Tidal Velocity Amplitude Research Articles

Flow separation, dipole formation, and water exchange through tidal straits

Abstract. We investigate the formation and evolution of dipole vortices and their contribution to water exchange through idealized tidal straits. Self-propagating dipoles are important for transporting and exchanging water properties through straits and inlets in coastal regions. In order to obtain a robust dataset to evaluate flow separation, dipole formation and evolution, and the effect on water exchange, we conduct 164 numerical simulations, varying the width and length of the straits as well as the tidal forcing. We show that dipoles form and start propagating at the time of flow separation, and their vorticity originates in the velocity front formed by the separation. We find that the dipole propagation velocity is proportional to the tidal velocity amplitude and twice as large as the dipole velocity derived for a dipole consisting of two point vortices. We analyze the processes creating a net water exchange through the straits and derive a kinematic model dependent on dimensionless parameters representing strait length, dipole travel distance, and dipole size. The net tracer transport resulting from the kinematic model agrees closely with the numerical simulations and provides an understanding of the processes controlling net water exchange.

Channel curvature effects on estuarine circulation in a highly stratified tropical estuary: The São Francisco river estuary (Brazil)

The effects of the channel curvature on estuarine circulation were assessed in the São Francisco river estuary. This estuary is a deltaic front system, with a highly stratified salinity distribution. The assessment was based on field experiments where currents and salinity and temperature were monitored during a semi-diurnal tidal cycle in two cross-sections with distinct curvatures (radii of 2 and 8 km). The least-squares method was used to determine the mean velocity (residual velocity), and the tidal velocity amplitude and phase for the M2 harmonic. The results for the M2 harmonic were compared to an analytical solution that considers a simplified momentum balance between barotropic pressure gradient and bottom friction. The model skill for these two results was 0.44 and 0.63 for sections #A and #B, respectively. The differences between these results indicate that there are other dynamically influential forces for the longitudinal momentum balance besides those included in the analytical solution. The baroclinic pressure gradient presented values two orders of magnitude greater than the bottom friction, demonstrating the dynamic importance of this force. Although centrifugal acceleration is a component of the lateral momentum balance, it also plays a role in the distribution of longitudinal currents in the cross-section, displacing the maximum currents towards the exit of the curve. In addition to the channel curvature effect, the influence of the bathymetry was also observed on the tidal amplitude in the cross-sections and on the depth-averaged residual longitudinal velocities along the sections, with the highest average speed in the shallowest areas of each section.

Cylinder wakes in shallow oscillatory flow: the coastal island wake problem

Topographic complexity on continental shelves is the catalyst that transforms the barotropic tide into the secondary and residual circulations that dominate vertical and cross-shelf mixing processes. Island wakes are one such example that are observed to significantly influence the transport and distribution of biological and physical scalars. Despite the importance of island wakes, to date, no sufficient, mechanistic description of the physical processes governing their development exists for the general case of unsteady tidal forcing. Controlled laboratory experiments are necessary for the understanding of this complex flow phenomenon. Here, three-dimensional velocity field measurements of cylinder wakes in shallow-water oscillatory flow are conducted across a parameter space that is typical of tidal flow around shallow islands. The wake form in steady flows is typically described in terms of the stability parameter $S=c_{f}D/h$ (where $D$ is the island diameter, $h$ is the water depth and $c_{f}$ is the bottom boundary friction coefficient); in tidal flows, there is an additional dependence on the Keulegan–Carpenter number $KC=U_{0}T/D$ (where $U_{0}$ is the tidal velocity amplitude and $T$ is the tidal period). In this study we demonstrate that when the influence of bottom friction is confined to a Stokes boundary layer the stability parameter is given by $S=\unicode[STIX]{x1D6FF}^{+}/KC$ where $\unicode[STIX]{x1D6FF}^{+}$ is the ratio of the wavelength of the Stokes bottom boundary layer to the depth. Three classes of wake form are observed with decreasing wake stability: (i) steady bubble for $S\gtrsim 0.1$; (ii) unsteady bubble for $0.06\lesssim S\lesssim 0.1$; and (iii) vortex shedding for $S\lesssim 0.06$. Transitions in wake form and wake stability are shown to depend on the magnitude and temporal evolution of the wake return flow. Scaling laws are developed to allow upscaling of the laboratory results to island wakes. Vertical and lateral transport depend on three parameters: (i) the flow aspect ratio $h/D$; (ii) the amplitude of tidal motion relative to the island size, given by $KC$; and (iii) the relative influence of bottom friction to the flow depth, given by $\unicode[STIX]{x1D6FF}^{+}$. A model of wake upwelling based on Ekman pumping from the bottom boundary layer demonstrates that upwelling in the near-wake region of an island scales with $U_{0}(h/D)KC^{1/6}$ and is independent of the wake form. Finally, we demonstrate an intrinsic link between the dynamical eddy scales, predicted by the Ekman pumping model, and the island wake form and stability.

