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Diffusion in Irish coastal waters
Estuarine, Coastal and Shelf Science (1997) 44 (Supplement A), 15-23
Diffusion in Irish Coastal Waters A.
J. Elliote, A. G.
Bar-r" and D. Kerman"
"Uniuersity of Wales, Bangor, Marine Science Laboratories, Menai Bridge, Anglesey LL59 5EY, U.K. bKirk McClure Morton, Elmwood House, 74 Boucher Road, Belfast BT12 6RZ, U.K. cMCS Iniernational, Lismoyle House, Merchants Road, Galway, Ireland
Results are presented from dye-diffusion experiments performed at a selection of near-shore and estuarine sites around the coastline of Ireland. The data from discrete dye releases have been analysed to derive along- and across-patch horizontal diffusivities, and estimates are presented of both Fickian and non-Fickian diffusion coefficients. Most of the sites were characterized by strong tidal currents so that the dye became vertically well-mixed soon after release. Consequently, the mixing was found to be tidally dominated and, to a reasonable approximation, the magnitude of the one-dimensional (radial spreading) horizontal diffusivity (m? s - 1) was equal to the tidal current speed (m s - 1). Wind-induced mixing was only of secondary importance, but this may be a consequence of the experiments having taken place during conditions of generally light winds. In general, the observed mixing rates agree with those © 1997 Academic Press Limited predicted by oceanic diffusion diagrams. Keywords: diffusion; diffusivity; tidal mixing; wind mixing; Ireland
Introduction Diffusion experiments, involving both discrete and continuous releases of rhodamine dye, are commonly performed to determine the mixing characteristics of a near-shore site. Parameterized as diffusion coefficients or diffusivities, the derived mixing characteristics are then used in numerical simulations to assess the likely impact of an effluent. Such studies usually concern the selection of a site for a sewage outfall or the modification of an existing industrial release. In recent years, however, the growth of the fish farm industry has prompted dye experiments that have been directed at determining mixing characteristics and flushing times as part of environmental impact assessment studies. Models of the diffusion from a continuous source in the sea often make use of the diffusion velocity concept (Bowden & Lewis, 1973), which relates the diffusivity linearly to the diffusion time, estimates of the diffusion velocity being derived by field trials. The weakness of the approach is that it is not clear how data collected during calm conditions can be used to predict the spreading during rougher weather. To overcome this, a diffusion model should isolate the spreading due to tide and wind so that the effects of weather can be quantified; such a technique would then lead to improved predictions of spreading during a range of tide and weather conditions. The goal of this study was to investigate the dependence of the mixing coefficients on the local environmental conditions, including such parameters as water depth, 0272-7714/97/44AOI5+09 $25.00/0
tidal current and wind speed. Empirical formulae relating the mixing to the tidal currents and wind have been derived for use in applied studies. This paper presents the results from a series of experimental releases conducted over a period of several years, and attempts to parameterize the mixing rates in terms of the ambient tidal currents and winds. The majority of the sites were in estuarine, embayment and lough locations, with a few sites being close to the shoreline on an open coastline (Figure 1 and Table 1). For completeness, one inland lough (Lough Neagh, the largest freshwater lake in the U.K. and Ireland) has been included in the dataset. At each site, discrete dye releases were usually made at times of both spring and neap tides, the experiments being repeated during ebb and flood conditions. Unfortunately, since the experiments were performed using a small sampling craft, the observations are biased towards low wind speed conditions. When analysing dye data, it is usual to make an analogy between the solution of the one-dimensional diffusion equation and the normal probability function. The one-dimensional diffusion equation, with a constant diffusivity K:
8C =~(K8C)
at ax
(1)
ax
has the solution: C(x, t)
=
(x
2
Q 1/2 exp - 4Kt ) (4nKt)
©
(2)
1997 Academic Press Limited
16 A.
J.
Elliott et al,
where Q is the amount of substance released at time t= O (Bowden, 1983). This has the same form as the normal probability function, or Gaussian curve, with variance r:? and mean f.l given by: 54.5°
f( x)
1 exp ( (T(2n) 1/ 2
(3)
provided that: 53.5°
(4) and the diffusive spreading is measured with respect to the centroid of the dye patch. Equation (4) is the method by which diffusivities are usually derived during the analysis of a dye dataset . When the diffusivity, K, is not a constant, it can be estimated (Bowden, 1983) from the time derivative of the variance using:
sz.s-
51.5°
1
d(T 2
K=--
(5)
2 dt
FIGURE 1. Locations of the dye releases.
TABLE 1. Summary of the dye-diffusion statistics
No .
Location
Date
s;
~
KH
a
m
H
W
U
Tide
1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10
Lough Foyle Cushendall Cushendall Lame Lough Lame Lough Belfast Lough Belfast Lough Strangford Lough Strangford Lough Dundalk Bay Dundalk Bay Courtown Courtown Kinsale Kinsale Bantry Bay Bantry Bay Westport Bay Westport Bay Achill Island Achill Island Donegal Harbour Lough Neagh
Aug 91 Nov-Dec 93 Mar-Nov 93 [un 90 Aug 93 Oct 93 Oct 93 [un 90 Aug 94 lui 92 Oct 92 Aug 94 Aug 94 Sep 93 Sep 93 Oct-Nov 90 Sep 90 Nov 90 Nov 90 Nov 94 Nov 94 Nov 89 May-Inn 93
4·49 1·30 1'92 7·16 0·05 0·37 0·33 1'19 2·13 0'63 0'5 0 3·72 4 '55 2 '85 7'17 3·80 2·09 1'36 1·05 0-53 0-15 3-18 0-29
0·35 0'06 0'07 0·66 0·01 0·03 0'04 0·16 0·19 0·11 0·09 0'25 0'19 0·22 0'40 0'23 0·21 0·19 0·15 0·23 0-02 0·10 0·04
1·25 0·28 0 ·38 2·17 0·02 0·10 0·12 0·44 0·63 0·26 0·21 0·96 0·94 0'79 1-69 0·93 0·66 0'51 0·39 0·35 0-05 0·56 0'11
0·00 0'01 0·00 0·52 0·00 0·02 0·00 0·02 0·01 0·07 0·12 0·00 0·01 0·03 0·07 1·05 0-06 0-06 0'30 0·00 0-00 7·65 0·01
1·76 1-44 1'70 1'21 2·51 1·38 1·45 1·43 1·59 1·23 1·14 1·86 1-63 1·47 1·45 1·07 1·33 1·34 1·11 1·81 2 -33 0·80 1·39
7·5 15'0 15'0 4'0 8·0 8·0 8·0 10'0 6·5 5·0 5'0 9·0 8·6 5'5 5·5 8·5 8·5 3'5 3'5 4'5 5-5 3-0 4·0
6 ·0 5·0 3 ·8 3 '0 3 ·3 3·6 1·8 3·0 5-8 2-0 5 ·0 4 ·0 4 ·4 4 ·0 4 ·0 4·0 4 ·0 3·0 3 ·0 4-4 5-5 2 ·0 6-2
0·55 0 '40 0·35 0·55 0·25 0·30 0 '25 0'50 0·12 0'30 0 '20 0·45 0 ·42 0 ·35 0'25 1·00 0·75 0 '40 0 ·25 0-35 0'35 0·65 0-00
S+4 S N N S S N $ +2 N
11
11 12 13
s
N S N
s
N S N S N S N
S
K." K y and K n are Fickiandiffusivitiesin m2 s - I, a and m definethe non-Fickian power law, H is water depth (m), W is wind speed (m s - I) and U is the representative tidal current (rn s 8+2=2 days after springs etc.; N =neaps.
I).
