Storm surge computations for the Irish Sea using a three-dimensional numerical model including wave–current interaction

Continental Shelf Research 18 (1998) 201 — 251

Storm surge computations for the Irish Sea using a three-dimensional numerical model including wave—current interaction J.E. Jones, A.M. Davies* Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead L43 7RA, Merseyside, England, U.K. Received 10 January 1997; in revised form 18 June 1997; accepted 19 June 1997

Abstract A brief overview of the mathematical development (with references to the literature for detail) of a high-resolution (of order 1 km grid) three-dimensional model of the wind-driven flow in the eastern Irish Sea, including wave—current interaction is presented. The model is applied to the computation of the storm surge of November 1977, and the importance of local wind fields compared to far-field winds upon the surge in the eastern Irish Sea is examined. The significant improvement in surge elevations at Liverpool produced by using a high-resolution grid is clearly demonstrated by comparing results with earlier coarse grid models. Further improvement obtained by including wave—current interaction effects is also demonstrated. Differences between surface and near-bed currents and the high degree of spatial variability in bottom currents during major storms in the eastern Irish Sea is also considered. ( 1998 Elsevier Science Ltd. All rights reserved

1. Introduction Over the last 20 years there have been a number of major numerical modelling investigations aimed at understanding the mechanisms influencing the intensity and temporal variability of storm surges in the Irish Sea. Initial investigations based upon an analysis of observed storm surges (Lennon, 1963), showed that major storm surges were produced by the intense wind fields associated with atmospheric depressions moving from the Atlantic across Britain. Early numerical modelling studies of storm surges (Heaps 1965; Heaps, and Jones, 1979) were based upon the two-dimensional vertically integrated hydrodynamic equations, although some three-dimensional

*Corresponding author. 0278—4343/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved PII S 0 2 78 — 43 4 3( 9 7 ) 00 0 62 - 9

202

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

modelling (Heaps and Jones, 1975) was also performed. In these calculations, due to limitations in computer power, the finite-difference grid was quite coarse (of order 12 km), and the model covered only the Irish Sea. In recent years with advances in computing technology (both speed and memory), three-dimensional models covering both the Irish Sea and the region beyond (Fig. 1) (Davies and Jones 1992), have been developed with reasonably fine grids in the horizontal (of order 7 km). Simulations of the November 1977 storm surge using the model of Davies and Jones (1992) showed that during major wind events, flow into the Irish Sea through the North Channel and the Celtic Sea region (Fig. 2a) was important in determining the intensity of the surge in Liverpool Bay. A detailed series of calculations (Davies and Lawrence, 1994a) using a finer grid (resolution of order 31 km) model of the Irish Sea forced by idealized stationary winds, and uniform open 2 boundary flows, clearly showed the influence upon different regions of the Irish Sea of flow through the Celtic Sea and North Channel, and winds over the Irish Sea. These calculations demonstrated the importance of the correct determination of the flow into the Irish Sea, and the spatial variability of winds over the Irish Sea, in determining the currents and elevations in the region. These calculations also showed the importance of a fine grid in the eastern Irish Sea, and the influence of wave—current interaction. Although detailed studies of meteorological forcing in shelf edge and in oceanic regions where stratification effects are important have been presented in the literature, the physics of these regions is different from that of the Irish Sea. In particular, the Irish Sea is a semi-enclosed shallow sea region with large tidal currents (up to 3 m s~1) rather than a shelf edge region which is influenced by an oceanic input. Also during major wind events in winter time the region remains well mixed. The shallow nature of the region (water depths on average of order 50 m) and the significant waves in the area, mean that wave—current interaction effects are important, as demonstrated in the idealized calculations of Davies and Lawrence (1994a, b). For a detailed discussion of wave—current interaction and related problems the reader is referred to A.G. Davies (1986, 1991), A.G. Davies et al. (1988) and the reviews of A.G. Davies (1990) and Huntley and Bowen (1990), and references therein. Observational evidence of its importance is given in Green et al. (1990), Spaulding and Isaji (1987). The effect of wind waves in very shallow water regions has been considered generally by Signell et al. (1990), and specifically for the Bristol Channel by Glorioso and Davies (1995). Also their effect upon currents in areas such as the eastern Irish Sea (Fig. 2b) was considered by Davies and Lawrence (1994b) using a very fine grid model of the eastern Irish Sea, developed previously by Aldridge and Davies (1993), driven by uniform wind fields. Davies and Lawrence (1994b) found that wave—current interaction effects were very significant in the very shallow regions, influencing both tidal (Davies and Lawrence, 1994c) and wind-induced currents. Also the improved resolution obtained

c Fig. 1. Finite-difference grid of the coarse grid west coast model, showing location of grid points used in the comparison.

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

203

204

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

Fig. 2. Bottom topography of the region with depths in metres.

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

205

Fig. 2. (Continued.)

by using a finer grid in the eastern Irish Sea was important in determining the wind-induced response of this region. In this paper, we begin by giving in Section 2 a brief overview of the threedimensional numerical model (with references to the literature for detail) and the form of the wave—current interaction method used. Since details of the wave—current

206

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

method have been presented elsewhere (Davies and Lawrence, 1994b); only the major points are given. Section 3 examines the overall dynamical structure of the November 1977 storm surge by looking at the meteorological situation as well as referring to earlier computations from the coarse grid model (Fig. 1), which are compared with observed surge elevations at a number of gauges. Section 4 commences with a surge calculation using only the limited area high-resolution model (grid resolution of order 1 km) of the eastern Irish Sea (Fig. 3) independently of the large area model. However, in a second calculation this model is run with input from the larger area model, interpolated in space and time along its open boundaries. From a comparison of these model results and by comparison with observed elevations and currents, it is possible to examine in a scientific manner how much of the surge is generated by wind fields acting over the Eastern Irish Sea (the internal surge) and how much is generated outside this area (the external surge). The effect of enhanced grid resolution upon the accuracy of the computed surge is also examined, together with the improvement produced by having a more detailed topography in the eastern Irish Sea. In Section 5 the sensitivity of the model to bottom friction parameterization is examined which leads to an investigation of the influence of wave—current interaction upon the computed surge. This is examined by comparing surge elevations computed with and without its inclusion in the fine mesh model of the eastern Irish Sea. In section 6, the horizontal variability of the flow is presented with a final section summarising the major results.

2. Three-dimensional numerical model 2.1. Hydrodynamic model Since the large-scale model covers a significant range of latitude (Fig. 1), the three-dimensional non-linear hydrodynamic equations in spherical coordinates were used in both regions (namely the coarse grid west coast model Fig. 1, and the fine grid eastern Irish Sea model Fig. 3). The equations contain the non-linear advective terms, and the horizontal eddy viscosity terms (Davies, 1981). However in view of the fine horizontal resolution used, and in order not to smooth detailed features of the flows in shallow water, the coefficient of horizontal eddy viscosity was set to zero. Details of the equations have been presented elsewhere (Davies, 1981; Davies and Jones, 1992; Davies and Lawrence, 1994a) and will not be repeated here. The numerical solution of the equations involves initially a transformation to a normalized sigma coordinate p"(z#f)/(h#f), where the z-coordinate defines a depth below the undisturbed surface and points downwards, h is the undisturbed depth of water and f is the elevation of the sea surface above the undisturbed level. This is followed by discretization in the vertical using an expansion in terms of functions, taken as the eigenfunctions of the eddy viscosity profile. By this means a continuous current profile is computed from sea surface to sea bed (Davies, 1986, 1987, 1993). A standard uniform finite-difference grid is used in the horizontal, and

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

207

Fig. 3. Finite-difference grid of the high-resolution eastern Irish Sea model.

a time-split method (Aldridge and Davies, 1993) is used to integrate the hydrodynamic equations through time. In the calculations the coefficient of vertical eddy viscosity k, is related to the flow field using k"a(s, /, t)t(p),

(1)

