Session H28: Waves II

Internal Wave Attractors: Topographic effects on wave reflection and energy propagation

It is well known that the angle of propagation of internal gravity waves depends on the ratio of the frequency of the waves to the buoyancy frequency of the stratification. This constraint on the direction of propagation causes the wavelength of an internal wave to either increase or decrease when it reflects from a sloping boundary. Such geometric focusing can lead to wave energy being enhanced in predictable regions of the ocean, giving rise to the possibility of an internal wave attractor where the energy is focused onto a limit cycle. In this paper we revisit internal wave attractors in a simple trapezoidal tank, exploring how the structure of the attractor is disrupted by replacing the sloping boundary with a staircase configuration comprising only horizontal and vertical surfaces. Simple ray tracing suggests an attractor cannot form in such a geometry despite the macroscopic shape remaining unchanged. We explore this configuration experimentally, varying the length scale of the individual steps, demonstrating an evolution towards the classical attractor as the step size is decreased.   

A Cloak of Invisibility Against Ocean Waves

In this talk we show that that floating objects in stratified fluids can be cloaked against broadband incident waves by properly architecting the bottom corrugations. The density of water in an ocean or a sea is typically not constant due to, mainly, variations of temperature and salinity. Stratified waters, besides regular surface waves, admit the so-called internal waves, which are gravity waves that propagate within the body of the water. The concept behind the presented scheme is based on nonlinear resonance of surface and interfacial waves with the bottom topography and is obtained due to the dispersive nature of gravity waves. Perfect cloaking against monochromatic waves can theoretically be achieved and was further investigated via a direct high-order spectral scheme. The presented cloak is the alignment of bottom corrugations only, and therefore is surface noninvasive. Cloaking in seas by bottom modifications may play a role in protecting near shore or offshore structures (buoys) and in creating shelter for fishermen during storms. In reverse it can result in disappearance and appearance of surface waves in areas where sandbars (or any other appreciable bottom variations) exist.   

An extended application for strongly nonlinear two-layer model

Strongly nonlinear internal wave models that have been developed in recent years have mostly been derived under long-wave, shallow-water assumptions. This talk will focus on assessing the applicability of these models in setups that go beyond their derivation hypotheses by comparisons with direct numerical simulations of near two-layer Euler-fluids' motion emanating from experimentally realizable initial conditions. By placing numerical filters that effectively truncate high Fourier modes, the ill-posedness associated with the model equations is resolved, allowing numerical time evolution studies to proceed. As wave profiles change dynamically, the numerical filters are designed adaptively. Compared with full Euler solutions, model evolutions show good agreement from small to moderate amplitude waves. Even for large amplitudes, when Kelvin-Helmholtz instability can occur, the primary waves are still captured in both amplitude and phase, demonstrating how an accurate filter implementation is capable of enhancing the models' predictive validity. Comparisons with two-layer Korteweg-de Vries (KdV) equation for the same initial conditions will also be presented, and advantages and shortcomings of the different models will be discussed.   

Bottom Boundary Layer Turbulence under an Internal Solitary Wave of Depression: Effects of Barotropic Current

The development of the bottom boundary layer (BBL) in the footprint of an internal solitary wave of depression is investigated by means of spectrally accurate numerical simulations in two and three dimensions. The focus is on the subsequent generation and ejection of vortices that potentially leads to bed sediment resuspension. Our preliminary study, based on two-dimensional simulations, has indicated that introduction of a barotropic current enables the formation of a robust separation bubble in a region of adverse pressure gradient over the bed. As a result, the production of packets of instability waves from the separation bubble, which evolve into vortices intermittently shed into the water column, is greatly enhanced. In the present study, the effect of barotropic current is investigated in detail for different wave amplitudes and stratification profiles with the wave Reynolds number up to the order of 10$^{5}$. This parametric study is performed in idealized, two-dimensional flow fields, in which fully-nonlinear, fully-nonhydrostatic wave field obtained by solving the DJL equation is adopted as a base state flow. Some preliminary results will be presented from three-dimensional Large Eddy Simulations which have been performed for a select set of governing parameters to explore the transition into turbulence in the BBL.   

