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.     

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 island wake generated by an elliptical tidal flow

An idealized numerical study of the influence of a tidal flow around an island has been undertaken with ROMS. The study focusses on coastal island wakes which are mainly controlled by elliptical tidal current flows on shallow shelves. This model is typical of some isolated continental shelf islands. The model is forced by a semi-diurnal barotropic inertia gravity wave imposed on the four open boundaries of a rectangular domain and its propagation results in an elliptical tidal flow within the domain in which the circular island lies. The influence of the surrounding island bathymetry and of the ellipse shape has been studied both in two and three dimensions. In the island vicinity, the residual circulation patterns over a tidal period show alongshore flow divergence along the major axis and convergence along the minor axis. A thin tidal ellipse (i.e. with a large ratio between major and minor axes) leads to strong eddy activity periods in the lee of the island during the flood and ebb phases, with eddy dissipation phases in between. By contrast, an almost round ellipse (axis ratio nearly 1) leads to vorticity filaments which continuously progress around the island without eddy shedding. The presence of a topographic slope in the vicinity of the island strengthens the eddy activity. This study suggests that the tidal current rotation favors the development of the eddy rotating in the same direction and weakens the development of the second eddy. In three dimensions with a surrounding bathymetry, an intense upwelling occurs in a large area in the lee of the island and the vertical velocities are stronger with thinner ellipses. With a flat bottom the vertical motions are almost fully generated by convergence and divergence of the secondary flow. With a varying bottom topography, the vertical motions come from a combination of this mechanism with convergence and divergence of the depth averaged flow.     

Wind effects on the lateral structure of density-driven circulation in Chesapeake Bay

The response of the density-driven circulation in the Chesapeake Bay to wind forcing was studied with numerical experiments. A model of the bay with realistic bathymetry was first applied to produce the density-driven flow under average river discharge and tidal forcing. Subsequently, four spatially uniform wind fields (northeasterly, northwesterly, southwesterly, and southeasterly) were imposed to examine the resulting cross-estuary structure of salinity and flow fields. In general, northeasterly and northwesterly winds intensified the density-driven circulation in the upper and middle reaches of the bay, whereas southeasterly and southwesterly winds weakened it. The response was different in the lower bay, where downwind flow from the upper and middle reaches of the bay competed with onshore/offshore coastal flows. Wind remote effects were dominant, over local effects, on volume transports through the bay entrance. However, local effects were more influential in establishing the sea-level slopes that drove subtidal flows and salinity fields in most of the bay. The effect of vertical stratification on wind-induced flows was also investigated by switching it off. The absence of stratification allowed development of Ekman layers that reached depths of the same order as the water depth. Consequently, bathymetric effects became influential on the homogeneous flow structure causing the wind-induced flow inside the bay to show a marked transverse structure: downwind over the shallow areas and upwind in the channels. In the presence of stratification, Ekman layers became shallower and the wind-induced currents showed weaker transverse structure than those that developed in the absence of stratification. In essence, the wind-driven flows were horizontally sheared under weak stratification and vertically sheared under stratified conditions.     

Multilayer shallow water flow using lattice Boltzmann method with high performance computing

A multilayer lattice Boltzmann (LB) model is introduced to solve three-dimensional wind-driven shallow water flow problems. The multilayer LB model avoids the expensive Navier–Stokes equations and obtains stratified horizontal flow velocities as vertical velocities are relatively small and the flow is still within the shallow water regime. A single relaxation time BGK method is used to solve each layer coupled by the vertical viscosity forcing term. To increase solution stability, an implicit step is suggested to obtain flow velocities. The main advantage of using the LBM is that after selecting appropriate equilibrium distribution functions, the LB algorithm is only slightly modified for each layer and retains all the simplicities of the LBM within the high performance computing (HPC) environment. The performance of the parallel LB model for the multilayer shallow water equations is investigated on CPU-based HPC environments using OpenMP. We found that the explicit loop control with cache optimization in LBM gives better performance on execution time, speedup and efficiency than the implicit loop control as the number of processors increases. Numerical examples are presented to verify the multilayer LB model against analytical solutions. We demonstrate the model’s capability of calculating lateral and vertical distributions of velocities for wind-driven circulation over non-uniform bathymetry.     