208

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

with t(p) a fixed viscosity profile in the vertical and a a coefficient determining the eddy viscosity magnitude, which varies with horizontal position (s, /), where s, /, are longitude and latitude, and with t the time. The coefficient a is related to water depth and east and north directed depth mean currents uN and, vN using a"K (uN 2#vN 2)1@2h 1

(2)

with K "0.0025 a constant coefficient (Bowden, 1978). 1 Along open boundaries a radiation condition was applied. In the case of the large-scale coarse grid west coast model (Fig. 1), the five dominant tidal constituents, namely M , S , N , K and O were introduced together with ‘far-field’ meteorologi2 2 2 1 1 cal forcing, as described in Davies and Jones (1992). The wind stress forcing over the region of the model varied with space and time (see Section 3). For the high-resolution eastern Irish Sea model (Fig. 3) the M , S , K , O , M and M tidal constituents, 2 2 1 1 4 6 taken from Jones and Davies (1996), were introduced along the open boundary together with the external surge derived from the large area model and interpolated on to the finer grid of the eastern Irish Sea model. Here the external surge is taken to mean the surge generated in the region outside the area covered by the eastern Irish Sea model. The surge is considered to be the difference between elevations and currents computed with tidal and meteorological forcing and those due to tidal forcing only. In one calculation, in order to study the internal surge only (here defined as the surge due to meteorological forcing over the eastern Irish Sea model), the boundary surge input was set to zero. (Further details of the calculations are given in the next section.) Along closed boundaries the normal component of the current was set equal to zero. In the high-resolution eastern Irish Sea model, grid boxes close to the coast can ‘flood and dry’ over the tidal cycle, and this was incorporated using the methods given in Aldridge and Davies (1993). [A review of methods to accomplish this can be found in Flather and Hubbert (1990).] For wind-driven flows, the surface stress is set to the externally specified east- and north-directed components of the wind stress, namely F , G , which during the course 4 4 of the storm event vary with horizontal position and time. A detailed discussion of the computation of the wind stresses F , G can be found in Davies and Jones (1992) and 4 4 will not be considered here. In essence, these wind stresses were computed using a quadratic drag law from an accurate hindcast surface wind field distribution specifically performed for the storm event (details of the wind field are presented later and will not be discussed here). The drag coefficient used in this computation depended upon wind speed (Smith and Banke, 1975). The computed wind stress distribution based upon three hourly analysed winds was interpolated through space and time to every grid point of the model, and every time step. Consequently, the wind stress used to force the model varied in magnitude both in the horizontal and through time. A number of calculations were performed, in order to compare differences in surge elevations derived using two- and three-dimensional models. In one calculation, to examine the influence of bed friction, surge elevations were computed using a vertically

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

209

integrated two-dimensional model in which the bed stress components F and B G were computed from the depth mean currents uN and vN using B F "kN ouN (uN 2#vN 2)1@2, G "kN olN (uN 2#vN 2)1@2 (3) B B with kN "0.180]10~2 being an appropriate drag coefficient to use in a two-dimensional model, and o denotes density. In the three-dimensional model the bed stress components F and G can be related B B to the near-bed currents u , v using a quadratic friction law; thus ) ) (4) G "kov (u2#v2 )1@2, F "kou (u2#v2 )1@2, ) ) B ) ) B ) ) where k is the drag coefficient. The value of k in a three-dimensional model depends upon the reference height at which the bottom currents are computed, and this in turn depends upon the vertical variation of eddy viscosity in the near-bed layer. In the limit the vertical eddy viscosity decreases to its molecular level at the sea bed and thus k goes to infinity in order to satisfy a no-slip condition for the bed currents. The relationship between friction coefficient k, near-bed viscosity and reference height is given in Davies (1990) and will not be discussed here. Tidal calculations using a flow-dependent eddy viscosity, the profile of which was constant in the vertical [i.e. t(p)"1 in equation (1)] with k"0.375]10~2 (i.e. a calculation in which the model did not resolve the near-bed region), although accurately reproducing tidal elevations, and flows in the upper part of the water column, failed to reproduce the near-bed currents. (This profile of viscosity was used by Davies and Jones (1992) in an earlier series of calculations of the storm surge of November 1977). However, in subsequent tidal calculations, Jones and Davies (1996) used an eddy viscosity profile [i.e. t(p)] in which viscosity was reduced linearly in a bed layer occupying 0.1 of the water depth to a value at the sea bed of 0.2 that in the upper part of the water column. With this reduction in viscosity, the model was able to resolve the near-bed layer, and hence the reference height at which the bed friction coefficient was evaluated was reduced with k increasing to 1.125]10~2. In essence, they found that by reducing viscosity in the near-bed layer, and increasing the friction coefficient they effectively reduced the reference height above the sea bed at which the bed stress was computed, giving an improvement in the accuracy of the bed currents computed with the model. This viscosity profile and bed friction coefficient are used in the three-dimensional calculations performed with the eastern Irish Sea model, and no attempt was made to change it to improve surge results, i.e. it was fixed at the value used in the accurate tidal solution. In the absence of wave—current interaction effects which increase the level of near-bed turbulence in shallow water (A.G. Davies et al., 1988; A.G. Davies, 1986, 1990, 1991; Grant and Madsen, 1979; Signell et al., 1990; Davies and Lawrence, 1994b, c; Davies and Jones, 1996) the drag coefficient is fixed at the value appropriate for the eddy viscosity profile. Obviously, a spatially varying friction coefficient could be used depending upon bed types and bed forms as in Davies and Lawrence (1995). Since there is some uncertainty in the exact location and type of bed, k could legitimately be varied to improve the surge result and the degree of wave—current

210

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

interaction. However, as our prime aim here is to examine the effects of viscosity profile in the near-bed layer and wave—current interaction rather than to obtain an accurate fit to the observed surge a specified drag coefficient was used in the absence of wave—current interaction. 2.2. Wave—current interaction model In shallow water regions, the wave orbital velocity is non-zero at the sea bed, and additional turbulence is generated in the near-bed region. This enhancement of turbulence leads to an increase in the drag coefficient k and hence modifies the bed stress in the three-dimensional model, when wind waves are present. A number of wave—current interaction models exist in the literature to take account of this. In previous calculations using idealized topography and winds (Davies and Jones, 1996), and idealized winds over the Irish Sea (Davies and Lawrence, 1994b, c) the model of Grant and Madsen (1979) in the form published by Signell et al. (1990) was used to take into account the enhancements in the bed stress due to wave—current interaction. The same method is used here in the high-resolution eastern Irish Sea model to take into account the enhanced bed stress (Calcn 5, Table 1) during the storm event of November 1977. Since wave—current interaction effects are important only in shallow near-coastal regions, they must be adequately resolved using fine grids (Davies and Lawrence, 1994b, 1995). For this reason, wave—current interaction effects were not considered in the coarse grid large area west coast model (Fig. 1) which could not resolve the near-shore shallow regions. Although the parameterization of wave—current interaction in the form applied by Signell et al. (1990) is used here to be consistent with the previous Irish Sea calculations, it is interesting to note that recently a significant number of wave—current interaction parameterization have appeared in the literature which are compared in Soulsby et al. (1993) and could, where appropriate, be used in the three-dimensional model described here.

