Copyright 1976. All rights reserved

MIXING AND DISPERSION IN ESTUARIES Hugo B. Fischer Professor of Civil Engineering, University of California, Berkeley, California 94720

INTRODUCTION

x8084

Prior to 1950, studies of estuaries were primarily observational. Rhodes ( 1950), for instance, described velocities and salinities in several estuaries along the eastern coast of the United States. Stommel & Farmer (1952) compiled data from 20 worldwide estuaries, ranging in size from the Moros in France (2.3 km long and a few meters deep) to the Straits of Juan de Fuca (100 km long and 350 m deep). Their report, although more than 20 years old, remains one of the more extensive compilations of salinity and velocity data in existence and, in addition, shows the wide range of water bqdies we call estuaries. Since the appearance in the early 1950s of Ketchum's work (1951a,b, 1955), of the unpublished manuscript On the Nature of Estuarine Circulation by Stommel & Farmer (1952), and of Pritchard's analyses of the salt balance in the James (1952 and later), researchers have used more analytic techniques to try to understand the process of mixing in estuaries and to quantify such aspects as residence times and pollutant concentrations.

It is not easy to agree on what an estuary is. Schubel (1971) listed 10 earlier definitions each of which he found unacceptable for some reason and settled on an eleventh definition by Pritchard (1967): "An estuary is a semi-enclosed coastal body of water which has a free connection with the open sea and within which sea water is measurably diluted with fresh water derived from land drainage. " Even this definition is inadequate for our purpose, because it excludes such estuaries as San Diego Bay where the fresh-water flow is less than the evaporation, but which can be treated like other estuaries with respect to mixing problems. For our purposes it may be more appropriate to say that estuaries are something like pornographyhard to define exactly, but we know one when we see one.

A number of writers (Stommel & Farmer 1952, Bowden 1967, Pritchard 1967, Schubel 1971, Dyer 1973) have given classification schemes for estuaries. In hydrodynamic terms all the schemes distinguish three major categories: sharply stratified estuaries, such as fjords and salt-wedge estuaries; partially stratified estuaries, in which there is a significant vertical-density gradient and vertical mixing is inhibited;

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108 FISCHER

and well-mixed estuaries. In this review we consider only the latter two categories; the reader interested in fjords should consult the review by Rattray (1967). We also omit discussion of the initial spreading of buoyant effluents, a subject reviewed recently by Stolzenbach & Harleman (1972) and Koh & Brooks (1975).

In this brief review we can hardly expect to cover even a small part of the estuarine literature. Book-length collections are given by Ippen (1966), Lauff (1967), Ward & Espey (1971), Gameson (1973), and Dyer (1973). Instead, we give an updated account of what we understand concerning the causes of dispersion in estuaries. Studies of mixing carried out through 1964 were reviewed by Bowden (1967). Although we mention some of these works briefly for completeness, the emphasis is on the period 1967-1974. We also mention current methods for predicting and modeling dispersion and how these are limited by our still incomplete understanding of the dispersion process.

Notation

Symbols used infrequently are defined where they appear. The following symbols are used throughout to refer to the major parameters characterizing an estuary:

A Cross-sectional area T Tidal period b Width u Velocity in the x-direction. Subscripts d Depth (the mean depth unless other- are defined where they appear.

wise stated) V rms tidal velocity D Longitudinal-dispersion coefficient Vf Qf/A 9 Acceleration of gravity Vo Peak flood-tide velocity Pr Tidal prism U Mean tidal velocity Qf Discharge of fresh water into the u* Shear velocity (r:/p)l/2. In tidal flows

estuary from tributaries the rms value is meant unless otherwise R Estuarine Richardson Number: stated.

R = !J.pgQJ/pbV3 p Density S Salinity. Subscripts are defined where !J.p Difference in density between water

they appear. entering the estuary from tributaries x,Y,Z Coordinate directions. x is along the and "ocean" water, where "ocean" refers

axis of the estuary, y is horizontal to whatever water body is outside the across the estuary, and z is vertically mouth of the estuary downward from the water surface. (J Tidal frequency (2n/T)

e, Transverse turbulent-mixing coefficient for mass

ev Vertical turbulent-mixing coefficient for mass

SIMPLE METHODS FOR PREDICTING POLLUTANT FLUSHING-THE EARLY WORK OF KETCHUM AND STOMMEL

An early concept of tidal flushing, called the tidal-prism concept, assumed that the volume of water entering the estuary during flood tide would be completely mixed

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MIXING AND DISPERSION IN ESTUARIES 109

with water already in the estuary. Ketchum (1951a,b) suggested instead that complete mixing should be assumed only within segments of length equal to the tidal excursion and gave a method for predicting the salinity in each segment. He applied his method to three quite different estuaries : Raritan River and Bay, New Jersey; Alberni Inlet, British Columbia; and Great Pond, Falmouth, Massachusetts. In each case the prediction agreed with observations. Ketchum's concept was rewritten in terms of the one-dimensional diffusion equation by Arons & Stommel (1951), who derived the longitudinal salinity distribution : S = So exp [F(I- L/x)]. in which So is salinity outside the mouth of the estuary, F is an empirical "flushing number," Lis the length of the estuary, and x is measured seaward from the head. This salinity distribution seems to be representative of many estuaries, but Arons and Stommel were not able to relate the value of F to observable channel properties.

Although Ketchum's method seemed promising, Stommel (1953) found that when it was applied to conditions in the Severn it gave a grossly incorrect result. He concluded : "In short, it does not appear likely that any good purpose can be served at present by making a priori suppositions about the turbulent mixing process." He suggested using the distribution of river water as an indicator and described a method for predicting the concentration of a pollutant undergoing a first-order decay, based on observations of the longitudinal distribution of salinity. Ketchum (1955) gave a similar method using his tidal-prism segmentation. Methodology based on Stommel's and Ketchum's work is still widely used by Sanitary Engineers. The results, found quickly and easily, often contain as much information as can be used in ecological studies.

To summarize, the concentration of a conservative pollutant assumed to be uniformly mixed throughout the cross section is C = MI/QJ at the discharge cross section. Herein, M is the mass of pollutant introduced per unit time, 1= (So- S)/So and S refers to the cross-sectional average salinity at the discharge cross section (presumably averaged over the tidal cycle, since cyclic fluctuations are not considered in the theory). The concentration seaward of the point of introduction is Clx/ I. where Ix is the local cross-sectional average of I; landward of the point of introduction the concentration is CSx/S. where Sx is the local cross-sectional average of salinity. Decaying pollutants always have concentrations less than those given for conservative pollutants; details of computational schemes for decaying pollutants are given in the referenced papers.

