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The tidal bore of the Garonne River (France) was investigated on 29 August, 31 August and 27 October 2015, during which instantaneous velocity measurements were performed continuously at high-frequency (200 Hz). The tidal bore occurrence had a marked effect on the flow field and turbulent Reynolds stress data, indicating large shear stresses, together with large and rapid fluctuations, during the bore passage and the early flood tide. Like many natural process flows, the tidal bore flow motion was dominated by coherent structure activities and turbulent events, with significant impact on the natural systems including in terms of sediment processes. Herein a new turbulent event analysis was developed for the highly-unsteady rapidly-varied tidal bore flow. The analysis was based upon basic concepts, in which turbulent bursting events were defined in terms of the instantaneous relative turbulent flux, and the method was extended to the rapidly-varied, highly-unsteady tidal bore motion. The turbulent event data showed relatively close results for most fluxes during the tidal bores. The event duration showed some tidal trend, with longer turbulent events immediately after the tidal bore passage, occurring simultaneously with major sediment erosion processes. The field data set and analyses suggested that a turbulent event analysis may be applicable to highly-unsteady rapidly-varied flows, providing quantitative details into the turbulent bursts that are responsible for major mixing and sedimentary processes.

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a $C^0$ finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution., 27 pages, 13 figures

The paper reports unique high-resolution observations of meteotsunami by a large array of oceanographic instruments deployed on the Atchafalaya Shelf (Louisiana, USA) in 2008 with the primary aim to study wave dissipation in muddy environments. The meteotsunami event on March 7, 2008, was caused by the passage of a cold front which was monitored by the NOAA NEXRAD radar. The observations of water surface elevations on the shelf show a highly detailed textbook picture of an undular bore (solibore) in the process of its disintegration into a train of solitons. The picture has a striking feature never reported before not only for the meteotsunamis but in other contexts of disintegration of a long-wave perturbation into a sequence of solitons as well—the persistent presence of a single soliton, well ahead of the solibore. Data analysis and simulations based on the celebrated variable-coefficient KdV (vKdV) equation first proposed by Ostrovsky and Pelinovsky (Izv Atmos Ocean Phys 11:37–41, 1975) explain the physics of this phenomenon and suggest that the formation of the lone soliton ahead of the solibore is very likely to be the result of the specific interplay of natural meteotsunami forcing and nearshore bathymetry. The analysis strongly suggests that the patterns of coexisting lone solitons and packets of cnoidal waves should be quite common for meteotsunamis. They were not observed before only because of the scarcity of high-resolution observations. The results highlight the effectiveness of the vKdV equation in providing understanding of the fundamental mechanisms of the complex natural phenomenon that would otherwise require computationally very expensive numerical models.

This study addresses the impact of nonlinear wave evolution on the resulting wave force values on a vertical wall. To this end, the problem of interaction between non-breaking water waves and a vertical wall over constant depth is investigated. The investigation is performed using a two-dimensional wave flume model which is based on the high-order spectral method. Wave generation is simulated at the flume entrance by means of the additional potential concept. This model enables to preserve full dispersivity. Therefore, the model enables to examine the role of nonlinear evolution in the formation of extreme wave force values on a vertical wall for a wide range of water depths. The results for the force exerted on a vertical wall are presented for shallow and deep water conditions. In shallow water, extreme wave force values occur due to the formation of an undular bore. In deep water, extreme wave forces have been obtained as a result of disintegration process of incident wave groups into envelope solitons. Multiple maximum force values have been detected for each of the highest run-up peaks. This phenomenon has been introduced in shallow water conditions and is extended here for deep water conditions.

In this work, a free surface flow simulator is developed in which Navier–Stokes equations using Marker and Cell (MAC) method in the framework of finite difference methodology have been solved. The free surface is tracked by the volume of fluid (VOF) method. The numerical code without free-surface is validated against the flow past a square cylinder. Three different free surface flows, i.e., dam break flow, two-dimensional cavity filling and undular bore, are studied to demonstrate the efficacy of the developed numerical model to simulate free surface flow. The numerical model used in the present work involves tracking of free surface of a single fluid in a two-fluid system. The parameters which can affect the interface orientation of the fluid is given as boundary condition at the interface. The inherent advantage of such numerical models is its ability to track free surface for high density and viscosity ratio fluids like air–water. The numerical model used in the present work is capable of solving such high density ratio two-fluid systems for which the effects due to surface tension are negligible. Results from all the problems are compared with earlier results available in literature.