The column headed ' Tide' denotes the time within the spring/neap cycle: S, springs;
Diffusion in Irish coastal waters
The mean (centroid) and variance of a dye-patch concentration can be estimated (Bowden & Lewis, 1973) by computing the integrated concentration, CA , across the patch which is given by: o:
CA =
.r Cdx
(6)
where C is the concentration along a dye transect. The centre of mass, xo' of the distribution is then computed as: 1
xo=C
00
f Cdx
(7)
A -cr'
Hence, a measure of the width of the patch can be obtained from the variance, a Z, of the dye distribut~on using:
K=B 2 t where B has the units of velocity and is called the diffusion velocity (Bowden et al., 1974). It is strictly more correct to consider K to be a function of patch size rather than of diffusion time (Taylor) 1959). However) for practical purposes, this is equivalent to the statement that K varies with time. Since a dye patch usually elongates with time) orthogonal axes can be defined in the along-patch and across-patch directions and the respective diffusivities J(..x and Ky estimated from the growth with time of the along- and across-patch concentration variances (a/ and ay Z) . The elongation of a dye patch is caused by the process of shear diffusion which is due to the interaction of a Fickian diffusion process with the advective effect of the tidal- or wind-driven currents. For the case of steady shear in the transverse direction (Sy=au/3y)) the along- and across-patch variances will grow initially (Okubo) 1967) according to:
(8) In a further analogy with the Gaussian curve, since 95% of the area beneath the normal probability curve is contained within ± 2a of the mean value, the length scale of a patch of diffusing substance is often defined to be 4a where a is estimated by integrating the concentration across the patch. From observations at different diffusion times, the dependence of a Z on time can be determined. Furthermore, if it is assumed that aZ=O when t=O and that the variance follows a power law of the form: (9)
then a and m are constants that can be determined (Bowden, 1983). From Equation (5), K will be given by: 1 2
I<:=-mat"'-l
(10)
so that K can be calculated from the power law dependence of a Z against diffusion time using the estimated values of a and m. A constant value of K corresponds to a Fickian diffusion process for which m 1 and a2 increase linearly with time as given by Equation (4). In practice, log (a) is tabulated against log (t) and the value of K is determined by least squares. It is frequently the case that the variance increases more rapidly than linearly with diffusion time, implying that K itself depends on time. Observations are often found to follow a growth law approximating to m=2, so that the diffusivity can be expressed in the form
=
17
(11) and:
ay 2= 2Ky t
(12)
Consequently) for times greater than: t l
=~(3Ky) l/Z
s, K."
(13)
the major dimension of the patch [given by Equation (11)] will be dominated by the influence of the shear. An equivalent expression can be derived for the case of a vertical shear process. Note that if Equation (13) represents a horizontal process) Kx=Ky and the critical time depends only on the magnitude of the shear. In coastal waters) there are two main cop-tenders for the source of the shear; it can be generated either by lateral current differences near the shore (with offshore water moving more rapidly than water near the coast) or by vertical shear generated by tidal currents and wind-forcing. If the process is dominated by vertical shear) it is likely that the pollutant will become vertically wellmixed in a time that is short compared to the duration of the total spreading. As a consequence) the mixing will enter the second stage of shear diffusion (Okubo & Carter, 1966; Okubo, 1967) when the variance along the major axis of the patch will be given by:
S ZH4 t a/=2.K.,t+ ~OK
(14)
z
where Sz=3u/3z, H is the water depth and K; is the vertical diffusivity. Equations (11-14) will be used
18 A.
J. Elliott et al.
to explore the observed spreading rates and the underlying processes.
50
Methods
40
Discrete releases were made at each site by discharging a known quantity of dye solution into the surface waters. The releases were effectively instantaneous. The dye patch was then surveyed using a small boat fitted with a chart recorder and a Turner fluorometer connected to a pump that sampled the near-surface seawater. The position of the sampling vessel was monitored continuously using a shore-based tracking system. Repeated surveys were made by traversing through the patch with a mixture of along- and across-patch transects, and the position and size of the patch during each survey were then assessed by contouring isolines of concentration. Although the time to make an individual survey was relatively short, the position of each patch was subsequently corrected for the effects of tidal advection during the sampling. Each patch was monitored for 2-5 h and was sampled several times during that period. Estimates of the currents were obtained by using a moored vessel equipped with a direct reading current meter, and wind speed was measured by a hand-held anemometer. The releases were usually made at times of both spring and neap tides, and during ebb and flood conditions. Sometimes several releases were made at each site (for example, 10 discrete releases were made in Westport Bay). Continuous releases were also made occasionally but they have not been included in the analysis. Figure 1 and Table 1 show the locations at which the discrete releases were made. The results for each site have been averaged in the preparation of Table 1 when there was more than one spring and neap release. Therefore, the values given in Table 1 are general parameters that are representative of each location. With the exception of Lough N eagh, the dye was observed to become well mixed in the vertical relatively soon after release. Ideally, the variance of the dye concentration of each patch in orthogonal along- and across-patch directions should be determined at a range of diffusion times. This can be achieved by integrating the dye concentration along each transect and deriving the statistics described by Equations (6-10). In practice, however, this may not be feasible if the data have been recorded in analogue form using a chart recorder. An alternative, and simpler, method is to estimate the length and width of the patch and then make the assumption that each length scale represents 4a where a is the standard deviation of the
~--------------------.,.
o
8
o
o o o
o 0
o
10
10
20
30
40
50
Plume width (m) 2. Agreement between the width of patches estimated using four times the standard deviation of the concentration derived by integration, and the width determined directly from the patch outline. The figure was derived using dye data collected in the North Sea (Morales et al., 1997).
FIGURE
concentration profile in the selected direction. Support for this simplification is shown in Figure 2 which presents a comparison between the widths of patches derived by integration of the dye concentration to estimate the variance and hence the standard deviation, plotted as the ordinate, and overall width of each patch estimated directly from a knowledge of the position of the patch outline, shown along the abscissa. The patch outline was defined by the locations at which the dye concentration had fallen to 1% of the peak value. The regression equation between the two parameters, assuming that the line of best fit passes through the origin, is given by (width) =4,3 x standard deviation. Consequently, the 4a approximation provides a realistic estimate of patch width. The along- and across-patch length scales were therefore determined by mapping the patch outline, converted to equivalent standard deviations (ax and a) which could be plotted against diffusion time, and then least squares was used to extract the estimates of along- and across-patch diffusivity (Kx and K y ) . Since modelling studies usually require a single estimate of diffusivity to represent the mixing rates in the directions of the model axes, which may have arbitrary orientation with respect to the along- and across-patch directions, a radial (one-dimensional) length scale was defined by:
Diffusion in Irish coastal waters 19 3.00
~
2.75
I-
2.50
l-
2.25
I--
o
._-----,----------------, o Patch 1 '" Patch 2 + Patch 3 x Patch 4 o Patch 5 ..,.. Patch 6 x x Patch 7 z Patch 8 Y Patch 9 x • Patch 10 + .x
~ 2.001-o
°
x
o
+
~ 2.00 r
x
0
...<
...<
1.75 I-
x
1.75 t-
1.50 I1.25
3.00 r - - - - - - . . , - - - - - - - - - - - - - - , Patch 1 '" Patch 2 2.75 r + Patch 3 x Patch4 x o Patch 5 2.50 r f------....-J o + x '" x +0
x
°
1.50 t-
~eo.5
I-
o
0
1.25 r
I
\
I
I
3.0
3.5
4.0
4.5
5.0
° <>
0
~eo.5
I
I
I
I
3.0
3.5
4.0
4.5
5.0
Log (t)
Log(t)
3. Standard deviation of the one-dimensional patch size, (JH' plotted against diffusion time for the Westport releases.
FIGURE
a H =(ax ay )112
~
0°
'"
'"
+
"'l' ",0
(15)
with the corresponding value of diffusivity, K H , being defined by Equations (4) or (5).
