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The Boundary Layer Character of Tidal Currents in the Eastern Irish Sea
Estuarine, Coastal and Shelf Science (2002) 55, 465–480 doi:10.1006/ecss.2001.0918, available online at http://www.idealibrary.com on
The Boundary Layer Character of Tidal Currents in the Eastern Irish Sea A. J. Elliott Centre for Applied Oceanography, University of Wales, Bangor, Marine Science Laboratories, Menai Bridge, Anglesey LL59 5EY, U.K. Received 19 March 2001 and accepted in revised form 12 October 2001 The vertical profile of the tidal currents observed by a ship borne ADCP at a site in the eastern Irish Sea where the water depth is 50 m was logarithmic with height above the bed in the lower 50–60% of the water column except for periods of about 1 h either side of slack water. The maximum height of the boundary layer reached 40 m during the flood tide. The shearing stress, determined by the method of Bowden et al. (1959), showed a tendency to be uniform with height in the lower half of the water column while the eddy viscosity increased linearly away from the bed, scaling on u*z where u* is the friction velocity. Both of these features are characteristic of the logarithmic profile region of a boundary layer flow. Estimates of zo, the roughness length, and CD, the bottom drag coefficient, were around 10 3 m and 2·410 3, consistent with the known character of the bed. The scatter in the derived boundary layer parameters was reduced significantly by averaging the data within 30-min intervals which suggests the presence of tidal eddies having periods comparable to the 5-min sampling period of the ADCP. Short period fluctuations in the flow were coherent over the lower 20 m of the water column and propagated vertically away from the bed. 2002 Elsevier Science Ltd. All rights reserved. Keywords: tides; currents; Irish Sea; bottom boundary layer; log-profile; shear stress; bed roughness
Introduction The computation of the vertical structure of tidal currents and the parameterisation of the associated turbulence and vertical mixing form the focus of many numerical studies. The development of the tidal boundary layer is routinely computed by threedimensional hydrodynamic codes in shelf sea simulations, yet the target of such studies is often directed towards an understanding of the seasonal and annual cycles (e.g. Holt & James, 1999). Moreover, although data for model validation may be collected at hourly and shorter time scales, the analysis often concentrates on much longer periodicities. Therefore, as part of a programme of model development for operational forecasting over the Irish and Celtic Seas, current data were collected with a relatively short sampling period so that the development of the tidal boundary layer could be documented at time scales ranging from minutes to hours. This paper reports on the observed character of the tidal boundary layer, the results from the operational model will be presented separately. During 31 August and 1 September 1999, the tidal currents were monitored at a site in the eastern Irish Sea (Figure 1) by an anchored ship fitted with a hull mounted ADCP. The system used a 5-min averaging interval and the vertical resolution was set at 2 m. The 0272–7714/02/090465+16 $35.00/0
sensor was mounted at 3·5 m below the waterline and the first data were recorded at a depth of 6·5 m below the surface. The measurements took place during a period of spring tides with light winds and low wave energy. The water was observed to be vertically wellmixed at the site (5332.8 N, 43.5 W) which is consistent with the strength of the tidal flow and the position of the site at the western limit of the fresh water stratification that occurs within Liverpool Bay (Czitrom & Simpson, 1998). The observations commenced about 1 h after local high water on 31 August and continued for 25 h. Since the ADCP was hull mounted on a surface vessel it was necessary to process the data to derive bed referenced currents. This was achieved by inspecting each velocity profile and finding the deepest bin containing valid data values. The centre of the next lowest bin was then assumed to correspond to the bed. This can introduce an error of 1 m into the height of the profile; the consequence of this error will be discussed in a later section. Figure 2 shows a time series of the east and north components of the flow at a height of 20 m above the bed. The 20 m flow was selected as it will approximate the depth-mean current in water that is 50 m deep (Prandle, 1982). For comparison in Figure 2(a) the tidal currents predicted by a depth-integrated two-dimensional tidal 2002 Elsevier Science Ltd. All rights reserved.
466 A. J. Elliott
50 m
53.6N
50 m
53.4
Liverpool Bay
25 m
Latitude
Anglesey 53.2 North Wales
N
53.0
52.8
52.6
50 m
–4.75
25 m
–4.50
–4.25
–4.00
–3.75
–3.50
–3.25
–3.00W
Longitude
F 1. The experimental site (+) in the eastern Irish Sea. The triangle near the coast of Anglesey marks the location of the measurements by Bowden et al. (1959). Water depths are contoured in metres.
model developed at the Proudman Oceanographic Laboratory (Elliott et al., 1992) are shown by dashed curves. This demonstrates the robustness of the tidal currents from the two-dimensional model which are used for oil spill forecasting and other practical simulations (Elliott, 1991). The observed currents were comparable in magnitude but led the simulated currents by a phase of about 20 which corresponds to an interval of 45 min. Figure 2(b) shows the water depth derived directly from the bottom echo of the ADCP signal, illustrating that the tidal range is around 7 m during spring tides. The depth derived from the ADCP bottom echo has a relatively coarse resolution, therefore a semi-diurnal fit to the observed depth was derived and will be used later in an analysis of the shearing stress. This is shown by the dashed curve in Figure 2(b). The agreement between the water depth derived directly from the ADCP bottom echo and by inspecting the velocity data in the near-bed cells will be discussed in a later paragraph. The tidal currents displayed an asymmetry between the ebb and flood directions. (Flood is defined as the period when the eastward component of the flow is
positive and sea level is rising in the eastern Irish Sea. Peak flood flow occurred around 23:00h (UTC) on 31 August and at 12:00h on 1 September.) The flood tide lasted about 1 h less than the ebb, but this was compensated for by a flow that was approximately 20% stronger. The tidal axis was aligned to within 8 of due east and the peak flood speed was of the order of 1·2 m s 1. Contour plots of the flow resolved along the major and minor axes of the tidal motion illustrated the vertical motion of the ship due to the tidal motion of the sea surface. At 10–12 and 22–24 h into the observations, the contoured velocities extended to a height of 44 m above the bed. To this figure must be added the 6·5 m of the ADCP below the water surface to give a total depth of water of 50·5 m. This can be compared to the depths of 49·5–49·8 m and 49·0–49·5 m provided by the bottom echo results (Figure 2) which were derived independently of the method that inspected the velocity data in the near-bottom data bins. The good agreement between the two methods suggests that the height origin for the velocity profiles was placed at the bed with an accuracy of 0·5–1·0 m.
Tidal currents in the Eastern Irish Sea 467 1.5 (a) 1.0
–1
(m s )
0.5 0.0 –0.5 –1.0 –1.5
31 Aug
1999
1 Sep
31 Aug
1999
1 Sep
50 49
(b)
48 (m)
47 46 45 44 43 42
F 2. (a) The east and north components of the flow at a height of 20 m above the bed (solid curves), also shown are the depth-mean currents simulated by the Proudman Oceanographic Laboratory tidal model (dashed curves). (b) The water depth measured by the bottom echo of the ADCP (solid curve) and the curve fitted to the data by tidal analysis (dashed curve).
Results The vertical structure of the tidal flow The results from a tidal analysis of the eastward component of the flow are presented in Figure 3(a). The analysis was made for a single semi-diurnal tidal constituent that could represent the combined effect of the M2 and S2 components. The amplitude of the current (solid curve) increases steadily up to a height of 35 m above the bed and has a value of about 1 m s 1 at mid-depths. The variability above 35 m is due to the vertical tidal motion of the sea surface which resulted in the uppermost data bins only containing data values during periods of high water. Figure 3(b) shows the vertical structure plotted against a logarithmic depth scale. A striking feature of the flow is the logarithmic increase (shown by a dashed line) in tidal speed from a height of 6 m up to 30 m above the bed. The speeds measured at heights of 2 m and 4 m above the bed clearly do not fit the linear relationship. This is to be expected as the data from downward looking ADCP sensors are unreliable in the bottom 10% of the water column. These two data points were therefore not used in the subsequent
analyses. The phase profiles presented in Figure 3 show a steady increase with height above the bed. The near-bed flow led the surface current by about 5 which corresponds to a surface lag of about 10 min. The variability of the results above a height of 35 m can be removed by using a sigma-level analysis that interpolates each vertical profile onto a grid that covers the region between the bed and the highest valid data bin. The results from such an analysis, which treated both the E–W and N–S components of the flow so that the tidal ellipse parameters and the rotary components could be derived, showed that the direction of the major axis was aligned at about 8 to the south of east at all depths. In addition, the flow was more strongly rectilinear towards the bed and the tidal motion was predominantly anti-clockwise in the bottom 10 m and clockwise in the upper part of the water column. This characteristic of the rotary components has been observed in the southern North Sea by Souza and Simpson (1996). The logarithmic increase in current speed with height above the bed suggests the presence of a boundary layer flow which can be described by the standard formulation (Bowden, 1978):
468 A. J. Elliott –1
–1
Amplitude (m s ) 0.0 0.2 50 (a)
0.4
0.6
0.8
Amplitude (m s ) 1.0
1.2
0.0 0.2 50 (b)
0.4
0.6
0.8
1.0
1.2
45 40 20 35
(m)
30
10
1
25 5
20 15 10
2
5 0 230
235
240
245
250
Phase (deg)
10
0
230
235
240
245
250
Phase (deg)
F 3. (a) Vertical structure of the amplitude (solid curve, m s 1) and phase (dashed curve, arbitrary degrees) of the tidal flow. (b) Same results plotted with a logarithmic depth scale.
