Explore the vast range of titles available.

Very long waves are generated when a ship moves across an appreciable depth change $\\unicode[STIX]{x0394}h$ comparable to the average and relatively shallow water depth $h$ at the location, with $\\unicode[STIX]{x0394}h/h\\simeq 1$ . The phenomenon is new and the waves were recently observed in the Oslofjord in Norway. The 0.5–1 km long waves, extending across the 2–3 km wide fjord, are observed as run-ups and run-downs along the shore, with periods of 30–60 s, where a wave height up to 1.4 m has been measured. The waves travelling with the shallow water speed, found ahead of the ships moving at subcritical depth Froude number, behave like a mini-tsunami. A qualitative explanation of the linear generation mechanism is provided by an asymptotic analysis, valid for $\\unicode[STIX]{x0394}h/h\\ll 1$ and long waves, expressing the generation in terms of a pressure impulse at the depth change. Complementary fully dispersive calculations for $\\unicode[STIX]{x0394}h/h\\simeq 1$ document symmetries of the waves at positive or negative $\\unicode[STIX]{x0394}h$ . The wave height grows with the ship speed $U$ according to $U^{n}$ with $n$ in the range 3–4, for $\\unicode[STIX]{x0394}h/h\\simeq 1$ , while the growth in $U$ is only very weak for $\\unicode[STIX]{x0394}h/h\\ll 1$ (the asymptotics). Calculations show good agreement with observations.

Laboratory measurements by PIV of strongly nonlinear irregular waves are compared to orbital velocity calculations of directional field waves. Nondimensional fluid velocities are obtained dividing by a local wave phase speed which is evaluated from a local wave number and wave slope. 30 laboratory waves have slope in the range 0.14–0.4. Nine among the waves have a slope of about 0.3, and breaking is measured in four among the waves. A backward breaking wave has local wave slope as small as 0.22. Wave induced acceleration is measured. Orbital velocity of two samples of directional field waves in storm conditions are calculated by nonlinear method and compared to the laboratory kinematics. The effect of the ocean surface current is accounted for. The field data include wave groups of local elevation gradient up to 0.3. The sample with the longest fetch exhibits nondimensional orbital velocities that are similar to the laboratory waves including levels up to breaking. In the other field sample, nondimensional fluid velocities are proportional to the local wave slope (up to 0.3), and velocities are well below breaking level. Key Points Local wave slope and phase speed interpret kinematics of steep irregular waves Laboratory waves exhibit breaking for wave slope of 0.3 Calculation of kinematics in GOTEX waves shows similarities with laboratory waves

A nonlinear calculation procedure for obtaining the wave kinematics from elevation measurements from field data is outlined in detail. A horizontal current is accounted for. The numerical calculations from the field data are compared to the kinematics of random waves obtained in laboratory measurements.

Internal solitary waves (ISWs) of large amplitude moving in the coastal ocean induce sizeable horizontal velocities above the sea bed. In turn, these give rise to instability and vortex formation in the bottom boundary layer (BBL), and sediment resuspension and concentration maintenance in the water column. We present two-dimensional laminar simulations in a numerical tank suitable for internal wave motion, including the processes of the BBL. The combined wave and vorticity field encounters a cloud of tracer particles near the bottom. The tracer particles are moved vertically because of the vorticity field during a first encounter. The reflected wave intercepts a second time with the tracer particles, which are then moved further vertically. Numerical experiments with a kinematic viscosity of 1/100 cm 2 s −1 or 1/1000 cm 2 s -1 are used to manipulate the scale of the Reynolds number at a moderate and great laboratory scale. The final vertical position of the tracer particles is found below a vertical level of approximately 0.23 times the water depth ( H ) after the second passage. The result is independent of the scale. This vertical position matches available field measurements of a summer benthic nepheloid layer reaching a height of 0.19 H . The laminar model predictions compare very well to the ISW-driven vortex formation measured in a three-dimensional laboratory wave tank. Convergence of the calculated vortex formation is documented.

