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.     

A model study of tidal distributions in the Celtic and Irish Sea regions determined with finite volume and finite element models

An unstructured mesh tidal model of the west coast of Britain, covering the Celtic Sea and Irish Sea is used to compare tidal distributions computed with finite element (FE) and finite volume (FV) models. Both models cover an identical region, use the same mesh, and have topography and tidal boundary forcing from a finite difference model that can reproduce the tides in the region. By this means, solutions from both models can be compared without any bias towards one model or another. Two-dimensional calculations show that for a given friction coefficient, there is more damping in the FV model than the FE model. As bottom friction coefficient is reduced, the two models show comparable changes in tidal distributions. In terms of mesh resolution, calculations show that for the M₂ tide, the mesh is sufficiently fine to yield an accurate solution over the whole domain. However, in terms of higher harmonics of the tide, in particular the M₆ component, its small-scale variability in near-shore regions which is comparable to the mesh of the model, suggests that the mesh resolution is insufficient in the near-coastal regions. Even with a finer mesh in these areas, without detailed bottom topography and a spatial varying friction depending on bed types and bed forms, which is not available, model skill would probably not be improved. In addition in the near-shore region, as shown in the literature, the solution is sensitive to the form of the wetting/drying algorithm used in the model. Calculations with a 3D version of the FV model show that for a given value of k, damping is reduced compared to the 2D version due to the differences in bed stress formulation, with the 3D model yielding an accurate tidal distribution over the region.     

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.     

Internal tide modelling and the influence of wind effects

Initially the development of shallow sea three-dimensional barotropic tidal models is briefly reviewed with a view to determining what were the key measurements that allowed progress in this field and rigorous model validation. Subsequently this is extended to a brief review of baroclinic tidal models to try to determine a “way forward” for baroclinic model development. The difficulty of high spatial variability, and wind influence are identified as possibly important issues that must be considered in validating baroclinic tidal models. These are examined using a three-dimensional unstructured grid model of the M₂ internal tide on the shelf edge region off the west coast of Scotland. The model is used to investigate the spatial variability of the M₂ internal tide, and associated turbulence energy and mixing in the region. Initial calculations are performed with tidal forcing only, with subsequent calculations briefly examining how the tidal distribution is modified by down-welling and up-welling favourable winds. Calculations with tidal forcing only, show that there is significant spatial variability in the internal tide and associated mixing in the region. In addition, these are influenced by wind effects which may have to be taken into account in any model validation exercise. The paper ends by discussing the comprehensive nature of data sets that need to be collected to validate internal tidal models to the same level currently attained with three dimensional barotropic tidal models.     

An idealized model of tidal dynamics in the North Sea: resonance properties and response to large-scale changes

An idealized process-based model is developed to investigate tidal dynamics in the North Sea. The model geometry consists of a sequence of different rectangular compartments of uniform depth, thus, accounting for width and depth variations in a stepwise manner. This schematization allows for a quick and transparent solution procedure. The solution, forced by incoming Kelvin waves at the open boundaries and satisfying the linear shallow water equations on the f plane with bottom friction, is in each compartment written as a superposition of eigenmodes, i.e. Kelvin and Poincaré waves. A collocation method is employed to satisfy boundary and matching conditions. First, the general resonance properties of a strongly simplified geometry with two compartments, representing the Northern North Sea and Southern Bight, are studied. Varying the forcing frequency while neglecting bottom friction reveals Kelvin and Poincaré resonance. These resonances continue to exist (but with lower amplification and a modified spatial structure) when adding the Dover Strait as a third compartment and separating the solutions due to forcing from either the north or the south only. Including bottom friction dampens the peaks. Next, comparison with tide observations along the North Sea coast shows remarkable agreement for both semi-diurnal and diurnal tides. This result is achieved with a more detailed geometry consisting of 12 compartments fitted to the coastline of the North Sea. Further simulations emphasize the importance of Dover Strait and bottom friction. Finally, it is found that a sea level rise of 1 m, uniformly applied to the entire North Sea, amplifies the M2-elevation amplitudes almost everywhere along the coast, with an increase of up to 8 cm in Dover Strait. Bed level changes of ±1 m, uniformly applied to the Southern Bight only, imply weaker changes, with changes in coastal M2-elevation amplitudes below 5 cm.     