where u* is the friction velocity, zo is the roughness length and is Von Karman’s constant (0·40). The data points in the height range from 6–20 m above the bed when fitted to (1) gave estimates for u* and zo of 0·031 m s 1 and 6·410 5 m. If the logarithmic profile is used to estimate the speed, u1, at a height 1 m above the bed then the quadratic drag coefficient can be derived by equating CDu12 and u*2. This produced an estimated for the drag coefficient of 1·7210 3. However, since a tidal analysis parameterises the envelope of the tidal flow these estimates should be regarded as extreme values and not as averages over the tidal cycle. Estimates of boundary layer flow parameters derived via equation (1) are sensitive to errors in the heights assigned to the velocity observations. To assess the impact of such errors the flow profile shown in Figure 3 was shifted vertically by 1 m as this represents the uncertainty in the position of the bed when determined using ADCP bins with a 2-m vertical resolution. The two estimates for u*, zo and CD derived in this manner had values of 91/109%, 26/
295%, and 77/127% of those quoted above. Thus, taking the ratio of the extreme values, u* varied by a factor of 1.20, zo varied by a factor of 11·4 and CD varied by a factor of 1·65. As a consequence, while the estimates of u* and CD are fairly robust, the values derived for zo should be regarded more cautiously. The time varying character of the boundary layer The profiles shown in Figure 3 represent the envelope of the tidal speed with height above the bed, and while the envelope suggests a logarithmic profile in the lower half of the water column this does not guarantee that instantaneous profiles will display this characteristic. The analysis was therefore repeated with the individual 5-min profiles by determining the goodness-of-fit to a logarithmic relationship between the heights of 6 m and 20 m above the bed (excluding the values at 2 m and 4 m for reasons detailed above). The results are presented in Figure 4 which shows the tidal asymmetry in the eastward component of the flow at a height of 20 m above the bed [Figure 4(a)] and the correlation coefficient [Figure 4(b)] which expresses the goodness-of-fit to equation (1). The correlation coefficient shows that the boundary layer
Tidal currents in the Eastern Irish Sea 469
1.2
(a)
–1
U20 (m s )
1.0 0.8 0.6 0.4 0.2 0.0 1.0
(b)
0.8
R
0.6 0.4 0.2 0.0 0.08
(c)
0.07 –1
u* (m s )
0.06 0.05 0.04 0.03 0.02 0.01 0.00 10
–1
(d)
Z0 (m)
10 10
–3
10
–4
10
–5
10 10
CD
–2
–6
–2
(e)
–3
10
10
–4
0
50
100
150 Sample no.
200
250
300
F 4. (a) Speed (m s 1) at z=20 m for the 5 min data. The dashed lines show the times of slack water. (b) The correlation coefficient, R, for the goodness-of-fit to a logarithmic profile between the heights of 6 m and 22 m above the bed. (c) The friction velocity, u* (m s 1), derived from the logarithmic profile. (d) The roughness length, zo (m). (e) The quadratic drag coefficient, CD.
470 A. J. Elliott
was well established up to a height of 20 m apart from 1 h before and 1·5 h after slack water. High correlation (>0·95) was achieved for periods of up to 3 h during the central portions of both the flood and ebb flow. The derivation of the boundary layer parameters was restricted to profiles when the mean current speed increased with height above the bed; this excluded the periods around slack water. For profiles that showed an increasing speed with height the parameters u*, zo and CD were estimated and they are presented in [Figure 4(c)–4(e)]. If the tidal flow consisted of a pure harmonic current at the semi-diurnal frequency, the friction velocity might be expected to vary in a comparable harmonic manner while the roughness length and the drag coefficient should be constant throughout the tidal period. However, the results shown in Figure 4, which show considerable scatter and are suggestive of envelopes rather than curves, are not consistent with such a hypothesis. The variability in the boundary layer parameters was reduced significantly by averaging the velocity at each depth within 30 min non-overlapping blocks (Figure 5). Repeating the least-squares logarithmic profile analysis with 30 min blocked data produced similar results for the tidal speed at a height of 20 m [Figure 5(a)] but increased the goodness-offit [Figure 5(b)] so that the correlation coefficient exceeded 0·98 during the central portion of the ebb and flood tide. This extremely close agreement to a logarithmic profile in the lower 50% of the water column is a striking feature of the data. The averaging process removed much of the variability in the derived estimates of the boundary layer parameters. Figure 5(c) suggests that the friction velocity increased asymmetrically during each ebb and flood period with a rapid rise after slack water followed by a more gradual decrease as the next slack approached. The value of both the roughness length and the drag coefficient increased between consecutive periods of slack water and the mean values of zo and CD derived from the 30 min averaged velocities were 8·810 4 m and 2·410 3. These values can be compared with those reported by Bowden et al. (1959) who measured the vertical current structure near to the north coast of Anglesey in water that was around 20 m deep and obtained values of 1·610 3 m and 3·510 3. Admiralty charts of the present site suggest that the sea bed is composed of sand, broken shells, gravel and pebbles. Heathershaw (1981) reported that the roughness length appropriate for such a bed varies between 0·3010 3 m (for sand and shells) and 3·010 3 m (for gravel and pebbles), while the appropriate drag coefficient lies in the range
2·410 3 to 4·710 3. In consequence, the values derived from the logarithmic layer analysis lie within the expected ranges. Lueck and Lu (1997) observed a tidal boundary layer that extended to a height of 20 m above the bed in a 30 m deep tidal channel. They estimated the thickness of the logarithmic layer by computing the correlation coefficient between u and ln(z) as a function of height above the bed. To do this they started with the lowest three data points and computed the agreement to a linear fit. If this value exceeded a specified value they added in the fourth data point and recalculated the fit. This process was repeated until a height was reached at which the fit was no longer significant at the chosen confidence level. The logarithmic layer thickness was then defined as the height of the uppermost data point for which a significant linear fit was achieved. A similar analysis was applied to the 30 min averaged data and the layer thickness was defined as the region within which the correlation exceeded 0·98. (This ensures a highly significant fit. For example at 20 m above the bed the regression would involve eight data points, for this number of points a regression value above 0·75 is significant at the 99% level.) The results are presented in Figure 6 which shows the thickness of the logarithmic layer and also the upper limit (dashed curve) of the ADCP observations. It is apparent that the layer extended to a height of 30–40 m above the bed and its thickness was greater during the stronger flood tide than during the ebb. During both of the flood periods Figure 6 suggests that the logarithmic layer extended over the whole portion of the water column that was sampled by the ADCP.
The vertical distribution of the shearing stress In a classic paper Bowden et al. (1959) described the derivation of the shearing stress from vertical profiles of the tidal currents at a site near the coast of Anglesey (Figure 1). Consider a co-ordinate system with z=0 at the sea bed and z positive with height above the bed. Suppose that the flow is rectilinear along the x axis so that rotational effects can be ignored. (This is a good approximation at the site of the observations.) The depth-resolving equation for motion in the x direction can be written as:
where N is the eddy viscosity. The pressure term associated with the surface slope is unknown but can
Tidal currents in the Eastern Irish Sea 471
1.2
(a)
–1
U20 (m s )
1.0 0.8 0.6 0.4 0.2 0.0 1.0
(b)
0.8
R
0.6 0.4 0.2 0.0 0.08 0.07
(c)
–1
u* (m s )
0.06 0.05 0.04 0.03 0.02 0.01 0.00 10
–1
(d)
Z0 (m)
10 10
–3
10
–4
10
–5
10 10
CD
–2
–2
(e)
10
10
–6
–3
–4
0
10
20
30
40
50
Sample no.
