Atle Jensen

Properties of large-amplitude internal waves

Properties of solitary waves propagating in a two-layer fluid are investigated comparing experiments and theory. In the experiments the velocity eld induced by the waves, the propagation speed and the wave shape are quite accurately measured using particle tracking velocimetry (PTV) and image analysis. The experiments are calibrated with a layer of fresh water above a layer of brine. The depth of the brine is 4.13 times the depth of the fresh water. Theoretical results are given for this depth ratio, and, in addition, in a few examples for larger ratios, up to 100:1. The wave amplitudes in the experiments range from a small value up to almost maximal amplitude. The thickness of the pycnocline is in the range of approximately 0.13{0.26 times the depth of the thinner layer. Solitary waves are generated by releasing a volume of fresh water trapped behind a gate. By careful adjustment of the length and depth of the initial volume we always generate a single solitary wave, even for very large volumes. The experiments are very repeatable and the recording technique is very accurate. The error in the measured velocities non-dimensionalized by the linear long wave speed is less than about 7{8% in all cases. The experiments are compared with a fully nonlinear interface model and weakly nonlinear Korteweg{de Vries (KdV) theory. The fully nonlinear model compares excellently with the experiments for all quantities measured. This is true for the whole amplitude range, even for a pycnocline which is not very sharp. The KdV theory is relevant for small wave amplitude but exhibit a systematic deviation from the experiments and the fully nonlinear theory for wave amplitudes exceeding about 0.4 times the depth of the thinner layer. In the experiments with the largest waves, rolls develop behind the maximal displacement of the wave due to the Kelvin{Helmholtz instability. The recordings enable evaluation of the local Richardson number due to the flow in the pycnocline. We nd that stability or instability of the flow occurs in approximate agreement with the theorem of Miles and Howard.

Breaking and broadening of internal solitary waves

Solitary waves propagating horizontally in a stratified fluid are investigated. The fluid has a shallow layer with linear stratification and a deep layer with constant density. The investigation is both experimental and theoretical. Detailed measurements of the velocities induced by the waves are facilitated by particle tracking velocimetry (PTV) and particle image velocimetry (PIV). Particular attention is paid to the role of wave breaking which is observed in the experiments. Incipient breaking is found to take place for moderately large waves in the form of the generation of vortices in the leading part of the waves. The maximal induced fluid velocity close to the free surface is then about 80% of the wave speed, and the wave amplitude is about half of the depth of the stratified layer. Wave amplitude is defined as the maximal excursion of the stratified layer. The breaking increases in power with increasing wave amplitude. The magnitude of the induced fluid velocity in the large waves is found to be approximately bounded by the wave speed. The breaking introduces a broadening of the waves. In the experiments a maximal amplitude and speed of the waves are obtained. A theoretical fully nonlinear two-layer model is developed in parallel with the experiments. In this model the fluid motion is assumed to be steady in a frame of reference moving with the wave. The Brunt-Vaisala frequency is constant in the layer with linear stratification and zero in the other. A mathematical solution is obtained by means of integral equations. Experiments and theory show good agreement up to breaking. An approximately linear relationship between the wave speed and amplitude is found both in the theory and the experiments and also when wave breaking is observed in the latter. The upper bound of the fluid velocity and the broadening of the waves, observed in the experiments, are not predicted by the theory, however. There was always found to be excursion of the solitary waves into the layer with constant density, irrespective of the ratio between the depths of the layers.

An experimental study of wave run-up at a steep beach

This paper presents experiments on run-up of strongly nonlinear waves on a beach of 10.54° inclination. Velocity fields are obtained by the PIV (particle image velocimetry) technique. Acceleration measurements are also attempted, but it is difficult to obtain useful results in every case. In addition, free-surface profiles are extracted from digital images and wave resistance probes. The investigation focuses on the dynamics of the early stages of the run-up, when steep fronts evolve in the vicinity of the equilibrium shoreline, but maximum run-up heights are also reported. Measurements on moderately nonlinear waves are compared to results from long-wave theories, including a numerical Boussinesq model and analytic shallow-water results from the literature. In particular the applicability of the long-wave theories is addressed. However, most attention is given to run-up of high incident solitary waves that are on the brink of breaking at the shoreline. In one case a temporarily slightly overturning wave front is found that neither develops into a plunger or displays appreciable spilling. This feature is discussed in view of measured velocity and acceleration patterns and with reference to the dam-break problem. Effects of scaling, as well as viscous damping, are also briefly discussed.

Shear-induced breaking of large internal solitary waves

The stability properties of 24 experimentally generated internal solitary waves (ISWs) of extremely large amplitude, all with minimum Richardson number less than 1/4, are investigated. The study is supplemented by fully nonlinear calculations in a three-layer fluid. The waves move along a linearly stratified pycnocline (depth h 2 ) sandwiched between a thin upper layer (depth h 1 ) and a deep lower layer (depth h 3 ), both homogeneous. In particular, the wave-induced velocity profile through the pycnocline is measured by particle image velocimetry (PIV) and obtained in computation. Breaking ISWs were found to have amplitudes (a 1 ) in the range a 1 > 2.24√h 1 h 2 (1 + h 2 /h 1 ), while stable waves were on or below this limit. Breaking ISWs were investigated for 0.27 0.86 and stable waves for L x /λ < 0.86. The results show a sort of threshold-like behaviour in terms of L x /λ. The results demonstrate that the breaking threshold of L x /λ = 0.86 was sharper than one based on a minimum Richardson number and reveal that the Richardson number was found to become almost antisymmetric across relatively thick pycnoclines, with the minimum occurring towards the top part of the pycnocline.

