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The development of various volume penalization techniques for use in modeling topographical features in the ocean is the focus of this paper. Due to the complicated geometry inherent in ocean boundaries, the stair-step representation used in the majority of current global ocean circulation models causes accuracy and numerical stability problems. Brinkman penalization is the basis for the methods developed here and is a numerical technique used to enforce no-slip boundary conditions through the addition of a term to the governing equations. The second aspect to this proposed approach is that all governing equations are solved on a nonuniform, adaptive grid through the use of the adaptive wavelet collocation method. This method solves the governing equations on temporally and spatially varying meshes, which allows higher effective resolution to be obtained with less computational cost. When penalization methods are coupled with the adaptive wavelet collocation method, the flow near the boundary can be well-resolved. It is especially useful for simulations of boundary currents and tsunamis, where flow near the boundary is important. This paper will give a thorough analysis of these methods applied to the shallow water equations, as well as some preliminary work applying these methods to volume penalization for bathymetry representation for use in either the nonhydrostatic or hydrostatic primitive equations.

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MPAS-Ocean is used to simulate an idealized, density-driven overflow using the dynamics of overflow mixing and entrainment (DOME) setup. Numerical simulations are carried out using three of the vertical coordinate types available in MPAS-Ocean, including z-star with partial bottom cells, z-star with full cells, and sigma coordinates. The results are first benchmarked against other models, including the MITgcm's z-coordinate model and HIM's isopycnal coordinate model, which are used to set the base case used for this work. A full parameter study is presented that looks at how sensitive overflow simulations are to vertical grid type, resolution, and viscosity. Horizontal resolutions with 50 km grid cells are under-resolved and produce poor results, regardless of other parameter settings. Vertical grids ranging in thickness from 15 m to 120 m were tested. A horizontal resolution of 10 km and a vertical resolution of 60 m are sufficient to resolve the mesoscale dynamics of the DOME configuration, which mimics real-world overflow parameters. Mixing and final buoyancy are least sensitive to horizontal viscosity, but strongly sensitive to vertical viscosity. This suggests that vertical viscosity could be adjusted in overflow water formation regions to influence mixing and product water characteristics. Lastly, the study shows that sigma coordinates produce much less mixing than z-type coordinates, resulting in heavier plumes that go further down slope. Sigma coordinate are less sensitive to changes in resolution but as sensitive to vertical viscosity compared to z-coordinates.