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Calculating Exchange Times in a Scottish Fjord Using a Two-dimensional, Laterally-integrated Numerical Model
Estuarine, Coastal and Shelf Science (2001) 53, 437–449 doi:10.1006/ecss.1999.0624, available online at http://www.idealibrary.com on
Calculating Exchange Times in a Scottish Fjord Using a Two-dimensional, Laterally-integrated Numerical Model P. A. Gillibrand FRS Marine Laboratory, P.O. Box 101, Victoria Road, Aberdeen, AB11 9DB, U.K. Received 4 December 1998 and accepted in revised form 13 September 1999 In order to assess the potential impact of pollutants, particularly soluble wastes discharged by the mariculture industry, on the fjordic sea loch environment in Scotland, simple management models have been developed which estimate steady-state concentrations based on the quantities of effluent released and the residence time of such material within a loch. These models make various simplifications about the hydrodynamic characteristics of Scottish sea lochs, the most important of which is the concept of an exchange time which parametrizes the rate at which pollutants are removed from the system. Exchange times for individual lochs are calculated using the tidal prism method, which has some well-known shortcomings. In this paper, a two-dimensional laterally-integrated circulation model is used to investigate the exchange characteristics of Loch Fyne and its sub-basins. By simulating the transport of a passive, conservative tracer, the turnover times for the loch, two sub-basins and various depth layers are calculated. By varying the starting time of the tracer simulations, the variability in the exchange times is examined. The results from the circulation model are compared with the estimates given by the tidal prism method. The results show that the tidal prism method consistently underestimates the exchange times, although the predicted times tend to lie within the range of the simulated times. Including a simple return flow factor into the tidal prism estimate leads to significant improvements in the comparison. Keywords: fjords; flushing; tidal prism; numerical model; Scotland
Introduction The fjordic sea lochs on the western seaboard of Scotland have long been considered as pristine environments, with minimal anthropogenic impact and exceptional water quality amongst European waters. In recent years, concern has grown over the increase of pollutants discharged into this environment. The growth of the mariculture industry, which is concentrated in the sea lochs of the west coast, has caused particular concern. The intensive nature of salmon farming inevitably leads to considerable discharges of soluble and particulate nutrients excreted by the fish. In addition, in Scotland the industry is plagued by infestations of sea lice, a parasite that causes distress and even mortality amongst salmon. The lice are removed by means of chemical medicines, some of which are administered as bath treatments and, following treatment, are subsequently discharged directly into the water column. The dispersion and removal of these chemicals from the local environment is a further issue tackled by industry regulators. Management models used by regulators to address the problems described above are necessarily simple 0272–7714/01/100437+13 $35.00/0
and quick to apply, given the frequency with which they are used (e.g. Gillibrand & Turrell, 1997). As such, the models use simplified concepts of the real hydrodynamics of these fjordic systems. One such concept is the use of a ‘ flushing time ’ to represent the exchange of water (and contaminants) within the sea loch with the coastal water outside. Flushing times have traditionally been calculated using the tidal prism method (Dyer, 1973) or variations thereof (Edwards & Sharples, 1986). In essence, pollutants are assumed to be removed by tidal exchange only, whereas in reality wind-forcing and the baroclinic circulation may enhance or inhibit such exchange. In addition to these complications, the tidal prism method involves some assumptions, namely those of complete mixing on each tide and of no pollutant return on the flood tide, which are accepted as being unrealistic of real conditions. However, tidal exchange is the most consistent exchange mechanism over both the longer term and on a daily basis, and is undoubtedly an important contributor to the removal of pollutants from these systems. The application of simplified management models to complex environments raised questions about the
438 P. A. Gillibrand
accuracy and reliability of tidal prism-derived flushing times and their appropriateness for some sea lochs, especially the larger ones. Particular concern was expressed about Loch Fyne, the largest sea loch in Scotland, which extends northwards from the Firth of Clyde in the south-west of the country and supports an annual production of about 6000 tonnes of farmed salmon. This paper seeks to address whether simple estimates of flushing time are appropriate for pollution management of the loch, and whether a sea loch with several basins can be treated as one system or whether each basin should be modelled separately. To this end, a two-dimensional, laterally-integrated numerical model, suitable for simulating the water circulation in fjords (Elliott et al., 1992; Gillibrand et al., 1995) has been applied to Loch Fyne and used to investigate the circulation and exchange of the system. The variability of turnover times (see later section for a definition of turnover time) of the three basins of Loch Fyne are considered separately, as are the turnover times of the near-surface layers and deep basin water. The mean turnover times for each basin are compared to the estimates given by the tidal prism method and suggestions for simple improvement in the latter are made. The study area Loch Fyne is situated in the south-west of Scotland, extending 64 km northwards from its mouth on the Firth of Clyde, with average width of almost 3 km (Figure 1). In the upper reaches above the outer sill, the width is generally less than 2 km. The loch has two sills, the first being 36 m deep and located about 24 km from the mouth, the second 42 m deep and about 36 km from the mouth. The two basins behind these sills have maximum depths of 64 m and 135 m, respectively. Between the outer sill and the mouth, water depths reach about 200 m. The tide in Loch Fyne is predominantly semidiurnal with a mean spring range of 3·1 m and a mean neap range of 1 m. The catchment area of the loch is 894 km2 which results in a mean annual freshwater runoff of 1340·4106 m3 or 42·5 m3 s 1 (Edwards & Sharples, 1986). Much of this freshwater comes from surrounding hills to the north and east, where the response to rainfall is very rapid and the river flow is therefore very sporadic. Loch Fyne has one of the smallest freshwater to tidal flow ratios in Scotland (Edwards & Sharples, 1986). For this study, Loch Fyne was sub-divided into several basins and layers (Figure 2). Basin 1 extends from the head of the loch to the inner sill, Basin 2
T 1. The surface areas (km2) and volumes (106 m3) of the three sub-basins and depth layers of Loch Fyne
Surface area 10 m layer volume Above sill depth volume Whole water column
Basin 1
Basin 2
Basin 3
42·13 392·57 1362·47 2409·57
77·47 708·09 2436·52 3550·08
176·16 1665·88 5759·42 9579·83
from the head to the outer sill and Basin 3 from the head to the mouth of the loch (i.e. Basin 3 is the whole loch). The exchange of different depth layers was also considered. In the first set of simulations, turnover times for the whole water column in each basin were calculated. In the second set, times for a 10 m deep near-surface layer were calculated. In the third set, the exchange of the water column above sill depth was simulated. Finally, tracer dispersion from the deep water (below sill depth) in the upper basin was briefly examined. The surface area and volume of each sub-basin and depth layer are given in Table 1.
Definition of turnover time Discussions of the various definitions of exchange times (e.g. flushing time, turnover time, half life) are given by Prandle (1984) and Luff and Pohlmann (1995). In this study, the turnover time, or ‘ equivalent flushing time ’ (Luff & Pohlmann, 1995), was adopted. The turnover time is defined as the time needed for the total mass of material within the area of interest to be reduced to a factor of e 1 (i.e. 0·37) of its original mass (Prandle, 1984). This is the same definition as used by Edwards and Sharples (1986) for their calculation of flushing times for 110 Scottish sea lochs. For each basin and depth layer in Loch Fyne, the turnover time, TE, was calculated by the tidal prism method, i.e.:
where V is the volume of the basin or layer at low water (106 m2), A is the surface area of the basin at mid-tide (km2) and R is the mean spring tidal range (m). The mean tidal range in north-west Scotland is typically a factor 0·7 of the mean spring range (Edwards & Sharples, 1986). The conversion factor 0·52 recognizes that there are 1·93 semi-diurnal tides per day.
Exchange times in a Scottish Fjord 439 56.3 N Loch Fyne 56.2 0
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F 1. Location of map of Loch Fyne. The main map shows the positions of the two sills and fish farms in production in 1998 are indicated by squares. The position of CTD Stations 5 and 14 are also indicated by the solid circles.
Description and implementation of the model The model used in this study is a two-dimensional, laterally integrated numerical model. Because most sea lochs are very narrow relative to their length, water movements across the loch are considered to be of secondary importance (the Rossby radius of deformation is much greater than the width of the loch). In such cases, two-dimensional models which resolve longitudinally and vertically are able to simulate the most important features of the circulation. Two-
dimensional models have been widely used to investigate circulation in fjordic estuaries (e.g. Dunbar & Burling, 1987; Lavelle et al., 1991; Stacey et al., 1991, 1995; Stacey & Pond, 1992). The basic model used in this study has been described by Elliott et al. (1992) and Gillibrand et al. (1995) and is described only briefly here. Full details of the model formulation implementation and calibration as used in this study can be found in Gillibrand and Turrell (in press). The model solves a standard set of laterally integrated equations describing the conservation of
440 P. A. Gillibrand Basin 3 Basin 2 Basin 1 0
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F 2. Longitudinal view of Loch Fyne showing the real seabed profile (dashed line) and the model approximation of it (solid line). The division of the loch into three basins is indicated by the vertical dashed lines. The 10 m surface layer and the water column above sill depth are denoted by the horizontal dashed lines.
momentum, volume, salinity and turbulent kinetic energy. The present model is fully non-linear, unlike that employed by Gillibrand et al. (1995) which had the non-linear advection terms removed. An equation of state derives the density distribution from the salinity. The turbulence closure sub-model is the level 2·5 scheme of Mellor and Yamada (1982), but with the length scale, I, specified following the method of Stacy et al. (1995). The vertical diffusion coefficients are calculated from the length scale, turbulent kinetic energy and stability functions as described by Mellor and Yamada (1982). Following Stacey et al. (1995), lower bounds were placed on the vertical diffusion coefficients. These bounds were dependent on the local stratification, with the minimum vertical diffusivity set by: Kvmin =a0N1·5
(2)
where N is the local Brunt-Va¨ isa¨ la¨ frequency and a0 is a constant (Stacy et al., 1995). A series of calibration simulations resulted in a value of a0 of 1·8 10 8 m2 s 5/2. This gave minimum values of Kv of
about 10 4 m2 s 1 in the deep basin and 5·0 10 7 m2 s 1 in the strongly stratified surface layer. The model is forced by along-channel wind stress, river runoff, sea surface oscillation and the salinity profile measured from the adjacent coastal ocean. The initial salinity distribution within the loch is specified, based on field data, and the governing equations then control the temporal evolution of salinity and velocity. The equations are solved on a cartesian grid using finite difference approximations (Gillibrand et al., 1995). In the present case, Loch Fyne is modelled using a grid consisting of 17 columns and a maximum of 19 rows. The vertical spacing between grid rows can be varied, thus allowing good vertical resolution near the surface without necessitating an excessive number of grid points to cover the deep basins. The water surface is allowed to pass through the surface grid rows (Hamilton, 1975; Gillibrand et al., 1995), so that the 3 m tide can be simulated without having to increase the spacing of the near surface rows. Thus, the grid itself is a variable and grid points are included and excluded from the grid as the water surface rises and falls. The time increment was 60 s. Coefficient
Exchange times in a Scottish Fjord 441
values such as the horizontal diffusivities, the friction coefficient and the wind drag coefficient are retained as specified by Gillibrand et al. (1995).
