Shallow Water Approximation Research Articles

The MEMENTO code for modeling of macroscopic melt motion in fusion devices

The MEMENTO (MEtallic Melt Evolution in Next-step TOkamaks) code is a new numerical implementation of the physics model originally developed for the MEMOS-U code with the objective to self-consistently describe the generation of melt and its subsequent large scale dynamics in fusion devices and to assess the damage of metallic reactor armor under powerful normal and off-normal plasma events. The model has been validated in multiple dedicated EUROfusion experiments. MEMENTO solves the heat and phase transfer problem coupled with the incompressible Navier–Stokes equations in the shallow water approximation for the thin liquid film over the solid metal and with the current propagation equations on a domain that features a time-evolving deforming metal-plasma interface. The code utilizes non-uniform and adaptive meshing along with sub-cycling in time facilitated by the AMReX open-source framework as well as AMReX’s built-in parallelization capabilities.

Model of bores interaction in the swash

This paper examines the impact of wave interactions in the inner-surf zone, including the swash, on shoreline excursion dynamics. Specifically, the study focuses on how the time lag between consecutive waves and the slope of the beachface affect shoreline dynamics. To investigate this, a laboratory experimental setup is developed to study the behaviour of two consecutive bore-type waves travelling over a fluid layer with constant depth, representing an idealized inner-surf zone, before impacting an inclined solid plane with slope β, representing the beachface. Bore waves are generated using two dam-break flow devices, with a controlled time lag Δt between them. This simplified physical model allows for the assessment of semi-theoretical predictive models based on shallow water approximation, resulting in ballistic-type models for shoreline motion. By combining these approaches, the study characterizes different flow regimes in the (Δt,β) parameter space. It is observed that varying Δt at a fixed β distinguishes four interaction regimes, either wave–wave interaction in the inner-surf zone or wave-swash interaction in the swash, with possibilities of merging or collision depending on wave orientation. In particular, increasing Δt leads to either bore–bore merging in the inner-surf, bore-runup merging in the swash, bore-backwash collision in the swash or bore–bore collision in the inner-surf. Bore–bore merging and bore-runup merging enhance runup, peaking when Δt is such that merging occurs near the transition from the inner-surf to the swash. On the contrary, bore-backwash collision results in additional dissipation of the second bore’s dynamics, leading to a reduced shoreline extension compared to that induced by the first bore. All processes within the swash, including the transition between bore-runup and bore-backwash regimes, as well as the enhancement and extra dissipation of the second bore’s swash excursion, exhibit some level of dependence on β. Overall, the results from this physical model help characterize and explain local mechanisms triggered by wave interaction near the shoreline. Validating this model’s relevance warrants specific attention through comparisons with field data. As an initial validation attempt, field data extracted from the literature are compared to the physical model, showing promising agreement.

A mathematical analysis of the Kakinuma model for interfacial gravity waves. Part II: justification as a shallow water approximation

We consider the Kakinuma model for the motion of interfacial gravity waves. The Kakinuma model is a system of Euler–Lagrange equations for an approximate Lagrangian, which is obtained by approximating the velocity potentials in the Lagrangian of the full model. Structures of the Kakinuma model and the well-posedness of its initial value problem were analysed in the companion paper [14]. In this present paper, we show that the Kakinuma model is a higher order shallow water approximation to the full model for interfacial gravity waves with an error of order $O(\delta _1^{4N+2}+\delta _2^{4N+2})$ in the sense of consistency, where $\delta _1$ and $\delta _2$ are shallowness parameters, which are the ratios of the mean depths of the upper and the lower layers to the typical horizontal wavelength, respectively, and $N$ is, roughly speaking, the size of the Kakinuma model and can be taken an arbitrarily large number. Moreover, under a hypothesis of the existence of the solution to the full model with a uniform bound, a rigorous justification of the Kakinuma model is proved by giving an error estimate between the solution to the Kakinuma model and that of the full model. An error estimate between the Hamiltonian of the Kakinuma model and that of the full model is also provided.

