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

Philip Hall & Alan M. Davies

Received: 30 November 2006 /Accepted: 24 January 2007 / Published online: 11 September 2007# Springer-Verlag 2007

Abstract A non-linear three-dimensional unstructured gridmodel of the M2 tide in the shelf edge area off the westcoast of Scotland is used to examine the spatial distributionof the M2 internal tide and its higher harmonics in theregion. In addition, the spatial variability of the tidally in-duced turbulent kinetic energy and associated mixing in thearea are considered. Initial calculations involve only tidalforcing, although subsequent calculations are performedwith up-welling and down-welling favourable winds toexamine how these influence the tidal distribution (partic-ularly the higher harmonics) and mixing in the region. Bothshort- and long-duration winds are used in these calcu-lations. Tidal calculations show that there is significantsmall-scale spatial variability particularly in the higher har-monics of the internal tide in the region. In addition, turbu-lence energy and mixing exhibit appreciable spatialvariability in regions of rapidly changing topography, withincreased mixing occurring above seamounts. Wind effectssignificantly change the distribution of the M2 internal tideand its higher harmonics, with appreciable differencesfound between up- and down-welling winds and long-and short-duration winds because of differences in mixingand the presence of wind-induced flows. The implicationsfor model validation, particularly in terms of energy transferto higher harmonics, and mixing are briefly discussed.

Keywords Finite-elementmodel .Windeffects . Internal tide

1 Introduction

Over the last 30 years, there has been significant progress incomputing tidal elevations and current profiles in shallowsea regions under homogeneous conditions. Initial calcula-tions were two dimensional and used coarse finite-differencegrids, which could not resolve near coastal regions. Con-sequently, the main focus was on the M2 tide (Flather 1976),rather than its higher harmonics that were produced by non-linear effects in shallow coastal regions. With advancesin computing power, finite-difference grids were refinedand three-dimensional models developed to examine cur-rent structure (Davies and Kwong 2000). Although finite-difference grids became progressively finer, the majority ofcalculations were based on a uniform grid and hence re-solution in nearshore regions was limited. To avoid thisproblem, boundary-fitted coordinates were developed toimprove resolution in nearshore regions.

An alternative approach to the finite-difference methodwas to use finite elements (e.g. Walters and Werner 1989;Werner 1995; Walters 2005; Fortunato et. al. 1997, 1999;Heniche et al. 2000; Ip et. al 1998; Jones and Davies 2006a).In these models the element size could be varied fromcoarse offshore to fine in nearshore regions. Consequently,the generation of higher harmonics of the tide in nearshoreregions could be accurately reproduced in large-area models(Jones and Davies, 2006b).

With increasing computing power, three-dimensionalmodels have been developed, and horizontal finite-differencegrids have been refined. In shallow sea regions where tidalmixing is strong and the water column remains well mixed,the stratification effects upon tidal turbulence are absent.Consequently, although tidal currents over a region mayhave been collected at different times, the periodic nature ofthe tides means that in essence, a synoptic data set can beproduced. In stratified regions, in particular those adjacent to

Ocean Dynamics (2007) 57:305–323DOI 10.1007/s10236-007-0117-2

Responsible editor: Phil Dyke

P. Hall (*) :A. M. DaviesProudman Oceanographic Laboratory,6 Brownlow Street,Liverpool L3 5DA, UKe-mail: phh@pol.ac.uk

topography (e.g. Dogger Bank, Proctor and James 1996)where stratification changes, the determination of tidalcurrents and associated internal tides as the barotropic tidepropagates over steep topography is more complex becauseof the time-evolving nature of the stratification.

Early internal tidal models used primarily analyticalapproaches (Craig 1987; Sherwin and Taylor 1989, 1990)and were of cross-sectional form. Subsequently, three-dimensional models using uniform grids were developed(e.g. Cummins and Oey 1997) and used to examine thegeneration of the internal tide in a range of geographicallocations. These models showed the importance of along-shelf topography, off-shelf sea mounts (Xing and Davies1998; Xing and Davies 1999) and how the shape of seamounts and ridges influenced internal-tide propagation.

The uniform nature of the grid resolution in these modelsmeant that the focus was the generation and propagation of theM2 internal tide in three dimensions rather than a study ofprocesses generating higher harmonics of the internal tide.Although, recently, a cross-section high-resolution finite-difference grid model was used to examine the generation ofhigher harmonics of the tide in the proximity of sea mounts(Lamb 2004).

So far, the majority of internal-tide calculations useduniform finite-difference models and considered the M2

tide. However, the shallow sea applications of finite-elementmodels clearly shows that this approach is ideal in a shelf-edge environment where both enhanced resolution isrequired at the shelf edge and at offshore locations, e.g.seamounts. Recently, calculations of the M2 internal tidehave been performed using finite elements to examine itsgeneration in shelf-edge regions and the effect of elementresolution (e.g. Hall and Davies 2005a). This involved acomparison with a fine-resolution (2.4 km across shelf and4.6 km along shelf) finite-difference internal-tide model(Xing and Davies 1998) and calculations with a range offinite-element meshes to compute the internal tide off thewest coast of Scotland. The region covered by the model,namely the Malin-Hebrides shelf, was identical to that usedby Xing and Davies 1998, as were water depths, the openboundary conditions and vertical stratification. By thesemeans, a rigorous comparison was possible, and the in-fluence of finite element resolution on the internal tide wasassessed. A comparison with limited measurements was alsopossible. Although the main features of the internal tidecomputed with the finite difference and fine finite elementmodel were comparable, it was evident that the intensity ofthe M2 internal tide in its generation region and its sub-sequent propagation was improved by using the finer meshgrid.

