Comparison of Observed (HF Radar and ADCP Measurements) and Computed Tides in the North Channel of the Irish Sea

1764 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31

ALAN M. DAVIES,PHILIP HALL,M.JOHN HOWARTH,PHILIP J. KNIGHT, AND ROSE J. PLAYER Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead, Merseyside, United Kingdom

(Manuscript received 8 March 1999, in ®nal form 12 September 2000)

ABSTRACT A three-dimensional high-resolution (grid of order 1 km) model of the North Channel of the Irish Sea, incorporating a one equation turbulence energy submodel to parameterize vertical mixing, is used to compute

the M2, S2, N2, K1, and O1 tides. Elevations and currents are compared with observations, with particular emphasis on a detailed comparison with current pro®les recorded by two acoustic Doppler current pro®lers (ADCPs) and HF radar measurements of the surface current. The comparison with the HF radar shows small-scale spatial variations in both modeled and observed currents superimposed upon the larger scale tidal currents. These small-scale changes appear to be associated with variations in bottom topography and can only be resolved by using ®ne-grid models or detailed measurements.

Computed M2 current pro®les derived from a multiconstituent calculation are in excellent agreement with

pro®les measured by the ADCPs. However, in a single constituent M2 calculation, the magnitude of the current is overpredicted. Increasing the bottom friction coef®cient to compensate for the absence of other constituents improves the accuracy of the bottom current, although the thickness of the turbulent boundary layer is underpredicted with a resulting overprediction of the surface current that cannot be corrected for by changing the bottom friction coef®cient.

1. Introduction surements exist in the region, although they are outside The North Channel connects the Irish Sea to the Heb- the area of the HF radar, and they have been used to rides Malin Shelf off the west coast of Scotland and is validate the model. an area of signi®cant exchange between the two regions The HF radar system deployed here is the Ocean Sur- during major storm events. Water depths in the area are face Current Radar (OSCR) details of which are given typically of the order of 100 m increasing to over 250 in Prandle et al. (1993) and Prandle (1991). Surface m in a deep area between Portpatrick and Orlock Point current vectors can be measured to a theoretical accuracy of about 4 cm sϪ1, and bins of the order of 1.2- (Fig. 1). Tidal currents are strong, with M2 velocities of order 100 cm sϪ1, and hence the area remains well km radial length. The range of the system is such that mixed. The variation in water depth gives rise to a sig- it gives an adequate coverage over the North Channel, ni®cant spatial variability in the magnitude of the tidal with a resolution comparable with the 1-km model grid. currents. To examine the variability of the tidal current The OSCR system was deployed for 418 days from involves a high-resolution three-dimensional (3D) nu- Portpatrick (the master site) and Crammag Head (the merical model (described here) and a detailed set of slave site) (Fig. 2). A 150-KHz broadband ADCP and tidal measurements, made using an HF radar system a pressure recorder were deployed in a seabed frame in deployed for 14 months with the location of the master the center of the channel (at 54Њ46ЈN, 5Њ25ЈW, water and slave as shown in Fig. 2. By this means the radar depth 142 m) (position A2) from 13 July 1993 to 28 could cover the full width of the North Channel at ap- October 1994. Currents were recorded in the vertical in proximately 54Њ48ЈN. An acoustic Doppler current pro- 23 5-m bins, from 12.5 m to 122.5 m above the seabed. ®ler (ADCP) was deployed to the north of the region The nominal accuracy of the current measurements ob- covered by the HF radar for 1 month (position A1 in tained from the ADCP was 0.75 cm sϪ1 (Howarth et al. Fig. 2), and a second for 15 months in the center of the 1995). A 150-KHz narrowband ADCP was deployed channel (position A2 in Fig. 2). Other tidal current mea- for 1 month (20 September±31 October 1993) at A1 (55Њ00ЈN, 05Њ30ЈW in 139 m of water) currents were recorded in thirteen 8-m bins from 14 to 110 m above Corresponding author address: Dr. Alan M. Davies, Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead L43 the bed. 7RA, United Kingdom. The three-dimensional model is based on Davies and E-mail: [email protected] Hall (1998, hereafter referred to as DH), which was used

᭧ 2001 American Meteorological Society

Unauthenticated | Downloaded 09/29/21 09:10 AM UTC JULY 2001 DAVIES ET AL. 1765 tide gauge measurements. Also shown is the location of ϩ . 2. Location of measurement sites, with x denoting current, and IG the HF radar Master and Slave, and the cross section X±Y. ADCPs are denoted A1 and A2. . 1. Bottom topography of the region and place names. F IG F

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FIG. 3. Finite difference grid of the model.

to compute the M 2 and M4 tides in the region, extended The objective of this paper is to use the HF radar to include the S 2, N 2, K1, and O1 tides and for com- measurements together with the detailed current pro®le parison with the HF radar and ADCP measurements. derived from the bottom-mounted ADCPs and any other Although these tidal constituents have been examined current measurements in the region to determine the over the whole shelf (Davies et al. 1997), the grid res- accuracy of the three-dimensional tidal ¯ow ®eld de- olution of that model (of order 12 km) was insuf®cient rived from the model, and to use the model and mea- to resolve the North Channel. Recent tidal calculations surements to examine the detailed spatial variability of in the eastern Irish Sea (Jones and Davies 1996) have the tidal currents in the area. To date, there have been shown that a 1-km grid is required to examine spatial no detailed comparisons of high-resolution three-di- variations in tidal currents (although elevations show mensional model computations against HF radar mea- signi®cantly less variability), suggesting that a model surements and ADCP current pro®les. These compari- with this resolution is required in the North Channel. sons show signi®cant spatial variability related to chang- Although such a model, due to limitations in computing, es in bottom topography and the limitation of coarse will be restricted in its geographical extent (Fig. 3), the grid models and comparison with point measurements. open boundaries are signi®cantly well removed from Subsequently calculations are performed with only the the area covered by the HF radar. Also, the form of the M2 tidal input but increased bottom friction to try to open boundary condition is such that the tidal current compensate for the frictional effects of other constitu- is uniform from sea surface to seabed and hence current ents. These calculations show that this approach is very structure, which is the most signi®cant test of the model limited and that the other constituents must be included is determined by the physics within it. in order to have the correct turbulence and hence vis-

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TABLE 1. Summary of parameters used in the calculations. cosity in the water column to reproduce the M 2 current pro®le. Calc Constituents k value In the next section we brie¯y outline the basics of 1 All 0.01 the model and the turbulence energy submodel used to 2 All 0.005 compute the coef®cient of momentum transfer (eddy 3 M2 only 0.01 viscosity). In subsequent sections comparisons with the 4 M2 only 0.005 data are presented.

