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Variations of longitudinal diffusivity in the Hudson Estuary
Estuarine and Coastal Marine Science
(1979)
8, 555-564
Variations of Longitudinal Diffusivity in the Hudson Estuary
Eric S. Posmentier Institute of Marine and Atmospheric Sciences, City College of New York, zjdth Street and Convent Avenue, New York, New York ZOO~Z, U.S.A.
and John M. Raymont Burns and Roe, Inc., 185 Crossways Park Drive, Woodbury, New York 1~797, U.S.A. Received 9June ~977 and in revisedform 24 April ~978
Keywords: diffusivity ; effective longitudinal; data; stratification;
estuarine circulation ; salinity numerical model; Hudson River
The coefficient of longitudinal diffusion for salt has been calculated from the distribution of salinity observed in the Hudson Estuary at nine different times during 1974. The salinity distribution appears to be quasi steady-state, and the diffusion coefficient is spatially constant between the Upper Bay and Verplanck. The diffusion coefficient varied in time by a factor of three. It was not well correlated with the stratification parameter. It was slightly less dependent on the freshwater discharge in the Estuary than on the tidal amplitude, which varies by a factor of nearly two between spring and neap tides. Salinities predicted by a model are slightly less accurate if the diffusion coefficient depends on the stratification parameter, than if the diffusion coefficient is kept constant. If the diffusion coefficient is a power function of both fresh water discharge and tidal amplitude, salinity predictions are significantly improved. These results suggest that density-induced, gravitational, vertical circulation does not dominate the longitudinal diffusion of salt in the Hudson Estuary. Transverse circulation may be at least as significant a salt transport mechanism as vertical circulation. The predictive reliability of a onedimensional, advective-diffusive model of the salinity distribution in the Hudson Estuary depends on a realistic, variable coefficient of longitudinal diffusion for salt. Furthermore, such a model cannot use the same coefficient to predict the distribution of other properties unless the combination of transport mechanisms for these other properties is the same as that for salt.
Introduction Diffusion coefficients are heavily relied upon by many numerical and analytical models of hydrodynamic processes. The accuracies to which they are known, the empirical justifications for their use, and even their definitions vary widely. This paper is concerned with the coefficient of longitudinal diffusion, D, for one specific water property (salt) in one specific estuary (the Hudson). This coefficient appears in the one-dimensional dispersion equation for salt, an equation which is widely used as a basis for numerical modeling of estuaries. In this introductory section D will be defined and conceptualized. The purposes of the ensuing sections 555 o3o2-3524/7g/o6o555+10
$02.00/o
Q 1979 Academic Press Inc. (London) Ltd.
556
E. S. Posmentier &J.
M. Raymont
are to present the basis for the calculation of D from data recently collected in the Lower Hudson Estuary, to relate the variations in D to variations in bulk parameters of the Hudson, to infer the relative importance of various transport mechanisms in the Hudson and to comment on the applicability of a one-dimensional advective-diffusive model to the prediction of the salinity distribution in the Hudson. At any instant and any point in an estuary, the longitudinal (x-direction) transport of salt is simply the product of the salinity (S) and the longitudinal velocity (V). The average longitudinal transport is defined by F= W’L,
(1)
where the brackets ( ) denote an average. The subscripts y, z, t, and T indicate an average taken with respect to the transverse coordinate, the depth, a statistical ensemble (assuming V and S to have non-deterministic, or turbulent, components), and time over a tidal cycle, respectively. It is conventional to distinguish between two parts of F by using the expression F=F,,+F,
(4
where F,,, the advective part of F, is defined by Fo= (W,,tr
W),,,,
(3)
and F,,, the diffusive part of F, is defined by Fe= W%T The coefficient of longitudinal
-w,ztT(v),,tr
(4)
diffusion can now be defined by D= -Fahwhztrlaxi
(5)
The first term in the brackets on the right side of equation (4) can be expressed as the sum of sixteen terms (each representing a different transport mechanism) by expanding each of the variables into the sum of an average and a deviation from the average, and iterating this expansion procedure once for each of the four subscripts. Equivalent expansion procedures have been used by many other investigators, for example, Fischer (1972). One of these sixteen terms is identical to the second term in the brackets on the right side of equation (4), which represents purely advective transport, so they cancel and purely advective transport does not contribute to F, or D. Four of the sixteen terms are identically zero, if it is assumed that the tidal average of the turbulent components of V and S are zero. The eleven remaining terms represent the eleven transport mechanisms comprising longitudinal diffusion. Since there is some inconsistency among various investigators in the use of the term ‘ longitudinal diffusion ’ , it should be pointed out that in this paper ‘longitudinal diffusion’ refers to the combination of all eleven of these transport mechanisms, which includes seven deterministic diffusion processes and four non-deterministic turbulent diffusion processes, and which exclude only purely advective transport. All eleven transport mechanisms are identified in the following list : Steady-state transverse circulation of the vertically-averaged variables. Steady-state vertical circulation of the laterally-averaged variables, partially due to the density-induced gravitational circulation. 3. Steady-state cellular circulation, similar to I and 2 but geometrically more complex. 4. Tidally-oscillating transverse circulation (or transversely circulating tidal components) of the vertically-averaged variables. I.
2.
