Analytical estimates of quasi-linear diffusion coefficients and electron lifetimes in the inner radiation belt

The behavior of trapped electrons in the Earth's radiation belts can be described in terms of relativistic quasi-linear diffusion by cyclotron-resonant plasma waves. Multidimensional dynamical simulations of the full belts require rapid and accurate evaluations of the diffusion coefficients. Recently developed approximations already allow replacing the integration over the wave-normal distribution by an evaluation at a carefully chosen point. An approximate, but useful, analytical formulation of diffusion coefficients is derived here in the particular limit of low frequency and moderately oblique waves. It accounts for both the integration over the wave-normal distribution and the summation over all the relevant n-harmonic resonances. Analytical estimates of the diffusion coefficients averaged over the bounce motion along a geomagnetic field line are also given. From them, a simplified analytical expression of the electron lifetimes is deduced. Detailed comparisons between the analytical formulas and the full numerical solutions are presented for different cases of resonant interactions of electrons with hiss waves inside the plasmasphere, demonstrating a fair agreement and providing a better understanding of the simulation results.