On the exact solutions of (magneto)hydrodynamic systems and the superposition principles of nonlinear helical waves

TL;DR

This paper systematically analyzes exact solutions and superposition principles in (magneto)hydrodynamic systems, identifying specific modes that eliminate nonlinearities and discussing implications for turbulence and magnetic phenomena.

Contribution

It introduces a unified approach to superposition principles in hydrodynamic models, identifying key modes that suppress nonlinearity and clarifying solutions in magnetohydrodynamics.

Findings

Mono-wavelength homochiral Beltrami mode satisfies superposition conditions.

XYz flow/wave mode can superpose to cancel nonlinearities.

Discussions on exact solutions relevant to turbulence and magnetic phenomena.

Abstract

The principles of restricted superposition of circularly polarized arbitrary-amplitude waves for several hydrodynamic type models are illustrated systematically with helical representation in a unified sense. It is shown that the only general modes satisfying arbitrary-amplitude superposition to kill the generic nonlinearity are the mono-wavelength homochiral Beltrami mode and the one-dimensional-two-component stratified vorticity mode, which we call the XYz flow/wave; while, there are other special superposition principles for some specific cases. We try to remark on the possible connections with the geo- and/or astro-physical fluid and magnetohydrodynamic turbulence issues, such as the rotating turbulence, dynamo and solar atmosphere turbulence, especially with the introduction of disorder locally frozen in some (randomly distributed) space-time regions. Recent disagreements about exact solutions of Hall and fully two-fluid magnetohydrodynamics are also settled down by such a treatment. This work complements, by studying the modes which completely kill the triadic interactions or the nonlinearities, previous studies on the thermalization purely from the triadic interactions, and in turn offers alternative perspectives of the nonlinearities.