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.     

The vertical structure of tidal currents and other oscillatory flows

Assuming a linearized equation of motion in the absence of stratification, the vertical structure of tidal currents is shown to be a function of two dimensionless parameters. The first of these, Y = (ω/E)1/2D, is analogous to an Ekman height with ω the tidal frequency, D the depth, and E the (constant) vertical eddy viscosity. The second parameter, J = (ωE)1/2[(8/3π)kŪ], introduces the effect of a quadratic bed stress through the bed-stress coefficient k and the depth-averaged velocityÜ. From these it is possible to illustrate the full range of possible vertical structure and to understand the basic scaling laws involved.By assuming E =aU¯D good agreement between theory and observation was found. With this assumption vertical structure reduces to a function of just one parameter, namely kS, where S =Ū2π/Dω is the Strouhal number.By resolving tidal current ellipses into clockwise and anticlockwise rotating components the original theory developed for recti-linear flow can be applied to fully three-dimensional flow. In this way, many of the observed characteristics of current structure in shallow seas may be explained.     

Effect of water depth and the bottom boundary layer upon internal wave generation over abrupt topography

The role of water depth and bottom boundary layer turbulence upon lee-wave generation in sill regions is examined. Their effect upon vertical mixing is also considered. Calculations are performed using a non-hydrostatic model in cross-section form with a specified tidal forcing. Initial calculations in deeper water and a sill height such that the sill top is well removed from the surrounding bed region showed that downstream lee-wave generation and associated mixing increased as bottom friction coefficient k increased. This was associated with an increase in current shear across the sill. However, for a given k, increasing vertical eddy viscosity A _v reduced vertical shear in the across sill velocity, leading to a reduction in lee-wave amplitude and associated mixing. Subsequent calculations using shallower water showed that for a given k and A _v, lee-wave generation was reduced due to the shallower water depth and changes in the bottom boundary layer. However, in this case (unlike in the deepwater case), there is an appreciable bottom current. This gives rise to bottom mixing which in shallow water extends to mid-depth and enhances the mid-water mixing that is found on the lee side of the sill. Final calculations with deeper water but small sill height showed that lee waves could propagate over the sill, thereby reducing their contribution to mixing. In this case, bottom mixing was the major source of mixing which was mainly confined to the near bed region, with little mid-water mixing.     

Tide-induced vertical suspended sediment concentration profiles: phase lag and amplitude attenuation

In tidal environments, the response of suspended sediment concentration (SSC) to the current velocity is not instantaneous, the SSC lagging behind the velocity (phase lag), and the amplitude of SSC variation decreasing with height above the bed (amplitude attenuation). In order to quantitatively describe this phenomenon, a one-dimensional vertical advection–diffusion equation of SSC is derived analytically for uniform unsteady tidal flow by defining a concentration boundary condition using a constant vertical eddy diffusivity and sediment settling velocity. The solution, in simple and straightforward terms, shows that the vertical phase lag increases linearly with the height above the bed, while the amplitude of the SSC variation decreases exponentially with the height. The relationship between the SSC and the normalized current velocity can be represented by an ellipse or a line, depending on the phase lag. The lag of sediment movement or “diffusion/settling lag” is the mechanism generating the phase lag effect. Field observations used for validation show that the theoretically predicted and the observed curves of the vertical SSC phase lag and amplitude attenuation show reasonable agreement. The procedure proposed in this paper substantially simplifies the modeling of suspended matter transport in tidal flows.     

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.     

