The Ginzburg-Landau Equation for Whistler Chorus in the Magnetosphere: Stability of Single-Mode Solutions

The analogy between free-electron lasers (laboratory devices which generate coherent light with tunable frequencies) and whistler wave-particle interactions in the magnetosphere has recently been extended to account for the effects of spatially dependent waves with a spectrum of frequencies. The whistler amplitude and phase were found to be governed by one of the most well-studied nonlinear equations in physics – the Ginzburg-Landau equation (GLE). While the GLE can be used to predict the complex nonlinear physics that arises from multi-mode interactions, it also permits condensation to single-mode solutions. In this study, we investigate the stability of single modes in the context of magnetospheric chorus. It has been shown that if the mode with the highest linear growth rate is unstable, then all other modes are unstable. Conversely, if the mode with the highest linear growth rate is stable, then there is always a band of modes surrounding it which are also stable, with modes outside the band being unstable. In both cases, the stability condition is given by an inequality that is well known in the GLE literature. Using perturbative methods, we reduce the complicated terms within the inequalities down to simple expressions in terms of the physical parameters of the system. Using this result, we show that the mode with the highest linear growth rate is always stable for magnetospheric chorus and derive a simple expression for the stability bandwidth. We find that the predicted bandwidth is consistent with in situ satellite observations of chorus modes and support our analytical calculations with numerical simulations of the GLE.