Development of a non-hydrostatic coastal ocean model using the discontinuous Galerkin method

Abstract

The primary objective of this study is to develop a non-hydrostatic coastal ocean model using the discontinuous Galerkin finite element method. A series of different discretisation approaches are thus developed and compared for the simulation of free surface and buoyancy-driven flows.

The first non-hydrostatic formulation presented in this thesis directly extends the shallow water equations based solver within the \emph{Thetis} coastal ocean model to include non-hydrostatic effects. With an edge-based pressure method, the total pressure is decomposed into hydrostatic and non-hydrostatic components, where the hydrostatic part is correlated to the free surface elevation. The depth-integrated momentum equations are solved for a provisional velocity field, which is subsequently corrected using the non-hydrostatic pressure that is obtained from the continuity equation satisfying the divergence-free condition for velocity. The resulting depth-integrated non-hydrostatic model is shown to accurately represent weakly dispersive wave processes.

Since the depth-integrated model has limitations for certain applications (e.g. for highly dispersive waves), a multi-layer non-hydrostatic model is developed by extending the depth-integrated model to a multi-layer system to improve its dispersion representation capabilities. The developed multi-layer model solves for layer-averaged momentum and inter-layer fluxes over the layers of the system. The resulting quasi-3D algorithm is shown to accurately simulate highly dispersive waves as well as having the ability to represent vertical flow structure.

Both the depth-integrated and multi-layer models, whose implementations both rely on 2D triangular meshes, are developed primarily with the target of efficiently and accurately simulating dispersive free surface waves. For the simulation of baroclinic buoyancy-driven flows, a three-dimensional non-hydrostatic model accounting for vertical mesh deformation is developed. The 3D model, utilising a prismatic mesh, is the direct non-hydrostatic extension of the existing 3D hydrostatic solver in Thetis, and is found to be favourable for stratified flows and mass transport since a relatively high vertical mesh resolution is generally required in order to appropriately resolve the driving density fields. Comparisons with the quasi-3D multi-layer model are also conducted based on several barotropic test cases, indicating that the multi-layer model is in turn favourable for dispersive free surface problems due to higher computational efficiency.

Based on a $\sigma$-coordinate transformation, a new three-dimensional non-hydrostatic model is subsequently developed, which still uses 3D prismatic meshes but solves the underlying equations on a fixed $\sigma$-coordinate mesh. With the selection of the low-order piecewise-constant \PzeroDG \,and piecewise-linear \PoneDG \,discretisations in the vertical for the velocity and pressure fields, respectively, the developed $\sigma$-coordinate model can naturally retain the wave dispersion characteristics of the multi-layer model, as confirmed through both mathematical derivation and numerical tests. Higher-order vertical discretisation choices can also be made which can reduce the number of vertical layers required for the accurate representation of wave dispersion. Model verification and validation indicate that the $\sigma$-coordinate model can accurately represent dispersive barotropic surface waves with as few as one vertical layer, and can simulate baroclinic internal waves with reasonable accuracy using relatively coarse mesh resolution. It is also demonstrated that consistency in the coupling of barotropic and baroclinic flows can be properly ensured.

Finally, the validated non-hydrostatic coastal ocean model is applied to the study of landslide-generated tsunami. A granular landslide model is developed, which is coupled with the non-hydrostatic model to form a two-layer system. Landslide motion in the lower layer is modelled using a depth-averaged formulation for a shallow subaerial debris flow, while wave generation and propagation in the upper ``layer'' are simulated using the $\sigma$-coordinate non-hydrostatic model. For the landslide a well-balanced vertex-based wetting and drying treatment is implemented for the Runge-Kutta discontinuous Galerkin method, while an implicit wetting and drying scheme is employed for the moving shoreline that is present in the upper layer free surface flow problem. The two-layer model is validated against a series of test cases, and is found to describe well the complex behaviour of the granular landslides from initiation to deposition, as well as the consequent wave generation and propagation.