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This paper presents multirate explicit time-stepping schemes for solving partial differential equations with discontinuous Galerkin elements in the framework of Large-scale marine flows. It addresses the variability of the local stable time steps by gathering the mesh elements in appropriate groups. The real challenge is to develop methods exhibiting mass conservation and consistency. Two multirate approaches, based on standard explicit Runge–Kutta methods, are analyzed. They are well suited and optimized for the discontinuous Galerkin framework. The significant speedups observed for the hydrodynamic application of the Great Barrier Reef confirm the theoretical expectations.

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.

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 thermodynam

Numerical modeling is now, along with experiment and theory, part of the scientific method. Numerical models for marine flows exist for more than forty years. These models have evolved dramatically, with improved numerical methods, and much more accurate modeling of unresolved phenomena. However, most of the mainstream marine models still rely on the old numerical paradigm, based on finite difference methods and structured grids. The development of new numerical models, based on state-of-the art numerical methods on unstructured grids, is now an area of active research. Those unstructured meshes allow a faithful representation of the coastlines and the choice of the resolution following guidelines from the physics and not from the numerics. This thesis fits within this research. It is part of the development of SLIM (http://www.climate.be/SLIM), the Second-generation Louvain-la-neuve Ice-ocean Model. This model uses finite element methods on unstructured meshes made up of triangles for two-dimensional modeling, and triangular prisms for three-dimensional modeling. Numerical aspects are considered in this work. First, a comparison of several finite-element discretizations of the two-dimensional shallow water equations is performed. Second, to handle flows on the sphere, a novel algorithm is described, based on local coordinate systems, that allows a discretization free from singularities. Finally, a prototype three-dimensional baroclinic model is presented. The model features a spatial discretization built upon discontinuous finite elements and a time integration performed with an implicit mode splitting., (FSA 3) -- UCL, 2010

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. (C) 2010 Elsevier B.V. All rights reserved.

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.

In recent years, unstructured-mesh models have gained attention due to their flexibility in representing complex topography and variable spatial resolution. The Second generation Louvain-la-Neuve Ice-ocean Model (SLIM, www.climate.be/slim) is an unstructured-mesh finite element model that is being developed at the Université catholique de Louvain (Louvain-la-Neuve, Belgium). The ocean model solves the shallow water equations by means of the discontinuous Galerkin finite element method. It features a 1D river model, 2D depth averaged model and a full 3D model. After a general presentation of SLIM and its main applications, we will further focus on sea-ice modeling. The sea-ice component of SLIM has representations of both dynamic and thermodynamic sea-ice processes and includes viscous-plastic rheology along with a complete parametrization of the atmospheric fluxes. Unstructured meshes, with their natural ability to fit boundaries and increase locally the mesh resolution, propose an alternative framework to capture the complex oceanic areas formed by coasts and islands. Such an example is illustrated by the numerous narrow straits constituting the Canadian Arctic Archipelago. A key point of unstructured meshes is that they allow the use of mesh adaptivity. A Lagrangian, adaptive sea ice model allowing the computational grid to move with the ice drift is currently being developed. In order to maintain a good quality of the mesh, the mesh has to be adapted during the simulation, involving particular mesh adaptation techniques. This Lagrangian version of the model has several interesting applications, such as the dynamical mesh refinement along any region of interest (e.g., the ice edge), buoys tracking, or the inclusion of material properties in the rheology. Among these applications, the ice age constitutes both an interesting diagnostic tool and parameter for determining the ice physical properties (albedo, strength, salinity, ...). There are basically two ways of mo

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

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 threedimensional 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.

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. (C) 2009 Elsevier Ltd. All rights reserved.

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 P-1(NC) - P-1 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. (C) 2008 Elsevier Ltd. All rights reserved.

Over the last decade, increasing efforts have been directed towards the development of marine models using unstructured meshes. Such models allow for an accurate representation of the coastlines (e.g., islands, narrow straits) and the bathymetry. The mesh can easily be refined in regions of interest or coarsened in those regions where the dynamics is less demanding. Finally, unstructured meshes set up in spherical geometry allow to avoid singularities at the poles, rendering those techniques potentially useful for global ocean modelling. We present the latest version of {\it SLIM} (Second-generation Louvain-la-Neuve Ice-ocean Model, \textsf{http://www.climate.be/SLIM}), a three-dimensional, baroclinic, unstructured-mesh, finite-element ocean model. Herein, the emphasis is put on the mode splitting, which treats implicitly the two-dimensional mode while the three-dimensional part is treated explicitly. This method allows the representation of fast gravity waves with a reasonable time step, and generates a system much faster to solve than a fully implicit three-dimensional approach. It is well known that classical implementations of mode splitting resort to a weak coupling between two- and three-dimensional modes which can be the cause of inconsistencies. We introduce a new coupling method in which both modes are solved simultaneously. The new approach enables to ensure consistency: the mode splitting algorithm can be viewed as a specific time-stepping scheme, rather than a spatial approximation. Efficient implicit-explicit time discretizations ({\it IMEX}) schemes are used, which ensure stability and an optimal computational time. The model is validated by its application to a set of three-dimensional baroclinic simulations, such as the well-known overflow DOME test-case.