Multiple time scales and pressure forcing in discontinuous Galerkin approximations to layered ocean models

Abstract This paper addresses some issues involving the application of discontinuous Galerkin (DG) methods to ocean circulation models having a generalized vertical coordinate. These issues include the following. (1) Determine the pressure forcing at cell edges, where the dependent variables can be discontinuous. In principle, this could be accomplished by solving a Riemann problem for the full system, but some ideas related to barotropic-baroclinic time splitting can be used to reduce the Riemann problem to a much simpler system of lower dimension. Such splittings were originally developed in order to address the multiple time scales that are present in the system. (2) Adapt the general idea of barotropic-baroclinic splitting to a DG implementation. A significant step is enforcing consistency between the numerical solution of the layer equations and the numerical solution of the vertically-integrated barotropic equations. The method used here has the effect of introducing a type of time filtering into the forcing for the layer equations, which are solved with a long time step. (3) Test these ideas in a model problem involving geostrophic adjustment in a multilayer fluid. In certain situations, the DG formulation can give significantly better results than those obtained with a standard finite difference formulation

A discontinuous Galerkin finite element method for quasi-geostrophic frontogenesis

In this thesis, a mixed continuous and discontinuous Galerkin finite element method is developed for the three-dimensional quasi-geostrophic equations, and is used to investigate the role that weather front formation plays in the transfer of energy to small scales that would produce a k. 5=3 energy spectrum as observed in the atmosphere. The quasi-geostrophic equations are used for computational efficiency and are found to be sufficient for producing simple fronts. Discontinuous Galerkin finite elements are used for the potential vorticity as continuous Galerkin methods perform poorly with advection dominated problems. The less dynamical vertical direction is discretised with finite difference to simplify the finite element method in the horizontal. Streamfunction boundary values are derived for free-slip boundary conditions in the three-dimensional model. The scheme is verified with numerical tests and is shown to converge at optimal rates until free-slip boundaries are introduced. Conservation of energy and enstrophy are shown numerically. Using the numerical method, a channel model simulation suggests that the bend up of fronts produced by a meandering zonal jet could be a viable mechanism for producing a k.5=3 regime

Pressure Forcing and Dispersion Analysis for Discontinuous Galerkin Approximations to Oceanic Fluid Flows

This paper is part of an effort to examine the application of discontinuous Galerkin (DG) methods to the numerical modeling of the general circulation of the ocean. One step performed here is to develop an integral weak formulation of the lateral pressure forcing that is suitable for usage with a DG method and with a generalized vertical coordinate that includes level, terrain-fitted, isopycnic, and hybrid coordinates as examples. This formulation is then tested, in special cases, with analyses of dispersion relations and numerical stability and with some computational experiments. These results suggest that the advantages of DG methods may significantly outweigh their disadvantages, in the settings tested here. This paper also outlines some other issues that need to be addressed in future work.Keywords: Ocean modeling, Multi-layer ocean models, Discontinuous Galerkin method, Shallow water equations, Well-balanced forcingKeywords: Ocean modeling, Multi-layer ocean models, Discontinuous Galerkin method, Shallow water equations, Well-balanced forcin

