The Influence of Eddy Viscosity Parameterization and Turbulence Energy Closure Scheme Upon the Coupling of Tidal and Wind Induced Currents

Estuarine, Coastal and Shelf Science (2001) 53, 415–436 doi:10.1006/ecss.1999.0623, available online at http://www.idealibrary.com on

The Influence of Eddy Viscosity Parameterization and Turbulence Energy Closure Scheme Upon the Coupling of Tidal and Wind Induced Currents A. M. Davies and J. Xing Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead, Merseyside L43 7RA, U.K. Received 20 November 1998 and accepted in revised form 7 September 1999 A three-dimensional model of the Irish Sea, and the shelf and shelf-edge region off the west coast of Scotland is used to examine the interaction of wind induced surface currents and M2 tidal currents which can lead to an increased near surface M4 tidal current in the region. Calculations are performed using uniform southerly and westerly wind stresses, with vertical eddy viscosity computed using either a simple flow dependent form of viscosity, or one related to the flow and surface wind stress or derived using a two-equation turbulence model. Resulting M4 tidal distributions computed with the various viscosity formulations are compared with each other. The spatial distribution the M4 tidal current in the case of the flow dependent viscosity, shows a significant enhancement of the surface M4 tide due to non-linear coupling between the surface shear and the tidally dependent viscosity. In deep water this is the dominant mechanism giving rise to an enhancement in near surface M4 current. In shallow water, however, where other non-linear processes are important in generating the M4 tide, namely changes in water depth and friction, these are significant as well as the non-linear surface shear effect, in changing the M4 tidal distribution. However, when an eddy viscosity formulation depending upon flow and surface wind stress is used, the non-linear coupling mechanism producing an enhanced M4 tidal surface current is reduced, and no increase in M4 surface current occurs. Calculations with the two equation turbulence model, do not produce an increase in the near surface M4 tide, as the near surface viscosity in this model is determined principally by the wind stress rather than the tidal current. Some conclusions concerned with developing a simple eddy viscosity model having similar characteristics to the turbulence energy model, and experimental data sets for validation are considered at the end of the paper.  2001 Academic Press Keywords: Irish Sea; hydrodynamic model; tidal currents; wind induced currents; eddy viscosity; turbulence energy; air-sea interaction

Introduction In recent years three-dimensional tidal models using simple flow dependent vertical eddy viscosity parameterizations have been shown to be very successful in reproducing tidal currents in shallow near coastal regimes (Lynch & Werner, 1987, 1990; Aldridge & Davies, 1993), limited area sea regions (e.g. the Irish Sea: Davies & Lawrence, 1994a; Georges Bank: Lynch et al., 1995; Gulf of Maine: Lynch & Naimie, 1993), and over continental shelf-edge regions (e.g. Foreman et al., 1993; Davies et al., 1997a, b). An inter-comparison of a spectral model (Davies, 1987; Lardner, 1990; Lardner & Song, 1992; Nøst, 1994) with a k-
turbulence model (Davies & Gerritsen, 1994) showed that a simple flow dependent eddy viscosity gave more accurate tidal current profiles in the Irish Sea than the two-equation k-
model. However, a recent inter-comparison (Xing & Davies, 1995a, b) using a two-equation q2
q2, formulation of the turbulence energy model (Blumberg & Mellor, 1987) 0272–7714/01/100415+22 $35.00/0

gave results of comparable accuracy to these obtained using a simple flow dependent eddy viscosity. The effects of enhancements in the bed stress due to additional turbulence associated with wind waves and additional flow due to wind driven circulation in shallow regions upon the spatial distribution of tidal elevations and currents has been shown by Davies and Lawrence (1994b) in the shallow Eastern Irish Sea, and demonstrated in an idealized North Sea region (represented by a semi-enclosed sloping rectangular basin) by Davies and Jones (1996). In these calculations the mechanism changing the spatial distribution of the tidal elevation is the increase in the bed stress associated with the wind waves, which is particularly important in shallow water, and the increase in the bed stress due to the wind driven flow. This change in the tidal elevation gives rise to a change in the spatial distribution of the depth mean tidal currents. Besides changes in tidal elevations and depth means currents, in a three-dimensional model the near bed tidal current is reduced due to enhanced frictional  2001 Academic Press

416 A. M. Davies and J. Xing

58°N

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F 1. Finite difference grid of the three-dimensional model, with positions (1)–(4) points at which the current profile is examined in detail.

effects (Aldridge & Davies, 1993), associated with wind waves and wind driven flows. Also, changes in the near surface tidal current profile and magnitude both at the fundamental and higher harmonic occur (Davies & Lawrence, 1995) when the eddy viscosity is related to the flow field. Changes in the near surface tidal current profile under conditions of uniform wind forcing, were investigated by Davies and Lawrence (1995) using a

spectral/functional model of the Irish Sea with a flow dependent viscosity. They found that the near surface M2 tidal current profile was modified by the wind, with a significant increase in the M4 tidal current near the sea surface. These effects could be explained, using a single point model in the vertical (Davies & Lawrence, 1994c) in terms of a non-linear coupling between a flow dependent eddy viscosity, the time variation of which was controlled by the tidal currents

The influence of eddy viscosity 417 59°N

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F 2. (a) Water depth distribution in the region covered by the model, showing location of 25, 50, 100, 120, 200, 600, 1000, 1400 and 1800 m contours; (b) geographical location of places named in the text.

which were particularly strong in shallow water, and the near-surface shear induced by the wind. These early investigations of changes in tidal current profile due to non-linear coupling with the wind were restricted to shallow water regions where bottom frictional effects and wind induced set up against the coastline could influence the solution. Also, the profile of eddy viscosity through the vertical was constant with a magnitude and time variations determined predominantly by the tidal currents. In a recent series of calculations aimed at examining the wind induced circulation over the whole shelf, Davies et al. (2001) also considered the non-linear coupling in deep water. The model used by them was, however, similar to that applied by Davies and Lawrence (1995), namely a spectral model with a flow dependent viscosity. Some evidence for non-linear coupling in deep water (namely the North Channel of the Irish Sea) was found by Hall and Davies (2001) using a turbulence closure model to determine eddy viscosity, although this coupling was not examined in detail. There is therefore a need to use a model covering a range of water depths with various parameterizations of eddy viscosity to examine this problem. In this paper we use a three-dimensional model of the Irish Sea and the sea region off the west coast of Scotland, extending beyond the shelf edge (Figure 1) to investigate if the non-linear coupling mechanism between tidal and wind driven currents found by

