Evanescent and inertial-like waves in rigidly-rotating odd viscous liquids

Abstract

Three-dimensional non-rotating odd viscous liquids give rise to Taylor columns and support {axisymmetric} inertial-like waves [\emph{J. Fluid Mech.}, vol. {973}, A30, (2023)]. When an odd viscous liquid is subjected to rigid-body rotation however, there arise in addition a plethora of other phenomena that need to be clarified. In this paper we show that three-dimensional incompressible or two-dimensional compressible odd viscous liquids, rotating rigidly with angular velocity $Ω$, give rise to both oscillatory and evanescent inertial-like waves or a combination thereof (which we call of mixed type), that can be \emph{non-axisymmetric}. By evanescent we mean that along the radial direction, typically when moving away from a solid boundary, the velocity field decreases exponentially. These waves precess in a prograde or retrograde manner with respect to the rotating frame. The oscillatory and evanescent waves resemble, respectively, the body and wall-modes observed in (non-odd) rotating Rayleigh-Bénard convection [\emph{J. Fluid Mech.}, vol. {248}, pp. 583-604 (1993)]. We show that the three types of waves (wall, body or mixed) can be classified with respect to pairs of planar wavenumbers $κ$ which are complex, real or a combination, respectively. Experimentally, by observing the precession rate of the patterns, it would be possible to determine the largely unknown values of the odd viscosity coefficients. This formulation recovers as special cases recent studies of equatorial or topological waves in two-dimensional odd viscous liquids which provided examples of the bulk-interface correspondence at frequencies $ω<2Ω$. We finally point out that the two and three-dimensional problems are formally equivalent. Their difference then lies in the way data propagate along characteristic rays in three dimensions.