Processes influencing suspended sediment movement on the Malin–Hebrides shelf

Continental Shelf Research 22 (2002) 2081–2113 Processes inﬂuencing suspended sediment movement on the Malin–Hebrides shelf Alan M. Davies*, Jiuxing ...

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Continental Shelf Research 22 (2002) 2081–2113

Processes inﬂuencing suspended sediment movement on the Malin–Hebrides shelf Alan M. Davies*, Jiuxing Xing Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead, Merseyside CH43 7RA, UK Received 25 April 2001; received in revised form 26 February 2002; accepted 17 May 2002

Abstract A three-dimensional baroclinic model, including suspended sediment transport, is used in cross-sectional form to examine processes (tidal, along-shelf current, wind-waves, and wind) inﬂuencing suspended sediment transport off the west coast of Scotland. Sub-grid scale processes are parameterized using a turbulence energy model. Calculations using appropriate sediment types and tidal forcing for the region, show that the barotropic tide alone essentially cannot suspend bottom sediment in the area. However, when an along-shelf ﬂow is added some sediment is suspended. An off-shelf net transport of sediment occurs in the bottom layer, with a weaker on-shelf transport in the surface region. This is consistent with recent measurements in the area which show the lack of tidal resuspension, and the importance of an along-shelf ﬂow in sediment suspension. Observations reveal a weak off-shelf ﬂow in the bottom boundary layer. The magnitude of this transport is increased by the presence of wind-waves, or by a reduction in the sediment settling velocity. Observations during a storm clearly show the importance of wind-waves in causing sediment resuspension particularly close to the shore. When density stratiﬁcation was included, internal tides were generated with associated regions of enhanced bottom friction velocity un and hence increased sediment resuspension. Although the dominant feature of the ﬂow is similar to that found with the barotropic tide, this is modiﬁed by smaller scale currents due to the presence of the internal tide. Summer observations indicate that strong internal waves can enhance sediment suspension. However from the limited number of observations and the high spatial variability of the internal tide found in the model, the importance of the internal tide is difﬁcult to assess from the measurements. When an upwelling favourable wind is added, the vertical density gradient in the near-bed region increases, which changes the internal tide and hence its effect upon sediment suspension and transport. The presence of an along-shelf ﬂow induced by the wind signiﬁcantly increases the bed stress leading to enhanced sediment resuspension. Sediment transport is dominated by the wind-driven ﬂow, with coastal upwelling moving sediment from the bottom boundary layer to the surface layer where it is advected off-shelf. Observations made during upwelling favourable winds, conﬁrm this off-shelf advection of sediment in the upper part of the water column. Conversely with a downwelling favourable wind, the thickness of the bottom boundary layer is increased, and in the shelf edge region sediment is suspended to a greater height in the water column. In the near-coastal area, under downwelling conditions, the sediment is conﬁned to the bottom boundary layer and is advected away from the shore in the near-bed region. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Malin–Hebrides shelf; Wind; Tide; Sediment; Waves; Stratiﬁcation

*Corresponding author. E-mail address: [email protected] (A.M. Davies). 0278-4343/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 7 8 - 4 3 4 3 ( 0 2 ) 0 0 0 7 5 - 4

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1. Introduction Although signiﬁcant research has been performed in shallow sea regions to examine processes, namely, tides, wind-induced currents and wind-waves giving rise to sediment movement, little work has been done in shelf edge areas. In shelf edge regions where density stratiﬁcation intersects the bottom topography, and the barotropic tide is strong, such as the Malin-Shelf area, internal tides are produced which can signiﬁcantly affect the distribution of bed shear stress and produce enhanced mixing both in the near-bed region and higher in the water column (Xing and Davies, 1998a, b). To date the main focus in internal tide research has been to examine the currents and the signiﬁcant internal displacements associated with the internal tide (Craig, 1987; New, 1988; New and Pingree, 1990; Sherwin and Taylor, 1989, 1990; Sherwin, 1988, 1991; Holloway, 1996, 1991; Lamb, 1994; Xing and Davies, 1996a, b, 1998a, b). Recently, modelling work looking in detail at how the internal tide and the mixing associated with it is inﬂuenced by the detailed shape of the bottom topography, the proﬁle of density and the parameterization of the mixing has been performed (Xing and Davies, 1998b). Also the nonlinear coupling between the internal tide and wind effects has been studied by Xing and Davies (1997), who showed that the wind could signiﬁcantly modify the internal tide. The fact that the internal tide changes the bed shear stress magnitude and distribution together with the vertical mixing in the shelf-edge region means that it will inﬂuence the sediment distribution in this area. To date very little work (e.g. Heathershaw, 1985; Flagg, 1988; Butman, 1988) has been done on this problem, although Heathershaw et al. (1987) used a cross-sectional two layer model to investigate how the internal tide changed the bed stress distribution in the shelf-edge region and its implication for sediment movement as bed load. Here we are concerned with its effect and the inﬂuence of stratiﬁcation upon the spatial and temporal variability of suspended sediment. In this paper, we deal with the development of a fully three-dimensional non-linear baroclinic

model which includes a suspended sediment module and hence can address the problem of sediment movement under the inﬂuence of the internal tide and wind forcing. The model is based upon the earlier three-dimensional model of Xing and Davies (1998b), which is extended here to include a sediment dynamics capability. A turbulence energy sub-model is included to compute the vertical mixing and the vertical diffusion of sediment. Solutions are obtained using a ﬁnite difference approach on a sigma co-ordinate in the vertical, with a reﬁned grid in the near-surface (for wind-driven ﬂows) and near-bed regions. By this means the bottom boundary layer, where sediment concentration is a maximum, can be resolved in detail, as can the surface wind-driven shear layer. Also the model can deal with an arbitrary bottom topography and vertical stratiﬁcation. Although extensive sediment measurements and modelling has been performed in shallow seas and where wind and wave effects are important (Heathershaw, 1981; Davies et al., 1988; Aldridge, 1996, 1997; Huntley et al., 1994; Huntley and Hazen, 1988; Gerritsen and Berentsen, 1998; Grant and Madsen, 1979, 1982, 1986; Madsen et al., 1993; Glenn and Grant, 1987; Green et al., 1995; Keen and Slingerland, 1993), the highly turbulent nature of these regions means that the water column remains well mixed. By contrast in the shelf-edge region, although internal tides give rise to a highly turbulent bottom boundary layer, the presence of the thermocline signiﬁcantly reduces the turbulence above this layer. As we will show this has a major inﬂuence upon sediment movement, which makes sediment transport distinctly different from that found in the shelf seas. Also at the shelf edge, sediment can be moved offshelf as water cascades down the shelf slope (Shapiro and Hill, 1997). This cascading of water is known to occur in the Malin-shelf edge region and to be important in sediment transport although this has never been quantiﬁed. This process is outside the scope of the present paper. The initiation of sediment suspension by strong currents associated with solitons produced by the internal tide is also an important process in shelf edge sediment movement (Johnson et al., 2001),

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although this cannot be included in the present hydrostatic model. Here we consider the role of a range of processes, namely barotropic and baroclinic tides, stratiﬁcation, along-shelf ﬂow, wind-wave effects and up/downwelling favourable winds upon the resuspension of sediment, and its advection with the associated spatial and temporal variability produced by these processes. Observations at the Hebrides shelf (McCandliss, 2000) show that sediment movement by horizontal advection is three orders of magnitude larger than the vertical. Consequently, it is essential to understand and quantify the role of various physical processes leading to advection in the region. Because of the diversity of processes occurring at the continental margins which produce exchange of particulate matter between shelf and ocean, it is important to understand the role of each in suspending and advecting sediment. The aim of this paper is to do this by using a range of calculations involving each physical process in turn. The Hebrides shelf edge is chosen because it is representative of many shelf edges. Also it has no complicated features such as submarine canyons, shallow banks, or ridges, which could invalidate results from a cross-sectional model, and conclusions based upon such a model. The region was also the focus of a 16-month long observational programme from August 1995 to February 1996 (the SES experiment). Although it is not the objective of the present paper to simulate the sediment distribution during this period (indeed this would be very difﬁcult given the lack of detailed knowledge of bed sediment distributions, types and settling velocities in the area), model results give some insight into which processes are giving the observed suspended sediment distributions obtained at the time of the measurements. Similarly, the observations give some conﬁdence in the model’s ability to represent the effect of these processes upon the movement of suspended sediment. Unlike shallow sea regions (e.g., the Irish Sea), where the spatial distribution and types of sediment are well known and the amount of sediment available for resuspension can be quantiﬁed, such information is not available at the Hebrides Shelf.

