Acquisition of the tide-induced Lagrangian residual current field by the PIV technique in the laboratory

In this paper, a new particle image velocimetry (PIV)-based measurement method is proposed to obtain the high-resolution tide-induced Lagrangian residual current field in the laboratory. A long gravity wave was generated to simulate the tide in a narrow tank full of water laden with PIV particles. Consecutive charge-coupled device (CCD) images were recorded with the studied layer illuminated with a laser beam. Two images separated by one tidal period were processed by applying the pattern-matching algorithm to get the horizontal tide-induced Lagrangian residual current field. The results coincide with sporadic results from the traditional surface-float tracing method, but with much higher spatial resolution and accuracy. Furthermore, it is found that the direct acquisition of the Lagrangian residual current may reduce the error at least by one order compared with those acquisition methods that require the detailed information of the tidal cycle.     

Laboratory experiment on the 3D tide-induced Lagrangian residual current using the PIV technique

The 3D structure of the tide-induced Lagrangian residual current was studied using the particle image velocimetry (PIV) technique in a long shallow narrow tank in the laboratory. At the mouth of the tank, a wave generator was used to make periodic wave which represents the tide movement, and at the head of the tank, a laterally sloping topography with the length of one fifth of the water tank was installed, above which the tide-induced Lagrangian residual current was studied. Under the weakly nonlinear condition in the present experiment setup, the results show that the Lagrangian residual velocity (LRV) field has a three-layer structure. The residual current flows inwards (towards the head) in the bottom layer and flows outwards in the middle layer, while in the surface layer, it flows inwards along the shallow side of the sloping topography and outwards along the deep side. The depth-averaged and breadth-averaged LRV are also analyzed based on the 3D LRV observations. Our results are in good agreement with the previous experiment studies, the analytical solutions with similar conditions and the observational results in real bays. Moreover, the volume flux comparison between the Lagrangian and Eulerian residual currents shows that the Eulerian residual velocity violates the mass conservation law while the LRV truly represents the inter-tidal water transport. This work enriches the laboratory studies of the LRV and offers valuable references for the LRV studies in real bays.     

Simulation of the Lagrangian tide-induced residual velocity in a tide-dominated coastal system: a case study of Jiaozhou Bay,China

The Eulerian residual transport velocity and the first-order Lagrangian residual velocity for weakly nonlinear systems have been used extensively in the past to depict inter-tidal mass transport. However, these could not explain the observed net surface sediment transport pattern in Jiaozhou Bay (JZB), located on the western Yellow Sea. JZB is characterized by strong tidal motion, complex topography and an irregular coastline, which are features of typical nonlinear systems. The Lagrangian residual velocity, which is applicable to general nonlinear systems, was simulated with the water parcel tracking method. The results indicate that the composition of the Lagrangian residual velocity at different tidal phases coincides well with the observed net surface sediment transport pattern. The strong dependence of water flushing time on the initial tidal phase can also be explained by the significant intra-tidal variation of the Lagrangian residual velocity. To investigate the hydrodynamic mechanism governing the nonlinearity of the M ₂ tidal system, a set of nonlinearity indexes were defined and analysed. In the surface layer, horizontal advection is the main contributor to the strong nonlinearity near the bay mouth, while in the bottom layer, the strong nonlinearity near the bay mouth may result from the vertical viscosity and horizontal advection.

     

Simulation of the Lagrangian tide-induced residual velocity in a tide-dominated coastal system: a case study of Jiaozhou Bay, China

The Eulerian residual transport velocity and the first-order Lagrangian residual velocity for weakly nonlinear systems have been used extensively in the past to depict inter-tidal mass transport. However, these could not explain the observed net surface sediment transport pattern in Jiaozhou Bay (JZB), located on the western Yellow Sea. JZB is characterized by strong tidal motion, complex topography and an irregular coastline, which are features of typical nonlinear systems. The Lagrangian residual velocity, which is applicable to general nonlinear systems, was simulated with the water parcel tracking method. The results indicate that the composition of the Lagrangian residual velocity at different tidal phases coincides well with the observed net surface sediment transport pattern. The strong dependence of water flushing time on the initial tidal phase can also be explained by the significant intra-tidal variation of the Lagrangian residual velocity. To investigate the hydrodynamic mechanism governing the nonlinearity of the M ₂ tidal system, a set of nonlinearity indexes were defined and analysed. In the surface layer, horizontal advection is the main contributor to the strong nonlinearity near the bay mouth, while in the bottom layer, the strong nonlinearity near the bay mouth may result from the vertical viscosity and horizontal advection.     

