Development of a finite element model for the modelling of topography induced internal wave behaviour in the coastal ocean

Abstract

It is universally accepted that the world’s oceans act as a large heat engine and play a critical role in shaping the earth’s climate. Understanding ocean circulation and the transport of heat throughout the global ocean is thus a necessary and challenging journey for the climate scientist. The physical oceanography community, in trying to construct better ocean models, are calling for improved representation of mesoscale processes and associated mixing. Processes specifically targeted are those occurring about the thermocline such as wave and vortical modes. Tidal currents over deep ocean and shallow coastal topographies are a strong source of internal waves and mixing. Today, ocean general circulation models are primarily based on the Reynolds averaged Navier-Stokes (RANS) equations with the hydrostatic pressure approximation and a finite difference numerical formulation. Mesoscale processes and mixing are represented through sub-grid scale parameterisations, sometimes by rather elementary forms of vertical diffusivity. Recently, an exciting shift has become apparent, with momentum towards ‘next generation’ ocean models with non-hydrostatic physics and finite element numerics. These models are still in their infancy for global application and hence the need remains for smaller higher resolution diagnostic models to examine mesoscale and small scale mixing processes in more detail and assist with their customised parameterisation. This was the impetus for pursuing the present research. This thesis describes the development of a hydrodynamic code for buoyancy influenced topographic flow. To represent bathymetry and density variation, a two-dimensional (x-z) modelling domain was chosen. The model is based on the steady state RANS equations, is non-hydrostatic, has a k-ε turbulence closure and employs the finite element method. A non-hydrostatic formulation is necessary in order to capture internal wave behaviour and wave-current interactions. The k-ε model calculates turbulence quantities with turbulence dissipation rate both a practical parameter and with potential to link with established spectral laws. A pressure split combined with a supporting spline routine for calculation of horizontal hydrostatic pressure gradients and an implicit implementation of a discontinuous Galerkin method flux in the element coefficient matrices for density transport are two novel approaches attempted in this study. Both theories were introduced with the aim to help overcome computational difficulties associated with buoyancy influenced flows. A number of model simulations have been performed for simple geometry problems at laboratory scale, including simulations of the widely used benchmark problem of the backward facing step. Model results are compared to two backward facing step experimental data sets and the model is shown to perform well, with good agreement in the key flow characteristic of reattachment point, demonstrating verification and validation of the code. Simulations are also conducted using scaled down topography of two coastal sills, to apply the model to a more natural topography. The flow field and density contours from the steady state simulations do not resemble sill field observations at the time of maximum sill flow and it is concluded that a transient model is required for the gradual build up of complex flow conditions that eventually lead to the observed flow features at maximum tidal flow.