The Lagrangian paths, horizontal Lagrangian drift velocity, $U_{L}$ , and the Lagrangian excess period, $T_{L}-T_{0}$ , where $T_{L}$ is the Lagrangian period and $T_{0}$ the Eulerian linear period, are obtained by particle tracking velocimetry (PTV) in non-breaking periodic laboratory waves at a finite water depth of $h=0.2~\\text{m}$ , wave height of $H=0.49h$ and wavenumber of $k=0.785/h$ . Both $U_{L}$ and $T_{L}-T_{0}$ are functions of the average vertical position of the paths, $\\bar{Y}$ , where $-1<\\bar{Y}/h<0$ . The functional relationships $U_{L}(\\bar{Y})$ and $T_{L}-T_{0}=f(\\bar{Y})$ are very similar. Comparisons to calculations by the inviscid strongly nonlinear Fenton method and the second-order theory show that the streaming velocities in the boundary layers below the wave surface and above the fluid bottom contribute to a strongly enhanced forward drift velocity and excess period. The experimental drift velocity shear becomes more than twice that obtained by the Fenton method, which again is approximately twice that of the second-order theory close to the surface. There is no mass flux of the periodic experimental waves and no pressure gradient. The results from a total number of 80 000 experimental particle paths in the different phases and vertical positions of the waves show a strong collapse. The particle paths are closed at the two vertical positions where  $U_{L}=0$ .

A nonlinear calculation procedure for obtaining the wave kinematics from elevation measurements from field data is outlined in detail. A horizontal current is accounted for. The numerical calculations from the field data are compared to the kinematics of random waves obtained in laboratory measurements.

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 h2) sandwiched between a thin upper layer (depth h1) and a deep lower layer (depth h3), 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 (a1) in the range $a_1\\,{>}\\,2.24\\sqrt{h_1h_2}(1+h_2/h_1)$, while stable waves were on or below this limit. Breaking ISWs were investigated for 0.27 < h2/h1 < 1 and 4.14 < h3/(h1 + h2) < 7.14 and stable waves for 0.36 < h2/h1 < 3.67 and 3.22 < h3/(h1 + h2) < 7.25. Kelvin–Helmholtz-like billows were observed in the breaking cases. They had a length of 7.9h2 and a propagation speed 0.09 times the wave speed. These measured values compared well with predicted values from a stability analysis, assuming steady shear flow with U(z) and ρ(z) taken at the wave maximum (U(z) horizontal velocity profile, ρ(z) density along the vertical z). Only unstable modes in waves of sufficient strength have the chance to grow sufficiently fast to develop breaking: the waves that broke had an estimated growth (of unstable modes) more than 3.3–3.7 times than in the strongest stable case. Evaluation of the minimum Richardson number (Rimin, in the pycnocline), the horizontal length of a pocket of possible instability, with wave-induced Ri < 14, (Lx) and the wavelength (λ), showed that all measurements fall within the range Rimin = −0.23Lx/λ + 0.298 ± 0.016 in the (Lx/λ, Rimin)-plane. Breaking ISWs were found for Lx/λ > 0.86 and stable waves for Lx/λ < 0.86. The results show a sort of threshold-like behaviour in terms of Lx/λ. The results demonstrate that the breaking threshold of Lx/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.

A three-dimensional two-layer, fully dispersive and strongly nonlinear interfacial wave model, including the interaction with a time-varying bottom topography, is developed. The method is based on a set of integral equations. The source and dipole terms are developed in series expansions in the vertical excursions of the interface and bottom topography, obtaining explicit inversion by Fourier transform. Calculations of strongly nonlinear interfacial waves with excursions comparable to the thinner layer depth show that the quadratic approximation of the method contains the essential dynamics, while the additional cubic terms always are small. Computations confirm the onset of wave train formation driven by topography, observed in experiments (Maxworthy, J. Geophys. Res., vol. 84(C1), 1979, pp. 338–346), depending on the Froude number and the topography height. Simulations of tidally driven three-dimensional internal wave formation show the formation of two wave trains per half tidal cycle for strong forcing and one wave train for weak forcing. Waves of both backward and forward curvature are calculated.

The instability and vortex shedding in the bottom boundary layer caused by internal solitary waves of depression propagating along a shallow pycnocline of a fluid are computed by finite-volume code in two dimensions. The calculated transition to instability agrees very well with laboratory experiments (Carr et al., Phys. Fluids, vol. 20, issue 6, 2008, 06603) but disagrees with existing computations that give a very conservative instability threshold. The instability boundary expressed by the amplitude depends on the depth $d$ of the pycnocline divided by the water depth $H$, and decays by a factor of 2.2 when $d/H$ is 0.21, and by a factor of 1.6 when $d/H$ is 0.16, and the stratification Reynolds number increases by a factor of 32. The instability occurs at moderate amplitude at large scale. The calculated oscillatory bed shear stress is strong in the wave phase and increases with the scale. Its non-dimensional magnitude at stratification Reynolds number 650 000 is comparable to the turbulent stress that can be extracted from field measurements of internal solitary waves of similar nonlinearity, moving along a pycnocline of similar relative depth.