Classical waves and instabilities using the minimalist approach

TL;DR

This paper introduces a minimalist approach to analyze classical wave phenomena and instabilities in fluid and magnetohydrodynamic flows using a single principal differential equation, simplifying the study of complex perturbations.

Contribution

It presents a unified formalism for studying wave instabilities in fluid dynamics and MHD, deriving analytical expressions for growth rates and instability ranges.

Findings

Unified principal equation for wave analysis.

Analytical growth rates for Kelvin-Helmholtz instability.

Influence of magnetic fields and compressibility on instabilities.

Abstract

The minimalist approach in the study of perturbations in fluid dynamics and magnetohydrodynamics involves describing their evolution in the linear regime using a single first-order ordinary differential equation, dubbed principal equation. The dispersion relation is determined by requiring that the solution of the principal equation be continuous and satisfy specific boundary conditions for each problem. The formalism is presented for flows in cartesian geometry and applied to classical cases such as the magnetosonic and gravity waves, the Rayleigh-Taylor instability, and the Kelvin-Helmholtz instability. For the latter, we discuss the influence of compressibility and the magnetic field, and also derive analytical expressions for the growth rates and the range of instability in the case of two fluids with the same characteristics.