2. The hydrodynamic model coef®cients, but only including the M2 tide in order to determine the in¯uence of frictional effects due to the Since details of the model are given in DH, they will other tidal constituents upon the M 2 tide. not be presented here. A radiation condition (Davies Besides comparing the amplitude and phase of the u 1986) involving only the depth mean current (hence no and ␷ components of velocity it is also useful to compare pro®le of the current was speci®ed) was applied at open their rotary components, namely with Rϩ a velocity vec- boundaries. On closed boundaries the normal compo- tor that rotates anticlockwise when viewed from above, nent of ¯ow was zero, and in shallow water a drying and RϪ the clockwise component. The phase of the an- condition was applied. At the sea surface the stress was ticlockwise and clockwise are denoted by ␾ϩ and ␾Ϫ zero, and a quadratic friction law was applied at the (Soulsby 1983). seabed. The boundary layer thickness of the anticlockwise Discretization in the vertical involved 25 irregularly (␦ϩ) and clockwise (␦Ϫ) components can also be com- spaced levels on a sigma coordinate, with enhanced res- puted, using (Soulsby 1983) olution in the near-bed region. A uniform ®nite differ- CU* CU* ence grid with a resolution of 1 km was used in the ␦ ϭ and ␦ ϭ (1) horizontal (Fig. 3). A time splitting approach was used ϩϪ␻ ϩ f ␻ Ϫ f to integrate the equations (see DH for details). with ␻ the tidal and f the Coriolis frequency, U the Although the calculation of tidal current pro®les using * bed friction velocity, and C a coef®cient. Based on ob- simple ¯ow-dependent eddy viscosity parameterizations served values of boundary layer thickness, Soulsby has proved very successful (Davies and Gerritsen 1994; (1983) suggests a value of C of order 0.1. Davies and Jones 1990; Davies et al. 1996, 1997), here we use a turbulence energy submodel so that we can examine the accuracy of the computed pro®le and its a. Semidiurnal tides (M 2, S 2, and N2) sensitivity to bottom frictional effects at the ADCP sites 1) SPATIAL TIDAL DISTRIBUTIONS (MOORINGS) where a detailed current pro®le is available in the vertical. Unfortunately no turbulence energy dissipation OVER THE WHOLE REGION rate measurements are available in this region. The tur- The computed M 2 cotidal chart (Calc 1) shows an bulence energy model adopted here has been shown amphidromic point between the Kintyre peninsula and (Xing and Davies 1995, 1996) to be as accurate as two the island of Islay, situated to the northeast of Ireland, equation models (Oey and Chen 1992; Luyten et al. with tidal elevation amplitudes increasing to the south 1996; Blumberg and Mellor 1987) at a fraction of the of this (Fig. 4a). A similar cotidal chart (not shown) is computational cost. The form of the model and appro- found for the S 2 and N2 tides, with S 2 tidal amplitudes priate boundary conditions are given in DH and are not about 3.5 times smaller than M2 and N 2 about 4 times repeated here. smaller. The distribution of the coamplitude and cophase

lines found for the M2, S 2, and N2 tides in both calculations 1 and 2, away from the amphidromic point, 3. Computed tidal distributions is in good agreement with other cotidal charts of the The tidal regime was determined by starting the mod- region based on model results (Davies and Jones 1992) el from an initial condition of zero elevation and motion and observations (George 1980; Howarth and Pugh with open boundary forcing at the M2, S 2, N 2, K1, and 1983; Howarth 1990). As found by DH, the exact lo- O1 frequencies. These constituents were chosen because cation of the amphidromic point is very sensitive to they are the major tides in the region and can be ac- small changes in the open boundary input to the model. curately separated from each other using 15 days in the Although the general features of the M 2 cotidal chart case of the M2 and S 2 tides and a month when the N 2 are in reasonable agreement with observations, a de- tide is included. In an initial calculation (Calc 1, Table tailed comparison (not presented) with measurements in 1), the bottom friction coef®cient k was set at 0.01, and the area (Fig. 2) shows that at positions A, D, E, G, and subsequently (Calc 2) k was reduced to 0.005, in order H in the vicinity of the amphidromic point there are to determine the in¯uence of the coef®cient of bottom some errors in the model in both calculations 1 and 2. friction upon the solution. Calculations were also per- These errors are typically an underprediction of the or- formed (Calc 3 and 4, Table 1) with these two friction der of 8 cm. Within the North Channel, at locations M,

Unauthenticated | Downloaded 09/29/21 09:10 AM UTC 1768 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 current ellipse at every 2 M fourth grid point. amplitude in centimeters (solid) and phase in degrees (dashed). (b) Distribution of the major and minor axis of the surface 2 M . 4. (a) Cotidal chart of the IG F

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N, and O, the model has a tendency to overpredict tidal alignment of the ellipse is reproduced by the model to amplitudes by about 5%, although the phase is in rea- within 3Њ, as subsequently we will show that there is a sonable agreement with observations (to within 7Њ). At discrepancy between model and measurements at ADCP positions P and Q in the center of the region covered (A1). A slightly more accurate result is obtained with by the HF radar there is good agreement with obser- k ϭ 0.01 compared with k ϭ 0.005. vations, although at location R situated close to the coast The orientation of the current ellipse at positions 22 the model overpredicts the amplitude (namely 115 cm and 0 is comparable to that found at location 14 and observed compared with 123 cm predicted, about a 7% 17, namely an orientation of about 120Њ that is deter- error), possibly due to poor grid resolution in this region. mined by the topography of the deep channel (Fig. 1).