Tidally-induced sediment transport patterns in the upper Bay of Fundy: A numerical study

The Minas Basin, the eastern end of the Bay of Fundy, is well known for its high tide ranges and strong tidal currents, which can be exploited to extract electricity power. The properties of the tidally-induced sediment transport in the Minas Basin, where significant changes in tidal processes may occur due to a recently proposed tidal power project, have been studied with a three-dimensional hydrodynamic model, an empirical bed load sediment transport model and surface sediment concentrations derived from the remotely-sensed images. The hydrodynamic model was evaluated against independent observational data, which include tidal elevation, tidal current (in the full water column and bottom layer), residual current profile and tidal asymmetry indicators. The evaluation shows that the model is in good agreement with the observations.The sediment transport includes two components, bed load and suspended particulate load. The bed load is calculated using the modelled bottom shear stress and the observed grain size data. The estimated features of bed load transport roughly agree with the observed patterns of the erosion and deposition in the Minas Basin and Cobequid Bay. The transport of the suspended load is estimated using the modelled velocity fields and the surface sediment concentration derived from remote-sensing images. The comparisons between the modelled results and the limited observations illustrate that the observed directions of suspended sediment transport are basically reproduced by the model. The modelled net suspended sediment input into the Minas Basin through Minas Passage is 2.4×10⁶ m³ yr^?1, which is comparable to the observed value of 1.6×10⁶ m³ yr^?1.The variations of the bed load and the suspended load in space and time are also presented. The total net transport, defined as the mean value of the sum of bed and suspended load transports during the tidal cycle, shows strong spatial variability. The magnitude of the transport flux ranges from 0.1 to 0.2 kg m^?1 s^?1 in Minas Channel and Minas Passage, 0.1 kg m^?1 s^?1 in Cobequid Bay, to 0.01 kg m^?1 s^?1 in the central Minas Basin and Southern Bight. In Minas Channel, the sediment transport follows the structure of the tidal residual circulation, which features a large anticlockwise gyre. The sediment in Minas Passage moves eastward and deposits into the central Minas Basin. However, the sediment from the eastern part of the Basin moves westward and deposits in the central Minas Basin as well. In the Cobequid Bay, sediment moves eastward and deposits in the upper bay.     

Local and remote response of the North Sea dynamics to morphodynamic changes in the Wadden Sea

The response of the tidal system in the southern North Sea to morphodynamic changes was investigated in a modelling study using fine resolution bathymetric observations available for 1982–2011. The Semi-implicit Cross-scale Hydroscience Integrated System Model (SCHISM) was set up for the different sets of bathymetries. One set of bathymetry was compiled from a large number of bathymetric measurements over many years, while the other two reflected bathymetry state in the area of Wadden Sea during 2000 and 2011, respectively. The temporal and spatial evolution of bathymetry was dominated by migration of tidal channels. The M4 tide showed larger sensitivity to bathymetric change in the Wadden Sea than the M2 tide, whereas the structure of the latter remained rather robust. The largest change of the tidal wave due to the differences in bathymetries was located off the North Frisian Wadden Sea. Traces of changes were also found far away from the regions of their origin because the tidal waves in the North Sea propagate the local disturbances basin-wide. This illustrated an efficient physical mechanism of teleconnectivity, i.e. effecting the local responses to the larger-scale or remote change of ocean bottom caused by erosion and deposition. The tidal distortion resulting from the relatively small bathymetric changes was substantial, particularly in the coastal zone. This is a manifestation of the nonlinear tidal transformation in shallow oceans and is crucial for the sediment transport and the morphodynamic feedback, because of the altered tidal asymmetry.     

Impact of tidal mixing with different scales of bottom roughness on the general circulation

The current study deals with a parameterization of diapycnal diffusivity in an ocean model. The parameterization estimates the diapycnal diffusivity depending on the location of tidal-related energy dissipation over rough topography. The scheme requires a bottom roughness map that can be chosen depending on the scales of topographic features. Here, we implement the parameterization on an ocean general circulation model, and we examine the sensitivity of the modeled circulations to different spatial scales of the modeled bottom roughness. We compare three simulations that include the tidal mixing scheme using bottom roughness calculated at three different ranges of spatial scales, with the largest scale varying up to 200 km. Three main results are discussed. First, the dependence of the topographic spectra with depth, characterized by an increase in spectral energy over short length scales in the deep ocean, influences the vertical profile of the diffusivity. Second, the changes in diffusivities lead to different equilibrium solutions in the Atlantic meridional overturning circulation and bottom circulation. In particular, the lower cell of the Atlantic overturning and the bottom water transport in the Pacific Ocean are stronger for stronger diffusivities at the corresponding basins and depths, and the strongest when using the small-scale roughness map. Third, a comparison of the density fields of the three simulations with the density field of World Ocean Atlas dataset, from which the models are initialized, shows that among the simulations with three different roughness maps, the one using small-scale bottom roughness map has the smallest density bias.

