Analytical and numerical analysis of tides and salinities in estuaries; part I: tidal wave propagation in convergent estuaries

Analytical solutions of the momentum and energy equations for tidal flow are studied. Analytical solutions are well known for prismatic channels but are less well known for converging channels. As most estuaries have a planform with converging channels, the attention in this paper is fully focused on converging tidal channels. It will be shown that the tidal range along converging channels can be described by relatively simple expressions solving the energy and momentum equations (new approaches). The semi-analytical solution of the energy equation includes quadratic (nonlinear) bottom friction. The analytical solution of the continuity and momentum equations is only possible for linearized bottom friction. The linearized analytical solution is presented for sinusoidal tidal waves with and without reflection in strongly convergent (funnel type) channels. Using these approaches, simple and powerful tools (spreadsheet models) for tidal analysis of amplified and damped tidal wave propagation in converging estuaries have been developed. The analytical solutions are compared with the results of numerical solutions and with measured data of the Western Scheldt Estuary in the Netherlands, the Hooghly Estuary in India and the Delaware Estuary in the USA. The analytical solutions show surprisingly good agreement with measured tidal ranges in these large-scale tidal systems. Convergence is found to be dominant in long and deep-converging channels resulting in an amplified tidal range, whereas bottom friction is generally dominant in shallow converging channels resulting in a damped tidal range. Reflection in closed-end channels is important in the most landward 1/3 length of the total channel length. In strongly convergent channels with a single forward propagating tidal wave, there is a phase lead of the horizontal and vertical tide close to 90^o, mimicking a standing wave system (apparent standing wave).     

A coupled analytical model for salt intrusion and tides in convergent estuaries

ABSTRACT

In this paper we develop a coupled analytical model for salinity and tidal propagation in estuaries where the cross-sectional area varies exponentially. A simple analytical model for tidal dynamics has been used to estimate the tidal excursion, which has an important influence on the salt intrusion process since it determines the extreme salinities (i.e. salinity distribution for high water slack and low water slack). The objective of the coupling is to reduce the number of calibration parameters, which subsequently strengthens the reliability of the salt intrusion model. Moreover, the coupling enables us to assess the potential impacts of external changes, both human-induced interventions (e.g. dredging) and natural changes (e.g. global sea level rise), on the salt intrusion process. In addition, the fully analytical solution for hydrodynamics allows immediate estimation of the tidally averaged depth and friction coefficient for given water level recordings and salinity measurements. This is particularly useful when a geometric survey is not available. The coupled model has been applied to six previously unsurveyed estuaries in Malaysia and the results show that the correspondence between analytical estimations and observations is very good. Thus, the coupled model proves to be a useful tool to obtain estimates of salt intrusion in estuaries based on a minimum amount of information required and for assessing the effect of human-induced or natural changes.

EDITOR D. Koutsoyiannis ASSOCIATE EDITOR B. Dewals     

Analytical study of the transverse distribution of along-channel and transverse residual flows in tidal estuaries

We present an analytical model to decompose complex along-channel and transverse residual flows into components induced by individual mechanisms. The model describes the transverse distribution of residual flows in tidally dominated estuaries. Scaling and perturbation techniques are used to obtain analytical solutions for residual flows over arbitrary across-channel bed profiles. The flows are induced by horizontal density gradients, tidal rectification processes, river discharge, wind, channel curvature and the earth's rotation. These rectification processes induce residual flows that are up-estuary to the right and down-estuary to the left of an estuarine channel (looking up-estuary in the northern hemisphere). The tidal rectification processes fundamentally change the transverse structure of along-channel residual flows in many tidal estuaries, as these processes cause the flows to be internally asymmetric about the mid-axis of the channel for relatively large tidal velocities, steep channels or narrow estuaries. In addition, velocity scales are derived from the analytical solutions to estimate the relative importance of the various residual flow mechanisms from estuarine parameters. A case study of a transect across the Upper Chesapeake Bay showed that important features of the residual flow observed in that transect are reproduced and explained by the analytical model. The velocity scales were able to identify the relevant residual flow mechanisms as well. The tidal rectification processes considered here result from advection of along-channel tidal momentum by Coriolis-induced transverse tidal currents.     