In this paper, the work presented in Hall and Davies2005a is extended to examine the spatial distribution of theshorter-wavelength higher harmonics of the internal tide

in the Malin-Hebrides shelf region. In addition, how theseharmonics and stratification in the shelf edge regionchanges in response to wind forcing is considered. Byusing a range of wind conditions, some insight as to therole meteorology has in modifying the internal tidal signalin different regions can be determined. In addition, theresults presented in Hall and Davies 2005a are extended toexamine the spatial variability of higher tidal harmonicsnamely, M4 and M6. As these are due to non-linear effects,they are a good guide as to how internal tidal models arerepresenting these processes. However, their short wave-length and associated variability is a major challenge bothin terms of grid resolution and collecting high-resolutionsynoptic data sets for model validation. The importance ofmixing in the lateral boundary layers of the ocean has beenshown to have important consequences for the computedcirculation produced by large-scale models (e.g. Samelson1998; Spall 2001). Because mixing is related to thebreaking of small-scale waves, the energy transfer fromthe M2 tide to the shorter-wave higher harmonics isimportant in this energy transfer to mixing. However, asthe present model is hydrostatic with a spatial resolution oforder kilometres, then mixing has to be parameterized usinga turbulence closure scheme. In this paper, Mellor–Yamada(see Blumberg and Mellor 1987 for detail) is used in thevertical and the scale-selective filter of Smagorinsky 1963in the horizontal. As shown in Hall and Davies 2005a, asthe finite element grid is reduced for a specified filterparameter C in the Smagorinsky formulation, a Gibbs-type oscillation occurs in the boundary layer close to to-pography. To remove this, the filter parameter C in theSmagorinsky formulation needs to be increased, therebysmoothing the boundary layer. In addition, the extent towhich wind-induced mixing in the lateral boundary layersinfluences internal-tide generation, particularly the higherharmonics, and how this diffuses away from the lateraloceanic boundaries is very important. However, in terms ofwind-forced motion in boundary layers, as shown by Halland Davies 2005b, it is necessary to choose an appropriatevalue of C in the Smagorinsky form of sub-grid scaleparameterization to adequately represent the wind-forcedboundary layer on a given grid. The need to use sub-gridscale parameterizations of mixing in these models, whichimpact upon boundary layer stability and thickness, meansthat it is difficult to get a rigorous numerically convergedsolution in the sense that the internal tide converges as thegrid scale is reduced because of the fact that the mixingchanges in these boundary layers and hence the densityfield in the region where the internal tide is produced. Forthis reason, the higher resolution grid and appropriatesub-grid scale mixing as used by Hall and Davies 2005a,in a successful simulation of the internal tide, is usedthroughout.

306 Ocean Dynamics (2007) 57:305–323

An alternative approach would be to use a full three-dimensional non-hydrostatic model on a very fine grid oforder 10 m, with minimum fixed constant diffusion coef-ficients, thereby avoiding the sub-grid parameterizationproblem. However, as shown by Berntsen et al. (submitted),this does not guarantee convergence as the grid is refinedunless a scale-selective filter and sub-grid scale mixingparameterization are included. In addition, such a calcula-tion would be computationally prohibitive. These problemsconcerning model convergence and mixing parameteriza-tion suggest that detailed internal-tide turbulence measure-ments are required in shelf-edge regions. However, asshown later, there is significant spatial variability in these,both of which are affected by the wind, suggesting thatlarge detailed synoptic measurements are required for mod-el validation. To the authors’ knowledge, this is the firsttime that the three-dimensional distribution of higherharmonics of the internal tides, associated turbulence in-tensities and the influence of wind effects upon them hasbeen presented in a shelf-edge region.

The order of the paper is such that the region consideredand the model are discussed in the next section. Subsequentsections describe the three-dimensional variability of the

higher harmonics of the internal tide. The response to windforcing and the resulting changes in tidal distribution areconsidered later in the paper. Major findings and implica-tions for data collection to validate internal tidal models aresummarized at the end.

2 The geographical region and internal tidal model

As the full three-dimensional equations and the turbulenceclosure model were presented in Xing and Davies 1997,they will not be repeated here, where only the main featuresof the equations are discussed. The hydrostatic approxima-tion is used in the model, and density is derived fromtemperature using the equation of state given in Xing andDavies 1998. The model is forced by the barotropic tideintroduced through the open boundary (Hall and Davies2005a), and the internal tide is generated as isotherms aremoved over the topography. The model is fully prognostic,with free surface elevation changing as the tide propagateson and off shelf. Vertical mixing of momentum and densityare parameterized using vertical eddy viscosity and dif-fusivity coefficients computed using a turbulence energy

Fig. 1 Bottom topography (water depths in m) and area covered by the model. Cross-sections where the influence of the wind upon the tide isexamined in detail are denoted by lines C2, D1n, D1c and D1s. Geographical locations of regions named in the text are also shown

Ocean Dynamics (2007) 57:305–323 307

closure namely, the Mellor and Yamada sub-model (e.g.Blumberg and Mellor 1987). As the form of the turbulencemodel and similar models is given elsewhere (e.g. Luytenet al.1996; Xing and Davies 1998), it will not be repeatedhere. As shown in Hall and Davies 2005b, the Smagorinsky1963 form of horizontal eddy viscosity used here is par-ticularly good at preventing the development of numericalinstabilities when wind-forced internal waves are generatedon an irregular grid. However, as discussed previously andshown in Hall and Davies 2005a, b), the filter parameter Cin the Smagorinsky formulation of viscosity must be chosenin an appropriate manner to match the grid resolution inboundary layers, and it does influence the extent of theselayers. The value and grid resolution used here is identicalto Hall and Davies 2005a, which yielded an accurate M2

internal tide.As the short-term modification of the internal tide be-

cause of wind forcing rather than its seasonal variation isexamined here, there is no applied surface heat flux. Asurface boundary condition on momentum and turbulenceof wind stress origin as in Xing and Davies 1997 is ap-plied at the sea surface with a quadratic friction law as in

Xing and Davies 1998 at the seabed. Identical barotropictidal forcing to that used in the finite-difference model(Xing and Davies 1998) and finite-element model (Halland Davies 2005a) was applied. The model uses a sigmacoordinate representation in the vertical, and 40 levelswere used in the calculations. Sigma levels were such thatenhanced resolution in the near-bed and near-surfaceregion (as in Xing and Davies 1998; Hall and Davies2005a) where the internal tide was largest was used in allcalculations.

The finite-element model (Fig. 1) is identical to thatused by Hall and Davies 2005a. The domain covers arange of water depths from the order of 10 m close to thecoast to the order of 3,000 m in the ocean. Water depthchanges rapidly at the shelf edge (approximately the200 m contour), as it does along the edge of the AntonDohrn Seamount and Hebrides Terrace Seamount. Asshown in Xing and Davies 1998 and Hall and Davies2005a, in these regions, there is appreciable internal-tidegeneration. Detailed studies using a range of unstructuredgrid approaches (Hall and Davies 2005b; Hall and Davies2005c) showed that it was necessary to have a fine grid

Fig. 2 The high-resolution (see Hall and Davies 2005a) finite-element grid used in the model

308 Ocean Dynamics (2007) 57:305–323

adjacent to the topography to accurately reproduce wind-forced internal waves generated in regions of topographicchange. This was achieved using the finite-element grid(Fig. 2) found by Hall and Davies 2005a to accuratelyreproduce the tide. By using the same mesh in the presentcalculations, a direct comparison with Hall and Davies2005a is possible, enabling the effect of the wind upon theM2 tide to be determined.

