Stratified Tidal Flow Research Articles

The Generation of Nonlinear Internal Waves in the South China Sea: A Three‐Dimensional, Nonhydrostatic Numerical Study

AbstractThe generation of nonlinear internal waves (NLIWs) by stratified tidal flow over an isolated topography has been a subject of research for many years. However, the enhanced complexity of the dynamics as a consequence of flow over two topographical features has been less studied and understood. Here, we report results of the generation of type‐A and type‐B NLIWs in the South China Sea that originate from a double‐ridge system. Based on a realistic three‐dimensional nonhydrostatic simulation, it is found that type‐A NLIWs are generated at the western ridge through the mechanism of internal tide release, while the generation of type‐B NLIWs occurs at the eastern ridge as a consequence of internal wave interaction. Associated with the ridge height, separation distance, and modulated tidal forcing, the generation of type‐B NLIWs is strongly modified by the resonant process between the two ridges. As a result, type‐A and type‐B NLIWs appear in the observations with alternating large and small amplitudes. In addition, with a detailed knowledge of the generation, it is also found that the different arrival time of these waves could be due to their different generation times. Finally, the reflection/transmission of type‐B NLIWs at the subcritical western ridge is suggested as a key feature of resonance. However, it is governed by hydraulically controlled flow instead of the ratio of the local topographic slope to the internal wave beam angle.

Cross-shore stratified tidal flow seaward of a mega-nourishment

The Sand Engine is a 21.5 million m3 experimental mega-nourishment project that was built in 2011 along the Dutch coast. This intervention created a discontinuity in the previous straight sandy coastline, altering the local hydrodynamics in a region that is influenced by the buoyant plume generated by the Rhine River. This work investigates the response of the cross-shore stratified tidal flow to the coastal protrusion created by the Sand Engine emplacement by using a 13 h velocity and density survey. Observations document the development of strong baroclinic-induced cross-shore exchange currents dictated by the intrusion of the river plume fronts as well as the classic tidal straining which are found to extend further into the nearshore (from 12 to 6m depth), otherwise believed to be a mixed zone. Estimates of the centrifugal acceleration directly after construction of the Sand Engine showed that the curvature effects were approximately 2 times stronger, suggesting that the Sand Engine might have played a role in controlling the cross-shore exchange currents during the first three years after the completion of the nourishment. Presently, the curvature effects are minute.

Generation of baroclinic tides over an isolated underwater bank

The interaction of stratified tidal flow with an isolated bank (Jones Bank in the Celtic Sea) was investigated numerically and using observational data collected during the 25th research cruise of the R/V James Cook in July 2008. Two scenarios of observed wave generation, one with pure tidal forcing and another with an extra background current, were reproduced numerically and compared against the observational data. It was found that the tidal currents alone produced subcritical conditions for the first‐mode internal wave generation and supercritical flow for the second‐mode waves. Under these conditions, the first‐mode waves with amplitudes up to 10 m freely radiated from the bank, gradually transforming into attenuating dissipative baroclinic bores. Due to supercritical conditions, the second‐mode waves were arrested at the lee side of the bank for 3 h where they grew in amplitude (up to 35 m) and after their release propagated as a series of second‐mode internal solitary waves. The additional background current that was measured near the bank in the middle of June 2008 radically changed the conditions of wave generation. A strong southeastern current arrested the first‐mode perturbations at the NE flank of the bank for 2 h, where they grew in amplitude to almost 40 m and disintegrated into packets of first‐mode internal solitary waves after their release. For the second‐mode perturbations, the flow was substantially supercritical for more than one fourth of a tidal period so that these waves were just washed away from the generation site without any visible amplification.

Numerical modeling of three-dimensional stratified tidal flow over Camarinal Sill, Strait of Gibraltar

[1] The baroclinic response to barotropic tidal forcing in the Camarinal Sill area, within the Strait of Gibraltar, is investigated with a three-dimensional, fully nonlinear, nonhydrostatic numerical model. The aim of numerical efforts was the assessment of three-dimensional effects, which are potentially significant in the area because of rather irregular bottom topography, variable background stratification, and complex structure of barotropic tides. Model results reveal a complex baroclinic response under relatively moderate flood tidal currents, which includes the formation of internal hydraulic jumps upstream of the sill, internal cross waves close to the channel walls, and a plunging pycnocline at the lee side of the sill crest. These structures exhibit significant cross-channel spatial dependence and may appear to be aligned together across the channel. This fact makes their identification difficult from the surface pattern captured by remote sensing images. Under strong barotropic forcing (spring tides) the upstream hydraulic jumps are shifted to the lee side of Camarinal Sill, where a single internal hydraulic jump is formed. Significant first- and second-mode hydraulic jumps are also generated near smaller secondary sills in Tangier basin, thus extending the occurrence of intense water mixing and energy dissipation to other zones of the strait.