Davies and Lawrence (1995) in the shallow Irish Sea still occurs in the deep water regions off the west coast of Scotland (Figure 2). Calculations are performed both with the simple flow dependent viscosity used by Davies and Lawrence (1995) and with viscosity determined from a two-equation q2
q2l turbulence energy model (Blumberg & Mellor, 1987) to examine if there are significant differences between these models in how wind driven and tidal current coupling occurs. Differences are found and a final series of calculations are performed with a modified viscosity representation to explain these differences. By using a model covering both deep and shallow regions it is possible to identify two distinct processes that can modify the M4 tidal currents. In deep water surface M4 tidal currents are modified by a coupling between a flow dependent viscosity and a surface shear produced by the wind, as predicted by the point model of Davies and Lawrence (1994c). However, in shallow water changes in water depth due to wind induced set up against the coast, and changes in bed friction also act to modify the tidal currents, both the fundamental and the higher harmonic. This gives rise to a significant spatial variability in the interaction of the tidal and wind driven flows, which cannot be accounted for by the simple point model of Davies and Lawrence (1994c) and explains the differences between this model and the results found by Davies and Lawrence (1995) in the eastern Irish Sea.

418 A. M. Davies and J. Xing

The formulation of the three-dimensional hydrodynamic model is briefly described in the next section, with subsequent parts of the paper dealing with the parameterization of vertical eddy viscosity, and results from the series of calculations, outlined above. The three-dimensional hydrodynamic model The three-dimensional equations for a homogeneous sea region expressed in transport form using a sigma coordinate () in the vertical are well-established and are given in Xing and Davies (1995a, b) and will not be repeated here. These equations are discretized in space using a staggered Arakawa C uniform finite difference grid in the horizontal with a variable grid in  co-ordinates in the vertical. The vertical diffusion term Av is computed either in terms of the flow field, or using a two-equation turbulence energy model (see later). A fully implicit time integration method is used to solve terms involving the vertical diffusion in order to avoid the use of a short time step when a fine grid is used in the vertical. Details of the method can be found in Davies and Jones (1991) and Xing and Davies (1995a). A two-time level numerical integration method (Davies, 1987) is used to advance the solution through time, in order to avoid problems associated with the leap-frog method.

An alternative to determining the vertical eddy viscosity Av directly from equation 1 or 2 is to compute it in terms of both a flow dependent viscosity Ac and one related to the wind friction velocity U*, thus: Am =Ac +Aw.

(3)

In the examples considered later we take in an analogous manner to equations (1) and (2): Ac =C1(u¯2 +v¯2)Y
c()

(4)

Aw =KoU*S*w().

(6)

In equations 4 and 5, c () is a profile of current dependent viscosity, with w() the corresponding profile for the wind induced viscosity, with Ko von Karman’s constant, U*s friction velocity and * a constant surface roughness length. Details of the form of c() and w() will be given later in connection with specific calculations. Eddy viscosity computed from a turbulence energy sub-model

Parameterization of vertical eddy viscosity Flow dependent eddy viscosity A simple flow dependent eddy viscosity Av which has been used very successfully to model tidal currents in shallow regions is given by: Av =C1(u¯2 +v¯2)Y
m()

Flow and wind stress dependent viscosity

An alternative, although computationally more expensive means of determining the eddy viscosity is to compute it from a turbulence energy model which involves prognostic equations for the turbulence energy and mixing length, namely:

(1)

where C1 is an observationally determined coefficient of order 0·0025, (Bowden & Hamilton, 1975; Bowden, 1978) with u¯ and v¯ the east and north components of the depth mean currents,
the thickness of the boundary layer, in shallow water taken as the water depth h, and m () the profile of eddy viscosity through the vertical. An alternative formulation in this
is related to the thickness of the bottom boundary layer, gives (Davies, 1991; Davies et al., 1997a, b):

with 2 a characteristic frequency of the motion, and C2 a constant.

and

In these equations q2 =2E with
The influence of eddy viscosity 419

depth;  is the elevation of the sea surface above the undisturbed level; z is the vertical co-ordinate increasing vertically upwards with z= the free surface and z=
h the sea bed; t is time. The wall proximity function W and the various coefficients Sq, B1, E1, are as defined in Blumberg and Mellor (1987). The vertical eddy viscosity is computed from: Av =,qSM,

(9)

with the algebraic form of the stability function SM identical to that used by Xing and Davies (1995a, b), and will not be presented here. Boundary conditions At land boundaries a condition of zero flow normal to the coastline is assumed, with a zero flux of both q2 and q2, at both land and open boundaries. Tidal energy enters the region through the open sea boundary where a radiation condition involving both elevations and depth mean currents is applied. (Details of the radiation condition and an overview of the open boundary problem can be found in the review article of Davies et al. (1997c), and further details will not be given here.) At the sea bed, a slip condition is applied in the hydrodynamic equations at a reference height zh above the sea bed, and the bed stress components FB, GB are related to the bottom currents uh, vh using a quadratic condition. In the turbulence energy sub-model, a boundary condition including the balance of the turbulence production, dissipation and diffusion formulated by Xing and Davies (1995b), is used at the bed, for the normal derivative of turbulence energy. The mixing length ,, tends to z0 at the sea bed. In the absence of wind forcing, the surface boundary condition is one of zero surface stress, and zero derivative of turbulence energy. In the case of wind driven flow the surface stress equals the applied wind stress and there is a source of surface turbulence with the length scale at the surface tending to a small surface roughness length zs (Hall & Davies, 2001). Tidal and wind driven currents computed using the flow dependent viscosity The three-dimensional numerical model used in the calculations (Figure 1) has a grid resolution of 1/15 in latitude and 1/20 in longitude, and covers both the Irish Sea (a shallow water region [Figure 2(a,b)] where the interaction of tidal and wind driven currents was studied previously (Davies & Lawrence, 1994a, b)

