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  • View in gallery

    Fields from a realistic numerical simulation of the Columbia River estuary and plume, from 19 Jul 2005. (a) Tidally averaged surface salinity is plotted, with the path of a thalweg section through the estuary shown as a white line. The short white lines on the map are locations of the sections used to calculate salt flux terms shown in later figures. (b) Tidally averaged salinity on the thalweg section; (c) river flow; and (d) rms depth-averaged current near the estuary mouth. This is at a time of moderate river flow and the transition from neap to spring tides.

  • View in gallery

    Terms in the Eulerian salt flux balance, averaged over a spring–neap cycle, at nine cross sections in the Columbia River simulation. (a) The along-channel structure of the section-averaged salinity. The gray lines show the extrema of the tidally averaged salinity. The dashed line shows one tidal excursion from the mouth. (b) The river flow and rms tidal velocity are plotted, along with the eddy diffusivity KH required for the tidal salt flux term. (c) Plot of terms in the salt balance, (2.6), vs along-channel position. (d) Plot of the thalweg depth and sectional area.

  • View in gallery

    The differential isohaline transport function −∂Q/∂s vs salinity, at the Columbia River section closest to the mouth, averaged over a spring–neap cycle. The gray line is the same function calculated from the Eulerian transport and salinity.

  • View in gallery

    TEF (a) salinity and (b) transport vs x, averaged over a spring–neap cycle (black lines). For comparison, the same properties calculated from the Eulerian-averaged properties are plotted in gray. The two most landward values of are zero, and so is not defined at those points in (a).

  • View in gallery

    (a) Contour plot of the isohaline transport function Q(x, s) averaged over the spring–neap cycle. The sense of the overturning circulation along Q contours is into the estuary for higher s and out for lower s. (b) Profiles of Q(s) at three different x values.

  • View in gallery

    Low-passed Eulerian salt flux and isohaline TEF terms plotted vs time for the section closest to the mouth: (a) river flow and tidal transport, (b) Eulerian salt flux terms, (c) TEF salinity, and (d) TEF transport. For reference, the dashed lines in (c),(d) show the same properties calculated from the Eulerian-averaged fields.

  • View in gallery

    Properties from the TEF analysis, for the section closest to the mouth, plotted vs river flow and tidal transport: (a),(b) salinity difference Δs = sinsout and (c)–(f) the transports.

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Calculating Estuarine Exchange Flow Using Isohaline Coordinates

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  • 1 University of Washington, Seattle, Washington
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Abstract

A method for calculating subtidal estuarine exchange flow using an isohaline framework is described, and the results are compared with those of the more commonly used Eulerian method of salt flux decomposition. Concepts are explored using a realistic numerical simulation of the Columbia River estuary. The isohaline method is found to be advantageous because it intrinsically highlights the salinity classes in which subtidal volume flux occurs. The resulting expressions give rise to an exact formulation of the time-dependent Knudsen relation and may be used in calculation of the saltwater residence time. The volume flux of the landward transport, which can be calculated precisely using the isohaline framework, is of particular importance for problems in which the saltwater residence time is critical.

River Influences on Shelf Ecosystems Program Contribution Number 52.

Corresponding author address: Parker MacCready, School of Oceanography, University of Washington, Box 355351, Seattle, WA 98195-5351. E-mail: pmacc@uw.edu

Abstract

A method for calculating subtidal estuarine exchange flow using an isohaline framework is described, and the results are compared with those of the more commonly used Eulerian method of salt flux decomposition. Concepts are explored using a realistic numerical simulation of the Columbia River estuary. The isohaline method is found to be advantageous because it intrinsically highlights the salinity classes in which subtidal volume flux occurs. The resulting expressions give rise to an exact formulation of the time-dependent Knudsen relation and may be used in calculation of the saltwater residence time. The volume flux of the landward transport, which can be calculated precisely using the isohaline framework, is of particular importance for problems in which the saltwater residence time is critical.

River Influences on Shelf Ecosystems Program Contribution Number 52.

