Exploring Nonlinear Drift Waves: Limiting Cases and Dynamics

TL;DR

This paper derives a comprehensive nonlinear drift wave equation that includes both ion vorticity and electron density effects, analyzing its limiting cases and solutions like solitons and vortices.

Contribution

It introduces a general nonlinear drift wave equation incorporating electron density perturbation and ion vorticity, extending the classic Hasegawa-Mima model.

Findings

The general equation reduces to the Hasegawa-Mima equation when electron density effects are neglected.

Under specific assumptions, the equation supports soliton and vortex solutions.

Limiting forms of the equation are analyzed using reductive perturbation methods.

Abstract

A general equation for drift waves is derived incorporating both nonlinear electron density perturbation and ion vorticity effects. It is emphasized that the well-known Hasegawa-Mima (HM) equation for drift waves [A. Hasegawa and K. Mima, Phys. Fluids 21, 87 (1978)] includes only the ion vorticity term and neglects nonlinear electron density contribution that naturally arises from the electrons Boltzmann response. If ion vorticity term is ignored, then the general nonlinear equation reduces to an equation which can give two-dimensional soliton solution under an appropriate coordinate transformation. Furthermore, under the assumption that the normalized electrostatic potential depends only on one spatial coordinate along the predominant propagation direction, i.e. $\Phi = \Phi(y)$, the equation reduces to one-dimensional KdV equation [H. Saleem, Phys. Plasmas 31, 112102 (2024)]. Conversely, if the nonlinear electron density term is artificially suppressed and a two-dimensional potential $\Phi = \Phi(x, y)$ is considered, the equation reduces to Hasegawa-Mima equation supporting dipolar vortex solution. Because the HM equation ignores nonlinear electron density term, it cannot support one- or two-dimensional soliton solutions. Finally, the limiting forms of the general nonlinear equation are also briefly discussed using the reductive perturbation method (RPM).