3. The meteorological situation The period 7—17 November 1977 is chosen for the numerical simulation because it contains two significantly different types of meteorological events, which have been characterized by Lennon (1963). The first is one in which the depression moved from west to east across the area to the north of Scotland and then on to northern Norway. This is the situation that occurred on the 11—12 November. The second, which is less common, is that in which the depression followed a more southerly track moving from off the west coast of Scotland and over the North Sea to Denmark, the situation that occurred between the 13 and 15 November. In view of the scientific interest in this period, a detailed meteorological analysis was performed from which wind stresses at three hourly intervals were derived over the region covered by the large area coarse grid west coast model (Fig. 1). The actual wind stresses used in the numerical model plotted at every fifth grid point of the large area model from 12 h 11 November to 06 h

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

211

12 November covering the first surge event are given in Fig. 4, with wind stresses from 00 h 14 November to 00 h 15 November covering the second event in Fig. 5. In the calculations with the numerical models these wind stresses were interpolated to each grid point of the model and through time. The time series of the wind stress at a point (54°N, 3°45@W) in the central Eastern Irish Sea is shown in Fig. 6 to indicate in more detail than is possible in Figs 4 and 5 the time variability of the wind stress. Consequently, in all calculations the models were forced by a spatially and temporally varying wind stress. The wind stresses used in the present work have also been used in storm surge computations in two earlier models: a coarse grid (14 km) two-dimensional model covering the Irish Sea only (Heaps and Jones, 1979), and a finer grid (7 km) threedimensional model of the Celtic and Irish Sea (Davies and Jones, 1992). The effects of refining the grid in the eastern Irish Sea (Fig. 3) can therefore be compared with results from these coarser grid models. Also using the high-resolution Eastern Irish Sea model (Fig. 3) with and without wave—current interaction, its effect upon surge elevations can be determined. Although both the two- and three-dimensional models were used to compute the storm surge for the full period from 7 to 17 November, here we will concentrate upon two major storm events, namely the storm that produced the surge that occurred at 00 h 13 November and the one that produced the surge peak at about 00 h 15 November. From an examination of the meteorological charts for the period given in Davies and Jones (1992), it is evident that even in the case of the large area coarse grid west coast model, (Fig. 1) currents in the region will be influenced by wind-induced currents generated outside the area covered by the model. Davies and Jones (1992) showed that these far-field effects could be taken into account by running the large area west coast model (Fig. 1) with observed surge elevations from Castletownsend and Newlyn linearly interpolated along the southern boundary of the model and observations at Malin imposed along the northern boundary of the west coast model (Fig. 1). Although they showed that calculations with these open boundary conditions could take account of ‘far-field’ effects the surge elevation at Liverpool was significantly underestimated in the model (Fig. 7), primarily because the coarse grid of the model could not resolve the region (compare Figs. 1 and 3). Considering initially the meteorological event which led to the first surge, it is evident that at 12 h 11 November 1977 there were strong winds from the southwest with wind stresses exceeding 0.5 Pa over large areas of the Irish Sea, and of this order in the eastern Irish Sea (Figs 4a and 6). Over the following 6 h the magnitude of these wind stresses increased (Figs 4b and 6) and by 00 h 12 November there were strong winds from the west over most of the Irish Sea with maximum wind stresses of order 1.5 Pa off Anglesey, decreasing to about 0.5 Pa in the north of the Eastern Irish Sea (Fig. 4c). Over the next 6 hs the wind stress diminished although it was still significant over the Celtic Sea (Fig. 4d). It is evident from Fig. 4a—d and Fig. 6, that the west coast region at this time was subjected to a significant spatially and temporally varying wind stress, characterized by winds from the southwest changing to westerly. In the case of the second surge event at 12 h 13 November 1977 a system of strong winds from the northwest was moving into the area and by 00 h 14 November

Fig. 4. Wind stress vectors at every fifth grid point of the numerical model at (a) 12 h 11 November 1977; (b) 18 h 11 November 1977; (c) 00 h 12 November 1977; (d) 06 h 12 November 1977.

212 J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

213

Fig. 4. (Continued.)

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

214

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

a region of wind stress exceeding 1.0 Pa had developed off the west coast of Scotland (Fig. 5a) although wind stresses over the Irish Sea were less than 0.5 Pa (Figs 5a and 6) at this time. Over the next 12 h the wind stress off the west coast of Scotland increased significantly reaching 2.0 Pa (Fig. 5b), although over the eastern Irish Sea they were still only of the order of 0.5 Pa (Figs 5b and 6). The effect of the strong wind stress off the northwest coast of Scotland was to drive water into the eastern Irish Sea, through the North Channel. Over the next 12 h the magnitude of this wind stress decreased with the direction of winds in the Irish Sea changing from winds from the west to winds from the northwest (Fig. 5c).

4. Relative importance of internal and external surge In an initial series of calculations the three-dimensional high-resolution model of the eastern Irish Sea (Fig. 3) was used to examine the relative importance of the surge entering the region through the open boundary (the external surge) and that generated by wind effects over the region covered by the model (the internal surge). The aim here is to gain some scientific understanding as to the mechanisms producing changes in surface elevation at various ports. Since the tides are important in determining the background level of viscosity and bed friction in the area, in all calculations the tidal input to the model was the same, and was taken from Jones and Davies (1996). The surge residual was obtained by subtracting a tide-only solution from one including tidal and meteorological forcing. This is consistent with the way in which the data were de-tided by subtracting a tidal time series computed from an harmonic analysis of the observed data from the data time series. By this means a surge time series including any tide—surge interaction effect is produced. This is a more rigorous test of the model’s ability to reproduce the surge and the non-linear interactions than comparing total (tide#surge) time series, which in the region considered here is dominated by tidal effects and subtle changes produced by non-linear effects could not be so clearly seen. Also the separation of tidal, surge and interaction effects is scientifically more interesting than a study aimed purely at reproducing elevations at ports. 4.1. The external surge (Calcn 1) In an initial calculation (Calcn 1, Table 1) wind forcing over the eastern Irish Sea was omitted and the model was forced by tidal input as described previously and storm surge input, both elevations and currents, extracted from the coarser grid model (Fig. 1) and linearly interpolated to every grid point of the open boundary of the higher-resolution model (Fig. 3). The time series of the observed and computed external surge at Douglas (Fig. 8a), shows that the external surge determined with the coarse grid west coast model (Fig. 1) and linearly interpolated on to the fine grid eastern Irish Sea model (Fig. 3) can accurately reproduce the major features of the observed surge at Douglas, which is close to the open boundary of the fine mesh model. Hence, this will be influenced more by boundary input to the model than winds

Fig. 5. As Fig 4, but at (a) 00 h 14 November 1977, (b) 12 h 14 November 1977, (c) 00 h 15 November 1977.

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251 215

216

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

Fig. 5. (Continued.)

over the eastern Irish Sea. The external surge elevations at Workington, Heysham and Hilbre appear to be significantly less than the observed surge with the major characteristics at Workington and Heysham being very similar to those found at Douglas, although at Hilbre the magnitude of the surge peaks is reduced by frictional effects, and higher frequencies are present due to non-linear effects. A similar change to that found at Hilbre is also evident at Liverpool with the peaks in the external surge well below those found in the observations, despite the improved resolution (compared to the west coast model) in this area. It is evident from Fig. 8a that the observed time series is incomplete at Heysham and Hilbre due to failure of the gauge near the surge peak. Also the gauge at Workington suffered slippage producing datum changes on two occasions; these were removed from the time series.

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

217

Fig. 6. Time series of the east and north components of the wind stress at the centre of the eastern Irish Sea.

Also shown (Fig. 8b) are time series of observed and computed u component of current at location (D) at three positions in the water column, namely p"0.48, 0.62 and 0.80, corresponding to heights of 8, 16 and 22 m above the bed in a depth of 42 m, i.e. mainly in the lower-half of the water column, and at one position at location (B), namely p"0.5 a height of 22 m above the sea bed (Howarth and Jones, 1981), (see Fig. 3 for location of observations). Only the u component is considered here because Howarth and Jones (1981) concluded that the magnitude of the v-component was so small that no reliance could be placed upon it. Comparing observed and externally computed surge currents (Fig. 8b) it is evident that neither varies very much in the vertical. This is to be expected since the measurements were made well above the bottom boundary layer where frictional effects would influence the solution and well below the direct wind-driven surface layer. Also the time variability in the computed currents found at (B) is similar to that at (D). This is to be expected since the two locations are in close proximity (Fig. 3) and only the surge entering through the open boundary is considered in the calculation. Comparing observed and computed east components of velocity it is evident that the computed current does not contain the high-frequency oscillations found in the observations. This is to be expected, since although the model is driven with wind stresses which have been interpolated through space and time to each grid point and time step of the model, the original data are based on an analysed data set having

218

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

Fig. 7. Time series of storm surge elevation (m) at various ports computed using the coarse grid west coast model. Table 1 Summary of parameters used in the calculation Calcn

Surge

Wave effects

Model

1 2 3

External Internal External and internal External and internal External and internal

No No No

3D 3D 3D

No

2D

Yes

3D

4 5

c Fig. 8. Time series of the external storm surge (Calcn 1, Table 1) computed using the fine grid eastern Irish Sea model, (a) elevations (m) (solid line observed, dashed computed) at various ports and (b) currents (cm s~1) (light line observed, heavy computed) at various current meters.

