Internal Solitary Waves Of Depression Research Articles

A spectral quadrilateral multidomain penalty method model for high Reynolds number incompressible stratified flows

SUMMARYA two‐dimensional quadrilateral spectral multidomain penalty method (SMPM) model has been developed for the simulation of high Reynolds number incompressible stratified flows. The implementation of higher‐order quadrilateral subdomains renders this model a nontrivial extension of a one‐dimensional subdomain SMPM model built for the simulation of the same type of flows in vertically nonperiodic domains (Diamessis et al., J. Comp. Phys, 202:298‐322, 2005). The nontrivial aspects of this extension consist of the implementation of subdomain corners, the penalty formulation of the pressure Poisson equation (PPE), and, most importantly, the treatment of specific challenges that arise in the iterative solution of the SMPM‐discretized PPE. The two primary challenges within the framework of the iterative solution of the PPE are its regularization to ensure the consistency of the associated linear system of equations and the design of an appropriate two‐level preconditioner. A qualitative and quantitative assessment of the accuracy, efficiency, and stability of the quadrilateral SMPM solver is provided through its application to the standard benchmarks of the Taylor vortex, lid‐driven cavity, and double shear layer. The capacity of the flow solver for the study of environmental stratified flow processes is shown through the simulation of long‐distance propagation of an internal solitary wave of depression in a manner that is free of numerical dispersion and dissipation. The methods and results presented in this paper make it a point of reference for future studies oriented toward the reliable application of the quadrilateral SMPM model to more complex environmental stratified flow process studies. Copyright © 2014 John Wiley & Sons, Ltd.

Internal solitary waves in Upper Lake Constance

The conceptual hypothesis that internal solitary waves (ISWs) in lakes are generated at fronts of basin-scale internal waves, propagate until they break at the terminal boundary and extend over the lake width in cross-propagation direction is tested using temperature and current profiles measured in Upper Lake Constance. Trains of ISWs of depression propagate in close association with the front of a Kelvin wave at almost constant phase speed from the main basin of Upper Lake Constance to the western shore of a sub-basin, Lake Uberlingen. The trough of the ISWs extends over the entire basin width in narrow Lake Uberlingen but not in the wider main basin. A substantial part of the ISWs propagating towards the western terminal boundary did not return, suggesting wave-energy loss due to wave breaking. Close to the shorelines parallel to their propagation direction the amplitudes of the ISWs are diminished. Temperature inversions suggest that wave breaking is occurring at these boundaries, but also in the open water where breaking events extend along the width of the ISW front. The heterogeneity of ISW breaking suggests that the associated mixing and vertical fluxes of plankton and nutrients across the thermocline are highly variable in space and time.

The Behaviors of Internal Solitary Waves near the Continental Shelf of South China Sea Inferred from Satellite Images

The behaviors of internal solitary waves (ISWs) near the continental shelf of SCS are analyzed. We interpret manifestations of four kinds of ISWs in MODIS images: mode-1 depression wave, bolus, mode-1 elevation wave, and mode-2 convex wave. The geographical distributions of these ISWs are investigated. Bolus and mode-1 elevation wave, the two evolution stages of polarity conversion process of depression ISW, only exists around 22˚N, which is in agreement with the sign of mode-1 nonlinear coefficient α1. The mode-2 convex waves generally follow the southern part of mode-1 depression ISW, where the steep terrain favors their generation. The information of a mode-1 depression ISW, such as amplitude and propagation speed, is also derived from the MODIS image.

Intrinsic Breaking of Internal Solitary Waves in a Deep Lake

Based on simulations with the Dubreil-Jacotin-Long (DJL) equation, the limiting amplitude and the breaking mechanisms of internal solitary waves of depression (ISWs) are predicted for different background stratifications. These theoretical predictions are compared to the amplitude and the stability of the leading internal solitary waves of more than 200 trains of ISWs observed in the centre of a sub-basin of Lake Constance. The comparison of the model results with the field observations indicates that the simulated limiting amplitude of the ISWs provides an excellent prediction of the critical wave height above which ISWs break in the field. Shear instabilities and convective instabilities are each responsible for about half of the predicted wave breaking events. The data suggest the presence of core-like structures within the convectively unstable waves, but fully developed and stable cores were not observed. The lack of stable trapped cores in the field can be explained by the results from dynamic simulations of ISWs with trapped cores which demonstrate that even slight disturbances of the background stratification cause trapped cores to become unstable.

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.