Results An example of the patch statistics is shown in Figure 3 which presents the results from the 10 patches released at Westport. Each patch is shown by a different symbol, and each group of symbols can be used to define the dependence of patch variance against diffusion time. The data can be fitted to either a Fickian process by forcing the line of best fit to have a slope of 0'5, or to a non-Fickian spreading law by assuming the power law given by Equation (9) and deriving the parameters a and m. For the present purposes, the data were sorted into two groups, corresponding to spring and neap tides, and the diffusivities were derived for each group (Table 1, Site 10). The parameters a and m in Table 1 are only given for the radial diffusivity K H . Note, however, that the radial variance increased as tl-34 during spring tides which is faster than the t 1 ' O growth that would be expected for a Fickian spreading process. For comparison, Figure 4 shows the dataset from Bantry Bay where five patches were released. This site showed a greater range between the spring and neap values, with spring diffusivities being twice the order of those observed at Westport.
4. Standard deviation of the one-dimensional patch size, (JH' plotted against diffusion time for the Bantry Bay releases.
FIGURE
The complete dataset, showing the dependence on patch size against diffusion time for all the sampled sites, is presented in Figure 5. This clearly shows that the radial variance increases substantially faster than Fickian. The dashed line shows the spreading law described by Okubo (1971) which has the form: 2
0·011
2'34
aH==~t
(16)
Although the present data appear to lie above the dashed line, inspection of the Okubo (1971) paper shows that Equation (16) also underestimates the spreading at short time scales when compared with his data. Consequently, the stronger mixing suggested by Figure 5 is not due to the stronger tidal currents in Irish waters in comparison to the mainly western Atlantic locations examined by Okubo, but appears to be a consistent feature of coastal diffusion at short time scales (minutes to hours). Figure 6 shows the relationship between the alongand across-patch diffusivities, and Ky. The dashed line shows the regression line that passes through the origin; this suggests that KJKy ':::! 12 and that the patches were typically 3·5 times longer than wide. If the elongation is due to the effects of tidal advection, with offshore or mid-channel water moving more rapidly than water near the shore, Equation (11) should represent the growth of the along-patch variance. If K E represents the effective along-patch Fickian diffusivity, one might expect that:
s;
20 A.
J. Elliott et
al.
3.00...------- - - - - - - - - - - - - - ,
2.75
I-
2.50
I-
2.25
I-
~0.5
8
7 1-
o
3
657
"S
4 1-
~ 2.00o
....:l
1.75 -
12
3o
0
2-
1.50 -
2
1.25 I11 I
3.0
3.5
4.0
4.5
5.0
0.1
Log (1')
I
I
I
I
I
0.2
0.3
0.4
0.5
0.6
0.7
/(y (m2s-1)
5. A compos ite plot showing the data from all releases. ---, Spreading law described by Okubo (1971) . 0 , all da ta. FIGURE
2 2KEt=3K)'S2t3
I
(17)
when t>t l [from Equation (13)]. Consequently, the effective along-patch diffusivity K E (equivalent to K x ) should be proportional to Ky. The relative importance of lateral and vertical shear diffusion processes can be estimated by assuming representative values for the lateral and vertical current shear. From Table 1, the typical current speed was 0·5 m s - 1 and, since most releases took place within 200 m of the shore line, the maximum value for 'Ou/'Oy is of the order 2·5 x 10 - 3 S - 1 . Similarly, since the typical water depth was around 8 m, 'Ou/'Oz can be estimated to be of the order 0·06 s - I . Table 1 can be used to obtain a mean value of 0·2 m 2 s - 1 for I<;, and K z can be estimated to be of the order 50 x 10 - 4 m 2 s - 1 (Talbot & Talbot, 1974) . This is broadly consistent with the vertical mixing times that were observed. In addition, Figure 5 can be used to obtain an estimate of 104 s for a typical value of diffusion time by considering the position of the centroid of the data points. If the patch elongation is due to lateral current shear, the process is likely to be a first-stage shear diffusion process for which the elongation along the major axis will be given by Equation (11). At a diffusion time of 104 s, this gives a value for the along-patch variance of 8·5 x 10 5 m 2 . Alternatively, if the spreading is dominated by vertical current shear,
6. Relationship betw een th e along-patch (K J and acro ss-patch (K) diffusivities (both in m 2 s - I). The numbers refer to the locations given in Table 1.
F IGURE
the elongation will follow the second-stage growth rate given by Equation (14) (in water 8 m deep with a K; value of 50 x 10 - 4 m 2 s - I, vertical mixing will take about 20 min). Neglecting the F ickian spreading term, the second term on the right-hand side of Equation (14) gives a value of about 5·3 x 10 5 m 2 which is comparable to the variance associated with the lateral shear. The effects of wind shear can be estimated by assuming that the surface water moves at 3% of the wind speed (Brown, 1991) . The mean wind speed from Table 1 is around 4 m s - I, consequently 'O u/'Oz will be of order 0·02 s - I. Wh en this estimate of the vertical current shear is substituted into Equation (14), it gives a value for the variance at 104 s of about 0·3 x 10 5 m 2 , suggesting that the effect of wind shear is only of secondary importance. However, these results should be interpreted with caution. The extent of the patch elongation due to the lateral current shear has been emphasized by choosing a relatively long diffusion time (10 3 s) since the variance grows as t 3 • In addition, studies of the spread of oil (Elliott, 1986 ) have suggested that the spreading rate near a boundary will be intermediate between a first- and secondstage process. Consequently, the horizontal current shear may not spread a patch as rapidly as suggested by Equations (11) and (17). Figure 7 shows the relationship between the onedimensional diffusivity, K H , and the prevailing tidal
Diffusion in Irish coastal waters
2.25 -
2.25 -
3
2.00 -
1.75 -
8
1.50 1.25 -
~
1.00 -
3
8
1.50 -
~
""
3
2.00 -
1.75 -
I'w
'"'
'f., 3
1
""
77
~
9
8
0.75 0.50 r-
10
ft
~ 7 8 9
1.00 -
10
0.50 -
5
6 2 4 4 1 3 11 1 0.2 0.4
5
12
12
6
0.25 f1~
1
1.25 -
0,75 -
9
5
2 11
0.25 _
6 4
4
1
I
1
1
1
0.6
0.8
1.0
1
2
I
3
3
~
11
13
I
I
I
4
5
6
W(m 8-1)
U(m 8- 1) FIGURE 7.
21
Dependence of the one-dimensional diffusivity (KH in m 2 s - 1) on tidal current (m s - I) and wind speed (m s - 1).
2.25
f-
2.00
f-
2.25 -
3
2.00
1.75 -
""
3
7 7
0.25 f1~
8
1.50 -
I'w
1
0.50 _
c-
1.75 -
8
1.50 -
0.75 -
3
9
8 10 12 10 11
1
4
r
r-
9
8 9
5
iq
0.50 - 2 5
2
6 6 I
~ 1.00
0.75 -
9
5
1
1.25-
12
10 10
0.25 - 2 6 6
4
13
2
and wind conditions. Although the scatter is considerable, the' data suggest that there is a higher correlation with the tidal current speed than with the wind speed. However, this may also reflect the limited range of wind speeds that was sampled; in contrast, tidal current speeds ranged from zero (Lough Neagh) to 1 m s - 1 (Bantry Bay). Ifvertical current shear is dominant and the water is vertically well mixed, the along-patch diffusivity can have the form: (18)
~
11
I
1
0.3
0.4
where u is the depth-mean current, H is water depth and a is a constant (Elder, 1959; Bowden, 1965). Thus the mixing will increase in a manner that is proportional both to current speed and to water depth, the dependence on water depth being a reflection of the increased efficiency of the mixing due to the larger eddies in deeper water. However, if an expression similar to Equation (11) underlies the patch elongation, K; will depend more strongly on the vertical current shear which will be of order u/H. Figure 8 shows plots of K H against uH and u/H. The radial diffusivity K H was selected because of its
22
A.
