A baroclinic discontinuous Galerkin finite element model for coastal flows

Numerical modelling of coastal flows is a challenging topic due to complex topography of the coastal zone, rapid flow dynamics and large density variations. Such phenomena are best simulated with unstructured grid models due to their highly flexible spatial discretisation. This article presents a three-dimensional discontinuous Galerkin finite element marine model. Discontinuous Galerkin spatial discretisation is combined with an explicit mode splitting time integration scheme. Slope limiters are introduced to guarantee high quality of the tracer fields in the presence of strong gradients. Free surface movement is accounted for by means of an Arbitrary Lagrangian Eulerian (ALE) moving mesh method. Water volume and tracers are conserved. The conservation properties and baroclinic adjustment under gravity are tested with numerical benchmarks. Finally, the model is applied to the Rhine river plume in an idealised setting.     

Accurate representation of geostrophic and hydrostatic balance in unstructured mesh finite element ocean modelling

Accurate representation of geostrophic and hydrostatic balance is an essential requirement for numerical modelling of geophysical flows. Potentially, unstructured mesh numerical methods offer significant benefits over conventional structured meshes, including the ability to conform to arbitrary bounding topography in a natural manner and the ability to apply dynamic mesh adaptivity. However, there is a need to develop robust schemes with accurate representation of physical balance on arbitrary unstructured meshes. We discuss the origin of physical balance errors in a finite element discretisation of the Navier–Stokes equations using the fractional timestep pressure projection method. By considering the Helmholtz decomposition of forcing terms in the momentum equation, it is shown that the components of the buoyancy and Coriolis accelerations that project onto the non-divergent velocity tendency are the small residuals between two terms of comparable magnitude. Hence there is a potential for significant injection of imbalance by a numerical method that does not compute these residuals accurately. This observation is used to motivate a balanced pressure decomposition method whereby an additional “balanced pressure” field, associated with buoyancy and Coriolis accelerations, is solved for at increased accuracy and used to precondition the solution for the dynamical pressure. The utility of this approach is quantified in a fully non-linear system in exact geostrophic balance. The approach is further tested via quantitative comparison of unstructured mesh simulations of the thermally driven rotating annulus against laboratory data. Using a piecewise linear discretisation for velocity and pressure (a stabilised P₁P₁ discretisation), it is demonstrated that the balanced pressure decomposition method is required for a physically realistic representation of the system.     

Semi-Lagrangian methods for a finite element coastal ocean model

Coastal ocean hydrodynamic models are subject to a number of stability constraints. The most important of these are the Courant–Friedrichs–Levy (CFL) constraint on gravity waves, a Courant (Cr) number constraint on advection, and a time step constraint on the vertical component of viscous stresses. The model described here removes these constraints using a semi-implicit approximation in time and a semi-Lagrangian approximation for advection. The accuracy and efficiency of semi-Lagrangian methods depends crucially on the methods used to calculate trajectories and interpolate at the foot of the trajectory. The focus of this paper is on evaluation of several new and old semi-Langrangian methods. In particular, we compare 3 methods to calculate trajectories (Runge–Kutta (RK2), analytical integration (AN), power-series expansion (PS)) and 3 methods to interpolate (local linear (LL), global linear (GL), global quadratic (GQ)) on unstructured grids. The AN and PS methods are both efficient and accurate, and the latter can be expanded in a straightforward manner to treat time-dependent velocity. The GQ interpolation method provides a major step in introducing efficient and accurate semi-Lagrangian methods to unstructured grids.     