F 5. Similar to Figure 4 but showing the results derived from the 30 min block-averaged velocities.
472 A. J. Elliott
From (2) we obtain
40 35 30
(m)
25
so that
20 15 10 5 0
5
10
15
20
25
30
35
40
45
50
on integrating from the bed to a height z. Hence
Block
F 6. Height of the logarithmic layer (m) during the 25 h of the observations. The dashed line shows the upper limit of the data coverage by the ADCP.
be estimated by integrating equation (2) between the bed and surface:
The second term on the LHS of equation (9) is equal to u*2 and the constant c can be determined by requiring the eddy stress at the bed to be equal to the stress associated with the friction velocity. Hence the shearing stress at a height z above the bed will be given by
which gives and the corresponding value of the eddy viscosity can be derived from where u is the depth-mean current which can be obtained by integrating the vertical profile or by approximation using the current speed at a height of 20 m above the bed. The second and third terms of the RHS of equation (4) are related to the bottom and surface stresses. The bottom stress can be estimated using
where u*2 is derived from the results shown in Figure 5. The surface stress was set to zero as the wind was light during the measurement period. Therefore the surface slope term can be derived using the expression
where the sign of u has been transferred to u* to discriminate between ebb and flood currents, and H was estimated using the dashed curve shown in Figure 2(b).
Of the three terms on the RHS of equation (10) the first term is defined at the bed and is then constant with height above the bed, the second term is an integral of the acceleration over the depth range from the bed up to height z and will therefore vary smoothly, while the final term is proportional to z. As a consequence the shearing stress can be expected to vary smoothly with height. In contrast, equation (11) involves division by the local vertical gradient in the horizontal current. This can at times be close to zero and can change sign due to small scale variability in the profile. As a consequence, the eddy viscosity profile may not be a smooth function of height above the bed, nor may it always have the expected positive sign. Figure 7 presents vertical profiles of the shear stress determined for each of the two ebb/flood periods (30 min blocks 1–23 and 24–50). The results of Bowden et al. (1959) showed the shear stress to be a
Tidal currents in the Eastern Irish Sea 473 –2
(N m ) –1.5 45 (a)
–1.0
–0.5
0.0
0.5 22
40
21
1.0
451 6 23
17
18 16
7
2.0
19
20
2
35
1.5
15
1418
30
(m)
25 20 15 10 5 0 –2
(N m ) –1.5 45 (b)
–1.0
–0.5
0.5 46
40 35
0.0
28 33 31
30 33
1.5
2.0
45 43
44 42
27 29 32
1.0
40
41
39
30
(m)
25 20 15 10 5 0
F 7. The computed vertical profiles of the shearing stress (N m2). The labels refer to the 30 min sample numbers shown in Figure 4.
maximum at the bed with a linear decrease towards the imposed surface boundary condition of zero stress. In the present case, however, only about 50% of the profiles show this form of profile; several of the profiles are suggestive of a fairly constant shear stress in the bottom 30 m of the water column. The asymmetry in the tidal flow is apparent in Figure 7, with the peak stress during the flood tide exceeding the ebb value by a factor of about 1·5.
The estimates of the eddy viscosity determined from equation (11) are presented in Figure 8. While the majority of the values are positive with a maximum value of about 0·5 m2 s 1 at mid-depth, some extreme and negative values were derived for the reasons discussed above. The dashed lines shown in Figure 8 represent the limiting values of the eddy viscosity that might be expected in a boundary layer flow for which N scales according to u*z where z
474 A. J. Elliott 2 –1
–1.00 45 (a)
–0.75
–0.50
40
(m s ) 0.00
–0.25
0.25
22
0.75
1.00
0.50
0.75
1.00
20 21 19 17
2
35
0.50
3
45 1413
6
18 16
15
28
30
(m)
25 20 15 10 5 0 2 –1
–0.75
–1.00 45 (b)
–0.50
(m s ) 0.00
–0.25
40 35
0.25
46 45 30
28
38
27 41 42 41 10 39 33
32
43 29 33
(m)
30 25 20 15 10 5 0
F 8. The computed eddy viscosities. The dashed lines show the envelope given by a u*z dependence.
represents the distance from the boundary and the amplitude of the tidal flow has been used to estimate the limiting value of u*. Figure 8 suggests that the dashed lines are envelopes to the values of N obtained in the bottom half of the water column; this supports
the suggestion that the flow is logarithmic in character with height above the bed over a substantial portion of the water column. The scatter shown in Figure 8 can be reduced if the logarithmic profile is assumed and the )u/)z term in
Tidal currents in the Eastern Irish Sea 475 2 –1
–1.00 45 (a)
–0.75
–0.50
–0.25
(m s ) 0.00
0.25
2212
40
0.50
0.75
20
19
18 17
2
3 6 4 16
35
14
1.00
15
7
8 13 9
0.25
0.50
30
(m)
25 20 15 10 5 0 2 –1
–1.00 45 (b)
–0.75
–0.50
–0.25
(m s ) 0.00
46
40 39
1.00
45 27 44 42
30
35
0.75
43 28 40
3313
4129
32
30
(m)
25 20 15 10 5 0
F 9. Similar to Figure 8 but using the derived logarithmic profile to estimate )u/)z in the denominator of (12).
the denominator of equation (11) is evaluated as u*/z. The resulting profiles are presented in Figure 9. The assumption of a logarithmic profile produces values of N that are positive and broadly similar to those derived using the local value of )u/)z especially in the lower portion of the water column.
Model simulations The one-dimensional closure scheme model described by Li and Elliott (1993) has been used to simulate the vertical structure of the flow and the associated shear stress. The governing equations are
476 A. J. Elliott
and
The eddy viscosity is calculated as N=qSm
(14)
where the length scale is specified using the method of Chen et al. (1988) which deals with the surface and bottom boundary layers separately but allows them to merge. The parameter Sm is a stability function related to the flux Richardson number (Mellor & Durbin, 1975) and the turbulent kinetic energy, q2/2, can be obtained from the solution of the Level 2 equation
the tidal cycle on both linear and logarithmic depth axes. Inspection of the profiles plotted against a logarithmic depth axis [Figure 10(b)] suggests that the logarithmic profile region of the flow extends up to a height of 20–30 m above the bed. The magnitude of the computed currents and stresses are comparable to those observed, but the numerical results lack the observed ebb/flood asymmetry due to the use of just M2 and S2 constituents. In agreement with the observations, the goodness-of-fit, between 0·7 m and 33 m above the bed, for a linear dependence between the simulated current and the logarithmic depth scale exceeded 0·99 for several of the profiles shown in Figure 10(b). However, the shearing stress distributions shown in Figure 10(b) display a near-bed maximum and a steady decrease towards a zero surface value. While this is compatible with the results of Bowden et al. (1959) it does not agree with about 50% of the profiles that are presented in Figure 7. Discussion and conclusions
when the water is vertically well-mixed. The horizontal pressure gradients were assumed to be of the form
where uT and vT are depth-averaged tidal velocity components. It is assumed that these can be expressed as (uT, vT)=(Hu cos(tGu), Hv cos(tG)) (17) where H and G are the amplitude and phase derived from a depth-averaged hydrodynamic model and is the tidal frequency. The numerical scheme used a sigma depth grid that had increased resolution near the top and bottom boundaries, and an implicit solution method was used to compute the flow structure. The amplitude and phase of the tidal currents at the experimental site were provided by the M2 and S2 constituents from the Proudman Laboratory 8 km model (R Proctor, pers. comm.) The model results are presented in Figure 10 for a value of zo of 510 3 m which represents a gravel/ pebble bed, comparable results being obtained for a zo of 510 4 m which represents a sand/shell bed. The figure shows the vertical structure of the eastward component of the current at hourly intervals within
The observations show that the tidal currents off the coast of Anglesey are strongly rectilinear in character and develop a logarithmic profile within 1.5 h after each slack water. This profile extends to a height of 30–40 m above the bed during the central portion of each ebb and flood tide. The apparent thickness of the logarithmic layer can be contrasted with earlier observations which had suggested that the constant stress portion of the boundary layer should not extend to a height of more than 10% of the water depth (e.g. Bowden et al., 1959), although more recent evidence (Lueck & Lu, 1997) has shown that significantly thicker logarithmic layers can be generated in tidal channels. The evidence for a logarithmic layer was supported by the computed shearing stress profiles which showed almost uniform values in the bottom 20–30 m of the water column for about 50% of the observed profiles. The log-profile was observed for most of the tidal cycle except for 1 h before and 1·5 h after slack water. The absence of a log-profile at such times, when the flow is non-steady, is a well documented characteristic of shallow water flow (e.g. Soulsby & Dyer, 1981). To examine the character of the high-frequency fluctuations in the measured currents, the 30-min block-averaged data were interpolated back onto the 5-min sampling interval and subtracted from the original data. The resulting time series for the currents at heights between 6 m and 20 m above the bed are shown in Figure 11. The high-frequency variability was of the order of 10% of the tidal speed and was most intense at the times of maximum flow. The
Tidal currents in the Eastern Irish Sea 477 –1
45
–1.0
–0.5
(m s ) 0.0
–1.0
–0.5
(m s ) 0.0
0.5
1.0
0.5
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(a)
40 35
(m)
30 25 20 15 10 5 –1
40 30
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20 1
(m)
10
5
–2
–1.0 40
–0.5
0.0
(N m ) 0.0
0.0
0.5
1.0
(c)
35 30
(m)
25 20 15 10
F 10. Results from a closure scheme model with zo =510 3 m. (a) Vertical profile of the east component of the tidal current (m s 1) at hourly intervals. (b) Same as (a) but plotted on a logarithmic depth scale. (c) The vertical profiles of the simulated shearing stress.
