A polar-harmonic unified gas-kinetic scheme for magnetized ion dynamics from cyclotron kinetics to the Hall-Pedersen constitutive limit

Abstract

Magnetized ion transport in weakly ionized plasmas ranges from gyroangle-dependent kinetics to Hall-Pedersen drift-diffusion as collisionality and magnetization vary. We develop a polar-harmonic unified gas-kinetic scheme (PH-UGKS) for the ion Vlasov-BGK equation in a uniform magnetic field. The scheme evolves the full ion distribution by coupling a conservative density update to exponential evolution of its nonequilibrium component. Exact collision-rotation integration in gyroangle Fourier space is combined with a time-averaged kinetic flux that incorporates spatial transport and electric acceleration, together with a compact Hall-Pedersen correction to the density flux. The scheme conserves ion number, and analysis establishes second-order temporal consistency and asymptotic preservation of the Hall-Pedersen density limit at fixed magnetization. Numerical tests reproduce ion Bernstein dispersion and Dory-Guest-Harris growth rates and resolve changes in the gyroharmonic spectrum as the collision-to-gyrofrequency ratio varies. The driven ion-flux response agrees with an independent characteristic-Volterra reference, including finite-frequency departures from the instantaneous Hall-Pedersen relation. In collisional tests, accurate responses are obtained with time steps far larger than both the collision time and the gyroperiod. Fixed-resolution density tests confirm convergence to the corresponding Hall-Pedersen discretization. The same kinetic formulation thus connects kinetic response and macroscopic transport without switching to a fluid solver or subcycling microscopic time scales.