Model simulations have been compared against current and salinity data obtained from Loch Fyne between November 1994 and February 1995 (Gillibrand & Turrell, in press). The model reproduced the tidal characteristics of the flow reasonably well, particularly in the water column above the sill. The observed time series of velocity in the upper basin were also well simulated. The general features of the observed salinity time series from the upper basin were also reproduced by the model, but the data showed more high frequency variability than was simulated. However, the final axial salinity field predicted by the model compared well with the observed field from February 1995. The lack of high frequency salinity variability in the model was probably due to the implementation of a minimum vertical diffusivity, which may have damped vertical salinity gradients. The minimum diffusivity may reflect mixing conditions on a basin-wide scale but ignore localized effects. Stigebrandt (1976) and others since (e.g. Stigebrandt & Aure, 1989) have proposed that internal tides in fjords may produce localized mixing which impacts the salinity field on a basin-wide scale through lateral density currents. Such effects cannot be reproduced by the model. The relatively accurate reproduction of the lowfrequency fluctuations in the salinity time series suggest that the variability is caused principally by local wind and runoff effects and not by external signals propagating into the fjord (because the open boundary salinity varied linearly from the November profile to the February profile). This implies that the dispersion of tracer in the model simulations (see below) will also be dominated by local effects and should therefore be reasonably well simulated by the model. Passive tracer simulation The turnover times were calculated by simulating the transport of a passive conservative tracer. The advection and diffusion of the tracer, C, was treated analogously to that of salinity, i.e.:
Depth (m)
Model performance
C = 0.0
C = 1.0
40
80
120
160 0
10
20
30
40
50
60
Distance from mouth (km)
F 3. Initial tracer concentrations for a simulation to calculate the exchange time of the water column above sill depth in Basin 1. Initial concentrations in the basin are 1·0, elsewhere in the loch concentrations are zero.
where x, z are the longitudinal and vertical co-ordinates, respectively (x positive towards the mouth, z positive downwards); u and w are the corresponding velocity components; B is the channel width; Kx and Kz are the horizontal and vertical diffusion coefficients, respectively; and Ps is a tracer source. The model was used to simulate the period 21 November 1994 to 23 February 1995, the only period when data were collected. To operate the model, the salinity field is held fixed at the values of 21 November while the velocity field is ‘ spun-up ’ for 3 days (18–21 November). The salinity is then allowed to evolve with the other variables over the 94 day simulation. Exchange times were calculated during this simulation. Initial tracer concentrations were specified as 1·0 in the basin and layer of interest and zero throughout the rest of the loch (see Figure 3 for an example). The simulation then progressed with the tracer concentrations held fixed until the start time of the exchange calculation was reached, at which point the tracer field was allowed to evolve according to Equation 3 for the remainder of the simulation. The start of the exchange simulation was stepped forward in 3-day increments, starting at 21 November, to provide a range of turnover times for each basin and give an indication of the variability of the parameter. The turnover time was taken as the time at which the tracer mass in the basin and depth layer of interest had decreased to 37% of its initial mass. For simulations using tracers to calculate exchange times, it is essential that the tracer is conserved by the model. Tests to establish this were performed by recording both the mass of tracer remaining within the loch and the cumulative mass of tracer transported out through the mouth. The sum of these variables
442 P. A. Gillibrand T 2. Simulated turnover times (days) of the three sub-basins for the whole water column. The turnover times calculated by the tidal prism method are included for comparison
100.0
Percentage mass
80.0
60.0
Mean Minimum Maximum Standard deviation Tidal Prism Method
40.0 37.0
Basin 1
Basin 2
Basin 3
34·5 26 40 3·44 14·0
30·9 24 37 3·15 11·2
13·9 10 19 2·53 13·3
20.0
TE 0
14
28
42
56
70
84
98
Time (days)
F 4. Results of the tracer conservation test, showing modelled time series of the mass of tracer remaining in the loch (solid line), the cumulative mass exiting the loch and the sum of the two. The masses are presented as percentages of the initial mass. The calculation of the exchange time, TE, based on the tracer mass falling below 37%, is illustrated. Also shown in the time series of Equation 6 with TE =16 days (closed diamonds).
should equal the initial tracer mass. Time series of these three variables (in percentage of the initial mass) taken from the first simulation, which lasted the full 94 days, are presented in Figure 4. The total mass remained at over 99% for the whole duration of the simulation. Modelling results Whole water column The simulated turnover times for the whole water column in the three basins are given in Table 2. For each basin approximately 21 simulations were performed, with the starting time stepped forward by three days each simulation (the exact number of simulations therefore depends on the turnover time itself). The mean, minimum, maximum and standard deviation for each basin are presented. From Table 2 it is evident that the turnover time increases as basin volume decreases. Basin 1 has a mean turnover time of 34·5 days, with values ranging from 26 to 40 days. Exchange times for Basin 2 vary from 24 to 37 days with a mean values of 30·9 days, and for the whole loch (Basin 3) the corresponding values are 10, 19 and 13·9 days. Standard deviations for all three basins are approximately 3 days. Thus,
although the volume of the whole loch is a factor of four greater than that of Basin 1 and a factor 2·7 greater than Basin 2, the mean turnover time of the whole loch is only 40% and 45%, respectively, of the times for the two basins. The open boundary of the whole loch consists of unpolluted ocean water and the exchange of effluent will therefore be quicker than for the internal basins where effluent can be reimported. Exchange of the outer basin is also not inhibited by a sill and, for these two reasons, it is perhaps not surprising that the turnover time for the whole loch is relatively short. Surface layer 10 m deep The second set of simulations, calculating the turnover time of a 10 m deep surface layer, were performed because management models are often applied to this near surface area. Based on energy considerations, Edwards and Sharples (1986) estimated that 88% of sea lochs had a surface layer 10 m deep or less and suggested that the flushing times calculated by the tidal prism method applied to this surface layer. The strong stratification regularly observed in sea lochs (e.g. Edwards & Edelsten, 1977; Allen, 1995; Gillibrand et al., 1995) is typically found in that depth range and effluents are discharged from fish farms into this near-surface layer. The simulated turnover times for a 10 m deep surface layer in the three basins are given in Table 3. For each basin, approximately 31 simulations were performed, and the mean, minimum, maximum and standard deviation for each basin are presented. The near-surface waters are exchanged much more rapidly than the deeper water and turnover times are much reduced. Mean simulated times for Basins 1, 2 and 3 are 3·7, 3·8 and 3·0 days, respectively. Maximum exchange times vary between 7–9 days but all three basins can turnover in as little as 1 day.
T 3. Simulated turnover times (days) of the three sub-basins for the 10 m deep surface layer. The turnover times calculated by the tidal prism method are included for comparison. The Return Flow Method results are the turnover times for Basins 1 and 2 calculated using Equation 4 Basin 2
Basin 3
3·7 1 8 1·91 2·5 3·7
3·8 1 9 2·19 2·5 3·7
3·0 1 7 1·48 2·5 —
T 4. Simulated turnover times (days) of the three sub-basins for the water column above sill depth (i.e. above 42 m). The turnover times calculated by the tidal prism method are included for comparison. The Return Flow Method results are turnover times for Basin 1 and 2 calculated using Equation 4 Basin 2
Basin 3
3 –1
Basin 1
River runoff (m s )
Mean Minimum Maximum Standard deviation Tidal Prism Method Return Flow Method
Water level (m)
–1
Basin 1
Wind velocity (m s ) Turnover time (days)
Exchange times in a Scottish Fjord 443
Mean Minimum Maximum Standard deviation Tidal Prism Method Return Flow Method
10·5 3 16 2·82 8·0 11·8
13·8 8 18 2·53 7·8 11·5
10·6 3 16 2·76 8·1 —
Water column above sill depth The simulated turnover times for the water column above sill depth in the three basins are given in Table 4. For each basin, typically 27 simulations were performed, the starting time stepped forward by 3 days each time, and the mean, minimum, maximum and standard deviation for each basin are presented. The exclusion of the deep water in Basin 1 from the simulations results in much reduced exchange times compared to those for the whole water column. Mean turnover times for the three basins are about 10–13 days, reductions on the whole water column simulations of 70% and 55% for Basins 1 and 2, respectively. Basin 3 exhibits smaller reductions of 25% minimum. Turnover times are 3 days for Basins 1 and 3, but 8 days for Basin 2. Maximum times are 16–18 days for all three basins. Variability of exchange times The variability in the exchange times for the water column above sill depth in the three basins is
18
(a)
12 6 0 15 10 5 0 –5 –10 –15 2.0
(b)
(c)
1.0 0.0 –1.0 –2.0 400
(d)
300 200 100 0 20 Nov 4 Dec 18 Dec 1 Jan 15 Jan 29 Jan 12 Feb 26 Feb
F 5. (a) Variability of the exchange times for the water column above sill depth in each basin. The turnover time is plotted against the start date of each simulation for the whole loch (solid line), Basin 2 (long dashes) and Basin 1 (short dashes). Also shown are the north/south (solid line positive northwards) and east/west (dashed line, positive eastwards) components of (b) wind velocity; (c) sea surface oscillation at the mouth; and (d) freshwater discharge at the head of the loch.
