A simple advection-dispersion model for the salt distribution in linearly tapered estuaries

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi: /2006jc003840, 2007 A simple advection-dispersion model for the salt distribution in linearly tapered estuaries Peter S. Gay 1 and James O Donnell 2 Received 27 July 2006; revised 30 November 2006; accepted 9 April 2007; published 18 July [1] We present a simple advection-dispersion model for the subtidal salt distribution in estuaries with linearly varying cross-sectional area and a nonzero net salt flux. A novel analytic solution allows investigation of the dependence of the curvature and gradient of the longitudinal salinity distribution on runoff, dispersion coefficient, and channel contraction or expansion. The model predicts that in estuarine segments that contract toward the fresher boundary, the salinity gradient is stronger than in a prismatic channel. When the dispersion coefficient is large compared to the salinity intrusion lengthscale, LR A 0 (the product of segment length and net volume flux divided by cross-sectional area at the ocean boundary), the curvature of the salt concentration may be negative, a characteristic not possible in uniform channel models. The main effect of up-estuary salt flux is to strengthen the salinity gradient. The model can be extended to multiple segments in order to simulate geometrically complicated estuaries. The model is employed to estimate an effective dispersion coefficient and to describe the salinity variation in the western 53 km of Long Island Sound where the cross section of the basin varies linearly. Using 8 years of monthly observations at seven stations we find that, since the curvature of the vertically averaged salinity is negative, the model and data are consistent only if the net volume flux and salt flux are toward the fresher boundary, the East River. Combining prior estimates of the magnitudes of the fluxes and their uncertainties with the model and salinity observations using a least squares approach, we estimate the dispersion coefficient for the Western Sound as 580 m 2 /s. Citation: Gay, P. S., and J. O Donnell (2007), A simple advection-dispersion model for the salt distribution in linearly tapered estuaries, J. Geophys. Res., 112,, doi: /2006jc Introduction [2] Direct measurement of the long-term and large-scale transport of material in estuaries has proven to be very difficult. This has made the use of scalar distributions (like temperature and salinity) to make quantitative estimates of fluxes for ecosystem and pollution transport models an appealing strategy. Several monitoring programs in the maor estuaries of the United States have now accumulated over a decade of salinity, temperature and water quality observations and these archives make transport parameter estimation practical. These data also allow the evaluation of ideas about how estuaries transport salt and other material and respond to changes in external forcing (winds, freshwater and heat inputs, sea-level variation and changes in properties of external waters). In work by Gay et al. [2004], the archive described by Kaputa and Olsen [2000] for Long Island Sound (LIS) was utilized in a zero-dimensional box model which allowed dispersive exchange between the 1 Integrative Oceanography Division, Scripps Institution of Oceanography, La Jolla, California, USA. 2 Department of Marine Sciences, University of Connecticut, Groton, Connecticut, USA. Copyright 2007 by the American Geophysical Union /07/2006JC estuary and the ocean in addition to a simple representation of a two-layered exchange flow. The salinity structure internal to the estuary, though clearly well represented in the data, was not exploited since only the spatial integrals were simulated. In this paper we extend their approach and develop a model of the longitudinal structure of the monthly mean vertical and lateral average salinity in western LIS and then use it to infer the magnitudes of transport parameters. [3] Models of the salt transport in two-dimensional, straight, prismatic channels have been frequently used to understand the axial salinity distribution in estuaries [Hansen and Rattray, 1965; Chatwin, 1976; Lerczak and Geyer, 2004; Hetland and Geyer, 2004]. MacCready [2004] adopted this geometry to study the dependence of the along-channel salinity profile (its gradient and curvature) on the channel width to depth ratio, the axial variation in dispersion, and the relative importance of gravitational circulation. These models have three common assumptions: (1) the gravitational circulation is the principal mechanism of up-estuary salt transport; (2) the salinity gradient is uniform in much of the estuary (the central regime); and (3) the salinity at the mouth remains fixed so that the influence of changes in runoff is to change the axial salinity gradient. However, Monismith et al. [2002] and Banas et al. [2004] have shown that in estuaries with strong tidal currents and complicated geometry, the salt transport due to gravitational circulation, and 1of12

2 that due to rectification of tidal frequency fluctuations and meteorologically forced motions [Fischer et al., 1979], can be effectively represented by the sum of a dispersion term and a mean advective transport. [4] In some circumstances lateral (across channel) circulation can have a significant influence on the longitudinal salt transport [Fischer et al., 1979] as a result of channel curvature [Dyer, 1989; Lerczak and Geyer, 2004], lateral variability in depth [Li and O Donnell, 1997; Li et al., 1998; Uncles, 2002], the presence of headlands and inlets [Geyer and Signell, 1992] and the Coriolis effect [Valle Levinson et al., 2003]. In the model we develop here we assume some, or all, of these processes are active at unresolved space scales and timescales and parameterize the effect of their interaction with the mean circulation by a dispersion coefficient. The estuary is assumed to converge or expand over its length and this is the only spatial scale resolved. Prescribing the cross sectional area to vary linearly, either as a consequence of width or depth changes, allows an analytic solution