1. Introduction
In estuaries, the exchange between ocean and river water is fundamentally important to the dynamics (Hansen and Rattray 1965) as well as biogeochemical processes such as nutrient fluxes, hypoxia, and contaminant transport (Sutherland et al. 2011). Exchange flow is not simply an advective process, because in order for exchange to occur, the incoming saltwater must be mixed with freshwater, as described by the Knudsen relation for estuarine exchange flow (Knudsen 1900). Therefore, the mixing of salinity is an essential ingredient of exchange flow.
Before examining in detail the relationship between exchange flow and mixing of salinity, it is important to establish a clear, quantitative definition of “mixing of salinity.” In the ocean turbulence community, the mixing of a tracer is defined by the tracer variance dissipation rate (Osborn and Cox 1972; Stern 1968; Nash and Moum 1999). This quantity was used by Burchard et al. (2009) to quantify the mixing of salinity in the Baltic Sea, and Becherer and Umlauf (2011) developed a temperature variance framework to study the mixing of temperature in lakes. Two recent studies have examined the relationship between exchange flow and the mixing of salinity in estuaries (Wang et al. 2017; MacCready et al. 2018). Wang et al. (2017) first quantified the estuarine volume-integrated salinity variance dissipation in an estuary based on the exchange flow, using the Hudson River estuary as a case study, based on the total exchange flow (TEF) transformation of the salt flux into salinity coordinates. While they established the relationship between exchange flow and salinity mixing, they did not frame the relationship in terms of the salinity variance equation. MacCready et al. (2018) also addressed the relationship between exchange flow and salinity variance dissipation, but using the conservation of salinity variance as a framework for the analysis. Using Knudsen relations (Knudsen 1900) to address the time-average regime, they derived a remarkably simple expression linking the exchange flow to the volume-integrated mixing of salinity and demonstrated its utility in a numerical simulation of an idealized estuary with constant river flow and a spring–neap modulation of tidal amplitude.
While the equations in MacCready et al. (2018) provide an integrated measure of the balance of salinity variance in an estuary, they do not address the mechanisms for the variations with river flow and spring–neap cycle and the mixing processes inside the estuary that actually accomplish this balance.
Total salinity variance can be decomposed into vertical and horizontal salinity variance, and vertical salinity variance can be used to represent the strength of stratification (Li et al. 2018). Stratification influences the strength of salinity variance dissipation, because it links salinity variance dissipation and turbulent buoyancy flux to turbulence production. Therefore, stratification (or vertical variance) links turbulence production to salinity variance dissipation, then to exchange flow through the salinity variance balance equations, and the relationships among them may shed light on the mechanisms of their variability with river flow and the spring–neap cycle.
In the present paper, to understand the mechanisms of their variability with river flow and tidal amplitudes in a realistic estuarine domain, we use the numerical model of the Hudson estuary to study the mixing processes inside the estuary, including salinity variance dissipation, turbulent buoyancy flux, and turbulence production, and how these mixing processes relate to exchange flow under different river conditions and spring–neap cycle in a partially stratified estuary.
The paper is organized as follows. In section 2, we briefly describe the numerical model of the Hudson estuary, the salinity variance balance equations, and the theoretical relationship among salinity variance dissipation, turbulent buoyancy flux, and turbulence production. In section 3, we examine the salinity variance balance in the Hudson estuary under steady state and study the variations of exchange flow, salinity variance dissipation, turbulent buoyancy flux, and turbulence production with river flow. In section 4, the influence of the spring–neap cycle on the variations of exchange flow, salinity variance dissipation, turbulent buoyancy flux, and turbulence production and the related mechanisms are studied. Section 5 presents the discussion and conclusions.
2. Methods
a. Numerical model of the Hudson estuary
The Hudson estuary model setup is identical to that used by Wang et al. (2017), which is based on a series of validated ROMS model studies (Warner et al. 2005a; Warner et al. 2010). The model consists of a 1133 × 530 × 16 curvilinear grid, including the New York Harbor, the Hudson estuary, and the East River. In the present paper, the estuarine region for analyzing is chosen from the Battery, that is, the mouth of the Hudson estuary, to the end of salt intrusion. Parameterizations for vertical eddy viscosity and diffusivity are determined using the k–ε turbulence closure scheme with the Canuto-A stability functions (Umlauf and Burchard 2005; Warner et al. 2005b). The horizontal diffusivity is set to be 0, so the horizontal mixing of salinity is neglected in the following analysis of this paper. To get the main features of tides in the Hudson estuary and study the balance between exchange flow and salinity variance dissipation for stationary forcing conditions, the ocean boundary is forced by M2 and S2 tidal constituents to simulate spring–neap variations. The river discharge in the Hudson estuary typically varies about from 150 to 2000 m3 s−1 between low and high discharge conditions. To study the response of the estuary to river flow under steady state, four cases are simulated with different constant river discharges at the river boundary, which are 200, 500, 1000, and 2000 m3 s−1 (Fig. 1).
(a) The Hudson estuary bathymetry, with the black line denoting the thalweg. The rightward direction in the figure indicates the north direction. (b)–(e) Model-generated spring–neap-averaged thalweg salinity with river discharge of 200, 500, 1000, and 2000 m3 s−1, respectively.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0032.1
b. The balance equations of estuarine salinity variance
Equations (2) and (3) reveal the balance between exchange flow and salinity variance dissipation integrated over the estuarine volume. To study how salinity variance is dissipated inside the estuary, we consider the relationship among salinity variance dissipation, turbulent buoyancy flux, turbulence production, and stratification, as discussed next.
c. Relationship among salinity variance dissipation, turbulent buoyancy flux, and turbulence production
d. Decomposition of salinity variance
3. Variations of exchange flow and mixing processes with river flow
In this section, the Hudson estuary model outputs are used to test the validity of Eq. (3) in representing the relationship between salinity variance dissipation and exchange flow in a realistic domain. We also study the influence of the variations of river flow on exchange flow and mixing processes.
