Abstract
Abstract. The discontinuous Galerkin (DG) finite element method is well suited for the modelling, with a relatively small number of elements, of three-dimensional flows exhibiting strong velocity or density gradients. Its performance can be highly enhanced by having recourse to r-adaptivity. Here, a vertical adaptive mesh method is developed for DG finite elements. This method, originally designed for finite difference schemes, is based on the vertical diffusion of the mesh nodes, with the diffusivity controlled by the density jumps at the mesh element interfaces. The mesh vertical movement is determined by means of a conservative arbitrary Lagrangian–Eulerian (ALE) formulation. Though conservativity is naturally achieved, tracer consistency is obtained by a suitable construction of the mesh vertical velocity field, which is defined in such a way that it is fully compatible with the tracer and continuity equations at a discrete level. The vertically adaptive mesh approach is implemented in the three-dimensional version of the geophysical and environmental flow Second-generation Louvain-la-Neuve Ice-ocean Model (SLIM 3D; www.climate.be/slim). Idealised benchmarks, aimed at simulating the oscillations of a sharp thermocline, are dealt with. Then, the relevance of the vertical adaptivity technique is assessed by simulating thermocline oscillations of Lake Tanganyika. The results are compared to measured vertical profiles of temperature, showing similar stratification and outcropping events.
Highlights
The vertical discretisation strategy of marine models has evolved drastically during the last five decades
The discontinuous Galerkin (DG) finite element method is well suited for the modelling, with a relatively small number of elements, of three-dimensional flows exhibiting strong velocity or density gradients
The objective of this work is to adapt the method to an unstructured-mesh discontinuous Galerkin (DG) finite element model, namely the three-dimensional version of the Second-generation Louvain-la-Neuve Ice-ocean Model (SLIM 3D; www.climate.be/slim)
Summary
The vertical discretisation strategy of marine models has evolved drastically during the last five decades. Other discretisations were developed, mainly inspired by the progress in atmospheric modelling, for which coordinates based on the pressure field were used (Sutcliffe, 1947; Eliassen, 1949; Phillips, 1957). In oceanography, such developments led to ρ or σ coordinates (Freeman et al, 1972; Owen, 1980; Bleck and Boudra, 1986; Nihoul et al, 1986; Blumberg and Mellor, 1987). This method is well suited for tracer transport in the ocean interior, which occurs mainly
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