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On the Mixing Processes in Estuaries: The Fractional Freshwater Method RevisitedAuthor(s): P. Regnier and J. P. O'KaneReviewed work(s):Source: Estuaries, Vol. 27, No. 4 (Aug., 2004), pp. 571-582Published by: Coastal and Estuarine Research FederationStable URL: http://www.jstor.org/stable/1353471 .Accessed: 23/11/2011 04:09

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Estuaries Vol. 27, No. 4, p. 571-582 August 2004

On the Mixing Processes in Estuaries: The Fractional Freshwater

Method Revisited

P. REGNIER1 and J. P. O'KANE2

1 Department of Earth Sciences-Geochemistry, Faculty of Geosciences, Utrecht University, Utrecht, The Netherlands

2 Department of Civil and Environmental Engineering, National University of Ireland, Cork

(UCC), Cork, Ireland

ABSTRACT: A mathematically transparent model for long-term solute dynamics, based on an oscillating reference frame, is applied to the analysis of the mixing process in estuaries. Classical tidally-averaged transport models for estuaries, all derived in some way from the Fractional Freshwater Method of Ketchum (1951) are reinterpreted in this framework. We demonstrate that in these models, the dispersion coefficients obtained from salinity profiles are not

always a good representation of the mixing intensity of other dissolved constituents. In contrast, the hypothesis of equal coefficients is always verified in our oscillating coordinate system, which is almost devoid of tidal harmonics. The math- ematical representation of the seaward boundary condition is also investigated. In the tidally-averaged Eulerian models, a fixed Dirichlet boundary condition is usually imposed, a condition that corresponds to an immediate, infinite dilution of the dissolved constituent beyond the fixed estuarine mouth. This mathematical representation of the estuarine-coastal zone interface at a fixed location is compared with the case of an oscillating location, which protrudes back and forth into the sea with the tide. Results demonstrate that the mathematical representation of the seaward boundary condition has a significant influence on the resulting mixing curves. We also show how to apply our approach to the prediction of mixing curves in real estuaries.

Introduction

The original models for predicting solute dis- persion date back to the early 1950s. Ketchum (1951, 1955) developed the Fractional Freshwater Method, a heuristic model for predicting along- channel distribution of pollutants in well-mixed es- tuaries. By heuristic, it is meant that it could not be derived directly from the fundamental laws of mass conservation in an estuary. Almost at the same time, Stommel (1953) proposed an alterna- tive method of calculating the solute distribution through the use of a time-average, one-dimension- al advection-diffusion equation. Several authors (Holley and Harleman 1965; Fisher et al. 1979; Chatwin and Allen 1985; Savenije 1992) have pointed out the difficulties in deriving such a time- average equation from an instantaneous mass bal- ance constrain. This led Fisher et al. (1979) to con- clude that in the view of the weaknesses of the as- sumptions required to derive such a tidally-average model, it was better to consider Stommel's equa- tion as an empirical one. Because of the apparent deficiencies in the time-average, advection-diffu- sion equation, an alternative, two-parameter inte-

* Corresponding author; address: Department of Earth Sci- ences-Geochemistry, Post Office Box 80021, 3508 TA Utrecht, The Netherlands; tele: +31-30-2535409; fax: +31-30-2535302; e-mail: pregnier@geo.uu.nl

gral equation model was developed in the late 1950s for the Thames estuary (Thames Report 1964). Neither the Thames integral equation itself, nor the critique of the advection-diffusion equa- tion in the Thames Report (1964), have received prominent attention since their publication. Only a few alternatives to the classical approach for modeling concentration distributions have been proposed, notable exceptions being the Lagrang- ian model of (Fisher 1972b; Fisher et al. 1979) and the Constant Volume Reference Frame approach of O'Kane (1980). A critical analysis of Ketchum and Stommel's models is essential, as they are at the foundation of modern estuarine geochemistry, in the analysis of mixing curves (e.g., Boyle et al. 1974; Liss 1976; Kaul and Froelich 1984; Yeats 1993; Shiller 1996).

Three key assumptions are explicitly or implicitly included in the heuristic solute transport models that form the basis of property versus salinity anal- ysis: the estuarine system must be at steady state, the flow and mixing processes are one-dimension- al, and the dispersion coefficient is identical for all dissolved constituents. The first two assumptions have been extensively investigated in Officer and Lynch (1981), Loder and Reichard (1981), and Regnier et al. (1998). Pritchard (Kinsman's notes of his lectures, unpublished), and more recently Dronkers (1982) and Chatwin and Allen (1985),

? 2004 Estuarine Research Federation 571

572 P. Regnier and J. P. O'Kane

have examined the conditions required for the pa- rameterization by a diffusion-like term of spatially- averaged turbulent processes in one-dimensional transport models of a single constituent. In this paper, it is assumed that these conditions are met and that an equation for the longitudinal mass bal- ance of scalar quantities can be established. In what follows, our analysis is primarily relevant to cross-sectionally, well-mixed estuaries.

Within a few decades, a large number of studies have advanced our understanding of the physical mechanisms underlying mass transport in estuaries (e.g., Fisher 1972a, 1976; Hughes and Rattray 1980; Dronkers and van de Kreeke 1986; Jay et al. 1997; MacCready and Geyer 2001). The method is generally based on the decomposition of the lon- gitudinal scalar flux of salinity along Eulerian or Lagrangian (e.g., isohaline surfaces) frames of ref- erence. Such decompositions have successfully been applied to both well-mixed and stratified es- tuaries.

The resulting effects of scalar fluxes on property versus salinity distributions have received much less attention. The constraints imposed by the hy- pothesis of equal diffusion coefficients for all dis- solved species (the third assumption in the heuris- tic model) have not been addressed carefully, in particular in cases when solutes are not subject to the same boundary constraints. This is typically the situation when constituent-specific, along-channel sources (e.g., point inputs, exchange of matter with the sediments) are affecting their estuarine distribution. Insufficient emphasis has also been given to the mathematical representation of the seaward boundary condition. In the case of tidally- averaged, advection-dispersion models, the latter is often prescribed as a Dirichlet boundary condi- tion, which implies immediate and indefinitely large dilution beyond the estuarine-coastal zone in- terface. This condition is an oversimplification and its effects on tidal mixing require further investi- gation.