The restrictions on this methodology are twofold. First, it does not account for fluctuations within the tidal cycle or for longer-term changes in tide, wind, and inflow of fresh water and pollutants. Second, it assumes complete cross-sectional mixing. Lack of stratification, although not a formal requirement, is an implicit one because stratification inhibi ts uniform mixing. The remainder of this review concerns efforts to remove these restrictions. First we discuss what is known about turbulent mixing and longitudinal dispersion, and then review briefly some of the more sophisticated models that have attempted to supplant the earlier work. It is worth noting at the outset, however, that if a salinity distribution is available as a tracer, and if stratification is not too strong, the methodology developed in the early 1950s is often the most suitable for engineering use.

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110 FISCHER

TURBULENT MIXING IN ESTUARIES

Turbulent mixing is defined as the effect of random velocity variations having length scales on the order of the width of the estuary or less, and time scales much less than the duration of a tidal cycle. Turbulence is not important in distributing a tracer along the longitudinal axis, because longitudinal dispersion results primarily from motions of larger scale. Turbulence is important in distributing material locally and because the rate of longitudinal dispersion is in part controlled by the interaction of turbulent mixing with the larger-scale motions. Turbulent-mixing coefficients, E for momentum and e for mass, are defined by the equations T = -pE ou/on and if = -e oC/on, in which T is shear stress, if is mass transport per unit time and per unit area, C is concentration, and n is a coordinate direction. E and e are sometimes called the "eddy viscosity" and "eddy diffusivity," respectively.

Vertical Mixing

In homogeneous open-channel flow the vertical-mixing coefficient is approximately ev = 0.07du* (Fischer 1973), where d is depth and u* is the shear velocity. In estuaries, however, vertical mixing is usually reduced by some degree of density stratification. Moreover, the effect of stratification probably depends on how the turbulence is generated, whether at a free shear layer between bodies of water of different densities or from bottom friction (Turner 1973, p. 117). In this section we review the few experimental measurements of turbulent mixing in real estuaries, with the understanding that their results are not wholly consistent and that a probable cause is the lack of differentiation between sources of turbulence.

Francis et al (1953) measured rates of vertical mixing in the Kennebec Estuary, Maine, and reported coefficients ranging from approximately 50 to 650 cm2 sec-I. Kent & Pritchard (1959) used data obtained in 1950 to compute vertical-mixing lengths in the James Estuary. In their analysis, one composite vertical profile is used to solve for the values of three unknown coefficients by matching an "observed" and a "theoretical" curve at three points in the vertical. The result led Pritchard (1960) to propose { -3 uz2(d-z)2 ev = 8.59 x 10

d3

+9.57 X 10-3 z(h-z)H exp (- 21tZ/L)} (1 +0.276Ri)-Z (1)

dT

in which u is the mean tidal velocity (T-1 S� lui dt)/ H is the height, T the period, and L the length of surface wind waves, and Ri = g(op/oz)/p(OU/OZ)2 is the gradient Richardson Number. Harleman & Ippen (1967) stated that this expression described their observations in a tidal flume reasonably well and that it is suitable for engineering use.

Bowden (1963) computed vertical-mixing coefficients for the Mersey ranging from

1 My interpretation ; Pritchard was not specific in his definition.

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MIXING AND DISPERSION IN ESTUARIES 111

about 5 cm2 sec- l at the surface to up to 71 cm2 sec-l at mid-depth. These results were in reasonable agreement with Munk & Anderson's (1948) suggestion that ev = eo (1 + 3.33Ri)- 3/2, in which eo is the value of ev for neutral stability, expected to be approximately 500 cm2 sec-l in the Mersey. Bowden computed Ri from an average over the tidal period of hourly values of ap/az and (au/az)2, a procedure slightly different from Pritchard's ; the range of Ri in the Mersey was from approximately 0.1 near the bottom to 1.0 at mid-depth. Bowden & Gilligan (1971) also noted that ratios of mixing coefficient for momentum to mixing coefficient for mass were in reasonable agreement with Munk and Anderson's suggestion that e/E = (1 + 3.33Ri)-3/2(1 + lORi)I/2. On the other hand, Pritchard (1971, p. 21) noted that equation (1) with Ri = 0 also gives the correct magnitude of ev in the Mersey.

Fisher (1972) used the governing equations to derive three dimensionless numbers, EvPUfL/g6.pd3, EvL/Ufh2, and Uf/UO, in which L is the length of salinity intrusion. Using data from previous experiments, he obtained reasonable correlations between these three groups. It is difficult to relate his results to previous ones, however, especially as he seems to have found that ev is independent of 6.p/p. Fisher did no new experiments, but it seems clear that further laboratory and field observations are essential to resolve the uncertainties of the previous studies.

Transverse Mixing

Even in steady flows of homogeneous density the factors controlling transverse mixing are poorly understood. In straight, rectangular channels the transversemixing coefficient et � 0.15du* (Fischer 1973), but in natural channels much higher values are sometimes observed. Ward (1974) extracted mixing coefficients from some previously published field data. His results for San Francisco Bay, Cordova Bay, British Columbia, and the Gironde Estuary, France, were, respectively, as follows: et/du* = 1.00, 0.42, and 1.03. A dye release in the Delaware Estuary (Fischer 1974a) gave et = 1 .2du* � 1 m2 sec- I . Ward also described laboratory experiments in a meandering oscillatory flow, in which the time-averaged value of et/du* varied from 0.66 to 1.70.

Transverse mixing in estuaries is probably caused in part by large-scale horizontal circulations induced by shoreline irregularities and secondary circulations. It may be that when sufficient experimental data are available et in well-mixed estuaries will be found to correlate better against the width and tidal velocity than against du*. Not much is known about the effect of a stable density gradient. Taylor (1931) noted that according to meteorological observations stratification will reduce the rate of transverse mixing just as.it reduces the rate of vertical mixing, but I know of no estuarine observations to confirm this remark.