fluctuations shown in Figure 11 suggest vertical coherence at time scales that extend from 30 min to an hour. Unfortunately, such periodicities cannot be inferred since the data will be contaminated by the effects of aliasing if tidal eddies were present at the
measurement site. However, the vertical coherence of the instantaneous fluctuations can be estimated from a depth-time contour plot of the fluctuations which suggests that the fluctuations originate at the bed and propagate upwards into the water column with a
478 A. J. Elliott 0.8
0.6
–1
(m s )
0.4
0.2
0.0
–0.2 0
5
10
15
20
25
Time (h)
F 11. The high-frequency current fluctuations (m s 1) at heights of 6–20 m above the bed. Consecutive profiles are shifted vertically by 0·1 m s 1 to avoid overlapping. The lowest curve (with an origin at 0·0 m s 1) shows the fluctuations at a height of 6 m, the uppermost curve (with an origin offset at 0·7 m s 1) corresponds to the fluctuations at a height of 20 m.
phase lag of about 10 min over the 14 m covered by the data. If the development of the boundary layer is considered as resulting from the diffusion of low momentum away from the bed then the eddy viscosity, N, can be estimated using (Fischer et al. (1979), equation (5.10(a))
where H is the thickness of the boundary layer after a time . By taking values of 20 m for H and 1·5 h for we obtain an estimate for N of 0·03 m2 s 1. This estimate therefore represents an average value over the bottom 20 m in the first hour after the turn of the tide when the currents will be relatively weak. From Figure 5(c) the value of u* at such times will be of order 0·01 m s 1, if this value is substituted into the expression u*z with z equal to 10 m then an estimate for the viscosity of 0·04 m2 s 1 is obtained. This is consistent with the result derived from equation (18) above. The viscosity estimates shown in Figures 8 and 9 at a height of 10 m above the bed exceed this value because the profiles are not shown around the time of slack water, and those that are shown represent the conditions during the periods of stronger flow. For a rotating flow, the Ekman layer is the near bed layer in which the dynamics is non-geostrophic due to the effect of friction (Bowden, 1978). It has a thickness defined as:
where f is the angular frequency. For a purely tidal flow, f should be replaced by where =2/ 12·423600 s 1. However, if the flow is considered to be the superposition of clockwise (C/W) and anticlockwise (A/C) rotary components (Soulsby, 1983), bottom mixed layers will be generated with thickness given by
and
where c is a constant with a value of about 0·08 for the eastern Irish Sea. If we assume a value for u* of 2·510 2 m s 1, the following estimates for the thickness of the boundary layer are obtained: LE =71 m, + =9 m and =94 m. These values for the thickness of the bottom layers associated with the rotary components are compatible with the character of the tidal ellipse and its sense of rotation discussed earlier. The tidal ellipse analysis suggested that the rotary flow was positive in the bottom 10 m and negative in the upper portion of the water column. Both the Ekman thickness and the clockwise layer
Tidal currents in the Eastern Irish Sea 479
thickness estimated above are consistent with the observed thickness of the logarithmic layer which filled a substantial fraction of the water column. It is possible that the variability revealed in Figure 4 is related to bottom generated eddies that are advected by the tidal flow. In a similar tidal regime in the southern North Sea, Nimmo Smith and Thorpe (1999) reported the ubiquitous nature of tidal eddies that had diameters comparable to the water depth (50 m) and which were advected at a speed comparable to the depth-mean current. The North Sea eddies were estimated to cover about 20–30% of the sea surface. If similar eddies were present at the Irish Sea site it would imply that each feature would spend about 1 min crossing the beam of the ADCP sensor and that 1 or 2 eddies would cross the beam during the 5-min sample interval. As a consequence, the 5-min averaged data would contain aliased signals that were related to the eddy field. The reduced variability that resulted from the 30 min averaging of the ADCP currents (compare Figures 4 and 5) suggests that the raw velocity data contained fluctuations at time scales that are short compared to 30 min. The increase in zo and CD displayed by the analysis of the averaged data [Figure 5(d) and (e)] during each ebb and flood period might not be associated with a change of the bed form during each tide but could instead reflect the time scale for the development of the tidal eddies after a slack water and the time taken for the eddies to fill the water column from the bed up to the surface. Nimmo Smith and Thorpe (1999) reported that in the southern North Sea tidal eddies were evident at the surface 100–150 min after a period of slack water. This is consistent with the time scale suggested by Figures 4 and 5 for the redevelopment of the boundary layer following the turn of the tide. Images of the sea bed taken at a site within a few kilometres of the present observations show that the bed is composed of coarse sand, pebbles and shell fragments (E. I. S. Rees, pers. comm.). This is consistent with the mean values derived for the roughness length. There is no evidence of sea bed ripples on the photographs, therefore it is unlikely that the variation within the tidal cycle in the roughness and bottom drag was associated with morphological changes to the bed. Why do the present results on the thickness of the tidal logarithmic boundary layer contrast with the findings of Bowden et al. (1959)? The maximum speed of the tidal currents at the present site were 1·2 m s 1, compared to a value of 0·7 m s 1 in Red Wharf Bay. In addition, the offshore site was in water depths of around 50 m while the position near the
coast of Anglesey had a depth of about 20 m. As a consequence, while the vertical shear would be comparable at the two sites the bottom stress at the time of maximum flow would be four times stronger at the present site. It is possible that the ubiquitous tidal eddies reported by Nimmo Smith and Thorpe (1999) in the southern North Sea, which can explain some of the characteristics of the observed offshore boundary layer, may not have been present at the nearshore site sampled by Bowden et al. (1959) because they are restricted to relatively deep offshore waters. The observed vertical structure of the shearing stress (Figure 7) contrasts with the profiles computed by a conventional turbulence closure model (Figure 10). One possible explanation is that the closure model lacks an imbedded sub-model to describe the evolution and growth upwards through the water column of the bottom-generated tidal eddies. Observations and model studies of such features is a topic deserving of further study.
Acknowledgements This study was supported by the EU INTERREG II programme as part of the ERIS project, and also by the Chief Scientist’s Group and the Rural Marine Division of MAFF within contract AE1021.
References Bowden, K. F. 1978 Physical problems of the benthic boundary layer. Geophysical Surveys 3, 255–296. Bowden, K. F., Fairbairn, L. A. & Hughes, P. 1959 The distribution of shearing stresses in a tidal current. Geophysical Journal of the Royal Astronomical Society 2, 288–305. Chen, D., Horrigan, S. G. & Wang, D.-P. 1988 The late summer vertical nutrient mixing in Long Island Sound. Journal of Marine Research 46, 753–770. Czitrom, S. P. R. & Simpson, J. H. 1998 Intermittent stability and frontogenesis in an area influenced by land runoff. Journal of Geophysical Research 103, 10,369–10,376. Elliott, A. J. 1991 EUROSPILL: oceanographic processes and NW European shelf databases. Marine Pollution Bulletin 22, 548–553. Elliott, A. J., Dale, A. C. & Proctor, R. 1992 Modelling the movement of pollutants in the UK shelf seas. Marine Pollution Bulletin 24, 614–619. Fischer, H. B., Imberger, J., List, E. J., Koh, R. C. Y. & Brooks, N. H. 1979 Mixing in Inland and Coastal Waters. Academic Press, New York, 483 pp. Heathershaw, A. D. 1981 Comparisons of measured and predicted sediment transport rates in tidal currents. Marine Geology 42, 75–104. Holt, J. T. & James, I. D. 1999 A simulation of the southern North Sea in comparison with measurements from the North Sea Project. Part 1: Temperature. Continental Shelf Research 19, 1087–1112. Li, Z. & Elliott, A. J. 1993 Modelling the vertical structure of tidal currents and temperatures in the UK shelf seas. Estuarine, Coastal and Shelf Science 36, 549–564.