to be derived and the effect of parameter choices to be conveniently assessed. In addition, estuaries with complicated geometries can be simulated by applying the model to segments with piecewise constant rates of expansion or contraction. [5] Exponentially tapered channel geometries have been used to model estuaries [Ianniello, 1979; Kranenburg, 1986; McCarthy, 1991; Brockway et al., 2006]. It has also been conectured that interactions among channel morphology and the hydrology result in an exponential taper [Dyer, 1997; Savenie, 2005]. Lewis and Uncles [2003] have used an exponentially tapered model to develop an analytical model to understand the dependence of the maximum salinity gradient and its location, and the curvature of the salinity profile, on the channel geometry, runoff and dispersion. Note that they investigate models with both spatially uniform dispersion and with dispersion increasing exponentially toward the mouth, but find that this latter form is inconsistent with observations. Their approach is similar in philosophy to ours, however, we prefer a linearly varying cross section since the use of a small number of such segments fits the complicated geometry of many estuaries better than a single exponential form. Multiple segment models also allow some of the effects of multiple and distributed freshwater sources to be simulated, which is not done in Savenie s [1986, 1993] model of an exponentially tapered estuary using two segments of differing taper. Our model also allows for a net salt flux through the estuary segment (which requires nonzero salinity at the up-estuary end of the segment) as well as the possibility of net volume transport at this up-estuary boundary apart from tributaries entering the segment (with negligible salinity), which is not done by either Lewis and Uncles [2003] or Savenie [1986, 1993]. This gives the ability to model more complicated salinity structure than the sigmoidal profiles presented by Savenie [1986]. If, as Savenie [2005] asserts, depth variation is negligible compared with width variations (there are many important exceptions, even in alluvial estuaries) then the effects of varying cross-sectional area can be interpreted as those of varying width. Also, for a uniform width channel there should be an interpretation of our results consistent with theoretical predictions of effects of varying depth. In a follow-up paper with application to the Chesapeake Bay, Delaware Bay and Long Island Sound we explore this issue further. [6] In section 2 of this paper, we develop the model for a single-segment linearly tapered estuarine channel with a river entering at one end and a connection to the ocean at the other, allowing for a net salt flux at both ends. Characteristics of the solution are described in section 3. The dependence of the salinity profile (i.e., the first and second derivatives of the cross-section average salinity and the salinity intrusion) on the channel taper parameter, dispersion coefficient and net salt flux in the estuary is examined. Equations are developed for the mean salinity of a segment and total salt content since these are useful in the development of flushing and residence timescales. In section 4, the value of the model is demonstrated by comparing predictions to observations determined in western LIS where the cross section varies rapidly and the curvature of the salinity field is negative. 2. Model Development [7] We model the axial salt transport as the sum of a spatially constant dispersion coefficient applied to the axial salinity gradient, and a cross-sectionally-averaged advective salt flux due to the mean flow. Using the coordinate system of Harleman [1966] and Kranenburg [1986], x = 0 is taken as the seaward (high salinity) end of the segment and x defined to increase toward the fresher boundary. Figure 1 depicts an idealized estuarine segment: The thick gray lines indicate the linear decrease of the cross-sectional area from A 0,atx =0, at the rate ax. Q s is the net flux of salt out of the up-estuary end of the segment. Q R is the volume flux of brackish water into the up-estuary end of the segment from regions of the estuary not included in the modeled segment. R is the sum of Q R and the freshwater input, Q T, from tributaries entering the segment in the region of its up-estuary boundary. The salinity at the seaward boundary is denoted s 0, and the salinity decreases in the direction of increasing x. [8] Following Hamilton and Rattray [1978], the crosssectionally integrated and time averaged (over several tidal periods) equation for conservation of salt, allowing for variation in channel cross section as well as distributed sources of freshwater along the channel, @x where A(x) is the cross-sectional area at axial location x. Note that s(x, t) is the cross-sectional average salinity times the density of seawater (s has units kg/m 3 ) and u is the cross-section average subtidal velocity in the axial direction (see Figure 1). This model represents the transport of salt resulting from the interaction of high-frequency (mainly tidal) motions with the slowly varying lateral and vertical salinity gradients by term, where K is the axial dispersion coefficient [Hunkins, 1981; Van de Kreeke and Zimmerman, 1990; Jay et al., 1997]. [9] We take the cross-sectional area of the region of the estuary between x = 0 and x = L to be Ax ðþ¼a 0 ax; ð1þ ð2þ 2of12