Variations of longitudinal dzj@sivity in the Hudson Estuary
557
5. Tidally-oscillating vertical circulation (or vertically circulating tidal components) of the laterally-averaged variables. 6. Cellular circulation of the tidal components of the variables, similar to 4 and 5 but geometrically more complex. 7. Tidal pumping (correlation between the tidal components of the cross-sectionally averaged variables). 8. Transverse turbulent eddies (correlation between the turbulent components of the vertically-averaged variables). 9. Vertical turbulent eddies (correlation between the turbulent components of the laterally-averaged variables). IO. Three-dimensional turbulence (the local, spatially incoherent correlation between the turbulent components of the variables), similar to 8 and 9, but geometrically more complex. I I. Turbulent pumping (correlation between the cross-sectional averages of the turbulent components of the variables). This is probably a negligible term. Several of these mechanisms have been studied and discussed extensively. A recent review of this subject was prepared by Fischer (1976). Because it represents a complex combination of eleven different salinity transport mechanisms, D in general may be sensitive to many gross properties of an estuary and the circulation, and to other more subtle characteristics, as well. However, if only one or two of the eleven salinity flux terms are relatively significant, D might have a simple dependence on only a few bulk parameters of the estuary and the circulation, or might even be nearly constant. Fortunately this is true of a number of actual estuaries and the one-dimensional dispersion equation for salt is therefore applicable to these estuaries. The following sections will empirically examine how these possibilities relate to the Hudson Estuary. Data acquisition
and analysis
A program of monthly surveys of the Hudson Estuary from the inner New York Bight to Saugerties, New York, was begun by the CUNY Institute of Marine and Atmospheric Sciences in December, 1973. The salinity observations on which this report is based were made during each of nine surveys in 1974 at each of six stations from 3 km south (seaward) of the Battery to 53 km north of the Battery. Other physical observations, and biological, chemical, and geological observations, included among routine station operations, have been reported elsewhere (Posmentier & Rachlin, 1976; Weiss et al., 1975). Salinity was determined from temperature and conductivity measurements made in situ by an Inter Ocean 513 D Probe. Raymont (1976) vertically averaged the salinity measurements. Since each survey acquired just one (S), datum at each station, tidal averaging was not attempted. Furthermore, since the difference between (S), and (S),, is largely detenninistic, tidal averaging is an inefficient use of data. Instead, Raymont (1976) determined S,(X), the vertically-averaged salinity at the time of low slack, based on the assumption of tidal advection. This treatment of the data was used successfully by Harleman et al. (1972). Its application to the data in this report was further justified post hoc by noting that semi-logarithmic plots of S, versus x had considerably less scatter than similar plots of S versus X. The former plots, S, versus X, were very well approximated by straight lines; i.e., the data of each separate survey are well-approximated by a function of the form SL(x,r)=SO(r)exp --u(r)x
(6)
558
E. S. Posmentier &J.
M. Raymont
The parameters a and So were determined by linear regression of In S, on X. Interpolated values of S,(x,r), which were calculated by using Equation (6) and the appropriate values of a and S,, are referred to in the remainder of this paper as ‘observed’ salinities. The flow rate Q=A ( V)xvtT where A is the cross-sectional area, was estimated by Raymont (1976) in the area of investigation by finding weighted averages of U.S.G.S. flow rate data at Green Island, in the Upper Hudson. To find D, use must be made of the one-dimensional advection-diffusion equation for salt: A as -d(ADg-ASV) aza7
where r is time, and the symbol ( ),,, r has been omitted from S and V for simplicity. However, the variable (S),,, r in (7) is not the same as the observed salinity S,(X), so S&),u, and S,, are not subject to analysis in the context of (7). To overcome this obstacle, a quantity d is defined as the distance from the position of any low-slack salinity to the position of the same tidally-averaged salinity; d is well-approximated by the half-tidal range. Thus, S&-47)=
>,a T
(8)
where 7 is time. Substituting (8) into (7) results in
dS&+d,T) ax
A as,(x -47)
a7
AS,
(x-d,~)V
= o I
In solving (9) for D, it may be assumed that ad/ax=o, which is substantiated by NOAA and aA/ax may be tidal current tables for the Hudson. Also, the terms BS,/ar,aV/ax, dropped from equation (9) on the basis of their second-order magnitudes relative to the remaining terms. The empirical finding that AS,/& is negligible is equivalent to the statement that a quasi steady state exists. This is consistent with Posmentier and Rachlin’s (1976) finding that the residence time of the Lower Hudson is one to two weeks, whereas flow variations are either seasonal or only a few days period. Since S satisfies both equation (6) and equation (9), then D must be spatially constant, and equation (9) takes the simpler form D --a2s,
ax2
v -=O ah.
(10)
ax
The simultaneous solution of equation (6) and (IO) is SL=35exPli-4x+xo)l
(11)
where the arbitrary coefficient has been set equal to 35x,, because this is a representative ocean salinity, and where a= -V/D,
and xO=(I/a)ln
(35/S,,)
(12 a, b)
Once a and S,, are determined by the linear regression of Ins, on x, as discussed above, the values of D and x,, may be found from equations (Iza, b). This procedure was followed by Raymont (1976) to find D and x0 for each of the nine surveys in 1974. The nine values of D and x0 are tabulated in Table I, together with the respective values of Q (flow rate), and of N (tidal range) obtained from NOAA tide tables. The values of D in Table I vary by a factor of 3, and Q and N vary by factors of 3 and 1.8 respectively, reflecting the rather wide range of conditions which occurred during the nine surveys.
Variations of longitudinal disfiuivity in the Hudson Estuary
Variations
5.59
of D
The results in Table I show that D, the coefficient of longitudinal diffusion in the Lower Hudson Estuary, varies by a factor of 3 among the nine surveys in 1974. Since the predictive reliability of a one-dimensional advective-diffusive model depends on the predictability of D, several attempts have been made to relate D to bulk parameters of estuaries and their circulation. This section presents three alternative hypotheses for the behavior of D, their consistency with the data in Table I, and the reliability of salinities predicted on the basis of each of these hypotheses. Comparisons among the hypotheses are discussed in Section IV. TABLE
I. Values of Hudson
Estuary
parameters
measured during
nine cruises in
1974
Date I974
19-20 Jan. 2-4 March 67 April 4 May 7 June 13 July 4-5 Oct. 6-7
z Nov. Dec.