J. Elliott et al.
8,---...,..----...,..----.....--------, 7
~
.-<
I",
Moreover, since the observed winds only varied between 2 and 6 m s - 1, the regression is dominated by the effects of current speed. It is interesting to note, from the intercepts of the contour lines with the abscissa, that at low wind speed, the magnitude of K H (m 2 s -1) was approximately equal to the speed of the tidal current (m s - 1).
!! '1:1
Cll
1.00
Cll
P<
0.9
Discussion
111
'1:1
~
2 1
0.2
0.4
0.6
0,8
1.0
Current speed (m S-I)
9. Multiple regression dependence of the onedimensional diffusivity (KH in m 2 s - 1) on tidal current (m s - 1) and wind speed (m s - 1). The solid curves show the parameterization given by Equation (19).
FIGURE
relevance to modelling studies and because it displays less variability than both K x and Ky. Neither plot shows convincing evidence in favour of the two vertical shear mechanisms (dependence on H or liB). The dependence of K H on tidal current and wind speed is shown in Figure 9. The least squares multiple regression equation has the form: K H=0'03+ 1·03u+ 0'04w
(19)
while the regression equation that is forced to pass through the origin is given by: (20)
The contour lines corresponding to Equation (19) are shown in Figure 9 superimposed on the data points. The small value of the intercept (0'03) in Equation (19) gives some confidence to the reality of an underlying physical relationship. If the values of K H were not dependent on current and wind speed, the regression curve would pass through the centroid of the data points but would lie roughly parallel to the (u,w) plane. Since the intercept in Equation (19) is small and the derivatives of K H with respect to u and ware in close agreement with those given by Equation (20), the data support the hypothesis that the mixing depends both on current and wind speed. However, the scatter in Figure 9 is considerable with the r 2 value of the multiple regression being only about 0'2.
Dye-diffusion observations are usually characterized by the randomness that results from the underlying ocean turbulence. As a consequence, the data are rarely , noise free' or explainable by simple physical arguments. This situation can be contrasted with the understanding of other phenomena such as tidal dynamics where a strong theoretical understanding can be supported by datasets that display a high signal-to-noise ratio. The considerable scatter that is shown in the patch statistics is no worse than that observed in similar datasets collected in other sea areas (e.g. Morales et al., 1997). However, the present analysis contains data collected at a range of coastal locations and the results will have application in applied studies that involve water quality or biological problems (e.g. the dispersal of larvae). The neap tide diffusivity values derived for Lame Lough (Table 1, Site 3) are anomalously high as they reflect the influence of advective rather than diffusive processes. During this experiment, the patch was released close to the bank which resulted in one side of the dye patch becoming attached to the shore while the offshore side was advected seaward by the ebbing tide. As a consequence, the patch was stretched by the tidal currents and the derived diffusivities represent an extreme case of shear diffusion. The observed spreading rates are broadly comparable with those computed via the oceanic diffusion diagrams (Okubo, 1971). To a first approximation, the one-dimensional diffusivity, K H , was dependent on the tides and had a magnitude equal to the current speed. However, this may have been a reflection on the generally low wind speed conditions under which the data were collected. In an analysis of North Sea dye data, collected during a series of continuous plume experiments at a site around 30 miles offshore and in water 50 m deep, Morales et at. (1997) derived across-patch diffusivities that were approximately 10 times smaller than those reported here. Consequently, there is a significant difference between open water releases in relatively deep water and the very near-shore conditions considered here. Modelling studies usually require a single value of diffusivity that can be used to forecast water quality
Diffusion in Irish coastal waters
near a coastal outfall. In such circumstances, the value to be used should be chosen with care. If the hydrodynamic model has high resolution and will resolve the on/offshore current gradients and the small-scale eddies induced by coastal topography, it is probably appropriate to use the across-patch diffusivity, Kyo Otherwise, the effects of shear diffusion may be included twice in the simulations. This will happen if the effective along-patch diffusivity, K x ' or the one-dimensional diffusivity, K H , are used in the simulations because they have both been derived by a processing method that will quantify the impact of a shear diffusion process. If the hydrodynamic model is accurate, it will reproduce the shear diffusion effect when the imbedded diffusion equation (probably of Fickian form) interacts with the advective currents. Such a model should therefore use a ' Fickian ' value for the diffusivity, and this is most likely to be represented by the across-patch spreading. Only in the case of a low-resolution model, which fails to resolve the details of the advective flow, would it be acceptable to use the K; or K H estimate of diffusivity. One surprising result of the analysis has been the generally low range of diffusivity values. This, coupled with the apparent dependence on tidal speed, suggests that comprehensive dye experiments are not required for estimating the mixing characteristics of a site. However, dye data are a useful source of information when the dye is used as an advective tracer. More effort should perhaps be devoted to simulating the movement and concentration within a dye patch, rather than using the dye data purely for the derivation of mixing coefficients. With the recent improvements in effluent treatment processes, it is likely that modelling results will depend more critically on parameters other than the value of diffusivity that is adopted. For example, the total loading at the outfall (i.e. the quantity of bacteria released) is likely to be the most critical parameter that will influence the water quality of the receiving waters. This can vary by orders of magnitude, depending on the level of treatment that is adopted, and is the single most important parameter that will influence model results. Next in importance will be the decay time scale (Tgo) that is assumed for
23
the destruction of the bacteria by the action of sunlight and exposure to seawater. If water quality close to an outfall is being considered, the initial dilution of the plume will significantly influence the local effluent concentrations. In comparison to these processes, the effects of diffusive mixing, when estimating the spread of a non-conservative pollutant, are likely to be small. Acknowledgements
The authors thank their colleagues for assistance with the data collection and processing. The authors are grateful to the clients of Kirk McClure Morton and MCS International for permission to publish these results. References Bowden, K. F. 1965 Horizontal mixing in the sea due to shearing current, Journal of Fluid Mechanics 21, 83-95. Bowden, K. F. 1983 Physical Oceanography of Coastal Waters. Ellis Horward, Chichester, 302 pp. Bowden, K. F. & Lewis, R. E. 1973 Dispersion in flow from a continuous source at sea. Water Research 7, 1705-1722. Bowden, K. F., Krauel, D. P. & Lewis, R. E. 1974 Some features of turbulent diffusion from a continuous source at sea. Advances in Geophysics 18A, 315-329. Brown, ]. 1991 The final voyage of the Rapaiti, a measure of sea-surface drift velocity in relation to the surface wind. Marine Pollution Bulletin 22, 37-40. Elder, J. W. 1959 The dispersion of marked fluid in turbulent shear flow. Journal of Fluid Mechanics 5, 544-560. Elliott, A. J. 1986 Shear diffusion and the spread of oil in the surface layers of the North Sea. Deutsche Hydrographische Zeitschrift 39, 113-137. Morales, R. A., Elliott, A. J, & Lune1, T. 1997 The influence of tidal currents and wind on mixing in the surface layers of the sea. Marine Pollution Bulletin (in press). Okubo, A. 1967 The effect of shear in an oscillatory current on horizontal diffusion from an instantaneous source. Limnology and Oceanology 1, 194-204. Okubo, A. 1971 Oceanic diffusion diagrams. Deep-Sea Research 18, 789-802. Okubo, A. & Carter, H. H. 1966 An extremely simplified model of the 'shear effect' on horizontal mixing in a bounded sea. Journal oj Geophysical Research 71, 5267-5270. Talbot, J. W. & Talbot, G. A. 1974 Diffusion in shallow seas and in English coastal and estuarine waters. Rapports et Proces-uerbaux des Reunions, Conseil International pour l'Exploration de la Mer 167, 93-110. Taylor, G. 1. 1959 The present position in the theory of turbulent diffusion. Advances in Geophysics 6, 101-112.
