Shallow Water Equations Research Articles

Resonant Excitation of Kelvin Waves by Interactions of Subtropical Rossby Waves and the Zonal Mean Flow

Abstract Equatorial Kelvin waves can be affected by subtropical Rossby wave dynamics. Previous research has demonstrated the Kelvin wave growth in response to subtropical forcing and the resonant growth due to eddy momentum flux convergence. However, the relative importance of the wave–mean flow and wave–wave interactions for the Kelvin wave growth compared to the direct wave excitation by the external forcing has not been made clear. This study demonstrates the resonant Kelvin wave excitation by interactions of subtropical Rossby waves and the mean flow using a spherical shallow-water model. The use of Hough harmonics as basis functions makes Rossby and Kelvin waves prognostic variables of the model and allows the quantification of terms contributing to their tendencies in physical and wave spaces. The simulations show that Kelvin waves are resonantly excited by interactions of Rossby waves and the balanced zonal mean flow in the subtropics, provided the Rossby and Kelvin wave frequencies, which are modified by the mean flow, match. The resonance mechanism is substantiated by analytical expressions. The Kelvin wave tendencies are caused by velocity and depth tendencies: The velocity tendencies due to the meridional advection of the zonal mean velocity can be outweighed by the zonal advection of the Rossby wave velocity or by the depth tendencies due to Rossby wave divergence. Identifying the resonant excitation mechanism in data should contribute to the quantification of Kelvin wave variability originating in the subtropics. Significance Statement This study seeks to understand how Kelvin waves, which are eastward-propagating disturbances in the tropical atmosphere, are connected to Rossby wave dynamics in the subtropics. Using idealized simulations, the Kelvin wave excitation is explained as a resonance effect due to interactions of Rossby waves and the zonal mean flow. The mechanism contributes to the understanding of atmospheric wave interactions and extratropical effects on the tropics. Further work searching for evidence of the new mechanism in atmospheric data may shed a new light on subseasonal variability in the tropics, such as the Madden–Julian oscillation.

A Consistent and Well‐Balanced Hybrid Weighted Essentially Non‐Oscillatory Scheme for Shallow Water Equations on Unstructured Meshes

ABSTRACTIn this article, a type of high‐order consistent and well‐balanced hybrid weighted essentially non‐oscillatory (WENO) scheme is proposed for shallow water equations with flat or non‐flat bottom on unstructured triangular meshes. The hybrid scheme presents a new consistent discretization format on the flux and the source term with the goal of obtaining a hybridization of the high‐order WENO scheme and linear scheme. According to the modified multi‐resolution analysis approach, we can select the more robust and accurate WENO reconstruction in the vicinity of discontinuities, and the less expensive linear reconstruction in the smooth regions. As a result, the suggested hybrid WENO scheme can acquire the capacity of saving the computing time cost while maintaining the excellent numerical features of the original WENO scheme. Eventually, some extensive and classical two‐dimensional numerical examples, including a tidal bore of an estuary with an irregular computation area are provided to validate the performance of this hybrid WENO scheme on triangular meshes in terms of accuracy order, exact conservation property, shock‐capturing, good resolution, and computational efficiency.

Simulation of Flood-Control Reservoirs: Comparing Fully 2D and 0D–1D Models

Flood-control reservoirs are often used as a structural measure to mitigate fluvial floods, and numerical models are a fundamental tool for assessing their effectiveness. This work aims to analyze the suitability of fully 2D shallow-water models to simulate these systems by adopting internal boundary conditions to describe hydraulic structures (i.e., dams) and by using a parallelized code to reduce the computational burden. The 2D results are also compared with the more established approach of coupling a 1D model for the river and a 0D model for the reservoir. Two test cases, including an in-stream reservoir and an off-stream basin, both located in Italy, are considered. Results show that the fully 2D model can effectively handle the simulation of a complex flood-control system. Moreover, compared with the 0D–1D model, it captures the velocity field and the filling/emptying process of the reservoir more realistically, especially for off-stream reservoirs. Conversely, when the basin is characterized by very limited flood dynamics, the two approaches provide similar results (maximum levels in the reservoir differ by less than 10 cm, and peak discharges by about 5%). Thanks to parallelization and the inclusion of internal boundary conditions, fully 2D models can be applied not only for local hydrodynamic analyses but also for river-scale studies, including flood-control reservoirs, with reasonable computational effort (i.e., ratios of physical to computational times on the order of 30–100).

