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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. [ABSTRACT FROM AUTHOR]

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. ) 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. [ABSTRACT FROM AUTHOR]

Abstract: We present a flux-limiting wetting–drying approach for finite-element discretizations of the shallow-water equations using discontinuous linear elements for the elevation. The key ingredient of the method is the use of limiters for generalized nodal fluxes. This method is implemented into the Second-generation Louvain-la-Neuve Ice-ocean Model (SLIM), and is verified against standard test cases. The method is further applied to the wetting and drying of sand banks in the Scheldt Estuary, which is located in northern Belgium and the southern Netherlands. The results obtained for both the benchmarks and the realistic problem illustrate the accuracy of the method in describing the hydrodynamics in the vicinity of dry areas. In particular, the method strictly conserves mass, and there is no transport through dry areas. [Copyright &y& Elsevier]

Abstract: Within the framework of ocean general circulation modeling, the present paper describes an efficient way to discretize partial differential equations on curved surfaces by means of the finite element method on triangular meshes. Our approach benefits from the inherent flexibility of the finite element method. The key idea consists in a dialog between a local coordinate system defined for each element in which integration takes place, and a nodal coordinate system in which all local contributions related to a vectorial degree of freedom are assembled. Since each element of the mesh and each degree of freedom are treated in the same way, the so-called pole singularity issue is fully circumvented. Applied to the shallow water equations expressed in primitive variables, this new approach has been validated against the standard test set defined by [Williamson, D.L., Drake, J.B., Hack, J.J., Jakob, R., Swarztrauber, P.N., 1992. A standard test set for numerical approximations to the shallow water equations in spherical geometry. Journal of Computational Physics 102, 211–224]. Optimal rates of convergence for the finite element pair are obtained, for both global and local quantities of interest. Finally, the approach can be extended to three-dimensional thin-layer flows in a straightforward manner. [Copyright &y& Elsevier]

A method for generating computational meshes for applications in ocean modeling is presented. The method uses a standard engineering approach for describing the geometry of the domain that requires meshing. The underlying sphere is parametrized using stereographic coordinates. Then, coastlines are described with cubic splines drawn in the stereographic parametric space. The mesh generation algorithm builds the mesh in the parametric plane using available techniques. The method enables to import coastlines from different data sets and, consequently, to build meshes of domains with highly variable length scales. The results include meshes together with numerical simulations of various kinds. [ABSTRACT FROM AUTHOR]

Abstract: The sea ice age is an interesting diagnostic tool because it may provide a proxy for the sea ice thickness and is easier to infer from observations than the sea ice thickness. Remote sensing algorithms and modeling approaches proposed in the literature indicate significant methodological uncertainties, leading to different ice age values and physical interpretations. In this work, we focus on the vertical age distribution in sea ice. Based on the age theory developed for marine modeling, we propose a vertically-variable sea ice age definition which gives a measure of the time elapsed since the accretion of the ice particle under consideration. An analytical solution is derived from Stefan’s law for a horizontally homogeneous ice layer with a periodic ice thickness seasonal cycle. Two numerical methods to solve the age equation are proposed. In the first one, the domain is discretized adaptively in space thanks to Lagrangian particles in order to capture the age profile and its discontinuities. The second one focuses on the mean age of the ice using as few degrees of freedom as possible and is based on an Arbitrary Lagrangian–Eulerian (ALE) spatial discretization and the finite element method. We observe an excellent agreement between the Lagrangian particles and the analytical solution. The mean value and the standard deviation of the finite element solution agree with the analytical solution and a linear approximation is found to represent the age profile the better, the older the ice gets. Both methods are finally applied to a stand-alone thermodynamic sea ice model of the Arctic. Computing the vertically-averaged ice age reduces by a factor of about 2 the simulated ice age compared to the oldest particle of the ice columns. A high correlation is found between the ice thickness and the age of the oldest particle. However, whether or not this will remain valid once ice dynamics is included should be investigated. In addition, the present study, based on thermodynamics only, does not support a single age-thickness functional relationship. [Copyright &y& Elsevier]

Abstract: Resolving the shoreline undulation due to tidal excursion is a crucial part of modelling water flow in estuaries and coastal areas. Nevertheless, maintaining positive water column depth and numerical stability has proved out to be a very difficult task that requires special attention. In this paper we propose a novel wetting–drying method in which the position of the sea bed is allowed to fluctuate in drying areas. The method is implemented in a Discontinuous Galerkin Finite Element Model (DG-FEM). Unlike most methods in the literature our method is compatible with fully implicit time-marching schemes, thus reducing the overall computational cost significantly. Moreover, global and local mass conservation is guaranteed which is crucial for long-term environmental applications. In addition consistency with tracer equation is also ensured. The performance of the proposed method is demonstrated with a set of test cases as well as a real-world application to the Scheldt Estuary. Due to the implicit time integration, the computational cost in the Scheldt application is reduced by two orders of magnitude. Although a DG-FEM implementation is presented here, the wetting–drying method is applicable to a wide variety of shallow water models. [ABSTRACT FROM AUTHOR]

Abstract: We report on the development and validation of a coupled two- and one-dimensional finite-element model for the Scheldt tributaries, river, estuary and region of fresh water influence (ROFI). The hydrodynamic equations are solved on a single, unstructured, multi-scale mesh stretching from the shelf break to the Scheldt tributaries. The tide is forced on the shelf break and propagates upstream in the riverine network. Upstream boundaries lie on sluices or outside of the region of tidal dominance where daily averaged discharges are imposed. Two-dimensional, depth-averaged shallow water equations are solved by means of the discontinuous Galerkin (DG) method over the marine and estuarine parts of the computational domain. In the rivers, however, one-dimensional equations are dealt with using the DG method with the addition of a technique to cope with confluence points. Model parameters are carefully calibrated, leading to the simulation of wind- and tide-forced flows that are in excellent agreement with available data. The diffusivity in the transport equation is calibrated using time series of salinity at various locations in the estuary. Finally, the Lagrangian residual transport in the estuary and the adjacent coastal zone is investigated. This work is a major step towards an integrated model for studying the dynamics of waterborne contaminants and the water renewal timescales in the Scheldt land-sea continuum. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

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. [Copyright &y& Elsevier]