Malin-Hebrides shelf and shelf edge

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. C12, PAGES 27,821-27,847, NOVEMBER 15, 1998 A three-dimensional model of internal tides on the Malin-Hebrides shelf and shelf edge Jiuxing Xing and Alan M. Davies Proudman Oceanography Laboratory, Bidston Observatory, Birkenhead, Merseyside, United Kingdom. Abstract. 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" x 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. 1. Introduction recent validation performed by Davies et al. [1997], and the discussion of model validation in Lynch and Davies [1995].) A In recent years there is an increasing interest in processes along general review of tidal modelling is given in Davies et al. [ 1997a, the continental shelf edge [Huthnance, 1995]. Among many b]. physical processes at the shelf edge, internal tides and waves are a To date, the majority of the models used to study the internal widespread phenomen and are found to be particularly energetic tides have been semi-analytical and have dealt with the linear along many shelf edges where barotropic (surface) tides are strong, inviscid solution [Baines, 1982; Craig, 1987; Sherwin and Taylor, for example, the Malin-Hebrides shelf edge [Sherwin, 1988], the 1990; New, 1988]. The early work of Baines [1982] and Craig Bay of Biscay [Pingree et al., 1984], the Australia north-west shelf [ 1987] identified the mechanisms generating internal tides and the [Holloway, 1984, 1996], off northern California [Rosenfeld, 1990], importance of vertical density gradient and shelf slope in and off Vancouver Island [Drakopoulos and Marsden, 1993]. determining the generation and propagation of the intemal fide. The Many of the above papers have reported that internal tides are semi-analytical nature of these models restricted their use to characterized by large variations over relatively small spatial and idealized topography and density fields and excluded nonlinear temporal spaces. Because of this, in contrasto the barotropic tides processes and the use of turbulence energy schemes to represent at the shelf and shelf edge which can be resolved with relatively subgrid scale mixing. coarse grids (of the order of 8 km) [e.g., Flather, 1987; Lynch and A finite difference model has been developed by Chuang and Naimie, 1993; Foreman et al., 1993; Proctor and Davies, 1996; Wang [ 1981 ] to examine in detail the influence of the topographic Xing and Davies, 1996a], a fine cross-shelf grid resolution is and density variations upon internal tides. The model was used by required to accurately determine the internal tide. In a recent paper Sherwin and Taylor [1990] to examine internal tides in the using a cross-shelf model, Xing and Davies [1997a] compared Malin-Hebrides shelf region. In their work, time dependency was numerical solutions computed using a range of horizontal and removed by considering a single harmonic, and the vertical eddy vertical grids with analytical solutions of Craig [1987]. They found viscosity term was neglected, although a linear damping coefficient that an accurate solution could be obtained using an acros shelf grid was included. By this means, a time-independent set of equations of the order of 3 km with a vertical grid of the order of 50 levels. was obtained, which were then discretized on a finite difference grid The need to use a fine grid and accurate representation of bottom to yield a set of linear equations. Although solutions for the topography and density field at the shelf slope, together with the lack nonlinear internal fides were obtained by Pingree et al. [ 1984] and of comprehensive data sets, means that models of the internal tides Maze [ 1987], they used two-layered models with density uniform in have not been developed and validated to the same level as each layer. barotropic tidal models. As we will show here, there are difficulties More recently, primitive equation models have been developed in validating these models in three dimensions to the same accuracy and applied in a two dimensional cross-shelform to examine used for barotropic tidal models. (The reader is referred to the internal fides [Lamb, 1994; Holloway, 1996; Xing and Davies, 1996b, 1997b, c]. Lamb [1994] solved the nonlinear, nonhydrostatic equations to examine intemal fides and waves across Copyright 1998 by the American Geophysical Union. an idealized finite amplitude bank edge. The model successfully reproduced some observed features, including the formation of a Paper number 98JC large depression and hydraulic jump over the bank edge during off /98/98JC bank flow and two on-bank propagating depressions every tidal 27,821

2 27,822 XING AND DAVIES: THREE-DIM]ENSIONAL INTERNAL TIDE MODEL period. Holloway [1996] applied the model of Blumberg and comparisons of the internal tide with measurements in the region Mellor [1987] to a region on the Australia north west shelf. were also made. A three-dimensional simulation of the barotropic Holloway showed a high degree of spatial variability in the tide along the West Coast of Vancouver Island has recently been amplitude and phase of internal tidal currents and vertical performed by Foreman and Thomson [1997], who examined in displacements. Xing and Davies [1996b] developed a numerical detail the seasonal changes in shelf waves in the region due to model suitable for studying the generation of internal tides and buoyancy and wind-driven shelf edge currents. However, to the applied the model to the Malin-Hebrideshelf in two-dimensional authors' knowledge, the application of a full three-dimensional cross-shelf form. They found that the nonlinear terms moved energy model to determine the influence of offshore seamounts upon the from the semidiurnal M 2 tide to the higher harmonics giving rise to internal tide at the shelf edge has not been investigated. an M 4 internal tide [Xing and Davies, 1996b] and ultimately, In this paper, we build on the previous work of Xing and Davies particularly as the horizontal diffusive term was reduced, a tendency [ 1997a, b, c], extending the two-dimensional cross-sectional model toward the formation of an internal bore. Also, the nonlinear terms to a full three-dimensional model and using it to examine the had an important influence on internal tidal current magnitudes and internal tide generation and propagation on the Malin-Hebrideshelf energy flux. and shelf edge using realistic bathymetry and semidiurnal M 2 Xing and Davies [1997c], using the same cross-shelf model, barotropic tidal forcing. The model used here has identical water examined the interaction of the internal tide with wind-induced depths and open boundary forcing to that of the barotropic model of currents at the shelf edge. Their calculationshowed that in the case Xing and Davies [1996a], although a finer vertical and horizontal of an upwelling favourable wind the density gradient in the near-bed grid is employed to resolve the internal tide. Calculations are region was increased, leading to a slight modification of the internal performed with idealized winter and summer stratification, primarily tide at the fundamental frequency with a significant increase in the as a process study to examine the sensitivity of the amplitude of the higher harmonics due to the increase of the three-dimensional internal tide to changes in stratification and the nonlinear terms. With a downwelling favourable wind, the complexity introduced by the presence of topographic features other amplitude of the current and internal displacement of the internal than the shelf edge, namely seamounts, in the region. Also, tide at the fundamental frequency is significantly reduced as a result comparisons are made with the current meter data presented by Xing of the change in the density field. This significant modification of and Davies [1996a] with a view to determining the nature of an the internal tide by modest downwelling wind events suggests that observational data set that needs to be collected for validating a accurately filtering out the internal tide from other effects by the three-dimensional model of internal tides. The remainder of the harmonic analysis of measurements may be difficult, which has paper is organized in sections, with the next section describing the implications for the validation of internal tidal models to the same numerical model. Section 3 presents briefly the computation of the degree as barotropic models. barotropic semidiurnal fide. In section 4, we consider differences in The problem of nonlinear interaction between tidal and the internal tide for two differentemperature (density) profiles. We meteorological forcing in a shallow homogeneous sea region, and also present some result showing the influence of seamounts upon the accuracy of tidal currents derived from the harmonic analysis of internal tides, with a concluding discussion in section 5. a short-period (of the order of 60-days) deployment was examined by Pugh and Vassie [ 1976] by comparing the analysis of a short- 2. The Numerical Model period current meter record with the analysis of a record of one-year duration. They found that for the M 2 tide an error of the order of +2 A three-dimensional, free surface, nonlinear primitive equation cm s ' was likely to occur in the short-period analysis. Xing and model in polar coordinates is used in this study. The model is based Davies [ 1997c] found the modification of the M 2 internal tide during on the previous work of Davies and Xing [ 1995] and Xing and downwelling wind events (the dominant wind in the region) to be Davies [ 1996a, b] and uses topography-following sigma coordinates. the result of a change in the near-bed density field. This suggests Since details of the hydrodynamic equations and numerical solution that the influence of meteorological forcing upon internal tides may method are given elsewhere, only the main features of the model be stronger than that found by Pugh and Vassie [1976] for and its numerical solution will be outlined here. barotropic tides in shallow water and that their error estimate should The horizontal diffusion terms in the model are parameterized in be regarded as a lower bound in the comparisons between model and terms of a biharmonic horizontal viscous term, rather than using the observations made later in this paper. In performing these Laplacian form which is common in a number of numerical models comparisons, the harmonic analysis of less than a 60-day period was [e.g., Blumberg and Melior, 1987]. The biharmonic formulation has disregarded. (An M 230 day-analysis tide based on a is commonly advantages in controlling computational noise without imposing regarded as acceptable in well-mixed shallow sea regions.) Also, unrealistically high damping on larger-scale features of the flow any measurements that were made under strong wind conditions [Heathershaw et al., 1994]. The biharmonic form of the horizontal were not used in model/data comparisons. The emphasis is placed diffusion is computed on sigma surfaces. For the temperature on data collected during the summer when wind events are less equation, thereibre, spurious diapycnal diffusion occurs. To reduce intense and frequent and the internal tide is stronger. This problem is discussed in the latter part of this paper. the error, temperature deviation rather than temperature itself is Almost all of the work discussed above was performed in a used. Since we are primarily concerned with a sensitivity study, the two-dimensional cross-shelf section, which does not take into density field is determined from the temperature field using a simple accounthe effects of the alongshore topographic gradient. Also, the equation of state [Xing and Davies, 1997a], although more complex barotropic tides which are the Ibrcing term driving the baroclinic equations can be used [Blumberg and Mellor, 1987] and pressure tide cannot be accurately modelled in these cross-sectional models. effects on density could also be included. Wang [ 1989] used a three-dimensional model to study internal tides Although the advection of momentum terms are evaluated using that were generated over the sill in the Strait of Gibraltar. The recent central differencing, which is used in many models [e.g., Blumberg paper of Cummins and Oey [ 1997] has made it possible to reduce and Mellor, 1987] to advect both momentum and other fields (e.g., some of the detail in our revised paper. Cummins and Oey [1997] density), propertie such as temperature and salinity, which often performed a detailed three-dimensional simulation of the barotropic have sharp gradients associated with them, are advected using the and baroclinic tides off Northern British Columbia, considering in Total Variation Diminish scheme [James, 1996]. The use of such particular the internal energy flux in the region. A number of an advection scheme was shown to be essential for internal tides in