Research on the Risk of Drilling Phases Based on the Development Model of Shallow-Water Subsea Trees

China is actively advancing offshore oil and gas exploration and development, focusing on addressing the technical challenges associated with resource extraction in shallow waters. The shallow-water subsea tree development model has gradually been applied in such environments, alleviating some construction difficulties. However, it still poses well control risks that require systematic analysis and quantitative evaluation. Given that the blowout preventer (BOP) is located on the platform and the shallow-water subsea tree is only used during certain drilling stages, this study divided the drilling process into two phases: the first three sections and the fourth section. Based on the “man–machine–material–environment” analytical framework and an improved system-theoretic process analysis (STPA), a control model for the construction phases was developed. Fault tree analysis (FTA) was then employed to identify comprehensively the potential risks from the platform to the wellbore in both phases. Subsequently, the decision-making trial and evaluation laboratory (DEMATEL) method were used to assess quantitatively the well control risks. Using the average weight as the evaluation criterion, high-risk factors exceeding the average weight in each phase were identified. The results indicate that in the shallow-water subsea tree development model, well control risks in the first three drilling sections primarily stem from human errors and equipment failures, while risks in the fourth section are mainly caused by damage to the subsea tree itself. The identified risk factors provide a theoretical basis for enhancing well control safety management in the shallow-water subsea tree development model.

Rigid lid limit in shallow water over a flat bottom

AbstractWe perform the so‐called rigid lid limit on different shallow water models such as the abcd Bousssinesq systems or the Green–Naghdi equations. To do so, we consider an appropriate nondimensionalization of these models where two small parameters are involved: the shallowness parameter and a parameter which can be interpreted as a Froude number. When the parameter tends to zero, the surface deformation formally goes to the rest state, hence the name rigid lid limit. We carefully study this limit for different topologies. We also provide rates of convergence with respect to and careful attention is given to the dependence on the shallowness parameter .

Closed-Boundary Reflections of Shallow Water Waves as an Open Challenge for Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have recently emerged as a promising alternative to traditional numerical methods for solving partial differential equations (PDEs) in fluid dynamics. By using PDE-derived loss functions and auto-differentiation, PINNs can recover solutions without requiring costly simulation data, spatial gridding, or time discretization. However, PINNs often exhibit slow or incomplete convergence, depending on the architecture, optimization algorithms, and complexity of the PDEs. To address these difficulties, a variety of novel and repurposed techniques have been introduced to improve convergence. Despite these efforts, their effectiveness is difficult to assess due to the wide range of problems and network architectures. As a novel test case for PINNs, we propose one-dimensional shallow water equations with closed boundaries, where the solutions exhibit repeated boundary wave reflections. After carefully constructing a reference solution, we evaluate the performance of PINNs across different architectures, optimizers, and special training techniques. Despite the simplicity of the problem for classical methods, PINNs only achieve accurate results after prohibitively long training times. While some techniques provide modest improvements in stability and accuracy, this problem remains an open challenge for PINNs, suggesting that it could serve as a valuable testbed for future research on PINN training techniques and optimization strategies.

Linear waves in shallow water over an uneven bottom, slowing down near the shore

The exact solutions of the system of equations of the linear theory of shallow water are discussed, representing traveling waves with specific properties for the time propagation, which is infinite when approaching the shore and finite when leaving for deep water. These solutions are obtained by reducing one-dimensional shallow water equations to the Euler–Poisson–Darboux equation with a negative integer coefficient before the lower derivative. The analysis of the wave field dynamics is carried out. It is shown that the shape of a wave approaching the shore will be differentiated a certain number of times, which is illustrated by a number of examples. When a wave moves away from the shore, its profile is integrated. The solutions obtained in the framework of linear theory are valid only for a finite interval of depth variation.

An Improved PINN Algorithm for Shallow Water Equations Driven by Deep Learning

Solving shallow water equations is crucial in science and engineering for understanding and predicting natural phenomena. To address the limitations of Physics-Informed Neural Network (PINN) in solving shallow water equations, we propose an improved PINN algorithm integrated with a deep learning framework. This algorithm introduces a regularization term as a penalty in the loss function, based on the PINN and Long Short-Term Memory (LSTM) models, and incorporates an attention mechanism to solve the original equation across the entire domain. Simulation experiments were conducted on one-dimensional and two-dimensional shallow water equations. The results indicate that, compared to the classical PINN algorithm, the improved algorithm shows significant advantages in handling discontinuities, such as sparse waves, in one-dimensional problems. It accurately captures sparse waves and avoids smoothing effects. In two-dimensional problems, the improved algorithm demonstrates good symmetry and effectively reduces non-physical oscillations. It also shows significant advantages in capturing details and handling complex phenomena, offering higher reliability and accuracy. The improved PINNs algorithm, which combines neural networks with physical mechanisms, can provide robust solutions and effectively avoid some of the shortcomings of classical PINNs methods. It also possesses high resolution and strong generalization capabilities, enabling accurate predictions at any given moment.