On the development and analysis of coupled surface–subsurface models of catchments. Part 3. Analytical solutions and scaling laws

The objective of this three-part work is to formulate and rigorously analyse a number of reduced mathematical models that are nevertheless capable of describing the hydrology at the scale of a river basin (i.e. catchment). Coupled surface and subsurface flows are considered. In this third part, we focus on the development of analytical solutions and scaling laws for a benchmark catchment model that models the river flow (runoff) generated during a single rainfall. We demonstrate that for catchments characterised by a shallow impenetrable bedrock, the shallow-water approximation allows a reduction of the governing formulation to a coupled system of one-dimensional time-dependent equations for the surface and subsurface flows. Asymptotic analysis is used to derive semi-analytical solutions for the model. We provide simple asymptotic scaling laws describing the peak flow formation, and demonstrate its accuracy through a comparison with the two-dimensional model developed in Part 2. These scaling laws can be used as an analytical benchmark for assessing the validity of other physical, conceptual or statistical models of catchments.

Accuracy and stability analysis of horizontal discretizations used in unstructured grid ocean models

One important tool at our disposal to evaluate the robustness of Global Circulation Models (GCMs) is to understand the horizontal discretization of the dynamical core under a shallow water approximation. Here, we evaluate the accuracy and stability of different methods used in, or adequate for, unstructured ocean models considering shallow water models. Our results show that the schemes have different accuracy capabilities, with the A- (NICAM) and B-grid (FeSOM 2.0) schemes providing at least 1st order accuracy in most operators and time integrated variables, while the two C-grid (ICON and MPAS) schemes display more difficulty in adequately approximating the horizontal dynamics. Moreover, the theory of the inertia-gravity wave representation on regular grids can be extended for our unstructured based schemes, where from least to most accurate we have: A-, B, and C-grid, respectively. Considering only C-grid schemes, the MPAS scheme has shown a more accurate representation of inertia-gravity waves than ICON. In terms of stability, we see that both A- and C-grid MPAS scheme display the best stability properties, but the A-grid scheme relies on artificial diffusion, while the C-grid scheme does not. Alongside, the B-grid and C-grid ICON schemes are within the least stable. Finally, in an effort to understand the effects of potential instabilities in ICON, we note that the full 3D model without a filtering term does not destabilize as it is integrated in time. However, spurious oscillations are responsible for decreasing the kinetic energy of the oceanic currents. Furthermore, an additional decrease of the currents’ turbulent kinetic energy is also observed, creating a spurious mixing, which also plays a role in the strength decrease of these oceanic currents.

Методы расчета растекания опасных и загрязняющих веществ по рельефу

The study considers and compares several methods of calculation of the process of stable liquid spread over a solid impermeable surface: the SPH method and the VOF method for the three-dimensional system of equations; the Lax-Wendroff method and the finite volume method for the two-dimensional «shallow water» system of equations. The «shallow water» systems of equations are obtained by averaging the full system of equations over depth assuming the absence of a vertical velocity component. The procedure of numerical calculation by the Lax-Wendroff method is provided in detail. The emphasis is made on considering shallow water approximation as such approximation enables obtaining a three-dimensional flow pattern in the framework of a two-dimensional model. The «shallow water» system of equations is provided taking into account the action of friction forces. All four calculation methods have been verified by using the data of a water dam breaking experiment in a 122-meter canal. The experiments recorded the dynamics of liquid propagation and water rise at different locations and moments. The comparison implies that the SPH method requires significant computational costs due to the need to specify a large number of particles in order to ensure acceptable accuracy. The VOF calculation method blurs the air-water interface, which hampers the determination of spill configuration and the free surface position. The dam breaking problem has been solved most accurately by two methods, namely the Lax-Wendroff method and the finite volume method, which has been confirmed by the quantitative assessment of error in calculations by various methods. It has been demonstrated that the shallow water approximation implemented by the Lax-Wendroff numerical method used in the study ensures accuracy sufficient to solve problems of spill over the terrain surface.

Equipment Safety in Renewable Energies Exploitation

This paper considers the dam-break problem in a horizontal smooth 1D channel, for the purpose of equipment safety in renewable energies exploitation. The fluid is water and it can be described by a Newtonian model, provided that the inertial effects be neglected versus the viscous ones in the momentum balance. Assuming the shallow water approximation, a non dimensional equation is built from the continuity and the Navier-Stokes equations in the limit of zero-inertia and it is solved analytically in two limits: short time and long time. These solutions are then combined into a single, universal model. Limitations of the model are examined by comparison to a converged finite difference numerical solution of the flow equation.