In this paper, the Fractional Freshwater Method and associated approaches are reinterpreted in the framework of the mathematically transparent mod- el for long-term solute dynamics in estuaries pro- posed in O'Kane and Regnier (2003). This rein- terpretation, which incorporates explicitly the ef- fects of tidal oscillation on modeled concentration distributions, will allow us to clarify the arguments that are at the foundation of the heuristic models. The influence of the seaward boundary condition on the mixing curves is also investigated, and the mathematical representation of the diffusion pro- cess in the presence of along-channel sources is discussed.

THE FRACTIONAL FRESHWATER METHOD

The Fractional Freshwater Method uses salt as an index of mixing of the water masses allowing the need to account for transport explicitly to be circumvented. In this method, the fraction of freshwater over any segment x along the longitu- dinal curvilinear axis of the estuary is given by

(Ss - Sx)

(Ss) (1)

where Ss is the salinity of undiluted seawater and Sx is the mean salinity in a given segment x of the estuary. The bar indicates that the method applies to tidally-averaged variables.

Let Co and So, respectively, denote the concen- trations of a conservative solute C and salinity at an along-channel source located at x0; m be the discharge rate of the conservative constituent, giv- en as units of concentration per unit time; and Qf be the river discharge. Ketchum maintained that the distribution of C satisfies the following two re- lationships

C= c Sx mf Sx ?S 0 -S ?S So Qf fo

upstream of the outfall and

C = Co -=f fx Qfo fx

(2)

(3)

downstream of the outfall. These relationships al- low for the prediction of the distribution of C from the observed values of salinity (Fig. 1). If the salin- ity is normalized in such a way that Ss = 1, then fx = 1 - Sx with Sx comprised between 0 and 1. In this case, Eqs. 2 and 3 reduce to Cx = m/Qf((l -

So)/SO)Sx for x < xo and Cx = m/Qf(l - Sx) for x > xo, respectively.

The Fractional Freshwater Method avoids explic- it computation of the dispersion coefficient D (see below). As pointed out by Pearson and Pearson (1965), these are taken into account implicitly through the comparison of salt and pollutant dis- persion.

An alternative method for the prediction and in- terpretation of flushing and pollution dispersion relies on the one-dimensional, time-averaged, ad- vection-dispersion equation (Stommel 1953). Ac- cording to Neal (1966), the Fractional Freshwater Method and the continuous models based on the time-averaged, advection-diffusion equation give essentially similar results. According to Holley and Harleman (1965), the instantaneous (i.e., tidally- resolved) advection-diffusion equation and the time-averaged, advection-diffusion equation give fundamentally different results. One purpose of

Mixing Processes in Estuaries 573

between the advective and dispersive transport av- eraged over one or more tidal cycles (Bowden 1963).

It is claimed that a similar equation can be writ- ten for any other conservative solute C:

aC a -- - ac sv Qf -AD- + AIsv ax ax ax /

normalized salinity

FIG. 1. The Fractional Freshwater Method applied to three point inputs located at a normalized salinity equal to 0.3, 0.6, and 0.8.

this paper is to bring to the attention of the mod- ern reader the resolution of this disagreement.

In the case of salt, the time-averaged, advection- diffusion equation reads, assuming steady state, as:

Qfas 1 a / _.as\ -----l__ AD - = 0 (4)

A ax Aax Ax/

where x denotes a point on the one-dimensional curvilinear axis of the estuary; A is the cross-sec- tional area defined as the volume of water per unit length of the x axis; Qf is the volumetric flow rate of the riverine source; and D is the non-negative longitudinal dispersion coefficient with a dimen- sion of length squared per unit time. The bar in- dicates that the tidal components have been re- moved by averaging over time. Equation 4 shows that the average net flux of salt through any cross section is equal to zero, a balance being established

where Isv is the mass rate of injection of the sub- stance per unit volume at the source location.

The dispersion coefficient is an unknown param- eter and is classically determined from the salinity distribution according to:

) Qf~S X a(s D= _ - -> 0 (6) A(aS/ax)' ax

When aS/ax = 0, D is not defined; when aS/ax is close to zero, D is not accurately determined by Eq. 6 and other parameterizations of D may be used. This problem may be partly resolved by de- fining the start and end of the estuary so that AS/ ax > 0. On this basis, a continuous model of pol- lutant transport can be set up using Eq. 5, provided that the freshwater discharge and boundary con- ditions for C are given. The latter selects one unique continuous solution among the infinitely many continuous distributions, which are consis- tent with Eq. 5. Dirichlet boundary conditions have generally been applied at both ends of the estuary. At the seaward boundary this corresponds to im- mediate infinite dispersion outside the domain of the model estuary.

THE TIME-AVERAGED CONSERVATION EQUATION IN AN OSCILLATING REFERENCE FRAME

We may expect tidal harmonics in all water qual- ity variables in a macrotidal estuary. Removing them from the advection-diffusion equation by time averaging to give Eq. 5 is problematic, as first pointed out by Holley and Harleman (1965). A better way is, first, to transform the tidally-resolved mass conservation equation from a rate balance in an Eulerian reference frame, (x, t), to a rate bal- ance in a new reference frame, (x', t), designed so that the tide is removed to the greatest extent pos- sible. In contrast to the advection-diffusion equa- tion in an Eulerian frame, the remaining harmon- ics can now be removed through the application of a time smoothing operator, to yield an accurate, time-averaged, solute equation and solution in the (x', t) reference frame.

The transformation from (x, t) to (x', t) is math- ematically transparent, physically based and re- moves almost all of the tidal harmonics that are present in the original equation and its solution (O'Kane and Regnier 2003). The tide is removed

0

0

0 e

_I (U

1

(5)

normalized distance

0

0.6 0

0

. 0.4

0

574 P. Regnier and J. P. O'Kane

because x' oscillates so as to maintain a constant volume of water, V, upstream of x' at all times. The tide can be reinserted in the solution with the in- verse transformation from (x', t) to (x, t) whenever it is required. The transformation and its inverse are implicitly defined by the equation V(x, t) = V(x', 0) where V is the volume function for the estuary, i.e., the volume of water upstream of x at any time, t.

