On determining the role of wind wave turbulence and grid resolution upon computed storm driven currents

Continental Shelf Research 20 (2000) 1825}1888

On determining the role of wind wave turbulence and grid resolution upon computed storm driven currents Alan M. Davies*, Simon C.M. Kwong, Roger A. Flather Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead, Merseyside, CH43 7RA, UK Received 11 November 1997; received in revised form 24 November 1998; accepted 20 September 1999

Abstract A three-dimensional coarse grid (of resolution 12 km) hydrodynamic model covering the European Continental shelf with eddy viscosity depending upon current intensity and windwave signi"cant height and period, together with a higher resolution (of order 0.924 km) limited area model of the sea region around the Shetland Islands are developed. These models are used to investigate the in#uence of a source of wind wave turbulence, and local grid re"nement upon the three-dimensional currents during a major storm, namely the storm of January 93. This period was chosen because it contains a number of storm events with signi"cant wave heights exceeding 10 m. Also, it is the time when the tanker Braer went aground on the Shetland Islands, and serves to illustrate the importance of high-resolution three-dimensional #ows in the region of an oil spill. The spatial distribution of the signi"cant wave height and period over the continental shelf at the time of the Braer spill is computed using the WAM wave model running on the same "nite di!erence grid as the continental shelf hydrodynamic model. This suite of models is used to investigate the spatial distribution of currents and waves at the time of the Braer oil spill on the Shetland Islands. The importance of including a highresolution grid in the region of the Shetland Isle and taking account of &far "eld' e!ects computed with the shelf wide model within the limited area model, together with the accuracy of local winds in determining the circulation in the vicinity of the Shetland Isle is considered. Calculations show that including an additional source of wave-dependent viscosity has a signi"cant in#uence on surface currents particularly in deep water. However, this is restricted to a high shear surface layer having a thickness of a few meters. Circulation "elds from the high-resolution model, show the importance of having accurate topography and a "ne grid in the region of the Shetland Isle, and input from the larger area

* Corresponding author. Tel.: #44-151-653-8633; fax: #44-151-653-6269. E-mail address: [email protected] (A.M. Davies). 0278-4343/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 2 7 8 - 4 3 4 3 ( 0 0 ) 0 0 0 5 2 - 2

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model. Details of the #ow "eld to the south of the Shetlands, and the time variability of an eddy in the region are shown to be in#uenced by the far "eld input to the area, which in turn depends upon the shelf wide wind "elds.  2000 Elsevier Science Ltd. All rights reserved. Keywords: European continental shelf; Shetlands; Storm currents; Wave turbulence; Oil spill; Hydrodynamic model

1. Introduction Although three-dimensional models of tidal currents over large areas such as the continental shelf (Davies, 1986a; Davies et al., 1997a, b) and limited high-resolution tidal models (Davies and Jones, 1996a; Jones and Davies, 1996) with a #ow dependent eddy viscosity, are now well established and validated, the problem of determining the wind-induced circulation still poses major di$culties. In the case of tidal models in homogeneous situations the vertical variation of the tidal current (current pro"le) is determined by bed frictional e!ects and the pro"le of bed generated turbulence in the vertical. In the case of wind-induced current pro"les in regions of signi"cant tidal currents, the current pro"le is determined by both the tidally produced turbulence at the sea bed, and meteorological-induced turbulence at the sea surface, a component of which is due to wave breaking (Davies, 1985). The magnitude of the wind forced surface current, its depth of penetration, and the associated wind waves during major storm events is of practical importance in two respects. Firstly, in terms of the forces on o!shore structures, these can be related to both surface currents and wind wave magnitude. A second important issue is how is oil dispersed from any signi"cant spillage that occurs during a major wind event. The movement of the oil is signi"cantly in#uenced by wind driven surface currents and the thickness of the surface wind driven layer (Samuels et al., 1982). Although signi"cant work has been done to examine the two-dimensional response of a sea region to atmospheric forcing (e.g. Gjevik, 1991; Gjevik and Merri"eld, 1993; Zhang and Li, 1996), three-dimensional calculations taking account of wind wave e!ects have only recently been performed (e.g. Davies and Lawrence, 1994, 1995; Zhang and Li, 1997; Davies et al., 1998). In shallow near coastal regions wind wave-induced turbulence at the sea bed is important and has been considered in three-dimensional calculations by Davies and Lawrence (1994, 1995), Li and Zhang (1997) and Zhang and Li (1997). However in deeper water this e!ect is of less important, although the e!ect of wave breaking at the surface, upon surface currents during major storms is signi"cant (Davies, 1985). This problem, together with the question as to how "ne a grid is required in the horizontal to resolve the wind-induced circulation, is addressed in detail in this paper. In a previous paper, Davies et al. (1998) used the 12 km grid, three-dimensional model of the shelf to examine the wind-induced #ow at the time of the Braer oil spill. The three-dimensional numerical model used in that calculation had a coarse "nite di!erence grid in the horizontal and therefore could not take account of the detailed #ow "eld in the region of the Shetland Islands where the tanker went aground. Also,

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no account of wind wave e!ects was considered in the model, although these are important in determining wind induced circulations and hence the fate of any oil spill. In this paper we use the WAM third generation wave model over the shelf to compute the spatial variability of signi"cant wave heights and periods during the "rst ten days of the January 1993 storm the time following the Braer oil spill. From this spatial and temporal variability of signi"cant wave height and period, a wavedependent eddy viscosity is computed which together with a wind and #ow-dependent viscosity is used in the three-dimensional #ow model to determine currents on the shelf. The e!ect of a wave-dependent eddy viscosity upon the currents is examined by comparing the #ow "elds with those of Davies et al. (1998) where this additional viscosity was not present. The principal objective here is to determine the in#uence of a wave-dependent source of viscosity upon the surface currents, and hence the importance of knowing the wave "eld at the time of major storm events, and the nature and accuracy of the near surface measurements that need to be made in order to validate the form of viscosity used in the model. It is not the intention here to try and predict oil movements or near surface currents, in that current measurements were limited to a small data set collected using H.F. Radar close to the Shetlands for a very limited period (not the period examined in detail here) and hence a comprehensive set of measurements for model validation is not available. However a detailed set of HF Radar measurements together with Acoustic Doppler Current Pro"ling measurements has recently been made in the North Channel of the Irish Sea during a major storm, and comparisons with this data using a high-resolution model (of order 1 km) of the region embedded within the shelf wide model have recently been made (Davies et al., 1998a). This comparison shows the importance of wind wave turbulence in determining surface currents, and the need for an accurate and detailed set of measurements for model validation. Unfortunately, such a data set was not available for the storm period considered here. The present study is aimed at identifying the importance of various processes and hence the type of measurements, e.g. wave, near surface currents, current shear, etc., which need to be made. The "nite di!erence grid of the three-dimensional shelf model (Fig. 1) is not su$ciently "ne to accurately resolve the region close to the Shetlands where the oil spill occurred, and to study this in more detail a high-resolution (grid spacing 0.924 km), three-dimensional limited area model of the Shetland region (Fig. 2) was also developed. (The bottom topography used in the models is shown in Figs. 3(a) and (b)). The Shetland model was forced, with both the predicted winds used over the whole shelf, and observed winds measured at a point on the Shetlands close to the oil spill, and assumed to be uniform over the whole region of the high-resolution model. By comparing wind stresses, and currents, in particular surface #ow computed from both observed and predicted winds it is possible to assess the level of accuracy of the #ow "elds, which are responsible for advecting the oil, using both predicted and observed winds. From this comparison it is possible to determine di!erences in surface currents from predicted winds and those based on local observations. The importance of an accurate knowledge of surface currents for oil spill modelling has been shown by Samuels et al. (1982), and its relation to the surface wind in oceanic conditions has been demonstrated by Hughes (1956), although in shallow seas bottom friction and

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Fig. 1. Finite di!erence grid of the three-dimensional shelf model.

coastal setup has an in#uence (Davies, 1985, 1986b). Also analytical studies using single point models, in in"nitely deep oceans (Dobroklonskiy, 1969; Dyke, 1977; Weber, 1981) have shown the importance of near surface viscosity in determining surface currents. Calculations are also performed using the Shetlands model with and without input from the shelf model in order to determine the importance of far "eld e!ects upon the circulation in the region of the Shetlands. The importance of high-resolution in the region of the oil spill, which occurred to the west of Queendale Bay situated at

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Fig. 2. Finite di!erence grid of the high-resolution Shetlands model.

the southern tip of the Shetland Islands (Fig. 2), will be considered by examining the spatial distribution of currents in the region. A brief description of the WAM wave model, and the three-dimensional hydrodynamic model, with references to the literature for detail, is presented in the next section, together with the form of eddy viscosity used in the hydrodynamic model. In subsequent sections the wind "eld and computed wave parameters over the shelf are described together with the resulting #ow "eld as are results from the Shetland model, together with a "nal conclusions section.

