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On the basis of the Saint-Venant equations for flowing granular matter, we study the various travelling waveforms that are encountered in chute flow for growing Froude number. Generally, for$Frone finds either a uniform flow of constant thickness or a monoclinal flood wave, i.e. a shock structure monotonically connecting a thick region upstream to a shallower region downstream. For$Fr>2/3$both the uniform flow and the monoclinal wave cease to be stable; the flow now organizes itself in the form of a train of roll waves. From the governing Saint-Venant equations we derive a dynamical system that elucidates the transition from monoclinal waves to roll waves. It is found that this transition involves several intermediate stages, including an undular bore that had hitherto not been reported for granular flows.

We consider free-surface flow over two localised obstacles using the framework of the forced Korteweg–de Vries equation in a suite of numerical simulations. Our main concern is with the transcritical regime when the oncoming flow has a Froude number close to unity. The flow behaviour can be characterised by the Froude number and the maximum heights of the obstacles. In the transcritical regime at early times, undular bores are produced upstream and downstream of each obstacle. Our main aim is to describe the interaction of these undular bores between the obstacles, and to find the outcome at very large times. We find that the flow development can be defined in three stages. The first stage is described by the well-known development of undular bores upstream and downstream of each obstacle. The second stage is the interaction between the undular bore moving downstream from the first obstacle and the undular bore moving upstream from the second obstacle. The third stage is the very large time evolution of this interaction, when one of the obstacles controls criticality. For equal obstacle heights, our analytical and numerical results indicate that either one of the obstacles can control flow criticality, that being the first obstacle when the flow is slightly subcritical and the second obstacle otherwise. For unequal obstacle heights the larger obstacle controls criticality. The results obtained here complement a recent numerical study using the fully nonlinear, but non-dispersive, shallow water equations.

Dam-break and lock-exchange flows are considered in a Boussinesq two-layer fluid system in a uniform two-dimensional channel. The focus is on inviscid ‘weak’ dam breaks or lock exchanges, for which waves generated from the initial conditions do not break, but instead disperse in a so-called undular bore. The evolution of such flows can be described by the Miyata–Camassa–Choi (MCC) equations. Insight into solutions of the MCC equations is provided by the canonical form of their long wave limit, the two-layer shallow water equations, which can be related to their single-layer counterpart via a surjective map. The nature of this surjective map illustrates that whilst some Riemann-type initial-value problems (dam breaks) are analogous to those in the single-layer problem, others (lock exchanges) are not. Previous descriptions of MCC waves of permanent form (cnoidal and solitary waves) are generalised, including a description of the effects of a regularising surface tension. The wave solutions allow the application of a technique due to El's approach, based on Whitham's modulation theory, which is used to determine key features of the expanding undular bore as a function of the initial conditions. A typical dam-break flow consists of a leftwards-propagating simple rarefaction wave and a rightward-propagating simple undular bore. The leading and trailing edge speeds, leading edge solitary wave amplitude and trailing edge linear wavelength are determined for the undular bore. Lock-exchange flows, for which the initial interface shape crosses the mid-depth of the channel, by contrast, are found to be more complex, and depending on the value of the surface tension parameter may include ‘solibores’ or fronts connecting two distinct regimes of long-wave behaviour. All of the results presented are informed and verified by numerical solutions of the MCC equations.

It is well-known that transcritical flow over a localized obstacle generates upstream and downstream nonlinear wavetrains. The flow has been successfully modelled in the framework of the forced Korteweg–de Vries equation, where numerical and asymptotic analytical solutions have shown that the upstream and downstream nonlinear wavetrains have the structure of unsteady undular bores, connected by a locally steady solution over the obstacle, which is elevated on the upstream side and depressed on the downstream side. Inthispaper we consider the analogous transcritical flow over a step, primarily in the context of water waves. We use numerical and asymptotic analytical solutions of the forced Korteweg–de Vries equation, together with numerical solutions of the full Eulerequations, to demonstrate that a positive step generates only an upstream-propagating undular bore, and a negative step generates only a downstream-propagating undular bore.