Dynamic links between shape of the eddy viscosity profile and the vertical structure of tidal current amplitude in bays and estuaries

Several field studies in bays and estuaries have revealed pronounced subsurface maxima in the vertical profiles of the current amplitude of the principal tidal harmonic, or of its vertical shear, over the water column. To gain fundamental understanding about these phenomena, a semi-analytical model is designed and analysed, with focus on the sensitivity of the vertical structure of the tidal current amplitude to formulations of the vertical shape of the eddy viscosity. The new analytical solutions for the tidal current amplitude are used to explore their dependence on the degree of surface mixing, the vertical shape of eddy viscosity in the upper part of the water column and the density stratification. Sources of surface mixing are wind and whitecapping. Results show three types of current amplitude profiles of tidal harmonics, characterised by monotonically decreasing shear towards the surface, “surface jumps” (vertical shear of tidal current amplitude has a subsurface maximum) and “subsurface jets” (maximum tidal current amplitude below the surface), respectively. The “surface jumps” and “subsurface jets” both occur for low turbulence near the surface, whilst additionally the surface jumps only occur if the eddy viscosity in the upper part of the water column decreases faster than linearly to the surface. Furthermore, “surface jumps” take place for low density stratification, while and “subsurface jets” occur for high density stratification. The physics causing the presence of surface jumps and subsurface jets is also discussed.     

Comparison of a simple planetary boundary layer model with measurements of a turbulent boundary layer under pack ice

A simple analytical theory which describes the motion in a turbulent planetary boundary layer near a rough sea bed by using a two-layer eddy viscosity model is presented. The vertical structure of the current in the boundary layer is presented, and comparisons are made with data fromMcPhee and Smith (1976, Journal of Physical Oceanography,6, 696–711) obtained from measurements of the turbulent boundary layer under drifting Arctic ice.     

Residual circulation and stratification in the Liverpool Bay region of freshwater influence

Wind and tidal straining are proposed as key mechanisms influencing the magnitude and timing of the horizontal flux of freshwater across regions of freshwater influence (ROFIs). Evidence for this hypothesis is presented in estimates of the tidally averaged residual current profile, obtained from 5 years of continuous acoustic doppler current profiler measurements in the Liverpool Bay ROFI. The modified horizontal Richardson number (R_x^wtR_{x}^{wt}), which includes both the tidal and the wind forcing, was assessed as a measure of stratification. R_x^wtR_{x}^{wt} was found to be a good indicator of the timing of the evolution and destruction of stratification, but was not as successful as an indicator of the magnitude of stratification, both enduring and periodic. The observed mean residual velocities are compared to those predicted by a classical solution, and the eddy viscosity (N _z ) is shown to be a control on differences between the observed and predicted circulation. Principal component analysis is used to show that the strongest residual currents occur when the water column periodically alternates between a well-mixed and stratified state, a consequence of straining, rather than simply related to the density gradient. Evidence of wind straining was found in the correspondence between the wind direction and the near surface and near bed residual current direction.     

On the Mechanism of the Secular Tidal Acceleration of the Solid Inner Core and the Viscosity of the Liquid Core

As is known, the secular deceleration of the Earth's diurnal rotation is explained mainly by the tidal friction in the ocean. Below we consider this mechanism in some detail, taking into account also elastic deformations of the mantle under the action of ocean loading and the interaction between the tide-generating body, ocean tidal wave, liquid outer core, and solid inner core. It is shown that elastic displacements of the core-mantle boundary under the action of ocean loading are of about the same amplitude and phase as the elastic loading displacements of the Earth's outer surface. As a result, side by side with the mechanism of secular deceleration of diurnal rotation of the mantle, there are also (1) the opposite mechanism of secular acceleration of diurnal rotation of the outer liquid core and of the solid inner core and (2) the mechanism of excitation of differential rotation in the liquid core. Taking these effects into account, we compare theoretical and modern observed data on the eastward drift of the solid inner core. It is shown that the best agreement may be obtained if the turbulent viscosity of the liquid core is about 2 × 10 ³ Poise     

A model of the wave and current boundary‐layer structure

A model of the wave and current boundary‐layer structure was developed using the k–ε turbulent closure model. The finite‐difference method was used to solve the governing equations. Vertical logarithmic grids and equal time steps were adopted. The following modelled simulations were obtained: (1) vertical profiles of wave velocity amplitude, eddy viscosity coefficient and turbulent kinetic energy with waves only; (2) vertical profiles of wave velocity amplitude, mean current velocity, eddy viscosity coefficient and turbulent kinetic energy with waves having a following current. To test the validity and the rationality of the present model, vertical profiles of modelled wave velocity amplitude and mean velocity were compared with corresponding experimental results available in the literature. Copyright © 2006 John Wiley & Sons, Ltd.     