     

Deep circulation driven by strong vertical mixing in the Timor Basin

The importance of deep mixing in driving the deep part of the overturning circulation has been a long debated question at the global scale. Our observations provide an illustration of this process at the Timor Basin scale of ～1000 km. Long-term averaged moored velocity data at the Timor western sill suggest that a deep circulation is present in the Timor Basin. An inflow transport of ～0.15 Sv is observed between 1600 m and the bottom at 1890 m. Since the basin is closed on its eastern side below 1250 m depth, a return flow must be generated above 1600 m with a ～0.15 Sv outflow. The vertical turbulent diffusivity is inferred from a heat and transport balance at the basin scale and from Thorpe scale analysis. Basin averaged vertical diffusivity is as large as 1 × 10^?3 m² s^?1. Observations are compared with regional low-resolution numerical simulations, and the deep observed circulation is only recovered when a strong vertical diffusivity resulting from the parameterization of internal tidal mixing is considered. Furthermore, the deep vertical mixing appears to be strongly dependent on the choice of the internal tide mixing parameterization and also on the prescribed value of the mixing efficiency.     

Mechanisms of bottom boundary fluxes in a numerical model of the Shetland shelf

Across-slope bottom boundary layer (BBL) fluxes on the shelf-edge connect this region to deeper waters. Two proposed ways in which across-slope BBL fluxes can occur, in regions that have a slope current aligned to the bathymetry, are the frictional veering of bottom currents termed the ‘Ekman drain’ and through local wind-forced downwelling (wind-driven surface Ekman flow with an associated bottom flow). We investigate the variability, magnitude and spatial scale of BBL fluxes on the Shetland shelf, which has a prominent slope current, using a high-resolution (～2 km) configuration of the MITgcm model. Fluxes are analysed in the BBL at the shelf break near the 200 m isobath and are found to have a seasonal variability with high/low volume transport in winter/summer respectively. By using a multivariate regression approach, we find that the locally wind-driven Ekman transport plays no explicit role in explaining daily bottom fluxes. We can better explain the variability of the across-slope BBL flux as a linear function of the speed and across-slope component of the interior flow, corresponding to an Ekman plus mean-flow flux. We estimate that the mean-flow is a greater contributor than the Ekman flux to the BBL flux. The spatial heterogeneity of the BBL fluxes can be attributed to the mean-flow, which has a much shorter decorrelation length compared to the Ekman flux. We conclude that both the speed and direction of the interior current determines the daily BBL flux. The wind does not explicitly contribute through local downwelling, but may influence the interior current and therefore implicitly the BBL fluxes on longer timescales.     

Regional ocean tide loading modelling around the Iberian Peninsula

We developed a new 1/12° resolution oceanic tide model in the complex region that surrounds the Iberian Peninsula. The model, named IBER01, allows us to obtain more accurate tidal loading computations for precise geodetic and gravimetric observations in this area. The modelling follows the scheme of data assimilation (coastal tide gauge, bottom pressure sensors and TOPEX/Poseidon altimetry) into a hydrodynamical model, which is based on two-dimensional barotropic depth averaged shallow-water equations. Detailed bathymetry data and quadratic bottom friction with a specific drag coefficient for the region have been considered. Improved ocean load maps for the Iberian Peninsula are obtained for eight harmonic constituents (Q1, P1, O1, K1, N2, M2, S2 and K2), after computing the load effect (Newtonian attraction and elastic contribution) using IBER01 and six present-day global oceanic tide models for comparison. The results achieved verify the quality of the new model. Our ocean loading computations reduce considerably the discrepancies between the theoretical Earth tide parameters and those from observations at the level of 0.3%.     

