Linear Shallow Water Equations Research Articles

An Analytical Study of Tsunamis Generated by Submarine Landslides

In this work, the problem of tsunamis generated by underwater landslides is addressed. Two new solutions are derived in the framework of the linear shallow water equations and linear potential wave theory, respectively. Those solutions are analytical (1 + 1D) and another is semi-analytical (2 + 1D). The 1 + 1D model considers a solid body sliding over a sloping beach at a constant speed, and the 2 + 1D model considers a solid landslide that moves at a constant velocity on a flat bottom. The solution 1 + 1D is checked numerically using a different finite scheme. The 2 + 1D model examines the kinematic and geometric features of the landslide at a constant ocean depth and its influence on the generation of tsunamis. Landslide geometry significantly influences run-up height. Our results reveal a power law relationship between normalized run-up and landslide velocity within a realistic range and a negative power law for the landslide length–thickness. Additionally, a critical aspect ratio between the length and width of the sliding body is identified, which enhances the tsunamigenic process. Finally, the results show that the landslide shape does not have a decisive influence on the pattern of tsunami wave generation and propagation.

The wave passage over a seamount without reflection: Analytical results using the generalized Carrier–Greenspan transform

A method for the transformation of linear shallow water equations based on a generalization of the Carrier–Greenspan transform, well known in the theory of wave rolling on a flat slope, is presented. Thanks to it, the initial equations for waves over arbitrary bathymetry are reduced to a wave equation, from which both the displacement of the free surface and the flow velocity can be obtained simultaneously. Exact solutions in the sum of traveling waves have been obtained for certain configurations of the seabed in the form of seamounts. It is shown that when moving from a more gentle slope to a sharper one, the waveform will integrate a certain number of times, and the maximum amplitude is not reached at the very top of the mountain.

Bragg reflection of linear shallow-water waves by an array of widely spaced trapezoidal bars on a fringing reef flat

A closed-form solution to the linear shallow-water wave equation for wave reflection by a finite periodic array of widely spaced trapezoidal bars on a fringing reef flat is presented. It is shown that, when the total number, height, or top width of the trapezoidal bars increases, the strength of the Bragg resonance increases but the downward shift of the resonance phase becomes more pronounced. When the width of the reef face increases or when the reef face changes from a concave profile to a convex profile, the strength of the Bragg resonance decreases and the downward shift of the resonance phase also becomes more pronounced. Especially, it is found that the location of trapezoidal bars plays an important role in the strength and occurrence condition of the Bragg resonant reflection. Finally, the optimal collocation of the location and width of trapezoidal bars is investigated. Considering both construction cost and reflecting capacity, the relative width is recommended to be within 0.4≤wb/d≤0.6 and the distance from the reef edge to trapezoidal bars is suggested to be equal to the spacing of trapezoidal bars.

Scattering of moving atmospheric pressure induced tsunamis by bathymetry and coastline

Tsunamis can be generated by a moving atmospheric pressure disturbance. The 2022 Tonga volcanic eruption and tsunami demonstrated the global relevance of such a tsunami generation mechanism. The generated locked and free waves are often scattered by bathymetric variations and coastlines, generating more free waves and resulting in a complex wave field. The physical processes involved can be described by the forced linear shallow water equations. Analytical solutions are sought after for simplified bathymetric variations and coastline configurations. In this paper, a moving atmospheric pressure front, which is uniform in the direction normal to its propagation direction, is considered. The water wave motion is assumed to start from the quiescent condition, so that the incident locked and free waves are related. However, since the wave scattering processes are linear, the solutions obtained in this paper can be used for analyzing scattering of the incident locked and free waves with uncorrelated amplitudes. For one-dimensional horizontal (1DH) problems, the water depth is either a constant or has a sudden change (i.e., a step) with or without a vertically walled coastline and the atmospheric pressure front always moves in the direction normal to the depth contours. In the case where a coastline is considered, the atmospheric pressure front can either move from the land to the sea or from the sea to the land. In the 2DH (two-dimensional horizontal) example, the wave field generated by a moving atmospheric pressure front, sweeping over the a circular island surrounded by a circular shelf, is investigated and discussed. For all the cases considered, analytical solutions in the integral form are obtained using the Fourier transform method. For the 1DH cases, analytical solutions are also presented in the form of infinite series, summing over infinite number of scattered and reflected waves from the bathymetric variations and coastlines. The new solutions reveal both the complexity of the wave scattering process and the significant differences between locked and free waves.