(m3 s-l) Flow rate 748 1130 1023 934
815 842 564 364 709
W-4
Tidal range 1’21 I.33 I.84 1.69 ' ‘43 I.04 I.34 1.50 I .36
D(m2 s-l) Equations (5,Ioa) 704 663 631 728 434 1340 so7 438 482
&(km) Equations (9,Iob) 9.64 13’99 2.90 4’04 4.28 15.76 12.46 3’98 0.69
Hypothesis A: constant D The simplest possible hypothesis for the behavior of D is that D is a constant subject to random measurement errors. This hypothesis will be examined here mainly as a control for comparison with Hypotheses B and C, although it should be mentioned that this simple hypothesis has remained in use even for the modeling of rather complex estuaries. A reasonable estimate of D for the Lower Hudson Estuary, under this hypothesis, is 3. the geometric mean of the nine values of D in Table I, which is 619 m2 s-l. Furthermore, it is assumed under Hypothesis A that S,=35% at x=X,,, the arithmetic mean of the nine values of the x,, in Table I, which is 7.52 km. The predicted value of S,, substituting Hypothesis A and equation (rza) into equation (I I), is thus
~L=35exp[~~/V)(Jc+r,)1
(13)
which is a function of x depending on the value of V at the time of a survey. The straight lines in Figure I illustrate the variation of S,, the predicted value of S from Equation II, with Q at x=0, 12, and 24 km. The points indicate the observed values of S, (from equation 6) and Q at x=0,12, and 24 km. The RMS salinity error is shown as a function of x in Figure 2. The mean-square value of the salinity errors at x=0,4,8, .. . 24 km inclusive is 16.58 (%$. This result, together with the assumptions discussed above, appears in column A of Table 2. Hypothesis B: stratification-dependent D Several different hypotheses have been studied and used in the prediction of D in verticallystratified estuaries in which vertical circulation (mechanisms 2 and 5) dominates longitudinal diffusion (for example, Harleman, 1966; Harleman & Abraham, 1966; O’Connor, 1962). For reasons discussed in the following section, these hypotheses are not entirely relevant to
560
E. S. Posmentier f3J. M. Raymont
30 20 -
IO 82 6cr, 4-
11
t
,
;
,2:
1000 II xl 600 800 4* (m3/s) Figure I, Observed vs predicted salinities under Hypothesis A. Dots, crosses, and triangles represent observed salinities corresponding to predicted salinity lines at xs=o, 12, and 24 km, respectively. 400
TABLE 2. Alternative hypotheses for the parameterization of D (diffusion coefficient) and x0 (position at which extrapolated salinity is 35y&,). The linear correlationsquared for the parametric function D is ~sc, and for xa it is Y’,.. The mean-square salinity error is (S-S)*
Hypothesis D d *0
A
B D=NB’aa(VS/NP)b 0.32 xo=zo
C D =dV=N’ 0.64 xo=gVhA
056 18.78
11.17
longitudinal diffusion in the Hudson Estuary, and therefore cannot be expected to compare very well with Hudson Estuary data. Nevertheless, such hypotheses are widely relied upon in modeling the Hudson Estuary, without having been systematically compared with Hypothesis A. It is of interest, therefore, to evaluate one such hypothesis (Harleman, 1966). Harleman (1966) presented theoretical arguments and data which show that, in a vertically stratified estuary, the effective D is greater than a hypothetical Dt corresponding to the longitudinal transport of salt only by non-deterministic turbulence. Dt is proportional to Pa The ratio D/D* is proportional to a power of a ‘Stratification Parameter’. Harleman (1966) defines this stratification parameter as the ratio of the turbulent energy dissipation per unit mass to the potential energy increase per unit mass, which is inversely proportional to VS/Ns (Harleman’s equation 12.38).
Variations of longitudinal diffusivity
in the Hudson Estuary
561
.xL (km) Figure 2. RMS error in salinity and C.
predictions
as a function
of xL for Hypotheses
A, B,
Based on these definitions and relationships, the hypothesis represented by D/N2’s=u(
VS/N2)b
(‘4)
follows, where a and b are constants which can be found by linear regression between the logarithms of the two terms in equation (14). By performing such a regression combining the nine pairs of terms in equation (14) evaluated at each value of x from o to 24 km, inclusive, in 4 km increments, values of a and b were found, and the correlation squared was 0.32. Only those three pairs of terms for which the observed salinity was less than 3x0 were omitted from the regression analysis. The hypothesis represented by equation (14) and the corresponding correlation squared appear in column B of Table 2. The regression line and the pairs of variables from x=0, 12, and 24 km are illustrated in Figure 3. (Using only the nine pairs of terms at x=0, the correlation squared improves slightly to 0.40.) 2000
c) 9i-i Il
600 400
200: 006 006 O-I
o-2
0.4
0.6 0.6 b0
vsx. N2 ( mr
1 Figure 3. Lqarithmic regression of diffusion enhancement (Vs/W), which is inversely proportional to the ‘Stratification regression line is the basis of Hypothesis B.
(D/W’*) against Parameter’. The
562
E. S. Posmentier &J.
M. Raymont
Substituting equation (14) into the Differential Equation (IO) a solution can be found for which S,(x,)=35. This solution for the predicted salinity is &(x,=
[
3sb- i c(x+xJ]
lib
(IS)
where C=
J/l -bN2’b-i)
(16)
It is immediately apparent that the predicted salinity (equation IS) is not consistent with the observed salinity (equation 6). This arises from the fact that equation (IS) is based on a hypothesis (equation 14) which is not consistent with the observation that D does not depend on x, and therefore does not depend explicitly on S. Thus, the correlation between the variables in equation (14), 0.32, is significantly less than unity not only because of random observational errors, but also because the formulation of equation (14) is not consistent with the longitudinal mixing przcess in the Hudson Estuary. The predicted salinities SL (from equation IS) are shown by the curves in Figure 4, and the observed salinities (from equation 6) are shown by the points, at x=0, 12, and 24 km. Since S, depends not only on x, but on the two independent parameters Q and N, it was convenient in constructing Figure 4 to defme an ‘effective flow rate’ Q* to use as the abscissa: Q*=Q(N/I.39,)2'b-t)/(l-b)
(17) The RMS salinity error is shown as a function of x in Figure 2. The mean-square value of the salinity errors at x=0,4,8, . . . 24 km inclusive is 18.78 (%,J2.This result, together with the assumptions discussed above, appear in column B of Table 2.
30
\ -Y=< .
20
IO 6 2, cr, 4
.
\
00 x 12 A 24 I
II
I 400
24
.
XL
2
I 600
*
I 600
.
I 1000
. II
Figure 4. Observed vs predicted salinities under Hypothesis B. Dots, crosses, and triangles represent observed salinities corresponding to predicted salinity lines at xL=o, IZ, and 24 km, respectively.
