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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.

A detailed field study was carried out on a tidal bore to document the turbulent processes and sediment entrainment which occurred. The measured bore, within the Arcins Channel of the Garonne River ( France), was undular in nature and was followed by well-defined secondary wave motion. Due to the local river geometry a collision between the Arcins channel tidal bore and the bore which formed within the main Garonne River channel was observed about 800m upstream of the sampling site. This bore collision generated a transient standing wave with a black water mixing zone. Following this collision the bore from the main Garonne River channel propagated 'backward' to the downstream end of the Arcins channel. Velocity measurements with a fine temporal resolution were complemented by measurements of the sediment concentration and river level. The instantaneous velocity data indicated large and rapid fluctuations of all velocity components during the tidal bore. Large Reynolds shear stresses were observed during and after the tidal bore passage, including during the 'backward' bore propagation. Large suspended sediment concentration estimates were recorded and the suspended sediment flux data showed some substantial sediment motion, consistent with the murky appearance of the flood tide waters. Copyright (C) 2015 John Wiley & Sons, Ltd.

The complexity of bora flow in the lee of Southern Velebit, Croatia, is investigated by means of high-resolution numerical simulations carried out with the US Naval Research Laboratory's (NRL) Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) model. The aim of this study is to identify reasons for the strong spatial variability of bora in the wider Zadar region and the uncharacteristically weak bora in the city of Zadar. The primary focus is put on the severe bora episode of 20 December 2004 during which a sodar (sonic detection and ranging) system was operated at Zemunik airport. Numerical results are verified against the available surface and upper-air observations. Upstream conditions governing dynamics of this particular bora event are characterized by the presence of a mean-state critical level at 3–4 km above mean sea-level (AMSL) defining the upstream bora layer. Low-level airflow over Zadar peninsula during this event displays marked temporal variability. During the morning, a highly three-dimensional and unsteady wake (i.e. region of reduced wind speed) is established over Zadar peninsula, downstream of a hydraulic jump that forms in the lee of, but not collocated with, highest portions of the Southern Velebit terrain. During its most extensive phase the wake scales the entire length of Zadar peninsula in the horizontal and extends up to the critical level in the vertical. A wake vortex develops as the flow reverses along the wake centreline. In the evening hours, an undular bore forms in the same location, with large-amplitude waves on top of it inducing boundary layer separation and rotors. Wake flow is affected both by changes in the upstream atmospheric structure and diurnal boundary-layer evolution. Sensitivity experiments show that the influence of Velebit's height is particularly strong, governing the onset and strength of bora. The terrain of Zadar peninsula exerts a strong influence on the characteristics of the developed bora flow, particularly when non-hydrostatic effects are significant. Boundary layer separation is particularly sensitive to downstream orography. Copyright © 2012 Royal Meteorological Society

This article discusses on the solution of the regularized long wave (RLW) equation, which is introduced to describe the development of the undular bore, has been used for modeling in many branches of science and engineering. A numerical method is presented to solve the RLW equation. The main idea behind this numerical simulation is to use the collocation and approximating the solution by radial basis functions (RBFs). To avoid solving the nonlinear system, a predictor-corrector scheme is proposed. Several test problems are given to validate the new technique. The numerical simulation, includes the propagation of a solitary wave, interaction of two positive solitary waves, interaction of a positive and a negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves and the development of an undular bore. The three invariants of the motion are calculated to determine the conservation properties of the algorithm. The results of numerical experiments are compared with analytical solution and with those of other recently published methods to confirm the accuracy and efficiency of the presented scheme.© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Transcritical flow over a localized obstacle generates upstream and downstream nonlinear wavetrains. In the weakly nonlinear long-wave regime, this flow has been modeled with the forced Korteweg-de Vries (fKdV) equation, where numerical simulations and asymptotic solutions have demonstrated that the upstream and downstream nonlinear wavetrains have the structure of unsteady undular bores, connected by a locally steady solution over the obstacle. Further, it has been shown that when the obstacle is replaced by a step of semi-infinite length, it is found that a positive step generates only an upstream-propagating undular bore, and a negative step generates only a downstream-propagating undular bore. This result suggests that for flow over a hole, that is a step down followed by a step up, the two wavetrains generated will interact over the hole. In this paper, this situation is explored by numerical simulations of the fKdV equation.