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Observations from the SES experiment showed that in general the bed varied from a greater percentage of mud off-shelf to coarser material and rock as the land was approached (McCandliss, 2000). However at any one location there was a range of sediment types with various settling velocities and critical bed shear stresses (unc ). Some of this, in particular material with a low settling velocity (less than 1.0 106 m s1) was due to phytoplankton blooms in the spring in the upper layers of the water column which had gradually settled to the bed as a ﬂuff layer. Other material was non-biological with a large range of settling velocity typically from 0.001 to 0.005 m s1 (McCandliss, 2000) and resided on the seabed. As the focus of this paper is the role of various physical processes in determining the resuspension of bed material and the advective processes inﬂuencing its movement we will concentrate on non-biological material with a bed source, and assume an average settling velocity ws of 0.0025 m s1 (which we will refer to as sediment type A, Table 1). To examine the inﬂuence of settling velocity we also consider a lower value of ws namely ws ¼ 0:001 m s1 (type B, Table 1). In the absence of detailed information on critical bed shear stress unc was taken everywhere as unc ¼ 0:67 cm s1, a typical value for the region. Also an inﬁnite source was assumed, although observations (McCandliss, 2000) suggested that at certain times (major wind events) at some locations where suspended sediment was measured the source was limited. A simple form of pick up function was used in the calculations (see later) with the same coefﬁcients used throughout. Since the focus of the present paper is into the processes inﬂuencing the temporal and spatial variability of suspended sediment, the use of an inﬁnite source of sediment, ﬁxed unc ; and the same pickup function at all locations is consistent with the aims of the paper. Obviously such an approach could not be used in a simulation model in which predicted sediment concentrations and bed distributions would be compared with observed. Such a simulation would require a detailed description of the full range of sediments occurring in the region, and their source limitations which is presently not available. The movement of sediment

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Table 1 Summary of parameters used in calculations Calc.

Tide

Along-shelf ﬂow

Stratiﬁcation

Waves

Wind

Sediment type

Relevant ﬁgures

1 2 3 4 5A 5B 6 7A 7B

Barotropic Barotropic Barotropic Barotropic Baroclinic Baroclinic Baroclinic Baroclinic Baroclinic

No Yes Yes No No No Yes No No

No No No Fixed Yes Yes Yes Yes Yes

No No WD No No No No No No

No No No No No No No Upwelling Downwelling

A A A A A B A A A

2, 3 4a,b 5a–c, 6 7 8a–c 8d 9a–d 10a–c 11a–c

WD is the waves with amplitude 1.0 m decaying close to shore and period 15 s. Sediment type A, unc =0.67 cm s1, ws ¼ 0:0025 m s1. Sediment type B, unc =0.67 cm s1, ws ¼ 0:001 m s1.

as bed load which is not considered here would also have to be taken into account. The formulation of the model is brieﬂy described in the next section, with more detail in Appendix A, with subsequent sections dealing with its application in cross-shelf form to the Malin– Hebrides shelf.

2. The model 2.1. The three-dimensional hydrodynamic model The three-dimensional model is based upon the earlier hydrodynamic model of Xing and Davies (1996a, b, 1998a, b), and consequently only a brief outline will be presented here. The continuity equation, momentum equations and transport equations for temperature and sediment concentration together with an appropriate equation of state are given for completeness in Appendix A. The eddy viscosity and diffusivity are computed using a turbulence energy sub-model, details of which are also given in Appendix A. In the calculations a ﬁnite difference grid of 300 m was used in the horizontal with 60 sigma levels in the vertical. This grid had enhanced resolution in the near-surface and near-bed regions in order to accurately resolve shear in these layers due to wind forcing and bed friction. To overcome the difﬁculty of calculating horizontal pressure

gradients and horizontal diffusion in sigma coordinates these terms are computed on a z coordinate. Then the pressure gradient and diffusion values are interpolated onto sigma co-ordinates. A time splitting method (Davies and Xing, 1995; Xing and Davies, 1996a, 1997) was used to integrate the equations through time, and the total variation diminishing (TVD) advection scheme was used for temperature and sediment advection. The TVD scheme has been shown previously (James, 1996; Xing and Davies, 1998a, b), to retain high concentration gradients when material is advected over steep topography. Provided there is sufﬁcient vertical grid resolution in the bottom boundary layer to deal with high sediment concentration gradients in this region when the vertical ﬂux is small and settling is large (e.g. heavy sediments), this method is ideal for sediment advection. The problem of having sufﬁcient resolution in the near-bed region to deal with heavy sediments is universal in sediment modelling and is independent of the advection method, although it does inﬂuence the accuracy of the solution. A single time step integration method was used rather than the leap-frog method (Blumberg and Mellor, 1987) to avoid the possibility of two divergent solutions, and the necessity to couple them, which is characteristic of the leapfrog method. Since extensive details of the numerical methods used in the model have been given before they will not be repeated here.

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Solutions were analyzed after a time integration of 15 days, by which time the effect of the initial conditions of zero free surface elevation and water motion had been removed. Although sediment erosion and deposition at the bed take place during this period, their effect upon bottom topography and hence water depths was negligible. 2.2. Wave–current interaction model In the calculations concerned with the inﬂuence of wind-waves at swell periods, of order 15–20 s, the wave amplitude Aw in the near-coastal region (depths of 100 m or less) was assumed to decay from its oceanic value A0 ; according to h 2 for hp100 m: ð1Þ Aw ¼ A0 100 The drag coefﬁcient in the quadratic friction law which related bed stress to bed current was modiﬁed using the wave–current interaction formulation of Davies and Lawrence (1994, 1995), based upon the work of Grant and Madsen (1979). Consequently, when wave effects are present the drag coefﬁcient increases, reﬂecting the additional turbulence due to the wind-waves. This has the effect of changing the ﬂow ﬁeld and turbulence and hence the computed bed stress and consequently the computed friction velocity unf due to the ﬂow. Under wave conditions there is an additional component of friction velocity unw due to the waves, with the total friction velocity un ¼ unf þ unw then being used to determine when sediment is suspended as given by Eq. (A.7) in Appendix A. Since unw depends upon the wave orbital velocity at the bed which is a function of wave period, amplitude and water depth, then for swell waves, unw often exceeds unf in water depths less than 200 m and hence the pick up of sediment (Eq. (A.7), Appendix A) can be signiﬁcantly increased by the presence of wind-waves.

3. Sediment transport on the Malin–Hebrides shelf Since our primary focus in this paper is to examine processes which inﬂuence the across-shelf movement of sediment under various conditions,

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we will apply the model in cross-sectional form with the same topography as used by Xing and Davies (1996b) to simulate the tide at 571N. As our aim is to understand the processes controlling suspended sediment movement rather than to reproduce sediment distributions which would require a detailed description of sediment types over the region (see earlier discussion) in the following calculations we will restrict our attention to two sediment types (see Table 1). As the pick up function (Eq. (A.7), Appendix A) is highly empirical and it is well known that the choice of a and n has a critical inﬂuence upon how much sediment is suspended, in all calculations we take unc ¼ 0:67 cm s1, a typical value for sediment in the region. In a study aimed at simulating sediment movement, a range of sediment types with different unc would be required. Also the role of a and n in determining sediment concentration would have to be examined and a more complex form of the pick up function used. Detailed information of this form is not available in the region. Besides ﬁxing unc at 0.67 cm s1 as discussed in the Introduction, we only consider two settling velocities namely ws ¼ 0:0025 and 0.001 m s1 in order to illustrate the role of settling velocity versus turbulence. A more detailed study of the role of these parameters upon shelf edge sediment movement induced by a single process e.g., the tide is clearly valuable and could possibly be related to measurements taken during calm periods. However our objective here is to in essence ﬁx the sediment type and assess the importance of a range of physical processes. 3.1. Barotropic tide (Calc. 1) In an initial calculation (Calc. 1, Table 1) the water in the region was assumed to be homogeneous, and tidal currents were induced by forcing at the M2 period at the off-shelf open boundary. (Besides summarizing the important parameters used in the various calculations, Table 1 gives the associated key ﬁgures.) A closed boundary was applied at the coastline (Fig. 1). This forcing was chosen to give an appropriate tidal current distribution in the region (Xing and

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offshore distance (km)

Fig. 1. Cross-shelf distribution of bottom topography, and location of Posn. (1), where current and sediment proﬁles together with diffusivities are examined in detail.

