A discontinuous finite element baroclinic marine model on unstructured prismatic meshes
 931 Downloads
 19 Citations
Abstract
We describe the space discretization of a threedimensional 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 higherorder discretization is straightforward.
Keywords
Internal Wave Discontinuous Galerkin Shallow Water Equation Discontinuous Galerkin Method Riemann Solver1 Introduction
Ocean models have reached a high level of complexity, and eddy resolving simulations are now much more affordable than in the past. However, mainstream models still fit into the same framework as the pioneering model published by Bryan (1969). This approach, which uses conservative finite differences on structured grids, approximates the coastlines as staircases and prevents flexible implementation of variable resolution. Yet, during the last 40 years, numerical methods have dramatically evolved. It is now time for ocean modeling to benefit from all those advances by developing ocean models using stateoftheart numerical methods on unstructured grids (Griffies et al. 2009).
Unstructured grid methods are mainly of two kinds: finite volumes and finite elements. In short, finite volumes were first developed for problems predominantly hyperbolic (i.e., dominated by waves or advective transport), while finite element methods were first developed for problems dominated by elliptic terms. The two research communities have evolved toward solutions that manage to treat efficiently both hyperbolic and elliptic problems. Unstructured grid marine modeling is an active area of research for coastal applications (Deleersnijder and Lermusiaux 2008). Indeed, the coastlines must be accurately represented, as they have a much stronger influence at the regional scale than at the global scale. Finite volume methods are now widely used, and models like FVCOM (Finite Volume Community Ocean Model) (Chen et al. 2003) have a large community of users. Many other groups are developing finite volume codes for ocean, coastal, and estuarine areas, such as Fringer et al. (2006), Ham et al. (2005), Stuhne and Peltier (2006), and Casulli and Walters (2000). Nonhydrostatic finite element methods are found in Labeur and Wells (2009) for smallscale problems. For largescale ocean modeling, continuous finite element methods are used in FEOM (Finite Element Ocean Model) (Wang et al. 2008a, b; Timmermann et al. 2009), and Imperial College Ocean Model (Ford et al. 2004a) relies on mesh adaptivity to capture the multiscale aspects of the flow (Piggott et al. 2008).
In the realm of finite difference methods, Arakawa’s C grid allows for a stable and relatively noisefree discretization of the shallow water equations and is now very popular for ocean modeling (Arakawa and Lamb 1977; Griffies et al. 2000). However, the search for an equivalent optimal finite element pair for the shallow water equations is still an open issue. Le Roux et al. (1998) gave the first review of available choices. More recent mathematical and numerical analysis of finite element pairs for gravity and Rossby waves are provided in Le Roux et al. (2007, 2008), Rostand and Le Roux (2008), and Rostand et al. (2008). Hanert et al. (2005) proposed to use the \(P_1^{\rm NC}\)–P _{1} pair, following Hua and Thomasset (1984). It appears that the \(P_1^{\rm NC}\)–P _{1} pair is a stable discretization, but its rate of convergence is suboptimal on unstructured grids (Bernard et al. 2008b). Following the same philosophy, the \(P_1^{\rm DG}\)–P _{2} pair was proposed by Cotter et al. (2009a). Such an element exhibits both stability and optimal rates of convergence for the Stokes problem and the wave equation (Cotter et al. 2009b). A review of the numerical properties of those pairs stabilized by interface terms can be found in Comblen et al. (2010b).
This paper focuses on the development of a marine model, called Secondgeneration LouvainlaNeuve Ice–ocean Model (SLIM^{1}) that should be able to deal with problems ranging from local and regional scales to global scales. In this model, we choose equalorder discontinuous interpolations for the elevation and velocity fields, as it allows us an efficient and easier implementation. Such an equalorder mixed formulation is stable as the interface stabilizing terms allows us to circumvent the Ladyzhenskaya–Babŭska–Brezzi conditions and to take advantage of the inherent good numerical properties of the discontinuous Galerkin (DG) methods for advection dominated processes. It also allows us to decouple horizontal and vertical dynamics, thanks to the blockdiagonal nature of the corresponding mass matrix. DG methods can be viewed as a kind of hybrid between finite elements and finite volumes. They enjoy most the strengths of both schemes while avoiding most of their weaknesses: Advection schemes take into account the characteristic structure of the equations, as for finite volume methods, and the polynomial interpolation used inside each element allows for a highorder representation of the solution. Moreover, no degree of freedom is shared between two geometric entities, and this high level of locality considerably simplifies the implementation of the method. Finally, the mass matrix is block diagonal, and for explicit computations, no linear solver is needed. We also observe a growing interest for the discontinuous Galerkin methods in coastal and estuarine modeling (Aizinger and Dawson 2002; Dawson and Aizinger 2005; Kubatko et al. 2006; Aizinger and Dawson 2007; Bernard et al. 2008a; Blaise et al. 2010). For atmosphere modeling, the highorder capabilities of this scheme are really attractive (Nair et al. 2005; Giraldo 2006), and the increasing use of DG follows the trend to replace spectral transform methods with local ones.