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2. Wave and three-dimensional hydrodynamic models 2.1. The WAM wave model The WAM spectral wave model, originally developed by Hasselmann and described by the WAMDI group (1988) has been applied in a wide range of problems (Komen et al., 1994; Wolf et al., 1988), and is used here to examine the spatial and temporal variability of the waves over the continental shelf during the storm of January 1993.

Fig. 3. (a). Bathymetry of the shelf region, with the 25, 50, 75, 100, 200, 300, 400, 500, 1000 and 2000 m contours marked on. (b) Bathymetry of the Shetland model, in metres (contour interval 10 m).

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Fig. 3. (Continued.)

The model contains terms for the generation of waves, their advection and dispersion, and wave}wave coupling together with wave dissipation through bottom friction and white-capping. Details of the model can be found in the literature (e.g. WAMDI group, 1988; Komen et al., 1994) and will not be repeated here. Calculations were performed over the region covered by the three-dimensional model (Fig. 1) with a source of swell introduced along the open boundary, approximating oceanic waves propagating into the area. Output from the model in terms of signi"cant wave height and period was saved at hourly intervals and used to compute a source of wave

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turbulence in the three-dimensional hydrodynamic model. This was parameterised using a coe$cient of eddy viscosity depending upon signi"cant wave height and period (Davies, 1985). 2.2. Three-dimensional hydrodynamic model The non-linear three-dimensional hydrodynamic equations are solved using a standard staggered "nite di!erence grid in the horizontal (Figs. 1 and 2) with a spectral/functional approach in the vertical, giving a continuous current pro"le from sea surface to sea bed. (A detailed description of the model and associated boundary conditions is given in Davies (1986a, b, 1987), with further detail and applications in Davies et al. (1998) and will not be repeated here.) The functional approach used here has also been applied to the calculation of wind induced #ows in a range of situations (Gordon, 1982; Gordon and Spaulding, 1987). The vertical di!usion of momentum in the model is parameterised using an eddy viscosity k, which is described in the next section. 2.3. Form of vertical eddy viscosity The vertical eddy viscosity is a combination of that due to wind waves k , 5 meteorological forcing k , and currents k , thus + ! k"k #k #k 5 + ! with k given by 5 k "C H/¹ (1) 5  1 + with C "0.028 a constant (Davies, 1985), and H and ¹ signi"cant wave height  1 + and period computed using the WAM model. The direct meteorological/wind stress contribution to the viscosity is given by Davies and Jones (1996b) k "i; j , (2) + H H where i"0.4 is Von Karman's constant, with ; wind friction velocity, and H j a background roughness with j "0.02 m used in all calculations, in order to H H compare results with those of Davies et al. (1998) derived without a speci"c source of wave-induced viscosity. The #ow-dependent viscosity, k is given by ! k "C (u #v )/u (3) !   with C "2.5;10\ a constant, and u "10\ s a typical frequency. This viscosity   formulation has been used previously (Davies, 1986a), and has been shown to be appropriate for tidal current simulations on the shelf (Davies and Aldridge, 1993), with u the order of the Coriolis frequency or the M tidal frequency. In Eq. (3) u and   v are the depth mean currents. The contribution to the viscosity from the waves, Eq. (1) is based upon laboratory and "eld measurements (Ichiye, 1967). It represents

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the downward transfer of momentum both by the turbulence and all small scale mixing processes, e.g. vertical velocity, wave breaking and even larger scale motions, e.g. Langmuir circulations, associated with the waves, which cannot be resolved in the model (see Davies (1985) who used such a wave-dependent form of viscosity in a number of idealized calculations). The depth of penetration of this wind wave turbulence is di$cult to determine, although based upon the depth of penetration of bubbles, it may exceed 4H (Thorpe, 1984, 1992) and such e!ects have been included in  shelf wide models (Davies et al., 1998a). The wind stress contribution is based upon classic boundary layer #ow, applied to the sea surface boundary, where j can be H regarded as a sea surface roughness length. This should probably be related to the form drag over a surface wind wave "eld, or some aspect of the wave spectrum (e.g. wave age) (Donellan, 1990). However, since such information was not available, a speci"ed value was assumed here. In shallow regions both wind induced and tidal currents can generate signi"cant turbulence at the sea bed which can reach the sea surface and hence contribute to the downward di!usion of the wind's momentum. The #ow-dependent eddy viscosity Eq. (3) is a simple parameterization of this process which has been very successful in tidal models (Davies, 1986a). 3. Computed waves and currents over the shelf 3.1. Form of the calculations In the "rst series of calculations the waves and currents were determined over the whole shelf using the "nite di!erence grid shown in Fig. 1, which has a grid spacing of 1/63;1/93 (approximately a 12 km resolution). In a subsequent series of calculations the high-resolution model was used. Initially, the wave "eld was determined by integrating the wave model forward in time from 00 h 1/1/93 with approximate hourly wind stress forcing (Fig. 4(a)) and swell input along the open boundary. Calculations were performed for a 10 day period with H and ¹ being saved at hourly intervals in 1 + order to determine k within the three-dimensional model. The computed spatial 5 distribution of signi"cant wave height and period appeared to be physically realistic and vary smoothly over the interior of the model, although close (to within three grid boxes) to the open boundary there appeared to be a mis-match with the swell input along the open boundary. This problem was investigated by running the model without a swell input, which gave a smooth solution in the vicinity of the open boundary with no signi"cant change in the interior. For completeness the results with a swell input are given here, although this has no signi"cant e!ect upon the solutions. In the #ow calculations the three-dimensional model was run with tidal forcing (15 tidal constituents), along the open boundaries of the model, and wind forcing applied at the sea surface. A parallel calculation with tidal forcing only was also performed, with the wind driven #ow being computed by subtracting this tidal solution from that due to tide and wind. It is necessary to include the tidal forcing in the wind forced calculation because in shallow water the tidal currents dominate the solution and are responsible for the background level of friction. Also turbulence produced by the tide

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has a signi"cant in#uence in shallow water in determining the rate at which the wind's momentum di!uses out of the surface layer (Provis and Lennon, 1983). A series of calculations was performed with and without an additional source of wave induced turbulence k in order to examine the in#uence of wave turbulence 5

Fig. 4. Spatial distribution at t"132 h of (a) wind stress vectors and contours (N m\), (b) signi"cant wave heights H (m) and periods ¹ (s), (c) surface current vectors (p"0.0, where p"z/h with z the vertical  + coordinate) computed using viscosity pro"le (A) with no wave input, (d) as (c) but with wave viscosity, (e) for mid-depth currents and contours of surface elevation, (f ) surface current vectors computed with viscosity pro"le (B) with no wave input, (g) as (f ) but with wave viscosity, (h) as (g) but just below the surface (i.e. p"0.05).

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Fig. 4. (Continued.)

upon surface currents. In all calculations the model was started from a state of rest at 00 h 1/1/93 and integrated forward in time with tidal and wind stress (Fig. 4(a)) forcing. In order to avoid exciting inertial oscillations in the surface layer where they can persist for many days the wind stress was initially applied gradually over a period of three inertial oscillations. During the "rst "ve days of the integration the wind stress over the northern North Sea was predominantly from the west, reaching a maximum of 1.6 N m\. (A detailed discussion of this is given in Kwong et al. (1997), hereafter referred to as KDF97 and will not be repeated here.) Over the next 12 h the region of

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Fig. 4. (Continued.)

maximum wind stress was concentrated o! the west coast of Scotland, and along the Norwegian coast with negligible winds to the south of these areas (Fig. 4(a)). In the series of results presented in the next section we will concentrate upon the storm period between the 5 January 1993, and the 11 January 1993. Results computed with the inclusion of a wind wave-dependent viscosity will be compared with those derived previously (Davies et al., 1998) without this term. For consistency the wind stress forcing in the model was identical to that used by Davies et al. (1998), using the approach described in Flather and Smith (1993).

