Session CZ: Waves II

Rotating Solitary Wave during Liquid Drainage

This work reports on the observation of a rotating solitary wave during liquid drainage from a cylindrical reservoir when shallow water conditions are reached. Using results obtained from a high-speed camera and image processing techniques, we discuss the mechanism leading to the formation of the solitary wave; and we examine its shape and speed as it propagates in two different cases, variable and constant water depths. The results were also compared to the KdV theory.   

Can non-propagating hydrodynamic solitons be forced to move?

Development of technologies based on localized states depends on our ability to manipulate and control these nonlinear structures. In order to achieve this, the interactions between localized states and control tools should be well modelled and understood. We present a theoretical and experimental study for handling non-propagating hydrodynamic solitons in a vertically driven rectangular water basin, based on the inclination of the system. Experiments have shown that tilting the basin induces non-propagating solitons to drift towards an equilibrium position through a relaxation process. Our theoretical approach is derived from the parametrically driven damped nonlinear Schr\"odinger equation which models the system. The basin tilting effect is incorporated as a spatially inhomogeneous linear correction on dissipation. A motion law for hydrodynamic solitons can be deduced from these assumptions. The model equation, which includes a constant speed and a linear relaxation term, nicely reproduces the motion observed in our experiment.   

Stationary solutions of the extended reduced Ostrovsky equation

The extended Ostrovsky equation describes large-amplitude internal oceanic waves affected by Earth's rotation, including an additional cubic term to take into account the effect of strong nonlinearities. Its reduced version, in which the small-scale, or Boussinesq, dispersion term is omitted, is relevant for the description of long internal waves in oceans, when their wavelengths are much larger than the basin depth. It may be of interest also for waves of other physical origin in nonlinear media with large-scale dispersion. In this work we present a systematic analysis of the stationary solutions to this extended reduced Ostrovsky equation and their categorization. Periodic and solitary solutions are constructed and their typical parameters are estimated for the natural oceanic conditions.   

Selection Rules for Internal Gravity Waves and Inertial Waves

Perturbation methods are used to calculate nonlinear interaction of waves, however most analyses skip the question as to whether the $0^\textnormal{th}$ order solutions exist. The dispersion relation for internal gravity waves does {\it not} relate the magnitude of the wave vector and its frequency, rather it relates the frequency and {\it direction} of the wave vector. Thus, spatially columnated beams of internal waves are made of a continuum of plane waves with different wavelengths, but the same frequency. For two parent beams to create a daughter, the plane waves within the parent and daughter beams must obey the triad condition (the spatial wave vector of the daughter equals the sum of the parents' vectors, and temporal frequency of the daughter equals the sum or difference of the parents' frequencies) and the dispersion relationship. Contrary to what is assumed implicitly, these conditions cannot always be satisfied. If they could, then the interaction of two beams of gravity waves would produce 8 daughter beams, consisting of two St~Andrew's crosses (each with 4 beams). The beams in one cross have a frequency equal to the sum of the frequencies of the parents and the beams in the other have a frequency equal to the difference. Most of these daughter beams cannot exist. We derive selection rules for the beams. We extend our analysis to a more generic set of waves.   

The Combined Effects of Light-wind and Surfactants on Spilling Breakers

Spilling breaking waves in the presence of light-winds and surfactants were studied experimentally in a wind-wave tank. The breaking waves were mechanically generated with a single wave maker motion that produces a weak spilling breaker in clean water without wind. Separate experiments were performed with the same wave maker motion and very low wind speeds in clean water and in water with various concentrations of Triton X-100 (a soluble surfactant). The crest-profiles of the waves along the center plane of the tank were measured with a cinematic laser-induced fluorescence technique. In clean water with a wind speed lower than 2.3 m/s (the minimum wind speed of wind-generated waves in our tank), the wave breaking is initiated with a bulge-capillary-ripple pattern. When the wind speed is above 2.3 m/s, wind waves are generated. These wind waves steepen on the front face of the crest of the mechanically generated waves and trigger breaking of these larger scale waves. In the presence of surfactants, the bulge-capillary-ripple pattern occurs at even higher wind speeds (3 m/s). Geometrical parameters describing the wave crest shape were found to scale with the wind speed to the third power.   

Laboratory Measurements of Droplets Generated by Breaking Water Waves

The production of droplets generated by breaking water waves greatly affects the heat, mass and momentum transfer between the atmosphere and the sea surface. In this study, the generation of droplets by single breaking water waves, was explored in a wave tank. Plunging breakers were generated from dispersively focused wave packets (average frequency 1.15 Hz) using a programmable wave maker. The profile histories of the breaking wave crests along the center plane of the tank were measured with a cinematic laser-induced fluorescence technique, while the droplet diameters and motions were measured with a double-pulsed cinematic shadowgraph technique. The two measurement systems were mounted on an instrument carriage that was set to move along the tank following the breaking crests. It was found that droplets are primarily generated when the wave's plunging jet generates strong turbulence during impact with the wave's front face and when large air bubbles, entrapped during the plunging process, rise to the free surface and pop.   

Effects of Rotation on Internal Solitary Waves

An internal solitary waves in a rotating system decays by radiation into longer Poincare waves. The radiation extinguishes the initial solitary wave in a finite time. Recent numerical and theoretical studies show that the radiated waves develop into a localized nonlinear internal wave packet, or envelop soliton, that persists for long times. The 13-meter LEGI-Coriolis platform in Grenoble, France was used to perform laboratory experiments designed to test these theoretical results. The experiments confirm theoretical predictions of the packet formation and characteristics including the phase speed of the carrier waves and packet group speed. In particular, the wave number of the carrier wave is found to be close to the linear wave with the maximum group velocity. The localized packet formation is, however, a consequence of nonlinearity. As the rotation rate is increased, the initial disturbance (produced from a dam-break) develops into the packet structure without first forming a distinct solitary wave. These nonlinear inertia-gravity wave packets are a natural outcome in a rotating system and do not require initiation by radiation decay of an internal solitary wave.   

Self-induced shear instabilities by large amplitude internal waves

Large amplitude internal waves in sharply stratified fluids can generate large shear flows supporting Kelvin-Helmholtz instabilities. This talk will present a combined theoretical and numerical study of this instability. Spectral analysis of the corresponding non-self adjoint Taylor-Goldstein type equation will be outlined and illustrated on a specific model of internal wave structure. Subsequently, we will focus upon the asymmetric intensification of the instability by weakly non-parallel shear perturbations along long wave shear profiles.   

The benthic boundary layer under fully-nonlinear internal solitary waves of depression

Long internal waves are common features on the continental shelf and in lakes, but their dissipation via benthic boundary layer drag is largely unknown, particularly when the wave amplitudes are large and boundary layer corrections based on linear theory are clearly invalid. In general, the wave-induced boundary layer experiences a continuous favorable-to-adverse variation of the pressure gradient, undergoes transition, may reach a strongly turbulent state, and frequently separates near the point of maximum adverse pressure gradient in the lee of the wave. In this study a model for fully-nonlinear solitary waves of depression in a two-layer stratification is employed as the inviscid base state, and a RANS solver with \textit{k-$\omega $} turbulence model is used to compute the stationary boundary layer under the wave. Local friction coefficients and eddy viscosities are computed in the footprint of the wave. Locations of boundary layer separation are computed as well as the integrated frictional drag over the region of attached boundary layer flow. Boundary layer characteristics are presented for a range of environmental conditions, Reynolds numbers, and surface roughness in an attempt to provide a quantitative measure of the frictional drag of long internal waves in realistic, shallow environs.