Corresponding author address: Parker MacCready, School of Oceanography, University of Washington, Box 355351, Seattle, WA 98195-5351. E-mail: pmacc@uw.edu

1. Introduction

Estuarine circulation and salinity patterns are the result of several competing factors: river flow pushes seaward, denser ocean water slides landward, and tidal currents stir and mix the two. In particular, the “exchange flow” or “gravitational circulation,” with deep inflow and shallow outflow, dominates the circulation structure of many estuaries. We have sought to understand this complex system in part by tidal averaging. Theories have been developed to predict the subtidal velocity and salinity fields (Hansen and Rattray 1965; Chatwin 1976). However, many fundamental questions remain regarding tidally averaged dynamics, particularly in systems with weak or periodic stratification (reviewed in MacCready and Geyer 2010).

From the point of view of salt conservation, estuarine topology is simple: a closed volume, except at river and ocean ends, and salt flux only occurs at the ocean end. Forming integrated water and salt budgets over this volume gives rise to the Knudsen (1900) relations. These state that the net tidally averaged salt flux in through the seaward section must equal the flux out through that section (in steady state). It has long been recognized that not all subtidal salt flux is due to the product of subtidal velocity and subtidal salinity. Not surprisingly, tidal currents interacting with bathymetry and other effects can give rise to tidally averaged salt flux (Hansen and Rattray 1965; Fischer 1976; Zimmerman 1986; Banas et al. 2004).

The method of calculating tidally averaged salt flux through an estuarine cross section has been developed by many researchers (e.g., Hughes and Rattray 1980; Dronkers and van de Kreeke 1986; Lerczak et al. 2006), and it is often found that the tidal salt flux can be comparable to that due to the exchange flow. The classical framework for calculating subtidal salt flux (described in detail in section 2) gives preference to a spatial, fundamentally Eulerian description of the flow. The exchange flow only exists in this formalism because it has spatial structure. A problem with this system is that it gives little information about the salinity being transported by the tidal salt flux term. In section 3, we explore a different framework for calculating tidally averaged salt flux through an estuarine cross section, using salinity instead of spatial position as the preferred sorting criterion. Applying this framework to a numerical simulation of the Columbia River estuary in section 4, we find that it gives a clearer view of the flux, transformation, and residence time of ocean water in the estuary. This leads to the definition of a new term, the “total exchange flow” (TEF), to describe the subtidal volume flux integrated over a salinity range. Results are discussed in section 5. This short paper does not provide a theory for predicting TEF and is meant primarily as a clear statement of the method of calculation of a quantity that would benefit from future theoretical work. The TEF may also prove useful more immediately as a means of calculating residence time in estuaries.

2. Eulerian salt flux decomposition

Here, we present the classical Eulerian method for decomposing the tidally averaged salt flux through an estuarine channel cross section. This gives rise to the river, exchange flow, and tidal fluxes. Following Lerczak et al. (2006), the subtidal salt flux F through a section is given by
e2.1
where 〈 〉 denotes a tidal low-pass filter, u is the along-channel velocity, s is salinity, and A is the area of integration. To make tidal changes in area less cumbersome in the analysis, we divide up A into a constant number of differential elements dA, which expand and contract tidally. Velocity, salinity, and salt flux are separated into three parts: (i) sectionally integrated/averaged and tidally averaged, (ii) sectionally varying and tidally averaged, and (iii) the sectionally and tidally varying remainder. The tidally averaged area properties are defined by
e2.2
The sectionally and tidally averaged velocity and salinity are then
e2.3
Note that u0 includes the Stokes drift and, excluding significant changes in subtidal estuarine volume, is related to the river flow volume flux QR by QR = −u0A0. As a sign convention, we assume u is positive up estuary (toward the river) but that QR is a positive quantity. The sectionally varying tidally averaged terms are defined by
e2.4
The classical exchange flow is u1, although a strict definition of the gravitational circulation would only include the vertically varying part of this. The spatially and tidally varying remainders are
e2.5
The subtidal salt flux may then be decomposed into three parts (river, exchange, and tidal) as
e2.6
where we have made use of the properties , , , and to eliminate six of the nine terms in (2.6). In general, the river flow term FR removes salt from the estuary, whereas those due to the exchange flow and tidal correlations, FE + FT, add salt. The volume-integrated salt budget is then
e2.7
An example of this Eulerian decomposition is presented in section 4.