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Fig. 4. (Continued.)

3.2. Specixc calculations In an initial series of calculations eddy viscosity was assumed to be constant in the vertical (referred to as viscosity pro"le A by Davies et al., 1998). In a "rst calculation (Calcn 1, Table 1), k was set equal to zero, although subsequently (Calcn 2, Table 1), 5 k was computed from Eq. (1). 5 At 12 h 6 January (t"132 h) a wind from the south west is evident o! the west coast of Scotland with a westerly wind o! the Norwegian coast (Fig. 4(a)) giving rise to

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Fig. 4. (Continued.)

a region (Fig. 4(b)) with signi"cant wave heights H of order 4 m and periods ¹ of 1 1 order 10 s in the northern North Sea. It is evident from Fig. 4(b) that the signi"cant wave height reaches a maximum (of order 7.0 m) o! the north-west of Norway, a region of strong (of order 0.8 N m\) wind stresses and a long fetch for westerly winds. The sheltering e!ect of the Shetland Island, which reduces the fetch for westerly winds can be clearly seen in Fig. 4(b), with a signi"cant local reduction in wave height and period to the east of the Shetlands. This local change in wave "eld is of some importance in any oil pollution model in

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Fig. 4. (Continued.)

that it will locally a!ect the Lagrangian motion of the oil due to the waves. (A discussion of this is beyond the scope of this paper, but it does illustrate the importance of detailed local wave predictions.) A more gradual decrease in signi"cant wave height and period between the northern North Sea and Southern Bight is clearly evident in Fig. 4(b). Surface currents in deep water (namely o! the shelf edge to the west of Scotland) computed without the inclusion of wave turbulence (Fig. 4(c)) are signi"cantly larger than those on the shelf, or those computed with a source of wave turbulence (Fig. 4(d)). The reason for this is that in deep water, the eddy viscosity determined from the #ow

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Fig. 4. (Continued.)

"eld is quite small, since the depth mean currents are small in this region, and hence surface currents and shear are large. Introducing a wave-dependent source of turbulence, signi"cantly in#uences the eddy viscosity in this area and reduces surface currents (Fig. 4(d)). In other areas, particularly in the central North Sea, where the wave amplitudes are less signi"cant (H "2.0 m) including the wave-dependent 1 viscosity produces a slight decrease in surface current (compare Figs. 4(d) and (c)) although the pattern of #ow does not change signi"cantly. The reason for the small change in currents in these areas produced by adding a source of wave turbulence, is that in this region the wind stress and wind waves are small, and the #ow is mainly

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Fig. 4. (Continued.)

driven by the north}south elevation pressure gradients (Fig. 4(e)) that have been produced as a result of the earlier westerly winds (Davies et al., 1998). The southerly #ow along the western side of the Norwegian Trench, with a northerly #ow within the trench which is evident at all depths (see Figs. 4(d) and (e)) is characteristic of the response of the North Sea to a westerly wind stress and the resulting elevation gradients (Fig. 4(e)). (See discussion in Dooley and Furnes, 1981; Furnes, 1980; Furnes et al., 1986.) The elevation gradients shown in Fig. 4(e), and the currents at mid-depth were not signi"cantly di!erent from those computed with no surface source of wave turbulence.

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Table 1 Calc.

Model area

Viscosity pro"le

Wave viscosity

Input from shelf model

Observed stresses

Interpolated stresses

1 2 3 4 5 6 7

Shelf Shelf Shelf Shelf Shetland Shetland Shetland

A A B B A A A

No Yes No Yes No No No

* * * * No Yes Yes

* * * * Yes Yes No

Yes Yes Yes Yes No No Yes

An alternative to applying a constant eddy viscosity in the vertical, is to assume that it reduces linearly in a surface layer of thickness 0.2h (with h the water depth) to a surface value of 0.02 k (termed viscosity pro"le B) with k determined from Eq. (1). This viscosity pro"le was used by Davies et al. (1998) without a surface source of wind wave-dependent viscosity. To compare the e!ects of a surface wave-dependent viscosity with those computed by Davies et al. (1998) we also consider currents computed with viscosity pro"le B. Surface currents computed with viscosity pro"le B with no source of wave turbulence (Calcn 3), show (Fig. 4(f )) physically unrealistically large (in many cases up to 3 m s\) surface currents in the region o! the west coast of Scotland and in the Northern North Sea. In these areas the signi"cant wave H is appreciable and when 1 a wave-induced surface source of turbulence is added (Calcn 4) then the magnitude of the surface currents are reduced (Fig. 4(g)). Although there are signi"cant di!erences in the spatial distributions of surface currents computed with viscosity pro"les (A) and (B) even when wave e!ects are included (cf. Figs. 4(d) and (g)), this di!erence is con"ned to the surface layer, in that below this layer at p"0.05 (where p"z/h is a normalized vertical coordinate, with z the vertical axis) there is no signi"cant di!erence in the #ows computed with either viscosity pro"le A or B (cf. Figs. 4(h) with (d)). Also, mid-depth currents and elevations gradients (not shown) determined in Calculations 1, 2, 3 or 4 were not signi"cantly di!erent. Over the next 12 h the magnitude of the wind stress increased o! the west coast of Scotland and by 00 h 7 January (t"144 h) a strong south westerly wind had developed in the region which subsequently moved northward, with the region of strongest winds by 12 h 7 January (t"156 h) situated to the north of Scotland (Fig. 5(a)). This strong wind "eld gave rise to large surface waves in the area o! the west coast of Scotland, with signi"cant wave heights exceeding 8 m (associated wave periods exceeding 10 s) in the region of strongest winds. To the north and south of this area wave heights decreased, although wave heights above 4 m still occurred in the region o! the Shetlands. Comparing Figs. 5(b) with 4(b) it is evident that the in#uence of the Shetlands upon the waves in its vicinity has been reduced. Previously (Fig. 4(a)), westerly winds had existed in the region of the Shetland, and the &sheltering' e!ects of

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the Shetlands on the wave "eld to the east had been a maximum. In the present case this e!ect is reduced as the dominant wind direction is from the south west, although it is still clear from Fig. 5(b), that the local wave "eld to the east of the Shetlands is di!erent from that found over the majority of the northern North Sea. Surface currents at 12 h 7 January (t"156 h) computed with no wind wave source of turbulence, and eddy viscosity pro"le A (Calcn 1, Table 1), show (Fig. 5(c)(i)) a region extending from the north of Ireland to the Faeroes with surface currents

Fig. 5. As Fig. 4 but at t"156 h, but 5(c)(i) surface current, 5(c)(ii) mid depth current, and (5(f )) surface current vectors computed with viscosity pro"le (B), including a source of wave-dependent viscosity.

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Fig. 5. (Continued.)

exceeding 1 m s\, with particularly strong currents in deep water, where the signi"cant wave height exceeds 6 m. Although currents at mid-depth (Fig. 5(c)(ii)) are much smaller, and in the region to the west of the Shetlands exhibit signi"cant spatial variability which will be discussed later in connection with the high-resolution model. Including a surface source of wind wave turbulence (Calcn 2, Table 1) decreases surface currents in the deep water region (Fig. 5(d)), although a region of strong current can be readily seen on the shelf o! the west coast of Scotland. A slight (of order 10 cm s\) reduction in surface current is evident over the northern North Sea

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Fig. 5. (Continued.)

although the spatial distribution of currents is not signi"cantly in#uenced by the additional source of turbulence due to the waves. Currents at mid-depth and sea surface elevation distributions computed with a wind wave source of turbulence (Fig. 5(e)) were not signi"cantly di!erent from those determined without this term, with both solutions showing a strong #ow along the west coast of Scotland, part of which passes to the west of the Shetlands, with some #ow passing to the south of the Shetlands and into the North Sea. Comparing surface (Fig. 5(d)) and mid-depth (Fig. 5(e)) currents at this time in the region of the Shetlands, it is evident that there are signi"cant di!erences in the #ow

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Fig. 5. (Continued.)

