Tidal-induced buoyancy flux and mean transverse circulation

A weakly nonlinear model is used to examine the mean transverse circulation (cross-isobath) driven by tidal-induced buoyancy flux. The mean Eulerian flows driven by both the barotropic and baroclinic tide are presented for a semi-infinite wedge. The mean flow driven by the barotropic tide is significant only near the apex where the thickness of the frictional boundary layer is comparable to the water depth. The mean flow there is characterized by a single-cell circulation with offshore flow near the bottom, and its magnitude can reach a few percentage or a significant fraction of the tidal velocity in oceanic applications. The mean flow driven by the baroclinic tide, on the other hand, is characterized by pairs of half-open (on the seaward side) counter-rotating cells, the number of which equals the vertical mode number. For a baroclinic tide propagating onshore, the mean flow near the top and bottom surfaces is always directed offshore and its magnitude can reach a large fraction of the tidal velocity. Taken together, the model thus predicts a mean offshore flow near the bottom while higher up in the water column the mean flow direction is less definite due to the contribution from different tidal components. The model results are consistent with some current measurements over the Georges Bank.     

Topographically generated steady currents in barotropic turbulence

Abstract

Steady currents develop in oceanic turbulence above topography even in the absence of steady forcing. Mesoscale steady currents are correlated with mesoscale topography with anticyclonic eddies above topographic bumps, and large scale westward flows develop when β is non-zero. The relationship between those two kinds of steady currents, as well as their dependence on various parameters, is studied using a barotropic quasi-geostrophic channel model. The percentage of steady energy is found to depend on the forcing, friction and topography in a non-monotonic fashion. For example, the percentage of steady currents grows with the energy level in the linear regime (low energies) and decreases when the energy level increases in the nonlinear regime (high energies). Mesoscale steady currents are the energy source for the steady westward flow U, and therefore U is the maximum when large scale and mesoscale currents are of the same order of magnitude. This happens when the ratio S of the large scale slope βH/f ₀ and the mesoscale rms topographic slope α is of order one. U decreases for both small and large values of S.     

The effects of alongshore variation in bottom topography on a boundary current—(topographically induced upwelling)

A theory which describes the constant f-plane flow of a steady inviscid baroclinic boundary current over a continental margin with a bathymetry that varies slowly in the alongshore but rapidly in the offshore directions is developed in the parameter regime (L_D/L)² ≤ Ro 1, where L_D is the internal deformation radius, L the horizontal length scale, and Ro the Rossby number. To lowest order in the Rossby number the flow is along isobaths with speed q_o = V_u(h,z)|Vh|/α, where V_u(h,z) is the upstream speed, α the upstream bottom slope at depth h, and Vh the bottom slope downstream at depth h. The lowest order flow produces a variation in the vertical component of relative vorticity along the isobath as the magnitude and direction of Vh vary in the downstream direction. The variation of vorticity requires a vertical as well as a cross-isobath flow at first order in the Rossby number. The first order vertical velocity is computed from the vorticity equation in terms of upstream conditions and downstream variations of the bathymetry. The density, pressure, and cross-isobath flow at first order in the Rossby number are then calculated. It is shown that in the cyclonic region of current (d/dh(V_u/α) > 0), if the isobaths diverge in the downstream direction ((∂/∂s)|Vh| < 0), then upwelling and onshore flow occur. The theory is applied to the northeastern Florida shelf to explain bottom temperature observations.     

Vertical structure of tidal current in a typically coastal raft-culture area

In this paper vertical structure of tidal current in a typically coastal raft-culture area is discussed by field measurement and a numerical model. The observations show that the vertical structure changed dramatically. A tidal surface boundary layer (SBL) is well formed due to the frictional effects induced by extensive, high-density suspended culture as surface obstruction. Both the aquaculture drag and the bottom friction are much higher than those in non-raft-culture areas, and show an obvious variation with tidal flow. The significant earlier ebbing and earlier flooding appear in the upper water column instead of the seabed. And the maximal phase lag is about 1 h within one tide cycle. A 1D hydrodynamic model was modified to include the SBL and parameterized with the field data. It replicated the observed velocity profile and was then used to investigate the impacts of varying culture density and bottom friction on the vertical tidal-current structure. Modeling results indicate that the surface current velocity was largely damped because culture activities enhanced the frictional effects on flow intensively. The magnitude and vertical structure of tidal current are determined together with aquaculture drag and bottom friction. In addition, the vertical velocity structure has a nonlinear trend along with culture density and bottom friction. This study is a theoretical foundation for optimizing aquaculture configuration through regulating culture density and species distribution.     