Herein, we provide the detailed description of the spatial discretization used in our discontinuous Galerkin finite element marine model SLIM, as well as an illustrative example of the ability of the model to represent complex baroclinic flows. Section 2 describes the partial differential equations considered. Sections 3 and 4 provide the numerical tools needed to derive an efficient stable and accurate discrete formulation. Section 5 details the discrete discontinuous formulation. Finally, in Section 6, we study the internal waves generated in the lee of an isolated seamount as computed with our model. In a companion paper, the time integration procedure will be discussed.
2 Governing equations
Largescale ocean models usually solve the hydrostatic Boussinesq equations for the ocean. As a result of the hydrostatic approximation, the vertical momentum equation is reduced to a balance between the pressure gradient force and the weight of the fluid. The conservation of mass degenerates into volume conservation, and the density variations are taken into account in the buoyancy term only. This set of equations is defined on a moving domain, as the seasurface evolves according to the flow.
 Horizontal momentum equation:$$ \begin{array}{rll} &&{\kern6pt}\frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{\nabla}_h \cdot {\left(\boldsymbol{u}\boldsymbol{u}\right)} + \frac{\partial (w\boldsymbol{u})}{\partial z} \! + \! f\mathbf{e}_z\wedge\boldsymbol{u} \! + \! \frac{1}{\rho_0}\boldsymbol{\nabla}_h p \! + \! g\boldsymbol{\nabla}_h \eta \\ && {\kern1pc} = \boldsymbol{\nabla}_h\cdot \left(\nu_h \boldsymbol{\nabla}_h \boldsymbol{u}\right) +\frac{\partial }{\partial z}\left( \nu_v \frac{\partial \boldsymbol{u}}{\partial z}\right), \end{array} $$(1)
 Vertical momentum equation:$$ \frac{\partial p}{\partial z}=g\rho(T,S)\qquad \mathrm{with}\qquad \rho=\rho_0+\rho^{\prime}(T,S). $$(2)
 Continuity equation:$$ \boldsymbol{\nabla}_h\cdot{\boldsymbol{u}}+\frac{\partial w}{\partial z}=0, $$(3)
 Freesurface equation:$$ \frac{\partial \eta}{\partial t}+\boldsymbol{u}^{\eta}\cdot\nabla_h \etaw^{\eta} = 0. $$(4)
 Tracer equation:$$ \begin{array}{rll} \frac{\partial c}{\partial t}+\boldsymbol{\nabla}_h\cdot{\left(\boldsymbol{u} c\right)}+\frac{\partial (wc)}{\partial z}&=&\boldsymbol{\nabla}_h\cdot{\left(\kappa_h\boldsymbol{\nabla}_h{c}\right)} \\ &&+\frac{\partial }{\partial z}\left(\kappa_v\frac{\partial c}{\partial z }\right), \end{array} $$(5)
Notations for the governing equations of the threedimensional baroclinic marine model
Coordinates and spatial operators  
x,y  Horizontal coordinates 
z  Vertical coordinate, pointing upwards with its origin at the sea surface at rest 
\(\boldsymbol{\nabla}_h\)  Horizontal gradient operator 
\(\boldsymbol{e}_z\)  Upward unit normal 
∧  Cross product symbol 
Material parameters or functions  
g  Gravitational acceleration 
ρ _{0}  Reference density 
f  Coriolis parameter 
h  Depth at rest 
ν _{ h }  Horizontal turbulent viscosity parameter 
ν _{ t }  Vertical turbulent viscosity parameter 
κ _{ h }  Horizontal turbulent diffusivity parameter 
κ _{ t }  Vertical turbulent diffusivity parameter 
Variables  
\(\boldsymbol{u}\)  Horizontal threedimensional velocity vector 
w  Vertical threedimensional velocity vector 
\(\boldsymbol{u}^\eta\)  Surface horizontal threedimensional velocity vector 
w ^{ η }  Surface vertical threedimensional velocity vector 
\(\boldsymbol{u}^{h}\)  Bottom horizontal threedimensional velocity vector 
w ^{ − h }  Bottom vertical threedimensional velocity vector 
η  Sea surface elevation 
p  Baroclinic pressure 
\(\boldsymbol{p}\)  Baroclinic pressure gradient 
c  Threedimensional tracer, can be S or T 
S  Salinity 
T  Temperature 
This set of equations defines the mathematical threedimensional baroclinic marine model and must be solved simultaneously with the suitable initial and boundary conditions. However, no models solve the primitive equations simultaneously. In practice, dynamic and thermodynamic equations are always decoupled.
3 Geometrical numerical tools

The computational domain evolves with time, and it is required to take into account the evolution of the domain in the discrete model. In Section 3.1, the standard ALE technique (arbitrary Lagrangian–Eulerian) implemented in the model is described.

Moreover, the computational domain lies on a sphere. In Section 3.2, we recall the algorithm that renders the model able to operate on any manifold including a sphere or a planar surface.
3.1 Arbitrary Lagrangian–Eulerian methods
3.2 Dealing with flows on the sphere
The model operates on arbitrarily shaped surfaces, including the sphere or plane surfaces, following Comblen et al. (2009). The basic idea of the procedure is that each local geometrical entity supporting vectorial degrees of freedom has its own Cartesian coordinate system. There are also coordinate systems associated with each vector test and shape functions for the horizontal velocity field. To supply a vectorial degree of freedom from a frame of reference to another, we only need to build local rotation operators.