Initial conditions corresponded to a state of rest withzero elevation and current, with the same temperature pro-file (Fig. 3) as used by Xing and Davies 1997, appliedeverywhere. This profile was identical to that used in Halland Davies 2005a, enabling comparisons to be made withearlier calculations. Tidal barotropic forcing at the M2 pe-riod was applied along the open boundary, and as found inHall and Davies 2005a, after six tidal cycles, a periodicbarotropic tide and quasi-periodic baroclinic tide wereestablished, which was harmonically analysed. Becauseboundary layer mixing occurred in the model and modifiedthe density field through time, this had the effect of slightlychanging the baroclinic tide with time. Hence, only a quasi-baroclinic tide was established. As Hall and Davies 2005adid not examine the higher harmonics of the tide, in aninitial calculation (calc. 1, Table 1), these are computed ina tide-only solution and presented in detail. As discussedpreviously, these are generated by non-linear effects, andhence the ability of the model to reproduce them is a criticaltest of how it can handle the cascade of energy to smallscales. However, as we will show, the short wavelength ofthese waves and their modification by wind effects suggestthat measurements to validate this aspect of the model willprove very difficult to make with a high level of accuracy.

The effect of wind forcing produced by a uniform steadywind upon the M2 tide and its harmonics was examined inlater calculations. A wind stress of 0.2 Pa as used in Xingand Davies 1997 was applied in these calculations. Thegeneration of strong inertial oscillations, which would per-sist in the solution and mask the steady wind-induced re-sponse, was avoided by increasing the wind stress with asine wave form corresponding to the first quarter of a sinecurve (namely 0 to π/2) for 12 h during the spin-up period.

3 Tidal distributions

As considered in Hall and Davies 2005a, barotropic tidalforcing produces a strong baroclinic tide that is separatedfrom the total solution by subtracting a barotropic tidecomputed by running the model in a homogeneous form(see Hall and Davies 2005a for detail). By this means,changes in the tidal current profile produced by viscouseffects that are significant in boundary layers are removedfrom those because of internal pressure gradients, thereby

leaving a true baroclinic signal. The alternative used inobservational work of computing the baroclinic tide bysubtracting the depth mean current does not take account ofviscous effects and will not produce a true baroclinic profilein the boundary layer. This is discussed later in connectionwith the M6 tide. By harmonically analysing the resultingbaroclinic tide, the residual current and the M2, M4 and M6

components of the baroclinic tide were determined. Asimilar approach was applied in the wind-forced case. In ahomogeneous tidal model, the residual current namely, thetidal residual and the higher harmonics M4 and M6, are dueto non-linear effects. In the stratified case, differential tidalmixing between the shelf and shelf edge leads to a shelf-edge front, with an across-shelf density gradient. Associatedwith this density gradient is an along-frontal flow in additionto any flow induced by wind forcing. However, this aspectof the solution is beyond the scope of the present paper.

Previous tidal calculations (Xing and Davies 1997; Halland Davies 2005a) showed that the presence of topographicfeatures such as the Anton Dohrn offshore seamount atabout 57.5°N (Fig. 1) influences the local distribution of the

Fig. 3 Temperature profile (°C) used in the calculations

Ocean Dynamics (2007) 57:305–323 309

M2 internal tide. In addition, Lamb 2004 showed that to-pographic features such as seamounts had an importantrole in generating higher harmonics. For these reasons, wewill initially examine the distribution of the M2 tide and itshigher harmonics in this area. To be consistent with Halland Davies 2005a, the same cross-sections namely, D1n,D1c and D1s (Fig. 1), are considered to determine the spa-tial distribution of the higher harmonics. However, becausethe M2 baroclinic tidal distribution at all cross-sections wasexamined in Hall and Davies 2005a, this component willonly be considered in detail at cross section D1n.

Considering initially the M2 baroclinic tidal current atcross-section D1n, it is evident from Fig. 4 that on the shelf,a surface and bottom intensification of M2 baroclinic cur-rent amplitude, which is indicative of a first mode internalwave, is evident, although higher modes with reduced am-plitude also occur. A bottom intensification of the internaltide at the shelf break, extending down the slope, is alsoapparent. These regions of internal tide generation areassociated with areas of critical slope, where the shelf slope∂h/∂x matches the internal wave slope S, given byS2¼ ω2�f 2

� ��N2�ω2� �

with ω the M2 tidal frequency, fCoriolis and N buoyancy frequencies, and an enhancedinternal tide is generated (see Vlasenko et al. 2005 fordetails). Obviously, in the model, the topography and hence∂h/∂x varies down the slope. Furthermore, because of bot-tom boundary layer mixing, N varies from its initial valuein a complex manner depending upon local tidal turbulenceintensity (see later), which depends upon both barotropicand baroclinic bottom currents. These changes in ∂h/∂x andS exhibit significant small-scale variability, and henceappreciable local changes in internal-tide intensity occur.In addition, there is internal-tide propagation from theAnton Dohrn seamount onto the shelf slope (see below andXing and Davies 1996a, b). Comparison at other cross-sections (not presented) did, however, show that the extentof this downslope region of the intensified bottom currentvaried from section to section in the region of the sea-mount. This is to be expected because ∂h/∂x shows sig-nificant variability in the seamount region and bottomturbulence, and hence N changes over relatively smalldistances (see later).

The variation in the magnitude of the M2 internal tideover small horizontal distances (see separation of cross-sections D1n, D1c and D1s in Fig. 2) in the Anton Dohrnregion is because both super-critical and sub-criticalinternal tides are generated along the shelf slope and onthe seamount sides, producing internal tides that propagateboth onto the shelf and into the ocean (see Xing andDavies 1996a, b for detail). As shown in Xing and Davies1996a, b), the exact location and topography of theseamount was responsible for the off-shelf distribution ofthe internal tide. Further to the south of the seamount (seecross-section C2), where the slope is steeper, the region ofmaximum M2 internal tide occurs down the slope in deeperwater (not shown). In the shelf break region, a surfaceintensification occurs, with a slight increase on the shelf atdepth, suggesting that there is little on-shelf propagation ofthe internal tide at cross-section C2.

