Shallow Water Equations Model Research Articles

Global tsunami modelling on a spherical multiple-cell grid

A model of shallow water equations (SWEs) on a spherical multiple-cell (SMC) grid of 4-level (2.5–5–10–20 km) spatial resolutions is used to simulate tsunami propagation on global ocean surfaces. The unstructured SMC grid retains rectangular cells of the longitude-latitude grid so that efficient finite-difference schemes could be used. It also supports multi-resolutions like mesh refinement to resolve small islands and coastline details while keeping models compact enough to fit into available computers. Two earthquake-induced tsunami cases are simulated and compared with available observations. Results indicate that the modelled tsunami arrival times agree well with observations while tsunami wave heights are underestimated, particularly the observed runups on remote coastal lands. Possible reasons for this underestimation include the smoothing scheme used to suppress numerical oscillations and the missing of initial kinetic energy input from the earthquakes. Another reason is the limitation of the SWEs to describe coastal bores and breaking waves in coastal waters. A possible tsunami scenario induced by a landslide in the Canary Islands is also simulated to assess its potential impact on Atlantic coastal regions. Model results indicate that this kind of tsunami may cause severe damage to local areas but its effects on far fields, like the UK and American coastal regions are small. As the initial landslide disturbance is overly simplified, this study does not give a true representation of a real landslide tsunami but rather a qualitative assessment of its impact on the Atlantic Ocean. More realistic initial condition and improved model representation of coastal processes are needed for further studies of this possible landslide hazard.

Compartmentalization strategy for coastal flooding mitigation with application to a Northern Adriatic site

A method of compartmentalization has been developed to address the safety concerns of urban areas from coastal flooding, especially in situations where reinforcing the first line of defence (dunes, levees, etc.) is challenging due to, for instance, environmental limitations. Various compartmentalization options can be considered and evaluated in order to govern the flooding. Each option should be modelled with a suited numerical tool to better define the alternatives, determine the minimum height of the elements that bound the compartments and verify their effectiveness for the preliminary design, e.g. for the environmental screening stage. The methodology is demonstrated for a real case study, the flooding of Lido di Volano that occurred on November 22, 2022, using a simplified shallow water equations model designed for GPU computing to simulate large-scale flooding scenarios. Two compartmentalization schemes have been identified, both of which involve raising the existing road level. The maximum water levels in different zones of the study area are assessed as a function of crest height. These schemes have proven to be effective in protecting the urban settlement from flooding. Overall, the proposed method of compartmentalization, supported by a real case study analysis, offers a systematic approach to enhance the safety and the resilience of urban coastal areas. This type of solution capitalises on the experience of the local managers and technicians and may represent a shared solution among the coastal governance actors.

A novel framework of the lattice Boltzmann model for multilayer shallow water systems

This study proposes a novel framework of the lattice Boltzmann model for multilayer shallow water equations, considering the mass and momentum exchanges between layers (LABMSWE+). Compared with the original LABMSWE model consisting of N two-dimensional lattice Boltzmann method for shallow water equation (LABSWE) models, the new model includes 1+N LABSWE models. The singular LABSWE model with unit relaxation time is introduced to update the total water depth, and thus, the layer water depths can be obtained explicitly through the fixed layer ratios. The N-layer LABSWE models with the multiple-relaxation-time operator evolve the layer velocities. These two modules are coupled by the total water depth and depth-averaged velocities. The constructed model avoids the freely variable layer thicknesses, which is considered as the main source of the instability. In addition, the mass exchanges enable this model to simulate vertical circulation flows, which are beyond the application of the LABMSWE model. Several numerical tests are then conducted to validate the proposed model. The results show that it exactly satisfies the C-property. In addition, the central difference scheme is more stable and accurate than the upwind and nonequilibrium schemes in the computing of the mass exchanges. The numerical results have an excellent agreement with analytical solutions and reference data, while some unstable and nonphysical results are obtained by the original LABMSWE model. Moreover, the computational time is about 40%–60% of that for the MIKE3, a finite volume solver for the three-dimensional shallow water equations by the Danish Hydraulic Institute.

