Vorticity Dynamics on a Sphere with Discrete Exterior Calculus *

Geophysical flows, such as atmospheric flows on planets, are generally approximated as incompressible, inviscid surface flows, and characterized by two-dimensional (2D) turbulence. Evaluating the effect of rotation, in terms of the non-dimensional Rossby number, is important because the planetary rotation forms large scale structures (such as Rossby waves), which affects the direct enstrophy and inverse energy cascades in 2D turbulence.  Exterior calculus deals with the calculus on differential geometries, hence differential forms, and provides an alternative to the vector calculus. Discrete exterior calculus (DEC) is numerical exterior calculus and deals with the discrete differential forms. DEC-based discretization methods satisfy discrete analogues of continuous operations of interest, and conservation of key physical quantities such as vorticity is a well-known feature of DEC-base methods. These positive attributes makes DEC an excellent choice for investigating vorticity dynamics. Moreover, the DEC discretization is independent of the coordinate system, and therefore suitable for investigating flows over curved surfaces. Presently, we investigate the effect of rotation on the vorticity dynamics of flows on a unit sphere with a DEC scheme (Jagad et al. Phys. Fluids 2021). We vary the Rossby number from infinity (non-rotating) to 1.30x10^-3. Our investigations reveal that rotation diminishes the 2D turbulence cascades although it does not cease the cascade completely, and the zonalization of structures with decreasing Ro is non-monotonic depending on the choice of initial conditions.