Laboratory experiments on waveform inversion of an internal solitary wave over a slope-shelf

Internal solitary waves (ISWs) have been detected in many parts of the world oceans, particularly over slope-shelf topography, on which signature of waveform inversion has been identified. The effects of these waves on engineering operations and ecological process have also been reported in the literature. This article reports the results of a series of numerical modeling and laboratory experiments on waveform evolution of a depression ISW in a nearly stratified two-layer fluid system, in which specific water depth ratios above the horizontal plateau of the trapezoidal obstacle were arranged to facilitate the occurrence of waveform inversion. Classifications of waveform instability (no instability, shear instability and overturning with breaking) on the slope are confirmed in the present laboratory study. Numerical results for waveform variation are also found in fair agreement with the laboratory measurements for cases without waveform inversion and minor internal breaking. Moreover, laboratory results revealed that the depth ratio of the stratified two-layer fluid above the plateau and the magnitude of the incident ISW were the two most important factors for promoting waveform inversion beyond a turning point, in addition to the requirement of a sufficient distance from the shoulder of the trapezoidal obstacle. These factors also influenced the outcome of the shoaling process, energy dissipation, internal wave breaking and turbulent mixing on the front slope, as well as the likelihood of waveform inversion on the horizontal plateau. Contrary to the common perception, it was also observed, at least from the results of the present laboratory experiments, that not all the incident ISWs of depression would produce waveform inversion on the plateau, where the upper layer was physical greater than the bottom layer, unless moderate incident wave was provided. The outcome might also be attributed to the limited distance from the shoulder onto the plateau in the present laboratory setup. However, once waveform inversion occurred on the plateau, it was found, among others, that: (1) the amplitude of the transmitted leading crest and trough might be as low as 30 and 20%, respectively, to the amplitude of the incident wave in depression; (2) the characteristic wavelength of the transmitted leading trough doubled while that of the crest was asymptotically one-half of the incident wavelength, despite the wide range variation in the depth ratios above the plateau; and (3) the transmitted potential wave energy of the leading crest contained 30% of the incident energy. Based on the results of present laboratory experiments, the range for the non-dimensional parameter α, which indicates the effect of nonlinearity and the promotion of waveform inversion on horizontal plateau, will be proposed.

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.

A study of internal solitary waves observed on the continental shelf in the northwestern South China Sea

Based on in-situ time series data from the acoustic Doppler current profiler (ADCP) and thermistor chain in Wenchang area, a sequence of internal solitary wave (ISW) packets was observed in September 2005, propagating northwest on the continental shelf of the northwestern South China Sea (SCS). Corresponding to different stratification of the water column and tidal condition, both elevation and depression ISWs were observed at the same mooring location with amplitude of 35 m and 25 m respectively in different days. Regular arrival of the remarkable ISW packets at approximately the diurnal tidal period and the dominance of diurnal internal waves in the study area, strongly suggest that the main energy source of the waves is the diurnal tide. Notice that the wave packets were all riding on the troughs and shoulders of the internal tides, they were probably generated locally from the shelf break by the evolution of the internal tides due to nonlinear and dispersive effects.

Internal solitary wave-induced flow over a corrugated bed

A combined numerical and experimental study of the propagation of an internal solitary wave (ISW) over a corrugated bed is presented, in which the amplitude and the wavelength of the corrugated bed, together with the wave amplitude and wave speed of the ISW, have been varied parametrically. Both ISWs of elevation and depression have been considered. The wave-induced currents over the corrugated bed cause flow separation at the apex of the corrugations and a sequence of lee vortices forms as a result. These vortices develop fully after the main wave has passed over the topographic feature, resulting in deformation of the overlying pycnocline and, in some instances, significant vertical mixing. It is found that the intensity of the vortex formation is dependent on both the amplitude and wavelength of the bottom topography. In the case of an ISW of depression, the generation of vertically (upward)-propagating vortices is seen to result in entrainment of fluid from a bottom boundary jet (Carr and Davies, Phys Fluids 18:016601, 2006), while, in the elevation case, a second mechanism is present to induce significant turbulent mixing in the water column. It occurs when the bottom corrugations reach into, or are very near, the pycnocline at rest. Large waves of elevation that are stable on approach to the corrugations exhibit evidence of a spatio-temporally developing shear instability as they interact with the bottom corrugation. The shear instability takes the form of billows that have a vertical extent that can reach 50% of the wave amplitude.

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.]

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.