A Global C-Staggered Composite Model for Shallow Water Equations with Latitude–Longitude Grid and Reductions in the Polar Regions

To develop a numerical method for global geophysical fluids, we usually need to choose a spherical grid and numerical approximations to represent the partial derivative equations. Some alternatives include the use of finite differences or finite volumes with latitude–longitude or reduced grids. Each of these cases has some advantages and also some limitations. This paper presents a comparison between two methods and describes a composite model using them side by side. The first is a well-known method for latitude–longitude grids and was used from 75[Formula: see text][Formula: see text]S until 75[Formula: see text][Formula: see text]N. The second is a recently developed scheme for reduced grids and was used only in the polar regions. The similarity between the two methods allows the use of small adaptations in their approximations to obtain consistency and mass conservation also in the transition between the two regions. The composite model combines advantages of the other two schemes and has a smaller computational cost. Numerical tests indicated order 2 of convergence, prevention of grid-imprinting errors, and avoidance of nonlinear instability. This model has numerical properties that may lead to efficient implementations with massive parallel computation.

Tsunami inundation and flow velocity within a partially sheltered structural array

ABSTRACT This work examines tsunami run-up processes within a building array partially sheltered by a high-elevation, finite length, seawall. Work was performed using numerical modeling of 2D shallow water equations with varying seawall length, wave height, time scale, and structure size. Inundation depth and flow velocity were output at many locations of interest throughout the structural array. The most important influence on results when compared to the no-wall case was the sheltering or clearance angle as defined from the end of the wall. Behind the wall, inundation depth and cross-shore velocity declined significantly as sheltering angles increased (more sheltered). Alongshore inundation velocity (parallel to the seawall) increased strongly with increasing sheltering angle. A longer wave time scale resulted in significant increase in inundation depth and velocity especially at locations sheltered by the wall. Larger structure size or blockage ratio also led to increased inundation depth and flow velocity. However, varying wave height gave little change in inundation and velocity patterns. These results will serve to provide a better understanding of the wave inundation within a coastal urban region in a tsunami event when the seawall is breached, as well as the influence of coastal structure size and spacing on the tsunami inundation.

Assessing Reliability, Resilience and Vulnerability of Water Supply from SuDS

In recent decades, the impacts of urbanization on the hydrological cycle have led to an increase in the frequency and magnitude of urban flooding events, and this is also amplified by the effects of climate change. Sustainable Drainage Systems (SuDS) provide a revolutionary change in this field, improving the sustainability and resilience of cities. This research explores the integration of different SuDS with the aim of significantly reducing both the flow volume and celerity of floods in a residual urban catchment area of the metropolitan city of Querétaro (Mèxico), where extreme rainfall frequently occurs. This catchment is a representative suburb of urban pressure and environmental degradation problems. Currently, managing storm water under climate uncertainty through a multi-disciplinary approach is a major concern in this urban area. A 1D–2D coupling model of shallow water equations, the finite volume method, an unstructured meshing method, and a hybrid parallel computing application defined the optimal configuration of SuDS at catchment scale to reduce the flood vulnerability in Querétaro. Specifically, in this paper, we explore the management issues of the proposed SuDS configuration that acts as a water resource system with multiple purposes. A generic simulation model called MODSIM was applied to simulate the designed urban drainage system under a balanced IPCC future climate scenario in terms of reliability, resilience and vulnerability against water scarcity. The proposed hierarchical Reliability–Resilience–Vulnerability approach appears to be effective in evaluating the system performance, showing that the complete satisfaction of non-essential water uses in Querétaro can be assured at a 65% rate of reliability for a large range of reservoir storage conditions.

Potential Sunda Strait tsunami hazard due to the current deformation of Anak Krakatau

