Search

Highlights • A multi-layer non-hydrostatic coastal ocean model is developed using the discontinuous Galerkin finite element method. • The model is shown to predict nearshore dispersive wave processes accurately using only a small number of vertical layers. • Use of arbitrary unstructured meshes allows for non-uniform resolution and accurate representation of complex geometries. • Automated code generation techniques are employed promoting code efficiency, readability, extensibility and maintainability. Abstract A multi-layer non-hydrostatic version of the unstructured mesh, discontinuous Galerkin finite element based coastal ocean model, Thetis, is developed. This is accomplished using the PDE solver framework, Firedrake, which is used to automatically produce the code for the discretised model equations in a rapid and efficient manner. The motivation for this work is a need to accurately simulate dispersive nearshore free surface processes. In order to resolve both frequency dispersion and non-linear effects accurately, additional non-hydrostatic terms are included in the layer-integrated hydrostatic equations, producing a form similar to the layered non-linear shallow water equations, but with extra vertical velocities at the layer interfaces. An implementation process is adopted to easily handle the inter-layer connection, i.e. the governing equations are transformed into a depth-integrated system through the introduction of depth-averaged variables. The model is verified and validated through comparisons against several idealised and experimentally-based test cases. All the comparisons demonstrate good agreement, showing that the developed non-hydrostatic model has excellent capabilities in representing coastal wave phenomena including shoaling, refraction and diffraction of dispersive short waves, as well as propagation, run-up and inundation of non-linear tsunami waves. [ABSTRACT FROM AUTHOR]

Numerical storm surge models are essential to forecasting coastal flood hazard and informing the design of coastal defences. However, such models rely on a variety of inputs, many of which carry uncertainty. An awareness and understanding of the sensitivity of model outputs with respect to those uncertain inputs is therefore essential when interpreting model results. Here, we use an unstructured-mesh numerical coastal ocean model, Thetis , and its adjoint, to perform a sensitivity analysis for a hindcast of the 5th/6th December 2013 North Sea surge event, with respect to the bottom friction coefficient, bathymetry and wind stress forcing. The results reveal spatial and temporal patterns of sensitivity, providing physical insight into the mechanisms of surge generation and propagation. For example, the sensitivity of the skew surge to the bathymetry reveals the protective effect of a sand bank off the UK east coast. The results can also be used to propagate uncertainties through the numerical model; based on estimates of model input uncertainties, we estimate that modelled skew surges carry uncertainties of around 5 cm and 15 cm due to bathymetry and bottom friction, respectively. While these uncertainties are small compared with the typical spread in an ensemble storm surge forecast due to uncertain meteorological inputs, the adjoint-derived model sensitivities can nevertheless be used to inform future model calibration and data acquisition efforts in order to reduce uncertainty. Our results demonstrate the power of adjoint methods to gain insight into a storm surge model, providing information complementary to traditional ensemble uncertainty quantification methods. • We calibrate an unstructured-mesh finite element storm surge model of the North Sea • We use an adjoint approach to perform sensitivity analysis for modelled skew surges • We compare spatial and temporal patterns of sensitivity with respect to three inputs • We demonstrate the physical insight available through numerical adjoint methods. [ABSTRACT FROM AUTHOR]

A σ -coordinate non-hydrostatic coastal ocean model is developed using the discontinuous Galerkin finite element method. With the selection of the low-order piecewise-constant P 0 DG and piecewise-linear P 1 DG discretisations in the vertical for the velocity and pressure fields, respectively, the proposed σ -coordinate model can naturally retain the wave dispersion characteristics of the widely-adopted multi-layer approach of Zijlema and Stelling (2005), which is demonstrated through both mathematical derivation and numerical tests. Under the finite element approach, higher-order vertical discretisation choices can also be readily made which can reduce the number of vertical layers required for the accurate representation of wave dispersion. The model is verified and validated through comparisons against a series of test cases with analytical solutions or experimental measurements. All the comparisons demonstrate good agreement, indicating that the proposed model can accurately represent dispersive barotropic surface waves with as few as one vertical layer, and can simulate baroclinic internal waves with reasonable accuracy using relatively coarse mesh resolution. It is also demonstrated that consistency in the coupling of barotropic and baroclinic flows can be properly ensured. • A σ -coordinate non-hydrostatic model is developed using the discontinuous Galerkin method. • The model can accurately represent dispersive barotropic surface waves with as few as one vertical layer. • The model can simulate baroclinic dynamics with reasonable accuracy using relatively coarse mesh resolution. • Theoretical links between the σ -formulation, the multi-layer approach and the ALE technique are established. [ABSTRACT FROM AUTHOR]

