Solitary Wave Of Depression Research Articles

Current and Density Observations of Packets of Nonlinear Internal Waves on the Outer New Jersey Shelf

AbstractClosely spaced observations of nonlinear internal waves (NLIWs) were made on the outer continental shelf off New Jersey in June 2009. Nearly full water column measurements of current velocity were made with four acoustic Doppler current profilers (ADCPs) that were moored about 5 km apart on the bottom along a line approximately normal to the bathymetry between water depths of 67 and 92 m. Density profiles were obtained from two vertical strings of temperature and conductivity sensors that were deployed near each of the interior ADCP moorings. In addition, a towed ScanFish provided profiles and fixed-level records of temperature and salinity through several NLIW packets near the moorings. Several case studies were selected to describe the propagation of the NLIWs. One to three solitary waves of depression were observed in five selected packets. There were also occurrences of multiple-phase dispersive wave packets. The average propagation speed corrected for advection of the observed waves was 0.51 ...

Current and Density Observations of Packets of Nonlinear Internal Waves on the Outer New Jersey Shelf

Abstract Closely spaced observations of nonlinear internal waves (NLIWs) were made on the outer continental shelf off New Jersey in June 2009. Nearly full water column measurements of current velocity were made with four acoustic Doppler current profilers (ADCPs) that were moored about 5 km apart on the bottom along a line approximately normal to the bathymetry between water depths of 67 and 92 m. Density profiles were obtained from two vertical strings of temperature and conductivity sensors that were deployed near each of the interior ADCP moorings. In addition, a towed ScanFish provided profiles and fixed-level records of temperature and salinity through several NLIW packets near the moorings. Several case studies were selected to describe the propagation of the NLIWs. One to three solitary waves of depression were observed in five selected packets. There were also occurrences of multiple-phase dispersive wave packets. The average propagation speed corrected for advection of the observed waves was 0.51 ± 0.09 m s−1. The waves were directed primarily shoreward (~northwestward) along the mooring line with average wavelengths and periods of about 300 m and 10 min, respectively. Wave amplitudes and energies decreased with decreasing water depth. The observed wave parameters can be locally described by a two-layer Korteweg–de Vries (KdV) model, except for the decreasing amplitudes, which may be due to shear-induced dissipation and/or bottom drag. The various complementary observations utilized in this study present a unique description of NLIWs.

Numerical simulation of internal solitary wave—induced reverse flow and associated vortices in a shallow, two-layer fluid benthic boundary layer

The wave-induced velocity and pressure fields beneath a large amplitude internal solitary wave of depression propagating over a smooth, flat, horizontal, and rigid boundary in a shallow two-layer fluid are computed numerically. A numerical ocean model is utilised, the set-up of which is designed and tuned to replicate the previously published experimental results of Carr and Davies (Phys Fluids 18(1):016,601–1–016,601–10, 2006). Excellent agreement is found between the two data sets and, in particular, the numerical simulation replicates the finding of a reverse flow along the bed aft of the wave. The numerically computed velocity and pressure gradients confirm that the occurrence of the reverse flow is a consequence of boundary layer separation in the adverse pressure gradient region. In addition, vortices associated with the reverse flow are seen to form near the bed.

Dynamics of steep two-dimensional gravity–capillary solitary waves

In this paper, the unsteady evolution of two-dimensional fully nonlinear free-surface gravity–capillary solitary waves is computed numerically in infinite depth. Gravity–capillary wavepacket-type solitary waves were found previously for the full Euler equations, bifurcating from the minimum of the linear dispersion relation. Small and moderate amplitude elevation solitary waves, which were known to be linearly unstable, are shown to evolve into stable depression solitary waves, together with a radiated wave field. Depression waves and certain large amplitude elevation waves were found to be robust to numerical perturbations. Two kinds of collisions are computed: head-on collisions whereby the waves are almost unchanged, and overtaking collisions which are either almost elastic if the wave amplitudes are both large or destroy the smaller wave in the case of a small amplitude wave overtaking a large one.