Modelling the propagation of an internal solitary wave across double ridges and a shelf-slope

Evolution of the internal solitary waves (ISWs) in the northern South China Sea (SCS) has recently attracted the attention of many oceanographers in Taiwan and the United States. These ISWs are believed to have been induced by a branch of the Kuroshio current over Luzon Strait, which propagates westward over two ridges in the Luzon Strait between Taiwan and the Philippines, and further onto the continental margin with a shelf-slope in the SCS. This paper presents some preliminary results for the evolution of a depression ISW across two triangular obstacles using numerical modelling and laboratory experiments. The experimental results confirm that the intervals and relative height between the two obstacles are important factors in the interaction of an ISW with the obstacles. However, in the case of the movement of an ISW of depression-type across the Luzon Strait, the effect of the two ridges on the characteristics of the ISW might be less significant than that from the shelf-slope, due to the variations in relative water depth. Results from numerical experiments also show that the amplitude of an ISW can be augmented once the wave commences its contact with a shelf-slope, where an internal hydraulic jump and wave breaking with vortex motion are evident in the laboratory experiments. Eventually, an ISW of depression-type could become an elevation-type at the edge of the continental shelf landwards beyond the turning point, where the upper layer is larger than the bottom layer in a stratified water column.

GENERATION OF INTERNAL SOLITARY WAVE BY GRAVITY COLLAPSE

Laboratory experiments were conducted to investigate the generation of an internal solitary wave (ISW) in a two-layer free surface fluid system in wave flume (12 m × 0.5 m × 0.7 m), which included wave channel and compartment. There is filled with stratified twolayer fluid system in the channel and small compartment, respectively. These two regions in the wave flume are separated by a movable vertical gate at one end of the flume for generating the ISW (internal solitary wave). An ISW generation is thus caused by gravity collapse upon raising the vertical gate. Given positive potential energies, an elevation-type ISW is followed by an anticlockwise overturning motion at the interface; on the other hand, an ISW of depression-type was generated by clockwise motion. This paper presented physical properties related to wave generation in a stratified fluid in laboratory. In the wave flume, the stable wave propagation, either elevation-type or depression-type ISW, is influenced by environmental condition. An ISW of depression transferred into unstable fluctuation instead of soliton feature whilst the upper layer thickness is greater than the lower layer thickness. Similarly, an ISW of elevation reduces its wave amplitude and causes trailing oscillation followed with the leading wave.

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.

Satellite observation of internal solitary waves converting polarity

On a SPOT‐3 visible image over the South China Sea continental shelf, two contrasting signatures of internal solitary waves (ISWs) are observed: darker stripes leading brighter stripes in the lower part, and brighter stripes leading darker stripes in the upper part. The authors suggest that the contrasting patterns are caused by ISWs of different polarities. A packet of ISWs is observed in the process of converting polarity due to bottom shoaling, both in the propagation direction and in the transect direction. The evolution process can be obtained by transferring information from the spatial domain to the temporal domain. The conversion process starts with the original depression ISWs passing the 160 m isobath. The evolution process can be divided into two phases: the first is the broadening of the original depression waves and the second is the appearance of the new‐born elevation waves. These two phases are verified using numerical simulation.

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.

Numerical Experiments on the Breaking of Solitary Internal Wavesover a Slope–Shelf Topography

A theoretical study of the transformation of large amplitude internal solitary waves (ISW) of permanent form over a slope–shelf topography is considered using as basis the Reynolds equations. The vertical fluid stratification, amplitudes of the propagating ISWs, and the bottom parameters were taken close to those observed in the Andaman and Sulu Seas. The problem was solved numerically. It was found that, when an intense ISW of depression propagates from a deep part of a basin onto the shelf with water depth Hs, a breaking event will arise whenever the wave amplitude am is larger than 0.4(Hs − Hm), where Hm is the undisturbed depth of the isopycnal of maximum depression. The cumulative effect of nonlinearity in a propagating ISW leads to a steepening and overturning of a rear wave face over the inclined bottom. Immediately before breaking the horizontal orbital velocity at the site of instability exceeds the phase speed of the ISW. So, the strong breaking is caused by a kinematic instability of the propagating wave. At the latest stages of the evolution the overturned hydraulic jump transforms into a horizontal density intrusion (turbulent pulsating wall jet) propagating onto the shelf. The breaking criterion of the ISW over the slope was found. Over the range of examined parameters (0.52° < γ < 21.8°, where γ is the slope angle) the breaking event arises at the position with depth Hb, when the nondimensional wave amplitude a = am/(Hb − Hm) satisfies the condition a ≅ 0.8°/γ + 0.4. If the water depth Hs on a shelf is less than Hb, a solitary wave breaks down before it penetrates into a shallow water zone; otherwise (at Hs > Hb) it passes as a dispersive wave tail onto the shelf without breaking.

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.