1. Introduction
An estuary is a bay off the ocean whose circulation and density structure are affected by (i) a source of buoyancy such as a river or net precipitation and (ii) mixing and momentum input due to tides or wind. These conditions give rise to the exchange flow, a persistent inflow of ocean water and outflow of brackish water at the mouth (Hansen and Rattray 1965; MacCready and Geyer 2010), shown schematically in Fig. 1. The exchange flow can be regarded as the volume of water entering the estuary whose salinity is altered by mixing before exiting. Recent research has shown that the physics driving the exchange flow can have surprising complexity (Geyer and MacCready 2014) with the momentum input of tides or wind being important. However, from the earliest analyses (Knudsen 1900; Pritchard 1954; Hansen and Rattray 1965; Chatwin 1976; Walin 1977) onward it has been clear that the creation of mixed water is of central importance. This is most clearly evident in the Knudsen relations (Burchard et al. 2018), and presented in section 2, which demonstrate that the volume flux of the exchange flow may be many times greater than that of the river flow, and this amplification is dependent on the creation of mixed water. While the dependence on mixing is implicit in the Knudsen relations, a direct connection to the rate of mixing by turbulence has been elusive, in part because the word “mixing” may have different definitions in different contexts. The purpose of this paper is to show how a specific definition of mixing, one defined relative to salinity variance, may be linked quantitatively to the exchange flow.
Sketch of an estuarine along-channel section, indicating isohalines (gray lines), turbulent mixing (curly arrows), and volume transports through the open boundaries. Conceptually the estuary takes in river and ocean waters at rates Qr and Qin, which are the water-mass end members with high salinity variance. Mixing fills the estuary with a gradation of salinities, all with lower variance, and some part of this mixed water is exported at rate Qout.
Citation: Journal of Physical Oceanography 48, 6; 10.1175/JPO-D-17-0266.1
Definitions of mixing
Most commonly, mixing in the oceans (Polzin et al. 1997; Wunsch and Ferrari 2004) and mixing in estuaries (Fischer 1976; Basdurak et al. 2017) is associated with eddy viscosity or diffusivity. However, this mixing measure is not suitable here, since in well-mixed estuaries, when salinity is already fully mixed, salinity mixing vanishes despite high values of eddy diffusivity. Therefore, many estuarine researchers treat mixing in reference to the vertical component of Reynolds-averaged turbulent salt flux (Peters and Bokhorst 2001; Li and Zhong 2009). In general, the vertical component of mixing dominates fluxes in geophysical situations because vertical density (or salinity) gradients are typically one or more orders of magnitude greater than horizontal gradients. The turbulent and advective salt fluxes through isohaline surfaces (Walin 1977; MacCready et al. 2002; Wang et al. 2017) have been used as one way to relate the exchange flow to mixing. The fundamental result of this line of inquiry was that the inflow of salt due to the exchange flow must be balanced on average by turbulent and advective salt flux through an isohaline. What has been missing is to connect this back to the Knudsen relations. Wang et al. (2017) explicitly treated mixing using the salinity variance, and the current work builds on that.
Another main tool for the analysis of mixing has been in terms of the buoyancy flux in budgets of potential energy (Simpson et al. 1990; Garvine and Whitney 2006; MacCready et al. 2009; Cessi et al. 2014; Biton and Gildor 2014) or available potential energy (MacCready and Giddings 2016). One disadvantage of this analysis method, as Burchard et al. (2009) point out, is that the sign of the buoyancy flux changes between shear-driven and convectively driven mixing, and it does not represent horizontal mixing processes at all. In addition, the volume-integrated results that relate buoyancy flux to the exchange flow depend on the depth of the interface between inflowing and outflowing branches of the exchange flow at the mouth. In energetic or wide estuarine systems, this depth is difficult to define because of tidal or spatial variability. We find below that these problems are avoided by use of the salinity variance to define mixing.
The dissipation of salinity variance, often designated as χs, has long been recognized by the ocean turbulence community as one of the most important quantities representing mixing in the ocean (Nash and Moum 2002; Oakey 1982). In fact for an ocean environment dominated by salinity rather than temperature variance, it is a direct and unambiguous representation of mixing (i.e., the destruction of scalar variance by molecular processes; Tennekes and Lumley 1972). Given the fundamental, unambiguous quality of χs as a measure of mixing, it suggests that the salinity variance equation may be the best framework for addressing mixing in environments like estuaries in which density variations are dominated by salinity. Indeed Stern (1968) noted that the input of salinity variance at global scales by evaporation and precipitation has to be balanced on average by the dissipation of salinity variance at molecular scales. A similar balance is obtained in estuaries, but the source of variance derives from the freshwater and ocean inflows.