Fig. 2. A ‘‘snap shot’’ at t ¼ 1=8 T (where T is the tidal period) of the cross-shelf sediment concentration, and tidal friction velocity (un ) in the region of the shelf break (between 40 and 100 km from the coast).

Davies, 1998b). A snapshot of the across-shelf variation of friction velocity (Fig. 2) shows a rapid increase in the region at the top of the shelf slope, as currents are intensiﬁed as the tide propagates from the deep ocean onto the shelf. On the shelf, frictional effects retard the ﬂow and un decreases from a maximum of about 0.9 cm s1 at the shelf break, in an almost linear manner towards the coast. The time variation of un over a tidal cycle (as illustrated at Posn. (1)) was small (Fig. 3), due

to the near circular nature of the current ellipse in the region. Also the maximum value of un which occurred near the shelf break was only slightly larger than the critical friction velocity unc (taken as 0.67 cm s1), and hence sediment suspension due to the barotropic tide is conﬁned to near the shelf-edge region (Fig. 2). Time series of current proﬁles of the u (acrossshelf) and v (along-shelf) components at Posn. (1) (Fig. 3), over two tidal cycles, show a uniform ﬂow above a frictionally retarded bottom boundary layer. The amplitude of the u component of order 25 cm s1, is slightly larger than the v component (of order 20 cm s1), with the time of maximum u corresponding to a near zero v velocity (i.e. a 901 phase shift between the two components). The nearly equal magnitude of u and v; and this phase shift, gives rise to a near circular current ellipse (see Appendix B). In the case of a circular current ellipse, the current magnitude is constant with time, and hence un does not vary. In the present case, the time variation of un is due to the fact that u and v have different magnitudes, and are not phase shifted by exactly 901. Since the shear production term in the near-bed region is the major source of turbulence energy and is constant in the case of a circular ellipse, the eddy viscosity shows an almost constant value close to the seabed. Higher in the water column, there are differences in the phase relation between u and v and hence the ellipse is less circular. Consequently, there is a time variation in the shear production of turbulence term, and hence in the eddy viscosity. Also the shear production in this

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Fig. 3. Time series at Posn. (1) over two tidal cycles of the u and v components of current (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity). Also shown is a time series of the associated un and eddy viscosity from Calc. 1.

region is less than at the bed, and the inﬂuence of vertical diffusion may also have an effect. In the barotropic calculation, it is clear that except close to the shelf break, the value of un is below the unc value of 0.67 cm s1 used in the calculation (Table 1) and hence there is little sediment suspension. Observational evidence from SES suggests that tidal resuspension was negligible (McCandliss, 2000), although the transmissometer observations upon which this conclusion was based were taken 7 m above the seabed. Obviously for the fraction of sediment having a lower unc ; and lower settling velocity (see later) than considered here this material would be suspended in the near-bed region. Computed tidal residuals at the shelf edge (Xing and Davies, 2001a, b) have a maximum off-shelf ﬂow at the bed of the order of 3 cm s1 and consequently there will be a longterm transport of this material off-shelf. Observations averaged over the whole SES period suggest (McCandliss, 2000) long-term off-shelf transport of material in the bottom boundary layer by currents of this order of magnitude.

3.2. Barotropic tide with an along-shelf flow (Calc. 2) Besides tidal currents at the shelf edge, wind stress or oceanic forcing can induce an along-shelf ﬂow in the region (Xing and Davies, 1999a; Davies and Xing, 2000). Time series of u and v current proﬁles (Fig. 4a) with identical tidal forcing as previously, but with the addition of an along-shelf ﬂow due to a body forcing term F =h; with F ¼ 0:1 Pa, and h water depth (see Xing and Davies (1999a) for details), show (Fig. 4a) a similar time variation as previously (Fig. 3), but with the v component increased by the order of 20 cm s1 (a typical along-shelf ﬂow in the region). The effect of this mean ﬂow is to increase the current magnitude and hence un which has one maximum and one minimum value in each tidal cycle. Although the minimum un is below unc ; its maximum exceeds this value and leads to signiﬁcant sediment erosion. At the bed maximum concentration occurs at the time of maximum erosion, although higher in the water column this

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occurs later in time, as it takes some time for sediment eroded at the bed to diffuse out of this layer. Limited observational evidence in the SES region shows that greatest sediment erosion is associated with the location of this along-shelf ﬂow, emphasising the importance of this ﬂow in producing sediment suspension. Time-averaged velocity, sediment concentration and cross-shelf ﬂux of sediment are shown in Fig. 4b. The along-shelf time-averaged ﬂow is

dominated by the ﬂow due to external forcing. In the presence of bottom friction, this ﬂow gives rise to a bottom Ekman layer, with an associated across-shelf ﬂow (Davies and Xing, 2000) which in the near bottom boundary layer is mainly offshore in response to the bottom Ekman dynamics. At the shelf break there is a small (of order 3 cm s1) residual ﬂow due to tidal rectiﬁcation. In shallow water bottom frictional effects give rise to shear in the tidal currents in the near-bed region

(a) Fig. 4. (a) Time series at Posn. (1) over two tidal cycles of the u and v components of current (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity). Also shown is a time series of the associated un ; sediment concentration and eddy viscosity from Calc. 2, (b) time averaged over a tidal cycle of cross-shelf and alongshelf currents (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity) across-shelf sediment concentration and transport from Calc. 2.

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(b) Fig. 4 (continued).

which through the non-linear terms (Tee, 1980, 1985, 1987) can produce a tidal residual. This component of the residual ﬂow was negligible compared to the contribution from the other effects. The across-shelf distribution of un (not shown) is similar to that found previously although its value is increased everywhere, particularly on the shelf by the presence of the along-shelf ﬂow. This increase in un leads to enhanced sediment suspension compared to previously, with the tidally averaged sediment concentration, showing a maximum at the shelf break, coinciding with the position of maximum un : Contours of timeaveraged (over a tidal period to remove oscillatory

tidal effects) across-shelf sediment show an offshelf transport in the bottom boundary layer due to the presence of the off-shelf residual, with a weak on-shelf transport near the surface (Fig. 4b). Results from this calculation and the earlier one, show that the barotropic tide alone can only suspend sediment (for the given sediment type and settling velocity used here) for a small fraction of the tidal cycle. The addition of an along-shelf ﬂow of oceanic origin or produced by wind forcing does however increase un to such an extent that signiﬁcant sediment erosion occurs. This sediment is then transported off-shelf in the near-bed region by the weak residual ﬂow in the bottom Ekman layer. In the case of an increased along-shelf ﬂow

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due to wind forcing, surface wind-waves will be signiﬁcant and their effect was examined in a subsequent calculation.