A three-dimensional finite-element model of wind effects upon higher harmonics of the internal tide

A non-linear three-dimensional unstructured grid model of the M₂ tide in the shelf edge area off the west coast of Scotland is used to examine the spatial distribution of the M₂ internal tide and its higher harmonics in the region. In addition, the spatial variability of the tidally induced turbulent kinetic energy and associated mixing in the area are considered. Initial calculations involve only tidal forcing, although subsequent calculations are performed with up-welling and down-welling favourable winds to examine how these influence the tidal distribution (particularly the higher harmonics) and mixing in the region. Both short- and long-duration winds are used in these calculations. Tidal calculations show that there is significant small-scale spatial variability particularly in the higher harmonics of the internal tide in the region. In addition, turbulence energy and mixing exhibit appreciable spatial variability in regions of rapidly changing topography, with increased mixing occurring above seamounts. Wind effects significantly change the distribution of the M₂ internal tide and its higher harmonics, with appreciable differences found between up- and down-welling winds and long- and short-duration winds because of differences in mixing and the presence of wind-induced flows. The implications for model validation, particularly in terms of energy transfer to higher harmonics, and mixing are briefly discussed.     

Shallow water tidal currents in close proximity to the seafloor and boundary-induced turbulence

Velocity measurements with vertical resolution 0.02 m were conducted in the lowest 0.5 m of the water column using acoustic Doppler current profiler (ADCP) at a test site in the western part of the East China Sea. The friction velocity u _* and the turbulent kinetic energy dissipation rate ε _wl(ζ) profiles were calculated using log-layer fits; ζ is the height above the bottom. During a semidiurnal tidal cycle, u _* was found to vary in the range (1–7) × 10⁻³ m/s. The law-of-the-wall dissipation profiles ε _wl(ζ) were consistent with the dissipation profiles ε _mc(ζ) evaluated using independent microstructure measurements of small-scale shear, except in the presence of westward currents. It was hypothesized that an isolated bathymetric rise (25 m height at a 50-m seafloor) located to the east of the measurement site is responsible for the latter. Calculation of the depth integrated internal tide generating body force in the region showed that the flanks of the rise are hotspots of internal wave energy that may locally produce a significant turbulent zone while emitting tidal and shorter nonlinear internal waves. This distant topographic source of turbulence may enhance the microstructure-based dissipation levels ε _mc(ζ) in the bottom boundary layer (BBL) beyond the dissipation ε _wl(ζ) associated with purely locally generated turbulence by skin currents. Semi-empirical estimates for dissipation at a distance from the bathymetric rise agree well with the BBL values of ε _mc measured 15 km upslope.     

Estimating depth‐averaged velocities in rough ­channels

Profiles of streamwise velocity obtained from North Boulder Creek, Colorado, typically are non‐logarithmic in form and exhibit the strong influence of form drag associated with coarse bed roughness. The spatially averaged profile is consistent with recent theoretical profile forms suggested for rough channels that are based on a partitioning of the total stress between a fluid part and a part associated with form drag on bed particles. Estimates of local depth‐averaged velocity using algorithms that are based on several measurements in the flow column improve with explicit Riemann averaging, versus simple averaging, of the measurements. Estimates based on a single‐point measurement at 0·6 of the flow depth, assuming a logarithmic or approximately logarithmic velocity profile, are the least reliable. Copyright © 2000 John Wiley & Sons, Ltd.     

Tidal currents in the central Great Barrier Reef

Observations at 8 sites in the outer central Great Barrier Reef show M₂, S₂, K₁, and O₁ tidal currents flow directly off-shelf (northeast), when the corresponding tide at Townsville is at zero height and falling, with typical amplitudes of 12, 6, 3, and 2 cm s^?1. On the slope (at 300 m depth), the vertically averaged long-shelf component was small. On the shelf, the eccentricity of the tidal ellipses decreases shoreward and the tidal ellipses rotate anticlockwise. The major axes of the tidal ellipses tilt left of cross-shelf, especially for the diurnal constituents. There is satisfactory agreement between the observed and modelled cross-shelf currents. The long-shelf velocity is sensitive to the long-shelf changes in amplitude and phase of the tide heights and high quality tidal data for open boundary conditions will be required if numerical models are to model these currents satisfactorily.     

Computing near coastal tidal dynamics from observations and a numerical model

A two-dimensional vertically integrated model of the North Sea is used to compute the distribution of M₂ and M₄ tidal elevations and currents over the region. Comparison of computed and observed elevations and currents in the area shows that the model can accurately reproduce the M₂ tide in the North Sea, although there are difficulties with the M₄ tide particularly in the northern North Sea.Comparison between model and a large number of observations collected in a shallow water region off the east coast of England, revealed that the model can accurately reproduce the tides even in near coastal regions, where model resolution problems can occur. Comparisons of computed and observed M₂ tidal energy fluxes in this region, show that model and observations agree to within the order of 10% (the error associated with the necessary interpolation of the observations in order to compute the energy flux).The problem of computing energy dissipation in the area by subtracting the energy fluxes into and out of the region is shown to be ill-conditioned in that the energy dissipation in the area is comparable to the error in the energy flux. Consequently for the sea region considered here it is not meaningful to compare this energy budget with energy dissipation due to bottom friction.Energy dissipation for the whole of the North Sea is computed using the numerical model and the geographical distribution of dissipation due to bottom friction is given for the M₂ tide.     