Multiple‐station neural network for modelling tidal currents across Shinnecock inlet,USA

This paper presents the development of a multiple‐station neural network for predicting tidal currents across a coastal inlet. Unlike traditional hydrodynamic models, the neural network model does not need inputs of coastal topography and bathymetry, grids, surface and bottom frictions, and turbulent eddy viscosity. Without solving hydrodynamic equations, the neural network model applies an interconnected neural network to correlate the inputs of boundary forcing of water levels at a remote station to the outputs of tidal currents at multiple stations across a local coastal inlet. Coefficients in the neural network model are trained using a continuous dataset consisting of inputs of water levels at a remote station and outputs of tidal currents at the inlet, and verified using another independent input and output dataset. Once the neural network model has been satisfactorily trained and verified, it can be used to predict tidal currents at a coastal inlet from the inputs of water levels at a remote station. For the case study at Shinnecock Inlet in the southern shore of New York, tidal currents at nine stations across the inlet were predicted by the neural network model using water level data located from a station about 70 km away from the inlet. A continuous dataset in May 2000 was used for the training, and another dataset in July 2000 was used for the verification of the neural network model. Comparing model predictions and observations indicates correlation coefficients range from 0·95 to 0·98, and the root‐mean‐square error ranges from 0·04 to 0·08 m s^?1 at the nine current locations across the inlet. Copyright © 2007 John Wiley & Sons, Ltd.     

The influence of asymmetries in overlying stratification on near-bed turbulence and sediment suspension in a partially mixed estuary

Data collected from the York River estuary demonstrate the importance of asymmetries in stratification to the suspension and transport of fine sediment. Observations collected during two 24-h deployments reveal greater concentrations of total suspended solids during the flood phase of the tide despite nearly symmetric near-bed tidal current magnitude. In both cases, tidally averaged net up-estuary sediment transport near the bed was clearly observed despite the fact that tidally averaged residual near-bed currents were near zero. Tidal straining of the along-channel salinity gradient resulted in a stronger pycnocline lower in the water column during the ebb phase of the tide and appeared to limit sediment suspension. Indirect measurements suggest that the lower, more intense, pycnocline on the ebb acted as a barrier, limiting turbulent length scales and reducing eddy diffusivity well below the pycnocline, even though the lower water column was locally well mixed. In order to more conclusively link changes in stratification to properties of near-bed eddy viscosity and diffusivity, longer duration tripod and mooring data from an additional experiment are examined, that included direct measurement of turbulent velocities. These additional data demonstrate how slight increases in stratification can limit vertical mixing near the bed and impact the structure of the eddy viscosity below the pycnocline. We present evidence that the overlying pycnocline can remotely constrain the vertical turbulent length scale of the underlying flow, limiting sediment resuspension. As a result, the relatively small changes in stratification caused by tidal straining of the pycnocline allow sediment to be resuspended higher in the water column during the flood phase of the tide, resulting in preferential up-estuary transport of sediment.Responsible Editor: Iris Grabemann     

Modeling lateral circulation and its influence on the along-channel flow in a branched estuary

A numerical modeling study of the influence of the lateral flow on the estuarine exchange flow was conducted in the north passage of the Changjiang estuary. The lateral flows show substantial variabilities within a flood-ebb tidal cycle. The strong lateral flow occurring during flood tide is caused primarily by the unique cross-shoal flow that induces a strong northward (looking upstream) barotropic force near the surface and advects saltier water toward the northern part of the channel, resulting in a southward baroclinic force caused by the lateral density gradient. Thus, a two-layer structure of lateral flows is produced during the flood tide. The lateral flows are vigorous near the flood slack and the magnitude can exceed that of the along-channel tidal flow during that period. The strong vertical shear of the lateral flows and the salinity gradient in lateral direction generate lateral tidal straining, which are out of phase with the along-channel tidal straining. Consequently, stratification is enhanced at the early stage of the ebb tide. In contrast, strong along-channel straining is apparent during the late ebb tide. The vertical mixing disrupts the vertical density gradient, thus suppressing stratification. The impact of lateral straining on stratification during spring tide is more pronounced than that of along-channel straining during late flood and early ebb tides. The momentum balance along the estuary suggests that lateral flow can augment the residual exchange flow. The advection of lateral flows brings low-energy water from the shoal to the deep channel during the flood tide, whereas the energetic water is moved to the shoal via lateral advection during the ebb tide. The impact of lateral flow on estuarine circulation of this multiple-channel estuary is different from single-channel estuary. A model simulation by blocking the cross-shoal flow shows that the magnitudes of lateral flows and tidal straining are reduced. Moreover, the reduced lateral tidal straining results in a decrease in vertical stratification from the late flood to early ebb tides during the spring tide. By contrast, the along-channel tidal straining becomes dominant. The model results illustrate the important dynamic linkage between lateral flows and estuarine dynamics in the Changjiang estuary.     

