Nonlinear Drift Waves Beyond the Hasegawa-Mima Model

Abstract: A general nonlinear equation for drift waves is derived, incorporating both the nonlinear electron density term (scalar nonlinearity) and the ion vorticity term (vector nonlinearity). It is noted that the Hasegawa-Mima (HM) equation (A. Hasegawa and K. Mima, The Physics of Fluids 21, 87 (1978)) includes only the ion vorticity term, while it omits the nonlinear electron density term, which is a dominant nonlinear effect. The general nonlinear equation reduces to the Korteweg-de Vries (KdV) equation (H. Saleem, Physics of Plasmas 31, 112102 (2024)) if the normalized electrostatic potential is assumed to depend only on a single spatial coordinate along the direction of predominant wave propagation, i.e., ( \Phi = \Phi(y) ). If the scalar nonlinearity is artificially neglected and a two-dimensional dependence of the electrostatic potential is assumed, ( \Phi = \Phi(x, y) ), the equation reduces to a Hasegawa-Mima-like equation, which admits a solution in the form of dipolar vortices. It is further pointed out that, in the presence of a Boltzmann density distribution for electrons, the HM equation was derived using the approximation ( \frac{\partial_t \tilde{n}i}{n{i0}} \simeq \partial_t \Phi ), whereas the more accurate expression ( \frac{\partial_t \tilde{n}i}{n{i0}} \simeq \partial_t \left( \Phi + \frac{1}{2} \Phi^2 \right) ) should be used.