High-Order Numerical Methods in Lake Modelling

The physical processes in lakes remain only partially understood despite successful data collection from a variety of sources spanning several decades. Although numerical models are already frequently employed to simulate the physics of lakes, especially in the context of water quality management, improved methods are necessary to better capture the wide array of dynamically important physical processes, spanning length scales from ~ 10 km (basin-scale oscillations) - 1 m (short internal waves). In this thesis, high-order numerical methods are explored for specialized model equations of lakes, so that their use can be taken into consideration in the next generation of more sophisticated models that will better capture important small scale features than their present day counterparts. The full three-dimensional incompressible density-stratified Navier-Stokes equations remain too computationally expensive to be solved for situations that involve both complicated geometries and require resolution of features at length-scales spanning four orders of magnitude. The main source of computational expense lay with the requirement of having to solve a three-dimensional Poisson equation for pressure at every time-step. Simplified model equations are thus the only way that numerical lake modelling can be carried out at present time, and progress can be made by seeking intelligent parameterizations as a means of capturing more physics within the framework of such simplified equation sets. In this thesis, we employ the long-accepted practice of sub-dividing the lake into vertical layers of different constant densities as an approximation to continuous vertical stratification. We build on this approach by including weakly non-hydrostatic dispersive correction terms in the model equations in order to parameterize the effects of small vertical accelerations that are often disregarded by operational models. Favouring the inclusion of weakly non-hydrostatic effects over the more popular hydrostatic approximation allows these models to capture the emergence of small-scale internal wave phenomena, such as internal solitary waves and undular bores, that are missed by purely hydrostatic models. The Fourier and Chebyshev pseudospectral methods are employed for these weakly non-hydrostatic layered models in simple idealized lake geometries, e.g., doubly periodic domains, periodic channels, and annular domains, for a set of test problems relevant to lake dynamics since they offer excellent resolution characteristics at minimal memory costs. This feature makes them an excellent benchmark to compare other methods against. The Discontinuous Galerkin Finite Element Method (DG-FEM) is then explored as a mid- to high-order method that can be used in arbitrary lake geometries. The DG-FEM can be interpreted as a domain-decomposition extension of a polynomial pseudospectral method and shares many of the same attractive features, such as fast convergence rates and the ability to resolve small-scale features with a relatively low number of grid points when compared to a low-order method. The DG-FEM is further complemented by certain desirable attributes it shares with the finite volume method, such as the freedom to specify upwind-biased numerical flux functions for advection-dominated flows, the flexibility to deal with complicated geometries, and the notion that each element (or cell) can be regarded as a control volume for conserved fluid quantities. Practical implementation details of the numerical methods used in this thesis are discussed, and the various modelling and methodology choices that have been made in the course of this work are justified as the difficulties that these choices address are revealed to the reader. Theoretical calculations are intermittently carried out throughout the thesis to help improve intuition in situations where numerical methods alone fall short of giving complete explanations of the physical processes under consideration. The utility of the DG-FEM method beyond purely hyperbolic systems is also a recurring theme in this thesis. The DG-FEM method is applied to dispersive shallow water type systems as well as incompressible flow situations. Furthermore, it is employed for eigenvalue problems where orthogonal bases must be constructed from the eigenspaces of elliptic operators. The technique is applied to the problem calculating the free modes of oscillation in rotating basins with irregular geometries where the corresponding linear operator is not self-adjoint

Numerical simulation of flooding from multiple sources using adaptive anisotropic unstructured meshes and machine learning methods

Over the past few decades, urban floods have been gaining more attention due to their increase in frequency. To provide reliable flooding predictions in urban areas, various numerical models have been developed to perform high-resolution flood simulations. However, the use of high-resolution meshes across the whole computational domain causes a high computational burden. In this thesis, a 2D control-volume and finite-element (DCV-FEM) flood model using adaptive unstructured mesh technology has been developed. This adaptive unstructured mesh technique enables meshes to be adapted optimally in time and space in response to the evolving flow features, thus providing sufficient mesh resolution where and when it is required. It has the advantage of capturing the details of local flows and wetting and drying front while reducing the computational cost. Complex topographic features are represented accurately during the flooding process. This adaptive unstructured mesh technique can dynamically modify (both, coarsening and refining the mesh) and adapt the mesh to achieve a desired precision, thus better capturing transient and complex flow dynamics as the flow evolves. A flooding event that happened in 2002 in Glasgow, Scotland, United Kingdom has been simulated to demonstrate the capability of the adaptive unstructured mesh flooding model. The simulations have been performed using both fixed and adaptive unstructured meshes, and then results have been compared with those published 2D and 3D results. The presented method shows that the 2D adaptive mesh model provides accurate results while having a low computational cost. The above adaptive mesh flooding model (named as Floodity) has been further developed by introducing (1) an anisotropic dynamic mesh optimization technique (anisotropic-DMO); (2) multiple flooding sources (extreme rainfall and sea-level events); and (3) a unique combination of anisotropic-DMO and high-resolution Digital Terrain Model (DTM) data. It has been applied to a densely urbanized area within Greve, Denmark. Results from MIKE 21 FM are utilized to validate our model. To assess uncertainties in model predictions, sensitivity of flooding results to extreme sea levels, rainfall and mesh resolution has been undertaken. The use of anisotropic-DMO enables us to capture high resolution topographic features (buildings, rivers and streets) only where and when is needed, thus providing improved accurate flooding prediction while reducing the computational cost. It also allows us to better capture the evolving flow features (wetting-drying fronts). To provide real-time spatio-temporal flood predictions, an integrated long short-term memory (LSTM) and reduced order model (ROM) framework has been developed. This integrated LSTM-ROM has the capability of representing the spatio-temporal distribution of floods since it takes advantage of both ROM and LSTM. To reduce the dimensional size of large spatial datasets in LSTM, the proper orthogonal decomposition (POD) and singular value decomposition (SVD) approaches are introduced. The performance of the LSTM-ROM developed here has been evaluated using Okushiri tsunami as test cases. The results obtained from the LSTM-ROM have been compared with those from the full model (Fluidity). Promising results indicate that the use of LSTM-ROM can provide the flood prediction in seconds, enabling us to provide real-time flood prediction and inform the public in a timely manner, reducing injuries and fatalities. Additionally, data-driven optimal sensing for reconstruction (DOSR) and data assimilation (DA) have been further introduced to LSTM-ROM. This linkage between modelling and experimental data/observations allows us to minimize model errors and determine uncertainties, thus improving the accuracy of modelling. It should be noting that after we introduced the DA approach, the prediction errors are significantly reduced at time levels when an assimilation procedure is conducted, which illustrates the ability of DOSR-LSTM-DA to significantly improve the model performance. By using DOSR-LSTM-DA, the predictive horizon can be extended by 3 times of the initial horizon. More importantly, the online CPU cost of using DOSR-LSTM-DA is only 1/3 of the cost required by running the full model.Open Acces

