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This paper proposes an accurate and robust meshless approach for the numerical solution of the nonlinear equal width equation. The numerical technique is applied for approximating the spatial variable derivatives of the model based on the localized radial basis function-finite difference (RBF-FD) method. Another implicit technique based on θ − weighted and finite difference methods is also employed for approximating the time variable derivatives. The stability analysis of the approach is demonstrated by employing the Von Neumann approach. Next, six test problems are solved including single solitary wave, fusion of two solitary waves, fusion of three solitary waves, soliton collision, undular bore, and the Maxwellian initial condition. Then, the L 2 and L ∞ norm errors for the first example and the I 1 , I 2 , and I 3 invariants for the other examples are calculated to assess accuracy of the method. Finally, the validity, efficiency and accuracy of the method are compared with those of other techniques in the literature.

An undular mesospheric bore event has been recorded over the western Himalayan region in O(1S) 557.7 nm airglow images on a clear and moonless night of 02 October 2018 using a multi-wavelength all-sky imager at Hanle, Leh Ladakh, India (32.77°N, 78.97°E). The bore has a prominent leading dark front followed by trailing waves and it propagates with a mean observed phase velocity of ~31 ± 5 m/s. It also shows a small-scale undulation and clockwise rotation in its phase front. In order to understand the evolution of the bore, vertical temperature profiles from the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument onboard TIMED satellite and HWM14 wind maps are used. SABER temperature shows a mesospheric inversion layer prior to the occurrence of the bore event which acted as a thermal duct layer to guide the propagation of the bore. The analyses also suggest that inhomogeneity in the duct depth at different parts of the bore’s horizontal extension could lead the small-scale undulations in the bore phase fronts. The same can also be responsible for the rotation of the bore. Additionally, the peak separation of the bore’s trailing waves suggests that large-scale gravity wave interaction with pre-existing thermal duct could be the potential source for the generation of the undular bore at mesospheric height. Furthermore, the present results indicate that neutral instabilities and weakening duct layer in the path of the bore propagation might have accelerated the faster dissipation of the bore’s energy and consequently suppress its long-distance horizontal propagation.

In this work the dispersive shock wave solution of a Boussinesq Benjamin-Ono (BBO) equation, the standard Boussinesq equation with dispersion replaced by nonlocal Benjamin-Ono dispersion, isderived. This dispersive shock wave solution is derived using two methods, dispersive shock wavefitting and from a simple wave solution of theWhitham modulation equations for the BBO equation.The first of these yields the two edges of the dispersive shock wave, while the second yields the complete dispersive shock wave solution. As the Whitham modulation equations could not be set in Riemann invariant form, the ordinary differential equations governing the simple wave are solved using a hybrid numerical method coupled to the dispersive shock fitting which provides a suitable boundary condition. The full dispersive shock wave solution is then determined, which is found to be in excellent agreement with numerical solutions of the BBO equation. This hybrid method is a suitable and relatively simple method to fully determine the dispersive shock wave solution of anonlinear dispersive wave equation for which the (hyperbolic) Whitham modulation equations are known, but their Riemann invariant form is not.

It is known from linear theory that bottom oscillations in uniform open-channel flow can produce resonant surface waves with zero group velocity and diverging amplitude (Tyvand and Torheim 2012). This resonance exists for Froude numbers smaller than one, at a critical frequency dependent on the Froude number. This resonance phenomenon is studied numerically in the time domain, with fully nonlinear free-surface conditions. An oscillatory 2D bottom source is started, and the local elevation at resonance grows until it may reach a saturation amplitude. Four waves exist at subcritical Froude numbers, where resonance represents the third and the fourth wave merging. In the zero-frequency limit, the dispersive second and fourth wave merge into a steady wave with finite group velocity and amplitude, and no other periodic waves exist. In the time-dependent nonlinear analysis at zero frequency, a transient undular bore may emerge as the dominating phenomenon.