The major difficulty associated with the coordi- nate transformation is the mathematical represen- tation of along-channel sources. Indeed, those sources that are fixed on the bank in the Eulerian reference frame now oscillate in (x', t). We have shown in O'Kane and Regnier (2003) that for most operational purposes, it is sufficient to distribute uniformly the mass injection rate of a source with- in the tidal excursion about the source location. This approximation was first used with Preddy's in- tegral equation mixing-model of the Thames es- tuary (Thames Report 1964). We have named this the Modulated Discharge since a uniform injection over the tidal excursion, strictly speaking, requires the discharge rate to be modulated in inverse ratio to the average velocity of the estuary. This is always the sense in which the word modulated is used in this paper. The tidal excursion is given by e* = e'(x) = min x'(x, t) - max x'(x, t). It is the set of values of x' over which any point x moves during half a tidal cycle. The averaged velocity is defined as (ax'/at)x, that is, the tidal velocity of point x in the oscillating reference frame. Under the Modu- lated Discharge approximation, we have shown that at steady state, a time-averaged, mass conser- vation equation, which contains j along-channel sources, obeys:

= AoD + AP' + e 0x' 0x' 0 x') e? (7)

where Q' is the freshwater inflow; A (x') is the time invariant estuarine cross-section that is devoid of tidal effects; D. = D'(A'/AN) is the dispersion coefficient describing the longer-term (the bar in- dicates tidally-averaged) mixing processes in the (x', t) frame; ihj is the along-channel source num- ber j, specified as units of mass per unit time in- jected at the fixed location x = xj; e* is the appro- priate tidal excursion for source j, i.e., the distance in the (x', t) frame over which the point discharge at x = xj oscillates; P' is the algebraic sum of all sources and sinks of C', each given as a positive or negative concentration per unit time; the bar over mj/ej* indicates that this term arises from time-av- eraging the oscillating sources in the (x', t) frame.

If it is relatively easy to specify realistic boundary conditions to Eq. 7 at the landward limit of the

estuary, the task at hand is significantly more com- plicated at the estuarine mouth since the mecha- nisms and intensity of mixing are generally poorly known in this area. Any comparison between field data and model results should then not be per- formed without a preliminary analysis of the un- derlying hypothesis implied by the choice of the seaward boundary condition. In our case, a dis- tinction can be made between the x-estuary and the x'-estuary. These are the views of the estuary seen from each of the two reference frames. In this paper, the following boundary conditions at the sea (L) will be applied to Eq. 7:

A Dirichlet condition is fixed at x' = L. Beyond x' = L there is an immediate, indefinitely large, dilution of the estuarine waters within the receiv- ing coastal waters. In this case, the boundary con- dition oscillates when viewed from the (x, t) frame. During the ebb tide, the x'-estuary protrudes into the sea as a plume of indeterminate shape, but of known volume. This condition removes the advec- tive flushing action of the tide at x = L in the x- estuary, displacing it seawards on the x-axis. Mate- rial in solution, which is flushed past x = L by advective tidal motion during the ebb, may re-en- ter the x-estuary again during the following flood.

A Dirichlet condition is fixed at x = L. Beyond x = L there is an immediate, indefinitely large, dilution of the estuarine waters within the receiv- ing coastal waters. This mathematical constraint is probably most relevant for situations where strong along-shore currents carry the estuarine water mas- ses away from the mouth during the ebb tide. In this case, the boundary condition oscillates when viewed in the (x', t) frame. As a result, new, un- diluted coastal waters enter the x'-estuary during each flood tide.

In contrast to the time-averaged, advection-dis- persion equation in (x, t), these two boundary con- ditions allow for the representation of a range of realistic physical conditions at the mouth. The analysis could be extended further by prescribing a Neumann boundary condition at x or x' = L. In this case, no dilution of the solutes in the receiving sea is assumed. However, this hypothesis is unre- alistic since without dispersion at the mouth, it is impossible to drive a salt flux landwards against the freshwater inflow.

APPLICATION TO A SIMPLIFIED MODEL ESTUARY

The Model for a Conservative Solute (P' = 0)

Analytical solutions to the time-average, steady state transport Eq. 7 can readily be obtained for a simplified model estuary. This model has the fol- lowing characteristics: the cross section is of con- stant area A; the tide only generates a horizontal,

Mixing Processes in Estuaries 575

along-channel movement of water with a constant tidal excursion e* everywhere; the diffusion coef- ficient D is constant throughout the estuary; and the freshwater through-flow Q is constant.

Based on the above assumptions, the mass con- servation equation, for a nonreactive solute, re- duces to:

6 ac'- a2C' (8) Q' -A

Q, _ AD,_ C = 'i. mj (8) ax' x'2 e*

STEADY-STATE SOLUTIONS

Closed form solution to the steady-state solute transport Eq. 8 can readily be obtained using Green's functions. Dirichlet boundary conditions were already applied to Eq. 8 in O'Kane (1980), but the constraints on the differential equation were specified at +oo. In this paper, more realistic solutions for systems of semi-infinite length are in- vestigated, through the application of the Method of Images (Lin and Segel 1988).

The problem at hand is the following: we need to find a continuous distribution C'(x') such that

AC' = p'(x') 0 - x' < L with

C'(0) = -i; C'(L) = -Y2

where A is the spatial differential operator:

ax' ax' 2

(9)

superposition since the differential operator A is linear. The latter is also used to decompose the solution of Eq. 9 into three simpler problems:

AC' = p'(x') with C'(0) = 0;

AC' = 0

AC' = 0

C'(L) =0 A

with C'(0) = y; C'(L) = 0 B

with C'(0) = 0; C'(L) = 2 C

(12)

Because we want to investigate the specific effects of along-channel sources, we neglect the back- ground concentrations resulting from the mass flow through the system boundaries, i.e., we as- sume y1 -= Y2 = 0.

THE SOLUTION TO PROBLEM 12A

The solution to this two-point Dirichlet bound- ary value problem in the case of one modulated discharge located at x = 0, is obtained in two steps.