Variations of longitudinal diflusivity
in the Hudson Estuary
563
HypothesisC: empirical D With no prior knowledge of the relative importance of the different mechanisms of longitudinal diffusion of salt in the Hudson Estuary, it is not unreasonable to hypothesize that D and exp(x,) both vary as powers of Q and N, according to the formulas in Table z, column C. The constants in these formulas were found by regression analysis. The correlation squared for D and exp (x,,) were 0.64 and 0.56, respectively. These correlations squared values are also shown in Table 2, column C. Based on Hypothesis C, the salinity can be predicted by
where 6 and 4, are each calculated from both Q and N by using the results of the regression analysis mentioned in the preceding paragraph. The RMS difference between S(from equation 5) and the observed S, (from equation 6) is shown as a function of x in Figure 2. The meansquare value of the salinity errors at x=0,4,8, . . . 24 km inclusive is 11.17 (“~Ja, which also appears in Table 2 under column C.
Discussion Under hypothesis A, the mean-square salinity error was 16*58(%,)~.This result is poor, and any model of the Hudson Estuary using D for D would have little predictive value. Hypothesis B results in an even greater mean-square salinity error, even though it is a more sophisticated model with a two-parameter estimator for D! However, this result (and the small value, 0.32, of yzl)) can be expected by noting the inconsistency between equations (IS) and (6), which is due in turn to the discrepancy between equation (12) and the observation that D is not dependent on x and therefore not explicitly dependent on S. Hypothesis B is intended for application to partially stratified estuaries. Posmentier & Rachlin (1976) h ave shown that the Hudson Estuary is well-mixed at some times and places, two-layered at others, and occassionally multi-layered. Hypothesis B is therefore not entirely applicable to the Hudson Estuary, and its poor agreement with Hudson Estuary data should not be an unanticipated result. Hypothesis B is predicated on vertical stratification, which leads to longitudinal diffusion by mechanisms 2,5,7,9 and I I. However, the transverse circulation mechanisms, particularly mechanism I, can be as significant or even an order of magnitude more significant than vertical circulation mechanisms in the Mersey Estuary (Fischer, 1972), which is hydrodynamically similar to the Hudson. The importance of transverse circulation to longitudinal diffusion in the Hudson Estuary is further supported by salinity data and by Landsat multispectral images (Posmentier, 1977). Since salinity transport by transverse circuIation is not necessarily expected to obey equation (12), the importance of this mechanism is another reason to anticipate that equations (14) and (IS) will not agree well with Hudson Estuary data. Hypotheses similar to hypothesis B have been advanced by others. Harleman & Abraham (1966) defined a dimensionless ‘Estuary Number’ from which both D and B (a distance similar in meaning to x0) can be found by using empirical regression constants. This Estuary Number was shown to vary directly as the stratification parameter, herein evahrated as a predictor of D and xs. It is interesting to note that if the regression equation for B (Harleman & Abraham, 1966) is substituted into their equation for D, it is found that D depends only on the tidal velocity, and not at all on the Estuary Number (or Q). This is in direct contradiction to the assumption used in obtaining O’Connor’s (1962) results for the regression
E. S. Posmentier NJ. M. Raymont
564
between D and Q, omitting the variations of tidal amplitude. As will be seen in the following paragraph, D is actually influenced significantly by both Q and the tidal velocity. Hypothesis C yields the best agreement between observed and predicted salinities. The values of e and f (defined in Table z column C) are such that the range of values of Q in Table I causes D to vary by a factor of 1.81, and the range of N causes D to vary by a factor of 1.86. The range of predicted values of x,, based on variations in Q alone and N alone is 14.1 km and 9.1 km, respectively. Thus, predicted salinities close to the Battery (x=0) depend more on Q than N, but salinities farther upstream depend more on N than on Q. Any conservative substance entering the estuary only from the sea or from upstream of Verplanck will be subject to longitudinal diffusion at the same rate as salt. However, this is not necessarily true of other substances. For example, since the vertical stratification of dissolved oxygen is different from that of salinity, their respective coefficients of longitudinal diffusion by vertical circulation are different.
Conclusions D, the coefficient of longitudinal
diffusion for salt in the Hudson Estuary between the Battery & Verplanck, varied by a factor of at least 3 during 1974, Variations of D do not correlate well with the stratification parameter. Multiple regression of D on both fresh water flow and tidal amplitude results in very good correlation, with the latter variable exhibiting slightly greater control over D. Transverse circulation may be as significant as the vertical (density-induced) circulation in the longitudinal diffusion of salt. Values of D may bear no relation to the longitudinal diffusion of estuarine water properties other than salt. The predictive reliability of any salinity intrusion model of the Hudson Estuary is strongly influenced by the selection of a predictor for D.
References Fischer,
H. B. 1972 Mass transport
Mechanics 53,
mechanisms
in partially
stratified
estuaries. JournaE of Fluid
671-687.
Fischer, H. B. 1976 Mixing and dispersion in estuaries. Annual Review of Fluid Mechanics 8, 107-133. Harleman, D. F. 1966 Diffusion processes in stratified flow. In Estuary and Coastline Hydrodynamics (Ippen, A. T. ed.). McGraw-Hill, New York, N.Y., 575-597. Harleman, D. R. F., & Abraham, G. 1966 One-dimensional analysis of salinity intrusion in the Rotterdam Waterway. Delft Hydrodyn. Lab. Publ. No. 44, 26 pp. of the O’Connor, D. J. 1962 Organic pollution of New York Harbor-theoretical considerations.gournuI
Water Pollution Control Federation 34, Posmentier,
905-919.
E. S. & Rachlin, J. W. 1976 Distribution
of salinity
and temperature
in the Hudson Estuary.
Journal of Physical Oceanography 6,775-777. Raymont, J. M. Jr. 1976 An Investigation of the tidal dynamics and diffusion processesof the Hudson Estuary. M.S. thesis, City College of New York, New York. Weiss, D., Rachlin, J. W., Coch, N. K. 1975 The Hudson Estuary In NEIGC Guidebook. Department of Earth and Planetary Sciences, City College of New York, New York, pp. 29-64. Harleman, D. R. F., Quinlan, A. V., Ditmars, J. D. & Thatcher, M. L. 1972 Application of the M.I.T. Transient Salinity Intrusion Model to the Hudson River Estuary. Ralph M. Parsons Laboratory Report 153, Massachussetts Institute of Technology, Cambridge, Mass., 92139.