illustrated in Figure 5(a). The turnover time given by each simulation is plotted against the start date of the simulation. The exchange times show some coherence between the three basins. For the first four weeks, the turnover times remain relatively steady, although those for Basin 1 exhibit more variability than for the other two basins. This period is followed by a sharp minimum in exchange times for all basins, after which results for Basins 2 and 3 show a gradual increase, while those for Basin 1 remain relatively stable. The strong minimum which occurred in all simulations starting on 30 December corresponded to a period of strong northerly winds which evidently greatly enhanced the removal of tracer from the system [Figure 5(b)]. Figure 5 shows that, although the
Exchange times The exchange times calculated using the tidal prism method (Equation 1) are included in Tables 2–4. A comparison with the mean simulated turnover times shows that, as expected, the tidal prism method consistently underestimates the exchange time. The reasons for this are well understood, and result largely from the assumptions of complete mixing on each tide and of no return flow on the flood tide (Dyer, 1973). When the full water column depth in Basins 1 and 2 of this fjordic estuary is considered, the tidal prism method gives estimated exchange times of less than 50% of the simulated times (Table 2). Moreover, the estimated times are less than the minimum simulated turnover times. However, the prism method gives a very good estimate of the simulated turnover time for the whole loch (Basin 3). This seems to be fortuitous and occurs partly because the model simulates no pollutant return on the flood tide at the mouth of the loch, and also because a large proportion of the loch volume lies seaward of Basin 3 (Table 1) and is therefore subject to open exchange with coastal waters. In these circumstances, the bulk of the tracer can be relatively rapidly exchanged and the turnover time is much reduced. This is illustrated in Figure 6, which shows the depth-mean tracer concentration along the loch after one turnover period. In the outer basin (0–24 km), concentrations have dropped rapidly to typically 0·0–0·2, but inside the outer sill concentrations increase quite sharply. Also shown in Figure 6 is the cumulative percentage volume of the loch as a function of distance from the mouth. The high proportion of the loch volume that lies outside the outer sill (approximately 60%) means that the tracer concentrations here dominate the mass calculation and the mean tracer concentration decreases to 0·37 despite the high levels remaining in the inner part of the loch. The underlying assumptions of the tidal prism method probably represent more accurately the dynamics of the outer basin of the loch. The tidal prism method performs poorly for Basins 1 and 2 because of the deep water below sill depth, from where the tracer is removed only slowly. Figure 7 shows tracer concentrations following one turnover
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F 6. Depth-mean tracer concentrations (—) following one turnover period (i.e. total mass is 37% of the initial mass) for the simulation with tracer initially distributed through the whole water column of Basin 3. The initial tracer concentration was 1·0. The cumulative loch volume, plotted as a percentage of the whole volume against distance from the mouth is also shown (– – –).
0.2
40 Depth (m)
Discussion
1.0
Cumulative volume (%)
calculated turnover times show ranges of 10–13 days for the various basins (Table 4), for the majority of the simulated period exchange times for each basin only varied by 5–6 days. However, meteorological events, such as strong down-fjord winds, can significantly modify the tidal exchange rates.
Concentration
444 P. A. Gillibrand
0.1
0.2 0.3 0.4 0.5
0.6 0.7 0.8
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F 7. Tracer concentrations following one turnover period (i.e. total mass is 37% of the initial mass) for the simulation with tracer initially distributed through the whole water column of Basin 1. The initial tracer concentration was 1·0.
period for the Basin 1 simulation. Although most of the tracer has been removed from the surface waters, concentrations remain high in the water below. This is a clear demonstration of why simple calculations such as the prism method cannot be used to estimate residence times in deep fjordic basins. These basins tend to be flushed out in rapid episodic events, the frequency of which can be weeks for some sea lochs and years for others (e.g. Edwards & Edelsten,
Exchange times in a Scottish Fjord 445
Intra-basin variability Exchange times are defined by the mean tracer concentration in the basin of interest. As a result, the basin may be said to be exchanged when parts of it still
Depth (m)
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1977; Allen, 1995; Gillibrand et al., 1995, 1996). In between renewal events, deep water properties tend to show little variability. During these intervals, variability and exchange is due mainly to vertical diffusion. Deep water renewal in Loch Fyne has not been directly observed, but it is likely to be an annual event possibly occurring during winter and stimulated by density maxima in the deep water in the Firth of Clyde (Edwards et al., 1986; Rippeth et al., 1995; Rippeth & Simpson, 1996). A series of simulations of tracer dispersion from the water column below sill depth in Basin 1 resulted in a mean turnover time of 51·8 days with a range of 46–55 days. This exchange is due to vertical diffusion and vertical advection and exhibits less variability than exchange of the surface layers. The simulated mean turnover times for a 10 m surface layer show good agreement with the tidal prism estimates for all three basins (Table 3). However, a substantial proportion of the tracer is being mixed and recirculated below the 10 m zone and remains in the system despite being excluded from the calculation of turnover time. On average after one turnover period, 36% of the initial tracer mass remained within the loch but outside the 10 m surface layer. Given that 37% remained within that surface layer, only 27% of the tracer had typically been completely removed and the system cannot really be considered exchanged. When the water column above sill depth is considered, the tidal prism estimates also show fairly good agreement with the simulated times. For Basins 1 and 3, the estimates not only lie within the range of simulated values but within one standard deviation of the mean. The improvement on the first set of simulations clearly stems from the removal of the deep isolated basin from the calculation. It is emphasized, however, that any tracer which enters the deep basin during the simulation, although remaining within the system, is not included in the calculation of turnover time. The turnover time refers exclusively to the mass within the water column above sill depth. As in the simulations of a 10 m surface layer, some of the tracer from above the sill has entered the deep water behind the sill, but in this case the mass involved is a much smaller proportion of the initial amount, typically 6%, and over half the initial mass has been completely removed from the system.
0.1 0.2
0.5 0.4
0.3 0.1 0.1 0.2 0.3
0.5 0.3
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F 8. Tracer concentrations following one turnover period (i.e. total mass is 37% of the initial mass) for the simulation with tracer initially distributed through the water column above sill depth of (a) Basin 3; and (b) Basin 1. The initial tracer concentration was 1·0.
exhibit relatively high concentrations. The extent to which this occurs can be controlled to a degree by an appropriate choice of basin. Figure 8 compares typical tracer distributions after one turnover time from the simulations for the water column above sill depth in Basins 1 and 3. When the whole loch is simulated, tracer concentrations at the head of the loch still exhibit values up to 0·9 [Figure 8(a)]. For Basin 1, maximum concentrations following one turnover period are considerably lower, although values up to 0·6 are still evident at the head of the loch [Figure 8(b)]. Clearly, the use of the total mass of tracer to determine turnover time exposes regions of the basin to the effects of internal variability. These localized maxima increase with the volume of the box being simulated. If such variation is not desired, a multiplebox model of the system would be required, with the consequent increase in complexity that such models entail. An acceptable balance between the required simplicity of box models and a reasonable degree of accuracy is the ultimate aim of management tools.
446 P. A. Gillibrand Stn 5
Stn 14 0 S2
S4
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λ
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F 9. Schematic showing the parameters required to calculate the reflux of material, , into an internal basin based on the method of Cokelet and Stewart (1985). S1 denote the salinities of the four layers. The longitudinal positions of CTD Stations 5 and 14 are marked.
Effect of recirculation In the case of Basin 2, the tidal prism estimate is about 60% of the simulated mean and lies outside the range of simulated values. This poor estimate is mainly the result of a strong return of tracer on the flood tide. Whereas for Basin 1 some of the return flow tends to enter the deep water below sill depth and is thus excluded from the exchange time calculation, at the entrance to Basin 2 all the returned tracer remains within the surface layer. The tidal prism method can be easily modified to account for the proportion, , of the flow returned on each flood tide by defining a modified turnover time as:
A suitable value for for these systems is difficult to estimate. Luketina (1998) cited data indicating values of between 0·0 and 0·5, although he was discussing the return of contaminants at the entrance to estuaries. At internal barriers in fjordic sea lochs, which form the entrances to sub-basins, the vertical recirculation that occurs above sills may be more important. One possible method lies with the simple calculation described by Cokelet & Stewart (1985) to determine the reflux of material at internal barriers in fjords. Based on the salinities, S1–S4, of the
above- and below-sill layers on either side of the barrier (Figure 9), the reflux fraction, , is given by:
The reflux fraction is controlled by the ratio S4/S3, the surface and bottom layer salinities in the upper basin (Figure 9). As S4 increases from zero and approaches S3, so also increases from zero (no return flow) to one (complete return flow which implies an infinite turnover time, Equation 4). For Loch Fyne, a single reflux fraction was estimated for the length of the sill based on two temperature and salinity surveys carried out in November 1994 and February 1995. Salinity profiles from the basins inside (CTD station 5) and outside (CTD station 14) the sill from the two surveys are presented in Figure 10. Surface and bottom layer values of salinity were extracted from the profiles and, using Equation 5, was estimated to be 0·45 and 0·19 for November and February, respectively. Taking the average of the two values, and inserting this value (=0·32) in Equation 4 gives the modified flushing times shown in Tables 3 and 4 for Basins 1 and 2. The comparison with the simulated values is significantly improved. For both basins and both depth layers, the modified tidal prism estimate lies within one standard deviation of the simulated mean turnover
Exchange times in a Scottish Fjord 447 Salinity
Salinity 24 0
26
28
30
34
32
10
10
20
20
30
30
40
40
50
50
60
70
30
34
32
60
70
80
80
90
90
100
100
110
110
130
28
(b)
Depth (m)
Depth (m)
(a)
120
26
24 0
S1 = 33.2 S2 = 32.4 S3 = 32.6 S4 = 31.9
120 5
130 14
140
S1 = 33.1 S2 = 32.1 S3 = 32.5 S4 = 30.2
5
14
140
F 10. Depth profiles of salinity at CTD stations 5 and 14 in (a) November 1994; and (b) February 1995. Profiles were obtained using a Neil Brown ‘ Smart ’ CTD. The values of S1–S4 used in the reflux calculation are indicated for each month.