Step 1

Using Green's functions, a solution that satisfies Eq. 12A is found for an infinite range of x', i.e., for the boundary conditions

C'(-oo) = C'(+oo) = 0 (13)

The solution for the system of infinite length con- sists of three terms:

(10)

and p'(x') is the algebraic sum of the mass fluxes from the along-channel sources. In Eq. 9, we have assumed that both boundary conditions are fixed in the oscillating reference frame x' (see further discussion below).

In the case of one Modulated Discharge located at x = 0, p' is equal to

m/e* for-e* < x' - 0 p'(x') = 0 (11)

0 otherwise

The tidal excursion is a constant in our simpli- fied model estuary. The mathematical support -e* ' x' - 0 of the function p' (x') may be interpreted as follows. The reference state of the model estuary is high water. It occurs everywhere at exactly the same time. At high water the Eulerian coordinate (x) and constant volume coordinate (x') are co- incident by definition. As the tide ebbs, an observ- er in the x' frame sees the discharge point located at position x = 0 on the bank move upstream over its tidal excursion, i.e., in the negative x' direction, starting at x' = 0 and reaching x' = -e* at low tide. Note that only one along-channel source is considered here. The solution can be generalized to any number of sources through the principle of

m D ____ [e(U/D)(x'+e*) _ e(U/D)x']

QUe* for x' -- -e*

C'(x') = m x' D -- + ?1 + -[1 Q e* Ue*

e- (U/D)x'] , (14)

for -e* ' x' < 0

for 0 x'

where U = Q/A. In Eq. 14, the over-bars above A, Q, U, and D can be omitted because of the con- stancy of these variables in the simplified model estuary.

Figure 2 compares the time-averaged Modulated Discharge solutions with the tidally resolved, exact transient solution for t -> oo (Eq. 2 with constant

Q, A, and D in O'Kane and Regnier [2003]). It is only necessary to show both results at high water since they oscillate without change in shape over the constant tidal excursion. The results show that the time-averaged Modulated Discharge approxi- mation gives an almost perfect fit to the exact, tid- ally-resolved, long-term transient solution. In this case, three assumptions were made: Reynolds

576 P. Regnier and J. P. O'Kane

1

.o 0 .N

U a C 0

? 'a

O

._

O

0.8

0.6

0.4

0.2

0

* tidally resolved, exact solution at large t _ time average solution with modulated discharge

-1.5 -1 -0.5 position (x'le*)

0 0.5

FIG. 2. Steady-state concentration profiles at high water ob- tained for a time-averaged modulated discharge and the exact transient solution at large t, for a discharge at x = 0. The po- sition is normalized with respect to the tidal excursion e*, and the value x'/e* = x/e* = 0 corresponds to the location of the along-channel source at high tide. The constant value of the normalized concentration downstream of the outfall (C' = 1) is consistent with the seaward boundary condition located at +00.

Rules for time averaging apply in the oscillating reference frame; the along-channel source can be represented as a modulated discharge, i.e., as a source uniformly distributed within the tidal ex- cursion; and the dispersion coefficient D is a con- stant (O'Kane and Regnier 2003).

Step 2

Using the Method of Images, the solution is fur- ther investigated for a system for which

C'(x' > L) = 0 or C'[x'(x > L, t)] = 0 (15)

while maintaining C' = 0 at -oc. As shown by the shape of the mixing curves presented below in the text, maintaining the landward boundary condi- tion at -oc is probably only a mild restriction pro- vided that the water movement is not physically limited by structures such as weirs or gates. The method of images will also show that the seaward boundary condition provides a necessary sink in the overall mass balance on the estuary.

The absence of the over-bar in the first bound- ary condition in Eq. 15 indicates no time-averaging is required. Time-averaging is essential in the sec- ond case since a boundary fixed in the x reference frame oscillates in the x' frame. We approximate the time-averaging with a modulation in space.

To find the solution to Eq. 12A that satisfies the first point boundary condition (subscript p-bc be- low in Eq. 17), place an image point sink of strength m/Q at the point x' = L. The solution of A Cp = m/Q for a point source at x' = 0 (i.e., an oscillating source with respect to the bank), is

(m/Q)e(U/D)x' for -co x' < 0

C (m/Q) otherwise

The over-bar is absent since no time averaging is required in this case. The solution for an image point sink at x' = L is simply -Cp(x' - L), namely minus Eq. 16 with its argument (x') replaced ev- erywhere with (x' - L) that shifts it by a distance L along the x' axis. Subtracting Cp(x' - L) from Eq. 14 for a modulated point source at x = 0 of strength +m/Q gives

Cp-bc(x') = C'(x') - Cp(x' - L) (17)

For x' ' L, Cp_bc(x') = 0 as required. The time- averaged modulated internal point source and the point sink fixed at the boundary at x' = L cancel to give an overall mass balance on the estuary. The effect of the boundary condition is to reduce the concentration everywhere within the estuary.

The second boundary condition in Eq. 15 is specified at x = L. It oscillates in the x' frame from x' = L to x' = L - e*. This oscillation must be removed by time averaging in order to define a steady-state solution. This may be done as follows. At high water the Eulerian coordinate (x) and con- stant volume coordinate (x') are co-incident by def- inition. As the tide ebbs, an observer in the x' frame sees the boundary located at position x = L on the bank move upstream over its tidal excur- sion, i.e., in the negative x' direction, starting at x' = L and reaching x' = L - e* at low tide. The appropriate image sink in this case is a uniformly distributed sink, running from x' = L to x' = L - e* within the e'-estuary, produced by time averag- ing the modulated boundary condition at x = L. The image concentration C' for a modulated source at x = L is given by equation (14), but with the argument x' replaced everywhere with (x' - L), which shifts all three parts of Eq. 14 by L in the positive x' direction as required. Subtracting this image from C'(x') in Eq. 14 gives the solution for the second of the two boundary conditions in Eq. 15 as

Cm_bc(x') = C (x) - C(X - L) (18)

Figure 3 illustrates the Method of Images for pre- scribing a zero Dirichlet boundary concentration at the estuarine mouth (x = L = 2e*) which is treated as a time-averaged, modulated boundary condition. The thin solid line is the normalized solute concentration resulting from a Modulated Discharge source at x = 0 subject to C' = 0 at ?oo (see Fig. 2 above). In order to prescribe a time- averaged concentration boundary condition at x = L, a mirror solution is sought for a unit modulated sink (dashed line), which is distributed over the tidal excursion at the seaward boundary, such that