Vertical structure of residual slope circulation driven by JEBAR and tides: an idealised model

The residual circulation over the continental slope, and in particular, its vertical structure, is analysed by means of an idealised hydrodynamic model. The model is based on the depth-dependent shallow-water equations under uniform along-isobath conditions and is forced by a prescribed meridional density gradient and tidal velocities. By means of expansion in the small Rossby number solutions are analysed for conditions representative for the continental slopes off the Hebrides and in the Bay of Biscay. The steady solution at zeroth order consists of a linear density-driven flow. At order a tidally rectified flow is found and a stationary flow due to self-interaction of the zeroth-order density-driven flow. At order ² the leading-order effect of the interaction between the zeroth-order density-driven flow and the tides is found: the ‘interaction current’. The solutions up to and including order ² constitute an along-isobath steady slope current which is comparable to field data. The slope current and the accompanying cross-shelf circulation depend strongly on the shelf and flow characteristics. For the Hebridean case the density forcing predominates, but for the Biscay case the tidal effects are of the same order of magnitude as the density effects. Under those conditions the interaction current is significant which implies that linear superposition of density and tidal effects differs from the non-linear combination of both. It is also shown that the depth-average of the interaction current differs essentially from the solution obtained from a depth-averaged model.     

Tides and mixing in the northwestern East China Sea Part I: Rotating and reversing tidal flows

Bottom-mounted ADV and ADCP instruments in combination with CTD profiling measurements taken along the Chinese coast of the East China Sea were used to study the vertical structure of temperature, salinity, and velocity in reversing tidal currents on a shallow inner shelf and in rotating tidal flows over a deeper sloping bottom of the outer shelf. These two regimes of barotropic tide affect small-scale dynamics in the lower part of the water column differently. The reversing flow was superimposed by seiches of ∼2.3 h period generated in semienclosed Jiaozhou Bay located nearby. As the tidal vector rotates over the sloping bottom, the height of the near-bottom logarithmic layer is subjected to tidal-induced variations. A maximum of horizontal velocity U_max appears at the upper boundary of the log layer during the first half of the current vector rotation from the minor to the major axis of tidal ellipse. In rotating tidal flow, vertical shear generated at the seafloor, propagated slowly to the water interior up to the height of U_max, with a phase speed of ∼5 m/h. The time-shifted shear inside the water column, relative to the shear at the bottom, was associated with periodically changing increases and decreases of the tidal velocity above the log layer toward the sea surface. In reversing flows, the shear generated near the bottom and the shear at the upper levels were almost in phase.     

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

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

Seasonal along-isobath geostrophic flows on the west Florida shelf with application to Karenia brevis red tide blooms in Florida's Big Bend

TOPEX/Poseidon/Jason1 (T/P/J) sea surface height (SSH) measurements along tracks 91 and 15, crossing the wide West Florida continental shelf (WFS), were used to estimate seasonal across-shelf SSH gradients. SSH gradients and the knowledge that geostrophic flow approximately follows the isobaths enable estimation of the seasonal along-isobath geostrophic flows. The calculated along-isobath geostrophic flows are southeastward from December to March and northwestward in June, August, and September. The along-isobath geostrophic component of the flow is most likely small during the remaining months and, thus, not discernable in T/P/J SSH measurements. In agreement with previous theoretical, modeling, and observational work, the mid-shelf seasonal surface flow appears to be driven largely by the seasonal along-shore wind stress. Theory for flow driven by seasonal heat flux suggests negligible flow near the surface and on the bulk of the shelf away from the shelf break.     

Tidally generated island wakes and surface water cooling over Izu Ridge

The Kuroshio current is well known for generating cold wakes behind islands over Izu Ridge in Northwestern Pacific. Observational data from the geostationary Himawari-8 satellite for 2015–2017 revealed the occurrence of cold waters during the period when the Kuroshio current flows away from the islands. With a focus on tidal currents, this study presents an investigation of dynamical processes responsible for the formation of areas with low sea surface temperature (SST) through the adoption of a high-resolution numerical ocean model for an event that happened in July 2017. Areas with cold water emerged only when tidal currents are included in the numerical model. The model results indicate the cold surface waters are formed in the vicinity of the islands because of upwelling and vertical mixing. Qualitative features of the cold water formation for each island are found to depend on its size, topography, and ambient currents. Near Kozu Island, the tidal excursion is large enough to cause eddy shedding. These shed eddies are stirred by tidal currents to extend the surface cooling effect to wider areas. Near Hachijo Island, a persistent wake is formed by the ambient northward current. Inclusion of tidal currents destabilizes the wake, and consequently leads to the formation of a low SST area, although no clear eddy shedding is detected. The flow patterns around the islands are classified using an additional non-dimensional parameter, defined as the ratio between tidal excursion and island diameter.