Convergence of estuarine channels

Tide-dominated coastal plain estuaries have typically up-estuary convergent tidal channels. Analysis of estuarine characteristics indicates a dependence of the convergence length on relative tidal amplitude, relative intertidal area and river flow velocity. In order to explain these relationships we investigate a condition for continuity of net sediment transport throughout the estuary, corresponding to morphodynamic equilibrium. We show, by using an analytical solution of the tidal equations, that this condition is equivalent to a condition on the convergence length. This condition is evaluated for 21 estuaries in different regions of the world. It appears that the convergence length determined in this way can explain observed convergence lengths for the considered set of estuaries. The dependence of the convergence length on different estuarine characteristics is analysed by solving the fully coupled hydro-morphodynamic equations. We show that this dependence limits the range of variation of the tidal velocity amplitude. The analysis provides insight in the morphological response of estuaries to human interventions. The condition can easily be evaluated to yield an estimate of this response.

Residual Transport of Suspended Material by Tidal Straining near Sloping Topography

AbstractTidal straining is known to have an important impact on the generation of residual currents and the transport of suspended material in estuaries and the coastal ocean. Essential for this process is an externally imposed horizontal density gradient, typically resulting from either freshwater runoff or differential heating. Here, it is shown that near sloping topography, tidal straining may effectively transport suspended material across isobaths even if freshwater runoff and differential heating do not play a significant role. A combined theoretical and idealized modeling approach is used to illustrate the basic mechanisms and implications of this new process. The main finding of this study is that, for a wide range of conditions, suspended material is transported upslope by a pumping mechanism that is in many respects similar to classical tidal pumping. Downslope transport may also occur, however, only for the special cases of slowly sinking material in the vicinity of slopes with a slope angle larger than a critical threshold. The effective residual velocity at which suspended material is transported across isobaths is a significant fraction of the tidal velocity amplitude (up to 40% in some cases), suggesting that suspended material may be transported over large distances during a single tidal cycle.

Analytical approach for predicting fresh water discharge in an estuary based on tidal water level observations

Abstract. As the tidal wave propagates into an estuary, the tidally averaged water level tends to rise in landward direction due to the density difference between saline and fresh water and the asymmetry of the friction. The effect of friction on the residual slope is even more remarkable when accounting for fresh water discharge. In this study, we investigate the influence of river discharge on tidal wave propagation in the Yangtze estuary with specific attention to residual water level slope. This is done by using a one-dimensional analytical model for tidal hydrodynamics accounting for the residual water level. We demonstrate the importance of the residual slope on tidal dynamics and use it to improve the prediction of the tidal propagation in estuaries (i.e. tidal damping, velocity amplitude, wave celerity and phase lag), especially when the influence of river discharge is significant. Finally, we develop a new inverse analytical approach for estimating fresh water discharge on the basis of tidal water level observations along the estuary, which can be used as a tool to obtain information on the river discharge that is otherwise difficult to measure in the tidal region.

Asymptotic behavior of tidal damping in alluvial estuaries

Tidal wave propagation can be described analytically by a set of four implicit equations, i.e., the phase lag equation, the scaling equation, the damping equation, and the celerity equation. It is demonstrated that this system of equations has an asymptotic solution for an infinite channel, reflecting the balance between friction and channel convergence. Subsequently, explicit expressions for the tidal amplitude and velocity amplitude are derived, which are different from the generally assumed exponential damping equation that follows from linearizing the friction term. Analysis of the asymptotic behavior demonstrates that exponential damping of the tidal amplitude is only correct for a frictionless wave or an ideal estuary (no damping). However, in estuaries with modest damping (near ideal) it provides a reasonable approximation. In natural estuaries, there is generally a need to take account of local variability of, e.g., depth and friction, by subdividing the estuary into multiple reaches. This is illustrated with an example of the Scheldt estuary, which has been gradually deepened for navigation purpose over the last half century. The analytical model is used to study the effect of this deepening on the tidal dynamics in the main navigation channel, demonstrating that the navigation channel will become “overamplified” when it reaches a depth larger than the critical depth. In the case of overamplification, a further increase of the depth reduces the amplification until critical convergence (condition for a frictionless standing wave) is reached asymptotically. Finally, based on the ratio between the tidal amplitude at the seaward boundary and the asymptotic tidal amplitude, estuaries can be classified into damped, amplified, or ideal estuaries, which is illustrated with 23 real estuaries.