Numerical studies of ice flow have consistently identified the grounding zone of outlet glaciers and ice streams (the region where ice starts to float) as crucial for predicting the rate of grounded ice loss to the ocean. Owing to the extreme environments and difficulty of access to ocean cavities beneath ice shelves, field observations are rare. Estimates of melt rates derived from satellites are also difficult to make near grounding zones with confidence. Therefore, numerical ocean models are important tools to investigate these critical and remote regions. The relative inflexibility of structured grid models means, however, that they can struggle to resolve these processes in irregular cavity geometries near grounding zones. To help solve this issue, we present a new nonhydrostatic unstructured mesh model for flow under ice shelves built using the Firedrake finite element framework. We demonstrate our ability to simulate full ice shelf cavity domains using the community standard ISOMIP+ Ocean0 test case and compare our results against those obtained with the popular MITgcm model. Good agreement is found between the two models, despite their use of different discretisation schemes and the sensitivity of the melt rate parameterisation to grid resolution. Verification tests based on the Method of Manufactured Solutions (MMS) show that the new model discretisation is sound and second-order accurate. A main driver behind using Firedrake is the availability of an automatically generated adjoint model. Our first adjoint calculations, of sensitivities of melt rate with respect to different inputs in an idealised grounding zone domain, are promising and point to the ability to address a number of important questions on ocean influence on ice shelf vulnerability in the future. • New model for ocean flow beneath ice shelves on flexible unstructured meshes. • 2nd-order accurate discretisation verified by Method of Manufactured Solution test. • Good agreement with MITgcm is found for the ISOMIP+ 3D test case. • Preliminary adjoint results show significant potential for ice-ocean applications. [ABSTRACT FROM AUTHOR]

Tsunami generated by submarine slides are arguably an under-considered risk in comparison to earthquake-generated tsunami. Numerical simulations of submarine slide-generated waves can be used to identify the important factors in determining wave characteristics. Here we use Fluidity, an open source finite element code, to simulate waves generated by deformable submarine slides. Fluidity uses flexible unstructured meshes combined with adaptivity which alters the mesh topology and resolution based on the simulation state, focussing or reducing resolution, when and where it is required. Fluidity also allows a number of different numerical approaches to be taken to simulate submarine slide deformation, free-surface representation, and wave generation within the same numerical framework. In this work we use a multi-material approach, considering either two materials (slide and water with a free surface) or three materials (slide, water and air), as well as a sediment model (sediment, water and free surface) approach. In all cases the slide is treated as a viscous fluid. Our results are shown to be consistent with laboratory experiments using a deformable submarine slide, and demonstrate good agreement when compared with other numerical models. The three different approaches for simulating submarine slide dynamics and tsunami wave generation produce similar waveforms and slide deformation geometries. However, each has its own merits depending on the application. Mesh adaptivity is shown to be able to reduce the computational cost without compromising the accuracy of results. [ABSTRACT FROM AUTHOR]

The ∼8.15 ka Storegga submarine slide was a large (∼3000 km 3 ), tsunamigenic slide off the coast of Norway. The resulting tsunami had run-up heights of around 10–20 m on the Norwegian coast, over 12 m in Shetland, 3–6 m on the Scottish mainland coast and reached as far as Greenland. Accurate numerical simulations of Storegga require high spatial resolution near the coasts, particularly near tsunami run-up observations, and also in the slide region. However, as the computational domain must span the whole of the Norwegian-Greenland sea, employing uniformly high spatial resolution is computationally prohibitive. To overcome this problem, we present a multiscale numerical model of the Storegga slide-generated tsunami where spatial resolution varies from 500 m to 50 km across the entire Norwegian-Greenland sea domain to optimally resolve the slide region, important coastlines and bathymetric changes. We compare results from our multiscale model to previous results using constant-resolution models and show that accounting for changes in bathymetry since 8.15 ka, neglected in previous numerical studies of the Storegga slide-tsunami, improves the agreement between the model and inferred run-up heights in specific locations, especially in the Shetlands, where maximum run-up height increased from 8 m (modern bathymetry) to 13 m (palaeobathymetry). By tracking the Storegga tsunami as far south as the southern North sea, we also found that wave heights were high enough to inundate Doggerland, an island in the southern North Sea prior to sea level rise over the last 8 ka. [ABSTRACT FROM AUTHOR]