Waves and Currents: Hawking radiation in the hydraulics laboratory?

Laboratory experiments were performed to test an analogy between Hawking radiation (the process by which black holes radiate energy) and the propagation of water waves against an adverse current. A streamlined obstacle was placed in a flume to create a region of high velocity. Long waves generated downstream of the obstacle were blocked by this region and converted to a pair of short waves. The group velocities of both the converted waves were downstream, but one of the converted waves retained an upstream phase velocity, whereas the other had a downstream phase velocity. These waves are shown to be analogous to Hawking radiation.   

Spatio-temporal characterization of Capillary Wave Turbulence

Wave Turbulence concerns the study of the statistical properties of a set of numerous non-linear interacting waves. The archetype of this phenomenon are waves on the surface of a fluid [1]. We report a space-time characterization of Capillary Wave Turbulence, produced in the laboratory at the air-water interface. The three-dimensional shape of the free interface is measured as a function of time, by using an optical method, the ``Diffusing Light Photography'' [2], associated with a fast camera. Linear and non-linear dispersion relations are extracted by the computation of the spatio-temporal power spectrum of wave amplitude. When Wave Turbulence regime is reached, we observe power-law spectra both in time and in space, whose exponents are in agreement with the theoretical predictions of Wave Turbulence theory~[3].\\[4pt] [1] A. C. Newell and B. Rumpf, Annu. Rev. Fluid Mech. 43, 59 (2011) [2] W. B. Wright, R. Budakian and S. J. Putterman, Phys. Rev. Lett. 76, 4528 (1996)\\[0pt] [3] V. Zakharov, V. L'vov and G. Falkovich (1992) ``Kolmogorov spectra of turbulence'' Springer-Verlag   

Direct numerical simulations of gravity-capillary wave turbulence

Direct numerical simulation of the full two phase Navier-Stokes equations, including surface tension are performed, using the code Gerris (Popinet, 2009), in order to investigate gravity-capillary wave turbulence. Wave turbulence concerns the study of the statistical and dynamical properties of a set of nonlinear interacting waves (Zakharov, 1992). Waves at the air-water interface, initially at rest, are excited at low wave-numbers and a stationary wave turbulence state is obtained after a time long enough (typically 30 periods of the wave forcing period). The space-time wave height power spectrum is calculated for both capillary and gravity waves regimes. The observed dispersion relation is in agreement with the theoretical one for linear gravity-capillary wave. The wave height power spectrum in the wave-number-space or in the frequency-space exhibit a power law and will be discussed with respects of weak turbulence theory (Zakharov, 2012). Finally the scaling of the spectrum with the injected power will be compared with theoretical and experimental works.   

An Experiment on Two-Dimensional Interaction of Solitary Waves in Shallow Water System

The dynamics of solitary waves in horizontally two-dimensional region is not yet well understood. Recently two-dimensional soliton interaction of Kadmotsetv-Petviashvili (KP) equation which describes the weakly nonlinear long wave in shallow water system has been theoretically studied (e.g. Kodama (2010)). It is clarified that the ``resonant'' interaction which forms Y-shaped triad can be described by exact solution. Li et al. (2011) experimentally studied the reflection of solitary wave at the wall and verified the theory of KP equation. To investigate more general interaction process, an experiment in wave tank using two wave makers which are controlled independently is carried out. The wave tank is 4m in length and 3.6 m in width. The depth of the water is about 8cm. The wavemakers, which are piston-type and have board about 1.5m in length, can produce orderly solitary wave which amplitude is 1.0-3.5cm. We observe newly generated solitary wave due to interaction of original solitary waves which have different amplitude and/or propagation direction. The results are compared with the aforementioned theory of KP equation.