3 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL 27,823 regions of strong tidal forcing and steep topography [Xing and Davies, 1996b, 1997a]. The vertical eddy viscosity and diffusivity in the model are calculated using a range of turbulence energy submodels [Xing and Davies, 1997b]. Using the model in cross-shelform, Xing and Davies [1997b] examined the sensitivity of the internal tide to a range of turbulence energy closure schemes. They found no significant differences in the results; a simple Richardson number dependent viscosity and diffusivity produced results comparable to those obtained with a two-equation turbulence closure model. The slight differences in the solution using different turbulence closure models was insignificant compared to those arising as a result of small variations (comparable with those resulting from observational error) in topography or density field. Also, loss of accuracy due to using an inadequate vertical resolution produced a larger difference. In this paper, a two-equation turbulence model, termed the 2.5 level model [Mellor and Yamada, 1974, 1982], with modification by Galperin et al. [ 1988], is used. This model, used by, among others, Blumberg and Mellor [ 1987], Oey and Chen [ 1992], and Xing and Davies [ 1996a], involves an equation for turbulence kinetic energy (TKE) and an equation for the length scale. Since the form of the model is well-documented in the literature, details will not be given here. 57øN - ==================================== :::::::::::::::::::::::::::::::::::::::::::::::::: HllllIlHl lg1 IHllllllllll 58øN 57øN 56øN - 10øW -8øW -6øW -4øW Figure lb. As Figure 1 a, but showing location of current meters. At the sea surface there is no applied wind stress or heat flux. C For the turbulenc energy model, there is no turbulence flux through b = br+ - ( - :r), (1) the sea surface. The length scale at the sea surface is given by a roughness length z.,., which is set arbitrarily to that at the bed. In the absence of a surface wind stress or surface heating, the value of the where C=(gH) m, g is gravitational acceleration, H is total water surface roughness does not substantially influence the current depth, b is the normal component of the depth mean current, and br profile. At the sea bed, a quadratic friction law was applied with zo and r are the given normal components of tidal current and bed roughness (set to 0.005m) with no temperature flux through the elevation, which is derived by interpolating from a larger area model sea bedø (R.A. Flather, manuscript in preparation, 1995). By using a For the TKE model, we use a boundary condition at the sea bed radiation condition, neither the elevation nor the velocity is clamped including the balance of the turbulence production, dissipation, and on the open boundary, and waves can propagate out of the region. diffusion given by Xing and Davies [1996a]. The model is forced For the baroclinic flow and temperature field, a flow relaxation along the open boundary with an identical barotropic tide and scheme [Martinsen and Engedahl, 1987] is applied along the open radiation boundary condition used by Xing and Davies [1996a] for boundary. In this application, the baroclinic velocity is seto zero the barotropic flow, namely' at the open boundary and the temperature is set to the initial field. Because of the presence of the steep bottom topography along the shelf edge, water depths change rapidly along the northern and I southern open boundaries, and further numerical damping is necessary to avoid the disturbance generated along these open boundaries propagating into the interior of the model. This was done i iiiii iii i :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: by progressively increasing the horizontal diffusion coefficient in... :==: :: H,! ; "-' the relaxation zone to a value at its outer edge that was several times 58øN - larger than in the interior area. Other open boundary conditions were also tested [Roed et al., 1995; Roed and Cooper, 1987], although none was so perfect that they gave no significant differences in the interior where observational data was available (see later). In essence, any imperfections in the open boundary condition led to small-scale perturbations in the large-scale solution within about six grid boxes of the boundary. These small-scale features did not affect the largescale motion, such as the M 2 tide; the consistent bias (namely an 56øN - underestimate of tidal amplitude and overestimate in tidal phase) was related to the bias in the open boundary forcing at the M2 frequency and was essentially independent of the boundary Illllll"ll[ 11[[32'.... m 37 condition (see later discussion on model accuracy). A staggered finite difference grid, which density and the two components of velocity are solved at different grid points (the so- - 10øW -8øW -6øW.4ow called Arakawa C grid), is used in the horizontal (Figure 1 a, b), with Figure la. Finite differencegridofthemodel showing thelocation grid spacing of 1/24" north-south by 1/24 "west-east, a grid of tidal gauges used in the comparison. resolution of 4.6 km by 2.5 kin. It is importanto note that the

4 27,824 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL 59øN 58øN Anton Dohrn- Seamount 57ON - Hebrides Terrace Seamount 58øN - region covered by the model is characterized by a range of water depths varying from 2000 m in the deep ocean to the order of 10 m in the near coastal region (Figure l c). Since the surface Kelvin wave travels with a speed of (gh) m, the model requires a very small time step for the external mode. Therefore a time-splitting method is used to solve the model equations in order to reduce the computational overhead. The time steps used in the calculation are 300 s for the depthdependent mode and 10 s for the depth mean mode. This shortime step is required by the Courant-Friedrichs-Lewy condition. A single time step procedure rather than the leapfrog scheme [Blumberg and Mellor, 1987] is used to avoid having to average the two solutions associated with the leapfrog method. The value of the biharmonic horizontal diffusion coefficient for momentum and temperature is set to 10 m4s -. 55øN 3. Barotropic Tides on the Malin-Hebrides Shelf and Shelf edge Figure lc. Bottom topography used in the model (all depths in meters, with contour intervals of 200 m in water depths larger than Recent work on the modelling of the semidiumal and diurnal barotropic tidal flow over the Malin-Hebrides shelf and shelf edge 200 m and 25 m in water depths less than 200 m). The lines marked has been reported by Proctor and Davies [1996] and Xing and C1, C2, C3, D1, and D2 denote cross-shelf sections used for Davies [1996a]. Both of these papers used the same model domain presenting some results. P 1 and P2 are two locations near the shelf as that used here, but employed a coarse resolution grid than the edge used for time series plots. one shown in Figure 1 a. In the case of Proctor and Davies [1996], the water was assumed to be homogeneous, therefore the model did not take into account the stratification effects, and eddy viscosity higher grid resolution is required in the cross-shelf (west-east) was computed as an algebraic function of the flow field. Xing and direction in order to resolve the fine-scale features of the internal tide in this direction and the fine details of the shelf slope topography [Xing and Davies, 1998] which are responsible for this variability. Cummins and Oey [ 1997], on the basis of typical shelf Davies [1996a] reported results using a turbulence energy closure model to parameterize vertical eddy viscosity. They examined the effects of stratification upon the barotropic tides due to its influence upon vertical eddy viscosity by performing two calculations. In an edge profiles and stratification, suggest a length scale for internal initial calculation they assumed a homogeneousea region, and in motion of 25 km on the shelf and 75 km in deep water. They found little change between solutions performed with 5-km and 2.5-km a subsequent calculation a fixed density field was specified. Internal tides were therefore excluded, and the difference between the two finite difference grids, and concluded that a 5-km grid was adequate. solutions was due to the effects of the density field upon the vertical Since we are concerned later with the influence of topographic features upon the internal tide, we wish to use as fine a grid as is computationally acceptable in the cross-shelf direction. In the eddy viscosity. Here we use a finer grid than that in earlier calculations and allow density evolution so that we can study internal tide generation and propagation and three-dimensional vertical there are 50 computational levels, with a highe resolution spatial variability produced by seamounts. in the near-surface and near-bottom boundary layer. These The results of the barotropic semidiurnal (M2) tide computed conclusions concerning horizontal and vertical resolution were using the fine-grid resolution model (Figures 1 a, b) are essentially based upon an extensive series of calculations [Xing and Davies, similar to those of previous work [Proctor and Davies, 1996; Xing 1997a] using a cross-shelf model with a range of grid resolutions and Davies, 1996a] and will only be briefly presented in this and comparing results with the analytical solution of Craig [ 1987]. section. In an initial calculation (Calc 1), a fixed winter Although the solution of Craig [ 1987] is for a cross-section model with a linear slope, it serves as an indication of the accuracy of the stratification was specified (i.e., a diagnostic), and hence internal tides could not be generated. The barotropic tides are obtained by solution that might be expected in a realistic three-dimensional integrating the model in time for 10 tidal cycles from a state of rest model. The general features of the internal tide could be reproduced with M 2 tidal forcing along the open boundary. The time series of in this cross-sectional model using as little as 20 levels in the vertical. However, to reproduce currents (assuming an accurate the last two tidal cycles are harmonically analyzed to determine the M 2 tidal amplitude and phase. The density field associated with the density and topography field) to within 2 cm s ', the accuracy we winter stratification did however reduce eddy viscosity in regions of would hope to achieve from the present model, 50 levels were used sharp vertical density gradient because of the buoyancy suppression in the vertical. In wintertime, when the seasonal thermocline is weaker and the internal tide is not so strong, an accurate solution term in the turbulence energy equations. The effect of such a change in viscosity upon tidal current profiles in shallow sea regions could probably be reproduced with a smaller number of levels. has been discussed by Davies [ 1993], Xing and Davies [ 1996a], and Cummins and Oey [1997] performed some three-dimensional M.J. Howarth [manuscript in preparation, 1996]. Both calculations calculations with both 20 and 40 levels in the vertical and reached the conclusion that the main features of the internal tide could be computed with 20 levels. Consequently, by using 50 levels in the vertical we would hope to achieve an accurate solution for the and measurement show that a reduction in eddy viscosity due to stable stratification produces a phase shift in the tide across the pycnocline, with tidal current magnitudes above the pycnocline comparable with their free stream flow values [Davies, 1993]. This various topographic features used here. The shelf and shelf edge effect of stratification upon tidal current profiles is independent of