Contours of the u-current amplitude at the M4 frequencyin the Anton Dohrn region show (Fig. 5a–c) an intensifi-cation at depth along the eastern side of the seamount atcross-section D1n (Fig. 5a). This location corresponds tothe area of strong M2 baroclinic tidal currents (Fig. 4),which show significant spatial variability. Consequently,the non-linear momentum advection terms, which transferenergy to higher harmonics, are a maximum in this region

Table 1 Summary of various calculations

Calc. Wind Duration,Wind Direction

1 Tide only2 S, Nth3 L, Nth4 S, Sth5 L, Sth

S and L denote short- and long-duration winds, with Nth and Sth winddirection towards North or South.

Longitude

Depth(m)

11°W

10°W

9°W

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

Fig. 4 Amplitude (cm s−1) of the M2 harmonic of the u-component ofthe baroclinic tide along cross-section D1n

310 Ocean Dynamics (2007) 57:305–323

and give rise to this strong M4 signal. Similarly, a localincrease in M4 tidal current is apparent at the sea surfaceand seabed to the east of the seamount and near the shelfbreak where there is an appreciable lateral gradient in theM2 tidal current (Fig. 4). Besides, the non-linear momen-tum advection terms, other terms such as time varying eddyviscosity (Davies and Lawrence 1994), can give rise tohigher harmonics of the tide, although these were appre-ciably less. In addition, quadratic friction can generatehigher harmonics, although this is the M6 rather than M4

component (see later).At cross-section D1c, the region of maximum M4 current

amplitude is located on the western side of the seamount(Fig. 5b) in the region of significant gradients in the M2

baroclinic tidal current (not shown but see Hall and Davies2005a). A weaker signal is evident (Fig. 5b) on the easternside because of the reduced intensity of the M2 tide in thisregion compared to previously (Fig. 4). A mid-water M4

maximum current is apparent at the top of the shelf slope(Fig. 5b) corresponding to a region of rapid M2 tidal currentvariation.

At the southern edge of the seamount (cross-section D1s),there is a local enhancement of the M4 tidal current on eitherside of the seamount and in the shelf-break region (Fig. 5c).These areas correspond to regions of enhanced M2 u-currentamplitude (not shown but see Hall and Davies 2005a). Atcross-section C2, there is a small (of order 3 cm s−1) M4 tidalcurrent amplitude at the shelf break and along the shelfslope (not shown) in regions where the M2 tidal current issignificant.

The distribution of the M6 tidal current amplitude alongcross-section D1n (Fig. 6) is comparable to that for the M4

tide (Fig. 5a), although its amplitude is about half that ofthe M4 tide (compare Figs. 5a and 6). Similarly, at the othercross-sections (not shown), the spatial distribution of theM6 tide is comparable to that for M4, although its amplitudeis about half. This suggests that it is a non-linear interactionin regions where the M2 internal tide is largest and has astrong spatial variability (significant gradient) that is re-sponsible for the generation of higher harmonics. Awayfrom the near-bed region, the non-linear momentum advec-tion terms, which transfer energy from the M2 to M4, alsomove energy to the M6 tide. However, in the near-bed re-gion, the quadratic friction term is the dominant mechanismand moves energy from the M2 to M6 frequency. By gen-erating an M6 baroclinic tide by removing the barotropiccomponent using a homogeneous calculation from the totaltide, it is possible to isolate the M6 baroclinic tide. Thealternative that would have to be used in the case of obser-vational data would be to subtract a depth mean M6 tidalcurrent from the total, which would clearly only partiallyremove the barotropic M6 tide. Consequently, a rigorousvalidation of the M6 component against observations would

be very difficult. In addition, because M6 is generated inthe near-bed region through bed friction and turbulence,detailed measurements of bed roughness and thicknessof the turbulent bottom boundary layer would be required(see later).

As the internal tide up-wells and down-wells over thesteep topography, tidal mixing occurs, and this influencesthe temperature field. Although the extent of the displace-ment of the temperature surfaces varies with depth belowthe surface, it is useful to examine its spatial variability at agiven depth. To this end, the model was run for a signif-icantly longer period of time (namely 20 tidal cycles) toallow tidal mixing and tidally produced turbulent kineticenergy (t.k.e.) to become established. In essence, tidal mix-ing in the bottom boundary layer changes the temperaturefield from its initial prescribed value to one which is con-sistent with the internal tidal distribution and the turbulentEkman lower layer.

Away from the Anton Dohrn Seamount and the shelf-edge region, a near uniform spatial distribution of temper-ature was found at 600 m (Fig. 7a). Because this depth isabove the top of the seamount, the reduction in temperatureis associated with enhanced vertical mixing above the sea-mount (see later). Enhanced up-welling and vertical mixingat the shelf-edge region to the north of 57°N lead to regionsof cold water, the across-shelf extent of which varies withposition (Fig. 7a). It is evident from Fig. 7a that it is largestin the region to the northeast of the seamount where Xingand Davies 1996a,b) showed there is a significant M2 tidalenergy flux both along and across the depth contours. In theregion of cross-section C2, there appears to be a region ofslightly warmer shelf-edge water at 600 m depth. As theshelf slope is different at this location compared to cross-section D1, this gives rise to a maximum internal tidal cur-rent at about 1,200 m depth rather than higher up the slope(of order 600 m) found farther north. Differences in loca-tion at the depth of maximum internal tidal current gen-eration will produce differences in boundary layer mixingthat will influence the temperature distribution. Although adetailed study of this is beyond the scope of this paper, wewill briefly consider the spatial variability of t.k.e.

At a depth of 1,200 m, there is more horizontal spatialvariability in the temperature field (not shown), associatedwith small-scale variations in mixing because of spatialchanges in tidally produced turbulence. Contours of log10t.k.e. along the shelf slope show (Fig. 7b) significant spatialvariability. To be consistent with the temperature distribu-tion plot, these contours correspond to a depth of 1,200 mbelow the surface, rather than a height above the sea bedthat is more appropriate in the case of bed-generated tur-bulence. Because it is not possible because of the highlynon-linear nature of the processes producing t.k.e. to par-tition it into a barotropic and baroclinic part, these contours

Ocean Dynamics (2007) 57:305–323 311

Longitude

Depth(m)

11¡W

10¡W

9¡W

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

Longitude

Depth(m)

11¡W

10¡W

9¡W

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

Longitude

Depth(m)

11¡W

10¡W

9¡W

0

1

2

3

4

5

6

7

8

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

a b

c

Fig. 5 Amplitude (cm s−1) of the M4 harmonic of the u-component of the baroclinic tide at cross-sections a D1n, b D1c and c D1s

312 Ocean Dynamics (2007) 57:305–323

represent total t.k.e. A major source of this on the upperpart of the shelf-slope region is due to the barotropic tide.