Experimentation and theory are used to study the long-term dynamics of a two-dimensional density current flowing into a two-layer stratified basin. When the initial Richardson number,$\hbox{\it Ri}_{\rho}^{\hbox{\scriptsize\it in}}$, characterizing the ratio of the background stratification to the buoyancy flux of the density current, is less than the critical value of$\hbox{\it Ri}_{\rho}^{*} \,{=}\, 21-27$, it is found that the density current penetrates the stratified interface. This result is ostensibly independent of slope for angles between 30° and 90°. If the current does not initially penetrate the interface, then it slowly increases the density of the top layer until the interfacial density difference is reduced sufficiently to drive penetration. The time scale for this to occur,$t_{p} \,{=}\, (\hbox{\it Ri}^{\hbox{\scriptsize\it in}}_{\rho} - \hbox{\it Ri}_{\rho}^{*}) L/B^{1/3}$, is explicitly a function of the buoyancy fluxBand the length of the basinL. The initial Richardson number,$\hbox{\it Ri}^{\hbox{\scriptsize\it in}}_{\rho}$, is a function of depth, the initial reduced gravity of the interface and a weak function of slope angle. In the absence of initial penetration for very steep slopes of 75° and 90°, we observe that penetrative convection at the interface leads to significant local entrainment. In consequence, the top layer thickens and the interfacial entrainment rate increases as the fifth power of the interfacial Froude number. In contrast, such a process is not observed at comparable interfacial Froude numbers on lower slopes of 30°, 45° and 60°, thereby demonstrating the important role of impact angle on penetrative convection. We attribute the increased interfacial entrainment by the steep density currents as the result of the transition from an undular bore to a turbulent hydraulic jump at the point where the density current intrudes. We discuss the applicability of the observed circulation to the stability of the Arctic halocline where we find$0.56\,{\lesssim}\, t_{p} \,{\lesssim}\,1.2$years for a range of contemporary oceanographic conditions.

The morning glory is a meteorological phenomenon which occurs in northern Australia and takes the form of a series of roll clouds. The morning glory is generated by the interaction of nocturnal seabreezes over Cape York Peninsula and propagates in a south-westerly direction over the Gulf of Carpentaria. In the present work, it is shown that the morning glory can be modelled by the resonant flow of a two-layer fluid over topography, the topography being the mountains of Cape York Peninsula. In the limit of a deep upper layer, the equations of motion reduce to a forced Benjamin–Ono equation. In this context, resonant means that the underlying flow velocity of the seabreezes is near a linear long-wave velocity for one of the long-wave modes. The morning glory is then modelled by the undular bore (simple wave) solution of the modulation equations for the Benjamin–Ono equation. This modulation solution is compared with full numerical solutions of the forced Benjamin–Ono equation and good agreement is found when the wave amplitudes are not too large. The reason for the difference between the numerical and modulation solutions for large wave amplitude is also discussed. Finally, the predictions of the modulation solution are compared with observational data on the morning glory and good agreement is found for the pressure jump due to the lead wave of the morning glory, but not for the speed and half-width of this lead wave. The reasons for this are discussed.

We derive a time-stepping method for unsteady fully nonlinear two-dimensional motion of a two-layer fluid. Essential parts of the method are: use of Taylor series expansions of the prognostic equations, application of spatial finite difference formulae of high order, and application of Cauchy's theorem to solve the Laplace equation, where the latter is found to be advantageous in avoiding instability. The method is computationally very efficient. The model is applied to investigate unsteady trans-critical two-layer flow over a bottom topography. We are able to simulate a set of laboratory experiments on this problem described by Melville & Helfrich (1987), finding a very good agreement between the fully nonlinear model and the experiments, where they reported bad agreement with weakly nonlinear Korteweg–de Vries theories for interfacial waves. The unsteady transcritical regime is identified. In this regime, we find that an upstream undular bore is generated when the speed of the body is less than a certain value, which somewhat exceeds the critical speed. In the remaining regime, a train of solitary waves is generated upstream. In both cases a corresponding constant level of the interface behind the body is developed. We also perform a detailed investigation of upstream generation of solitary waves by a moving body, finding that wave trains with amplitude comparable to the thickness of the thinner layer are generated. The results indicate that weakly nonlinear theories in many cases have quite limited applications in modelling unsteady transcritical two-layer flows, and that a fully nonlinear method in general is required.

