Axisymmetric inertial modes in a spherical shell at low Ekman numbers

Abstract

We investigate the asymptotic properties of axisymmetric inertial modes propagating in a spherical shell when viscosity tends to zero. We identify three kinds of eigenmodes whose eigenvalues follow very different laws as the Ekman number $E$ becomes very small. First are modes associated with attractors of characteristics that are made of thin shear layers closely following the periodic orbit traced by the characteristic attractor. Second are modes made of shear layers that connect the critical latitude singularities of the two hemispheres of the inner boundary of the spherical shell. Third are quasi-regular modes associated with the frequency of neutral periodic orbits of characteristics. We thoroughly analyse a subset of attractor modes for which numerical solutions point to an asymptotic law governing the eigenvalues. We show that three length scales proportional to $E^{1/6}$, $E^{1/4}$ and $E^{1/3}$ control the shape of the shear layers that are associated with these modes. These scales point out the key role of the small parameter $E^{1/12}$ in these oscillatory flows. With a simplified model of the viscous Poincaré equation, we can give an approximate analytical formula that reproduces the velocity field in such shear layers. Finally, we also present an analysis of the quasi-regular modes whose frequencies are close to $\sin (\pi /4)$ and explain why a fluid inside a spherical shell cannot respond to any periodic forcing at this frequency when viscosity vanishes.

38 pages, 25 figures, to appear in J. Fluid Mechanics

References in corpus (1)

• Axisymmetric inertial modes in a spherical shell at low Ekman numbers

Cited by in corpus (16)

• First evidence of inertial modes in $\gamma$ Doradus stars: The core rotation revealed
• Theory of solar oscillations in the inertial frequency range: Linear modes of the convection zone
• Axisymmetric inertial modes in a spherical shell at low Ekman numbers
• Fossil field decay due to nonlinear tides in massive binaries
• Libration-driven inertial waves and mean zonal flows in spherical shells
• Theory of solar oscillations in the inertial frequency range: Amplitudes of equatorial modes from a nonlinear rotating convection simulation
• Spectral theory of pseudo-differential operators of degree 0 and application to forced linear waves
• The coupling between inertial and rotational eigenmodes in planets with liquid cores
• The effects of nonlinearities on tidal flows in the convective envelopes of rotating stars and planets in exoplanetary systems
• Resonant tidal responses in rotating fluid bodies: global modes hidden beneath localized wave beams
• Zonal flow in a resonant precessing cylinder
• Acoustic and inertial modes in planetary-like rotating ellipsoids
• Spectroscopic detection of Altair's non-radial pulsations
• Kinematic dynamos in triaxial ellipsoids
• Mathematics of internal waves in a 2D aquarium
• Internal shear layers generated by a vertically oscillating cylinder in unbounded and bounded rotating fluids