3 Figure 1. Sketch of the model geometry. The ocean is to the left of x = 0 and the low-salinity end is to the right. The segment is of length L. Thick gray lines indicate the linear decrease of the cross-sectional area from A 0 at x =0,atthe rate ax. Q s is the net flux of salt out of the up-estuary end of the segment. Q R is the volume flux of brackish water into the up-estuary end of the segment. R is the sum of Q R and the freshwater input, Q T, from tributaries entering the segment at its up-estuary boundary. The salinity at the seaward boundary is denoted s 0, and the salinity decreases in the direction of increasing x. where A 0 is the cross-sectional area at the seaward boundary of the segment, x = 0. Note that this formulation implies that a positive value of the channel taper parameter, a, represents a decrease in the cross-sectional area of the channel in the up-estuary direction as is sketched in Figure 1. Of course, negative values of a are possible. The value of a must be restricted such that A 0 > al, where L is the length of the segment, to avoid negative cross-sectional areas. [10] Integrating equation (1) from an axial location x to the up-estuary boundary of the segment, x = L, Z L x Asdx þ ðausþ L ðausþ x x : ð3þ [11] The first term on the left is the rate of change of the total amount of salt within the region of the estuary between x and L. The net volume flux in the estuary is Au. Denoting R as the sum of the volume flux of brackish water (Q R in Figure 1) and the volume flux from freshwater tributaries (Q T ) into the segment at the up-estuary end of the segment (x = L), equation Z L x L ðrsþ L ¼ ðrsþ x : ð4þ [12] Note that we have adopted the convention that the net volume flux through the estuary segment is in the direction of decreasing x (toward the ocean) when R >0. However, there are regions of estuaries such as in western LIS in which the mean volume transport is thought to be landward and this requires that R <0. [13] Equation (4) simply expresses that the rate of change of the amount of salt up-estuary of the section at x is determined by the difference in the salt flux at the upstream boundary, L, and that crossing x. were both zero at L, then the rate of change of the amount of salt stored upstream of x is determined by the flux at x. Note that in most situations < 0 so that the net up-estuary salt flux is the result of the competition between dispersion and advection. When these balance, or sum to the net salt flux, a steady state pertains. [14] Since we intend to apply the model to estuaries, and segments of estuaries, where there may be a nonzero salt flux at L we write the terms on the left of equation (4) as Q s, so that in a steady state equation (4) becomes Q s ¼ ðrsþ x x where Q s represents the net flux of salt through the estuary segment from the ocean, and the dispersion coefficient, K,is taken as a constant throughout the segment. Salt storage within a segment could easily be included in equation (5) by a simple redefinition of Q s. [15] Adopting the dimensionless variables ex x L and es s s 0, where s 0 is, for example, the salinity at the seaward boundary of the segment, and defining the parameters fq s Q s Rs 0, ea al A 0 and ~K KA 0 LR, then equation (2) and equation (5) imply ~Q s ¼ ~s ð1 ~a~x Þ~K d~s d~x : [16] The first term on the right of equation (6) represents advection by the mean flow and, as a result of the continuity constraint, the coefficient of ~s is independent of ~x. The second term on the right represents the effects of dispersion and the derivative of ~s is multiplied by the cross-sectional area which varies with ~x. Consequently, the ratio of the two terms therefore varies along the channel at a rate determined by ~a~k. It is also important to appreciate that the choice of scales requires the existence of a mean advective flux (R 6¼ 0), a negative salinity < 0) which implies an input of freshwater, and dispersion (K > 0). These are, obviously, essential features of estuaries [Fischer, 1976]. In addition, the scaled dispersion is positive, ~K > 0, except in the case that the net volume transport is into the estuary at the seaward boundary so that R < 0. There is evidence that this is the case in western LIS [Blumberg and Pritchard, 1997]. [17] The scales we have adopted for the parameters Q s, a, and K have straightforward physical interpretations. Rs 0,used to normalize Q s is the advective salt flux at the seaward boundary. A 0 is the cross-sectional area at the seaward boundary so ~a is simply the fractional convergence or expansion of the cross-sectional area over the length of the estuary segment. Recognizing KA 0 R = L D as the spatial scale determined by the advection and dispersion coefficient magnitudes for a constant cross-section channel, or the salinityintrusion length of Zimmerman [1988], then ~K = L D /L is simply the ratio of the advection-dispersion lengthscale to the scale imposed by the geometry. ~K can also be interpreted as the ratio of an advective timescale T A LA 0 R [Hetland ð5þ ð6þ 3of12

4 Figure 2. Examples of the model predicted crosssectional average salt concentrations distributions, ~s(~x), for net salt flux values (a) ~Q s = 0 and (b) ~Q s = 0.7. In both graphs the black lines show three solutions with a normalized dispersion coefficient of ~K x = 0.55 and crosssectional area convergence rates of ~a = 0.9 (dash-dotted line) 0 (solid line) and 0.9 (dashed line). Note that smaller values of ~a result in larger values of ~s for all ~x and all values of other parameters. The gray families of curves show solutions for ~K x = 5.55 and the same three of values of ~a. and Geyer, 2004] and a dispersive timescale T D L2 K [Geyer and Signell, 1992], i.e., ~K T A T D (the Peclet number [Zimmerman, 1988]). 3. Steady State Solution and Its Properties [18] Using separation of variables and applying the boundary condition ~s = ~s 0 at ~x = 0, it can be shown that the linear inhomogeneous ordinary differential equation, equation (6), has the solution ~s ðþ¼~s ~x 0 ð1 ~a~x Þ 1=~a ~K 1 þ ~Q s ~Q s : ð7þ [19] If the salinity at the seaward end of the segment, s 0,is used to normalize s, then the boundary condition is ~s(~x =0)= 1, but we retain a more general formulation so that the equation can be normalized by some other value. It is straightforward to demonstrate using lim ~a!0 (1 ~a~x) 1/~a ~K = exp( ~x/~k), that the uniform channel case (~a! 0) is consistent with the exponential solution of Zimmerman [1988]. [20] Figure 2a shows examples of the structure of ~s(~x) for ~s 0 =1,~K = 0.55 and ~Q s = 0 (no net salt flux through the domain) using thin black lines. The thin solid line represents the solution for a uniform channel, a = 0, which is simply an exponential decay. For an estuary segment in which the cross section decreases toward the freshwater source with ~a = 0.9, the solution is shown by the thin dashed line. Near the mouth (~x = 0), these two solutions are