Nonlinear waves in the terrestrial quasi-parallel foreshock

TL;DR

This study investigates high frequency nonlinear waves in the terrestrial foreshock region using the derivative nonlinear Schrödinger equation, comparing numerical solutions with spacecraft observations to understand wave dynamics.

Contribution

It demonstrates the applicability of the DNLS equation to model observed nonlinear wave trains in the foreshock, emphasizing the importance of canonical representation and pseudo-potential analysis.

Findings

Large amplitude nonlinear wave trains observed above proton cyclotron frequency

Wave phase speed approximates local Alfvén speed

Numerical solutions align with observed waveforms after filtering slow variations

Abstract

We study the applicability of the derivative nonlinear Schr\"{o}dinger (DNLS) equation, for the evolution of high frequency nonlinear waves, observed at the foreshock region of the terrestrial quasi-parallel bow shock. The use of a pseudo-potential is elucidated and, in particular, the importance of canonical representation in the correct interpretation of solutions in this formulation is discussed. Numerical solutions of the DNLS equation are then compared directly with the wave forms observed by Cluster spacecraft. Non harmonic slow variations are filtered out by applying the empirical mode decomposition. We find large amplitude nonlinear wave trains at frequencies above the proton cyclotron frequency, followed in time by nearly harmonic low amplitude fluctuations. The approximate phase speed of these nonlinear waves, indicated by the parameters of numerical solutions, is of the order of the local Alfv\'{e}n speed.