made in the near-bed region (Huthnance et al., 2002). 3.4. Internal tide effects

3.3. Barotropic tide with an along-shelf flow and wind-wave effects (Calc. 3) In this calculation (Calc. 3), the parameters are as previously, but wind-wave effects (wave amplitude A0 ¼ 1 m decreasing in water depths below 100 m, as described previously (Eq. (1) and period 15 s) were added. The effect of adding the waves is to signiﬁcantly increase the value of un at the shelf edge and on the shelf (Fig. 5a). Since the wave orbital velocity at the bed, for a given wave period, decreases rapidly with water depth then in deep water (depths below 300 m) un is not affected. The enhanced friction velocity over a tidal cycle (Fig. 5b) compared to the previous calculations without wave effects, leads to an increase in sediment concentration, at Posn. (1) (compare Figs. 5b and 4a). Although, the magnitude of the u component of velocity is only slightly inﬂuenced (of order 0.3 cm s1, Fig. 6) by the change in un ; the magnitude of the v component reduces from a maximum of 40 cm s1 in the upper part of the water column to a value of 36 cm s1, a change of order 4 cm s1 (Fig. 6). The reason for this appears to be the increased bed stress on the shelf, particularly in the near-shore region where water is shallow. This reduces the magnitude of the along-shelf ﬂow from that found without wave effects (Fig. 6). This reduction is largest in the shallow region although its effect does reach the shelf edge. This reduction in current in shallow water due to the presence of wind-waves was also found by Davies and Lawrence (1995) in the eastern Irish Sea. The presence of wave effects is to signiﬁcantly increase the sediment concentration on the shelf, and hence the off-shelf export of sediment in the bottom boundary layer (compare Figs. 5c and 4b, noting in particular differences in contour interval). The signiﬁcant increase in the on-shelf sediment concentration and increase in un due to wave effects is supported by observations taken in February 1996 when simultaneous measurements of near-bed sediment concentrations and un were

3.4.1. Fixed stratification (Calc. 4) In the previous series of calculations, the water was assumed to be homogeneous. However, measurements show that stratiﬁcation effects are important and that internal tides are generated in the region (Xing and Davies, 1998b). Before examining the inﬂuence of the internal tide, we brieﬂy consider the role of a speciﬁed stratiﬁcation (taken as that given by Xing and Davies, 1996b) upon the computed tidal proﬁle and bed shear stress. By this means an internal tide cannot be generated, but the stratiﬁcation acts to suppress the vertical diffusion of sediment in the region of the thermocline. Consequently, the inﬂuence of stratiﬁcation upon vertical diffusion can be separated from internal tide effects. The presence of vertical density stratiﬁcation suppresses tidally generated turbulence and hence viscosity in the water column. (Compare time series of eddy viscosity in Figs. 7 and 3). The stratiﬁcation effect results in conﬁning the region of tidal current shear close to the bed with a uniform ﬂow above this region (Compare Figs. 7 and 3). The amplitude of both the u and v components of velocity decrease in the near-bed region, below those found without stratiﬁcation effects, although their phase difference remains the same, leading to a near circular current ellipse. This gives a similar time variation of un to that found previously, although with a reduced magnitude (see Figs. 7 and 3). Since un is less than unc ; sediment is not suspended. In essence, the presence of a ﬁxed stratiﬁcation leads to a slight reduction in near-bed stress, and as previously, no sediment suspension. 3.4.2. Time-varying stratification (Calc. 5A) In this calculation, the density ﬁeld besides inﬂuencing the vertical distribution of mixing is advected by the ﬂow, and hence internal pressure gradients are produced giving rise to an internal tide. A ‘‘snap shot’’ of the internal tide stream function (note the depth mean current has been

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(a)

(b) Fig. 5. (a) A ‘‘snap shot’’ at t ¼ 1=8 T; of the across-shelf variation of friction velocity un ; for (i) currents only (solid bold line) (ii) currents with the addition of wind-waves (solid light line) (Calc. 3), (b) time series at Posn. (1) of u and v components of current (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity), associated friction velocity un (bold line without waves, lighter line with waves), sediment concentration and eddy viscosity over two tidal cycles from Calc. 3, (c) time average over a tidal cycle of cross-shelf and along-shelf currents (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity), across-shelf sediment concentration and transport from Calc. 3.

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(c) Fig. 5 (continued).

subtracted for clarity) at t ¼ 3=8 T (Fig. 8a(i)) shows a downwelling at the shelf edge associated with a clockwise circulation cell, with alternate clockwise and anticlockwise cells on the shelf, corresponding to a mode 1 internal tide. As the tide propagates on t ¼ 3=8 T and off t ¼ 7=8 T, the shelf (Fig. 8a(ii)) the circulation within each cell changes. Associated with this circulation are regions of enhanced and reduced bed stress and hence un : The spatial pattern of un at two moments in the tidal cycle is shown in Fig. 8a(i)–(ii), with the associated pattern of sediment suspension following the position of maximum un : This spatial distribution and its time variation is signiﬁcantly different to that found in the barotropic tidal case, and produces regions of enhanced bed stress where

the sediment can be eroded. The up and downwelling associated with the circulation cells of internal tidal origin can lead to regions, the position of which varies with time, of enhanced suspension and deposition. A ‘‘snap shot’’ of the across-shelf temperature ﬁeld at a time of shelf edge downwelling (Fig. 8a(i)), shows an upward displacement of isotherms at the shelf edge, with an adjacent downward displacement on the shelf. The upward displacement of the isotherm is the integrated effect of the upwelling circulation that occurred prior to this time. Half a period later (Fig. 8a(ii)), the 9.01C isotherm has moved off the shelf and down the slope due to the reversed shelf edge circulation at this time (Fig. 8a(ii)). On the shelf,

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Fig. 6. Time average over a tidal cycle of the difference (Calc. 3–Calc. 2) in across-shelf and along-shelf velocity (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity) due to the presence of wind-waves.

Fig. 7. Time series (over two tidal cycles) of u and v components of current (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity), associated friction velocity un ; and eddy viscosity at Posn. (1) from Calc. 4 (tide with ﬁxed stratiﬁcation).

tidal turbulence gives rise to a well-mixed bottom boundary layer, with the thermocline at approximately 70 m below the surface. In the region of the

shelf edge, the presence of the bottom tidal front, and associated across-shelf pressure gradients, leads to an along-shelf residual ﬂow (Ou and

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(ai)

(aii) Fig. 8. (a) Across-shelf variation of temperature, stream function (with solid line positive) sediment concentration and friction velocity un at (i) t ¼ 3=8 T; and (ii) t ¼ 7=8 T with T denoting tidal period (Calc. 5A), (b) time series over two tidal cycles, at Posn. (1) of departure of the density ﬁeld Dst (kg m3) from its background level, u and v current proﬁles (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity), eddy viscosity, sediment concentration, and friction velocity from Calc. 5A, (c) time-averaged (over a tidal cycle) contours of u and v components of velocity (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity) sediment concentration and transport in the shelf edge region from Calc. 5A, (d) time-averaged (over a tidal cycle) contours of concentration and transport in the shelf edge region with ws reduced to ws ¼ 0:001 m s1 (Calc. 5B).

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2095

(b) Fig. 8 (continued).

Maas, 1986, 1988; Chen and Beardsley, 1995) in addition to that produced by the external forcing. The presence of this thermocline, and internal pressure gradients associated with it, signiﬁcantly inﬂuence tidal current proﬁles in the vertical (compare Figs. 8b and 3). In essence, the tidal current above the thermocline is decoupled from that below, and there is a phase shift in the vertical. This also inﬂuences the phase difference between the u and v components of velocity, leading to a more rectilinear ﬂow in the near-bed region, where previously the ellipse was more circular. This change in ellipse properties increases the time variation in un (compare Figs. 8b with 3),

giving rise to times when it exceeds the unc value of 0.67 cm s1 used here, and sediment is suspended (Fig. 8b). The phase shift in the current across the thermocline gives rise to signiﬁcant shear at times of maximum on-shelf ﬂow, leading to a mid-water increase in viscosity at these times (see the 103 contour in Fig. 8b). The time-averaged across-shelf ﬂow shows (Fig. 8c) a region of off-shelf ﬂow along the shelf slope in the vicinity of the top of the slope, where there is a strong internal tidal circulation (Fig. 8a). Adjacent to this, on the shelf in the near-bed region there is an on-shelf residual current. These currents are due to internal tide rectiﬁcation (Xing

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(c)

(d) Fig. 8 (continued).