Non-linear channel–shoal dynamics in long tidal embayments

The dynamics of finite-amplitude bed forms in a tidal channel is studied with the use of an idealized morphodynamic model. The latter is based on depth-averaged equations for the tidal flow over a sandy bottom. The model considers phenomena on spatial scales of the order of the tidal excursion length. Transport of sediment mainly takes place as suspended load. The reference state of this model is characterized by a spatially uniform M₂ tidal current over a fixed horizontal bed. The temporal evolution of deviations from this reference state is governed by amplitude equations: these are a set of non-linear equations that describe the temporal evolution of bed forms. These equations are used to obtain new morphodynamic equilibria which may be either static or time-periodic. Several of these bottom profiles show strong similarity with the tidal bars that are observed in natural estuaries. The dependence of the equilibrium solutions on the value of bottom friction and channel width is investigated systematically. For narrow channels (width small compared to the tidal excursion length) stable static equilibria exist if bottom friction is slightly larger than r_cr. For channel widths more comparable to the tidal excursion length, multiple stable steady states may exist for bottom friction parameter values below r_cr. Regardless of channel width, stable time-periodic equilibria seem to emerge as the bottom friction is increased.Responsible Editor: Jens Kappenberg     

Simulation of the formation of Ushant thermal front

The formation of the Ushant thermal front off northwestern France is simulated numerically with a one-dimensional model of the vertical thermal structure of the sea. This simulation is derived from a Niiler and Kraus (1977, Modelling and prediction of the upper layers of the ocean, pp. 143–172) mixed layer model in which bottom friction is introduced. The model inputs are the meteorological parameters measured at Ushant and the barotropic tidal currents computed with a numerical model (Mariette et al., 1982, Oceanologica Acta, 5, 149–159). Both very fine time steps (Δt = 1h) and spatial steps (horizontal mesh size: Δx = Δy = 2nmi; vertical step: Δz = 1m) are used. This study is intended to be mainly concerned with the processes acting during the formation of the Ushant front. Comparison of the model results with satellite pictures and in situ measurements shows good agreement. We point out that biological blooms could occur according to a spring-neap cycle, during the thermal front formation. The prevailing effect of local processes such as tidally induced bottom friction and energy budget at the sea surface on the thermal evolution of the Mer d'Iroise is then emphasized. During the formation of the front, these effects seem to be more important than the advective processes which are not considered in this work.     

A test of the influence of tidal stream polarity on the structure of turbulent dissipation

Vertical mixing by the tides plays a key role in controlling water column structure over the seasonal cycle in shelf seas. The influence of tidal stirring is generally well represented as a competition between surface buoyancy input and the production of turbulent kinetic energy (TKE) by frictional stresses, a competition which is encapsulated in the Qh/u³ criterion. An alternative control mechanism arises from the limitation of the thickness of the bottom boundary layer due to the effects of rotation and the oscillation of the flow. Model studies indicate that, for conditions typical of the European shelf seas, the energy constraint exerts the dominant control but that for tidal streams with large positive polarisation (i.e. anti-clockwise rotation of velocity vector), some influence of rotation in limiting mixing should be detectable. We report here measurements of flow structure (with ADCPs) and turbulent dissipation (FLY Profiler) made at two similar locations in the Celtic Sea which differ principally in that the tidal currents rotate in opposite senses with approximately equal magnitude (polarity P=±0.6). A clear contrast was observed between the two sites in the vertical structure of the currents, the density profile and the rate of dissipation of TKE. At the positive polarity (PP) site (P≈+0.6), the bottom boundary layer in the tidal flow was limited to ∼20 mab (metre above the bed) and significant dissipation from bottom boundary friction was constrained within this layer. At the negative polarity (NP) site (P≈−0.6), the dominant clockwise rotary current component exhibited a velocity defect (i.e. reduction relative to the free stream) extending into the upper half of the water column while significant dissipation was observed to penetrate much further up the water column with dissipation levels ∼10^−4.5 W m⁻³ reaching to the base of the pycnocline at 70–80 mab. These contrasting features of the vertical distribution of dissipation are well reproduced by a 1-D model when run with windstress and tidal forcing and using the observed density profile. Model runs with reversed polarity at the two sites, support the conclusion that the observed contrast in the structure of tidal velocity, dissipation and stratification is due to the influence of tidal stream polarity. Increased positive polarity reduces the upward penetration of mixing which allows the development of stronger seasonal stratification, which, in turn, further inhibits vertical mixing.     