The lack of accurate S 2 and N 2 tidal observations in However at position 20 in a shallower near-coastal re- the region of the amphidrome prevents an accurate com- gion the orientation is of the order of 90Њ. In this shal- parison being performed in this area. However at po- lower region the model accurately (to within 2 cm sϪ1) sitions covered by the HF radar (e.g., locations P, Q, reproduces the near-sea-surface and near-bed current and R) there is reasonable agreement between model magnitude with k ϭ 0.005, but with a reduction in ac- and observation. In essence, for these tidal constituents curacy when k ϭ 0.01. the model has a similar bias to that found for the M 2 In the case of the S 2 and N 2 components at the ma- constituent, namely to overpredict the amplitude of the jority of locations the model's accuracy is comparable tide. to that found for the M 2 tide. However, at location 8, Surface tidal current ellipses (major and minor axis) the model accurately reproduces the S 2 and N 2 current at every fourth grid point of the model for the M 2 tide ellipse, in marked contrast to the results found for the are given in Fig. 4b (Calc 1). An area of maximum tidal M2 tide. As this point is close to the western open bound- currents occurs between the northern coast of Ireland ary where the tidal energy ¯ux enters the model, this and the southern end of the Kintyre peninsula, with the suggests that there may be an error in the distribution direction of ¯ow aligned with the topography. A similar of the M2 input along the boundary. The fact that the spatial distribution of the tidal currents was found for M2 tide is accurately reproduced farther south is prob- the S 2 and N 2 tides (not shown), although S 2 currents ably due to the correct overall input of M 2 energy into were about 2.5 times smaller than M 2, and N2 about the region. Davies and Hall found that tides in the north- four times smaller. western part of the region were very sensitive to small Although the overall spatial distribution of the ellip- changes in boundary input. ses is not in¯uenced by the value of k, the critical test of the model's accuracy is the extent to which it can 2) DETAILED COMPARISON OF VERTICAL reproduce current pro®les, and the spatial variability of DISTRIBUTIONS AT ADCP SITES currents in the region. To be consistent with DH, which did not involve a comparison with ADCP data (which Besides this comparison with current meters in the was not available at that time), the present dataset is region, a detailed comparison at two ADCP sites, A1 divided into two parts: non-ADCP data and the two and A2, was made. Pro®les of the semimajor and semi- ADCP datasets, positions marked A1 and A2 in Fig. 2. minor axes, amplitudes, orientation, and phase of the Considering initially the non-ADCP data, to be consis- current ellipse together with the clockwise and anti- tent with the numbering scheme of DH, the additional clockwise rotary components computed with k ϭ 0.01 dataset at 54.79ЊN, 5.29ЊW has been numbered (0, 1, are given in Fig. 5. The computed pro®les at A1 are 2) with other datasets as numbered in DH. shown as a continuous line, with the observations (13 A detailed comparison of computed and observed cur- bins in the vertical) denoted by a symbol. At location rents (not presented) showed that at positions in the A2 computed pro®les are denoted by the dashed line Clyde Sea the model (both Calc 1 and 2) had a tendency with observations (23 bins in the vertical) denoted by to slightly overpredict the current, suggesting that in the symbols. Also shown at this position is the value of the model too much tidal energy enters the Clyde Sea. surface current measured by the HF radar. This ®gure At location 8, close to the amphidromic point, the and a detailed comparison of computed and observed model tends to underpredict both the major and minor current values shows that at site A1 the M 2 semimajor axis of the ellipse, although the more circular nature of axis near the surface (of order 76 cm sϪ1) is accurately the ellipse in this region compared with the rectilinear reproduced by the model, with the near-bed current be- ¯ows found farther south is reproduced by the model. ing underestimated by about 2 cm sϪ1. This suggests At location 17, close to ADCP (A1) the model re- vertical shear, denoted by ⌬A (the difference in the semi- produces the near-surface current but underestimates the major axis), between near-surface and near-bed currents shear in the vertical (taken here as the difference be- is being slightly over estimated; that is, ⌬Aobs ϭ 19 cm Ϫ1 Ϫ1 tween near-surface and near-bed current), although the s , ⌬Acalc ϭ 21 cm s . For k ϭ 0.005 the near-surface change in the semiminor axis and sense of rotation (not current is accurately computed but the near-bottom cur- presented) is reproduced, together with the alignment rent is overestimated by 2 cm sϪ1. The magnitude of of the current ellipse. It is important to note that the the semiminor axis near the surface is slightly overes-

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FIG. 5. Pro®les of the semimajor, semiminor, orientation and phase of the M2, S2, N2 current

ellipses at A1 and A2. Also shown are the rotary components Rϩ and RϪ. Observations at A1 are shown by the open square symbol, with model results as the continuous line. At A2 observations timated in both calculations [observed value 5.4 cm sϪ1 compared to other observations in the region, namely lo- compared with 7.3 cm sϪ1 (Calc 1) and 6.8 cm sϪ1 (Calc cation 17, where the observed orientation was 129Њ near 2)]. However its reduction through depth and subse- the surface. This suggests that there may be an error in quent increase in the near-bed region together with the the alignment of the ellipse recorded by the ADCP in- change in sense of rotation (assigned to the sign of the strument or a local feature that produces a change in ellipse semiminor axis) is reproduced, as is the phase of the orientation. ellipse (Fig. 5). In the case of k ϭ 0.005, the semiminor At position A2, the ADCP was deployed for over a year axis close to bed is 4.6 cm sϪ1, a value less than that and hence a more accurate tidal analysis could be per- found with k ϭ 0.01. formed. The computed near-surface M2 semimajor axis is The major discrepancy at location A1 is in the orien- in excellent agreement with the observed (measured with tation of the semimajor axis, namely 139Њ observed com- the HF radar) and the ADCP, with an observed value of pared with 123Њ computed. The observed value is large 92.0 cm sϪ1 compared to a computed value of 90.3 cm

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FIG.5.(Continued) are denoted by the cross symbol, with surface values measured by the HF radar denoted by an open diamond symbol, and model results with the dashed line. sϪ1, an error of less than 2% (Fig. 5), although below the axis and its phase are accurately reproduced in the model. surface layer the comparison with the ADCP data, suggests Also at position A2, there is close agreement between the that the M2 semimajor axis of the near-surface tidal current orientation of observed and computed tidal current ellipse. is underestimated by 2 cm sϪ1 (k ϭ 0.01) and 4 cm sϪ1 Figure 5 clearly shows that there is a rapid reduction in (k ϭ 0.005) while the near-bed value is accurately repro- the magnitude of the semimajor and semiminor axes in duced with k ϭ 0.01 and overestimated by 3 cm sϪ1 when the very near-bed region below the bottom ADCP mea- k ϭ 0.005. Consequently the model does not accurately surement. As we will show later in connection with the reproduce the shape of the M2 current pro®le at location M2 only calculations (Calc 3 and 4) this near-bed variation A2, and has a tendency to underpredict the vertical shear. is related to bottom frictional effects and measurements These results clearly show that the change in current mag- in this high shear near-bed area are a critical test of the nitude from near-surface to near-bed is signi®cantly in¯u- model's accuracy. enced by the coef®cient of bed friction. As at location A1, A comparable absolute level of accuracy of the semi- Ϫ1 the main features of the vertical variation of the semimajor major axis for the S2 and N2 tides, namely 1 cm s to

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FIG.5.(Continued)

that found for the M2 tides is found at locations A1 and a uniform variation in the vertical not found in the ob- A2. The change in magnitude of the semiminor axis, sense servations at A1. Similarly at location A2 both near-sur- of rotation and phase for S2 and N2 was reproduced by face and near-bed semimajor axis of the current ellipse is the model although the difference in orientation of the accurately reproduced in the model. At A2, for the S2 tide ellipse between computed and observed found for the M2 the computed near-surface semimajor axis (Fig. 5) and its tide also occurred for the S2 and N2 tides. Also, the com- surface value are in excellent agreement with the ADCP puted semiminor axis of the N2 ellipse in the upper part data and HF radar measurement, with model and mea- of the water column is overpredicted in the model at lo- surements showing a uniform increase in tidal current cation A1. The observed values are however quite small magnitude close to the surface. However for N2 there is (less than 2 cm sϪ1) and, although the ADCP has a nominal a difference between computed and measured (HF radar) accuracy of 0.7 cm sϪ1, there would appear to be an error surface semimajor axis (Fig. 5). The HF radar measure- in the observations at this location. ment suggests a rapid decrease in the near-surface current At ADCP site A2, both model and observations exhibit not found in the model. This difference in surface current