Experiments on focusing unidirectional water waves

Results of four groups of experiments involving transient, mechanically generated water waves in a narrow wave tank are described. The purpose of these experiments was to investigate the limitations of the validity of linear theory predictions of the spatiotemporal structure of the surface elevation in focal regions. For unidirectional surface gravity waves, focusing occurs as a result of long waves overtaking short waves. Surprisingly, in our measurements, nonlinear effects are stronger in deep water than in intermediate depth water and are stronger in nonfocusing wave trains than in focusing wave trains. These trends can be explained by the observation that the dominant source of nonlinear interaction in our measurements was the Benjamin-Feir instability, which acts only over a limited duration in focusing wave trains, only in wave trains whose bandwidth is narrow, and only in deep water. Under conditions in which the Benjamin-Feir instability does not act (as is expected to be the case in the ocean), predictions that take into account amplitude-dependent dispersion but otherwise neglect nonlinear effects are in good agreement with measurements for wave trains with (ka)max slightly in excess of 0.30.

Convectively induced shear instability in large amplitude internal solitary waves

Laboratory study has been carried out to investigate the instability of an internal solitary wave of depression in a shallow stratified fluid system. The experimental campaign has been supported by theoretical computations and has focused on a two layered stratification consisting of a homogeneous dense layer below a linearly stratified top layer. The initial background stratification has been varied and it is found that the onset and intensity of breaking are affected dramatically by changes in the background stratification. Manifestations of a combination of shear and convective instability are seen on the leading face of the wave. It is shown that there is an interplay between the two instability types and convective instability induces shear by enhancing isopycnal compression. Variation in the upper boundary condition is also found to have an effect on stability. In particular, the implications for convective instability are shown to be profound and a dramatic increase in wave amplitude is seen for a ...

Optimization of acceleration measurements using PIV

Two methods are presented for extracting accelerations from velocity fields measured by a two camera PIV system. One method traces imaginary particles between the velocity fields, while the other employs linear regression for the direct estimation of the velocity gradients, which are combined to produce the total acceleration. The effect of experimental and discretization errors is discussed and analysed and a simple error parameter for the acceleration computations is identified. Also a filtering technique for the removal of short scale noise in the velocity fields is given. The methods are applied to three different applications, namely Stokes waves, a nearly spilling crest in an irregular wave train and a solitary wave. In all applications, errors and scattering in the accelerations are discussed by comparison to analytical or numerical solutions.

Runup and boundary layers on sloping beaches

The present study is devoted to discrepancies between experimental and theoretical runup heights on an inclined plane, which have occasionally been reported in the literature. In a new study on solitary wave-runup on moderately steep slopes, in a wave tank with 20 cm water depth, detailed observations are made for the shoreline motion and velocity profiles during runup. The waves are not breaking during runup, but they do break during the subsequent draw-down. Both capillary effects and viscous boundary layers are detected. In the investigated cases the onshore flow is close to the transitional regime between laminar and turbulent boundary layers. The flow behaviour depends on the amplitude of the incident wave and the location on the beach. Stable laminar flow, fluctuations (Tollmien-Schlichting waves), and formation of vortices are all observed. Comparison with numerical simulations showed that the experimental runup heights were markedly smaller than predictions from inviscid theory. The observed and computed runup heights are discussed in the context of preexisting theory and experiments. Similar deviations are apparent there, but have often been overlooked or given improper physical explanations. Guided by the absence of turbulence and irregular flow features in parts of the experiments we apply laminar boundary layer theory to the inundation flow. Outer flows from potential flow models are inserted in a nonlinear, numerical boundary layer model. Even though the boundary layer model is invalid near the moving the shoreline, the computed velocity profiles are found to compare well with experiments elsewhere, until instabilities are observed in the measurements. Analytical, linear boundary layer solutions are also derived both for an idealized swash zone motion and a polynomial representation of the time dependence of the outer flow. Due to lacking experimental or theoretical descriptions of the contact point dynamics no two-way coupling of the boundary layer model and the inviscid runup models is attempted. Instead, the effect of the boundary layer on the maximum runup is estimated through integrated losses of onshore volume transport and found to be consistent with the differences between inviscid theory and experiments.

Modelling of run up of steep non-breaking waves

Abstract A Navier–Stokes solver is used to examine steep waves as they run up a steep beach (10.54°). The volume of fluid method (VOF) is used to model the free surface. Comparison with experimental results shows reasonable overall agreement in the prediction of the free-surface, velocities and accelerations within the flow. A spurious feature at the free-surface was found which does reduce the quality of the results. For a steep wave we see the transition from a steep wave front to a smooth run-up tongue at the beach that is in qualitative agreement with experiment.

Experiments and computation of onshore breaking solitary waves

This is a combined experimental and computational study of solitary waves that break on-shore. Velocities and accelerations are measured by a two-camera PIV technique and compared to theoretical values from an Euler model with a VOF method for the free surface. In particular, the dynamics of a so-called collapsing breaker is scrutinized and the closure between the breaker and the beach is found to be akin to slamming. To the knowledge of the authors, no velocity measurements for this kind of breaker have been previously reported.