time. Although it is difficult to draw significant conclusions from these very limited results, the inclusion of a return flow element into the calculation of turnover time should certainly be expected to improve the estimate. Generalized application of TE The turnover time, as defined by Equation 1, is derived from the relationship: M=M0et/TE
(6)
where M is the tracer mass and M0 is the tracer mass at time t=0 (Prandle, 1984). Thus, when
t=TE,M=M0e 1. Theoretically, therefore, the simple calculation in Equation 1 provides a time scale for the complete dispersion of a tracer. The removal of tracer from the system is essentially exponential in character (see Figure 4). However, defining TE using the initial fall in tracer mass to 37% as indicated leads to an overestimation of the exchange rate at lower concentrations. In Figure 4, the graph of Equation 6, with TE =16 days, is plotted. Following the initial 16 day period which is used to define TE, the predicted tracer mass gradually deviates from the simulated time series. This deviation results from the increasing importance of mixing in the exchange process, as opposed to advection which
448 P. A. Gillibrand
dominates the initial stages. By defining the turnover time when advective processes are dominant, mixing is effectively excluded from the calculation, and caution must be exercised before applying this time scale to the complete exchange of the system. Concluding remarks This study has used a two-dimensional numerical model to investigate the exchange times of a fjordic Scottish sea loch and its sub-basins. The results from the model simulations were compared to estimates of the turnover time calculated using the simple tidal prism method. The main results of the study are summarized as follows: (1) The tidal prism method cannot predict the exchange rate of the deep isolated basin water and should be limited to estimating the exchange of water above sill depth. When this is done, the method tends to provide estimates that lie within one standard deviation of the simulated mean. (2) Although effluents tend to be discharged into a near-surface layer, mixing rapidly transports the material into deeper water. Thus, the turnover of the water column above sill depth, which is open to exchange with the adjacent ocean, should be estimated. This approach is more logical than using a rather arbitrary 10 m deep surface layer and may make the method more applicable to other Scottish fjords. (3) Single box models using simple calculations to approximate exchange should be carefully applied to either whole systems or sub-basins of those systems. Variability of exchange on a sub-basin scale may mean that substantial regions of a basin have not been adequately flushed following nominal turnover periods. The extent of the variability can be minimized by a suitable choice of basin. (4) The tidal prism method can be improved by including a return flow factor. At internal sills, applying the reflux calculation described by Cokelet and Stewart (1985) appears to improve estimates of the turnover time. For the internal sill of Loch Fyne, reflux factors varied between 0·19 and 0·45, with a mean value of 0·32. (5) At the mouth of the system less information about return flows is usually available. Luketina (1998) cited reports suggesting 0·5 might be a suitable default value for the return flow at the entrance to coastal estuaries. For management purposes, rather than selecting a single rather arbitrary
value, applying two values of at the entrance, say =0·0 and 0·5, could be used to provide a range of likely exchange times of the system. Management models using simple concepts such as an exchange time need to be applied quickly to assist with rapid decision making. As such, the complexity of multi-box models is avoided whenever possible. The model simulations described in this paper, although specific to Loch Fyne, suggest that a single exchange time, carefully applied to the water column above sill depth, can be an acceptable first-order approximation of contaminant exchange. For some of the larger Scottish sea lochs, however, individual basins may need to be treated separately to avoid the effects of sub-basin variability. In the particular case of Loch Fyne, although the system may be managed using the exchange time for the water column above sill depth along the full length of the loch, it will be necessary to model separately the potential impact in the upper basin alone if localized pollutant impacts are to be avoided. Acknowledgements This work was stimulated by discussions with Anne Henderson and Peter Singleton of the Scottish Environment Protection Agency. Ruediger Lang carried out some preliminary model simulations and calculations of the flushing time as part of his MSc dissertation. 2001 British Crown Copyright References Allen, G. L. 1995 Inflows, mixing and the internal tide of Upper Loch Linnhe. PhD Thesis, University of Wales, Bangor. 125 pp. Cokelet, E. E. & Stewart, R. J. 1995 The exchange of water in fjords: the efflux/reflux theory of advective reaches separated by mixing zones. Journal of Geophysical Research 90 (C4), 7287–7306. Dunbar, D. A. & Burling, R. W. 1987 A numerical model of stratified circulation in Indian Arm, British Columbia. Journal of Geophysical Research 92 (C12), 13075–13105. Dyer, K. R. 1973 Estuaries: a physical introduction. Wiley, London, 140pp. Edwards, A., Baxter, M. S., Ellett, D. J., Martin J. H. A., Meldrum D. T. & Griffiths C. R. 1986 Clyde Sea hydrography. Proceedings of the Royal Society of Edinburgh 90, 67–83. Edwards, A & Edelsten, D. J. 1977 Deep water renewal of Loch Etive: a three basin Scottish fjord. Estuarine and Coastal Marine Science 5, 575–593. Edwards, A. & Sharples, F. 1986 Scottish sea lochs: a catalogue. Scottish Marine Biological Association/Nature Conservancy Council. Elliott, A. J., Gillibrand, P. A. & Turrell, W. R. 1992 Tidal mixing near the sill of a Scottish sea loch. In Dynamics and exchanges in estuaries and the coastal zone. Coastal and Estuarine Studies 40,
Exchange times in a Scottish Fjord 449 (Prandle, D. ed.) American Geophysical Union, Washington DC, pp. 35–56. Gillibrand, P. A. & Turrell, W. R. 1997 The use of simple models in the regulation of the impact of fish farms on water quality in Scottish sea lochs. Aquaculture 159, 33–46. Gillibrand, P. A. & Turrell, W. R. In press. Observations and model simulations of water circulation in a Scottish Fjord. Marine Laboratory Aberdeen Report, Fisheries Research Services, Aberdeen. Gillibrand, P. A., Turrell, W. R. & Elliott, A. J. 1995 Deep water renewal in the upper basin of Loch Sunart, a Scottish fjord. Journal of Physical Oceanography 25, 1488–1503. Gillibrand, P. A., Turrell, W. R., Moore, D. C. & Adams, R. D. 1996 Bottom water stagnation and oxygen depletion in a Scottish sea loch. Estuarine, Coastal and Shelf Science 43, 217–235. Hamilton, P. 1975 A numerical model of the vertical circulation of tidal estuaries and its application to the Rotterdam Waterway. Geophysical Journal of the Royal Astronomical Society 40, 1–21. Lavelle, J. W., Cokelet, E. D. & Cannon, G. A. 1991 A model study of density intrusions into and circulation within a deep silled estuary: Puget Sound. Journal of Geophysical Research 96, 16779–16800. Luff, R. & Pohlmann, T. 1995 Calculation of water exchange times in the ICES-boxes with a eulerian dispersion model using a half-life time approach. Deutsche Hydrographic Zeitschrift 47, 287–299. Luketina, D. 1998 Simple tidal prism models revisited. Estuarine, Coastal and Shelf Science 46, 77–84.
Mellor, G. L. & Yamada, T. 1982 Development of a turbulence closure model for geophysical fluid problems. Reviews of Geophysics and Space Physics 20, 851–875. Prandle, D. A. 1984 A modelling study of the mixing of 137Cs in the seas of the European continental shelf. Philosophical Transactions of the Royal Society of London 310, 407–436. Rippeth, T. P. & Simpson, J. H. 1996 The frequency and duration of episodes of complete vertical mixing in the Clyde Sea. Continental Shelf Research 16, 933–947. Rippeth, T. P., Midgley, R. P. & Simpson, J. H. 1995 The seasonal cycle of stratification in a Scottish fjord. In Ecology of Fjords and Coastal Waters. (Skjotdal, H. R., Hopkins, C., Erikstad, K. E. & Leinaas, H. P. eds). Elsevier Science B.V., Amsterdam, pp. 85– 92. Stacey, M. W. & Pond, S. 1992 A numerical model of the internal tide in Knight Inlet, British Columbia. Atmosphere-Ocean 30, 382–418. Stacey, M. W., Pond, S. & Cameron, A. N. 1991 The comparison of a numerical model to observations of velocity and scalar data in Burrard Inlet and Indian Arm, British Columbia. Journal of Geophysical Research 96, 10777–10790. Stacey, M. W., Pond, S. & Nowak, Z. P. 1995 A numerical model of the circulation in Knight Inlet, British Columbia, Canada. Journal of Physical Oceanography 25, 1037–1062. Stigebrandt, A. 1976 Vertical diffusion driven by internal waves in a sill Fjord. Journal of Physical Oceanography 6, 486–495. Stigebrandt, A. & Aure, J. 1989 Vertical mixing in basin waters of Fjords. Journal of Physical Oceanography 19, 917–926.