and the deep water area off the west coast of Scotland where this process has not been examined using the range of viscosity parameterizations described. It is evident from Figure 2(a), that the water depths change significantly in the model, from depths of the order of 10 m in the Eastern Irish Sea, to a deep area of order 250 m in the North Channel of the Irish Sea, with depths exceeding 1000 m in the oceanic region off the west coast of Scotland [Figure 2(a,b)]. Tide only solution In order to determine the extent to which the M2 and M4 tide are influenced by non-linear effects produced by the wind, it is necessary to initially establish the tidal regime in the region in the absence of meteorological forcing and then to examine how this is modified by non-linear effects. In an initial calculation (calculation A, Table 1), using the flow dependent eddy viscosity (equation 4), with
=h, (which we will term an hU dependent viscosity) and eddy viscosity profile m constant in the vertical, the model was forced with both M2 and M4 tidal input to the radiation condition along all open boundaries, and integrated forward in time until a periodic solution, unaffected by the initial conditions (of the order of 5 days) was obtained, which was then harmonically analysed for the M2 and M4 tides. Although it is not the aim of this paper to investigate the physics of the tide and tidal currents in this region, as this has been considered previously (Xing & Davies, 1995b, 1996), it is necessary to briefly examine the major features of the tide in the area in order to understand the mechanisms giving rise to interactions between the tidal and wind generated currents. The computed M2 cotidal chart [Figure 3(a)], shows an amphidromic point off the SE corner of Ireland, and in the North Channel of the Irish Sea, in good agreement with previous observations (Robinson, 1979; Pugh, 1981; Howarth, 1990), with tidal amplitudes increasing in Liverpool Bay. A decrease in amplitude within the North Channel, with subsequent increase off the west coast of Scotland, again in good agreement with previous observations and numerical model studies (Xing & Davies, 1995b, 1996), is evident in Figure 3(a). The M4 cotidal chart [Figure 3(b)] shows a rapid increase in the M4 tide in the eastern Irish Sea with a degenerate amphidromic point between the Isle of Man and Anglesey [Figure 2(b), 3(b)]. These features and tidal amplitudes are in good agreement with observations (Robinson, 1979) and results from other numerical modelling studies. The spatial distribution of the M2 tidal current ellipse (plotted at every second grid point) at sea

420 A. M. Davies and J. Xing T 1. Summary of parameters used in the various calculations Calculation A B C D E F G H I J

Wind forcing No 0·2 N m
2 (South) 0·2 N m
2 (West) 1·0 N m
2 (South) 1·0 N m
2 (West) No 0·2 N m
2 (South) 0·2 N m
2 (South) 0·2 N m
2 (South) 0·2 N m
2 (South)

Viscosity formulation Flow Flow Flow Flow Flow

dependent (hU) dependent (hU) dependent (hU) dependent (hU) dependent (hU) q2
q2, q2
q2, Flow dependent (hU) Flow dependent (U2) Flow dependent (U2)

surface and sea-bed [Figure 4(a)], shows tidal current magnitudes of the order of 5 cm s
1 in the ocean increasing rapidly from ocean to shelf, particularly in the region of the shelf-edge, where the tidal current ellipse is near circular, changing to a north–south oriented ellipse off the west coast of Scotland. Within the North Channel the major axis of the ellipse is aligned with the channel, with the sense of rotation changing to west–east in the eastern Irish Sea, with a north–south alignment in the southern Irish Sea, and a region of near zero tidal currents to the west of the Isle of Man. This spatial distribution of tidal current ellipse magnitudes and orientations is in good agreement with observations and earlier numerical modelling studies (Xing & Davies, 1995a, b). Comparing surface and bottom current ellipses it is evident that the magnitude of the bed currents is reduced by frictional effects. The spatial distribution of M4 surface and bed tidal current ellipses [Figure 4(b)] shows that in deep water the M4 tidal current is negligible, although in the shallow water regions of the eastern Irish Sea, and in the vicinity of headlands the M4 tidal current is significant (of order up to 20 cm s
1). Comparing Figure 4(b(i)) and Figure 4(b(ii)), it is evident that in shallow water, bottom friction reduces the current in the near bed region, particularly in shallow regions. Tidal and moderate (0·2 N m
2) wind stress forcing In a subsequent calculation (Table 1, calculation B) the same tidal forcing was applied along the open boundary, with an imposed uniform southerly wind stress of 0·2 N m
2 applied over the whole region. In order to avoid inertial oscillations being excited by suddenly applying the wind stress, the wind was increased linearly over three inertial periods. As in the previous calculation, an hU dependent eddy viscosity was used with eddy viscosity profile constant from sea surface to sea-bed. After five days the solution was

Viscosity profile

Wind dependent viscosity

Constant (A) Constant (A) Constant (A) Constant (A) Constant (A) — — Constant (A) Constant (A) Varying (B)

No No No No No — — Yes Yes Yes

harmonically analysed for the M2 and M4 tides. Computed M2 and M4 co-tidal charts (not shown), were not significantly different from those found in the tide-only calculation. Also, there were no significant differences in the M2 tidal current ellipses. However, the surface M4 current ellipse showed a major increase in magnitude in the deep water regions, with the major axis of the ellipse aligned with the wind direction [Figure (5)]. In shallow water regions such as the eastern Irish Sea the major axis of the ellipse changes from an east–west alignment to a nearly north–south orientation in some locations. Although the effect of the wind stress on average is to increase the surface M4 tidal current. In a number of regions a decrease in the surface current is evident, in particular in the area due west of Anglesey. The reason for this will be considered later. Comparing M4 tidal currents, just below the sea surface (=0·06), computed with the wind, with those determined previously, it is evident that in deep water the region of enhanced M4 tide is restricted to the near surface layer, although in shallow water regions such as the eastern Irish Sea changes are evident. Also, M4 bottom currents computed with and without wind forcing show similar spatial distributions in all water depths. However, a detailed comparison showed that in the case of tidal- and wind-forced flow the bed current, particularly in shallow regions, was slightly reduced below that for tidal forcing only, due to frictional effects. The reason for the increased magnitude of the surface M4 tidal current when the wind is present, and its spatial variability, can be appreciated from the calculations of Davies and Lawrence (1994c) using a single point model in the vertical. They showed, for both a hU dependent viscosity and that computed using equation 5 (which we will term a U 2 dependent viscosity) that the mechanism enhancing the surface M4 tidal current was the non-linear interaction between the sheared surface layer, and the time

