Numerical wave propagation for the triangular P1DGP1_{DG}-P2P2 finite element pair

Inertia-gravity mode and Rossby mode dispersion properties are examined for discretisations of the linearized rotating shallow-water equations using the $P1_{DG}$-$P2$ finite element pair on arbitrary triangulations in planar geometry. A discrete Helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations. In the $f$-plane case, this decomposition is used to obtain decoupled equations for the geostrophic modes, the inertia-gravity modes, and the inertial oscillations. As has been noticed previously, the geostrophic modes are steady. The Helmholtz decomposition is used to show that the resulting inertia-gravity wave equation is third-order accurate in space. In general the \pdgp finite element pair is second-order accurate, so this leads to very accurate wave propagation. It is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency $f$, and which do not propagate. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised $P1_{DG}$-$P2$ equations in the $\beta$-plane case, resulting in a Rossby wave equation which is also third-order accurate.Comment: Revised version prior to final journal submissio

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

Mixed finite elements for numerical weather prediction

We show how two-dimensional mixed finite element methods that satisfy the conditions of finite element exterior calculus can be used for the horizontal discretisation of dynamical cores for numerical weather prediction on pseudo-uniform grids. This family of mixed finite element methods can be thought of in the numerical weather prediction context as a generalisation of the popular polygonal C-grid finite difference methods. There are a few major advantages: the mixed finite element methods do not require an orthogonal grid, and they allow a degree of flexibility that can be exploited to ensure an appropriate ratio between the velocity and pressure degrees of freedom so as to avoid spurious mode branches in the numerical dispersion relation. These methods preserve several properties of the C-grid method when applied to linear barotropic wave propagation, namely: a) energy conservation, b) mass conservation, c) no spurious pressure modes, and d) steady geostrophic modes on the $f$-plane. We explain how these properties are preserved, and describe two examples that can be used on pseudo-uniform grids: the recently-developed modified RT0-Q0 element pair on quadrilaterals and the BDFM1-\pdg element pair on triangles. All of these mixed finite element methods have an exact 2:1 ratio of velocity degrees of freedom to pressure degrees of freedom. Finally we illustrate the properties with some numerical examples.Comment: Revision after referee comment

Geostrophic balance preserving interpolation in mesh adaptive shallow-water ocean modelling

The accurate representation of geostrophic balance is an essential requirement for numerical modelling of geophysical flows. Significant effort is often put into the selection of accurate or optimal balance representation by the discretisation of the fundamental equations. The issue of accurate balance representation is particularly challenging when applying dynamic mesh adaptivity, where there is potential for additional imbalance injection when interpolating to new, optimised meshes. In the context of shallow-water modelling, we present a new method for preservation of geostrophic balance when applying dynamic mesh adaptivity. This approach is based upon interpolation of the Helmholtz decomposition of the Coriolis acceleration. We apply this in combination with a discretisation for which states in geostrophic balance are exactly steady solutions of the linearised equations on an f-plane; this method guarantees that a balanced and steady flow on a donor mesh remains balanced and steady after interpolation onto an arbitrary target mesh, to within machine precision. We further demonstrate the utility of this interpolant for states close to geostrophic balance, and show that it prevents pollution of the resulting solutions by imbalanced perturbations introduced by the interpolation

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

Seamless cross-scale modeling with SCHISM

We present a new 3D unstructured-grid model (SCHISM) which is an upgrade from an existing model (SELFE). The new advection scheme for the momentum equation includes an iterative smoother to reduce excess mass produced by higher-order kriging method, and a new viscosity formulation is shown to work robustly for generic unstructured grids and effectively filter out spurious modes without introducing excessive dissipation. A new higher-order implicit advection scheme for transport (TVD2) is proposed to effectively handle a wide range of Courant numbers as commonly found in typical cross-scale applications. The addition of quadrangular elements into the model, together with a recently proposed, highly flexible vertical grid system (Zhang et al., A new vertical coordinate system for a 3D unstructured-grid model. Ocean Model. 85, 2015), leads to model polymorphism that unifies 1D/2DH/2DV/3D cells in a single model grid. Results from several test cases demonstrate the model\u27s good performance in the eddying regime, which presents greater challenges for unstructured-grid models and represents the last missing link for our cross-scale model. The model can thus be used to simulate cross-scale processes in a seamless fashion (i.e. from deep ocean into shallow depths). Published by Elsevier Ltd

Discontinuous finite-element methods for two- and three-dimensional marine flows

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, 201

A finite element method for solving the shallow water equations on the sphere

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

Practical evaluation of five partly discontinuous finite element pairs for the non-conservative shallow water equations

This paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second-order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non-conforming linear elements for both velocities and elevation (p(1)(NC)-P-1(NC),) is presented, giving optimal rates of convergence in all test cases. P-1(NC)-P-1 and P-1(DG)-P-1 mixed formulations lack convergence for inviscid flows. P-1(DG)-P-2 pair is more expensive but provides accurate results for all benchmarks. P-1(DG)-P-1(DG) provides an efficient option, except for inviscid Coriolis-dominated flows, where a small lack of convergence is observed. Copyright (C) 2009 John Wiley & Sons, Ltd