The global linear system of discrete equations is then formulated in terms of the vector degrees of freedom expressed in their own frame of reference. To build and assemble the local matrix corresponding to an element, we first fetch all the needed vectorial degrees of freedom into the coordinate system of this element, and then we compute the local matrix or vector. We then apply rotation matrices to this matrix so that its lines and columns are expressed in the frame of reference of the corresponding test and shape functions, respectively. The transformation of the local linear system can be expressed in terms of _{ x } P _{ ξ } and _{ ξ } P _{ x }, the rotation matrices from and to the frame of reference in which the integration is performed, respectively (Comblen et al. 2009).
Similarly, to assemble local matrices and vectors corresponding to the integral over an interface between two elements with different coordinate systems, we first fetch all the information in the frame of reference of the interface, then compute the integral, and fetch back the lines and columns of the matrices in the frame of reference of the corresponding test and shape functions. All the curvature treatment is embedded in the rotation matrices, and the discrete equations are expressed exactly as if the domain was planar.
With such a procedure, it is possible to solve the equations on the sphere, circumventing completely any possible singularity problem. For notational convenience, the full discrete formulation will be presented within a Cartesian framework, but it is important to note that the model is fully implemented to operate on the sphere.
4 Discontinuous Galerkin methods

Choosing the way to compute the discrete values at the interelement interfaces is the critical ingredient to obtain a stable and accurate discrete formulation in the framework of the DG methods. The discrete fields are dualvalued at the interelement interfaces. For the advective fluxes at these interfaces, the values of the variables are obtained on Riemann solvers applied to the hyperbolic terms of the model. Details about the Riemann solvers are given in Section 4.1.

Incorporating the diffusion operators inside a DG formulation also requires special care. We use the Symmetric Interior Penalty Galerkin (SIPG) technique to accommodate diffusion operators. Moreover, the mathematical formulation exhibits anisotropic diffusion and the algorithm is adapted by computing the interior penalty coefficients on a virtual stretched geometry. The methodology used is summarized in Section 4.2.
Summary of the finite element spaces used for each field. Triangular linear elements are noted P _{1} while vertical linear elements are noted L _{1}. The superscript DG stands for Discontinuous Galerkin
Field 
 Finite element space 

Free surface elevation  η  \(P_1^{\rm DG}\) 
Horizontal threedimensional velocity vector  \(\boldsymbol{u}\)  \(P_1^{\rm DG}\times L_1^{\rm DG}\) 
Vertical threedimensional velocity  w  \(P_1^{\rm DG}\times L_1^{\rm DG}\) 
Threedimensional tracer  c  \(P_1^{\rm DG}\times L_1^{\rm DG}\) 
Density deviation  ρ′  \(P_1^{\rm DG}\times L_1\) 
Baroclinic pressure gradient  \(\boldsymbol{p}\)  \(P_1^{\rm DG}\times L_1\) 
4.1 Riemann solvers
A twodimensional set of barotropic discrete equations can be obtained by the vertical depth integration (or the algebraic stacking of the resulting lines and columns of the global system) of the threedimensional set of baroclinic discrete equations. The basic idea of this procedure is to define the lateral interface in the discrete threedimensional baroclinic equations in such a way that the corresponding twodimensional discretization by depth integration is a robust stable formulation. In particular, the use of the integral freesurface equation (Eq. 7) and the selection of the discrete spaces should lead to a stable and accurate corresponding twodimensional discrete problem. Here, the resulting corresponding discrete problem is close to the discretization of the shallow water equations with \(P_1^{\rm DG}\) shape functions for the twodimensional velocities and elevation. A robust formulation can be derived for this problem, following Comblen et al. (2010b).
The key ingredient of a stable and accurate DG formulation is the choice of the definition for a common value for the variables along the interfaces. It is necessary to define adequately these common values with a Riemann solver. For nonlinear problems, it can be quite complicated to compute the exact Riemann solver, and approximate Riemann solvers are usually resorted to. For the shallow water equations, approximate Riemann solvers are deduced from the conservative form of the equations (LeVeque 2002; Toro 1997; Comblen et al. 2010b).
In this work, we use the Riemann solver of the linear shallowwater equations (Eqs. 14 and 15) for the terms corresponding to surface gravity waves (i.e., elevation gradient in the twodimensional, depthaveraged momentum equation and velocity divergence in the continuity equation (Comblen et al. 2010a)). In the threedimensional problem, i.e., in the momentum equation and in the active tracers equations (typically temperature and salinity), we add to the mean flux the jump of the variables multiplied by an upper bound on the second fastest wave, which is the sum of the fastest internal wave phase speed and the advection velocity. Determining the phase speed of the fastest internal wave is not easy for a complicated stratification profile.
In our examples, we simply use values of γ deduced from some numerical experiments, by a trial and error procedure. The selection of the coefficient γ is a key ingredient of the numerical technique. In fact, in our computations, we select an ad hoc coefficient that is selected with some physical intuition. However, it is not a robust procedure and this can prevent the model from being used if there are no general solutions for it. In other words, this problem is not fully solved but the importance of Riemann solvers in shallow water models were discussed in Comblen et al. (2010b).