. . . I . . . I . . I . . I . I I I I . I I I I

Mixing Processes in Estuaries 577

.Y E 0.5-

0

o 0

o -0.5 -

_ \

-1 ' ' ' ' I , , I,,, "

-2 -1 0 1 2 position (x'/e*)

FIG. 3. The method of images. Thin line: modulated unit source at x = 0, C' (x'); dashed line: modulated unit sink at x/ e* = L/e* = 2, -C'(x' - L); and thick line: superposition of modulated unit sink and source solutions, C' (x')- C' (x'- L).

the superposition of the unit sink and source dis- tributions makes the concentration C' at x' -> L equal to zero (thick solid line). Figure 3 shows the resulting solution for a seaward boundary that is fixed with respect to the estuarine bank, i.e., an oscillating boundary condition in the (x', t) frame.

The Salinity Distribution. Because we want to in- vestigate the distribution of C'(x') versus S'(x'), a continuous salinity distribution is also required. The problem at hand in this case is

AS' = 0 with

S'(-oo) = 0; S'(x' = L) = So If an oscillating boundary condition with respect to the bank is assumed at the mouth, the solution to this second-order ordinary differential equation is equal to

Sp (x') = Soe (UD)(x'-L) for -oo <_ x' < L (20)

which shows that the salinity profile computed un- der the assumptions of the simplified model estu- ary (e.g., constant diffusion coefficient and cross sections) follows a simple exponential function. The advantage of the simplified model estuary, in contrast to the Green's functions obtained for real estuaries presented below, is that it leads to closed- form solutions, which are sufficient to substantiate our claims concerning the mixing processes in es- tuaries.

PREDICTING CONSERVATIVE MIXING CURVES IN REAL ESTUARIES

In the previous section, closed-form solutions to predict the effect of along-channel sources on the mixing curves in a simplified model estuary have been proposed. Here, the concept is extended to estuaries of general shape (i.e., varying A, D, and

e*) for the case of a Modulated Discharge. A Dir- ichlet boundary condition fixed at x' = L is con- sidered here.

In a manner similar to the simplified model es- tuary, application of the theory of Green's func- tions results in a solution made of three parts (O'Kane 1980). In this case, a general solution is sought, which is expressed directly with respect to salinity instead of x'. If x0 denotes the position of the fixed along-channel source, the solution reads, for a tide labeled at high water slack:

C'(x') =

m S(x') x-e* 1 - S() dy')

Q' e* X S(y')

for x' xo- e*

m *<[1-S(x')][x' -(xo - e*)]

X 1 - S(y') + S(x') S(y) dy'

for x - e* x' - xo

m [1 - S(x')] for xo ?< x'

(A)

(B)

(C)

(21)

The function describing the distribution of a con- servative solute injected through an along-channel source located at x0 in a finite estuary of general shape is proportional to the freshwater dilution augmented by a function of the salinity concentra- tion. In Eq. 21A, C' is linearly proportional to S' in the same way as the landward part of the solu- tion obtained by the Fractional Freshwater Method (Eq. 2). In contrast to the Freshwater method, the slope of this line is now equal to the weighted av- erage slope of all lines comprised between y' = xo and y' = x0 - e*. The bounds on C' follow from the maximum and minimum values of the inte- grand in Eq. 21A. For any value of x' - x0 - e*, we have:

C' - ( 1 S (x0 ~) > C'(x') CUB -Q,S (x') S'(xo - e*) Q A

m 1 - S'(xo) > - S' (x') -SCLB Q' s' ( Xo) - LB (22)

The seaward part of the solution (Eq. 21C) is iden- tical to the solution of the Fractional Freshwater Method downstream of the outfall location (Eq. 3). Equation 21B is a combination of parts A and C in such a way that the equation forms a smooth tran- sition between them.

Inspection of the solution shows that the model

578 P. Regnier and J. P. O'Kane

n

0.6 -

O ' ' ' ' I ' ' , I I ' ' ' , 0 I , J , I l , 2 -2 -1 0 1 2

position (x'/e*)

FIG. 4. Normalized concentration profiles versus normalized distance obtained in (x', t) for a modulated discharge. The in- fluence on the choice of the seaward boundary condition is investigated. Thin line: boundary condition at x' = L oscillating with respect to the bank; and thick line: boundary condition fixed with respect to the bank at x = L, modulated and distrib- uted over its tidal excursion upstream.

can be applied to any real estuary provided that: the river discharge is known; longitudinal salinity profiles are available; the tidal excursion about each along-channel source is known; and the po- sition and intensity of the along-channel sources are identified. A solution in the form of Eq. 21 is then obtained for each of the along-channel sourc- es, the overall effect of several sources on the mix- ing curves being equal to a superposition, or linear combination, of their individual solutions (i.e., the solutions are simply added together, weighted by their individual mass discharge rates).

Equation 21 has no closed form solution in terms of elementary functions. The integrals ap- pearing in Eq. 21 can readily be evaluated numer- ically, using the widely available numerical integra- tors implemented in computational mathematics software such as Matlab, Mathematica, or Maple (we used Maple in our case).

As an illustration of the method, conservative mixing curves are calculated for a typical macro- tidal estuary, the Western Scheldt (Belgium-Neth- erlands), with the following characteristics: the model is limited to the first 100 km of the estuary, a distance that corresponds roughly to the maxi- mum extent of the salt intrusion; the river dis- charge is set to a typical value of 70 m3 s-1; the salt model of Savenije (1992) is applied to compute the along-channel distribution of salt; and two along channel sources, located at x = 50 and 80 km, are considered. In both cases, the intensity of the discharge is assigned a value of 50,000 kg d-1. The tidal excursion e*, computed in O'Kane and Regnier (2003), is roughly constant over this por-

0.1 0.2 normalized salinity

0.3

FIG. 5. Concentration-salinity mixing curves about an along- channel source for a modulated discharge. Note the nonlinear concentration-salinity relationship within a tidal excursion about the source location.

tion of the estuary with a value close to 14 km for a spring tide.

