June 19, 1996

A recent paper by David Luketina (to be published in Estuarine, Coastal and Shelf Science) correctly points out that there is an error in the flushing formulation presented in our 1992 paper, "MODEL FOR ESTIMATING TIDAL FLUSHING OF SMALL EMEiAYMENTS". It seems I did not correctly account for the influence of the net inflow on the tidal prism, such that the expression for the effective flow rate due to the tide should be

P rather than Q = - as in the paper T

Thus, expanding eq. 3 in our paper would result in

T, = v P for - :

P I + (1 +b) - T

2

and

v T = - P I I

for - I - T 2

which is identical to eq. 4 for no net inflow (I = O), but significantly different for nonzero net inflow. This also means that eq. 6 in our paper should be modified by substituting the revised expression for Q presented above. There is a more sophisticated expression for correcting Q if an allowance for the changing duration of ebb relative to flood is made, presented by Luketina, but I'm not convinced it's worth pursuing for the level of sophistication of our model.

Larry Sanford

are very likely to be present during any given interval of several days, which is often the critical period for adequate flushing to maintain water quality. However, the wind is not as regular or as predictable as the astronomical tide. For this reason, the more conservative estimate of flushing character- istics can be realized by considering the most limited flushing condition, that of the astronomical tide acting alone.

Tidal flushine is nroduced throueh reoeated exchanee of the intertidal u L

volume, or tidgl pAsm, between the embayment and Tts receiving water bodv. Water entering the embavment on flood tide fills the intertidal volume

~ ~

unt6 high tide is rea'ched. ~ h i s n e u , water mixes with existing water in the embayment. As the tide falls, the intertidal volume of water flows out of the embayment as an ebb tidal current. Some fraction of this water is lost by advection or mixing into the receiving water body, and the remainder returns to the embayment on the following flood tide. Tidal flushing there- fore denends not onlv on the embavment geometrv and the tidal ranee. but also on'the fraction 6f effluent water that Feturns, br the return flow Tactor. Models based on this flushing mechanism are known as tidal nrism flushing - models.

Tidal nrism flushing models have been in use for some time. but the origin

tech&ue withokt furtherreference, implying g&eral faLiliari&. For some applications, there are more specific references available. For example, Callahan (1981) and van de Kreeke (1983) discuss the use of tidal prism models for estimating residence time in marina basins, and they provide a number of references in the engineering literature.

A substantial amount of confusion has arisen over the proper use of tidal prism models. This confusion has centered on three issues: the validity of the approach when mixing is incomplete, the proper formulation of the model, and the determination of an appropriate value for the return flow factor. We address the last two issues in this paper; we assume that the embayment is small enough and that currents are vigorous enough to ensure complete internal mixing.

Rapid thorough internal mixing requires that there is insufficient vertical stratification to inhibit vertical mixing over the tidal cycle. It also requires either that the basin is small relative to a typical tidal excursion, which may be calculated on the basis of inlet currents, or that there is some other regular mode of internal circulation and/or source of internal mixing. Details of basin morphology that help determine the extent of internal mixing are beyond the scope of this paper [but see, e.g., Nece et al. (1976) and SCCC (Coastal 1985)l. Embayments that are not well mixed are likely to exhibit internal gradients in concentration, with areas remote from the connection to the receiving water body flushing more slowly than the rate predicted by tidal prism mdels .

- An additional im~ortant assumution is that the receiving water bodv is

itself well flushed, dr of a sufficie&ly large volume to effecfively diluteihe water exiting the basin over periods longer than a tidal cycle. It is in this regard only that we invoke additional mixing and circulation in the receiving water body, such as might be driven by the wind or waves. If this is not the case, embayment water will begin to materially affect the properties of the receiving water body over several tidal cycles, and the basic tidal prism flushing formulation will he invalidated. The embayment and receiving water

dependent decay rate k in units of time-' such that concentration will decay at a rate kc independent of flushing.

A true tidal prism model is discontinuous. Water is ejected from an embayment only on ebb tide and new water enters an embayment only on flood tide, hence mixing with ambient waters outside the emhayment and decreases in c are discontinuous at tidal time scales. However, van de Kreeke (1983) showed that under most circumstances the discontinuous tidal prism model is well approximated (within 5%) by a model with continuous inflow and outflow at a volume flow rate of Q = PIT. The equivalent tidal mass flux of tracer out of the basin is then Qc, in units of mass per unit time, with a return flux of bQc. Tracer in the ambient waters outside the basin also enters at s rate (1 - b)Qc,,bi,,,, such that the net outflow of tracer mass from the basin due to tidal flushing is (1 - b)Q(c - cambi,,J. Similarly, the net outflow of tracer mass due to the inflow I is I(c - c,).