4.2 Symmetric interior penalty Galerkin methods
In the realm of discontinuous Galerkin methods, various discretizations of the Laplacian operator are reviewed in, e.g., Arnold et al. (2002). Two of them are especially popular: the interior penalty methods (Riviere 2008) and the local DG method (Cockburn and Shu 1998).
Finally, the diffusion terms are split into a horizontal part and a vertical part, preventing the implementation of rotated diffusion tensors as described for instance by Redi (1982). Such rotated diffusion is not implemented in the current SLIM model. Distinct viscosity and diffusivity coefficients are chosen to represent the different effects of each of the many unresolved physical processes. Both the diffusion operator and the mesh can be anisotropic. A simple procedure consists in virtually stretching the mesh in the vertical direction so that we recover an isotropic diffusion in the deformed geometry. The mesh is not really modified, but the local interior penalty coefficients are chosen in such a way that they correspond to an isotropic diffusion on the modified mesh.
5 Discrete DG finite element formulations
For the sake of completeness, we provide here the full weak DG finite element formulations for each equation of the SLIM model (Eqs. 1, 3, 5, 6, and 7) using the numerical tools described in both previous sections. The discrete formulations can be then derived by replacing the test functions and the solution by the corresponding DG polynomial approximation.
5.1 Horizontal momentum equation
 Horizontal advection:$$ \begin{array}{rll} \sum\limits_{e=1}^{N_e} \bigg[&& <\boldsymbol{\nabla}_h \widehat{\boldsymbol{u}} : \boldsymbol{u} \boldsymbol{u}>_{e} + \ll \widehat{\boldsymbol{u}} \cdot \left\{\boldsymbol{u}\right\} \left\{\boldsymbol{u}\right\} \cdot \boldsymbol{n}_h\gg_{e} \\ &+& \ll \gamma \left[\boldsymbol{u}\right]\cdot \widehat{\boldsymbol{u}} \gg_{e}\bigg] \end{array} $$
 Vertical advection:$$ \begin{array}{rll} \sum\limits_{e=1}^{N_e} \bigg[&& < \frac{\partial \widehat{\boldsymbol{u}}}{\partial z}\cdot (ww_z)\boldsymbol{u}>_{e} \\ &+& \ll\widehat{\boldsymbol{u}} \cdot (ww_z)^\text{down} \ \boldsymbol{u}^\text{upwind} \ n_z\gg_{e}\bigg] \end{array} $$
 Elevation gradient:$$ \sum\limits_{e=1}^{N_e} \bigg[  <\boldsymbol{\nabla}_h \cdot \widehat{\boldsymbol{u}} \ g \eta>_{e} + \ll g \eta^\text{riemann} \ \widehat{\boldsymbol{u}} \cdot \boldsymbol{n}_h\gg_{e} \bigg] $$
 Horizontal diffusion:$$ \begin{array}{rll} \sum\limits_{e=1}^{N_e} \bigg[&& <\nu_h \left(\boldsymbol{\nabla}_h \widehat{\boldsymbol{u}}\right) : \left( \boldsymbol{\nabla}_h \boldsymbol{u}\right)^T>_{e} \\ &+& \ll\nu_h \widehat{\boldsymbol{u}} \cdot \left\{\boldsymbol{\nabla}_h \boldsymbol{u}\right\}\cdot \boldsymbol{n}\gg_{e}\\ &+& \ll \nu_h \boldsymbol{\nabla}_h\widehat{\boldsymbol{u}}\cdot \boldsymbol{n} \cdot \left[\boldsymbol{u}\right]\gg_{e} +\sigma \ll \nu_h \widehat{\boldsymbol{u}} \cdot \left[\boldsymbol{u}\right]\gg_{e} \bigg] \end{array} $$
 Vertical diffusion:$$ \begin{array}{rll} \sum\limits_{e=1}^{N_e} \bigg[ && < \nu_v\frac{\partial \widehat{\boldsymbol{u}}}{\partial z} \cdot \frac{\partial \boldsymbol{u}}{\partial z}>_{e} + \ll \nu_v \widehat{\boldsymbol{u}} \cdot \left\{\frac{\partial \boldsymbol{u}}{\partial z}\right\}n_z\gg_{e}\\&+&\ll \nu_v \frac{\partial \widehat{\boldsymbol{u}}}{\partial z}n_z \cdot \left[\boldsymbol{u}\right]\gg_{e} +\,\sigma \ll \nu_v \widehat{\boldsymbol{u}} \cdot \left[\boldsymbol{u}\right]\gg_{e} \bigg] \end{array} $$
 Baroclinic pressure gradient and Coriolis:$$ \sum\limits_{e=1}^{N_e} \bigg[ <\widehat{\boldsymbol{u}} \cdot \frac{\boldsymbol{p}}{\rho_0}>_{e} + f\mathbf{e}_z\wedge\boldsymbol{u} \ \bigg] $$
5.2 Vertical momentum equation
5.3 Continuity equation
In the lateral interface, we use \(\boldsymbol{u}^\text{riemann}\) because the discrete twodimensional integral free surface equations will be obtained by aggregating the threedimensional discrete continuity equations. In fact, the discrete procedure mimics the algebra performed to obtain the integral freesurface equation (Eq. 7) by integrating the equation of continuity (Eq. 3) and substituting the impermeability conditions at both the sea bed and the sea surface. The sea bed impermeability is already included in the discrete formulation of the continuity equation, and the seasurface condition will be incorporated by the motion of the free surface. Finally, let us recall that when we deduce \(\boldsymbol{u}^\text{riemann}\) and \(\eta^\text{riemann}\) with the exact Riemann solver of the linear shallow water equations, we use the fact that the depth integration of the momentum equation coupled with the free surface equation has to degenerate into a stable and an accurate \(P_1^{\rm DG}P_1^{\rm DG}\) discretization of