Results MIXING IN A SIMPLIFIED MODEL ESTUARY

Figures 4 and 5 show the steady-state solute pro- files obtained with respect to distance and salinity for the simplified model estuary in (x', t). The in- fluence of the seaward boundary condition and the effects of along-channel sources can now read- ily be investigated.

The Choice of the Seaward Boundary Condition

Figure 4 compares the solute distribution ob- tained for the boundary condition C' = 0 at either x = L or x' = L. Note that in both cases, the Mod- ulated Discharge provides a smooth concentration profile within the whole estuarine domain. Results show that a boundary condition fixed at x' = L and oscillating in the (x, t) frame leads to higher solute concentrations within the estuary than a boundary condition fixed with respect to the bank at x = L.

The Along-channel Source

Figure 5 shows the mixing curve at low salinity, in the vicinity of the source input at x' = 0. An oscillating boundary condition at the mouth (i.e., fixed at x' = L) is considered here, but the dynam- ics would be similar if a fixed boundary condition in (x, t) were applied to both concentration and salinity profiles. The analysis of the mixing curve indicates that within a tidal excursion about the source location, the concentration-salinity relation- ship shows a smooth transition with a definite cur- vature, i.e., the profile is nonlinear within e*. This

c 0.8

8 0.6 0 0 ? 0.4

r 0.2

Modulated discharge

e* D 0

1 A

Mixing Processes in Estuaries 579

1

0.75

'0.5 (0

0.25

0t

HWS

/50000 kg/d 50000 kg/d

- 1 , , , , , I , , , I I , ,I, , I I , I I I, I I ,

0.2 0.4 0.6 normalized distance

.-

0

normalized salinity

FIG. 6. Application of the concentration-salinity mixing model to the Western Scheldt estuary. Normalized along-chan- nel salt distribution at high water slack (HWS) versus normal- ized distance. Effect of two along-channel sources located at x0 = 0.2 (thin solid line) and 0.5 (dashed line) on concentration- salinity mixing curves. The respective normalized salinities at HWS for these two along-channel sources are approximately 0.35 and 0.75. The thick solid line was obtained by linear su- perposition of the two individual solutions.

behavior will be investigated below, taking into ac- count the fact that the Modulated Discharge pro- vides a very good approximation of the exact, tid- ally resolved distribution (Fig. 2).

MIXING IN A REAL ESTUARY

Figure 6 shows the along-channel distribution of salt computed with the model of Savenije (1992). The individual and combined effects of the two along-channel sources on the mixing curves are shown. Results were obtained by numerical inte- gration of Eq. 21 for each source and indicate that, as in the simplified model estuary, the peaks in concentration are smoothed by the tidal oscilla-

tion. They also demonstrate that the position of the along-channel source has a significant effect on the resulting maximum concentration in the pro- file; the further downstream the source is located, the weaker its effect on the concentration-salinity mixing curves. The linear superposition of solu- tions yields a complex (geometrically nonlinear) conservative profile, which takes into account the combined effects of all along-channel sources.

Discussion THE SEAWARD BOUNDARY CONDITION

Results show that a boundary condition fixed at x' = L and oscillating in the (x, t) frame signifi- cantly reduces the apparent dispersion beyond the mouth at x = L and decreases the dilution and transport processes within the estuary. This leads to significantly higher concentrations in the estu- ary than in the case of a boundary condition fixed with respect to the bank at x = L. The latter as- sumes infinite dispersion of solute right at the es- tuarine mouth.

Subtracting Eq. 18 from Eq. 17 gives the differ- ence between the two possible boundary condi- tions

Cp_bc (x) - _bc (x) = C'(x' - L)

-Cp (x' - L) (23)

Since C' (x') - Cp (x') = Cp (x') for all values of x',

we conclude that

Cp-bc (X) > Cm-bc (X ) (24)

The distributed sink from x' = L to x' = L - e* produced by the modulated boundary condition at x = L, leads to smaller concentrations within the e'-estuary, in comparison with those produced by the point boundary condition at x' = L. This is in accordance with expectation.

THE ALONG-CHANNEL SOURCE

Ketchum and Stommel's heuristic models as- sume that through an averaging procedure of some kind, the high-frequency tidal oscillatory convection has been removed. This result prompts us to recast Ketchum's model in the (x', t) frame for which time averaging is accurate and mathe- matically transparent. This reinterpretation is rea- sonable because the observed salinity data used to construct the mixing index in the Fractional Fresh- water Method are uniquely defined versus position and must be corrected to an equivalent tidal po- sition. This correction is in fact equivalent to a mathematical transformation from the (x, t) to the (x', t) frames of references (O'Kane and Regnier 2003). Salinity data collected at various times dur- ing a tidal cycle, collapse onto a single curve when

0.8

580 P. Regnier and J. P. O'Kane

c 0.8 - /

8 0.6- /

O - /

\

0

= 0.2- , / \ 0.21

-2 -1 0 1 2 position (x'/e*)

FIG. 7. Normalized concentration profiles versus normalized distance obtained from the Fractional Freshwater Method and the mathematically transparent time-averaged solute equation in (x', t).

plotted in the (x', t) frame instead of the (x, t) frame. This has been verified in the case of a num- ber of estuaries, the Thames being the first (Thames Report 1964).

The comparison between the Fractional Fresh- water Method and the time-averaged Modulated Discharge for the simplified model estuary allows one to quantify the respective transport and mix- ing processes about along-channel sources. Figure 7 shows that the Fractional Freshwater Method ap- plied to the salt profile given by Eq. 20 significantly underestimates mixing in the vicinity of along- channel sources. These results demonstrate that Ketchum's intuitive results are correct, provided they are reinterpreted in the oscillating reference frame. The relative motion of this frame smears each point input so that it is continuously distrib- uted along the oscillating estuary with an intensity equal to its differential residence function (O'Kane and Regnier 2003). There are two inte- grable singularities in this function, one at each end of the tidal excursion, when the tide turns. An acceptable approximation is provided by the Mod- ulated Discharge case; the tidally-averaged, smeared input is simply approximated by the in- tensity m/e* within the tidal excursion e* about the point of discharge where the mass rate of in- jection is m. According to Fig. 7, direct time aver- aging in (x, t) would in fact require an ad-hoc in- crease in the dispersion coefficient about each along-channel source. This would facilitate mov- ing, in some way, the curve (thin solid line) ob- tained for the Fractional Freshwater Method to- wards the exact steady-state distribution (thick dashed line) obtained for the Modulated Dis- charge. In fact, it is easily demonstrated that the solution predicted by the Fractional Freshwater

I

c 0.8

8 0.6 c 0

3 0.4

| 0.2 r 0.2

- - - - Modulated discharge Fractional Freshwater Method

.. I7 -

D-:j E-

0.3 0.1 0.2 normalized salinity

FIG. 8. Normalized concentration profiles versus normalized salinity (mixing curves) obtained from the Fractional Freshwa- ter Method and the mathematically transparent time-average solute equation in (x', t).