Adding these four factors (source, decay, tidal flushing, and net flow) together, we write an equation governing the time rate of change of c as

dc S - = - - I . . . . . . . . . . . . kc -- (I - b)Q (c - c a,bic "3 - - (C - c,) (1) dt V V V

where t denotes time; and the mass flux terms have been converted to concentration units by normalizing by the average basin volume.

There are two important special cases of (1). The first is the initial-value problem for determining the flushing rate of the embayment. For simplicity, we assume that c,,,,,,, = c, = O for this case, that the source has shut off (S = 0) , and that the tracer is conservative (k = 0). Then

Solving (2) subject to the initial condition c = c, at t = 0 yields

c = c,, exp [-:I, where Tf = (1 - b)Q + I

In the case of tidal flushing alone ( I = 0 ) , which is the one that we consider here, the expression of Tf simplifies to

reexpanding Q into its component terms. The factor T,! which has the units of time, is the e-folding time of the solution; the e-foidmg time for flushing is also called the residence time in both the engineering (van de Kreeke 1983) and oceanoera~hic (Knauss 1978) literatures. . .

.4 scuond in~port;int u;trr ( e . g . . for water quality c;tlculutions) is the steady- st;rtc solution far a nonc<,nsrr\ativc traccr with a con5tanr internal source S. In this case, the governing equation reduces to

- = . . . . . . . . . . . . . . - - - S -- (1 - b)Q(c c-,,) Z(c c,) Vkc O (5)

The solution to this equation is simply

Tidal height ------------- --

- FIG. 2. Extremes of Phase between Channel Tidal Current and Coastal Tidal Cur- rent. (a) 90" Out of Phase, Coastal Tide is Progressive Wave; ( b ) in Phase, Coastal Tide is Standing Wave

Referring again to Fig. 1, we consider a flow emerging from the channel on ebb tide with velocity

where u is positive on flood tide according to the convcntion of Fig. 1; and u, = amplitude of the tidal current in the channel. Conservation of mass dictates that

,= n2 27Tr P = A ~ ~ ~ , , u ~ sin y dr = A,,,,u,T . . . . . . . . . . . . . . . . . . . . . . . .

7T (8)

Thus

The alongshore tidal velocity on the coast outside the embayment is

2111 u = U sin - z where u is positive in the ebb tide direction according to the convection of Fig. 1; and U = amplitude of the tidal current along the coast; U may be determined either by direct measurement, or by referring to published tidal current tables [e.g., NOAA (Tidal 1990)l. We assume a constant depth h at the channel entrance and all along the coast. The effective width of the plume at the instant it exits the channel is given by B, = A,,,,Ih. The effluent plume travels a tidal excursion X, = UTIn down the coast during the ebb half of the tidal cycle before it reverses direction and heads back toward the channel mouth on flood tide.

The effluent plume emerging from the channel mouth is effectively a horizontal discharge jet. If the momentum of the jet is low relative to the

momentum of the alongshore flow, we might expect the plume to remain attached to the coast. However, if the momentum of the jet is relatively high, we might expect the plume to separate from the coast and move into open water. The behavior of such a discharge jet emerging into a transverse flow was studied extensively in the laboratory by Carter (1969), and we will make use of his results here. Following Carter's notation, we define the ratio of channel to coastal velocity as

which remains constant over the tidal cycle because of the covarying cur- rents. Carter found that the velocity of the jet quickly adjusts to the velocity of the coastal current. The width of the plume also adjusts to conserve mass as the velocity changes, so that the plume width at the adjustment point is B = Bur, neglecting any mixing that occurs in the adjustment time. Carter defined the position of the adjustment point to be a distance x, down the coast, with the axis of the plume a distance x,! offshore. He found that x, and y,, appropriately scaled by the channel w~dth B,, are strong functions of r. His results are summarized in Fig. 3.

pared with X, and can be ignored; but y , cannot be iGored,'because the cross-shore length scales are much smaller than the alongshore length scales. We have derived a two-part algebraic fit t o they, data in Fig. 3, and have added a lower limit set by y, = Bl2 (the plume center line cannot be closer