Existence and uniqueness of the global conservative weak solutions for a cubic Camassa-Holm type equation

The extended cubic Camassa-Holm equation can be derived as asymptotic model from shallow-water approximation to the 2D incompressible Euler equations. This model encompasses both quadratic and cubic nonlinearities and the solution of the extended cubic Camassa-Holm equation possible development of singularities in finite time. This paper is devoted to the continuation of solutions to the extended cubic Camassa-Holm equation beyond wave breaking, and the global existence and uniqueness of the Hölder continuous energy conservative solutions for the Cauchy problem of the extended cubic Camassa-Holm type equation are investigated. An equivalent semilinear system was first introduced by a new set of independent and dependent variables, which can resolve all singularities due to possible wave breaking. Returning to the original variables, depending continuously on the initial data, the existence of the global conservative weak solutions can be obtained. Moreover, by analyzing the evolution of the quantities u and v=2arctan⁡ux along each characteristic, the uniqueness of the global conservative solutions for the Cauchy problem with general initial data u0∈H1(R) was proved.

On the Issue of Generation of Hydrodynamic Waves in the Shovi Gorge (Georgia) due to a Collapse on Glacier Tbilisa

The occurrence of a glacial mudflow in the Shovi gorge on August 3, 2023 was caused by a destructive process on the Tbilisa glacier. Based on the existing picture of the spread of the mudflow, it is possible to carry out an analysis regarding the rheological qualities of the water-saturated soil mass that forms its basis. It is also possible to approximately estimate the parameters of various types of hydrodynamic waves, the generation of which was probably possible during the propagation of the debris flow. It seems that such a problem can be satisfactorily solved only in the case of a correct assessment of the values of dimensionless criteria for the flow of debris flow, which are convenient quantitative characteristics that simplify the analysis of the qualitative consequences of applying the principle of hydrodynamic similarity. For example, in the case of approximating the bed of a debris flow with a rectangular channel, you can use the results of well-known analytical solutions obtained under simplifying assumptions that are valid for certain intervals of variation in the Reynolds and Froude similarity numbers. In particular, in the shallow water approximation, in the case of sufficiently large Reynolds numbers, when the fluid flow is highly turbulent, the Froude parameter allows one to fairly correctly simulate changes in the flow regime that occur as a result of the generation of hydrodynamic waves. The negative effects that often accompany the process of propagation of wave disturbances largely depend on the type of these waves. Therefore, in the case of a mudflow in the Shovi gorge, it seems quite realistic, for example, the generation of the so-called rolling hydrodynamic waves, which could well have been among the probable causes that determined the catastrophic scale of the tragic event.

Manipulation of self-organized multi-vortical states in active magnetic roller suspensions

Ensembles of active magnetic colloids demonstrate complex collective behavior and self-organization when driven out of equilibrium by external magnetic fields. The interplay between magnetic and hydrodynamic interactions often result in spontaneous self-assembly and the emergence of localized vortices in ensembles of ferromagnetic rollers. We report experimental and computational study of the self-organized multi-vortical state in ensembles of magnetic rollers under the imposed external confining potential realized by patterned substrates. We explore the behavior of the system on a substrate patterned with shallow wells ordered in a lattice with square symmetry and lattice spacing smaller than the wells’ diameter. We demonstrate, that ensembles of active magnetic rollers evolve in response to changes in the geometry of the confining lattice from a globally correlated state characterized by anti-ferromagnetic vortex ordering to a state with rapidly changing particle flows and self-organized vortical states with a lattice constant larger than one of the confining pattern. Our experimental observations are accompanied by numerical simulations based on phenomenological coarse grained particle dynamics coupled to Navier–Stokes hydrodynamics in shallow water approximation. The reported results provide insights into the collective behavior of active magnetic rollers under confinement and suggest strategies for the manipulation of collective dynamic states in active magnetic liquids.

