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Simple Tidal Prism Models Revisited
Estuarine, Coastal and Shelf Science (1998) 46, 77–84
Simple Tidal Prism Models Revisited D. Luketina School of Civil and Environmental Engineering, University of New South Wales, Sydney 2052, Australia Received 22 January 1996 and accepted in revised form 6 January 1997 Simple tidal prism models for well-mixed estuaries have been in use for some time and are discussed in most text books on estuaries. The appeal of this model is its simplicity. However, there are several flaws in the logic behind the model. These flaws are pointed out and a more theoretically correct simple tidal prism model is derived. In doing so, it is made clear which effects can, in theory, be neglected and which can not. ? 1998 Academic Press Limited Keywords: estuary; flushing; residence time; tidal prism
Introduction Simple tidal prism models have been in use for a considerable period of time. Text books such as Officer (1976), Dyer (1973) and Harleman (1966) refer to tidal prism theory. Ketchum (1951) used the simple tidal prism model as the starting point for the development of his estuarine box model. It is interesting that none of these authors gives a reference to the development of the simple tidal prism model. Further, none of these authors presents the theory behind the simple tidal prism model. This is surprising given the usefulness of the model in estimating residence times and concentrations of dissolved and suspended materials in well-mixed estuaries and embayments (for example, see Sanford et al., 1992). This paper examines the logic behind the simple tidal prism model, and discusses which effects can be neglected and which can not. This is certainly important from a pedagogical perspective. Also, in some instances, the correctly derived simple tidal prism model can give significantly different results from the classical simple tidal prism model. This paper uses the definition of Cameron and Pritchard (1963) that ‘ An estuary is a semi-enclosed coastal body of water which has a free connexion with the open sea and within which sea water is measurably diluted with fresh water derived from land drainage ’. A well-mixed or vertically homogeneous estuary, as referred to in this paper, is one which has negligible vertical and lateral gradients of salt. Thus salt flux only occurs longitudinally. A well-mixed estuary may be caused by wind mixing or tidal stirring. 0272–7714/98/010077+08 $25.00/0/ec970235
The relative effect of the river inflow to the tidal flow is typically expressed by:
where P is the tidal prism and VR =QRT is the river inflow during one tidal cycle where QR is the river flow rate and T is the tidal period. The tidal prism is defined as the volume of water contained in an estuary or embayment between the low and high tide levels. If the river inflow and tidal forcing are dominating the dynamics of an estuary, a simple classification scheme results. Provided that an estuary is not of the fjord type, an estuary is likely to be vertically well mixed when R¦0·1 if tidal stirring is the dominant mixing process (Schultz & Simmons, 1957). Other classification schemes are discussed in Fischer (1976). Note that in this case, the length L of the estuary needs to be sufficiently short so that the estuary is also horizontally homogeneous, thereby satisfying the study’s definition of well mixed. This paper first discusses the classical simple tidal prism model as typically presented. Following this, a more theoretically correct tidal prism model is derived by correcting some mistakes that were made in deriving the classical tidal prism model. The next part of the paper then extends the tidal prism model by taking into account return flows and the lags of the ebb and flood flows. Finally, a continuous form of the tidal prism model with additional source and decay terms is derived. Before proceeding, it should be noted that tidal prism models are a gross simplification of actual estuarine mixing. Further, the basis of this paper is an ? 1998 Academic Press Limited
78 D. Luketina VR
P
River
V, S
S=0
VR + VP VP Ocean
Ebb tide
Substituting equation 3 into equation 4 gives the freshwater fraction f as:
Flood tide S = So
F 1. An estuary represented by a single box where V is the low tide volume and P is the tidal prism. The outflow and inflow volumes are shown for the ebb and flood tides.
examination of theoretical models. As such, comments relating to the relative validity of the models should be seen in this context. Future work will involve testing the models using field data and numerical models. The ‘ classical ’ simple tidal prism model as typically presented—model A Officer (1976) basically explains the simple tidal prism model as: In this method it is assumed that on the flood tide the volume of seawater Vp entering the estuary is entirely of oceanic salinity ó and that it is completely mixed with a corresponding volume of fresh water VR as measured over the entire tidal cycle. It is further assumed that this entire quantity of mixed water is completely removed from the estuary on the ebb tide . . .
Note that So is used here to denote the salinity in the ocean. A sketch of this process is as per Figure 1 where the volume of water entering the estuary on the flood tide is denoted Vp. For an estuary in steady state, the amount of salt crossing the ocean boundary of the box must be zero over a tidal cycle. Hence, conservation of salt yields:
where S is the salinity in the estuary at high tide, assuming a well-mixed estuary, and So is the salinity in the ocean. Further, it is assumed that the tidal prism is given by:
Substituting this into equation 1 gives the salinity as:
If it is assumed that the estuarine water is a mixture of seawater and (fresh) river water, then the average fraction of fresh water by volume must then be:
A steady release of a miscible passive pollutant released at the head of the estuary is assumed to dilute at the same rate as the fresh water entering the estuary. If the concentration of river water and pollutant in the estuary are denoted by Cfw and C (kg m "3), respectively, then:
where the subscript ‘ source ’ denotes the concentration at the source. The freshwater concentration Cfw in the estuary is given by:
Substituting equation 5 into equation 7 gives the freshwater concentration in the estuary as:
The concentration of the pollutant at its source is given by:
if it is assumed that the pollutant is released at a rate of W kg s "1 and is fully mixed across the upstream (river) end of the prism or box. Substituting equations 8 and 9 into equation 6 and noting that the concentration of fresh water at its source Cfw source must be simply the density ñfw of fresh water leads to:
where Wfw is the mass rate of supply of fresh water and is simply given by Wfw =ñfw QR. The subscript A denotes that CA is the concentration calculated using model A. Equation 10 shows that, according to the classical simple tidal prism model—model A, the dilution of a steady source only depends upon the tidal period and the tidal prism. A more correct tidal prism model—model B As before, the estuary is represented by a single box; however, the ebb and flood phases will now be
Simple tidal prism models revisited
VR /2
P
River
V
VR /2 River
P V
VR /2 + VP Ocean
VP – VR /2 Ocean
Ebb tide Overall reduction in volume = VP
Flood tide Overall increase in volume = VP
79
Following the procedure used in deriving equation 4 and making use of equation 12 gives the average fraction of fresh water f as:
Following a similar procedure to that used in deriving equations 8–10 one obtains:
F 2. An estuary represented by a single box where V is the low tide volume and P is the tidal prism. The outflow and inflow volumes are shown for the ebb and flood tides.