5 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL 27,825 58øN 57øN, resulting a strong cross-shelf tidal current, which is largely responsible for the generation of the internal tide. This is discussed in more detail later in the paper. The M 2 tidal current ellipse near the seabed has a similar spatial distribution, although the magnitude of the near-seabed current is significantly reduced because of the bottom friction effects. A detailed comparison of computed and observed M 2 tidal elevation amplitude and phase at a number of coastal and shelf edge 57øN locations (see Figure l a for the location of these positions) are presented in Table 1. The overall prediction of the M 2 tidal elevation amplitude and phase are reasonable good. The RMS errors for the amplitude and phase are 8.3 cm and 16.50, 56øN respectively. The error in amplitude is comparable with the 10-cm error found by Davies et al. [1985] in a two-dimensional model of the North Sea, although the error of in phase is much larger than the 30 error found by them. The 10-cm error in amplitude founded by - 10ø13/ -8øW -6øW -4ø13/ Davies et al. [1985] can be attributed to a small error in open boundary elevation data in that model. The importance of an accurate specification of the tide along the open boundary was i i i confirmed by a sensitivity study of open boundary forcing which revealed that computed tides in the interior of the model were sensitive to small changes in certain regions of the open boundary. The enhanced accuracy of the limited area North Sea model of Davies et al. [1985] and Davies [1976] can be attributed to an accurate knowledge of the M 2 tide on the open boundary and that the open boundary of the model was small compared to the region 57øN./ covered. Also, the open boundary in these models was adjusted to yield an accurate solution. In the present calculation, the model has a large open boundary and the tidal forcing is not well defined (see later). Also, the decision was made here to use the same boundary 56øN forcing as Proctor and Davies [ 1996] and Xing and Davies [ 1996a] so that comparisons could be made with these models, despite the known limitations in the boundary forcing (namely an underestimation of tidal elevation amplitudes and an overestimation of phases). [,!, - 10øW -8øW -6øW There is a slight improvement of the computed amplitude in the present model compared with the previous work of Proctor and Figure 2. Cotidal chart of M 2 barotropic tide computed using the Davies [1996] and Xing and Davies [1996a]. (There is an error in numerical model: (a) tidal amplitude (cm),(b) tidal phase (deg). the computed M 2 tidal elevation amplitude and phase at location No. 30 in Table 2a of Xing and Davies [1996a].) This improvement is probably because of the finer grid resolution, which improves effects produced by the internal pressure gradients associated with internal tides. The computed M 2 cotidal chart shown in Figure 2, is similar to that computed by Xing and Davies [1996a], using the same model domain but a coarser horizontal grid resolution, and shows the tidal phase increasing northward, characteristic of a Kelvin wave propagating along a coastline. In Figures 3a and 3b we presenthe distribution of the major and minor axis of the M 2 tidal current ellipse at the sea surface (o=0.0) and near the seabed (o=-1.0), plotted at every fourth grid point. Again, these results are similar to those of Xing and Davies [1996a]. From the figures it is evident accuracy in the near-shore region. An example of the improvement in accuracy at coastal gauges due to a finer grid can be seen by comparing the tides on the shelf computed by Kwong et al. [1997] (RMS errors for M2of 12.7 cm and 17.3 ø) with a grid resolution of 12 km with those of Sinah and Pingtee [1997], who used a significantly fine grid, which reduced the errors to 10 cm and 3 ø. Also, the more scale-selective form of the horizontal diffusion in the present model (by using a biharmonic form of the horizontal diffi]sion) reduces the smoothing (compared to the Laplacian form used previously) of the long-wave part of the solution (namely the tidal solution) while still removing the short waves that can give rise that the strongestidal currents occur in the region of the North to nonlinear instabilities. The size of these RMS errors is an Channel, where current magnitudes are of the order of 2 m s - near indication of the overall accuracy of the solution, which is affected the surface, and in the coastal region between the Hebrides Islands and the west coast of the Scottish mainland. Along the shelf edge by the bias to underestimate the M 2 elevation amplitude and overestimate the tidal phase, as can be seen from the values of Ah (the dashed line in the figures is 200 m water depth and represents and Ag (Table 1). These systematic errors are probably associated the location of the shelf edge), the tidal current changes from a few cm s ' on the deeper water side of the shelf edge to about 20 cm s ' on the shallower water side over a short distance. Also, the major axis of the M 2 tidal ellipse are nearly perpendicular to the isobath with the open boundary input to the model, derived by interpolating from a larger area model, (R.A. Flather, manuscript in preparation, 1995), which had a tendency to underestimate the M 2 elevation amplitude and overestimate the phase in the region considered here. along the shelf edge, in particular, in the region between 55øN and Also, it is clear from Table 1 that large tidal elevation amplitude and

6 IIIIIIIIII ttttt ß ß 27,826 XING AND DAVIES: THREE-DIMENSIONAL INTERN TIDE MODEL a 58øN /////////////////////// /////////////////////.I //////////////// ///////////// ///////////// ////////// II// ////// 11 II/ ///teltill III /111////11 ttt i I I 1.0 rns - 57øN t11111t111 IIIIIIIIII t + IIIIIIIIII 56øN..... x ½++,,,, ttt!ttltlii - 10øW -8øW -6øW -4øW øN 57øN 56øN.6ow.4ow Figure 3. Computed major and minor axis of the M2 barotropic tidal current ellipse at every fourth grid point: (a) at sea surface (o=0.0), (b) close to the seabed (o=-1.0), and (c) vectors of M 2 tidal energy flux. Also given are contours of the forcing function F (Nm'2) from (2), with (d) contours of F, and (e) contours of Fy. (Contours are 10, 20, 50, 100, and 150 Nm-2.) The dashed line is the 200 m water depth, representing the shelf edge.

7 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL 27,827 Table 1. Comparison of computed and observed M 2 elevation tidal amplitude Near the North Channel, the agreement between computed and ha(cm ) and phase g (deg) andifference Ah (cm), Ag (deg) at a number of observed tidal elevation amplitude and phase is lessatisfactory coastal and offshorelocations, to etherwithrmserrors. because of a number of factors, such as the open boundary Location Observed Calc 1 Differences Number ha g ha g Ah Ag RMS e ors phas errors occur at a number of locations, such as No. 20 and No. 24. This may relate to the local factors which the model cannot resolve. Cummins and Oey [ 1997] obtained absolut errors for the M2 tide of 4.1 cm in their model of tides off northern British Columbia, and they make the point that it is permissible to neglect comparisons with observations in regions which could not be resolved with their grid. Here we have used all available data, although in many locations it might be sensible to disregard the results because of limited model resolution. The fact that many of the coastal gauges along the west coast of Scotland are situated in small inlets which the model cannot resolve (Figure l a) and the bias in the open boundary input to the model explain the rather large RMS errors associated with the solution. The bias in the open boundary input can be readily removed (by a slight modification of the input values); however, as our main aim here is a study of three-dimensional effects upon the internal tide, and to be consistent with earlier solutions, we have not attempted to improve the accuracy of the model by revising the open boundary. condition, local coastal geometry, and being close to the tidal amphidromic point. Nevertheless, the computed tidal elevation amplitude and phase at a number of locations along the shelf edge, such as Nos. 14, 16, 19, and 26, give very satisfactory results. Since the shelf edge area is the primary region of internal tide generation, the main topic of this paper, these result suggest that the barotropic tide is sufficiently accurate in these regions to provide an appropriate barotropic forcing for the baroclinic internal tide. A detailed comparison of observed and computed M 2 tidal current amplitude and phase for the u and v components at a number of current meter sites is presented in Tables 2a and 2b, with Table 3 giving the associated ellipse parameters. These tables present results from various calculations, with (Calc 2 and 3, see later) and without (Calc 1) including internal motion in the model. As discussed by Xing and Davies [1996a], many factors, such as the background level of friction due to oceanic flow and other tidal constituents (namely S2, N2, K etc.), are not included in the model. The neglect of these additional sources of turbulence must be borne in mind in the intercomparison of the model and observation. Also, in the case of the calculations including intemal tides (Calc 2 and 3), the lack of a precise knowledge of the three-dimensional density field will influence the validity of model and data intercomparisons in regions where the internal tide is important. As discussed previously, the modification of the internal tide by meteorological effects suggests that the amplitude and phase of the M 2 tidal current derived from a harmonic analysis of a 60-day current time series may contain a significantly larger error than that found from a comparable analysis of tidal currents in shallow water [Pugh and Vassie, 1976]. Of particular importance in the later discussion of internal tides are those locations near the shelf edge (locations T to E in Figure lb). In general, the model gives satisfactory results with a few exceptions, such as at location N the computed u velocity (20 cm s 't) at the lower part of the water column being larger than the observed (about 12 cm s't), and at locations I, J, and K the computed velocity is too small compared with the observed. The reasons for this will be discussed later in connection with internal tides. 4. Internal Tide Generation Along the Malin-Hebrides Shelf Edge 4.1. Introduction To understand the spatial distribution of the internal tidal magnitude in the shelf edge region, it is first essential to examine the spatial distribution of the barotropic tidal energy flux in the area (Figure 3c), since this is the forcing term producing the baroclinic tide. Energy flux vectors (Figure 3c) in deep water show a flux to the north, which is consistent with the tidal propagation given in the cotidal charts (Figures 2a,b), with a cross-shelf energy flux occurring between 55.5øN and 57.5øN, suggesting thathe barotropic tidal forcing for intemal tide generation is a maximum in this region. Apart from the barotropic tidal forcing, the two fundamental factors influencing the internal tide generation are the vertical and horizontal variation of the stratification and the bottom topography. According to Baines [ 1982], the amplitude of the forcing function (F) for the intemal tide in a cross-slope (x-z) section (F x) and along slope (y-z) section (F¾) are given by

8 27,828 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TiDE MODEL Table 2a. Comparison of observed and computed amplitude (cm s 'l) and phase ug(degrees) of the u component of the M tidal current at a number of depths and locations. Rig Observed Calc 1 Calc 2 Calc 3 h, z/h Number uh Ug uh Ug u Ug u Ug m U V T T Q P S S S R R R O O L L L N N M M M W W I I J J K K H G G E F C C C A A D B B B

9 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL 27,829 Table 2b, Comparison of observed and computed amplitudevn(cm s 'z) and phase vg(degrees) of the v component of the M tidal current at a number of depths and locations. Rig Observed Calc 1 Calc 2 Calc 3 h, z/h Number vn Vg vn Vg vn Vg vn Vg m U V T T Q P S S S R R R O O L L L ol N N M M M W W I I J o J K K o H G G E F C ø C C A A D B B B

10 27,830 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL Table 3. Comparison of observed and computed semimajor axim(cm s-l), orientation 0(degrees), and rotation R of the M current ellipse at a number of depths and locations. Rig Observed Calc 1 Calc 2 Calc 3 h z/h Number / 0 R / 0 R / 0 R / 0 R m U V T T Q P s s s R R R O O L L L N N M M M W ø W I I ø0.27 J J K K H G G E F C C C A A D B B B

