Upper Limit on Outer Radiation Belt Electron Flux Based on Dynamical Equilibrium

In the Earth's radiation belts, an upper limit on the electron flux is expected to be imposed by the Kennel-Petschek mechanism, through the generation of exponentially more intense whistler-mode waves as the trapped flux increases above this upper limit, leading to fast electron pitch-angle diffusion and precipitation into the atmosphere. Here, we examine a different upper limit, corresponding to a dynamical equilibrium in the presence of energetic electron injections and both pitch-angle and energy diffusion by whistler-mode chorus waves. We first show that during sustained injections, the electron flux energy spectrum tends toward a steady-state attractor resulting from combined chorus wave-driven energy and pitch-angle diffusion. We derive simple analytical expressions for this steady-state energy spectrum in a wide parameter range, in agreement with simulations. Approximate analytical expressions for the corresponding equilibrium upper limit on the electron flux are provided as a function of the strength of energetic electron injections from the plasma sheet. The analytical steady-state energy spectrum is also compared with maximum electron fluxes measured in the outer radiation belt during several geomagnetic storms with strong injections, showing a good agreement at 100-600 keV.