FREQUENCY DISPERSION IN DEPTH-INTEGRATED MODELS THROUGH MACHINE LEARNING SURROGATES

Boussinesq- type wave models have the accuracy to resolve wave propagation in coastal zones, having the ability to capture nearshore dynamics that include both nonlinear and dispersive effects for relatively short waves. The accuracy of Boussinesq type models over their counterparts which utilize the non- linear shallow water (NLSW) equations provides a clear advantage in studying nearshore processes. However, the computational expense of finding the Boussinesq solution over the NLSW solution hinders fast and/ or real time simulation using Boussinesq type models.

Applications of physics informed neural operators

We present a critical analysis of physics-informed neural operators (PINOs) to solve partial differential equations (PDEs) that are ubiquitous in the study and modeling of physics phenomena using carefully curated datasets. Further, we provide a benchmarking suite which can be used to evaluate PINOs in solving such problems. We first demonstrate that our methods reproduce the accuracy and performance of other neural operators published elsewhere in the literature to learn the 1D wave equation and the 1D Burgers equation. Thereafter, we apply our PINOs to learn new types of equations, including the 2D Burgers equation in the scalar, inviscid and vector types. Finally, we show that our approach is also applicable to learn the physics of the 2D linear and nonlinear shallow water equations, which involve three coupled PDEs. We release our artificial intelligence surrogates and scientific software to produce initial data and boundary conditions to study a broad range of physically motivated scenarios. We provide the source code, an interactive website to visualize the predictions of our PINOs, and a tutorial for their use at the Data and Learning Hub for Science.

On Tsunami Waves Induced by Atmospheric Pressure Shock Waves After the 2022 Hunga Tonga‐Hunga Ha'apai Volcano Eruption

AbstractEmploying a linear shallow water equation (LSWE) model in the spherical coordinates, this paper investigates the tsunami waves generated by the atmospheric pressure shock waves due to the explosion of the submarine volcano Hunga Tonga–Hunga Ha'apai on 15 January 2022. Using the selected 59 atmospheric pressure records in the Pacific Ocean, an empirical atmospheric pressure model is first constructed. Applying the atmospheric pressure model and realistic bathymetric data in the LSWE model, tsunami generation and propagation are simulated in the Pacific Ocean. The numerical results show clearly the co‐existence of the leading locked waves, propagating with the speed of the atmospheric pressure waves (∼1,100 km/hr), and the trailing free waves, propagating with long gravity ocean wave celerity (∼750 km/hr). During the event, tsunamis were reported by 41 Deep‐ocean Assessment and Reporting of Tsunamis (DART) buoys in the Pacific Ocean, which require corrections because of the occurrence of atmospheric pressure waves. The numerically simulated tsunami arrival time and the amplitudes of the wave crest and trough of the leading locked waves compare reasonably well with the corrected DART measurements. The comparisons for the trailing waves are less satisfactory, since free waves could also have been generated by other tsunami generation mechanisms, which have not been considered in the present model, and by the scattering of locked waves over changing bathymetry. In this regard, the numerical results show clearly that the deep Tonga trench (∼10 km) amplifies the trailing waves in the Southeast part of the Pacific Ocean via the Proudman resonance condition.

Free Long-Wave Transformation in the Nearshore Zone through Partial Reflections

Abstract Long waves play an important role in coastal inundation and shoreline and dune erosion, requiring a detailed understanding of their evolution in nearshore regions and interaction with shorelines. While their generation and dissipation mechanisms are relatively well understood, there are fewer studies describing how reflection processes govern their propagation in the nearshore. We propose a new approach, accounting for partial reflections, which leads to an analytical solution to the free wave linear shallow-water equations at the wave-group scale over general varying bathymetry. The approach, supported by numerical modeling, agrees with the classic Bessel standing solution for a plane sloping beach but extends the solution to arbitrary alongshore uniform bathymetry profiles and decomposes it into incoming and outgoing wave components, which are a combination of successively partially reflected waves lagging each other. The phase lags introduced by partial reflections modify the wave amplitude and explain why Green’s law, which describes the wave growth of free waves with decreasing depth, breaks down in very shallow water. This reveals that the wave amplitude at the shoreline is highly dependent on partial reflections. Consistent with laboratory and field observations, our analytical model predicts a reflection coefficient that increases and is highly correlated with the normalized bed slope (bed slope relative to wave frequency). Our approach shows that partial reflections occurring due to depth variations in the nearshore are responsible for the relationship between the normalized bed slope and the amplitude of long waves in the nearshore, with direct implications for determining long-wave amplitudes at the shoreline and wave runup.