11 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL 27,831 Fx= P,,Qx N2z dh Since we are considering idealized temperature distributions, the across-shelf and along-shelf distributions of the forcing function F x co t h 2 dx can be readily computed. Contours of the across-shelf (taken as and west-east) component of F (namely Fx) at a depth of 50 m below the surface (the depth at which N 2 is a maximum; see Figure 4 in both F = P"Q' 'N2z dh (2) winter and summer) show a region of maximum F, at the shelf edge Y coth 2 dy at approximately 56.8øN (Figure 3d). (Only the contour plot for summer stratification is shown, with winter stratification showing where Qx is the cross-shelf volume flux and Q¾ the along-slope a similar spatial distribution, since the vertical variation of N 2 is volume flux due to the barotropic tide, N is the buoyancy tyequency, similar in both cases, although in winter it is reduced by a factor of h is water depth, dh/dx is the cross-shelf topography slope, dh/dy the about 4.) It is interesting to note that besides the region of along-shelf topography slope, co, the tidal frequency, and P0 a intensified F along the shelf edge, it also increases on the western reference density. Equation (2) states that the internal tide forcing and eastern sides of the Anton Dohrn Seamount. Contours of the function is proportional to the above-mentioned three factors. along-shelf (north-south) component (F 0 also show (Figure 3e) that Contour plots of Fx and F,, are given in Figures 3d and 3e and are this component of F is increased in the region of the seamount, discussed later. where it is larger than the west-east component because of the larger In order to gain insight into the generation of the internal tide barotropic north-south component of the tidal energy flux in this along the shelf edge region, it is desirable to perform numerical region. Although at 56.8øN the west-east component of F (namely experiments with two different structures of stratification, namely F 0 dominates the shelf edge, further north at 58.6øN where the a weakly stratified surface layer, which simulates the winter or shelf slope is gentler, the north-south component (Fv) dominates. transitional season, and a strongly stratified surface layer, which These contour plots of F clearly show significant spatial variability simulates the summer stratification. Figures 4a and 4b show the in the forcing function in the region, although it is clear that internal profiles of the temperature and corresponding buoyancy frequency tide generation will be a maximum along the shelf edge at 56.8øN, for both cases. The main difference between the two temperature and that some internal tide will be generated over the Anton Dohrn profiles is in the top 100 m of the surface layer. In the weakly Seamount. stratified case, the temperature gradient in this surface layer is 0.01 On the basis of the summer profiles N 2 shown in Figure 4, and "C/m, and in the strongly stratified case, the temperature gradient in the shelf slope in the shelf edge region, idealized calculations the surface layer is 0.05 "C/m. Farther down the water column in suggest that a super critical (off-shelf propagating) internal tide will the deeper water, the temperature gradients for the two cases are be generated along the upper part of the shelf slope, although farther identical and smaller than those in the surface layer. This deeper down the shelf slope a subcritical internal tide can be generated. change in temperature represents the permanent temperature This together with the spatial variability found in F, suggests that structure in this area [Ellett et al. 1986]. There is no initial the internal tides will show significant horizontal variability in the horizontal temperature structure in the model in order to minimize region with both on-shelf and off-shelf propagation. (This will be the complexity of the internal tidal results. Without a detailed discussed later in the paper.) survey of the three-dimensional density field, such detail cannot be Although, as stated above, the primary aim here is to study the included. For this reason we also consider only the vertical changes in the internal tide due to seasonal changes in stratification variation in temperature, whereas a detailed simulation would using two different (summer and winter) stratifications, we will also require a knowledge of salinity. As we will show, even under the compare computed currents with the set of observations given in idealized conditions considered here the three-dimensional Tables 2a, 2b, and 3 and with Calc 1 (barotropic tide) in order to variability of the internal tide is very significant. determine at which locations discrepancies between model results -200 T p, rofiles ß 0 ''' I ''' I '/... "1 ''' -200 N^2 (10"'-6) 0 ', e..,.,...,. ;,,,.,.,.' ' OO OO ' I I O0 5O Figure 4. Vertical profiles (a) temperature ("C) and (b) square of buoyancy frequency (s-2) used in the numerical calculations. Solid lines indicate the weakly stratified case and dotted lines indicate the strongly stratified case.

12 27,832 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL and observations may be due to the internal tide. The influence of motion. The internal current, or baroclinic current as it is sometimes seamounts, namely the Anton Dohrn and Hebrides Terrace referred, is defined in the same way. The "tide" and "current" refer seamounts (see Figure lc) is also considered. In summary, the to the total tide or current (a solution including both internal and experiments of a weakly (strongly) stratified case are referred to as barotropic tide). Calc 2 (Calc 3), and the experiment using modified bottom In Figure 5, we present a snapshot of the internal current at the topography (strongly stratified) is referred to as Calc 4. sea surface (o=0.0) after a 5 day model integration when the internal tide has been established. From the figure, it is evident that strong internal tidal currents occur along the shelf edge, in particular, in the 4.2. Results of a Weakly Stratified Case region between 56.5øN and 57øN. As shown previously (Figure 3d), (Calc 2, Winter Stratification) this is the region where the cross shelf component of F is a maximum. Over the shelf, the internal tidal currents are much The calculation of the internal tide is obtained by integrating the smaller, except in the region near the North Channel. The tidal model from a state of rest with M 2 tidal forcing along the open dissipation (both the bottom friction and internal friction) is the boundary for 10 tidal cycles. Unlike the barotropic solution obtained main reason for the rapid decay of the internal tide on the shelf. The in the last section, strictly speaking, an absolutely periodic solution strong internal tidal current near the North Channel may not be cannot be achieved since the density field is now evolving. realistic. In this area, although the slope of the bottom topography However, 10 tidal cycles (over 5 days) is the time scale needed to is much gentler than that along the shelf edge, the barotropic tide is establish the internal tidal structure. The internal tides are much stronger. In reality, the water in this region tends to be determined from this solution by subtracting the solution without homogeneous due to strong tidal mixing, and a tidal front divides including internal motion (barotropic tide). Alternatively we can the stratified and homogeneousea regions [Howarth, 1982]. These define the internal tide as the depth-dependent mode in the model local factors have not been taken into account in the present study solution since the depth mean and depth-dependent mode are aimed primarily at the shelf edge. Also model resolution may be computed separately. It is important to note that there are slight insufficient in this region to accurately representhe flow in the differences in these solutions in that in the barotropic model, current channel between locations 39 and 40 (Figure la). Therefore care structure can be generated by viscous forces alone, whereas in the should be taken in interpreting results near the North Channel area, baroclinic model, both viscous and internal pressure gradients which is outside the scope of the present study. Over the deeper generate current structure. In the deep water region (water depth water side of the shelf edge, the internal tidal current has clearly larger than 100 m), the internal tides defined this way are very close been influenced by the Anton Dohrn and Hebrides Terrace to those defined by subtracting two different model runs since seamounts, and strong baroclinic currents over the two seamounts frictional effects are small. However, in the shallow water region can be seen in Figure 5. The influence of the seamounts upon the (water depth less than 100 m) where bottom friction is strong, the internal tide will be discussed later in this paper. effects of the bottom friction should be taken into account to obtain the internal tide. To avoid confusion, in this and the next sections, we define "internal tide" as the difference between the computed tide including internal motion and that without including internal 58øN 57øN 56øN Figure 5. The internal tidal current (defined as the difference between the computed current including internal motion and that without internal motion) at the sea surface for the weakly stratified case after a 5-day model integration. The dashed lines denote water depths of (left to right ) 2000 m, 1000 m and 200 m. -4ow The time series of temperature, current velocities (total velocity u and v, which can be regarded as cross-shelf and along-shelflow, respectively), and turbulent eddy viscosity at two locations near the shelf edge are presented in Figure 6a and b. The two locations (P1 and P2), shown in Figure 1 c, have a water depth of 520 m and 125 m, respectively. The temperature contours clearly show the asymmetry over the tidal cycles, in particular at location P2. In a previous calculation [Xing and Davies, 1996b], using a cross-shelf "slice" model, the tide was decomposed into its various harmonics. Results of this decomposition showed that some of the asymmetry in the temperature field over a tidal cycle could be attributed to the generation of higher harmonics due to the nonlinear nature of the internal tide, although, as shown by Hibiya [1986], a linear model can generate waves of a complex shape due to tidal flow over a sill. The tidal phase difference over the water depth can be seen in the current velocity plots. In the case of the deep-water location, the along-shelf velocity contours show a nearly 150-degree phase difference over the depth, which has been observed over the continental shelf edge. At the shallower-water position, where the internal tide is much smaller, there is little phase difference except in the bottom boundary layer. Another important feature shown in the time series plots is the strong near-surface and bottom currents, which result in strong turbulence mixing near the bottom boundary. However, outside the bottom boundary layer, the internal vertical mixing generated by the internal tide seems to be small. There is little difference between the computed amplitude and phase of the M 2 tidal elevation with and without including internal motion in the model. Therefore the cotidal chart for this calculation will not be presented here. The distribution of the major and minor axis of the M 2 tidal current ellipse (barotropic plus baroclinic) is shown in Figure 7a (at the sea surface) and in Figure 7b (near the seabed). Compared with the barotropic tidal current ellipse shown

13 XING AND DAVIES' THREE-DIMENSIONAL INTERNAL TIDE MODEL 27,833 Tem,perotur, e,( C,), -200 '-400-6OO I I I,,,, visc Io m^2/s OO -- = -6OO bt6btll i= Figure 6a. Contours of temperature ("C), u and v components of velocity (cm s'l), and eddy viscosity (log 0 m2/s) over two M2 tidal cycles at location P 1 (see Figure 1 c) for the weakly stratified case. in Figure 3, it is evidenthat there is a stronger tidal current along evident in the case of the results including the internal tide, but the the shell' edge and on the deep-water side of the shelf edge. computed v velocity is smaller in the lower part of the water colunto. A detailed comparison of computed and observed amplitude and At location N, with water depth of 145 m, we see that the tidal phase of the u and v components of velocity and current ellipse current increasesignificantly from that at M, although the two parameters are given in Tables 2 and 3. Compared with the locations are within a few grid boxes of each other. In the upper part observations, the computed results including internal tides (Calc 2) of the water column, both results (Calc 1 and Calc 2) agree very are only marginally better than results that do not include internal well with the observations. However, in the lower part of the water fides (Calc 1). As mentioned previously, the data in the tables were column, the u velocity computed withouthe internal tide (Calc 1) chosen primarily from periods when the internal tides were largest significantly overestimates the observations (20 cm s ' versus 11 - (the summer, see later). In wintertime, with reduced vertical 14 cm s'l). The results with the internal tide (Calc 2) improve stratification, the internal tidal signal is much weaker, and hence slightly (17 cm s-l). At locations I, J, and K, as mentioned before, tidal current profiles are not substantially different from those found the barotropic calculation underestimates the velocity significantly. under homogeneous conditions. Consequently, a significant overall The calculation including the internal tide show some, although no improvement in the agreement with observations would not be significant, improvement. expected. However, it is interesting to examine changes at locations To understand why including the internal tide could improve near the shelf edge and away from the open boundaries, namely, results at one location but produce little change at other locations, locations M, N, W, J, K, H, E, etc. in Table 2 where the internal tide it is necessary to examine its spatial variability. In Figure 8, we may have an effect. At location M, where the water depth is over present plots at three cross-shelf sections (C1, C2, and C3; see 1600 m, the observed amplitude of the cross-shelf (u) component of Figure lb) of the amplitude of the cross-shelf internal tidal current velocity is small (< 2 cm s"), and the amplitude of the along-shelf (u component of velocity), internal tidal displacement, and internal component (v) of velocity is about 5-8 cm s -. The computed results tidal energy density. (both with and without the internal tide) are comparable with the The internal displacement computed by observations. The computed tidal current withouthe internal tide w (barotropic tides) shows no vertical structure for both amplitude and n = --, (3) to t phase. However, the amplitude of the observed u component of current increases the lower part of the water column. This is also where w is the amplitude of the vertical velocity (in z coordinate),