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

219

220

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

Fig. 8. (Continued.)

a spatial resolution of order 50 km and 3 h in time. Consequently the analysed data set does not contain any of the short period wind events (‘gustiness’), which were recorded at Liverpool at the time (Heaps and Jones, 1979), or any small-scale horizontal divergences or convergences in the wind stress, both of which will cause

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

221

high-frequency oscillations in the currents. Also the observed velocity maximum of order 15 cm s~1 occurring at 18 h 11 November is not reproduced in the model, although the velocity of !10 cm s~1 occurring at 06 h 12 November is reproduced. However, the velocity maximum of !20 cm s~1 at 18 h on 14 November is underestimated. 4.2. The internal surge (Calcn 2) In a subsequent calculation (Calcn 2, Table 1), no effect of the external surge was introduced into the region, through the radiation condition along the open boundary, although a tidal input was maintained. Consequently, the surge in the area was produced only by the time and space varying winds over the region. The computed internal surge elevation at Douglas (Fig. 9a) was near zero, with that at Workington significantly less than that due to the external surge (cf. Figs 9a and 8a), although at Heysham and Hilbre the internal surge is a major part of the total surge. At Liverpool the internal surge has a significant contribution to the surge peak that occurs just after midnight on 12 November, although not to the surge peak prior to this. Similarly, the surge peak at about 18 h 14 November has a contribution from the internal surge, although there is no significant peak in the earlier internal surge at 06 h 14 November corresponding to the peak in the external surge (Fig. 8a). The time series of the internal surge, particularly at locations such as Heysham, can be directly related to the time series of the wind stress over the region (Fig. 6), in that the first peak at approximately 00 h 12 November in the internal surge at Heysham has a similar short duration to that found in the easterly component of the wind stress, with the smaller peak, although of longer duration, internal surge at about 00 h 15 November corresponding to the wind stress at this time. Time series of the east component of the computed internal surge current, at locations (D) and (B) at all depths shows that it is significantly smaller than the external component of current. (cf. Figs 9b and 8b). This demonstrates that at these locations the major contribution to the surge current is external to the eastern Irish Sea. This point is investigated more fully in connection with the total surge. 4.3. The total surge (Calcn 3) Time series of surge elevations computed with both wind forcing over the region and the external surge interpolated along the open boundary (Table 1, Calcn 3), are given in Fig. 10a. Comparing Fig. 10a with Figs 8a and 9a, it is evident that at Douglas the total surge is not significantly different from the external surge, for the reasons described previously, with the surge at Workington, and Heysham being a combination of external and internal surges. At shallow water locations such as Hilbre and Liverpool where both internal and external contributions of the surge are significant and the non-linear effects are appreciable, it is more difficult to combine linearly the two components of the surge to give the total, in that as shown by Davies and Lawrence (1994a, b) additional wind-induced currents modify frictional levels in a non-linear manner which significantly influences the surge. A linear combination of

222

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

223

Fig. 9. (Continued.)

b Fig. 9. Time series of the internal storm surge (Calcn 2, Table 1) computed using the fine grid eastern Irish Sea model, (a) elevations (m) (solid line observed, dashed computed) at various ports, and (b) currents (cm s~1) (light line observed, heavy computed) at various current meters.

224

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

225

Fig. 10. (Continued.)

b Fig. 10. Time series of the total (internal and external) storm surge (Calcn 3, Table 1) computed using the fine grid eastern Irish Sea model, (a) elevations (m) (solid line observed, dashed computed) at various ports, and (b) currents (cm s~1) (light line observed, heavy computed) at various current meters.

226

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

external and internal surges to produce the total surge was however successful in deep water. It is evident that at Heysham and Hilbre there is a significant phase error in the time of the surge maximum. As stated previously the instruments at these sites subsequently failed, and prior to this there may have been recording errors in time producing a phase shift. The close proximity of Hilbre and Liverpool, and the fact that the time of the surge was reproduced at Liverpool in part supports this, and emphasises the difficulty of obtaining accurate measurements during extreme events. Comparing the computed surge time series at Liverpool in Fig. 10a, with that determined previously (Davies and Jones, 1992) using the coarser grid west coast model (Fig. 1), it is evident that the finer grid of the present model has significantly increased the magnitude of the surge peaks, leading to an improvement in accuracy of the surge maximum although there still appears at the end of the period to be a shift in datum between the observations and the model. The computed east component of the total surge current at positions (D) and (B) at all depths (Fig. 10b) are overestimated at 06 h 12 November, although the model reproduces the mean value, although not the oscillations in the maximum east-current at 18 h 14 November. Some evidence of the westerly outflow at about 06 h 12 November 1977 in the opposite direction to the wind forcing is evident in the currents at all depths. Since this outflow is pressure driven, although modified by local wind events, it appears as the westward current of 10 cm s~1 in the computed external surge (Fig. 8b). Peaks in the east component of the current at 12 h 12 November and 00 h 13 November, (Fig. 10b) appear to be associated with variations in the wind stress (Fig. 6) and the external surge (Fig. 8b). The easterly pressure-driven flow which occurs at about 18 h 14 November to 06 h 15 November is evident at locations D and B. The model appears to reproduce accurately the magnitude and duration of this flow. The fact that the model does not reproduce all the high-frequency variations in the observed currents may in part be due to the meteorological forcing used which is based on three hourly values. Also the external surge which is introduced across the open boundary of the model, and which clearly has a significant influence upon the current time series (Fig. 8b) was derived from the coarse grid model with three hourly meteorological forcing. These comparisons with observed surge elevations and currents suggest that the surge elevation can be improved by refining the grid, to give better resolution in the near-coastal region. This improvement must be partly due to the fact that the model can now adequately resolve the main channels particularly those leading to Liverpool which are of the order of 1 km across. It is unclear whether further grid refinement would lead to further improvement as the model response may now be limited by the accuracy of the inputs such as boundary surge elevations or wind stresses. To the authors’ knowledge most storm surge comparisons have not yet examined the effect of successive grid refinements. Also with each grid refinement there may need to be an appropriate improvement in the resolution of the wind and wave inputs, hence any improvement seen in the surge elevations could be related to these improved inputs

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

227

rather than grid refinement. In the present model, the same wind stresses are being used as in DJ92 as well as Heaps and Jones (1979) albeit interpolated to a finer grid. However in the case of currents in offshore locations, these may require a more detailed meteorological forcing in space and time.

