Variational integrators for the rotating shallow water equations

Brecht, Rüdiger (2021) Variational integrators for the rotating shallow water equations. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

The numerical simulation of the Earth’s atmosphere plays an important role in developing our understanding of climate change. The atmosphere and ocean can be seen as a shallow fluid on the globe; here, we use the shallow water equations as a first step to approximate these geophysical flows. Then, the numerical model can only be accurate if it has good conservation properties, e.g. without conserving mass the simulation can not be physical. Obtaining such a numerical model can be achieved using numerical variational integration. Here, we have derived a numerical variational integrator for the rotating shallow water equations on the sphere using the Euler–Poincaré framework. First, the continuous Lagrangian is discretized; then, the numerical scheme is obtained by computing the discrete variational principle. The conservational properties and accuracy of the model are verified with standard test cases. However, in order to obtain more realistic simulations, the shallow water equations need to include physical parametrizations. Thus, we introduce a new representation of the rotating shallow water equations based on a stochastic transport principle. Then, benchmarks are carried out to demonstrate that the spatial part of the stochastic scheme preserves the total energy. The proposed random model better captures the structure of a large-scale flow than a comparable deterministic model. Furthermore, to be able to carry out long term simulations we extend the discrete Euler–Poincaré framework with a selective decay. The selective decay dissipates an otherwise conserved quantity while conserving energy. We apply the new framework to the shallow water equations to dissipate the potential enstrophy. Then, we carry out standard benchmarks to demonstrate the conservation properties. We show that the selective decay resolves more small scales compared to a standard dissipation.

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