In some tsunami waves travelling over the ocean, such as the one approaching the eastern coast of Japan in 2011, the sea surface of the ocean is depressed by a small metre-scale displacement over a multi-kilometre horizontal length scale, lying in front of a positive elevation of comparable magnitude and length, which together constitute a down-up wave. Shallow water theory shows that the latter travels faster than the former, leading to an interaction, whose description is the issue addressed in this paper, using model equations of the Korteweg–de Vries type. First, we re-examine the undular bore solutions of the Korteweg–de Vries equation which describe how an initial depression wave deforms into a depression rarefaction wave followed by an undular bore of large elevation waves riding on this depression. Then we develop a new extended Korteweg–de Vries equation some of whose solutions can be used to describe the interaction of an elevation wave chasing a depression wave. These show that the two waves coincide at a given position and time producing a maximum elevation. Typically this amplitude is larger than the initial displacement magnitude by a factor which can be as large as two, which may explain anomalous elevations of tsunamis at particular positions along their trajectories. It is physically significant that for these small amplitude waves, no wave breaking occurs and there is no excess dissipation. Then, following the transition, the elevation wave moves ahead of the depression wave and the distance between them increases either linearly or logarithmically with time. The implications for how these down-up tsunami waves reach the shoreline are considered.

The paper presented a smoothed particle hydrodynamics (SPH) method to study the three-dimensional (3D) tidal bore scenarios. The SPH method is a mesh-free particle modeling technique that can track the large deformation of free surfaces in a straightforward and accurate way. Two benchmark cases of the tidal bore propagation were computed and compared with the experimental results. The first one is related to the undular and breaking bores in a regular open channel, and the second one considers the undular bore passing through the contraction of bridge piers. Physical laboratory experiments have also been carried out to validate the numerical investigations. The comparisons of both the free surface profile and velocity field demonstrated that the SPH technique could provide a very promising tool to simulate tidal bore phenomena in engineering practice. The work is the first to systematically explore the potentials of mesh-free SPH modeling approach in predicting the tidal bore features under 3D flow conditions.

A tidal bore is a series of waves propagating upstream as the tidal flow turns to rising, and the bore front corresponds to the leading edge of the tidal wave in a funnel shaped estuarine zone with macro-tidal conditions. Some field observations were conducted in the tidal bore of the Garonne River on 7 June 2012 in the Arcins channel, a few weeks after a major flood. The tidal bore was a flat undular bore with a Froude number close to unity: Fr1 = 1.02 and 1.19 (morning and afternoon respectively). A key feature of the study was the simultaneous recording of the water elevation, instantaneous velocity components and suspended sediment concentration (SSC) estimates, together with a detailed characterisation of the sediment bed materials. The sediment was some silty material (d50 ≈ 13 μm) which exhibited some non-Newtonion thixotropic behaviour. The velocity and SSC estimate were recorded simultaneously at high frequency, enabling a quantitative estimate of the suspended sediment flux at the end of the ebb tide and during the early flood tide. The net sediment flux per unit area was directed upstream after the bore, and its magnitude was much larger than that at end of ebb tide. The field observations highlighted a number of unusual features on the morning of 7 June 2012. These included (a) a slight rise in water elevation starting about 70 s prior to the front, (b) a delayed flow reversal about 50 s after the bore front, (c) some large fluctuations in suspended sediment concentration (SSC) about 100 s after the bore front and (d) a transient water elevation lowering about 10 min after the bore front passage. The measurements of water temperature and salinity showed nearly identical results before and after the tidal bore, with no evidence of saline and thermal front during the study.

To study the hydrodynamics of tidal bore, a physical modeling study is carried out in a rectangular flume with considerations of the tidal bore heights, the propagation speeds, the tidal current velocities, the front steepness, and the bore shapes. After the validation with the field observations, the experimental results are analyzed, and it is shown that: (1) the greater initial ebb velocity or the larger initial water depth impedes the tidal bore propagation, (2) the maximum bore height appears at an initial ebb velocity in the range of 0.5 m/s-1.5 m/s, (3) when the Froude number exceeds 1.2, an undular bore appears, after it exceeds 1.3, a breaking bore occurs, and after it exceeds 1.7, the bore is broken.

In this paper, numerical solutions for the Rosenau-Korteweg-de Vries equation are studied by using the subdomain method based on the sextic B-spline basis functions. Numerical results for five test problems including the motion of single solitary wave, interaction of two and three well-separated solitary waves of different amplitudes, evolution of solitons with Gaussian and undular bore initial conditions are obtained. Stability and a priori error estimate of the scheme are discussed. A comparison of the values of the obtained invariants and error norms for single solitary wave with earlier results is also made. The results show that the present method is efficient and reliable.

In this work, we construct the lumped Galerkin approach based on cubic B-splines to obtain the numerical solution of the generalized regularized long wave equation. Applying the von Neumann approximation, it is shown that the linearized algorithm is unconditionally stable. The presented method is implemented to three test problems including single solitary wave, interaction of two solitary waves and development of an undular bore. To prove the performance of the numerical scheme, the error norms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{2}$$\end{document}L2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L_{\infty}}$$\end{document}L∞ and the conservative quantities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I_{1}}$$\end{document}I1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I_{2}}$$\end{document}I2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I_{3}}$$\end{document}I3 are computed and the computational data are compared with the earlier works. In addition, the motion of solitary waves is described at different time levels.