5. Sensitivity to bottom friction parameterization and wave-current interaction effects In the previous series of calculations, the eastern Irish Sea model was used to examine the relative importance of external and internal surges upon surge elevations and currents in the Irish Sea. In this section, we consider the sensitivity of surge elevations to the parameterization of bottom friction in that model. In all calculations, the tidal and surge input to the open boundary condition and meteorological forcing were identical to those considered previously; however, we examine two forms of bottom friction. Since the majority of operational storm surge models are twodimensional (we initially consider the two-dimensional form of the model derived by vertically integrating the equations and expressing the bed stress in terms of the depth mean current [Eq. (3)] (Table 1, Calcn 4), with an appropriate two-dimensional friction coefficient k6 "0.0018 which accurately reproduced tidal elevations in the region. Subsequently, the model is again used in three-dimensional form but with the bottom friction taking account of wave—current interaction effects (Table 1, Calcn 5). 5.1. Total surge computed with a two-dimensional model (Calcn 4) Although a three-dimensional model is necessary to examine wind-induced currents in the majority of storm surge calculations the main emphasis is on changes in sea surface elevation and two-dimensional models are used. To examine if a similar improvement in surge prediction due to improvements in grid resolution is possible with a two-dimensional model, the previous calculation was repeated with a twodimensional high-resolution model, covering the eastern Irish Sea (Fig. 3). Comparing time series of elevations computed with the two-dimensional model (Fig. 11) with those determined previously (Fig. 10a), it is evident that there is no significant difference at Douglas, which is to be expected since the surge here is dominated by the external surge. Also no significant differences are apparent at Workington. However, at Heysham, the surge peaks at 00 h 12 November and 18 h 14 November computed with the two-dimensional model are the order of 20 cm less than those computed with the three-dimensional model due to difference in frictional damping in shallow water in the two models. Similar reductions in the surge peak and slight changes in the time variation of the surge peak are evident at Hilbre and Liverpool, with the two-dimensional model tending to reduce the magnitude of the surge peak at Liverpool and shift it slightly in time (cf. Figs. 11 and 10a). Although comparable accuracy in tidal elevations computed with two- and threedimensional models (Jones and Davies, 1996) can be obtained by using appropriate drag coefficients, this is more difficult in storm surge models. This is primarily because in near-coastal regions, there are significant differences in direction between surface

228

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

229

and bed surge currents. These differences often lead to smaller depth mean than bed currents and consequently differences in bed stress. For example, in the eastern Irish Sea, a wind from the west drives a surface flow towards the land, sea surface elevations increase giving a significant opposing near bed current with a small depth mean current (Davies and Lawrence, 1994b, 1995). A consequence of this is that when the bed stress is computed from the depth mean current with an appropriate drag coefficient, it is significantly smaller than that found in a three-dimensional model where the bed stress is computed from the bottom current. Since one of the major balances in near-shore regions is between elevation gradient and the difference between surface stress (specified by the wind stress) and bed stress, then differences in bed stress between two- and three-dimensional models can influence computed surge elevations (Davies and Jones, 1993). 5.2. Total surge computed with the three-dimensional model including wave—current interaction effects. In a final calculation (Calcn 5, Table 1) the earlier computation with the threedimensional model (Calcn 3, Table 1) was repeated but with the bottom friction coefficient in Eq. (4) no longer specified, but determined by the level of wave—current interaction, which was parameterized as described previously (Davies and Lawrence, 1994b, 1995), using the method of Signell et al. (1990). A significant wave height of 1.5 m with a period of 8.0 s and varying wave friction factor depending upon water depth (Davies and Lawrence, 1994a—c, 1995) was used in these calculations. These values were primarily chosen to be consistent with Davies and Lawrence (1994a—c, 1995), who examined the influence of wave—current interaction with uniform winds upon steady-state solutions. Here we wish to examine their effect in a time evolving situation. Also these parameters are appropriate for the significant storm conditions considered here (Carter, 1982; Draper, 1992) and were used throughout the period. Since the wave—current interaction formulation used in the model can deal with an arbitrary spatial and time variation in the wave field, it would be possible to include an arbitrary wave-distribution, which could be adjusted within limits corresponding to our lack of knowledge of the wave fields at the time, to ensure an optimal correspondence between observed and computed surge elevation. However, since our prime aim here is to examine the influence of wave effects, a fixed wave height and period was used in order to clearly illustrate their influence. Time series of surge elevation computed taking account of changes in the bottom friction coefficient due to wave—current interaction (Fig. 12), at Douglas, are not significantly different from those found previously (cf. Fig. 12 with Fig. 10a). This is to be expected since at this location the surge is primarily produced by the external surge entering the eastern Irish Sea. Also at this position the water is deep (of order b Fig. 11. Time series of the total (internal and external) storm surge elevations (m) (solid line observed, dashed computed) at various ports, computed using the two-dimensional fine grid eastern Irish Sea model (Calcn 4, Table 1).

230

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

231

40—50 m) and hence wave—current interaction effects at the bed will be small (Davies and Lawrence, 1994a—c). At Workington there are some slight differences, particularly in the surge peaks on 14 November, where the amplitude of the peaks is increased (cf. Figs. 12 and 10a). Similarly, at Heysham, the surge peak at about 00 h 12 November, is increased improving the agreement with observations with an increase in surge peaks on the 14 November, and a similar response at Hilbre. At Liverpool the surge peak at about 00 h 12 November increases by the order of 30 cm when wave—current interaction effects are included leading to an improvement in the accuracy of the surge, with a similar change and improvement on the 14 November. The increased amplitude of the oscillations produced when wave—current interaction effects are included is due to the modification of the bottom friction coefficient which changes the non-linear interaction of tide and surge due to frictional effects. Also as shown by Davies and Lawrence (1994b, c, 1995) increased frictional effects due to the wind-induced current and wave—current interaction can modify the tide. Consequently, when the residual is computed by subtracting a tide-only solution from the tide and surge solution, some tidal signal remains in the residual together with the effects of the non-linear interaction (Davies and Lawrence, 1994b, c, 1995). As discussed previously the extent to which the model can reproduce these is a more sensitive test of the model than its ability to reproduce total elevations. Time series of currents at rigs B and D computed with wave—current interaction effects included were not significantly different from those found previously. This is to be expected since these moorings are in deep water (of order 40 m) where wave—current interaction is small, and hence is unlikely to have any effect. However, as shown by Davies and Lawrence (1994b, c, 1995), in shallow water wave—current interaction can have a significant influence upon current profiles. The improvement in computed surge elevations at Liverpool can be readily appreciated from the comparison (Fig. 13) of elevations computed using a coarse grid model (Heaps and Jones, 1979) where the intensity of the surge peak was omitted, to that computed with the present model which shows an intense surge peak, the magnitude of which is increased when wave—current interaction effects are included. To finalize the comparison of surge elevations we briefly consider root-meansquare errors determined over the whole period from calculations with the highresolution eastern Irish Sea model. As discussed previously, a number of gauges failed and r.m.s. errors are available only from Douglas, Workington and Liverpool over the whole period and are presented in Table 2, for the two- and three-dimensional calculations, and that including wave—current interaction. From the Table it is evident that the r.m.s. error is reduced particularly at Liverpool when the three-dimensional model is used. However, including wave—current interaction effects leads to an increase in r.m.s. error. This increase particularly at Liverpool is to a certain extent b Fig. 12. Time series of the total (internal and external) storm surge elevations (m) (solid line observed, dashed computed) at various ports, computed with the three-dimensional fine grid eastern Irish Sea model (Calcn 5, Table 1) taking into account wave—current interaction.

232

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

Fig. 13. Time series of storm surge elevation at Liverpool computed by (a) Heaps and Jones (1979), (b) Davies and Jones (1992), (c) using the fine grid eastern Irish Sea model, and (d) as (c) but including wave—current interaction, (solid line observed, dashed computed).

misleading in that in order to examine the effects of the waves, a fixed wave field appropriate to the time of maximum wind was maintained over the whole period. A consequence of this is that wave effects at times of low to moderate winds were exaggerated [cf. computed curves (c) and (d) in Fig. 13] and this will have artificially increased the r.m.s. error. It is however clear from Fig. 13 [curves (c) and (d)] that the storm surge peak is increased when wave—current interaction effects are included leading to an improvement in accuracy of the surge peak. In the previous series of calculations (Calcns 1—4), we have examined time series of elevations and currents at only a limited number of locations. However, in order to

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

233

Table 2 Root mean square (r.m.s.) errors for the period determined with the two dimensional (2D), three dimensional (3D), and three dimensional model including wave—current interaction (3D#W/C) Port

Douglas Workington Liverpool

Computed r.m.s. error (m) 2D

3D

3D#W/C

0.118 0.349 0.287

0.115 0.344 0.245

0.108 0.351 0.257

understand the dynamics of the wind-induced flow field in the eastern Irish Sea during a major storm event it is important to examine the spatial variability of elevations and currents. Spatial distributions of these can be compared with the previous coarse grid results of Davies and Jones (1992), and the influence of enhanced horizontal resolution upon current variability can be determined.

