Weak turbulence theory of the nonlinear evolution of the ion ring distribution

The nonlinear evolution of an ion ring instability in a low β magnetospheric plasma is considered. The evolution of the two-dimensional ring distribution is essentially quasilinear. Ignoring nonlinear processes the timescale for the quasilinear evolution is the same as for the linear instability 1/1τQL~γLτQL~γL. However, when nonlinear processes become important, a new timescale becomes relevant to the wave saturation mechanism. Induced nonlinear scattering of the lower-hybrid waves by plasma electrons is the dominant nonlinearity relevant for plasmas in the inner magnetosphere and typically occurs on the timescale 1/1τNL~ω(M/m)W/WnTnTτNL~ω(M/m)W/WnTnT, where W is the wave energy density, nT is the thermal energy density of the background plasma, and M/Mmm is the ion to electron mass ratio, which has the consequence that the wave amplitude saturates at a low level, and the timescale for quasilinear relaxation is extended by orders of magnitude.