Diffusion in Irish Coastal Waters A.
J. Elliote, A. G.
Bar-r" and D. Kerman"
"Uniuersity of Wales, Bangor, Marine Science Laboratories, Menai Bridge, Anglesey LL59 5EY, U.K. bKirk McClure Morton, Elmwood House, 74 Boucher Road, Belfast BT12 6RZ, U.K. cMCS Iniernational, Lismoyle House, Merchants Road, Galway, Ireland
Results are presented from dye-diffusion experiments performed at a selection of near-shore and estuarine sites around the coastline of Ireland. The data from discrete dye releases have been analysed to derive along- and across-patch horizontal diffusivities, and estimates are presented of both Fickian and non-Fickian diffusion coefficients. Most of the sites were characterized by strong tidal currents so that the dye became vertically well-mixed soon after release. Consequently, the mixing was found to be tidally dominated and, to a reasonable approximation, the magnitude of the one-dimensional (radial spreading) horizontal diffusivity (m? s - 1) was equal to the tidal current speed (m s - 1). Wind-induced mixing was only of secondary importance, but this may be a consequence of the experiments having taken place during conditions of generally light winds. In general, the observed mixing rates agree with those © 1997 Academic Press Limited predicted by oceanic diffusion diagrams. Keywords: diffusion; diffusivity; tidal mixing; wind mixing; Ireland
Introduction Diffusion experiments, involving both discrete and continuous releases of rhodamine dye, are commonly performed to determine the mixing characteristics of a near-shore site. Parameterized as diffusion coefficients or diffusivities, the derived mixing characteristics are then used in numerical simulations to assess the likely impact of an effluent. Such studies usually concern the selection of a site for a sewage outfall or the modification of an existing industrial release. In recent years, however, the growth of the fish farm industry has prompted dye experiments that have been directed at determining mixing characteristics and flushing times as part of environmental impact assessment studies. Models of the diffusion from a continuous source in the sea often make use of the diffusion velocity concept (Bowden & Lewis, 1973), which relates the diffusivity linearly to the diffusion time, estimates of the diffusion velocity being derived by field trials. The weakness of the approach is that it is not clear how data collected during calm conditions can be used to predict the spreading during rougher weather. To overcome this, a diffusion model should isolate the spreading due to tide and wind so that the effects of weather can be quantified; such a technique would then lead to improved predictions of spreading during a range of tide and weather conditions. The goal of this study was to investigate the dependence of the mixing coefficients on the local environmental conditions, including such parameters as water depth, 0272-7714/97/44AOI5+09 $25.00/0
tidal current and wind speed. Empirical formulae relating the mixing to the tidal currents and wind have been derived for use in applied studies. This paper presents the results from a series of experimental releases conducted over a period of several years, and attempts to parameterize the mixing rates in terms of the ambient tidal currents and winds. The majority of the sites were in estuarine, embayment and lough locations, with a few sites being close to the shoreline on an open coastline (Figure 1 and Table 1). For completeness, one inland lough (Lough Neagh, the largest freshwater lake in the U.K. and Ireland) has been included in the dataset. At each site, discrete dye releases were usually made at times of both spring and neap tides, the experiments being repeated during ebb and flood conditions. Unfortunately, since the experiments were performed using a small sampling craft, the observations are biased towards low wind speed conditions. When analysing dye data, it is usual to make an analogy between the solution of the one-dimensional diffusion equation and the normal probability function. The one-dimensional diffusion equation, with a constant diffusivity K:
8C =~(K8C)
at ax
(1)
ax
has the solution: C(x, t)
=
(x
2
Q 1/2 exp - 4Kt ) (4nKt)
©
(2)
1997 Academic Press Limited
16 A.
J.
Elliott et al,
where Q is the amount of substance released at time t= O (Bowden, 1983). This has the same form as the normal probability function, or Gaussian curve, with variance r:? and mean f.l given by: 54.5°
f( x)
1 exp ( (T(2n) 1/ 2
(3)
provided that: 53.5°
(4) and the diffusive spreading is measured with respect to the centroid of the dye patch. Equation (4) is the method by which diffusivities are usually derived during the analysis of a dye dataset . When the diffusivity, K, is not a constant, it can be estimated (Bowden, 1983) from the time derivative of the variance using:
sz.s-
51.5°
1
d(T 2
K=--
(5)
2 dt
FIGURE 1. Locations of the dye releases.
TABLE 1. Summary of the dye-diffusion statistics
No .
Location
Date
s;
~
KH
a
m
H
W
U
Tide
1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10
Lough Foyle Cushendall Cushendall Lame Lough Lame Lough Belfast Lough Belfast Lough Strangford Lough Strangford Lough Dundalk Bay Dundalk Bay Courtown Courtown Kinsale Kinsale Bantry Bay Bantry Bay Westport Bay Westport Bay Achill Island Achill Island Donegal Harbour Lough Neagh
Aug 91 Nov-Dec 93 Mar-Nov 93 [un 90 Aug 93 Oct 93 Oct 93 [un 90 Aug 94 lui 92 Oct 92 Aug 94 Aug 94 Sep 93 Sep 93 Oct-Nov 90 Sep 90 Nov 90 Nov 90 Nov 94 Nov 94 Nov 89 May-Inn 93
4·49 1·30 1'92 7·16 0·05 0·37 0·33 1'19 2·13 0'63 0'5 0 3·72 4 '55 2 '85 7'17 3·80 2·09 1'36 1·05 0-53 0-15 3-18 0-29
0·35 0'06 0'07 0·66 0·01 0·03 0'04 0·16 0·19 0·11 0·09 0'25 0'19 0·22 0'40 0'23 0·21 0·19 0·15 0·23 0-02 0·10 0·04
1·25 0·28 0 ·38 2·17 0·02 0·10 0·12 0·44 0·63 0·26 0·21 0·96 0·94 0'79 1-69 0·93 0·66 0'51 0·39 0·35 0-05 0·56 0'11
0·00 0'01 0·00 0·52 0·00 0·02 0·00 0·02 0·01 0·07 0·12 0·00 0·01 0·03 0·07 1·05 0-06 0-06 0'30 0·00 0-00 7·65 0·01
1·76 1-44 1'70 1'21 2·51 1·38 1·45 1·43 1·59 1·23 1·14 1·86 1-63 1·47 1·45 1·07 1·33 1·34 1·11 1·81 2 -33 0·80 1·39
7·5 15'0 15'0 4'0 8·0 8·0 8·0 10'0 6·5 5·0 5'0 9·0 8·6 5'5 5·5 8·5 8·5 3'5 3'5 4'5 5-5 3-0 4·0
6 ·0 5·0 3 ·8 3 '0 3 ·3 3·6 1·8 3·0 5-8 2-0 5 ·0 4 ·0 4 ·4 4 ·0 4 ·0 4·0 4 ·0 3·0 3 ·0 4-4 5-5 2 ·0 6-2
0·55 0 '40 0·35 0·55 0·25 0·30 0 '25 0'50 0·12 0'30 0 '20 0·45 0 ·42 0 ·35 0'25 1·00 0·75 0 '40 0 ·25 0-35 0'35 0·65 0-00
S+4 S N N S S N $ +2 N
11
11 12 13
s
N S N
s
N S N S N S N
S
K." K y and K n are Fickiandiffusivitiesin m2 s - I, a and m definethe non-Fickian power law, H is water depth (m), W is wind speed (m s - I) and U is the representative tidal current (rn s 8+2=2 days after springs etc.; N =neaps.
I).