LONGITUDINAL DISPERSION IN ESTUARIES

This section discusses a number of mechanisms that act separately but concurrently to produce longitudinal dispersion in estuaries. The river that feeds fresh water to the estuary is a source of both buoyancy and momentum (although in this review we consider only "weakly flowing" rivers and neglect the input of momentum), and

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112 FISCHER

the buoyancy input drives an internal circulation. The tide interacting with the bathymetry induces a variety of residual and transient currents, and the wind drives both local currents and, in some cases, currents of relatively large extents and durations. Each of these types of current and their effects will be discussed later in this section. At the outset, however, it seems useful to attempt a synthesis of all the dispersion mechanisms, and to show that their effect is additive, by making a general decomposition of the salinity and velocity distributions. Details of the analysis and some drawings helpful for visualization have been given by Fischer (1972a).

We denote a tidal-cycle average by ( ) and a cross-sectional average by -, and write

U(x. y. z. t) = Uo(X) + UI(X. t) + us(x. y. z)+ u'(x. y. z. t)

+ neglected turbulent fluctuations (2)

in which Uo = , UI = u-uo. and Us = -uo. The salinity S is written as a similar sum. By virtue of the definitions, (Ul> = 0, iis = 0, ii' = 0, (u') = 0, and (&) = 0, where IX is any cross product of the form UOSI' etc. The salt balance for an estuary in steady state becomes

+ (u'S')}.

Net seaward "trapping" Residual shear effect + advection currents: nonsteady wind

Gravitational + pumping+wind

(3)

On the right-hand side the first term is the net seaward transport by the mean outflow, and the remaining three terms that maintain the balance are landward transport by the various mechanisms discussed in subsequent sections. Equation (3) is somewhat oversimplified, because A (the cross-sectional area) is assumed constant throughout the tidal cycle; a more general theory treating the tidal-cycle variation of A does not seem possible, because the definitions of the components of U and S require a constant cross section. The temporal variation of A in the Columbia River was treated by Hansen (1965). Writing A = is substantially larger than (A>uo. Since Qf = uo + (AI UI >, both terms are included in equation (3); other terms involving AI. all of which Hansen found to be of small magnitude in the Columbia, have been dropped.

It can also be useful to subdivide Us and u' into transverse and vertical components. The transverse distribution is defined as the variation across the estuary of the vertical average, for instance, us,(y) = d-I S� us(y. z) dz. where d is the local depth. We can also define a vertical component, usv• as the deviation from the vertical mean. Then within a cross section u.(y, z) = us,(y) +usv(y. s). Similar definitions can be made for u'. Ss. and S'. and averages of cross products of the form usrSsv etc can all be shown to be zero. Typical profiles of Us, and Usv are shown in Figure 1 . This further decomposition is made so that in reviewing different mechanisms we can discuss the effects of transverse and vertical circulations separately. By

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MIXING AND DISPERSION IN ESTUARIES 113

transverse circulations we mean the effects of veIocity distributions in the horizontal x-y plane resulting from components such as us!, and by vertical circulations we mean the effects of velocity distributions in vertical x-z planes.

Dispersion by Gravitational Circulation "Gravitational circulation" is the name given to the residual internal circulation driven by the density variation. The residual circulation is actually caused by at least four factors: the density distribution, inertial and frictional effects, the earth's rotation, and wind. There have been a number of studies, however, in which currents were predicted and observations were analyzed entirely in terms of gravitational effects. Application of any of these studies to real estuaries is limited by their failure to consider more than one factor; nevertheless, the studies compose a substantial portion of the literature on estuaries and deserve review and understanding.

Several attempts have been made to quantify the gross effects of the buoyancy input from the river. Schultz & Simmons (1957) suggested that the ratio Qf T/P, determines the degree of stratification; when the ratio is greater than one they said the estuary would normally be highly stratified. Harleman & Abraham (1966) defined an "Estuary Number" E = P, Ua/gdUf T. Fischer (1972a; see also Turner 1973, p. 158) introduced an "Estuarine Richardson Number," R = tipgQf/pbU3, which is the ratio of the input of buoyancy per unit width to the effect of the tidal current. Larger values of R should correspond to increasingly stratified estuaries. Rand E are approximately related (assuming proportionalities between P,/bdT. U. and U 0) by R = E-1(tip/ p). Harleman & Ippen (1967) state that the transition from

x

z

Figure 1 Typical profiles of u., and us •.

y

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114 FISCHER

strongly stratified to well-mixed estuaries is in the range 0.03 < E < 0.3; since for most estuaries /lpJp � 0.025, this corresponds approximately to 0.08 < R < 0.8.

In the remainder of this section we discuss field observations, laboratory observations, and analyses of gravitational circulation. We shall see that R is a helpful indicator of the state of stratification of an estuary. Unfortunately, however, we do not find any direct correlation between the magnitude of R and the importance of gravitational circulation to the total dispersion process. This is because mixing in estuaries is determined in part by the bathymetry, and no combination of purely external inputs can wholly describe the process.

OVERMIXING At the outset it seems important to mention a restriction on the magnitude of the gravitational circulation that may occur because of throttling at the estuary mouth. Any particularly narrow section (but usually the inlet or mouth) may act as a hydraulic control on an internal two-layer flow, because the composite internal Froude number cannot exceed unity (Stommel & Farmer 1953). If mixing within the estuary is vigorous the dilution of the tributary flow may be limited by inflow of saline water through the control section. Stommel and Farmer called this situation "overmixmg," and performed a simple experiment to demonstrate the phenomenon. There do not seem to have been any experiments using geometries typical of real estuaries or tidal flows, however, and all of the studies mentioned in the remainder of this section assume implicitly that no control section exists. Stommel and Farmer mentioned five estuaries they thought might be overmixed, including New York Harbor. To the writer's knowledge, however, not many engineering studies have paid attention to this concept.