"eld (both magnitude and direction) which will be discussed later in terms of the high-resolution Shetlands model and the movement of the oil spill. Surface currents (not shown) computed using viscosity pro"le (B), with no surface source of wind wave viscosity (Calcn 3, Table 1) were very large, of order up to 5 m s\, in the region to the north and west of Scotland. Currents of this magnitude appear physically unrealistic. However, it is clear that in this area waves are large (H  exceeds 6 m), and when a source of wave viscosity was introduced the surface current over the ocean region and the deep northern North Sea was signi"cantly reduced

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Fig. 5. (Continued.)

(Fig. 5(f )), although in the shallower regions there are no substantial changes since in these areas the wave "eld is small, and current eddy viscosity dominates. As previously a comparison of currents just below the surface layer (p"0.05) with surface currents (Fig. 5(f )) showed that the strong surface currents were con"ned to the near surface layer. Between 12 h 7 January and 06 h 9 January the area of maximum wind stress over the northern North Sea decreased, and was replaced by one in which the region of maximum wind stress moved from o! the southwest corner of England, over the Irish

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Fig. 5. (Continued.)

Sea and central North Sea. A detailed description of this is given in Kwong et al. (1997) (KDF97) and will not be repeated here. By 06 h 9 January (t"198 h) an area of south westerly wind stress over the Irish Sea and Central North Sea, with Southerly winds over the Norwegian Trench (Fig. 6(a)) had been established. A region where signi"cant wave heights exceeded 5 m is clearly evident (Fig. 6(b)) in the area between Scotland and Norway where the wind stress is a maximum (Fig. 6(a)) with wave periods of the order of 8.0 s. Wave heights decrease away from this region with a nearly linear decrease moving south into the shallower Southern Bight of the North

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Fig. 5. (Continued.)

Sea where wave heights are of the order of 2.0 m. A decrease in wave height in the shallower near coastal regions is also evident, with a reduction in wave period (Fig. 6(b)). Some local changes in wave height are also evident near the Shetlands. Surface currents in the North Sea o! the east coast of Scotland, computed with viscosity pro"le A and no source of wave viscosity (Calcn 1, Table 1) show strong (exceeding 1 m s\) surface currents, aligned with the wind direction (Fig. 6(c)) in the region of maximum wind stress. In the deep northern North Sea the current is aligned at about 30 to the right of the wind direction with a northerly out#ow in the

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Norwegian Trench. The existence of an out#ow in the Norwegian Trench under these conditions is con"rmed by measurements (Dooley and Furnes, 1981) and analytical models (Furnes, 1980). Increasing turbulence by adding a source of wave-dependent viscosity (Calcn 2, Table 1) reduces surface currents (not shown) over the central and northern North Sea region where maximum wind stress forcing and surface waves are present, although over the Southern Bight of the North Sea and in the Norwegian Trench surface currents are not changed signi"cantly. The reason for the small change

Fig. 6. As Fig. 4 but at t"198 h, but, (d) surface current vectors computed with viscosity pro"le (B) with no wave input, (e) as (d) but with wave viscosity.

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Fig. 6. (Continued.)

in the Southern Bight is that tidal turbulence dominates in this area. Also both in this region and in the Norwegian Trench, the northerly #ow is driven by the local elevation pressure gradients which have been setup as a response to the wind forcing. The depth mean currents and surface elevations computed with and without a surface source of wind wave turbulence were not signi"cantly di!erent, suggesting that the wave turbulence a!ects only the surface layer. Surface currents computed with viscosity pro"le B without any source of wind wave turbulence (Calcn 3, Table 1), are signi"cantly stronger (Fig. 6(d)) over the central North Sea than those found previously. At this particular time the strongest winds are

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Fig. 6. (Continued.)

over the central North Sea (Fig. 6(a)) rather than as previously where winds in this region were weak. In these circumstances even in shallow water where the current viscosity is large, in the absence of surface wave turbulence, the wind's momentum cannot di!use to depth. Adding a surface source of wave turbulence (Fig. 6(e)), produces a major reduction in the surface current in the deep water regions where the #ow-dependent viscosity is small with a signi"cant reduction in the North Sea (cf. Figs. 6(e) and (d)), where the #ow-dependent viscosity is much larger. However, the surface current is larger than that found with pro"le A. As in previous calculations, the region of high currents is con"ned to the near surface layer, with currents below this

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Fig. 6. (Continued.)

layer (p"0.05) not signi"cantly di!erent from those found previously at the sea surface with pro"le A and a source of wave turbulence. The region of maximum wind stress show in Fig. 6(a), subsequently moved to the north-east and by 18 h/9/January was replaced by a region of westerly to southwesterly winds over the northern North Sea. The resulting computed #ow "elds are described in KDF97 and will not be discussed here, where we will concentrate upon the e!ects of a subsequent build up of winds over the Celtic Sea region. Between 06 h 10 January (t"222 h) and 18 h 10 January (t"234 h) the west coast and Celtic Sea

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Fig. 6. (Continued.)

area was subjected to wind stresses of up to 3 N m\ (Fig. 7(a)). These wind stresses produced signi"cant wave heights o! the west coast of Ireland which exceed 8 m (Fig. 7(b)) with wave periods of the order of 9 s, with comparable wave heights and periods in the Celtic Sea. To the north of Scotland and over the North Sea where the wind stress was not so intense (Fig. 7(a)) signi"cant wave heights were of the order of 4 m, decreasing to 2 m in the Southern Bight of the North Sea, where wave periods were the order of 5 s (Fig. 7(b)). As in the previous calculations a region of local wave variations existed to the east of the Shetlands.

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Surface currents, (computed using viscosity pro"le A (Calcn 1, Table 1) with no source of wave turbulence) o! the west coast of Scotland and within the Celtic Sea, are in essence aligned with the wind "eld with areas of maximum surface current corresponding to areas of maximum wind stress (Fig. 7(c)). In regions of maximum wind stress the surface current can exceed 2 m s\, although over the southern North Sea, surface currents are below 0.1 m s\.

Fig. 7. As Fig. 4 but at t"222 h, but just plots (a), (b) and (c), with (d) surface current vectors computed with viscosity pro"le (B) including a source of wave-dependent viscosity.

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Fig. 7. (Continued.)

Surface currents computed with the addition of wave viscosity (Calcn 2, Table 1) (not shown) were reduced in the area of large waves with current magnitudes being reduced in some areas by 0.5 m s\, although the direction of surface currents was essentially unchanged. Over areas such as the North Sea the resulting elevation gradients produced by earlier storm events are primarily responsible for driving the #ow in these regions and surface currents are not signi"cantly a!ected by wavedependent viscosity which is smaller in these regions. As previously surface currents computed using viscosity pro"le B, without the addition of a wave-dependent viscosity (Calcn 3, Table 1) appeared physically unreal-

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Fig. 7. (Continued.)

istic (exceeding 5 m s\) in the deep water regions o! the west coast of Scotland and in the Celtic Sea. Adding a source of wave-dependent viscosity (Calcn 4, Table 1) was found to signi"cantly reduce the magnitude of the surface currents (Fig. 7(d)), although they are still large in the surface layer in deep water o! the west coast of Scotland. A detailed examination of the surface #ows showed that the high sheared layer, on the shelf is the order of less than 5 m in thickness, with currents, essentially aligned with the wind "eld, but with reduced magnitude occurring below this layer. Over the next 6 h the region of maximum wind stress moved over the North Sea (Fig. 8(a)), with stresses over the central North Sea of the order of 0.8 N m\, and

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Fig. 7. (Continued.)

exceeding 1.2 N m\ over the northern North Sea, and o! the west coast of Scotland. Associated with this change in the region of maximum wind stress, the area of maximum wave height (where wave heights exceeded 11 m) moved northward to a position just to the west of the Hebrides (Fig. 8(b)). It is interesting to note that in this case, the wind direction in the region of the Shetlands was from the south east, and this gave a region of rapid local variations in the wave "eld on the western side of the Shetlands. Also, with this wind direction the contours of signi"cant wave height which

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were previously (Fig. 7(b)) aligned to "rst order west}east across the North Sea, are now aligned north}south. As in previous calculations, surface currents computed without a wave-dependent surface eddy viscosity (Calcn 1, Table 1) are large (of the order of 2 m s\) in the region to the north west of Scotland and over the central northern North Sea (Fig. 8(c)). However, the magnitude of these surface currents (not shown) is dramati-

Fig. 8. Spatial distribution at t"228 h of (a) wind stress vectors and contours (N m\), (b) Signi"cant wave heights H (m) and periods ¹ (s), (c) surface current vectors (p"0.0) computed using viscosity pro"le (A) 

with no wave input, (d) for mid-depth currents and contours of surface elevation.

