Rogue waves in shallow water

Most of the processes resulting in the formation of unexpectedly high surface waves in deep water (such as dispersive and geometrical focusing, interactions with currents and internal waves, reflection from caustic areas, etc.) are active also in shallow areas. Only the mechanism of modulational instability is not active in finite depth conditions. Instead, wave amplification along certain coastal profiles and the drastic dependence of the run-up height on the incident wave shape may substantially contribute to the formation of rogue waves in the nearshore. A unique source of long-living rogue waves (that has no analogues in the deep ocean) is the nonlinear interaction of obliquely propagating solitary shallow-water waves and an equivalent mechanism of Mach reflection of waves from the coast. The characteristic features of these processes are (i) extreme amplification of the steepness of the wave fronts, (ii) change in the orientation of the largest wave crests compared with that of the counterparts and (iii) rapid displacement of the location of the extreme wave humps along the crests of the interacting waves. The presence of coasts raises a number of related questions such as the possibility of conversion of rogue waves into sneaker waves with extremely high run-up. Also, the reaction of bottom sediments and the entire coastal zone to the rogue waves may be drastic.     

A primitive-variable Riemann method for solution of the shallow water equations with wetting and drying

A Riemann flux that uses primitive variables rather than conserved variables is developed for the shallow water equations with nonuniform bathymetry. This primitive-variable flux is both conservative and well behaved at zero depth. The unstructured finite-volume discretization used is suitable for highly nonuniform grids that provide resolution of complex geometries and localized flow structures. A source-term discretization is derived for nonuniform bottom that balances the discrete flux integral both for still water and in dry regions. This primitive-variable formulation is uniformly valid in wet and dry regions with embedded wetting and drying fronts. A fully nonlinear implicit scheme and both nonlinear and time-linearized explicit schemes are developed for the time integration. The implicit scheme is solved by a parallel Newton-iterative algorithm with numerically computed flux Jacobians. A concise treatment of characteristic-variable boundary conditions with source terms is also given. Computed results obtained for the one-dimensional dam break on wet and dry beds and for normal-mode oscillations in a circular parabolic basin are in very close agreement with the analytical solutions. Other results for a forced breaking wave with friction interacting with a sloped bottom demonstrate a complex wave motion with wetting, drying and multiple interacting wave fronts. Finally, a highly nonuniform, coastline-conforming unstructured grid is used to demonstrate an unsteady simulation that models an artificial coastal flooding due to a forced wave entering the Gulf of Mexico.     

Monitoring near-shore shingle transport under waves using a passive acoustic technique

Passive acoustic techniques have been used to measure shingle (gravel) sediment transport in very shallow water, near the wave breaking zone on a beach. The experiments were conducted at 1:1 scale in the Large Wave Flume, Grosse Wellen Kanal (GWK) at Hannover, Germany. The frequency spectrum induced by shingle mobilized under breaking waves can be distinguished from other ambient noise, and is found to be independent of water depth and wave conditions. The inverse relationship between centroid frequency and representative grain size is shown to remain valid in shallow water wave conditions. Individual phases of onshore and offshore transport can be identified. Analysis of the acoustic frequency spectrum provides insight into the mechanics of phase-resolved shingle transport.     

Long wave run-up on random beaches

The estimation of the maximum wave run-up height is a problem of practical importance. Most of the analytical and numerical studies are limited to a constant slope plain shore and to the classical nonlinear shallow water equations. However, in nature the shore is characterized by some roughness. In order to take into account the effects of the bottom rugosity, various ad hoc friction terms are usually used. In this Letter, we study the effect of the roughness of the bottom on the maximum run-up height. A stochastic model is proposed to describe the bottom irregularity, and its effect is quantified by using Monte Carlo simulations. For the discretization of the nonlinear shallow water equations, we employ modern finite volume schemes. Moreover, the results of the random bottom model are compared with the more conventional approaches.     

Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method

The wave equation for linear shallow water waves propagating over a varying bottom topography has the same form as that for p-polarized electromagnetic waves in inhomogeneous dielectric media. The role played by the dielectric permittivity in the case of electromagnetic waves is played by the inverse water depth. We apply the invariant imbedding theory of wave propagation, which has been developed mainly to study the electromagnetic wave propagation, to linear shallow water waves in the special case where the depth depends on only one coordinate. By comparing the numerical result obtained using our method, when the depth profile is linear, with an exact analytical formula, we demonstrate that our method is numerically reliable. The invariant imbedding method can be used in studying the influence of complicated bottom topography on the propagation of shallow water waves, in a numerically exact manner. We illustrate this by considering the case where a periodic modulation is added to a linear depth profile. Bragg scattering due to the periodic component competes with the tsunami effect due to the linear depth variation. This competition is seen to generate interesting physical effects. We also consider a ridge-type bottom topography and examine the resonant transmission phenomenon associated with the Fabry–Perot effect.     

Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method

The wave equation for linear shallow water waves propagating over a varying bottom topography has the same form as that for p-polarized electromagnetic waves in inhomogeneous dielectric media. The role played by the dielectric permittivity in the case of electromagnetic waves is played by the inverse water depth. We apply the invariant imbedding theory of wave propagation, which has been developed mainly to study the electromagnetic wave propagation, to linear shallow water waves in the special case where the depth depends on only one coordinate. By comparing the numerical result obtained using our method, when the depth profile is linear, with an exact analytical formula, we demonstrate that our method is numerically reliable. The invariant imbedding method can be used in studying the influence of complicated bottom topography on the propagation of shallow water waves, in a numerically exact manner. We illustrate this by considering the case where a periodic modulation is added to a linear depth profile. Bragg scattering due to the periodic component competes with the tsunami effect due to the linear depth variation. This competition is seen to generate interesting physical effects. We also consider a ridge-type bottom topography and examine the resonant transmission phenomenon associated with the Fabry-Perot effect.     

A model for the horizontal directionality of breaking wave noise in the surf zone

A model is presented for the horizontal directionality of the noise generated by individual breaking waves in the surf zone. The model is based on the interaction between sound radiated by ringing bubbles created in a breaking wave crest and the population of acoustically quiescent bubbles left on the seaward side of the wave in the shallow water waveguide. The effect of the quiescent bubbles is to absorb sound from all but the very ends of the breaking crest, resulting in the formation of "acoustic hot-spots." The model calculations are in good agreement with observations of the horizontal directionality of noise from individual breaking waves in the surf zone.     

Unstructured finite volume discretisation of bed friction and convective flux in solute transport models linked to the shallow water equations

The finite volume discretisation of the shallow water equations has been the subject of many previous studies, most of which deal with a well-balanced conservative discretisation of the convective flux and bathymetry. However, the bed friction discretisation has not been so profusely analysed in previous works, while it may play a leading role in certain applications of shallow water models. In this paper we analyse the numerical discretisation of the bed friction term in the two-dimensional shallow water equations, and we propose a new unstructured upwind finite volume discretisation for this term. The new discretisation proposed improves the accuracy of the model in problems in which the bed friction is a relevant force in the momentum equation, and it guarantees a perfect balance between gravity and bed friction under uniform flow conditions. The relation between the numerical scheme used to solve the hydrodynamic equations and the scheme used to solve a scalar transport model linked to the shallow water equations, is also analysed in the paper. It is shown that the scheme used in the scalar transport model must take into consideration the scheme used to solve the hydrodynamic equations. The most important implication is that a well-balanced and conservative scheme for the scalar transport equation cannot be formulated just from the water depth and velocity fields, but has to consider also the way in which the hydrodynamic equations have been solved.     