The column headed ' Tide' denotes the time within the spring/neap cycle: S, springs;
Diffusion in Irish coastal waters
The mean (centroid) and variance of a dye-patch concentration can be estimated (Bowden & Lewis, 1973) by computing the integrated concentration, CA , across the patch which is given by: o:
CA =
.r Cdx
(6)
where C is the concentration along a dye transect. The centre of mass, xo' of the distribution is then computed as: 1
xo=C
00
f Cdx
(7)
A -cr'
Hence, a measure of the width of the patch can be obtained from the variance, a Z, of the dye distribut~on using:
K=B 2 t where B has the units of velocity and is called the diffusion velocity (Bowden et al., 1974). It is strictly more correct to consider K to be a function of patch size rather than of diffusion time (Taylor) 1959). However) for practical purposes, this is equivalent to the statement that K varies with time. Since a dye patch usually elongates with time) orthogonal axes can be defined in the along-patch and across-patch directions and the respective diffusivities J(..x and Ky estimated from the growth with time of the along- and across-patch concentration variances (a/ and ay Z) . The elongation of a dye patch is caused by the process of shear diffusion which is due to the interaction of a Fickian diffusion process with the advective effect of the tidal- or wind-driven currents. For the case of steady shear in the transverse direction (Sy=au/3y)) the along- and across-patch variances will grow initially (Okubo) 1967) according to:
(8) In a further analogy with the Gaussian curve, since 95% of the area beneath the normal probability curve is contained within ± 2a of the mean value, the length scale of a patch of diffusing substance is often defined to be 4a where a is estimated by integrating the concentration across the patch. From observations at different diffusion times, the dependence of a Z on time can be determined. Furthermore, if it is assumed that aZ=O when t=O and that the variance follows a power law of the form: (9)
then a and m are constants that can be determined (Bowden, 1983). From Equation (5), K will be given by: 1 2
I<:=-mat"'-l
(10)
so that K can be calculated from the power law dependence of a Z against diffusion time using the estimated values of a and m. A constant value of K corresponds to a Fickian diffusion process for which m 1 and a2 increase linearly with time as given by Equation (4). In practice, log (a) is tabulated against log (t) and the value of K is determined by least squares. It is frequently the case that the variance increases more rapidly than linearly with diffusion time, implying that K itself depends on time. Observations are often found to follow a growth law approximating to m=2, so that the diffusivity can be expressed in the form
=
17
(11) and:
ay 2= 2Ky t
(12)
Consequently) for times greater than: t l
=~(3Ky) l/Z
s, K."
(13)
the major dimension of the patch [given by Equation (11)] will be dominated by the influence of the shear. An equivalent expression can be derived for the case of a vertical shear process. Note that if Equation (13) represents a horizontal process) Kx=Ky and the critical time depends only on the magnitude of the shear. In coastal waters) there are two main cop-tenders for the source of the shear; it can be generated either by lateral current differences near the shore (with offshore water moving more rapidly than water near the coast) or by vertical shear generated by tidal currents and wind-forcing. If the process is dominated by vertical shear) it is likely that the pollutant will become vertically wellmixed in a time that is short compared to the duration of the total spreading. As a consequence) the mixing will enter the second stage of shear diffusion (Okubo & Carter, 1966; Okubo, 1967) when the variance along the major axis of the patch will be given by:
S ZH4 t a/=2.K.,t+ ~OK
(14)
z
where Sz=3u/3z, H is the water depth and K; is the vertical diffusivity. Equations (11-14) will be used
18 A.
J. Elliott et al.
to explore the observed spreading rates and the underlying processes.
50
Methods
40
Discrete releases were made at each site by discharging a known quantity of dye solution into the surface waters. The releases were effectively instantaneous. The dye patch was then surveyed using a small boat fitted with a chart recorder and a Turner fluorometer connected to a pump that sampled the near-surface seawater. The position of the sampling vessel was monitored continuously using a shore-based tracking system. Repeated surveys were made by traversing through the patch with a mixture of along- and across-patch transects, and the position and size of the patch during each survey were then assessed by contouring isolines of concentration. Although the time to make an individual survey was relatively short, the position of each patch was subsequently corrected for the effects of tidal advection during the sampling. Each patch was monitored for 2-5 h and was sampled several times during that period. Estimates of the currents were obtained by using a moored vessel equipped with a direct reading current meter, and wind speed was measured by a hand-held anemometer. The releases were usually made at times of both spring and neap tides, and during ebb and flood conditions. Sometimes several releases were made at each site (for example, 10 discrete releases were made in Westport Bay). Continuous releases were also made occasionally but they have not been included in the analysis. Figure 1 and Table 1 show the locations at which the discrete releases were made. The results for each site have been averaged in the preparation of Table 1 when there was more than one spring and neap release. Therefore, the values given in Table 1 are general parameters that are representative of each location. With the exception of Lough N eagh, the dye was observed to become well mixed in the vertical relatively soon after release. Ideally, the variance of the dye concentration of each patch in orthogonal along- and across-patch directions should be determined at a range of diffusion times. This can be achieved by integrating the dye concentration along each transect and deriving the statistics described by Equations (6-10). In practice, however, this may not be feasible if the data have been recorded in analogue form using a chart recorder. An alternative, and simpler, method is to estimate the length and width of the patch and then make the assumption that each length scale represents 4a where a is the standard deviation of the
~--------------------.,.
o
8
o
o o o
o 0
o
10
10
20
30
40
50
Plume width (m) 2. Agreement between the width of patches estimated using four times the standard deviation of the concentration derived by integration, and the width determined directly from the patch outline. The figure was derived using dye data collected in the North Sea (Morales et al., 1997).
FIGURE
concentration profile in the selected direction. Support for this simplification is shown in Figure 2 which presents a comparison between the widths of patches derived by integration of the dye concentration to estimate the variance and hence the standard deviation, plotted as the ordinate, and overall width of each patch estimated directly from a knowledge of the position of the patch outline, shown along the abscissa. The patch outline was defined by the locations at which the dye concentration had fallen to 1% of the peak value. The regression equation between the two parameters, assuming that the line of best fit passes through the origin, is given by (width) =4,3 x standard deviation. Consequently, the 4a approximation provides a realistic estimate of patch width. The along- and across-patch length scales were therefore determined by mapping the patch outline, converted to equivalent standard deviations (ax and a) which could be plotted against diffusion time, and then least squares was used to extract the estimates of along- and across-patch diffusivity (Kx and K y ) . Since modelling studies usually require a single estimate of diffusivity to represent the mixing rates in the directions of the model axes, which may have arbitrary orientation with respect to the along- and across-patch directions, a radial (one-dimensional) length scale was defined by:
Diffusion in Irish coastal waters 19 3.00
~
2.75
I-
2.50
l-
2.25
I--
o
._-----,----------------, o Patch 1 '" Patch 2 + Patch 3 x Patch 4 o Patch 5 ..,.. Patch 6 x x Patch 7 z Patch 8 Y Patch 9 x • Patch 10 + .x
~ 2.001-o
°
x
o
+
~ 2.00 r
x
0
...<
...<
1.75 I-
x
1.75 t-
1.50 I1.25
3.00 r - - - - - - . . , - - - - - - - - - - - - - - , Patch 1 '" Patch 2 2.75 r + Patch 3 x Patch4 x o Patch 5 2.50 r f------....-J o + x '" x +0
x
°
1.50 t-
~eo.5
I-
o
0
1.25 r
I
\
I
I
3.0
3.5
4.0
4.5
5.0
° <>
0
~eo.5
I
I
I
I
3.0
3.5
4.0
4.5
5.0
Log (t)
Log(t)
3. Standard deviation of the one-dimensional patch size, (JH' plotted against diffusion time for the Westport releases.
FIGURE
a H =(ax ay )112
~
0°
'"
'"
+
"'l' ",0
(15)
with the corresponding value of diffusivity, K H , being defined by Equations (4) or (5).