(1979)
8, 555-564
Variations of Longitudinal Diffusivity in the Hudson Estuary
Eric S. Posmentier Institute of Marine and Atmospheric Sciences, City College of New York, zjdth Street and Convent Avenue, New York, New York ZOO~Z, U.S.A.
and John M. Raymont Burns and Roe, Inc., 185 Crossways Park Drive, Woodbury, New York 1~797, U.S.A. Received 9June ~977 and in revisedform 24 April ~978
Keywords: diffusivity ; effective longitudinal; data; stratification;
estuarine circulation ; salinity numerical model; Hudson River
The coefficient of longitudinal diffusion for salt has been calculated from the distribution of salinity observed in the Hudson Estuary at nine different times during 1974. The salinity distribution appears to be quasi steady-state, and the diffusion coefficient is spatially constant between the Upper Bay and Verplanck. The diffusion coefficient varied in time by a factor of three. It was not well correlated with the stratification parameter. It was slightly less dependent on the freshwater discharge in the Estuary than on the tidal amplitude, which varies by a factor of nearly two between spring and neap tides. Salinities predicted by a model are slightly less accurate if the diffusion coefficient depends on the stratification parameter, than if the diffusion coefficient is kept constant. If the diffusion coefficient is a power function of both fresh water discharge and tidal amplitude, salinity predictions are significantly improved. These results suggest that density-induced, gravitational, vertical circulation does not dominate the longitudinal diffusion of salt in the Hudson Estuary. Transverse circulation may be at least as significant a salt transport mechanism as vertical circulation. The predictive reliability of a onedimensional, advective-diffusive model of the salinity distribution in the Hudson Estuary depends on a realistic, variable coefficient of longitudinal diffusion for salt. Furthermore, such a model cannot use the same coefficient to predict the distribution of other properties unless the combination of transport mechanisms for these other properties is the same as that for salt.
Introduction Diffusion coefficients are heavily relied upon by many numerical and analytical models of hydrodynamic processes. The accuracies to which they are known, the empirical justifications for their use, and even their definitions vary widely. This paper is concerned with the coefficient of longitudinal diffusion, D, for one specific water property (salt) in one specific estuary (the Hudson). This coefficient appears in the one-dimensional dispersion equation for salt, an equation which is widely used as a basis for numerical modeling of estuaries. In this introductory section D will be defined and conceptualized. The purposes of the ensuing sections 555 o3o2-3524/7g/o6o555+10
$02.00/o
Q 1979 Academic Press Inc. (London) Ltd.
556
E. S. Posmentier &J.
M. Raymont
are to present the basis for the calculation of D from data recently collected in the Lower Hudson Estuary, to relate the variations in D to variations in bulk parameters of the Hudson, to infer the relative importance of various transport mechanisms in the Hudson and to comment on the applicability of a one-dimensional advective-diffusive model to the prediction of the salinity distribution in the Hudson. At any instant and any point in an estuary, the longitudinal (x-direction) transport of salt is simply the product of the salinity (S) and the longitudinal velocity (V). The average longitudinal transport is defined by F= W’L,
(1)
where the brackets ( ) denote an average. The subscripts y, z, t, and T indicate an average taken with respect to the transverse coordinate, the depth, a statistical ensemble (assuming V and S to have non-deterministic, or turbulent, components), and time over a tidal cycle, respectively. It is conventional to distinguish between two parts of F by using the expression F=F,,+F,
(4
where F,,, the advective part of F, is defined by Fo= (W,,tr
W),,,,
(3)
and F,,, the diffusive part of F, is defined by Fe= W%T The coefficient of longitudinal
-w,ztT(v),,tr
(4)
diffusion can now be defined by D= -Fahwhztrlaxi
(5)
The first term in the brackets on the right side of equation (4) can be expressed as the sum of sixteen terms (each representing a different transport mechanism) by expanding each of the variables into the sum of an average and a deviation from the average, and iterating this expansion procedure once for each of the four subscripts. Equivalent expansion procedures have been used by many other investigators, for example, Fischer (1972). One of these sixteen terms is identical to the second term in the brackets on the right side of equation (4), which represents purely advective transport, so they cancel and purely advective transport does not contribute to F, or D. Four of the sixteen terms are identically zero, if it is assumed that the tidal average of the turbulent components of V and S are zero. The eleven remaining terms represent the eleven transport mechanisms comprising longitudinal diffusion. Since there is some inconsistency among various investigators in the use of the term ‘ longitudinal diffusion ’ , it should be pointed out that in this paper ‘longitudinal diffusion’ refers to the combination of all eleven of these transport mechanisms, which includes seven deterministic diffusion processes and four non-deterministic turbulent diffusion processes, and which exclude only purely advective transport. All eleven transport mechanisms are identified in the following list : Steady-state transverse circulation of the vertically-averaged variables. Steady-state vertical circulation of the laterally-averaged variables, partially due to the density-induced gravitational circulation. 3. Steady-state cellular circulation, similar to I and 2 but geometrically more complex. 4. Tidally-oscillating transverse circulation (or transversely circulating tidal components) of the vertically-averaged variables. I.
2.