Study of a novel range-dependent propagation effect with application to the axial injection of signals from the Kaneohe source

A novel range-dependent propagation effect occurs when a source is placed on the seafloor in shallow water with a downward refracting sound speed profile, and sound waves propagate down a slope into deep water. Under these conditions, small grazing-angle sound waves slide along the bottom downward and outward from the source until they reach the depth of the sound channel axis in deep water, where they are detached from the sloping bottom and continue to propagate outward near the sound channel axis. This "mudslide" effect is one of a few robust and predictable acoustic propagation effects that occur in range-dependent ocean environments. As a consequence of this effect, a bottom mounted source in shallow water can inject a significant amount of acoustic energy into the axis of the deep ocean sound channel that can then propagate to very long ranges. Numerical simulations with a full-wave range-dependent acoustic model show that the Kaneohe experiment had the appropriate source, bathymetry, and sound speed profiles that allows this effect to operate efficiently. This supports the interpretation that some of the near-axial acoustic signals, received near the coast of California from the bottom mounted source located in shallow water in Kaneohe Bay, Oahu, Hawaii, were injected into the sound channel of the deep Pacific Ocean by this mechanism. Numerical simulations suggest that the mudslide effect is robust.     

A splitting approach for the fully nonlinear and weakly dispersive Green–Naghdi model

The fully nonlinear and weakly dispersive Green–Naghdi model for shallow water waves of large amplitude is studied. The original model is first recast under a new formulation more suitable for numerical resolution. An hybrid finite volume and finite difference splitting approach is then proposed, which could be adapted to many physical models that are dispersive corrections of hyperbolic systems. The hyperbolic part of the equations is handled with a high-order finite volume scheme allowing for breaking waves and dry areas. The dispersive part is treated with a classical finite difference approach. Extensive numerical validations are then performed in one horizontal dimension, relying both on analytical solutions and experimental data. The results show that our approach gives a good account of all the processes of wave transformation in coastal areas: shoaling, wave breaking and run-up.     

Berkhoff approximation in a problem on surface gravity wave propagation in a basin with bottom irregularities

The problem of the propagation of small-amplitude surface gravity waves in a basin of constant mean depth with one- and two-dimensional bottom roughness is solved in the framework of the Berkhoff model by a mean-field method. In both cases the solutions obtained are compared with the solutions of sets of exact linearized equations of the hydrodynamics of an incompressible fluid. The comparison of the exact and approximate mean-field attenuation coefficients has shown that the Berkhoff approximation is appropriate for the solution of this problem in the case of shallow water for an arbitrary correlation length of bottom irregularities and in the case of arbitrary depth and large-scale irregularities. An explanation is given for the limits of applicability of the Berkhoff approximation which are connected with the weak variability of the vertical structure of the wave field in shallow water and in a basin with large-scale depth fluctuations. The mean-field attenuation coefficients reach their maximum values in the region k_oh_o≥1 (where k_o is the wavenumber of the surface gravity wave in a basin of constant depth h_o). The location of these maxima is practically independent of the correlation length of the bottom irregularities. For the case of one-dimensional irregularities the effect of bottom roughness on the surface gravity wave velocity is investigated. It is shown that the surface wave in a basin with an uneven bottom may propagate more slowly, as well as faster than the wave in a basin with an even bottom, depending on the relations between the wavelength, depth and correlation length of the bottom imperfections.     

Collective bubble oscillations as a component of surf infrasound

Plunging surf is a known generator of infrasound, though the mechanisms have not been clearly identified. A model based on collective bubble oscillations created by demise of the initially entrained air pocket is examined. Computed spectra are compared to infrasound data from the island of Kauai during periods of medium, large, and extreme surf. Model results suggest that bubble oscillations generated by plunging waves are plausible generators of infrasound, and that dynamic bubble plume evolution on a temporal scale comparable to the breaking wave period may contribute to the broad spectral lobe of dominant infrasonic energy observed in measured data. Application of an inverse model has potential to characterize breaking wave size distributions, energy, and temporal changes in seafloor morphology based on remotely sensed infrasound.     