FIELD OBSERVATIONS OF GRAVITATIONAL CIRCULATION Observations of gravitational circulation in partially stratified estuaries have been described by Pritchard (1952, 1954) in the James, and by Bowden (1963, 1965), Bowden & Sharif El Din (1966), and Bowden & Gilligan (1971) in the Mersey. Pritchard reported more than 5000 velocity and s�inity samples taken in the Chesapeake Bay system ; the most concentrated grouping was in the James River and was analyzed to provide a study of the salt balance. Pritchard analyzed the data on the assumption that both salinity and velocity could be assumed constant in the lateral direction; thus the salt balance for an element of height dz,length dx, and spanning the entire width b is given by

as a a b at = - ax (buS) - az

(bwS) (4)

in which u and w are velocities in the x and z directions. Pritchard divided u, w, and S into their residual values and the deviations therefrom (i.e., w = < w > + w* etc) and equation (4) averaged over the tidal cycle to give

as a a 1 a 1 a at

= - ax - az - b ax {b} - b az {b}. (5)

Pritchard called the first two terms on the right-hand side "advective" transport, and the second two terms "diffusive" transport, since he apparently thought of

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MIXING AND DISPERSION IN ESTUARIES 1 15

116 FISCHER

WES flume except that the vertical-roughness strips projected upward from the bottom instead of being attached to the sides, and a tidal generator was added at the fresh-water end so that the apparent length of the flume could be varied. In addition to the previous parameters, the Chezy roughness coefficient, mean depth, and flume length were varied; experiments were made varying each parameter systematically while holding the other parameters at base values. 2

Rigter reported only the length of salinity intrusion at the flume bottom at low tide, so his results cannot be compared exactly with the WES experiments; however, there were no obvious dissimilarities. Rigter presented a correlation of a dimensionless length of salinity intrusion vs a Froude number based on ebb-tide velocity, a Froude number based on river-discharge velocity, and the ratio of the flume length to the resonance length for the tidal wave.

Fischer (1974b), in a discussion of Rigter's paper, suggested that the length of the flume should have little to do with salinity intrusion or dispersion except insofar as it modified tidal velocities. He found that both the Delft and WES results correlated reasonably well against a modification of R in which the mean velocity was replaced by the shear velocity, R' = I:!pgQ,/pbu!. He also showed that the length of salinity intrusion in the Delft flume was consistent with his (1972a) prediction that the longitudinal-dispersion coefficient should be proportional to depth to the first power.

Daniels (1974) reported the results of eight runs in a nearly rectangular tidal flume 600 ft long and 11 ft wide. Roughness was provided only by rocks on the bottom; in contrast to the Delft and WES findings, a continuous stratification was obtained using bottom roughness alone. The depths and tidal amplitudes were in the range of the previous experiments. Although the width-to-depth ratio and the method of adding roughness were quite different from the previous experiments, the lengths of salinity intrusion were similar and the results were in reasonable agreement with Fischer's plot of the Delft and WES data using R' as the independent parameter.

The major limitation of all experimental results reported to date has been that the chaimels were rectangular. In a later section we note that transverse gravitational circulation might produce quite different results in nonrectangular channels. Thus, although Fischer's (1974b) synthesis of the salinity flume data may be adequate for rectangular channels, it may have limited application to real, nonrectangular estuaries.

ANALYTICAL STUDIES OF GRAVITATIONAL CIRCULATION Abbott (1960) studied the production of drift currents in estuaries by the action of the convective accelerations of the tidal motion and the extra pressure gradient due to the salinity gradient. He obtained the dimensionless number, K = (gdf/2(dp/dx)/(Ju�p. Small values of K were thought to indicate that drift currents were caused by the nonlinear convective terms in the equation of motion, while large values would indicate that drift

2 NOTE ADDED IN PROOF Results of some additional studies in the Delft flume have recently been given by G. Abraham, M. Karelse, and W. Lases, Proc. Congr. Int. Assoc. Hydraul. Res., 1 6th, 1975, 3: 275-83. Comparisons were made of the effect of different types of roughness. Salinity intrusion lengths were similar in aU runs, but the interface between fresh and saline water was much sharper in the runs with bottom roughness alone than in the others.

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MIXING AND DISPERSION IN ESTUARIES 117

currents were caused by the salinity gradient. Examples were : Thames, K = 0.8; Mersey, K = 7; James, K = 12. Bowden (1967), however, criticized the theoretical analysis of convective accelerations and noted that the result for the Thames is in conflict with the model results of Inglis & Allen (1957).

Abbott also showed that for large values of K the direction of the drift current just above the bed is landward or seaward depending on whether the quantity j;d(op/ox) is greater than or less than p o(/ox, where ( is the mean tide level of the water surface.

Rattray & Hansen (1962) and Hansen & Rattray (1965) gave analytical solutions for two-dimensional gravitational circulation. They divided the estuary into an outer, a central, and an inner zone ; the central zone, discussed primarily here, is the region where the salinity gradient is essentially constant. All of the solutions assume a rectangular, laterally homogeneous estuary, and use Pritchard's (1956) simplification of the equations of motion,

loP 0 ( ou) p ox = oz Eo oz

and

loP -- =g p oz '

in which P is pressure. The laterally averaged equations for water and salt are

o(bu)/ox+fJ(bw)/oz = 0 and

(6)

(7)

(8)

(9)

Hansen and Rattray thought of ex as a horizontal turbulent diffusivity. As has already been mentioned, the contribution of turbulent diffusion to the longitudinal transport of salt in an estuary is negligible. The analysis makes sense, however, if Ex is looked on as the local manifestation of all the processes that distribute salt in an estuary, other than the vertical circulation explicitly treated by the analysis.

Similarity solutions to the above equations were found, yielding dimensionless salinity and velocity distributions. An important result was that, in the absence of wind; the salinity and velocity profiles are determined by two independent parameters ; thus attempts to relate either salinity or velocity variations to only one external parameter must be suspect. The solution also shows that even though an estuary may appear to be well mixed vertically, there may still be an important gravitational circulation.

The solution includes a fourth parameter, v, calculable from the other three by an equation given in the paper. Hansen & Rattray (1966) showed that this fourth parameter is the constant in the equation

oS Ex Ox = vUfS; (10)

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118 FISCHER

ex as/ax represents the transport of salt landward by all processes other than the vertical gravitational circulation, and Vf S represents the total salt transport seaward. The difference is the salt flux due to the gravitational circulation, and hence (1- v) is the fraction of salt flux due to this circulation (Hansen and Rattray called the remainder, vVfS, the "diffusive fraction" following Pritchard's termin.ology; more exactly, it might be called the "nongravitational fraction"). v can be related to the relative salinity difference and the ratio of surface-to-mean velocities at a cross section by the relationships given in the paper; the results are shown in Figure 2, in which bS is the difference between the surface and bottom salinity and s is the surface residual velocity. Thus in theory a few observations of velocity and salinity at a cross section can be used to determine the relative amount of salt flux by gravitational circulation vs other mechanisms.