6. Overview of space–time variability for total surge with wave–current interaction 6.1. Surface elevation, surface and bed current distributions In order to understand in detail the time-varying response of the eastern Irish Sea under wind conditions, it is necessary to examine contours of surface elevation (Fig. 14a—e), and surface and bed currents (Fig. 15a—e), together with the associated meteorological forcing, computed with the high-resolution eastern Irish Sea model. During the first major storm event (Fig. 4), there were strong southwesterly winds (the convention here for winds, is that used in meteorology namely the direction from which the wind comes), that caused sea level to rise in the Eastern Irish Sea (Fig. 14a). Comparing sea-surface elevation contours with those computed previously (Davies and Jones, 1992, hereafter referred to as DJ92) it is evident that in the near shore region the elevations are slightly (of order 5 to 10 cm) higher in the present model than in DJ92, with a very rapid increase in regions such as the Solway, Morecambe Bay and Liverpool Bay which could not previously (DJ92) be resolved. Obviously, although the resolution of the present model is much finer in these regions than that used in DJ92, it cannot resolve all the features, and to do so would require a very fine grid. Flather and Hubbert (1990) using a limited area model of Morecambe Bay found that in many regions a grid resolution of 1 km was insufficient, and a grid of 100 m still could not resolve some features in near coastal regions, where ‘wetting and drying’ occurs. Surface currents (at every third grid point of the model), Fig. 15a(i) in offshore regions, show a nearly uniform flow to the northeast with currents in some areas exceeding 50 cm s~1, with a patch of currents of reduced magnitude (less than 10 cm s~1) to the north of the Isle of Man. Significantly larger spatial variability is

234

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

235

Fig. 14. (Continued.)

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

b Fig. 14. Storm surge elevation contours at (a) 18 h 11/Nov, (b) 00 h 12/Nov, (c) 06 h 12/Nov (d) 12 h 14/Nov, (e) 18 h 14/Nov.

236

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

Fig. 14. (Continued.)

Fig. 15. As Fig. 14, but for (i) surface and (ii) bed, storm surge currents.

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251 237

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

Fig. 15. (Continued.)

238

239

Fig. 15. (Continued.)

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

Fig. 15. (Continued.)

240

241

Fig. 15. (Continued.)

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

242

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

evident here, particularly in near coastal regions and within the Solway Estuary, Morecambe Bay and Liverpool Bay than found previously (DJ92). A near-coast easterly flow is apparent along the Welsh coast with a northward flow in the shallow water regions of Liverpool Bay. When examining the spatial variability of currents (in near-shore regions, e.g. Solway Estuary, Morecambe Bay) it is important to note that they have been plotted at only every third grid point of the model, and consequently circulation features which appear here showing currents changing significantly from one grid point to another are in fact more adequately resolved in the model than it appears here. However, since the model contains the non-linear terms which move energy from the large-scale motion to the shortest scale resolvable in the model, and these terms are particularly important in shallow water, then grid scale features will inevitably be present in such regions however fine the model grid. If the model contains artificial numerical smoothing or explicit horizontal eddy viscosity, then these grid scale features can be removed. In the present series of calculations the horizontal viscous terms were omitted in order to enable the detailed features of the flow to be examined. However, a significantly smoother flow field which could be resolved on the grid would have been obtained by using the Smagorinsky (1963) form of the horizontal viscosity which increases horizontal viscosity in high shear regions. We now examine the spatial variability of currents at the sea bed, taken as the height above the rigid bed (where the current is zero) at which the slip condition [equation (4)] has been applied. This reference height is typically 100 cm above the rigid bed. Currents at the sea bed [Fig. 15a(ii)] are reduced by frictional effects from those found at the surface, with typical values the order of 10 cm s~1. Over the majority of the region the bed currents flow westward, although in the region off the northeast coast of the Isle of Man some spatial variability of the bottom current is evident associated with small-scale variations in bottom topography in this region (Fig. 2). Similarly, significant spatial variability in bottom currents (on a scale that could not be resolved by DJ92) is evident in the near-shore regions. The predominantly offshore flows shown in Fig. 15a(ii) are mainly driven by the pressure gradients, associated with the rise of sea level along the coast (Fig. 14a). Although the major features of the flow field could be reproduced by the coarse grid of DJ92, this model could not reproduce the small-scale spatial variability [Fig. 15a(ii)] found in the bed currents computed with the finer grid model. At 00 h 12 November 1977 a predominantly westerly wind was blowing over the eastern Irish Sea, producing the east—west surface elevation gradients given in Fig. 14b. Comparing Fig. 14b with the corresponding figure in DJ92, it is evident that although DJ92, gave the major features of the elevation gradient, the coarser grid of that model could not reproduce the detailed variations shown in Fig. 14b, particularly local increases in surface elevation in the near coastal region. Surface currents [Fig. 15b(i)] show a uniform eastward flow on shore, with regions where the current exceeds 50 cm s~1, with a more complex flow pattern in shallow water regions such as the Solway, details of which could not be resolved in DJ92, but are evident here. Currents at the bed [Fig. 15b(ii)], show significantly greater spatial variability than that found by DJ92. In particular, the coarser grid model of DJ92 did not resolve the

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

243

direct wind-driven eastward flow along the Welsh coast, and the complex circulation pattern in Liverpool Bay. Also the significant spatial variability in the bottom currents to the northeast of the Isle of Man, and the inflow into Morecambe Bay was not resolved in DJ92, which in essence gave the impression of a westward, pressuredriven flow at the sea bed, with none of the spatial variability shown here. By 06 h 12 November the wind’s magnitude over the eastern Irish Sea had decreased, with a corresponding reduction in sea level (Fig. 14c). Although the sea—level contours (Fig. 14c) show similar features to those given in DJ92, their spatial variability is much larger, with surface currents [Fig. 15c(i)] showing more variability than those computed in DJ92. In particular, DJ92 could not reproduce the details of the wind-driven flow along the Welsh coast and in Liverpool Bay and Morecambe Bay which are evident here. Currents at the sea bed [Fig. 15c(ii)] at this time, of weak winds over the Irish Sea, show a major westward pressure-driven flow out of the eastern Irish Sea, as sea levels decrease. A nearly uniform flow is evident in the high—resolution model [Fig. 15c(ii)], the main features of which except in the Solway and Morecambe Bay were reproduced in DJ92. This suggests that the coarser grid model of DJ92 could reproduce pressure-driven bottom currents, but not the detail of the wind-driven flow. The depression and associated winds which gave rise to the first surge event moved away during 12 November 1977, to be replaced at 12 h 14 November 1977 by a system of strong northwesterly winds (Fig. 5) blowing over the eastern Irish Sea. Surface currents induced by these winds, exhibit a nearly uniform wind-driven south-eastward flow over the eastern Irish Sea [Fig. 15d(i)], causing sea levels to rise to the order of 0.3 m in the Liverpool Bay region (Fig. 14d). The major features (although not the finer detail) found in the spatial distribution of surface currents and elevation contours away from near shore regions are not significantly different from those found by DJ92. Bottom currents [Fig. 15d(ii)] exhibit a westward pressure-driven flow out of Liverpool Bay, with outflows from the Solway Firth and Morecambe Bay. As the wind stress over the region increased, sea levels rose and by 18 h 14 November 1977 sea levels had exceeded 0.6 m in Liverpool Bay (Fig. 14e) with very rapid increases in near-shore regions such as the Solway, Morecambe Bay and in the vicinity of Liverpool. Although the coarse grid model of DJ92 could reproduce the major features shown in Fig. 14e, the local near-shore intensification was not resolved. Surface currents at this time [Fig. 15e(i)] show a region of strong (up to 50 cm s~1) eastward currents to the east of the Isle of Man, with weaker (of order 10 cm s~1) currents to the north and south of this region. These main features of the flow were reproduced by the coarser grid model used in DJ92. However, the model of DJ92 failed to show the strong flow to the east along the north coast of Wales, or the details of the circulation patterns in the Solway Firth or Liverpool Bay, due to the lack of grid resolution in these regions. Surface and bottom currents [Fig. 15e(i)—(ii)] show that within the Solway Firth, a shallow water region (depths below 20 m) the wind-driven current extends from sea surface to sea bed, with water at this time entering the estuary at all depths causing sea-surface elevations to rise. As previously, the major features of the flow field shown in Fig. 15e(ii), are comparable to these given in DJ92, although the detailed spatial

