescence measurements that spanned a wide range of thruster geometries, operating conditions and applied magnetic fields. The new expression for the anomalous frictional heating reported here has been derived from the same kinetic theory and shows Q_ea^R is a function of the lower-hybrid wave properties at maximum growth and the drifts that drive it. Compared to phenomenological models that take in to account anomalous heating in the electron fluid energy equation through R_eα in the work term E∙j_e, the kinetic formulation yields as much as 4 higher frictional heating.