In this article, we study solitary-wave solutions of the nonlinear Benjamin–Bona–Mahony–Burgers(BBM–Burgers) equation based on a lumped Galerkin technique using cubic B- spline finite elements for the spatial approximation. The existence and uniqueness of solutions of the Galerkin version of the solutions have been established. An accuracy analysis of the Galerkin finite element scheme for the spatial approximation has been well studied. The proposed scheme is carried out for four test problems including dispersion of single solitary wave, interaction of two, three solitary waves and development of an undular bore. Then we propose a full discrete scheme for the resulting IVP. Von Neumann theory is used to establish stability analysis of the full discrete numerical algorithm. To display applicability and durableness of the new scheme, error norms L 2 , L ∞ and three invariants I 1 , I 2 and I 3 are computed and the acquired results are demonstrated both numerically and graphically. The obtained results specify that our new scheme ensures an apparent and an operative mathematical instrument for solving nonlinear evolution equation.

Undular bores, also termed dispersive shock waves, generated by an initial discontinuity in height as governed by two forms of the Boussinesq system of weakly nonlinear shallow water wave theory, the standard formulation and a Hamiltonian formulation, two related Whitham-Boussinesq equations and the full water wave equations for gravity surface waves are studied and compared. It is found that the Whitham-Boussinesq systems give solutions in excellent agreement with numerical solutions of the full water wave equationsfor the positions of the leading and trailing edges of the bore up until the onset on modulational instability. TheWhitham-Boussinesq systems, which are far simpler than the full water wave equations, can then be used to accurately model surface water wave undular bores. Finally, comparisons with numerical solutions of the full water wave equations show that the Whitham-Boussinesq systems give a slightly lower threshold for the onset of modulational instability in terms of the height of the initial step generating the undular bore.

We investigate the removal of a dense bottom layer by a gravity current, via Navier–Stokes simulations in the Boussinesq limit. The problem is governed by a dimensionless thickness parameter for the bottom layer, and by the ratio of the density differences between bottom layer, gravity current and ambient fluids. A quasisteady gravity current forms that propagates along the interface and displaces some of the dense bottom fluid, which accumulates ahead of the gravity current and forms an undular bore or a series of internal gravity waves. Depending on the ratio of the gravity current front velocity to the linear shallow-water wave velocity, we observe the existence of different regimes, characterized by small-amplitude waves or by a train of steep, nonlinear internal waves. We develop a semiempirical model that provides reasonable estimates of several important flow properties. We also formulate a more sophisticated, self-contained model based on the conservation principles for mass and vorticity that does not require empirical closure assumptions. This model is able to predict such quantities as the gravity current height and the internal wave or bore velocity as a function of the governing dimensionless parameters, generally to within approximately a 10 accuracy. An energy budget analysis provides information on the rates at which potential energy is converted into kinetic energy and then dissipated, and on the processes by which energy is transferred from the gravity current fluid to the dense and ambient fluids. We observe that the energy content of thicker and denser bottom layers grows more rapidly.

To calculate tsunami forces on coastal structures it is of great importance to determine the shape of the tsunami front reaching the coast. Based on literature reviews, analytical reasoning, video footage, and numerical modelling it is concluded that both the continental shelf slope and the bay geometry have a significant influence on the transformation of a tsunami wave near the coastline. After conducting 1D and 2DH wave simulations, a distinction is made between three types of tsunami waves; a non-breaking front (surging), a breaking front and an undular bore breaking front. Tsunami waves transform into these three wave types over a steep continental shelf, an intermediate sloped continental shelf, and a gentle sloped continental shelf, respectively. A new tsunami surf-similarity parameter is proposed to quantitatively predict the type of wave at the coastline, which was validated based on observations during the 2011 Tohoku Earthquake and Tsunami., Journal of Coastal and Hydraulic Structures, Vol. 1 (2021)

It is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent.