3. Isohaline salt flux decomposition

An alternate way to look at subtidal estuarine salt flux is to average the transport as a function of salinity instead of as a function of spatial position within the section. Adopting salinity as a coordinate allows us to better keep track of the flux of specific types of water, as was done by Döös and Webb (1994) when they analyzed the meridional circulation across the Antarctic Circumpolar Current in potential density coordinates. The approach is also intrinsic to the “transformed Eulerian mean” used to clarify eddy fluxes in atmosphere and ocean flows (Vallis 2006). MacCready and Geyer (2001) and MacCready et al. (2002) used an isohaline coordinate system to analyze water mass transformations in an idealized numerical simulation of an estuary. There the use of isohaline coordinates highlighted the patterns by which the estuary created midsalinity water during springs and ejected it during neaps. Hetland (2005) used this approach to analyze mixing in a river plume. MacDonald (2006) used tidally averaged fluxes through a section binned by salinity (and temperature) class to calculate flushing time in Mt. Hope Bay. The approach has also been used to calculate tidally averaged fluxes in other systems by MacDonald and Horner-Devine (2008; Fraser River), Gregg and Pratt (2010; Hood Canal), and Ralston et al. (2010; Merrimack River). The main goal of this paper is to directly compare the results of an isohaline salt flux decomposition to that of an Eulerian decomposition to understand their differences and to make inferences about estuarine function.

The tidally averaged volume flux of water with salinity greater than s is defined as
e3.1
where As is the tidally varying portion of the cross section with salinity greater than s. Here, Q can be defined for any cross section and for all salinity 0 ≤ ssocn. We will refer to Q(x, s, t) as the isohaline transport function. It is similar to the transport streamfunctions used to describe the meridional overturning circulation of the global ocean, but with salinity as one coordinate. Because of the way As is defined, it follows that Q(scon) = 0. In the limit s = 0, the integral (3.1) covers the whole cross section and so Q(0) = −QR. To find the volume flux in a specific salinity class, we evaluate
e3.2
The minus sign in (3.2) means that a positive value of −∂Q/∂s corresponds to inflow in a given salinity class. In practice, one typically calculates −∂Q/∂s as the tidally averaged transport in a series of finite salinity bins, essentially a histogram of transport versus salinity, which is then integrated to find Q.
There are several other integral properties we may define. As a group, we will refer to these as the TEF, defined in part by
e3.3
where, for example, “in” means that we only count −∂Q/∂s in the integral when its sign brings water into the estuary. In our coordinate system, Qout would be negative in sign. We refer to Qin,out as the total exchange flow, because they represent the flux of water into and out of the estuary due to all processes (tidal and subtidal) that occur in distinct salinity classes. Note that, if there is flux in and out of the same size and in the same salinity class, then −∂Q/∂s would count that as zero. An example of this would be a subtidal recirculation eddy in a part of the estuary with constant salinity. Another example would be purely tidal advection back and forth of a “frozen” salinity field. Both do not show up in −∂Q/∂s.
In terms of volume conservation (excluding subtidal changes in estuarine volume), we may surmise
e3.4
The salt flux due to TEF is given by
e3.5
It is also useful to define the flux-weighted salinities that characterize the inflow and outflow. These are given by
e3.6
and we define their difference as Δssinsout. The full set of TEF terms is Qin,out and sin,out. Total salt conservation may be written as
e3.7
Simple manipulation of volume (3.4) and salt (3.7) conservation yields the time-dependent version of the Knudsen relations,
e3.8
The Knudsen relations are typically derived assuming steady, two-layer flow and do not apply in situations where FT is important. The TEF formulation avoids this problem, and the expressions (3.8) are exact. Total salt conservation may also be written as
e3.9
where we have used integration by parts and the boundary condition Q(scon) = 0 to derive the second equality. For steady salt balance, (3.9) states that Q integrates to zero over salinity space.
For comparison, similar expressions may be defined for the Eulerian salt flux decomposition, using
e3.10
where the superscript Eu stands for Eulerian, to indicate that all the fields are time averaged in near-constant spatial positions, as in section 2. The area is the portion of the cross section over which the tidally averaged salinity s0 + s1 is greater than s. From (3.10), we may then define , , and ΔsEu exactly as done in (3.3) and (3.6). Note, however, that the resulting quantities will not in general satisfy (3.8), because they lack the contribution of the tidal salt flux.