14 _. 27,834 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL o - d.o... Temperoture ( C ) 0-5o -loo -9.0 f o i... o.o o I i [ [ ,. visc -5O -100 / i Iiiiiiiiii v ii illlllll. ( 1 iiiiiiii \ / i I IIIIIII / i IIIIIii ' / J,11,1111 I -- ///////'; LL k\.. _: -5O -loo 4-150,,, bt6btl 1 i=5 i Figure 6b. As in Figure 6a, but at location P [ and to t is the M 2 tidal frequency. It is importan to note that (3) is only a linear approximation to the internal displacement and is in essence a scaled vertical velocity with the scaling factor co t. It is used here for convenience to give an indication of vertical displacements, which are commonly reported in observational papers. As Holloway [1996] pointed out, this definition of the with u i, v,, w i the three components of the internal current velocity, and the angle brackets indicate a time average over an M 2 tidal cycle. The amplitudes of the cross-shelf component of internal tidal current (Figure 8a) clearly show the bottom intensification of the internal tide. Differences in the topography profile among the three vertical displacement may be influenced by a barotropi component, cross-shelf sections result in the differences in the details of the Zt,, given by internal tidal structure for each crossection. For example, at cross u dh section C1 the maximum internal tidal current occurs at a water co, dx (4) depth of about 700 m below the surface, where the topography slope is large. At C2, the shelf edge slope is larger and the strong slope Alternatively, the displacement of the internal tide can be calculated occurs at about m below the sea surface, and therefore the by maximum internal tidal current occurs at a shallower depth. W- W h The distribution of the vertical internal tidal displacement (Figure n -, (5) COt 8b) shows a very large displacement (maximum over 40 m) near the shelf edge. The strongest displacement occurs at section C2. Also, where w h is the amplitude of the vertical velocity for the barotropic at this section the strong displacement occurs further offshore at tide. The results computed using the above definition have some, about 10.5"W (over 20m), which is consistent with the strong but not significant, difference over the shelf edge. To be consistent internal tidal current. The distribution of the internal tidal energy with previous work [Xing and Davies, 1997b, c], we use the density (Figure 8c) reflects that of the internal tidal current definition given by (3). amplitude and displacement. The internal tidal energy density is computed as a sum of the The high degree of spatial variability shown in Figure 8a, internal tidal kinetic energy and the potential energy of internal suggests that the contribution of the internal tide to tidal current elevation [Sherwin and Taylor, 1990]; thus, the intemal tidal energy structure is quite sensitive to the exact location where measurements density assuming a linear approximation is given by are made. If measurements are made in a location where the internal tide is not significant, then the difference between model results and Ei = < measurements will not change when stratification effects are 2 Pø(Ui + vi + wi )> + ' PøN ' (6)

15 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL 27,835 58øN 57øN 1.0 ms" Snapshots of the internal tidal current at the same time as that given previously are presented at three vertical levels, namely, at the sea surface (Figure 9a), 100 m below the sea surface (Figure 9b), and near the seabed (Figure 9c). Compared with the weakly stratified case, the internal tidal current is much stronger along the shelf edge (of the order of 15 cm s ' near the surface). This increased amplitude can be attributed to the larger forcing function F in summer than winter because of the increase of N 2 by a factor of 4 in the surface layer. The contours of F presented previously clearly show that the internal tide is primarily generated on the shelf a o u 56øN - 10øW -8øW -6øW -4øW I I I 58ON - 57øN 56øN Figure 7. Computed major and minor axis of the M 2 tidal current ellipse, including the intemal tide (weakly stratified case). (a) At sea surface, and (b) near the seabed. lo 8 W (deg,) u amp. (cm/s) 0 I ' U,' x included, whereas modelled results at a nearby location where internal tides are important will change significantly. Also, results from cross-sectional models [Xing and Davies, 1997b, c] show that small changes in density field and details of topography in some locations produce a significant change in the distribution of the internal tide. Consequently, a detailed three-dimensional knowledge of these is required. - 1 ooo 4.3 Results of a Strongly Stratified Case (Calc 3, Summer Stratification) In this case, we shall see that a strong sea surface temperature gradient has a profound influence on the internal tide generation and Figure 8. Contours propagation, not only in the surface layer, but also throughout velocity (cm s' ), almost the whole water column along the shelf edge and on the internal tide energy shelf x=110 lo 8 of (a) the amplitude of the internal tidal u (b) intemal tide vertical displacement (m), and (c) density (Jm" ) along three cross-shelf sections, C1, C2, and C3, as shown in Figure lc (weakly stratified case).

16 _ 27,836 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL b o displacement amp. (m), 1 C 0 KE+PE (d/m"3), -looo max= lo w (deg) displacement amp. (m) KE+PE I -5oo - 1 ooo ['. max C displacement amp. (m), I KE+PE.(J/m^5) -5OO looo m x= t mox= Figure 8. (continued) slope and in the region of the seamounts, which helps to explain the value of the first internal wave mode, based upon the density profile significant spatial variability seen in Figure 9. Also, from an used in the model. analysis of the vertical density profile and shelf slope, the internal To examine in more detail the cross-shelf variation of the internal tide generated the upper part of the shelf slope is supercritical and tide and the role of the seamounts in internal tide generation, it is propagates out into the ocean. However the internal tide generated instructive to plot contours of the internal tidal current amplitude on the lower part of the shelf slope is subcritical, and can propagate and displacement at the M 2 frequency at cross section D1 through both off and on shelf. The magnitude of the internal tide the Anton Dohrn Seamount and cross section D2 through the propagating onto the shelf is reduced significantly toward the Hebrides Terrace Seamount (Figure l c). Contours of the amplitude coastal region because of turbulence dissipation, both vertical and of the internal tidal current and displacement together with the phase horizontal, as well as the bottom friction. The internal tidal wave of the M 2 internal tidal surface current along cross section D1 wavelength is about 45 km, which is consistent with the theoretical (Figure 10a) show a region of enhanced tidal current amplitude in

17 XING AND DAVIES: THREE-DIMeNSIONAL INTERNAL TIDE MODEL 27,837 the surface layer on the shelf side of the shelf slope, with further regions of enhanced surface current to the west of the shelf slope and in the proximity of the seamount. Regions of maximum internal tidal displacement are clearly evident near the top of the shelf slope and further down the slope. Also regions of enhance displacement occur on the western and eastern side of the Anton Dohm Seamount, with a region of maximum displacement in excess of 34 m occurring at a water depth of about 450 m above the Anton Dohrn Seamount. A second local maximum (amplitudes exceeding 18 m) is evident at about 10øW in a water depth of 1400 m. (The maximum barotropic contribution to these displacements computed from (4) is less than 5%; hence the main contribution is from the internal tide.) These spatial distributions of current and displacement amplitude, together with the contour plots of F x, and an analysis of the slope of bottom topography and the local characteristic slope for internal waves Ci [Craig, 1987] given by - 10øW -8øW -6øW -4øW %-f Ci =,- 2 - '-2 (7) N -% b 58øN 57øN 56øN 58øN 57øN 56ON - ', '..:x '?,' - 10øW -8øW -6øW -4øW (withf the Coriolis parameter) suggesthat along this cross section, both supercritical and subcritical internal tides are generated along the shelf slope and on the sides of the seamount, producing internal tides which propagate both onto the shelf and into the ocean. The plot of the phase of the u component of the surface current shows the phase increasing to the west and east of the seamount. (The abrupt decrease in phase to the east of the seamount is due to the phasexceeding 360ø.) The phase reaches a maximum at a position about 10.2øW and then decreases toward the shelf edge with a minimum phase at the top of the shelf break and an increase phase to the east of this. This phase plot, together with the spatial distributions of tidal current amplitude and phase, suggests that the tide is generated both along the slope and on the sides of the seamount, and can propagate both toward the shelf and the ocean from these generation points. A discussion of the role of the seamount in internal tide production is presented later. In the case of the second cross section (Figure 10b) through the Hebrides Terrace Seamount, contours of current amplitude show off shelf regions of intensified surface current, suggesting internal tide propagation from the shelf edge into the ocean, although there is also an internal tidal signal on the shelf. Regions of enhanced bed current amplitude and intemal displacement (Figure 10b) are clearly evident along the shelf slope at a depth of the order of 1000 m and in the vicinity of the Hebrides Terrace Seamount. A region of enhanced vertical displacement occurs approximately above the Hebrides Terrace Seamount at a depth of about 600 m. As in the previous cross-sectional plot of the phase of the u component of surface current, this maximum is approximately midway between the top of the seamount and the top of the shelf slope, and decreases to the west and east of this, although an increase on the shelf is evident. The influence of the seamount upon this will be discussed later in the paper. Time series of temperature, u and v components of current velocities (total velocity), and eddy viscosity over two tidal cycles at two locations near the shelf edge (see Figure 1 c) are presented in Figure 1 la for location P1 (water depth 520 m) and Figure 1 lb for - 10øW -8øW -6øW -4øW Figure 9. The internal tidal current after a 5-day model integration: (a) at the sea surface (o=0.0),(b) 100 m below the sea surface, and (c) near the seabed (o=-1.0)(strongly stratified case).

