Alan M. Davies

A three‐dimensional model of internal tides on the Malin‐Hebrides shelf and shelf edge

In this paper, we describe a three-dimensional baroclinic sea model and its application to the computation of the internal tide over a shelf and shelf edge region. The model covers the Malin-Hebrides shelf and shelf edge with water depths ranging from a few meters near shore to over 2000 m in the deep sea. A fine horizontal grid resolution of 1/24° × 1/24° (about 2.4 km east-west and 4.6 km north-south grid spacing) and 50 vertical computational levels enables us to examine the internal tide generation at the shelf edge and its propagation. Numerical calculations illustrate the generation of the internal tide over the shelf edge and its propagation toward both the shelf and deep sea in a strongly stratified surface layer (the case of summer stratification). In a weakly stratified surface layer case (winter stratification), the internal tides generated at the shelf edge are much weaker and are dissipated away from the shelf edge region, in particular over the shallow shelf because of strong tidal mixing. A comparison of observed and computed tidal currents is made under a range of stratified conditions, and this shows the difficulty in rigorously validating three-dimensional internal tidal models. The model also indicates that the Anton Dohrn and Hebrides Terrace seamounts have an important influence upon the internal tide propagation in the region.

A Three-Dimensional Baroclinic Model of the Irish Sea: Formation of the Thermal Fronts and Associated Circulation

A high-resolution, three-dimensional baroclinic shelf sea model is developed and applied to the determination of the various processes influencing the generation and position of thermal fronts and the associated circulation in the Irish Sea. The model has a horizontal grid resolution of approximately 3.6 km by 3.0 km and can resolve the thermal fronts in the region, in particular in the western Irish Sea, although the grid is not sufficiently fine to resolve small-scale features along these fronts produced by baroclinic instabilities. Using meteorological forcing from Dublin Airport (located close to the center of the region) and tidal forcing, a series of numerical experiments were performed to examine the processes influencing the location and dynamics of thermal fronts and associated circulation in the Irish Sea. The results show a reasonable qualitative agreement with observations of the thermal stratification and the associated circulation in the western Irish Sea. The ability of the model to reproduce the main features of the frontal structures in the region having been established, it is used to examine the interannual variability of the density flow field. Calculations using meteorological forcing from different years show that the western Irish Sea fronts and the associated cyclonic circulation are persistent features, although exact details of the fronts and their times of formation and breakdown show a large interannual variability. Model results also reveal a patch of thermal stratification in the northern part of the eastern Irish Sea, with northward thermal-density-driven currents in this region.

THE EFFECT OF WIND DIRECTION AND MIXING UPON THE SPREADING OF A BUOYANT PLUME IN A NON-TIDAL REGIME

Abstract A three-dimensional baroclinic model, incorporating an accurate density advection scheme, and a range of turbulence closure models which can account for wind-wave turbulence at the sea surface, is used to examine the horizontal spreading and vertical mixing of a surface buoyant plume under a range of idealized wind stresses and bottom topography variations. Initial calculations show that the spread of the plume and its vertical mixing are very sensitive to the values of vertical viscosity and diffusivity. These coefficients are initially determined with a two-equation turbulence energy model, although subsequent calculations show that similar results can be obtained using a one equation model provided suitable stability functions are chosen for viscosity and diffusivity. Calculations assuming a constant water depth show that the offshore spread of the plume is greatest from an along shore upwelling favourable wind which gives rise to an offshore spread as a surface buoyant jet. An along shore downwelling favourable wind slightly reduces the offshore spread. In the case of a sloping bottom the offshore extent is reduced by the bottom slope, with the surface wind driven flow being a maximum in the near shore region. The magnitude of the surface current is sensitive to assumed values of the surface roughness length, and in the region of the plume shows a significant spatial variability due to a combination of a surface wind driven flow and that produced by the plume outflow. Suggestions as to how a proposed experiment involving a shore based HF Radar and a salinity survey in the region of the Ebro plume may be used to validate a physically realistic model of the region are made.

