Finite element methods, grid refinement, and boundary currents in geophysical modeling

Most of today’s dynamical cores of weather and climate models are based on finite difference, hybrid finite volume/finite difference, or spectral discretization methods. In the first part of this thesis, we evaluate a new finite element approach for the discretization of the equations of motion in geophysical modeling. Finite element methods offer a local, higher order representation of the physical fields and allow the use of unstructured grids and grids with variable resolution. We use a new finite element that combines a continuous second order representation for the scalar field with a discontinuous first order representation for the velocity field, to develop a shallow-water model on a rotating sphere. This specific choice of a low-order element is attractive, since it has the remarkable property of being able to represent the geostrophic balance and fulfill the Ladyzhenskaya-Babuska-Brezzi-condition, which is a necessary condition for convergence in finite element modeling. In summary, we present a stable model setup and certify the new finite element approach to offer very promising properties for use in dynamical cores of global weather or climate models, in all tests performed. We propose the use of the stereographic projection to introduce spherical geometry to global finite element models. In the second part of this thesis, we analyze the influence of grid refinement on fundamental features of geophysical modeling. Through the study of the transition of waves between coarse and fine parts of a grid, and the influence of grid refinement on the representation of geostrophic balance and turbulent cascades, we investigate possible sources of errors for applications of grid refinement in ocean and atmosphere modeling. Furthermore, we investigate improvements that are possible through grid refinement, by evaluating simulations of flow over topography, local wave patterns, and western boundary currents. We find that the improvements possible with local grid refinement justify the risk of a spurious reflection and scattering of waves, in the given geophysical setup. In the final part of this thesis, we study boundary currents and boundary separation in finite element models. We evaluate the influence of local resolution, eddy viscosity, the grid structure, and the boundary conditions to the numerical representation of boundary currents, and try to identify proper criteria to detect boundary separation points in ocean modeling, for no-slip and free-slip boundary conditions, and steady and unsteady flows. To find these criteria, we study the physical fields along the coast line, and evaluate classical and recent theories for flow separation in Fluid Dynamics. Out of all of the evaluated criteria to detect separation points on no-slip boundaries, the two separation criteria by Prandtl work best. For free-slip boundaries, the two separation criteria by Lekien and Haller turned out to be the best choice. Dies ist ein kleines Zahnrad in einem großen Getriebe. Niemand weiß, ob eine Drehung eine kleine oder eine große Wirkung entfalten wird.

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