Unauthenticated | Downloaded 09/29/21 09:10 AM UTC JULY 2001 DAVIES ET AL. 1773 tide, along the line XY for (a) semimajor axis, 2 M . 6. Across-channel variation of the IG F (b) orientation of the ellipse,values (c) (dot-dashed), phase and of the water ellipse; depth with (solid) observed values in (dashed), meters. computed

Unauthenticated | Downloaded 09/29/21 09:10 AM UTC 1774 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 for phase. A Њ . X for amplitude and 5 1 Ϫ tide (a) semimajor axis, (b) orientation of the ellipse, and (c) phase of the ellipse. 2 M . 7. Location of HF radar cells, with the difference between computed and observed IG F The scale of thethe differences side between of observed the and cell computed corresponds is to such an that error a of displacement 5 to cm s values, denoted by afor horizontal the displacement of the line that runs vertically through each cell scale bar is added for clarity, with location of A2 marked with an

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FIG. 8. Distribution of the major and minor axis of the M2 surface current ellipse from radar measurements.

(of less than 2 cm sϪ1) is probably within the error bar surements are available in a limited region (Fig. 2). The associated with the HF radar measurement. across channel variation in surface current amplitude, Pro®les of the amplitude and phase of the anticlock- orientation, and phase together with water depths along wise (Rϩ) and clockwise (RϪ) components are given in the line XY are shown in Fig. 6. The computed M 2 Fig. 5 for k ϭ 0.01. The amplitude of the (Rϩ) com- semimajor axis (Fig. 6a) increases from zero at the Irish ponent of the M 2 tide is approximately constant in the coast, to a maximum about 5 km from the coast, then upper part of the water column, with a rapid decrease decreases farther offshore to about 90 cm sϪ1 at 23 km in the near-bed region (bottom 10%). The thickness of from the Irish coast, in the deep channel (depth 220 m). the bottom boundary layer (i.e., the region where the The computed semimajor axis agrees with observations pro®le departs from its free stream value) is much thin- to within 2 cm sϪ1 away from the coastline (i.e., between ner for the anticlockwise component because of the (␻ 10 and 30 km from the Irish coast). Close to the Irish ϩ f ) term in the denominator [see Eq. (1)] than for the coast both model and measurements change over short clockwise component. This is because the (␻ Ϫ f ) term distances possibly due to changes in the bottom topog- in the denominator is small at the latitude of the ob- raphy. This region is at the limit of the range of the HF servations, f ϭ 0.12 ϫ 10Ϫ3 sϪ1, which is close to the Ϫ3 Ϫ1 radar; also there are dif®culties in the model in resolving M2 frequency of wM2 ϭ 0.14 ϫ 10 s , and hence the clockwise boundary layer occupies the whole water col- Copeland Island. Close to the Scottish coast the mag- umn. Both the amplitude and phase (except at position nitude of the computed currents rapidly decreases as the A1 due to differences in ellipse orientation) of the com- water depth shallows. This decrease is not observed, puted clockwise and anticlockwise components of the possibly due to inaccuracies in the HF radar or excessive model damping in shallow water. M2 and S 2 tide are in good agreement with the observations. However for the N 2 tide the agreement is not Good agreement (to within about 6Њ) occurs over the so good, possibly due to errors in the observations as- majority (between 10 km and the Scottish coast) of the sociated with this smaller component. channel in computed ellipse orientation, see Fig. 6b. Also there appear to be some discrepancies between However, close (to within 10 km) to the Irish coast the the computed and observed surface values of the rotary observations show signi®cant changes in the ellipse ori- components of the tide. With the HF radar for some entation over short (of order 3 km) distances, possibly properties suggesting a rapid change in the near-surface due to inaccuracies in the measurements. The model layer compared with the ADCP measurements at depth. does not show such large changes in orientation, al- However, these differences are of the order of the errors though there are some variations associated with chang- in the instruments, and in the harmonic analysis method es in the bottom topography. Even in the deep water due to noise in the record. regions (between 20 and 25 km), both observations and model show small-scale variations in ellipse amplitude 3) DETAILED SPATIAL DISTRIBUTIONS IN THE and orientation superimposed upon a smoother larger REGION COVERED BY THE HF RADAR scale across-channel variation, which appear to be as- (SEMIDIURNAL TIDES) sociated with small changes in the bottom topography The ADCP measurements are insuf®cient to examine (Figs. 6a,b). Such variations could not be detected by the horizontal variability for which the HF radar mea- conventional current meter observations or a coarser

Unauthenticated | Downloaded 09/29/21 09:10 AM UTC 1776 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 ent ellipse at every fourth grid tide. 1 K point for the . 9. (a) Cotidal chart, amplitude in centimeters (solid) and phase in degrees (dashed). (b) Distribution of the major and minor axis of the surface curr IG F

Unauthenticated | Downloaded 09/29/21 09:10 AM UTC JULY 2001 DAVIES ET AL. 1777 grid model. This suggests that comparisons of point differences are within Ϯ6cmsϪ1. A similar distribution current measurements and coarse grid models may be to that found for the M2 semimajor axis, occurs for the subject to signi®cant errors due to lack of resolution and orientation of the ellipse [Fig. 7b]. In this case a phase not inadequacies in the model physics. error of 5Њ corresponds to a displacement from the center Excellent agreement in phase (to within 3Њ) is evident to the side of the cell. This ®gure shows that model and (Fig. 6c) at distances greater than 4 km from the Irish measurements agree to within 5Њ over the majority of the coast. Close to the Irish coast, the computed phase de- region except close to the island of Copeland. This is creases below zero, producing the discontinuity as this con®rmed by the histogram of errors (Table 2), which value is subtracted from 360Њ. In this region the obser- shows a slight bias to overpredict the orientation of the vations show signi®cant spatial variability, possibly due ellipse. The current ellipse phase errors, [Fig. 7c] (con- to changes in bottom topography and limitations in HF vention as before), show that the only region of signif- radar accuracy. icant phase error is close to Copeland. This is con®rmed