244

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

variability of the flow shown here, particularly in the near-shore region could not be reproduced in DJ92. The set of elevation contours and current vectors shown here, together with the corresponding meteorological situations, shows a complex time and spatially evolving flow field, produced by wind events over the Irish Sea and the effect of forcing produced by flows entering the region to the north and south of the Isle of Man. Changes in sea-surface elevation gradients in response to the magnitude and alignment of the wind stress with the coast have a significant influence on bottom currents. The comparison with the earlier coarse grid results of DJ92, shows that although the earlier model could reproduce the major features of the flow, a significant amount of the details of the spatial variability of the currents, was omitted. Naturally, if a more detailed bottom topographic data set were available, then a more refined finitedifference grid would be justified, and more detailed horizontal variability in the currents, particularly the bed currents would emerge. Also from the current comparisons shown earlier, besides improved accuracy in the hydrodynamic model, it is probably necessary to have a more detailed description of the spatial variability and the higher-frequency time variability of the wind, before detailed comparisons can be made with currents. The fact that a significant proportion of the time variability of the currents in the eastern Irish Sea is generated outside the region means that it may be necessary to consider the high-frequency variability and short space scale variations not only on the scale of the Irish Sea, where there are a number of meteorological stations, but possibly over a larger area with the associated difficulties of obtaining accurate meteorological data. 6.2. Maximum eddy viscosity and bottom friction coefficient distributions Since the eddy viscosity is a function of the flow field, it varies with horizontal position and time, and in order to examine the change in its magnitude due to the wind-induced flow, it is useful to plot contours of the instantaneous maximum viscosity that occurred over the period 7—17 November due to the tide alone (Fig. 16a) and the tide and meteorologically induced current (Fig. 16b). Contours of maximum tidally induced viscosity (Fig. 16a) show viscosity values of less than 0.02 m2 s~1 in the near-coastal region where water depths are shallow, increasing to values of the order of 0.14 m2 s~1 to the north and south of the Isle of Man, where water depths are about 50 m and tidal currents are strong [of order 1 m s~1; Aldridge and Davies (1993)]. A rapid increase in eddy viscosity is evident in Morecambe Bay associated with an isolated deep water region, namely the Lune Deep (water depths exceeding 50 m). Contours of maximum eddy viscosity computed with tidal and wind forcing (Fig. 16b) show a significant increase in viscosity magnitude in shallow near-coastal areas, with the region in which viscosity values are below 0.02 m2 s~1 confined to a near—coastal band, due to the increase in current magnitude produced by the wind—induced flow in the near shore region. Maximum viscosity values to the north and south of the Isle of Man reach values of the order of 0.2 m2 s~1 due to the increased current in these areas produced by the wind-induced flow as presented previously (Fig. 15). Contours of maximum bed stress due to the tide alone (Fig. 17a),

Fig. 16. (a) Contours of maximum eddy viscosity (m2 s~1) due to the tide, (b) Contours of maximum eddy viscosity (m2 s~1) due to the tide and wind driven flow.

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251 245

246

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

Fig. 17. (a) Contours of maximum bed stress (N m~2) due to the tidal currents, (b) Contours of maximum bed stress (N m~2) due to tidal and wind induced currents, (c) Contours of maximum bed stress (N m~2) due to tidal and wind induced currents, allowing for an increased friction coefficient due to wave—current interaction, (d) Contours of maximum friction coefficient (k]102) due to wave—current interaction.

247

Fig. 17. (Continued.)

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

248

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

show regions to the north and south of the Isle of Man where the maximum bed stress due to the tidal current exceeds 2 N m~2. To the east of the Isle of Man off the west coast of England there is a region of weak semi-diurnal tide (see Aldridge and Davies, 1993 for details) where the bed stress is less than 0.75 N m~2, with the bed stress increasing rapidly in the shallow coastal estuaries where tidal currents exceed 1 m s~1. Instantaneous maximum bed stress contours during the surge of November 1977 (Fig. 17b), show that in the regions to the north and south of the Isle of Man the maximum bed stress is increased from over 2 N m~2 (the tidal bed stress), to over 4 N m~2, due to the addition of the wind-induced current, with the areas where the maximum bed stress exceeds 2 N m~2 covering the regions to the northeast and southeast of the Isle of Man. Maximum bed stresses in the area to the east of the Isle of Man which previously were the order of 0.5 N m~2, are increased to 1.25 N m~2 due to the additional stress produced by the wind-induced bed currents in this area (Fig. 15). Calculations of the maximum bed stress allowing for increases in the bottom drag coefficient k due to wave—current interaction, show similar patterns in off-shore regions to those found previously (cf. Fig. 17c with 17b). However, in shallow water areas where wave—current interaction lead to an increase in the friction coefficient the maximum bed stress is increased (Fig. 17c). Increases in the friction coefficient k (which in the absence of wave—current interaction has a value of k"0.01125"1.125]10~2) can be readily appreciated from Fig. 17d. This figure shows the maximum value that k reached during the surge period, due to enhancements produced by wave—current interaction. In deep water regions to the northeast and southwest of the Isle of Man wave current interaction is small and k is below 1.5]10~2 (Fig. 17d). However, in shallow near-coastal regions where wave—current interaction is significant, k exceeds 2]10~2 (Fig. 17d) and rapidly increases to values of 5]10~2 in very near coastal areas (Fig. 17d). This increase in k accounts for the increase in bed stress computed from the bed currents, when wave—current interaction is included. Contours of maximum eddy viscosity and bed stress show that there is significant spatial variability in these parameters in the region, particularly in the near coastal area, which can only be accurately resolved with a fine grid model. The increase in the bed stress in near-coastal regions due to wave—current interaction is particularly important in problems concerned with sediment transport where an accurate knowledge of the maximum instantaneous bed stress is required. 7. Concluding remarks The high-resolution, bottom boundary layer resolving model of the eastern Irish Sea previously used by Jones and Davies (1996) to simulate the tides in the region, has been applied to a study of the storm surge of November 1977. A brief description (with references to the literature for detail) of the model has been given in the first part of the paper together with its extension to take account of storm-driven currents and modifications of the bottom friction coefficient due to wave—current interaction. A comparison of observed and computed surge elevations at ports in the eastern Irish Sea, has shown that at Douglas the November 1977 storm surge elevation is

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

249

mainly determined by the flow into the eastern Irish Sea with wind fields over the eastern Irish Sea having a minor contribution. The accurate prediction of the surge at Douglas shows that the larger area coarser grid west coast model can determine the surge at this location. The comparison of the observed and computed surge at Liverpool, shows that both the external surge entering the eastern Irish Sea and the internal surge generated by winds over the region are of equal importance in determining the surge at this location. From the comparison of the computed surge elevation at Liverpool determined previously by Heaps and Jones (1979) using a two-dimensional coarse grid (of order 12 km) model and the three-dimensional intermediate grid (of order 7 km) of Davies and Jones (1992), with the present high resolution (of order 1 km), it is evident that such a high-resolution grid is required to improve the accuracy of predicted storm surges at Liverpool. Also storm surge predictions of a comparable accuracy, although with some slight differences in the time series of elevations, can be made with a two-dimensional model using the same high-resolution grid, provided the bottom friction coefficient is adjusted to reflect the different formulation of bed stress. From the comparison of observed and computed storm surge currents, it does not appear that the model can reproduce the high-frequency fluctuations in the currents. This may be due to the fact that the wind stress used to drive the model is based on three hourly winds. Calculations to investigate this problem associated with current prediction are presently in progress. Although the general features of the spatial distribution of elevations and currents computed with the fine grid model are comparable with those computed using the coarser grid model of Davies and Jones (1992), significantly greater spatial variability is evident in the near-bed currents, suggesting that fine grid models are particularly important when comparing observed and computed currents, besides the accurate determination of elevations at ports such as Liverpool. Taking into account the wave—current interaction in the three-dimensional model improves the surge prediction and could also be incorporated in a two-dimensional model with the appropriate parameterization. Wave—current interaction effects are important in the near-shore region, and will depend upon bed types. This suggests that besides a fine grid it will be necessary to include details of bed types in the near shore region together with an accurate prediction of waves and meteorological forcing.