FIG. 3. Data of Carter (1969), Showing Observed Coordinates x, and y, as Func- tions of r (Points) and Composite Best-Fit Line (12)

to the coast than one-half of the plume width). This lower limit is reached when r < 0.6 or it, < 0.6U. Thus, we express y, as

& = 1.97r1 575 for r > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12a) Bo

= 1,05r2488 for 0.6 5 r ,S 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . B"

(12b)

'm 0.5r for r < 0.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 2 ~ ) Bo

Based on the foregoing considerations, we model the coastal effluent plume as a finite-width line source for the diffusion of tracer into coastal waters, with initial width B at a distance y , away from the coast. We define the origin of our y-axis to be at the center of the plume, with the positive y-direction toward the coast such that the coast is at y = y,. We assume that the plume is well mixed over depth, and that diffusion is greatly over- shadowed by advection in the alongshore (x) direction; this is the "spreading disk" model of Okubo and Pritchard (1969). Under these assumptions, diffusion acts primarily across the plume (in the y-direction).

The basin water that exits the channel at the beginning of ebb is in coastal waters for a time r' = Z(T12) = 7, where t' is defined as the time since leaving the channel mouth. This initial effluent water travels the farthest downcoast and back and is most affected by diffusion. Water that exits at any time after the beginning of ebb travels less distance downcoast and returns to the channel sooner, and so is less affected by diffusion. Thus, each portion of the effluent plume has a unique t' associated with it when it arrives back at the channel mouth, and we can express the tracer-con- centration distribution from which the returning water is drau.n as a function of B and i' alone. Assuming a constant eddy diffusivity and combining equations 1.36 and 4.62 of Csanady (1973). we write

B B - - B 2 y , + Y - ZY,

c(t', y) = 0.5c,,$, (ed + erf - f l u

+ erf f l u

B - - y + 2?,,,

+ erf 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

<2n 1 (13)

where c,,,, = tracer concentration in the plume at t' = 0; n = (2KJr')05; and K , = cross-shore eddy diffusivity. We assume that c,,,, is a constant set by cariditions at the end of the previous flood tide; if the water in the basin is well mixed at the end of the flood tide, there is no reason for the con- centration in the outflow to change through the ebb tide. The first two terms on the RHS of (13) represent diffusion with no coastline, and the last two terms account for the presence of the coastline through the method of images.

The water that flows into the channel during the flood tide must be drawn from a band of width B immediately adjacent to the ebb tide coast; the width is set by conservation of mass, as before, and the return band is confined to the coast because on flood tide the channel behaves as a sink, with no jet-like momentum characteristics. The position of this band is not

642

coincident with the position of the effluent plume unless y, = Bf2. The average tracer concentration returned to the channel (c,,,) may be calculated by integrating the concentration distribution given in (13) over the width of the return flow band adjacent to the coast and over the age of the effluent plume, or

Get . . . . . . . . . . . . . . . . . . . . . . . . . ~ ( r ' , y) dy df ' TB z = o y=y,-5 (14)

The return flow factor is then simply b = c,,Jc,,,,. In many instances, the effluent plume from an emhayment may pass

beyond a point and move out into open water. If we allow our model plume to enter open water after a distance x,,, < Xo down the coast, then the water that just reaches x,,,, in half of a tidal cycle does so at time f ' =

to,,,, where

The portion of the plume that travels beyond x,,,, remains out of the vicinity of the channel mouth at time I' > 2t,,,,, and portion of the plume that does not reach x,,,, remains out for f ' < 2t0,,,. The correct expression for dif- fusion without a coast at y = y, is

so that the total returned concentration may be approximated as

. . . . . . . . . . . . . . . . . . . . . . . . . . +I" i = z ~ ~ ~ . , I'" Y = y n i - ~ cQpe.(t', y) dt' dy (17)

in the case of a partially open coast. The horizontal eddy diffusivity K, may be written as

K, = Cu*h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)

where C = an empirical constant; and u* = shear velocity; ur = (7,/p)05, where T~ = the turbulent bottom boundary shear stress, p = density of the water. Fischer et al. (1979) found values of C that range from 0.15 for straight rectangular channels, through 0.6 for natural rivers, to approxi- mately 1 for estuaries. Csanady (1973) gives a value of C = 0.23. In any case, the value of Cis likely to vary substantially from location to location depending on site-specific factors (e.g., boundary irregularities, secondary circulations). We choose a simple unweighted average of the values just quoted, C = 0.5. The value of u* is related to the tidal velocity on the coast, u, through a drag coefficient that depends primarily on the ratio of the bottom roughness to the depth. The ratio uiur can vary from less than 10 in shallow water with a rough bottom to greater than 30 in deep water with a smooth bottom (Smith 1975). A commonly used value for shallow,