similar. However, for larger values of x, the narrowing of the channel and the consequent reduction in the magnitude of the coefficient of the axial gradient in equation (6) requires that the gradient be more negative in order for the up-estuary dispersive flux to balance the advective transport. For comparison the predicted salt concentration in an estuary with a cross section that increases in ~x with ~a = 0.9 is shown by the thin dashdotted line in Figure 2a. Near ~x = 0 the solution is again very similar to the ~a = 0 case (thin black solid line), but for larger values of ~x the axial gradient is substantially less negative since the channel expansion increases the coefficient of the first derivative in equation (6). [21] The general character of the model solutions are quite different for large values of ~K. The thick gray lines in Figure 2a show ~s(~x) computed for ~K = 5.55 and ~Q s =0. The solid line represents the uniform channel case for reference. Though this solution appears to be linear, it is really a declining exponential, but since the domain is only 20% of the decay scale (~K), the exponential character of the solution cannot be discerned. The dashed gray line in Figure 2a shows ~s(~x) computed for ~K = 5.55 and ~Q s = 0 in a converging estuary segment with ~a = 0.9. Comparison of this curve with the solid gray line demonstrates the effect of the reduction of the channel cross section with ~x for large ~K. The salinity gradient at all ~x is more negative than in the uniform channel case but for large ~K the axial gradient becomes more negative as ~x increases while the opposite is the case for smaller ~K. The negative curvature in this solution for ~s makes it qualitatively quite different from those shown by the black lines. The gray dash-dotted line in Figure 2a shows the solution for the same values of ~K and ~Q s, but for ~a = 0.9, i.e., increasing cross section with ~x.as expected, the axial gradients are reduced relative to the prismatic channel and the curvature of ~s is positive. The curvature must vanish somewhere between ~a = 0.9 and ~a = 0.9. We determine this point using equation (9) below. [22] To illustrate the effect of the net salt flux, Q s, we show in Figure 2b the same range of solutions as in Figure 2a, but with ~Q s = 0.7. This value is chosen since it is in the range of values appropriate for a simulation of segments of estuaries like LIS (as a whole) through which there is a net salt flux toward fresher waters. For western LIS, into which there is negligible runoff and in which the net volume flux and salt flux are both toward fresher waters, ~Q s < 0. The general characteristics of the solution are similar to the ~Q s = 0 case but the rate of decline of ~s with axial distance ~x is significantly larger, especially for small ~K. Note that the solutions shown in Figure 2b with ~K = 0.55 (black lines) show ~s decaying to zero within the estuary segment while the gradient remains nonzero. This is a physically unrealizable situation. It arises because the length scale was chosen to represent the geometry and not the transport parameters. A simple redefinition of the scales could eliminate this artifact. [23] Consideration of the variety of solutions shown in Figure 2a and 2b shows that the introduction of a simple linear variation in channel cross section (and a geometric length scale) to the advection-dispersion model of the salt distribution in an estuary leads to a much richer set of structures than the exponential decrease that occurs in a uniform, prismatic channel. To further illustrate how the gradient and curvature of the sectionally averaged axial 4of12

5 Figure 3. Nondimensional gradient of the salinity profile as a function of taper (~a) and dispersion (~K x ) parameter at (a and b) ~x = 0.2 and (c and d) ~x = 0.9. The net salt flux parameter values are ~Q s =0in Figures 3a and 3c and ~Q s = 0.7 in Figures 3b and 3d. salinity profile are determined by the parameters, we differentiate equation (7) to obtain the gradient and ¼ ~s 0 1 þ ~Q s ð1 ~a~x Þ ð 1 ~a ~K Þ=~a~K ; ð8þ 2 2 ¼ 1 ~a ~s 0 1 þ ~Q s ~K ð1 ~a~x Þ ð 1 2~a ~K Þ=~a~K : ð9þ ~K 2 [24] It is clear from equation (8) that the salinity gradient is always negative if ~K > 0 and, for ~a~k = 1, is constant throughout the estuary. In addition, the first factor on the right causes the magnitude of the salinity gradient to increase with the net salt flux, ~Q s. Equation (9) shows that the sign of the curvature of the salinity distribution is determined by the sign of the first factor and the model suggests that negative curvature can only occur in estuary segments that converge (~a > 0) and have ~a ~K > 1. The curvature vanishes for ~a = 1 ~K, which is 0.18 for ~K = 5.55 as in Figure 2a. Recall that it is necessary for ~a 1 to avoid negative cross-sectional areas, so that for 0 ~K 1, there cannot be negative curvature. [25] The dependence of the salinity gradient on ~a and ~K is complicated since they appear in equation (9) both separately and as a product. To qualitatively illustrate their effects, Figure 3 displays the on ~K and ~a at ~x = 0.2 (Figures 3a and 3b) and ~x = 0.8 (Figures 3c and 3d), for cases in which the salt flux through the estuary segment is ~Q s = 0 (Figures 3a and 3c) and ~Q s = 0.7 (Figures 3b and 3d). Near the ocean, for ~x =0.2,the salinity gradient decreases monotonically as ~K increases for all ~a. Nearer the river (~x = 0.8), the gradient appears to be much more sensitive to the value of ~a than in higher-salinity locations. At lower salinities, the gradient is most sensitive to the value of ~K when ~K < 1, i.e., when the salinityintrusion length, L D, is less than the segment length Salinity Intrusion [26] Another useful model prediction is the salinity intrusion distance, or the distance from the mouth to a particular isohaline, s ~ I. This can be expressed using equation (7) as 0 ~x I ¼ 1 s ~I þ ~s 0 Q ~ s ~a ~s 0 1 þ ~Q s! ~a ~K 1 A: ð10þ [27] For a prismatic channel, this is more conveniently evaluated as ~x I (~a! 0) = ~Kln( ~s 0ð1þ~Q s Þ ). In Figure 4 we ~s I þ~s 0 ~Q s display the dependence of ~x I on ~K and ~a for the isohalines s ~ I = 2/30 (Figures 4a and 4b) and ~s I = 0.5 (Figures 4c and 4d). For simplicity we set ~s 0 = 1, and contrast the ~Q s =0 (Figures 4a and 4c) and ~Q s = 0.7 (Figures 4b and 4d) cases. Note that equation (10) will predict ~x I > 1 for some isohalines and choices of ~Q s, ~K and ~a. Since we restrict the model domain to 0 ~x 1, ~x I = 1 is the maximum contour value shown in Figure 4. For combinations of ~K and ~a to the right and below the ~x I = 1 contour for a particular isohaline, the salinity at ~x = 1 is greater than the isohaline value, i.e., ~s(x =1)>~s I and the isohaline value is not found within the modeled reach of the estuary. 5of12