Breaking of shoaling internal solitary waves

The breaking of fully nonlinear internal solitary waves of depression shoaling upon a uniformly sloping boundary in a smoothed two-layer density field was investigated using high-resolution two-dimensional simulations. Our simulations were limited to narrow-crested waves, which are more common than broad-crested waves in geophysical flows. The simulations were performed for a wide range of boundary slopes S ∈ [0.01, 0.3] and wave slopes extending the parameter range to weaker slopes than considered in previous laboratory and numerical studies. Over steep slopes (S ≥ 0.1), three distinct breaking processes were observed: surging, plunging and collapsing breakers which are associated with reflection, convective instability and boundary-layer separation, respectively. Over mild slopes (S ≤ 0.05), nonlinearity varies gradually and the wave fissions into a train of waves of elevation as it passes through the turning point where solitary waves reverse polarity. The dynamics of each breaker type were investigated and the predominance of a particular mechanism was associated with a relative developmental time scale. The breaking location was modelled as a function of wave amplitude (a), characteristic wave length and the isopycnal length along the slope. The breaker type was characterized in wave slope (Sw = a/Lw, where Lw is a measure of half of the wavelength) versus S space, and the reflection coefficient (R), modelled as a function of the internal Iribarren number, was in agreement with other studies. The effects of grid resolution and wave Reynolds number (Rew) on R, boundary-layer separation and the evolution of global instability were studied. High Reynolds numbers (Rew ~ 104) were found to trigger a global instability, which modifies the breaking process relative to the lower Rew case, but not necessarily the breaking location, and results in a ~ 10 % increase in R, relative to the Rew ~ 103 case.

Observation of Axisymmetric Solitary Waves on the Surface of a Ferrofluid

We report the first observation of axisymmetric solitary waves on the surface of a cylindrical magnetic fluid layer surrounding a current-carrying metallic tube. According to the ratio between the magnetic and capillary forces, both elevation and depression solitary waves are observed with profiles in good agreement with theoretical predictions based on the magnetic analogue of the Korteweg-de Vries equation. We also report the first measurements of the velocity and the dispersion relation of axisymmetric linear waves propagating on the cylindrical ferrofluid layer that are found in good agreement with theoretical predictions.

Spectral Stability of Stationary Solutions of a Boussinesq System Describing Long Waves in Dispersive Media

We study the spectral (in)stability of one-dimensional solitary and cnoidal waves of various Boussinesq systems. These systems model three-dimensional water waves (i.e., the surface is two-dimensional) with or without surface tension. We present the results of numerous computations examining the spectra related to the linear stability problem for both stationary solitary and cnoidal waves with various amplitudes, as well as multipulse solutions. The one-dimensional nature of the wave forms allows us to separate the dependence of the perturbations on the spatial variables by transverse wave number. The compilation of these results gives a full view of the two-dimensional stability problem of these one-dimensional solutions. We demonstrate that line solitary waves with elevated profiles are spectrally stable with respect to one-dimensional perturbations and long transverse perturbations. We show that depression solitary waves are spectrally stable with respect to one-dimensional perturbations, but unstable with respect to transverse perturbations. We also discuss the instability of multipulse solitary waves and cnoidal-wave solutions of the Boussinesq system.

Statistics of acoustic intensity fluctuations caused by non-linear internal waves of elevation in shallow water in the South China Sea.

In April 2007, a 6-day long field experiment was conducted to measure internal waves and their acoustic effects at the edge of the continental shelf in the Northern South China Sea. Very large internal solitary waves of depression originating in the Luzon Strait generate trains of internal elevation waves as they encounter a subaqueous bluff at the shelf edge in 120-m water depth. The acoustic transect, from the moored acoustic source transmitting 400-Hz m-sequences to two vertical line arrays moored at 3- and 6-km ranges, was oriented roughly parallel to the crests of the passing internal elevation waves, so as to observe the strong horizontal refraction effects which occur under such conditions. Statistics of and correlations between the internal wave characteristics and their associated effects on acoustic propagation are presented. [Work supported by ONR.]

Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions

High-order asymptotic series are obtained for two- and three-dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known sech 2 solution of the Korteweg–de Vries equation; in three dimensions, the first term is the rational lump solution of the Kadomtsev–Petviashvili equation I. The two-dimensional series is used (with nine terms included) to investigate how small surface tension affects the height and energy of large-amplitude waves and waves close to the solitary version of Stokes’ extreme wave. In particular, for small surface tension, the solitary wave with the maximum energy is obtained. For large surface tension, the two-dimensional series is also used to study the energy of depression solitary waves. Energy considerations suggest that, for large enough surface tension, there are solitary waves that can get close to the fluid bottom. In three dimensions, analytic solutions for the high-order perturbation terms are computed numerically, and the resulting asymptotic series (to three terms) is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension. Finally, the above asymptotic method is applied to the Benney–Luke (BL) equation, and the resulting asymptotic series (to three terms) is verified to agree with the solitary-wave solution of the BL equation.