A tidal bore is a natural phenomenon usually occur in bays with large tidal waves. Sometimes this large tidal inflow are channeled deep into a river. In Indonesia, this natural phenomenon is found in the Kampar River, which is known as the tidal bore Bono. Sometimes, these tidal bore phenomena disappear, as happened to the Mascaret, tidal bore on the River Seine France. Through an understanding of the formation of tidal bore mechanism, there is hope that the tidal bore Bono in Kampar River can be preserved. In this paper, the occurrence of tidal bore Bono is simulated using the non-hydrostatic Saint-Venant equation under a staggered grid formulation. To test the accuracy of the implementation, several scenarios of hydraulic jumps were simulated first. The numerical results have shown to quantitatively confirm the analytical formula of bore height and velocity, two parameters that are important to characterize a bore wave. Further, by adopting a model that incorporates the non-hydrostatic pressure, our simulation shows the appearance of an undular bore accompanying the shock front. Finally, by using tidal current data measured along Kampar River estuary, our simulation that employs the actual river topography can show the appearance of tidal bore Bono. Our simulations were shown to be in fair agreement with the measurement.

The transformation of long waves&mdash, such as tsunamis and storm surges&mdash, evolving over a continental shelf is investigated. We approach this problem numerically using a pseudo-spectral method for a higher-order Euler formulation. Solitary waves and undular bores are considered as models for the long waves. The bathymetry possesses a periodic ridge-valley configuration in the alongshore direction which facilitates a means by which we may observe the effects of refraction, diffraction, focusing, and shoaling. In this scenario, the effects of wave focusing and shoaling enhance the wave amplitude and phase speed in the shallower regions of the domain. The combination of these effects leads to a wave pattern that is atypical of the usual behavior seen in linear shallow-water theory. A reciprocating behavior in the amplitude on the ridge and valley for the wave propagation causes wave radiation behind the leading waves, hence, the amplitude approaches a smaller asymptotic value than the equivalent case with no lateral variation. For an undular bore propagating in one dimension over a smooth step, we find that the water surface resolves into five different mean water levels. The physical mechanisms for this phenomenon are provided.

In this work a new numerical scheme is proposed in which Haar wavelet method is coupled with finite difference scheme for the solution of a nonlinear partial differential equation. The scheme transforms the partial differential equation to a system of algebraic equations which can be solved easily. The technique is applied to equal width equation in order to study the behaviour of one, two, three solitary waves, undular bore and soliton collision. For efficiency and accuracy of the scheme, L2 and L∞ norms and invariants are computed. The results obtained are compared with already existing results in literature. Keywords: Wavelet, Finite differences, Solitary waves

The results of field observations of an internal undular bore that were performed in a coastal zone of constant depth in the Sea of Japan are presented. A hydrodynamic model of undular bores is discussed according to which the recorded disturbances of the water medium are an experimental prototype of strongly nonlinear (intense) internal undular bores on the pycnocline of shelf waters of Peter the Great Bay with an intensity close to the limit.

We develop a general approach to the description of dispersive shock waves (DSWs) for a class of nonlinear wave equations with a nonlocal Benjamin-Ono type dispersion term involving the Hilbert transform. Integrability of the governing equation is not a pre-requisite for the application of this method which represents a modification of the DSW fitting method previously developed for dispersive-hydrodynamic systems of Korteweg-de Vries (KdV) type (i.e. reducible to the KdV equation in the weakly nonlinear, long wave, unidirectional approximation). The developed method is applied to the Calogero-Sutherland dispersive hydrodynamics for which the classification of all solution types arising from the Riemann step problem is constructed and the key physical parameters (DSW edge speeds, lead soliton amplitude, intermediate shelf level) of all but one solution type are obtained in terms of the initial step data. The analytical results are shown to be in excellent agreement with results of direct numerical simulations., Comment: 25 pages, 5 figures

