Classical Waves and Instabilities Using the Minimalist Approach

The minimalist approach in the study of perturbations in fluid dynamics and magnetohydrodynamics consists in describing their evolution in the linear regime using a single first-order ordinary differential equation, dubbed principal equation. The dispersion relation is found from the requirement the solution of the principal equation to be continuous and satisfy certain boundary conditions in each specific 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 and Kelvin-Helmholtz instabilities. For the latter we discuss the influence of compressibility and the magnetic field, and also derive analytical expressions for the growth rates in the case of two fluids with the same characteristics.