A non-hydrostatic algorithm for free-surface ocean modelling

An original implementation of a non-hydrostatic, free-surface algorithm based on a pressure correction method is proposed for ocean modelling. The free surface is implemented through an explicit scheme combined with a mode-spitting method but the depth-averaged velocity and the position of the free surface are updated at each non-hydrostatic iteration. The vertical momentum equation is also integrated up to the surface enabling a natural and accurate treatment of the surface layer. The consistent specification of the numerical schemes provides balanced transfers of potential and kinetic energy. This algorithm is well-suited for implementation as a non-hydrostatic kernel on originally hydrostatic free-surface ocean models such as Symphonie (http://poc.obs-mip.fr/pages/research_topics/modelling/symphonie/symphonie.htm) for which it has originally been developed.Energy balances associated with the propagation of short surface waves and solitary waves are presented for two dedicated well-documented configurations over closed domains. The buoyancy flux, the work rate of the pressure force together with the power of the advective terms are evaluated and discussed for the generation and the propagation of these two types of waves. The dissipation rate is in particular shown to be several orders of magnitude smaller than the work rates of the hydrostatic and non-hydrostatic pressure forces confirming the necessity for the exchanges of energy to be numerically balanced. The algorithm is subsequently applied to the complex generation of non-linear solitary internal waves by surface tides over Georges Bank, in the Gulf of Maine. The generation and the propagation of the observed non-linear and non-hydrostatic features in this region are correctly reproduced.     

A tree-based solver for adaptive ocean modelling

The development of an adaptive (in space and time) ocean model from an existing adaptive finite-volume Navier–Stokes model is described. A flexible and efficient quadtree spatial discretisation is used which requires collocation of all variables (i.e. an A-grid discretisation). We demonstrate that the use of an approximate projection method allows for implicit damping of instabilities generally associated with the A-grid, at the expense of a relatively small amount of numerical energy dissipation, while accurately preserving dispersive properties and geostrophic balance. The finite-volume formulation also maintains second-order spatial accuracy at all solid boundaries. Test cases demonstrate the efficacy of the adaptive ocean model, and the advantages it has in terms of efficient representation of multi-scale behaviour within a single model. The model is freely available as open-source code.     

Boussinesq-type modelling using an unstructured finite element technique

A model for solving the two-dimensional enhanced Boussinesq equations is presented. The model equations are discretised in space using an unstructured finite element technique. The standard Galerkin method with mixed interpolation is applied. The time discretisation is performed using an explicit three-step Taylor–Galerkin method. The model is extended to the surf and swash zone by inclusion of wave breaking and a moving boundary at the shoreline. Breaking is treated by an existing surface roller model, but a new procedure for the detection of the roller thickness is devised. The model is verified using four test cases and the results are compared with experimental data and results from an existing finite difference Boussinesq model.     

Application of a coupled discontinuous–continuous Galerkin finite element shallow water model to coastal ocean dynamics

A coupled discontinuous–continuous Galerkin (DG–CG) shallow water model is compared to a continuous Galerkin generalized wave-continuity equation (GWCE) based model for the coastal ocean, whereby local mass imbalance typical of GWCE-based solutions is eliminated using the coupled DG–CG approach. Two mass imbalance indicators for the GWCE-based model are presented and analyzed. The indicators motivate discussion on the suitability of using a GWCE-based model versus the locally conservative coupled DG–CG model. Both realistic and idealized test problems for tide, wind, and wave-driven circulation form the basis of the study. For the problems studied, coupled DG–CG solutions retain the robustness of well-documented solutions from GWCE-based models and also capture the dynamics driven by small-scale, highly advective processes which are problematic for GWCE-based models. Issues associated with the coupled DG–CG model are explored, including increased cost due to increased degrees of freedom, the necessary application of slope limiters, as well as the actual coupling process.     

Analysis of stiffened shell for ships and ocean structures by finite element method

Static analysis of stiffened shells has been carried out using an eight-noded isoparametric element for the shell and a three-noded curved beam element for the stiffener. A same displacement function is used for the shell and the stiffener elements. A modified technique has been followed to analyse the shell, which is an improvement over the degenerated shell concept. The stiffness matrix of the curved beam element is generated irrespective of its position and orientation within the shell element. The stiffness matrix of the stiffener is then transferred to all the nodes of the shell element. Numerical examples of stiffened shells with concentric and eccentric stiffeners have been analysed and the results presented together with those available in published literature.     