Abstract The tsunami Sunda Strait event at 2018 is a prove that volcanic activity may trigger a devastating and unpredictable tsunami. Several findings indicate that this catasrophe generated by flank collapse caused by the instability of Gunung Anak Krakatau (AK). Other processes of volcanic activity that cause tsunamis include pyroclastic flows, subaerial and submarine landslides, underwater explosions, blasts, and caldera collapse. In the 1883, Krakatau Mount produced the largest tsunami event, and it is apparent that AK is one of threats in Sunda Strait that may potentially produce tsunami in the future since it is still active until now. Some latest research shows there are two submarine landslide threats near Gunung Anak Krakatau; in the northeast part of AK and an elliptical landslide source in the west part of AK with the estimated landslide volume about 0.014 km3 and 0.6 km3, respectively. Those threats will be simulated by shallow water equation model to obtain the tsunami wave height near the source and at crucial areas along Sunda Strait as well as its tsunami time arrival. The first scenario is sourced from a new deposit from the 2018 eruption, and the latter is a change of slopes bathymetry due to volcanic or seismic activity. It is observed that the highest elevation of tsunami from the first scenario reached only 2.5 cm at Tanjung Lesung, Banten and 80 cm at the nearest island, Panjang Island. The second scenario has the height of 50 cm at Labuan, Banten and 70 cm at Panjang Island. The tsunami time travel at the surrounding islands is ranging between 0.6 - 5.8 minutes (scenario 1) and 0.7-4.9 minutes (scenario 2). Both scenario reached the inexpensive device of sea level monitoring at the east side of Rakata after 5 minutes and the height is 30 cm and 1.6 cm respectively for scenario 1 and 2.

Application of 1D model for overland flow simulations on 2D complex domains

High computational demand limits the applications of two-dimensional (2D) shallow water equation models for high resolution overland flow simulations, while one-dimensional (1D) models can achieve higher computing efficiency. This study applied a 1D hydrodynamic model to surrogate 2D models for overland flow simulations on complex 2D domains. With one dimension reduced, the surrogate model simulation would have acceptable accuracy with much higher computing efficiency. The surrogating is fulfilled through mimicking 2D models in mesh generations, so that the 1D channel network is generated in such a way that it geometrically covers the whole domain without overlapping and intersection, and hydrologically follows the steepest slopes. Several benchmark cases on 2D domains in both laboratory and field scales with complex geometry, where no 1D models have been ever applied, are used to compare the 1D and 2D model simulations. The comparisons demonstrate that the 1D model does have potentials to efficiently simulate overland flow on 2D complex domains with accuracy comparable to 2D models.

Evaluating hydrodynamics and implications to sediment transport for tidal restoration at Swan Cove Pool, Virginia

Coastal restoration projects are significantly important in coastal ecosystems as wetland losses accelerate. This study investigates tidal hydrodynamics and the potential impacts of a waterway opening on an existing roadway to restore and revitalise salt marshes in a small estuarine system in Virginia, USA. A depth-integrated, discontinuous Galerkin shallow-water equations model (DG-SWEM) is applied for astronomic tide simulation. The model employs a high-resolution unstructured mesh with a minimum element size of less than one meter and resolves complex tidal flows in the entire barrier island system, including the existing culvert gate and canal system. Compared to the existing system, the water exchange increased dramatically (flushing time also dramatically decreased) under the opened scenarios regardless of the opening width (22.9-, 30.5-, and 38.1-m width). By increasing the opening width, peak velocity through the proposed opening decreased 30–40%, and the maximum shear stress was reduced by more than half. The high-resolution model represented complex tidal flows, including eddies, and assisted in striking a balance of water-exchange capability, opening stability, and minimising their potential erosions for the opening design. Besides, erosion and sediment transport potentials for suspended sediments were estimated using bed shear stress and a Lagrangian particle tracking module. Such proxy modelling approach allows for the impact assessment of civil engineering and ecological waterworks in complex and highly damped tidal flow areas and is readily transferrable to other like systems (e.g. causeway construction, causeway cutting, biota passageways, and inlet modification).

On the data assimilation of initial distribution for 2-dimensionalshallow-water equation model

Abstract The utilization of data assimilation (DA) techniques is prevalent in marine meteorology for the purpose of estimating the complete state of the system. This is done to address the practical limitations associated with measurement data, which can only be specified at finite number of discrete points within a limited domain. We develop an efficient DA algorithm to reconstruct the initial state of the shallow-water equations (SWE) within a 2-dimensional rectangular domain using sparse spatial measurement data. Our algorithm takes into account both the complete Coriolis force and the ocean bottom topography in the SWE model, resulting in accurate recovery of the initial status. After establishing the uniqueness of the solution to the nonlinear SWE with appropriate boundary conditions, we proceed to establish the conservation laws for the suitably defined energy quantity for this traveling wave system. This generalization of the known conservation laws for the simplified SWE system, which ignores the Coriolis force and topography, allows us to reveal the influence of nonconstant sea floor topography on wave propagation. In order to restore the initial state through the minimization of a cost functional using DA techniques, we proceed by deriving the adjoint problem for our iteration process. Additionally, we establish a discrete scheme for the governing equations in the Arakawa C-grid framework, from which we rigorously derive the error associated with energy conservation in discrete form. The numerical implementations are also provided to validate our proposed scheme through the verification of energy conservation and the reconstruction effect of the initial state for various configurations.