On the vertical structure of the Rhine region of freshwater influence

An idealised three-dimensional numerical model of the Rhine region of fresh water influence (ROFI) was set up to explore the effect of stratification on the vertical structure of the tidal currents. Prandle’s dynamic Ekman layer model, in the case of zero-depth-averaged, cross-shore velocities, was first used to validate the response of the numerical model in the case of barotropic tidal flow. Prandle’s model predicted rectilinear tidal currents with an ellipse veering of up to 2%. The behaviour of the Rhine ROFI in response to both a neap and a spring tide was then investigated. For the given numerical specifications, the Rhine plume region was well mixed over the vertical on spring tide and stratified on neap tide. During spring conditions, rectilinear tidal surface currents were found along the Dutch coast. In contrast, during neap conditions, significant cross-shore currents and tidal straining were observed. Prandle’s model predicted ellipse veering of 50%, and was found to be a good indicator of ellipticity magnitude as a function of bulk vertical eddy viscosity. The modelled tidal ellipses showed that surface currents rotated anti-cyclonically whereas bottom currents rotated cyclonically. This caused a semi-diurnal cross-shore velocity shearing which was 90° out of phase with the alongshore currents. This cross-shore shear subsequently acted on the horizontal density gradient in the plume, thereby causing a semi-diurnal stratification pattern, with maximum stratification around high water. The same behaviour was exhibited in simulations of a complete spring–neap tidal cycle. This showed a pattern of recurring stratification on neaps and de-stratification on springs, in accordance with observations collected from field campaigns in the 1990’s. To understand the increase in ellipticities to 30% during neaps and the precise shape of the vertical ellipse structure, stratification has to be taken into account. Here, a full three-dimensional numerical model was employed, and was found to represent the effect of de-coupling of the upper and lower layers due to a reduction of mixing at the pycnocline.     

A Note on the Estimation of Eddy Diffusivity and Dissipation Length in Low Winds over a Tropical Urban Terrain

— Urban terrain poses a challenge for modeling air pollutant diffusion. In tropics, because of the dominant low wind speed environment, the importance of understanding the turbulence diffusion is even more critical, and uncertain. The objective of this study is to estimate the vertical eddy diffusivity of an urban, tropical atmosphere in low–wind speeds. Turbulence measurements at 1 Hz were made at 4-m level over an urban terrain with a roughness length of 0.78 m during winter months. Eddy diffusivity is estimated from spectral quantities of the turbulence data involving turbulent kinetic energy (E) and its dissipation rate (?). The spectral information of the vertical velocity fluctuations is used to estimate the vertical length scale which provides information on the eddy diffusivity. In addition, the product of friction velocity and the vertical length scale has been used to non-dimensionalize the eddy diffusivity, which is shown to increase with increasing instability. Using the eddy diffusivity (K) estimates from the E – ? approach, a relation is suggested for the mixing length based eddy diffusivity models of the form: K = c _w .[2.5 ? 0.5(z/L)], where z is the measurement height, L is the Obukhov length, and c _w has an average value close to 1 for unstable and near 0.5 for stable conditions for the urban terrains.     

