Inertial waves in a differentially rotating spherical shell

TL;DR

This paper studies inertial waves in a differentially rotating spherical shell, revealing new mode types, broader frequency ranges, and behaviors at corotation resonances, with implications for fluid dynamics in astrophysical contexts.

Contribution

It introduces the analysis of inertial waves in differentially rotating shells, identifying new mode behaviors and stability properties not seen in solid-body rotation models.

Findings

Existence of two types of inertial modes depending on hyperbolic domain coverage.

Broader frequency range for inertial wave propagation compared to solid-body rotation.

Modes with thin shear layers scaling with E^{1/3} and E^{1/2} in viscosity.

Abstract

We investigate the properties of small-amplitude inertial waves propagating in a differentially rotating incompressible fluid contained in a spherical shell. For cylindrical and shellular rotation profiles and in the inviscid limit, inertial waves obey a second-order partial differential equation of mixed type. Two kinds of inertial modes therefore exist, depending on whether the hyperbolic domain where characteristics propagate covers the whole shell or not. The occurrence of these two kinds of inertial modes is examined, and we show that the range of frequencies at which inertial waves may propagate is broader than with solid-body rotation. Using high-resolution calculations based on a spectral method, we show that, as with solid-body rotation, singular modes with thin shear layers following short-period attractors still exist with differential rotation. They exist even in the case of a full sphere. In the limit of vanishing viscosities, the width of the shear layers seems to weakly depend on the global background shear, showing a scaling in E^{1/3} with the Ekman number E, as in the solid-body rotation case. There also exist modes with thin detached layers of width scaling with E^{1/2} as Ekman boundary layers. The behavior of inertial waves with a corotation resonance within the shell is also considered. For cylindrical rotation, waves get dramatically absorbed at corotation. In contrast, for shellular rotation, waves may cross a critical layer without visible absorption, and such modes can be unstable for small enough Ekman numbers.