The main features of the spatial variability of the S 2 by the histogram of phase errors (Table 2), which shows tide are comparable to those found for the M2 tide. The that at the majority (over 600 bins) the phase is computed across-channel variation of the S 2 phase is comparable to within Ϯ10Њ. The distribution (not shown) of the dif- to that found for the M 2 tide although there is a clear ference in the magnitude of anticlockwise (hac) and bias in the model to overpredict the phase by the order clockwise (hc) components of the surface current are of 10Њ, associated with errors in the phase of the open similar to those found for the semimajor axis, with the boundary forcing. The spatial distribution of the N 2 anticlockwise (Rϩ) having a bias to be underpredicted semimajor axis, orientation, and phase (not shown) is in the model and the clockwise (RϪ) slightly overpre- similar to that found for the M 2 and S 2 tides. dicted. The distribution of errors in the anticlockwise The spatial distribution of differences between the component of the phase (gac) (Table 2) shows a bias to observed and computed semimajor axis of the surface underpredict the phase, which is also found although to

M2 tidal current at the HF radar cells are shown in Fig. a lesser extent in the clockwise component (gc). 7a. The cells located in the deep water channel (Fig. 1) For the S2 component of the tide, the distribution of (depths exceed 200 m) are shaded in Fig. 1. The location errors in the semimajor axis (not shown) is comparable to of the measurement is at the center of the cell, with the that found for M2. The histogram of errors (Table 2) shows difference between observed and computed correspond- (unlike for the M2 tide) that there is no signi®cant bias in ing to an eastward displacement of the line for a positive the magnitude of the S2 semimajor axis. The spatial dis- error and a westward displacement for a negative error. tribution of the orientation and phase (not shown) of the

The magnitude of the error is such that a displacement S2 tidal currents are comparable with those found for the to the side of the cell corresponds to an error of 5 cm M2 tide. However, over the whole region there is a sig- Ϫ1 s . ni®cantly larger error (of over 5Њ) in the phase of the S2 Figure 7a shows that at the most northeasterly cells tide than that found for the M2 tide, as shown in the computed and observed values differ by about 15 cm histograms (Table 2). There is a bias to underpredict the Ϫ1 s , although this error decreases on moving offshore. amplitude of the anticlockwise component of the S2 tide Over the majority of the region observed and computed and overpredict the clockwise component (Table 2), in a Ϫ1 semimajor axis agree to within 5 cm s . Away from similar manner to that found for the M2 tide. Also the the coast the magnitude of the semimajor axis of the phase of the anticlockwise component is underpredicted Ϫ1 Ϫ1 M2 tide is of order 85 cm s , and hence a 5 cm s and that of the clockwise component is overpredicted. This error corresponds to a 6% error. However, in the south- is slightly different to that found for the M2 tide where west farthest away from the HF radar, close to Copeland both were underpredicted and may be associated with Island there are signi®cant differences between model slight differences in the phase of the open boundary input and observations. This may in part be due to errors in for the two constituents. the HF radar signal in this area and dif®culties in the For the N2 tide the current ellipse errors (not shown) model resolving the ¯ow near the island. Current ellip- were comparable to those found for the M2 and S2 tide. ses derived from the HF radar measurements (Fig. 8) The histogram of errors (Table 2) shows there is a slight show some irregularity in the region of Copeland Island. bias to overpredict the current, although at the majority Also the noise level in the signal received by the HF of locations it is accurately computed to within Ϯ2cm radar was signi®cantly larger in this region. A detailed sϪ1. discussion of how HF Radar accuracy is reduced with Despite the various biases in the model, it is evident distance from the transmitter is beyond the scope of the that it can accurately reproduce the main features in the paper but is discussed in Howarth et al. (1995) and semidiurnal surface tidal currents in the region covered by Prandle (1991). the HF radar (Fig. 7 and Table 2). Also the vertical var- The histogram of errors based upon 652 cells where iation is in good agreement with the observations made HF radar measurements exist (Table 2) shows that there at ADCP sites A1 and A2. That these sites are well re- is a slight bias to underestimate the tidal current mag- moved from the open boundary of the model and no tidal nitude by 2 cm sϪ1 although at the majority of locations current pro®le was speci®ed along the open boundary sug-

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FIG. 10. Pro®les of the semimajor, semiminor, orientation and phase of the O1 and K1 current

ellipse at ADCP sites A1 and A2. Also shown are the rotary components Rϩ and RϪ. Observations at A1 are shown by the open square symbol, with model results as the continuous line. At A2

gests that the physics within the model is suf®cient to tide (not shown) exhibiting a similar pattern. The com- enable it to accurately reproduce the semidiurnal tidal cur- puted charts from both calculations 1 and 2 are in good rent pro®les in the region. agreement with those based on observations (Howarth 1990) and a west coast model (Davies and Jones 1992). Comparison with observations gave rms errors for both b. Diurnal tides (K1 and O1) calculations 1 and 2 of 1.4 cm and 11Њ for K1 and1cm 1) SPATIAL TIDAL DISTRIBUTIONS OVER THE and 6Њ for O . Surface tidal current ellipses for the K WHOLE REGION 1 1 tide (Fig. 9b) (O1 ellipses not given) show a similar

The K1 cotidal chart does not show any signi®cant spatial distribution to that obtained for the semidiurnal spatial variability over the area (Fig. 9a), with the O1 tide although the current magnitude is signi®cantly less.

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FIG. 10. (Continued) observations are denoted by the cross symbol, with surface values measured by the HF radar denoted by an open diamond symbol, and model results with the dashed line.

2) DETAILED COMPARISON OF VERTICAL the nominal accuracy (of about 0.75 cm sϪ1) of the DISTRIBUTIONS AT ADCP SITES instrument. However at location A2, below the near- surface layer, the magnitude of the semimajor axis de- A detailed comparison of the K1 tidal current with measurements is not possible at A1 where only a month creases uniformly with depth, with model results in rea- of data exists. At A2, there was over a year of mea- sonable agreement with observations, allowing for errors in the observations. Also both model and measurements although the pro®le of both the K1 and O1 tidal currents showed some unrealistic variability in the surements show an increase in magnitude of the vertical. At A1 and A2 there are some physically un- semiminor axis near the seabed, with orientation agree- realistic oscillations in the observed O1 pro®le (Fig. 10), ing to within 2Њ although there are differences in the which are not shown in the model, suggesting some phase. Both calculations 1 and 2 show similar vertical inaccuracy in the measurements. The magnitude of the pro®les with slightly (of the order of 0.1 cm sϪ1) stron- semimajor axis is only the order of 2 cm sϪ1, close to ger bottom currents in the case of calculation 2.