A.M. Davies, J. Xing / Continental Shelf Research 22 (2002) 2081–2113

and Davies, 2001a). On the shelf there are adjacent regions of on-shelf/off-shelf residual current near mid-depth, with an opposite residual ﬂow in the surface layer. These are associated with the circulation cells found in the internal tide. Since sediment is conﬁned to the bottom boundary layer, and positions where un exceeds unc ; namely the shelf edge and one on-shelf location, then it is only the residual ﬂow ﬁeld at these points that can contribute to sediment movement (see the /ucS contours in Fig. 8c). Since in both cases the residual ﬂow is off-shelf, then the tidally averaged sediment transport is also off-shelf. This calculation shows that the internal tide can lead to regions of enhanced bed stress above that found with the barotropic tide and that in these areas there is an off-shelf sediment transport in the near-bed region. The very limited set of near-bed suspended sediment measurements made during the summer period as part of the SES experiment did not detect persistent sediment re-suspension by the internal tide. One reason for this could be the lack of very near-bed measurements. Also the high degree of spatial variability shown in the internal tide (see Xing and Davies 1999b to see how this is related to small scale topography) in the model suggests that measurements would have to be made with a high spatial resolution to ensure that the position of enhanced bed stress was instrumented. On the one occasion that a large internal tide was observed at a time of sediment measurements it lead to a signiﬁcant increase in suspended sediment concentration (McCandliss, 2000). 3.4.3. Time-varying stratification with a reduced settling velocity (ws ¼ 0:001 m s1) (Calc. 5B) In a subsequent calculation (Calc. 5B), the settling velocity was reduced from that found previously in order to examine its inﬂuence upon the across-shelf transport of sediment. Limited measurements of sediment type and settling velocity in the region suggest (see Introduction) a mixture of sediments with different fractions having different settling velocities. Although changing the settling velocity does not inﬂuence the across-shelf residual ﬂow, or the location of areas where sediment suspension can occur (compare Figs. 8c and d), it does mean that the sediment can

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be moved to a greater height above the bed (Fig. 8d) and spends longer in the water column. As in the previous calculation the time-average sediment transport in the near-bed region is offshelf (Fig. 8d), however in the upper part of the water column there is an on-shelf transport which did not occur previously. This calculation clearly shows that the settling velocity is particularly important in determining the sediment transport rate. This inﬂuence of settling velocity upon the advection of sediment was clearly observed during the SES experiment. 3.5. Internal tide with an along-shelf flow (Calc. 6) In the initial series of calculations (described earlier), the presence of an along-shelf ﬂow upon sediment transport due to a barotropic tide was examined. Here the inﬂuence of an identical slope current upon sediment transport produced by an internal tide is considered. The across-shelf variation of friction velocity un at t ¼ 3=8 T, and t ¼ 7=8 T (Fig. 9a(i)–(ii)), shows that the presence of an along slope current signiﬁcantly increases un at t ¼ 7=8 T compared to previously (Fig. 8a(ii)), while at t ¼ 3=8 T, it is reduced on the shelf but increased at the shelf edge (compare 8a(i) and 9a(i)). The reason for these changes is that although the along-shelf current increases the v component of current when the tidal ﬂow is to the north, the oscillatory nature of the tide is such that when it is ﬂowing to the south, this reduces the v component which will affect un : This can be clearly seen from a comparison of the v component of velocity proﬁles in Figs. 8b and 9b. Without a shelf-edge ﬂow the near-bed v component oscillates from 10 to about +10 cm s1. When the along-shelf ﬂow which has a bottom current of about +10 cm s1 is added, v changes from 0 to about 20 cm s1 over the tidal cycle. Besides changes in total current amplitude being produced by the addition of a shelf edge current, changes in local vorticity associated with crossshelf shear in the along-shelf ﬂow can inﬂuence the off-shelf propagation of the internal tide. In the case of a super-inertial internal wave its propagation from its generation point depends upon the difference between its frequency and the local

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(ai)

(aii) Fig. 9. (a) Across-shelf variation of sediment concentration and corresponding friction velocity un ; at (i) t ¼ 3=8 T and (ii) t ¼ 7=8 T due to an internal tide and an along-shelf ﬂow (Calc. 6), (b) time series at Posn. (1) of departure of density ﬁeld Dst (kg m3) from its background level, u and v current proﬁles (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity), eddy viscosity, sediment concentration, and friction velocity (un ) over two tidal cycles (Calc. 6), (c) time average over a tidal cycle of cross-shelf (u) and along-shelf (v) currents (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity) sediment concentration and transport from Calc. 6, (d) across-shelf variation of temperature from Calc. 6.

inertial frequency (Dale et al., 2001). In the case of an internal tidal wave propagating into a sheared ﬂow there is a local effective frequency oeff ¼ o þ Z=2; with o the tidal frequency and Z vorticity associated with the ﬂow. If Z=2 is sufﬁciently large and negative that oeff is less than the inertial frequency then the internal wave is trapped (Kunze, 1985; Kunze and Sanford, 1984) and energy can accumulate. If the change in oeff is not so large then it can modify the propagation of the internal wave (Mooers, 1975; Lerczak et al., 2001). In the present calculation, the across-shelf change in the along-shelf ﬂow is insufﬁcient to produce a major change in the internal tide. However the internal tide propagation has changed slightly due to the along-shelf ﬂow. In the present case, it appears that changes in friction and propagation of the internal tide produced by the additional

along-shelf current slightly modify the u velocity (compare Figs. 8b and 9b). The phase relation between the u and v components of velocity is such that they are both near zero close to the beginning (t ¼ 0:15 T) of the tidal cycle, giving the near zero un found in Fig. 9b, which then rises and falls during the cycle. This time variation is signiﬁcantly larger than found previously and is due to the additional along-shelf ﬂow. As earlier (Fig. 8b) at times of maximum un there is maximum sediment suspension in the near-bed region, with increased vertical eddy viscosity and diffusivity due to the greater turbulence produced at the bed. This causes the sediment to diffuse out of the nearbed region, with maximum sediment concentration at height occurring later in the tidal cycle in an analogous manner to that found in the barotropic case (Fig. 4a). The signiﬁcant increase in un and

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(b) Fig. 9 (continued).

diffusivity at t ¼ 7=8 T is the reason for the enhanced sediment concentration over the whole shelf region at this time (Fig. 9a(ii)). Although the same external forcing as in the barotropic case is used to drive an along-shelf ﬂow, the difference in intensity of tidal mixing across the shelf edge leads to horizontal density gradients (Fig. 9d) which as discussed previously force additional along-shelf ﬂows. These currents modify those found in the barotropic case (Fig. 4b) leading to the greater spatial variability shown in Fig. 9c. Changes in the density ﬁeld and frictional effects due to the presence of the along-

shelf ﬂow, modify the time-averaged across-shelf ﬂow (Fig. 9c) compared to that found earlier (Fig. 8c). However, the basic features, namely an off-shelf current close to the bed, and on-shelf ﬂow higher in the water column are still present (Fig. 9c). For the reasons discussed earlier, the time-averaged concentration, and off-shelf transport in the bottom boundary layer are higher than found previously (Fig. 8c), with an enhanced onshelf transport above the near-bed region. This calculation and the earlier one, clearly show that an along-shelf ﬂow signiﬁcantly inﬂuences the sediment suspension and transport for

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(c)

(d) Fig. 9 (continued).

both barotropic and baroclinic tides. The increased transport in the baroclinic case is appreciably more than in the barotropic calculation, and shows both an off-shelf transport at the

bed and on-shelf transport higher in the water column. This transport is consistent with the longterm transport derived from SES measurements (McCandliss, 2000).