6 Figure 4. Nondimensional salinity intrusion to (top) s/s 0 = 2/30 and (bottom) s/s 0 = 0.5 as a function of taper (~a = al ) and dispersion (~K A 0 x = K xa 0 LR ) parameters for ~Q s = 0 (left frames) and for ~Q s = 0.7 (right frames). ~Q s = Q s Rs 0.Q s is the net flux of salt out of the up-estuary end of the segment which is zero for estuaries like CB and DB in which the salinity goes to zero at the up-estuary end. [28] In all four frames in Figure 4 it is clear that the intrusion distance ~x I increases as ~K increases for all values of ~a. This is consistent with the uniform channel results of Zimmerman [1988]. Figure 4 also shows that the up-estuary location of an isohaline is less sensitive to the rate of channel cross-section variation when ~a < 0 (i.e., the cross section increases with ~x). Note, however, that in this range of ~a the isohaline intrusion distance varies much more rapidly with ~K than in a converging channel. For example, Figure 4a shows that the 2/30 isohaline is located at ~x I 0.4 for ~K = 0.1 and ~a = 1.0. In this expanding channel an increase of the dispersion coefficient to ~K = 0.2 causes the isohaline to move to ~x I 1. In contrast, for a converging channel with ~a = 1.0 the same adustment of ~K only causes ~x I to increase from 0.3 to 0.4. The position of the 0.5 isohaline shown in Figure 4c is further up the estuary but has the same increased sensitivity to ~K at smaller ~a. [29] The effect of the geometry can be understood by consideration of equation (6) which requires that the difference in the advective and dispersive salt fluxes is a constant which in most estuaries is zero. When the coefficient of the derivative in equation (6), (1 ~a~x)~k, is larger, a less negative salinity gradient can balance the advective flux term. Obviously, when the scaled salinity is fixed at unity at ~x = 0 and the magnitude of the gradient is reduced, all isohalines are located further up the estuary. The effect of increasing ~K is to reduce the along channel salinity gradient and move isohalines up the channel. Since (1 ~a~x) multiplies the dispersion coefficient, decreasing ~a effectively amplifies the effect of increasing ~K. This interpretation of the model solutions suggests that changes in dispersion coefficient, resulting from seasonal wind or runoff variations for example, should be more difficult to detect in observations of the salinity intrusion in converging segments of estuarine channels than in prismatic or diverging segments where the sensitivity to ~K is amplified. [30] The influence of a net salt flux on the intrusion of salt is demonstrated in Figures 4b and 4d for ~Q s = 0.7. A positive net flux can arise when the dispersion of salt from the ocean overwhelms the export by advection. For particular values of ~K and ~a this requires d~s d~x be more negative and the intrusion of isohalines is consequently reduced. For example, Figure 4a shows that for ~a = 1 and ~K = 1, the 2/30 isohaline intrusion distance is approximately ~x I 0.9 when ~Q s = 0. When the net salt flux is increased to ~Q s = 0.7, Figure 4b shows that the intrusion is reduced to ~x I Mean Salt Concentration [31] The total amount of salt contained in the estuary or segment is useful in evaluating residence times for salt or freshwater, and therefore we note here that the total amount of salt in the estuary segment, ~s = R 1 0 (1 ~a~x)~s(~x) d~x, is 1 þ ~Q s ~S ¼ ~s ~a 1=~K þ 2~a ð Þ 2þ1=~a ~K ~Q s 1 ~a! : 2 ð11þ Though this expression appears to be badly behaved as ~a approaches 0 and 1/2~K, careful examination of the limits shows that ~s varies smoothly with ~a and is consistent with the result obtained for a prismatic channel. 6of12

7 Figure 5. Dependence of the area mean salt concentration ~s in an estuary segment as a function of ~a and ~K for (a) ~Q = 0 and (b) ~Q = 0.7. The dependence of ~s on~q s for (c) ~a = 0.5 (solid lines) and 0.5 (dashed lines) and ~K = 0.7 (black lines) and 3.5 (gray lines) and for (d) ~K = 0.7 (black lines) and 3.5 (gray lines). [32] In the geometry we have adopted, the volume of a segment, normalized by A 0 L, depends upon ~a as ~V ¼ 1 ~a 2 : ð12þ [33] The mean salt concentration is therefore 1 þ ~Q h s h~si ¼ 1 ð1 ~a 1 ~a 2 1= ~K þ 2~a Þ 1=~a ~Kþ2 i ~Q s : ð13þ [34] The dependence of the mean salinity in the estuary, h~si, on~a and ~K is shown in Figures 5a and 5b for ~Q s =0 and ~Q s = 0.7 respectively.h~si appears to be relatively insensitive to ~a in both examples. However, it is also clear that larger values of ~K result in a substantial increase in the average salt content in the estuary for both values of ~Q s. Larger ~K values result from an increased up-estuary dispersion coefficient (larger K) or by reducing the dilution of the ocean water by decreasing the volume flux (smaller R). [35] Figure 5c shows the dependence of h~si on ~Q s for ~a = 0.5 (solid lines) and 0.5 (dashed lines), and ~K = 0.7 (black lines) and 3.5 (gray lines). The pairs of black and gray lines lie close together because h~si is relatively insensitive to ~a. As is evident in equation (13), for all combinations of ~a and ~K, h~si decreases linearly with increasing ~Q s and this is demonstrated in Figure 5c. Note also that estuaries with larger ~K contain more salt. [36] In the absence of net evaporation, a mean salinity of the estuary that exceeds that at the ocean boundary is physically unrealistic. Inspection of equation (13) reveals that ~Q s must therefore be restricted such that ~Q s > 1 inthe case that ~K >0or,if~K < 0, then ~Q s < 1. In essence, this means that the net salt