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 fixed (as opposed to free) upper boundary condition.

Exact Solitary Water Waves with Vorticity

The solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field. The main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity. Each solution has a supercritical parameter value and decays exponentially at infinity. The proof is based on a generalized implicit function theorem of the Nash–Moser type. The first approximation to the surface profile is given by the “KdV” equation. With a supercritical value of the surface tension coefficient, a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity.

Laboratory observations on internal solitary wave evolution on steep and inverse uniform slopes

Laboratory experiments were conducted to investigate the evolution of internal solitary waves of depression or elevation type reflecting from steep slope in a stratified two-layer fluid system. Environmental settings considered in these experiments included the upper and lower layer thickness, the difference in interface levels and inclination of uniform slope, etc. Physical phenomenon and dynamic mechanism of wave fluctuations are discussed in the course of wave breaking and evolution on the slope. Based on the experimental results available, criteria governing wave evolutions are proposed for an internal wave of depression or elevation encountering uniform slope from normal to inverse configuration. A mirror-image model is hypothesized to provide a generic description of the physical consequences leading to wave breaking and mixing on a wide range of uniform slopes. In addition, mixing efficiency resulting from wave breaking was found to approach a maximum value of 25% when the characteristic length ratio was around 0.5, with a reduction in efficiency for slopes on either side of this peak.

The motion of an internal solitary wave of depression over a fixed bottom boundary in a shallow, two-layer fluid

The motion of a large-amplitude internal solitary wave of depression over a fixed bottom boundary in a shallow, two-layer fluid is investigated experimentally. Measurements of the velocity fields close to the bottom boundary are presented to illustrate the generation of an unsteady boundary jet along the bed. The formation of the jet, the structural characteristics of which show striking similarities with those predicted by recent numerical model studies by Diamessis and Redekopp [J. Phys. Oceanogr. (in press)], is attributed to boundary layer separation in the adverse pressure gradient region of the wave-induced flow.

Modelling the motion of an internal solitary wave over a bottom ridge in a stratified fluid

A series of laboratory experiments has been carried out to investigate the passage of an internal solitary wave of depression over a bottom ridge, in a two-layer fluid system for which the upper and lower layer is linearly-stratified and homogeneous respectively. Density, velocity and vorticity fields induced by the wave propagation over the ridge have been measured simultaneously at three locations, namely upstream, downstream and over the ridge crest, for a wide range of model parameters. Results are presented to show that wave breaking may occur for a sufficiently large wave amplitude and a strong ridge blockage factor, with accompanying mixing and overturning. Density field data are presented (i) to illustrate the overturning and mixing processes that accompany the wave breaking and (ii) to quantify the degree of mixing in terms of the wave and ridge parameters. For weak encounters, good agreement is obtained between the laboratory experimental results (velocity and vorticity fields induced by the wave propagation) and the predictions of a recently-developed fully nonlinear theory. Discrepancies between theory and experiment are discussed for cases in which breaking and mixing occur.

On interfacial gravity-capillary solitary waves of the Benjamin type and their stability

Symmetric solitary waves on the interface between two fluid layers are discussed for the case when the upper fluid is bounded above and the lower fluid is infinitely deep. In this flow geometry, under the conditions that the fluid densities are nearly equal and interfacial tension is relatively large, Benjamin [J. Fluid Mech. 245, 401 (1992)] derived a weakly nonlinear long-wave evolution equation which admits a new kind of elevation solitary wave that features decaying oscillatory tails. We present computations of solitary waves of this type and examine their stability to small perturbations on the basis of the full Euler equations using numerical methods. Computed solitary-wave profiles are found to be in good agreement with solutions of the Benjamin equation even for Weber and Froude numbers well outside the formal range of validity of weakly nonlinear long-wave analysis. Furthermore, our computations reveal that, unless the upper fluid is very light relative to the lower fluid, moderately steep interfacial solitary waves behave qualitatively as predicted by the Benjamin equation: elevation waves with a tall center crest, akin to those found by Benjamin, are stable and coexist with depression solitary waves (not reported by Benjamin) which are unstable. As expected, for low enough density ratio, these two solution branches exchange stabilities and computed profiles resemble those of free-surface gravity-capillary solitary waves on deep water.

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.