Modeling spreading with degassing using anisotherm viscoplastic multiphase shallow water approximation

Modeling spreading with degassing using anisotherm viscoplastic multiphase shallow water approximation

The Problem of Effective Evacuation of the Population from Floodplains under Threat of Flooding: Algorithmic and Software Support with Shortage of Resources

Extreme flooding of the floodplains of large lowland rivers poses a danger to the population due to the vastness of the flooded areas. This requires the organization of safe evacuation in conditions of a shortage of temporary and transport resources due to significant differences in the moments of flooding of different spatial parts. We consider the case of a shortage of evacuation vehicles, in which the safe evacuation of the entire population to permanent evacuation points is impossible. Therefore, the evacuation is divided into two stages with the organization of temporary evacuation points on evacuation routes. Our goal is to develop a method for analyzing the minimum resource requirement for the safe evacuation of the population of floodplain territories based on a mathematical model of flood dynamics and minimizing the number of vehicles on a set of safe evacuation schedules. The core of the approach is a numerical hydrodynamic model in shallow water approximation. Modeling the hydrological regime of a real water body requires a multi-layer geoinformation model of the territory with layers of relief, channel structure, and social infrastructure. High-performance computing is performed on GPUs using CUDA. The optimization problem is a variant of the resource investment problem of scheduling theory with deadlines for completing work and is solved on the basis of a heuristic algorithm. We use the results of numerical simulation of floods for the Northern part of the Volga-Akhtuba floodplain to plot the dependence of the minimum number of vehicles that ensure the safe evacuation of the population. The minimum transport resources depend on the water discharge in the Volga river, the start of the evacuation, and the localization of temporary evacuation points. The developed algorithm constructs a set of safe evacuation schedules for the minimum allowable number of vehicles in various flood scenarios. The population evacuation schedules constructed for the Volga-Akhtuba floodplain can be used in practice for various vast river valleys.

Traveling internal waves in a two-layer shallow medium with variable bathymetry and current: Surface and near-bottom flows

In the linear approximation, we examine the one-dimensional problem of long internal wave propagation without reflection in a stationary two-layer flow with a free boundary in a channel of variable depth and width, in the limiting cases of surface and near-bottom currents. We use the shallow-water approximation and assume that liquids in the layers are ideal, immiscible, and having a small relative density difference, and that the flow is only in one of the layers. To filter fast processes (such as surface gravity waves), the flow velocity is assumed to have the same order of magnitude as the speed of internal waves. Both surface and near-bottom flows are divided into three classes, in accordance with conditions providing wave propagation without reflection. The properties of the flows in each of the classes are examined, and it is shown that the global reflectionless near-bottom flows do exist. The main results and conclusions are illustrated by a few particular solutions. A detailed comparison with the first part of the study concerning the flows with currents in both layers made it possible to clarify and refine some of the results obtained there, and to formulate questions for further investigation. The results obtained may be of interest for understanding of those natural processes in surface and near-bottom flows, which are significantly influenced by internal waves.

Time-Dependent Modelling of the Wave-Induced Vibration of Ice Shelves

This work presents an investigation of ice shelf vibration using a model based on shallow water approximation. The study focused on the effect of changes in the draft on the vibration of the ice shelf and presents both time-domain and frequency-domain results. The model used a radiation condition for energy propagation into the ice shelf. Furthermore, an energy balance relation was derived to investigate the energy flow within the system. Results show that changes in the draft can significantly impact the ice shelf’s vibration and that the energy flow within the system is affected by the geometry of the ice shelf. Results are presented for the interaction of wave packets in the time domain with the ice shelf. These show that energy is reflected and transmitted by the ice shelf and that the motion of the wave packets is very different in the ice shelf than in the open water. Overall, this study provides insight into the dynamics of ice shelf vibration and highlights the importance of considering changes in the draft and using the time domain when modelling these phenomena.

Modelling Neutron-Star Ocean Dynamics

We revisit the calculation of mode oscillations in the ocean of a rotating neutron star, which may be excited during thermonuclear X-ray bursts. Our present theoretical understanding of ocean modes relies heavily on the traditional approximation commonly employed in geophysics. The approximation elegantly decouples the radial and angular sectors of the perturbation problem by neglecting the vertical contribution from the Coriolis force. However, as the implicit assumptions underlying it are not as well understood as they ought to be, we examine the traditional approximation and discuss the associated mode solutions. The results demonstrate that, while the approximation may be appropriate in certain contexts, it may not be accurate for rapidly rotating neutron stars. In addition, using the shallow-water approximation, we show analytically how the solutions that resemble r-modes change their nature in neutron-star oceans to behave like gravity waves. We also outline a simple prescription for lifting Newtonian results in a shallow ocean to general relativity, making the result more realistic.