Seasonal variation of the <Emphasis Type="Italic">M</Emphasis><Subscript>2</Subscript> tide

The seasonal cycle of the main lunar tidal constituent M ₂ is studied globally by an analysis of a high-resolution ocean circulation and tide model (STORMTIDE) simulation, of 19 years of satellite altimeter data, and of multiyear tide-gauge records. The barotropic seasonal tidal variability is dominant in coastal and polar regions with relative changes of the tidal amplitude of 5–10 %. A comparison with the observations shows that the ocean circulation and tide model captures the seasonal pattern of the M ₂ tide reasonably well. There are two main processes leading to the seasonal variability in the barotropic tide: First, seasonal changes in stratification on the continental shelf affect the vertical profile of eddy viscosity and, in turn, the vertical current profile. Second, the frictional effect between sea-ice and the surface ocean layer leads to seasonally varying tidal transport. We estimate from the model simulation that the M ₂ tidal energy dissipation at the sea surface varies seasonally in the Arctic (ocean regions north of 60°N) between 2 and 34 GW, whereas in the Southern Ocean, it varies between 0.5 and 2 GW. The M ₂ internal tide is mainly affected by stratification, and the induced modified phase speed of the internal waves leads to amplitude differences in the surface tide signal of 0.005–0.0150 m. The seasonal signals of the M ₂ surface tide are large compared to the accuracy demands of satellite altimetry and gravity observations and emphasize the importance to consider seasonal tidal variability in the correction processes of satellite data.     

Influence of compound bedforms on hydraulic roughness in a tidal environment

The effect exerted by the seabed morphology on the flow is commonly expressed by the hydraulic roughness, a fundamental parameter in the understanding and simulation of hydro- and sediment dynamics in coastal areas. This study quantifies the hydraulic roughness of large compound bedforms throughout a tidal cycle and investigates its relationship to averaged bedform dimensions. Consecutive measurements with an acoustic Doppler current profiler and a multibeam echosounder were carried out in the Jade tidal channel (North Sea, Germany) along large compound bedforms comprising ebb-oriented primary bedforms with superimposed smaller secondary bedforms. Spatially averaged velocity profiles produced log-linear relationships which were used to calculate roughness lengths. During the flood phase, the velocity profiles were best described by a single log-linear fit related to the roughness created by the secondary bedforms. During the ebb phase, the velocity profiles were segmented, showing the existence of at least two boundary layers: a lower one scaling with the superimposed secondary bedforms and an upper one scaling with the ebb-oriented primary bedforms. The drag induced by the primary bedform during the ebb phase is suggested to be related to flow expansion, separation, and recirculation on the downstream side of the bedform. Three existing formulas were tested to predict roughness lengths from the local bedform dimensions. All three predicted the right order of magnitude for the average roughness length but failed to predict its variation over the tidal cycle.     

Variability of internally generated turbulence in an estuary,from 100 days of continuous observations

We present detailed observations of internally generated turbulence in a sheared, stratified natural flow, as well as an analysis of the external factors leading to its generation and temporal variability. Multi-month time series of vertical profiles of velocity, acoustic backscatter (0.5 Hz), and turbulence parameters were collected with two moored acoustic Doppler current profilers (ADCPs) in the Hudson River estuary, and estuary-long transects of water density were collected 30 times. ADCP backscatter is used for visualization of coherent turbulent structures and evaluation of surface wave biases to the turbulence measurements. Benefits of the continuous long-term turbulence record include our capturing: (1) the seasonality of turbulence due to changing riverflow, (2) hysteresis in stratification and turbulence over the fortnightly cycle of tidal range, and (3) intermittent events such as breaking internal waves. Internal mixing layers (IMLs) are defined as turbulent regions above the logarithmic velocity layer, and the bottom boundary layer (BBL) is defined as the continuously turbulent range of heights above the bed. A cross-correlation analysis reveals how IML and BBL turbulence vary with stratification and external forcing from tidal range, river flow, and winds. Turbulence in both layers is maximal at spring tide and minimal when most stratified, with one exception—IML turbulence at a site with changing channel depth and width is maximal at times of maximum stratification and freshwater input.