John Grue

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.

A fast method for fully nonlinear water-wave computations

A fast computational method for fully nonlinear non-overturning water waves is derived in two and three dimensions. A corresponding time-stepping scheme is developed in the two-dimensional case. The essential part of the method is a fast converging iterative solution procedure of the Laplace equation. One part of the solution is obtained by fast Fourier transform, while another part is highly nonlinear and consists of integrals with kernels that decay quickly in space. The number of operations required is asymptotically O(N logN), where N is the number of nodes at the free surface. While any accuracy of the computations is achieved by a continued iteration of the equations, one iteration is found to be sucient for practical computations, while maintaining high accuracy. The resulting explicit approximation of the scheme is tested in two versions. Simulations of nonlinear wave elds with wave slope even up to about unity compare very well with reference computations. The numerical scheme is formulated in such a way that aliasing terms are partially or completely avoided.

Wave forces on three-dimensional floating bodies with small forward speed

A boundary-integral method is developed for computing first-order and mean second-order wave forces on floating bodies with small forward speed in three dimensions. The method is based on applying Greens theorem and linearising the Green function and velocity potential in the forward speed. The velocity potential on the wetted body surface is then given as the solution of two sets of integral equations with unknowns only on the body. The equations contain no water-line integral, and the free-surface integral decays rapidly. The Timman-Newman symmetry relations for the added mass and damping coefficients are extended to the case when the double-body flow around the body is included in the free-surface condition. The linear wave exciting forces are found both by pressure integration and by a generalised far-field form of the Haskind relations. The mean drift force is found by far-field analysis. All the derivations are made for an arbitrary wave heading. A boundary-element program utilising the new method has been developed. Numerical results and convergence tests are presented for several body geometries. It is found that the wave exciting forces are presented for several body geometries. It is found that the wave exciting forces and the mean drift forces are most influenced by a small forward speed. Values of the wave drift damping coefficient are computed. It is found that interference phenomena may lead to negative wave drift damping for bodies of complicated shape.

Evolution equations for strongly nonlinear internal waves

This paper is concerned with shallow-water equations for strongly nonlinear internal waves in a two-layer fluid, and comparison of their solitary solutions with the results of fully nonlinear computations and with experimental data. This comparison is necessary due to a contradictory nature of these equations which combine strong nonlinearity and weak dispersion. First, the Lagrangian (Whitham’s) method for dispersive shallow-water waves is applied to derivation of equations equivalent to the Choi–Camassa (CC) equations. Then, using the Riemann invariants for strongly nonlinear, nondispersive waves, we obtain unidirectional, evolution equations with nonlinear dispersive terms. The latter are first derived from the CC equations and then introduced semiphenomenologically as quasistationary generalizations of weakly nonlinear Korteweg–de Vries and Benjamin–Ono models. Solitary solutions for these equations are obtained and verified against fully nonlinear computations. Comparisons are also made with available observational data for extremely strong solitons in coastal zones with well expressed pycnoclines.

On the breaking of internal solitary waves at a ridge

An experimental laboratory study has been carried out to investigate the propagation of an internal solitary wave of depression and its distortion by a bottom ridge in a two-layer stratified fluid system. Wave profiles, density fields and velocity fields have been measured at three reference locations, namely upstream, downstream and over the ridge. Experiments have been performed with wave amplitudes in the range 0.2– 1.9 times the depth of the upper layer, and a ratio between the lower and the upper layer in the range 3.0–8.5. The ridge slope was varied from 0.1 to 0.33 and the maximum ridge height was two-thirds of the thicker fluid layer. Over the ridge, the flow has been classified into: (i) cases when the bottom ridge has little influence on the propagation and spatial structure of the internal solitary wave, (ii) cases where the internal solitary wave is significantly distorted by the blocking effect of the ridge (though no wave breaking occurs), and (iii) cases for which the internal solitary wave is broken as it encounters and passes over the bottom ridge. A detailed description of the processes leading to wave breaking is given. Breaking has been found to take place when the fluid velocity in the lower layer exceeds 0.7 of a local nonlinear wave speed, defined at the top of the ridge. The breaking condition is also expressed in terms of the amplitude of the incident wave, the layer thickness ratio and the relative height of the ridge. The wave breaking can be determined from the input parameters of the experiment. The transmitted waves have been found to always consist of a leading pulse (solitary wave) followed by a dispersive wavetrain. The (solitary) wave amplitude is significantly reduced only when breaking takes place at the ridge. Internal waves of mode two are generated in cases with strong breaking.

WAVE RADIATION AND WAVE DIFFRACTION FROM A SUBMERGED BODY IN A UNIFORM CURRENT

Radiation and diffraction of free-surface waves due to a submerged body in a uniform current is considered. The fluid layer is infinitely deep and the motion is two-dimensional. Applying the method of integral equations, the radiation problem and the diffraction problem for a submerged circular cylinder are examined. For small speed U of the current a forced motion of a given frequency will give rise to four waves. It is shown, however, that, for a circular cylinder, an incoming harmonic wave gives rise to two waves only. Depending on the frequency, the new generated wave may be considered as a transmitted or a reflected wave. The mean second-order force is computed. For the radiation problem the first-order damping force is also obtained. It is shown that, for some values of the parameters, the damping force is negative. This result is closely related to the fact that a harmonic wave travelling upstream with a phase velocity less than U conveys negative energy downstream. The forces remain finite as U σ/ g (σ ≡ the frequency, g ≡ the acceleration due to gravity) approaches ¼.

Nonlinear water waves at a submerged obstacle or bottom topography

Nonlinear diffraction of low-amplitude gravity waves in deep water due to a slightly submerged obstacle is studied experimentally in a wave channel and theoretically. The obstacle is either a circular cylinder or a rectangular shelf. The incoming waves (with wavelength λ) undergo strong nonlinear deformations at the obstacle when the wave amplitude is finite. An infinite number of superharmonic waves are then introduced to the flow. Their wavelengths far a way from the obstacle are λ/4, λ/9, λ/16,..., due to the dispersion relation being quadratic in the wave frequency

A method for computing unsteady fully nonlinear interfacial waves

We derive a time-stepping method for unsteady fully nonlinear two-dimensional motion of a two-layer fluid. Essential parts of the method are: use of Taylor series expansions of the prognostic equations, application of spatial finite difference formulae of high order, and application of Cauchys theorem to solve the Laplace equation, where the latter is found to be advantageous in avoiding instability. The method is computationally very efficient. The model is applied to investigate unsteady trans-critical two-layer flow over a bottom topography. We are able to simulate a set of laboratory experiments on this problem described by Melville & Helfrich (1987), finding a very good agreement between the fully nonlinear model and the experiments, where they reported bad agreement with weakly nonlinear Korteweg–de Vries theories for interfacial waves. The unsteady transcritical regime is identified. In this regime, we find that an upstream undular bore is generated when the speed of the body is less than a certain value, which somewhat exceeds the critical speed. In the remaining regime, a train of solitary waves is generated upstream. In both cases a corresponding constant level of the interface behind the body is developed. We also perform a detailed investigation of upstream generation of solitary waves by a moving body, finding that wave trains with amplitude comparable to the thickness of the thinner layer are generated. The results indicate that weakly nonlinear theories in many cases have quite limited applications in modelling unsteady transcritical two-layer flows, and that a fully nonlinear method in general is required.

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.