Double-diffusive instabilities in rotating flows: pseudo-hermiticity and exceptional points

The Prandtl number, i.e. the ratio of the fluid viscosity to a diffusivity parameter of other physical nature such as thermal diffusivity or ohmic dissipation, plays a decisive part for the onset of instabilities in hydrodynamic and magnetohydrodynamic flows. The studies of many particular cases suggest a significant difference in stability criteria obtained for the Prandtl number equal to 1 from those for the Prandtl number deviating from 1. We demonstrate this for a circular Couette flow with a radial temperature gradient andfor a differentially rotating viscous flow of electrically conducting incompressible fluid subject to an external azimuthal magnetic field. Furthermore, in the latter case we find that the local dispersion relation is governed by a pseudo-Hermitian matrix both in the case when the magnetic Prandtl number, Pm, is Pm = 1 and in the case when Pm = −1. This implies that the complete neutral stability surface contains three Whitney umbrella singular points and two mutually orthogonal intervals of self-intersection. At these singularities thedouble-diffusive system reduces to a marginally stable G-Hamiltonian system. The role of double complex eigenvalues (exceptional points) stemming from the singular points in exchange of stability between modes is demonstrated.