considered separately. It is assumed that the ebb and flood phases are of equal duration. Seawater entering the estuary on the flood tide is assumed to mix fully with the water in the estuary prior to the ebb tide. Here, the volume of water entering or leaving the estuary on the flood or ebb tide, respectively, in the absence of river flow is denoted as VP. Then, with the addition of a river flow QR =VRT, a volume of VP-VR/2 of water must enter the estuary from the ocean during the flood phase in order to satisfy continuity. The volume leaving the estuary and entering the ocean on the ebb tide will be VP +VR/2. This is demonstrated in Figure 2. For an estuary in steady state, the amount of salt crossing the ocean boundary of the box must be zero over a tidal cycle. Hence:
where, as before, S is the salinity in the estuary at high tide, assuming a well-mixed estuary, and So is the salinity in the ocean. Now, the overall reduction in the volume of an estuary over an ebb tide, from Figure 2, can be seen to be Vp. Thus Vp must be equal to the volume of the tidal prism P so that equation 11 becomes:
Both relationships for the salinity (i.e. equations 3 and 12) indicate that S=So, as expected, for the case of zero flow. However, only equation 12 indicates that the salinity S is equal to zero when the tidal prism is equal to the volume of river water that enters the estuary over half of a tidal cycle. This is exactly the criterion that Ketchum (1951) attempted to apply, and that Dyer and Taylor (1973) applied for a box or prism to consist entirely of fresh water.
where the subscript ‘ B ’ denotes model B. Unlike the classical model (model A), equation 14 indicates that the concentration resulting from a steady release of a miscible passive pollutant into an estuary depends upon the river flow rate QR. Similarly to the classical model, equation 14 shows that the pollution concentration will reduce as the tidal range and river flow rate increase. As the tidal range approaches zero (i.e. P vanishes), equation 14 suggests that C=2W/QR when, quite clearly, the concentration should be C=W/QR (Fischer et al., 1979) with the estuary behaving as a large river (ignoring gravitational circulation here). The reason for this discrepancy is that equation 11 implicitly assumes that the flow between the ocean and the estuary occurs for an equal duration in either direction. This is not true if there is a river flow, as will be shown in the following section.
An even more rigorous tidal prism model—model C In this section, model B is extended by incorporating the duration of the ebb and flood flows and variation in the level of flushing at the mouth of the estuary. A sketch of an estuary is shown in Figure 3 where the instantaneous flow between the estuary and the ocean is denoted as Q. The river inflow to the estuary QR is assumed to be steady. The continuity equation for an estuary is given by:
where Pi is the instantaneous volume between the free surface and the low tide level, t is time and it is assumed that the water is incompressible. As pointed out previously, for the entire estuary to be well mixed, it is necessary that the estuary is relatively short. This means that there is unlikely to be much
80 D. Luketina
River S=0
Pi V, S
Q
0.2
Ocean S = So
F 3. An estuary represented by a single box where V is the low tide volume and Pi is the instantaneous volume between the free surface and the low tide level. The flow QR is assumed to be steady while Q is an instantaneous flow rate which can be of either sign.
0.0
where P is the tidal prism and the tidal amplitude is assumed to vary sinusoidally with period T. Note that the form of equation 16 is such that high tides occur at t=(n+0·25)T and low tides at t=(n+0·75)T where n is an integer. Substituting equation 16 into equation 15, simplifying and rearranging results in the flow Q being given by:
The change from ebb to flood flow and vice versa occurs when Q is equal to zero. Setting Q=0 in equation 17 gives:
where to now specifies the times at which Q is zero. One can replace to by:
where te and tf are the times at which the ebb and flood flows start in the absence of any river flow and ôe and ôf are the time lags introduced by the presence of river flow. Substituting equation 19 into equation 18 and solving for the lags yields:
for the normalized lags for the ebb and flood flows respectively where è is defined by:
τe /T
–0.1 –0.2 0.0
tidal attenuation within the estuary and Pi can be approximated by:
τf /T
0.1
τ /T
QR
0.2
0.4 0.6 QRT/πP
0.8
1.0
F 4. Normalized lag for the flood and ebb flows as a function of the normalized river flow QRT/ðP. T 1. Start and finish times for the first flood and ebb flows plus the duration of these flows (note that tf "te =T/2) Flow cycle
Start
Finish
Duration
Ebb 1 Flood 1
te +ôe tf +ôf
tf +ôf T+te +ôe
T/2+ôf "ôe T/2"ôf +ôe
Thus the ebb and flood flow lags depend upon the ratio R which is the ratio of the river inflow over a tidal cycle VR =QRT to the tidal prism P. This is shown graphically in Figure 4 from which it can be seen that the flood flow is increasingly delayed as the river flow rate is increased. On the other hand, the ebb flow occurs earlier. The duration of the ebb and flood flows can be determined as per Table 1. The normalized durations of the ebb and flood flows are plotted in Figure 5 where it can be seen that, as the normalized river flow QRT/ðP increases, the duration of the flood flow decreases and the duration of the ebb flow increases. Eventually, when QRT/ðP reaches unity, there is no flood flow and seawater does not enter the estuary (assuming that there is no gravitational circulation). As expected, when there is no river flow, the ebb and flood flows occur for equal periods of time. Conservation of mass can now be applied to determine the high tide salinity within the estuary. However, before proceeding, an implicit assumption that was made earlier will be raised. That is that the salinity in the ocean adjacent to the estuary is the same as the normal oceanic value So. This can only occur if all of the water that exits from an estuary on an ebb tide is advected away before the flood tide commences. In practice, some fraction of the water that enters the
1
1.0
0.8
0.8
0.4
flood
0.6 0.4 0.2
0.2 0
81
0 0.25 0.5 0.75 0.99
ebb 0.6
S/So
Duration/T
Simple tidal prism models revisited
0.2
0.4 0.6 QRT/πP
0.8
1.0
F 5. Normalized durations of the flood and ebb flows as a function of the normalized river flow QRT/ðP.
