Mathematics Seminar – 03/06/20

Abstract: The commonly employed models for the global ocean are composed of a coupled system of equations with different dynamics.  One of the main challenges in their numerical solution is the presence of multiple time-scales, where different processes (e.g., external and internal gravity waves, advection, vertical mixing) take different times to be completed.  In order to predict the underlying slow dynamics of ocean currents and stratification, time-stepping methods need to be able to take large steps not restricted by the fast transient processes, while maintaining both stability and accuracy.

Towards this goal, we develop time discretization methods based on exponential integrators in the context of the E3SM model.  Mainly, we focus on the dynamics, and consider a stacked rotating shallow-water ocean model.  The method is based on a splitting of the forcing term into a linear rotating multi-layer wave-operator and a non-linear residual, capturing the advective forces.  Solution strategies for the linear part are based on skew-adjoint Krylov methods.  The resulting exponential integrators can take large time steps, independent of the speed of internal and external gravity waves.  Additionally, the vertically coherent structure of the fastest waves can be used to compress the wave operator into a few vertical modes.  In a special case, employing a reduction only to the barotropic component, we obtain a method with similar features to the well-known split-explicit method.  Numerical experiments show that the methods are stable over decade-long simulation horizons and accurately reproduce solution statistics.