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Also shown in Fig. 10 is the surface current measured over most of the region there is excellent agreement by the HF radar at position A2. A comparison with (Fig. 11a). subsurface ADCP values shows some signi®cant chang- The change in orientation along line XY of the com- es in the near-surface current that are not found in the puted O1 tidal current ellipse (Fig. 11b) is similar to model or in the measured pro®les from the ADCP in- that found for the M2 tide. This is to be expected since strument. However, these differences lie within the error the direction of tidal ¯ow in the region is determined bars associated with the instruments. by the topography. The observed orientation of the O1 ellipse, however, shows signi®cantly greater variability

than found for M2, presumably re¯ecting a larger error 3) DETAILED SPATIAL DISTRIBUTIONS IN THE in the measured orientation of this component, due to REGION COVERED BY THE HF RADAR (DIURNAL its smaller magnitude. Similarly the phase (Fig. 11c) of TIDES) the observed O1 tidal ellipse shows signi®cantly greater spatial variability than that found in the M tide, al- Due to dif®culties in separating the K tide from the 2 1 though it is in good agreement with the observations S1 and P1 tide, and hence possible errors in the observed (Fig. 11c). K1 tide, we will con®ne our attention to the O1 surface The change in the observed amplitude, orientation tidal currents. The spatial distribution of errors in the and phase of the K1 tide along line XY (not shown) is semimajor axis of the O1 current ellipse (not presented) signi®cantly noisier than that found for O1, particularly showed that over the majority of the region it is less in the region close to the Irish coast. However away Ϫ1 than 0.5 cm s , although its magnitude away from the from this region, model and observations agree to the coast is less than 2.0 cm sϪ1. In the area close to Cope- same level of accuracy as that found for the O1 tide. land, at the limit of the HF radar, errors of up to 10 cm These comparisons suggest that in the region covered Ϫ1 s are present. The errors histogram (Table 2) shows by the HF radar, a rigorous comparison of observed and that there is no signi®cant bias in the O1 results, although computed diurnal tidal currents is not possible because for K1 there is a bias to overestimate the tidal current the magnitude of these currents is comparable with the amplitude. However, for both O1 and K1 the observations accuracy of the radar system and the measurement ca- and measurements agree to within Ϯ2cmsϪ1 at over pabilities of the ADCP instrument. However, on average 600 bins (Fig. 9b). The plot of orientation errors (not there does not appear to be any major differences be- shown) revealed that there are signi®cant errors in the tween observations and calculations, except for the K1 region close to the Scottish coast, with very large errors tidal currents in the area at the limit of the HF radar's at the limit of the range of the HF radar. The ellipse range. orientation errors histogram for O1 (Table 2) shows that there is a bias to overestimate the orientation of the tidal 4. Single constituent run (M 2 only) ellipse, although for K1 there is no signi®cant bias. This difference in bias is surprising in that the orientation of In the previous series of calculations, the M 2 tide was the ellipse is primarily dictated by the geometry of the determined in combination with other tides. Hence fric- channel and should be the same for all tidal constituents, tional effects, both bottom friction and internal friction as was the case for the M 2, S 2, and N 2 tides. The de- (vertical eddy viscosity), took into account additional parture in the case of K1 is probably due to measurement turbulence due to the other tides. In many models (Da- errors and those occurring in the harmonic analysis pro- vies et al. 1996; Xing and Davies 1996; Sinha and Pin- cedure. gree 1997) calculations are performed with only the M2 The distribution of O1 phase errors (not shown) is tide. To investigate what effect the other constituents comparable to that given for the orientation errors. The have upon the computed M 2 tide, the previous calcu- histogram of O1 phase errors (Table 2) does not show lation was performed with the same bottom drag co- a signi®cant bias, with the majority of errors con®ned ef®cient (k ϭ 0.01) but with only the M2 tidal forcing to the range of Ϯ10Њ, although for the K1 tide the error (Calc 3). distribution is much wider and there are a signi®cant Although the distribution of co-amplitude and co- number of points where the phase error exceeds 90Њ. phase lines, together with current ellipses was compa- A smooth variation in the computed values of the rable with that found previously (Calc 1), it was clear amplitude of the semimajor axis of the O1 tidal currents that at locations A, D, and H, the tidal elevation am- along line XY (Fig. 11a) occurs in the region away from plitude is slightly (on average of the order of 2 cm) the Irish coast with some small-scale variation close to higher than that found in calculation 1, suggesting a the coast, re¯ecting depth changes in this region. The slight change in the location of the M 2 amphidromic observed O1 tidal currents show signi®cantly greater point. spatial variability, particularly in the region adjacent to At the majority of positions where the M2 current Ϫ1 the Irish coast, although the small magnitude of the O1 exceeds 10 cm s the semimajor axis increases by the Ϫ1 Ϫ1 currents (of order 2 cm s ) and the error in the HF order of 2 cm s when an M 2 only calculation is per- radar may give rise to this spatial variability. However formed (Calc 3). However at location 20, the surface

Unauthenticated | Downloaded 09/29/21 09:10 AM UTC JULY 2001 DAVIES ET AL. 1781 component of the observed (dashed) and calculated (dot-dashed) 1 O (m) (solid) along cross section XY. h . 11. Spatial variations of (a) the semimajor axis, (b) orientation, and (c) phase of the IG F current ellipses fortide the and water depth

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FIG. 12. Pro®les of the semimajor, semiminor, orientation, and phase of the M2 current ellipse

and components Rϩ and RϪ at sites A1 and A2. Observations at A1 are shown by the open square symbol, with model results as the continuous line. At A2 observations are denoted by the cross symbol, with surface values measured by the radar denoted by an open diamond symbol, and model results with the dashed line. current increases from 82 to 88 cm sϪ1 (observed value of the ADCP) is slightly (of the order of 2 cm sϪ1) un- 85 cm sϪ1), with bed current increasing from 66 to 71 derestimated in calculation 1, the slight increase in cal- cm sϪ1 (observed value 69 cm sϪ1) suggesting that fric- culation 3 produces an excellent agreement with ADCP tional effects are slightly underestimated in the M2 only observations (Fig. 12) although the computed surface cur- calculation (Calc 3). rent exceeds that measured by the HF radar. Also, the A similar increase (of the order of 2 cm sϪ1) occurs in increase in the bed current in calculation 3 leads to an the semimajor axis of the M2 near surface current at ADCP increased error. location A1 (compare Figs. 12 and 5) leading to a reduc- These calculations suggest that, when M2 is computed tion in the accuracy of the surface current. However at in the absence of the other tidal constituents, the reduction location A2, where the near-surface current (the upper bin in frictional effects due to their omission leads to a slight

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TABLE 2. Distribution of the number of radar cells where the difference (observed Ϫ computed) of the semimajor axis A, amplitude of anticlockwise hac, amplitude of clockwise hc, or orientation ␪, phase ␾, or phase of anticlockwise gac or clockwise gc lies between Ϯ2cm sϪ1 Ϯ4cmsϪ1 ...Ϯ10 cm sϪ1,orϮ10Њ, Ϯ20Њ,...,Ϯ50Њ. (Values outside the range are not shown.)