estuary during the flood tide is made up of water that left the estuary on the previous ebb tide. This fraction is known as the return flow factor b (Sanford et al., 1992). A negligible return flow factor (i.e. close to zero) can only result if the receiving water is well flushed—typically by alongshore currents (Fischer et al., 1979; Sanford et al., 1992). Making use of the return flow factor, the mass of salt leaving the estuary on the ebb me and flood mf flows are:
Note that mf is negative as salt must enter the estuary on the flood flow. Implicit in the formulations of equations 22 and 23 is that the flow Q, through the mouth of the estuary, is relatively uniform. Relatively uniform, means that all fluid is travelling in the same direction. However, estuaries and embayments with wide mouths are often subject to asymmetries with regard to the flood and ebb flow patterns (Bruun et al., 1978; Wolanski & Imberger, 1987). At times, ebb and flood flows can coexist. This means there can be salt transport despite the net flow Q being zero. We can evaluate the mass of salt Äm that leaves the estuary over a tidal cycle by summing equations 22 and 23. The integration can then be carried out by substituting equations 17, 20 and 21 and noting that te =T/4 and tf =3T/4 because of the form assumed earlier for the tidal behaviour (i.e. equation 16) to give:
0.0
0.1
0.2
0.3 0.4 QRT/πP
0.5
0.6
0.7
F 6. Normalized steady-state high tide salinity S/So as a function of the normalized river flow QRT/ðP for the classical tidal prism model—model A (short dashes), the improved tidal prism model—model B (long dashes) and the even more rigorously derived tidal prism model—model C (solid lines), where values of b are as indicated.
where, as before, è=cos "1 (QRT/ðP). For an estuary in steady state Äm=0. Setting Äm=0 in equation 24 results in a steady-state salinity of:
The three equations derived for the steady-state high tide salinity (i.e. equations 3, 12 and 25) are shown plotted in Figure 6. Before proceeding further, one needs to recognize that the estuary must be well mixed for the tidal prism models presented here to be valid. A typical estuary will be well mixed provided that R¦0·1 and can be considered to be partially mixed when R~0·25 (Schulz & Simmons, 1957). Wind stirring can result in increased mixing so that an estuary in a windy location could be well mixed for values of R exceeding 0·1. Another factor which should be considered is the density of the water in the river; a river with a significant sediment load can easily have a combined water plus suspended sediment density as great as that of seawater. This means that mixing will not be inhibited by density differences when compared to the case of clean river water entering an estuary. Thus there is another scenario where a well-mixed estuary could result for values of R exceeding 0·1. Although possible, a reasonably wellmixed estuary will rarely occur for R>0·25. For this reason, Figure 6 has been replotted so that the maximum value of R is 0·25 (this corresponds to R/ð=QRT/ðP=0·08). Looking at Figure 7, for the case where the return flow factor b is zero, it is clear
82 D. Luketina 1.0 0.25 0
S/So
0.8 0.6
0.5
0.75
0.4 0.99 0.2
0
0.01
0.02
0.03
0.04 0.05 QRT/πP
0.06
0.07
0.08
F 7. Normalized steady-state high tide salinity S/So as a function of the normalized river flow QRT/ðP for the classical prism model—model A (short dashes), the improved tidal prism model—model B (long dashes) and the even more rigorously derived tidal prism model—model C (solid lines), where values of b are as indicated. Unlike Figure 6, this plot has values of QRT/ðP limited to the range in which a reasonably well-mixed estuary may occur. Note that the long dashes (model B) coincide with the solid line (model C) for which b=0.
that there is little practical difference between the different models. This indicates that the effect of the river flow on varying the duration of the ebb and flood flows can be neglected. However, the effect of the return flow factor is considerable. Reported values of b include 0·23 for Indian River Bay, Delaware (Sanford et al., 1992) and range from 0 to 0·33 for San Francisco Bay (Nelson & Lerseth, 1972). In the absence of any data, it has been suggested that the return flow factor be set to 0·5 (USEPA, 1985). Note that if suitable field measurements are available, Figure 7 can be used to determine the return flow factor b for a well-mixed estuary. Alternately, the physically based method of Sanford et al. (1992) can be used to estimate b. For the case of QRT/ðP°1 it can be shown that sin è=(1-(QRT/ðP)2)1/2~1 so that the salinity S as given by equation 25 is well approximated by:
This equation should be good compromise between complexity and accuracy. Note that for the case where b=0, equation 26 is identical to equation 12 (model B).
steady state or equilibrium. While this is true for salinity, it is not necessarily true for other substances. This section extends model C so that it is a continuous unsteady model which incorporates additional source or sink terms. If one makes the same assumption used in deriving equation 26 from equation 25 (i.e. that QRT/ðP°1 so that sin è~1 and è~ð/2), the mass of any substance that leaves the estuary during a tidal cycle can be derived in an analogous manner to equation 24 to give:
where C is the high tide concentration in the estuary, Co is the background concentration in the ocean, CR is the concentration in the river, W (kg s "1) is a source term of negligible volume flux within the estuary and k (s "1) is a decay rate. The first three terms on the right-hand side of equation 27 can be derived from equation 24. The last three terms represent inflow of the substance from the river, sources within the estuary and decay of the substance, respectively. The rate of change of concentration C in the estuary is then given by:
The case of an instantaneous release of a conservative substance within the estuary can be modelled by setting W=0, CR =Co =0, and k=0 to give:
Solving this for the initial condition C=Ci at t=0 results in:
A continuous model—model D The preceding models are based upon the assumption that, in a tidally averaged sense, the estuary is in
where the subscript ‘ D ’ denotes model D, trD is the residence time and is given here by:
Simple tidal prism models revisited 1.0 0.10 0.20 0.8
0.30 0.40 0.50
0.6
b
0.60 0.70
0.4
0.80 0.90
0.2
1.00 0.0 0.00
0.02
0.04 QRT/πP
0.06
0.08
F 8. Contours of the ratio of trA/trD (the ratio of the residence time based on the classical tidal prism method—model A, so that given by equation 31—model D) as a function of the normalized river flow QRT/ðP and the return flow factor b. The same contours also apply to CA/CD which is the ratio of the concentration resulting from a continuous source based on the classical tidal prism method—model A, to that given by equation 35—model D.