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3.6. Internal tide with upwelling and downwelling favourable winds 3.6.1. Upwelling favourable wind (Calc. 7A) In this calculation, the tidal forcing, stratiﬁcation, and sediment type were as in Calc. 5A. Also there was no externally forced along-shelf mean ﬂow, apart from an upwelling favourable alongshelf wind stress of 0.2 Pa applied at the sea surface. This wind stress produced an upwelling of the isotherms at the shelf edge and increased the mixing in the surface layer over the whole region and the bottom boundary layer on the shelf, leading to a sharp thermocline between approximately 60 km from the coast and the shelf edge (Fig. 10a). In the near-shore region, the enhanced mixing produced a homogeneous water column.

2101

The effect of the wind is to signiﬁcantly modify the on-shelf and shelf break density ﬁeld compared to that used previously (compare Figs. 10a and 8a(i)), leading to a modiﬁcation of the internal tide on the shelf. Although the internal tide changes, the pattern of clockwise and anticlockwise circulation cells, with a similar structure and spacing is still evident on the shelf. Also the wind stress enhances the friction velocity on the shelf and hence the sediment suspension in this region (Fig. 10a) due to a wind-induced current to the south on the shelf (Figs. 10b and c). As in the case of adding an along-shelf ﬂow, the wind-induced ﬂow on the shelf adds a mean ﬂow to the south onto the v component of the oscillatory ﬂow as can be seen from the time series at Posn. (1). When combined with the u component this gives the time series of the friction velocity shown in Fig. 10b.

(a) Fig. 10. (a) Across-shelf variation of sediment concentration, corresponding un ; temperature distribution and stream function (with solid line positive) due to an internal tide and upwelling favourable wind (Calc. 7A), at t ¼ 3=8 T; (b) time series at Posn. (1) of departure of density ﬁeld Dst (kg m3) from its background level, u and v current proﬁles (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity) eddy viscosity, sediment concentration, and friction velocity (un ) over two cycles (Calc. 7A), (c) time average over a tidal cycle of cross-shelf (u) and along-shelf (v) currents (with solid contours indicating a positive velocity, the zero line denoted by a dotted and dashed contour, and dashed line a negative velocity) sediment concentration and transport (Calc. 7A).

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(b) Fig. 10 (continued).

Unlike the addition of a pressure driven alongshelf ﬂow, which does not substantially inﬂuence the stratiﬁcation, the wind enhances the surface mixing leading to a change in stratiﬁcation and this inﬂuences the proﬁle of the u and v components of current. This causes the region of nearbed maximum current to move closer to the seabed (Fig. 10b), which has the effect of changing the vertical eddy viscosity and diffusivity in the nearbed region. This inﬂuences the diffusion of sediment close to the bed. The wind also increases turbulence in the near surface layer. However since sediment does not reach this region at the shelf

edge, it has little effect here, although in shallower regions sediment can diffuse into the surface layer (see later). As in the case of the internal tide with an alongshelf ﬂow, the increased friction velocity above that found with the internal tide alone, leads to an increased sediment suspension at times of maximum un (Fig. 10b). However the time variation of un is such that it remains above unc for a longer period and hence sediment is suspended throughout the tidal cycle, unlike the case of an along-shelf ﬂow (compare Figs. 9b and 10b). The fact that the density stratiﬁcation is closer to the bed, and hence

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(c) Fig. 10 (continued).

near-bed diffusivity is lower than previously does however prevent the sediment diffusing so far away from the bed (compare Figs. 9b and 10b). The time-averaged along-shelf ﬂow (Fig. 10c), shows a wind-driven ﬂow to the south, superimposed upon which there is an along-shelf ﬂow due to the cross-shelf density gradients shown in Fig. 10a. The across-shelf ﬂow shows an up welling at the shelf edge with an on-shelf ﬂow in the near-bed region on the shelf. At the surface the wind drives a predominantly off-shelf current. Both surface and near-bed currents show signiﬁcant spatial variability due to the internal tide. The coastal boundary produces up welling in this region with sediment being advected into the surface layer and thereby transported towards

the shelf edge. This produces a signiﬁcantly higher sediment concentration and off-shelf transport (see /ucS plot Fig. 10c) of sediment in the surface layer than in the earlier calculations. The signiﬁcant increase in near-shore suspended sediment concentration under upwelling favourable winds is consistent with measurements made during similar wind conditions in February 1996 as part of the SES experiment. Also the off-shelf transport in the surface layer found in the model is consistent with measurements made under upwelling favourable winds (McCandliss, 2000). Although a qualitative comparison is not possible due to the large range of sediment types that were observed to be in suspension, and the fact that many were source limited, comparison with

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measurements showed a similar spatial variation in the suspended sediment. This comparison cannot be carried too far, in that in the present case we are considering a steady-state response to a constant wind stress, while observations were made during a period of rapidly increasing and decreasing wind. However both model and measurements show the importance of upwelling favourable winds in increasing near-shore sediment concentrations, and off-shelf advection of sediment in the surface layer. 3.6.2. Downwelling favourable wind (Calc. 7B) The effect of a change of wind direction with all other parameters as previously was to produce a downwelling circulation in shallow water. As warmer surface water was advected below colder water, vertical mixing occurred leading to an essentially well-mixed water column at the coast (not shown) while on the shelf there was a sharp thermocline and well-mixed bottom layer. At the shelf break there is a well-mixed bottom boundary layer, below a near surface thermocline (see

Fig. 11a and Dst time series in Fig. 11b). The vertical stratiﬁcation close to the coast is signiﬁcantly reduced from that found with an upwelling wind or no wind. However, between the shelf edge, and approximately 60 km off-shore, there is a sharp thermocline and an associated internal tide (Fig. 11a). The across-shelf distribution of the friction velocity at t ¼ 3=8 T, shows a smaller value with an associated reduced sediment concentration than that found with the upwelling wind (compare Figs. 10a and 11a). This is because the change in density ﬁeld due to the downwelling wind alters the phase of the internal tide. This change, together with the difference in direction of the along-shelf wind-induced ﬂow associated with the downwelling wind modiﬁes the magnitude and time of maximum un as can be seen in Fig. 11b from the time series of velocity at Posn. (1). The effect of the downwelling wind at Posn. (1) is to move the position of the thermocline closer to the surface, producing a tidal current proﬁle in the near-bed region comparable to that in the

(a) Fig. 11. (a) As in Fig. 10a, but for a downwelling favourable wind (Calc. 7B), (b) As in Fig. 10b, but for a downwelling favourable wind (Calc. 7B), (c) As in Fig. 10c, but for a downwelling favourable wind (Calc. 7B).

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Fig. 11 (continued).

homogeneous case. A phase shift at the level of the thermocline at about 50 m below the surface is evident in the current proﬁles. The v component is intensiﬁed by the along-shelf wind-driven ﬂow. The increase in bottom currents and change in phase relation between the u and v components compared with the upwelling wind case, leads to an increased maximum un which has the effect of increasing the sediment concentration at Posn. (1) in the near-bed region (compare Figs. 10b and 11b). Also the fact that the thermocline in the downwelling case, at this location has moved

closer to the surface, leads to an enhancement in the eddy viscosity and diffusivity in the near-bed region. This enhanced diffusivity enables the sediment to diffuse higher into the water column. The time variability of the top of the sediment layer (taken as the 0.2 mg l1 contour) closely follows the vertical position of rapid decrease in diffusion (the region where viscosity decreases from 102 to 105). This suggests that the location of this turbulence level determine the thickness of the sediment boundary layer. As the position of the thermocline changes with the on-shelf, off-shelf propagation of the tide, so does the region of

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(c) Fig. 11 (continued).

reduced diffusion, which separates the wind forced turbulent surface layer from the tidally produced turbulence in the bottom layer. Although the direction of the time-averaged v velocity is different from the upwelling case, its magnitude and spatial variation are comparable (compare Figs. 10c and 11c) with small differences due to the ﬂow induced by the different acrossshelf density gradients. The across-shelf velocity shows an on-shelf ﬂow in the surface layer, the thickness of which is slightly less than that found with the upwelling wind, due to the proximity of the thermocline to the surface. Below this layer there is an off-shelf ﬂow. The presence of an internal tidal residual contributes to the smallscale spatial variability found in the ﬂow. Also the

on-shelf surface ﬂow, and downwelling at the coast, prevents any sediment in the near-shore region being carried into the surface layer. As shown at Posn. (1), stable stratiﬁcation at the shelf edge inhibits sediment diffusion away from the near-bed region. Consequently in the downwelling case the sediment is concentrated close to the bed (Fig. 11c), where there is an off-shelf sediment transport, with a small on-shelf transport above this layer at the shelf edge. These calculations clearly show signiﬁcant differences in the spatial pattern of on-shelf and off-shelf sediment transport between upwelling and downwelling winds, which is consistent with SES measurements (McCandliss, 2000) made under a range of wind conditions.