Subgrid scale modelling of transport processes.

Consideration of stabilisation techniques is essential in the development of physical models if they are to faithfully represent processes over a wide range of scales. Careful application of these techniques can significantly increase flexibility of models, allowing the computational meshes used to discretise the underlying partial differential equations to become highly nonuniform and anisotropic, for example. This exibility enables a model to capture a wider range of phenomena and thus reduce the number of parameterisations required, bringing a physically more realistic solution. The next generation of fluid flow and radiation transport models employ unstructured meshes and anisotropic adaptive methods to gain a greater degree of flexibility. However these can introduce erroneous artefacts into the solution when, for example, a process becomes unresolvable due to an adaptive mesh change or advection into a coarser region of mesh in the domain. The suppression of these effects, caused by spatial and temporal variations in mesh size, is one of the key roles stabilisation can play. This thesis introduces new explicit and implicit stabilisation methods that have been developed for application in fluid and radiation transport modelling. With a focus on a consistent residual-free approach, two new frameworks for the development of implicit methods are presented. The first generates a family of higher-order Petrov-Galerkin methods, and the example developed is compared to standard schemes such as streamline upwind Petrov-Galerkin and Galerkin least squares in accurate modelling of tracer transport. The dissipation generated by this method forms the basis for a new explicit fourth-order subfilter scale eddy viscosity model for large eddy simulation. Dissipation focused more sharply on unresolved scales is shown to give improved results over standard turbulence models. The second, the inner element method, is derived from subgrid scale modelling concepts and, like the variational multiscale method and bubble enrichment techniques, explicitly aims to capture the important under-resolved fine scale information. It brings key advantages to the solution of the Navier-Stokes equations including the use of usually unstable velocity-pressure element pairs, a fully consistent mass matrix without the increase in degrees of freedom associated with discontinuous Galerkin methods and also avoids pressure filtering. All of which act to increase the flexibility and accuracy of a model. Supporting results are presented from an application of the methods to a wide range of problems, from simple one-dimensional examples to tracer and momentum transport in simulations such as the idealised Stommel gyre, the lid-driven cavity, lock-exchange, gravity current and backward-facing step. Significant accuracy improvements are demonstrated in challenging radiation transport benchmarks, such as advection across void regions, the scattering Maynard problem and demanding source-absorption cases. Evolution of a free surface is also investigated in the sloshing tank, transport of an equatorial Rossby soliton, wave propagation on an aquaplanet and tidal simulation of the Mediterranean Sea and global ocean. In combination with adaptive methods, stabilising techniques are key to the development of next generation models. In particular these ideas are critical in achieving the aim of extending models, such as the Imperial College Ocean Model, to the global scale