83
One can compare the flushing times given by equations 31 and 32 by examining their ratio trA/trD which is in the term within the square brackets in equation 31. The ratio is plotted in Figure 8 as a function of the return flow factor b and the normalized river flow rate QRT/ðP. Looking at Figure 8, it is clear that the classical tidal prism method—model A, when compared to tidal prism model D, significantly underestimates the residence time for the larger values of the return flow factor b. When b is zero, however, the classical tidal prism method can overestimate the residence time as the normalized river flow QRT/ðP increases. However, for typical values of the return flow factor and the normalized river flow in a well mixed estuary, it is likely that the classical tidal prism method—model A, will underestimate the residence time. Another case which can be readily solved is that of a constant release of pollutant into the estuary. In this case we must eventually achieve steady state so that dC/dt=0 and we obtain:
This can be rearranged to give the steady-state concentration as:
When comparing this equation with that obtained by Sanford et al. (1992) it should be noted that the volume V used by Sanford et al. (1992) is a mid-tide volume (equivalent to V+P/2 in the terminology used here). Furthermore, Sanford et al. (1992) defined their concentration as a mid-tide value. Once these differences are taken into account, the residence time evaluated by Sanford et al. (1992) only differs from that given by equation 31 due to the effects of river flow not being correctly incorporated into their formulation. The definition of the residence time that is used here is different from that for the flushing time tF which is usually defined as the time needed for the river flow to replace all of the river water in the estuary (Officer, 1976). The residence time is a more appropriate time scale than the flushing time, as a pollutant can still leave an estuary or embayment when there is little or no river flow. The residence time using the classical tidal prism method, trA is (Dyer, 1973):
For the case of a continuous release of a conservative substance in the estuary with Co =CR =0, the above equation reduces to:
The concentrations given by equations 10 (the classical tidal prism method—model A), 32 and 35 can be compared by examining their ratio CA/CD. This ratio ends up being the term within the square brackets in equation 35. So that, as expected, CA/CD =tA/trD. The ratio is plotted in Figure 8 as a function of the return flow factor b and the normalized river flow rate QRT/ðP. Figure 8 shows that the classical tidal prism method—model A, significantly underestimates the concentration for the larger values of the return flow factor b. When b is zero, however, the classical tidal prism method can overestimate the concentration as the normalized river flow QRT/ðP increases. However,
84 D. Luketina
for typical values of the return flow factor and the normalized river flow in a well-mixed estuary, it is likely that the classical tidal prism method will underestimate the concentration resulting from a continuous source of a conservative substance. Cases other than the instantaneous release and continuous release, as solved above, can be solved numerically using equation 28. In doing so, it should be noted that equation 28 was derived for the high tide concentration at which time the estuary is assumed to be well mixed. The concentrations indicated by equation 28 for times other than high tide will give an indication of the average concentration within the estuary. However, equation 28 can not be expected to resolve details on time scales of less than the tidal period T or spatial scales of less than the length L of the estuary. However, the additional complications inherent in doing this make the use of the simple tidal prism model (model D) fairly attractive.
estimates the concentration for larger values of the return flow factor b. When b is zero (one of the assumptions of the classical method), the classical tidal prism method can overestimate the concentration as the normalized river flow QRT/ðP increases. However, for typical values of the return flow factor and the normalized river flow in a well-mixed estuary, it is likely that the classical tidal prism method will underestimate concentrations. It is recommended that model D (the continuous model) be typically used as a starting point for investigations using simple tidal prism models.
Summary and conclusions
References
The classical tidal prism method (model A), based upon what has been typically presented, has never been properly explained and is not formulated correctly. The method can be easily reformulated by correctly taking into the effects of the river flow. These are two-fold. First, the river flow affects the volume of ocean water that enters the estuary on each tidal cycle. Second, the river flow causes the flood flow to be lagged and the ebb flow to occur earlier than would occur without the presence of the river flow. However, it has been shown, for the normalized river flows that are likely to occur in a well-mixed estuary, that the lagging effect can be neglected. The tidal prism model was then extended by incorporating a return flow factor which, for typical values, plays a significant role in the residence time of the estuary. Finally, a continuous model was developed for the concentration which incorporated source and decay terms. Implicit assumptions in all of the models are that the estuary is fully mixed at high tide and that the flow between the estuary and ocean consists of fluid which is all travelling in the same direction. All concentrations are, strictly speaking, high tide values. Based upon comparisons with a more theoretically correct tidal prism method (model D), the classical tidal prism method (model A) significantly under-
Acknowledgements This paper was written while the author was on study leave at the Department of Civil Engineering, Canterbury University. The reviewers are thanked for their useful comments and suggestions.