Fully nonlinear extensions of Boussinesq equations are derived to simulate surface wave propagation in coastal regions. By using the velocity at a certain depth as a dependent variable (Nwogu 1993), the resulting equations have significantly improved linear dispersion properties in intermediate water depths when compared to standard Boussinesq approximations. Since no assumption of small nonlinearity is made, the equations can be applied to simulate strong wave interactions prior to wave breaking. A high-order numerical model based on the equations is developed and applied to the study of two canonical problems: solitary wave shoaling on slopes and undular bore propagation over a horizontal bed. Results of the Boussinesq model with and without strong nonlinearity are compared in detail to those of a boundary element solution of the fully nonlinear potential flow problem developed by Grilli et al. (1989). The fully nonlinear variant of the Boussinesq model is found to predict wave heights, phase speeds and particle kinematics more accurately than the standard approximation.

Bore propagation near the shoreline, the transition from bore to wave run-up, and the ensuing run-up motion on a uniformly sloping beach are investigated experimentally. As a bore approaches the shoreline, the propagation speed first decelerates by compressing its wave form and then suddenly accelerates at the shoreline. Although this behaviour is qualitatively in agreement with the inviscid shallow-water wave prediction (often called the ‘bore collapse’ phenomenon), unlike the genuine bore-collapse phenomenon, the acceleration is caused by the ‘momentum exchange’ process, i.e. collision of the bore against the initially quiescent water along the shoreline. Owing to this momentum exchange, a single bore motion degenerates into two successive run-up water masses; one involves a turbulent run-up water motion followed by the original incident wave motion. The transition process from undular bore to wave run-up appears to be different from that of a fully developed bore. The bore front overturns directly onto the dry beach surface, and the run-up is characterized by a thin splashed-up flow layer.

The form taken by a finite amplitude internal seiche in a shallow two-layer fluid within a long narrow container is akin to a shock wave in a gas-filled tube; an undular bore or surge is present which reflects back and forth along the length of the container. The internal surge in Loch Ness is sometimes observed to retain its amplitude over several seiche periods, and a resonance with the wind appears possible. This idea is explored by developing a theoretical model and by making a laboratory experiment, but difficulties are encountered in estimating the size of the parameters which characterize the natural phenomenon.

If a long wave of elevation travels in shallow water it steepens and forms a bore. The bore is undular if the change in surface elevation of the wave is less than 0·28 of the original depth of water. This paper describes the growth of an undular bore from a long wave which forms a gentle transition between a uniform flow and still water. A physical account of its development is followed by the results of numerical calculations. These use finite-difference approximations to the partial differential equations of motion. The equations of motion are of the same order of approximation as is necessary to derive the solitary wave. The results are in general agreement with the available experimental measurements.

A solution for the weak bore is found in which the mean profile is dominated by viscosity, so that the velocity variation is given essentially by a quasi-uniform Poiseuille flow. It is found that such a transition between flows of different depths is possible provided the Froude number is less than 1·58. The possibility of superposing an inviscid perturbation on such a flow is then investigated. Under favourable circumstances the effect of this perturbation is to add to the profile of the free surface a term which decays exponentially in front of the bore, but is oscillatory behind it.

Nonlinear effects on the refraction of water waves are discussed. The existence of conjugate solutions for wave fields, one set of which corresponds to the ‘anomalous’ refraction solutions of Peregrine & Ryrie (1983) provides the stimulus for consideration of jumps in wave-field properties. Just such a jump is described by Yue & Mei (1980) for a case where near-linear waves are reflected with small deflection by a rigid wall.Wave jumps between conjugate solutions appear to be possible for finite-amplitude wavetrains. These are examined and the structure of wave jumps in the near-linear, small-deflection case is elucidated. They have a structure directly analogous to that of an undular bore on shallow water with surface tension (‘hydraulic analogy’). Numerical results of Yue & Mei (1980) provide valuable guidance and confirmation.Linear waves are reflected at a caustic, and can be described with Airy functions. Although equivalent weakly nonlinear solutions exist, the results from reflection by a wall and from use of the hydraulic analogy show that, unlike the caustics of linear theory, nonlinear caustics should not be considered in isolation. Caustic cusps, or wave focusing, must be considered, unless bed topography has discontinuities. A qualitative discussion of focusing based on the behaviour of unsteady waves in the hydraulic analogy shows that wave jumps can be expected. The relationship to linear theory is also put in perspective. Nonlinearity causes the linear rays to split into two sets of characteristics. The splitting of a ray focus leads to two wave jumps.Consideration of the case of a semi-infinite beach shows that anomalous refraction is most unlikely to occur because there is an offshore influence of the beach on the wave field which changes the incident wave conditions to prevent anomalous refraction.