Analytical investigation of trapped waves over a submerged exponential ridge

Based on the linear shallow-water equations, analytical solutions for trapped waves over an asymmetric oceanic ridge with an exponential cross-sectional profile are presented. The solutions can become those for a symmetric ridge by setting the same topographic parameter for both sides, in which there are both symmetric and asymmetric trapped waves. The dispersion relationships imply that the wavenumber is related to not only the frequency and water depth at the top of the ridge but also the cross-section profile parameters and the mode number. With increasing frequency, the phase velocity decreases rapidly and approaches the classical shallow-water wave velocity for fixed water depth at the top and profile parameters. Meanwhile, for increasing angular frequency, the energy velocity first increases sharply from less than the shallow-water wave velocity and then decreases gently toward it. For an asymmetric ridge, the amplitude profile is confined to a narrower region on the steeper side and spread in a wider region on the less-steep side. Finally, the analytical solutions are applied to reveal the trapping effect of the Hawaii ridge on the 2011 Tohoku Tsunami in the North Pacific.

Assessment of Run-Up Height and Arrival Time of 365 AD Tsunami at Lampedusa Island, Italy

The eastern Mediterranean Sea is a subduction zone between the African plate and the Eurasian plate that has high seismicity. The tsunami of July 21, 365 AD was one of the most severe in the region, and its impact was estimated to have extended to the islands around the Aegean Sea, Cyprus, southern Calabria, Sicily, Malta, and Lampedusa. However, an analysis of tsunami waves reaching Lampedusa Island has not been conducted. Therefore, this study will present the run-up height and arrival time of tsunami waves near the Lampedusa coast. The methods applied are linear and non-linear shallow water equations by inputting the rupture parameters of the 365 AD earthquake. We used Delft Dashboard and Delft3D software to simulate tsunami wave propagation offshore. The simulation results indicate that an earthquake with a magnitude of approximately Mw 8.5 triggered tsunami waves with a maximum run-up height of 2.02 m in nearshore Lampedusa. The travel time for the maximum run-up height was 3.5 hours after the generation of the first tsunami wave in the western Hellenic Arc, Greece. We infer that seawater most likely inundated the coast of Lampedusa during the tsunami earthquake in 365 AD.

On multilevel Monte Carlo methods for deterministic and uncertain hyperbolic systems

In this paper, we evaluate the performance of the multilevel Monte Carlo method (MLMC) for deterministic and uncertain hyperbolic systems, where randomness is introduced either in the modeling parameters or in the approximation algorithms. MLMC is a well known variance reduction method widely used to accelerate Monte Carlo (MC) sampling. However, we demonstrate in this paper that for hyperbolic systems, whether MLMC can achieve a real boost turns out to be delicate. The computational costs of MLMC and MC depend on the interplay among the accuracy (bias) and the computational cost of the numerical method for a single sample, as well as the variances of the sampled MLMC corrections or MC solutions. We characterize three regimes for the MLMC and MC performances using those parameters, and show that MLMC may not accelerate MC and can even have a higher cost when the variances of MC solutions and MLMC corrections are of the same order. Our studies are carried out by a few prototype hyperbolic systems: a linear scalar equation, the Euler and shallow water equations, and a linear relaxation model, the above statements are proved analytically in some cases, and demonstrated numerically for the cases of the stochastic hyperbolic equations driven by white noise parameters and Glimm's random choice method for deterministic hyperbolic equations.

Analytical and numerical investigation of edge waves near a vertical breakwater over a convex bottom

Vertical breakwaters are commonly built on the shore, and their existence must influence the character of nearshore dynamics. Analytical solutions based on linear-shallow water equation are derived for edge waves on a convex exponential profile which may be used to idealize the actual bottom profile with a vertical breakwater at the shoreline. The free surface elevation of the edge waves is described using the first kind of the Bessel functions of order ν. The amplitude profiles are similar to those of edge waves on a sloping beach but confined to a narrower offshore domain near the vertical breakwater. The dispersion relationship is derived from the no-flux boundary condition at the vertical breakwater. The wave number is not only related to the frequency but also the water depth at the vertical breakwater and the profile parameter. The group velocity increases rapidly to the maximum value with the increase of frequency, then decreases slowly and finally approaches the classic shallow water wave celerity of the depth at the vertical breakwater. The possible reason for this distinct behavior may be because dynamic characteristics of edge waves are highly dependent on the profile. Moreover, an extensively validated Boussinesq wave model is used to conduct numerical experiments for edge waves induced by surface bumps. The numerical results are consistent with the proposed solutions, confirming the validity of the new analytical solutions.