M3

nearshore situations is 10 (Fischer et al. 1979), but we choose a slightly more consewative value of 14, which roughly corresponds to a I0 cm bottom roughness in a 2.5 m deep water column. We also need a constant value of u* for (IS), which we set equal to the root-mean-square (RXIS) value over the tidal cycle. Thus

and

K,. = 0.5u*,,,h = 0.025Uh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (20)

Eq. (20) completes the formulation of the return flow factor model. Required input parameters are the tidal period T, the intertidal volume of the basin P, the entrance channel cross-sectional area A,,,, the coastal water depth h, the magnitude of the coastal tidal current U, and the distance downcoast to open water x,,,, (if applicable). All of the other required values can be calculated from the equations presented earlier. The problem is considerably simplified if we assume that c,,, = 1, so that b = c,,, as evaluated from either (14) or (17). Note that the integrals in (14) and (17) cannot be evaluated analytically, but must be calculated numerically. The numerical evaluation is, however, very simule and fast.

We reiterate that our return flow factor hodel was derived for only the case of tidally covarying channel and coastal currents. Any phase &f£erence between the-channel &d coastal currents should decrease the return flow factor, with a phase difference of 90" resulting in b = 0. A possible method to account for phase differences between the channel and coastal currents, which we present without discussion, is to use an interpolation such as

bA. = [ - sin (g)] b

where b,, ,= revised estimate of the return flow factor; and A+ = phase difference m radians.

BEHAVIOR OF RETURN FLOW FACTOR UNDER VARY~NG CONDITIONS

Neglecting the effect of an open coastline, the return flow factor b is a function of three length scales and one time scale. The time scale is the tidal period T, which may be considered fixed for a given location. The length scales are the initial coastal width of the effluent plume B, the distance offshore of the plume center line y,, and a characteristic diffusion length scale, which we define as the value of a at the end of the tidal cycle.

It is trivial to show that b may be expressed entirely as a function of two nondimensional numbers, which we choose as y J B and u,iB. The rela- tionship

was evaluated through successive model runs [solving (14)] and is plotted in Fig. 4.

The behavior of the return flow factor in Fig. 4 is easily explained. The parameter y,/B represents the ratio of the center-line distance offshore to the initial plume width. Its minimum value is 0.5, which corresponds to the effluent plume directly at the coastal boundary. This yields the highest return flow factor for a given value of uJB , since the hand of returning water on flood tide is coincident with the effluent plume. As yJB increases for a given value of u,JB the plume moves away from the coast, the returning waterband begins to include tracer-free water inshore of the effluent plume, and the return flow factor decreases. The parameter crJB represents the effect of lateral diffusion relative to the initial plume width. A t low values of yJB, when the plume is at or near the coast, increasing u d B corresponds to more effective diffusion of the tracer away from the location of the returning water band and a decreasing return flow factor. At higher values of y,/B, when the plume has separated from the coast, increasing udB can actually increase the return flow factor by causing greater diffusion of tracer back into the returning water band.

For many configurations of interest, the effluent plume will never leave the coast: and y,/B = 0.5. This is because, as previously stated, the velocity

FIG. 4. Contours of Value of Return Flow Factor b Determined by Solving (14) Repeatedly

FIG. 5. Return Flow Factor as Function of v,IB Only, for g,lB = 0.5, Applicable when v, c 0.6U

in the channel must be greater than 60% of the velocity on the coast for the plume to separate. In this case b does not depend on y,lB, and may he expressed as a function of uJB alone. This dependence is plotted in Fig. 5, which clearly shows the rapid decrease in b with increasing nJB.

Field studies of tidal flushing are obvious candidates for comparison to the model resented here. However. manv ~ublished field studies concen- ~ ~~ , . trittr. prim:;ril! on tlu\hinp ~.hsr:tctcri>tics u,ithin 3n crnh;t!rn~~nt, ;ind do not ~ l i \ , ~ muzl~ inform;cri<~n on the, phts~cal environmer~t out\iJc in the rccci\ inr h e r body. Flow and mixing outside the embayment are primary dete; minants of the return flow factor in our model. Thus, much of the tidal flushing literature is not appropriate for direct comparison to our model.