Results An example of the patch statistics is shown in Figure 3 which presents the results from the 10 patches released at Westport. Each patch is shown by a different symbol, and each group of symbols can be used to define the dependence of patch variance against diffusion time. The data can be fitted to either a Fickian process by forcing the line of best fit to have a slope of 0'5, or to a non-Fickian spreading law by assuming the power law given by Equation (9) and deriving the parameters a and m. For the present purposes, the data were sorted into two groups, corresponding to spring and neap tides, and the diffusivities were derived for each group (Table 1, Site 10). The parameters a and m in Table 1 are only given for the radial diffusivity K H . Note, however, that the radial variance increased as tl-34 during spring tides which is faster than the t 1 ' O growth that would be expected for a Fickian spreading process. For comparison, Figure 4 shows the dataset from Bantry Bay where five patches were released. This site showed a greater range between the spring and neap values, with spring diffusivities being twice the order of those observed at Westport.
4. Standard deviation of the one-dimensional patch size, (JH' plotted against diffusion time for the Bantry Bay releases.
FIGURE
The complete dataset, showing the dependence on patch size against diffusion time for all the sampled sites, is presented in Figure 5. This clearly shows that the radial variance increases substantially faster than Fickian. The dashed line shows the spreading law described by Okubo (1971) which has the form: 2
0·011
2'34
aH==~t
(16)
Although the present data appear to lie above the dashed line, inspection of the Okubo (1971) paper shows that Equation (16) also underestimates the spreading at short time scales when compared with his data. Consequently, the stronger mixing suggested by Figure 5 is not due to the stronger tidal currents in Irish waters in comparison to the mainly western Atlantic locations examined by Okubo, but appears to be a consistent feature of coastal diffusion at short time scales (minutes to hours). Figure 6 shows the relationship between the alongand across-patch diffusivities, and Ky. The dashed line shows the regression line that passes through the origin; this suggests that KJKy ':::! 12 and that the patches were typically 3·5 times longer than wide. If the elongation is due to the effects of tidal advection, with offshore or mid-channel water moving more rapidly than water near the shore, Equation (11) should represent the growth of the along-patch variance. If K E represents the effective along-patch Fickian diffusivity, one might expect that:
s;
20 A.
J. Elliott et
al.
3.00...------- - - - - - - - - - - - - - ,
2.75
I-
2.50
I-
2.25
I-
~0.5
8
7 1-
o
3
657
"S
4 1-
~ 2.00o
....:l
1.75 -
12
3o
0
2-
1.50 -
2
1.25 I11 I
3.0
3.5
4.0
4.5
5.0
0.1
Log (1')
I
I
I
I
I
0.2
0.3
0.4
0.5
0.6
0.7
/(y (m2s-1)
5. A compos ite plot showing the data from all releases. ---, Spreading law described by Okubo (1971) . 0 , all da ta. FIGURE
2 2KEt=3K)'S2t3
I
(17)
when t>t l [from Equation (13)]. Consequently, the effective along-patch diffusivity K E (equivalent to K x ) should be proportional to Ky. The relative importance of lateral and vertical shear diffusion processes can be estimated by assuming representative values for the lateral and vertical current shear. From Table 1, the typical current speed was 0·5 m s - 1 and, since most releases took place within 200 m of the shore line, the maximum value for 'Ou/'Oy is of the order 2·5 x 10 - 3 S - 1 . Similarly, since the typical water depth was around 8 m, 'Ou/'Oz can be estimated to be of the order 0·06 s - I . Table 1 can be used to obtain a mean value of 0·2 m 2 s - 1 for I<;, and K z can be estimated to be of the order 50 x 10 - 4 m 2 s - 1 (Talbot & Talbot, 1974) . This is broadly consistent with the vertical mixing times that were observed. In addition, Figure 5 can be used to obtain an estimate of 104 s for a typical value of diffusion time by considering the position of the centroid of the data points. If the patch elongation is due to lateral current shear, the process is likely to be a first-stage shear diffusion process for which the elongation along the major axis will be given by Equation (11). At a diffusion time of 104 s, this gives a value for the along-patch variance of 8·5 x 10 5 m 2 . Alternatively, if the spreading is dominated by vertical current shear,
6. Relationship betw een th e along-patch (K J and acro ss-patch (K) diffusivities (both in m 2 s - I). The numbers refer to the locations given in Table 1.
F IGURE
the elongation will follow the second-stage growth rate given by Equation (14) (in water 8 m deep with a K; value of 50 x 10 - 4 m 2 s - I, vertical mixing will take about 20 min). Neglecting the F ickian spreading term, the second term on the right-hand side of Equation (14) gives a value of about 5·3 x 10 5 m 2 which is comparable to the variance associated with the lateral shear. The effects of wind shear can be estimated by assuming that the surface water moves at 3% of the wind speed (Brown, 1991) . The mean wind speed from Table 1 is around 4 m s - I, consequently 'O u/'Oz will be of order 0·02 s - I. Wh en this estimate of the vertical current shear is substituted into Equation (14), it gives a value for the variance at 104 s of about 0·3 x 10 5 m 2 , suggesting that the effect of wind shear is only of secondary importance. However, these results should be interpreted with caution. The extent of the patch elongation due to the lateral current shear has been emphasized by choosing a relatively long diffusion time (10 3 s) since the variance grows as t 3 • In addition, studies of the spread of oil (Elliott, 1986 ) have suggested that the spreading rate near a boundary will be intermediate between a first- and secondstage process. Consequently, the horizontal current shear may not spread a patch as rapidly as suggested by Equations (11) and (17). Figure 7 shows the relationship between the onedimensional diffusivity, K H , and the prevailing tidal
Diffusion in Irish coastal waters
2.25 -
2.25 -
3
2.00 -
1.75 -
8
1.50 1.25 -
~
1.00 -
3
8
1.50 -
~
""
3
2.00 -
1.75 -
I'w
'"'
'f., 3
1
""
77
~
9
8
0.75 0.50 r-
10
ft
~ 7 8 9
1.00 -
10
0.50 -
5
6 2 4 4 1 3 11 1 0.2 0.4
5
12
12
6
0.25 f1~
1
1.25 -
0,75 -
9
5
2 11
0.25 _
6 4
4
1
I
1
1
1
0.6
0.8
1.0
1
2
I
3
3
~
11
13
I
I
I
4
5
6
W(m 8-1)
U(m 8- 1) FIGURE 7.
21
Dependence of the one-dimensional diffusivity (KH in m 2 s - 1) on tidal current (m s - I) and wind speed (m s - 1).
2.25
f-
2.00
f-
2.25 -
3
2.00
1.75 -
""
3
7 7
0.25 f1~
8
1.50 -
I'w
1
0.50 _
c-
1.75 -
8
1.50 -
0.75 -
3
9
8 10 12 10 11
1
4
r
r-
9
8 9
5
iq
0.50 - 2 5
2
6 6 I
~ 1.00
0.75 -
9
5
1
1.25-
12
10 10
0.25 - 2 6 6
4
13
2
and wind conditions. Although the scatter is considerable, the' data suggest that there is a higher correlation with the tidal current speed than with the wind speed. However, this may also reflect the limited range of wind speeds that was sampled; in contrast, tidal current speeds ranged from zero (Lough Neagh) to 1 m s - 1 (Bantry Bay). Ifvertical current shear is dominant and the water is vertically well mixed, the along-patch diffusivity can have the form: (18)
~
11
I
1
0.3
0.4
where u is the depth-mean current, H is water depth and a is a constant (Elder, 1959; Bowden, 1965). Thus the mixing will increase in a manner that is proportional both to current speed and to water depth, the dependence on water depth being a reflection of the increased efficiency of the mixing due to the larger eddies in deeper water. However, if an expression similar to Equation (11) underlies the patch elongation, K; will depend more strongly on the vertical current shear which will be of order u/H. Figure 8 shows plots of K H against uH and u/H. The radial diffusivity K H was selected because of its
22
A.