A model/data comparison for shallow-water reverberation

A bottom-scattering model based on sediment small scatterers, single scattering approximation is presented, which is combined with a normal-mode-based reverberation model. The combined model (for total reverberation) is compared with measurements of shallow-water reverberation from the 2001 Asian Sea International Acoustic Experiment (ASIAEX) in the East China Sea. Reverberation intensities as a function of time and frequency are compared with theoretical predictions with reasonable agreement. The effects of the rough sea surface on the reverberation are also discussed.     

A multilevel method for finite volume discretization of the two-dimensional nonlinear shallow-water equations

In this article we propose and implement a multilevel method to simulate the solution of the two-dimensional nonlinear shallow-water equations. The multilevel method is based on a central-upwind finite volume scheme and uses new incremental unknowns which enable to preserve the numerical conservation of the scheme. The method is tested and analyzed on two and three levels of discretization on different test cases and turns out to furnish a good solution of the problems while saving CPU time.     

A finite element model for wave refraction and diffraction

A two-dimensional hybrid finite element method is developed to study the scattering of water waves by an island and to calculate wave forces and moments on offshore structures. The offshore structure, which could be either semi-submerged or fully extended in the water, is assumed to be stationary. The numerical model is based on the mild-slope equation. It can be applied to both long-wave and short-wave problems. A special treatment for the problem with the semi-submerged structure is introduced. Comparisons are given with existing analytical solutions and other numerical results. The present model is shown to be an efficient and accurate method for the solution of wave refraction and diffraction problems.     

A three-dimensional semi-implicit unstructured grid finite volume ocean model

A new three-dimensional semi-implicit finite-volume ocean model has been developed for simulating the coastal ocean circulation, which is based on the staggered C -unstructured non-orthogonal grid in the horizontal direction and z -level grid in the vertical direction. The three-dimensional model is discretized by the semi-implicit finite-volume method, in that the free-surface and the vertical diffusion are semi-implicit, thereby removing stability limitations associated with the surface gravity wave and vertical diffusion terms. The remaining terms in the momentum equations are discretized explicitly by an integral method. The partial cell method is used for resolving topography, which enables the model to better represent irregular topography. The model has been tested against analytical cases for wind and tidal oscillation circulation, and is applied to simulating the tidal flow in the Bohai Sea. The results are in good agreement both with the analytical solutions and measurement results.     

A coastal ocean model of semi-implicit finite volume unstructured grid

A two-dimensional coastal ocean model based on unstructured C-grid is built, in which the momentum equation is discretized on the faces of each cell, and the continuity equation is discretized on the cell. The model is discretized by semi-implicit finite volume method, in that the free surface is semi-implicit and the bottom friction is implicit, thereby removing stability limitations associated with the surface gravity wave and friction. The remaining terms in the momentum equations are discretized explicitly by integral finite volume method and second-order Adams-Bashforth method. Tidal flow in the polar quadrant with known analytic solution is employed to test the proposed model. Finally, the performance of the present model to simulate tidal flow in a geometrically complex domain is examined by simulation of tidal currents in the Pearl River Estuary.     

解浅水方程的集中质量有限元方法

本文给出三角形单元集中质量有限元解浅水波方程的方法以及该方法在IBM-PC微机上的应用实例并取得较好结果。     

An alternative coordinate system for solving finite difference ocean models

An alternative ‘compressed’ coordinate system for representing finite difference grids in ocean models is presented and compared with the traditional Cartesian method. The alternative method represents any arbitrary three-dimensional domain as a one-dimensional vector. Advantages of using this method include the exclusion of all land ‘dry’ points from the computational grid, leading to savings in the memory used and increased execution speed where domains have a low ratio of wet to total computational cells. The Cartesian and compressed models are applied to a variety of domains to assess the performance of each in terms of memory use and computational speed.     