A numerical procedure is proposed for the solution of the Boussinesq equations in rectangular channels over a horizontal bed. The Boussinesq equations account for non-hydrostatic effects in free-surface flows. The proposed approach is a predictor-corrector procedure where the hyperbolic part of the equations is treated using a higher-order estimate of the variable slope of the Godunov-type, MUSCL scheme. The proposed slope estimate is third-order accurate on irregular grids and fourth-order accurate on regular grids. The dispersive, non-hydrostatic terms are treated using a semi-implicit discretization. A stability analysis of the proposed predictor-corrector procedure shows that optimal accuracy is achieved when the implicitation parameter is set equal to 0.5. This analysis is confirmed by numerical experiments, whereby the propagation of a solitary wave and the development of an undular bore are reproduced and compared successfully with the available analytical solutions. The proposed method is shown to be of a good level of accuracy without being too sensitive to the numerical parameters. Its applicability in practical problems is illustrated by a comparison with laboratory experiments. Copyright (C) 2008 John Wiley & Sons, Ltd.

In the late Jurassic period, about 142 million years ago, an asteroid hit the shallow paleo- Barents Sea, north of present-day Norway. The geological structure resulting from the impact is today known as the Mjlnir crater. The present work attempts to model the generation and the propagation of the tsunami from the Mjolnir impact. A multi-material hydrocode SOVA is used to model the impact and the early stages of tsunami generation, while models based on shallow-water theories are used to study the subsequent wave propagation in the paleo-Barents Sea. We apply several wave models of varying computational complexity. This includes both three-dimensional and radially symmetric weakly dispersive and nonlinear Boussinesq equations, as well as equations based on nonlinear ray theory. These tsunami models require a reconstruction of the bathymetry of the paleo- Barents Sea. The Mjolnir tsunami is characteristic of large bolides impacting in shallow sea; in this case the asteroid was about 1.6 km in diameter and the water depth was around 400 m. Contrary to earthquake- and slide-generated tsunamis, this tsunami featured crucial dispersive and nonlinear effects: a few minutes after the impact, the ocean surface was formed into an undular bore, which developed further into a train of solitary waves. Our simulations indicate wave amplitudes above 200 m, and during shoaling the waves break far from the coastlines in rather deep water. The tsunami induced strong bottom currents, in the range of 30-90 km/h, which presumably caused a strong reworking of bottom sediments with dramatic consequences for the marine environment.

A numerical simulation of the regularized long wave (RLW) equation is obtained using a high-order compact difference method, based on the fourth-order compact difference scheme in space and the fourth-order Runge–Kutta method in time integration. The method is tested on the problems of propagation of a solitary wave, interaction of two positive solitary waves, interaction of a positive and a negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves, the development of an undular bore and the solitary waves induced by boundary motion. The three invariants of the motion are calculated to determine the conservation properties of the algorithm. L2 and L∞ error norms are used to measure differences between the exact and numerical solutions. The results obtained by proposed method are compared with those of other recently published methods and shown to be more accurate and efficient.

Field observations imply that accumulation and advection of invertebrate larvae and other plankton by propagating fronts provides an efficient mechanism for the cross-shelf transport and subsequent coastal settlement of larvae and plankton. Gravity currents propagating into a heavier, uniform-density fluid are known to have near-surface velocities u that exceed the propagation speed c of the gravity current nose. It is shown by simple dam-break laboratory experiments and through corresponding numerical solutions of two-dimensional stratified flow and Lagrangian particle advection that this characteristic of gravity currents can lead to accumulation in the gravity current nose of particles that originate well behind the gravity current head. However, efficient particle accumulation and transport with the speed c only occurs for particles with sufficient vertical buoyancy or vertical swimming capability. Weakly buoyant or neutral particles are swept down and back away from the gravity current head by the circulation in this region. A dam break into an ambient fluid with stratification (two-layered) could result in the transport of some particles that are initially trapped in the forepart of the leading disturbance, an undular bore. However, this situation does not result in significant accumulation of particles at the leading edge of the disturbance. The numerical solutions show that particles that originate in the dense fluid ahead of the gravity current are much less likely to be accumulated at the gravity current nose. A simple scaling criterion for the minimum vertical buoyancy velocity for efficient accumulation and transport of buoyant particles based on the gravity current and particle characteristics is developed and compares favorably with the numerical results. The implications of these results for transport of larvae in coastal environments are discussed. The dynamics of marine benthic invertebrate populations are greatly affected by larval transport. For near-shore populations, larval transport prior to settlement includes at least three processes: cross-shore advection, because larvae are generally very small and cannot control their cross-shore position by swimming alone; larval behavior, because by swimming vertically, larvae might be able to exploit some depth-dependent cross-shore flows; and accumulation by convergent circulation in propagating features, because cross-shore advection of larvae is often dependent on the retention of larvae at the propagating features. Several hypotheses of onshore larval transport, namely surface slicks over linear internal waves (Shanks 1983), internal bore warm fronts (Pineda 1994), and upwelling relaxation fronts (Farrell et al. 1991), invoke concentration of larvae at propagating features. These hypotheses state that larvae become concentrated at convergent features aligned parallel to the coastline and that larval transport occurs as these linear features propagate shoreward. The interplay between transport and accumulation is poorly understood; issues include (1) the dependency of transport on accumulation, and of accumulation on larval behavior, and (2) whether the frontal larvae originate onshore or off