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Fig. 8. (Continued.)

cally reduced in the region o! the west coast of Scotland when a source of wavedependent viscosity is added. In other areas such as the Southern Bight of the North Sea this additional viscosity has a smaller though signi"cant e!ect, since waves are not so intense in this region, and the #ow induced viscosity is larger than in deep water. However, as shown previously when the region of strong wind forcing occurs over shallow water, it is necessary to add a source of wave-dependent viscosity in order to di!use the wind's momentum out of the surface layer. The dominance of the south-westerly component of the wind at this time gives rise to a west}east elevation gradient along the west coast of Scotland which drives a

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Fig. 8. (Continued.)

north going #ow at depth (Fig. 8(d)). Observational evidence (Hill and Simpson, 1986) and tracer studies (McKay et al., 1986) exist to support these #ow "elds and the importance of elevation gradients produced by the wind in driving them. Similarly pressure gradients due to the wind determine the exchange between the Irish Sea and the west coast of Scotland through the North Channel (Brown and Gmitrowicz, 1995). In the southern part of the North Sea the north going #ow from the Southern Bight and along the west coast of Denmark, reduces sea surface elevations in the region of the Wash. A comparison of Fig. 8(d) with a similar "gure computed by Davies et al.

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Fig. 8. (Continued.)

(1998) without a surface source of wave viscosity showed no signi"cant di!erences in elevations or mid-depth currents. Surface currents computed using viscosity pro"le (B) (Calcn 3, Table 1) without a surface source of wave turbulence (not shown) exceeded 10 m s\, and even when a surface source of wave-dependent viscosity was added (Calcn 4, Table 1) surface currents to the north-west of Scotland still appeared physically unrealistic. As in previous calculations these extreme currents were restricted to a near surface boundary layer, with currents below this layer (not shown) exhibiting more realistic values.

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This series of calculations using the large area coarse grid model clearly shows that during the "rst ten days in January wind stresses over the shelf in the region o! the west coast of Scotland and the northern North Sea were particularly strong reaching the order of 3 N m\. These wind stresses produced large amplitude waves with signi"cant wave heights above 8 m, and wave periods exceeding 8 s. In the majority of cases the winds were from the west or south-west, and hence the region to the east of the Shetlands was sheltered from these waves, and an area of reduced signi"cant wave height was found. Di!erences in signi"cant wave height between the west and east coast of the Shetlands depending on wind direction is an important consideration in determining the Lagrangian movement of the oil associated with Stokes drift due to the surface waves. Also di!erences in the in#uence of wave turbulence (depending upon H and ¹ ) upon the depth of the surface wind driven layer can be important in 

oil spill prediction. Surface currents computed with a source of wave-induced viscosity had a reduced magnitude and appeared more physically realistic than those computed without this term. In the case of an eddy viscosity pro"le in which viscosity was reduced in the surface layer, an additional source of wave viscosity was particularly important in producing physically sensible currents at times of major winds, and under these conditions the constant viscosity pro"le may be more realistic. The calculations considered here, suggest that if the eddy viscosity decreases in the near surface layer to a small value, as in a classic `law of the walla for a rigid boundary, and no additional wave-dependent source of viscosity is included, the surface currents appear physically unrealistic. The addition of an additional source of wave-dependent viscosity, intended to parameterize the vertical transfer of momentum due to small-scale processes associated with wind waves signi"cantly reduces the near surface currents, and appears to yield more physically realistic near surface currents. The exact form of this parameterization is di$cult to determine, although it should be related to parameters such as H and ¹ (Ichiye, 1967; Davies, 1985) with 

measurements (Thorpe, 1984, 1992) showing that the depth of penetration is the order of four times the signi"cant wave height. In the case of the wind stress-dependent viscosity, the e!ective roughness length j is di$cult to determine, although recent measurements (Stacey and Pond, 1997) H and detailed comparison of surface currents with surface measurements in the North Channel of the Irish Sea (Davies et al., 1998b), have suggested values as large as 1 m. The use of a j of this order in the present calculations would reduce the surface H currents below those given here, although without precise measurements of surface current of the form given in Davies et al. (1998b) the exact value of j cannot be H determined. The use of parameterizations of viscosity in terms of #ow "elds may be limited in conditions of major storms, and in a detailed comparison of computed surface currents with measurements, Davies et al. (1998b) used a turbulence energy closure model with a source of wind wave turbulence at the sea surface. Such a source of turbulence has a signi"cant e!ect on the downward transfer of momentum, and as suggested by Thorpe (1995) will contribute to the di!usion of oil in the surface layer, although this is outside the scope of this paper.

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Calculations using the large area model have clearly shown the signi"cant temporal and spatial variability of currents induced by the wind, and the role of bottom topography and coastal in#uence in determining the #ow directions. An accurate determination of the spatial variability of currents, and from this the computation of a horizontal di!usion coe$cient related to horizontal shear (Smagorinsky, 1963) is particular important in determining the di!usion of oil. Obviously in determining the fate of any oil spill it will be necessary to have an accurate representation of coastal and topographic features in the region of the spill together with accurate meteorological data. These points are considered in the next section in connection with the limited area high-resolution Shetland model.

4. Winds and 6ows in the Shetland region In the previous series of calculations we examined the wind driven currents over the continental shelf using the larger area model, and the in#uence of wind wave produced viscosity upon the surface currents. In this section we will use the high-resolution limited area Shetlands model, and consider the e!ects of running this model alone without an open boundary input from the larger area model, and subsequently the in#uence of open boundary input interpolated from the larger area model. Di!erences in computed currents using both the predicted meteorological forcing employed previously in the large area model, but now interpolated on to every grid point of the Shetland model, and observed winds taken from Sumburgh applied at every point will also be examined. The e!ects of enhanced resolution in the region of the Shetlands upon the #ows will be considered by comparing currents computed with the highresolution and shelf wide models. Since the wave data was only available on the coarser grid model, a source of wave viscosity will not be included in the limited area model, and eddy viscosity will be assumed constant in the vertical. (A source of wave viscosity can be included in the high-resolution model, as can an arbitrary pro"le of eddy viscosity. Here, in order to compare results with Calcn 1, the eddy viscosity was constant in the vertical and a surface source of wave turbulence was not included. However including this additional physics would have the same e!ect as in the earlier calculation). In all calculations tidal e!ects were incorporated in the high-resolution model by including a tidal input along the open boundary of the model determined using the same 15 tidal constituents as in the large area model. To obtain some insight into the spatial variability and magnitudes of the semi-diurnal tides and the higher harmonics in the region, Figs. 9(a) and (b), show the semi-major and semi-minor axis of the M and M tide at every third grid point of the model. It is evident from Fig. 9(a), that   the strongest tidal currents occur o! the southern tip of the Shetlands, where there is a near recti-linear #ow with M tidal currents reaching the order of 150 cm s\.  However, to the west and east of this tidal current strength decreases to the order of 50 cm s\. The presence of strong tidal currents, with a signi"cant change in direction at the southern tip of the Shetlands, means that the advective terms which are responsible for generating higher harmonics, (e.g. the production of the M tide from 