Wave Riemann description of friction terms in unsteady shallow flows: Application to water and mud/debris floods

In this work, the source term discretization in hyperbolic conservation laws with source terms is considered using an approximate augmented Riemann solver. The technique is applied to the shallow water equations with bed slope and friction terms with the focus on the friction discretization. The augmented Roe approximate Riemann solver provides a family of weak solutions for the shallow water equations, that are the basis of the upwind treatment of the source term. This has proved successful to explain and to avoid the appearance of instabilities and negative values of the thickness of the water layer in cases of variable bottom topography. Here, this strategy is extended to capture the peculiarities that may arise when defining more ambitious scenarios, that may include relevant stresses in cases of mud/debris flow. The conclusions of this analysis lead to the definition of an accurate and robust first order finite volume scheme, able to handle correctly transient problems considering frictional stresses in both clean water and debris flow, including in this last case a correct modelling of stopping conditions.     

磁化水的磁化机理研究

研究了搅拌状态下三种磁化水的物理性质.得到水的表面张力系数、黏度和密度随磁感应强度和磁化时间均呈现多极值增减变化关系.该实验规律虽亦有多篇文献报道,但至今尚无合理解释.本文从水的分子结构和氢键的特点出发,综合应用电磁学、结构化学和热学的相关理论对水的磁化机理作了进一步研究,对磁化水表面张力系数、黏度和密度的多极值现象给出了解释.研究认为,当将水体置于200 mT以上的磁场中磁化时,电子受到磁场作用或扰动,状态发生变化,影响了电子间的相互作用,使部分氢键断裂.由于形成氢键的条件易于满足,液态水在一定温度下氢 关键词： 磁化水 磁化机理 表面张力 氢键     

From the paddle to the beach – A Boussinesq shallow water numerical wave tank based on Madsen and Sørensen’s equations

This article describes a one-dimensional numerical model of a shallow-water flume with an in-built piston paddle moving boundary wavemaker. The model is based on a set of enhanced Boussinesq equations and the nonlinear shallow water equations. Wave breaking is described approximately, by locally switching to the nonlinear shallow water equations when a critical wave steepness is reached. The moving shoreline is calculated as part of the solution. The piston paddle wavemaker operates on a movable grid, which is Lagrangian on the paddle face and Eulerian away from the paddle. The governing equations are, however, evolved on a fixed mapped grid, and the newly calculated solution is transformed back onto the moving grid via a domain mapping technique. Validation test results are compared against analytical solutions, confirming correct discretisation of the governing equations, wave generation via the numerical paddle, and movement of the wet/dry front. Simulations are presented that reproduce laboratory experiments of wave runup on a plane beach and wave overtopping of a laboratory seawall, involving solitary waves and compact wave groups. In practice, the numerical model is suitable for simulating the propagation of weakly dispersive waves and can additionally model any associated inundation, overtopping or inland flooding within the same simulation.     

浅海近程混响的振荡现象*

研究浅海近程混响特性对于评估和提高主动声纳性能具有重要意义。多次浅海混响实验显示,近程混响强度存在稳定的振荡现象,脉宽基本对振荡的幅度和周期没有影响。为解释这一现象,本文基于射线理论和小斜率近似给出了浅海近程混响模型,仿真与实测数据结果基本吻合。数值仿真结果表明：海底反射声场对单站声纳接收到回声信号的贡献远小于海底近垂向大掠射角散射声场的作用;混响强度振荡现象是海底近程散射声场的多途现象造成的,并由此给出了振荡周期与海深及收发深度的关系。     