18 27,838 XING AND DAViES: THREE-DIMENSIONAL INTERNAL TIDE MODEL [ t; b o u o. (cm/s) -SO0 -looo OO w (U g) w (U g) sp/ am d,splacement amp (m) _,ooo max 4g 2 10 w ( ) w ( s) 1 0 [)hose at sea surface phase at sea surface 200 2oo lo W Figure 10. Contours along (a) cross section D 1, and (b) cross section D2, of internal tidal current amplitude (cm s 'l) and displacement (m). Also shown is the phase of the surface current. Locations of D1 and D2 are shown in Figure 1 c (strongly stratified case). lo w (a g) location P2 (water depth 125 m). At location P1, the temperature viscosity and difthsivity, there is also a large eddy viscosity beneath changes from 14øC in a surface well-mixed layer having a thickness the strongly stratified surface layer, centred at about 150 m depth, of order 20 m to the order of 9.5øC at a depth of about 90 m below due to the combination of weak stratification (which reduces the the sea surface. Associated with this thermocline is a region of density suppression of the turbulence) and velocity shear (which is velocity shear in both the across-shelf (taken as u component) and the main production term for the turbulence). The maximum of the the along-shelf (taken as v component) velocity. Compared with the eddy viscosity and diffusivity (not shown here, but similar to the weakly stratified case, there is a significant increase of the along- eddy viscosity) over 1 m 2 s 'l, which should have a significant shelf component of velocity (v) near the bottom boundary layer; but influence on the biological productivity. near the surface, the v component of velocity has a similar value to At location P2, we see a stronger asymmetric structure in the that of the weakly stratified case. There is a large phase difference temperature field over the tidal cycle than that in the weakly in the v component of velocity over the water depth (almost 180ø). stratified case. The tidal current velocity (both u and v) has a In contrasto the along-shelf velocity (v), the cross-shelf component maximum centred at about 80 m below the sea surface, just above of velocity (u) has a strong intensificationear the surface layer, and the bottom boundary layer. Compared with the weakly stratified is twice as large as in the weakly stratified case. However, the u case, there is a significant increase of the along-shelf component of velocity near the bottom has only a slight increase. In addition to a velocity, in particular, a strong tidal residual velocity of over 10 cm strong bottom turbulence boundary layer and large bottom eddy -is near the sea surface (the upper 10 m of the water column).

19 _ XING AND DAVIES' THREE-DIMENSIONAL INTERNAL TIDE MODEL 27,839 o Temeeroture ( C ) v cm/s -. [ J/ - t J , o visc Io s (m^2/s) O bt4bt11 i= Figure 11a. Contours of temperature CC), u and v components of velocity (cm s' ), and eddy viscosity (log 0 m2/s) over two M 2 tidal cycles at location P 1 (see Figure 1 c) for the strongly stratified case. The results of two different stratifications (Calc 2 and Calc 3, see The computed major and minor axis of the M 2 tidal current ellipse at the sea surface and near the seabed are presented in Figure Table 4) show that at cross-shelf sections C1 and C3, the maximum 12. Compared with the results without the internal tide (Figure 3) u amplitude, displacement, and internal tidal energy density have and the weakly stratified case (Figure 7), the current ellipse relatively smaller changes as a result of the different stratification than at cross section C2. This is because the bottom topography distributions and magnitudes confirm thathere are strong internal tides propagating both on the shelf and off the shelf. The cross-shelf gradient and maximum of the internal tide occur in deeper water edge current is stronger than that in the weakly stratified case, as ( m) while the density difference for the two cases only seen in Figure 11 a. occurs in the top 100 m surface layer. At cross-shelf section C2, The distributions of the cross-shelf component of internal current there is nearly a 50% increase of the maximum cross-shelf velocity amplitude, vertical displacement, and internal tidal energy density amplitude due to the stronger surface stratification (16.0 cm s ' at C1, C2, and C3 are presented in Figure 13. These results should versus 10.9 cm s-t). This results in an increase four times the be compared with the results of the weakly stratified case in Figure internal tidal energy density. However, the maximum internal tidal 8. A summary of the comparison in terms of the maximum values displacement, which occurs in deeper water, only changes by 0.9 m of the u velocity amplitude, displacement, and the energy density for (i.e., 44.8 m versus 43.9 m) the three sections and different calculations is given in the Figures These spatial distributions clearly show (in a comparison of 8 and 13, and summarized in Table 4. It must be pointed out that Figure 3a and 8a) that with summer stratification the magnitude of the maximum value is not necessarily the best parameter for the the u component of current in the surface layer is significantly comparison, but it should give some insight into the influence of the increased, with substantial changes in the near-bed region along the topography profiles on the internal tide. From these values (Table slope. This clearly demonstrates that to rigorously validate the 4), it is evident thathe strongest internal tides occur along cross model, a comprehensive instrument deployment in the surface layer section C2, which is consistent with the fact that the strong and down the slope during the summer period is necessary. topographic slope and strong cross-shelf component of the Comparing observed and computed current components (Table barotropic tide (Figure 3c), together with a maximum in the 2a, b) and ellipse parameters (Table 3) it is evidenthat at the contours of F (Figure 3d), occur along this section. majority of locations there is very little difference between the

20 . 27,840 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL o Temperoture,(C')... i,,, i o loo o o o.o O -5O -IO0 -IO0-150,,, I I I ,... i bt4btl 1 i=,.3 Figure 11b. As in Figure 1 la, but at location P results obtained with typical summer and winter stratification. At location R, the near-bed u current (o = -0.84) is slightly reduced by summer stratification with a slight increase in the surface current. A similar, although slightly greater, change in the v component of current between summer and winter stratification also occurs at this location (Table 2b), which significantly increases the difference in the vertical between near-surface and near-bed current. However, at nearby location S there is no change in the near-bottom current and I t//,x////////// ttt// //.+ t/// t//+ \1 {t// 58øN --./ + {tllt 57øN 1.0 ms ' 56øN - 10øW -8øW -4øW Figure 12. Computed major and minor axis of the M2 tidal current ellipse, including the internal tide (strongly stratified case), (a) at sea surface; (b) near the seabed.

21 .. XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL, 27,841 a o u omp. (cm/s) bøl _ dis placernent amp. (rn) lo 8 w (deg) 0 u am. cm, s -5OO max= u omp. (cm/s) dispmocement omp t'k' ''_... '"-: ' ' o ' ' lo o q max=40 0 Figure 13. Contours of (a) the amplitude of the internal tidal u velocity (cm s- ), (b) internal tide vertical displacement (m), and (c) internal tide energy density (Jm'3) along three cross-shelf sections, C1, C2, C3, as shown in Figure 1 c (strongly stratified case). only a slight increase the surface current between summer and decreasing from l 7. l cm s -t to 13.1 cm s ' between calculations with winter stratification. Farther south along the shelf edge at locations winter (Calc 2) and summer (Calc 3) stratification. With the M and N, the intensification of the tidal current from deep oceanic summer stratification, both the computed u and v components of location M (h=1614 m) to shallower location N (h=145 m) is clearly current and the current ellipses (Table 3) are in good agreement with evident. At location M in deep water, the difference in tidal current observations. amplitude and phase between calculations with winter (Calc 2) and Farther south at locations I, J, and K, as discussed previously, the summer (Calc 3) stratification is small. However, at location N the barotropic tidal current is underestimated in the model, and hence amplitude of the u component of near-surface current (o = ) the magnitude of the barotropic tidal forcing, which is required to increases from 15.6 cm s ' to 18.1 cm s ' with current at depth produce the baroclinic tide, will not be correct.

22 27,842 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL O -ooof, /t o KE +PE (J/m^5) lo 8 Kœ+Pœ J m^3 since our primary aim here is to understand in a qualitative manner the influence of the seamounts, such a change which does not modify the shelf slope or deep-water topography is a reasonable approach. (It is important to note that the seamounts were not removed by assuming a constant topography seaward of the base of the shelf slope, as this would significantly and artificially (in the sense of modifying the ocean depths) affect the propagation of the internal tide.) The shelf, ocean, and shelf edge topography is identical to the previous calculations. Only the strongly stratified case is presented here. The only difference between this calculation and the strongly stratified calculation (Calc 3) is the removal of the two seamounts. A snapshot of the internal tidal current (Figure 14) at the same time as those presented previously reveals a significant change over the deeper-water side of the shelf edge. Compared with results of the realistic bottom topography case (Calc 3, Figure 9), it is evident that the spatially varying currents that previously appeared over the seamount topography are replaced by a more spatially uniform internal tide propagating toward the ocean. The internal tide is approximately parallel to the shelf edge topography, suggesting that in this case they could be modelled by cross-section slice models. -5oo To examine in detail the influence of removing the seamounts upon the magnitude and propagation of the internal tide, it is instructive to compare contours of current amplitude, displacement, and phase of the surface current along cross sections DI and D2 (Figures 15a, 15b) with those obtained previously (Figures 10a, - looo 10b). A comparison of u current amplitudes and phase of the surface L (compare Figures 15a 10a) shows a similar distribu of current amplitudes on the shelf and at the top of the shelf slope, -15oo 1 o s with similar changes in the phase of the surface current between 10øW and the shelf. Although there is a slight intensification of surface current at 11 øw in the absence of the seamount, suggesting -5OO KE +PE an off-shelf propagation of the internal tide (away from its generation point at the shelf slope), the magnitude of the internal tidal current in this region is significantly less than that found with the seamount. Also, the fairly regular change in phase of the surface current in the off-shelf direction, which occurs in the absence of the seamount, is significantly changed by the presence of the seamount. Similar comparisons of the internal displacements with and without the seamount confirm that the presence of the seamount significantly max= 9 1 Table 4. Comparison of the maximum of the u velocity amplitude (U ),displacement (rim) and internal tidal energy density (Em) at three crossshelf sections (C1, C2. and C3) Figure 13. (continued) 4.4. Results Using Modified Bottom Topography (Calc 4, Strongly Stratified Case) Calculations using the three-dimensional model. In this calculation, we deliberately remove the Anton Dohrn and Hebrides Terrace seamounts from the bathymetry data. This was done by removing the decrease in sea depth associated with each of the seamounts and linearly interpolating water depths in the region of the seamount to give a smoothly varying topography comparable to that found away from the seamount. Obviously, any such modification of the topography is artificial and subjective; however, Cross Sections Calc 2 cm/s m J/m 3 C C C Calc 3 C C C Calc 4 C C C