However, as shown by Xing and Davies 1996a, b atdepth, along shelf slopes, it is a maximum in near-bed re-gions where the internal tide is generated. The strong max-imum at 1,200 m, in the shelf-edge region (Fig. 7b) at thelocation of strong internal tides at cross-section C2 showsthat this is associated with the internal-tide production inthis area, at this depth. As discussed above, internal-tideproduction was evident in the across-slope distribution ofu-current amplitude at this depth. The significant spatialdistribution of t.k.e. along the shelf slope region, shown inFig. 7b, illustrates that there will be appreciable variation inturbulence intensity measurements from one transect toanother. In addition, a high-resolution model mesh resolu-tion both across and along the shelf slope is required.

Appreciable small-scale variations in t.k.e. are evidentaround the Anton Dohrn Seamount, with the region of en-hanced turbulence to the north (Fig. 7b), associated with theshallower water in this area. A similar feature is just evidenton top of the Hebrides Terrace Seamount (Fig. 7b).

To examine the depth and lateral variation of t.k.e. inmore detail, cross-section plots of log10 t.k.e. were com-puted along cross-sections D1n, D1c, D1s (Fig. 8a–c) andC2, at the time corresponding to the internal-tide contoursshown previously. Contours of t.k.e. at cross-section D1n,

exhibit (Fig. 8a) enhanced turbulence in the near-bed re-gion, particularly at the shelf break and at depth along theeastern edge of the seamount where internal-tide generationis a maximum (Fig. 4a). At cross-section D1c, the region ofmaximum t.k.e. is situated at depth on the west side of theseamount (Fig. 8b) in the region of strong M2 internal-tideproduction. A region of intensified turbulence is evident atthe shelf break with patches of intensified turbulence downthe shelf slope in areas corresponding to internal-tide gen-eration. Similar patches of intensified turbulence occur onboth the western and eastern side of the seamount and alongthe shelf slope at cross-section D1s (Fig. 8c) in regions ofenhanced internal tide. At cross-section C2, where theinternal tide at the shelf break is small, near-bed t.k.e. (notshown) is small and of limited lateral extent.

The effect of increased turbulence energy in theseregions is to enhance near-bed mixing thereby reducingthe buoyancy suppression of turbulence in the near-bedregion. A consequence of this is that in some areas, near-bed turbulence increases with time. This time increase wasfound to be particularly large and extend away from thenear-bed region at cross-sections D1s and C2. A compar-ison of the time variation of t.k.e. at cross-section D1s (notpresented) shows that the intensity and lateral extent ofturbulent energy along the seamount slope and in the shelfedge region where the internal tide is a maximum rapidlygrows with time. Similarly, at cross-section C2 (notshown), there is a rapid increase at a depth of 1,200 m,associated with the strong internal tide in this region. Asdiscussed previously in connection with Fig. 7b, at thisdepth, the maximum turbulence energy in the shelf edgeregion occurs at cross-section C2. The fact that theseregions of maximum t.k.e. occur at locations of maximuminternal tidal current amplitude shows that in the shelf-slope region there will be other maxima at different posi-tions and depths. Consequently, in the shelf-slope region,there will be considerable small-scale variation in thelocation of strong t.k.e. regions. This clearly shows thatany measurement strategy aimed at determining turbulenceenergy data sets for model validation will require high-resolution sampling.

4 Wind-forcing effects

4.1 Wind towards the north (a down-welling favourable wind)

To determine the influence of wind effects upon the internaltide, calculations were performed with winds of short andlong duration towards the north (calcs. 2 and 3) and thesouth (calcs. 4 and 5). A summary of the calculations isgiven in Table 1. After the initial ramping-up period, afixed time and space invariant wind stress of 0.2 Pa was

Longitude

Depth(m)

11°W

10°W

9°W

.0

.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

Fig. 6 As Fig. 4, but for the M6 harmonic at cross-section D1n

Ocean Dynamics (2007) 57:305–323 313

used in all calculations, with open boundary barotropic tidalforcing as in calc. 1.

The effect of short-term wind forcing was enhancedmixing in the surface layer with an associated weakening ofthe thermocline in the near-surface (of order 100 m) layer.However, below this layer, there was little change in thetidally averaged (over an M2 tidal cycle) temperature field.

To determine the effect of wind upon tidal u-currentamplitude along cross-section D1n, differences in ampli-tude with and without wind forcing were computed foreach constituent (Fig. 9a–c). Contours of this difference forthe M2 internal tide away from the shelf show (Fig. 9a) thatthe change in the M2 internal tide is mainly restricted to thesurface layer where the temperature and hence density fieldhas been changed by the wind. The largest effect occurs inthe surface layer above the seamount where the currentbecause of the internal tide was a maximum (Fig. 4). On the

shelf, besides changes in the M2 internal tide at the surface,there are some changes at depth. This is due to enhancedmixing arising from an increase in bottom turbulencebecause of wind-induced bottom currents in shallow water.On average, these changes were of the order of 5 cm s−1

(Fig. 9a), compared with the M2 internal tidal currentamplitude (Fig. 4) of order above 20 cm s−1. Namely, achange of more than 20%. Changes in the u-currentamplitude of the M2 internal tide at other cross-sectionswere not appreciably different to that shown in Fig. 9a.

For the M4 component, the most significant modifica-tions (Fig. 9b) occur in the surface layer particularly in theshelf-break region where the M2 tidal current (Fig. 4) islargest. As discussed previously, the M4 is produced by thenon-linear interaction of the M2 tide with itself. Conse-quently, changes in M4 will be largest in regions where thelargest changes occur in M2 namely, the surface layer.