the twodimensional shallow water equations. Therefore, as the discrete freesurface equation will be obtained by aggregating the discrete continuity equation, it is mandatory to use \(\boldsymbol{u}^\text{riemann}\) here, to have it in the resulting freesurface equation. As a last remark, it is also important to emphasize that the vertical velocity is not a prognostic field. It is a byproduct used to deduce the vertical advection terms in the momentum and tracer equations. Elevation, velocities, and tracer are prognostic fields. An accurate DG discretization of our set of equation should be such that these fields are computed with an optimal accuracy, i.e., p + 1 convergence rate if order p shape functions are used. It is not the case for vertical velocity. Vertical velocity is not smooth because it results from the integration of the divergence of the horizontal velocity. It behaves similarly to the volume term from an advection term integrated by parts: It is not smooth, but it does not impair the optimal convergence of the other fields. Indeed, at the discrete level, it is easily seen that, if the prisms are straight, the vertical velocity lives in a discrete space that is piecewise constant in the horizontal direction rather than linear. The smoothness of tracer and horizontal velocity field is recovered as usual in DG, using the interface term, acting as a penalty term.
5.4 Freesurface equation
5.5 Tracer equation
 Horizontal advection:$$ \begin{array}{rll} \sum\limits_{e=1}^{N_e} \bigg[ &&<\boldsymbol{\nabla}_h{\hat{c}}\cdot\boldsymbol{u} c>_{e} + \ll \hat{c} \left\{c\right\} \boldsymbol{u}^\text{riemann}\cdot \boldsymbol{n}_h\gg_{e} \\&+& \ll \hat{c} \gamma \left[c\right] \gg_{e}\bigg] \end{array} $$
 Vertical advection:$$ \begin{array}{rll} \sum\limits_{e=1}^{N_e} \bigg[ && < \frac{\partial \hat{c}}{\partial z} \ (ww_z)c>_{e} \\ &+& \ll\left(\hat{c} \ (ww_z)^\text{down} c^\text{upwind}\right) n_z\gg_{e}\bigg] \end{array} $$
 Horizontal diffusion:$$ \begin{array}{rll} \sum\limits_{e=1}^{N_e} \bigg[&& <\kappa_h \left(\boldsymbol{\nabla}_h \hat{c}\right) \cdot \left( \boldsymbol{\nabla}_h c \right)>_{e} + \ll\kappa_h \hat{c} \left\{\boldsymbol{\nabla}_h c\right\} \cdot \boldsymbol{n}\gg_{e} \\ &+&\ll \kappa_h \boldsymbol{\nabla}_h\hat{c}\cdot \boldsymbol{n} \cdot \left[c\right]\gg_{e} +\sigma \ll \kappa_h \hat{c} \left[c\right]\gg_{e} \bigg] \end{array} $$
 Vertical diffusion:$$ \begin{array}{rll} \sum\limits_{e=1}^{N_e} \bigg[ && < \kappa_v\frac{\partial \hat{c}}{\partial z} \frac{\partial c}{\partial z}>_{e} + \ll \kappa_v \hat{c} \left\{\frac{\partial c}{\partial z}\right\} n_z\gg_{e}\\ &+&\ll \kappa_v \frac{\partial \hat{c}}{\partial z} n_z \left[c\right]\gg_{e} +\sigma \ll \kappa_v \hat{c} \left[c\right]\gg_{e} \bigg] \end{array} $$
To ensure consistency, it is mandatory that the discrete advection term degenerates to the continuity equation when a unit tracer concentration is considered (White et al. 2008). Therefore, the same discrete space must be used for both c and w. Moreover, we must use the same velocity approximations in the interface terms \(\boldsymbol{u}^\text{riemann}\) for the horizontal advection and \(w^\text{down}\) for the vertical advection. But the interface value for the tracer concentration c at the interface is not constrained by consistency considerations: Upwind or centered values can be used. The additional term from the Lax–Friedrichs solver does not impair consistency as it is exactly nil for a constant tracer. The bottom boundary conditions must also be compatible, this being ensured by suppressing the boundary terms for advection at the sea bottom.
5.6 Validation and mesh refinement analysis
6 Numerical results
We simulate the internal waves in the lee of a moderately tall seamount. The simulation of such a complex flow can be considered as a relevant test case. Complicated phenomena can be observed in the wake of mountains, such as internal wave structures and vortex streets (Chapman and Haidvogel 1992, 1993; Ford et al. 2004b). Such a problem was simulated with threedimensional baroclinic finite difference (Huppert and Bryan 1976; Chapman and Haidvogel 1992, 1993), finite volume (Adcroft et al. 1997), and finite element models (Ford et al. 2004b; Wang et al. 2008a, b). If the seamount is small enough, a complicated structure of standing internal waves can develop in the wake of the seamount. Chapman and Haidvogel (1993) provide a detailed numerical study of internal lee waves trapped over isolated Gaussianshaped seamounts. Such a testcase is also used by Ford et al. (2004b) to assess the qualities and drawbacks of their model. With our threedimensional baroclinic marine model SLIM, we simulate the internal lee waves for a seamount whose height is 30% of the total depth.