Calculating Exchange Times in a Scottish Fjord Using a Two-dimensional, Laterally-integrated Numerical Model P. A. Gillibrand FRS Marine Laboratory, P.O. Box 101, Victoria Road, Aberdeen, AB11 9DB, U.K. Received 4 December 1998 and accepted in revised form 13 September 1999 In order to assess the potential impact of pollutants, particularly soluble wastes discharged by the mariculture industry, on the fjordic sea loch environment in Scotland, simple management models have been developed which estimate steady-state concentrations based on the quantities of effluent released and the residence time of such material within a loch. These models make various simplifications about the hydrodynamic characteristics of Scottish sea lochs, the most important of which is the concept of an exchange time which parametrizes the rate at which pollutants are removed from the system. Exchange times for individual lochs are calculated using the tidal prism method, which has some well-known shortcomings. In this paper, a two-dimensional laterally-integrated circulation model is used to investigate the exchange characteristics of Loch Fyne and its sub-basins. By simulating the transport of a passive, conservative tracer, the turnover times for the loch, two sub-basins and various depth layers are calculated. By varying the starting time of the tracer simulations, the variability in the exchange times is examined. The results from the circulation model are compared with the estimates given by the tidal prism method. The results show that the tidal prism method consistently underestimates the exchange times, although the predicted times tend to lie within the range of the simulated times. Including a simple return flow factor into the tidal prism estimate leads to significant improvements in the comparison. Keywords: fjords; flushing; tidal prism; numerical model; Scotland
Introduction The fjordic sea lochs on the western seaboard of Scotland have long been considered as pristine environments, with minimal anthropogenic impact and exceptional water quality amongst European waters. In recent years, concern has grown over the increase of pollutants discharged into this environment. The growth of the mariculture industry, which is concentrated in the sea lochs of the west coast, has caused particular concern. The intensive nature of salmon farming inevitably leads to considerable discharges of soluble and particulate nutrients excreted by the fish. In addition, in Scotland the industry is plagued by infestations of sea lice, a parasite that causes distress and even mortality amongst salmon. The lice are removed by means of chemical medicines, some of which are administered as bath treatments and, following treatment, are subsequently discharged directly into the water column. The dispersion and removal of these chemicals from the local environment is a further issue tackled by industry regulators. Management models used by regulators to address the problems described above are necessarily simple 0272–7714/01/100437+13 $35.00/0
and quick to apply, given the frequency with which they are used (e.g. Gillibrand & Turrell, 1997). As such, the models use simplified concepts of the real hydrodynamics of these fjordic systems. One such concept is the use of a ‘ flushing time ’ to represent the exchange of water (and contaminants) within the sea loch with the coastal water outside. Flushing times have traditionally been calculated using the tidal prism method (Dyer, 1973) or variations thereof (Edwards & Sharples, 1986). In essence, pollutants are assumed to be removed by tidal exchange only, whereas in reality wind-forcing and the baroclinic circulation may enhance or inhibit such exchange. In addition to these complications, the tidal prism method involves some assumptions, namely those of complete mixing on each tide and of no pollutant return on the flood tide, which are accepted as being unrealistic of real conditions. However, tidal exchange is the most consistent exchange mechanism over both the longer term and on a daily basis, and is undoubtedly an important contributor to the removal of pollutants from these systems. The application of simplified management models to complex environments raised questions about the
438 P. A. Gillibrand
accuracy and reliability of tidal prism-derived flushing times and their appropriateness for some sea lochs, especially the larger ones. Particular concern was expressed about Loch Fyne, the largest sea loch in Scotland, which extends northwards from the Firth of Clyde in the south-west of the country and supports an annual production of about 6000 tonnes of farmed salmon. This paper seeks to address whether simple estimates of flushing time are appropriate for pollution management of the loch, and whether a sea loch with several basins can be treated as one system or whether each basin should be modelled separately. To this end, a two-dimensional, laterally-integrated numerical model, suitable for simulating the water circulation in fjords (Elliott et al., 1992; Gillibrand et al., 1995) has been applied to Loch Fyne and used to investigate the circulation and exchange of the system. The variability of turnover times (see later section for a definition of turnover time) of the three basins of Loch Fyne are considered separately, as are the turnover times of the near-surface layers and deep basin water. The mean turnover times for each basin are compared to the estimates given by the tidal prism method and suggestions for simple improvement in the latter are made. The study area Loch Fyne is situated in the south-west of Scotland, extending 64 km northwards from its mouth on the Firth of Clyde, with average width of almost 3 km (Figure 1). In the upper reaches above the outer sill, the width is generally less than 2 km. The loch has two sills, the first being 36 m deep and located about 24 km from the mouth, the second 42 m deep and about 36 km from the mouth. The two basins behind these sills have maximum depths of 64 m and 135 m, respectively. Between the outer sill and the mouth, water depths reach about 200 m. The tide in Loch Fyne is predominantly semidiurnal with a mean spring range of 3·1 m and a mean neap range of 1 m. The catchment area of the loch is 894 km2 which results in a mean annual freshwater runoff of 1340·4106 m3 or 42·5 m3 s 1 (Edwards & Sharples, 1986). Much of this freshwater comes from surrounding hills to the north and east, where the response to rainfall is very rapid and the river flow is therefore very sporadic. Loch Fyne has one of the smallest freshwater to tidal flow ratios in Scotland (Edwards & Sharples, 1986). For this study, Loch Fyne was sub-divided into several basins and layers (Figure 2). Basin 1 extends from the head of the loch to the inner sill, Basin 2
T 1. The surface areas (km2) and volumes (106 m3) of the three sub-basins and depth layers of Loch Fyne
Surface area 10 m layer volume Above sill depth volume Whole water column
Basin 1
Basin 2
Basin 3
42·13 392·57 1362·47 2409·57
77·47 708·09 2436·52 3550·08
176·16 1665·88 5759·42 9579·83
from the head to the outer sill and Basin 3 from the head to the mouth of the loch (i.e. Basin 3 is the whole loch). The exchange of different depth layers was also considered. In the first set of simulations, turnover times for the whole water column in each basin were calculated. In the second set, times for a 10 m deep near-surface layer were calculated. In the third set, the exchange of the water column above sill depth was simulated. Finally, tracer dispersion from the deep water (below sill depth) in the upper basin was briefly examined. The surface area and volume of each sub-basin and depth layer are given in Table 1.
Definition of turnover time Discussions of the various definitions of exchange times (e.g. flushing time, turnover time, half life) are given by Prandle (1984) and Luff and Pohlmann (1995). In this study, the turnover time, or ‘ equivalent flushing time ’ (Luff & Pohlmann, 1995), was adopted. The turnover time is defined as the time needed for the total mass of material within the area of interest to be reduced to a factor of e 1 (i.e. 0·37) of its original mass (Prandle, 1984). This is the same definition as used by Edwards and Sharples (1986) for their calculation of flushing times for 110 Scottish sea lochs. For each basin and depth layer in Loch Fyne, the turnover time, TE, was calculated by the tidal prism method, i.e.:
where V is the volume of the basin or layer at low water (106 m2), A is the surface area of the basin at mid-tide (km2) and R is the mean spring tidal range (m). The mean tidal range in north-west Scotland is typically a factor 0·7 of the mean spring range (Edwards & Sharples, 1986). The conversion factor 0·52 recognizes that there are 1·93 semi-diurnal tides per day.
Exchange times in a Scottish Fjord 439 56.3 N Loch Fyne 56.2 0
5
10 km
56.1
Sill (42 m)
56.0
55.9
Sill (36 m)
14
55.8
Firth of Clyde
55.7 –5.5
–5.4
–5.3
–5.2
–5.1
–5.0
–4.9
–4.8
–4.7
F 1. Location of map of Loch Fyne. The main map shows the positions of the two sills and fish farms in production in 1998 are indicated by squares. The position of CTD Stations 5 and 14 are also indicated by the solid circles.
Description and implementation of the model The model used in this study is a two-dimensional, laterally integrated numerical model. Because most sea lochs are very narrow relative to their length, water movements across the loch are considered to be of secondary importance (the Rossby radius of deformation is much greater than the width of the loch). In such cases, two-dimensional models which resolve longitudinally and vertically are able to simulate the most important features of the circulation. Two-
dimensional models have been widely used to investigate circulation in fjordic estuaries (e.g. Dunbar & Burling, 1987; Lavelle et al., 1991; Stacey et al., 1991, 1995; Stacey & Pond, 1992). The basic model used in this study has been described by Elliott et al. (1992) and Gillibrand et al. (1995) and is described only briefly here. Full details of the model formulation implementation and calibration as used in this study can be found in Gillibrand and Turrell (in press). The model solves a standard set of laterally integrated equations describing the conservation of
440 P. A. Gillibrand Basin 3 Basin 2 Basin 1 0
40
Depth (m)
80
120
160
200
0
10
20
30
40
50
60
Distance from mouth (km)
F 2. Longitudinal view of Loch Fyne showing the real seabed profile (dashed line) and the model approximation of it (solid line). The division of the loch into three basins is indicated by the vertical dashed lines. The 10 m surface layer and the water column above sill depth are denoted by the horizontal dashed lines.
momentum, volume, salinity and turbulent kinetic energy. The present model is fully non-linear, unlike that employed by Gillibrand et al. (1995) which had the non-linear advection terms removed. An equation of state derives the density distribution from the salinity. The turbulence closure sub-model is the level 2·5 scheme of Mellor and Yamada (1982), but with the length scale, I, specified following the method of Stacy et al. (1995). The vertical diffusion coefficients are calculated from the length scale, turbulent kinetic energy and stability functions as described by Mellor and Yamada (1982). Following Stacey et al. (1995), lower bounds were placed on the vertical diffusion coefficients. These bounds were dependent on the local stratification, with the minimum vertical diffusivity set by: Kvmin =a0N1·5
(2)
where N is the local Brunt-Va¨ isa¨ la¨ frequency and a0 is a constant (Stacy et al., 1995). A series of calibration simulations resulted in a value of a0 of 1·8 10 8 m2 s 5/2. This gave minimum values of Kv of
about 10 4 m2 s 1 in the deep basin and 5·0 10 7 m2 s 1 in the strongly stratified surface layer. The model is forced by along-channel wind stress, river runoff, sea surface oscillation and the salinity profile measured from the adjacent coastal ocean. The initial salinity distribution within the loch is specified, based on field data, and the governing equations then control the temporal evolution of salinity and velocity. The equations are solved on a cartesian grid using finite difference approximations (Gillibrand et al., 1995). In the present case, Loch Fyne is modelled using a grid consisting of 17 columns and a maximum of 19 rows. The vertical spacing between grid rows can be varied, thus allowing good vertical resolution near the surface without necessitating an excessive number of grid points to cover the deep basins. The water surface is allowed to pass through the surface grid rows (Hamilton, 1975; Gillibrand et al., 1995), so that the 3 m tide can be simulated without having to increase the spacing of the near surface rows. Thus, the grid itself is a variable and grid points are included and excluded from the grid as the water surface rises and falls. The time increment was 60 s. Coefficient
Exchange times in a Scottish Fjord 441
values such as the horizontal diffusivities, the friction coefficient and the wind drag coefficient are retained as specified by Gillibrand et al. (1995).