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4. Concluding discussion Processes (stratiﬁcation, tides, along-shelf currents, wind-waves and winds) controlling suspended sediment dynamics in the shelf edge region off the west coast of Scotland have been examined using a three-dimensional non-linear model, applied in cross-shelf form. The vertical diffusion coefﬁcients in the model were derived from a turbulence energy sub-model. Initial calculations with barotropic forcing at the M2 period, applied at the off-shore boundary, produced near circular current ellipses, with bed stress and hence un showing little variation in time. Tidal velocities were a maximum in the region of the shelf break, and decayed towards the coast. For the sediment unc chosen in this study (typical of shelf edge regions) only un at the shelf edge exceeded unc ; and hence sediment was only suspended in this area. The effect of an alongshelf ﬂow was to increase un above unc at times when the tidal current enhanced this ﬂow, thereby leading to sediment suspension. The time-average (over a tidal period) across-shelf transport, showed an off-shelf transport in the bottom layer with a weaker on-shelf transport at the surface. The lack of sediment suspension due to the tide is consistent with measurements in the region as is the importance of the along-shelf ﬂow in suspending sediment. The long-term off-shelf transport of sediment in the bottom boundary layer due to residual currents of the order of 3 cm s1 is consistent with the time-averaged velocity (of order 5 cm s1) and sediment transport observed during the SES experiment. Calculations in which wind-wave effects were included signiﬁcantly enhanced the value of un ; and sediment suspension, although the across-shelf residual ﬂow changed very little with a larger change in the along-shelf wind-driven component. This enhanced sediment suspension lead to an increased off-shelf transport in the near-bed region. Since the major increase in un was in the near-shore region, the amplitude and period of the waves in this area can have a signiﬁcant effect on the across- and off-shelf transport of sediment. Enhanced sediment resuspension in the near-

2107

coastal region and on the shelf as a result of wave activity was clearly observed during the February 1996 storm event. Calculations involving stratiﬁcation effects, showed that increased un and regions of upwelling and downwelling associated with the internal tide lead to increased sediment suspension. The spatial distribution of across-shelf time-averaged current and hence sediment transport showed signiﬁcantly greater variability, associated with the wavelength of the internal tide, than in the barotropic case. A reduction in settling velocity enabled the near-bed turbulence to diffuse the sediment to greater heights in the water column, and thereby increased the time-averaged sediment transport. Due to a lack of measurements in the summer time, and the fact that the nearest bed transmissometer measurement was 7 m above the bed, the role of internal tides in transporting sediment could not be fully investigated using measurements from the SES period. However on the one occasion that a large internal tide was observed at the same time sediment measurements were made it led to signiﬁcant sediment movement. The presence of an along-shelf current in combination with the internal tide, lead to an increase in maximum un and associated vertical diffusion which increased sediment concentration in the near-bed region. This produced an enhanced off-shelf transport in the bottom boundary layer. With upwelling favourable winds, the near-bed temperature gradient and that across the thermocline increased, leading to stronger internal tides on the shelf, although at the coast the water was well mixed. Sediment which was suspended close to the coast, was advected into the surface layer by the vertical velocity in this region, and moved offshore in the wind-driven surface layer. The importance of upwelling favourable winds in transporting sediment off-shore in the upper part of the water column was clearly evident during the SES period. A downwelling wind modiﬁed the stratiﬁcation in a different manner to that found with the upwelling wind, and consequently changed the internal tide (Xing and Davies, 1997) and hence the time variability and magnitude of un which together with the corresponding change in the near-bed diffusion coefﬁcients, inﬂuenced

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sediment suspension. Although this process leads to enhanced sediment suspension, the upward advection of sediment in the near-coastal region found with an upwelling wind, was replaced by downwelling and hence sediment did not reach the surface layer. In the shelf edge region, a thicker bottom boundary layer was created than in the upwelling case. In this layer, the upward diffusion of sediment was inhibited by a reduction of turbulence at the level of the thermocline, with the thickness of the layer changing over a tidal cycle in response to changes in location of the thermocline as the internal tide propagated onto and off the shelf. The sediment concentration in the wind-driven ﬂow problems was signiﬁcantly larger than in the case of the internal tide alone, primarily because of the enhanced bed friction velocity due to the along-shelf ﬂow produced by the wind. This series of calculations clearly showed the importance of stratiﬁcation, the internal tide, along-shelf ﬂow, wave effects, and upwelling/ downwelling winds in determining suspended sediment advection in the region off the west coast of Scotland. The signiﬁcant spatial and temporal variability found in the suspended sediment concentrations computed with the model, particularly those produced by the internal tide suggests that a detailed set of water column and near-bed measurements under a range of conditions is required to understand the various processes and for model validation. Here, we have only considered one sediment with a speciﬁed unc and settling velocity which is uniformly distributed over the region. An inﬁnite source of sediment is assumed. Our aim has been to illustrate the various shelf-edge processes inﬂuencing suspended sediment movement. To proceed to a simulation of sediment movement would require a detailed knowledge of the distribution of various sediment types which is not currently available. Although the topography used here corresponds to a cross-section of the Hebrides shelf, the idealized nature of the forcing and sediment is such that the results should be generally applicable to sediment movement on any shelf edge subject to comparable physical processes.

Acknowledgements This work was funded in part by the EU under project INTAS99-1600. Valuable discussions with Dr. McCandliss concerning measurements made in the region during SES are very much appreciated. The authors are indebted to Mr. R.A. Smith for help in preparing diagrams and Mrs. L. Parry, C. Burke and L. Ravera for typing the paper.

Appendix A For completeness, the full three-dimensional model in Cartesian co-ordinates is presented here. The continuity equation, momentum equations, and transport equations for temperature and sediment concentration in transport form using s coordinates where s ¼ ðz zÞ=H with s ¼ 0 at the sea surface and s ¼ 1 at the seabed, are given by qz þr qt

Z

0

~ ðH V Þ ds ¼ 0;

ðA:1Þ

1

qHu ~ Þ þ qHuo þ r ðHuV qt qs qz þ BPFx fHn ¼ gH qx 1q qHu Km þ 2 þ HFu ; H qs qs qHv ~ Þ þ qHno þ r ðHvV qt qs qz þ BPFy þ fHu ¼ gH qy 1q qHv Km þ 2 þ HFv H qs qs

ðA:2Þ

ðA:3Þ

with o determined diagnostically from the threedimensional continuity equation. The pressure P at any depth s is given by qP ¼ rgH: qs

ðA:4Þ

A.M. Davies, J. Xing / Continental Shelf Research 22 (2002) 2081–2113

The time evolution of temperature T is given by qHT ~ Þ þ qHTo þ r ðHT V qt qs 1 q qHT Kv ¼ 2 þ HFT ; H qs qs

ðA:5Þ

with a corresponding equation for sediment concentration C given by qHC ~ Þ þ qHCðo ws =HÞ þ r ðHC V qt qs 1 q qHC ¼ 2 Kv þ HFc : H qs qs