Method is exactly the same as the one obtained in (x', t) if one assumes that the along-channel source oscillates along the estuarine bank maintaining a constant upstream volume of water, an unrealistic situation.

The results also show that the continuous func- tions representing S'(x') and C'(x') are linearly proportional to each other in the Fractional Fresh- water Method (Fig. 8, thin solid line). The math- ematically transparent solution obtained for the Modulated Discharge case in the simplified model estuary proves that the distribution of C' (x') with- in one tidal excursion about the outfall is in fact a smooth nonlinear function of S' (x') (Fig. 8, thick dashed line). Similar nonlinear transition is ob- tained in the case of a real estuary (Fig. 6b). These results differ markedly from the intuitive analysis of mixing curves, which imposes a peak in concen- tration at the source location and demonstrates that the Fractional Freshwater Method systemati- cally underestimates the concentration at any point upstream of the outfall location.

A measure of the underestimation may be ob- tained for a real estuary by comparing the solute concentration calculated at position x0 - e* (i.e., one tidal excursion upstream of the outfall loca- tion) with both approaches. In the Fractional Freshwater Method, Eq. 2 gives, using a normal- ized salinity scale:

m 1 - S(x0) CFm(xo - e*) = -S(x0 - e*) S (x0)

Q f (25)

Equation 25 is equivalent to the lower bound of C', CLB in Eq. 22. The value of the upper bound of C', CUB, at x0 - e* is given by:

II ......

Mixing Processes in Estuaries 581

CUB(XO - e*) = Q-[1 - S(x0 - e*)] (26) Qf

Because CFFM = CLB, the difference between CUB and C'LB is a measure of the underestimation as- sociated to the Fractional Freshwater Method:

m S(x0) - S(x0o - e*) m ASTE

Qf S(xo) Qf S(xo)

where ASTE is the salinity gradient within a tidal excursion upstream of the outfall. Equation 27 shows that the error introduced by the Fractional Freshwater Method will increase whenever ASTE is increasing. Large values of ASTE are to be expected when a large tidal excursion is combined with a steep salinity gradient within e*. Eq. 27 shows that, for the same ASTE, the error will increase as one moves upstream within the estuary.

Exact calculations of the error require numeri- cal solution of Eq. 21. Alternatively, if one assumes as a first approximation that the average slope of C'(x') for x' c x0 - e* is simply the average of the lower and upper bounds on C', then the differ- ence in concentration at x0 - e* obtained with the average slope and the slope obtained with the Frac- tional Freshwater Method is AC = A/2. Compari- son of the results obtained using such an approx- imation with those obtained numerically shows that it is indeed a reasonable assumption. The ratio AC: CFFM quantifies the underestimation of the Fractional Freshwater Method in a way that does not depend on the ratio m/Qf. For example, a val- ue of ASTE of 5%o for an outfall located at S(x0) = 10%o leads to an underestimation of approximate- ly 70%. Moving the outfall seaward at S(x0) = 20%o reduces the underestimation to approxi- mately 40% for the same value of ASTE.

Our results cast a new light on the key assump- tion that the sectionally and tidally averaged con- centration of solute is transported in the same way as the concentration of salinity that surrounds it. This assumption is only valid in an oscillating con- stant-volume reference frame, the (x', t) frame in this paper, and not in the (x, t) frame. The expla- nation of the non-equivalence of mixing by disper- sion, in the classical one-dimensional transport models, with and without tidal averaging, is to be found in the application of the Reynolds Rules for averages in the (x, t) frame. Indeed, in this case, the non-null product of the fluctuations associated with time averaging in (x, t) must be part of the diffusion coefficient determined from observed sa- linity distributions, because all the transport modes, which are not explicitly taken into account in the time-averaged differential equation, are fold- ed within the dispersion term (Regnier at al. 1998;

O'Kane and Regnier 2003). Our results prove that in the presence of along-channel sources, the short-term fluctuations in C(x, t) and S(x, t) as- sociated with the oscillatory convection, are not linearly correlated in an Eulerian frame. Conse- quently, the tidally-averaged diffusion coefficients for salt and any other solute are not identical in this case.

Conclusions

In this paper, steady-state solutions to the advec- tion-dispersion equation describing the long-term solute dynamics in an oscillating reference frame have been proposed. Salt profiles and solute dis- tributions subject to along-channel sources were obtained for a well-mixed estuary combining Green's functions and the Method of Images. The comparison of solutions that satisfy either fixed or oscillating Dirichlet boundary conditions at the sea demonstrate that the mathematical representation of the mixing process outside of the estuarine do- main influences significantly the resulting distri- bution of a solute in the estuary.

The solutions of our analytical model were also used to provide an analysis of the mixing curves. In particular, attention was paid to the hypothesis of identical mixing for all solutes in estuaries. We have demonstrated that this hypothesis, which is explicitly or implicitly formulated in tidally aver- aged transport models using a fixed reference frame, is not verified when solutes are subject to along-channel sources. We have argued that the in- equality in the dispersion coefficients finds its ori- gin in the parameterization of the Reynolds stress- es arising from the time averaging procedure. In classical models, the latter are included in a mixing term, which is determined from salinity profiles. One may expect that the Reynolds stresses associ- ated to the respective solutes are not mathemati- cally related, because the equation describing the instantaneous distribution of dissolved constituents are not subject to identical boundary constraints. As demonstrated in O'Kane and Regnier (2003), this difficulty is circumvented in our case because the oscillating reference frame removes the Reyn- olds stresses arising from the tidal oscillatory con- vection. This means that the mixing processes as- sociated with tidal pumping are not included in the dispersion coefficients used to predict the long-term solute dynamics in the (x', t) frame. The claim of identical dispersion coefficients for all sol- utes discharging into an estuary can only be sub- stantiated for tidally-averaged solute transport models using an oscillating reference frame. It is explained by means of an example how such mod- els can be implemented in a real estuary.