The most notable recent application of the salinity variance equation with relevance to estuaries is the paper by Burchard and Rennau (2008) in which they use the conservation of salinity variance as a means of quantifying numerically induced mixing in coastal ocean models. More relevant to the results presented here, however, is the application of the salinity variance equation by Burchard et al. (2009) to identify the extreme spatial and temporal variability of mixing in the Baltic Sea. In addition, Wang et al. (2017) use the total exchange flow (TEF) method to relate the exchange flow to volume-integrated mixing in a model of the Hudson River estuary.
2. Equation development
3. Model estuary results
We use a numerical simulation of an idealized estuary-shelf system (Figs. 2 and 3) forced with a steady river flow and two tidal constituents that give rise to a spring–neap cycle. This particular configuration was designed to be as simple as possible while still spanning a wide range of estuarine parameter space, from well mixed to strongly stratified on the Geyer and MacCready (2014) parameter space diagram (their Fig. 6). In fact, the parameters that form the axes of that diagram were used to choose the simulation parameters. While this does not give an exhaustive exploration of parameter space (e.g., fjords are neglected), it serves as a useful first test of the use of the variance budget. The simulation is done using the Regional Ocean Modeling System (ROMS; Shchepetkin and McWilliams 2005; Haidvogel et al. 2000), which solves the hydrostatic, incompressible, Reynolds-averaged momentum and tracer conservation equations with a terrain-following vertical coordinate and a free surface. The horizontal domain is a spherical grid extending from longitude −1° to 3° and latitude 44° to 46°N. The grid resolution is 1000 m in the estuary and stretches to 5000 m at the boundaries. The model is forced with 1500 m3 s−1 river flow at the eastern end and is initialized with constant salinity of 35 (and zero in the river). Radiation boundary conditions and nudging to initial salinity are used on the open ocean boundaries to maintain the ocean salinity and allow the river plume to exit the domain. The only other forcing is a tidal sea surface height variation on the open boundaries, at M2 and S2 frequencies, with amplitudes of 0.75 and 0.25 m, respectively. This gives rise to a pronounced spring–neap cycle. There are 40 vertical layers, and vertical mixing is parameterized using k–ε (turbulent kinetic energy dissipation) with the Canuto-A stability functions (Umlauf and Burchard 2005). Bottom drag followed a quadratic stress law with drag coefficient 3 × 10−3. The model was run for 3 months with a baroclinic time step of 30 s and had established a nearly repeating spring–neap cycle after the first month. The third month was used for all analysis presented here. Snapshots (history files) were saved hourly, and averages of terms, including salt fluxes, over each hour were also saved. These allowed for construction of budgets with near-perfect conservations of volume and salt, although as we will see salinity variance is still subject to numerical mixing. A 24–24–25 Godin filter (Emery and Thomson 1998) was used for all tidal averaging.
Model (top) surface salinity and (middle; bottom) sections, during highly stratified, neap tide conditions. (top) The black lines show the locations of the sections, with “o” marking the zero distance of the middle and bottom panels. The model domain extends well beyond the limits shown in (top). The dimensions of the whole domain are 315 km by 223 km, and the dimensions of the portion shown are 118 km by 45.5 km. The volume of integration for the analysis goes from the estuary mouth to 1.5° landward. The time of this snapshot corresponds to day 12 of the time series plots.
Citation: Journal of Physical Oceanography 48, 6; 10.1175/JPO-D-17-0266.1
As in Fig. 2, but during well-mixed, spring tide conditions. The time of this snapshot corresponds to day 19 of the time series plots.