Variations of longitudinal dzj@sivity in the Hudson Estuary
557
5. Tidally-oscillating vertical circulation (or vertically circulating tidal components) of the laterally-averaged variables. 6. Cellular circulation of the tidal components of the variables, similar to 4 and 5 but geometrically more complex. 7. Tidal pumping (correlation between the tidal components of the cross-sectionally averaged variables). 8. Transverse turbulent eddies (correlation between the turbulent components of the vertically-averaged variables). 9. Vertical turbulent eddies (correlation between the turbulent components of the laterally-averaged variables). IO. Three-dimensional turbulence (the local, spatially incoherent correlation between the turbulent components of the variables), similar to 8 and 9, but geometrically more complex. I I. Turbulent pumping (correlation between the cross-sectional averages of the turbulent components of the variables). This is probably a negligible term. Several of these mechanisms have been studied and discussed extensively. A recent review of this subject was prepared by Fischer (1976). Because it represents a complex combination of eleven different salinity transport mechanisms, D in general may be sensitive to many gross properties of an estuary and the circulation, and to other more subtle characteristics, as well. However, if only one or two of the eleven salinity flux terms are relatively significant, D might have a simple dependence on only a few bulk parameters of the estuary and the circulation, or might even be nearly constant. Fortunately this is true of a number of actual estuaries and the one-dimensional dispersion equation for salt is therefore applicable to these estuaries. The following sections will empirically examine how these possibilities relate to the Hudson Estuary. Data acquisition
and analysis
A program of monthly surveys of the Hudson Estuary from the inner New York Bight to Saugerties, New York, was begun by the CUNY Institute of Marine and Atmospheric Sciences in December, 1973. The salinity observations on which this report is based were made during each of nine surveys in 1974 at each of six stations from 3 km south (seaward) of the Battery to 53 km north of the Battery. Other physical observations, and biological, chemical, and geological observations, included among routine station operations, have been reported elsewhere (Posmentier & Rachlin, 1976; Weiss et al., 1975). Salinity was determined from temperature and conductivity measurements made in situ by an Inter Ocean 513 D Probe. Raymont (1976) vertically averaged the salinity measurements. Since each survey acquired just one (S), datum at each station, tidal averaging was not attempted. Furthermore, since the difference between (S), and (S),, is largely detenninistic, tidal averaging is an inefficient use of data. Instead, Raymont (1976) determined S,(X), the vertically-averaged salinity at the time of low slack, based on the assumption of tidal advection. This treatment of the data was used successfully by Harleman et al. (1972). Its application to the data in this report was further justified post hoc by noting that semi-logarithmic plots of S, versus x had considerably less scatter than similar plots of S versus X. The former plots, S, versus X, were very well approximated by straight lines; i.e., the data of each separate survey are well-approximated by a function of the form SL(x,r)=SO(r)exp --u(r)x
(6)
558
E. S. Posmentier &J.
M. Raymont
The parameters a and So were determined by linear regression of In S, on X. Interpolated values of S,(x,r), which were calculated by using Equation (6) and the appropriate values of a and S,, are referred to in the remainder of this paper as ‘observed’ salinities. The flow rate Q=A ( V)xvtT where A is the cross-sectional area, was estimated by Raymont (1976) in the area of investigation by finding weighted averages of U.S.G.S. flow rate data at Green Island, in the Upper Hudson. To find D, use must be made of the one-dimensional advection-diffusion equation for salt: A as -d(ADg-ASV) aza7
where r is time, and the symbol ( ),,, r has been omitted from S and V for simplicity. However, the variable (S),,, r in (7) is not the same as the observed salinity S,(X), so S&),u, and S,, are not subject to analysis in the context of (7). To overcome this obstacle, a quantity d is defined as the distance from the position of any low-slack salinity to the position of the same tidally-averaged salinity; d is well-approximated by the half-tidal range. Thus, S&-47)=
>,a T
(8)
where 7 is time. Substituting (8) into (7) results in
dS&+d,T) ax
A as,(x -47)
a7
AS,
(x-d,~)V
= o I
In solving (9) for D, it may be assumed that ad/ax=o, which is substantiated by NOAA and aA/ax may be tidal current tables for the Hudson. Also, the terms BS,/ar,aV/ax, dropped from equation (9) on the basis of their second-order magnitudes relative to the remaining terms. The empirical finding that AS,/& is negligible is equivalent to the statement that a quasi steady state exists. This is consistent with Posmentier and Rachlin’s (1976) finding that the residence time of the Lower Hudson is one to two weeks, whereas flow variations are either seasonal or only a few days period. Since S satisfies both equation (6) and equation (9), then D must be spatially constant, and equation (9) takes the simpler form D --a2s,
ax2
v -=O ah.
(10)
ax
The simultaneous solution of equation (6) and (IO) is SL=35exPli-4x+xo)l
(11)
where the arbitrary coefficient has been set equal to 35x,, because this is a representative ocean salinity, and where a= -V/D,
and xO=(I/a)ln
(35/S,,)
(12 a, b)
Once a and S,, are determined by the linear regression of Ins, on x, as discussed above, the values of D and x,, may be found from equations (Iza, b). This procedure was followed by Raymont (1976) to find D and x0 for each of the nine surveys in 1974. The nine values of D and x0 are tabulated in Table I, together with the respective values of Q (flow rate), and of N (tidal range) obtained from NOAA tide tables. The values of D in Table I vary by a factor of 3, and Q and N vary by factors of 3 and 1.8 respectively, reflecting the rather wide range of conditions which occurred during the nine surveys.
Variations of longitudinal disfiuivity in the Hudson Estuary
Variations
5.59
of D
The results in Table I show that D, the coefficient of longitudinal diffusion in the Lower Hudson Estuary, varies by a factor of 3 among the nine surveys in 1974. Since the predictive reliability of a one-dimensional advective-diffusive model depends on the predictability of D, several attempts have been made to relate D to bulk parameters of estuaries and their circulation. This section presents three alternative hypotheses for the behavior of D, their consistency with the data in Table I, and the reliability of salinities predicted on the basis of each of these hypotheses. Comparisons among the hypotheses are discussed in Section IV. TABLE
I. Values of Hudson
Estuary
parameters
measured during
nine cruises in
1974
Date I974
19-20 Jan. 2-4 March 67 April 4 May 7 June 13 July 4-5 Oct. 6-7
z Nov. Dec.