A flushing study performed at Bay Colony Marina in Indian River Bay, Del., is an exception. The location and physical configuration of Bay Colony Marina are shown in Figs. 6 and 7. A dye study performed in June 1990 included measurements of dilution within the marina, measurements of dilution of the plume of dye outside the marina, and measurements of tidal height variation over the period of the flushing study [by EA Engineering, Science, and Technology, Inc. (Bay 1990)]. Data for the current outside Bay Colony Marina were not collected simultaneonsly, but current and tidal height data were collected directly across Indian River Bay as part of a separate (unpublished) study performed by one of the writers (Boicourt) in April-May 1988 (see Fig. 7 for location). These current data were used to estimate the amplitude of the coastal tidal current outside Bay Colony Marina; U in the model. The average tidal ranges measured in these two

646

FIG. 7. Indian River Bay, Delaware, Showing Location of Bay Colony Marina, 5 ft (1.5 m) MLW Depth Contour, Location (0) of Mooring of Current Meter in 1988 Study, and Measured Surface Distribution of Dye Dilution from Bay Colony Marina on June 15, 1990

studies were identical, and the coastal morphologies of the two sites are very similar, so we feel confident in using the tidal current speeds measured in 1988 as proxies for the tidal currents during the 1990 Bay Colony dye study. Furthermore, analysis of the 1988 data showed that the tide in Indian River Bay is a standing wave, so the basin currents and coastal currents should be in phase and our model should he directly relevant.

The writers compiled a set of average tidal characteristics of Bay Colony Marina and the adjacent waters of Indian River Bay by combining the results of the 1988 and 1990 studies and utilizing the definitions given previously. These characteristics are listed in Table 1. The effluent plume should he against the coast, since r < 0.6, so that y,/B = 0.5; and uJB = 4.3. If the plume were against the coast for its entire tidal excursion outside the marina, we could use Fig. 5 to estimate b = 0.30. The value of b predicted by our model in this case is slightly smaller at b = 0.23, since the effluent plume from Bay Colony enters open water 1,800 ft (550 m) downcoast of the inlet mouth; hence dilution of the plume is greater than in an entirely coastally bounded case. The corresponding residence time of water in the Bay Colony Marina is predicted to be Tf = 2.47 days from (4).

In the Bay Colony Marina field study, diluted Rhodamine WT dye was injected continuously into the marina basin at four injection points June 11-15. 1990. Inventories of the total mass of dve in the marina were mea- sured by extensive sampling at high water on June 15, 17, and 19. and converted to average concentration by dividing by the marina volume. These

data were analyzed to evaluate dilutio'n in the marina, treating the June 15 concentration as the initial value for normalization of the subsequent con- centrations. The results are presented in Fig. 8 for comparison to (4), using the return flow factor model prediction of b = 0.23. Model results with b

649

TABLE 1. Average Tidal Characteristics Of Bay Colony Marina and Adjacent In- dian River Bay

Variable name

('1 A R P H h V T A,,,. BO U xopm XO uo r B 01

Model, b = 0.23

Model. b = 0 .5

Days Aner Release

FIG. 8. Results of June 1990 Dye Flushing Study and Model-Predicted Flushing Behavior Using both Predicted Return Flow Factor of 0.23 and Assumed Return Flow Factor of 0.5

Value

U.S. customaiy units (2)

8.2 acres = 3.57 x IN sq ft 2.1 ft 7.5 x IN cu ft 7.7 ft 5.0 ft 2.75 x 106 cu ft 12.42 hr 700 sq fl 140 ft 0.57 fiisec 1,XW ft 8,110 ft 0.075 ftlscc 0.13 18.5 ft 80 ft

Sf units (3)

3.32 x 1fY rn2 0.64 m 2.1 x 10d mi 2.3 m 1.5 m 7.79 x 1V m' 44,712 s 65 rn2 43 rn 0.17 m/s 550 rn 2,470 m 0.023 m / s 0.13 5.6 rn 24 rn

= 0.5 are also presented for comparison; settine b = 0.5 is often recom- mended when there is no other hasis for estimahg the return flow factor (SCCC: Guidelines 1983; USEPA: Coastal 1985). The model clearly does a good, though slightly conservative, job at predicting both basic flushing behavior and an appropriate value for the return flow factor. Note that assuming a return flow factor of 0.5 substantially overpredicts the residence time in this case.

