Subcritical growth of electron phase-space holes in planetary radiation belts

The discovery of long-lived electrostatic coherent structures with large-amplitude electric fields ($1 \leq E \leq 500 $ mV/m) by the Van Allen Probes has revealed alternative routes through which planetary radiation belts' acceleration can take place. Following previous reports showing that small phase-space holes, with $q\phi /T^c_e\simeq 10^{-2}-10^{-3}$, could result from electron interaction with large-amplitude whistlers, we demonstrate one possible mechanism through which holes can grow nonlinearly (i.e. $\gamma \propto \sqrt{\phi}$) and subcritically as a result of momentum exchange between hot and cold electron populations. Our results provide an explanation for the common occurrence and fast growth of large-amplitude electron phase-space holes in the Earth's radiation belts.