 Seamount ratio

δ = 0.3
 Rossby number

\(\displaystyle Ro=\frac{U}{fL} = 0.2\)
 Reynolds number

\(\displaystyle Re=\frac{UL}{\nu_h} = \text{1,000}\)
 Burger number

\(\displaystyle Bu = \frac{NH}{fL} = \sqrt{\frac{g}{\rho_0}\frac{\partial \rho}{\partial z}}\frac{H}{fL} = 1\)
The only numerical parameter that has to be selected in the threedimensional baroclinic model is the jump penalty parameter γ in the Lax–Friedrichs solver. For this problem, we select γ = 4 m s^{ − 1}. This parameter must be an upper bound of the phase speed of the fastest wave. From the linear theory, we know that with the prescribed stratification, the maximum phase speed of an internal wave is about c = 1 m s^{ − 1}, so that the fastest threedimensional phenomenon propagates at c + U ≈ 1.5 m s^{ − 1}. For discontinuous linear elements combined with the secondorder explicit Runge–Kutta timestepper (Chevaugeon et al. 2007) used in this simulation, the relevant CFL conditions leads us to select a time step of 30 s.
7 Conclusions
The spatial discretization of a threedimensional baroclinic freesurface marine model is introduced. This model relies on a discontinuous Galerkin method with a mesh of prisms extruded in several layers from an unstructured twodimensional mesh of triangles. As the prisms are vertically aligned, the calculation of the vertical velocity and the baroclinic pressure gradient can be implemented in an efficient and accurate way. All discrete fields are defined in discontinuous finite element spaces, to take advantage of the good numerical properties of the discontinuous Galerkin methods for advection dominated problems and for wave problems. To be able to use the Riemann solver of the shallow water equations, the discretization of the threedimensional horizontal momentum and the continuity equations are defined in such a way that their discrete integration along the vertical axis provides a stable \(P_1^{\rm DG}P_1^{\rm DG}\) formulation of the shallow water equations. Therefore, we can stabilize the discrete equations by using the exact Riemann solver of the linear shallow water for the gravity waves. For internal waves, an additional stabilizing term is derived from a Lax–Friedrichs solver. In the baroclinic dynamics, the vertical velocity acts as a source term, while the role of the approximate Riemann solver is to penalize interelement jumps to recover optimal accuracy. Consistency is ensured. The model is able to advect exactly a tracer with a constant concentration, meaning that the discrete transport term is compatible with the continuity equation.
In conclusion, we use a threedimensional finite element baroclinic freesurface model to represent accurately the complex structure of the internal waves in the lee of an isolated seamount, using either linear or quadratic shape functions. The model does not yet handle internally supercritical flows that occur for instance when internal waves break or in steed gravity currents. Including a limiting strategy to handle shockwaves would be the next required step. Finally, the second key ingredient for an efficient threedimensional marine model is the definition of a good time integration procedure. This will be the topic of the second part of this contribution.
Footnotes
Notes
Acknowledgements
Sébastien Blaise and Jonathan Lambrechts are research fellows with the Belgian Fund for Research in Industry and Agriculture (FRIA). Richard Comblen is a research fellow with the Belgian National Fund for Scientific Research (FNRS). Eric Deleersnijder is a research associate with the Belgian National Fund for Scientific Research (FNRS). The research was conducted within the framework of the Interuniversity Attraction Pole TIMOTHY (IAP 6.13), funded by the Belgian Science Policy (BELSPO), and the programme ARC 04/09316, funded by the Communauté Française de Belgique.