Benchmark on the numerical simulations of the hydrodynamic and morphodynamic evolution of tidal channels and tidal inlets

A correct understanding of the hydrodynamics and morphodynamics of tidal basins is of fundamental importance for the fate of the Venice Lagoon, Italy. If on one hand, the development of sophisticated numerical models is called for in order to reproduce the complexity of the mechanisms governing the morphodynamic evolution of many natural environments, including lagoons, on the other hand, a clear knowledge of the reliability and limits of the results provided by these models is crucial in order to establish the condition under which they can be safely applied. To this aim, researchers involved in numerical modeling in the framework of the recent Corila research programmes, agreed to perform an accurate comparison of results provided by three different numerical models, applying them to the test case offered by the experimental investigations performed under controlled conditions by Tambroni et al. (2005a). Here, we consider the following numerical models: (i) a 2D finite element hydrodynamic model coupled with a 2D finite volume morphodynamic model (5 and 3); (ii) a 2D finite element morphodynamic model (Ferrarin et al., 2008); (iii) a 2D depth-averaged model for the inlet region, coupled with a 1D model for the channel (Tambroni et al., 2005b). A first set of simulations concerns the fixed bed case and shows that all the models provide similar results: in particular, they are able to predict the observed free surface oscillations satisfactorily, while comparison with the measured velocity field is less satisfactory. Moreover, as far as the flow field at the inlet is concerned, the models describe accurately the potential flow into the channel during the flood phase, while they are not able to adequately reproduce the occurrence of the fine structure of the shear layers shed by the inlet edges during the ebb phase. This limit is related to the shallow water character of the models. As for the morphodynamics, the long term equilibrium configurations of the bottom of the channel and of the near inlet region show qualitative agreement with the experimental observations, although in this case the differences between the results provided by the distinct numerical approaches are more marked.     

The effect of tidal asymmetry and temporal settling lag on sediment trapping in tidal estuaries

Over decades and centuries, the mean depth of estuaries changes due to sea-level rise, land subsidence, infilling, and dredging projects. These processes produce changes in relative roughness (friction) and mixing, resulting in fundamental changes in the characteristics of the horizontal (velocity) and vertical tides (sea surface elevation) and the dynamics of sediment trapping. To investigate such changes, a 2DV model is developed. The model equations consist of the width-averaged shallow water equations and a sediment balance equation. Together with the condition of morphodynamic equilibrium, these equations are solved analytically by making a regular expansion of the various physical variables in a small parameter. Using these analytic solutions, we are able to gain insight into the fundamental physical processes resulting in sediment trapping in an estuary by studying various forcings separately. As a case study, we consider the Ems estuary. Between 1980 and 2005, successive deepening of the Ems estuary has significantly altered the tidal and sediment dynamics. The tidal range and the surface sediment concentration has increased and the position of the turbidity zone has shifted into the freshwater zone. The model is used to determine the causes of these historical changes. It is found that the increase of the tidal amplitude toward the end of the embayment is the combined effect of the deepening of the estuary and a 37% and 50% reduction in the vertical eddy viscosity and stress parameter, respectively. The physical mechanism resulting in the trapping of sediment, the number of trapping regions, and their sensitivity to grain size are explained by careful analysis of the various contributions of the residual sediment transport. It is found that sediment is trapped in the estuary by a delicate balance between the M ₂ transport and the residual transport for fine sediment ( $\emph{w}_s=0.2$  mm s^???1) and the residual, M ₂ and M ₄ transports for coarser sediment ( $\emph{w}_s=2$  mm s^???1). The upstream movement of the estuarine turbidity maximum into the freshwater zone in 2005 is mainly the result of changes in tidal asymmetry. Moreover, the difference between the sediment distribution for different grain sizes in the same year can be attributed to changes in the temporal settling lag.     

Empirical observations and numerical modelling of tides,channel morphology,and vegetative effects on accretion in a restored tidal marsh

Tidal marshes form at the confluence between estuarine and marine environments where tidal movement regulates their developmental processes. Here, we investigate how the interplay between tides, channel morphology, and vegetation affect sediment dynamics in a low energy tidal marsh at the Paul S. Sarbanes Ecosystem Restoration Project at Poplar Island. Poplar Island is an active restoration site where fine-grained material dredged from navigation channels in the upper Chesapeake Bay are being used to restore remote tidal marsh habitat toward the middle bay (Maryland, USA). Tidal currents were measured over multiple tidal cycles in the inlets and tidal creeks of one marsh at Poplar Island, Cell 1B, using Acoustic Doppler Current Profilers (ADCP) to estimate water fluxes throughout the marsh complex. Sediment fluxes were estimated using acoustic backscatter recorded by ADCPs and validated against total suspended solid measurements taken on site. A high-resolution geomorphic survey was conducted to capture channel cross sections and tidal marsh morphology. We integrated simple numerical models built in Delft3d with empirical observations to identify which eco-geomorphological factors influence sediment distribution in various channel configurations with differing vegetative characteristics. Channel morphology influences flood-ebb dominance in marshes, where deep, narrow channels promote high tidal velocities and incision, increasing sediment suspension and reducing resilience in marshes at Poplar Island. Our numerical models suggest that accurately modelling plant phenology is vital for estimating sediment accretion rates. In-situ observations indicate that Poplar Island marshes are experiencing erosion typical for many Chesapeake Bay islands. Peak periods of sediment suspension frequently coincide with the largest outflows of water during ebb tides resulting in large sediment deficits. Ebb dominance (net sediment export) in tidal marshes is likely amplified by sea-level rise and may lower marsh resilience. We couple field observations with numerical models to understand how tidal marsh morphodynamics contribute to marsh resilience. © 2019 John Wiley & Sons, Ltd.     