A three‐dimensional model of diurnal and semidiurnal tides on the European shelf

A three-dimensional hydrodynamic model of the continental shelf is used to examine the sensitivity of computed diurnal (K1 and O1) and semidiurnal (M2, S2, and N2) tidal elevations and currents to changes in eddy viscosity profile, friction coefficient, and water depth. The influence of the tide-generating potential is also considered. The model uses a standard finite difference grid in the horizontal, with a functional approach in the vertical, thereby giving a continuous tidal current profile from sea surface to seabed. The magnitude of the vertical eddy viscosity depends upon the flow field. Comparisons are made between model-computed profiles and harmonic analyses of 278 current time series, at a range of locations in varying water depths, with particular emphasis on locations where there are current measurements at a number of points in the vertical. A detailed examination of these comparisons shows that frictional effects in the near-bed region are particularly important in determining near-bed shear and that an accurate specification of water depth is critical for tidal current magnitude. The spatial distribution of computed current ellipses shows that the semidiurnal currents are strongest in shallow water, with the diurnal currents exhibiting an intensification in the shelf edge region. The inclusion of the equilibrium tide is particularly important for the correct determination of the magnitude of the K1 tidal current as well as K1 elevations in the North Sea. The detailed comparisons show that the magnitude of the tidal currents is correctly determined to within ±5 cm s−1 at 220 locations, with the model accurately (to within ±2 cm s−1) reproducing the vertical variation (the most sensitive test of a three-dimensional model) of the tidal current at most locations.

On using turbulence energy models to develop spectral viscosity models

Abstract This paper briefly deals with the mathematical formulation of turbulence energy models using a finite difference grid through the vertical, and the alternative formulation using a modal approach in terms of continuous functions in the vertical. In the modal model the modes are taken as eigenfunctions of a prescribed viscosity profile. Calculations show that computed tidal current profiles determined from the turbulence energy model are quite sensitive to the formulation of mixing length within the model. Profiles computed using the modal model with a fixed viscosity profile (the time-averaged viscosity profile derived from the turbulence energy model), the magnitude of which is related to the flow, are able to reproduce current profiles of both the fundamental and higher harmonics to the same level of accuracy as those determined with the turbulence energy model. By this means an “enhanced” modal model can be developed having the same characteristics as a turbulence energy model. The significantly greater computational efficiency of the modal model (a factor of five reduction in memory and 10 in computer time) coupled with the uncertainty in mixing length formulation of the turbulence energy model, suggests that modal models are valuable alternatives to, and at present show comparable accuracy with, turbulence energy models in predicting tidal current profiles.

Application of Turbulence Energy Models to the Computation of Tidal Currents and Mixing Intensities in Shelf Edge Regions

Abstract The major steps in the formulation of a three-dimensional shelf edge model using a sigma coordinate system in the vertical are briefly described. Vertical diffusion of momentum is parameterized using a range of turbulence closure models, and results are compared with earlier calculations using a simple flow-related viscosity. The influence of the magnitude of horizontal eddy viscosity upon spatial variability at the shelf edge is examined. The model is applied to the calculation of M2 and O1 tidal elevations and currents in the shelf edge region off Scotland (the Hebrides shelf), and comparisons are made with data collected in the region. Computed M2 tidal elevations and currents are shown to be in adequate agreement with observations, with no significant differences between tidal profiles computed with the turbulence energy models or the earlier calculations using a simple viscosity model. The magnitude of the horizontal eddy viscosity does not appear to affect this component of the tide. Tidal ...

Formulation of a variable-function three-dimensional model, with applications to the M2 and M4 tide on the North-West European Continental Shelf

Abstract The major steps in the mathematical formulation of a three-dimensional (3D) tidal model using a variable modal approach to represent current profiles in the vertical are presented in the initial section of this paper. Using this method the number of functions through the vertical can be increased locally where enhanced resolution is required, while ensuring mass and momentum conservation, and reducing computational time using a time-splitting technique. Calculations of the M2 and M4 tidal elevations and currents on the North-West European Continental Shelf are performed, initially with bed stress determined from the depth mean current, a hybrid 2D–3D model, the advantages of which are discussed in the paper, and subsequently computing the bed stress from the bottom current (a full 3D model). Computations are made using a range of viscosity formulations and bed-friction coefficients. A rigorous comparison of computed currents against the harmonic analysis of observations from 278 measured time-series is performed in order to determine the accuracy of the computed currents, which were found to agree to within ± 5 cm s−1 with observations at over 200 locations.

