The Ginzburg-Landau Model of Magnetospheric Chorus: Instabilities and Mode Condensation

Abstract

The analogy between free-electron lasers (FELs) - laboratory devices which generate intense coherent light with tunable frequencies - and whistler wave-particle interactions in the magnetosphere has recently been extended to account for waves with spatially dependent amplitudes and a spectrum of frequencies. The whistler was found to be governed by one of the most well-studied nonlinear equations in physics, the Ginzburg-Landau equation (GLE), which can be used to predict the complex nonlinear physics of multi-mode interactions. In this study, we focus on the single-mode solutions of the GLE and investigate their propagation and stability in the context of magnetospheric chorus. As with FELs, there are two types of instabilities, the Benjamin-Feir instability, where all single modes are unstable, and the Eckhaus instability, where there is a band of stable modes, but all modes outside of the band are unstable. Both stability conditions are given by well known inequalities in the GLE literature. For whistler-mode chorus, we analytically reduce the inequalities to simple expressions and show that, to the extent that the GLE represents magnetospheric chorus, it is Benjamin-Feir stable. We also derive the width of the Eckhaus stability band. We find that the predicted bandwidth is consistent with in situ satellite observations and support our analytical calculations with numerical simulations of the GLE. Our simulations demonstrate the robustness of the stable modes, the evolution from unstable modes to stable ones, and the tendency for mode condensation, whereby a noisy spectrum of modes tends to relax to a single stable mode.