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.     

A numerical study of the barotropic tides and tidal energy distribution in the Indonesian seas with the assimilated finite volume coastal ocean model

The tides and tidal energetics in the Indonesian seas are simulated using a three-dimensional finite volume coastal ocean model. The high-resolution coastline-fitted model is configured to better resolve the hydrodynamic processes around the numerous barrier islands. A large model domain is adopted to minimize the uncertainty adjacent to open boundaries. The model results with elevation assimilation based on a simple nudge scheme faithfully reproduced the general features of the barotropic tides in the Indonesian Seas. The mean root-mean-square errors between the observed and simulated tidal constants are 2.3, 1.1, 2.4, and 1.5 cm for M₂, S₂, K₁, and O₁, respectively. Analysis of the model solutions indicates that the semidiurnal tides in the Indonesian Seas are primarily dominated by the Indian Ocean, whereas the diurnal tides in this region are mainly influenced by the Pacific Ocean, which is consistent with previous studies. Examinations of tidal energy transport reveal that the tidal energy for both of the simulated tidal constituents are transported from the Indian Ocean into the IS mainly through the Lombok Strait and the Timor Sea, whereas only M₂ energy enters the Banda Sea and continues northward. The tidal energy dissipates the most in the passages on both sides of Timor Island, with the maximum M₂ and K₁ tidal energy transport reaching about 750 and 650 kW m^–1, respectively. The total energy losses of the four dominant constituents in the IS are nearly 338 GW, with the M₂ constituent dissipating 240.8 GW. It is also shown that the bottom dissipation rate for the M₂ tide is about 1–2 order of magnitudes larger than that of the other three tidal components in the Indonesian seas.     

Internal tides in the central Mediterranean Sea: observational evidence and numerical studies

Ocean Dynamics - Internal tides are studied in the central Mediterranean Sea using observational data and numerical experiments. Both numerical results and observations indicate that the baroclinic...     

Morphological controls in sandy estuaries: the influence of tidal flats and bathymetry on sediment transport

The morphodynamics of shallow, vertically well-mixed estuaries, characterised by tidal flats and deeper channels, have been investigated. This paper examines what contributes to flood/ebb-dominant sediment transport in localised regions through a 2D model study (using the TELEMAC modelling system). The Dyfi Estuary in Wales, UK has been used as a case study and, together with idealised estuary shapes, shows that shallow water depths lead to flood dominance in the inner estuary whilst tidal flats and deep channels cause ebb dominance in the outer estuary. For medium sands and with an artificially ‘flattened’ bathymetry (i.e. no tidal flats), the net sediment transport switches from ebb-dominant to flood-dominant where the parameter a/h (local tidal amplitude ÷ local tidally averaged water depth) exceeds 1.2. Sea level rise will reduce this critical value of a/h and also reduce the ebb-directed sediment transport significantly, leading to a flood-dominated estuarine system. A similar pattern, albeit with greater transport, was simulated with tidal flats included and also with a reduced grain size. This suggests that analogous classifications for flood/ebb asymmetry of the tide in estuaries as a whole may not represent the local sediment transport in sufficient detail. Through the Dyfi simulations, the above criterion involving a/h is shown to be complicated further by augmented flow past a spit at the estuary mouth which gives rise to a self-maintaining scour hole. Simulations of one year of bed evolution in an idealised flat-bottomed estuary, including tidal flow past a spit, recreate the flood/ebb dominance on either side of the spit and the formation of a scour hole in between. The erosion rate at the centre of the hole is reduced as the hole deepens, suggesting the establishment of a self-maintaining equilibrium state.     