     

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

Abstract

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

Experiments with a numerical model of coastal currents and tidal mixing fronts

A three-dimensional numerical model is used to simulate the development of disturbances on shelf-sea coastal currents and fronts. The model, which has a free surface, uses a finite difference grid ☐ scheme based on sigma coordinates. It has a semi-implicit scheme for the barotropic flow and a hydrid advection scheme to retain sharp fronts. The results demonstrate that (i) eddy formation follows changes at the inflow of a coastal current, (ii) a simple radiation boundary condition at the outflow produces nearly identical results for different outflow boundary positions, (iii) eddy growth, with matching behaviour of surface and bottom fronts, follows a small displacement on a tidal mixing front and (iv) effects of friction and mixing can significantly alter the behaviour of the front and the relative strength of the cyclonic and anticyclonic eddies formed.     

A large class of non-zonal oceanic flows satisfying the arnold-blumen sufficient condition for stability

Abstract

Recently Andrews has discussed an example of a topographically-forced non-zonal now satisfying the Arnold-Blumen sufficient condition for stability. At large distances from the topographic centre this flow becomes purely zonal and westward. After underlining the richness of solutions of the Andrews model, the present paper goes on to show that Andrews' technique can be applied successfully to a model where the vertical profile of static stability resembles those found in the ocean. In particular we obtain a large class of hydrodynamic stable flows, forced by the bottom topography, for continuously stratified fluids (two layers each with uniform Brunt-Väisälä frequency).     

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.     

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     

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 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.     

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

Abstract

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

Flow patterns at the Chesapeake Bay entrance

The spatial and temporal variability of water entering and leaving the Chesapeake Bay estuary was determined with a spatial resolution of 75 m. The four cruises during which the observations were made took place under different conditions of freshwater discharge, tidal phase, and wind forcing. The tidal variability of the flows was dominated by the semidiurnal constituents that displayed greatest amplitudes and phase lags near the surface and in the channels that lie at the north and south sides of the entrance. The subtidal variability of the flows was classified into two general scenarios. The first scenario occurred during variable or persistently non-southwesterly winds. Under these conditions there was surface outflow and bottom inflow in the two channels, inflow over the shoal between the two channels, and possible anticyclonic gyre formation over the shoal. The flow pattern in the channels was produced by gravitational circulation and wind forcing. Over the shoal it was caused by tidal rectification and wind forcing. The second scenario occurred during persistently southwesterly winds. The anticyclonic gyre over the shoal vanished suggesting that wind forcing dominated the tidal rectification mechanism over the shoal, while gravitational circulation and wind forcing continued to cause the flows in the channels. In both scenarios, most of the volume exchange took place in the channels.     

Improved bounds on linear instability of barotropic zonal flow within the shallow water equations

Here we develop mathematical results to describe the location of linear instability of a parallel mean flow within the framework of the shallow water equations; growth estimates of near neutral modes (for disturbances subcritical with respect to gravity wave speed) in the cases of non-rotating and rotating shallow water. The bottom topography is taken to be one-dimensional and the isobaths are parallel to the mean flow. In the case of a rotating fluid, the isobaths and the mean flow are assumed to be zonal. The flow is front-like: there is a monotonic increase of mean flow velocity. Our results show that for barotropic flows the location of instabilities will be a semi-ellipse region in the complex wave velocity plane, that is based on the wave-number, Froude number, and depth of the fluid layer. We also explore the instability region for the case of spatially unbounded mean velocity profiles for non-rotating shallow water.     

The impact of the spatial variability in bottom roughness on tidal dynamics and energetics,a case study: the M 2 surface tide in the North European Basin

A modified version of the 3D finite-element hydrostatic model QUODDY-4 is used to quantify the changes in the dynamics and energetics of the M ₂ surface tide in the North European Basin, induced by the spatial variability in bottom roughness. This version differs from the original one, as it introduces a module providing evaluation of the drag coefficient in the bottom boundary layer (BBL) and by accounting for the equilibrium tide. The drag coefficient is found from the resistance laws for an oscillatory rotating turbulent BBL over hydrodynamically rough and incompletely rough underlying surfaces, describing how the wave friction factor as well as other resistance characteristics depend on the dimensionless similarity parameters for the BBL. It is shown that the influence of the spatial variability in bottom roughness is responsible for some specific changes in the tidal amplitudes, phases, and the maximum tidal velocities. These changes are within the model noise, while the changes in the averaged (over a tidal cycle) horizontal wave transport and the averaged dissipation of barotropic tidal energy may be of the same orders of magnitude as are the above energetic characteristics as such. Thus, contrary to present views, ignoring the spatial variability in bottom roughness at least in the North European Basin is only partially correct: it is valid for the tidal dynamics, but is liable to break down for the tidal energetics.