582 P. Regnier and J. P. O'Kane

LITERATURE CITED

BOWDEN, K. F. 1963. The mixing processes in a tidal estuary. International Journal of Air and Water Pollution 7:343-356.

BOYLE, E., R. COLLIER, A. T. DENGLER, J. M. EDMOND, A. C. NG, AND R. F. STALLARD. 1974. On the chemical mass-balance in estuaries. Geochimica Cosmochimica Acta 38:341-364.

CHATWIN, P. C. AND C. M. ALLEN. 1985. Mathematical models of dispersion in rivers and estuaries. Annual Review Fluid Mechan- ics 17:119-149.

DRONKERS,J. 1982. Conditions for gradient-type dispersive trans- port in one-dimensional, tidally averaged transport models. Estuarine Coastal and Shelf Science 14:599-621.

DRONKERS, J. AND J. VAN DE KREEKE. 1986. Experimental deter- mination of salt intrusion mechanisms in the Volkerak estu- ary. Netherlands Journal of Sea Research 20:1-19.

FISHER, H. B. 1972a. Mass transport mechanisms in partially stratified estuaries. Journal of Fluid Mechanics 53:671-687.

FISHER, H. B. 1972b. A Lagrangian Method for Predicting Pol- lutant Dispersion in Bolinas Lagoon, Marin County, Califor- nia. U.S. Geological Survey Professional Paper Washington, D.C. 582-B.

FISHER, H. B. 1976. Mixing and dispersion in estuaries. Annual Review of Fluid Mechanics 8:107-133.

FISHER, H. B., E. J. LIST, R. C. Y. KOH, J. IMBERGER, AND N. H. BROOKS. 1979. Mixing in Inland and Coastal Waters. Academ- ic Press, London, U.K.

HOLLEY, E. R. AND D. R. F. HARLEMAN. 1965. Dispersion of Pol- lutants in Estuary Type Flows. Report No. 74, Hydrodynamics Laboratory. MIT, Cambridge, Massachusets.

HUGHES, F. W. AND M. RATTRAYJR. 1980. Salt fluxes and mixing in the Columbia River estuary. Estuarine Coastal and Marine Science 10:479-493.

JAY, D. A., R.J. UNCLES, J. LARGIER, W. R. GEYER,J. VALLINO, AND W. R. BOYNTON. 1997. Estuarine scalar flux estimation revis- ited: A commentary on recent developments. Estuaries 20: 262-280.

KAUL L. W. AND P. N. FROELICH. 1984. Modeling estuarine nu- trient geochemistry in a simple system. Geochimica Cosmochim- ica Acta 48:1417-1433.

KETCHUM, B. H. 1951. The exchanges of fresh and salt water in tidal estuaries. Journal of Marine Research 10:18-38.

KETCHUM, B. H. 1955. Distribution of coliform bacteria and oth- er pollutant in tidal estuaries. Sewage and Industrial Wastes 27: 1288-1296.

LIN, C. C. AND L. A. SEGEL. 1988. Mathematics Applied to De-

terministic Problems in the Natural Sciences. SIAM, Phila- delphia, Pennsylvania.

LIss, P. S. 1976. Conservative and non-conservative behavior of dissolved constituents during estuarine mixing, p. 93-130. In J. D. Burton and P. S. Liss (eds.), Estuarine Chemistry. Aca- demic Press, London, U.K.

LODER, T. C. AND R. P. REICHARD. 1981. The dynamics of con- servative mixing in estuaries Estuaries 4:64-69.

MACCREADY, P. AND W. R. GEYER. 2001. Estuarine salt flux through an isohaline surface. Journal of Geophysical Research 106:11629-11637.

NEAL, V. T. 1966. Predicted flushing times and pollution distri- bution in the Columbia river estuary, p. 1463-1480. In Amer- ican Society of Civil Engineers, Proceedings of the 10th Con- ference of Coastal Engineers, Tokyo, Japan.

OFFICER, C. B. AND D. R. LYNCH. 1981. Dynamics of mixing in estuaries. Estuarine Coastal Shelf Science 12:525-533.

O'KANE, J. P. 1980. Estuarine Water Quality Management. Pit- man, London, U.K.

O'KANE, J. P. AND P. REGNIER. 2003. A mathematically transpar- ent low-pass filter for tidal estuaries. Estuarine Coastal and Shelf Science 57:593-603.

PEARSON, C. R. AND J. R. A. PEARSON. 1965. A simple method for predicting the dispersion of effluent in estuaries, p. 50- 56. In Symposium No. 9: New Chemical Engineering Prob- lems in the Utilization of Water. American Institute of Chemical Engineers, London, U.K.

REGNIER P., A. MOUCHET, R. WOLLAST, AND F. RONDAY. 1998. A discussion of methods for estimating residual fluxes in strong tidal estuaries. Continental Shelf Research 18:1543-1571.

SAVENIJE, H. H. G. 1992. Rapid assessment techniques for salt intrusion in alluvial estuaries. Ph.D. Dissertation, IHE Delft, Delft, The Netherlands.

SHILLER, A. M. 1996. The effect of recycling traps and upwelling on estuarine chemical flux estimates. Geochimica Cosmochimica Acta 60:3177-3185.

STOMMEL, H. 1953. Computation of pollution in a vertically mixed estuary. Sewage Industrial Wastes 25:1061-1071.

THAMES REPORT. 1964. Effects of Polluting Discharges on the Thames Estuary. Water Pollution Research Technical Note No. 11. HMSO, London, U.K.

YEATS, P. A. 1993. Input of metals to the North Atlantic from two large Canadian estuaries. Marine Chemistry 43:201-209.

Received, December 3, 2002 Revised, December 1, 2003

Accepted, January 21, 2004