Model simulations have been compared against current and salinity data obtained from Loch Fyne between November 1994 and February 1995 (Gillibrand & Turrell, in press). The model reproduced the tidal characteristics of the flow reasonably well, particularly in the water column above the sill. The observed time series of velocity in the upper basin were also well simulated. The general features of the observed salinity time series from the upper basin were also reproduced by the model, but the data showed more high frequency variability than was simulated. However, the final axial salinity field predicted by the model compared well with the observed field from February 1995. The lack of high frequency salinity variability in the model was probably due to the implementation of a minimum vertical diffusivity, which may have damped vertical salinity gradients. The minimum diffusivity may reflect mixing conditions on a basin-wide scale but ignore localized effects. Stigebrandt (1976) and others since (e.g. Stigebrandt & Aure, 1989) have proposed that internal tides in fjords may produce localized mixing which impacts the salinity field on a basin-wide scale through lateral density currents. Such effects cannot be reproduced by the model. The relatively accurate reproduction of the lowfrequency fluctuations in the salinity time series suggest that the variability is caused principally by local wind and runoff effects and not by external signals propagating into the fjord (because the open boundary salinity varied linearly from the November profile to the February profile). This implies that the dispersion of tracer in the model simulations (see below) will also be dominated by local effects and should therefore be reasonably well simulated by the model. Passive tracer simulation The turnover times were calculated by simulating the transport of a passive conservative tracer. The advection and diffusion of the tracer, C, was treated analogously to that of salinity, i.e.:
Depth (m)
Model performance
C = 0.0
C = 1.0
40
80
120
160 0
10
20
30
40
50
60
Distance from mouth (km)
F 3. Initial tracer concentrations for a simulation to calculate the exchange time of the water column above sill depth in Basin 1. Initial concentrations in the basin are 1·0, elsewhere in the loch concentrations are zero.
where x, z are the longitudinal and vertical co-ordinates, respectively (x positive towards the mouth, z positive downwards); u and w are the corresponding velocity components; B is the channel width; Kx and Kz are the horizontal and vertical diffusion coefficients, respectively; and Ps is a tracer source. The model was used to simulate the period 21 November 1994 to 23 February 1995, the only period when data were collected. To operate the model, the salinity field is held fixed at the values of 21 November while the velocity field is ‘ spun-up ’ for 3 days (18–21 November). The salinity is then allowed to evolve with the other variables over the 94 day simulation. Exchange times were calculated during this simulation. Initial tracer concentrations were specified as 1·0 in the basin and layer of interest and zero throughout the rest of the loch (see Figure 3 for an example). The simulation then progressed with the tracer concentrations held fixed until the start time of the exchange calculation was reached, at which point the tracer field was allowed to evolve according to Equation 3 for the remainder of the simulation. The start of the exchange simulation was stepped forward in 3-day increments, starting at 21 November, to provide a range of turnover times for each basin and give an indication of the variability of the parameter. The turnover time was taken as the time at which the tracer mass in the basin and depth layer of interest had decreased to 37% of its initial mass. For simulations using tracers to calculate exchange times, it is essential that the tracer is conserved by the model. Tests to establish this were performed by recording both the mass of tracer remaining within the loch and the cumulative mass of tracer transported out through the mouth. The sum of these variables
442 P. A. Gillibrand T 2. Simulated turnover times (days) of the three sub-basins for the whole water column. The turnover times calculated by the tidal prism method are included for comparison
100.0
Percentage mass
80.0
60.0
Mean Minimum Maximum Standard deviation Tidal Prism Method
40.0 37.0
Basin 1
Basin 2
Basin 3
34·5 26 40 3·44 14·0
30·9 24 37 3·15 11·2
13·9 10 19 2·53 13·3
20.0
TE 0
14
28
42
56
70
84
98
Time (days)
F 4. Results of the tracer conservation test, showing modelled time series of the mass of tracer remaining in the loch (solid line), the cumulative mass exiting the loch and the sum of the two. The masses are presented as percentages of the initial mass. The calculation of the exchange time, TE, based on the tracer mass falling below 37%, is illustrated. Also shown in the time series of Equation 6 with TE =16 days (closed diamonds).
should equal the initial tracer mass. Time series of these three variables (in percentage of the initial mass) taken from the first simulation, which lasted the full 94 days, are presented in Figure 4. The total mass remained at over 99% for the whole duration of the simulation. Modelling results Whole water column The simulated turnover times for the whole water column in the three basins are given in Table 2. For each basin approximately 21 simulations were performed, with the starting time stepped forward by three days each simulation (the exact number of simulations therefore depends on the turnover time itself). The mean, minimum, maximum and standard deviation for each basin are presented. From Table 2 it is evident that the turnover time increases as basin volume decreases. Basin 1 has a mean turnover time of 34·5 days, with values ranging from 26 to 40 days. Exchange times for Basin 2 vary from 24 to 37 days with a mean values of 30·9 days, and for the whole loch (Basin 3) the corresponding values are 10, 19 and 13·9 days. Standard deviations for all three basins are approximately 3 days. Thus,
although the volume of the whole loch is a factor of four greater than that of Basin 1 and a factor 2·7 greater than Basin 2, the mean turnover time of the whole loch is only 40% and 45%, respectively, of the times for the two basins. The open boundary of the whole loch consists of unpolluted ocean water and the exchange of effluent will therefore be quicker than for the internal basins where effluent can be reimported. Exchange of the outer basin is also not inhibited by a sill and, for these two reasons, it is perhaps not surprising that the turnover time for the whole loch is relatively short. Surface layer 10 m deep The second set of simulations, calculating the turnover time of a 10 m deep surface layer, were performed because management models are often applied to this near surface area. Based on energy considerations, Edwards and Sharples (1986) estimated that 88% of sea lochs had a surface layer 10 m deep or less and suggested that the flushing times calculated by the tidal prism method applied to this surface layer. The strong stratification regularly observed in sea lochs (e.g. Edwards & Edelsten, 1977; Allen, 1995; Gillibrand et al., 1995) is typically found in that depth range and effluents are discharged from fish farms into this near-surface layer. The simulated turnover times for a 10 m deep surface layer in the three basins are given in Table 3. For each basin, approximately 31 simulations were performed, and the mean, minimum, maximum and standard deviation for each basin are presented. The near-surface waters are exchanged much more rapidly than the deeper water and turnover times are much reduced. Mean simulated times for Basins 1, 2 and 3 are 3·7, 3·8 and 3·0 days, respectively. Maximum exchange times vary between 7–9 days but all three basins can turnover in as little as 1 day.
T 3. Simulated turnover times (days) of the three sub-basins for the 10 m deep surface layer. The turnover times calculated by the tidal prism method are included for comparison. The Return Flow Method results are the turnover times for Basins 1 and 2 calculated using Equation 4 Basin 2
Basin 3
3·7 1 8 1·91 2·5 3·7
3·8 1 9 2·19 2·5 3·7
3·0 1 7 1·48 2·5 —
T 4. Simulated turnover times (days) of the three sub-basins for the water column above sill depth (i.e. above 42 m). The turnover times calculated by the tidal prism method are included for comparison. The Return Flow Method results are turnover times for Basin 1 and 2 calculated using Equation 4 Basin 2
Basin 3
3 –1
Basin 1
River runoff (m s )
Mean Minimum Maximum Standard deviation Tidal Prism Method Return Flow Method
Water level (m)
–1
Basin 1
Wind velocity (m s ) Turnover time (days)
Exchange times in a Scottish Fjord 443
Mean Minimum Maximum Standard deviation Tidal Prism Method Return Flow Method
10·5 3 16 2·82 8·0 11·8
13·8 8 18 2·53 7·8 11·5
10·6 3 16 2·76 8·1 —
Water column above sill depth The simulated turnover times for the water column above sill depth in the three basins are given in Table 4. For each basin, typically 27 simulations were performed, the starting time stepped forward by 3 days each time, and the mean, minimum, maximum and standard deviation for each basin are presented. The exclusion of the deep water in Basin 1 from the simulations results in much reduced exchange times compared to those for the whole water column. Mean turnover times for the three basins are about 10–13 days, reductions on the whole water column simulations of 70% and 55% for Basins 1 and 2, respectively. Basin 3 exhibits smaller reductions of 25% minimum. Turnover times are 3 days for Basins 1 and 3, but 8 days for Basin 2. Maximum times are 16–18 days for all three basins. Variability of exchange times The variability in the exchange times for the water column above sill depth in the three basins is
18
(a)
12 6 0 15 10 5 0 –5 –10 –15 2.0
(b)
(c)
1.0 0.0 –1.0 –2.0 400
(d)
300 200 100 0 20 Nov 4 Dec 18 Dec 1 Jan 15 Jan 29 Jan 12 Feb 26 Feb
F 5. (a) Variability of the exchange times for the water column above sill depth in each basin. The turnover time is plotted against the start date of each simulation for the whole loch (solid line), Basin 2 (long dashes) and Basin 1 (short dashes). Also shown are the north/south (solid line positive northwards) and east/west (dashed line, positive eastwards) components of (b) wind velocity; (c) sea surface oscillation at the mouth; and (d) freshwater discharge at the head of the loch.
