Structure-preserving discretisation of incompressible fluids on the rotating sphere
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The equations of incompressible fluids on the rotating sphere are a fundamental model for numerous physical phenomena. The most notable examples are large scale dynamics of oceanic and atmospheric flows. These equations possess an infinite family of conserved quantities known as Casimirs. This family has profound effects on the energy transfer mechanisms across scales of motion, e.g., the double energy cascade, absent in three dimensions, theorised by Kraichnan. It is therefore natural to demand preservation of Casimirs in the numerical algorithm used for the simulation for the discrete system. In this work we will present recent developments in geometric integration for rotating fluids in two dimensions. In particular, we will illustrate a recently developed efficient and scalable Lie-Poisson integrator for flows on the rotating sphere. We will present a geometric integrator which conserves the Casimirs of the system at a modest computational cost. The construction of such scheme, the main numerical algorithms and their parallelisation on modern supercomputing facilities will be discussed. Finally, an application to the spectrum of homogeneous two-dimensional turbulence with and without rotation will be discussed.