Contemporary hydrodynamics and morphological change of a microtidal estuary: a numerical modelling study

Contemporary hydrodynamics and morphological change are examined in a shallow microtidal estuary, located on a wave-dominated coast (Port Stephens, NSW, Australia). Process-based numerical modelling is undertaken by combining modules for hydrodynamics, waves, sediment transport and bathymetry updates. Model results suggest that the complex estuarine bathymetry and geometry give rise to spatial variations in the tidal currents and a marked asymmetry between ebb and flood flows. Sediment transport paths correspond with tidal asymmetry patterns. The SE storms significantly enhance the quantities of sediment transport, while locally generated waves by the westerly strong winds also are capable of causing sediment entrainment and contribute to the delta morphological change. The wave/wind-induced currents are not uniform with flow over shoals driven in the same direction as waves/winds while a reverse flow occurring in the adjacent channel. The conceptual sediment transport model developed in this study shows flood-directed transport occurs on the flood ramp while ebb-directed net transport occurs in the tidal channels and at the estuary entrance. Accretion of the intertidal sand shoals and deepening of tidal channels, as revealed by the model, suggest that sediment-infilling becomes advanced, which may lead to an ebb-dominated estuary. It is likely that a switch from flood- to ebb-dominance occurs during the estuary evolution, and the present-day estuary acts as a sediment source rather than sediment sink to the coastal system. This is conflictive to the expectation drawn from the estuarine morphology; however, it is consistent with previous research suggesting that, in an infilling estuary, an increase in build-up of intertidal flats/shoals can eventually shift an estuary towards ebb dominance. Thus, field data are needed to validate the result presented here, and further study is required to investigate a variety of estuaries in the Australian area.     

A study of tides and currents in Cook Strait, New Zealand

Greater Cook Strait (GCS) lies between the North and the South Islands of New Zealand. Its location at the convergence of the Pacific and Indo-Australian tectonic plates leads to interesting bathymetry with an adjacent shallow shelf and deep ocean trench as well as numerous crossing faults and complex shoreline geometry. Our purpose in this study is to examine tides and currents in GCS and, in particular, identify the major forcing mechanisms for the residual currents. Toward this end, we use an unstructured-grid numerical model to reproduce the tides and currents, verify these results with observations and then use the model to separate the various forcing mechanisms. The physical forcing includes nonlinear generation from tides and tidal currents, differences in sea level between the Pacific Ocean and Tasman Sea boundaries, density variations, wind stress and river discharge into GCS. Each of these mechanisms is important in different areas.     

Lattice Boltzmann simulations of the transient shallow water flows

A two-dimensional lattice Boltzmann model (LBM) is presented for transient shallow water flows. The model is based on the shallow water equations coupled with the large eddy simulation model. In order to obtain accurate results efficiently, a multi-block lattice scheme is applied at the area where a local finer grid is needed for strong change in physical variables. The model is verified by applying to five cases with transient processes: (a) a tidal wave over steps; (b) a perturbation over a submerged hump; (c) partial dam break flow; (d) circular dam break flow; (e) interaction between a dam break surge and four square cylinders. The objectives of this study are to validate the two-dimensional LBM in transient flow simulation and provide the detailed transient processes in shallow water flows.     

