Effects of Rotation on Vorticity Dynamics on a Sphere with Discrete Exterior Calculus

TL;DR

This study explores how rotation affects vorticity dynamics on a sphere using Discrete Exterior Calculus, revealing complex behaviors in vortex evolution and cascade processes across different rotation rates.

Contribution

It introduces a DEC-based numerical scheme to analyze vorticity dynamics on a rotating sphere, highlighting non-monotonic zonalization effects and partial suppression of enstrophy cascade.

Findings

Rotation reduces forward enstrophy cascade.

Zonalization varies non-monotonically with Rossby number.

Rotation does not fully arrest enstrophy cascade.

Abstract

We investigate incompressible, inviscid vorticity dynamics on a rotating unit sphere using a Discrete Exterior Calculus (DEC) scheme. For a prescribed initial vorticity distribution, we vary the rate of rotation of the sphere from zero (non-rotating case, which corresponds to infinite Rossby number (Ro)) to 320 (which corresponds to Ro = $1.30 \times 10^{-3}$), and investigate the evolution with time of the vorticity field. For the non-rotating case, the vortices evolve into thin filaments due to so-called forward/direct enstrophy cascade. At late times an oscillating quadrupolar vortical field emerges as a result of the inverse energy cascade. Rotation diminishes the forward cascade of enstrophy (and hence the inverse cascade of energy) and tend to align the vortical structures in the azimuthal/zonal direction. Our investigation reveals that, for the initial vorticity field comprising of intermediate-wavenumber spherical harmonics, the zonalization of the vortical structures is not monotonic with ever decreasing Rossby numbers and the structures revert back to a non-zonal state below a certain Rossby number. On the other hand, for the initial vorticity field comprising of intermediate to large-wavenumber spherical harmonics, the zonalization is monotonic with decreasing Rossby number. Although, rotation diminishes the forward cascade of enstrophy, it does not completely cease/arrest the cascade.