ðA:6Þ

A simple equation of state was used to convert temperature into density, namely r ¼ r0 ½1 bðT T0 Þ ; with b ¼ 0:0002=1C and T0 a reference temperature corresponding to r0 : ~ ¼ ðu; vÞ and ðu; v; oÞ are In these equations, V the velocity components corresponding to the ðx; y; sÞ coordinates: r is density; T is the temperature; C the concentration, H ¼ h þ z is the total water depth; z is the elevation of the sea surface above the undisturbed level h; z is the water depth increasing vertically upwards with z ¼ z the free surface and z ¼ h the seabed; f is the Coriolis parameter, g is the gravitational acceleration; t is time; Fu ; Fv ; FT and Fc are horizontal diffusions for the momentum, temperature and sediment concentration, with BPFx ; BPFy the Baroclinic Pressure Force terms. The vertical eddy viscosity and diffusivity are denoted by Km ; Kv with ws the settling velocity which depends upon sediment size. At the off-shore boundary a radiation condition was applied, while at the land boundary the normal component of ﬂow was set to zero. In the calculations involving the tide, the vertical derivative of velocity was zero at the sea surface, while for wind-driven ﬂows, the surface stress was set equal to the wind stress. At the seabed a quadratic friction law was applied. For temperature the vertical normal component was zero at sea surface and seabed. A similar boundary condition was applied to sediment concentration at the surface. At the bed a pick up function was deﬁned by Kv qC ¼ aðu2n =u2nc 1Þn H qs

ðA:7Þ

2109

with n ¼ 3=2 and a set at 0.001 if un > unc ; otherwise a ¼ 0: Here un is friction velocity, and unc its critical value, depends upon grain size and the cohesive nature of the sediment. The eddy viscosity and diffusivity used in the hydrodynamic model were computed using a turbulence energy sub-model (Blumberg and Mellor, 1987; Oey and Chen, 1992a, b; Luyten et al., 1996; Baumert and Radach, 1992). This involved a predictive equation for q2 ¼ 2E (where E is turbulence energy), namely qHq2 ~ Þ þ q ðHq2 oÞ þ r ðHq2 V qt ( qs ) 2Km qu 2 qv 2 þ þ Dq ¼ qs qs H 1 q qHq2 q3 H Sq qc þ 2G þ 2 H qs b1 c qs

ðA:8Þ

with G ¼ Kv ðqb=qsÞ; accounting for the suppression of turbulence by buoyancy, where b ¼ gðr r0 Þ=r0 is buoyancy with r density and r0 a background average density, and Kv a diffusion coefﬁcient for density, and Dq ; the horizontal diffusion of turbulence. A horizontal diffusion coefﬁcient of 5 m2 s1 was used in all calculations for momentum and density, and b1 ¼ 16:6; Sq ¼ 0:2 are speciﬁed constants. The diffusion coefﬁcients for momentum Km and density Kv in a stratiﬁed ﬂuid are computed from Km ¼ lqSM ; Kv ¼ lqSH

ðA:9Þ

with SM and SH given by algebraic expressions (Xing and Davies, 1996a). In the calculations described later the sediment concentration was assumed to be so small that it did not affect the density which was determined by the temperature ﬁeld. The mixing length c was computed using an algebraic expression (Xing and Davies, 1996a) namely 1 1 c ¼ 1= þ ðA:10Þ c1 c2 with c1 and c2 determined from c1 ¼ KðH þ sH þ z0 Þexp ðb1 sÞ;

ðA:11Þ

c2 ¼ KðsH þ zs Þ

ðA:12Þ

2110

A.M. Davies, J. Xing / Continental Shelf Research 22 (2002) 2081–2113

and K ¼ 0:4 Von Karman’s constant, b1 an empirical coefﬁcient, with z0 the bed roughness length, and zs a surface roughness length, which controls the value of c at the sea surface. At lateral boundaries, no normal ﬂux of turbulence energy is assumed, while at the sea surface a source of turbulence related to the wind stress is speciﬁed (Craig, 1996; Xing and Davies, 2001b). Similarly at the bed a source of turbulence is assumed, related to the bed stress (Xing and Davies, 2001b).

(a)

Appendix B Here we brieﬂy illustrate how the phase relationship between the two velocity components of the fundamental (u1 ; v1 ) and higher harmonic (u2 ; v2 ) of the tidal current inﬂuence the time variation of un : Initially, we will consider for completeness two simple cases where the result is obvious, and then examine the more complex situation of a fundamental and higher harmonic in both the u and v direction. In all cases an amplitude of unity is assumed for the fundamental and 0.5 for the higher harmonic. B.1. Case 1. Oscillatory currents in the u and v direction at the fundamental frequency When u and v are in phase this corresponds to a rectilinear ﬂow with the current magnitude ðu2 þ v2 Þ1=2 and hence un having two maxima, and two zero values per tidal cycle (Fig. 12a). A similar time variation occurs when they are at 1801 difference in phase. As the phase shift varies from 01 to 1801 the nature of the current ellipse changes from rectilinear to circular with a 901 phase difference, in which case current magnitude and hence un is constant (Fig. 12b), returning to rectilinear as 1801 is approached. B.2. Case 2. Oscillatory currents in the u direction at the fundamental and higher harmonic In the case in which the two harmonics are in phase Fig. 13a, current magnitude exhibits two maximum values of equal magnitude over the tidal cycle. As the phase changes the pattern changes to

(b) Fig. 12. Time series of u velocity (dotted), v velocity (dashed), and magnitude ðu2 þ v2 Þ1=2 (solid line), for (a) u; v in phase, (b) u; v; with a 901 phase shift, over a tidal cycle.

a single maximum, with two smaller maximum values Fig. 13b (phase difference of 901) returning to a situation of two maxima but at different times in the cycle (Fig. 13c, phase difference of 1801). B.3. Case 3. Oscillatory currents in the u and v direction at both the fundamental and higher harmonic When the u and v components are in phase in both the fundamental and higher harmonic, and both harmonics are in phase, namely rectilinear ﬂow in both harmonics in the same direction, then the time variation of current magnitude is identical to that given in Fig. 13a, although its magnitude is increased by the presence of the v component of velocity. A phase shift of 901 between u1 ; v1 giving a circular current ellipse at the fundamental (Fig. 14b) or a phase shift of 901 between u2 ; v2 (a circular current ellipse at the higher harmonic) (Fig. 14c), gives rise to a current magnitude time series with multiple maxima and minima. In the case (not shown) of a 901 phase shift in both u1 ; v1 and u2 ; v2 ; then circular current ellipses occur in both the fundamental and the higher harmonic, and a constant current magnitude is produced

A.M. Davies, J. Xing / Continental Shelf Research 22 (2002) 2081–2113

(a)

(b)

(c)

2111

(a)

(b)

(c)

Fig. 13. Time series of fundamental frequency u1 (dotted) and higher harmonic u2 (dashed) and magnitude (solid line) for (a) u1 and u2 in phase, (b) u1 and u2 with a 901 phase shift, (c) u1 and u2 with a 1801 phase shift.

Fig. 14. As in Fig. 13 with (a) u1 ; v1 in phase (rectilinear ﬂow) and u2 ; v2 in phase, with no phase shift between fundamental and higher harmonic (b) as (a) but with u1 ; v1 phase shifted by 901 (circular ellipse), (c) as (a) but with u2 ; v2 phase shifted by 901 (circular ellipse).

irrespective of the phase relation between the fundamental and its higher harmonic. This series of simple plots shows that un can have a complex time variation, or in some cases can be constant, depending upon the phase relationship between the u and v components of currents at both the fundamental and the higher harmonic. (This also has implications when trying to linearize bottom friction (Hunter, 1975)). Obviously for sediment movement the critical determining factor is the direction of ﬂow at the time that un exceeds unc : The classic example given in the literature is for rectilinear ﬂow with the fundamental (M2 tide) and its higher harmonic (M4 tide) phase shifted by 901 (Fig. 13b) producing a strong ﬂood, when there is a possibility of unc

being exceeded and a weaker but longer ebb. This would obviously give a preferred direction for sediment movement. Interestingly, as discussed in connection with Fig. 14, in two dimensions this time variation in un due to a fundamental and its higher harmonic can change and in the limit can give a time independent un :

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Processes influencing suspended sediment movement on the Malin–Hebrides shelf

Processes influencing suspended sediment movement on the Malin–Hebrides shelf

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