Numerical computation of the tidally induced Lagrangian residual current in a model bay

With a depth-averaged numerical model, the tidally induced Lagrangian residual current in a model bay was studied. To correctly reflect the long-term mass transport, it is appropriate to use the Lagrangian residual velocity (LRV) rather than the Eulerian residual velocity (ERV) or the Eulerian residual transport velocity (ETV) to describe the residual current. The parameter κ, which is defined as the ratio of the typical tidal amplitude at the open boundary to the mean water depth, is considered to be the indicator of the nonlinear effect in the system. It is found that the feasibility of making the mass transport velocity (MTV) approximate the LRV is strongly dependent on κ. The error between the MTV and the LRV tends to increase with a growing κ. An additional error will come from the various initial tidal phases due to the Lagrangian drift velocity (LDV) when κ is no longer small. According to the residual vorticity equation based on the MTV, the Coriolis effect is found to influence the residual vorticity mainly through the curl of the tidal stress. A significant difference in the flow pattern indicates that the LRV is sensitive to the bottom friction in different forms.     

Residual vorticity induced by the action of tidal currents in combination with bottom topography in the Southern Bight of the North Sea

Abstract

Results are presented of calculations on the generation of residual vorticity by tidal currents over the bottom topography of the Southern Bight of the North Sea. A typical order of magnitude is 10^?6 to 10^?7 s ^?1. This is compared with current measurements on calm days, when similar magnitudes are found. At windspeeds less than about 5 m/s tidal generation of residual vorticity is important; at higher windspeeds wind effects begin to dominate. Our results are relevant in understanding the spatial variability of residual currents, because a non-zero vorticity implies the existence of horizontal gradients in the residual current field.     

A numerical study of horizontal dispersion in a macro tidal basin

Tidal circulation in Cobscook Bay, a macro tidal basin, is simulated using the three-dimensional, nonlinear, finite element ocean model, QUODDY_dry. Numerical particles are released from various transects in the bay at different tidal phases and tracked for several tidal cycles. Initially, nearby particles in the main tidal channel experience a great deal of spreading and straining, and after a few tidal cycles, they are separated in different parts of the bay. The fundamental mechanism for particle dispersion is the chaotic advection that arises from long tidal excursions passing through many residual eddies. A loosely correlated, inverse relationship between the two dimensionless parameters, ν (the ratio of the residual current to the tidal current) and λ (the ratio of the tidal excursion to the main topographic scale), can be constructed for large values of ν. Several Lagrangian statistical measures are used to quantify and distinguish dispersion regimes in different parts of Cobscook Bay. It is found that the effective Lagrangian dispersion coefficient can be estimated using the product of the magnitude of residual currents and the tidal excursion.     

Difference between the Lagrangian trajectories and Eulerian residual velocity fields in the southwestern Yellow Sea

The responses to tidal and/or wind forces of Lagrangian trajectories and Eulerian residual velocity in the southwestern Yellow Sea are investigated using a high-resolution circulation model. The simulated tidal harmonic constants agree well with observations and existing studies. The numerical experiment reproduces the long-range southeastward Eulerian residual current over the sloping bottom around the Yangtze Bank also shown in previous studies. However, the modeled drifters deployed at the northeastern flank of the Yangtze Bank in the simulation move northeastward, crossing over this strong southeastward Eulerian residual current rather than following it. Additional sensitivity experiments reveal that the influence of the Eulerian tidal residual currents on Lagrangian trajectories is relatively weaker than that of the wind driven currents. This result is consistent with the northeastward movement of ARGOS surface drifters actually released in the southwestern Yellow Sea. Further experiments suggest that the quadratic nature of the bottom friction is the crucial factor, in the southwestern Yellow Sea, for the weaker influence of the Eulerian tidal residual currents on the Lagrangian trajectories. This study demonstrates that the Lagrangian trajectories do not follow the Eulerian residual velocity fields in the shallow coastal regions of the southwestern Yellow Sea.     