Numerical simulation of open channel basaltic lava flow through topographical bends

In this study, we utilised computational fluid dynamics to investigate the behaviour of open-channel basaltic lava flows navigating bends on shield volcanoes. Our focus was on understanding the relationship between flow velocity, rheology, and bend geometry. Employing a simple Force Balance Model (FBM), which considers the equilibrium between hydrostatic pressure and centrifugal force, we accurately approximated the changes in the height of the lava’s free surface through various bend geometries. Our analysis includes examining the influence of channel depth, width, and bend radius on the flow, revealing that variations in these parameters significantly affect the flow’s vertical displacement. Additionally, the bend sector angle emerged as a critical factor, indicating a minimum angle necessary for the flow to fully develop before exiting the bend.Further, we assessed the applicability of the Shallow Water Equations (SWE) for modelling the inertial displacement of the lava flow in bends, finding a good fit. The study extended to comparing the FBM’s predictions of the tilt angle of the flow’s free surface with the SWE results, showing notable agreement under specific conditions, particularly at a bend angle of 90 degrees. The impact of fluid density was also considered, revealing that density is a contributing factor to the development of the wetted line in the bend, a factor that is not captured by the simple FBM model. Finally, we explored different rheologies akin to natural lava flows, such as viscoplastic flow, and determined that factors like yield stress, consistency index, and power law index have a small impact on the flow behaviour in a steady-state condition within a bend.

Modeling sediment movement in the shallow-water framework: A morpho-hydrodynamic approach with numerical simulations and experimental validation

This work presents a morpho-hydrodynamic model and a numerical approximation designed for the fast and accurate simulation of sediment movement associated with extreme events, such as tsunamis. The model integrates the well-established hydrostatic shallow-water equations with a transport equation for the moving bathymetry that relies on a bedload transport function. Subsequently, this model is discretized using the path-conservative finite volume framework to yield a numerical scheme that is not only fast but also second-order accurate and well-balanced for the lake-at-rest solution. The numerical discretization separates the hydrodynamic and morphodynamic components of the model but leverages the eigenstructure information to evolve the morphologic part in an upwind fashion, preventing spurious oscillations. The study includes various numerical experiments, incorporating comparisons with laboratory experimental data and field surveys.

Physics‐Informed Neural Networks for the Augmented System of Shallow Water Equations With Topography

AbstractPhysics‐informed neural networks (PINNs) are gaining attention as an alternative approach to solve scientific problems governed by differential equations. This work aims at assessing the effectiveness of PINNs to solve a set of partial differential equations for which this method has never been considered, namely the augmented shallow water equations (SWEs) with topography. Differently from traditional SWEs, the bed elevation is considered as an additional conserved variable, and therefore one more equation expressing the fixed‐bed condition is included in the system. This approach allows the PINN model to leverage automatic differentiation to compute the bed slopes by learning the topographical information during training. PINNs are here tested for different one‐dimensional cases with non‐flat topography, and results are compared with analytical solutions. Though some limitations can be highlighted, PINNs show a good accuracy for the depth and velocity predictions even in the presence of non‐horizontal bottom. The solution of the augmented system of SWEs can therefore be regarded as a suitable alternative strategy to deal with flows over complex topography using PINNs, also in view of future extensions to realistic problems.

A High-Order Well-Balanced Discontinuous Galerkin Method for Hyperbolic Balance Laws Based on the Gauss-Lobatto Quadrature Rules

In this paper, we develop a high-order well-balanced discontinuous Galerkin method for hyperbolic balance laws based on the Gauss-Lobatto quadrature rules. Important applications of the method include preserving the non-hydrostatic equilibria of shallow water equations with non-flat bottom topography and Euler equations in gravitational fields. The well-balanced property is achieved through two essential components. First, the source term is reformulated in a flux-gradient form in the local reference equilibrium state to mimic the true flux gradient in the balance laws. Consequently, the source term integral is discretized using the same approach as the flux integral at Gauss-Lobatto quadrature points, ensuring that the source term is exactly balanced by the flux in equilibrium states. Our method differs from existing well-balanced DG methods for shallow water equations with non-hydrostatic equilibria, particularly in the aspect that it does not require the decomposition of the source term integral. The effectiveness of our method is demonstrated through ample numerical tests.