We have also compared our model to the formulation suggested by the U.S. Environmental Protection Agency (USEPA) (Coastal, 1985), which is based on a discontinuous approach. As long as the same return flow factor is used in both formulations, rhe predicted residence times are less than 5% different; the USEPA formulation always gives a slightly higher value. Van de Kreeke (1985) shows that the approximate equivalence of the two ap- proaches becomes invalid for Tp'T < 1.5, but this limit is fairly restrictive and not likelv to be a maior concern excent undcr conditions in which a ~

continuous approach is cl&rly wrong (i.e., when the tidal prismis as large as the average volume of the basin). Under most circumstances. therefore. .~ ~

the continugus-flow approximation and resultant expression f i r residence time are as accurate as needed, given all of the other uncertainties involved in the simple tidal prism approach. Errors in predicted tidal flushing due to inappropriate specification of a return flow factor overwhelm the differ- ences between the various formulations.

COMMENT

Presentations of tidal prism flushing models have often either ignored retum flow entirely (Dyer 1973; Callaway 1981) or suggested that the retum flow factor be set to 0.5 in the absence of better information (SCCC: Guide- lines 1985; USEPA: Coastal 1985). In fact, predictions of tidal flushing are quite sensitive to the chosen value of the return flow factor. The main point of the analysis presented here is that the return flow factor depends on the fate of the tidal prism when it is outside the emhayment. In the absence of direct measurements of return flow, it should be possible to model that fate and to arrive at a more rational estimate of a return flow factor, which is what we have attempted.

We have chosen to model the effluent plume that spreads evenly down- coast from the embavment mouth on ebb tide when the tidal currents in t h ~ un~h;i)riic~nt .ind aimg the i03,t co\iir!.. .411! d~r)drturc frdm thi, \IIICII! r . o \ ; ~ r \ ~ n ~ ,itu.ltion nil1 rend to I L ~ U ~ C thc r ~ t u r n tlon i;tc~or M'c hil\c nlads a nuGbg of assumptions about the behavior of the plume and its rate of mixine with coastal waters on the basis of empirical formulations. some of \\ 11ich:irc dcrivc~l iro111 lith<~r:it~r). cxpcr~nic~it; onl! \\'c I13b.c ; I I \ I I :~s\umcJ that the IranJ of \vatr.r that rr,turn\ 1,) t11c 111drin.1 ci,ii be ~lcitrlv dcl111~~1cd. All of these assumptions are open to testing and revision on an individual basis. For example. the considerablv more complex plume dilution model of l'ail!. and Sa) rc I'J7hj migl~t inurc, i~r.~ur.iIcIv 111<)dcI (he plunic dilurion ccint~rur~ ;,I the unc l ~) fch t , tide. i t .tpr~rurlrl.ltc input r i~lllc* il~llld \p~.~.ifiud. Alternately, a site-specific value 3d.r the lateral diffusivity might 6e deter- mined by field dilution studies using dye tracers.

We have also ignored the possible effects of winds, waves, and steady currents on the return flow factor. These effects may he more important on open coasts than in the relatively sheltered environment of tidally dom- inated estuaries. However, it may be argued that the effects of winds, waves,

and currents will usually result in a return flow flower than the tidal estimate presented here.

Steady, nontidal alongshore currents can be foroffshore currents, and can affect the return flow faanisms. The first, which always acts to reduce the ra steady current added to tidally reversing currents aafter the tide turns to flood and before the tide turambient, nonplume water is directly adjacent to the these intervals there is no effluent plume water reall. Increasing the strength of the steady current relincreases the duration of these intervals, until theytide, when the steady current equals the maximumand the return flow factor is reduced to zero.

The second mechanism through which steady cuflow depends on the direction of the alongshore cuis in the ebb direction along the coast, the effluenthe return band of water is wider than in the stricsame conservation-of-mass requirements cited prevcumstances this results in more of the effluent plumthe return hand and a tendency for the return However, this increase is offset by a compensatingdue to the first mechanism just discussed, resultchange. When the current is in the flood directiowider, the return band narrower, and in most circufactor will be decreased even more than the decreaanism.