Hansen and Rattray called Figure 2 a "stratification-circulation diagram," and suggested that it would be a useful way to classify estuaries. They divided the plot into four main zones, or types, as shown in the figure. Type I estuaries are those in which there is no reversal of the residual current; thus, there is no upstream salt transport by gravitational circulation. Type II estuaries have a flow reversa:I at depth; upstream salt transfer is partly caused by gravitational circulation and partly by other mechanisms. Type III estuaries are distinguished from Type II estuaries by the dominance of gravitational circulation in accounting for over"99% of the upstream salt transfer. Hansen and Rattray divided each of these three types into two subtypes, (a) and (b), corresponding to bS/S less than and greater than 0.1,

s/Uf

HANSEN AND RATTRAY'S STRATIFICATION - CIRCULATION DIAGRAM

Figure 2 Hansen and Rattray's Stratification Circulation Diagram, with lines of constant R added.

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MIXING AND DISPERSION IN ESTUARIES 1 19

respectively. They also defined a Type IV to include salt-wedge estuaries (although the underlying analysis would not apply to this type of estuary).

In the James River, Hansen and Rattray's diagram predicted a value of v of the order of 0.01 to 0. 1, which is in agreement with Pritchard's belief that the salt balance is maintained almost entirely by the gravitational circulation. Bowden & Gilligan (1971) plotted the results of four measuring stations in the Mersey on the stratification-circulation diagram. For each station the plotted points for a variety of fresh-water discharges fell approximately on a straight line corresponding to a constant value of v (although the scatter is substantial) ; values of v computed from the observed data were within the range of values given from the stratificationcirculation diagram. They also found that the values of OS/S and '/Uj computed from Hansen and Rattray's theory were in reasonable agreement with their observations.

Finally, Hansen and Rattray suggested that it should be possible to relate the parameters of the stratification-circulation diagram, 88/8 and

120 FISCHER

with Rasmussen and Hinwood's findings; Imberger studied a laminar gravitational flow in a box cavity and found that the flow in the central region was independent of the boundary conditions but was completely determined by the heat and mass fluxes imposed.

THE TRANSVERSE GRAVITATIONAL CIRCULATION Fischer (1972a) suggested that in addition to the two-dimensional gravitational circulation in the x-z plane already analyzed, a nonrectangular cross section would contain a transverse circulation in the x-y plane. The computed circulation was landward in the deeper parts of the cross section and seaward in the shallower parts. The concept is illustrated in Figure 3; 3A. shows the salt balance as postulated by Pritchard for the James, and

r NET DOWNSTREAM TRANSPORT NEAR

\7 ________________ . THE SURFACE . ?

�- �--"'- < VERTICA L ADVECTION --t t t t AND DIFFUSION

� > :;�4W*� �ri§'�= �"" NET UPSTREAM TRANSPORT �h.«�� NEAR THE BOTTOM (A)

TRANSVERSE ADVECTION AND DIFFUSION NET UPSTREAM TRANSPORT IN THE DEEPS

( B) Figure 3 Two views of a salt balance maintained by gravitational circulation. fA) The vertical circulation envisaged by Pritchard in the James; (B) a three-dimensional circulation in a nonrectangular channel.

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MIXING AND DISPERSION IN ESTUARIES 121

3B shows what might be the important mechanisms for the salt balance in a nonrectangular channel. A marked fluid particle will circulate as shown in 3B if the time scale for vertical mixing is much less than that for transverse mixing, because in that case the particle will average the vertical velocity and move at a velocity determined by its transverse location.

For the Mersey estuary, using Bowden and Gilligan's data and estimating a value of Ilt = 0.5 m2 sec-i, Fischer found that the ratio of time scales for transverseto-vertical mixing had a value of 18; he calculated that the transverse gravitational circulation should contribute about 90% of the magnitude of the dispersion coefficient. Laboratory experiments now in progress in a flume with a wide trapezoidal channel have confirmed that the shape of the cross section has a major effect on the length of salinity intrusion.3

SUMMARY OF THE STUDIES OF GRAVITATIONAL CIRCULATION Laboratory and analytical studies to date have been limited mostly to consideration of circulation in the vertical (x-z) plane. The WES and Delft flume studies depended on the artifice of vertical bars to induce turbulence throughout the depth and reproduce the continuous density gradients observed in such estuaries as the James and the Mersey. In addition, the flumes used in both studies had small ratios of width to depth. For both reasons it would not be surprising if the results of the flume studies had little relevance to salinity intrusion in natural estuaries. The analysis of Hansen and Rattray is similarly limited by an assumption of transverse homogeneity, and also by the assumption that the residual current is entirely gravitational in origin, whereas much of the observed residual current may come from inertial

"pumping" as described in the next section. It would seem useful for researchers to direct their efforts towards a three-dimensional understanding of gravitational circulation, and towards an understanding of the interplay between the buoyancy input from the river and the momentum input from the tide in generating the distribution of currents in real estuaries.

Dispersion by Tide and Wind

TRAPPING AND PUMPING Schijf & Schonfeld (1953) proposed that salt intrusion in the Dutch estuaries is caused by what they called a "storing-basin" mechanism. They envisaged a series of basins apart from the main stream, which might represent real embayments or simply tidal shoals. The basins would fill and empty according to local tidal stage. Since in most estuaries high and low tide precede slack water, there would be a part of each tidal cycle in which the basins would both fill and empty while the flow in the main stream maintained its direction. Thus the basins would trap part of any constituent carried by the main stream, and would spread it along the stream axis. Schijf and Schonfeld estimated that the dispersion coefficients caused by this mechanism about equaled that observed in Dutch estuaries (i.e. about 102-103 m2 sec- i).

A mechanism similar to Schijf and Schonfeld's was sketched by Pritchard (1960)

3 Personal communication from Mr. J. Hamrick, a graduate student at ·the University of California, Berkeley. A doctoral dissertation is in progress.

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122 FISCHER

and analyzed by Okubo (1973). Okubo envisioned a series of "traps" along the channel sides, caused by shoreline irregularities. Exchange between the traps and the main channel was diffusive, with a time scale of k-1• The primary difference between Okubo's analysis and Schijf and Schonfeld's concept is that Okubo did not

I '& J

NET CIRCULATION PATTERN

DEL T A OUTflOW - 1800 CFS

TIDAL EXCHANGE RATIO - 0.24

SCALE • IS,OOO CF5 � 10.000 (F5 � 5000 CF5

1000 CFS

dire-(lion 01110"'-

Figure 4 The distribution of residual currents in San Francisco Bay, as computed by a two-dimensional numerical program (after Nelson & Lerseth 1972).