480 A. J. Elliott Lueck, R. G. & Lu, Y. 1997 The logarithmic layer in a tidal channel. Continental Shelf Research 17, 1785–1801. Mellor, G. L. & Durbin, P. A. 1975 The structure and dynamics of the ocean surface mixed layer. Journal of Physical Oceanography 5, 718–728. Nimmo Smith, W. A. M. & Thorpe, S. A. 1999 Surface effects of bottom-generated turbulence in a shallow sea. Nature 400, 251–254. Prandle, D. 1982 The vertical structure of tidal and other oscillatory flows. Continental Shelf Research 2, 191–207.
Souza, A. J. & Simpson, J. H. 1996 The modification of tidal ellipses by stratification in the Rhine: ROFI. Continental Shelf Research 16, 997–1007. Soulsby, R. L. 1983 The bottom boundary layer of shelf seas. In: Physical Oceanography of Coastal and Shelf Seas (Johns, B., ed.) Elsevier, 189–266. Soulsby, R. L. & Dyer, K. R. 1981 The form of the near-bed velocity profile in a tidally accelerating flow. Journal of Geophysical Research 86, 8,067–8,074.
The Boundary Layer Character of Tidal Currents in the Eastern Irish Sea A. J. Elliott Centre for Applied Oceanography, University of Wales, Bangor, Marine Science Laboratories, Menai Bridge, Anglesey LL59 5EY, U.K. Received 19 March 2001 and accepted in revised form 12 October 2001 The vertical profile of the tidal currents observed by a ship borne ADCP at a site in the eastern Irish Sea where the water depth is 50 m was logarithmic with height above the bed in the lower 50–60% of the water column except for periods of about 1 h either side of slack water. The maximum height of the boundary layer reached 40 m during the flood tide. The shearing stress, determined by the method of Bowden et al. (1959), showed a tendency to be uniform with height in the lower half of the water column while the eddy viscosity increased linearly away from the bed, scaling on u*z where u* is the friction velocity. Both of these features are characteristic of the logarithmic profile region of a boundary layer flow. Estimates of zo, the roughness length, and CD, the bottom drag coefficient, were around 10 3 m and 2·410 3, consistent with the known character of the bed. The scatter in the derived boundary layer parameters was reduced significantly by averaging the data within 30-min intervals which suggests the presence of tidal eddies having periods comparable to the 5-min sampling period of the ADCP. Short period fluctuations in the flow were coherent over the lower 20 m of the water column and propagated vertically away from the bed. 2002 Elsevier Science Ltd. All rights reserved. Keywords: tides; currents; Irish Sea; bottom boundary layer; log-profile; shear stress; bed roughness
Introduction The computation of the vertical structure of tidal currents and the parameterisation of the associated turbulence and vertical mixing form the focus of many numerical studies. The development of the tidal boundary layer is routinely computed by threedimensional hydrodynamic codes in shelf sea simulations, yet the target of such studies is often directed towards an understanding of the seasonal and annual cycles (e.g. Holt & James, 1999). Moreover, although data for model validation may be collected at hourly and shorter time scales, the analysis often concentrates on much longer periodicities. Therefore, as part of a programme of model development for operational forecasting over the Irish and Celtic Seas, current data were collected with a relatively short sampling period so that the development of the tidal boundary layer could be documented at time scales ranging from minutes to hours. This paper reports on the observed character of the tidal boundary layer, the results from the operational model will be presented separately. During 31 August and 1 September 1999, the tidal currents were monitored at a site in the eastern Irish Sea (Figure 1) by an anchored ship fitted with a hull mounted ADCP. The system used a 5-min averaging interval and the vertical resolution was set at 2 m. The 0272–7714/02/090465+16 $35.00/0
sensor was mounted at 3·5 m below the waterline and the first data were recorded at a depth of 6·5 m below the surface. The measurements took place during a period of spring tides with light winds and low wave energy. The water was observed to be vertically wellmixed at the site (5332.8 N, 43.5 W) which is consistent with the strength of the tidal flow and the position of the site at the western limit of the fresh water stratification that occurs within Liverpool Bay (Czitrom & Simpson, 1998). The observations commenced about 1 h after local high water on 31 August and continued for 25 h. Since the ADCP was hull mounted on a surface vessel it was necessary to process the data to derive bed referenced currents. This was achieved by inspecting each velocity profile and finding the deepest bin containing valid data values. The centre of the next lowest bin was then assumed to correspond to the bed. This can introduce an error of 1 m into the height of the profile; the consequence of this error will be discussed in a later section. Figure 2 shows a time series of the east and north components of the flow at a height of 20 m above the bed. The 20 m flow was selected as it will approximate the depth-mean current in water that is 50 m deep (Prandle, 1982). For comparison in Figure 2(a) the tidal currents predicted by a depth-integrated two-dimensional tidal 2002 Elsevier Science Ltd. All rights reserved.
466 A. J. Elliott
50 m
53.6N
50 m
53.4
Liverpool Bay
25 m
Latitude
Anglesey 53.2 North Wales
N
53.0
52.8
52.6
50 m
–4.75
25 m
–4.50
–4.25
–4.00
–3.75
–3.50
–3.25
–3.00W
Longitude
F 1. The experimental site (+) in the eastern Irish Sea. The triangle near the coast of Anglesey marks the location of the measurements by Bowden et al. (1959). Water depths are contoured in metres.
model developed at the Proudman Oceanographic Laboratory (Elliott et al., 1992) are shown by dashed curves. This demonstrates the robustness of the tidal currents from the two-dimensional model which are used for oil spill forecasting and other practical simulations (Elliott, 1991). The observed currents were comparable in magnitude but led the simulated currents by a phase of about 20 which corresponds to an interval of 45 min. Figure 2(b) shows the water depth derived directly from the bottom echo of the ADCP signal, illustrating that the tidal range is around 7 m during spring tides. The depth derived from the ADCP bottom echo has a relatively coarse resolution, therefore a semi-diurnal fit to the observed depth was derived and will be used later in an analysis of the shearing stress. This is shown by the dashed curve in Figure 2(b). The agreement between the water depth derived directly from the ADCP bottom echo and by inspecting the velocity data in the near-bed cells will be discussed in a later paragraph. The tidal currents displayed an asymmetry between the ebb and flood directions. (Flood is defined as the period when the eastward component of the flow is
positive and sea level is rising in the eastern Irish Sea. Peak flood flow occurred around 23:00h (UTC) on 31 August and at 12:00h on 1 September.) The flood tide lasted about 1 h less than the ebb, but this was compensated for by a flow that was approximately 20% stronger. The tidal axis was aligned to within 8 of due east and the peak flood speed was of the order of 1·2 m s 1. Contour plots of the flow resolved along the major and minor axes of the tidal motion illustrated the vertical motion of the ship due to the tidal motion of the sea surface. At 10–12 and 22–24 h into the observations, the contoured velocities extended to a height of 44 m above the bed. To this figure must be added the 6·5 m of the ADCP below the water surface to give a total depth of water of 50·5 m. This can be compared to the depths of 49·5–49·8 m and 49·0–49·5 m provided by the bottom echo results (Figure 2) which were derived independently of the method that inspected the velocity data in the near-bottom data bins. The good agreement between the two methods suggests that the height origin for the velocity profiles was placed at the bed with an accuracy of 0·5–1·0 m.
Tidal currents in the Eastern Irish Sea 467 1.5 (a) 1.0
–1
(m s )
0.5 0.0 –0.5 –1.0 –1.5
31 Aug
1999
1 Sep
31 Aug
1999
1 Sep
50 49
(b)
48 (m)
47 46 45 44 43 42
F 2. (a) The east and north components of the flow at a height of 20 m above the bed (solid curves), also shown are the depth-mean currents simulated by the Proudman Oceanographic Laboratory tidal model (dashed curves). (b) The water depth measured by the bottom echo of the ADCP (solid curve) and the curve fitted to the data by tidal analysis (dashed curve).