J. Elliott et al.
8,---...,..----...,..----.....--------, 7
~
.-<
I",
Moreover, since the observed winds only varied between 2 and 6 m s - 1, the regression is dominated by the effects of current speed. It is interesting to note, from the intercepts of the contour lines with the abscissa, that at low wind speed, the magnitude of K H (m 2 s -1) was approximately equal to the speed of the tidal current (m s - 1).
!! '1:1
Cll
1.00
Cll
P<
0.9
Discussion
111
'1:1
~
2 1
0.2
0.4
0.6
0,8
1.0
Current speed (m S-I)
9. Multiple regression dependence of the onedimensional diffusivity (KH in m 2 s - 1) on tidal current (m s - 1) and wind speed (m s - 1). The solid curves show the parameterization given by Equation (19).
FIGURE
relevance to modelling studies and because it displays less variability than both K x and Ky. Neither plot shows convincing evidence in favour of the two vertical shear mechanisms (dependence on H or liB). The dependence of K H on tidal current and wind speed is shown in Figure 9. The least squares multiple regression equation has the form: K H=0'03+ 1·03u+ 0'04w
(19)
while the regression equation that is forced to pass through the origin is given by: (20)
The contour lines corresponding to Equation (19) are shown in Figure 9 superimposed on the data points. The small value of the intercept (0'03) in Equation (19) gives some confidence to the reality of an underlying physical relationship. If the values of K H were not dependent on current and wind speed, the regression curve would pass through the centroid of the data points but would lie roughly parallel to the (u,w) plane. Since the intercept in Equation (19) is small and the derivatives of K H with respect to u and ware in close agreement with those given by Equation (20), the data support the hypothesis that the mixing depends both on current and wind speed. However, the scatter in Figure 9 is considerable with the r 2 value of the multiple regression being only about 0'2.
Dye-diffusion observations are usually characterized by the randomness that results from the underlying ocean turbulence. As a consequence, the data are rarely , noise free' or explainable by simple physical arguments. This situation can be contrasted with the understanding of other phenomena such as tidal dynamics where a strong theoretical understanding can be supported by datasets that display a high signal-to-noise ratio. The considerable scatter that is shown in the patch statistics is no worse than that observed in similar datasets collected in other sea areas (e.g. Morales et al., 1997). However, the present analysis contains data collected at a range of coastal locations and the results will have application in applied studies that involve water quality or biological problems (e.g. the dispersal of larvae). The neap tide diffusivity values derived for Lame Lough (Table 1, Site 3) are anomalously high as they reflect the influence of advective rather than diffusive processes. During this experiment, the patch was released close to the bank which resulted in one side of the dye patch becoming attached to the shore while the offshore side was advected seaward by the ebbing tide. As a consequence, the patch was stretched by the tidal currents and the derived diffusivities represent an extreme case of shear diffusion. The observed spreading rates are broadly comparable with those computed via the oceanic diffusion diagrams (Okubo, 1971). To a first approximation, the one-dimensional diffusivity, K H , was dependent on the tides and had a magnitude equal to the current speed. However, this may have been a reflection on the generally low wind speed conditions under which the data were collected. In an analysis of North Sea dye data, collected during a series of continuous plume experiments at a site around 30 miles offshore and in water 50 m deep, Morales et at. (1997) derived across-patch diffusivities that were approximately 10 times smaller than those reported here. Consequently, there is a significant difference between open water releases in relatively deep water and the very near-shore conditions considered here. Modelling studies usually require a single value of diffusivity that can be used to forecast water quality
Diffusion in Irish coastal waters
near a coastal outfall. In such circumstances, the value to be used should be chosen with care. If the hydrodynamic model has high resolution and will resolve the on/offshore current gradients and the small-scale eddies induced by coastal topography, it is probably appropriate to use the across-patch diffusivity, Kyo Otherwise, the effects of shear diffusion may be included twice in the simulations. This will happen if the effective along-patch diffusivity, K x ' or the one-dimensional diffusivity, K H , are used in the simulations because they have both been derived by a processing method that will quantify the impact of a shear diffusion process. If the hydrodynamic model is accurate, it will reproduce the shear diffusion effect when the imbedded diffusion equation (probably of Fickian form) interacts with the advective currents. Such a model should therefore use a ' Fickian ' value for the diffusivity, and this is most likely to be represented by the across-patch spreading. Only in the case of a low-resolution model, which fails to resolve the details of the advective flow, would it be acceptable to use the K; or K H estimate of diffusivity. One surprising result of the analysis has been the generally low range of diffusivity values. This, coupled with the apparent dependence on tidal speed, suggests that comprehensive dye experiments are not required for estimating the mixing characteristics of a site. However, dye data are a useful source of information when the dye is used as an advective tracer. More effort should perhaps be devoted to simulating the movement and concentration within a dye patch, rather than using the dye data purely for the derivation of mixing coefficients. With the recent improvements in effluent treatment processes, it is likely that modelling results will depend more critically on parameters other than the value of diffusivity that is adopted. For example, the total loading at the outfall (i.e. the quantity of bacteria released) is likely to be the most critical parameter that will influence the water quality of the receiving waters. This can vary by orders of magnitude, depending on the level of treatment that is adopted, and is the single most important parameter that will influence model results. Next in importance will be the decay time scale (Tgo) that is assumed for
23
the destruction of the bacteria by the action of sunlight and exposure to seawater. If water quality close to an outfall is being considered, the initial dilution of the plume will significantly influence the local effluent concentrations. In comparison to these processes, the effects of diffusive mixing, when estimating the spread of a non-conservative pollutant, are likely to be small. Acknowledgements
The authors thank their colleagues for assistance with the data collection and processing. The authors are grateful to the clients of Kirk McClure Morton and MCS International for permission to publish these results. References Bowden, K. F. 1965 Horizontal mixing in the sea due to shearing current, Journal of Fluid Mechanics 21, 83-95. Bowden, K. F. 1983 Physical Oceanography of Coastal Waters. Ellis Horward, Chichester, 302 pp. Bowden, K. F. & Lewis, R. E. 1973 Dispersion in flow from a continuous source at sea. Water Research 7, 1705-1722. Bowden, K. F., Krauel, D. P. & Lewis, R. E. 1974 Some features of turbulent diffusion from a continuous source at sea. Advances in Geophysics 18A, 315-329. Brown, ]. 1991 The final voyage of the Rapaiti, a measure of sea-surface drift velocity in relation to the surface wind. Marine Pollution Bulletin 22, 37-40. Elder, J. W. 1959 The dispersion of marked fluid in turbulent shear flow. Journal of Fluid Mechanics 5, 544-560. Elliott, A. J. 1986 Shear diffusion and the spread of oil in the surface layers of the North Sea. Deutsche Hydrographische Zeitschrift 39, 113-137. Morales, R. A., Elliott, A. J, & Lune1, T. 1997 The influence of tidal currents and wind on mixing in the surface layers of the sea. Marine Pollution Bulletin (in press). Okubo, A. 1967 The effect of shear in an oscillatory current on horizontal diffusion from an instantaneous source. Limnology and Oceanology 1, 194-204. Okubo, A. 1971 Oceanic diffusion diagrams. Deep-Sea Research 18, 789-802. Okubo, A. & Carter, H. H. 1966 An extremely simplified model of the 'shear effect' on horizontal mixing in a bounded sea. Journal oj Geophysical Research 71, 5267-5270. Talbot, J. W. & Talbot, G. A. 1974 Diffusion in shallow seas and in English coastal and estuarine waters. Rapports et Proces-uerbaux des Reunions, Conseil International pour l'Exploration de la Mer 167, 93-110. Taylor, G. 1. 1959 The present position in the theory of turbulent diffusion. Advances in Geophysics 6, 101-112.