Reduction in wave shoaling over a linear transition bottom using a porous medium

Wave shoaling, which involves an increase in wave amplitude due to changes in water depth, can damage shorelines. To mitigate this damage, we propose using porous structures such as mangrove forests. In this study, we use a mathematical model to examine how mangroves, acting as a porous breakwater, can reduce wave shoaling amplitude. Shallow water equations are used as the governing equations and are modified to account for the presence of porous media. To measure the wave reduction generated by the porous media, the wave transmission coefficient is estimated using analytical and numerical approaches. The separation of variables method and the staggered finite volume method are utilized for each approach, respectively. The numerical results are then validated against the previously obtained analytical solutions. We then vary the friction and porosity parameters, influenced by the presence and extent of porous media, to evaluate their effectiveness in reducing wave shoaling.

Resolution of the bidimensional shallow water equations with horizontal temperature gradients by a well Balanced Roe scheme

This work presents a numerical resolution of the two-dimensional Ripa system using Roe scheme. A discretization of the source term satisfying the conservation property is presented. The mathematical model is based on the ordinary shallow water equations in merging the horizontal temperature gradient. The numerical approach employs unstructured grids and integrates Runge–Kutta method and the minmod limiter to achieve second order accuracy in both time and space domains. A strategy of mesh adaptation based on temperature gradient is implemented to refine the study domain and gain in computational cost. Various numerical tests are conducted to verify the effeciency of the numerical method.

Turbulent models of shallow-water equations-based smoothed particle hydrodynamics

The depth-averaged models such as those based on the shallow water equations (SWEs) are commonly used to simulate the large-scale flows with engineering importance. The smoothed particle hydrodynamics (SPH) approach has been documented to solve the SWEs due to its mesh-free superiority in treating the free surfaces and wet-dry boundaries. However, nearly all SWE-SPH models were developed without a turbulent model, which seriously limited the model applications where the flows are complex and where the turbulent parameters are explicitly needed. For the first time, this paper includes a depth-averaged turbulent k̂-ε̂ model in the SWE-SPH solver, making the model more capable of treating the turbulent flows in the practical field. For comparison purpose, a sub-particle-scale turbulent model widely adopted in three-dimensional (3D) SPH was also included in the present SWE-SPH scheme. To evaluate the performance of the two proposed turbulent SWE-SPH models, various open channel flows of increasing complexity were simulated, and the SPH computations were compared with the reported data in the literature. Through the analysis of results for a rough riverbed, L-shaped and sudden expansion channels, it is demonstrated that the present turbulent SWE-SPH models are equipped with good robustness and accuracy in capturing the shallow water turbulent dynamics, with the potential to be used in practical river and coastal flows. In summary, there are two distinct novelties in the proposed work. First, the mesh-free numerical modeling technique SPH is used to solve the shallow water equations, which enable the model to work in large engineering field through simple and effective tracking of free surfaces and wet-dry boundaries. Second, the proposed research expands the shallow water SPH modeling technique by including robust turbulence simulation capacity. The newly developed model can address more challenging engineering scenarios such as the sediment and pollutant transports when the flow turbulence plays an important role and where the turbulent parameters are explicitly required in the relevant transport equations.

Physics-informed neural networks for inversion of river flow and geometry with shallow water model

The river flow transports sediment, resulting in the formation of alternating sandbars in the riverbed. The underlying physics is characterized by the interaction between flow and river geometry, necessitating an understanding of their inseparable relationship. However, the dynamics of river flow with alternating sandbars are hard to understand due to the difficulty of measuring flow depth and riverbed geometry during floods with current technology. This study implements an innovative approach utilizing physics-informed neural networks (PINNs) to estimate important hydraulic variables in rivers that are difficult to measure directly. The method uses sparse yet obtainable flow velocity and water level data. The governing equations of motion, continuity, and the constant discharge condition based on the mass conservation principle are integrated into the neural network as physical constraints. This approach enables the completion of sparse velocity fields and the inversion of flow depth, riverbed elevation, and roughness coefficients without requiring direct training data for these variables. Validation was performed using model experiment data and numerical simulations derived from these experiments. Results indicate that the accuracy of the estimations is relatively robust to the number of training data points, provided their spatial resolution is finer than the wavelength of the sandbars. The inclusion of mass conservation as a redundant constraint significantly improved the convergence and accuracy of the model. This PINNs-based approach, using measurable data, offers a new way to quantify complex river flows on alternating sandbars without significant updates to conventional methods, providing new insights into river physics.