flux should not exceed the product of the mean volume flux and the boundary salinity. Figure 5d shows that the mean salinity of the estuary segment decreases as the salt flux decreases (becomes more negative) for all combinations of ~a = 0.5 (solid lines) and 0.5 (dashed lines), and ~K = 0.7 (black lines) and 3.5 (gray lines). This is a consequence of the influence of the magnitude of the net salt flux on the spatial gradient. As ~Q s increases (becomes less negative), the gradient of ~s has to become more negative which, assuming the salinity at the mouth stays fixed, reduces the average salt concentration in the segment. As in Figure 5c, the channel geometry has only a weak influence on h~si and the salinity of the estuary is higher when the magnitude of the dispersion coefficient is greater Parameter Estimation [37] A practical application of the model is the empirical estimation of model parameters from observations of the salinity distribution in an estuary for application in biogeochemical and ecological models. Use of equation (13) would allow use of off-axis data in evaluating hsi, but is complicated by the appearance of ~K in both the exponent and the denominator. It is possible to use equation (7) to estimate both ~K and ~Q s, however, the problem is nonlinear and the inverse methods are complicated. By rearranging equation (7) as ln ~s þ ~s 0 Q ~ s ¼ 1 ln 1 ~a~x ~s 0 þ ~s 0 ~Q s ~a~k ð Þ ð14þ it is clear that if an independent estimate of ~Q s and its uncertainty is available from other studies (such as Gay et 7of12

8 Figure 6. Western Long Island Sound. Stations with letters are sampled twice monthly by the CT-DEP. Stations 3 and 2 are discussed in the text. At the lower left is New York City and the East River and the region to the right of station F3 is the Stratford Shoal and Middle Ground. The Housatonic River and New Haven harbor can be seen at the upper right corner of the figure. al. [2004] for LIS), or if we assume that ~Q s = 0 as is appropriate for a closed estuary, then the dispersion coefficient can be determined for specific ~a by conventional regression. Note that for ~a = 0 the exponential solution (see comment following equation (7)) should be used to derive an appropriate analog of equation (14). 4. Application to Western LIS [38] In most estuaries the lateral and vertical average time mean velocity at the seaward boundary is seaward and the salt flux is landward, i.e., both R and Q s are nonnegative, and theories like those of Uncles [2002], Brockway et al. [2006], and MacCready [2004] apply to this situation. However, in western LIS (Figure 6) there is thought to be a mean volume transport toward the East River implying R < 0 throughout western LIS, at the same time that the East River is the dominant source of freshwater west of the Stratford Shoal due to an exchange flow [Blumberg and Pritchard, 1997]. This requires that both the dimensionless parameters ~K and ~Q s be negative. Salinity measurements throughout the water column in western LIS have been collected regularly by the Connecticut Department of Environmental Protection (CT-DEP) since 1991 [see Kaputa and Olsen, 2000]. The availability of this data set and the unusual salt transport characteristics of the area make it an ideal location to test the validity and utility of our model. Note that the orientation of Figure 6, with salinity increasing toward the right, is opposite that shown in Figure 1. [39] We define the origin of the coordinate system, x =0, to be slightly west of Stratford Shoal at station F3 (see Figure 6). The western boundary, x = L, is taken as station A4 which is approximately 53 km to the west. Additional sources of freshwater to this segment of LIS are thought to be small and are neglected. Using bathymetry observations to compute the cross sectional area we estimate ~a = 0.9 (see Figure 7). The cross-sectional area at station F3 is estimated to be 0.47 km 2. The dimensional taper is then a = 8.0 m 2 /m. Blumberg and Pritchard [1997] combined an analysis of current meter observations with a numerical circulation model of the East River to estimate the volume transport through the East River in upper (toward LIS) and lower (toward New York Harbor) layers with uncertainties of 100 m 3 /s. The net transport, which we use here, p ffiffi is their difference, R = 310 m 3 /s, with uncertainty 2 (100) = 140 m 3 /s. Gay et al. [2004] combined the Blumberg and Pritchard [1997] estimate and an inverse model of the LIS salt budget to estimate the salt export through the East River as Q s = 11,000 ± 4,000 kg/s. [40] We use seven of the CT-DEP stations (A4, B3, C1, C2, D3, E1 and F3; see Figure 8) in our analysis. We vertically averaged the profiles to estimate the cross-sectional 8of12

9 Figure 7. Normalized cross-sectional area as function of axial distance from station F3 in Western LIS (solid line) and linear approximation (dashed line). average salt concentration, s, at each station location, x, for each month in the time period from January 1995 to December Since the lateral and tidal frequency fluctuations in salt concentration are not resolved by the CT-DEP data, the uncertainty associated with s must be inferred using other observations. There have been few near-synoptic across-sound sections in the area, however, a short series of transects were obtained by J. O Donnell et al. (manuscript in preparation, 2007) near station C1. The four stations labeled 2, 2a, 2b, and 3 in Figure 6 were occupied between 15 and 17 times during the 42 hours period following 18:00 on 24 July This cruise track produced 16 complete sections with the time of samples differing by less than 2 hours. We vertically and laterally averaged these observations to produced a time series of cross-sectional mean salt concentration, s. The variance of Figure 8. Stars and error bars denote time- and depth-averaged salinity, and standard deviation of the mean, at axial CT-DEP stations in western LIS (F3 at left and A4 at right). Thick line denotes depthaveraged salinity predicted using the model of western LIS and the fitted dispersion coefficient of 580 m 2 /s. The upper and lower (thinner) lines are based on adding or subtracting the uncertainty of the dispersion coefficient, 30 m 2 /s. 9of12