Net sediment transport in tidal basins: quantifying the tidal barotropic mechanisms in a unified framework

Net sediment transport in tidal basins is a subtle imbalance between large fluxes produced by the flood/ebb alternation. The imbalance arises from several mechanisms of suspended transport. Lag effects and tidal asymmetries are regarded as dominant, but defined in different frames of reference (Lagrangian and Eulerian, respectively). A quantitative ranking of their effectiveness is therefore missing. Furthermore, although wind waves are recognized as crucial for tidal flats’ morphodynamics, a systematic analysis of the interaction with tidal mechanisms has not been carried out so far. We review the tide-induced barotropic mechanisms and discuss the shortcomings of their current classification for numerical process-based models. Hence, we conceive a unified Eulerian framework accounting for wave-induced resuspension. A new methodology is proposed to decompose the sediment fluxes accordingly, which is applicable without needing (semi-) analytical approximations. The approach is tested with a one-dimensional model of the Vlie basin, Wadden Sea (The Netherlands). Results show that lag-driven transport is dominant for the finer fractions (silt and mud). In absence of waves, net sediment fluxes are landward and spatial (advective) lag effects are dominant. In presence of waves, sediment can be exported from the tidal flats and temporal (local) lag effects are dominant. Conversely, sand transport is dominated by the asymmetry of peak ebb/flood velocities. We show that the direction of lag-driven transport can be estimated by the gradient of hydrodynamic energy. In agreement with previous studies, our results support the conceptualization of tidal flats’ equilibrium as a simplified balance between tidal mechanisms and wave resuspension.     

On the exact shape of the horizontal profile of a topographically rectified tidal flow

Abstract

Starting from the nonlinear shallow water equations of a homogeneous rotating fluid we derive the equation describing the evolution of vorticity by a fluctuating bottom topography of small amplitude, using a multiple scale expansion in a small parameter, which is the topographic length scale relative to the tidal wave length. The exact response functions of residual vorticity for a sinusoidal bottom topography are compared with those obtained by a primitive perturbation series and by harmonic truncation, showing the former to be invalid for small topographic length scales and the latter to be only a fair approximation for vorticity produced by planetary vortex stretching. In deriving the exact shape of the horizontal residual velocity profile at a step-like break in the bottom topography, it is shown that the Lagrangian profile only exists in a strip having the width of the amplitude of the tidal excursion at both sides of the break, and that it vanishes outside that interval. Moreover, in the limit of small amplitude topography at least, it vanishes altogether for the generation mechanism by means of planetary vortex stretching. The Eulerian profile is shown to extend over twice the interval of the Lagrangian profile both for production by vortex stretching and by differential bottom friction. These finite intervals over which the residual velocity profiles exist for a step-like topography are not reproduced by harmonic truncation of the basic equation. This method gives exponentially decaying profiles, indicating spurious horizontal diffusion of vorticity. In terms of orders of magnitude, the method of harmonic truncation is reliable for residual velocity produced by vortex stretching but it overestimates the residual velocity produced by differential bottom friction by a factor 2.     

3D analytical solution to the tidally induced Lagrangian residual current equations in a narrow bay

The 3D first-order Lagrangian residual velocity (LRV) equation is established, and its analytical solution is obtained in a narrow bay. The results show clearly the 3D structure of the first-order LRV. When the exponential bottom profile is assumed, the upper half layer of the water flows in through the deep channel from the open boundary directly to the head of the bay. Then the water will return to the area surrounding the lower half of the inflow area. The downwelling area is located mainly at the deep channel, while the upwelling area occupies both sides of the bay. The inter-tidal water transport, obtained by integrating the 3D first-order LRV through the water column, has a pattern similar to the previous study in which the 2D depth-averaged Lagrangian residual current equations were solved. The inter-tidal water transport is used to analyze the water exchange, and it is found that the water exchange at different cross sections increases smoothly with the distance between the cross sections and the head of the bay until about one wavelength. It is also found that the pattern of the breadth-averaged Lagrangian residual current varies with the length of the bay if a non-flat bottom profile is used. The depth-integrated LRV and the breadth-averaged LRV are mainly determined by the different terms of the tidal body force, with the former determined by the bottom friction related term and the latter by the eddy viscosity related term. When the bay is longer than one wavelength, different results in the outer bay can be observed.     