Results The vertical structure of the tidal flow The results from a tidal analysis of the eastward component of the flow are presented in Figure 3(a). The analysis was made for a single semi-diurnal tidal constituent that could represent the combined effect of the M2 and S2 components. The amplitude of the current (solid curve) increases steadily up to a height of 35 m above the bed and has a value of about 1 m s 1 at mid-depths. The variability above 35 m is due to the vertical tidal motion of the sea surface which resulted in the uppermost data bins only containing data values during periods of high water. Figure 3(b) shows the vertical structure plotted against a logarithmic depth scale. A striking feature of the flow is the logarithmic increase (shown by a dashed line) in tidal speed from a height of 6 m up to 30 m above the bed. The speeds measured at heights of 2 m and 4 m above the bed clearly do not fit the linear relationship. This is to be expected as the data from downward looking ADCP sensors are unreliable in the bottom 10% of the water column. These two data points were therefore not used in the subsequent
analyses. The phase profiles presented in Figure 3 show a steady increase with height above the bed. The near-bed flow led the surface current by about 5 which corresponds to a surface lag of about 10 min. The variability of the results above a height of 35 m can be removed by using a sigma-level analysis that interpolates each vertical profile onto a grid that covers the region between the bed and the highest valid data bin. The results from such an analysis, which treated both the E–W and N–S components of the flow so that the tidal ellipse parameters and the rotary components could be derived, showed that the direction of the major axis was aligned at about 8 to the south of east at all depths. In addition, the flow was more strongly rectilinear towards the bed and the tidal motion was predominantly anti-clockwise in the bottom 10 m and clockwise in the upper part of the water column. This characteristic of the rotary components has been observed in the southern North Sea by Souza and Simpson (1996). The logarithmic increase in current speed with height above the bed suggests the presence of a boundary layer flow which can be described by the standard formulation (Bowden, 1978):
468 A. J. Elliott –1
–1
Amplitude (m s ) 0.0 0.2 50 (a)
0.4
0.6
0.8
Amplitude (m s ) 1.0
1.2
0.0 0.2 50 (b)
0.4
0.6
0.8
1.0
1.2
45 40 20 35
(m)
30
10
1
25 5
20 15 10
2
5 0 230
235
240
245
250
Phase (deg)
10
0
230
235
240
245
250
Phase (deg)
F 3. (a) Vertical structure of the amplitude (solid curve, m s 1) and phase (dashed curve, arbitrary degrees) of the tidal flow. (b) Same results plotted with a logarithmic depth scale.
where u* is the friction velocity, zo is the roughness length and is Von Karman’s constant (0·40). The data points in the height range from 6–20 m above the bed when fitted to (1) gave estimates for u* and zo of 0·031 m s 1 and 6·410 5 m. If the logarithmic profile is used to estimate the speed, u1, at a height 1 m above the bed then the quadratic drag coefficient can be derived by equating CDu12 and u*2. This produced an estimated for the drag coefficient of 1·7210 3. However, since a tidal analysis parameterises the envelope of the tidal flow these estimates should be regarded as extreme values and not as averages over the tidal cycle. Estimates of boundary layer flow parameters derived via equation (1) are sensitive to errors in the heights assigned to the velocity observations. To assess the impact of such errors the flow profile shown in Figure 3 was shifted vertically by 1 m as this represents the uncertainty in the position of the bed when determined using ADCP bins with a 2-m vertical resolution. The two estimates for u*, zo and CD derived in this manner had values of 91/109%, 26/
295%, and 77/127% of those quoted above. Thus, taking the ratio of the extreme values, u* varied by a factor of 1.20, zo varied by a factor of 11·4 and CD varied by a factor of 1·65. As a consequence, while the estimates of u* and CD are fairly robust, the values derived for zo should be regarded more cautiously. The time varying character of the boundary layer The profiles shown in Figure 3 represent the envelope of the tidal speed with height above the bed, and while the envelope suggests a logarithmic profile in the lower half of the water column this does not guarantee that instantaneous profiles will display this characteristic. The analysis was therefore repeated with the individual 5-min profiles by determining the goodness-of-fit to a logarithmic relationship between the heights of 6 m and 20 m above the bed (excluding the values at 2 m and 4 m for reasons detailed above). The results are presented in Figure 4 which shows the tidal asymmetry in the eastward component of the flow at a height of 20 m above the bed [Figure 4(a)] and the correlation coefficient [Figure 4(b)] which expresses the goodness-of-fit to equation (1). The correlation coefficient shows that the boundary layer
Tidal currents in the Eastern Irish Sea 469
1.2
(a)
–1
U20 (m s )
1.0 0.8 0.6 0.4 0.2 0.0 1.0
(b)
0.8
R
0.6 0.4 0.2 0.0 0.08
(c)
0.07 –1
u* (m s )
0.06 0.05 0.04 0.03 0.02 0.01 0.00 10
–1
(d)
Z0 (m)
10 10
–3
10
–4
10
–5
10 10
CD
–2
–6
–2
(e)
–3
10
10
–4
0
50
100
150 Sample no.
200
250
300
F 4. (a) Speed (m s 1) at z=20 m for the 5 min data. The dashed lines show the times of slack water. (b) The correlation coefficient, R, for the goodness-of-fit to a logarithmic profile between the heights of 6 m and 22 m above the bed. (c) The friction velocity, u* (m s 1), derived from the logarithmic profile. (d) The roughness length, zo (m). (e) The quadratic drag coefficient, CD.
470 A. J. Elliott
was well established up to a height of 20 m apart from 1 h before and 1·5 h after slack water. High correlation (>0·95) was achieved for periods of up to 3 h during the central portions of both the flood and ebb flow. The derivation of the boundary layer parameters was restricted to profiles when the mean current speed increased with height above the bed; this excluded the periods around slack water. For profiles that showed an increasing speed with height the parameters u*, zo and CD were estimated and they are presented in [Figure 4(c)–4(e)]. If the tidal flow consisted of a pure harmonic current at the semi-diurnal frequency, the friction velocity might be expected to vary in a comparable harmonic manner while the roughness length and the drag coefficient should be constant throughout the tidal period. However, the results shown in Figure 4, which show considerable scatter and are suggestive of envelopes rather than curves, are not consistent with such a hypothesis. The variability in the boundary layer parameters was reduced significantly by averaging the velocity at each depth within 30 min non-overlapping blocks (Figure 5). Repeating the least-squares logarithmic profile analysis with 30 min blocked data produced similar results for the tidal speed at a height of 20 m [Figure 5(a)] but increased the goodness-offit [Figure 5(b)] so that the correlation coefficient exceeded 0·98 during the central portion of the ebb and flood tide. This extremely close agreement to a logarithmic profile in the lower 50% of the water column is a striking feature of the data. The averaging process removed much of the variability in the derived estimates of the boundary layer parameters. Figure 5(c) suggests that the friction velocity increased asymmetrically during each ebb and flood period with a rapid rise after slack water followed by a more gradual decrease as the next slack approached. The value of both the roughness length and the drag coefficient increased between consecutive periods of slack water and the mean values of zo and CD derived from the 30 min averaged velocities were 8·810 4 m and 2·410 3. These values can be compared with those reported by Bowden et al. (1959) who measured the vertical current structure near to the north coast of Anglesey in water that was around 20 m deep and obtained values of 1·610 3 m and 3·510 3. Admiralty charts of the present site suggest that the sea bed is composed of sand, broken shells, gravel and pebbles. Heathershaw (1981) reported that the roughness length appropriate for such a bed varies between 0·3010 3 m (for sand and shells) and 3·010 3 m (for gravel and pebbles), while the appropriate drag coefficient lies in the range
2·410 3 to 4·710 3. In consequence, the values derived from the logarithmic layer analysis lie within the expected ranges. Lueck and Lu (1997) observed a tidal boundary layer that extended to a height of 20 m above the bed in a 30 m deep tidal channel. They estimated the thickness of the logarithmic layer by computing the correlation coefficient between u and ln(z) as a function of height above the bed. To do this they started with the lowest three data points and computed the agreement to a linear fit. If this value exceeded a specified value they added in the fourth data point and recalculated the fit. This process was repeated until a height was reached at which the fit was no longer significant at the chosen confidence level. The logarithmic layer thickness was then defined as the height of the uppermost data point for which a significant linear fit was achieved. A similar analysis was applied to the 30 min averaged data and the layer thickness was defined as the region within which the correlation exceeded 0·98. (This ensures a highly significant fit. For example at 20 m above the bed the regression would involve eight data points, for this number of points a regression value above 0·75 is significant at the 99% level.) The results are presented in Figure 6 which shows the thickness of the logarithmic layer and also the upper limit (dashed curve) of the ADCP observations. It is apparent that the layer extended to a height of 30–40 m above the bed and its thickness was greater during the stronger flood tide than during the ebb. During both of the flood periods Figure 6 suggests that the logarithmic layer extended over the whole portion of the water column that was sampled by the ADCP.