Ocean tides and resonance

Tidal currents and surface amplitudes are calculated globally for the dominating diurnal and semidiurnal constituents using an established tidal model under a range of altered bathymetry. The purpose is to evaluate if the well-known amplification of the global tides during the Last Glacial Maximum (LGM) is related to changed propagation properties for the tidal wave or to changed damping due to removal of shelf seas. The response of the tides and tidal dissipation to future sea-level rise is also discussed. The tides in the present and LGM oceans were simulated first, followed by runs where the present day bathymetry was used but the shelf seas removed by the introduction of vertical walls or where sea level is allowed to rise. Previously reported results regarding tidal amplitudes and dissipation rates are reproduced in the control runs. The runs without shelf seas show significantly enhanced tidal amplitudes in the North Atlantic, whereas sea-level rise of 5 m above present levels show a significant shift in the amphidromic points on a local and regional scale but had a limited effect on the open ocean tides. Simulations with very large sea-level rise show a significantly decreased global tidal dissipation, whereas experiments without friction in present-day shallow water display results similar to those with no shelf seas. The results all point towards changing damping properties due to the removal of shelf seas as being the mechanism behind the LGM amplification, and they imply the importance of implementing future sea-level changes properly in tidal simulations.     

Numerical study of circulation and temperature-salinity distributions in the Bras d’Or Lakes

The Bras d’Or Lakes (BdOL) are a large, complex and virtually land-locked estuary in central Cape Breton Island of Nova Scotia and one of Canada’s charismatic ecosystems, sustaining ecological and cultural communities unique in many aspects. The BdOL comprise two major basins, many deep and shallow bays, several narrow channels and straits and a large, geologically complex watershed. Predictive knowledge of the water movement within the estuary is a key requirement for effective management and sustainable development of the BdOL ecosystem. A three-dimensional (3D) primitive-equation ocean circulation model is used to examine the estuary’s response to tides, winds and buoyancy forcing associated with freshwater runoff in a series of numerical experiments validated with empirical data. The model results generate intense, jet-like tidal flows of about 1 m s^?1 in the channels between the basins and connecting them to the ocean and relatively weak tidal currents in other regions, which agrees well with previous observations and numerical results. Wind forcing and buoyancy forcing associated with river runoff play important roles in generating the significant sub-tidal circulations in the estuary, including narrow channels, deep basins and shallow bays. The circulation model is also used to reconstruct the 3D circulation and temperature-salinity distributions in the summer months of 1974, when current and hydrographic measurements were made at several locations. The sub-tidal circulation in the estuary produced by the model is characterised by wind and barometric set-up and set-down in different sections of the system, and a classic two-layer estuarine circulation in which brackish, near-surface waters flow seaward from the estuary into the Atlantic Ocean, and deep salty waters flow landward through the major channel. The model results reproduce reasonably well the overall features of observed circulation and temperature-salinity fields made in the BdOL in 1974 but generally underestimate the observed currents and density stratification. The model discrepancies reflect the use of spatially mean wind forcing and spatially and monthly mean surface heat flux and the inability of the coarse model horizontal resolution (～500 m) to resolve narrow channels and straits.     

Flow structure at a trifurcation near a North Florida inlet

Barrier island estuarine systems are common along the East and Gulf coasts of Florida. While some information regarding these systems is available in report form, detailed observational studies of their hydrodynamic properties are scarce in existing literature. Hydrography and current velocity were observed at a tidally driven coastline trifurcation, adjacent to the St. Augustine Inlet, Florida, in the Guana–Tolomato–Matanzas Estuary. Data were collected over nearly a semidiurnal period on February 2, 2006. The domain is well mixed and convergence fronts appear aligned with bathymetry. Eighty-six percent of the tidal variability in the study area is explained by the semidiurnal harmonic, which propagates through the system as a quasi-standing wave. The mean flow structure at the inlet (inflow in channel and outflow over shoals) governs intra-estuarine communication and is consistent with theoretical residual flows produced by a standing tidal wave. The governing force balance is between advective acceleration and the barotropic pressure gradient. The mean flow structure across the inlet might be explained by both Li and O’Donnell's [2005. The effect of channel length on the residual circulation in tidally dominated channels. Journal of Physical Oceanography 35, 1826–1840] analytical model, and Stommel and Farmer's [1952. On the nature of estuarine circulation. Woods Hole Oceanographic Institute, Woods Hole, Massachusetts, Ref. 52–51, 52–63, 52–88] source–sink analog. Flow characteristics for St. Augustine Inlet are compared with Beaufort Inlet, North Carolina; North Inlet, South Carolina; and Sand Shoal Inlet, Virginia. While these systems share similar characteristics, a common subtidal flow structure is not evident.     