Analytical and numerical analysis of tides and salinities in estuaries; part II: salinity distributions in prismatic and convergent tidal channels

Estuaries, commonly, are densely populated areas serving the needs of the inhabitants in multiple ways. Often the interests are conflicting and decisions need to be made by the local managers. Intake of fresh water for consumption, agricultural purposes or use by industries may take place within a region not far landward of the limit of salt intrusion. Human interventions (e.g. deepening of the navigation channels) or climate changes (sea level rise, reduction of the river discharge) can bring these intake locations within the reach of saline or brackish water and consequently endanger their function. To support policy and managerial decisions, a profound knowledge of processes associated with the salinity structure in estuaries is required. Although nowadays advanced numerical three-dimensional models are available that are able to cope with the complexity of the physics there is still a need for relatively simple tools for quick-scan actions in a pre-phase of a project or for instructive purposes. The analytical model described in this paper may serve these needs. It computes the maximum salinity distribution using the dispersion coefficient in the mouth as the only model parameter. The model has been calibrated using observational data in a large number of estuaries and experimental data in a tidal flume. The dispersion coefficient was successfully related to geometric and hydrodynamic parameters resulting in an expression that can be used for convergent estuaries as well as prismatic channels, see Eqs. 25a and 25b. Application of the model in a predictive mode showed its promising capabilities. Comparison with three-dimensional numerical models indicates that the channel geometry in the estuary mouth largely influences dispersive processes. The analytical model for salt intrusion may be used in combination with the analytical model for tidal propagation in convergent estuaries and tidal channels by Van Rijn (part I). In this way, input is obtained on the tidal velocity amplitude and the Chezy roughness following calibration of this model on tidal amplitudes along the estuary.

Transverse structure of tidal and residual flow and sediment concentration in estuaries

An analytical and a numerical model are used to understand the response of velocity and sediment distributions over Gaussian-shaped estuarine cross-sections to changes in tidal forcing and water depth. The estuaries considered here are characterized by strong mixing and a relatively weak along-channel density gradient. It is also examined under what conditions the fast, two-dimensional analytical flow model yields results that agree with those obtained with the more complex three-dimensional numerical model. The analytical model reproduces and explains the main velocity and sediment characteristics in large parts of the parameter space considered (average tidal velocity amplitude, 0.1–1 m s − 1 and maximum water depth, 10–60 m). Its skills are lower for along-channel residual flows if nonlinearities are moderate to high (strong tides in deep estuaries) and for transverse flows and residual sediment concentrations if the Ekman number is small (weak tides in deep estuaries). An important new aspect of the analytical model is the incorporation of tidal variations in the across-channel density gradient, causing a double circulation pattern in the transverse flow during slack tides. The gradient also leads to a new tidally rectified residual flow component via net advection of along-channel tidal momentum by the density-induced transverse tidal flow. The component features landward currents in the channel and seaward currents over the slopes and is particularly effective in deeper water. It acts jointly with components induced by horizontal density differences, Coriolis-induced tidal rectification and Stokes discharge, resulting in different along-channel residual flow regimes. The residual across-channel density gradient is crucial for the residual transverse circulation and for the residual sediment concentration. The clockwise density-induced circulation traps sediment in the fresher water over the left slope (looking up-estuary in the northern hemisphere). Model results are largely consistent with available field data of well-mixed estuaries.

Equilibrium and cross-sectional stability of tidal inlets: application to the Frisian Inlet before and after basin reduction

The focus of this paper is on the adaptation of the inlet cross-sectional area after basin reduction. To calculate the new equilibrium and degree of stability, use is made of a method first proposed by Escoffier. As part of this study his largely empirical approach is expanded and given a physical basis. The inlet channel is divided in an entrance and interior section. Sand enters the entrance section on the flood and is removed by the ebb tidal current. On annually average basis, the volume of sand entering the inlet is assumed to be a constant fraction of the littoral drift. A theoretical expression for the ebb tidal sand transport as a function of the tidal velocity amplitude and a length dimension of the cross-section is derived. An important ingredient in the stability analysis is the empirical cross-sectional area versus tidal prism relationship for inlets in equilibrium and the related equilibrium velocity. A physical justification for this relationship is presented. As a measure for cross-sectional stability, a stability index, S, is introduced. For S≥1, the inlet is very stable. Stability increases with increasing values of S. For inlets that are very stable, the value of S can be related to P/ V, i.e. the ratio of tidal prism and volume of littoral drift. A method to determine the time scale for the entrance section to return to a new equilibrium after a change in basin surface area is presented. To illustrate its practical use, the Escoffier method is applied to the Frisian Inlet before and after basin reduction. In 1969, the basin area and tidal prism were reduced by approximately 30%. Based on the Escoffier method, the original equilibrium cross-sectional area of 22,000 m 2 is expected to reduce to 15,500 m 2. The adaptation time scale is estimated to be 30 years. Even after basin reduction, the inlet remains very stable with S=31 and P/ V=585.