10 the series was and this provides an estimate of the error associated with not resolving tidal variability. To estimate the additional uncertainty introduced by lateral structure we computed the variance of s s 2a, the difference between the cross-section mean and the vertical mean at station 2a (the station closest to C2). We obtained and take the square root of the sum of these, 0.13, as our estimate of the uncertainty in the measurements s. Note that it is dominated by the errors introduced by the failure of the surveys to resolve the tidal frequency variability. [41] We use a least squares approach to estimate K and evaluate whether the model and observations are consistent [Gay et al., 2004; Wunsch, 1996; Press et al., 1987]. With s 0 taken as the value at the seaward boundary of the segment (here station F3), then equation (14) can be rewritten in dimensional form as R ln 1 axi A 0 K i; ¼ ; ð15þ aln R s i; þ Q s = R s 0; þ Q s where the subscript i represents the stations E1, D3, C2, C1, B3, and A4, and indicates the monthly averages of surveys undertaken between 1995 and The expected uncertainty in the dispersion coefficient estimate resulting from the uncertainty in the parameters and observations on the right of equation (15) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is estimated for each station and each month by s Ki, = Pk s k K i;, where we have used s p, with p ={R, Q s,s, s 0 }, to represent the uncertainty in the quantity p, or s 2 K i; 2 32 R s i; þ Q s ln 6 R s 0; þ Q s K 2 i; 0 ln R 1 s i; þ Q s R s 0; þ Q s s 0; s i; ¼ B þ R R s 0; þ Q s R s i; þ Q s A þ s 2 R s 0; þ Q s R s i; þ Q s R 2 þ s 2 s R s i; þ Q ; s 2 s 2 R R 2 Q s þ s 2 s 0 R s 0; þ Q s ð16þ [42] Note that the uncertainties in the geometric parameters are neglected. In evaluating equation (16) we use s s = s s0 = 0.13, s Qs = 4000 kg/s and s R = 140 m 3 /s, and the terms in brackets are evaluated using the unweighted mean salinity at each station (i.e., averaged over all months for each station i). [43] The dispersion coefficient for western LIS is computed as a weighted mean of all the elements of K i,, using the notation of Wunsch [1996], as K ¼ E T R 1 1E nn E T R 1 nn y; ð17þ where E is a column vector of ones with I * J rows; R nn is a diagonal matrix with elements proportional to the expected variance s 2 K i, ; and y is a column vector representing the elements of K i, as y =[K 1,1, K 1,2... K 2,1, K 2,2... K 6,8*12 ] T. If our model, the Blumberg and Pritchard [1997] East River volume flux estimate, and the Gay et al. [2004] salt flux estimate are consistent with the observations and associated P error estimates, then the cost function c 2 = I P J i¼1 ¼1 (K i, K) 2 2 /s Ki, should have a Chi-square distribution with degrees of freedom, n = I J 1 [Wunsch, 1996; Press et al., 1987]. When n is large (^20) we expect that c 2 n and the standard deviation s c 2 p ffiffiffiffiffi 2n. An autocorrelation of the six time series represented by the observations, y (for each of the six stations west of F3), suggests that the degrees of freedom should be less than half of I J 1. In addition, owing to a variety of sampling problems, salinity profiles at all stations were not obtained p ffiffiffiffiffi on every monthly survey. We estimate n 230 and 2n = 15. [44] Evaluation of equation (17) results in a long-term mean dispersion coefficient for western LIS of K fit = 580 ± 2 30 m 2 /s with c fit = 228 compared with the expected value of n = 230 ± 15. This appears to be an excellent fit. Figure 8 supports this conclusion, and shows the time-mean depth-averaged salinity structure predicted by equation (7) using K = K fit, R = 310 m 3 /s, Q s = 11,000 kg/s and ~s 0 = 1. The vertical lines are located at the scaled station positions and show the time mean, and the standard deviation, of the vertical average salinity observations scaled by the value at station F3. The agreement of the model and data is clearly excellent. The estimate of K is in the middle of the range expected in large estuaries [Fischer, 1976; Thorpe, 2005] and the standard deviation of K fit is quite small. [45] In addition to parameter estimation, the model can be used to evaluate whether other proposed East River flux estimates are consistent with the large archive of salinity observations. For example, Geyer and Chant [2006] recently questioned the direction, as well as the magnitude, of the mean volume transport through the East River. If we hypothesize that the volume flux is the negative of the Blumberg and Pritchard [1997] estimate (R = +310 m 3 /s) while the salt flux is consistent with the Gay et al. [2004] estimate (11,000 kg/s) and the salinity observations, then we obtain K fit = 4290 ± 30 m 2 /s with c 2 fit = 11,000. This is a very high value for the dispersion coefficient and a very poor description of the salinity field. Clearly, this is unlikely and the hypothesis must be reected. 5. Discussion and Conclusion [46] We have developed a simple model for the longitudinal structure of the salinity field in tidal estuaries. The principal novelties of the model are that the channel cross section may vary linearly along the channel and there may be a net salt flux through the estuary. The model has a solution of analytic form so that the sensitivity of important estuary characteristics (e.g., the derivatives of the salinity distribution, the mean salt content and the intrusion of isohalines) to parameter choices can be conveniently investigated. We find that convergence or divergence of the channel can dramatically change the salt distribution. Figure 2 shows that when an estuary narrows rapidly with distance from the ocean (~a = 0.9), and dispersion is relatively strong ( ~K = 5.55), instead of the exponential 10 of 12