In this article, consideration is given to weak bores in free-surface flows. The energy loss in the shallow-water theory for an undular bore is thought to be due to upstream oscillations that carry away the energy lost at the front of the bore. Using a higher-order dispersive model equation, this expectation is confirmed through a quantitative study which shows that there is no energy loss if dispersion is accounted for.

In the study, differential quadrature methods based on cubic B-splines are proposed. In order to test the accuracy and the efficiency of the present methods, some numerical solutions of the Regularized Long Wave Equation are simulated. Wave generation, traveling solitary wave, interaction of the two solitary waves and undulation solutions of the RLW equation are studied. Obtained simulations are compared with some earlier works. Lowest three conserved quantities are computed for all test problems.

The process of tidal bore passing through bridge piers is numerically simulated in an open channel using a pure Lagrangian method, i.e. Smooth Particle Hydrodynamics (SPH). Due to its “gridless” feature, SPH is intrinsically well-suited for the complex hydrodynamic simulation when the fluid encounters a structure. The behavior of bores propagation along the channel is illustrated herein and includes the bore's free surface properties and flow field, and in particular, the intricate behavior of unsteady flow during the passage of tidal bore. The generation of shock waves between the structures can be clearly observed in the numerical experiment. Intense vortical structures behind the bore front are generated as the bore propagate through the constriction.

A tidal bore may occur in a macro-tidal estuary when the tidal range exceeds 4.5–6 m and the estuary bathymetry amplifies the tidal wave. Its upstream propagation induces a strong mixing of the estuarine waters. The propagation of undular tidal bores was investigated herein to study the effect of bridge piers on the bore propagation and characteristics. Both the tidal bore profile and the turbulence generated by the bore were recorded. The free-surface undulation profiles exhibited a quasi-periodic shape, and the potential energy of the undulations represented up to 30% of the potential energy of the tidal bore. The presence of the channel constriction had a major impact on the free-surface properties. The undular tidal bore lost nearly one third of its potential energy per surface area as it propagated through the channel constriction. The detailed instantaneous velocity measurements showed a marked effect of the tidal bore passage suggesting the upstream advection of energetic events and vorticity “clouds” behind the bore front in both channel configurations: prismatic and with constriction. The turbulence patches were linked to some secondary motions and the proposed mechanisms were consistent with some field observations in the Daly River tidal bore. The findings emphasise the strong mixing induced by the tidal bore processes, and the impact of bridge structures on the phenomenon.

A tidal bore is a natural and fragile phenomenon, which is of great importance for the ecology of an estuary. The bore development is closely linked with the tidal range and the river mouth shape, and its existence is sensitive to any small change in boundary conditions. Despite their ecological and cultural value, little is known on the flow field, turbulent mixing and sediment motion beneath tidal bores. Indeed, some striking features can be highlighted in two-dimensional simulations, such as large velocity fluctuations and flow recirculation structures. Using Large Eddy Simulation method, the numerical results emphasised the complicated turbulent structures and their unsteadiness under a tidal bore.

The role of turbulence is investigated in dam-break flows, where a finite volume of fluid is released from a compartment into a long, rectangular channel. After a sudden removal of the lock gate, a gravity current, undular bore, or solitary wave develops, depending on the ambient fluid height in the channel. The temporal evolution of the moving front has been measured and evaluated. It was observed that the dilution using a very small amount (a few weight ppm) of a long chain polymer (polyethylene-oxide) in the fluid strongly affected flow properties. Pronounced drag reduction has been found in dry bed flows (whereas the polymer increased the viscosity of the fluid). The presence of a few mm-thick ambient fluid layer in the channel effectively destroyed drag reduction, in spite of the fact that strong turbulence was obvious and the propagation velocity of the front was almost unchanged.

A tidal bore is a hydrodynamic discontinuity propagating upstream in an estuarine zone with a funnel shape as the tide starts rising under spring tidal conditions. The transient sediment motion beneath tidal bores was investigated in laboratory under controlled flow conditions by measuring simultaneously the fluid and sediment particle velocities. Although no sediment transport was observed in the initially steady flow and in undular bores, a transient sheet flow motion was observed beneath the breaking bores. The sediment transport was initiated during the passage of the bore roller toe by the large longitudinal pressure gradient force, and the sediment particles were subjected to large horizontal accelerations. About 5 % of all particles were accelerated in excess of 1 g. The sediments were advected upstream with an average velocity close to the instantaneous fluid velocity. The time evolution of instantaneous particle velocity for each trajectory was analysed, using the starting point of particle trajectory corresponding to the entrainment point, and the end point to the particle stoppage point. The present data provided some quantitative data in terms of force terms acting on sediment particles beneath a tidal bore and their trajectory characteristics.