Anomalous sea surface reverberation scale model experiments

Low frequency sea surface sound backscattering from approximately 100 Hz to a few kHz observed from the 1960s broadband measurements using explosive charges to the Critical Sea Test measurements conducted in the 1990 s is substantially higher than explained by rough sea surface scattering theory. Alternative theories for explaining this difference range from scattering by bubble plumes/clouds formed by breaking waves to stochastic scattering from fluctuating bubble layers near the sea surface. In each case, theories focus on reverberation in the absence of the large-scale surface wave height fluctuations that are characteristic of a sea that produces bubble clouds and plumes. At shallow grazing angles, shadowing of bubble plumes and clouds caused by surface wave height fluctuations may induce first order changes in the backscattered signal strength. To understand the magnitude of shadowing effects under controlled and repeatable conditions, scale model experiments were performed in a 3 m x 1.5 m x 1.5 m tank at the Technical University of Denmark. The experiments used a 1 MHz transducer as the source and receiver, a computer controlled data acquisition system, a scale model target, and a surface wave generator. The scattered signal strength fluctuations observed at shallow angles are characteristic of the predicted ocean environment. These experiments demonstrate that shadowing has a first order impact on bubble plume and cloud scattering strength and emphasize the usefulness of model scale experiments for studying underwater acoustic events under controlled conditions.     

陆架斜坡海域上坡波导环境中声能量急剧下降现象及其定量分析

深度较浅的声源其辐射声波在陆架斜坡海域上坡传播时,在斜坡顶端会出现声能量急剧下降现象.利用射线声学模型分析了造成这一现象的原因,并根据抛物方程声场模型计算的深海和浅海平均传播损失定义了"声能量急剧下降距离",定量分析了声源位置对该现象的影响.结果表明:声源深度对"声能量急剧下降距离"影响较大,而声源与斜坡底端水平距离对其影响较小;当声源深度变大时,部分掠射角较小的声线最终能够达到斜坡顶端,致使"声能量急剧下降距离"增大,继续增加声源深度,将导致上坡声能量急剧下降现象消失.利用抛物方程声场模型对陆架斜坡海域上坡声传播进行数值仿真,结合"声能量急剧下降距离"的定义,计算并比较了声源位置不同时该距离的变化,数值计算结果验证了理论分析.     

Theoretical analysis and experimental results of interference striation pattern of underwater target radiated noise in shallow water waveguide

Waveguide invariant in shallow water is an attractive topic in recent three decades. The interference phenomena of direct wave of radiated noise of underwater target and reflection wave from sea surface and sea bottom can be considered as a typical case of shallow water waveguide.The interference striation pattern of direct wave and its reflection is the effective and comprehensive figure for better understanding the essence of shallow water waveguide invariant.The theoretical analysis of interference phenomena generated by direct wave of radiated noise of underwater target and its reflection wave from sea surface and sea bottom is presented in this paper.It is shown that the interference wave resulted by sea surface reflection will produce striation pattern centered at high frequency band.But the interference of nulling frequency resulted by sea bottom reflection may be at low frequency or high frequency, it strongly depends on the acoustic behavior of sea bottom.The relationship between main parameters of interference striation pattern and target,receiver,and environment is derived.It is shown that the interference striation have the shape of hyperbolic curve.The equation set of the hyperbolic curve and its asymptotic line is presented.The at sea experiment carried out in South China sea shows some interesting results.A part of data processing results are illustrated in this paper.The results expressed in this paper show that the interference striation pattern can be used,in some conditions,as a potential means for target recognition.     

浅海小掠射角的海底界面声反向散射模型的简化

在浅海,尤其是负梯度声速剖面和海面较为平静的浅海波导,海底界面反向散射是浅海混响的主要来源.经验散射模型只适用于分析浅海混响平均强度衰减特性,而基于物理机理建立的反向散射模型克服了这一缺陷,但同时也引入了其受地声模型约束的问题.本文结合了海底反射系数的三参数模型,对浅海远场海底反向散射模型进行了简化,以减少地声模型的输入参数.理论分析了海底反射系数的相移参数可以描述海底对声场的散射作用,无需任何海底地声参数的先验知识.通过对海底反向散射模型近似简化,结果表明在临界角附近和甚小掠射角范围内的海底粗糙界面反向散射模型的角度特性和强度特性受海底沉积层的影响不同:在临界角附近,海底反向散射的角度特性受海底反射系数的相移参数加权,而其散射系数则近似与相移参数无关;对于甚小掠射角,海底反向散射的角度特性近似与海底反射系数的相移参数无关,其散射系数则近似与相移参数的4次方成正比.