23 XING AND DAVIES: THREE-DIMeNSIONAL INTERNAL TIDE MODEL 27,843 58øN 57øN 56øN 58øN 57øN 56øN 58øN 57øN... :,,,,,t,< < /) _ / :.,; : ';..,,, :..... ' ///// Lg ZL ;%'_,,, ] i.,,,,..... //.". t/,,... l... 1 ;. X' :..., 0.1 m s '.....:,,, Z ' ';... ' _ øW -8øW -6øW -4øW influences the magnitude and propagation of the internal tide in the off-shelf region. A similar comparison at cross section D2 (compare Figures 15b and 10b) shows that the off-shelf intensification which occurs in the region of the seamount disappears when the seamount is removed. In this case, the seamount is closer to the shelf edge, and small changes in current amplitude and internal displacement, together with changes in phase of the surface current, also occur at the top of the shelf slope and on the shelf. The influence of the seamountopography on the internal tide is further explored in the cross-section plots of the amplitude of the M= component of the cross-shelf velocity (Uh), vertical displacement (rl), and internal tidal energy density (E ) at section C2 (Figure 16) and the maximum values of Uh, rl, and E h in Table 4 (denoted Urn, rh,,, E,,,) (Calc 4). It is evident from Table 4 and Figure 16 that the seamount's influence is not limited to the vicinity of the seamount. Along the shelf edge, where the maximum of the internal tidal amplitude occurs, there is a significant decrease of the internal tidal intensification. For example, at section C2 the maximum of Uh (Urn) reduces from 16 cm s 'l to 12.7 cm s 'l, maximum displacement changes from 44.8 m to 32.6 m, and maximum internal tidal energy E, changes from 23 J/m to 15.9 J/m 3. Obviously, the exact value of U,, r/,, and E,, will depend on the detailed gradient of the shelf slope in the region where the seamount was removed, and as stated previously, such a removal by its nature is artificial. However, these changes are indicative of the influence of the seamounts. This calculation suggests that in this region the seamounts have an important influence upon the internal tide and consequently not only must the gradient of topography and density field be accurately determined at the shelf edge, but also in the vicinity of the seamounts Calculations using a cross-shelf (x-z) slice model. In order to gain some insight into the extent to which an internal tide is generated a seamount in isolation and the seamount's influence on the on and off-shelf propagation of the internal tide, it is instructive to set up a cross-shelf slice model. The shelf slope topography is typical of that found in the three-dimensional model, with the seamount located on the shelf slope having a height approximately the average of that of the Anton Dohm Seamount and the Hebrides Terrace Seamount, and hence may be regarded as a typical height. Although such a model is more idealized than the three-dimensional model presented previously, it does enable the role of the seamount upon the on-shelf and off-shelf propagation of the internal tide to be clearly illustrated in isolation from the northsouth propagation of the tide. Contours of u current amplitude (Figure 17a) show an internal tide on the shelf with an increase in near-bed current magnitude at the top of the shelf slope. An increase current in the near-bed region at approximately 60 km offshore is also evident. Increased bed currents also occur on the eastern and western side of the seamount, with the strongest bed currents on the eastern side. A region of reduced current magnitude is evident above the seamount with an intensification surface current magnitude at about 170 km offshore. Contours of current amplitude and displacement without the seamount (Figure 17b) at the top of the slope and on the shelf are 56øN Figure 14. The internal tidal currents after a 5-day model integration: (a) at the sea surface (o = 0.0), (b) 100 m below the sea surface, and (c) near the seabed ( o = - 1.0) (for the case withouthe Anton Dohrn and Hebrides Terrace seamounts).

24 27,844 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL 0 d,spiocement omp (m) q g' - o d,splocement omp (m) I ( - ' ' -looo - 15oo max= max=26 7 I 10 ' 1 I 0 phose ot see surfoce phose ot seo surfoce o w (de9) 0 lo w (de9) Figure 15. Contours along (a) cros section D1, and (b) cros section D2, of internal tidal current amplitude (cm s -1) and displacement (m). Also shown is the phase of the surface current for the case without the Anton Dohrn and Hebrides Terrace seamounts. Locations D1 and D2 are shown in Figure lc. not significantly different to those found previously (compare Figures 17b and 17a), although at depths of the order of 1000 m along the slope there is a significant change in the contours of the current amplitude. However, current contours at this depth at the In order to examine the seamount in isolation, water depths to the east of the seamount were fixed at the maximum value found in this area, namely approximately 1700 m, thereby removing the influence of the slope. Contours of u current amplitude (Figure 17c) show an offshore location where the seamount was previously situated intensification of near-bed currents to the west of the seamount (approximately 140 km offshore) are much smoother and comparable to that found previously with the seamount and slope significantly differento those found previously. The intensification (compare Figures 17c and 17a). A similar intensification, although of the surface current at approximately 160 km is still present, although with a reduced magnitude, to that found with the seamount. with current magnitude below that found with the slope, is evident in the near-bed currents to the east of the seamount. The reduction in current magnitude on this side of the seamount when the slope is The change in u current amplitude in this series of calculations removed is probably due to the fact that when the slope is present, produced by removing the seamount are comparable to those found previously in the three-dimensional model, suggesting that the cross-sectional model is an appropriate tool to use in examining the internal tidal energy generated along the shelf slope, besides propagating toward the shelf, propagates offshore down the slope (Figure 17a) and influences the near-bed current on the eastern side seamount in isolation. To examine the effect of the seamount in of the seamount. Regions of intensified surface current are evident more detail, the previous cross-shelf slice model calculation was repeated with the slope removed and only the seamount remaining. to the west of the seamount at approximately 160 km offshore and to the east of the seamount at approximately 80 km offshore, and to

25 . XING AND DAVIF. S: THREE-DIMENSIONAL INTERNAL TIDE MODEL 27,845 a o may explain the complex variations in surface current magnitude in the presence of slope and seamount. This series of calculations using both the three-dimensional and two-dimensional cross-section models, together with the contour plots of F x and F, and tidal energy flux vectors, suggests thathe seamounts can generate a local internal tide and can influence the magnitude and distribution of the internal tide away from the shelf slope t max= w (d ) a -looo b o 2-15oo I km td7s3 M lo 8 ß,, KE+PE,(J/m^3) z, ß ß ooo '" , ' I L ' ' ' km td7s3m M2 mox= 15.9 u amp. (cm/s) Figure 16. Contours of (a) the amplitude velocity (cm s'l), (b) internal tide vertical displacement (m), and (c) w** of the internal tidal u -5oo - OW,,._. internal tide energy density (Jm '3) along the cross-shelf section C2 T - ooo Hebrides (marked in Terrace Figure seamounts. lc). For the case without the Anton Dohrn and a lesser extent at about 120 km offshore, associated with internal ' tide propagation on either side of the seamount as the direction of the tidal flow reverses. The difference in location of these two regions of intensified surface current relative to the top of the seamount arises from the asymmetric nature of the seamount. The km region of intensified surface current at 160 km goes some way to explaining why the surface current at this location is at a maximum when both the seamount and the slope are present. Also, the existence of a number of surface maxima to the east of the seamount td7sams M2 Figure 17. Contours of internal tidal current amplitude (cm s -1) along an idealized shelf edge cross section for (a) slope and seamount, (b) slope alone, and (c) seamount alone.

26 27,846 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL T1DE MODEL 5. Conclusion 5.1. Discussion of Results In this paper, we presented a three-dimensional, nonlinear, primitive equation model and its application to the Malin-Hebrides shelf and shelf edge, off the northwest coast of Scotland, where water depth ranges from a few meters near shore to over 2000 m in the deep sea. To first order, internal tides are generated along the shelf edge of the region as a result of the interaction between steep topography and strong barotropic tides in stratified conditions and propagate both on shelf and off shelf. However, a numerical calculation shows that the Anton Dohrn and Hebrides Terrace seamounts have important influences upon the propagation of the internal tide in the deep water which complicate the spatial distribution in the area. The calculations presented in this paper clearly show the importance of the variation in seasonal stratification upon the propagation of the internal tide in the region. The cross-shelf component of the barotropic semidiurnal M 2 tidal current changes from a few cm s - on the deeper side of the shelf edge to about 20 cm s ' on the shallower side of the shelf edge over a short distance. The model, with a fine grid resolution and without including internal tides, gave barotropic tidal elevation and currents which were not significantly different from those obtained previously [Proctor and Davies, 1996; Xing and Davies, 1996a] using identical forcing and topography, although a coarser grid. This suggests that the barotropic tide can be adequately resolved in this region on the present grid, which is confirmed by the comparison with observations. The primary reason for employing a finer grid here has been to use the model to examine the possible spatial variation and strength of internal tides under different stratification conditions. In a calculation with a weakly stratified surface layer, which represents a transitional or winter density condition, the internal tide generated along the shelf edge is much weaker than that found in summer, since the magnitude of the forcing function F x in the shelf edge region is about a quarter of that occurring in the summer because of the reduction of the maximum value of N 2. stratification is included. The important question that this raises is how to obtain a sufficiently large data set to rigorously validate a three-dimensional internal tidal model in the same way as has been done for barotropic tides [e.g., Davies et al., 1997]. Kwong et al. [ 1997] used over 290 current harmonic analysis data sets to validate a continental shelf wide model. This would require a synoptic set (because of changes in seasonal stratification) of measurements over the region of sufficient duration to perform a harmonic analysis with corresponding detailed measurement of the density field and bottom topography. The model calculations clearly show that the period of summer stratification is the time when the largest internal tides would be generated and that instrument deployments in the top 100 m and in the bottom boundary layer near the top of the shelf edge would be optimal, although measurements along the shelf and in the vicinity of the seamounts would also be required. A detailed survey to provide the necessary initial density field for the model would also be required. The difficulties due to meteorological effects found by Pugh and Vassie (1976) in accurately determining the tidal current amplitude and phase in shallow water from 60-day current meter deployments may be increased because of modifications of the internal tide by wind events [Xing and Davies, 1997c]. This suggests that longperiod deployments are required. Also, the influence of small-scale topographic features upon the spatial variability of the tides [Xing and Davies, 1998] implies an intense measurement program which, together with meteorological influences, suggests that a rigorous validation of internal tidal currents to the same level achieved for the barotropic tide may be very difficult. In this paper, we have not included the frictional influence of other tidal constituents, in particular, the S 2, N 2, K, and O tides. However, Cummins and Oey [1997] found that these constituents did not have a significant effect upon the M 2 internal tide. Also, Kwong et al. [ 1997] showed that these tides did not significantly influence the M 2 barotropic tide in the region. Hence our decision not to include them appears justified. (To have included these constituents would have involved running the present model for over 30 days, a computationally prohibitive task with the present However, with a strongly stratified surface layer, which represents grid resolution.) a summer stratification, internal tides generated along the shelf edge A more serious problem may be the influence of a slope current are much stronger. In this case, an analysis of the ratio of internal of oceanic origin upon the internal tide. However, without a wave speed and the shelf slope [Craig, 1987], suggested that both detailed knowledge of this flow it is difficult to include its effect within the model. subcritical and supercritical internal tides could be generated at various positions along the slope, with the subcritical tide giving rise to both on-shelf and off-shelf propagating internal tides. In the case Acknowledgments. The authors are indebted to the reviewers for of the weak stratification, the maximum vertical eddy viscosity and a number of good suggestions, to A. Bain for some computational diffusivity occur in the bottom boundary layer. However, with assistance, to R.A. Smith lbr help in preparing the diagrams, and to strong stratification, in addition to the intense strong bottom L. Ravera for help in text preparation. boundary mixing, strong turbulence mixing occurs beneath the stratified surface layer near the shelf edge due to the shear References production of turbulence in this region [Xing and Davies, 1997b], which may have important biological implications. Baines, P.G., On internal tide generation models, Deep Sea Res. Part A., 29, , Implications for Experimental Design Blumberg, A.F., and G.L. Melior, A description of a three-dimensional coastal ocean circulation model, in Three-dimensional coastal ocean The significant differences in the cross-shelf variability of the models, edited by N.S. Heaps, pp. 1-16, AGU, Washington, D.C., internal tide from one cross section to another and the spatial Chuang, W., and D.P. Wang., Effects of density front on the generation and changes in the forcing functions F x and Fy clearly show the propagation of internal tides, J. Phys. Oceanogr., 11, , three-dimensional nature of the internal tide in the region, with the Craig, P.D., Solutions for intemal tide generation over coastal topography, J. Mar. Res., 45, , final calculations showing the complexity introduced into the Cummins, P.F., and L.-Y. Oey, Simulation of barotropic and baroclinic tides internal tide distribution by the presence of the seamounts. At a offnorthem British Columbia, J. Phys. Oceanogr., 27, , limited number of locations, in particular position N, there are Davies, A.M., A numerical model of the North Sea and its use in choosing significant differences due to seasonal stratification, and as shown locations for the deployment of offshore tide gauges in JONSDAP '76 here, the computed solution is improved when the summer oceanographic experiment, Dtch. Hydrogr. Z., 29, 11-24, 1976.