Longitude12°W 11°W 9 °W °W

Latitude

55°N

56°N

58°N

59°N

6.7

6.8

6.9

7.0

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8.0

8.1

8.2

8.3

8.4

8.5

8.6

8

Longitude

12°W 11°W 9°W °W

Latitude

55°N

56°N

58°N

59°N

-6.4

-6.2

-6.0

-5.8

-5.6

-5.4

-5.2

-5.0

-4.8

-4.6

-4.4

-4.2

-4.0

-3.8

-3.6

-3.4

-3.2

-3.0

-2.8

-2.6

-2.4

8

a b

Fig. 7 a Horizontal temperature distribution (°C) at a depth of 600 m below the surface, b horizontal distribution of log10 t.k.e. (m2 s−2) at a depth

of 1,200 m below the surface. Both were derived from a long tidal integration

314 Ocean Dynamics (2007) 57:305–323

Longitude

Depth(m)

11˚W

10˚W

9˚W

-6.4

-6.2

-6.0

-5.8

-5.6

-5.4

-5.2

-5.0

-4.8

-4.6

-4.4

-4.2

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-3.8

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-3.4

-3.2

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

Longitude

Depth(m)

11¡W

10¡W

9¡W

-6.4

-6.2

-6.0

-5.8

-5.6

-5.4

-5.2

-5.0

-4.8

-4.6

-4.4

-4.2

-4.0

-3.8

-3.6

-3.4

-3.2

-3.0

-2.8

-2.6

-2.4

-2.2

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

a b

Longitude

Depth(m)

11¡W

10¡W

9¡W

-6.4

-6.2

-6.0

-5.8

-5.6

-5.4

-5.2

-5.0

-4.8

-4.6

-4.4

-4.2

-4.0

-3.8

-3.6

-3.4

-3.2

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

c

Fig. 8 Cross-sections of log10 t.k.e. (m2 s−2) at cross-sections a D1n, b D1c and c D1s from a short tidal integration

Ocean Dynamics (2007) 57:305–323 315

Typically, the changes in M4 are only of the order of1 cm s−1 (Fig. 9b), although this is significant in the surfacelayer where the M4 tidal current is only about 3 cm s−1

(Fig. 5a). At depth, along the eastern side of the seamount,the M4 changes by about 1.4 cm s−1. In this area, the M4

current is appreciable (up to 14 cm s−1; Fig. 5a), and thischange reflects the small shift in M2 tidal current maximumat depth (Fig. 4) in this region because of the smalldifference in the temperature field produced by the wind. Atother cross-sections (not shown), comparable changes tothe M4 tidal current amplitude occur in the surface layer,with small changes at depth in regions where the M4 tidalcurrent is a maximum.

The wind-induced difference in the M6 tidal currentalong cross-section D1n (Fig. 9c) shows a comparabledistribution in the surface layer to that computed for M4,although with a significant (of order 50%) reduction inmagnitude. Similarly, at depth, the region of changecorresponds to that found for the M4 tidal current (compareFig. 9b and c). A study of the M6 tide shows that in thisregion, the M6 arises from M4 and M2 interaction, withthese changes reflecting differences in the M2 distribution.As for M4, at other cross-sections (not shown), the M6

changes in the surface layer are comparable to those shownin Fig. 9c, with modifications at depth occurring in regionswhere the M6 current is a maximum.

A consequence of applying a short-duration windforcing is that the major change in the density field onlyoccurred in the surface layer. However, when a wind stressof 0.2 Pa was applied for 30 tidal cycles, Xing and Davies1997 found that there was appreciable mixing along theshelf slope region, which affected the generation and off-shelf propagation of the internal tide. As shown in Xing andDavies 1997, this influenced the M2 tide at depth. Toexamine this in more detail, the previous calculation wasrepeated with the wind applied for 20 tidal cycles (calc. 3).

The result of applying the wind stress for a longer periodis appreciably more mixing in shallow water than found inthe tide-only calculation together with a weakening of thesurface thermocline (Fig. 10a) compared to the short-windperiod calculation. It is evident from Fig. 10a that at depth,particularly in the regions adjacent to the sea bed, enhancedmixing has changed the local density gradient. This is inpart due to the wind field, but tidal mixing also contributedto the mixing at depth in slope regions because of theextended time integration.

Contours of the time-averaged (over a tidal cycle)differences in the temperature distribution at 200 mbetween that with tide and wind and tide alone are givenin Fig. 10b. In the case of a down-welling favourable wind,in a coastal region, and as found by Xing and Davies 1997at the shelf edge, warm surface water down-wells to depthand gives rise to an increase in temperature. Some shelf-

edge regions of increased temperature at depth are clearlyevident in Fig. 10b, although elsewhere, the temperature isreduced. In the deep ocean, as expected, there is little changein temperature. The existence of patches of increased/decreased temperature in the shelf-edge region shows thatthe three-dimensional temperature response to wind forcingis more complex than that found by Xing and Davies 1997 ina cross-sectional model. In three dimensions, the internal tidecan propagate both across and along the shelf edge. Inaddition, the vertical velocity associated with the wind-induced flow is more complex than in a cross-sectionalmodel. This leads to greater spatial variability in the internaltidal signal and the associated mixing of tidal and windorigin. In addition, along-shelf flows of wind origin canadvect water from one region of tidal mixing to another, aprocess that is not included in a cross-sectional model.

As in the short-wind period calculation, contours of thedifference in u-current amplitude along cross-section D1nfor the M2, M4 and M6 components were determined. Forthe M2 tide, it is evident (Fig. 11a) that there is nosubstantial change on the shelf. However, an appreciablechange in M2 baroclinic tidal current amplitude (Fig. 11a)has occurred in the surface layer above the shelf break andat the top of the shelf slope associated with the change instratification. In addition, near the foot of the slope, anappreciable M2 internal tide is produced (Fig. 11a) associ-ated with the change in stratification. In addition, a stronginternal tide was produced along the western side of theseamount (Fig. 11a), a region where previously (Fig. 4)there was no internal tidal signal. The reason for theappearance of the internal tide in these areas is due to alocal change in stratification arising from prolonged tidaland wind-induced mixing. Comparable changes in thestratification at depth at other cross-sections D1c and D1s(not shown) gave similar changes.

The modification of the M2 internal tide gives rise to asimilar pattern of change in the M4 component (Fig. 11b).However, the magnitude of the change (Fig. 11b) is small(of order up to a maximum of 2.5 cm s−1). As in the short-duration wind case (calc. 2, Table 1), there is an appreciablechange in the M4 near-surface tidal current amplitudebecause of wind mixing in the surface layer affecting theM2 tide. In addition, there is a small M4 tidal signal at depthalong the shelf slope and on the western side of theseamount in the regions where there is an M2 internal tidalsignal in the longer-duration wind case. Similar large-scalechanges in the M4 current amplitude occurred at othercross-sections (not shown). Although the large-scale distri-bution was similar, the exact location and magnitude variedbetween cross-sections because of differences in topogra-phy and mixing.