Quantifying the Contributions of Tidal Straining and Gravitational Circulation to Residual Circulation in Periodically Stratified Tidal Estuaries

Abstract This numerical modeling study quantifies for the first time the contribution of various processes to estuarine circulation in periodically stratified tidal flow under the impact of a constant horizontal buoyancy gradient. The one-dimensional water column equations with periodic forcing are first cast into nondimensional form, resulting in a multidimensional parameter space spanned by the modified inverse Strouhal number and the modified horizontal Richardson number, as well as relative wind speed and wind direction and the residual runoff. The along-tide momentum equation is then solved for the tidal-mean velocity profile in such a way that it is equated to the sum of the contributions of tidal straining (resulting from the temporal correlation between eddy viscosity and vertical shear), gravitational circulation (resulting from the depth-varying forcing by a constant horizontal buoyancy gradient), wind straining, and depth-mean residual flow (resulting from net freshwater runoff). This definition of tidal straining does not only account for tidal asymmetries resulting from horizontal buoyancy gradients but also from wind straining and residual runoff. For constant eddy viscosity, the well-known estuarine circulation analytical solution with polynomial residual profiles is directly obtained. For vertically parabolic and constant-in-time eddy viscosity, a new analytic solution with logarithmic residual profiles is found, showing that the intensity of the gravitational circulation scales with the horizontal Richardson number. For scenarios with realistic spatially and temporally varying eddy viscosity, a numerical water column model equipped with a state-of-the-art two-equation turbulence closure model is applied to quantify the individual contributions of the various processes to estuarine circulation. The fundamental outcome of this study is that, for irrotational flow with periodic stratification and without wind forcing and residual runoff, the tidal straining is responsible for about two-thirds and gravitational circulation is responsible for about one-third of the estuarine circulation, proportionally dependent on the horizontal Richardson number, and weakly dependent on the Strouhal number. This new and robust result confirms earlier estimates by H. Burchard and H. Baumert, who suggested that tidal straining is the major generation mechanism for estuarine turbidity maxima. However, a sensitivity analysis of the model results to details of the turbulence closure model shows some uncertainty with respect to the parameterization of sheared convection during flood. Increasing down-estuary wind straining and residual runoff reduce the quantitative contribution of tidal straining. For relatively small horizontal Richardson numbers, the tidal straining contribution to estuarine circulation may even be reversed by down-estuary wind straining.

A dedicated tidal stream atlas of the stratified tidal flows near Stonecutters Bridge, Hong Kong, based on 3D numerical simulations with HLES

Construction works at the Stonecutters Bridge crossing the Rambler Channel in Hong Kong involve bridge deck lift operations with dynamically positioned barges. Nautical simulations for marine safety assessment and the programming of the bridge deck lift operations require reliable predictions of the local currents. To that purpose, we designed a detailed hydrodynamic Delft3D-FLOW model for simulating 3D stratified tidal flow and larger-scale eddy-dynamics using Horizontal Large Eddy Simulation (HLES). This model was nested in the 3D hydrodynamic model of the Pearl River estuary and the Hong Kong waters, which has been extensively calibrated and validated in other studies in the Hong Kong area. The results of the detailed model show good agreement with dedicated field measurements for both the dry and wet seasons. Based on tidal analysis of the model results, predictions for the water levels and currents could be made that include the complex flow patterns and (tidally driven) eddies and recirculations. For easy prediction of the flow conditions in the future, a digital tidal stream atlas (Delft-STREAM) was set up that is capable of predicting these flow fields. This tool provided a valuable contribution to the programming and execution of the marine works of Stonecutters Bridge. The new development in this study is the inclusion of complex 3D flow patterns into a digital tidal stream atlas, capable of predicting detailed flow fields for construction planning purposes.

