02 January 2018

Discrete Conservation Properties for Shallow Water Flows using Mixed Mimetic Spectral Elements


This papers explores the use of a new method for the solution of the rotating shallow water equations. This new method exactly preserves the conservation of various moments and balance relations in the discrete form with arbitrarily high order accuracy. As such it mitigates against various baises over long time integrations, and is therefore a strong candidate for use in modelling the full climate system.


The conservation and error convergence properties of the mixed mimetic spectral element method make it a strong candidate for modeling geophysical dynamics over long time integrations with high accuracy and minimal biases. In particular it should be well suited to the modeling of both gravity waves on fast time scales as Rossby waves on slow time scales due to the conservation of mass and energy in the case of gravity waves and the conservation of vorticity and the preservation of geostrophic balance in the case of Rossby waves.


A mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element histopolation functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension. These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as higher moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exact spatial integration. The method possesses a weak form statement of geostrophic balance due to the compatible nature of the solution spaces and arbitrarily high order spatial error convergence.

David Lee
Los Alamos National Laboratory