The action of surface waves along the coast outalso tend to reduce the return flow factor predictedto increased mixing and diffusion. This increased either by surface-wave-breaking in the surf zone, obottom roughness due to wave-current boundary land Madsen 1979; Yoon et al. 1990). In the few cwhich increased diffusion actually increases the recrease is only slight and the values of the return floto very small changes in predicted residence times

In sum, the return flow faetor and tidal residenshould be conservative estimates in almost all cirof our model at predicting the basic flushing charaMarina, Del.. further areues that exact reproductiomay not he required focthe model to b e a useful tflnw factors and flushine times. In other words. c~.. ~ - ~ ~ - damental processes that control plume behavior apa good estimate of the return flow factor, just adiscontinuous tidal prism model by a continuous mexchange rate is sufficient for a good estimate ofcomplexity andlor attempts to match site-specific under some circumstances, but we believe that appropriate for a wide class of problems requiriestimates of flushing in tidal embayments, and tharesults in return for a minimum of effort.

actor that is equal to or

ced by winds, waves, or ctor through two mech- eturn flow factor, is that lways results in intervals ns to ebb, during which channel opening. During turned to the channel at ative to the tidal currents occupy the entire flood opposing tidal current

rrents can affect return rrent. When the current t plume is narrower and tly tidal case due to the iously. Under most cir-

e being contained within flow factor to increase. decrease in return flow ing in only a small net n, the effluent plume is mstances the return flow se due to the first mech-

side the embayment wilt here in most cases, due

mixing may be produced r by increased apparent

ayer interactions (Grant ases plotted in Fig. 4 in turn flow factor, the in- w factor are low, leading . ce times predicted here

cumstances. The success cteristics of Bay Colony

n of effluent plume detail ool for estimating return onsideration of the fun- pears to he sufficient for s approximation of the odel with an equivalent basin flushing. Further detail might be required the present approach is ng simple, conservative t it is likely to yield good

A simple model was developed for tidal flushing of a small, well-mixed , embayment. The governing equation and the forms of the solutions are versions of the classic tidal orism flushing formulation. aooroximating the discontinuous tidal prism flbshing mechzhism as an equlvv'alent stead;-ex- change flow. For eeneralitv, the governing equation was written for anon- conservative trace; with <first-Gder conceniration-dependent decay rate and an internal source, allowing both a flushing-rate expression and a steady- state concentration expression to he derived. The results are very similar to previously published formulations, given equal values of the input pa- rameters. Most of the required values are readily derived from basin ge- ometry and known or predictable tidal characteristics. However, the return flow factor, to which the results are particularly sensitive, is not derivable from basin characteristics alone. The return flow factor has been ignored or specified by assumption in previous formulations.

The present model goes beyond previous formulations by developing an explicit model for the return flow factor for the case of covarying tidal currents in the embayment channel and along the coast outside. This model considers the fate of the plume of effluent water that exits the embayment mouth and flows downcoast on ebb tide, mixing with tracer-free coastal waters. The plume flows back toward the embayment mouth on flood tide, hut only a fraction of the tracer mass that exited the embayment is drawn back in. This fraction is identically the return flow factor.

The return flow factor model was solved for a range of representative cases, and the results were plotted for general reference. The value of the return flow factor predicted by the model was shown to depend on both the relative momentum of the embayment channel and coastal currents and the amount of cross-olume diffusion that occurs outside the embayment. Jct-ltkc cltsr~nel ck~rr&ts ,UI cbh 11dc iil tcl rap111 r n ~ x i t ~ g I)[ ~ I I L I I I C 1cit1~,11 :ind ct,;ist;ll wilterr u\u;~ll! tend 1 ~ 1 dc,.re;c<c tllc. r~~turtr flow f.lrt#lr. I:hcur~i(lili c,f plutne u.itcr\ into opcn rv:trr.r h ~ y r n d the c t~d of a point or \harp hcncl. pl12,~. ~ltff~.rc,n~-c\ h~.tuccn the cnlbit!.n1~111 ch:tiulel currsnts :i11J i t i ~ cc~itrti~l current,. and thc ctfecrs of \r ind, iind H . ~ I \ C I \I~ouId :11w ~ lcc~e i~ \ r . the rc.tt~rt~ flow factor in most cases.

The flushing model and return flow factor submodel were compared to the results of a dye study performed in a marina basin in Indian River Bay, Del., utilizing additional data collected nearby on the magnitude of tidal currents. The model re dictions and the marina basin dye study results compared favorably, &th the model yielding a slightly cbnservaiive pre- diction of flushing time. Setting the return flow factor to 0.5 as is often supgested for cases in which better information is lacking. substantiallv - ov&xedicted flushing time.

