W. K. Melville

ON LONG NONLINEAR INTERNAL WAVES OVER SLOPE-SHELF TOPOGRAPHY

An experimental and theoretical study of the propagation and stability of long nonlinear internal waves over slope-shelf topography is presented. A generalised Korteweg-de Vries (KdV) equation, including the effects of nonlinearity, dispersion, dissipation and varying bottom topography, is formulated and solved numerically for single and rank-ordered pairs of solitary waves incident on the slope. The results of corresponding laboratory experiments in a salt-stratified system are reported. Very good agreement between theory and experiment is obtained for a range of stratifications, topography and incident-wave amplitudes. Significant disagreement is found in some cases if the effects of dissipation and higher-order (cubic) nonlinearity are not included in the theoretical model. Weak shearing and strong breaking (overturning) instabilities are observed and found to depend strongly on the incident-wave amplitude and the stratification on the shelf. In some cases the instability of the lowest-mode wave leads to the generation of a second-mode solitary wave. The application of these findings to the prediction and interpretation of field data is discussed.

Transcritical two-layer flow over topography

The evolution of weakly-nonlinear two-layer flow over topography is considered. The governing equations are formulated to consider the effects of quadratic and cubic nonlinearity in the transcritical regime of the internal mode. In the absence of cubic nonlinearity an inhomogeneous Kortewegae Vries equation describes the interfacial displacement. Numerical solutions of this equation exhibit undular bores or sequences of Boussinesq solitary waves upstream in a transcritical regime. For sufficiently large supercritical Froude numbers, a locally steady flow is attained over the topography. In that regime in which both quadratic and cubic nonlinearity are comparable, the evolution of the interface is described by an inhomogeneous extended Kortewegde Vries (EKdV) equation. This equation displays undular bores upstream in a subcritical regime, but monotonic bores in a transcritical regime. The monotonic bores are solitary wave solutions of the correspondmg homogeneous EKdV equation. Again, locally steady flow is attained for sufficiently large supercritical Froude numbers. The predictions of the numerical solutions are compared with laboratory experiments which show good agreement with the solutions of the forced EKdV equation for some range of parameters. It is shown that a recent result of Miles (1986), which predicts an unsteady transcritical regime for single-layer flows, may readily be extended to two-layer flows (described by the forced KdV equation) and is in agreement with the results presented here. Numerical experiments exploiting the symmetry of the homogeneous EKdV equation show that solitary waves of fixed amplitude but arbitrary length may be generated in systems described by the inhomogeneous EKdV equation.

Microwave backscatter and acoustic radiation from breaking waves

An experimental study of the microwave backscatter and acoustic radiation from breaking waves is reported. It is found that the averaged microwave and acoustic measurements correlate with the dynamics of wave breaking. Both the mean-square acoustic pressure and the backscattered microwave power correlate with the wave slope and dissipation, for waves of moderate slope (S < 0.28). The backscattered power and the mean-square pressure are also found to correlate strongly with each other. As the slope and wavelength of the breaking wave packet is increased, both the backscattered power and the mean-square pressure increase. It is found that a large portion of the backscattered microwave power precedes the onset of sound production and visible breaking. This indicates that the unsteadiness of the breaking process is important and that the geometry of the wave prior to breaking may dominate the backscattering. It is observed that the amount of acoustic energy radiated by an individual breaking wave scaled with the amount of mechanical energy dissipated during breaking. These laboratory results are compared to the field experiments of Farmet & Vagle (1988), Crowther (1989) and Jessup et al. (1990).

On interfacial solitary waves over slowly varying topography

The propagation of long, weakly nonlinear interfacial waves in a two-layer fluid of slowly varying depth is studied. The governing equations are formulated to include cubic nonlinearity, which dominates quadratic nonlinearity in some parametric neighbourhood of equal layer depths. Numerical solutions are obtained for an initial profile corresponding to either a single solitary wave or a rank-ordered pair of such waves incident in a monotonic transition between two regions of constant depth. The numerical solutions. supplemented by inverse-scattering theory, are used to investigate the change of polarity of the incident waves as they pass through a ‘turning point ’ of approximately equal layer depths. Our results exhibit significant differences from those reported by Knickerbocker & Newell (1980), which were based on a model equation. In particular, we find that more than one wave of reversed polarity may emerge.