Highlights: [•] Ocean circulation beneath ice shelves is simulated using an unstructured-mesh. [•] We represent a smoothly varying ice base in the presence of a vertical ice front. [•] We implemented ice shelf/ocean interaction in the context of finite-element method. [Copyright &y& Elsevier]

Large tsunamis can be generated by submarine slides, but these events are rare on human timescales and challenging to observe. Experiments and numerical modelling offer methods to understand the mechanisms by which they generate waves and what the potential hazard might be. However, to fully capture the complex waveform generated by a submarine slide, the slide dynamics must also be accurately modelled. It is computationally difficult to model both a three-dimensional submarine slide whilst simultaneously simulating oceanic-scale tsunamis. Past studies have either coupled localised models of the slide generation to oceanic-scale tsunami simulations or simplified the slide dynamics. Here, we present a new methodology of model coupling that generates the wave in the ocean-scale model via boundary-condition coupling of a two-dimensional dynamic slide simulation. We verify our coupling methodology by comparing model results to a previous simulation of a tsunamigenic slide in the Gulf of Mexico. We then examine the effect of slide deformation on the risk posed by hypothetical submarine slides around the UK. We show the deformable submarine slide simulations produce larger waves than the solid slide simulations due to the details of acceleration and velocity of the slide, although lateral spreading is not modelled. This work offers a new methodology for simulating oceanic-scale tsunamis caused by submarine slides using the output of a two-dimensional, multi-material simulation as input into a three-dimensional ocean model. This facilitates future exploration of the tsunami risk posed by tsunamigenic submarine slides that affect coastlines not normally prone to tsunamis. • Numerical modelling framework simulates tsunami waves generated by deformable submarine slides. • Predicted waves agree with numerical model for waves generated by submarine slide in Gulf of Mexico. • Used methodology to simulate waves from hypothetical submarine slides around the UK. • We show how slide speed, acceleration, deformation and direction affect the wave. [ABSTRACT FROM AUTHOR]

The unstructured mesh, discontinuous Galerkin finite element discretisation based coastal ocean model, Thetis , has been extended to include non-hydrostatic (buoyancy-driven and free surface) dynamics. Two alternative approaches to achieve this are described in this work. The first (a 3D based algorithm) makes use of prismatic element based meshes and uses a split-step pressure projection method for baroclinic and barotropic modes, while the second (a 2D based algorithm) adopts a novel multi-layer approach to convert a 3D problem into a combination of multiple 2D computations with only 2D triangle meshes required. Model development is carried out at high-level with the Firedrake library, using code generation techniques to automatically produce low-level code for the discretised model equations in an efficient and rapid manner. Through comparisons against several barotropic/baroclinic test cases where non-hydrostatic effects are important, the implemented approaches are verified and validated, and the proposed algorithms compared. Depending on whether the problems are dominated by dispersive, baroclinic or barotropic features, recommendation are given over the use of full 3D or multi-layer 2D based approaches to achieve optimal computational accuracy and efficiency. It is demonstrated that while in general the 2D approach is well-suited for barotropic problems and dispersive free surface waves, the 3D approach is more advantageous for simulating baroclinic buoyancy-driven flows due in part to the high vertical resolution typically required to represent the active tracer fields. • A non-hydrostatic discontinuous finite element coastal ocean model is presented. • Two non-hydrostatic extensions are compared showing respective advantages. • Both algorithms are validated using several barotropic/baroclinic test cases. • The model development is carried out using a code generation framework, Firedrake. • The model is able to provide accurate reproduction of surface and exchange flows. [ABSTRACT FROM AUTHOR]