Amplitude (cm sϪ1) Constituent Ϫ10 Ϫ8 Ϫ6 Ϫ4 Ϫ2 ϩ2 ϩ4 ϩ6 ϩ8 ϩ10

M2 A 6 9 9 42 95 166 97 80 30 20 ␪ 0 0 4 17 526 68 19 10 4 1 ␾ 0 0 0 0 443 159 22 10 8 4 hac 1 2 3 5 33 92 193 160 70 28 gac 9 4 13 16 47 523 24 1 0 0 hc 20 27 79 116 138 135 47 21 21 9 gc 0 0 0 6 183 405 33 16 6 3

S2 A 6 8 19 52 231 236 54 23 4 7 ␪ 0 2 2 37 492 89 21 7 2 0 ␾ 0 3 8 279 306 25 10 7 9 1 hac 1 4 6 8 54 449 113 9 6 2 gac 3 5 8 19 18 99 447 36 10 1 hc 2 11 41 132 300 143 17 1 1 0 gc 3 5 9 145 414 30 22 9 9 3

N2 A 2 2 20 81 391 123 24 7 1 0 ␪ 0 2 9 54 439 120 17 6 1 0 ␾ 0 1 1 5 454 132 31 17 4 2 hac 0 1 2 7 120 496 23 3 0 0 gac 7 7 16 19 69 428 84 10 2 2 hc 0 1 4 164 438 40 5 0 0 0 gc 0 0 1 5 205 347 41 33 10 5

K1 A 0 0 0 0 519 86 26 7 8 4 ␪ 6 12 14 55 181 154 78 39 7 3 ␾ 7 10 50 80 170 87 42 14 2 8 hac 0 0 0 0 297 329 15 10 1 0 gac 6 10 22 50 93 174 102 49 29 14 hc 0 0 0 0 550 86 11 5 0 0 gc 18 47 53 98 95 73 57 22 10 4

O1 A 0 0 0 0 302 287 41 15 6 1 ␪ 8 22 35 164 305 64 14 4 2 1 ␾ 4 4 11 61 248 204 67 10 3 2 hac 0 0 0 0 319 311 20 2 0 0 gac 0 5 7 17 97 213 147 71 31 9 hc 0 0 0 0 324 306 22 0 0 0 gc 1 3 11 31 106 220 140 44 25 10

M4 A 0 0 3 28 120 429 50 11 7 0 ␪ 48 68 63 64 51 51 36 21 18 14 ␾ 37 49 47 29 20 22 24 16 15 12 hac 0 0 0 5 197 419 23 5 1 0 gac 11 18 13 19 20 16 14 9 21 22 hc 0 0 2 35 234 333 42 3 2 1 gc 28 27 21 28 34 34 22 26 33 39 Phase (deg) Ϫ50 Ϫ40 Ϫ30 Ϫ20 Ϫ10 ϩ10 ϩ20 ϩ30 ϩ40 ϩ50

Ϫ1 (of order 2 cm s ) increase in the amplitude of the M2 compared to an M2 only calculation. The difference in current particularly in the upper part of the water column. thickness of the boundary layer and the reduction in shear In essence the increased turbulence and, hence, viscosity can be seen by comparing the amplitude of the clockwise produced by the other tidal constituents increases the thick- component of current in the multiconstituent run (Fig. 5), ness of the bottom boundary layer and, hence, reduces the with that obtained in the single constituent run (Fig. 12). vertical shear and the magnitude of the surface current In the M2 only calculation (Fig. 12) the increase in the

Unauthenticated | Downloaded 09/29/21 09:10 AM UTC 1784 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 amplitude of the clockwise component with height above A detailed series of comparisons of a comprehensive the bed is overpredicted, whereas in the multiconstituent range of turbulence models in the Irish Sea was per- run a more gradual change occurs producing a closer formed by Davies and Gerritsen (1994) and Xing and agreement with observations (Fig. 5). To examine the in- Davies (1995, 1996). These suggested that there was no ¯uence of bottom friction coef®cient calculation 3 was signi®cant advantage in using a two equation turbulence repeated with k ϭ 0.005 (calc 4). model compared with a ¯ow-dependent viscosity. They

Although decreasing k did not effect the M 2 tidal also found that the one-equation model appeared to be distribution, the tidal amplitude at locations A, D, and a suitable compromise between the two-equation model E were increased with an associated change in phase, and a ¯ow-dependent viscosity. suggesting a shift in the amphidromic point. As found In this paper the earlier model with a one equation in calculation 2, movement of the amphidromic point turbulence energy submodel (DH) has been used to ex- due to frictional effects produces changes in the tidal amine the M2, S 2, N 2, K1, and O1 tides in the region. amplitude at location K, which decreases to 51 cm from A slightly increased set of current measurements is the previous value of 55 cm (Calc 3). In the southern available for comparison, in particular the two ADCP part of the region, from location M onward, changing sets. These provide measurements closer to the bed than k has little effect on the tidal amplitude. Also tidal el- those used by DH. Also, the set of HF radar measure- evation amplitudes and phase within the Clyde Sea do ments in the region is now available for validating the not appear to be signi®cantly in¯uenced by these chang- surface tidal current computed with the model. es in friction. Comparison with previous calculations and obser- The semimajor axis of the ellipse in the upper part vations in the area showed that the model could repro- of the water column at the majority of locations is not duce the major features of the M 2, S 2, N2, K1, and O1 signi®cantly in¯uenced by the changes in k described tides. A detailed comparison of computed surface cur- here. However, at location O the semimajor axis of the rents with HF radar measurements showed that the M 2 bottom current is increased by 3 cm sϪ1 (compared to tidal currents (the amplitude of which were of the order Calc 3). Similarly, at location 20 the bed current is of 100 cm sϪ1) could be accurately (to within Ϯ4cm increased from 71 cm sϪ1 (k ϭ 0.01) to 76 cm sϪ1 (k sϪ1) reproduced at 400 out of 652 bins. Also ellipse ϭ 0.005), although as at other locations there is no orientation was accurately (to within Ϯ10Њ) reproduced signi®cant change in surface current. at 594 bins, with the phase having a similar level of At ADCP site A1, the decrease in k has no in¯uence accuracy at over 600 bins. A comparable level of ac- on the surface current, although the bottom current is curacy was found for the S 2 and N2 tidal components, Ϫ1 increased by 3 cm s . This is different from the change although for S 2 there was a phase error of 10Њ, associated in current structure between the single and multicon- with the phase of the open boundary forcing. The great- stituent run, where the in¯uence of additional constit- est difference between model and measurements was uents was to reduce surface currents with a small change found to occur at the limit of the HF radar range and in bottom currents. A similar change in current pro®le may in part be associated with measurement errors. between multiconstituent and M 2 only with k ϭ 0.005 A detailed comparison of the M 2, S 2, and N2 surface and 0.01 was found at location A2. tidal currents with the HF radar showed small-scale var- These comparisons suggest that although bottom fric- iations in the ellipse semimajor axis and orientation as- tion can be increased or decreased in an M2 only run sociated with small changes in bottom topography. to compensate for the errors in bottom stress due to the These features were found in both the observations and absence of the other constituents, this cannot correct for the model, and could only be detected because of the the additional turbulence in the water column due to the high sampling resolution of the HF radar and the ®ne other constituents, and hence errors in the currents in grid of the model. This demonstrates that comparisons the surface layer, and in the pro®le of the amplitude of between a coarser grid model and point measurements, the clockwise component. rather than the spatial coverage provided by the HF Radar, could yield errors not due to poor physics in the model but due to limitations in resolution. In essence 5. Concluding remarks because of the ®nescale spatial variability found in the In an earlier paper (DH) the high-resolution model computed and the HF radar measurements, point values described here was used to simulate the M2 tide in the are of limited use. Also to match the detailed measure- North Channel region. A range of parameterizations of ments provided by the radar, a more detailed and ac- eddy viscosity from a turbulence energy model, to a curate set of bottom topography and open boundary simple ¯ow-dependent formulation were used in that forcing conditions would be very valuable. A ®ner grid model. Comparisons with current meter data in the re- and more detailed topography in the region of the Irish gion (in general measurements were well above the near- coast would be very valuable if the accuracy of the radar bed high shear layer) showed that an accurate solution in this region could be improved. could be obtained with all formulations of eddy vis- In the case of the diurnal species, the current mag- cosity considered in DH. nitude was much lower than that of the semidiurnal