Parameterization of ocean wave-induced mixing processes for finite water depth

Three dimensional wave-induced mixing plays an important role in shallow water area. A quite direct approach through the Reynolds average upon characteristic length scale is proposed to parameterize the horizontal and vertical shallow water mixing. Comparison of finite depth case with infinite depth results indicates that the difference of the wave-induced mixing strength is evident. In the shallow water condition, the infinite water depth approximation overestimates the mixing strength in the lower layers. The nonzero horizontal wave-induced mixing presents anisotropic property near the shore. The Prandtl’s mixing length theory underestimated the wave-induced mixing in the previous studies.     

A diapycnal diffusion algorithm for isopycnal ocean circulation models with special application to mixed layers

An algorithm is presented for solving the one-dimensional diffusion equation for density, written in terms of density (or a like surrogate) as the independent variable. The algorithm maintains nonnegative layer thicknesses, the premise of the transformation to density as the independent coordinate, under certain restrictions. Near-zero thickness layers can be maintained at the boundaries to accommodate future inflation in response to heating from the boundary. Layers can shrink to near-zero thickness in response to cooling from the boundary. A slight modification of the algorithm permits layers to have diffusion coefficients which differ by orders of magnitude. This provides a natural framework for a surface mixed layer in an isopycnal model, in which the mixed layer is distinguished as a zone of very high turbulent diffusivity overlying an ocean interior of much smaller turbulent diffusivity. The “mixed layer” may be an aggregation of several isopycnal layers rather than just one. A substantial jump in density at the mixed layer base can be represented by several near-zero thickness isopycnal layers. The specification of the thickness of the mixing zone, i.e., the mixed layer depth, is external to the algorithm. An illustration is given using a Kraus–Turner-type specification.     

A three-dimensional hydrostatic model for coastal and ocean modelling using a generalised topography following co-ordinate system

A three-dimensional hydrostatic model is presented that combines a generalised vertical co-ordinate system with an efficient implicit solution technique for the free surface. The model is capable of maintaining high resolution in the surface and/or bottom boundary layers as well as dealing with steep topography. Horizontal diffusion is calculated using the Smagorinsky formulation and a k–ε turbulence model is used in the vertical. In addition the model uses higher-order advection routines. An important aspect in three-dimensional models is the choice of vertical discretisation. If one is mostly interested in problems which are governed by boundary layer flows, a terrain following or sigma co-ordinate system seems attractive. This paper focuses on the development of a generalised sigma-type grid in a three-dimensional hydrostatic model. The generalised grid offers a wide range of possibilities including grid refinement toward the bed or surface, a mixed layer transformation, and a constant layer transformation where the lowermost or uppermost grid cells can be specified to have a constant height above the bed or below the surface. A number of tests are presented which show that the model is capable of simulating both shallow nearshore, estuarine flows as well as large-scale geophysical flows. These include an extreme flooding event in the shallow North Sea and the Odden ice tongue formation in the Greenland Sea.     

利用有限元方法求解双曲型缓坡方程

本文提出了一种双曲型缓坡方程的有限元计算方法 ,在建立有限元积分方程时通过在造波线处加入脉动源项来实现内部造波 ,并在开边界处利用阻尼层吸波 ,减少了在边界处由于数值处理引起的误差。数值计算结果与实测值吻合良好。本方法可用于大区域波浪场的计算中     

Non-hydrostatic finite element model for coastal wave processes

This paper presents the application of the depth-integrated non-hydrostatic finite element model, CCHE2D-NHWAVE (Wei and Jia, 2014), for simulating several types of coastal wave processes. Specifically, the model is applied to (1) predict the swash zone hydrodynamics involving wave bore propagation, (2) resolve wave propagation, breaking, and overtopping in fringing reef environments, (3) study the vegetation effect on wave height reduction through both submerged and emergent vegetation zones using the drag force term technique, and (4) simulate tsunami wave breaking in the nearshore zone and inundation in the coastal area. Satisfactory agreement between numerical results and benchmark data shows that the non-hydrostatic model is capable of modeling a wide range of coastal wave processes. Furthermore, thanks to its simple numerical formulation, the non-hydrostatic model also demonstrates a better computation efficiency when comparing with other numerical models.