4. Application to the Columbia River estuary

To explore the consequences of the Eulerian and isohaline methods of salt flux decomposition, we will apply both to a numerical simulation of the Columbia River estuary. This is a realistic hindcast of April–September 2005 done as part of the River Influences on Shelf Ecosystems (RISE) project, Hickey et al. (2010). The model setup is described in MacCready et al. (2009) and extensively validated against observations in Liu et al. (2009). The model used is the Regional Ocean Modeling System 2.2 (ROMS; Haidvogel et al. 2000). In the estuary, it has ~400-m horizontal resolution, 20 vertical levels, and the minimum depth is set at 3 m (no wetting and drying). The model forcing includes realistic variation of river flow, tides, atmospheric wind stress and heat flux, and ocean boundary conditions.

The Columbia River estuary has intense tidal currents (up to 3 m s−1 at the surface at maximum ebb), strong river flow (3000–10 000 m3 s−1), and its salt structure varies from salt wedge to well mixed (Jay and Smith 1990). The length of the salt intrusion is 20–50 km. Analyzing Eulerian salt flux across a well-instrumented section near the mouth, Hughes and Rattray (1980) found that more than half of the up-estuary salt flux was due to FT.

A snapshot of tidally averaged model salinity is shown in Fig. 1, along with time series of river and tidal forcing fields. Salt flux was analyzed at nine cross sections. Each section lay exactly on u grid points of the model, and nearby salinity values were interpolated to these locations. The sections are not exactly aligned with the local cross-channel direction, but this should make little difference to the section-integrated fluxes. Results of the Eulerian decomposition are shown in Fig. 2. The fields are averaged over the spring–neap cycle centered on the time shown in Fig. 1. The classical salt flux terms, versus along-channel distance, are plotted in Fig. 2c, and FT dominates over FE in the outer part of the estuary. This is not surprising, because this region is within one tidal excursion of the mouth, and the Stommel and Farmer (1952) tidal pumping mechanism is probably active there. Using a Fickean parameterization of tidal salt flux, , requires a large value of KH, O(3000 m2 s−1) near the mouth (Fig. 2b).

Fig. 1.
Fig. 1.

Fields from a realistic numerical simulation of the Columbia River estuary and plume, from 19 Jul 2005. (a) Tidally averaged surface salinity is plotted, with the path of a thalweg section through the estuary shown as a white line. The short white lines on the map are locations of the sections used to calculate salt flux terms shown in later figures. (b) Tidally averaged salinity on the thalweg section; (c) river flow; and (d) rms depth-averaged current near the estuary mouth. This is at a time of moderate river flow and the transition from neap to spring tides.

Citation: Journal of Physical Oceanography 41, 6; 10.1175/2011JPO4517.1

Fig. 2.
Fig. 2.

Terms in the Eulerian salt flux balance, averaged over a spring–neap cycle, at nine cross sections in the Columbia River simulation. (a) The along-channel structure of the section-averaged salinity. The gray lines show the extrema of the tidally averaged salinity. The dashed line shows one tidal excursion from the mouth. (b) The river flow and rms tidal velocity are plotted, along with the eddy diffusivity KH required for the tidal salt flux term. (c) Plot of terms in the salt balance, (2.6), vs along-channel position. (d) Plot of the thalweg depth and sectional area.