The influence of eddy viscosity 421 (a)

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F 3. Computed cotidal chart at (a) M2 frequency; and (b) M4 frequency.

dependent nature of the surface eddy viscosity which was related to the time varying magnitude of the depth mean current, the time dependency of which depended upon the tidal current. In essence, a circular tidal current ellipse did not produce a time varying viscosity since the current magnitude was constant,

whereas a rectilinear tide showed a significant time variation, depending upon tidal current magnitude. Although the maximum shear in the wind forced surface current is in the direction of the wind, and hence the major axis of the M4 due to the non-linear interaction will in general be in this direction, the spatial

422 A. M. Davies and J. Xing (a(ii))

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F 4. Spatial distribution of (a) M2 current ellipses at (i) sea surface and (ii) sea-bed; (b) M4 current ellipses at (i) sea surface and (ii) sea-bed. Scale as shown with length and number of lines indicating magnitude and alignment of the current ellipse.

variability in the magnitude and nature (near rectilinear or near circular) of the current ellipse, and its orientation relative to the wind direction (Davies & Lawrence, 1994c) explains some of the variability found in Figure 5(a) in deep water, although as we will show, other processes have an influence in shallow water.

The sensitivity of the alignment and magnitude of the M4 tidal current ellipse due to the non-linear interaction of the wind induced surface shear and the M2 tidal current can be readily appreciated by considering the surface spatial distribution computed with a westerly wind of 0·2 N m
2 (Figure 6).

The influence of eddy viscosity 423 (a)

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F 5. Spatial distribution of M4 current ellipses from an harmonic analysis of the model forced with a southerly wind stress of 0·2 N m
2 and the M2 and M4 tide at (a) sea surface (=0·0); (b) =0·06; and (c) sea-bed (=1·0).

Comparing Figure 6 with Figure 5(a) it is evident that in the deep water regions the major axis is aligned with the wind direction. Off the west coast of Scotland at about 56·2N, 8W, a region of near zero M4 surface tidal currents exists in both solutions. It is interesting to note that this location corresponds to a region of small near circular M2 surface current ellipses [Figure 4(a)] where the time variation of the

eddy viscosity will be a minimum and hence the coupling with the wind induced shear will be small. From a comparison of the calculated M4 surface currents in these deep-water regions off the west coast of Scotland, from the single point model (Davies & Lawrence, 1994c), namely the major axis of the M4 current ellipse aligned with the wind field, with magnitude related to M2 tidal current strength and nature

424 A. M. Davies and J. Xing

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F 6. As Figure 5, but a westerly wind stress of 0·2 N m
2, and only the surface ellipses.

of the M2 current ellipse, it is clear that the solution is more appropriate than in the Irish Sea. Previously in the Irish Sea region, Davies and Lawrence (1994a) used a three-dimensional model to examine the coupling at a limited number of points. Based upon this limited set of points they found significant spatial variation in the coupling, which could not be readily explained. From a comparison of M4 Irish Sea surface ellipses [Figures 6 and 5(a)] it is evident that in the shallow Irish Sea region the major axis of the surface M4 ellipse under wind conditions cannot be simply related to the direction of the wind. Also, depending upon wind direction, the location of the position where the smallest M4 surface currents are found changes. For instance, the region of near zero currents found to the west of Anglesey in the case of the southerly wind does not occur with the westerly wind. The reason for the more complex behaviour of the M4 surface tidal current ellipses seen here and found by Davies and Lawrence (1994a) in the shallower Irish sea than that which occurs off the west coast of Scotland is due to a number of processes. The simplest process to understand is that the M4 surface ellipses shown in Figure 5(a) and Figure 6, is a linear combination of the M4 arising from the M2 tide without wind, [which is essentially zero in deep water but shows significant variation in the Irish Sea; Figure 4(b(i))] and that due to surface interaction between the wind shear and

the M2 tide, the process considered by Davies and Lawrence (1995c) using a point model. However, besides this process, as we will show, the additional bed stress due to the wind induced currents changes the spatial distribution of the M2 and M4 tides. A comparison of the M4 co-tidal chart computed with wind forcing (not shown) with that computed without wind forcing [Figure 3(b)] showed some differences in amplitude and phases which we will consider in more detail in the next section. As in the south wind case, the region of enhanced M4 tidal current strength in deep water is restricted to the surface layer, with M4 currents below this region (not shown) not significantly different from those found without wind forcing [Figure 4(b(i))], although in shallow water the effect is still evident. The M4 tidal currents at the bed (not shown) were only slightly different from those found without wind forcing [Figure 4(b(ii))], although as we will show subsequently these do change in shallow water with stronger winds. Tidal and strong (1·0 N m
2) wind stress forcing In this section in order to gain more insight into the mechanisms producing the spatial variability of the resulting M4 tidal currents, particularly in the shallow regions of the Irish Sea, we examine the coupling under conditions of a stronger wind stress. The M4 cotidal chart derived from a harmonic analysis of the solution computed with a southerly wind stress of 1 N m
2 (calculation D, Table 1) shows [Figure 7(a)] a decrease in M4 tidal elevations in shallow water regions such as the eastern Irish Sea [compare Figure 7(a) with Figure 3(b)] with a slight change in phase (of order 20). The reason for this is that the wind induced flow changes surface elevations [Figure 7(b)] [an increase of 0·5 m is evident in the Solway which is significant when the water depth is the order of a few meters, in an area of significant M4 generation; Figure 3(b)]. Besides changes in surface elevation, the wind induced flow gives rise to currents of the order of 25 cm s
1 in the eastern Irish Sea, [illustrated here by the wind driven bed current; Figure 7(c)] which influence the M4 tide in the region, through frictional effects. Although the M4 tide in the surface layer over the whole region is increased above that found previously by the increase in wind stress, below this region the M4 tide in deep water is not affected except in the Irish Sea region where frictional effects produce a change in the near-bed region [compare Figure 7(d) and 4(b(ii))]. It is important to note that this change in the M4 tidal currents in the near-bed region is significantly larger

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(c) 40

(b) 28

58°N

12 16

32

58°N

44

8

36 28 24

0.25 m s−1

57°N

40

39

24 20

0

57°N

4

28

4 36 0

56°N

56°N

32

20 24

55°N

24

36 32

20

20

12

54°N

54°N

8 4 00

0

53°N

55°N

64 36 48 52 44 40 16 28 16 20 24

53°N

4

4 8 4 8

–10°W

–8°W

–6°W

–10°W

–4°W

–8°W

–6°W

–4°W

(d) 58°N

0.2 m s−1

57°N

56°N

55°N

54°N

53°N

–10°W

–8°W

–6°W

–4°W

F 7. (a) M4 computed cotidal chart; (b) residual surface elevation (cm); (c) wind-driven bed currents; (d) M4 current ellipses at the bed, derived from an harmonic analysis of the model forced with the M2 and M4 tide and a southerly wind of 1·0 N m
2.