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MIXING AND DISPERSION IN ESTUARIES 123

specify the rise and fall of the tide as the exchange mechanism; he assumed that the exchange rate would be constant rather than synchronized with the tide.

Okubo found that for a uniform velocity of flow in the main channel of velocity u = Uo cos at and a uniform distribution of trapS along the sides having a ratio of trap volume to channel volume of r, the effective longitudinal diffusivity is given by

D' u� D = --+ ----=--'---,-

l+r s(1+r)2(1+r+a/k)' (11)

where D' is the longitudinal diffusivity in the main channel itself. Taking the Mersey as an example, Uo = 1.5 m sec- I, and a = 1.4 x 10-4 sec-1; if we assume reasonable values of trap-volume ratio r = 0.1 and characteristic exchange time k-1 = 104 sec, D = 0.9D'+360 m2 sec-l. The second term is equal to the larger of two values of effective diffusivity reported by Bowden and Gilligan. Thus it appears that the trapping mechanism alone may account for longitudinal dispersion in the Mersey.

"Pumping" refers to residual circulations induced by the interaction of the tidal wave with the bathymetry. Analytical solutions for two-dimensional circulation in the vertical plane have been given by Johns (1970) and McGregor (1971). Probably much more important, however, are the larger-scale residual-transverse circulations frequently noted in curving and braided channels (Rhodes 1950). As one example, Figure 4 shows residual circulations in San Francisco Bay, as computed by a twodimensional numerical program (Nelson & Lerseth 1972). Another example is the residual circulations observed in a hydraulic model of the Mersey by Price & Kendrick (1963, Figure 5). The initial Mersey tests had no ocean salinity, but a strong transverse circulation was observed; when ocean salinity was added it induced the gravitational landward flow along the bottom, as expected, but the transverse circulation could still be perceived superimposed on the vertical circulation. In the latter case the net transverse circulation would be partly gravitational and partly "pumped," but in the initial case, with no gradients of density present, all of the residual circulation would be "pumped."

One cause of "pumping" is deflection of tidal currents by the earth's rotation. Pritchard (1952), for instance, notes that in the Chesapeake Bay system the salinity is consistently higher along the left banks of the channels. To date there do not seem to have been any analytical studies of the magnitude of the pumped current due to the earth's rotation. The writer once set up a tidal current in a straight, rectangular channel on a rotating table; a residual transverse circulation was observed, but not measured because of the small size of the equipment. It would be useful to repeat this experiment with better equipment.

The Shear Effect

The "shear effect" refers to the dispersive action of the velocity profile ; a tracer mixed throughout the cross section will be dispersed longitudinally because it is carried faster by currents at the surface and in the center of the channel than at the bottom and near the banks. Taylor (1954) showed that in a pipe the longitudinal-

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10

1.0 1------+-----+------+-----+-1

0.1 1-------+------+--�,..c:....--_I__----___1

0.001 0.1 1.0

y Figure 5 The effect of the period of oscillation on the longitudinal-dispersion coefficient caused by the shear effect. y is the ratio of the dispersion coefficient in a flow osciJIating with period T to the dispersion coefficient in the same flow as T -- 00.

dispersion coefficient is given by D = 10. 1ru*, where r is the radius. Holley & Harleman (1965) arbitrarily replaced the pipe radius by the hydraulic radius H = r/2 and claimed that the result D = 20.2Hu* would apply to estuaries. Subsequent writers have introduced the Manning "n" to obtain D = 77niiH5/6, and have misnamed this the "Taylor result." Harleman (1971a) suggested that twice this value matches some observations in the constant-density portions of real estuaries.

Bowden (1965) found that in a current oscillating with amplitude Uo [i.e. with u = uo(y, z) sin (2ntjT)] D would be half that in an equivalent flow at a steady velocity Uo. Okubo (1967) showed, however, that the effect of the oscillation depends on the ratio of the period T to the time scale for cross-sectional mixing Te· Bowden's result was correct for Te � T; for larger values of Te Figure 5, obtained by Holley, Harleman & Fischer (1970), shows the ratio of D to its value in the same flow but with an infinitely long period of oscillation.4 The analysis leading to Figure 5 was based on an in-phase velocity distribution, u = u(y) sin at; however, unpublished numerical work has shown that introducing a velocity distribution with a cross-sectional phase lag, u = u(y) sin {a[t+k(y)]}, as is typical in real estuaries, has little effect on the result.

4 At this writing (April 1975) S. Fukuoka and P. C. Chatwin have submitted manuscripts to the Journal of Fluid Mechanics that extend Holley et aI's results, using different methods of analysis.

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MIXING AND DISPERSION IN ESTUARIES 125

In estuaries Tc has the value d2/s. for vertical mixing and b2/s, for transverse mixing. The effects of vertical shear are usually limited because the depth is small relative to the width, and the shear effect is proportional to the square of the characteristic length. The effects of transverse shear are usually limited because for transverse shear TITe is usually small. The maximum contribution of the shear effect is found when TITe = 0.7, and results in a maximum-dispersion coefficient in a tidal flow of approximately 50 m2 sec - 1 (Fischer 1969) ; in most estuaries, however, the effect is probably much less.

Mixing by Wind

It is not easy to quantify the effect of winds on estuarine mixing. Breaking wind waves obviously induce local vertical mixing, and the wind drives a local surface current that leads to a vertical circulation. Wu (1969) analyzed the enhancement of the vertical-shear effect by wind-driven current, but since this effect is small the enhancement by wind is probably unimportant. Studies of wind-induced mixing in lakes have shown that ambient stratification is a determining factor (Csanady 1975) ; presumably the same is true in estuaries, but the addition of a tidal current makes an analytical solution unlikely.

In many cases the most important effect of the wind is to drive a transverse circulation similar to the "topographic gyres" in lakes described by Csanady. When a constant wind blows over a basin of variable depth a vertical.ly averaged current is induced, flowing with the wind in the shallows and against the wind in the deeps. Some numerical computations of dispersion caused by tidal flows with wind vs without wind are given by Fischer (1970). Figure 6 shows a topographic gyre observed in a laboratory basin with a triangular cross section.

Example Computations for Northern San Francisco Bay

It is of interest to try to determine the relative importance of the various mechanisms in different estuaries. Unfortunately, no simple set of dimensionless numbers can tell us which mechanisms to look for in a given case. In bar-built estuaries, such as along the Gulf Coast of the United States, wind-driven currents probably predominate. In some deep estuaries, such as perhaps Puget Sound, Washington, gravitational currents may predominate. In this section we review some computations for northern San Francisco Bay by Fischer & Dudley (1975) that suggest that in this bay trapping and pumping are the important mechanisms.