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tides. Close to the Irish coast, at the limit of the HF K1 and O1 tides in the Celtic and Irish Seas. Progress in Ocean- radar, the accuracy of the measured currents appeared ography, Vol. 29, Pergamon, 197±234. ÐÐ, and H. Gerritsen, 1994: An intercomparison of three-dimen- to be signi®cantly reduced due to the lower signal to sional tidal hydrodynamic models of the Irish Sea. Tellus, noise ratio, particularly in the K1 component of the tide. 46A(2), 200±221. However, away from the Irish coast there was excellent ÐÐ, and P. Hall, 1998: The sensitivity of tidal current pro®les in agreement between modeled and computed currents. the North Channel of the Irish Sea to the parameterization of momentum diffusion. Contin. Shelf Res., 18, 357±404. The comparison of tidal current pro®les with those mea- ÐÐ, S. C. M. Kwong, and R. A. Flather, 1996: Formulation of a sured with the ADCP showed that the model could ac- variable function three dimensional model, and computation of curately reproduce the vertical variation of the current the M2 tide and overtides on the European Shelf. Contin. Shelf for the semidiurnal species, although the magnitude of Res., 17, 165±204. the diurnal species was so low that the measured pro®le ÐÐ, ÐÐ, and ÐÐ,1997: A three-dimensional model of diurnal and semi-diurnal tides on the European shelf. J. Geophys. Res., exhibited physically unrealistic oscillations. (Oceans), 102, 8625±8656. Comparison of the M 2 current pro®le with ADCP George, K. J., 1980: Anatomy of an amphidrome. Hydrogr. J., 18, measurements showed excellent agreement in the mul- 5±12. ticonstituent calculation, although in an M only cal- Howarth, M. J., 1990: Atlas of tidal elevations and currents around 2 the British Isles. Dept. of Energy, Offshore Technology Report culation the model overpredicted the current. Although OTH 89 293, 16 pp. increasing the bottom friction coef®cient led to a re- ÐÐ, and D. T. Pugh, 1983: Observations of tides over the continental duction in bottom current, and hence a closer agreement shelf of north-west Europe. Physical Oceanography of Coastal with observations, the surface current was still over- and Shelf Seas, B. Johns, Ed., Elsevier, 135±188. ÐÐ, A. J. Harrison, P. J. Knight, and R. J. J. Player, 1995: Mea- predicted. This suggests that, although an enhanced bot- surement of Net Flow through a Channel. Proceedings of the tom friction coef®cient can in part take account of ad- IEEE Fifth Working Conference on Current Measurement, S. P. ditional friction due to the other tides, it cannot correct Anderson, Ed., W. S. Sullwold, 121±126. for the additional turbulence they produce in the water Jones, J. E., and A. M. Davies, 1996: A high resolution three dimensional model of the M2,M4,M6,S2,N2,K1 and O1 tides column, and hence the thicker bottom boundary layer in the eastern Irish Sea. Estuarine Coastal Shelf Sci., 42, 311± and reduced surface current. This suggests that the tur- 346. bulence closure scheme used in the model can accurately Luyten, P. J., E. Deleersnijder, J. Ozer, and K. G. Ruddick, 1996: account for tidally produced turbulence provided the Presentation of a family of turbulence closure models for strat- major tidal constituents are included within the model. i®ed shallow water ¯ows and preliminary application to the Rhine out¯ow region. Contin. Shelf Res., 16, 101±130. Oey, L.-Y., and P. Chen, 1992: A model simulation in the Northeast Acknowledgments. The authors are indebted to Mr. Atlantic Shelves and Seas. J. Geophys. Res., 97, 20 087±20 115. R. A. Smith for help in preparing diagrams and Mrs. L. Prandle, D., 1991: A new view of near-shore dynamics based on observations from HF Radar. Progress in Oceanography, Vol. Ravera and Mrs. C. Burke for typing the paper. 27, Pergamon, 403±438. ÐÐ, S. G. Loch, and R. Player, 1993: Tidal ¯ow through the Straits of Dover. J. Phys. Oceanogr., 23, 23±37. REFERENCES Sinha, B., and R. D. Pingree, 1997: The principal lunar semi diurnal

tide and its harmonics: Baseline solutions for M2 and M4 con- Blumberg, A. F., and G. L. Mellor, 1987: A description of a three- stituents on the north-west European continental shelf. Contin. dimensional coastal ocean circulation model. Three-Dimensional Shelf Res., 17, 1321±1366. Coastal Ocean Models, N. S. Heaps, Ed., Coastal and Estuarine Soulsby, R. L., 1983: The bottom boundary layer of shelf seas. Phys- Sciences, No. 4, Amer. Geophys. Union, 1±16. ical Oceanography of Coastal and Shelf Seas, B. Johns, Ed., Davies, A. M., 1986: A three-dimensional model of the northwest Elsevier, 189±266.

European continental shelf with application to the M4 tide. J. Xing, J., and A. M. Davies, 1995: Application of three dimensional Phys. Oceanogr., 16, 797±813. turbulence energy models to the determination of tidal mixing ÐÐ, and J. E. Jones, 1990: Application of a three-dimensional tur- and currents in a shallow sea. Progress in Oceanography, Vol. bulence energy model to the determination of tidal currents on 35, Pergamon, 153±205. the northwest European shelf. J. Geophys. Res., 95, 18 143±18 ÐÐ, and ÐÐ,1996: Application of a range of turbulence energy

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