Three‐dimensional radiation belt simulations in terms of adiabatic invariants using a single numerical grid

Abstract

The dynamics of electrons in the Earth's radiation belts can be described by the Fokker‐Planck equation, which includes radial diffusion, and local energy and pitch angle diffusion. Previously, the Versatile Electron Radiation Belt (VERB) code utilized two grids to solve the Fokker‐Planck equation; one grid, which keeps the first and second adiabatic invariants constant, was used for the computation of the radial diffusion, and the other grid, orthogonal in energy and pitch angle at each fixed radial distance, was used for the computation of energy diffusion, pitch angle diffusion, and mixed energy and pitch angle diffusion. At each time step, the results were interpolated between the two grids. In the current work, we present a new method for numerical solution of the Fokker‐Planck diffusion equation written in terms of adiabatic invariants using a single numerical grid. The one‐grid solution allows us to eliminate the time‐consuming interpolation. The solution in terms of adiabatic invariants implicitly accounts for conservation of phase space density during adiabatic magnetic field fluctuations. The Fokker‐Planck equation written in terms of adiabatic invariants is easier to solve with a realistic magnetic field configuration, compared to the Fokker‐Planck equation written in terms of energy and pitch angle. To validate the approach, we compare a 2‐D simulation at fixed L* in terms of the adiabatic invariants (μ, K) with a simulation on a grid orthogonal in energy and pitch angle. Both simulations produce essentially the same result. Accurate simulations on the (μ, K) grid require resolution of 601 × 601 points, while only 101 × 101 grid points are sufficient for performing an accurate simulation on the grid orthogonal in energy and pitch angle. We propose a new invariant V ≡ μ ⋅ (K + 0.5)2 with the new grid orthogonal in (V, K, L*), that is more convenient for 2‐D and 3‐D simulations. The 2‐D and 3‐D simulations on the (V, K, L*) grid produce the same result as the two‐grid method with a grid resolution of only 101 × 101 points in V and K.