A discontinuous finite element baroclinic marine model on unstructured prismatic meshes: I. Space discretization

We describe the space discretization of a three-dimensional baroclinic finite element model, based upon a discontinuous Galerkin method, while the companion paper (Comblen et al. 2010a) describes the discretization in time. We solve the hydrostatic Boussinesq equations governing marine flows on a mesh made up of triangles extruded from the surface toward the seabed to obtain prismatic three-dimensional elements. Diffusion is implemented using the symmetric interior penalty method. The tracer equation is consistent with the continuity equation. A Lax–Friedrichs flux is used to take into account internal wave propagation. By way of illustration, a flow exhibiting internal waves in the lee of an isolated seamount on the sphere is simulated. This enables us to show the advantages of using an unstructured mesh, where the resolution is higher in areas where the flow varies rapidly in space, the mesh being coarser far from the region of interest. The solution exhibits the expected wave structure. Linear and quadratic shape functions are used, and the extension to higher-order discretization is straightforward

Timing and placing samplings to optimally calibrate a reactive transport model: exploring the potential for <i>Escherichia coli</i> in the Scheldt estuary

For the calibration of any model, measurements are necessary. As measurements are expensive, it is of interest to determine beforehand which kind of samples will provide the maximum of information. Using a criterion related to the Fisher information matrix, it is possible to design a sampling scheme that will enable the most precise model parameter estimates. This approach was applied to a reactive transport model (based on SLIM) of Escherichia coli in the Scheldt Estuary. As this estuary is highly influenced by the tide, it is expected that careful timing of the samples with respect to the tidal cycle will have an effect on the quality of the data. The timing and also the positioning of samples were optimised according to the proposed criterion. In the investigated case studies the precision of the estimated parameters could be improved by up to a factor of ten, confirming the usefulness of this approach to maximize the amount of information that can be retrieved from a fixed number of samples

High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations

An innovating approach is proposed to solve vectorial conservation laws on curved manifolds using the discontinuous Galerkin method. This new approach combines the advantages of the usual approaches described in the literature. The vectorial fields are expressed in a unit non-orthogonal local tangent basis derived from the polynomial mapping of curvilinear triangle elements, while the convective flux functions are written is the usual 3D Cartesian coordinate system. The number of vectorial components is therefore minimum and the tangency constraint is naturally ensured, while the method remains robust and general since not relying on a particular parametrization of the manifold. The discontinuous Galerkin method is particularly well suited for this approach since there is no continuity requirement between elements for the tangent basis definition. The possible discontinuities of this basis are then taken into account in the Riemann solver on inter-element interfaces.The approach is validated on the sphere, using the shallow water equations for computing standard atmospheric benchmarks. In particular, the Williamson test cases are used to analyze the impact of the geometry on the convergence rates for discretization error. The propagation of gravity waves is eventually computed on non-conventional irregular curved manifolds to illustrate the robustness and generality of the method

A discontinuous finite element baroclinic marine model on unstructured prismatic meshes: II. Implicit/explicit time discretization

We describe the time discretization of a three-dimensional baroclinic finite element model for the hydrostatic Boussinesq equations based upon a discontinuous Galerkin finite element method. On one hand, the time marching algorithm is based on an efficient mode splitting. To ensure compatibility between the barotropic and baroclinic modes in the splitting algorithm, we introduce Lagrange multipliers in the discrete formulation. On the other hand, the use of implicit–explicit Runge–Kutta methods enables us to treat stiff linear operators implicitly, while the rest of the nonlinear dynamics is treated explicitly. By way of illustration, the time evolution of the flow over a tall isolated seamount on the sphere is simulated. The seamount height is 90% of the mean sea depth. Vortex shedding and Taylor caps are observed. The simulation compares well with results published by other authors

Design of a sampling strategy to optimally calibrate a reactive transport model: Exploring the potential for Escherichia coli in the Scheldt Estuary.

For the calibration of any model, measurements are necessary. As measurements are expensive, it is of interest to determine beforehand which kind of samples will provide maximal information. Using a criterion related to the Fisher information matrix as a measure for information content, it is possible to design a sampling scheme that will enable the most precise parameter estimates. This approach was applied to a reactive transport model (based on the Second-generation Louvain-la-Neuve Ice-ocean Model, SLIM) of Escherichia coli concentrations in the Scheldt Estuary. As this estuary is highly influenced by the tide, it is expected that careful timing of the samples with respect to the tidal cycle can have an effect on the quality of the data. The timing and also the positioning of samples were optimised according to the proposed criterion. In the investigated case studies the precision of the estimated parameters could be improved by up to a factor of ten, confirming the usefulness of this approach to maximize the amount of information that can be retrieved from a fixed number of samples. Precise parameter values will result in more reliable model simulations, which can be used for interpretation, or can in turn serve to plan subsequent sampling campaigns to further constrain the model parameters.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Design of a sampling strategy to optimally calibrate a reactive transport model: Exploring the potential for <i>Escherichia coli</i> in the Scheldt Estuary

For the calibration of any model, measurements are necessary. As measurements are expensive, it is of interest to determine beforehand which kind of samples will provide maximal information. Using a criterion related to the Fisher information matrix as a measure for information content, it is possible to design a sampling scheme that will enable the most precise parameter estimates. This approach was applied to a reactive transport model (based on the Second-generation Louvain-la-Neuve Ice-ocean Model, SLIM) of Escherichia coli concentrations in the Scheldt Estuary. As this estuary is highly influenced by the tide, it is expected that careful timing of the samples with respect to the tidal cycle can have an effect on the quality of the data. The timing and also the positioning of samples were optimised according to the proposed criterion. In the investigated case studies the precision of the estimated parameters could be improved by up to a factor of ten, confirming the usefulness of this approach to maximize the amount of information that can be retrieved from a fixed number of samples. Precise parameter values will result in more reliable model simulations, which can be used for interpretation, or can in turn serve to plan subsequent sampling campaigns to further constrain the model parameters

Capturing the residence time boundary layer - Application to the Scheldt Estuary.

At high Peclet number, the residence time exhibits a boundary layer adjacent to incoming open boundaries. In a Eulerian model, not resolving this boundary layer can generate spurious oscillations that can propagate into the area of interest. However, resolving this boundary layer would require an unacceptably high spatial resolution. Therefore, alternative methods are needed in which no grid refinement is required to capture the key aspects of the physics of the residence time boundary layer. An extended finite element method representation and a boundary layer parameterisation are presented and tested herein. It is also explained how to preserve local consistency in reversed time simulations so as to avoid the generation of spurious residence time extrema. Finally, the boundary layer parameterisation is applied to the computation of the residence time in the Scheldt Estuary (Belgium/The Netherlands). This timescale is simulated by means of a depth-integrated, finite element, unstructured mesh model, with a high space-time resolution. It is seen that the residence time temporal variations are mainly affected by the semi-diurnal tides. However, the spring-neap variability also impacts the residence time, particularly in the sandbank and shallow areas. Seasonal variability is also observed, which is induced by the fluctuations over the year of the upstream flows. In general, the residence time is an increasing function of the distance to the mouth of the estuary. However, smaller-scale fluctuations are also present: they are caused by local bathymetric features and their impact on the hydrodynamics