Bruun, P., Mehta, A. & Johnsson, I. 1978 Stability of Tidal Inlets—Theory and Engineering. Elsevier, Amsterdam. Cameron, W. M. & Pritchard, D. W. 1963 Estuaries. In The Sea, Vol. 2 (Hill, M. N., ed.). Wiley, New York. Dyer, K. R. 1973 Estuaries: a Physical Introduction. Wiley, London. Dyer, K. R. & Taylor, P. A. 1973 A simple, segmented prism model of tidal mixing in well-mixed estuaries. Estuarine and Coastal Marine Science 1, 411–418. Fischer, H. B. 1976 Mixing and dispersion estuaries. Annual Review of Fluid Mechanics 8, 107–134. Fischer, H. B., List, E. J., Koh, R. C., Imberger, J. & Brooks, N. H. 1979 Mixing in Inland and Coastal Waters. Academic Press, New York. Harleman, D. R. F. 1966 Pollution in estuaries. In Estuary and Coastline Hydrodynamics, Engineering Society Monographs, Chap. 14. McGraw-Hill, New York. Ketchum, B. H. 1951 The exchange of fresh and salt waters in tidal estuaries. Journal of Marine Research 10, 18–38. Nelson, A. W. & Lerseth, R. J. 1972 A Study of Dispersion Capability of San Francisco Bay-Delta Waters. California Department of Water Resources, Sacramento, California. Officer, C. B. 1976 Physical Oceanography of Estuaries (and Associated Coastal Waters). Wiley, London. Sanford, L. P., Boicourt, W. C. & Rives, S. R. 1992 Model for estimating tidal flushing of small embayments. ASCE Journal of Waterway, Port, Coastal and Ocean Engineering 118 (6), 635–655. Schultz, E. A. & Simmons, H. B. 1957 Fresh water–salt water density currents, a major cause of siltation in estuaries. US Army Corps Engineers Communications in Tidal Hydraulics, Technical Bulletin 2, 28 pp. USEPA 1985 Coastal Marinas Assessment Handbook. United States Protection Agency, Region IV, Atlanta, Georgia. Wolanski, E. & Imberger, J. 1987 Friction-controlled selective withdrawal near inlets. Estuarine, Coastal and Shelf Science 24, 327–222.
Simple Tidal Prism Models Revisited D. Luketina School of Civil and Environmental Engineering, University of New South Wales, Sydney 2052, Australia Received 22 January 1996 and accepted in revised form 6 January 1997 Simple tidal prism models for well-mixed estuaries have been in use for some time and are discussed in most text books on estuaries. The appeal of this model is its simplicity. However, there are several flaws in the logic behind the model. These flaws are pointed out and a more theoretically correct simple tidal prism model is derived. In doing so, it is made clear which effects can, in theory, be neglected and which can not. ? 1998 Academic Press Limited Keywords: estuary; flushing; residence time; tidal prism
Introduction Simple tidal prism models have been in use for a considerable period of time. Text books such as Officer (1976), Dyer (1973) and Harleman (1966) refer to tidal prism theory. Ketchum (1951) used the simple tidal prism model as the starting point for the development of his estuarine box model. It is interesting that none of these authors gives a reference to the development of the simple tidal prism model. Further, none of these authors presents the theory behind the simple tidal prism model. This is surprising given the usefulness of the model in estimating residence times and concentrations of dissolved and suspended materials in well-mixed estuaries and embayments (for example, see Sanford et al., 1992). This paper examines the logic behind the simple tidal prism model, and discusses which effects can be neglected and which can not. This is certainly important from a pedagogical perspective. Also, in some instances, the correctly derived simple tidal prism model can give significantly different results from the classical simple tidal prism model. This paper uses the definition of Cameron and Pritchard (1963) that ‘ An estuary is a semi-enclosed coastal body of water which has a free connexion with the open sea and within which sea water is measurably diluted with fresh water derived from land drainage ’. A well-mixed or vertically homogeneous estuary, as referred to in this paper, is one which has negligible vertical and lateral gradients of salt. Thus salt flux only occurs longitudinally. A well-mixed estuary may be caused by wind mixing or tidal stirring. 0272–7714/98/010077+08 $25.00/0/ec970235
The relative effect of the river inflow to the tidal flow is typically expressed by:
where P is the tidal prism and VR =QRT is the river inflow during one tidal cycle where QR is the river flow rate and T is the tidal period. The tidal prism is defined as the volume of water contained in an estuary or embayment between the low and high tide levels. If the river inflow and tidal forcing are dominating the dynamics of an estuary, a simple classification scheme results. Provided that an estuary is not of the fjord type, an estuary is likely to be vertically well mixed when R¦0·1 if tidal stirring is the dominant mixing process (Schultz & Simmons, 1957). Other classification schemes are discussed in Fischer (1976). Note that in this case, the length L of the estuary needs to be sufficiently short so that the estuary is also horizontally homogeneous, thereby satisfying the study’s definition of well mixed. This paper first discusses the classical simple tidal prism model as typically presented. Following this, a more theoretically correct tidal prism model is derived by correcting some mistakes that were made in deriving the classical tidal prism model. The next part of the paper then extends the tidal prism model by taking into account return flows and the lags of the ebb and flood flows. Finally, a continuous form of the tidal prism model with additional source and decay terms is derived. Before proceeding, it should be noted that tidal prism models are a gross simplification of actual estuarine mixing. Further, the basis of this paper is an ? 1998 Academic Press Limited
78 D. Luketina VR
P
River
V, S
S=0
VR + VP VP Ocean
Ebb tide
Substituting equation 3 into equation 4 gives the freshwater fraction f as:
Flood tide S = So
F 1. An estuary represented by a single box where V is the low tide volume and P is the tidal prism. The outflow and inflow volumes are shown for the ebb and flood tides.
examination of theoretical models. As such, comments relating to the relative validity of the models should be seen in this context. Future work will involve testing the models using field data and numerical models. The ‘ classical ’ simple tidal prism model as typically presented—model A Officer (1976) basically explains the simple tidal prism model as: In this method it is assumed that on the flood tide the volume of seawater Vp entering the estuary is entirely of oceanic salinity ó and that it is completely mixed with a corresponding volume of fresh water VR as measured over the entire tidal cycle. It is further assumed that this entire quantity of mixed water is completely removed from the estuary on the ebb tide . . .