Response of large-scale coastal basins to wind forcing: influence of topography

Because wind is one of the main forcings in storm surge, we present an idealised process-based model to study the influence of topographic variations on the frequency response of large-scale coastal basins subject to time-periodic wind forcing. Coastal basins are represented by a semi-enclosed rectangular inner region forced by wind. It is connected to an outer region (represented as an infinitely long channel) without wind forcing, which allows waves to freely propagate outward. The model solves the three-dimensional linearised shallow water equations on the f plane, forced by a spatially uniform wind field that has an arbitrary angle with respect to the along-basin direction. Turbulence is represented using a spatially uniform vertical eddy viscosity, combined with a partial slip condition at the bed. The surface elevation amplitudes, and hence the vertical profiles of the velocity, are obtained using the finite element method (FEM), extended to account for the connection to the outer region. The results are then evaluated in terms of the elevation amplitude averaged over the basin’s landward end, as a function of the wind forcing frequency. In general, the results point out that adding topographic elements in the inner region (such as a topographic step, a linearly sloping bed or a parabolic cross-basin profile), causes the resonance peaks to shift in the frequency domain, through their effect on local wave speed. The Coriolis effect causes the resonance peaks associated with cross-basin modes (which without rotation only appear in the response to cross-basin wind) to emerge also in the response to along-basin wind and vice versa.     

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.     

Long-term model of planimetric and bathymetric evolution of a tidal lagoon

This model is based on the concept of transport concentration, defined as the time-averaged concentration in a given location of a lagoon, which determines the long-term net transport of sediments as the sum of a dispersive and an advective flux. Dispersive net flux of sediments is due to the alternate components of the tidal flow, while the advective net flux of sediments is due to the residual (Eulerian) component of the tidal, fluvial and littoral flow and possibly to the asymmetry between flow and ebb tide.     

On the interaction between a radiatively damped planetary wave and the zonally averaged circulation in the middle atmosphere

The interaction between a planetary wave damped by cooling to space and the zonally averaged circulation in the middle atmosphere is examined for a steady-state situation in middle latitudes. Quasi-geostrophic scaling of Type 2 is assumed (i.e. the space scales are planetary and the superrotation is small).A set of mean equations is derived for this scaling which is complementary to the set of perturbation equations previously studied. The mean equations show that a planetary wave induces a mean meridional circulation which is balanced by an eddy momentum forcing function and a mean diabatic heating which is balanced by an eddy heat flux forcing function. The vertical eddy fluxes enter the forcing at the same order as the horizontal eddy fluxes.An analytical wave solution is found for the case of an atmosphere in constant superrotation. The eddy fluxes and forcing functions are evaluated for this special case. It is found that they are very sensitive to the values of the radiative damping coefficient and the superrotation. Since the damping coefficient depends on the ozone concentration and the intensity of the solar ultraviolet flux, the results suggest that changes in these quantities can strongly modify the wave-mean flow interaction in the middle atmosphere. Possible implications for climate change are discussed.     

Behaviour of a hydrodynamic finite element model

A finite element model which solves the vertically integrated momentum and continuity equations is described. Linear triangular elements are used to describe the geometry and parameter variations. The Galerkin method of weighted residuals is employed to cast the equations in a form amenable to numerical solution. The model is based on a fully implicit formulation using finite differences for the temporal derivatives.Means of evaluating the non-linear terms of the governing equations are described, and model results are presented for a frictionless tidal channel. The example is chosen such that the non-linearities have a large influence on the solution, and as a result the linearization scheme significantly affects the model's behaviour.Suppression of the non-linear instabilities generated by the convective terms in the momentum equations is examined for the case of flow around a 180° bend. Both the imposition of artificially high roughness coefficients and the use of an effective eddy viscosity are examined in terms of their ability to damp the oscillations which arise for this example.Finally, model results are presented for a case study involving determination of remedial measures to improve flow conditions at a river outfall in Southern Ontario.     

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).