426 A. M. Davies and J. Xing

than that found previously with the smaller wind stress [compared Figure 7(d) and Figure 5(iii)]. From this comparison it is evident that the orientation of the major axis of the bed ellipse has changed in the Liverpool Bay region and in the area of the north and north-east of the Isle of Man. Significant changes in the ellipse are evident in the region between Ireland and Anglesey associated with changes in frictional effects produced by the additional wind driven flow, and changes in surface elevations. This is a different mechanism to that described previously which involved a non-linear coupling in the surface layer. Here we are concerned with changes produced by frictional effects in the bottom boundary layer and effects due to changes in water depth. In a subsequent calculation (calculation E, Table 1) a westerly wind stress of 1·0 N m
2 was applied. The associated M4 co-tidal chart [Figure 8(a)] shows an increase in M4 tidal elevation in the eastern Irish Sea, and a change in the distribution of co-amplitude and co-phase lines both in this region and between Ireland and Anglesey [compare Figure 8(a) and Figure 3(b)]. This effect is produced by the change in surface elevation due to the wind [Figure 8(b)]. This change in surface elevation, although comparable to that found with the southerly wind, has a different spatial distribution [compare Figure 8(b) and 7(b)]. Also, the additional wind induced currents [illustrated here with the bottom current; Figure 8(c)] exhibit a different circulation pattern to that found with the southerly wind [compare Figure 8(c) and 7(c)] although current magnitudes are similar. The change in surface elevation influences the non-linear term involving the total water depth in the continuity equation (which is important for M4 generation in the area; Davies & Lawrence, 1994a), with the addition of a strong wind induced flow in shallow water affecting the bottom friction term. These effects are important in shallow water, although in deep water they do not influence the M4 tide. The different distributions of surface elevation and wind driven flows found with the different wind directions explains why there are differences in the distribution of the M4 tidal currents in shallow water under various wind conditions. As previously, the M4 tide in the surface layer increases significantly with an associated change in the bottom layer in shallow water [compare Figure 8(d) with Figure 4(b(ii))]. From this comparison it is evident that the M4 tidal current increases in the Irish Sea and in the region between Ireland and Anglesey. The orientation of the current ellipses also changes, however, as can be seen from Figures 7(d) and 8(d), the orientation in the shallow eastern Irish Sea is significantly different between the two wind cases,

suggesting that in these shallow water regions, besides changes in the current ellipse associated with frictional effects during strong wind events which modify the co-tidal chart, non-linear coupling with the wind can affect currents in the near bed layer. Tidal and wind driven currents computed using the viscosity from the turbulence energy model Tide only solution In an initial calculation (calculation F, Table 1) the model was forced with M2 and M4 tidal input to the radiation condition. This input was identical to that used previously, although in this case the eddy viscosity was computed using the q2
q2, turbulence energy sub-model. The model was integrated forward in time for an identical period to that in the previous series of calculations before being harmonically analysed for the M2 and M4 tides. Although it is not our intention here to study in detail differences in the M4 tide determined with turbulence energy and eddy viscosity models (that problem has been considered in Xing & Davies, 1995b), it is however essential to determine the spatial distribution of the M4 tide computed with the turbulence model. This is necessary so that we can consider changes in the tide computed with this model due to wind effects. Although the spatial distribution of the cotidal charts and M2 current ellipses were not significantly different from those found with the flow dependent viscosity, there were some slight differences in the spatial distributions of surface and bed M4 current ellipses (Figure 9). Comparing Figure 9(a,b) with Figure 4(b(i),(ii)), it is evident that in deep water both models give a negligible M4 tidal current, although in the shallow eastern Irish Sea, a similar spatial distribution is found in both calculations, with the q2
q2, model giving slightly larger M4 tidal currents. A similar reduction in magnitude between surface and bed M4 currents computed with the q2
q2, model to that found with the flow dependent viscosity is clearly evident. Tidal and moderate wind stress forcing (0·2 N m
2) In order to examine differences in the coupling of wind induced shear and time dependent eddy viscosity, (and hence the generation of a near-surface M4 current) between the two models, it is necessary to repeat calculation B (Table 1) using the q2
q2, model. In an initial calculation (calculation G, Table 1) the model was forced by both tidal input and a southerly wind stress of 0·2 N m
2.

The influence of eddy viscosity 427 phase (deg)

amplitude (cm)

(a)

320

28

280

58°N

0

4

58°N 240

57°N

57°N 4

200

200 8

56°N

56°N

160

8 20

24

0

54°N

40

4

80

16 12 8 4

80

0 24 20800 280 140

280

80

40

54°N

55°N

40

N

4

55°N

8

160

4

4

8

53°N

53°N 16

km 0 60 –10°W

–8°W

–6°W

–10°W

–4°W

–8°W

–6°W

–4°W

(c)

16 20 24

12

8

0

4

(b)

8

4

58°N

58°N

28

28

55°N 2 32 8

20 24

54°N 40 44 48 3

36 32

6

4

2 28 4

16

8

55°N

56°N

24

0

16

8

12

4

56°N

54°N

0.25 m s−1

57°N

57°N

28

53°N

–10°W

–8°W

–6°W

53°N

–10°W

–4°W

–8°W

–6°W

(d) 58°N

0.2 m s−1

57°N

56°N

55°N

54°N

53°N

–10°W

–8°W

–6°W

–4°W

F 8. (a–d) As Figure 7, but with a westerly wind of 1·0 N m
2.