27 XING AND DAVIES: THREE-DIMENSIONAL INTERNAL TIDE MODEL 27,847 Davies, A.M., Numerical problems in simulating tidal flows with a friction Maze, R., Generation and propagation of nonlinear internal waves induced by velocity dependent eddy viscosity and the influence of stratification, Int. the tide over a continental slope, Cont. She.lfRes., 7, , J. Num. Meth. Fluid., 16, , Melior, G.L., and T. Yamada, A hierarchy of turbulence closure models for Davies, A.M., and J. Xing, An inter-comparison and validation of range of planetary boundary layers, J. Atmos. Sci., 31, , turbulence energy schemes used in three dimensional tidal models, in Mellor, G.L., and T. Yamada, Development of a turbulence closure model for Quantitative Skill Assessment.[br Coastal Ocean Models, Coastal and geophysical fluid problem, Rev. Geophys., 20, , Estuarine Stud., Vol. 47, edited by D.R. Lynch, and A.M. Davies, pp.71 - New, A.L., Internal tidal mixing in the Bay of Biscay, Deep Sea Res., Part A, 95, AGU, Washington, DC., , , Davies, A.M., J. Sauvel, and J. Evans, Computing near coastal dynamics from Oey, L-Y., and P. Chen, A model simulation in the Northeast Atlantic Shelves observations and a numerical model, Cont. She.If Res., 4, , and Seas, J. Geophys. Res., 97, 20,087-20,115, Davies, A.M., J.E. Jones, and J. Xing, A review of recent development in Pingtee, R.D., D.K. Griffiths, and G.T. Mardell, The structure of the internal tidal hydrodynamic modelling, Part 1, Spectral models, J. Hydraul. Eng., tide at the Celtic Sea shelf break, J. Mar. Biol. Assoc. U.K., 64, , 123, , 1997a Davies, A.M., J.E. Jones, and J. Xing, A review of recent development in Proctor, R., and A.M. Davies, A three dimensional hydrodynamic model of tidal hydrodynamic modelling, Part 2, Turbulence energy models, J. fides off the north-west coast of Scotland, J. Mar. Syst., 7, 43-66, Hydraul. Eng., 123, , 1997b. Pugh, D.T., and J.M. Vassie, Tide and surge propagation offshore in the Davies, A.M., S.C.M. Kwong and R.A. Flather, A three-dimensional model Dowsing region of the North Sea, Dtch. Hydrogr. Z., 29, , of diurnal and se nidiumal tides on the European shelf, J. Geophys. Res., Rosenfeld, L.K., Baroclinic semidiurnal tidal currents over the continental 102, , shelf offnorthem Califomia, J. Geophys. Res., 95, 22,153-22,172, Drakopoulos, P.G., and R.F. Marsden, The internal tide off the west coast of R0ed, L.P., and C.K. Cooper, A study of various open boundary conditions Vancouver Island, J. Phys. Oceanogr., 23, , for wind forced barotropic numerical ocean models, in Three-dimensional Ellerr, D.J., A. Edwards, and R. Bowers, The hydrography of the Rockall Models of Marine and Estuarine Dynamics, Elsevier Oceanogr. Ser. Vol. Channel - an overview, Proc. R. Soc. Edinburgh., 88B, 61-81, , edited by J.C.J. Nichoul and B.M. Jamart, pp , Elsevier, Flather, R.A., A numerical model investigation of tides and diurnal-period New York, continental shelf waves along Vancouver Island, J. Phys. Oceanogr., 18, R0ed, L.P., B. Hackett, B. Gjevik, and L.I. Eide, A review of the Metocean , Modeling Project (MOMOP), Part 1, Model Comparison Study, in Foreman, M.G.G., and R.E. Thomson, Three-dimensional model simulations Quantitative Skill Assessment.[br Coastal Ocean Models, Coastal of tides and buoyancy currents along the west coast of Vancouver Island, Estuarine Stud. Vol. 47, edited by, D.R. Lynch and A.M. Davies, pp J. Phys. Oceanogr., 27, , , AGU, Washington, D.C., Foreman, M.G.G., R.F. Henry, R.A. Walters, and V.A. Ballantyne, A finite Sinha, B., and R.D. Pingtee, The principal lunar semi-diurnal tide and its element model for tides and resonance along the north coast of British harmonics: Baseline solutions for M 2 and M 4 constituents on the North- Columbia, J. Geophys. Res., 98, , West European Continental She![ Con. She.lf Res., 17, , Galperin, B., L.H. Kantha, S. Hassid, and A. Rosati, A quasi-equilibrium Sherwin, T.J., Analysis of an internal tide observed on the Malin Shelf, north turbulent energy model for geophysical flows, J. Atmos. Sci., 45, 55-62, of Ireland, J. Phys. Oceanogr., 18, , Sherwin, T.J., and N.K. Taylor, Numerical investigations of linear internal Heathershaw, A.D., J. Small, and C.E. Stretch, Frictional formulations in tide generation in the Rockall Trough, Deep Sea Research., 37, numerical ocean models and their effect on simulated acoustic fields, J , Phys. Oceanogr., 24, , Wang. D.P., Model of mean and tidal flows in the Strait of Gibraltar, Deep Hibiya, T., Generation mechanism of internal waves by tidal flow over a sill, Sea Res., 36, , J. Geophys. Res., 91, , Xing, J., and A.M. Davies, Application of turbulence models to the Howarth, M.J., Non-tidal flow in the North Channel of the Irish Sea, in computation of tidal currents and mixing intensities shelf edge regions, Hydrodynamics of Semi-enclosed She.If Seas, edited by J.C.J. Nihoul, J. Phys. Oceanogr., 26, , 1996a. pp Elsevier, New York, Xing, J., and A.M. Davies, Processes influencing the internal tide, its higher Holloway, P.E., On the semi-diurnal internal tide at a shelf-break region on harmonics, and tidally induced mixing on the Malin-Hebrides Shelf, Prog. the Australia north-west shelf, J. Phys. Oceanogr. 14, , Oceanogr., 38, , 1996b. Holloway, P.E., A numerical model of internal tides with application to the Xing, J., and A.M. Davies, Formation of a three-dimensional shelf edge Australian north west shelf, J. Phys. Oceanogr., 26, 21-37, model and its application to internal tide generation, Cont. She.If Res., I8, Huthnance, J.M., Circulation, exchange and water masses at the ocean , 1997a. margin: the role of physical processes at the shelf edge, Prog. Oceanogr. Xing, J., and A.M. Davies, Application of a range of turbulencenergy 35, , James, I.D., Advection schemes for shelf sea model, J. Mar. Syst., 8, , Kwong, S.C.M., A.M. Davies, and R.A. Flather, A three-dimensional model of the principal tides on the European Shelf, Prog. Oceanogr., 39, , Lamb, K.G., Numerical experiments of internal wave generation by strong tidal across a finite amplitude bank edge, J. Geophys. Res., 99, , Lynch, D.R. and A.M. Davies, Quantitative Skill Assessment.[br Coastal Ocean models. Coastal Estuarine Stud., Vol. 47, AGU, Washington, D.C., Lynch, D.R., and C.E. Naimie, The M 2 tide and its residual on outer banks of the Gulf of Maine, J. Phys. Oceanogr., 23, , Martinsen, E.A., and H. Engedahl, Implementation and testing of a lateral boundary schemes as an open boundary condition in a barotropic ocean lnodel, Coastal Eng., 1I, , models to the computation of the intemal tide, Int. J. Numer. Meth. Fluids, 26, , 1997b. Xing, J., and A.M. Davies, The influence of wind effects upon internal tides in shelf edge regions, J. Phys. Oceanogr., 27, , 1997c. Xing, J., and A.M. Davies, The influence of topographic features and density fronts upon the internal tides in shelf edge regions, Int. J. Numer. Methods Fluids, in press, A.M. Davies and J. Xing, Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead, Merseyside L43 7RA UK. ( pol.ac.uk; (Received July 31, 1996; revised June 17, 1998; accepted June 22, 1998.)