Jiuxing Xing

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.

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 ...

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.

Application of a range of turbulence energy models to the determination of M4 tidal current profiles

Abstract A fully nonlinear, three-dimensional hydrodynamic model of the Irish Sea, using a range of turbulence energy sub-models, is used to examine the influence of the turbulence closure method upon the vertical variation of the current profile of the fundamental and higher harmonics of the tide in the region. Computed tidal current profiles are compared with previous calculations using a spectral model with eddy viscosity related to the flow field. The model has a sufficiently fine grid to resolve the advection terms, in particular the advection of turbulence and momentum. Calculations show that the advection of turbulence energy does not have a significant influence upon the current profile of either the fundamental or higher harmonic of the tide, although the advection of momentum is important in the region of headlands. The simplification of the advective terms by only including them in their vertically integrated form does not appear to make a significant difference to current profiles, but does reduce the computational effort by a significant amount. Computed current profiles both for the fundamental and the higher harmonic determined with a prognostic equation for turbulence and an algebraic mixing length formula, are as accurate as those determined with a two prognostic equation model (the so called q 2 – q 2 l model), provided the mixing length is specified correctly. A simple, flow-dependent eddy viscosity with a parabolic variation of viscosity also performs equally well.

Application of three dimensional turbulence energy models to the determination of tidal mixing and currents in a shallow sea

Abstract A three dimensional hydrodynamic model, using a sigma coordinate grid in the vertical, with subgrid scale diffusion represented using a range of turbulence energy closure schemes, is used to examine M 2 tidal elevations and currents in the Irish Sea. Changes in turbulence energy intensity in particular surface turbulence with variations in mixing length formulation is also considered. A no-slip condition is applied at the sea bed and the sensitivity of tidal elevations and particular tidal currents to changes in bottom roughness length and formulation of mixing length are examined; also detailed comparisons are made with measurements. From these comparisons it is evident that the bed roughness has a major influence upon tidal elevations, and current profiles in the near bed region. Near bed currents are also influenced by the form of the mixing length close to the sea bed, although its value in the upper part of the water column is of less importance. However, the vertical variation of turbulent energy, in particular its surface value, is very sensitive to the mixing length. Elevations computed with the three dimensional model with various closure assumptions are compared with those from a 2D-vertically integrated hydrodynamic model. In general the two dimensional model appears to reproduce elevations slightly more accurately than the three dimensional model. A ‘hybrid’ three dimensional model using an identical bed stress formulation to that in a two dimensional model is used as a means to compute currents and turbulence energy intensities while maintaining the same elevation distribution as that computed with a two dimensional model.

Processes influencing tidal mixing in the region of sills

A non-hydrostatic model in cross sectional form with idealized topography representing a sill and forced by a barotropic tidal flow is used to examine the role of a hydraulic transition and internal lee waves in determining mixing in the sill region. Calculations using smooth topography show that unsteady lee waves are generated on the sill slope during flood tide. These waves propagate toward the sill when the tide reverses leading to enhanced mixing in the sill region. The addition of small scale topography on the leeside of the sill changes the lee wave distribution with an associated increase in mixing depending upon wavelength and amplitude of topography. Calculations show the importance of the hydraulic transition, lee waves and small scale topography in determining mixing in sill regions compared to smooth large scale topography.

Processes influencing the internal tide, its higher harmonics, and tidally induced mixing on the Malin-Hebrides Shelf

Abstract A three dimensional prognostic hydrodynamic model with a turbulence energy submodel is used to examine the spatial distribution of the internal tide in the shelf edge region off the west coast of Scotland. In the initial series of calculations presented here the model is used in a cross-shelf form. The model contains all the non-linear terms necessary to examine the generation of the higher harmonics of the tide, and the generation of very short waves on the density interface near the shelf-break. Initial calculations assuming a homogeneous sea region show a thick turbulent bottom boundary layer generated by bottom friction, which reduces the tidal current in the near-bed region, although above this the flow is essential inviscid with no vertical variation. Calculations with a specified vertical stratification reduce the thickness of the bottom turbulent layer with an associated change in current profile. In the case of specified stratification no internal tides are generated, however when the density field is allowed to evolve a significant internal tide is generated, with an associated internal shear giving rise to enhanced mixing in regions of largest shear. Results from the model show that higher harmonics of the internal tide, and short internal waves are generated above the shelf-break, where horizontal mixing processes are important. The magnitude of these short internal waves increases with increasing tidal forcing, and also depends upon the size of the horizontal diffusion term. Regions of intense surface and bed turbulence associated with the internal tide are found in the shelf-break area, suggesting that turbulence energy measurements in this location may be a good test of turbulence closure models. However the significant spatial variability associated with these regions shows that an intense measurement campaign is required to avoid significant undersampling.

Processes influencing wind-induced current profiles in near coastal stratified regions

Cross sectional and single point models of wind-induced currents in a stratified sea coastal region are used to determine the role of internal waves due to various wind periods upon mixing and current profiles. The limitations of a point model in reproducing current profiles are also examined. Calculations are performed with both fixed vertical diffusion coefficients and those derived from a turbulence closure model. Wind forcing at sub-inertial, inertial and super-inertial frequencies together with a wind impulse are considered. Results show that the offshore extent of the coastal boundary layer where mixing occurs depends upon the frequency of the wind forcing. The extent of this region and offshore propagation of internal waves influences the offshore variability of current profiles and the extent to which they can be reproduced by a single point model. Calculations with periodic wind forcing show that within the coastal boundary layer a point model cannot reproduce the current profile due to the internal pressure gradients associated with the internal wave field. However outside this region, current profiles from a point model are in good agreement with those computed with the cross section model when a barotropic pressure gradient forcing proportional to the local wind forcing is included. In the case of transient wind forcing, the inertial period dominates and determines the offshore extent of the coastal boundary layer. The transient nature of the response is such that although the single point model, outside the coastal region can reproduce the surface current due to direct wind forcing, the current at depth is not reproduced. The offshore extent of the coastal boundary layer computed with the turbulence energy model is comparable to that found with fixed diffusion coefficients although in some cases there are important differences which influence current profiles.