–4°W

428 A. M. Davies and J. Xing (a)

(a) 0.2 m s−1

58°N

58°N

57°N

57°N

56°N

56°N

55°N

55°N N

N

54°N

53°N

54°N

53°N

km 0 60 –10°W

–8°W

0.2 m s−1

0

–6°W

–10°W

–4°W

km 60

–8°W

–6°W

–4°W

–8°W

–6°W

–4°W

(b)

(b) 0.2 m s−1 58°N

58°N

57°N

0.2 m s−1

57°N

56°N

56°N

55°N

55°N

54°N

54°N

53°N

53°N

–10°W

–8°W

–6°W

–4°W

F 9. Spatial distribution of (a) surface and (b) sea-bed M4 tidal current ellipse computed with the q2
q2l model.

The computed distribution of the M4 tidal current ellipses at the sea surface [Figure 10(a)] does not show the increased magnitude in the near surface layer found in deep water with the flow dependent viscosity [Figure 9(a)]. However, in shallow water there are some small changes in the magnitude of the M4 tidal current, with in general a slight reduction [compare Figure 10(a) and Figure 9(a)] when the southerly wind stress forcing is included, due to the

–10°W

F 10. Spatial distribution of M4 current ellipses from an harmonic analysis of the model (using the q2
q2l sub-model to compute viscosity) forced with a southerly wind stress of 0·2 N m
2 and the M2 and M4 tide at (a) sea surface (=0·00) and (b) sea-bed (=1·0).

increased friction in the shallow water associated with the wind driven flow. Also, frictional effects have slightly reduced the near-bed currents. To understand why there are differences between the coupling of tidal and wind driven currents in the turbulence energy model compared with the flow

The influence of eddy viscosity 429

dependent eddy viscosity model it is instructive to examine time series of currents, turbulence energy and viscosity at four positions (locations 1 to 4 in Figure 1, although only locations 1 and 2 are considered in detail) computed with the q2
q2, model with and without wind forcing. Time series computed without wind forcing Figure 11(a) at position 1 in deep water (h=370 m) shows that above a high shear bottom boundary layer there is little shear in the water column, with the time series of turbulence energy showing some time variability related to the time of maximum shear production of turbulence, with some phase shift with height above the bottom boundary layer associated with the time taken for the turbulence to diffuse upwards. The mixing length in the calculations (not shown) is a maximum at about mid-water (see profiles in Xing & Davies, 1995a) and decays to zero near the sea surface. A consequence of this is that the eddy viscosity which is a product of the mixing length and turbulence energy is a maximum at mid-depth and decays in the surface layer. In shallow water (position 2, h=42 m) where the tidal currents are stronger, in particular the u-component, a near rectilinear flow is evident, with turbulence energy reaching a maximum at the sea-bed at times of maximum current shear, with corresponding maxima (although phase shifted) higher in the water column. As at the deep water location the product of turbulence energy and mixing length gives rise to a maximum eddy viscosity at mid-water with a small value in the surface layer. These time series, together with time series from positions 3 and 4 (not presented), show that in all water depths the surface eddy viscosity decreases to a small value with maximum viscosity at about midwater. In deep water with smaller tidal currents the time variability of the eddy viscosity appears to be much smaller than in shallow water where some time variation of viscosity occurs in the surface layer. These calculations do however suggest that the eddy viscosity should decrease in the surface layer in a tidal flow. The reason why tidal calculations using a constant viscosity profile with viscosity magnitude related to the flow field give tidal profiles of comparable accuracy to those found with turbulence energy models (Xing & Davies, 1995a), is that the vertical derivative of the tide is near zero in the surface layer and hence its product with near-surface viscosity is small. Consequently the magnitude and profile of nearsurface eddy viscosity is not important in a tidal problem. However, as shown by Davies and Lawrence

(1994c) using a single point model in the vertical, when a wind stress is applied to an oscillatory tidal flow, even when the tidal eddy viscosity decreases in the surface layer, the fact that the wind produces surface shear means that the product of the viscosity and the current shear is important and can lead to an enhancement of the M4 surface current even with a low surface viscosity. To understand why this coupling does not occur in the turbulence energy model it is necessary to consider time series produced with tidal and wind forcing [Figure 11(b)]. Considering initially position 1 in deep water. It is evident from Figure 11(b) that the addition of a southerly wind stress increases the v component of surface current with rotational effects causing some change to the u component, giving rise to shear in the surface layer which with the surface source of wind friction velocity dependent turbulence leads to an increase in surface turbulence energy and a corresponding change in viscosity. A similar effect is found at position 2, in shallow water, where the wind driven current dominates the v velocity in the surface layer, although below his layer the tidal signal is still evident. The time series of viscosity at this location still shows some time variability due to the strong u component of the tide, although this has been reduced by that produced by the wind component of velocity. The fact that there is still some time variability of viscosity in the surface region of shear in the v component of velocity means that time varying viscosity and vertical shear proposed by Davies and Lawrence (1994c) for producing coupling between the tidal current and the wind and moving energy to other harmonics still exists, although it is much less than that proposed by the simple model of Davies and Lawrence (1994c), where a surface wind source of turbulence was neglected, although a reduction in tidal eddy viscosity was considered. Although this surface coupling may well exist in shallow water regions, the change it produces in the M4 tide may be difficult to separate from the other effects in shallow water due to changes in water level and friction produced by the wind induced currents. Tidal currents computed using a flow dependent and wind dependent viscosity In the previous series of calculations using the q2
q2, model, no enhancement of the M4 surface tidal current occurred in deep water compared with that found with the hU dependent viscosity and predicted by Davies and Lawrence (1994c) using both an hU dependent and a U2 dependent viscosity with

430 A. M. Davies and J. Xing Position 1

(a)

–1

–1

u (cm s )

0

–100 Depth (m)

–100 Depth (m)

v (cm s )

0

–200

–200

–300

–300

0.5

8

–4

4

4

8

4

8

–400 0.0

4

1.0 1.5 t T–1 –1 2 tke log (m s )

0

–400 0.0

2.0

1.5

2.0

tT 2 –1 visc log (m s )

0

–3.0

–100 –6

.0

Depth (m)

–200

–300

–2.0

–200

–300 –2.0

–5.