(m3 s-l) Flow rate 748 1130 1023 934
815 842 564 364 709
W-4
Tidal range 1’21 I.33 I.84 1.69 ' ‘43 I.04 I.34 1.50 I .36
D(m2 s-l) Equations (5,Ioa) 704 663 631 728 434 1340 so7 438 482
&(km) Equations (9,Iob) 9.64 13’99 2.90 4’04 4.28 15.76 12.46 3’98 0.69
Hypothesis A: constant D The simplest possible hypothesis for the behavior of D is that D is a constant subject to random measurement errors. This hypothesis will be examined here mainly as a control for comparison with Hypotheses B and C, although it should be mentioned that this simple hypothesis has remained in use even for the modeling of rather complex estuaries. A reasonable estimate of D for the Lower Hudson Estuary, under this hypothesis, is 3. the geometric mean of the nine values of D in Table I, which is 619 m2 s-l. Furthermore, it is assumed under Hypothesis A that S,=35% at x=X,,, the arithmetic mean of the nine values of the x,, in Table I, which is 7.52 km. The predicted value of S,, substituting Hypothesis A and equation (rza) into equation (I I), is thus
~L=35exp[~~/V)(Jc+r,)1
(13)
which is a function of x depending on the value of V at the time of a survey. The straight lines in Figure I illustrate the variation of S,, the predicted value of S from Equation II, with Q at x=0, 12, and 24 km. The points indicate the observed values of S, (from equation 6) and Q at x=0,12, and 24 km. The RMS salinity error is shown as a function of x in Figure 2. The mean-square value of the salinity errors at x=0,4,8, .. . 24 km inclusive is 16.58 (%$. This result, together with the assumptions discussed above, appears in column A of Table 2. Hypothesis B: stratification-dependent D Several different hypotheses have been studied and used in the prediction of D in verticallystratified estuaries in which vertical circulation (mechanisms 2 and 5) dominates longitudinal diffusion (for example, Harleman, 1966; Harleman & Abraham, 1966; O’Connor, 1962). For reasons discussed in the following section, these hypotheses are not entirely relevant to
560
E. S. Posmentier f3J. M. Raymont
30 20 -
IO 82 6cr, 4-
11
t
,
;
,2:
1000 II xl 600 800 4* (m3/s) Figure I, Observed vs predicted salinities under Hypothesis A. Dots, crosses, and triangles represent observed salinities corresponding to predicted salinity lines at xs=o, 12, and 24 km, respectively. 400
TABLE 2. Alternative hypotheses for the parameterization of D (diffusion coefficient) and x0 (position at which extrapolated salinity is 35y&,). The linear correlationsquared for the parametric function D is ~sc, and for xa it is Y’,.. The mean-square salinity error is (S-S)*
Hypothesis D d *0
A
B D=NB’aa(VS/NP)b 0.32 xo=zo
C D =dV=N’ 0.64 xo=gVhA
056 18.78
11.17
longitudinal diffusion in the Hudson Estuary, and therefore cannot be expected to compare very well with Hudson Estuary data. Nevertheless, such hypotheses are widely relied upon in modeling the Hudson Estuary, without having been systematically compared with Hypothesis A. It is of interest, therefore, to evaluate one such hypothesis (Harleman, 1966). Harleman (1966) presented theoretical arguments and data which show that, in a vertically stratified estuary, the effective D is greater than a hypothetical Dt corresponding to the longitudinal transport of salt only by non-deterministic turbulence. Dt is proportional to Pa The ratio D/D* is proportional to a power of a ‘Stratification Parameter’. Harleman (1966) defines this stratification parameter as the ratio of the turbulent energy dissipation per unit mass to the potential energy increase per unit mass, which is inversely proportional to VS/Ns (Harleman’s equation 12.38).
Variations of longitudinal diffusivity
in the Hudson Estuary
561
.xL (km) Figure 2. RMS error in salinity and C.
predictions
as a function
of xL for Hypotheses
A, B,
Based on these definitions and relationships, the hypothesis represented by D/N2’s=u(
VS/N2)b
(‘4)
follows, where a and b are constants which can be found by linear regression between the logarithms of the two terms in equation (14). By performing such a regression combining the nine pairs of terms in equation (14) evaluated at each value of x from o to 24 km, inclusive, in 4 km increments, values of a and b were found, and the correlation squared was 0.32. Only those three pairs of terms for which the observed salinity was less than 3x0 were omitted from the regression analysis. The hypothesis represented by equation (14) and the corresponding correlation squared appear in column B of Table 2. The regression line and the pairs of variables from x=0, 12, and 24 km are illustrated in Figure 3. (Using only the nine pairs of terms at x=0, the correlation squared improves slightly to 0.40.) 2000
c) 9i-i Il
600 400
200: 006 006 O-I
o-2
0.4
0.6 0.6 b0
vsx. N2 ( mr
1 Figure 3. Lqarithmic regression of diffusion enhancement (Vs/W), which is inversely proportional to the ‘Stratification regression line is the basis of Hypothesis B.
(D/W’*) against Parameter’. The
562
E. S. Posmentier &J.
M. Raymont
Substituting equation (14) into the Differential Equation (IO) a solution can be found for which S,(x,)=35. This solution for the predicted salinity is &(x,=
[
3sb- i c(x+xJ]
lib
(IS)
where C=
J/l -bN2’b-i)
(16)
It is immediately apparent that the predicted salinity (equation IS) is not consistent with the observed salinity (equation 6). This arises from the fact that equation (IS) is based on a hypothesis (equation 14) which is not consistent with the observation that D does not depend on x, and therefore does not depend explicitly on S. Thus, the correlation between the variables in equation (14), 0.32, is significantly less than unity not only because of random observational errors, but also because the formulation of equation (14) is not consistent with the longitudinal mixing przcess in the Hudson Estuary. The predicted salinities SL (from equation IS) are shown by the curves in Figure 4, and the observed salinities (from equation 6) are shown by the points, at x=0, 12, and 24 km. Since S, depends not only on x, but on the two independent parameters Q and N, it was convenient in constructing Figure 4 to defme an ‘effective flow rate’ Q* to use as the abscissa: Q*=Q(N/I.39,)2'b-t)/(l-b)
(17) The RMS salinity error is shown as a function of x in Figure 2. The mean-square value of the salinity errors at x=0,4,8, . . . 24 km inclusive is 18.78 (%,J2.This result, together with the assumptions discussed above, appear in column B of Table 2.
30
\ -Y=< .
20
IO 6 2, cr, 4
.
\
00 x 12 A 24 I
II
I 400
24
.
XL
2
I 600
*
I 600
.
I 1000
. II
Figure 4. Observed vs predicted salinities under Hypothesis B. Dots, crosses, and triangles represent observed salinities corresponding to predicted salinity lines at xL=o, IZ, and 24 km, respectively.