Simulation of the Full‐Process Dynamics of Floating Vehicles Driven by Flash Floods

AbstractFlash flooding has become more prominent under climate change, threatening people's life and property. Post‐event investigations of recent events emphasize the role of floating debris, such as vehicles, in exacerbating damage. Few modeling methods and tools have been developed to simulate the full‐process dynamics of floating debris driven by large‐scale flood waves in real world. In this work, a fully coupled model is developed for simulating the full‐process interactive movements of vehicles driven by flash flood hydrodynamics, from entrainment, transport to deposition. The proposed coupled modeling system consists of a finite volume shock‐capturing hydrodynamic model solving the 2D shallow water equations and a 3D discrete element method (DEM) model. The proposed two‐way coupling approach estimates the hydrostatic and hydrodynamic forces acting on solid objects using the water depth and velocity predicted by the hydrodynamic model; the resulting counter forces on the fluid flow are then considered by adding extra source terms in the hydrodynamic model. A multi‐sphere method is further embedded in the DEM model to better represent vehicle shapes. New calculation modules are further implemented to represent the vehicle entrainment, contact and stopping motions. The coupled model is applied to reproduce a flash flood event hit Boscastle in the UK in 2004. Over 100 vehicles were moved and carried downstream by the highly transient flood flow. The model well predicts the hydrodynamics, interactive transport process and the final locations of vehicles. The proposed coupled model provides a new tool for simulating large‐scale flash flooding processes, including debris dynamics.

Local Time-Stepping for the Shallow Water Equations using CFL Optimized Forward-Backward Runge-Kutta Schemes

Local Time-Stepping for the Shallow Water Equations using CFL Optimized Forward-Backward Runge-Kutta Schemes

An operational discontinuous Galerkin shallow water model for coastal flood assessment

Hydrodynamic modeling for coastal flooding risk assessment is a highly relevant topic. Many operational tools available for this purpose use numerical techniques and implementation paradigms that reach their limits when confronted with modern requirements in terms of resolution and performances. In this work, we present a novel operational tool for coastal hazards predictions, currently employed by the BRGM agency (the French Geological Survey) to carry out its flooding hazard exposure studies and coastal risk prevention plans on International and French territories. The model, called UHAINA (wave in the Basque language), is based on an arbitrary high-order discontinuous Galerkin discretization of the nonlinear shallow water equations with SSP Runge–Kutta time stepping on unstructured triangular grids. It is built upon the finite element library AeroSol, which provides a modern C++ software architecture and high scalability, making it suitable for HPC applications. The paper provides a detailed development of the mathematical and numerical framework of the model, focusing on two key-ingredients : (i) a pragmatic P0 treatment of the solution in partially dry cells which garantees efficiently well-balancedness, positivity and mass conservation at any polynomial order; (ii) an artificial viscosity method based on the physical dissipation of the system of equations providing nonlinear stability for non-smooth solutions. A set of numerical validations on accademic benchmarks is performed to highlight the efficiency of these approaches. Finally, UHAINA is applied on a real operational case of study, demonstrating very satisfactory results.

Including dynamic capacity impact in an improved model for optimal flood control operation in river-type reservoirs

Flood control operations in river-type reservoirs are significantly affected by both the dynamic reservoir capacity and flood propagation within the reservoir area. However, the traditional reservoir flood control optimal operation (RFCOO) model is usually based on the static capacity method, which may bring large errors in scheduling calculations for river-type reservoirs. To address this issue, an improved model, the reservoir dynamic capacity flood control optimal operation (RDCFCOO) model, that includes the influence of the dynamic reservoir capacity and flood propagation, was developed to improve the accuracy of flood regulation calculations in river-type reservoirs. This improved model includes a 1D unsteady flow simulation and a rederived objective function based on the maximum peak clipping criterion, and expresses the state transformation equations using the de Saint-Venant equations. Furthermore, a multi-core parallel DP (PDP) algorithm that incorporates reduction of state dimensionality (RSD) was developed to effectively derive optimal operation schemes. The improved model and algorithm developed herein were validated through an application to the Xiangjiaba Reservoir, China. The results show that: (1) By adding the item (qiw) to the objective function and substituting the de Saint-Venant equations for the water balance equation, the RDCFCOO model developed in this paper can account for the impact of dynamic capacity and flood propagation. As a result, it exhibits a more accurate and detailed flood control process that was closer to the actual conditions of river-type reservoirs compared to the RFCOO model; (2) By removing the invalid discrete state points from the search space and fully utilizing multi-core processors, the proposed PDP with RSD can significantly improve the computational efficiency of the DP in solving RDCFCOO problems. The improved model and algorithm can effectively improve the calculation accuracy of flood control optimal scheduling in river-type reservoirs and provide more information for decision-makers.