Session AZ: Waves I

Numerical and Experimental Study of Scott Russell's Solitary Waves

The motion of solitary waves on the free surface of a layer of water is studied. The waves are generated by a moving bump placed at the bottom or a pressure source on the surface. The problem is first discussed using a model equation, called force Kortweg-de Vries (FKdV) equation. Then, such forced waves are studied experimentally using a water tank with a moving bump at the bottom. The results from the FKdV equation match very well with those from the experiments if the solitary wave is not near the wave of maximum amplitude. Finally, it is shown that the solitary wave observed by Scott Russel in 1834 is just one of the forced solitary waves presented here.   

Observation of odd and even two-dimensional standing solitary waves in water

By means of the parametric excitation of water waves in a Hele- Shaw cell, we report the existence of two new types of highly localized, standing solitary water waves. They are respectively of odd and even symmetries. Both patterns oscillate subharmonically with the forcing frequency. The even pattern resembles the oscillon originally recognized at the surface of a vertically vibrated layer of brass beads [1]. The odd pattern has apparently never been observed before in any media.\\[4pt] [1] P. B. Umbanhowar, F. Melo, F. and H. Swinney, Nature (London) 382, 793 (1996)   

Singularity formation in a model of shallow water wave equations

We will present numerical solutions from initial value calculations of a model of shallow water wave equation. For small data, numerical solution develops singularity in a finite time. Driven by the structure of solutions, we carry out analysis based on numerical results to prove singularity formation. Numerical and theoretical results will be shown.   

Generation of the maximum breaking wave amplitude by means of unidirectional wave focusing

This work overviews existing methods and proposes new approach for the wave breaking generation in wave tanks. Due to dispersive nature of surface gravity waves, a unidirectional wave packet can be pre-programmed to focus its energy at a desired location in space and time. In this work, frequency modulated packets were generated by means of a single flat paddle hinged at the bottom. Two methods proposed within our approach reveal strong nonlinear wave envelope modulation in high amplitude regimes. Wave packets with high steepness were found to deviate from linear expectations by shifting their energy towards the leading edge. If left uncorrected, such modulation leads to defocusing of the wave energy, causing breaking waves to be less energetic and to appear prior to the desired location. Various empirical corrections were tested to account for the modulation, among which only the phase shift correction for the second method proved to be successful. Based on this finding, a wave generation procedure is established, which allows to simulate large wave breaking events and their interaction with various structures and vessels.   

Experimental study of interfacial waves induced by surface waves in muddy water

A peculiar feature has been observed in a laboratory tank with monochromatic surface waves propagating in muddy water with a thin layer of clear surface water: a quasi-stable set of interfacial waves that appear as longitudinally-oriented rotating tubes at the mud-water interface. These ``interfacial tubes" are spatially periodic and temporally subharmonic structures whose direction of apparent rotation alternates with each passing surface wave crest. Rotation results from coupled upwelling and downwelling of clear surface water and muddy water below. The interfacial tubes appear to be standing nonlinear interfacial waves that result from a three-wave interaction involving a surface wave train and two interfacial wave trains. This is believed to be the first documented observation of this phenomenon in its nonlinear form. The topics covered in this presentation are relevant to the study of internal wave generation, wave damping and nearshore mixing processes.   

Near-critical reflection of nonlinear obliquely incident internal wave beam from a slope

The reflection of internal gravity waves from sloping boundaries is believed to contribute significantly to vertical mixing in the ocean. This mechanism is likely to be enhanced when a wave is incident at an angle to the horizontal that is close to the slope of the boundary, given that the amplitude of the reflected wave becomes infinite according to linear inviscid theory if the angle of incidence exactly matches the slope. To clarify the role of nonlinear effects in this resonance, the reflection of a nonlinear wave beam of finite cross-section is analyzed by a matched-asymptotics approach, exploiting the fact that, near the critical angle, the reflected disturbance is confined to a thin boundary layer in an ``inner'' region close to the slope. Unlike prior studies, which assume that incident waves approach the boundary in a plane normal to the isobaths, here the oncoming wave is oblique. This gives rise to an alongslope mean flow component that is equally strong to the upslope induced mean flow, and the evolution of the reflected wave is fully nonlinear, in sharp contrast to the case of normal incidence where nonlinear effects are minor. The theoretical predictions are discussed in connection with related numerical and experimental results.   