Comparison between predicted and observed sand waves and sand banks in the North Sea

For the first time a prediction model of regular morphological patterns on the seabed was tested against observations of sand wave and sand bank occurrence in the entire North Sea. The model, which originates from first physical principles, predicts this occurrence via two dimensionless parameters on the basis of the water depth, the tidal velocity amplitude, the level of zero intercept above the seabed (z0), and a viscosity variation parameter ε. The latter two quantities were varied in a number of predictions for the entire North Sea, whereas for the first two, local values were used. The range of realistic values of ε and z0 was large enough to let these two parameters distinguish between the possible (combinations of) bed forms, as is shown in the shallower southern bight of the North Sea. The results were more sensitive to variations in z0 than in ε. A slightly more detailed approach focused on sand waves only in the southern North Sea and estimated local values for z0 using depth information. Quantification of the results showed that the model was able to predict the contours of the sand wave patches, but it could not account for the absence of the bed features within this area. The type of bed deposit partly explains the smaller‐scale variation. The work confirms the validity of the theoretical bed form prediction model and verifies the hypothesis that the large‐scale seabed features are formed as free instabilities of tide‐topography interactions.

Acoustic Doppler Current Profiler measurements of tidal phase and amplitude in Cook Strait, New Zealand

Strong tidal flows are observed in Cook Strait which separates the North and South Islands of New Zealand. The high velocities within the 30 km wide Strait result from a 140° phase difference in the M 2 tide between the ends of the Strait. Extraordinarily 135° of this phase difference occurs over just 40 km in the narrowest section of the Strait. Measurements from a ship mounted Acoustic Doppler Current Profiler (ADCP) over a single tidal cycle are used to determine the horizontal and vertical variation of tidal phase and amplitude in the Strait. Results show that tidal velocity amplitude ranges from 70 cm s −1 on the west of the Strait to 140 cm s −1 on the east. There was little amplitude variation over most of the water column. The eastern side of the Strait led the west by 20°. Near bottom velocity led surface velocity by approximately 10° due to the effect of bottom friction on the oscillating flow. Results from a subsequent 1 month deployment of ADCPs on the same line as the ship track are used to hindcast the semi-diurnal tide on the day of the shipboard measurements. The shipboard measured semi-diurnal tidal amplitude and phase agree extremely well with the hindcast composite of the three largest tidal constituents. Thus shipboard measurements over a single tidal cycle were able to accurately determine the horizontal and vertical variation of phase and amplitude of the semi-diurnal tide in Cook Strait.

Tidal and residual flows in the Western Dutch Wadden Sea I: Numerical model results

A two-dimensional numerical model, which includes tidal basins in the western Dutch Wadden Sea and a part of the adjacent North Sea, is used to study tidal and tidally-driven residual flows and elevations. The model is verified by comparing observed and computed water elevations in some stations and transport rates through the tidal inlets. The consequence of topographical and geometrical differences for the terms in the governing equations is discussed by comparing the magnitude of these terms in some characteristic grid points, viz. the open sea, inlet and basin channel, in which all terms are decomposed along and perpendicular to the dominant current direction. Averaging over a tidal period shows that compared to the time-dependent equations an important shift takes place in the relative influence of the different terms, mainly caused by the increased influence of the advective term. The resulting tidally-driven residual flow field is interpreted as a combination of a constant volume transport (1 to 2% of the tidal transport in the tidal inlets) between connected tidal basins and isolated residual eddies (with velocities of 10 to 15% of the tidal velocity amplitude and a typical length scale of between 3 and 10 km). Observations confirm that the drop in residual elevations in tidal inlets and the rise in tidal basins are phenomena characteristic of all tidal basins.

Control of the homopause level

This paper reviews the hypothesis that internal gravity wave breaking in the factor responsible for upper atmosphere turbulence generation and control of the number density at the homopause level. Available data from Mars and Earth support the hypothesis, originally introduced by Lindzen, that breaking thermal tides are responsible for the necessary turbulence. The lowest-order, equatorially trapped, westward-propagating diurnal tidal mode is the primary cause on the Earth, and lower-order westward-propagating semidiurnal modes are the primary cause on Mars. The frequencies and vertical wavenumbers of the responsible modes determine the eddy diffusion coefficients; energy density is only indirectly involved by determining which modes can break. A general scale relationship between tidal heating and velocity amplitude can be used to assess the breaking potential of tidal modes, and is used here as the basis for arguing that the number density at the Martian homopause is likely to have been stable over much of geologic time. Difficulties with this approach arise in the cases of Venus and Titan because of their peculiar zonal winds. For Jupiter and Saturn, internal gravity waves generated by nontidal mechanisms probably control the homopause.