The vertical distribution of the shearing stress In a classic paper Bowden et al. (1959) described the derivation of the shearing stress from vertical profiles of the tidal currents at a site near the coast of Anglesey (Figure 1). Consider a co-ordinate system with z=0 at the sea bed and z positive with height above the bed. Suppose that the flow is rectilinear along the x axis so that rotational effects can be ignored. (This is a good approximation at the site of the observations.) The depth-resolving equation for motion in the x direction can be written as:
where N is the eddy viscosity. The pressure term associated with the surface slope is unknown but can
Tidal currents in the Eastern Irish Sea 471
1.2
(a)
–1
U20 (m s )
1.0 0.8 0.6 0.4 0.2 0.0 1.0
(b)
0.8
R
0.6 0.4 0.2 0.0 0.08 0.07
(c)
–1
u* (m s )
0.06 0.05 0.04 0.03 0.02 0.01 0.00 10
–1
(d)
Z0 (m)
10 10
–3
10
–4
10
–5
10 10
CD
–2
–2
(e)
10
10
–6
–3
–4
0
10
20
30
40
50
Sample no.
F 5. Similar to Figure 4 but showing the results derived from the 30 min block-averaged velocities.
472 A. J. Elliott
From (2) we obtain
40 35 30
(m)
25
so that
20 15 10 5 0
5
10
15
20
25
30
35
40
45
50
on integrating from the bed to a height z. Hence
Block
F 6. Height of the logarithmic layer (m) during the 25 h of the observations. The dashed line shows the upper limit of the data coverage by the ADCP.
be estimated by integrating equation (2) between the bed and surface:
The second term on the LHS of equation (9) is equal to u*2 and the constant c can be determined by requiring the eddy stress at the bed to be equal to the stress associated with the friction velocity. Hence the shearing stress at a height z above the bed will be given by
which gives and the corresponding value of the eddy viscosity can be derived from where u is the depth-mean current which can be obtained by integrating the vertical profile or by approximation using the current speed at a height of 20 m above the bed. The second and third terms of the RHS of equation (4) are related to the bottom and surface stresses. The bottom stress can be estimated using
where u*2 is derived from the results shown in Figure 5. The surface stress was set to zero as the wind was light during the measurement period. Therefore the surface slope term can be derived using the expression
where the sign of u has been transferred to u* to discriminate between ebb and flood currents, and H was estimated using the dashed curve shown in Figure 2(b).
Of the three terms on the RHS of equation (10) the first term is defined at the bed and is then constant with height above the bed, the second term is an integral of the acceleration over the depth range from the bed up to height z and will therefore vary smoothly, while the final term is proportional to z. As a consequence the shearing stress can be expected to vary smoothly with height. In contrast, equation (11) involves division by the local vertical gradient in the horizontal current. This can at times be close to zero and can change sign due to small scale variability in the profile. As a consequence, the eddy viscosity profile may not be a smooth function of height above the bed, nor may it always have the expected positive sign. Figure 7 presents vertical profiles of the shear stress determined for each of the two ebb/flood periods (30 min blocks 1–23 and 24–50). The results of Bowden et al. (1959) showed the shear stress to be a
Tidal currents in the Eastern Irish Sea 473 –2
(N m ) –1.5 45 (a)
–1.0
–0.5
0.0
0.5 22
40
21
1.0
451 6 23
17
18 16
7
2.0
19
20
2
35
1.5
15
1418
30
(m)
25 20 15 10 5 0 –2
(N m ) –1.5 45 (b)
–1.0
–0.5
0.5 46
40 35
0.0
28 33 31
30 33
1.5
2.0
45 43
44 42
27 29 32
1.0
40
41
39
30
(m)
25 20 15 10 5 0
F 7. The computed vertical profiles of the shearing stress (N m2). The labels refer to the 30 min sample numbers shown in Figure 4.
maximum at the bed with a linear decrease towards the imposed surface boundary condition of zero stress. In the present case, however, only about 50% of the profiles show this form of profile; several of the profiles are suggestive of a fairly constant shear stress in the bottom 30 m of the water column. The asymmetry in the tidal flow is apparent in Figure 7, with the peak stress during the flood tide exceeding the ebb value by a factor of about 1·5.
The estimates of the eddy viscosity determined from equation (11) are presented in Figure 8. While the majority of the values are positive with a maximum value of about 0·5 m2 s 1 at mid-depth, some extreme and negative values were derived for the reasons discussed above. The dashed lines shown in Figure 8 represent the limiting values of the eddy viscosity that might be expected in a boundary layer flow for which N scales according to u*z where z
474 A. J. Elliott 2 –1
–1.00 45 (a)
–0.75
–0.50
40
(m s ) 0.00
–0.25
0.25
22
0.75
1.00
0.50
0.75
1.00
20 21 19 17
2
35
0.50
3
45 1413
6
18 16
15
28
30
(m)
25 20 15 10 5 0 2 –1
–0.75
–1.00 45 (b)
–0.50
(m s ) 0.00
–0.25
40 35
0.25
46 45 30
28
38
27 41 42 41 10 39 33
32
43 29 33
(m)
30 25 20 15 10 5 0
F 8. The computed eddy viscosities. The dashed lines show the envelope given by a u*z dependence.
represents the distance from the boundary and the amplitude of the tidal flow has been used to estimate the limiting value of u*. Figure 8 suggests that the dashed lines are envelopes to the values of N obtained in the bottom half of the water column; this supports
the suggestion that the flow is logarithmic in character with height above the bed over a substantial portion of the water column. The scatter shown in Figure 8 can be reduced if the logarithmic profile is assumed and the )u/)z term in
Tidal currents in the Eastern Irish Sea 475 2 –1
–1.00 45 (a)
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2212
40
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20
19
18 17
2
3 6 4 16
35
14
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8 13 9
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46
40 39
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45 27 44 42
30
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43 28 40
3313
4129
32
30
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25 20 15 10 5 0
F 9. Similar to Figure 8 but using the derived logarithmic profile to estimate )u/)z in the denominator of (12).
the denominator of equation (11) is evaluated as u*/z. The resulting profiles are presented in Figure 9. The assumption of a logarithmic profile produces values of N that are positive and broadly similar to those derived using the local value of )u/)z especially in the lower portion of the water column.