Tidal flow over three-dimensional topography in a stratified fluid

Our laboratory experiments and numerical simulations of stratified tidal flow past model topography (a half sphere on a horizontal plane) reveal several three-dimensional flow features, including an unexpected flow perpendicular to the forcing plane (the vertical plane through the center of the sphere, in the direction of the oscillating tide). This perpendicular flow has a time-independent component and a component oscillating at twice the tidal frequency. Our results show that the time-independent part of the perpendicular flow forms a large-scale horizontal circulation, which could enhance material transport and mixing near bottom topography in the oceans. In addition, for small forcing amplitude we find that the azimuthal dependence of the internal wave field is described by the functional form cos ϕ, as predicted by linear inviscid theory. At higher forcing amplitude, the internal wave energy is more concentrated in the forcing direction. Finally, we observe a wave intensity asymmetry in the polar direction and explain the asymmetry using a geometrical argument.

Numerical modelling of stratified tidal flow over a fjord sill

Field observations of tidally driven stratified flow in the sill area of Knight Inlet (British Columbia) revealed a very complicated structure, which includes solitary waves, upstream bifurcation, hydraulic jump and mixing processes. Recent observations suggest that the flow instabilities on the plunging pycnocline at the lee side of the sill may contribute to solitary wave generation through a subharmonic interaction. The present study reports on a series of numerical experiments of stratified tidal flow in Knight Inlet performed with the help of a fine resolution fully non-linear non-hydrostatic numerical model. The model reproduces all important stages of the baroclinic tidal dynamics observed in Knight Inlet. Results demonstrate that solitary waves are generated apart from the area of hydrodynamic instability. Accelerating tidal flux forms a baroclinic hydraulic jump just above the top of the sill, whereas the bifurcations and zones of shear instabilities are formed downstream of the sill. The first baroclinic mode having the largest velocity escapes from the generation area and propagates upstream, disintegrating further into a packet of solitary waves reviling the classical “non-subharmonic” mechanism of generation. The remaining part of the disturbance (slow baroclinic modes) is arrested by tidal flow and carried away to the lee side of the obstacle, where shear instability, billows and mixing processes are developed. Some sensitivity runs were performed for different value of tidal velocity.

Analysis of Supercritical Stratified Tidal Flow in a Scottish Fjord

Abstract The baroclinic tidal regime of the fjord Loch Etive (Scotland) is studied. Analysis is performed on the basis of both in situ data and numerical simulations, with the use of a fully nonlinear nonhydrostatic fine-resolution model. It was found that the crest of the sill that separates Loch Etive into inner and outer parts is subjected to a supercritical flow regime with maximum Froude numbers in excess of 5. Strong supercritical conditions lead to the formation of flow separation just above the sill’s crest. As is inherent to jet-type fjordic systems, this, in turn, leads to a weak nonlinear baroclinic wave response. On the other hand, observations and numerical results also revealed the presence of propagating internal tidal waves with amplitudes up to 10 m, several kilometers from the constriction. It is shown that these baroclinic tidal waves are excited during the ebb phase over a section of the inner flank of the sill, at a depth below 30 m, where the local Froude number is substantially less than unity. Thus Loch Etive simultaneously exhibits both weak nonlinear response due to strong supercritical conditions with flow separation over the sill and a significant linear baroclinic tidal response due to the deeper flank of the sill. In this respect, exhibiting both jet- and wave-type behavior, Loch Etive can be referred to as a “hybrid type” fjord: the authors suggest that many jet-type fjords will also generate a significant baroclinic response and should correctly fit into this new category.

Upstream Internal Hydraulic Jumps

Abstract In stratified tidal flow over a sill, the character of the upstream response is determined by a Froude number Fs based on the stratification near the surface. This is distinguished from the Froude number governing the response in the neighborhood of the sill crest, which is based on the weak density step associated with a flow bifurcation. For moderate values of Fs, the upstream response consists of nonlinear waves or a weak undular bore. For larger values of Fs, a strong, quasi-stationary, internal hydraulic jump dominates the upstream response. At sufficiently large values of Fs, the upstream bore is swept downstream and lost. Acoustic backscatter and velocity data are presented for the case of a strong internal bore or gravity current in a tidally modulated sill flow. Numerical simulations with varying near-surface stratification are presented to illustrate the upstream responses at different values of Fs. The theory of two-layer hydraulic flows is invoked to account for the development of the upstream jump.