Traveling internal waves in a two-layer shallow medium with variable bathymetry and current

In the linear approximation, we examine one-dimensional problems of long internal wave propagation in a stationary flow of two-layer fluids with free boundary in a channel of variable depth and width. We use the shallow-water approximation and assume that liquids in the layers are ideal, immiscible, and have a small relative density difference inherent to natural currents. The conditions the flow must satisfy for wave propagation without reflection are found and analyzed. It is shown that there are three classes of such flows, and the characteristic properties of each of them are studied and compared with those found earlier in a similar problem for surface waves. A general analysis of the problem is illustrated by a few particular solutions. The results obtained can be of interest for understanding natural phenomena in which internal waves play a significant role.

A numerical experiment on a new piston-type wavemaker: Shallow water approximation

In this paper, we present a numerical experiment on a new piston-type wavemaker using a recently proposed piston-type wavemaker theory. The theory was primarily derived from the classical Boussinesq equation, based on the Pseudo-parameter Iteration Method (PIM). We first use the new theory to observe low amplitude propagating solitary waves and cnoidal waves, whereby we see the workability and validity of the theory. We further succeed in obtaining the graph of a mean power characteristic, in the range of 0.4 ≤kh≤ 1.0, of the new piston-type wavemaker from the new theory.

Modeling Spreading with Degassing Using Anisotherm Viscoplastic Multiphase Shallow Water Approximation

Modeling Spreading with Degassing Using Anisotherm Viscoplastic Multiphase Shallow Water Approximation

Performance and limits of a shallow-water model for landslide-generated tsunamis: from laboratory experiments to simulations of flank collapses at Montagne Pelée (Martinique)

SUMMARYWe investigate the dynamics and deposits of granular flows and the amplitude of landslide-generated water waves using the HySEA depth-averaged shallow-water numerical model, both at laboratory and field scales. We evaluate the different sources of error by quantitatively comparing the simulations with (i) new laboratory experiments of granular collapses in different conditions (dry, immersed, dry flow entering water) and slope angles and (ii) numerical simulations made with the SHALTOP code that describes topography effects better than most depth-averaged landslide-tsunami models. For laboratory configurations, representing the limits of the shallow-water approximation in such models, we show that topography and non-hydrostatic effects are crucial. When topography effects are accounted for empirically—by artificially increasing the friction coefficient and performing non-hydrostatic simulations—the model is able to reproduce the granular mass deposit and the waves recorded at gauges located at a distance of more than two to three times the characteristic dimension of the slide with an error ranging from 1 to 25 per cent depending on the scenario, without any further calibration. Taking into account this error estimate, we simulate landslides that occurred on Montagne Pelée volcano, Martinique, Lesser Antilles as well as the generated waves. Multiple collapse simulations support the assumption that large flank collapses on Montagne Pelée likely occurred in several successive subevents. This result has a strong impact on the amplitude of the generated waves and thus on the associated hazards. In the context of the ongoing seismic volcanic unrest at Montagne Pelée volcano, we calculate the debris avalanche and associated tsunamis for two potential flank-collapse scenarios.

Lock-exchange problem for Boussinesq fluids revisited: Exact shallow-water solution

An exact solution to the lock-exchange problem, which is a two-layer analogue of the classical dam-break problem, is obtained in the shallow-water (SW) approximation for two immiscible fluids with slightly different densities. The problem is solved by the method of characteristics using analytic expressions for the Riemann invariants. The obtained solution, which represents an inviscid approximation to the high-Reynolds-number limit, is, in general, discontinuous containing up to three hydraulic jumps which are due to either multivaluedness or instability of the continuous SW solution. Hydraulic jumps are resolved by applying the Rankine–Hugoniot conditions for the SW mass and generalized momentum conservation equations. The latter contains a free parameter α which defines the relative contribution of each layer to the interfacial pressure gradient. We consider a solution for α=0, which corresponds to both layers affecting the interfacial pressure gradient with equal weight coefficients. This solution is compared with the solutions resulting from the application of the classical Benjamin's front condition as well as the circulation conservation condition, which correspond to α=−1 and α→∞, respectively. The SW solution reproduces all principal features of 2D numerical solution for viscous fluids. The gravity current speed is found to agree well with experimental and numerical results when the front acquires the largest stable height which occurs at α=5−2. We show that two-layer SW equations for the mass and generalized momentum conservation can describe interfacial waves containing hydraulic jumps in a self-contained way without external closure conditions.