Note that So is used here to denote the salinity in the ocean. A sketch of this process is as per Figure 1 where the volume of water entering the estuary on the flood tide is denoted Vp. For an estuary in steady state, the amount of salt crossing the ocean boundary of the box must be zero over a tidal cycle. Hence, conservation of salt yields:
where S is the salinity in the estuary at high tide, assuming a well-mixed estuary, and So is the salinity in the ocean. Further, it is assumed that the tidal prism is given by:
Substituting this into equation 1 gives the salinity as:
If it is assumed that the estuarine water is a mixture of seawater and (fresh) river water, then the average fraction of fresh water by volume must then be:
A steady release of a miscible passive pollutant released at the head of the estuary is assumed to dilute at the same rate as the fresh water entering the estuary. If the concentration of river water and pollutant in the estuary are denoted by Cfw and C (kg m "3), respectively, then:
where the subscript ‘ source ’ denotes the concentration at the source. The freshwater concentration Cfw in the estuary is given by:
Substituting equation 5 into equation 7 gives the freshwater concentration in the estuary as:
The concentration of the pollutant at its source is given by:
if it is assumed that the pollutant is released at a rate of W kg s "1 and is fully mixed across the upstream (river) end of the prism or box. Substituting equations 8 and 9 into equation 6 and noting that the concentration of fresh water at its source Cfw source must be simply the density ñfw of fresh water leads to:
where Wfw is the mass rate of supply of fresh water and is simply given by Wfw =ñfw QR. The subscript A denotes that CA is the concentration calculated using model A. Equation 10 shows that, according to the classical simple tidal prism model—model A, the dilution of a steady source only depends upon the tidal period and the tidal prism. A more correct tidal prism model—model B As before, the estuary is represented by a single box; however, the ebb and flood phases will now be
Simple tidal prism models revisited
VR /2
P
River
V
VR /2 River
P V
VR /2 + VP Ocean
VP – VR /2 Ocean
Ebb tide Overall reduction in volume = VP
Flood tide Overall increase in volume = VP
79
Following the procedure used in deriving equation 4 and making use of equation 12 gives the average fraction of fresh water f as:
Following a similar procedure to that used in deriving equations 8–10 one obtains:
F 2. An estuary represented by a single box where V is the low tide volume and P is the tidal prism. The outflow and inflow volumes are shown for the ebb and flood tides.
considered separately. It is assumed that the ebb and flood phases are of equal duration. Seawater entering the estuary on the flood tide is assumed to mix fully with the water in the estuary prior to the ebb tide. Here, the volume of water entering or leaving the estuary on the flood or ebb tide, respectively, in the absence of river flow is denoted as VP. Then, with the addition of a river flow QR =VRT, a volume of VP-VR/2 of water must enter the estuary from the ocean during the flood phase in order to satisfy continuity. The volume leaving the estuary and entering the ocean on the ebb tide will be VP +VR/2. This is demonstrated in Figure 2. For an estuary in steady state, the amount of salt crossing the ocean boundary of the box must be zero over a tidal cycle. Hence:
where, as before, S is the salinity in the estuary at high tide, assuming a well-mixed estuary, and So is the salinity in the ocean. Now, the overall reduction in the volume of an estuary over an ebb tide, from Figure 2, can be seen to be Vp. Thus Vp must be equal to the volume of the tidal prism P so that equation 11 becomes:
Both relationships for the salinity (i.e. equations 3 and 12) indicate that S=So, as expected, for the case of zero flow. However, only equation 12 indicates that the salinity S is equal to zero when the tidal prism is equal to the volume of river water that enters the estuary over half of a tidal cycle. This is exactly the criterion that Ketchum (1951) attempted to apply, and that Dyer and Taylor (1973) applied for a box or prism to consist entirely of fresh water.
where the subscript ‘ B ’ denotes model B. Unlike the classical model (model A), equation 14 indicates that the concentration resulting from a steady release of a miscible passive pollutant into an estuary depends upon the river flow rate QR. Similarly to the classical model, equation 14 shows that the pollution concentration will reduce as the tidal range and river flow rate increase. As the tidal range approaches zero (i.e. P vanishes), equation 14 suggests that C=2W/QR when, quite clearly, the concentration should be C=W/QR (Fischer et al., 1979) with the estuary behaving as a large river (ignoring gravitational circulation here). The reason for this discrepancy is that equation 11 implicitly assumes that the flow between the ocean and the estuary occurs for an equal duration in either direction. This is not true if there is a river flow, as will be shown in the following section.
An even more rigorous tidal prism model—model C In this section, model B is extended by incorporating the duration of the ebb and flood flows and variation in the level of flushing at the mouth of the estuary. A sketch of an estuary is shown in Figure 3 where the instantaneous flow between the estuary and the ocean is denoted as Q. The river inflow to the estuary QR is assumed to be steady. The continuity equation for an estuary is given by:
where Pi is the instantaneous volume between the free surface and the low tide level, t is time and it is assumed that the water is incompressible. As pointed out previously, for the entire estuary to be well mixed, it is necessary that the estuary is relatively short. This means that there is unlikely to be much
80 D. Luketina
River S=0
Pi V, S
Q
0.2
Ocean S = So
F 3. An estuary represented by a single box where V is the low tide volume and Pi is the instantaneous volume between the free surface and the low tide level. The flow QR is assumed to be steady while Q is an instantaneous flow rate which can be of either sign.
0.0
where P is the tidal prism and the tidal amplitude is assumed to vary sinusoidally with period T. Note that the form of equation 16 is such that high tides occur at t=(n+0·25)T and low tides at t=(n+0·75)T where n is an integer. Substituting equation 16 into equation 15, simplifying and rearranging results in the flow Q being given by:
The change from ebb to flood flow and vice versa occurs when Q is equal to zero. Setting Q=0 in equation 17 gives:
where to now specifies the times at which Q is zero. One can replace to by:
where te and tf are the times at which the ebb and flood flows start in the absence of any river flow and ôe and ôf are the time lags introduced by the presence of river flow. Substituting equation 19 into equation 18 and solving for the lags yields:
for the normalized lags for the ebb and flood flows respectively where è is defined by:
τe /T
–0.1 –0.2 0.0
tidal attenuation within the estuary and Pi can be approximated by:
τf /T
0.1
τ /T
QR
0.2
0.4 0.6 QRT/πP
0.8
1.0
F 4. Normalized lag for the flood and ebb flows as a function of the normalized river flow QRT/ðP. T 1. Start and finish times for the first flood and ebb flows plus the duration of these flows (note that tf "te =T/2) Flow cycle
Start
Finish
Duration
Ebb 1 Flood 1
te +ôe tf +ôf
tf +ôf T+te +ôe
T/2+ôf "ôe T/2"ôf +ôe
Thus the ebb and flood flow lags depend upon the ratio R which is the ratio of the river inflow over a tidal cycle VR =QRT to the tidal prism P. This is shown graphically in Figure 4 from which it can be seen that the flood flow is increasingly delayed as the river flow rate is increased. On the other hand, the ebb flow occurs earlier. The duration of the ebb and flood flows can be determined as per Table 1. The normalized durations of the ebb and flood flows are plotted in Figure 5 where it can be seen that, as the normalized river flow QRT/ðP increases, the duration of the flood flow decreases and the duration of the ebb flow increases. Eventually, when QRT/ðP reaches unity, there is no flood flow and seawater does not enter the estuary (assuming that there is no gravitational circulation). As expected, when there is no river flow, the ebb and flood flows occur for equal periods of time. Conservation of mass can now be applied to determine the high tide salinity within the estuary. However, before proceeding, an implicit assumption that was made earlier will be raised. That is that the salinity in the ocean adjacent to the estuary is the same as the normal oceanic value So. This can only occur if all of the water that exits from an estuary on an ebb tide is advected away before the flood tide commences. In practice, some fraction of the water that enters the
1
1.0
0.8
0.8
0.4
flood
0.6 0.4 0.2
0.2 0
81
0 0.25 0.5 0.75 0.99
ebb 0.6
S/So
Duration/T
Simple tidal prism models revisited
0.2
0.4 0.6 QRT/πP
0.8
1.0
F 5. Normalized durations of the flood and ebb flows as a function of the normalized river flow QRT/ðP.