Derivation of fundamental resonant period in width-varying semi-closed basins using modified SWE

The natural periods of water basin have a critical role in causing a resonance phenomenon. When the external waves coming into the basin possess similar periods, an increment in the wave amplitude might happen and give rise to serious damage to the surrounding environment. Many of the previous studies focused on finding the resonant period in basins with rectangular width. However, only a few have addressed the problem of basins with varying widths. Using a different analytical approach, we obtained the fundamental resonant period of a rectangular and triangular semi-closed basin. The analytical solution is obtained from modified linear shallow water equations using the separation of variables method. This new approach is simpler yet powerful in deriving the desired period. Furthermore, a modification of the staggered finite volume method is also proposed to find the periods numerically. The proposed scheme can be suitably applied to solve the discussed problem since it is conservative, robust, and free from damping errors. The results show that our analytical solutions align with those obtained from potential flow theory and the developed numerical scheme. Furthermore, we also compared the general characteristics of the resonance phenomenon that occurs in basins with constant and varying width. Since basins with non-constant width have lower fundamental resonant periods, the generated wave elevations are higher than those in constant-width basins.

TEMPORAL AND SPATIAL VARIATIONS IN TSUNAMI CODA ENERGY IN MEDIUM- TO LONG-TERM DECAY PROCESSES

Temporal and spatial variations in tsunami coda waves in medium- to long-term decay processes were investigated based on theoretical analysis and observations. In this paper, medium- and long-term were defined as approximately τ ≤ t ≤ 2τ and t ≥ 2τ, respectively, where t is elapsed time from the earthquake origin time, and τ is the coda energy e-folding decay time. In the medium- to long-term decay processes, the energy and amplitude of tsunami coda waves may become small so that the energy dissipation due to turbulent friction can be negligible compared to the directional spectrum of coda waves. It was found that the energy balance equation without dissipation for the directional spectrum, D(f,θ), should be approximately satisfied in a quasi-steady state, and the quantity CCgD(f,θ) should hold for any two points, where C and Cg are the wave phase speed and group velocity, respectively. The latter relationship indicates the inverse proportion of the tsunami coda energy to the water depth, and the similarity of the frequency spectrum of coda waves between any two points. The water depth dependency of tsunami coda was confirmed over a wide frequency range by using the 2011 Tohoku Earthquake Tsunami data. However, the tsunami coda energy with periods of 30–70 min did not follow the h-1 law, because continental seiche and edge waves were generated at Kochi, and the amplitude of tsunami coda waves at other stations did not show exponential decay in time. The estimated frequency spectra of coda waves at depths larger than 100 m were found to be inversely proportional to frequency squared, f2, for a wide frequency range. The f-2 law of frequency spectrum was theoretically derived by using linear shallow water equations with linear friction and dispersion terms. The effects of linear friction and dispersion on the frequency spectrum were deduced from the spectral solutions.

Integrating tsunami simulations in web applications using BROWNI, an open source client-side GPU-powered tsunami simulation library

Tsunami simulation software is a key component of state-of-the-art early warning systems but the inherent complexities in phases of installation, execution, pre and post-processing prevent their use in other areas of risk management such as communication and education. Recent advances in software and computational capacities such as the efficiency of GPU computing and the ubiquity of web browsers bring new opportunities to bridge the gap between expert and non-expert users. Here we present a Javascript library to enable a web browser to facilitate gathering and analyzing data from tsunami simulations, by means of interactive and efficient visualizations. At its core, the library uses WebGL, the browser’s standard 3D graphics API, to run GPU accelerated computations of a tsunami model. A far-field tsunami model is implemented (linear shallow water equations discretized on spherical coordinates), and its implementation is validated against real tsunami observations, and benchmarked with two other tsunami software-packages. Two software platforms that use this library are presented to illustrate the powerful applications that can be developed for risk communication and education. These applications are characterized by their interactivity and fast computations, which enable users to focus on the understanding of the phenomena of tsunami propagation and iterate quickly to assess different scenarios and potential implications to tsunami risk management. Some limitations on this approach are discussed, in aspects such as scalability, performance, multi-threading and batch-processing, that can be relevant for other users. In our experience, the before mentioned benefits very well compensate the discussed limitations for this kind of applications. The library has an open source license, and is meant to be imported without modifying its source code to facilitate the creation of new applications as the ones herein presented.

Analytical and numerical investigation of trapped ocean waves along a submerged ridge

Abstract

Asymptotic, Convergent, and Exact Truncating Series Solutions of the Linear Shallow Water Equations for Channels with Power Law Geometry

The present study was originally motivated by some intriguing exact solutions for waves propagating in nonuniform media. In particular, for special depth profiles reflected waves did not appear and...