As in the short-duration calculation, the largest wind-induced modification of the M6 tidal current arises in the

316 Ocean Dynamics (2007) 57:305–323

Longitude

Depth(m)

11¡W

10¡W

9¡W

0

1

2

3

4

5

6

7

8

9

10

11

12

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

Longitude

Depth(m)

11¡W

10¡W

9¡W

.0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

Longitude

Depth(m)

11¡W

10¡W

9¡W

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

a b

c

Fig. 9 Contours of the difference in baroclinic tidal amplitude (cm s−1) between the wind (0.2 Pa towards the North) and tidal solution and a tide-only solution for short-wind-duration forcing (calc. 2) at cross-section D1n, for a M2, b M4 and c M6 frequency

Ocean Dynamics (2007) 57:305–323 317

surface layer (Fig. 11c). At depth, the largest change occursin the slope region where the modification of the M2 andM4 tide has been greatest. As considered previously, thisshows that M6 is produced by the non-linear interaction ofM2 and M4 in these areas, although changes in bottomfriction because of changes in M2 will have an effect.

As shown in the short-wind-forced calculation, theintroduction of a wind stress influences mixing and henceturbulence energy. How this is modified by the longer-duration wind was examined in terms of contours of t.k.e.from cross-section D1n (Fig. 11d). To determine to whatextent the presence of a long (20 tidal cycle) wind stress of0.2 Pa towards the north has upon the turbulence energydistribution, contours of t.k.e. were plotted at the variouscross-sections (Fig. 11d). The presence of the surface windstress gives rise to an enhanced layer of surface and on-shelf turbulence at cross-section D1n (Fig. 11d) that wasnot present in the tidal solution. In addition, the distributionof t.k.e. in the seamount and shelf-slope region changesbecause of wind-induced mixing and correspondingchanges in the density field. This modification of thedensity field affects the location and intensity of the internaltide (Fig. 11a) giving rise to the differences that are evidentbetween Figs. 11d and 8a. A detailed study (not presented)based on other cross-sections shows similar changes inmixing and t.k.e. distribution from one cross-section toanother. However, the spatial variability of the mixing fromone cross-section to another appears to be enhanced by thepresence of the wind.

This series of calculations shows that the distribution ofthe internal tide and its higher harmonics is modified by amodest wind stress of 0.2 Pa. With short-duration forcing,this is confined primarily to the surface layer. However, onthe 10-day timescale, it modifies the stratification alongshelf slopes and in particular those associated with off-shelfseamounts.

4.2 Wind towards the south (an up-welling favourablewind)

In this section, results are presented from calculationsperformed with both a short-duration (calc. 4) and longer-duration (calc. 5) up-welling favourable wind stresses of0.2 Pa. As mentioned previously, temperature contoursalong various cross-sections (not presented) show enhancedsurface mixing, comparable to those found in calc. 2. Someslight differences in the region of topographic slopes wereevident because of differences in mixing with an up-wellingwind compared to a down-welling one. The change in M2

tidal current amplitude (Fig. 12) was not significantlydifferent from that found with the down-welling wind(compare Figs. 9a and 12) except for an increase inintensity and lateral extent in the surface u-currentamplitude at about 10°W. This is due to changes in shelfslope stratification, which influences the generation regionand subsequent propagation of the M2 internal tide.Comparable changes to those shown in Fig. 12 occurredat other cross-sections. For the M4 and M6 components (not

Fig. 10 a Temperature contours (°C) at cross-section D1n from a tide and wind (0.2 Pa towards the North) integration, b changes in temperature(°C) at 200 m for a tide and north wind (0.2 Pa) minus a tide-only calculation, c as b but for a south wind

318 Ocean Dynamics (2007) 57:305–323

0

Longitude

Depth(m)

11W

10W

9W

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

40

42

44

46

48

2200

2000

1800

1600

1400

1200

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Longitude

Depth(m)

11W

10W

9W

.0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

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2.2

2.4

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2000

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1200

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Longitude

Depth(m)

11W

10W

9W

.00

.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

2200

2000

1800

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1400

1200

1000

800

600

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2000

a b

c d

Longitude

Depth(m)

11¡W

10¡W

9¡W

-6.4

-6.2

-6.0

-5.8

-5.6

-5.4

-5.2

-5.0

-4.8

-4.6

-4.4

-4.2

-4.0

-3.8

-3.6

-3.4

-3.2

-3.0

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

Fig. 11 As Fig. 9, but for a long integration (calc. 3) with tide and wind (0.2 Pa towards the North), for a M2, b M4 and c M6 tide with d log10 t.k.e.(m2 s−2)

Ocean Dynamics (2007) 57:305–323 319

shown), the dominant features of the distribution andmagnitudes of the changes were not appreciably differentto those shown in Fig. 9b,c.

As to be expected, there is appreciably more mixing inthe slope regions, in the 10-day wind-forced calculation(calc. 5). Calculations showed that wind forcing givesenhanced mixing in the slope region that subsequentlyspreads further into the ocean. Consequently, this change indensity influences both internal tide generation and prop-agation. As in the north wind case, contours of temperaturechange produced by the presence of the wind (Fig. 10c)show significant spatial variability in the shelf-edge region.As this wind direction is up-welling favourable, then across-sectional model such as that of Xing and Davies1997 would give cooler water in the shelf edge region.Such regions are clearly evident in Fig. 10c with themajority of regions where warming occurred previously(Fig. 10b) replaced with cooler regions (Fig. 10c) becauseof the change in wind direction. As before, there is nosignificant change in the deep ocean. However, at about57.5°N in both wind cases, there is an increase in watertemperature, which arises from enhanced vertical mixing of

tidal and wind origin at this location. The differences inmixing and density advection because of change in winddirection explains why there are differences in the M2

internal tide under an up-welling compared to down-welling favourable wind (compare Figs. 11a and 13a).

For the M4 component, the modification because of theup-welling wind shows (Fig. 13b) a larger change at depthand at a different position along the shelf slope thanpreviously (Fig. 11b). In addition, the surface distributionand that on the western side of the seamount has beenmodified (Fig. 13b). As discussed previously, these changesarise because of the different distribution of the M2 tidecomputed with an up-welling rather than down-wellingfavourable wind. Similar modifications to the M4 tide werefound at other cross-sections.