Tidal energy fluxes and dissipation on the European continental shelf

The spatial distribution of the energy flux, energy dissipation, current magnitude, and surface elevation amplitude of the major tidal constituents over the northwest European shelf are examined in detail. These distributions are obtained from the harmonic analysis of a 6 month three-dimensional simulation of 28 tidal constituents. The model currents are validated by a comparison of computed tidal harmonics against those derived from a harmonic analysis of up to 278 current time series. A similar comparison is performed for tidal elevations based on 257 tide gauges. Calculations show that for the most significant tidal constituents (e.g., M2, S2 and K2) there is a major energy flux in the deep water along the shelf edge off the northwest of Scotland, with some energy leaking onto the shelf and into the North Sea. A second source of energy is across the shelf edge at the southern end of the Celtic Sea, with this energy flux propagating into the Irish Sea and southern North Sea, where the majority of the energy is dissipation in the shallow regions. Significantly different distributions are found for the diurnal and shallow water constituents, and the spatial distributions of energy flux and dissipation of the various constituents are considered. The accuracy of separating tidal current harmonics using model data of less than a synodic period is examined with reference to the S2 and K2 tide. Calculations suggest that the accuracy of computed currents comparable to those obtained from observations can be obtained from an analysis of a 60 day period compared with the synodic period for S2 and K2 of 182 days, a significant saving in computer time.

The Influence of Wind Effects upon Internal Tides in Shelf Edge Regions

The interaction of the internal tide with wind-induced currents in the shelf edge region off the west coast of Scotland is studied using a baroclinic shelf edge model. The model is used in cross-shelf form with a horizontal finite-difference grid of the order of 0.6 km and 50 sigma levels in the vertical to study the modification of the internal tide produced by upwelling and downwelling winds. Horizontal mixing in the model is parameterized using either the Laplacian form of the horizontal diffusion or the biharmonic form and the sensitivity of the solution to both forms is examined. Coefficients for the vertical diffusion of momentum and density are determined using either an algebraic expression involving the Richardson number or from a two-equation turbulence energy submodel. Calculations show that in the case of an upwelling-favorable wind the density gradient in the near-bed region is increased leading to a slight modification (compared to the tide only solution) of the internal tide at the fundamental frequency with significant increases in amplitude of the higher harmonics due to the increase in the nonlinear terms produced by the increase in the density gradient. With a downwelling-favorable wind the amplitude of the current and internal displacement of the internal tide at the fundamental frequency are significantly reduced due to the change in the density field in the region of internal tidal production. This also leads to a reduction in the amplitude of the higher tidal harmonics. By using a fine grid in the horizontal, the coefficients in the horizontal diffusion terms were set at a minimum and no significant difference in solutions computed with the Laplacian or biharmonic diffusion terms was found. Similarly there are no differences in the major features of the flow field computed with the various parameterizations of the vertical diffusion, although there are some differences in the magnitude of the diffusion coefficients.

A three-dimensional model of the principal tides on the European shelf

Abstract A three-dimensional hydrodynamic model of the continental shelf is used to examine the spatial distribution, both elevations and currents, of 28 of the major tides in the region. Tidal constituents in the diurnal and semi-diurnal band together with higher harmonics and long period tides are used to illustrate the magnitude and spatial variability of the tides over the shelf. Comparisons are made between model computed profiles and harmonic analyses of 278 current time series, at a range of locations in varying water depths, with particular emphasis on locations where there are current measurements at a number of points in the vertical. Observed and computed tidal elevation amplitudes and phases are also compared at up to 257 shore-based and off-shore tide gauges. These detailed comparisons are used to determine any bias in the model and the accuracy of predicted elevations and currents derived with the model. Results of these comparisons show that although the diurnal and semi-diurnal tidal elevations and currents can be accurately reproduced using a 12 km grid, the error in the higher harmonics is significantly larger. The fact that the model contains all the non-linear terms (advection, quadratic friction, wave drift, time varying viscosity) necessary to generate these harmonics, together with a full set of tidal harmonics in order to have the correct level of friction, suggests that potentially it should reproduce these constituents. However, a rigorous comparison was found to be very difficult, because of the rapid variation of the higher harmonics in shallow water which cannot be adequately resolved in the model, and possible inaccuracy in the measurements of the small currents associated with the higher harmonics.