A numerical study of ship-induced bores is carried out with a focus on soliton generation in different supercritical regimes, and the transition from breaking solitons to a pure bore measured in Gourlay's (2001) laboratory experiments. The fully nonlinear Boussinesq model, FUNWAVE-TVD, with appropriate dissipation schemes, is able to simulate ship-induced broken bores with amplitudes and periods consistent with the measured data. Two breaking dissipation schemes with different breaking criteria (a viscosity dissipation scheme with the surface slope breaking criterion, and a shock-capturing dissipation scheme with the wave height or surface elevation breaking criteria) are examined in a series of numerical experiments. It is found that both models predict soliton generation and the general trend of the transition from breaking solitons at lower Froude numbers to a pure bore at higher Froude numbers. Some differences are identified between the two different breaking schemes, including breaking locations, soliton propagation speed, amplitudes of wakes behind the ship, and the critical Froude number for the transition from a broken undular bore to a pure bore. In general, the viscosity-type breaker with the surface slope criterion performs more consistently with the laboratory observation compared with the shallow water equation-based shock-capturing breaker.

In this study, the numerical behavior of the one-dimensional Regularized Long Wave (RLW) equation has been sought by the Strang splitting technique with respect to time. For this purpose, cubic B-spline functions are used with the finite element collocation method. Then, single solitary wave motion, the interaction of two solitary waves and undular bore problems have been studied and the effectiveness of the method has been investigated. The new results have been compared with those of some of the previous studies available in the literature. The stability analysis has also been taken into account by the von Neumann method.

A series of physical model experiments of tsunami inundation over an idealized industrial site located in a coastal area were performed in this study. Two types of tsunami-like waves with markedly different characteristics were generated. The first was a single long wave and the second was a long wave with soliton fission (undular bore). This study measured the water surface elevation and velocity, as well as the pressures at several heights for some building and tank models that were installed over the coastal area. An experimental benchmark dataset for the tsunami inundations and impacts over a complex topography were produced. This investigation shows that a wide spatial variance of the maximum inundation depth was observed. This was caused by the effects of the surrounding structures, especially the blocking effects due to the structures. The location where the effects occurred depends on the blocking fraction as well as the Froude number of inflows. This study also investigates the relationship between the pressures and neighboring flow profiles. It was determined that under the inundation flow condition without soliton fissions and wave breaking, the maximum pressures can be predicted by the Asakura-type model from the neighboring flow profiles. On the other hand, for the bore condition, the Cross-type model is more appropriate for predicting the maximum pressures. The log-normal standard deviations of the differences between the measured maximum pressures and the predicted pressures are also shown. This information can be useful when performing a probabilistic damage analysis or analyzing the fragility of buildings.

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.

This study documents the evolution of an impressive, largely undular bore triggered by an MCS-generated density current on 20 June 2015, observed as part of the Plains Elevated Convection at Night (PECAN) experiment. The University of Wyoming King Air with profiling nadir- and zenith-viewing lidars sampled the south-bound bore from the time the first bore wave emerged from the nocturnal convective cold pool and where updrafts over 10 m s−1 and turbulence in the wave’s wake were encountered, through the early dissipative stage in which the leading wave began to lose amplitude and speed. Through most of the bore’s life cycle, its second wave had a higher or equal amplitude relative to the leading wave. Striking roll clouds formed in wave crests and wave energy was detected to about 5 km AGL. The upstream environment indicates a negative Scorer parameter region due to flow reversal at midlevels, providing a wave trapping mechanism. The observed bore strength of 2.4–2.9 and speed of 15–16 m s−1 agree well with values predicted from hydraulic theory. Surface and profiling measurements collected later in the bore’s life cycle, just after sunrise, indicate a transition to a soliton.

In the present paper, a numerical method is proposed for the numerical solution of Rosenau-KdV equation with appropriate initial and boundary conditions by using collocation method with septic B-spline functions on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To check accuracy of the error norms L2 and L∞ are computed. Interaction of two and three solitary waves are used to discuss the effect of the behavior of the solitary waves during the interaction. Furthermore, evolution of solitons is illustrated by undular bore initial condition. These results show that the technique introduced here is suitable to investigate behaviors of shallow water waves.

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

Two implicit and two explicit schemes preserving the local momentum conservation laws exactly on any time–space region are proposed for the regularized long-wave equation. With appropriate boundary conditions, the schemes will be energy- and mass-preserving globally. Combining with the momentum conservation laws, we obtain the priori estimates of the numerical solution and the error estimates in l ∞ norm for the proposed implicit schemes. Numerical experiments show the excellent performance of the proposed schemes and the numerical results coincide with the theoretical ones.