A depth-dependent study of the topographic rectification of tidal currents

Abstract

A depth-dependent model for the topographic rectification of tidal currents in a homogeneous rotating fluid is used to examine the dependence of the rectified mean flow on various tidal, topographic and frictional parameters. Friction is parameterized through a vertically-uniform, time-independent vertical eddy viscosity and a bottom stress law applied near the top of the constant stress layer. The model neglects the interaction of mean and tidal currents, assumes uniformity along isobaths, and is closed with the assumption of zero depth-averaged mean flow across isobaths.

In the limit of depth-independence, the model reduces to that considered by Huthnance (1973) and Loder (1980) which, for weak friction, favours anticyclonic mean circulation around shallow regions and Lagrangian flow which is significantly reduced from the Eulerian. With the inclusion of vertical structure, the magnitude of the anticyclonic flow is amplified suggesting that depth-independent models may underestimate the along-isobath flow. For strong friction the direction of the mean flow depends on the orientation of the tidal ellipse relative to the isobaths. The depthindependent model again underestimates the magnitude of the along-isobath flow, but this can be offset with an appropriate reduction of the bottom friction coefficient.

The cross-isobath mean flows are one to two orders of magnitude weaker than the along-isobath flows and generally have more vertical structure. There is also a significant Stokes drift in the cross-isobath direction. Although there is some tendency for the cross-isobath mean bottom current to be down the cross-isobath mean pressure gradient, it appears that it is not generally possible to infer this current from depth-independent models.     

Realistic modelling of shelf-estuary regions

A very high-resolution modelling configuration was created for the estuary of Baía de Todos os Santos – BTS, Brazil (300 to 400 m), and adjacent coastal waters (600 to 1200 m). The adoption of a multi-corner domain approach allowed the variable spatial resolution required to resolve the shelf, the bay and their interactions. Seven years were simulated using realistic oceanic, atmospheric and riverine forcing. Model validation was done against observations showing the model skill to reproduce the thermohaline field, the tidal currents, as well as the variability of the free surface at tidal and sub-tidal time scales. The results provide the first representation of the tidal wave propagation along the bay, in terms of maps of amplitudes, phases and ellipses of the barotropic currents for the main tidal constituents. By analysing the residual currents at different depths, in terms of averages over the simulation period, several prominent structures were identified and named: (i) Salvador eddy (up to 0.2 m s⁻¹); (ii) St Antonio current (up to 0.45 m s⁻¹); (iii) Salvador current (up to 0.5 m s⁻¹); (iv) Itaparica eddy (up to 0.2 m s⁻¹); (v) Ilha dos Frades southern eddy (up to 0.1 m s⁻¹); and (vi) Ilha dos Frades northern eddy (up to 0.2 m s⁻¹). The model set-up proved to be highly efficient and robust simulating the BTS shelf-estuary region and such an approach may be suitable to other estuarine systems.

     

Tidally induced upwelling off Yangtze River estuary and in Zhejiang coastal waters in summer