–5.0

0

–400 0.0

–5.0

0.5

1.0 t T–1

Position 2

–3.0

1.5

–400 0.0

2.0

0.5

1.0 t T–1

–1

1.5

2.0

–1

u (cm s )

0

v (cm s )

0 –10 Depth (m)

–10 –20 –6 0

–30

20

–40

1.0 1.5 t T–1 –1 2 tke log (m s )

–30 –40

4

–50 0.0

2.0

0.5

0

–0

.3 –0

0

0

0.5

–4

–50 0.0

40

40 20

60

–40

8

–20

–8

Depth (m)

1.0 –1

–100

Depth (m)

0.5

.3

4

8

–8

1.0 1.5 t T–1 2 –1 visc log (m s )

2.0

–5.0

–0

–2.0

–50 0.0

Depth (m) –0

0.5

1.0 t T–1

–0 –0 .4 .3

–0 .4 –0 .3

.3

–0. 4

–30 –0 .4

Depth (m)

.3 –0

1.5

–20 –30 –40

2.0

–2

–10

–20

–40

.0

.4

–10

–50 0.0

–4.0

–4.0

0.5

–4.0

1.0 t T–1

–2.0 –4.0 –3.0

1.5

2.0

F 11. (a) F 11. (a) Time series of u and v components of velocity (cm s
1), turbulence energy log10 (m2 s
2), and viscosity log10 (m2 s
1) at positions (1) and (2) computed with the turbulence energy model. (b) As (a) but with the addition of a uniform southerly wind stress of 0·2 N m
2.

The influence of eddy viscosity 431 Position 1

–1

8 4 0

12

4

8

0 4 0

0 4

4

–1

u (cm s ) 0

0

4

(b)

v (cm s ) 8 4

–1

2

–100 Depth (m)

Depth (m)

8

–100

–200

–200

–300

–300

1.0 1.5 t T–1 –1 2 tke log (m s )

–400 0.0

2.0

0

0

4

4

0.5

4

–8

4

8

–400 0.0

0.5

1.0 1.5 t T–1 2 –1 visc log (m s )

2.0

–2.0

–4.0

–100 Depth (m)

Depth (m)

–100 –5.0

–200

–200

–300

–300 –5.0

–400 0.0

0.5

Position 2

–2.0–3.0 –4.0

–4.0

–4.0

1.0 t T–1

1.5

–400 0.0

2.0

0.5

1.0 t T–1 –1 v (cm s )

–1

u (cm s )

0

0

20 12 16 8

1.5

2.0

16

20

4 0

–10

–10 Depth (m) 40

60

–50 0.0

0.5

0

1.0 1.5 t T–1 –1 2 tke log (m s )

0

0.5

1.0 1.5 t T–1 2 –1 visc log (m s )

2.0

–2.0

–10

4

1.0 t T–1

–0 .4 –0 .3

–0

0.5

–0.

.3

.3

–0

.3

–0

1.5

–20 –30 –40

2.0

–50 0.0

F 11. (b)

–2 .0

Depth (m)

.4 –0

–30 –0.4

Depth (m)

4 8

–4

4

–50 0.0

2.0

–20

–50 0.0

–30 –40

–10

–40

–20

20

20

40

0

40

–4

–40

20

Depth (m)

4

–30

12

–20

–4.0

–4.0

0.5

–4.0 –3.0

–4.0

1.0 t T–1

1.5

2.0

432 A. M. Davies and J. Xing

viscosity either constant or increasing/decreasing in the surface layer. From the results of the q2–q2, model it was concluded that an important effect that was missing from the Davies and Lawrence (1994c) model was the inclusion of a surface source of wind dependent viscosity. To examine if this was correct calculation B was repeated but with a number of forms of viscosity. Initially the hU dependent form of viscosity was used with no decrease in tidal viscosity in the surface layer (viscosity profile A) with the addition of a wind source of viscosity equation 6, with w () unity at the sea surface and decreasing to zero over a depth of wind penetration given by CU*/f with C of order 0·2, giving a depth of penetration of the order of 25 m for the parameters considered here, with Aw =0·0028 m2 s
1 (comparable to that found in the turbulence model). Although this form and magnitude of viscosity was used in all calculations considered here since it depends upon U* then both Aw and the depth of penetration increase if U* increases. Calculations (calculation H, Table 1) using this parameterization of viscosity still gave significant M4 tidal currents in deep water because the surface tidal viscosity with an hU dependent formulation dominated the solution in deep water. To avoid this the calculation was repeated with a U 2 dependent viscosity (calculation I, Table 1) which in deep water corresponds to using
instead of h in equation (7) and is more physically justifiable in deep water. However, this formulation still gave an enhanced M4 tidal current in the surface layer, which could only be removed by using an eddy viscosity profile (Profile B) in which tidal viscosity decreased in a surface layer of order 0·1h to a value of the order of 0·001Ac (a value comparable to that found in the turbulence energy model for tidal flows). The spatial distribution of M4 surface tidal current ellipses (Figure 12) computed using the U 2 dependent eddy viscosity with viscosity profile B and a wind dependent source of viscosity (calculation J, Table 1) does not show a significant enhancement of the M4 tidal current ellipse in deep water where ellipses are comparable to those found with the q2
q2, model, although in shallow water regions, in particular the eastern Irish Sea, there are some differences in the alignment of the current ellipses. The reason for this will be discussed when we examine the time series at position 1. To understand why the M4 surface tidal ellipse in deep water is comparable with that found previously with the turbulence energy model it is necessary to consider the time series of viscosity and current at position 1 [compare Figures 11(b) and 13]. It is