Influence of variable salinity conditions in a tidal creek on riparian groundwater flow and salinity dynamics

While recent studies have revealed that tidal fluctuations in an estuary significantly affect groundwater flows and salt transport in the riparian zone, only seawater salinity in the estuary has been considered. A numerical study is conducted to investigate the influence of estuarine salinity variations on the groundwater flow and salt dynamics in the adjacent aquifer to extend our understanding of these complex and dynamic systems. Tidal salinity fluctuations (synchronous with estuary stage) were found to alter the magnitude and distribution of groundwater discharge to the estuary, which subsequently impacted on groundwater salinity patterns and residence times, especially in the riparian zone. The effects of salinity fluctuations were not fully captured by adopting a constant, time-averaged estuarine salinity. The modelling analysis also included an assessment of the impact of a seasonal freshwater flush in the estuary, similar to that expected in tropical climates (e.g. mean estuary level during flood significantly greater than average), on adjacent groundwater flow and salinity conditions. The three-month freshwater flushing event temporarily disrupted the salt distribution and re-circulation patterns predicted to occur under conditions of constant salinity and tidal water level fluctuations in the estuary. The results indicate that the salinity variations in tidal estuaries impact significantly on estuary–aquifer interaction and need to be accounted for to properly assess salinity and flow dynamics and groundwater residence times of riparian zones.     

Convection in a rotating magnetic system and taylor's constraint part II,numerical results

Abstract

Results are presented of a numerical study of marginal convection of electrically conducting fluid, permeated by a strong azimuthal magnetic field, contained in a circular cylinder rotating rapidly about its vertical axis of symmetry. To this basic state is added a geostrophic flow U_G (s), constant on geostrophic cylinders radius s. Its magnitude is fixed by requiring that the Lorentz forces induced by the convecting mode satisfy Taylor's condition. The nonlinear mathematical problem describing the system was developed in an earlier paper (Skinner and Soward, 1988) and the predictions made there are confirmed here. In particular, for small values of the Roberts number q which measures the ratio of the thermal to magnetic diffusivities, two distinct regions can be recognised within the fluid with the outer region moving rapidly compared to the inner. Otherwise, conditions for the onset of instability via the Taylor state (U_G 0) do not differ significantly from those appropriate to the static (U_G = 0) basic state. The possible disruption of the Taylor states by shear flow instabilities is discussed briefly.     

Impact of tidal power dams upon tides and environmental conditions in the Sea of Okhotsk

The transformation of natural tidal sea-level and currents is studied resulting from large-scale tidal power plant (TPP) dams in bays of the Sea of Okhotsk (SO). Some effects due to this transformation are estimated based on predictive modelling and a number of expected changes in amplitudes and phases, and spectral composition of tidal oscillations are described. Changes of morphometric properties of basins change the character of tidal motions even on significant distance from a dam. That is why, it is impossible to estimate this impact as usual boundary-value problems. The problem is solved based on “impedance” conditions on the open boundary of the model area, allowing to take into account the radiation of the additional perturbations induced by both waves reflected from the dam and nonlinear effects inside the area. In general, the transformation effects are proportional to the dam size and depend essentially on the dam location, the creation of which can change dissipative and resonance properties of the bays. The changes in tidal energetics of SO due to the dam construction are also considered to show noticeable reconstruction of horizontal energy fluxes and changes in the energy dissipation. Possible environmental consequences are related mainly to the transformation of tidal currents.     

Turbulence and mixing in a freshwater-influenced tidal bay: Observations and numerical modeling

In situ observations and numerical simulations of turbulence are essential to understanding vertical mixing processes and their dynamical controls on both physical and biogeochemical processes in coastal embayments. Using in situ data collected by bottom-mounted acoustic Doppler current profilers(ADCPs) and a free-falling microstructure profiler, as well as numerical simulations with a second-moment turbulence closure model, we studied turbulence and mixing in the Xiamen Bay, a freshwater-influenced tidal bay located at the west coast of the Taiwan Strait. Dynamically, the bay is driven predominantly by the M2 tide, and it is under a significant influence of the freshwater discharged from the Jiulong River. It is found that turbulence quantities such as the production and dissipation rates of the turbulent kinetic energy(TKE) were all subject to significant tidal variations, with a pronounced ebb-flood asymmetry. Turbulence was stronger during flood than ebb. During the flooding period, the whole water column was nearly well mixed with the depth-averaged TKE production rate and vertical eddy viscosity being up to 5?10?6 W kg?1 and 2?10?2 m2 s?1, respectively. In contrast, during the ebb strong turbulence was confined only to a 5?8 m thick bottom boundary layer, where turbulence intensity generally decreases with distance from the seafloor. Diagnosis of the potential energy anomaly showed that the ebb-flood asymmetry in turbulent dissipation and mixing was due mainly to tidal straining process as a result of the interaction between vertically shared tidal currents and horizontal density gradients. The role of vertical mixing in generating the asymmetry was secondary. A direct comparison of the modeled and observed turbulence quantities confirmed the applicability of the second-moment turbulence closure scheme in modeling turbulent processes in this weakly stratified tidally energetic environment, but also pointed out the necessity of further refinements of the model.     