Variations of longitudinal diflusivity
in the Hudson Estuary
563
HypothesisC: empirical D With no prior knowledge of the relative importance of the different mechanisms of longitudinal diffusion of salt in the Hudson Estuary, it is not unreasonable to hypothesize that D and exp(x,) both vary as powers of Q and N, according to the formulas in Table z, column C. The constants in these formulas were found by regression analysis. The correlation squared for D and exp (x,,) were 0.64 and 0.56, respectively. These correlations squared values are also shown in Table 2, column C. Based on Hypothesis C, the salinity can be predicted by
where 6 and 4, are each calculated from both Q and N by using the results of the regression analysis mentioned in the preceding paragraph. The RMS difference between S(from equation 5) and the observed S, (from equation 6) is shown as a function of x in Figure 2. The meansquare value of the salinity errors at x=0,4,8, . . . 24 km inclusive is 11.17 (“~Ja, which also appears in Table 2 under column C.
Discussion Under hypothesis A, the mean-square salinity error was 16*58(%,)~.This result is poor, and any model of the Hudson Estuary using D for D would have little predictive value. Hypothesis B results in an even greater mean-square salinity error, even though it is a more sophisticated model with a two-parameter estimator for D! However, this result (and the small value, 0.32, of yzl)) can be expected by noting the inconsistency between equations (IS) and (6), which is due in turn to the discrepancy between equation (12) and the observation that D is not dependent on x and therefore not explicitly dependent on S. Hypothesis B is intended for application to partially stratified estuaries. Posmentier & Rachlin (1976) h ave shown that the Hudson Estuary is well-mixed at some times and places, two-layered at others, and occassionally multi-layered. Hypothesis B is therefore not entirely applicable to the Hudson Estuary, and its poor agreement with Hudson Estuary data should not be an unanticipated result. Hypothesis B is predicated on vertical stratification, which leads to longitudinal diffusion by mechanisms 2,5,7,9 and I I. However, the transverse circulation mechanisms, particularly mechanism I, can be as significant or even an order of magnitude more significant than vertical circulation mechanisms in the Mersey Estuary (Fischer, 1972), which is hydrodynamically similar to the Hudson. The importance of transverse circulation to longitudinal diffusion in the Hudson Estuary is further supported by salinity data and by Landsat multispectral images (Posmentier, 1977). Since salinity transport by transverse circuIation is not necessarily expected to obey equation (12), the importance of this mechanism is another reason to anticipate that equations (14) and (IS) will not agree well with Hudson Estuary data. Hypotheses similar to hypothesis B have been advanced by others. Harleman & Abraham (1966) defined a dimensionless ‘Estuary Number’ from which both D and B (a distance similar in meaning to x0) can be found by using empirical regression constants. This Estuary Number was shown to vary directly as the stratification parameter, herein evahrated as a predictor of D and xs. It is interesting to note that if the regression equation for B (Harleman & Abraham, 1966) is substituted into their equation for D, it is found that D depends only on the tidal velocity, and not at all on the Estuary Number (or Q). This is in direct contradiction to the assumption used in obtaining O’Connor’s (1962) results for the regression
E. S. Posmentier NJ. M. Raymont
564
between D and Q, omitting the variations of tidal amplitude. As will be seen in the following paragraph, D is actually influenced significantly by both Q and the tidal velocity. Hypothesis C yields the best agreement between observed and predicted salinities. The values of e and f (defined in Table z column C) are such that the range of values of Q in Table I causes D to vary by a factor of 1.81, and the range of N causes D to vary by a factor of 1.86. The range of predicted values of x,, based on variations in Q alone and N alone is 14.1 km and 9.1 km, respectively. Thus, predicted salinities close to the Battery (x=0) depend more on Q than N, but salinities farther upstream depend more on N than on Q. Any conservative substance entering the estuary only from the sea or from upstream of Verplanck will be subject to longitudinal diffusion at the same rate as salt. However, this is not necessarily true of other substances. For example, since the vertical stratification of dissolved oxygen is different from that of salinity, their respective coefficients of longitudinal diffusion by vertical circulation are different.
Conclusions D, the coefficient of longitudinal
diffusion for salt in the Hudson Estuary between the Battery & Verplanck, varied by a factor of at least 3 during 1974, Variations of D do not correlate well with the stratification parameter. Multiple regression of D on both fresh water flow and tidal amplitude results in very good correlation, with the latter variable exhibiting slightly greater control over D. Transverse circulation may be as significant as the vertical (density-induced) circulation in the longitudinal diffusion of salt. Values of D may bear no relation to the longitudinal diffusion of estuarine water properties other than salt. The predictive reliability of any salinity intrusion model of the Hudson Estuary is strongly influenced by the selection of a predictor for D.
References Fischer,
H. B. 1972 Mass transport
Mechanics 53,
mechanisms
in partially
stratified
estuaries. JournaE of Fluid
671-687.
Fischer, H. B. 1976 Mixing and dispersion in estuaries. Annual Review of Fluid Mechanics 8, 107-133. Harleman, D. F. 1966 Diffusion processes in stratified flow. In Estuary and Coastline Hydrodynamics (Ippen, A. T. ed.). McGraw-Hill, New York, N.Y., 575-597. Harleman, D. R. F., & Abraham, G. 1966 One-dimensional analysis of salinity intrusion in the Rotterdam Waterway. Delft Hydrodyn. Lab. Publ. No. 44, 26 pp. of the O’Connor, D. J. 1962 Organic pollution of New York Harbor-theoretical considerations.gournuI
Water Pollution Control Federation 34, Posmentier,
905-919.
E. S. & Rachlin, J. W. 1976 Distribution
of salinity
and temperature
in the Hudson Estuary.
Journal of Physical Oceanography 6,775-777. Raymont, J. M. Jr. 1976 An Investigation of the tidal dynamics and diffusion processesof the Hudson Estuary. M.S. thesis, City College of New York, New York. Weiss, D., Rachlin, J. W., Coch, N. K. 1975 The Hudson Estuary In NEIGC Guidebook. Department of Earth and Planetary Sciences, City College of New York, New York, pp. 29-64. Harleman, D. R. F., Quinlan, A. V., Ditmars, J. D. & Thatcher, M. L. 1972 Application of the M.I.T. Transient Salinity Intrusion Model to the Hudson River Estuary. Ralph M. Parsons Laboratory Report 153, Massachussetts Institute of Technology, Cambridge, Mass., 92139.