MASNUM wave-tide-circulation coupled numerical model (MASNUM coupled model, hereinafter) is de-veloped based on the Princeton Ocean Model (POM). Both POM and MASNUM coupled model are ap-plied in the numerical simulation of the upwelling off Yangtze River estuary and in Zhejiang coastal waters in summer. The upwelling mechanisms are analyzed from the viewpoint of tide, and a new mechanism is proposed. The study suggests that the tidally inducing mechanism of the upwelling in-cludes two dynamic aspects: the barotropic and the baroclinic process. On the one hand, the residual currents induced by barotropic tides converge near the seabed, and upwelling is generated to maintain mass conservation. The climbing of the residual currents along the sea bottom slope also contributes to the upwelling. On the other hand, tidal mixing plays a very important role in inducing the upwelling in the baroclinic sea circumstances. Strong tidal mixing leads to conspicuous front in the coastal waters. The considerable horizontal density gradient across the front elicits a secondary circulation clinging to the tidal front, and the upwelling branch appears near the frontal zone. Numerical experiments are de-signed to determine the importance of tide in inducing the upwelling. The results indicate that tide is a key and dominant inducement of the upwelling. Experiments also show that coupling calculation of the four main tidal constituents (M2, S2, K1, and O1), rather than dealing with the single M2 constituent, im-proves the modeling precision of the barotropic tide-induced upwelling.     

The Estimation of the Cavity Effect by Higher Degree Finite Element Approximation

The measurements of tide-induced tilts and deformations are usually performed in underground cavities (mostly man-made). Therefore, the data obtained must be debugged of the influence of the cavity itself (cavity effect) and of the effect of elastic parameter inhomogeneities in the vicinity of the point of measurement (geological effect), if they are to he presented. Since these effects are reflected as a modulation of tidal waves (in both amplitude and phase), the only way to perform this can be modelling of the strain and stress field around the cavity as precisely as possible. The finite element method (F.E.M.) seems to be a very useful tool for this purpose. To justify some conclusions in this paper, a number of numerical runs of the F.E.M. models of the tide-induced strains and stresses (in two dimensions) were performed. A higher degree of approximation (up to the value of 7) was used in these calculations. To estimate the cavity effect, the correction factors (relative changes of particular strain components due to the presence of the cavity) were determined and plotted in the cavity's close neighbourhood. The distribution of the deformation energy (and also the work of external forces) over the whole domain of interest is given, and the conclusions for subsequent error estimation are drawn. An attempt was made to compare the real tilt data to the theoretical deduced from the numerical model.     

Analytical solution for the tidally induced Lagrangian residual current in a narrow bay

In a weakly nonlinear tidal system, the depth-averaged equations for the first-order Lagrangian residual velocity (LRV) are deduced systematically. For the case of a narrow bay, the equations are solved analytically and the results for a specific bottom profile are discussed in detail. According to the pattern of the first-order LRV, the bay can be divided into three parts, namely an inner part, a transitional zone, and an outer part. For the given depth profile, the streamline of the first-order LRV for a shorter bay is a part of that for a longer bay. The first-order LRV depends on a nondimensional parameter that combines the influences of the bottom friction coefficient, the tidal period and the averaged water depth. The form of the bottom friction also has a significant influence on the first-order LRV. The second-order LRV, i.e., the Lagrangian drift, is analytically solved and shows dependence on the initial tidal phase. The LRV differs from the Eulerian residual transport velocity both quantitatively and qualitatively. It is demonstrated that the residual currents obtained according to other definitions may cause misunderstanding of the mass transport in water exchange applications.     

Seasonal circulation and influence factors of the Bohai Sea: a numerical study based on Lagrangian particle tracking method

Seasonal circulation of the Bohai Sea (BS) in 1992 was investigated using Lagrangian particle tracking method. The hydrography of the BS was simulated based on an unstructured grid, finite-volume, three-dimensional primitive equation ocean model. With the use of the unstructured triangular grid, the model can easily fit the irregular coastal boundary of the BS. The simulated tides, tidal current, and thermohaline field agreed well with the observations. The transport of particles has three-dimensional structure in the BS. Compared with central Bohai and Bohai Strait, the differences of particles’ transportation between surface and bottom layer in three bays are small. The circulation in the summer is stronger than that in the winter, with the average residual velocity in the surface layer being about 3.7 cm/s during the summer while only 1.8 cm/s during the winter. Using the same model, several well-designed numerical experiments were performed to investigate the effect of oceanic tide, river discharge, wind stress, and thermal stratification on the circulation. It is shown that winds play an important role in the circulation of the BS during both the winter and the summer. Density circulation is important during the summer; however, it is negligible during the winter. River runoff only affects the area around the river mouth. Compared with wind and thermohaline effect, the contribution of tides is small during the summer, and the circulation under only M₂ tidal constituent could not reflect the actual circulation of the BS.     