The dominant features of the distribution of the changein M6 u-current amplitude (Fig. 13c) are comparable tothose found with a down-welling wind (Fig. 11c) and onaverage are small, of the order of 0.5 cm s−1. Although thelarge-scale features are the same, on the smaller scale, thereare some differences because of the differences in thedistributions of the M2 and M4 tide. As discussed earlier,the M6 tide arises primarily from the interaction betweenthese components. At other cross-sections, the differencesin tidal distribution are comparable to those shown at cross-section D1n.

The calculations presented here show that on the longertimescale, wind effects substantially modify the M2

component of the internal tide and its higher harmonics.Furthermore, the effect upon the internal tide is differentunder up-welling compared to down-welling wind forcing.Changes in the internal-tide distribution arise not only inthe surface layer but at depth where the wind field hasmodified the density field in the region of internal-tidegeneration. In addition to changes in the M2 component, thehigher harmonics are modified to a different extent depend-ing on wind direction.

5 Conclusions

The three-dimensional calculations presented here extendearlier work (Xing and Davies 1997), which used a simplecross-sectional finite-difference model to study the influ-ence of the wind. Such a slice model could not account forthe along-shelf variation of topography or for the presenceof seamounts. In addition, by using a proven finite-elementmodel (Hall and Davies 2005a) with its ability to refine thegrid in regions of rapid topographic change, it is possible toresolve the wind-induced mixing in the slope regions. As inHall and Davies 2005a, the model was forced with the M2

barotropic tide. However, unlike in that paper where theemphasis was on the influence of grid resolution upon the

Longitude

Depth(m)

11°W

10°W

9°W

0

1

2

3

4

5

6

7

8

9

10

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

Fig. 12 Contours of the difference in baroclinic tidal amplitude (cm s−1)between wind (0.2 Pa towards the South) and tidal solution and a tide-only solution for short-wind-duration forcing (calc. 4) at cross-sectionD1n for the M2 period

320 Ocean Dynamics (2007) 57:305–323

Longitude

Depth(m)

11¡W

10¡W

9¡W

0

2

4

6

8

10

12

14

16

18

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22

24

2200

2000

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Depth(m)

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10¡W

9¡W

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.2

.4

.6

.8

1.0

1.2

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1.8

2.0

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2000

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1600

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Longitude

Depth(m)

11¡W

10¡W

9¡W

.00

.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

2200

2000

1800

1600

1400

1200

1000

800

600

400

2000

a b

c

Fig. 13 As Fig. 9, but for a long duration wind (0.2 Pa towards the South; calc. 5), for a M2, b M4 and c M6 frequency

Ocean Dynamics (2007) 57:305–323 321

M2 tide and higher harmonics were not examined, here theyare considered in detail. In essence, the focus of the tide-only calculations was an examination of the spatialdistribution of the M4 and M6 internal tide. Calculationsshowed that with realistic topography and viscous effects,there was no well-defined point for the M2 tide generationor well-defined beams along which it propagated. This wasrather different than the isolated seamount results of Lamb2004 who showed narrow well-defined beams for M2

propagation with higher harmonics generated in regionswhere these beams intersected. This could give rise to mid-water generation of higher harmonics. The absence ofnarrow beams in the present calculations meant that higherharmonics were a maximum in the shelf slope region or oneither side of the seamount, where the M2 current and itshorizontal derivative were a maximum. Similarly, in thesurface layer where the M2 internal tide was reflectedgiving rise to local patches of internal tide current, with anassociated horizontal gradient, regions of enhanced surfacecurrent amplitude at the higher harmonics occurred.

The absence of narrow beams of tidal energy in thepresent calculation is probably due to using realistictopography and including viscous and diffusive effects.As shown by Vlasenko et al. (2005), higher vertical modesare very rapidly damped when viscous and diffusive effectsare included. This leads to appreciable broadening of thetidal beams and a reduction in interaction between them.The application of realistic topography and an observeddensity profile means that there is no longer a single sourceof internal tide generation as would occur with constantbuoyancy frequency and slope.

Calculated turbulence energy distributions and associat-ed mixing show that a region of cooler water occurs abovethe Anton Dohrn seamount. Such enhanced mixing aboveseamounts has been observed in a number of areas (e.g.Kunze and Toole 1997).

Short timescale calculations showed that the wind onlymodified the near-surface stratification and the change tothe M2 internal tidal current and its higher harmonics wasrestricted to the surface layer. Longer timescale calculationsshowed that both the tide and wind modified the temper-ature field at depth, giving rise to a modification of thedensity field and a change in the distribution of the internaltide and its harmonics.

On the short timescale, changes in the M2 internal tideand its higher harmonics produced with up-welling favour-able winds were comparable to those found with the down-welling favourable wind. This similarity is because on theshort timescale, it is the mixing in the surface layer that hasthe greatest effect, and this is not significantly influencedby wind direction. On the longer timescale, there was moremixing in the slope regions, the extent of which dependedon wind direction. Consequently, the M2 internal tide and

its higher harmonics together with boundary layer turbu-lence and mixing were changed substantially more.

The calculations presented here clearly show thatchanges in the density field in regions of steep topographywill have an impact upon the spatial distribution of the M2

internal tide and its harmonics. In the present calculations,these changes were produced by wind forcing. However,along-shelf flows of oceanic origin will also influence thedensity field in the region of internal-tide generation.Effects such as this together with meteorological forcingwill lead to significant variability of the M2 internal tidalsignal. Consequently, non-linear interaction processes thatgive rise to higher harmonics and a cascade to turbulenceand mixing will be affected. This has implications in termsof trying to partially validate these non-linear effects byseeing to what extent internal tidal models can reproducethe energy transfer to higher harmonics. In addition, itimpacts upon the spatial intensity required for datacollection and model validation, using higher tidal harmon-ics. Similarly, the small-scale variability of t.k.e. suggeststhat high spatial sampling is required to examine itsdistribution. Because its intensity is strongly related to theinternal-tide distribution in the bottom boundary layer,which depends on local topography and stratification, thesealso need to be measured in detail. The fact that the densityfield and hence the internal tide is modified by meteoro-logical forcing suggests that it is important to obtainsynoptic data sets for model validation. This is verydifferent to the barotropic case where a comprehensivedata set could be collected by sampling at different times.

Acknowledgements The authors are indebted to Mrs. L. Parry fortyping the paper and Mr. R.A. Smith for help in figure production.Access to bottom topography and open boundary forcing wereprovided by Dr. J. Xing and are gratefully acknowledged. Access tothe QUODDY code via the website is much appreciated.

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