evident that away from the direct wind driven surface layer the two models show comparable tidal flows. However, in the surface layer the u-component of current computed with the viscosity model is slightly (of order 2 cm s
1) larger than that found previously with the q2
q2, model with a comparable change in the v component of velocity. The surface viscosity (calculation J) is in essence independent of time and is determined by the wind stress. The magnitude of this viscosity is comparable to that found in the turbulence model. The fact that the surface wind stress determines the surface viscosity, which with a steady wind is constant, means that in this case there is no time variation in the surface viscosity and hence the mechanism proposed by Davies and Lawrence (1994c) for coupling surface wind shear and tidal currents through a time varying viscosity does not exist. It is, however, evident from Figure 13, that below the surface layer the eddy viscosity magnitude changes with the tidal current. In deep water this does not have a significant effect, although in shallow water (position 2) the time variation of eddy viscosity in the near-bed layer will be different to that found in the turbulence energy model and this will influence the M4 tide in shallow water as shown in Figure 12. Concluding remarks In this paper we have used a three-dimensional model of the sea region off the west coast of Scotland and the adjacent ocean to examine the non-linear coupling between wind induced currents and near surface shear and M2 tidal currents, and the resulting enhancement of the M4 surface current. The work extends that of Davies and Lawrence (1994a, b, c, 1995), who examined the problem in the Irish Sea (a shallow region) using a functional model with a flow dependent viscosity, and a single-point model. The major extension presented here is to compare the results using both a simple flow related viscosity and one computed using a turbulence energy model, in both shallow (the eastern Irish Sea), and deep (the region to the west of Scotland), and to produce a modified eddy viscosity formulation which explains these differences. Calculations using the flow dependent form of the viscosity, with a finite difference grid in the vertical, confirm the results found by Davies and Lawrence (1994a, c) using a spectral model, namely that an enhanced M4 surface current can be generated as a product of surface shear and the flow dependence of the near surface viscosity. By examining the spatial distribution of the M4 surface currents over the whole Irish Sea, rather than at a limited number of points, as was done by Davies and Lawrence (1994a), and for

The influence of eddy viscosity 433

58°N

57°N

0.2 m s−1

56°N

55°N

N

54°N

0

km

60

53°N

–10°W

–8°W

–6°W

–4°W

F 12. Spatial distribution of major and minor axis of the M4 tidal current ellipse computed with the viscosity model with a surface source of wind turbulence and a wind stress of 0·2 N m
2.

both moderate and high wind stresses (Davies & Lawrence, 1994a, only considered one) it is possible to appreciate the significant spatial changes due to non-linear interaction in the surface layer between the wind and the tide (the process considered by Davies & Lawrence, 1994c). Changes in the M4 tide due to other processes, namely changes in sea level and bed friction, which were not examined by Davies and Lawrence (1994c), have been shown to have an effect in shallow water. Results from the viscosity model and turbulence energy model presented here, (particularly under strong wind forcing) suggest that these other processes produce the significant changes in the M4 tide in the Irish Sea found by Davies and Lawrence (1994a) in their three-dimensional model. By considering the M4 generation at the sea surface in deep water, where the M4 tidal current is negligible,

the non-linear coupling between the tidal current and surface shear can be studied in isolation from other effects (frictional and water depth) which influence the M4 tide in shallow regions. The comparison of the enhanced M4 surface currents in deep water, for various wind directions, shows that the major axis aligns with the direction of surface shear (in essence the wind direction), with a magnitude depending upon the time variation of the viscosity, as predicted by the single-point model of Davies and Lawrence (1994c). Calculations using the q2
q2! model do not show any enhancement in the M4 surface current when wind forcing is applied. The reason for this is that the form of the mixing length used in this model is such that it decreases to a small value in the surface layer in the tide-only calculation, and hence the surface eddy

434 A. M. Davies and J. Xing Position 1

–1

–1

)

16

8

4

12

4

0

8

u (cm s 16 12

0 4

4 0

40

0

8

) 8 4

4

0

–100 Depth (m)

–100 Depth (m)

v (cm s

0 8

0

–200

–200

–300

–300

0.5

4

12 8

–400 0.0

1.0 t T–1

1.5

4

2

–1

visc log (m s

0.5

1.0 t T–1

1.5

2.0

)

0

–2.0

–3.0

–3.0

4

4

–400 0.0

2.0

4

8

4

Depth (m)

–100

–200

–300

–400 0.0

1.5

2.0 –1

) 8

v (cm s

0

4

4

0

16

)

8

–1

u (cm s

0

1.0 t T–1

16

Position 2

0.5

–10

–10

4

0

Depth (m)

60

4

8

0

–30

0

–4

Depth (m)

12 4

–30

–20

8

–20

–40

–40 –50 0.0

0.5

1.0 t T–1

1.5

–50 0.0

2.0 2

–1

visc log (m s

0

0.5

1.0 t T–1

1.5

2.0

)

Depth (m)

–10 –20 –30 –40 –50 0.0

0.5

1.0 t T–1

1.5

2.0

F 13. As Figure 11(b) but determined using the viscosity model with a surface source of wind stress of 0·2 N m
2.

The influence of eddy viscosity 435

viscosity is small and shows little time variability unlike in the case of the flow dependent viscosity. These differences in surface viscosity do not influence the tidal flow computed with either model since in the surface layer the tidal current shear is near zero and hence its product with the surface viscosity is also negligible. (This explains why Xing & Davies (1995b) found little difference in tidal current profiles computed with flow dependent eddy viscosity or that derived from a q2
q2! model.) However, when the wind shear is present the non-linear interaction which occurs with the flow dependent eddy viscosity does not occur with the q2
q2! model because any increase in surface turbulence energy or mixing length giving rise to an enhanced surface viscosity is determined by the time variation of the wind stress and not the tidal flow. In subsequent calculations comparable results to those obtained with the q2
q2! model were derived using a flow dependent eddy viscosity model in which the profile of eddy viscosity in the surface layer in the absence of winds was reduced to a small value. In the presence of winds an additional source of turbulence is present at the surface, with viscosity magnitude depending upon wind friction velocity and decaying with distance away from the sea surface to a depth related to the depth of frictional influence of the wind which again depends upon friction velocity. In deep water the model showed similar features to the q2
q2! model, namely no enhancement of M4 surface currents due to the wind. In shallow water both models showed significant changes in the M4 tidal currents due to wind effects although there were differences in the M4 computed with both models. In shallow water the bed-generated turbulence can reach the sea surface and is responsible for determining the position of tidal fronts, consequently in shallow water there is the possibility of coupling between wind driven current shear and tidal currents. However, in shallow seas the change in water depth produced by sea surface elevation increase due to the wind, and changes in bed friction due to wind induced flows and wave-current interaction effects will affect the M4 tidal currents. Both models show that the M4 tide will change in shallow regions, and observational evidence (Pugh & Vassie, 1976; Provis & Lennon, 1983) exists to support variations of tidal currents in shallow regions. Also, observational data from H F radar (Aldridge, pers. comm.) exists to support some change in the M4 tidal current during wind events. However, observational data sets to quantify the change and distinguish the mechanisms (surface coupling, bed friction) are not available. From the results presented here it is

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