Citation: Journal of Physical Oceanography 41, 6; 10.1175/2011JPO4517.1

The volume flux through the mouth section using the isohaline analysis, −∂Q/∂s, is plotted versus salinity in Fig. 3. This reveals that the average inflow happens over a narrow salinity range near oceanic values. The outflow occurs over a much larger salinity range, broadly distributed from 0 to 30. This is indicative of the mixing that occurs within the estuary. For comparison, −∂QEu/∂s is also plotted. It exhibits similar patterns, but with smaller magnitude and over a decreased salinity range.

Fig. 3.
Fig. 3.

The differential isohaline transport function −∂Q/∂s vs salinity, at the Columbia River section closest to the mouth, averaged over a spring–neap cycle. The gray line is the same function calculated from the Eulerian transport and salinity.

Citation: Journal of Physical Oceanography 41, 6; 10.1175/2011JPO4517.1

It appears that the total exchange flow is like an enhanced version of the Eulerian exchange flow. This interpretation is reinforced in Fig. 4, where the along-channel distribution of sin,out and Qin,out and their Eulerian versions are plotted. The isohaline exchange is generally larger than its Eulerian counterpart and transports a greater range of salinity values. A notable result in Fig. 4b is that the along-channel distribution of Qin,out is much smoother than that of . Moreover, the magnitude of Qin,out grows monotonically from river to ocean ends. To calculate the average TEF transport terms over the spring–neap cycle in Fig. 4, we simply average Qin,out over that time period. However, to ensure that the averaged salinity terms satisfy the salt balance, they are calculated, for example, as
e4.1
where denotes averaging over any period longer than tidal. It is these averages that are plotted in Fig. 4.
Fig. 4.
Fig. 4.

TEF (a) salinity and (b) transport vs x, averaged over a spring–neap cycle (black lines). For comparison, the same properties calculated from the Eulerian-averaged properties are plotted in gray. The two most landward values of are zero, and so is not defined at those points in (a).

Citation: Journal of Physical Oceanography 41, 6; 10.1175/2011JPO4517.1

The isohaline transport function Q(x, s) is plotted in Fig. 5. The Q contours of most outflowing water slope downward toward the mouth, indicating that outgoing water is being continually made saltier. This process requires turbulent mixing. Likewise, all inflowing water has Q contours sloping up toward the river end, indicating turbulent freshening. Notably, Q(x, s) is a relatively smooth surface. The maximum value of Q at any x location is equal to Qin at that section, and as seen in Fig. 4b its magnitude increases monotonically toward the mouth. If this were not the case, it would imply a closed recirculation cell in Q(x, s) space, which in turn would require a physically implausible increase of salinity of inflowing water.

Fig. 5.
Fig. 5.

(a) Contour plot of the isohaline transport function Q(x, s) averaged over the spring–neap cycle. The sense of the overturning circulation along Q contours is into the estuary for higher s and out for lower s. (b) Profiles of Q(s) at three different x values.

Citation: Journal of Physical Oceanography 41, 6; 10.1175/2011JPO4517.1

So far the results presented have been averaged over a complete spring–neap period. A sense of the temporal variation of terms is presented in Fig. 6, where low-passed terms are plotted versus time over the full 6-month duration of the simulation. Forcing terms are plotted in Fig. 6a. The run had about a dozen spring–neap cycles and a factor of 3 variation in QR. The Eulerian up-estuary salt flux terms tend to change from tidal-dominated during springs to exchange dominated during neaps (Fig. 6b). The TEF inflowing salinity tends to be very steady, with both sin and staying close to oceanic values (Fig. 6c). In contrast, sout and vary with river flow and tides. On average, ΔsEu is about two-thirds the size of Δs. The difference between isohaline and Eulerian versions is most extreme for the transports: is only about a quarter of Qin. Moreover, although tends to increase during neaps, Qin increases during springs. Also of interest, Qin shows very little variation with QR and, except for the spring–neap variation, holds remarkably steady at about 5000 m3 s−1 over the whole run.

Fig. 6.
Fig. 6.

Low-passed Eulerian salt flux and isohaline TEF terms plotted vs time for the section closest to the mouth: (a) river flow and tidal transport, (b) Eulerian salt flux terms, (c) TEF salinity, and (d) TEF transport. For reference, the dashed lines in (c),(d) show the same properties calculated from the Eulerian-averaged fields.

