Numerical study of viscous modes in a rotating spheroid

Abstract

The motion of an incompressible, viscous rotating fluid contained in an oblate spheroidal container is studied by a direct numerical simulation in an appropriate spheroidal coordinate system and in the linear approximation. The behaviour of a few eigenmodes is investigated as a function of the eccentricity e of the container, for aEkman number E = 10 -5 . Viscous effects are evidenced through internal shear layers, the spatial structure of which strongly depends on the eccentricity. In particular, for the spin-over mode, a resonance occurs around a critical value e c ≈ 0.50, where the decay rate strongly deviates from the predicted theoretical variation. This resonance is discussed in relation with the accidental coincidence between the spin-over frequency and two other frequencies corresponding to the (8,1,5) and (14,1,9) inertial eigenmodes.

Kinematic dynamos in spheroidal geometries

“…The method can also be applied to thermal convection and magnetoconvection problems. For nonhomoeoidal shells such as confocal shells, non-homoeoidal toroidal-poloidal fields are required, but derivation of the toroidal-poloidal spectral equations is more difficult (for the confocal case, see [33,34]). Work on time-stepping the KDP is in progress.…”

An EBE finite element method for simulating nonlinear flows in rotating spheroidal cavities

“…denotes the complex conjugate of the preceding term, v lm (r ) and w lm (r ) are a function of r satisfying the required boundary condition on the bounding surface of a spherical cavity, and Y m l ( , ) is the spherical harmonics of degree l. For the spherical problem, the boundary geometry of the container is automatically consistent with the nature of spherical polar coordinates, and, consequently, the spectral mathematical formulation/implementation is straightforward. Spheroidal boundary geometry, however, cannot be accommodated by spherical polar coordinates and, as a result, either a coordinate transformation that maps the spheroidal domain into the spherical domain [16] or complicated oblate spheroidal coordinates [17] must be used in the spectral approximation for nonlinear flows confined in the spheroidal cavities. A major numerical disadvantage of the spheroidal spectral method, in addition to the fact that the Legendre transform is computationally inefficient and severely limits the effectiveness of the method on modern parallel computers, is that the coordinate transformation or oblate polar spheroidal coordinates must lead to more complicated governing equations that make numerical implementation more difficult.…”

“…A major numerical disadvantage of the spheroidal spectral method, in addition to the fact that the Legendre transform is computationally inefficient and severely limits the effectiveness of the method on modern parallel computers, is that the coordinate transformation or oblate polar spheroidal coordinates must lead to more complicated governing equations that make numerical implementation more difficult. Moreover, there apparently exist numerical instabilities in the spheroidal spectral approximation using spherical harmonics when the eccentricity of a rotating spheroid is high, 0.6 [17]. It is hence desirable to find an alternative numerical method, which is non-spectral and can be readily implemented on modern parallel computers, for efficiently solving the problem of fluid dynamics in rotating spheroids.…”

The Viscous and Ohmic Damping of the Earth's Free Core Nutation

“…This assumption does not necessarily always hold because other eigenmodes might interact with uniform vorticity flows as demonstrated by Triana et al. (2019) (see also Rogister & Valette (2009) and Schmitt (2006)). The resulting flow in the interior can thus deviate considerably from solid body rotation.…”