The equal width equation is solved by a Petrov–Galerkin method using quadratic B-spline spatial finite elements. A linear recurrence relationship for the numerical solution of the resulting system of ordinary differential equations is obtained via a Crank–Nicolson approach involving a product approximation. The motion of solitary waves is studied to assess the properties of the algorithm. The development of an EW undular bore is investigated and compared with that of the RLW bore. © 1997 John Wiley & Sons, Ltd.

The RLW equation is solved by a least-squares technique using linear space-time finite elements. In simulations of the migration of a single solitary wave this algorithm is shown to have higher accuracy and better conservation than a recent difference scheme based on cubic spline interpolation functions. In addition, for very small amplitude waves (≤ 0.09) it has higher accuracy than an approach using quadratic B-spline finite elements within Galerkin's method. The development of an undular bore is modelled.

A B-spline finite element method is used to solve the regularized long wave equation numerically. This approach involves a Galerkin method with quadratic B-spline finite elements so that there is continuity of the dependent variable and its first derivative throughout the solution range. Time integration of the resulting system of ordinary differential equations is effected using a Crank-Nicolson approximation. Standard problems are used to validate the algorithm, which is then used to model the smooth development of an undular bore.

SUMMARY In this paper the formation and structure of an atmospheric undular bore is examined with the aid of a numerical model and theoretical analysis. The numerical model is first used to simulate a density current propagating into an unstratified environment. A detailed comparison of the model results with existing theory shows that for a given pressure head the current can travel at a greater speed than that predicted by irrotational theory. When the density current is allowed to propagate into a low-level stable layer it is found that for certain parameter regimes an undular bore forms ahead of the current. The results for two different types of stable layers are examined. For a layer with a step profile of potential temperature, comparison is made with both classical bore theory and the theory of density currents in two-layer fluids and encouraging agreement is found. When a linear profile is specified in the stable layer it is found that energy-conserving solutions can be obtained. The numerical fields are then compared with measurements of the morning glory, an undular bore observed in north-eastern Australia, and again agreement is good. Finally, preliminary results of model simulations with stratification above the stable layer are discussed.

SUMMARY Observations made during an expedition in October 1980 to study ‘morning glory’ wind squalfs in the Gulf of Carpentaria region of northern Australia are described. During a ten day period in Burketown, north Queensland, six squalls were documented on consecutive mornings (8-1 3 October), including two moving from the south, an apparently unusual direction. The unique data gathered from.the latter confirm that these have the same character as their north-easterly counterparts; that is, they are undular bores. The origin of southerly glories remains uncertain, but, at least, in some cases, appears to be. linked with the northeast advance of a frontal trough across central Australia. While the data are equivocal, it seems possible that the motion of the front into the developing nqcturnal inversion could generate waves, or hydraulic-type disturbances, which run along the inversion layer, ahead of the front, as proposed by Tepper (1950). The efficacy of such a mechanism is demonstrated by a simple laboratory experiment. Evidence is presented which suggests that morning glory type events occur elsewhere in the world, sometimes as precursors to cold fronts. Observations in the Gulf of Carpentaria region show that they are effective triggers of deep convection when they advance into air with a sufficient degree of conditional instability.