Exploring Nonlinear Drift Waves: Limiting Cases and Dynamics

TL;DR

The paper addresses nonlinear drift-wave dynamics in magnetized inhomogeneous plasmas and identifies a missing nonlinear electron-density contribution in the HM equation. It derives a general two-dimensional nonlinear drift-wave equation that includes both the Boltzmann electron response term and ion vorticity effects, in normalized form: $∂_t(Φ + 0.5 Φ^2 - ∇_{⊥}^2 Φ) + (∇_{⊥} Φ × z)·x + (L_n/ρ_s)(∇_{⊥} Φ × z · ∇_{⊥}) ∇_{⊥}^2 Φ = 0$. Under reductive perturbation and stretched coordinates, the equation reduces to a pure KdV-type form with soliton solutions when the vorticity term is small, and to a HM-like dipolar-vortex equation when the scalar nonlinearity is neglected. The results indicate that the scalar nonlinear electron-density term generally dominates for ρ_s^2 k_⊥^2 < 1, signaling a need to revise drift-wave models in space, astrophysical, and laboratory plasmas. The work provides explicit soliton and vortex structures and demonstrates the relevance of the full general equation for nonlinear drift-wave dynamics.

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. $Φ= Φ(y)$, 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 $Φ= Φ(x, y)$ 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).

Figures (1)

• Figure 1: The 3D plot of negative electrostatic potential $\Phi$ vs moving coordinate $\chi$ is shown for $M=0.1$ and $\lambda = 0.1$.