11 decay of salinity that is predicted in a prismatic channel, the curvature of salinity is negative as is observed in western LIS. [47] A central assumption of our model is that the dispersion coefficient, ~K, is a constant. Fischer et al. [1979] suggested that it should scale as the width of the channel since it sets the maximum scale of lateral shear in the longitudinal currents. However, observational evidence to support this prediction is sparse. Recently, Austin [2004] noted that in Chesapeake Bay the dispersion coefficient appears to decrease toward the mouth in the wider, southern region of the Bay. He suggests that this is due to the decreased subtidal cross-section average velocities in the vicinity of the mouth. It is clear from Fischer et al. s [1979] development that the lateral scale of the circulation should determine the magnitude of K. However, in nature there are multiple lateral length scales set by the dynamics, for example the Rossby radius, the bathymetry and the coastal geometry. Lewis and Uncles [2003] found that allowing K to increase toward the mouth in their exponentially tapered model did not explain the observed salinity structure as well as a uniform K. Since there is no firm theoretical reason to adopt a variable K model and to minimize complexity, we choose K equal to a constant and rely on the statistical evaluation of the model performance to test whether this assumption (as well as other assumptions made in our model) can be shown to be inconsistent with observations. [48] When the model is applied to western LIS in conunction with the data of Kaputa and Olsen [2000], the East River volume flux estimates of Blumberg and Pritchard [1997] and the salt flux estimates of Gay et al. [2004], we find that the model and data are consistent with a dispersion coefficient of 580 ± 30 m 2 /s. There has been some recent speculation about the direction of the transport in the East River [Geyer and Chant, 2006]. This model demonstrates that the negative curvature in the observed salinity distribution in western LIS is only possible when the salt and volume fluxes are toward New York Harbor. Sensitivity experiments also demonstrate that positive fluxes yield model salinity distributions that are statistically inconsistent with the observations. [49] The approach developed in this paper facilitates estimation of timescales relevant to various ecosystem and engineering concerns in estuaries, and should prove a useful tool to augment and extend work by Dettman [2001] and others involved in the management of estuaries. Budgets of oxygen, nitrogen, carbon, etc., are complicated by distributions of sources and sinks that differ from those of salt and fresh water. The segmented model approach, using a dispersion coefficient characteristic of each segment based on the salt budget analysis, provides a foundation for modeling these more complicated tracers. The combination of linearly tapered, uniform-dispersion-coefficient segments allows the development of models of nutrient distributions in geometrically complicated estuaries and coastal regions with attached estuaries. Notation a decrease in cross-sectional area with axial distance form the mouth. A cross-sectional area. A 0 cross-sectional area at seaward boundary of a segment. i station index. month index. K axial dispersion coefficient. K fit long-term mean dispersion coefficient fit to observations. L length of segment. Q R net volume flux of brackish water into the up-estuary boundary. Qs net salt flux out of the up-estuary boundary. Q T freshwater input from tributaries at the up-estuary boundary. R net volume flux out of the seaward boundary. R nn row-scaling matrix. s cross-section averaged salt concentration. s I isohaline defining the salinity intrusion. s 0 salinity at the seaward end of a segment. S total amount of salt in a segment. hsi mean salinity of a segment. t time. T A advective timescale. T D dispersive timescale. u cross-section average velocity in the axial direction. V volume of a segment. x axial coordinate measured up-estuary from the seaward end of a segment. x I salinity intrusion distance to a given isohaline. s p uncertainty in the quantity p, with p ={R, Q s,s, s 0 }. [50] Acknowledgments. We are grateful to the NOAA, Coastal Services Center, for their support of our work through the LISICOS program and to the Connecticut Department of Environmental Protection for providing us with their data. We also thank C. A. Edwards and J. A. Lerczak who provided helpful comments on an early version of this manuscript. The paper was completed while J. O Donnell was visiting T. Rippeth, John Simpson, and D. Bowers of the School of Ocean Sciences, University of Wales, Bangor. He is grateful for their advice and hospitality and for the kindness of the residents of Ynys Mon. References Austin, J. A. (2004), Estimating effective longitudinal dispersion in the Chesapeake Bay, Estuarine Coastal Shelf Sci., 60, Banas, N. S., B. M. Hickey, P. MacCready, and J. A. Newton (2004), Dynamics of Willapa Bay, Washington: A highly unsteady, partially mixed estuary, J. Phys. Oceanogr., 33, Blumberg, A. F., and D. W. Pritchard (1997), Estimates of the transport through the East River, New York, J. Geophys. Res., 102, Brockway, R., D. Bowers, A. Hoguane, V. Dove, and V. Vassele (2006), A note on salt intrusion in funnel-shaped estuaries: Application to the Incomati estuary, Mozambique, Estuarine Coastal Shelf Sci., 66, 1 5. Chatwin, P. C. (1976), Some remarks on the maintenance of the salinity distribution in estuaries, Estuarine Coastal Mar. Sci., 4, Dettman, E. H. (2001), Effect of water residence time on annual export and denitrification of nitrogen in estuaries: A model analysis, Estuaries, 24, Dyer, K. R. (1989), Sediment processes in estuaries: future research requirements, J. Geophys. Res., 94, 14,327 14,339. Dyer, K. R. (1997), Estuaries: A Physical Introduction, 195 pp., John Wiley, Hoboken, N. J. Fischer, H. B. (1976), Mixing and dispersion in estuaries, Annu. Rev. Fluid Mech., 8, Fischer, H. B., E. G. List, R. C. Y. Koh, J. Imberger, and N. H. Brooks (1979), Mixing in Inland and Coastal Waters, 483 pp., Elsevier, New York. Gay, P. S., J. O Donnell, and C. A. Edwards (2004), Exchange between Long Island Sound and adacent waters, J. Geophys. Res., 109, C06017, doi: /2004jc Geyer, W. R., and R. Chant (2006), The physical oceanography processes in the Hudson River estuary, in The Hudson River Estuary, edited J. S. Levinton, pp , Cambridge Univ. Press, New York. 11 of 12