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the M tide) are appreciable in the region. These terms give rise to the signi"cant (of  order 15 cm s\) M tidal currents in the area (Fig. 9(b)). Although the tidal current  will signi"cantly in#uence the spatial distribution of vertical eddy viscosity in the region, the oscillatory nature of the #ow means that it does not transport material, which is moved by the residual #ow. As previously the wind driven residual #ow in the region was determined by subtracting a tidal solution from one with tidal and meteorological forcing. As described previously in connection with the large area model between 1200 h 6 January (t"132 h) (Fig. 4(a)) and 1200 h 7 January (t"156 h) (Fig. 5(a)) the wind stress increased over the Shetlands. At 1200 h 6 January 93 (t"132 h), the computed wind stress over the Shetlands (Fig. 10(a)) was small, of the order of 0.2 N m\, and there was no substantial di!erence between this and the wind stress based on measured winds (Fig. 10(a)). [The convention used in Fig. 10(a) is that the wind stress derived by interpolating the stress used in the large area model is plotted at every third model grid point. Since this wind stress is derived from interpolation it will have a di!erent value at every grid point, although as we will show its spatial variability is small. The wind stress derived from the observed wind at Sumburgh is also shown as the stress vector placed for convenience in the centre of the Shetlands. When using this observed wind stress as meteorological forcing to the model this wind stress vector is applied at every model grid point (i.e. a spatially uniform, although varying in time wind stress is applied)]. Residual surface currents computed using the Shetlands model with the observed winds (Calc 5, Table 1) without any input from the larger area model show a small direct wind driven surface current (Fig. 10(b)), with no signi"cant current below the top 5 m. In a subsequent calculation (Calcn 6, Table 1) the Shetland model was again run with the observed wind stress, but in this case changes in sea surface elevation and wind driven currents determined with the large area model at this time Fig. 4(c) were interpolated onto the open boundary and used as a `far "elda input to the Shetland model. Currents induced by wind forcing and this input from the large area model exhibited only a small vertical variation between surface currents and those at depth (not shown) in marked contrast to the high surface shear layer with little #ow below, that was found when the Shetlands model was only forced by local winds. A detailed comparison of surface currents derived with a `far "elda input with those based on local wind forcing only (Fig. 10(b)), showed that to the west of the Shetland introducing an input from the large area model reduced the surface current, and a signi"cant westerly #ow is present o! the southern tip of the Shetlands which was not present previously. The reason for this is that at this time the large area model produces a west}east pressure gradient in the region of the Shetlands with an associated in#ow to the Shetland limited area model along its eastern boundary with an out#ow along its western boundary giving a #ow to the west in the region of the Shetlands. This pressure driven westerly #ow in the region of the Shetlands, reduces the magnitude of the surface wind driven #ow to the west of the Shetlands and produces the westerly #ow to the south of the Shetlands. A detailed comparison of surface currents computed with the limited area model with those determined from the large area model

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Fig. 9. Spatial distribution of the semi-major and semi-minor axis of (a) the M tide, (b) the M tide, at the   sea surface, at every third grid point of the model.

showed that the large area model could reproduce the main features of the #ow in the region of the Shetlands but not the small-scale features. However, to reproduce details of the #ow the high-resolution model is required with an open boundary input from

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Fig. 9. (Continued.)

the large area model. In particular, some of the detailed #ow at the southern tip of the Shetlands and in the near coastal region in particular the strong #ow in the vicinity of the island situated o! the east coast of the Shetlands at 60.33N is not resolved in the

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Fig. 10. Distribution at t"132 h computed with the Shetlands model of (a) wind stress, (b) surface current (p"0.0) without a far "eld input.

coarse grid model. The small value of the wind stress at this time and the close agreement between observed and computed stresses means that there was no signi"cant di!erence between #ows computed with the observed and interpolated stresses (Calcn 7, Table 1). Comparing interpolated wind stresses with those based on observations at t"144 h (Fig. 11(a)) it is evident that the interpolated stresses are larger than the observed and the e!ect of this upon computed #ows will be discussed subsequently. At

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Fig. 10. (Continued.)

this time wind stresses were increasing in the region of the Shetlands as the weather system situated o! the west coast of Scotland (Fig. 4(a)) moved northward (Fig. 5(a)) giving stresses of the order of 0.8 N m\ over the Shetlands (Fig. 11(a)). Forcing the model with this wind stress alone (Calcn 5, Table 1) without any `far "elda #ow from the larger model gave signi"cant surface currents with little #ow at depth. Comparing surface currents derived using observed wind stresses and those based on interpolated stresses (Calcn 7, Table 1), including in both cases open boundary input from the large area model, showed that the currents derived from interpolated

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Fig. 11. As Fig. 10 but at t"144 h, with (a) wind stress, (b) surface current with a far "eld input using interpolated wind stresses.

stresses (Fig. 11(b)) were about 20% larger than those based on observed winds (re#ecting the di!erences in the wind stresses) although the spatial patterns were similar with both calculations showing a change in current magnitude and direction in the near coastal region. In both cases currents at depth below the surface layer were small, and hence at this time only oil in the surface layer would be transported by the residual #ow. The fact that the surface currents have di!erent magnitudes at this time means that the advection of oil in the surface layer will be di!erent. Also, at this time it

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Fig. 11. (Continued.)

appears that the local forcing is slightly more important than that produced by `far "elda e!ects. Over the next 12 h the wind stress increases over the Shetlands and by t"156 h a uniform northerly wind stress of magnitude 1 N m\ exists in the region (Fig. 12(a)). From Fig. 12(a), it is evident that the interpolated wind stress is not signi"cantly di!erent from that derived from the observed winds. Forcing the limited area model with the observed stress (Calcn 5, Table 1) without a far "eld input to the model from the shelf region, produces the surface current shown in (Fig. 12(b)). This current is

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con"ned to the surface layer with a region of near zero current below. Comparing these currents with those computed including input from the larger area model, at sea surface Fig. 12(c)(i) and mid depth Fig. 12(c)(ii) it is evident that in the region to the north of 603N currents at mid-depth are small (Fig. 12(c)(ii)). However, in the region to the south of this there is a #ow to the east which enters through the western boundary of the Shetland model with currents intensifying in the shallow water region

Fig. 12. As Fig. 10 but at t"156 h, with (a) wind stress, (b) surface current without a far "eld input, (c) (i) surface current with a far "eld input, (c) (ii) mid-depth (p"0.5) with a far "eld input. Also shown are contours of residual elevation (m).

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Fig. 12. (Continued.)

(Fig. 12(c)(ii)) to the south of the Shetlands. Also given in Fig. 12(c)(ii) are contours of residual elevation which show a north}south elevation di!erence of the order of 6 cm, in the region of strong (of order 50 cm s\) residual #ows o! the southern top of the Shetland Islands. Although these residual currents are strong, they are approximately half the magnitude of the M tidal current amplitude (of order 150 cm s\ Fig. 9(a)) in  the region. However, since the tidal #ow is oscillatory there will be time periods when the tidal current magnitude is below that of the wind driven #ow. During such periods

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Fig. 12. (Continued.)

it will be the wind driven current which determines the viscosity at depth and hence the rate of di!usion of the wind's momentum out of the surface layer. To understand the di!erences in the #ows computed with the Shetland model when driven by wind forcing alone (Calcn 5, Table 1) and wind forcing plus a far "eld input from the shelf model it is conceptually useful to divide the #ow into two parts. The "rst, a local wind driven current which is con"ned to the surface layer and in essence is given in Fig. 12(b). The second component of the #ow, produced by far "eld e!ects, is a uniform current through the whole water column (although it is reduced by

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Fig. 12. (Continued.)

frictional e!ects at the bed) which in essence is shown in Fig. 12(c)(ii). The surface current (Fig. 12(c)(i)) is therefore determined by local wind e!ects in the region north of 603N, (since at this time input from the shelf model has little or no in#uence in this area) and is to "rst order a linear combination of Figs. 12(b) and (c)(ii) to the south of 603N. In this area the `far "elda #ow is in essence to the east with current magnitude increasing to the south of the Shetland. Adding this #ow to that shown in Fig. 12(b), increases the eastward component of the #ow, and in particular the magnitude of the #ow to the south of the Shetland, as observed in Fig. 12(c)(i).