References
 Adcroft A, Hill C, Marshall J (1997) Representation of topography by shaved cells in a height coordinate ocean model. Mon Weather Rev 95:2293–2315CrossRefGoogle Scholar
 Aizinger V, Dawson C (2002) A discontinuous Galerkin method for twodimensional flow and transport in shallow water. Adv Water Resour 25:67–84CrossRefGoogle Scholar
 Aizinger V, Dawson C (2007) The local discontinuous Galerkin method for threedimensional shallow water flow. Comput Methods Appl Mech Eng 196:734–746CrossRefGoogle Scholar
 Arakawa A, Lamb VR (1977) Computational design of the basic dynamical processes of the UCLA general circulation model. In: Chang J (ed) Methods in computational physics. Academic, New York, pp 173–265Google Scholar
 Arnold DN, Brezzi F, Cockburn B, Marini LD (2002) Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal 39:1749–1779CrossRefGoogle Scholar
 Bernard PE, Remacle JF, Legat V, (2008a) Boundary discretization for high order discontinuous Galerkin computations of tidal flows around shallow water islands. Int J Numer Methods Fluids 59:535–557CrossRefGoogle Scholar
 Bernard PE, Remacle JF, Legat V (2008b) Modal analysis on unstructured meshes of the dispersion properties of the \(P_1^{NC}P_1\) pair. Ocean Model 28:2–11CrossRefGoogle Scholar
 Blaise S, de Brye B, de Brauwere A, Deleersnijder E, Delhez EJM, Comblen R (2010) Capturing the residence time boundary layer—application to the Scheldt Estuary. Ocean Dyn 60:535–554CrossRefGoogle Scholar
 Bryan K (1969) A numerical method for the study of the circulation of the world ocean. J Comput Phys 4:347–376CrossRefGoogle Scholar
 Casulli V, Walters RA (2000) An unstructured grid, threedimensional model based on the shallow water equations. Int J Numer Methods Fluids 32:331–348CrossRefGoogle Scholar
 Chapman DC, Haidvogel DB (1992) Formation of Taylor caps over a tall isolated seamount in a stratified ocean. Geophys Astrophys Fluid Dynam 64:31–65CrossRefGoogle Scholar
 Chapman DC, Haidvogel DB (1993) Generation of internal lee waves trapped over a tall isolated seamount. Geophys Astrophys Fluid Dynam 69:33–54CrossRefGoogle Scholar
 Chen C, Liu H, Beardsley RC (2003) An unstructured grid, finitevolume, threedimensional, primitive equations ocean model: applications to coastal ocean and estuaries. J Atmos Ocean Technol 20:159–186CrossRefGoogle Scholar
 Chevaugeon N, Hillewaert K, Gallez X, Ploumhans P, Remacle JF (2007) Optimal numerical parametrization of discontinuous Galerkin method applied to wave propagation problems. J Comput Phys 223:188–207CrossRefGoogle Scholar
 Cockburn B, Shu C (1998) The local discontinuous Galerkin finite element method for timedependent convectiondiffusion systems. SIAM J Numer Anal 35:2440–2463CrossRefGoogle Scholar
 Comblen R, Blaise S, Legat V, Remacle JF, Deleersnijder E, Lambrechts J (2010a) A discontinuous finite element baroclinic marine model on unstructured prismatic meshes. Part II: implicit/explicit time discretization. Ocean Dyn doi: 10.1007/s1023601003574
 Comblen R, Lambrechts J, Remacle JF, Legat V (2010b) Practical evaluation of five partdiscontinuous finite element pairs for the nonconservative shallow water equations. Int J Numer Methods Fluids 73:701–724Google Scholar
 Comblen R, Legrand S, Deleersnijder E, Legat V (2009) A finite element method for solving the shallow water equations on the sphere. Ocean Model 28:12–23CrossRefGoogle Scholar
 Cotter CJ, Ham DA, Pain CC (2009a) A mixed discontinuous/continuous finite element pair for shallowwater ocean modelling. Ocean Model 26:86–90CrossRefGoogle Scholar
 Cotter CJ, Ham DA, Pain CC, Sebastian Reich S (2009b) LBB stability of a mixed Galerkin finite element pair for fluid flow simulations. J Comput Phys 228:336–348CrossRefGoogle Scholar
 Dawson C, Aizinger V (2005) A discontinuous Galerkin method for threedimensional shallow water equations. J Sci Comput 22–23:245–267CrossRefGoogle Scholar
 Deleersnijder E, Beckers JM (1992) On the use of the sigmacoordinate in regions of large bathymetric variations. J Mar Syst 3:381–390CrossRefGoogle Scholar
 Deleersnijder E, Lermusiaux PFJ (2008) Multiscale modeling; nestedgrid and unstructuredmesh approaches. Ocean Dyn (special issue) 58:335–498Google Scholar
 Ford R, Pain CC, Piggott M, Goddard A, de Oliveira CR, Umpleby A (2004a) A nonhydrostatic finite element model for threedimensional stratified oceanic flows, part I: model formulation. Mon Weather Rev 132:2816–2831CrossRefGoogle Scholar
 Ford R, Pain CC, Piggott M, Goddard A, de Oliveira CR, Umpleby A (2004b) A nonhydrostatic finite element model for threedimensional stratified oceanic flows, part II: model validation. Mon Weather Rev 132:2832–2844CrossRefGoogle Scholar
 Fringer OB, Gerritsen M, Street RL (2006) An unstructuredgrid, finitevolume, nonhydrostatic, parallel coastal ocean simulator. Ocean Model 14:139–173CrossRefGoogle Scholar
 Giraldo FX (2006) Highorder trianglebased discontinuous Galerkin methods for hyperbolic equations on a rotating sphere. J Comput Phys 214:447–465CrossRefGoogle Scholar