Further refinements of the model might incorporate a more real~st~c plume model, variable topography, and cross-shore shear in the coastal tidal cur- rents. Further test in^ against field data should be performed first, however, to determine whether such refinements are truly necessary.

This work was supported partially under a contract between the first two authors and Townsends, Inc., Millshoro, Del., and partially under Nation31 Science Foundation grant numbers OCE-8800767 and BSR-8814272; UM- CEES Contribution No. 2277.

652

APPENDIX I. REFERENCES

Ray Colony Marina flurhiii,. study. (1990). EA Engineering, Science, and Technol- .. . ogy, lnc:, ~parks,'Md. - -

Callaway, R. J. (1981). "Flushing study of South Beach Marina, Oregon." J Wtr~iy . Port. Coast., and Oc.; Div., ASCE, 107(2); 47-58.

Carter, H. H. (1969). "A preiiminary report on the characteristics of a heated jet discharged horizontally into a transverse current. Part 1: Constant depth." Tech. Report 61, Ref 69-14. Chesapeake Bay Institute, Johns Hopkins University, Bal- timore, Md.

Coastal Marinas Assessment Handbook. (1985). United States Environmental Pro- tection Agency (USEPA) Region IV; Atlanta; Fa.

Csanady, G. T. (1973). Turbulrntdi f f~ion in the environment. D . Reidel, Dordrecht, Holland.

Dyer, K. R. (1973). Estuaries: A physical introduction. John Wiley and Sons, New York, N.Y.

Fischer, H. B., List, E. J. , Koh, R.C.Y., Imbergei, J.; and Brooks, N. H. (1979). Mixing in inland and coastal tvaters. Academic Press, New York. 483 p.

Grant, W. D.; and Madsen, 0. S. (1979). "Combined wave and current interaction with a rough bottom." J. Geophys. Re,?., 84(C4), 1797-1808.

Guidelines for preparation of coastal marina report. (1985). South Carolina Coastal Council (SCCC), Charleston; S.C.

Harleman, D.R.F. (1966). "Pollution in estuaries." In Estuary and coastline hy- dronamicr, A. T. Ippen, ed. McGraw-Hill Book Company, Inc., New York. pp. 67n-647 - - - .. . . .

Knauss, J. A. (1978). Introduction to physical ocearii~graphy. Prentice-Hall, Engle- wood Cliffs, N.J.; 338 pp.

Nece, R. E., Falconer, R. A, , and Tsutsumi, T. (1976). "Planform influence on flushing and circulation in small harbors." 15th Conf. on Coastal Engrp., ASCE. . . New Ybrk, N.Y.

Okubo, A,, and Pritchard, D. W. (1969). "Summary of our present knowledge of the physical processes of mixing in the ocean and coastal waters, and a set of practical ~uidelines for the application of existing diffusion equations in the prep- aration of nuclear safety evaluations of the use of nuclear power sources in the sea." US. Atomic Energy Commission Report No. NYO-3109-40; Ref. 69-1. Ches- apeake Bay Institute, Johns Hopkins University, Baltimore, Md.

Paily, P. P., and Sayre, W. W. (1978). "Model for shore-attached thermal plumes in rivers." J. of the Hydrunl. Div., ASCE, 104(5), 709-723.

Smith, I. R. (1975). Turbulence in lakes and rivers; Scientific Publication No. 29. Freshwater Biological Association. Oxford, England.

Tidal current tables 1991; Atlantic Coast of North America. (1990). National Oceanic and Atmospheric Administration (NOAA), Riverdale, Md.

van de Kreeke, J. (1983). "Residence time: Application to small boat basins." J. Waterways, Port, Coartal and Ocean Engin. Div., ASCE, 109(4): 416-428.

Yoon, S . B., and Liu, P. L:F. (1990). "Effects of opposing waves on momentum jets." J. Wrnvy. Po% Coast., and Oc. Engrg., 116(5), 545-558.

APPENDIX II. NOTATION

The following symbols are used in this paper:

A = average surface area of embayment basin; A,,,, = average cross-sectional area of embayment channel a t connection

point; B = width of plume after adjustment to coastal velocity;

B, = effective width of plume as it leaves embayment channel; b = return flow factor;

b, = return flow factor adjusted for tidal phase difference;