Geometrical Optics Stability Analysis of Rotating Visco-Diffusive Flows

Geometrical optics stability analysis has proven effective in deriving analytical instability criteria for 3D flows in ideal hydrodynamics and magnetohydrodynamics, encompassing both compressible and incompressible fluids. The method models perturbations as high-frequency wavelets, evolving along fluid trajectories. Detecting local instabilities reduces to solving ODEs for the wave vector and amplitude of the wavelet envelope along streamlines, with coefficients derived from the background flow. While viscosity and diffusivity were traditionally regarded as stabilizing factors, recent extensions of the geometrical optics framework have revealed their destabilizing potential in visco-diffusive and multi-diffusive flows. This review highlights these advancements, with a focus on their application to the azimuthal magnetorotational instability in magnetohydrodynamics and the McIntyre instability in lenticular vortices and swirling differentially heated flows. It introduces new analytical instability criteria, applicable across a wide range of Prandtl, Schmidt, and magnetic Prandtl numbers, which still remains beyond the reach of numerical methods in many important physical and industrial applications.