Thetis-SWAN: A Python-interfaced wave–current interactions coupled system

Wave–Current Interactions (WCI) emerge in nearshore coastal areas, prompting the development of coupled modelling systems to simulate these phenomena. We present a new multi-scale parallelised Python-interfaced WCI coupled system adopting a component-based approach enabling model-component integration without inhibiting their respective development. The underlying principles emphasise model equitability, flexibility and language interoperability. The hybrid model comprises the spectral wave model SWAN and the 2-D shallow-water equation model, Thetis. The coupling is performed through the Basic Model Interface. The coupled WCI model is the first to employ a Python interface, while maintaining the efficiency of different lower-level compiled programming languages, Fortran for SWAN and C for Thetis. We discuss the system implementation, architecture, and underlying physics considered. The coastal waters of Duck, NC, serve as a practical demonstration in simulating WCI. We then elaborate on the rationale for the coupled system design to inform the development of coupled modelling frameworks for environmental systems.

An automatic mesh generator for coupled 1D–2D hydrodynamic models

Abstract. Two-dimensional (2D), depth-averaged shallow water equation (SWE) models are routinely used to simulate flooding in coastal areas – areas that often include vast networks of channels and flood-control topographic features and/or structures, such as barrier islands and levees. Adequately resolving these features within the confines of a 2D model can be computationally expensive, which has led to coupling 2D simulation tools to less expensive one-dimensional (1D) models. Under certain 1D–2D coupling approaches, this introduces internal constraints that must be considered in the generation of the 2D computational mesh used. In this paper, we further develop an existing automatic unstructured mesh generation tool for SWE models, ADMESH+, to sequentially (i) identify 1D constraints from the raw input data used in the mesh generation process, namely the digital elevation model (DEM) and land–water delineation data; (ii) distribute grid points along these internal constraints, according to feature curvature and user-prescribed minimum grid spacing; and (iii) integrate these internal constraints into the 2D mesh size function and mesh generation processes. The developed techniques, which include a novel approach for determining the so-called medial axis of a polygon, are described in detail and demonstrated on three test cases, including two inland watersheds with vast networks of channels and a complex estuarine system on the Texas, USA, coast.

Aceh's tsunami wave evolution and its interaction with hybrid protection structure

The 2004 Aceh tsunami tragedy was one of the most catastrophic occurrences, resulting in damage and severe casualties in multiple countries. This study proposes a hybrid coastal protection system made up of mangroves, a sea dike, a trench, or a combination of the three structures to prevent similar devastation in the future. This system is expected to reduce the tsunami wave height, thus lowering their potential damage. The tsunami wave propagation is reproduced using a nonlinear shallow water equation model. To construct a numerical scheme, a staggered grid finite volume method is implemented. This scheme is then validated using several benchmark tests. Once validated, the computational results are compared to experimental data collected at the Laboratory of Coastal Dynamics in Yogyakarta, Indonesia. Both the physical and numerical models use the downscaled Aceh tsunami waves and real bathymetry. Several scenarios of structures combination are presented with the aim to determine the most effective combination. A sensitivity analysis is also conducted to support the results.

Multiresolution approximation for shallow water equations using summation-by-parts finite differences

Abstract We present spatial approximation for shallow water equations on a mesh of multiple rectangular blocks with different resolution in Cartesian geometry. The approximation is based on finite-difference operators that fulfill Summation By Parts (SBP) property – a discrete analogue of integration by parts. The solution continuity conditions between mesh blocks are imposed in a weak form using Simultaneous Approximation Terms (SAT) method.We show that the resulting discrete divergence and gradient operators are anti-conjugate. The important consequences are the discrete analogues for mass and energy conservation laws along with the proof of stability for linearized equations. The numerical shallow water equations model based on the presented spatial approximation is tested using problems with meteorological context. Test results prove high-order accuracy of SBP-SAT discretization. The interfaces between mesh blocks of different resolution produce no significant noise. The local mesh refinement is shown to have positive effect on the solution both locally inside the refined region and globally in the dynamically coupled areas.