Acknowledgements The authors are indebted to R.A. Smith for preparing the diagrams and to L. Ravera and L. Parry for typing the text.

References Aldridge, J.N., Davies, A.M., 1993. A high resolution three-dimensional hydrodynamic tidal model of the Eastern Irish Sea. Journal of Physical Oceanography 23, 207—224.

250

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

Bowden, K.F., 1978. Physical problems of benthic boundary layer. Geophysical Surveys 3, 255—296. Carter, D.J.T., 1982. Prediction of wave height and period for a constant wind velocity using the Jonswap results. Ocean Engineering 9, 17—33. Davies, A.M., 1981. Three-dimensional hydrodynamic models, 1. A homogeneous ocean-shelf model; 2, A stratified model of the northern North Sea. In: Saetre, R., and Mork, M., (Eds.), The Norwegian Coastal Current, vol. 2, Bergen University, pp. 370—426. Davies, A.G., 1986. A numerical model of the wave boundary layer. Continental Shelf Research 6, 715—739. Davies, A.G., Soulsby, R.L., King, H.L., 1988. A numerical model of the combined wave and current bottom boundary layer. Journal of Geophysical Research 93, 491—508. Davies, A.G., 1990. A model of the vertical structure of the wave and current bottom boundary layer. In: Davies, A.M., (Ed.), Modeling Marine Systems, vol. 2, CRC Press, Boca Raton, FL, pp. 263—297. Davies, A.G., 1991. Transient effects in wave—current boundary layer flow. Ocean Engineering 18, 75—100. Davies, A.M., 1986. A three-dimensional model of the northwest European continental shelf with application to the M tide. Journal of Physical Oceanography 16, 797—813. 4 Davies, A.M., 1987. Spectral models in continental shelf sea oceanography. In: Heaps, N.S., (Ed.), Threedimensional Coastal Ocean Models, American Geophysical Union, Washington, DC (Coastal and Estuarine Sciences, No.4), pp. 71—106. Davies, A.M., 1990. On the importance of time varying eddy viscosity in generating higher tidal harmonics. Journal of Geophysical Research 95, 20 287—20 312. Davies, A.M., 1993. A bottom boundary layer-resolving three-dimensional tidal model: a sensitivity study of eddy viscosity formulation. Journal Physical Oceanography 23, 1437—1453. Davies, A.M., Jones, J.E., 1992. A three-dimensional wind driven circulation model of the Celtic and Irish Seas. Continental Shelf Research 12, 159—188. Davies, A.M., Jones, J.E., 1993. On improving the bed stress formulation in storm surge models. Journal of Geophysical Research 98, 7023—7038. Davies, A.M., Jones, J.E., 1996. The influence of wind and wind wave turbulence upon tidal currents: Taylor’s problem in three-dimensions with wind forcing. Continental Shelf Research 16, 25—99. Davies, A.M., Lawrence, J., 1994a. The response of the Irish Sea to boundary and wind forcing: results from a three-dimensional hydrodynamic model. Journal of Geophysical Research 99, 22 665—22 687. Davies, A.M., Lawrence, J., 1994b. Examining the influence of wind and wind-wave turbulence on tidal currents, using a three-dimensional hydrodynamic model including wave—current interaction. Journal of Physical Oceanography 24, 2441—2460. Davies, A.M., Lawrence, J., 1994c. Modelling the non-linear interaction of wind and tide: its influence on current profiles. International Journal for Numerical Methods in Fluids 18, 163—188. Davies, A.M., Lawrence, J., 1995. Modelling the effect of wave—current interaction on the three-dimensional wind driven circulation of the Eastern Irish Sea. Journal of Physical Oceanography 25, 29—45. Draper, L., 1992. Wave climate of the British Isles. OTH 89 303 (ISBN 011 4130 92 2), HMSO, London. Flather, R.A., Hubbert, K.P., 1990. Tide and surge models for shallow water — Morecambe Bay revisited. In: Davies, A.M., (Ed.), Modeling Marine Systems, vol 1, CRC Press, Boca Raton, FL, pp. 135—166. Glorioso, P.D., Davies, A.M., 1995. The influence of eddy viscosity formulation, bottom topography, and wind wave effects upon the circulation of a shallow bay. Journal of Physical Oceanography 25, 1243—1264. Grant, W.D., Madsen, O.S., 1979. Combined wave and current interaction with a rough bottom. Journal of Geophysical Research (Oceans) 84, 1797—1808. Green, M.O., Rees, J.M., Pearson, N.D., 1990. Evidence for the influence of wave—current interaction in a tidal boundary layer. Journal of Geophysical Research 95, 9629—9644. Heaps, N.S., 1965. Storm surges on a continental shelf. Philosophical Transactions of the Royal Society of London A257, 351—383. Heaps, N.S., Jones, J.E., 1975. Storm surge computations for the Irish Sea using a three-dimensional numerical model. Memoires de la Socie& te& Royale des Sciences Lie& ge 7, 289—333. Heaps, N.S., Jones, J.E., 1979. Recent storm surges in the Irish Sea. In: Nihoul, J.C.J. (Ed.), Marine Forecasting, Proceedings of the 10th International Liege Colloquium on Ocean Hydrodynamics, 1978, Elsevier, Amsterdam, pp. 285—319.

J.E. Jones, A.M. Davies/Continental Shelf Research 18 (1998) 201—251

251

Howarth, M.J., Jones, J.E., 1981. A comparison of numerical model and observed currents during a storm surge period. Estuarine Coastal and Shelf Science 12, 655—663. Huntley, D.A., Bowen, A.J., 1990. Modeling sand transport on continental shelves. In: A.M. Davies, (Ed.) Modeling Marine Systems, CRC Press, Boca Raton, FL., pp. 221—254. Jones, J.E., Davies, A.M., 1996. A high resolution three-dimensional model of the M , M , M , S , N , K , 2 4 6 2 2 1 and O tides in the eastern Irish Sea. Estuarine Coastal and Shelf Science 42, 311—346. 1 Lennon, G.W., 1963. The identification of weather conditions associated with the generation of major storm surges on the west coast of the British Isles. Quarterly Journal of the Royal Meteorological Society 89, 381—394. Signell, R.P., Beardsley, R.C., Graber, H.C., Capotondi, A., 1990. Effect of wave—current interaction on wind-driven circulation in narrow, shallow embayments. Journal of Geophysical Research 95, 9671—9678. Spaulding, M.L., Isaji, T., 1987. Three-dimensional continental shelf hydrodynamic model including wave current interaction. pp 405—426 In: Nihoul, J.C.J., Jamart, B.M., (Eds.), Three-dimensional Models of Marine and Estuarine Dynamics, Elsevier, Amsterdam, 629pp. Smagorinsky, J., 1963. General circulation experiments with the primitive equations, I. The basic experiment. Monthly Weather Review 91, 99—164. Smith, S.D., Banke, E.G., 1975. Variation of the sea surface drag coefficient with wind speed. Quarterly Journal of the Royal Meteorological Society 101, 665—673. Soulsby, R.L., Hamm, L., Klopman, G., Myrhaug, D., Simons, R.R., Thomas, G.P., 1993. Wave—current interaction within and outside the bottom boundary layer. Coastal Engineering 21, 41—69.