Citation: Journal of Physical Oceanography 41, 6; 10.1175/2011JPO4517.1

Many of the same dependencies are evident for the isohaline analysis in property–property plots (Fig. 7). It is clear that most properties depend on both river flow and tides. The sign of the trends for Δs, increasing with QR and decreasing with QTrms, seems consistent with conventional wisdom. It makes sense that −Qout increases with river flow because it has to be the sum of QR and Qin. Perhaps the greatest mystery is what sets the relatively steady value of Qin at 5000 m3 s−1. It does not appear to increase as as predicted by classical theories (reviewed in MacCready and Geyer 2010) and instead increases with tidal strength approximately as 0.15 × QTrms. At a more landward section (not shown) close to the dashed line in Fig. 2a where the tidal flux no longer dominates, there is almost no dependence of Qin on QTrms, and Qin decreases very weakly with QR. We leave it for future research to weave these patterns into a coherent parameterization.

Fig. 7.
Fig. 7.

Properties from the TEF analysis, for the section closest to the mouth, plotted vs river flow and tidal transport: (a),(b) salinity difference Δs = sinsout and (c)–(f) the transports.

Citation: Journal of Physical Oceanography 41, 6; 10.1175/2011JPO4517.1

5. Summary: Why do this?

In this paper, we have explored an alternate method for quantifying estuarine salt fluxes and exchange flow. Instead of using the classical Eulerian decomposition (e.g., Lerczak et al. 2006) we opt for calculating fluxes in an isohaline framework (e.g., MacDonald 2006). It is found that tidally averaged exchange flow is stronger and has a bigger difference between inflowing and outflowing salinities in the isohaline version.

The Eulerian analysis has the advantage that parts of it may be predictable through known theory. For example, we may use a theory based on the Hansen and Rattray (1965) solutions to predict u1 and s1 (e.g., Ralston et al. 2008). A disadvantage of the Eulerian approach is that current theories aimed at predicting the size of FT are developed in terms of KH and do not tell us about what salinity classes are transported by tidal correlations.

One advantage of the isohaline analysis is that its terms map directly into an exact version of the Knudsen relation. Another advantage is that it may be used readily in the calculation of residence times (MacDonald 2006), as
e5.1
Estimation of this for the Columbia River example gives times of about one day. Another interesting quantity which may be calculated is a “river amplification factor,”
e5.2
This could be useful for characterization of different estuarine systems, because it is a dimensionless expression of the exchange flow, a defining estuarine property. For the Columbia River simulation, αR ≈ 1. For an arrested salt wedge, it would be zero. In Puget Sound at Admiralty Inlet, it may be ~20 (Cokelet et al. 1991; Babson et al. 2006). Because of its connection to the residence time, αR may be a better way to characterize different estuaries than Hansen and Rattray’s (1965) “nu,”
e5.3
the “diffusive” fraction of up-estuary salt flux.

Finally, I believe that predicting Qin and associated isohaline properties represents an important goal for estuarine physics. What sets the size of Qin, its temporal variability, and its along-channel distribution? The apparent control of Qin by tides instead of the river in Figs. 7c,d, and its monotonic variation with x in Fig. 4b give some hope that there is a robust theory waiting to be found. Because TEF terms are made of integrals, they tend to combine all the physical processes that cause estuarine exchange. How can we parse them to get back to useful theoretical predictions? One way would be to focus on the tidal variation of the properties that cause transport in the salinity classes near sin or sout. Known physical processes (gravitational circulation, tidal pumping, tidal straining) may have very different temporal or spatial patterns of velocity and area (meaning cross-sectional area in a salinity class near sin,out). By focusing on the TEF salinity class, we may find which mechanism controls the salt flux on a section.

Acknowledgments

I wish to thank Rocky Geyer, Stephen Monismith, and Mark Stacey for formative discussions of the total exchange flow concept at the Estuarine Dynamics Summit in March 2010. Two anonymous reviewers contributed substantially to the final manuscript. This work was supported by NSF Grants OCE 0239089 and OCE 0849622.

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