Model simulations The one-dimensional closure scheme model described by Li and Elliott (1993) has been used to simulate the vertical structure of the flow and the associated shear stress. The governing equations are
476 A. J. Elliott
and
The eddy viscosity is calculated as N=qSm
(14)
where the length scale is specified using the method of Chen et al. (1988) which deals with the surface and bottom boundary layers separately but allows them to merge. The parameter Sm is a stability function related to the flux Richardson number (Mellor & Durbin, 1975) and the turbulent kinetic energy, q2/2, can be obtained from the solution of the Level 2 equation
the tidal cycle on both linear and logarithmic depth axes. Inspection of the profiles plotted against a logarithmic depth axis [Figure 10(b)] suggests that the logarithmic profile region of the flow extends up to a height of 20–30 m above the bed. The magnitude of the computed currents and stresses are comparable to those observed, but the numerical results lack the observed ebb/flood asymmetry due to the use of just M2 and S2 constituents. In agreement with the observations, the goodness-of-fit, between 0·7 m and 33 m above the bed, for a linear dependence between the simulated current and the logarithmic depth scale exceeded 0·99 for several of the profiles shown in Figure 10(b). However, the shearing stress distributions shown in Figure 10(b) display a near-bed maximum and a steady decrease towards a zero surface value. While this is compatible with the results of Bowden et al. (1959) it does not agree with about 50% of the profiles that are presented in Figure 7. Discussion and conclusions
when the water is vertically well-mixed. The horizontal pressure gradients were assumed to be of the form
where uT and vT are depth-averaged tidal velocity components. It is assumed that these can be expressed as (uT, vT)=(Hu cos(tGu), Hv cos(tG)) (17) where H and G are the amplitude and phase derived from a depth-averaged hydrodynamic model and is the tidal frequency. The numerical scheme used a sigma depth grid that had increased resolution near the top and bottom boundaries, and an implicit solution method was used to compute the flow structure. The amplitude and phase of the tidal currents at the experimental site were provided by the M2 and S2 constituents from the Proudman Laboratory 8 km model (R Proctor, pers. comm.) The model results are presented in Figure 10 for a value of zo of 510 3 m which represents a gravel/ pebble bed, comparable results being obtained for a zo of 510 4 m which represents a sand/shell bed. The figure shows the vertical structure of the eastward component of the current at hourly intervals within
The observations show that the tidal currents off the coast of Anglesey are strongly rectilinear in character and develop a logarithmic profile within 1.5 h after each slack water. This profile extends to a height of 30–40 m above the bed during the central portion of each ebb and flood tide. The apparent thickness of the logarithmic layer can be contrasted with earlier observations which had suggested that the constant stress portion of the boundary layer should not extend to a height of more than 10% of the water depth (e.g. Bowden et al., 1959), although more recent evidence (Lueck & Lu, 1997) has shown that significantly thicker logarithmic layers can be generated in tidal channels. The evidence for a logarithmic layer was supported by the computed shearing stress profiles which showed almost uniform values in the bottom 20–30 m of the water column for about 50% of the observed profiles. The log-profile was observed for most of the tidal cycle except for 1 h before and 1·5 h after slack water. The absence of a log-profile at such times, when the flow is non-steady, is a well documented characteristic of shallow water flow (e.g. Soulsby & Dyer, 1981). To examine the character of the high-frequency fluctuations in the measured currents, the 30-min block-averaged data were interpolated back onto the 5-min sampling interval and subtracted from the original data. The resulting time series for the currents at heights between 6 m and 20 m above the bed are shown in Figure 11. The high-frequency variability was of the order of 10% of the tidal speed and was most intense at the times of maximum flow. The
Tidal currents in the Eastern Irish Sea 477 –1
45
–1.0
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F 10. Results from a closure scheme model with zo =510 3 m. (a) Vertical profile of the east component of the tidal current (m s 1) at hourly intervals. (b) Same as (a) but plotted on a logarithmic depth scale. (c) The vertical profiles of the simulated shearing stress.
fluctuations shown in Figure 11 suggest vertical coherence at time scales that extend from 30 min to an hour. Unfortunately, such periodicities cannot be inferred since the data will be contaminated by the effects of aliasing if tidal eddies were present at the
measurement site. However, the vertical coherence of the instantaneous fluctuations can be estimated from a depth-time contour plot of the fluctuations which suggests that the fluctuations originate at the bed and propagate upwards into the water column with a
478 A. J. Elliott 0.8
0.6
–1
(m s )
0.4
0.2
0.0
–0.2 0
5
10
15
20
25
Time (h)
F 11. The high-frequency current fluctuations (m s 1) at heights of 6–20 m above the bed. Consecutive profiles are shifted vertically by 0·1 m s 1 to avoid overlapping. The lowest curve (with an origin at 0·0 m s 1) shows the fluctuations at a height of 6 m, the uppermost curve (with an origin offset at 0·7 m s 1) corresponds to the fluctuations at a height of 20 m.
phase lag of about 10 min over the 14 m covered by the data. If the development of the boundary layer is considered as resulting from the diffusion of low momentum away from the bed then the eddy viscosity, N, can be estimated using (Fischer et al. (1979), equation (5.10(a))
where H is the thickness of the boundary layer after a time . By taking values of 20 m for H and 1·5 h for we obtain an estimate for N of 0·03 m2 s 1. This estimate therefore represents an average value over the bottom 20 m in the first hour after the turn of the tide when the currents will be relatively weak. From Figure 5(c) the value of u* at such times will be of order 0·01 m s 1, if this value is substituted into the expression u*z with z equal to 10 m then an estimate for the viscosity of 0·04 m2 s 1 is obtained. This is consistent with the result derived from equation (18) above. The viscosity estimates shown in Figures 8 and 9 at a height of 10 m above the bed exceed this value because the profiles are not shown around the time of slack water, and those that are shown represent the conditions during the periods of stronger flow. For a rotating flow, the Ekman layer is the near bed layer in which the dynamics is non-geostrophic due to the effect of friction (Bowden, 1978). It has a thickness defined as:
where f is the angular frequency. For a purely tidal flow, f should be replaced by where =2/ 12·423600 s 1. However, if the flow is considered to be the superposition of clockwise (C/W) and anticlockwise (A/C) rotary components (Soulsby, 1983), bottom mixed layers will be generated with thickness given by
and
where c is a constant with a value of about 0·08 for the eastern Irish Sea. If we assume a value for u* of 2·510 2 m s 1, the following estimates for the thickness of the boundary layer are obtained: LE =71 m, + =9 m and =94 m. These values for the thickness of the bottom layers associated with the rotary components are compatible with the character of the tidal ellipse and its sense of rotation discussed earlier. The tidal ellipse analysis suggested that the rotary flow was positive in the bottom 10 m and negative in the upper portion of the water column. Both the Ekman thickness and the clockwise layer
Tidal currents in the Eastern Irish Sea 479
thickness estimated above are consistent with the observed thickness of the logarithmic layer which filled a substantial fraction of the water column. It is possible that the variability revealed in Figure 4 is related to bottom generated eddies that are advected by the tidal flow. In a similar tidal regime in the southern North Sea, Nimmo Smith and Thorpe (1999) reported the ubiquitous nature of tidal eddies that had diameters comparable to the water depth (50 m) and which were advected at a speed comparable to the depth-mean current. The North Sea eddies were estimated to cover about 20–30% of the sea surface. If similar eddies were present at the Irish Sea site it would imply that each feature would spend about 1 min crossing the beam of the ADCP sensor and that 1 or 2 eddies would cross the beam during the 5-min sample interval. As a consequence, the 5-min averaged data would contain aliased signals that were related to the eddy field. The reduced variability that resulted from the 30 min averaging of the ADCP currents (compare Figures 4 and 5) suggests that the raw velocity data contained fluctuations at time scales that are short compared to 30 min. The increase in zo and CD displayed by the analysis of the averaged data [Figure 5(d) and (e)] during each ebb and flood period might not be associated with a change of the bed form during each tide but could instead reflect the time scale for the development of the tidal eddies after a slack water and the time taken for the eddies to fill the water column from the bed up to the surface. Nimmo Smith and Thorpe (1999) reported that in the southern North Sea tidal eddies were evident at the surface 100–150 min after a period of slack water. This is consistent with the time scale suggested by Figures 4 and 5 for the redevelopment of the boundary layer following the turn of the tide. Images of the sea bed taken at a site within a few kilometres of the present observations show that the bed is composed of coarse sand, pebbles and shell fragments (E. I. S. Rees, pers. comm.). This is consistent with the mean values derived for the roughness length. There is no evidence of sea bed ripples on the photographs, therefore it is unlikely that the variation within the tidal cycle in the roughness and bottom drag was associated with morphological changes to the bed. Why do the present results on the thickness of the tidal logarithmic boundary layer contrast with the findings of Bowden et al. (1959)? The maximum speed of the tidal currents at the present site were 1·2 m s 1, compared to a value of 0·7 m s 1 in Red Wharf Bay. In addition, the offshore site was in water depths of around 50 m while the position near the
coast of Anglesey had a depth of about 20 m. As a consequence, while the vertical shear would be comparable at the two sites the bottom stress at the time of maximum flow would be four times stronger at the present site. It is possible that the ubiquitous tidal eddies reported by Nimmo Smith and Thorpe (1999) in the southern North Sea, which can explain some of the characteristics of the observed offshore boundary layer, may not have been present at the nearshore site sampled by Bowden et al. (1959) because they are restricted to relatively deep offshore waters. The observed vertical structure of the shearing stress (Figure 7) contrasts with the profiles computed by a conventional turbulence closure model (Figure 10). One possible explanation is that the closure model lacks an imbedded sub-model to describe the evolution and growth upwards through the water column of the bottom-generated tidal eddies. Observations and model studies of such features is a topic deserving of further study.
Acknowledgements This study was supported by the EU INTERREG II programme as part of the ERIS project, and also by the Chief Scientist’s Group and the Rural Marine Division of MAFF within contract AE1021.
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