Development of the global tsunami forecasting system considering the dynamic interaction of tide-tsunami around the Korean Peninsula

Tsunamis are extreme natural events that pose a significant threat to coastal communities, making a comprehensive understanding of tsunami propagation mechanisms necessary for forecasting and evacuation purposes. While previous forecasting models have successfully examined several factors influencing tsunami propagation, the impact of the dynamic interaction between tides and tsunamis has yet to be investigated thoroughly. The Yellow Sea is characterized by high tidal elevations and strong tidal currents, which can accelerate the tsunami impacts on the Korean coasts. This study developed a regional tide-tsunami interaction model based on the shallow water equation model to quantitatively investigate the dynamic tide-tsunami interaction and evaluate its influence on tsunami propagation and amplification mechanism. High-resolution numerical tests were conducted for two worst-case tsunami scenarios that occurred in the Korean Peninsula, including the 2010 Chilean tsunami (far-field forecasting) and the 2011 Tohoku tsunami (near-field forecasting). The performance of the numerical model was validated utilizing the observational tide data collected along the Korean coasts. The numerical model effectively reproduces the horizontal distribution of instantaneous free surface displacement and velocity. The results reveal that the dynamic tide-tsunami interaction induced by these tsunamis generally reduces the water level and velocity in the ocean while amplifying these quantities as the tsunamis approach the coastal regions. However, due to the complex and arbitrary features of the topography, the impact of the dynamic tide and tsunami interaction on water elevation and velocity is inconsistent even compared with measurements from the adjacent tidal gauges, which suggests that the dynamic interaction can play an opposite role during the propagation and amplification process. Furthermore, the different arrival times of tsunamis along the Korean coasts are dominated by the corresponding phase of the local tidal currents that develop in each region.

A Surrogate Model for Shallow Water Equations Solvers with Deep Learning

A Surrogate Model for Shallow Water Equations Solvers with Deep Learning

Coping with geometric discontinuities in porosity-based shallow water models

The use of classic two-dimensional (2D) shallow water equations (SWE) for flooding simulation in complex urban environments is computationally expensive, due to the need of refined meshes for the representation of obstacles and building. Aiming to reduce the computational burden, a class of sub-grid SWE models, where small-scale building features are preserved on relatively coarse meshes by means of macroscale porosity parameters, has been recently introduced in the literature. Among the other porosity-based models, the single porosity (SP) model is relevant because the corresponding one-dimensional (1D) Riemann problem is the building block for the construction of many porosity-based numerical schemes. Like the Riemann problem connected to mathematical models such as the SWE with variable bed elevation and the 1D Euler equations in contracting pipes, the SP Riemann problem may exhibit multiple solutions for certain initial conditions. This ambiguity can be solved by restoring the microscale information of the 2D SWE model that is lost at the SP macroscale. In the present paper, we disambiguate the solutions' multiplicity by systematically comparing the solution of the SP Riemann problem at local porosity discontinuities with the corresponding 2D SWE numerical solutions in contracting channels. An additional result of this comparison is that the SP Riemann problem should incorporate an adequate amount of head loss when strongly supercritical flows past sudden porosity reductions occur. An approximate Riemann solver, able to pick the physically congruent solution among the alternatives and equipped with the required head loss amount, shows promising results when implemented in a 1D single porosity finite volume scheme.

Techniques to embed channels in finite element shallow water equation models

Two new and novel techniques to efficiently represent channels in large-scale finite element shallow water equation models are introduced. The first enables the discontinuous representation of bathymetric depth by allowing nodes along a discontinuity to have two land surface elevations, one representing the bottom of the discontinuity and the other the top of the discontinuity. Elemental integration proceeds using the nodal depth corresponding to whether the element is on the upper or lower side of the discontinuity and enables efficient treatment of steep sided features in a two-dimensional horizontal discretization. The second technique consolidates nodal equations at paired nodes across a channel. Doing this eliminates cross-channel variability and consequently eliminates the across-channel Courant-Friedrichs-Lewy stability constraint on the model time step. Together the two techniques allow channels of various sizes as well as other instances of steep topography to be embedded seamlessly, efficiently and in a fully coupled manner within an otherwise two-dimensional horizontal spatial discretization. This new capability is especially useful in the interface zone subject to both coastal and hydrological influences as it effectively captures bi-directional channelized flow, bi-directional flow between the channel and the floodplain, and coupled flow when the channel and floodplain are fully submerged. This paper describes the methodological development and presents a series of tests that verify the performance of this novel solution approach.