 Griffies SM, Adcroft AJ, Banks H, Böning CW, Chassignet EP, Danabasoglu G, Danilov S, Deleersnijder E, Drange H, England M, FoxKemper B, Gerdes R, Gnanadesikan A, Greatbach RJ, Hallberg RW, Hanert E, Harrison MJ, Legg SA, Little CM, Madec G, Marsland S, Nikurashin M, Piran A, Simmons HL, Schröter J, Treguier BLSAM, Toggweiler JR, Tsujino H, Vallis GK, White L (2009) Problems and prospects in largescale ocean circulation models. OceanObs’09. http://www.oceanobs09.net/blog/?p=88
 Griffies SM, Böning C, Bryan FO, Chassignet EP, Gerdes R, Hasumi H, Hirst A, Treguier AM, Webb D (2000) Developments in ocean climate modeling. Ocean Model 2:123–192CrossRefGoogle Scholar
 Ham DA, Pietrzak J, Stelling GS (2005) A scalable unstructured grid 3dimensional finite volume mode for the shallow water equations. Ocean Model 10:153–169CrossRefGoogle Scholar
 Hanert E, Roux DYL, Legat V, Deleersnijder E (2005) An efficient Eulerian finite element method for the shallow water equations. Ocean Model 10:115–136CrossRefGoogle Scholar
 Haney RL (1991) On the pressure gradient force over steep topography in sigma coordinate ocean models. J Phys Oceanogr 21:610–619CrossRefGoogle Scholar
 Hua BL, Thomasset F (1984) A noisefree finite element scheme for the twolayer shallow water equations. Tellus 36A:157–165CrossRefGoogle Scholar
 Huppert HE, Bryan K (1976) Topographically generated eddies. DeepSea Res 23:655–679Google Scholar
 Kubatko EJ, Westerink JJ, Dawson C (2006) hp discontinuous Galerkin methods for advection dominated problems in shallow water flows. Comput Meth Appl Mech Eng 196:437–451CrossRefGoogle Scholar
 Labeur RJ, Wells GN (2009) Interface stabilised finite element method for moving domains and free surface flows. Comput Meth Appl Mech Eng 198(5–8):615–630CrossRefGoogle Scholar
 Le Roux DY, Rostand V, Pouliot B (2007) Analysis of numerically induced oscillations in 2D finiteelement shallowwater models part I: inertia–gravity waves. SIAM J Sci Comput 29:331–360CrossRefGoogle Scholar
 Le Roux DY, Rostand V, Pouliot B (2008) Analysis of numerically induced oscillations in 2D finiteelement shallowwater models part II: free planetary waves. SIAM J Sci Comput 30:1971–1991CrossRefGoogle Scholar
 Le Roux DY, Staniforth A, Lin CA (1998) Finite elements for shallowwater equation ocean model. Mon Weather Rev 126:1931–1951CrossRefGoogle Scholar
 LeVeque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge texts in applied mathematics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
 Nair RD, Thomas SJ, Loft RD (2005) A discontinuous Galerkin global shallow water model. Mon Weather Rev 133:876–888CrossRefGoogle Scholar
 Pedlosky J (2003) Waves in the ocean and atmosphere. Springer, New YorkGoogle Scholar
 Piggott MD, Gorman GJ, Pain CC, Allison PA, Candy AS, Martin BT, Wells MR (2008) A new computational framework for multiscale ocean modelling basid on adapting unstructured meshes. Int J Numer Methods Fluids 56:1003–1015CrossRefGoogle Scholar
 Redi MH (1982) Oceanic isopycnal mixing by coordinate rotation. J Phys Oceanogr 12:1154–1158CrossRefGoogle Scholar
 Riviere B (2008) Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. Frontiers in mathematics, vol 35. SIAM, PhiladelphiaGoogle Scholar
 Rostand V, Le Roux D (2008) Raviart–Thomas and Brezzi–Douglas–Marini finite element approximations of the shallowwater equations. Int J Numer Methods Fluids 57:951–976CrossRefGoogle Scholar
 Rostand V, Le Roux D, Carey G (2008) Kernel analysis of the discretized finite difference and finite element shallowwater models. SIAM J Sci Comput 31:531–556CrossRefGoogle Scholar
 Shahbazi K (2005) An explicit expression for the penalty parameter of the interior penalty method. J Comput Phys 205:401–407CrossRefGoogle Scholar
 Stuhne GR, Peltier WR (2006) A robust unstructured grid discretization for 3dimensional hydrostatic flows in spherical geometry: a new numerical structure for ocean general circulation modeling. J Comput Phys 213:704–729CrossRefGoogle Scholar
 Timmermann R, Danilov S, Schröter J, Böning C, Sidorenko D, Rollenhagen K (2009) Ocean circulation and sea ice distribution in a finite element global sea ice–ocean model. Ocean Model 27(3–4):114–129CrossRefGoogle Scholar
 Toro E (1997) Riemann solvers and numerical methods for fluid dynamics, a practical introduction. Springer, BerlinGoogle Scholar
 Wang Q, Danilov S, Schröter J (2008a) Comparison of overflow simulations on different vertical grids using the finite element ocean circulation model. Ocean Model 20:313–335CrossRefGoogle Scholar
 Wang Q, Danilov S, Schröter J (2008b) Finite element ocean circulation model based on triangular prismatic elements, with application in studying the effect of topography representation. J Geophys Res 113Google Scholar
 White L, Legat V, Deleersnijder E (2008) Tracer conservation for threedimensional, finiteelement, freesurface, ocean modeling on moving prismatic meshes. Mon Weather Rev 136:420–442CrossRefGoogle Scholar
 Williamson DL, Drake JB, Hack JJ, Jakob R, Swarztrauber PN (1992) A standard test set for numerical approximations to the shallow water equations in spherical geometry. J Comput Phys 102:211–224CrossRefGoogle Scholar