On the stability of Kelvin waves

We consider the evolution of weakly nonlinear dispersive long waves in a rotating channel. The governing equations are derived and approximate solutions obtained for the initial data corresponding to a Kelvin wave. In consequence of the small nonlinear speed correction it is shown that weakly nonlinear Kelvin waves are unstable to a direct nonlinear resonance with the linear Poincare’ modes of the channel. Numerical solutions of the governing equations are computed and found to give good agreement with the approximate analytical solutions. It is shown that the curvature of the wavefront and the decay of the leading wave amplitude along the channel are attributable to the Poincark waves generated by the resonance. These results appear to give a qualitative explanation of the experimental results of Maxworthy (1983), and Renouard, Chabert d’Hibres & Zhang (1987). The evolution of weakly nonlinear dispersive long waves in a rotating fluid has been the subject of some controversy in recent years. In the absence of rotation and weak transverse effects the subject is well developed, a result of the applicability of the Korteweg-de Vries (KdV) equation and its related equations to problems of onedimensional propagation, with the well-known solitary-wave solutions. However, Maxworthy (1983) drew attention to the problems associated with rotation in oceanographic applications to sea straits and the continental shelves where the transverse scales of the topography are not negligible when compared to the Rossby radius. Maxworthy conducted experiments on the second-mode waves evolving from the collapse of a mixed region in a stratified fluid. The more important features of these experiments were the curvature of the wavefronts (in contrast to the corresponding straight-crested linear Kelvin waves) and their dissipation along the channel, the latter being attributed to vertical radiation of inertial waves. In a very careful theoretical investigation Grimshaw (1985) derived evolution equations for weakly nonlinear, long internal waves in continuously stratified fluids and found that at least two cases could be separated. Strong rotation: the Rossby radius is at most comparable with the wavelength, and the effects of rotation are separable from the effects of weak nonlinearity and dispersion. The wave decays exponentially across the channel with the evolution along the channel described by a KdV equation. Weak rotation : the Rossby radius is greater than the wavelength, and the effects of rotation are not separable. The evolution equation is a rotationmodified Kadomtsev-Petviashvili (KP) equation. In no case could Grimshaw explain the wave front curvature in the absence of dissipation in a channel of finite width. He suggested that the observed curvature could be due to wave transience

Geostrophic adjustment in a channel : nonlinear and dispersive effects

We consider the general problem of geostrophic adjustment in a channel in the weakly nonlinear and dispersive (non-hydrostatic) limit. Governing equations of Boussinesq- type are derived, based on the assumption of weak nonlinear, dispersive and rotational effects, both for surface waves on a homogeneous fluid and internal waves in a two-layer system. Numerical solutions of the Boussinesq equations are presented, giving examples of the geostrophic adjustment in a channel for two different kinds of initial disturbances, both with non-zero perturbation potential vorticity. The timescales of rotational separation (that is, the separation of the Kelvin and Poincar6 waves due to their dispersive properties) and that of nonlinear evolution are considered, with particular concern for the resonant interactions of nonlinear Kelvin waves and linear Poincar6 waves described by Melville, Tomasson & Renouard (1989). A parameter measuring the ratio of the two timescales is used to predict when the free and forced Poincar6 waves may be separated in the solution. It also distinguishes the cases in which the linear solutions are valid for the rotational separation from those requiring the full Boussinesq equations. Finally, solutions for the evolution of nonlinear internal waves in a sea strait are presented, and the effects of friction on the wavefront curvature of the nonlinear Kelvin waves are briefly considered.

Evolution of weakly nonlinear short waves riding on long gravity waves

A nonlinear Schrodinger equation, describing the evolution of a weakly nonlinear short gravity wavetrain riding on a longer finite-amplitude gravity wavetrain, is derived. This equation is then used to predict the steady envelope of the short wavetrain relative to the long wavetrain. It is found that approximate analytical solutions agree very well with numerical solutions over a realistic range of wave steepness. The solutions are compared with corresponding studies of the modulation of linear short waves by Longuet-Higgins & Stewart (1960) and Longuet-Higgins (1987). We find that the effect of the nonlinearity of the short waves is to increase the modulation of their wavenumber, significantly reduce the modulation of their amplitude, and reduce the modulation of their slope when compared with the predictions of Longuet-Higgins (1987) for linear short waves on finite-amplitude long waves. The question of the stability of these steady solutions remains open but may be addressed by solutions of this nonlinear Schrodinger equation.

The surface velocity field in steep and breaking waves

Coincident simultaneous measurements of the surface displacement and the horizontal velocity at the surface of steep and breaking waves are presented. The measurements involve a novel use of laser anemometry at the fluctuating air-water interface and clearly show the limitations of surface displacement measurements in characterizing steep and breaking wave fields. The measurements are used to examine the evolution of the surface drift velocity, spectra, wave envelopes, and forced long waves in unstable deep-water waves. Preliminary results of this work were reported by Melville & Rapp (1983).

Three-dimensional instabilities of nonlinear gravity-capillary waves

Le systeme de valeurs propres pour le probleme de stabilite est engendre par une methode de Galerkin. Les instabilites se developpent au voisinage des courbes de resonance lineaire

Flow past a constriction in a channel: a modal description

We consider the waves generated by transcritical flow past a constriction in a channel, or by ships or surface pressure distributions travelling at transcritical speeds. The two-dimensionality of the upstream advancing nonlinear waves, which has been observed both experimentally and numerically by several authors, is described by a modal decomposition of the flow response. We show that the lowest transverse mode may evolve nonlinearly, leading to a two-dimensional response upstream, with the higher transverse modes swept downstream. This description is supported by comparing the initial evolution of the solutions to the corresponding linear and nonlinear problems. Averaging across the channel demonstrates that the three-dimensional problem may be related to the corresponding two-dimensional problem with an additional effective forcing coming from the nonlinear coupling of the higher modes to the lowest two-dimensional mode. This coupling leads to a dependence of the upstream solutions on the channel width as well as the Froude number. Solutions are also obtained for two-layer fluids in which cubic nonlinearity is also important. The inclusion of cubic nonlinearity permits the generation of twodimensional fronts upstream, and demonstrates that the transition from three- to two-dimensional solutions upstream is not specific to Boussinesq solitary waves.