Coupling of one- and two-dimensional finite elements for the computation of tidal flow in estuaries

A new method for the numerical computation of tidal flow in estuaries is demonstrated. One- and two-dimensional finite elements are directly coupled in a single model requiring the solution of only one common set of equations for all unknown parameters in each time step. The utility of the method is demonstrated with the computation of the tide in the Elbe Estuary. The computed water levels are compared with measured hydrographs at different gauges on the Elbe River.     

Longitudinal variation in lateral trapping of fine sediment in tidal estuaries: observations and a 3D exploratory model

This study investigates the longitudinal variation of lateral entrapment of suspended sediment, as is observed in some tidal estuaries. In particular, field data from the Yangtze Estuary are analysed, which reveal that in one cross-section, two maxima of suspended sediment concentration (SSC) occur close to the south and north sides, while in a cross-section 2 km down-estuary, only one SSC maximum on the south side is present. This pattern is found during both spring tide and neap tide, which are characterised by different intensities of turbulence. To understand longitudinal variation in lateral trapping of sediment, results of a new three-dimensional exploratory model are analysed. The hydrodynamic part contains residual flow due to fresh water input, density gradients and Coriolis force and due to channel curvature-induced leakage. Moreover, the model includes a spatially varying eddy viscosity that accounts for variation of intensity of turbulence over the spring-neap cycle. By imposing morphodynamic equilibrium, the two-dimensional distribution of sediment in the domain is obtained analytically by a novel procedure. Results reveal that the occurrence of the SSC maxima near the south side of both cross-sections is due to sediment entrapment by lateral density gradients, while the second SSC maximum near the north side of the first cross-section is by sediment transport due to curvature-induced leakage. Coriolis deflection of longitudinal flow also contributes the trapping of sediment near the north side. This mechanism is important in the upper estuary, where the flow due to lateral density gradients is weak.     

Horizontal patterns of water temperature and salinity in an estuarine tidal channel: Ria de Aveiro

This work presents results from two complementary and interconnected approaches to study water temperature and salinity patterns in an estuarine tidal channel. This channel is one of the four main branches of the Ria de Aveiro, a shallow lagoon located in the Northwest coast of the Iberian Peninsula. Longitudinal and cross-sectional fields of water temperature and salinity were determined by spatial interpolation of field measurements. A numerical model (Mohid) was used in a 2D depth-integrated mode in order to compute water temperature and salinity patterns. The main purpose of this work was to determine the horizontal patterns of water temperature and salinity in the study area, evaluating the effects of the main forcing factors. The field results were depth-integrated and compared to numerical model results. These results obtained using extreme tidal and river runoff forcing, are also presented. The field results reveal that, when the river flow is weak, the tidal intrusion is the main forcing mechanism, generating saline and thermal fronts which migrate with the neap/spring tidal cycle. When the river flow increases, the influence of the freshwater extends almost as far as the mouth of the lagoon and vertical stratification is established. Results of numerical modelling reveal that the implemented model reproduces quite well the observed horizontal patterns. The model was also used to study the hydrology of the study area under extreme forcing conditions. When the model is forced with a low river flow (1 m³ s⁻¹) the results confirm that the hydrology is tidally dominated. When the model is forced with a high river flow (1,000 m³ s⁻¹) the hydrology is dominated by freshwater, as would be expected in such an area.     

Experimental study of the velocity of density currents in convergent and divergent channels

The head velocity of the density current in the convergent and divergent channel is a key parameter for evaluating the extent to which suspended material travels, and for determining the type and distribution of sediment in the water body. This study experimentally evaluated the effects of the reach degree of convergence and divergence on the head velocity of the density current. Experiments were conducted in the flume with 6.0 m long, 0.72 m width and 0.6 m height. The head velocity was measured at three convergent degrees （-8°; -12°; -26°）, at three divergent degrees （8°; 12°; 26°） and two slopes （0.009, 0.016） for various discharges. The measured head velocity of the density current is compared with the head velocity of the density current in the constant cross section channel. Based on non-dimensional and statistical analysis, relations as linear multiple regression are offered for predicting head velocity of the density current in the convergent, divergent and constant cross section channel. Also the results of this research show that for the same slope and discharge, the head velocity of the density current in the convergent and divergent channel are greater and less than the head velocity of the constant cross section, respectively.