Stability of steep gravity–capillary solitary waves in deep water

The stability of steep gravity–capillary solitary waves in deep water is numerically investigated using the full nonlinear water-wave equations with surface tension. Out of the two solution branches that bifurcate at the minimum gravity–capillary phase speed, solitary waves of depression are found to be stable both in the small-amplitude limit when they are in the form of wavepackets and at finite steepness when they consist of a single trough, consistent with observations. The elevation-wave solution branch, on the other hand, is unstable close to the bifurcation point but becomes stable at finite steepness as a limit point is passed and the wave profile features two well-separated troughs. Motivated by the experiments of Longuet-Higgins & Zhang (1997), we also consider the forced problem of a localized pressure distribution applied to the free surface of a stream with speed below the minimum gravity–capillary phase speed. We find that the finite-amplitude forced solitary-wave solution branch computed by Vanden-Broeck & Dias (1992) is unstable but the branch corresponding to Rayleigh’s linearized solution is stable, in agreement also with a weakly nonlinear analysis based on a forced nonlinear Schrödinger equation. The significance of viscous effects is assessed using the approach proposed by Longuet-Higgins (1997): while for free elevation waves the instability predicted on the basis of potential-flow theory is relatively weak compared with viscous damping, the opposite turns out to be the case in the forced problem when the forcing is strong. In this régime, which is relevant to the experiments of Longuet-Higgins & Zhang (1997), the effects of instability can easily dominate viscous effects, and the results of the stability analysis are used to propose a theoretical explanation for the persistent unsteadiness of the forced wave profiles observed in the experiments.

Internal tidal bores and bottom nepheloid layers

During a littoral optical environment project field study in shallow water off Oceanside, California, large spikes in optical attenuation (dominated by suspended particles) were observed in a near-bottom mooring. Using moored current meters together with aircraft remote imaging overflights and profiling anchor stations, we established that the spikes were associated with the trailing edges of internal solitary waves of depression of tidal frequency. Each trailing edge had a bore-like face with strong convergent bottom currents followed by a rank-ordered packet of solitons. The near bottom optical attenuation spikes were correlated with these bottom convergent currents at the trailing edge and were not correlated with high bottom stress. Higher optical attenuation was also found near surface at the leading edge of the internal solitary wave of depression. We suggest that in a similar manner to the accumulation and shoreward transport of particles in a surface convergence area (e.g., slick lines), the bottom convergence area at the trailing edge also accumulates and transports particles shoreward. Bottom convergence must be associated with vertical motions which can help in lifting the nepheloid layer off the bottom. This scenario has implications for near bottom optical searches, for transport of non-buoyant zooplankton and for sediment redistribution.

Experiments on mixing due to internal solitary waves breaking on uniform slopes

The shoaling and breaking of an internal solitary wave of depression on a uniform slope were studied experimentally. The waves were generated with as large an amplitude as possible while minimizing mixing at the generation site, thus maximizing the amount of energy propagating onto the slope in the experiment. Various bottom slopes, fluid layer thickness ratios, and density ratios were investigated. The mechanism leading to breaking was examined with flow visualization and particle image velocimetry. Since the layer thickness ratio primarily controls the length (LW) of the solitary wave (for a given amplitude a), it is found that the ratio of LW and the characteristic length of the slope LS determines the amount of energy reflected from the slope. The mixing efficiency of the breaking event, defined as the ratio of the increase of potential energy divided by the amount of wave energy lost at the slope, peaks at a maximum of 25% when LW/LS = 0.5, with a decrease in efficiency for points on either side of this peak value.

Numerical study of surface solitary wave and Kelvin solitary wave propagation in a wide channel

The Petrov-Galerkin finite-element method is used to solve the unified Kadomtsev-Petviashvili equation [Chen and Liu, J. Fluid Mech. 288 (1995) 383]. Numerical experiments have been focused on studying the effect of slowly varying topography on the propagation of surface solitary waves in a stationary channel and Kelvin solitary waves in a rotating channel. We find that in the absence of rotation, an oblique incident solitary wave propagating over a three-dimensional shelf in a straight wide channel will eventually develop into a series of uniform straight-crested solitary waves, together with a train of small oscillatory waves moving upstream. With proper phase shifts, the shapes of these final two-dimensional solitary waves coincide with those of solitary waves emerging from a corresponding normal incident solitary wave propagating over the corresponding two-dimensional shelf. In a two-layered rotating channel, the variation of topography does not have much effect on the propagation of a Kelvin solitary wave of depression, whereas it can have a significant influence on the propagation of a Kelvin solitary wave of elevation. Explanations for these numerical findings are given.