estuary during the flood tide is made up of water that left the estuary on the previous ebb tide. This fraction is known as the return flow factor b (Sanford et al., 1992). A negligible return flow factor (i.e. close to zero) can only result if the receiving water is well flushed—typically by alongshore currents (Fischer et al., 1979; Sanford et al., 1992). Making use of the return flow factor, the mass of salt leaving the estuary on the ebb me and flood mf flows are:
Note that mf is negative as salt must enter the estuary on the flood flow. Implicit in the formulations of equations 22 and 23 is that the flow Q, through the mouth of the estuary, is relatively uniform. Relatively uniform, means that all fluid is travelling in the same direction. However, estuaries and embayments with wide mouths are often subject to asymmetries with regard to the flood and ebb flow patterns (Bruun et al., 1978; Wolanski & Imberger, 1987). At times, ebb and flood flows can coexist. This means there can be salt transport despite the net flow Q being zero. We can evaluate the mass of salt Äm that leaves the estuary over a tidal cycle by summing equations 22 and 23. The integration can then be carried out by substituting equations 17, 20 and 21 and noting that te =T/4 and tf =3T/4 because of the form assumed earlier for the tidal behaviour (i.e. equation 16) to give:
0.0
0.1
0.2
0.3 0.4 QRT/πP
0.5
0.6
0.7
F 6. Normalized steady-state high tide salinity S/So as a function of the normalized river flow QRT/ðP for the classical tidal prism model—model A (short dashes), the improved tidal prism model—model B (long dashes) and the even more rigorously derived tidal prism model—model C (solid lines), where values of b are as indicated.
where, as before, è=cos "1 (QRT/ðP). For an estuary in steady state Äm=0. Setting Äm=0 in equation 24 results in a steady-state salinity of:
The three equations derived for the steady-state high tide salinity (i.e. equations 3, 12 and 25) are shown plotted in Figure 6. Before proceeding further, one needs to recognize that the estuary must be well mixed for the tidal prism models presented here to be valid. A typical estuary will be well mixed provided that R¦0·1 and can be considered to be partially mixed when R~0·25 (Schulz & Simmons, 1957). Wind stirring can result in increased mixing so that an estuary in a windy location could be well mixed for values of R exceeding 0·1. Another factor which should be considered is the density of the water in the river; a river with a significant sediment load can easily have a combined water plus suspended sediment density as great as that of seawater. This means that mixing will not be inhibited by density differences when compared to the case of clean river water entering an estuary. Thus there is another scenario where a well-mixed estuary could result for values of R exceeding 0·1. Although possible, a reasonably wellmixed estuary will rarely occur for R>0·25. For this reason, Figure 6 has been replotted so that the maximum value of R is 0·25 (this corresponds to R/ð=QRT/ðP=0·08). Looking at Figure 7, for the case where the return flow factor b is zero, it is clear
82 D. Luketina 1.0 0.25 0
S/So
0.8 0.6
0.5
0.75
0.4 0.99 0.2
0
0.01
0.02
0.03
0.04 0.05 QRT/πP
0.06
0.07
0.08
F 7. Normalized steady-state high tide salinity S/So as a function of the normalized river flow QRT/ðP for the classical prism model—model A (short dashes), the improved tidal prism model—model B (long dashes) and the even more rigorously derived tidal prism model—model C (solid lines), where values of b are as indicated. Unlike Figure 6, this plot has values of QRT/ðP limited to the range in which a reasonably well-mixed estuary may occur. Note that the long dashes (model B) coincide with the solid line (model C) for which b=0.
that there is little practical difference between the different models. This indicates that the effect of the river flow on varying the duration of the ebb and flood flows can be neglected. However, the effect of the return flow factor is considerable. Reported values of b include 0·23 for Indian River Bay, Delaware (Sanford et al., 1992) and range from 0 to 0·33 for San Francisco Bay (Nelson & Lerseth, 1972). In the absence of any data, it has been suggested that the return flow factor be set to 0·5 (USEPA, 1985). Note that if suitable field measurements are available, Figure 7 can be used to determine the return flow factor b for a well-mixed estuary. Alternately, the physically based method of Sanford et al. (1992) can be used to estimate b. For the case of QRT/ðP°1 it can be shown that sin è=(1-(QRT/ðP)2)1/2~1 so that the salinity S as given by equation 25 is well approximated by:
This equation should be good compromise between complexity and accuracy. Note that for the case where b=0, equation 26 is identical to equation 12 (model B).