illustrated in Figure 5(a). The turnover time given by each simulation is plotted against the start date of the simulation. The exchange times show some coherence between the three basins. For the first four weeks, the turnover times remain relatively steady, although those for Basin 1 exhibit more variability than for the other two basins. This period is followed by a sharp minimum in exchange times for all basins, after which results for Basins 2 and 3 show a gradual increase, while those for Basin 1 remain relatively stable. The strong minimum which occurred in all simulations starting on 30 December corresponded to a period of strong northerly winds which evidently greatly enhanced the removal of tracer from the system [Figure 5(b)]. Figure 5 shows that, although the
Exchange times The exchange times calculated using the tidal prism method (Equation 1) are included in Tables 2–4. A comparison with the mean simulated turnover times shows that, as expected, the tidal prism method consistently underestimates the exchange time. The reasons for this are well understood, and result largely from the assumptions of complete mixing on each tide and of no return flow on the flood tide (Dyer, 1973). When the full water column depth in Basins 1 and 2 of this fjordic estuary is considered, the tidal prism method gives estimated exchange times of less than 50% of the simulated times (Table 2). Moreover, the estimated times are less than the minimum simulated turnover times. However, the prism method gives a very good estimate of the simulated turnover time for the whole loch (Basin 3). This seems to be fortuitous and occurs partly because the model simulates no pollutant return on the flood tide at the mouth of the loch, and also because a large proportion of the loch volume lies seaward of Basin 3 (Table 1) and is therefore subject to open exchange with coastal waters. In these circumstances, the bulk of the tracer can be relatively rapidly exchanged and the turnover time is much reduced. This is illustrated in Figure 6, which shows the depth-mean tracer concentration along the loch after one turnover period. In the outer basin (0–24 km), concentrations have dropped rapidly to typically 0·0–0·2, but inside the outer sill concentrations increase quite sharply. Also shown in Figure 6 is the cumulative percentage volume of the loch as a function of distance from the mouth. The high proportion of the loch volume that lies outside the outer sill (approximately 60%) means that the tracer concentrations here dominate the mass calculation and the mean tracer concentration decreases to 0·37 despite the high levels remaining in the inner part of the loch. The underlying assumptions of the tidal prism method probably represent more accurately the dynamics of the outer basin of the loch. The tidal prism method performs poorly for Basins 1 and 2 because of the deep water below sill depth, from where the tracer is removed only slowly. Figure 7 shows tracer concentrations following one turnover
100
0.8
80
0.6
60
0.4
40
0.2
20
8
0
16 24 32 40 48 Distance from mouth (km)
56
0 64
F 6. Depth-mean tracer concentrations (—) following one turnover period (i.e. total mass is 37% of the initial mass) for the simulation with tracer initially distributed through the whole water column of Basin 3. The initial tracer concentration was 1·0. The cumulative loch volume, plotted as a percentage of the whole volume against distance from the mouth is also shown (– – –).
0.2
40 Depth (m)
Discussion
1.0
Cumulative volume (%)
calculated turnover times show ranges of 10–13 days for the various basins (Table 4), for the majority of the simulated period exchange times for each basin only varied by 5–6 days. However, meteorological events, such as strong down-fjord winds, can significantly modify the tidal exchange rates.
Concentration
444 P. A. Gillibrand
0.1
0.2 0.3 0.4 0.5
0.6 0.7 0.8
80
0.9
120 160 0
10
20
30
40
50
60
Distance from mouth (km)
F 7. Tracer concentrations following one turnover period (i.e. total mass is 37% of the initial mass) for the simulation with tracer initially distributed through the whole water column of Basin 1. The initial tracer concentration was 1·0.
period for the Basin 1 simulation. Although most of the tracer has been removed from the surface waters, concentrations remain high in the water below. This is a clear demonstration of why simple calculations such as the prism method cannot be used to estimate residence times in deep fjordic basins. These basins tend to be flushed out in rapid episodic events, the frequency of which can be weeks for some sea lochs and years for others (e.g. Edwards & Edelsten,
Exchange times in a Scottish Fjord 445
Intra-basin variability Exchange times are defined by the mean tracer concentration in the basin of interest. As a result, the basin may be said to be exchanged when parts of it still
Depth (m)
0.2.3 0 .4 0 .5 0 0.6
0.8
0.6 0.5
0.
0.7
3
0.4
0.1
40
0.6 0.5 0.4 0.3 0.2
80 0.1
120 160
(a) 0
10
20
30
40
60
50
Distance from mouth (km) 0.2
40 Depth (m)
1977; Allen, 1995; Gillibrand et al., 1995, 1996). In between renewal events, deep water properties tend to show little variability. During these intervals, variability and exchange is due mainly to vertical diffusion. Deep water renewal in Loch Fyne has not been directly observed, but it is likely to be an annual event possibly occurring during winter and stimulated by density maxima in the deep water in the Firth of Clyde (Edwards et al., 1986; Rippeth et al., 1995; Rippeth & Simpson, 1996). A series of simulations of tracer dispersion from the water column below sill depth in Basin 1 resulted in a mean turnover time of 51·8 days with a range of 46–55 days. This exchange is due to vertical diffusion and vertical advection and exhibits less variability than exchange of the surface layers. The simulated mean turnover times for a 10 m surface layer show good agreement with the tidal prism estimates for all three basins (Table 3). However, a substantial proportion of the tracer is being mixed and recirculated below the 10 m zone and remains in the system despite being excluded from the calculation of turnover time. On average after one turnover period, 36% of the initial tracer mass remained within the loch but outside the 10 m surface layer. Given that 37% remained within that surface layer, only 27% of the tracer had typically been completely removed and the system cannot really be considered exchanged. When the water column above sill depth is considered, the tidal prism estimates also show fairly good agreement with the simulated times. For Basins 1 and 3, the estimates not only lie within the range of simulated values but within one standard deviation of the mean. The improvement on the first set of simulations clearly stems from the removal of the deep isolated basin from the calculation. It is emphasized, however, that any tracer which enters the deep basin during the simulation, although remaining within the system, is not included in the calculation of turnover time. The turnover time refers exclusively to the mass within the water column above sill depth. As in the simulations of a 10 m surface layer, some of the tracer from above the sill has entered the deep water behind the sill, but in this case the mass involved is a much smaller proportion of the initial amount, typically 6%, and over half the initial mass has been completely removed from the system.
0.1 0.2
0.5 0.4
0.3 0.1 0.1 0.2 0.3
0.5 0.3
80
0.4
0.2 0.1
120 160
(b) 0
10
20
30
40
50
60
Distance from mouth (km)
F 8. Tracer concentrations following one turnover period (i.e. total mass is 37% of the initial mass) for the simulation with tracer initially distributed through the water column above sill depth of (a) Basin 3; and (b) Basin 1. The initial tracer concentration was 1·0.
exhibit relatively high concentrations. The extent to which this occurs can be controlled to a degree by an appropriate choice of basin. Figure 8 compares typical tracer distributions after one turnover time from the simulations for the water column above sill depth in Basins 1 and 3. When the whole loch is simulated, tracer concentrations at the head of the loch still exhibit values up to 0·9 [Figure 8(a)]. For Basin 1, maximum concentrations following one turnover period are considerably lower, although values up to 0·6 are still evident at the head of the loch [Figure 8(b)]. Clearly, the use of the total mass of tracer to determine turnover time exposes regions of the basin to the effects of internal variability. These localized maxima increase with the volume of the box being simulated. If such variation is not desired, a multiplebox model of the system would be required, with the consequent increase in complexity that such models entail. An acceptable balance between the required simplicity of box models and a reasonable degree of accuracy is the ultimate aim of management tools.
446 P. A. Gillibrand Stn 5
Stn 14 0 S2
S4
40
Depth (m)
λ
80
S3 S1
120
160
200
0
10
20
30
40
50
60
Distance from mouth (km)
F 9. Schematic showing the parameters required to calculate the reflux of material, , into an internal basin based on the method of Cokelet and Stewart (1985). S1 denote the salinities of the four layers. The longitudinal positions of CTD Stations 5 and 14 are marked.
Effect of recirculation In the case of Basin 2, the tidal prism estimate is about 60% of the simulated mean and lies outside the range of simulated values. This poor estimate is mainly the result of a strong return of tracer on the flood tide. Whereas for Basin 1 some of the return flow tends to enter the deep water below sill depth and is thus excluded from the exchange time calculation, at the entrance to Basin 2 all the returned tracer remains within the surface layer. The tidal prism method can be easily modified to account for the proportion, , of the flow returned on each flood tide by defining a modified turnover time as:
A suitable value for for these systems is difficult to estimate. Luketina (1998) cited data indicating values of between 0·0 and 0·5, although he was discussing the return of contaminants at the entrance to estuaries. At internal barriers in fjordic sea lochs, which form the entrances to sub-basins, the vertical recirculation that occurs above sills may be more important. One possible method lies with the simple calculation described by Cokelet & Stewart (1985) to determine the reflux of material at internal barriers in fjords. Based on the salinities, S1–S4, of the
above- and below-sill layers on either side of the barrier (Figure 9), the reflux fraction, , is given by:
The reflux fraction is controlled by the ratio S4/S3, the surface and bottom layer salinities in the upper basin (Figure 9). As S4 increases from zero and approaches S3, so also increases from zero (no return flow) to one (complete return flow which implies an infinite turnover time, Equation 4). For Loch Fyne, a single reflux fraction was estimated for the length of the sill based on two temperature and salinity surveys carried out in November 1994 and February 1995. Salinity profiles from the basins inside (CTD station 5) and outside (CTD station 14) the sill from the two surveys are presented in Figure 10. Surface and bottom layer values of salinity were extracted from the profiles and, using Equation 5, was estimated to be 0·45 and 0·19 for November and February, respectively. Taking the average of the two values, and inserting this value (=0·32) in Equation 4 gives the modified flushing times shown in Tables 3 and 4 for Basins 1 and 2. The comparison with the simulated values is significantly improved. For both basins and both depth layers, the modified tidal prism estimate lies within one standard deviation of the simulated mean turnover
Exchange times in a Scottish Fjord 447 Salinity
Salinity 24 0
26
28
30
34
32
10
10
20
20
30
30
40
40
50
50
60
70
30
34
32
60
70
80
80
90
90
100
100
110
110
130
28
(b)
Depth (m)
Depth (m)
(a)
120
26
24 0
S1 = 33.2 S2 = 32.4 S3 = 32.6 S4 = 31.9
120 5
130 14
140
S1 = 33.1 S2 = 32.1 S3 = 32.5 S4 = 30.2
5
14
140
F 10. Depth profiles of salinity at CTD stations 5 and 14 in (a) November 1994; and (b) February 1995. Profiles were obtained using a Neil Brown ‘ Smart ’ CTD. The values of S1–S4 used in the reflux calculation are indicated for each month.