Spatio-temporal variability of currents over a mobile dune field in the Dover Strait

A one month field campaign featuring two spring–neap tide cycles and three strong storms has been performed in a mobile dune area located in the central part of the Dover Strait. These dunes are known to move in a complex manner as their migration direction varies in space and time (Le Bot et al., 2000, Le Bot, 2001, Le Bot and Trentesaux, 2004). In order to gain some insights into the dune motion processes we present an analysis of the spatio-temporal variability of currents in the area emphasizing the relative influence of tides and storms. A total of eight different hydro-meteorological regimes have been distinguished during the experiment duration. The analysis of the currents measurements at five locations in the area shows that the eight hydro-meteorological regimes induce very different current responses at the bottom. The residual tidal currents exhibit a significant spatial variability both in direction and in intensity. A numerical model of tidal currents over the Dover Strait confirms the strong spatio-temporal variability of the residual tidal currents featuring three singular points. Amongst them, a saddle point is located just south of the I-dune at the convergence of opposite direction residual tidal currents. The wind-induced currents are almost uniform in space, their intensity and direction however strongly depends on the wind regime and thus on time. The mean total current feature a spatial pattern which can be tidal of wind-induced currents dominated, or either in balance, depending on the regime considered. At the PERMOD campaign time scale, the total current is dominated by the residual tidal current. These results proved to give valuable insights to explain the complex dynamics of dune motion observed in this area by Le Bot et al., 2000, Le Bot, 2001, Le Bot and Trentesaux, 2004 at short and long time scales.     

Temporal observations of rip current circulation on a macro-tidal beach

A field experiment was conducted on a high energy macro-tidal beach (Perranporth, UK) to examine rip current dynamics over a low-tide transverse bar/rip system in response to changing tide and wave conditions. Hydrodynamic data were collected using an array of in situ acoustic doppler current meters and pressure transducers, as well as 12 GPS-tracked Lagrangian surf zone drifters. Inter-tidal and sub-tidal morphology were measured through RTK-GPS and echo-sounder surveys. Data were collected for eight consecutive days (15 tides) over a spring-neap tidal cycle with tidal ranges of 4–6.5 m and offshore significant wave heights of 1–2 m and peak periods of 5–12 s.     

Tidal and residual currents over abrupt deep-sea topography based on shipboard ADCP data and tidal model solutions for three popular bathymetry grids

The response of tidal and residual currents to small-scale morphological differences over abrupt deep-sea topography (Seine Seamount) was estimated for bathymetry grids of different spatial resolution. Local barotropic tidal model solutions were obtained for three popular and publicly available bathymetry grids (Smith and Sandwell TOPO8.2, ETOPO1, and GEBCO08) to calculate residual currents from vessel-mounted acoustic Doppler current profiler (VM-ADCP) measurements. Currents from each tidal solution were interpolated to match the VM-ADCP ensemble times and locations. Root mean square (RMS) differences of tidal and residual current speeds largely follow topographic deviations and were largest for TOPO8.2-based solutions (up to 2.8 cm?s^?1) in seamount areas shallower than 1,000 m. Maximum RMS differences of currents obtained from higher resolution bathymetry did not exceed 1.7 cm?s^?1. Single depth-dependent maximum residual flow speed differences were up to 8 cm?s^?1 in all cases. Seine Seamount is located within a strong mean flow environment, and RMS residual current speed differences varied between 5 % and 20 % of observed peak velocities of the ambient flow. Residual flow estimates from shipboard ADCP data might be even more sensitive to the choice of bathymetry grids if barotropic tidal models are used to remove tides over deep oceanic topographic features where the mean flow is weak compared to the magnitude of barotropic tidal, or baroclinic currents. Realistic topography and associated flow complexity are also important factors for understanding sedimentary and ecological processes driven and maintained by flow–topography interaction.