Citation: Journal of Physical Oceanography 48, 6; 10.1175/JPO-D-17-0266.1
The estuary stratification varies from highly stratified (Fig. 2) to well mixed (Fig. 3) over the spring–neap cycle. The effects of this time variation on the section-integrated transports and stratification are shown in TEF terms in Fig. 4, which covers the third month of the simulation. To form the TEF terms, we start with hourly values of velocity and salinity on a cross section of the estuary, in this case one near the mouth. The volume transport through this section in each model grid cell is binned according to its hourly salinity, in this case using 1000 salinity bins between 0 and 35. The transport in each salinity class is then tidally averaged. From this tidally averaged transport we determine at each time a “dividing salinity” above which the tidally averaged transport is landward and below which the transport is seaward (see appendix). Integrating the transport on either side of this dividing salinity gives us Qin and Qout. We may also form the average salt flux of the in- and outflow streams by integrating the transport times the salinity of each bin on either side of the dividing salinity. Then dividing these by Qin or Qout gives us Sin and Sout. The same procedure may be generalized to find the flux-weighted average of other quantities, as is done for salinity variance below. The volume-integrated salt budget (Fig. 4a) is dominated by the in- and outflow of the exchange flow. The timing of spring and neap are given by the tidal forcing, expressed as the hourly volume transport through the mouth (Fig. 4b). The volume transports (Fig. 4c) show that the exchange flow is amplified by up to 7 times over the river flow, so in this sense the system is clearly acting like an estuary. The stratification at the mouth, quantified as the transport-weighted salinities of the exchange flow, shows strong temporal variation, going from ΔS ≅ 13 to 1. The neap-tide restratification and increase in volume-integrated salt are consistent with common understanding. The exchange flow is weak during well-mixed spring tide conditions, and, at that time, the system is losing net salt at a rate close to the river flow times the average salinity.
Terms from the TEF budget. (a) The full, volume-integrated salt budget, and (b) the tidal transport at the mouth, giving the timing of spring and neap forcing. (c) The TEF volume transport terms and (d) the transport-weighted TEF salinities. The small variation of Qr in (c) is due to subtidal variation of surface height landward of the section where it is calculated. The estuary develops an exchange flow that is many times greater than the river flow. During well-mixed conditions, the exchange flow decreases markedly, and the salinity difference of incoming and outgoing streams becomes small. Times of peak spring and neap tides are marked with “S” and “N” on this and subsequent figures.
Citation: Journal of Physical Oceanography 48, 6; 10.1175/JPO-D-17-0266.1
The time history of the terms in the variance budget [(5)] is shown in Fig. 5a. There is significant time variability of the variance budget, with net rate of increase driven by advection early in neaps and loss driven by mixing at the neap-to-spring transition. Splitting the advection up into its three components (Fig. 5b), the deep inflow of ocean water is the main driver of the early-neap increase, aided by a relatively steady contribution from the river source of freshwater. The outflow at the mouth of brackish water is always a loss term for the net variance. During spring, the exchange of variance at the mouth adds up to almost nothing, and the only source is from the river. The volume-average variance remains relatively steady (Fig. 5c), indicating that much of the variance is contained in the along-channel salinity gradient and not in the vertical stratification.
(a) Terms in the salinity variance budget [(5)], where 
Citation: Journal of Physical Oceanography 48, 6; 10.1175/JPO-D-17-0266.1
One useful by-product of the variance budget is that we may use the error, computed as the residual of the budget, as an estimate of the numerical mixing (Burchard and Rennau 2008; Wang et al. 2017) in the model (Fig. 5a). The error arises because the resolved mixing, calculated as 
Turning to the mixing side of the budget, the resolved mixing and the full mixing (i.e., including numerical mixing) are plotted versus time in Fig. 6a. They have very similar shapes, both peaking at the neap-to-spring transition when strong stratification encounters strong turbulence. As expected, the full mixing is always greater than the resolved mixing. The approximate mixing term in (6) is also plotted and is found to overestimate the peak mixing by about the same amount that the resolved mixing underestimates it. However, the most important result we can get from the approximate budget in (6) is the portion SinSoutQr after averaging over the large spring–neap variation. Averaging over the complete time series using the method described in MacCready (2011), we find Sin = 33.36 and Sout = 25.37. The estimated average mixing is SinSoutQr = 1.27 × 106 (g kg−1)2 m3 s−1, only about 12% greater than the record-mean full mixing (including numerical mixing), which is 1.13 × 106 (g kg−1)2 m3 s−1. This level of error in (7) is surprisingly small given the order-one variation of actual mixing over the spring–neap cycle, suggesting that (7) is a reasonable long-time mean estimate of mixing in estuaries. These mixing values are about 68% of the theoretical maximum 
(a) Three versions of the tidally averaged net mixing. The green “resolved mixing” is M, the negative of the same curve in Fig. 5a. The magenta curve, full mixing, is defined as M + numerical mixing, the mixing based on all other terms in the variance budget. The approximate mixing based on TEF terms (light blue) is equal to the right side of (6). All three estimates are similar, with large spring–neap variation. (b) Two versions of the net advection of variance, where 
Citation: Journal of Physical Oceanography 48, 6; 10.1175/JPO-D-17-0266.1
4. Discussion and conclusions
The primary result of this work is the relation in (7), which states that the long-term average mixing in an estuary is approximately given by SinSoutQr, or alternatively SinΔSQin. The significance of this is that it unambiguously relates the mixing to the exchange flow of an estuary. The relationship relies on a specific definition of mixing, one that appears in the variance budget, as opposed to other possible definitions such as the buoyancy flux from the mechanical energy budget. Using an idealized numerical simulation, we have explored this estimate in detail, showing in particular that it only applies to long-term averages. We have limited our exploration to volume integrals of the variance; however, there is potentially much more to be gained in terms of understanding the system function by further decomposition. For example, the full variance can be separated into parts due to horizontal and vertical salinity gradients (Li et al. 2018); to good approximation only the vertical part is subject to mixing.