Modeling the effects of large-scale interior headland restoration on tidal hydrodynamics and salinity transport in an open coast, marine-dominant estuary

The effects of large-scale interior headland restoration on tidal hydrodynamics and salinity transport in an open coast, marine dominant estuary (Grand Bay, Alabama, U.S.A) are investigated using a two-dimensional model, the Discontinuous-Galerkin Shallow Water Equations Model (DG-SWEM). Three restoration alternatives are simulated for present-day conditions, as well as under 0.5 m of sea level rise (SLR). Model results show that the restoration alternatives have no impact on tidal range within the estuary but change maximum tidal velocities by ±5 cm/s in the present-day scenarios and by ±7 cm/s in the scenarios with 0.5 m of SLR. Differences in average salinity concentrations for simulated tropical and frontal seasons show increases and decreases on the order of 2 pss in the embayments surrounding the restoration alternatives; differences were larger (on the order of ±4 pss) for the scenarios with 0.5 m of SLR. There were minimal changes in average salinity outside of the estuary and no changes offshore. The size and position of the alternatives played a role in the salinity response as a result of changing the estuarine shoreline geometry and affecting the fetch within the bay. SLR was more impactful in increasing exposure to low salinity values (i.e., less than 5 pss) than the presence of the restoration alternatives. Overall, the modeled results indicate that these large-scale restoration actions have limited and localized impacts on the hydrodynamics and salinity patterns in this open coast estuary. The results also demonstrate the nonlinear response of salinity to SLR, with increases and decreases in the maximum, mean and minimum daily salinity concentrations from present-day conditions. This nonlinear response was a result of changes in the directions of the residual currents, which affected salinity transport.

A hybrid 1D-2D Lagrangian solver with moving coupling to simulate dam-break flow

In dam-break problems, two-dimensional (2D) models on the vertical plane appropriately capture the non-hydrostatic pressure distribution but with a high computational cost. This study presents a hybrid Lagrangian solver of one-dimensional (1D) shallow water flow and a 2D large-eddy simulation with a sub-particle scale to simulate dam-break flows over frictional beds. The proposed 1D-2D hybrid scheme is a useful trade-off between the accuracy of the 2D models on the vertical plane and the computational efficiency of 1D shallow water equations models. The configuration of sub-domains is established so that the 1D solves the regions with dominantly hydrostatic pressure, while the 2D solver accounts for the non-hydrostaticity. A moving coupling method is introduced based on two-layer shallow water equations with a uniform velocity distribution that constructs two sub-domains, including inner and outer zones in the section coupling model (SCM), or upper and lower layers within the layer coupling model (LCM). Then, the region with high vertical accelerations is substituted by the 2D model in the SCM or mixed SCM-LCM in which the overlapping zone and boundary interface move with time, based on the 1D flow feature. Also, a new method is proposed to deal with irregular boundary interfaces in the LCM while satisfying the momentum transfer. The present hybrid model is advantageous as it meets mass conservation, and there is no need for typical inflow/outflow boundary conditions at the coupling interfaces for simulating dam-break waves. Moving particle simulation (MPS) is utilized to solve the Lagrangian equations, which is advantageous in dealing with free-surface problems with large deformation. The model validation reveals that the hybrid results of water depths are in very good agreement with observed data and those obtained from the 2D model for selected dam-break flows with the ratio of initial depths downstream to upstream up to 0.4. In this respect, the average root-mean-square-error (RMSE) is computed at 8.09 mm in the SCM, 8.16 mm in the mixed SCM-LCM, and 7.48 mm in the pure 2D model, while both hybrid models save the computational time by 55% compared to the 2D model. Moreover, the mixed SCM-LCM preserves the nonlinearity of the wavefront over long times, whereas the SCM converts the wavefront into a shockwave considering the same number of fluid particles.