Northern San Francisco Bay consists of a series of large bays, the largest being San Pablo and Suisun Bays, joined by narrower channels. As can be seen in Figure 4, a one-dimensional analysis can only be a first approximation. Nevertheless, some reasonable results can be obtained by assuming a uniform channel described as follows : Qf (typical summer low flow) = 3400 cfs ; A = 27,000 ft2 ; d = 25 ft, tidal amplitude = 6 ft, peak ebb-flow tidal velocity = 3.5 ft sec- I ; and rms u* = 0.25 ft sec - I . Under these conditions the observed length of salinity intrusion is approximately 70 miles, which suggests a longitudinal-dispersion coefficient in the order of 1400 ft2 sec - I.

Fischer's (1974b) graph synthesizing the results of the Delft and the WES flume experiments predicts, for the parameter values just given, a salinity intrusion on

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126 FISCHER

Figure 6 A rotational current caused by a uniform wind blowing over a basin of variable depth. The wind is blowing from the top of the picture towards the bottom at a velocity of 4.5 m sec - '. The basin is 0.3 cm deep on the left side, sloping uniformly to 5 cm deep on the right side. Four dye plumes can be seen, each released from near the bottom. The plumes show that the flow is in the form of one nearly circular rotor extending throughout the basin. (The photograph is from an experiment by Mr. R. Spigel, a graduate student at the University of California, Berkeley.)

the order of only 3 miles. Hansen and Rattray's diagram (Figure 2), using observations at four stations in San Pablo and Suisun Bays, yields values of v of approximately 0.7. Okubo's trapping result [equation (11)] , with the rather arbitrary assumption that r = 0.05 and with k- I = 22,500 sec, gives D = 1490 fe sec - I. The pumped circulation of approximately 10,000 cfs in San Pablo Bay, shown in Figure 4, clearly contributes to the salinity intrusion, as would circulations presumably generated by the strong afternoon westerly winds common in San Pablo Bay during the summer.

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MIXING AND DISPERSION IN ESTUARIES 127

These computations suggest that vertical gravitational circulation is unlikely to be responsible for the observed length of salinity intrusion, whereas the combined effects of "trapping," "pumping," and wind are entirely capable of producing the observed results. A similar conclusion would probably hold in many geometrically similar estuaries, but it is not yet possible to classify all estuaries on the basis of which mechanisms are important. Numerical models, treated in the next section, are particularly adept at describing trapping and pumping, but much less adept at describing gravitational circulation. That may be why numerical models have been used with reasonable success to study San Francisco Bay, but have sometimes been less reliable in other estuaries. It is also one reason why it is useful to begin a study of an estuary by evaluating which mechanisms are important.

THE USE OF MODELS FOR MIXING STUDIES

In this section we briefly review the fundamentals of some model techniques, making no attempt to review all the recent model studies but stressing the capabilities and limitations of existing methods.

The One-Dimensional Differential E quation

The simplest "model" for longitudinal dispersion is the equation

a a a ( ac) at

(AC) + ax « u)AC) = ax . AD ax (12)

in which C is the tidal-cycle averaged concentration of a conservative constituent and the time derivative means the change from one tidal cycle to the next. Equation (12) has also been used with A and C representing values observed at high or low slack tide. The computed values of D will, of course, depend on the definitions of A and C, but otherwise the slack-tide and tidal-cycle averaged analyses are the same. The assumptions required to obtain (12) from the three-dimensional constituent equation are discussed by Okubo (1964).

Equation (12) can also be used to describe changes within a tidal cycle, using u and C in place of (ii) and (C>. Analytical solutions to the intracycle and cycleavetaged equations and a review of observations and suggested formulas for D have been given by Harleman (1971a). Harleman claimed that the value of D to be used in intracycle numerical computations should be much smaller than that observed by cycle averaging, because the intracycle equation includes in the advective term some of the processes expressed in the dispersive term in cycle averaging (in particular, (U1Sl »). On the other hand, in a channel of constant cross section a dispersing cloud of material increases its variance at the rate 2D regardless of whether it is advected back and forth during the tidal cycle or not. Apparently the value of D to be used in a numerical model depends on exactly how the dispersive mechanisms, most importantly the trapping, are represented in the model.

There have been several attempts to derive one-dimensional models not based on the differential equation. Preddy (1954) developed a model based on the probability distribution of the motion of particles during a tidal cycle. Di Toro (1969) extended the model by increasing the generality of the probability distribution. Although his

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128 FISCHER

physical reasoning is not clear, he claimed that the distribution of marked particles within an estuary would adopt the most random possible distribution consistent with the constraints, and obtained reasonable agreement with observations by maximizing the "entropy" of the system. Fischer (1972b) described what he called a "Lagrangian" model that follows the motion of water particles ; the model was verified by observations of the movement of a dye tracer in a small estuary.

The obvious limitation of the one-dimensional models is their one-dimensionality. Complete cross-sectional mixing from a side discharge requires a time on the order of 0.4h2/et • which may often be on the order of days. For instance, in a reach of the Delaware Estuary approximately 1300 m wide Fischer (1974a) found that complete mixing of a dye discharged near one bank would require approximately 10 days. Clearly the one-dimensional model would not give a good description of the spread of this dye. In many cases, material discharged into an estuary is flushed to the ocean in less time than is required for transverse mixing, and the one-dimensional equation is irrelevant.

Multidimensional Numerical Models

Much of the current work on estuarine mixing is towards improving the computational aspects of two- and three-dimensional numerical models. Computational abilities are improving so quickly that no review can be timely for long, nor is space available herein to discuss the many recent papers. In the future, numerical models will undoubtedly be an important tool in dispersion studies, as they are now even in their relatively primitive state. It seems worth noting, however, that most of the recent models have not been adequately verified by experiments. No matter how great the computational abilities are, the writer thinks that improvement in numerical models will also require a better understanding of the underlying processes, as well as the values of the required coefficients. This is particularly true of the effects of stratification and gravitational circulation, which have not yet been described numerically in any reliable way.