When tsunami waves propagate into shallow and gentle continental shelves, the effects of submarine topography and wave dispersion lead to the appearance of undular bore. This kind of wave is quite different from the initial tsunami wave in the waveform, and its amplitude is enhanced significantly. The undular bore will cause great and repeated runups on the beach, which brings larger destruction to the coastal region. In order to investigate the characteristics of the propagation of undular bore, this paper simulates the evolution from the sinusoidal long wave to the undular bore and analyzes the influencing factors. The beach slope has an important effect on the development of undular bores in terms of the undulation form, amplitude, wave length and occurrence time. The influence of bottom friction and submarine topography on the undulations and soliton fissions are discussed. These bottom conditions will lead to great change of the waveform and amplitude of undular bores.

The hydrodynamic characteristics, including local and convective accelerations as well as pressure gradient in the horizontal direction, of the external stream of an undular bore propagating on a 1:20 sloping beach are experimentally studied. A bore with the water depth ratio of 1.70 was generated downstream of a suddenly lifted gate. A high-speed particle image velocimetry was employed to measure the velocity fields during the run-up and run-down motions. The time series of free surface elevation and velocity field/profile of the generated bore, comprising a pure bore accompanied by a series of dispersive leading waves, are first demonstrated. Based on the fast Fourier transform (FFT) and inverse FFT (IFFT) techniques, the free surface elevation of leading waves and the counterpart of pure bore are acquired separately at a specified measuring section (SMS), together with the uniform horizontal velocity of the pure bore. The effect of leading-wave-induced velocity shift on the velocity profiles of the generated bore are then evaluated at the SMS. To understand the calculation procedure of accelerations and pressure gradient, three tabulated forms are provided as illustrative examples. Accordingly, the relationships among the partially depth-averaged values of the non-dimensional local acceleration, convective acceleration, total acceleration and pressure gradient of the generated/pure bore acquired at the SMS versus the non-dimensional time are elucidated. The trends in the non-dimensional accelerations and pressure gradient of the external stream of generated bore are compared with those of the pure bore. During the run-up motion from the instant of arrival of the bore front to the moment of the peak level at the SMS, continuous decrease in the onshore uniform horizontal velocity, and successive deceleration of the pure bore in the onshore direction are evidenced, exhibiting the pure bore under the adverse pressure gradient with decreasing magnitude. However, the pure bore once ridden by the leading waves is decelerated/accelerated spatially and accelerated/decelerated temporally in the onshore direction during the rising/descending free surface of each leading wave. This fact highlights the effect of pre-passing/post-passing of the leading wave crest on the velocity distribution of generated bore. It is also found that, although the leading waves have minor contribution on the power spectrum of the free surface elevation as compared with that of the pure bore, the leading waves do play an important role on the magnitudes of both accelerations and pressure gradient. The largest magnitude of the acceleration contributed by the leading waves is around 26 times the counterpart contributed by the pure bore. Further, during the run-down motion right after the moment for the peak level of the bore, a linear increase in the magnitude of the offshore uniform horizontal velocity and a constant local acceleration with increasing time are both identified. The partially depth-averaged value of the non-dimensional pressure gradient is equal to a small negative constant (&minus, 0.0115) in the offshore direction, indicating that the bore is subject to a constant favorable pressure gradient.

International audience; This paper reports a combined observational and numerical study of wave transformation over barrier reefs. The field instrumentation, which consists in a cross-shore network of pressure sensors and one high resolution velocity profiler, has been deployed for more than two months over the Ouano reef barrier, New Caledonia. The combined analysis of observations and numerical simulations shows that the reef barrier acts as an efficient, but depth-dependent, filter for incoming wave energy. The data analysis reveals the presence of very low frequency standing wave patterns due to reflection either at the reef barrier inner boundary or at the lagoon shore. Another striking feature is the development of undular bore trains over the reef flat.