Stratified Tidal Flow over a Bump

Abstract The interaction of a stratified flow with an isolated topographic feature can introduce numerous disturbances into the flow, including turbulent wakes, internal waves, and eddies. Measurements made near a “bump” east of Race Rocks, Vancouver Island, reveal a wide range of phenomena associated with the variable flow speeds and directions introduced by the local tides. Upstream and downstream flows were observed by placing two acoustic Doppler current profilers (ADCPs) on one flank of the bump. Simultaneous shipboard ADCP surveys corroborated some of the more striking features. Froude number conditions varied from subcritical to supercritical as the tidal velocities varied from 0.2 to 1.5 m s−1. During the strong ebb, when the moored ADCPs were located on the lee side, a persistent full-water-depth lee wave was detected in one of the moored ADCPs and the shipboard ADCP. However, the placement of the moorings would suggest that, by the time it appears in the moored ADCP beams, the lee wave has been swept downstream or has separated from the bump. Raw ADCP beam velocities suggest enhanced turbulence during various phases of the tide. Many of the three-dimensional flow characteristics are in good agreement with laboratory studies, and some characteristics, such as shear in the bottom boundary layer, are not.

On boundary–layer separation and internal wave generation at the Knight Inlet sill

This paper reports results of numerical simulations of stratified tidal flow over the Knight Inlet sill. A non–hydrostatic, two–dimensional model is used, which incorporates a no–slip bottom boundary condition through the use of a vertical eddy viscosity/diffusivity parametrization that is non–zero only near the bottom. In inviscid model runs, a large lee wave is rapidly formed, which quickly breaks, leading to the formation of a high–drag state and a strong downslope jet in the early stages of the ebb tide. The use of a no–slip bottom boundary condition results in boundary–layer separation from near the top of the sill. This significantly reduces the amplitude of the lee wave during the initial stages of the flow development. For most model runs, a large lee wave is ultimately formed and the separation point moves down the lee of the sill to a position immediately downstream of the lee wave. The transition to this high–drag state is significantly delayed compared with inviscid model runs. Weakened stratification immediately above the sill, inclusion of an eddy viscosity/diffusivity above the bottom and a pool of dense water on the downstream (seaward) side of the sill can all contribute to a delay in the transition to a high–drag state, and can eliminate it entirely. For one model run using a vertical eddy viscosity parametrization above the bottom, a reduction of the vertical diffusivity eliminated the formation of a high–drag state. This suggests that at least in some cases entrainment into the lee wave can cause its growth and result in the formation of a high–drag state.

Application of different turbulence closure model for stratified tidal flows and salinity in an estuarine system

Transient stratification in estuaries reflects competition between the stratifying influences of the vertical gravitational circulation and longitudinal density gradient by vertical shear, set against the mixing influence of, principally, tidally generated turbulence. A vertical (laterally averaged) two-dimensional model of an estuary, using seven different parameterizations of vertical mass and momentum mixing coefficients from the literature, is used to make general predictions about the nature of the time-dependent stratification, velocity field and salinity in an estuary. The downstream boundary, at the river mouth, is an M 2 tide with amplitude being half of the mean tidal range to force the model runs for numerical experiments. The results show the Mellor and Yamada scheme products larger stratification, density gradients and have less vertical mixing. Model calibration and verification is performed to use parameterization of mixing coefficients against observational data of salinity. The root-mean-square (RMS) errors and mean absolute errors are used as qualitative and quantitative criteria. The results show Park and Kuo scheme performs best. Mellor and Yamada scheme over-predict the amplitude of the stratification signal. The other schemes, such as Thompson, Lehfeldt and Bloss, Pacanowski and Philander, and Munk and Anderson, over-predict the maximum salinity.

Comparing the performance of the Mellor‐Yamada and the κ‐ε two‐equation turbulence models

The aim of this paper is to systematically compare κ‐ε and Mellor‐Yamada two‐equation turbulence models. Both models include prognostic equations for turbulent kinetic energy and a length scale related parameter which are used to calculate eddy viscosities and vertical diffusivities. The results from laboratory experiments, using mixed and stratified flows, are simulated in order to systematically compare and calibrate the models. It is shown that the Monin‐Obukhov similarity theory is well represented in both models. The models are used to simulate stratified tidal flow in the Irish Sea, and the results show that the κ‐ε models generally predict a larger phase lag between currents and turbulent dissipation, in the bottom boundary layer, than the Mellor‐Yamada models. The comparison between the model results and field measurements, of the rate of dissipation of turbulent kinetic energy, shows that both models require modification through the inclusion of an internal wave parameterization in order that they are able to correctly predict the observed levels of turbulent dissipation. As the main result, it is shown that the choice of the stability functions, which are used as proportionality factors for calculating the eddy viscosity and diffusivity, has a stronger influence on the performance of the turbulence model than does the choice of length scale related equation.