steady state or equilibrium. While this is true for salinity, it is not necessarily true for other substances. This section extends model C so that it is a continuous unsteady model which incorporates additional source or sink terms. If one makes the same assumption used in deriving equation 26 from equation 25 (i.e. that QRT/ðP°1 so that sin è~1 and è~ð/2), the mass of any substance that leaves the estuary during a tidal cycle can be derived in an analogous manner to equation 24 to give:
where C is the high tide concentration in the estuary, Co is the background concentration in the ocean, CR is the concentration in the river, W (kg s "1) is a source term of negligible volume flux within the estuary and k (s "1) is a decay rate. The first three terms on the right-hand side of equation 27 can be derived from equation 24. The last three terms represent inflow of the substance from the river, sources within the estuary and decay of the substance, respectively. The rate of change of concentration C in the estuary is then given by:
The case of an instantaneous release of a conservative substance within the estuary can be modelled by setting W=0, CR =Co =0, and k=0 to give:
Solving this for the initial condition C=Ci at t=0 results in:
A continuous model—model D The preceding models are based upon the assumption that, in a tidally averaged sense, the estuary is in
where the subscript ‘ D ’ denotes model D, trD is the residence time and is given here by:
Simple tidal prism models revisited 1.0 0.10 0.20 0.8
0.30 0.40 0.50
0.6
b
0.60 0.70
0.4
0.80 0.90
0.2
1.00 0.0 0.00
0.02
0.04 QRT/πP
0.06
0.08
F 8. Contours of the ratio of trA/trD (the ratio of the residence time based on the classical tidal prism method—model A, so that given by equation 31—model D) as a function of the normalized river flow QRT/ðP and the return flow factor b. The same contours also apply to CA/CD which is the ratio of the concentration resulting from a continuous source based on the classical tidal prism method—model A, to that given by equation 35—model D.
83
One can compare the flushing times given by equations 31 and 32 by examining their ratio trA/trD which is in the term within the square brackets in equation 31. The ratio is plotted in Figure 8 as a function of the return flow factor b and the normalized river flow rate QRT/ðP. Looking at Figure 8, it is clear that the classical tidal prism method—model A, when compared to tidal prism model D, significantly underestimates the residence time for the larger values of the return flow factor b. When b is zero, however, the classical tidal prism method can overestimate the residence time as the normalized river flow QRT/ðP increases. However, for typical values of the return flow factor and the normalized river flow in a well mixed estuary, it is likely that the classical tidal prism method—model A, will underestimate the residence time. Another case which can be readily solved is that of a constant release of pollutant into the estuary. In this case we must eventually achieve steady state so that dC/dt=0 and we obtain:
This can be rearranged to give the steady-state concentration as:
When comparing this equation with that obtained by Sanford et al. (1992) it should be noted that the volume V used by Sanford et al. (1992) is a mid-tide volume (equivalent to V+P/2 in the terminology used here). Furthermore, Sanford et al. (1992) defined their concentration as a mid-tide value. Once these differences are taken into account, the residence time evaluated by Sanford et al. (1992) only differs from that given by equation 31 due to the effects of river flow not being correctly incorporated into their formulation. The definition of the residence time that is used here is different from that for the flushing time tF which is usually defined as the time needed for the river flow to replace all of the river water in the estuary (Officer, 1976). The residence time is a more appropriate time scale than the flushing time, as a pollutant can still leave an estuary or embayment when there is little or no river flow. The residence time using the classical tidal prism method, trA is (Dyer, 1973):
For the case of a continuous release of a conservative substance in the estuary with Co =CR =0, the above equation reduces to:
The concentrations given by equations 10 (the classical tidal prism method—model A), 32 and 35 can be compared by examining their ratio CA/CD. This ratio ends up being the term within the square brackets in equation 35. So that, as expected, CA/CD =tA/trD. The ratio is plotted in Figure 8 as a function of the return flow factor b and the normalized river flow rate QRT/ðP. Figure 8 shows that the classical tidal prism method—model A, significantly underestimates the concentration for the larger values of the return flow factor b. When b is zero, however, the classical tidal prism method can overestimate the concentration as the normalized river flow QRT/ðP increases. However,
84 D. Luketina
for typical values of the return flow factor and the normalized river flow in a well-mixed estuary, it is likely that the classical tidal prism method will underestimate the concentration resulting from a continuous source of a conservative substance. Cases other than the instantaneous release and continuous release, as solved above, can be solved numerically using equation 28. In doing so, it should be noted that equation 28 was derived for the high tide concentration at which time the estuary is assumed to be well mixed. The concentrations indicated by equation 28 for times other than high tide will give an indication of the average concentration within the estuary. However, equation 28 can not be expected to resolve details on time scales of less than the tidal period T or spatial scales of less than the length L of the estuary. However, the additional complications inherent in doing this make the use of the simple tidal prism model (model D) fairly attractive.
estimates the concentration for larger values of the return flow factor b. When b is zero (one of the assumptions of the classical method), the classical tidal prism method can overestimate the concentration as the normalized river flow QRT/ðP increases. However, for typical values of the return flow factor and the normalized river flow in a well-mixed estuary, it is likely that the classical tidal prism method will underestimate concentrations. It is recommended that model D (the continuous model) be typically used as a starting point for investigations using simple tidal prism models.
Summary and conclusions
References
The classical tidal prism method (model A), based upon what has been typically presented, has never been properly explained and is not formulated correctly. The method can be easily reformulated by correctly taking into the effects of the river flow. These are two-fold. First, the river flow affects the volume of ocean water that enters the estuary on each tidal cycle. Second, the river flow causes the flood flow to be lagged and the ebb flow to occur earlier than would occur without the presence of the river flow. However, it has been shown, for the normalized river flows that are likely to occur in a well-mixed estuary, that the lagging effect can be neglected. The tidal prism model was then extended by incorporating a return flow factor which, for typical values, plays a significant role in the residence time of the estuary. Finally, a continuous model was developed for the concentration which incorporated source and decay terms. Implicit assumptions in all of the models are that the estuary is fully mixed at high tide and that the flow between the estuary and ocean consists of fluid which is all travelling in the same direction. All concentrations are, strictly speaking, high tide values. Based upon comparisons with a more theoretically correct tidal prism method (model D), the classical tidal prism method (model A) significantly under-
Acknowledgements This paper was written while the author was on study leave at the Department of Civil Engineering, Canterbury University. The reviewers are thanked for their useful comments and suggestions.
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