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The reason why the #ow north of 603N can be reproduced by the local wind but not that to the south can be readily understood from Fig. 5(c)(ii). This "gure shows a northerly mid-depth current that in essence #ows parallel to the western and eastern boundaries of the model to the north of 603N, with an eastward #ow (the Fair Isle current) entering the region to the south of this. The intensity of this #ow depending very much upon wind e!ects o! the west coast of Scotland (Davies et al., 1998). The fact that a model driven by local winds can at certain times reproduce the #ows north of 603N, although the #ow to the south of this is critically dependent particularly in the region to the south of the Shetlands upon the magnitude of the far "eld input interpolated onto the open boundary of the model, suggests that generalizations as to the successes of local area models, or the importance of `far "elda inputs to oil spill prediction models depends very much upon where the spill occurs and the dominant winds at the time. As previously the high-resolution model shows signi"cantly larger spatial variability in the #ow "elds than that in the coarser grid model, particularly in the near coastal region and this is important in determining where the oil ponds. Variations in magnitude of the surface current are also evident especially in near coastal regions associated with local gradient currents which modify the direct wind driven surface current (the wind drift current). These variations are important in determining the di!usion of oil in the surface layer. Over the following 12 h the wind stress magnitude over the area of the Shetlands increased and its direction changed to a spatially uniform south-westerly wind stress. Surface currents (Fig. 13(a)) computed with input from the large area model show a region of increased (relative to currents computed without an input from the large area model) surface current magnitude to the south}west of the southern end of the Shetland with a decrease in surface current to the south east. This variation was not present in the surface currents considered previously. This e!ect can be readily understood by examining currents at mid-depth (Fig. 13(b)) and the associated local elevation gradients which arise through the boundary forcing from the larger area model and local topographic e!ects. Fig. 13(b) shows a cyclonic gyre in the area to the south of the Shetland, with local elevation gradients producing gradient currents #owing in the same direct as the wind driven #ows to the south west of the southern tip of the Shetland, increasing the surface currents in this area, with a return #ow to the east o! the eastern coast of the southern part of the Shetlands against the direct wind driven currents. This #ow enhances surface currents to the south}west of the Shetland and reduces the surface current to the south}east at this time. Comparing currents at depth with and without (not shown) boundary forcing showed that this gyre is produced by the boundary forcing and local topography. This gyre should not be confused with the gyre in the residual tidal currents in the region, which has been removed when the tidal solution was subtracted from the total solution to produce the non-tidal residual shown here. To understand how this gyre in the wind driven residual #ow contributes to the total current at this time, it is interesting to examine the instantaneous tidal #ow (Fig. 13(c)) and the total #ow (Fig. 13(d)). The instantaneous tidal current shows (Fig. 13(c)) a uniform #ow in the region with maximum tidal currents of the order of 100 cm s\ to the south of the

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Shetland Island. The addition of the meteorologically induced residual #ow (Fig. 13(b)) which to the south-west of Shetland Island is signi"cantly larger (of order 50 cm s\) than the tidal #ow in this region (of order 20 cm s\), is to change the direction of the total current in this area, at this time (Fig. 13(d)), although elsewhere (e.g. in the northern part of the model), the #ow is dominated by the instantaneous tidal current (cf. Figs. 13(b)}(d)). In the region of the gyre in the meteorologically

Fig. 13. (a) Surface current using the observed wind stress with a far "eld input, (b) as (a) but for mid-depth (p"0.5) current, at t"168 h, (c) instantaneous tidal currents (p"0.5) at t"168 h, (d) total currents for mid depth (p"0.5) at t"168 h.

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Fig. 13. (Continued.)

induced currents (Fig. 13(b)), their magnitude and spatial distribution compared with that in the tidal #ow is such that the total #ow (Fig. 13(d)) still shows a gyre in the region, although with weak currents to the east of the gyre. Obviously, as the tidal #ow reverses and the magnitude and geographical distribution of the meteorological forced residual changes the contribution of each to the total instantaneous current will change. To examine the persistence of the gyre, mid depth currents from Calcn 6 (Table 1) are shown at t "174 h (Fig. 14(a)). Comparing Figs. 14(a) with 13(b), it is evident that at t"174 h the magnitude of the western boundary input has reduced,

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Fig. 13. (Continued.)

and this leads to a decrease in the currents associated with the gyre. Subsequently at t"180 h the boundary input has decreased to such an extent that the currents associated with the gyre are small (below 10 cm s\), and the position of the gyre has moved eastward as the current strength has decreased. A detailed examination (based upon residual current vectors at 3 h intervals, not presented), of the magnitude and persistence of the gyre over the period t"162}174 h showed that the gyre started to appear close to the southern tip of the Shetland Isle at t"162 h. Subsequently the magnitude of the currents associated with it increased, and its position moved to the east at t"165 h. Subsequently, current magnitudes

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Fig. 13. (Continued.)

increased and it moved further south to the position shown in Fig. 13(b). Over the following 3 h (t"171 h) the spatial extent of the eddy increased, although the magnitude of the currents decreased, until at t"174 h, current magnitudes and spatial extent had decreased (Fig. 14(a)). This circulation in the meteorologically induced residual #ow should not be confused with the persistent tidal residual #ow in the region which was removed when the total solution was de-tided. However, the mechanism producing this gyre, namely the non-linearities associated with the #ow in the region of the headland, will be

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similar to those that produce an appreciable tidal residual in the area. A detailed discussion of this is however outside the scope of this paper. At time t"198 h, a uniform southerly wind stress exists in the region of the Shetlands which combined with local elevations gradients produces a surface #ow to the north-east (Fig. 14(b)). Unlike in the previous case the surface current does not show an intensi"cation to the south of Shetlands, with a reduction to the east, but rather to "rst order, a uniform #ow in the region (Fig. 14(b)), without the gyre

Fig. 14. (a) Mid-depth (p"0.5) at t"174 h with a far "eld input, (b) surface current with a `far "elda input at t"198 h, (c) surface current with a `far "elda input, (d) mid-depth current with a `far "elda input, at t"234 h.

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Fig. 14. (Continued.)

circulation at depth found previously. Although current vectors at mid-depth were plotted at 6 h intervals up to t"234 h no signi"cant gyre was found. The fact that the gyre is only present at certain times, suggests that its existence depends very much upon the magnitude and direction of the #ow "eld, which is determined by large-scale meteorological e!ects. This was con"rmed by subsequent calculations described later in the paper. Although the spatial variability of currents found in the Shetland region in the large area model (Fig. 6(c)) shows similar features to those found using the high-resolution

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1883

Fig. 14. (Continued.)

model, the coarser grid model cannot reproduce the detail variations in the near shore region found in the high-resolution model. The location and magnitude of the residual eddy at depth and its in#uence upon the surface current is particularly interesting and very relevant to the residual movement of oil in the near-"eld region surrounding its release point. This can be further illustrated by considering two times when westerly and south westerly winds were present namely, t"234 and 240 h. At t"234 h the surface current computed with the observed winds and far "eld e!ects, (Fig. 14(c)) shows to "rst order an easterly #ow in the region of the Shetlands

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Fig. 14. (Continued.)

with an area of reduced #ow in the near coastal region on the western side of the southern part of the Shetlands, and enhanced #ow on the eastern side of the Shetlands. Considering the current at depth (Fig. 14(d)) it is evident that there is a strong easterly in#ow into the region from the southern part of the western boundary with current strength and elevation gradients increasing at the southern tip of the Shetlands as the water shallows (Fig. 3(b)). The local elevation gradients drive a #ow to the south on the western side of the island, with a #ow to the north on the eastern side. With these gradient currents enhancing the #ow to the south of the Shetlands.

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Surface currents 6 h later (namely t"240 h) (not shown) although exhibiting similar patterns to those found previously (Fig. 14(c)) do not show a region of enhanced surface current to the east of the Shetlands, but rather an area of reduced currents. Currents at depth at this time (not shown) are similar to those given in Fig. 14(d), with an in#ow into the region along the southern part of the western boundary, although the elevation gradients and the resulting #ows to the south-east of the Shetlands are di!erent, with a circulation eddy to the east of the Shetlands which was not present previously (Fig. 14(d)) and this has modi"ed the surface currents at this time. From a detailed examination of the #ow "eld throughout the period, the magnitude of the eddy and its exact location is particularly sensitive to the magnitude and spatial extent of the easterly #ow which enters the model through the southern part of the western boundary. Results from the large area model clearly show that the #ow paths to the west of the Shetlands are signi"cantly in#uenced by the topography and the time history and spatial variations of wind "elds on the shelf. Consequently, the magnitude and spatial distribution of the #ow along the western boundary of the limited area model may be di$cult to predict with a high level of accuracy which will in#uence the position of the eddy and hence may in#uence the near-"eld movement of the oil from the Braer spill.