The Use of the NAMI-DANCE Computational Complex on the Problem of Tsunami Waves

Mathematical models that are applicable to the simulation of the generation and propagation of tsunami waves from different sources, that is, underwater earthquakes, submarine landslides, and meteotsunamis, are described. The models are based on the well-known nonlinear shallow-water theory and its dispersion generalizations in two horizontal dimensions. Long wave dispersion related to the finiteness of water depth increases the order of the initial equations and, as a consequence, the time of computation. For this reason, physical dispersion in the investigations presented in this paper is replaced by numerical dispersion owing to a special choice of spatial and temporal steps. The numerical scheme for solving the shallow water equations is based on the leapfrog method. The equations are solved in spherical coordinates fixed to the rotating Earth with allowance for dissipative effects in the near-bottom layer with the use of the developed NAMI-DANCE code. For waves of seismic origin, the initial conditions for solving the hydrodynamic equations are taken from the solution of an elasticity theory problem to describe earthquake evolution (the Okada solution). In the case of a meteotsunami, atmospheric factors are taken into account as external forces and included in the right-hand sides of shallow water equations. Finally, generation of tsunami waves by submarine landslides is considered within the framework of a two-layer model with a lower viscous layer modeling the motion of the landslide. Two kinds of boundary conditions are used: free wave passage through open boundaries (in straits, etc.) using linear shallow-water equations and full reflection on the coast or in the near-shore area. Some test problems (benchmarks) on which the computational complex was verified are listed. The difficulties in the representation of tsunami characteristics due to bad data on the bottom bathymetry and topography of the land are mentioned. The possibility of using the developed code for the analysis of tsunami action on coasts and constructions is also discussed.

ON A STUDY OF STAGGERED LEAPFROG SCHEME FOR LINEAR SHALLOW WATER EQUATIONS

In this paper, we present an analytical and numerical study of staggered leapfrog scheme for linear shallow water equation. It is shown that the scheme is stable when Courant number < 1, has second order accurate in both time and space, and there is no damping error in this scheme. We implement the scheme to simulate standing wave in a closed basin to show that the surface motions stay zero in a node and have constant amplitude at the antinode. For an external force given into the basin, it will induce a resonance, which cause the wave amplitude is getting bigger at the position of antinode. Moreover, we simulate a wave in a tidal basin, and show that the model has infinite spin up time. For a linear shallow water equation with linear friction, it is shown that the model has finite spin up time.

Global Dissipation Models for Simulating Tsunamis at Far-Field Coasts up to 60 hours Post-Earthquake: Multi-Site Tests in Australia

At far-field coasts the largest tsunami waves may occur many hours post-arrival, and hazardous waves may persist for more than one day. Such tsunamis are often simulated by nesting high-resolution nonlinear shallow water models (covering sites of interest) within low-resolution reduced-physics global-scale models (to efficiently simulate propagation). These global models often ignore friction and are mathematically energy conservative, so in theory the modelled tsunami will persist indefinitely. In contrast, real tsunamis exhibit slow dissipation at the global-scale with an energy e-folding time of approximately one day. How strongly do these global-scale approximations affect nearshore tsunamis simulated at far-field coasts? To investigate this we compare modelled and observed tsunamis at sixteen nearshore tide-gauges in Australia, generated by the following earthquakes: $M_{w}9.5$ Chile 1960; $M_{w}9.2$ Sumatra 2004; $M_{w}8.8$ Chile 2010; $M_{w}9.1$ Tohoku 2011; and $M_{w}8.3$ Chile 2015. Each tsunami is represented using multiple published source models, to prevent bias in any single source from dominating the results. Each tsunami is simulated for 60 hours with a nested global-to-local model. On nearshore grids we solve the nonlinear shallow water equations with Manning-friction, while on the global grid we test three reduced-physics propagation models which combine the linear shallow water equations with alternative treatments of friction: 1) frictionless; 2) nonlinear Manning-friction; and 3) constant linear-friction. Compared with data, the frictionless global model well simulates nearshore tsunami maxima for $\simeq$ 8 hours after tsunami arrival, and Manning-friction gives similar predictions in this period. Constant linear-friction underestimates the size of early arriving waves. As the simulation duration is increased from 36 to 60 hours, the frictionless model increasingly overestimates observed wave heights, whereas models with global-scale friction work relatively well. The constant linear-friction model can be improved using delayed-linear-friction, where propagation is simulated with an initial frictionless period (12 hours herein). This prevents systematic underestimation of early wave heights. While nonlinear Manning-friction offers comparably good performance, a practical advantage of the linear-friction models herein is that solutions can be computed, to high accuracy, via a simple transformation of frictionless solutions. This offers a pragmatic approach to improving unit-source based global tsunami simulations at late times.