time. Although it is difficult to draw significant conclusions from these very limited results, the inclusion of a return flow element into the calculation of turnover time should certainly be expected to improve the estimate. Generalized application of TE The turnover time, as defined by Equation 1, is derived from the relationship: M=M0et/TE
(6)
where M is the tracer mass and M0 is the tracer mass at time t=0 (Prandle, 1984). Thus, when
t=TE,M=M0e 1. Theoretically, therefore, the simple calculation in Equation 1 provides a time scale for the complete dispersion of a tracer. The removal of tracer from the system is essentially exponential in character (see Figure 4). However, defining TE using the initial fall in tracer mass to 37% as indicated leads to an overestimation of the exchange rate at lower concentrations. In Figure 4, the graph of Equation 6, with TE =16 days, is plotted. Following the initial 16 day period which is used to define TE, the predicted tracer mass gradually deviates from the simulated time series. This deviation results from the increasing importance of mixing in the exchange process, as opposed to advection which
448 P. A. Gillibrand
dominates the initial stages. By defining the turnover time when advective processes are dominant, mixing is effectively excluded from the calculation, and caution must be exercised before applying this time scale to the complete exchange of the system. Concluding remarks This study has used a two-dimensional numerical model to investigate the exchange times of a fjordic Scottish sea loch and its sub-basins. The results from the model simulations were compared to estimates of the turnover time calculated using the simple tidal prism method. The main results of the study are summarized as follows: (1) The tidal prism method cannot predict the exchange rate of the deep isolated basin water and should be limited to estimating the exchange of water above sill depth. When this is done, the method tends to provide estimates that lie within one standard deviation of the simulated mean. (2) Although effluents tend to be discharged into a near-surface layer, mixing rapidly transports the material into deeper water. Thus, the turnover of the water column above sill depth, which is open to exchange with the adjacent ocean, should be estimated. This approach is more logical than using a rather arbitrary 10 m deep surface layer and may make the method more applicable to other Scottish fjords. (3) Single box models using simple calculations to approximate exchange should be carefully applied to either whole systems or sub-basins of those systems. Variability of exchange on a sub-basin scale may mean that substantial regions of a basin have not been adequately flushed following nominal turnover periods. The extent of the variability can be minimized by a suitable choice of basin. (4) The tidal prism method can be improved by including a return flow factor. At internal sills, applying the reflux calculation described by Cokelet and Stewart (1985) appears to improve estimates of the turnover time. For the internal sill of Loch Fyne, reflux factors varied between 0·19 and 0·45, with a mean value of 0·32. (5) At the mouth of the system less information about return flows is usually available. Luketina (1998) cited reports suggesting 0·5 might be a suitable default value for the return flow at the entrance to coastal estuaries. For management purposes, rather than selecting a single rather arbitrary
value, applying two values of at the entrance, say =0·0 and 0·5, could be used to provide a range of likely exchange times of the system. Management models using simple concepts such as an exchange time need to be applied quickly to assist with rapid decision making. As such, the complexity of multi-box models is avoided whenever possible. The model simulations described in this paper, although specific to Loch Fyne, suggest that a single exchange time, carefully applied to the water column above sill depth, can be an acceptable first-order approximation of contaminant exchange. For some of the larger Scottish sea lochs, however, individual basins may need to be treated separately to avoid the effects of sub-basin variability. In the particular case of Loch Fyne, although the system may be managed using the exchange time for the water column above sill depth along the full length of the loch, it will be necessary to model separately the potential impact in the upper basin alone if localized pollutant impacts are to be avoided. Acknowledgements This work was stimulated by discussions with Anne Henderson and Peter Singleton of the Scottish Environment Protection Agency. Ruediger Lang carried out some preliminary model simulations and calculations of the flushing time as part of his MSc dissertation. 2001 British Crown Copyright References Allen, G. L. 1995 Inflows, mixing and the internal tide of Upper Loch Linnhe. PhD Thesis, University of Wales, Bangor. 125 pp. Cokelet, E. E. & Stewart, R. J. 1995 The exchange of water in fjords: the efflux/reflux theory of advective reaches separated by mixing zones. Journal of Geophysical Research 90 (C4), 7287–7306. Dunbar, D. A. & Burling, R. W. 1987 A numerical model of stratified circulation in Indian Arm, British Columbia. Journal of Geophysical Research 92 (C12), 13075–13105. Dyer, K. R. 1973 Estuaries: a physical introduction. Wiley, London, 140pp. Edwards, A., Baxter, M. S., Ellett, D. J., Martin J. H. A., Meldrum D. T. & Griffiths C. R. 1986 Clyde Sea hydrography. Proceedings of the Royal Society of Edinburgh 90, 67–83. Edwards, A & Edelsten, D. J. 1977 Deep water renewal of Loch Etive: a three basin Scottish fjord. Estuarine and Coastal Marine Science 5, 575–593. Edwards, A. & Sharples, F. 1986 Scottish sea lochs: a catalogue. Scottish Marine Biological Association/Nature Conservancy Council. Elliott, A. J., Gillibrand, P. A. & Turrell, W. R. 1992 Tidal mixing near the sill of a Scottish sea loch. In Dynamics and exchanges in estuaries and the coastal zone. Coastal and Estuarine Studies 40,
Exchange times in a Scottish Fjord 449 (Prandle, D. ed.) American Geophysical Union, Washington DC, pp. 35–56. Gillibrand, P. A. & Turrell, W. R. 1997 The use of simple models in the regulation of the impact of fish farms on water quality in Scottish sea lochs. Aquaculture 159, 33–46. Gillibrand, P. A. & Turrell, W. R. In press. Observations and model simulations of water circulation in a Scottish Fjord. Marine Laboratory Aberdeen Report, Fisheries Research Services, Aberdeen. Gillibrand, P. A., Turrell, W. R. & Elliott, A. J. 1995 Deep water renewal in the upper basin of Loch Sunart, a Scottish fjord. Journal of Physical Oceanography 25, 1488–1503. Gillibrand, P. A., Turrell, W. R., Moore, D. C. & Adams, R. D. 1996 Bottom water stagnation and oxygen depletion in a Scottish sea loch. Estuarine, Coastal and Shelf Science 43, 217–235. Hamilton, P. 1975 A numerical model of the vertical circulation of tidal estuaries and its application to the Rotterdam Waterway. Geophysical Journal of the Royal Astronomical Society 40, 1–21. Lavelle, J. W., Cokelet, E. D. & Cannon, G. A. 1991 A model study of density intrusions into and circulation within a deep silled estuary: Puget Sound. Journal of Geophysical Research 96, 16779–16800. Luff, R. & Pohlmann, T. 1995 Calculation of water exchange times in the ICES-boxes with a eulerian dispersion model using a half-life time approach. Deutsche Hydrographic Zeitschrift 47, 287–299. Luketina, D. 1998 Simple tidal prism models revisited. Estuarine, Coastal and Shelf Science 46, 77–84.
Mellor, G. L. & Yamada, T. 1982 Development of a turbulence closure model for geophysical fluid problems. Reviews of Geophysics and Space Physics 20, 851–875. Prandle, D. A. 1984 A modelling study of the mixing of 137Cs in the seas of the European continental shelf. Philosophical Transactions of the Royal Society of London 310, 407–436. Rippeth, T. P. & Simpson, J. H. 1996 The frequency and duration of episodes of complete vertical mixing in the Clyde Sea. Continental Shelf Research 16, 933–947. Rippeth, T. P., Midgley, R. P. & Simpson, J. H. 1995 The seasonal cycle of stratification in a Scottish fjord. In Ecology of Fjords and Coastal Waters. (Skjotdal, H. R., Hopkins, C., Erikstad, K. E. & Leinaas, H. P. eds). Elsevier Science B.V., Amsterdam, pp. 85– 92. Stacey, M. W. & Pond, S. 1992 A numerical model of the internal tide in Knight Inlet, British Columbia. Atmosphere-Ocean 30, 382–418. Stacey, M. W., Pond, S. & Cameron, A. N. 1991 The comparison of a numerical model to observations of velocity and scalar data in Burrard Inlet and Indian Arm, British Columbia. Journal of Geophysical Research 96, 10777–10790. Stacey, M. W., Pond, S. & Nowak, Z. P. 1995 A numerical model of the circulation in Knight Inlet, British Columbia, Canada. Journal of Physical Oceanography 25, 1037–1062. Stigebrandt, A. 1976 Vertical diffusion driven by internal waves in a sill Fjord. Journal of Physical Oceanography 6, 486–495. Stigebrandt, A. & Aure, J. 1989 Vertical mixing in basin waters of Fjords. Journal of Physical Oceanography 19, 917–926.