5. Concluding remarks In this paper we have extended an earlier (Davies et al., 1998) three-dimensional model of the shelf to take account of a surface source of wind wave-induced turbulence, parameterized in terms of a coe$cient of eddy viscosity depending upon signi"cant wave height and period. This model together with the WAM wave model has been used to study the spatial distribution of wind waves and three-dimensional currents over the shelf for the period 5}10 January 1993, the time of the Braer oil spill at the southern tip of the Shetland Isle. Calculations using the WAM model have shown that signi"cant wave heights in the northern North Sea at the time the oil tanker Braer went aground exceeded 8 m in the open sea conditions to the west of the Shetlands. This area was exposed to strong westerly to southerly winds with stresses exceeding 2 N m\. On the eastern side of the Shetlands, the wave heights were signi"cantly less, due to the sheltering e!ect of the Shetlands. A series of computations was performed using the shelf model with and without a surface source of wave-dependent viscosity using two viscosity pro"les. In one pro"le viscosity was constant in the vertical and in a second its value decreased in the surface layer. Computed surface currents were signi"cantly reduced when a source of wave-dependent eddy viscosity was added, particularly in the case where viscosity decreased in the surface layer. At high wind speeds and signi"cant wave heights exceeding 8 m, surface currents computed with a constant eddy viscosity and a wavedependent viscosity appeared more physically correct than those derived with no wave-dependent viscosity, and viscosity decreasing in the surface layer. These

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calculations clearly show that a surface source of viscosity depending upon wave e!ects has a signi"cant in#uence upon the surface currents, suggesting that accurate measurements of wind speed, wave parameters and near surface shears (Collar and Vassie, 1978) at times of major wind events would be very valuable in determining appropriate formulations of surface eddy viscosity to use in these models, with near surface shear measurements being essential for model validation. Unfortunately, no such measurements were available for the period considered here. However, recent measurements of surface currents during a major wind event, using an HF Radar system and current pro"les from an Acoustic Doppler Current pro"ler have shown the importance of additional wave related sources of turbulence (Davies et al., 1998b). Although the major features of the #ow over the shelf could be resolved with the large-scale model, comparison of wind driven #ows in the area of the Shetlands, showed that a "ne resolution model was required to resolve #ow details. The "ne scale features of the #ow in the region of the Braer oil spill, in particular the existence and location of a local gyre would be particularly important in determining the movement of the oil. Detailed spatial measurement in the area of the gyre and the region of the open boundary of the Shetlands model where the in#ow which produces the gyre is located would be valuable in determining the spatial extent of the gyre and the exact nature of the in#ow in#uencing this. Calculations using the limited area Shetlands model, clearly showed the importance of high-resolution in this region, and the need to take account of `far "elda #ows entering the region generated by winds over the whole shelf. Without the link to a shelf wide model the limited area model would not reproduce currents in the region to the south of the Shetlands. Also calculations showed the sensitivity of surface currents to local meteorological conditions, which need to be accurately determined in order to derive local #ow "elds. Although the primary aim here has not been a detailed simulation of the period of the oil spill, or the movement of the oil itself, but rather a processes study to examine the in#uences of a wave-dependent viscosity, and model resolution upon #ows, it is clear that such factors are important in accurately predicting the movement of oil. This suggests that further modelling studies complemented by critical measurements are required before a rigorous validation of the #ow "elds under major wind events is possible. Acknowledgements The authors are indebted to Robert Smith for help in preparing the diagrams and to Linda Ravera for typing the paper. References Brown, J., Gmitrowicz, E.M., 1995. Observations of the transverse structure and dynamics of the low frequency #ow through the North Channel of the Irish Sea. Continental Shelf Research 15, 1133}1156. Collar, P.G., Vassie, J.M., 1978. Near-surface current measurement from a surface following data buoy (DB1). II. An harmonic and residual analysis of current meter records. Ocean Engineering 5, 291}308.

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Davies, A.M., 1985. A three-dimensional model of wind induced #ow in a sea region. Progress in Oceanography 15, 71}128. Davies, A.M., 1986a. A three-dimensional model of the north west European Continental Shelf, with application to the M tide. Journal of Physical Oceanography 16, 797}813.  Davies, A.M., 1986b. Application of a spectral model to the calculation of wind induced currents. Continental Shelf Research 5, 579}610. Davies, A.M., 1987. Spectral models in Continental Shelf Sea Oceanography. In: Heaps, N.S. (Ed.), Three-Dimensional Coastal Ocean Models. A.G.U., Washington, USA. pp. 71}106. Davies, A.M., Aldridge, J., 1993. A numerical model study of parameters in#uencing tidal currents in the Irish Sea. Journal of Geophysical Research 98, 7049}7067. Davies, A.M., Jones, J.E., 1996a. Sensitivity of tidal bed stress distributions, near bed currents, overtides and tidal residuals to frictional e!ects in the Eastern Irish Sea. Journal of Physical Oceanography 12, 2553}2575. Davies, A.M., Jones, J.E., 1996b. The in#uence of wind and wind wave turbulence upon tidal currents: Taylor's problem in three-dimensions with wind forcing. Continental Shelf Research 16, 25}99. Davies, A.M., Kwong, C.M., Flather, R.A., 1997a. Formulation of a variable function three-dimensional model, with application to the M and M tide on the North-west European Continental Shelf.   Continental Shelf Research 17, 165}204. Davies, A.M., Kwong, C.M., Flather, R.A., 1997b. A three-dimensional model of diurnal and semi-diurnal tides on the European shelf. Journal of Geophysical Research 102, 8625}8656. Davies, A.M., Kwong, C.M., Flather, R.A., 1998. A three-dimensional model of wind driven circulation on the shelf: application to the storm of January, 1993. Continental Shelf Research 18, 289}340. Davies, A.M., Lawrence, J., 1994. A three-dimensional model of the M tide in the Irish Sea: the importance  of open boundary conditions and in#uence of wind. Journal of Geophysical Research 99, 16197}16227. Davies, A.M., Lawrence, J., 1995. Modelling the e!ect of wave-current interaction on the three-dimensional wind-driven circulation of the Eastern Irish Sea. Journal of Physical Oceanography 25, 29}45. Davies, A.M., Jones, J.E., Kwong, S.C.M., Xing, J., 1998a. E!ects of waves in both the surface and bottom boundary layers. IMA Special Volume on Wind over Waves. Salford University. Davies, A.M., Hall, P, Howarth. M.J., Knight, P., Player, R.R., 1998b. A detailed comparison of measured and modelled wind driven currents in the North Channel of the Irish Sea, submitted for publication. Dobroklonskiy, S.V., 1969. Drift currents in the sea with an exponentially decaying eddy viscosity coe$cient. Oceanology 9, 19}25. Donellan, M.A., 1990. Air-Sea interaction. In: LeMehaute, B., Hanes, D.M. (Eds.), The Sea: Ocean Engineering Science 9a. Wiley, New York, pp. 239}292. Dooley, H.D., Furnes, G.K., 1981. In#uence of the wind "eld on the transport of the northern North Sea. In: Saetre. R., Mork, M. (Eds.), The Norwegian Coastal Current. Vol. 1. University of Bergen 795 pp. Dyke, P.P.G., 1977. A simple ocean surface layer model. Rivista Italiana di Geo"sica 4, 31}34. Flather, R.A., Smith, J., 1993. Recent progress with storm surge models } results for January and February, 1993. Proceeding MAFF Conference of River and Coastal Engineers, University of Loughborough 5}7 July, Ministry of Agriculture Fisheries and Food, pp. 6.2.1}6.2.16. Furnes, G.K., 1980. Wind e!ects in the North Sea. Journal of Physical Oceanography 10, 978}984. Furnes, G.K., Hackett, B., Saetre, R., 1986. Reto#ection of Atlantic water in the Norwegian Trench. Deep-Sea Research 33A, 247}265. Gjevik, B., 1991. Simulation of shelf sea response due to travelling storms. Continental Shelf Research 11, 139}166. Gjevik, B., Merri"eld, M.A., 1993. Shelf-sea response to tropical storms along the west coast of Mexico. Continental Shelf Research 13, 25}47. Gordon, R.L., 1982. Coastal ocean current response to storm winds. Journal of Geophysical Research 87, 1939}1951. Gordon, R.B., Spaulding, M.L., 1987. Numerical simulations of the tidal- and wind-driven circulation in Narragansett Bay. Estuarine, Coastal and Shelf Science 24, 611}636. Hill, A.E., Simpson, J.H., 1986. Low frequency variability of the Scottish coastal current induced by along-shore pressure gradients. Estuarine, Coastal and Shelf Sciences 27, 163}180.

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