Vorticity dynamics with Discrete Exterior Calculus

  • Jagad, Pankaj (KAUST)
  • Parsani, Matteo (KAUST)

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Two-dimensional turbulence, wherein there is direct cascade of enstrophy to smaller scales and inverse cascade of energy to the larger scales, is a paradigm for geophysical flows such as atmospheric flows on the earth and other planets. We investigate two dimensional turbulence on a rotating sphere with a discrete exterior calculus (DEC) scheme [1]. DEC retains at the discrete level many of the identities of its continuous counter part, and is a structure preserving numerical method. The DEC discretizations con- serve secondary quantities such as kinetic energy and circulation to machine precision for inviscid flows [2]. Moreover, it is coordinate independent, and convenient to use for simulating flows on surfaces. The governing equations comprises of incompressible Euler equations in a rotating reference frame. The flow evolves from a given initial vorticity distribution into thin filaments, and the small scales merge and form larger coherent structures at late times (see Fig. 1). We employ different initial conditions, and investigate the effect of initial spherical harmonic wavenumbers on the forward enstrophy and inverse energy cascades. We analyze the vorticity/energy power spectrum, spectral distribution of vorticity/energy power, and probability density measure of vorticity/energy. Our investigation reveals that rotation diminishes the cascade, however, the flows comprising of smaller scales experience less diminishing effect.