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Highlights • A multi-layer non-hydrostatic coastal ocean model is developed using the discontinuous Galerkin finite element method. • The model is shown to predict nearshore dispersive wave processes accurately using only a small number of vertical layers. • Use of arbitrary unstructured meshes allows for non-uniform resolution and accurate representation of complex geometries. • Automated code generation techniques are employed promoting code efficiency, readability, extensibility and maintainability. Abstract A multi-layer non-hydrostatic version of the unstructured mesh, discontinuous Galerkin finite element based coastal ocean model, Thetis, is developed. This is accomplished using the PDE solver framework, Firedrake, which is used to automatically produce the code for the discretised model equations in a rapid and efficient manner. The motivation for this work is a need to accurately simulate dispersive nearshore free surface processes. In order to resolve both frequency dispersion and non-linear effects accurately, additional non-hydrostatic terms are included in the layer-integrated hydrostatic equations, producing a form similar to the layered non-linear shallow water equations, but with extra vertical velocities at the layer interfaces. An implementation process is adopted to easily handle the inter-layer connection, i.e. the governing equations are transformed into a depth-integrated system through the introduction of depth-averaged variables. The model is verified and validated through comparisons against several idealised and experimentally-based test cases. All the comparisons demonstrate good agreement, showing that the developed non-hydrostatic model has excellent capabilities in representing coastal wave phenomena including shoaling, refraction and diffraction of dispersive short waves, as well as propagation, run-up and inundation of non-linear tsunami waves. [ABSTRACT FROM AUTHOR]

A σ -coordinate non-hydrostatic coastal ocean model is developed using the discontinuous Galerkin finite element method. With the selection of the low-order piecewise-constant P 0 DG and piecewise-linear P 1 DG discretisations in the vertical for the velocity and pressure fields, respectively, the proposed σ -coordinate model can naturally retain the wave dispersion characteristics of the widely-adopted multi-layer approach of Zijlema and Stelling (2005), which is demonstrated through both mathematical derivation and numerical tests. Under the finite element approach, higher-order vertical discretisation choices can also be readily made which can reduce the number of vertical layers required for the accurate representation of wave dispersion. The model is verified and validated through comparisons against a series of test cases with analytical solutions or experimental measurements. All the comparisons demonstrate good agreement, indicating that the proposed 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. • A σ -coordinate non-hydrostatic model is developed using the discontinuous Galerkin method. • The model can accurately represent dispersive barotropic surface waves with as few as one vertical layer. • The model can simulate baroclinic dynamics with reasonable accuracy using relatively coarse mesh resolution. • Theoretical links between the σ -formulation, the multi-layer approach and the ALE technique are established. [ABSTRACT FROM AUTHOR]

The unstructured mesh, discontinuous Galerkin finite element discretisation based coastal ocean model, Thetis , has been extended to include non-hydrostatic (buoyancy-driven and free surface) dynamics. Two alternative approaches to achieve this are described in this work. The first (a 3D based algorithm) makes use of prismatic element based meshes and uses a split-step pressure projection method for baroclinic and barotropic modes, while the second (a 2D based algorithm) adopts a novel multi-layer approach to convert a 3D problem into a combination of multiple 2D computations with only 2D triangle meshes required. Model development is carried out at high-level with the Firedrake library, using code generation techniques to automatically produce low-level code for the discretised model equations in an efficient and rapid manner. Through comparisons against several barotropic/baroclinic test cases where non-hydrostatic effects are important, the implemented approaches are verified and validated, and the proposed algorithms compared. Depending on whether the problems are dominated by dispersive, baroclinic or barotropic features, recommendation are given over the use of full 3D or multi-layer 2D based approaches to achieve optimal computational accuracy and efficiency. It is demonstrated that while in general the 2D approach is well-suited for barotropic problems and dispersive free surface waves, the 3D approach is more advantageous for simulating baroclinic buoyancy-driven flows due in part to the high vertical resolution typically required to represent the active tracer fields. • A non-hydrostatic discontinuous finite element coastal ocean model is presented. • Two non-hydrostatic extensions are compared showing respective advantages. • Both algorithms are validated using several barotropic/baroclinic test cases. • The model development is carried out using a code generation framework, Firedrake. • The model is able to provide accurate reproduction of surface and exchange flows. [ABSTRACT FROM AUTHOR]