With respect to the time variability of mixing, this simulation indicates that variance is greatest approaching the neap tide, because of the combination of strong advective input of variance associated with high-salinity inflow and weak mixing. The intensity of mixing increases markedly with increased stratification, even during the weaker turbulence conditions of the neap tide, and the variance drops sharply during this strongly stratified period. The analysis by Wang et al. (2017) of the dissipation of variance in a realistic model of the Hudson estuary shows a similar phasing of mixing, with the peak occurring several days after the neap tide. Estuaries with stronger or weaker stratification would likely show different phase relationships among the terms in the variance budget.
In terms of consequences to the conditions on the shelf, the salinity of river plumes depends entirely on the amount of mixing that happens in the estuary. This analysis, and especially (7), gives a quantitative way to estimate the amount of mixing that gave rise to any given river plume.
One interesting use of the mixing defined here is that it clearly relates to mixing efficiency (understood as buoyancy flux divided by net turbulence production in a mechanical energy budget). In our simulation, the smallest mixing occurred during springs, when the estuary became vertically well mixed, and in this case, the mixing efficiency has dropped to near zero. This provides an interesting elaboration of the idea of “overmixing” (Stommel and Farmer 1953; Hetland 2010), where the hydraulic control of the exchange flow led to a limit on the amount of mixing that could be supported landward of the control point. In our simulations, as in many estuaries, the regulation of mixing is more dynamic, being strongly modulated in time by the spring–neap cycle. This could be quantified across systems as an added dimension for parameter space diagrams (Geyer and MacCready 2014) in which the net mixing of a system is compared to the maximum possible mixing that the system could support.
This research benefited substantially from discussions with Tao Wang and Elizabeth Brasseale. The work was supported by the National Science Foundation through Grants OCE-1736242 to PM and OCE-1736539 to WRG and by the German Research Foundation through Grants TRR 181 and GRK 2000 to HB. The result in (7) was derived independently by WRG and HB.
APPENDIX
Note on the TEF Calculation Method
The well-mixed periods of the spring–neap cycle revealed an important flaw in the standard way of calculating TEF terms. In the usual TEF calculation, the transport is first binned by salinity class and then tidally averaged. Then one calculates Qin, for example, by adding up the transport of all salinity bins for which the transport is positive (MacCready 2011). For reasonably stratified flow, this is relatively insensitive to the number of salinity bins used. However, in well-mixed water the transport in salinity classes can be noisy, and with more salinity bins this noise can result in transport that changes sign many times over the range of salinity classes. When masking by the sign of transport to calculate Qin and Qout, this noise is aliased into increased values of the TEF transports, and the calculated transports grow with increasing numbers of salinity bins. This issue was noted in Lemagie and Lerczak (2015) during times of low stratification. To fix this problem we employ a different method for calculating TEF quantities, one also used in Wang et al. (2017). The strategy is to use the relatively smooth variation of the isohaline transport function, called Q(s) in MacCready (2011, his Fig. 5), which is the integral over salinity of the transport in salinity classes. We integrate from the high-salinity end and then find the salinity at which the transport function is a maximum. Then Qin is the integral of transport in salinity bins above this salinity, and Qout is the integral of transport in fresher salinity bins. The same dividing salinity is used for calculation of the salt flux. Experiments showed that the TEF quantities calculated using this procedure were insensitive to the number of salinity bins over a range from 100 to 1000 (1000 used here).
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