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     

Development of a Lagrangian sediment model to reproduce the bathymetric evolution of the Mersey Estuary

The development and application of a Lagrangian particle-tracking model to simulate sediment transport in the Mersey Estuary (UK) is described. Each of the particles (up to a million in total) is advected horizontally by the 3-D tidal currents. Related vertical movements are: (1) downwards by settlement at a prescribed velocity w _s and (2) both upwards and downwards by a distance related to the vertical eddy diffusivity. In a novel departure from traditional practice, where initial distributions of surficial sediments are specified, all particles are introduced at the seaward boundary of the model. Provenance studies indicate surficial sediments are overwhelmingly of marine origin. For the predominant fine sediments, ‘cyclical convergence’ in suspended sediment concentrations is approximated after about two spring-neap tidal cycles. Comparisons are shown between the suspended sediment concentrations and net deposition rates computed by this model against observed values and earlier computations utilising both 1-D and 3-D Eulerian models. While all of these results are in broad agreement, the flexibility of the Lagrangian approach for simulating flocculation, consolidation and mixed sediments illustrates its future potential.     

Surface current variability in the Keum River Estuary (South Korea) during summer 2002 as observed by high-frequency radar and coastal monitoring buoy

High-frequency (HF) radar observations of surface currents were conducted for 3 months during summer 2002 in the Keum River estuary. A comparison between HF radar-derived currents and directly measured ones form a buoy showed that the regression slope is close to 1 and the correlation coefficient greater than 0.86, with an RMS difference less than 13 cm/s which is less than 17% of the tidal current. This fairly good agreement allows us to use HF radar observation in investigating the surface flow and circulation in this tidal-current-dominant coastal-plume area. To examine the spatial variation in tidal current characteristics, as well as currents associated with non-tidal forcing, the HF radar-derived currents were separated into tidal and sub-tidal frequency currents. The overall pattern of M₂-current ellipse distribution in the study area showed a counterclockwise rotation, with the offshore maximum current direction to the northeast. Eccentricity, the direction of maximum current, and the phase of net motion of the ellipse changed near the estuary mouth and near the gap of the Saemangeum reclamation tide dyke due to the complex coastal geometry and the out-flowing jet during the ebb period.     

A comparison of biohydrodynamic interaction within mangrove and saltmarsh boundaries

Mangrove forests and saltmarshes are recognized for their roles in wave and current attenuation, although a comparison of in situ observations between woody and herbaceous plants is needed in order to understand the different mechanisms of bio‐physical interaction within coastal wetlands. The aim of our study was to compare the mechanisms of flow reduction and energy dissipation by mangrove trees and saltmarsh grass in a subtropical area where tidal currents dominate. Fieldwork was conducted to measure the hydrodynamic processes occurring at the boundaries between a bare mudflat and vegetated tidal flat, as the flow transitions from a bare mudflat to either mangrove or saltmarsh. Synchronous acoustic Doppler velocimetry (ADV) measurements at three sites revealed that the mangrove was more effective than the saltmarsh grass at flow reduction. In addition, a considerable rotation in flow direction was observed as the flow entered the mangrove trees, while rotation was considerably less pronounced within the saltmarsh edge. The mechanism for this difference was explained through a combination of changes in drag force and eddy viscosity over the two vegetation types. Although overall the mangrove was observed to dissipate energy more effectively than the saltmarsh, the relative efficiency of the vegetation at dissipating turbulent energy was found to vary with the maximum water level of tidal cycle. When the maximum water level remained below the mangrove canopy bottom (‘bio‐line’), the energy dissipation ability of the mangrove was relatively low, as a result of the presence of rigid, sparse trunks rather than denser saltmarsh grass found near the bed; when the maximum water level was sufficiently high to reach the mangrove canopy, the ability of the mangrove to dissipate energy was significantly increased, becoming more effective than the saltmarsh grass. Copyright © 2016 John Wiley & Sons, Ltd.