Hydraulic Models

Hydraulic models of estuaries normally follow the Froude scaling law, u; = Yr, where subscript r means the ratio of model-to-prototype quantity, and Y is a vertical dimension. To allow use of economically sized models and a turbulent Reynolds number, the vertical length scale Yr is usually much greater than the horizontal length scale Lr. Distortions of 10 to 1, such as Lr = 1/1000 and Yr = 1/100, are common. The models are adjusted to give the proper representation of tidal velocity and stage by addition of vertical metal roughness strips. The vertical strips also distribute the generation of turbulence over the depth, and induce vertical currents along the faces of the strips. Harleman (1971b) suggested that the vertical strips are essential to reproduce the correct vertical distribution of salinity in partially stratified models. Collar & Mackay (1973) reported that in a model with only bottom roughness, vertical mixing was insufficient to obtain a proper salinity profile, and it was necessary to increase the vertical transport by bubbling air from the bottom. It seems likely that the strips play an empirically

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MIXING AND DISPERSION IN ESTUARIES 129

essential role in reproducing the salinity distribution in models, but their role is by no means clear.

It is generally thought that after velocities and stages have been adjusted the longitudinal salinity distribution will automatically assume that of the prototype, if one uses a salinity ratio of unity. The Delaware River Model, for instance, was shown to produce proper salinity distributions for a wide range of river inflows (US Army 1956). Not many other data have been published to support this view, however, and the writer has been told privately that in some models it has been necessary to use river discharges other than those observed in the prototype in order to obtain the correct salinity intrusion.

Scaling relationships for models are given by Harleman (1971b) and Ackers & Jaffery (1973). Usually in distorted models turbulent-mixing coefficients do not scale properly. If ev and e, can be assumed to be proportional to du*, it can be shown that distorted models should underdisperse vertically and overdisperse transversely. Crickmore (1972) observed this result in a comparison of dye-dispersion studies in the prototype and in a model of a coastal power station site near Heysham, England. The presence of the vertical metal strips undoubtedly changes the nature of the turbulence in the model, however. Fischer & Hanamura (1975) found that in flow past strips e, = O.l iua, where a is the width of the strips. Thus, it may be possible to choose the width of the strips to obtain reasonable scaling of transverse mixing, but not of vertical mixing.

Simmons & Herrmann (1970) showed a close agreement between dye-dispersion results in model and prototype in San Diego Bay. The result could have been fortuitious, since a theory of similitude of longitudinal dispersion by Fischer & Holley (1971) predicts that San Diego Bay has just the right combination of parameters to give proper similitude. Fischer and Holley's theory considers only the shear effect, however, and is based on Il, oc du* ; it seems more likely that pumping and chopping are the major mechanisms for dispersion in San Diego Bay, and that these mechanisms were adequately modeled.

In summary, there is no total agreement on how well hydraulic models simulate mixing. Advective motions, especially the pumping and chopping mechanisms, are modeled reasonably well if the model is properly calibrated ; diffusive processes generally are not. With respect to the modeling of the zone near an outfall, Crickmore remarked, "If turbulent diffusion is thought to be significant in any effluent problem, the physical model's contribution is perhaps better limited to the provision of information on mean flow paths prevailing throughout the tide." On the other hand, large-scale processes such as salinity intrusion are often properly modeled. We may conclude that hydraulic models, so long as their restrictions are borne in mind, are a useful tool for problems involving three-dimensionality, complex boundaries, or stratification, for which hardly any other tools are adequate.

Field Studies of Dye Dispersion

Verification of models requires that some tests be conducted in real estuaries. Various methods for conducting field tests, and some results from off the coasts of Great Britain, are described by Talbot (1973). The most efficient tracer seems to be

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Rhodamine WT dye, a fluorescent dye that has a particularly low rate of decay and absorption on sediments, and can be measured in dilutions of as much as 1010. Even then, very large quantities of dye are required for most tests.

Kilpatrick & Cummings (1972) describe the results of a 24.8-hr injection of 750 lb of Rhodamine WTdye into Port Royal Sound, South Carolina. Concentrations were monitored for 41 days, and the principle of superposition was used to compute expected concentrations from a continuous discharge. Their report describes current methodology for large-scale dye studies ; the degree of difficulty is exemplified by the statement in the acknowledgments that at one time or another the study required the services of "most of the personnel of the U.S. Geological Survey in South Carolina." Other reports of relatively large-scale dye-dispersion studies are those of Hetling & O'Connell (1966) in the Potomac, Simmons & Herrmann (1970) in San Diego Bay, and Nelson & Lerseth (1972) in San Francisco Bay.

CONCLUDING REMARKS

How much has been learned in the 10 years since Bowden's review of work through 1964? Important analytical contributions include Hansen and Rattray's twoparameter system of classification, and Okubo's demonstration of the importance of the trapping mechanism. New laboratory data include the Delft experiments. The capabilities of hydraulic models for pollution studies are now better understood. Computer-modeling capability has increased greatly. On the other hand, neither hydraulic nor computer models can be applied to estuaries with the generality and dependability we might like. The computer models, especiallY, have not been tested as well as they could be with modem data-gathering techniques. Our knowledge of the factors determining rates of turbulent mixing is inadequate. It is still not clear how well the two-dimensional analyses of gravitational circulation can describe real estuaries without including transverse circulations. It is not yet

possible to look at a given estuary, compute the values of some appropriate dimensionless parameters, and say with certainty which mass-transport mechl'lnisms are the most important or what factors control the intrusion of salinity. In short, the past 10 years have produced much new theory, but not enough laboratory or field verification to know how well the new theory works. The engineer, faced with a demand to predict the effects of new diking, dredging, or rerouting of river flows, is still short of a 4ependable and tested methodology. Two things are needed : firstly, continued improvements in what remains our relatively primitive understanding of salinity intrusion and dispersion in estuaries, and, secondly, a greater emphasis on laboratory and field testing designed specifically to evaluate the new theories and numerical models.

ACKNOWLEDGMENTS

I am indebted to Drs. G. Abraham, P. C. Chatwin, 1. Imberger, and E. A. Prych, each of whom studied a draft of this paper and provided many thoughtful comments. I also wish to express my thanks to Mr. G. C. Cox, who made available the original of Figure 4, and to the National Science Foundation (Grant GI-34932) and the

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Water Resources Center of the University of California, ·who have supported my recent studies of estuarine circulation. Special thanks are due Dr. H. Snyder, Director of the Water Resources Center, who approved the request that a paper on estuaries be written in a village in the mountains of New Mexico.

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