Modeling Density and Turbulence in Stratified Tidal Medium

A finite-element model in a two-dimensional vertical plane has been developed to predict the flow, density, and turbulence fields of a stratified tidal medium. An algebraic stress/flux model has been used as turbulence closure. The flow, density, and turbulence fields are coupled explicitly in a three-level, semiimplicit time discretization. Model predictions have been compared with an experimental study performed in the tidal flume of Delft Hydraulics. Both the numerical model and the experimental observations have been used to study some of the basic characteristics of turbulence and turbulent transport mechanisms in a stratified tidal flow.

Bottom shear stresses and velocity profiles in stratified tidal planetary boundary layer flow from similarity theory

The paper presents the bottom shear stresses and velocity profiles in stratified tidal planetary boundary layer flow by using similarity theory. For a given seabed roughness length, free stream current velocity components, frequency of tidal oscillation, Coriolis parameter and stratification parameter, the maximum bottom shear stress is determined for flow conditions in the rough, smooth and transitional smooth-to-rough turbulent regime. Further, the direction of the bottom shear stresses and the velocity profiles are given. Comparison is made with data from field measurements of time-independent as well as tidal planetary boundary layer flow for neutral conditions, and the agreement between the predictions and the data is generally good. Further, an example of application for stable stratification is given, and qualitatively the predictions show, as expected, that the bottom shear stress and the thickness of the boundary layer become smaller for stable than for neutral stratification. Other features of the tidal planetary boundary layer flow are also discussed.

A simple model of mixing in stratified tidal flows

We use a simple model of vertical mixing in a stratified tidal flow to study flows that might be observed in shallow estuaries. In the model, flow is driven by a specified surface pressure gradient due to barotropic tidal motions. The model incorporates two means for accomplishing vertical mixing: (1) shear instability and (2) bottom mixed‐layer deepening by turbulence produced at the estuary bottom and in the shear layer at the top of the mixed layer. We discuss model results that show how the combined action of shear instability and tidal “stirring” accomplishes vertical mixing. The interaction between the instabilities that develop because of the substantial tidal timescale velocity shear caused by stratification is the key feature of mixing. Shear instability weakens the stability of the mixed layer, allowing stirring to efficiently entrain lighter fluid into the mixed layer. The importance of shear is found to depend on the stability of the water column and on the relative importance of frictional forces.

Differential model of the hydrophysical fields' vertical structure in a stratified tidal flow over a shelf

This paper studies the influence of shelf tidal flows on the structure of surface and near-bottom turbulent boundary layers.

Internal Wave Activity in Rotterdam Waterway

A variety of internal wave mechanisms is observed in the partially mixed Rotterdam Waterway estuary due to the interaction between stratified tidal flow and topography. The topographically generated waves include lee waves, resonant trapped waves and critical layers, freely propagating waves, breaking internal hydraulic jumps at the banks, and rotor formation. Acoustic images and profiles of density and velocity are presented. The acoustic images are explained on the basis of basic internal wave theory. Mathematical models tend to ignore smallscale internal wave activity as a source of turbulent kinetic energy. The paper investigates whether it is justified to do so on the scale of the topography. On the basis of an estimate of the internal wave‐induced production of turbulent kinetic energy it is concluded that for the observed internal waves this source of energy may not always be neglected when compared with the production by the background flow, in particular when first‐mode short internal waves are i...

Generation of internal waves in Massachusetts Bay

Observations of stratified tidal flow over a submarine ridge indicate that the large amplitude and asymmetry of the topography are critical in determining the type of flow response. The observations consist of hydrographic, current meter, and echo‐sounding data and are used as a basis for inferring the generation mechanism of previously observed trains of internal waves. The flow response on ebb and flood tide is categorized according to the Froude number classification of sill flows proposed by Farmer and Smith (1980a). The calculated Froude numbers, response length scale, and duration differ markedly between the two phases of the tide due to the asymmetry of the bank. The development of two features in the acoustic imagery related to wave formation is followed.