Laboratory experiments on internal wave reflection and absorption at a simulated oceanic pycnocline

Laboratory experiments have been performed to investigate the reflection of an internal wave beam with a ``pycnocline'' layer situated below an unstratified layer in order to simulate observed oceanic processes. An oscillating cylinder was used to generate wave beams in the well-known ``St. Andrew's Cross'' pattern that interacted with the pycnocline. The internal waves were observed and the incident and reflected amplitudes measured using the synthetic schlieren technique. In virtually all instances, near-perfect reflection or near-complete absorption at the pycnocline was observed, depending on the value of the pycnocline density gradient. The data indicate the existence of a transition from reflection to absorption that is a function of the ratio of the maximum BV frequency in the pycnocline to the BV frequency of the stratified layer.   

The Near-Field Internal Wave Field Generated by a Sphere Moving in a Stratified Fluid

High resolution numerical simulations of a sphere traveling horizontally at constant speed at high Reynolds number through a uniformly stratified fluid are shown to compare well with previous laboratory experimental measurements of the drag and the internal wave field. The results of these detailed numerical studies are used to test and revise source distribution parameterizations of the near-field waves that have been used in analytical studies based on linear theory. Such parameterizations have been shown to be useful in initializing ray-tracing schemes that can be used efficiently to compute wave propagation through realistic oceans with variable background properties.   

Horizontal transport of Lagrangian particles by basin-scale internal waves in a continuously-stratified circular lake

Horizontal transport of Lagrangian particles under the influence of wind-generated, basin-scale internal waves in a circular lake is studied by employing field solutions of the linear hydrostatic model and by simulations of the weakly-nonlinear, weakly-nonhydrostatic evolution model for continuous stratification subject to wind stress forcing over a basin of uniform depth. Both the azimuthal mode-one Kelvin and the gravest Poincare waves of the first two vertical eigenmodes are accounted for in the specification of the advection field. It is found that Kelvin waves play the dominant role in along-shore transport. Although vertical mode-two (V2) waves are usually considerably smaller than those of vertical mode-one (V1), yet V2 Kelvin waves possess sufficient ability in driving along-shore transport for a distance comparable with that of V1-only transport. This arises because of the longer residence time of disparately slow V2 Kelvin waves confined in the vicinity of the basin perimeter. Poincare waves, on the other hand, play a dominant role in off-shore transport, stretching and squeezing the particle cloud in radial directions with fast, near-inertial frequencies. These transport features are compared for different wind forcing strengths and different horizontal scale of the basin.   

Weak compressibility of surface wave turbulence

Clustering of matter on the surface of lakes and pools and of oil slicks and seaweed on the sea surface is well-known empirically but there is no theory that describes it. Since surface flows are always compressible, such a theory should be based on the description of the development of density of inhomogeneities in a compressible flow. We studied the growth of small-scale inhomogeneities in the density of particles floating in weakly nonlinear small-amplitude surface waves. Despite the small amplitude, the accumulated effect of the long-time evolution may produce a strongly inhomogeneous distribution of the floaters: density fluctuations grow exponentially with a small but finite exponent. We have shown that the exponent is of sixth or higher order in wave amplitude. As a result, the inhomogeneities do not form within typical time scales of the natural environment. Thus the turbulence of surface waves is weakly compressible and alone it cannot be a realistic mechanism of the clustering of matter on liquid surfaces. However if besides waves there are also currents, the interplay of waves with currents, might be in some cases responsible for the patchiness of the floaters.