By averaging over a spring–neap cycle, the time-averaged regime under different river flow conditions is assessed. As river discharge increases, the salt intrusion becomes shorter and stratification becomes stronger (Fig. 1), as expected from classical estuarine theory (MacCready and Geyer 2010). The TEF calculations of exchange flow and flux-weighted salinities (averaged over the spring-neap cycle) for the different river discharge cases are shown in Figs. 2a and 2b. While the outflow volume Qout and outflow salinity Sout vary strongly with river flow QR, the inflow volume flux Qin and inflow salinity Sin are almost invariant with river flow. A similar result is also found by MacCready (2011) in application to the Columbia River estuary. The relationships among these variables are consistent with the Knudsen relations.
Dependence of exchange flow and advective variance fluxes on river discharge, including (a) spring–neap-averaged inflow volume flux Qin (orange dots) and outflow volume flux −Qout (blue triangles) at the mouth for four different river discharges. (b) Spring–neap-averaged flux-weighted inflow salinity Sin and outflow salinity Sout at the mouth. (c) Magnitudes of the terms in Eq. (2), including advective salinity variance flux driven by exchange flow at the mouth (orange dots and blue triangles) and by river flow (gray plus signs), and net salinity variance input (green asterisks), which is equal to the dissipation term in Eq. (2).
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0032.1
Using TEF analysis, the advective salinity variance fluxes at the open boundaries are quantified (Fig. 2c). When calculating the estuarine volume-averaged variables, the estuarine volume is chosen from the estuarine mouth to the river end where no salt reaches. Therefore, for different river flow conditions, because the length of salt intrusion differs, the estuarine volume for calculating is different. As river discharge QR increases, the incoming salinity variance flux 
As Eq. (2) shows, the net input of salinity variance balances the estuarine volume-integrated salinity variance dissipation under steady-state conditions. Here we define the volume-integrated salinity variance dissipation quantified with the convergence of the advective terms in Eq. (2) as the full salinity variance dissipation (green asterisks in Fig. 3). Because horizontal and molecular mixing are neglected in the model, the spring–neap averaged and estuarine volume-integrated model-resolved dissipation due to vertical mixing can be quantified with 
Comparison of the full, model-resolved and approximate estuarine volume-integrated salinity variance dissipation. The green asterisks indicate the full salinity variance dissipation quantified with the convergence of the advective terms in Eq. (2), which are the same as the green asterisks in Fig. 2. It includes both the model-resolved dissipation due to physical mixing and unresolved dissipation due to numerical mixing. The gray triangles indicate the model-resolved salinity variance dissipation due to physical mixing, which are quantified with the model-resolved diffusivity Kz. The orange dots indicate the approximate salinity variance dissipation quantified with Eq. (3).
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0032.1
The integrated salinity variance dissipation can also be approximated using the simple relation in Eq. (3) with the values of Qin, Sin, and QR. This approximation (orange dots in Fig. 3) slightly overestimates the exactly full salinity variance dissipation (green asterisks in Fig. 3), revealing that under steady-state conditions, exchange flow and salinity variance dissipation in the Hudson estuary roughly satisfy the simple relationship as Eq. (3) shows. The errors mainly come from the approximations of 
Next we consider the influence of river flow on the mixing mechanisms. As shown in Eq. (9), the magnitude of salinity variance dissipation depends on the magnitudes of shear production and stratification inside the estuary. Using Eqs. (5) and (8), the estuarine volume-integrated and spring–neap-averaged shear production 
Spring–neap-averaged magnitudes of shear production, turbulent buoyancy flux, and salinity variance dissipation under different river discharge conditions. Both of the estuarine volume-integrated and volume-averaged values are shown. The estuarine volume-averaged vertical salinity variance 
4. Variations of exchange flow and mixing with spring-neap tides
a. Balance of salinity variance during the spring–neap cycle
Whereas the spring–neap-averaged result represents a simple balance between advective inputs and dissipation of salinity variance, the intensity of the various terms varies considerably through the spring-neap cycle due to the temporal variations of total salinity variance [the left-hand-side term in Eq. (2)]. In this section, the model results with intermediate river discharge conditions 500 m3 s−1 are used as an example to analyze the spring–neap variations. The spring–neap variations under the other river discharge conditions are similar to the results with river discharge of 500 m3 s−1. The tidally averaged values in the following analysis are all obtained through 33-h low-pass filtering.
Using Eq. (10), the spring–neap variations of tidally averaged total, vertical, and horizontal salinity variance are quantified (Fig. 4). Total variance peaks just after neap tides, with a minimum just after spring tides. Horizontal variance shows a similar but much more subtle spring–neap variation, but vertical variance shows marked spring–neap variation, almost vanishing during spring tides and contributing most of the increase of total salinity variance during neap tides.
Spring–neap variations of the estuarine volume-integrated and tidally averaged total, vertical, and horizontal salinity variance inside the estuary under a 500 m3 s−1 river discharge condition.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0032.1
Total salinity variance is supplied by the advective terms at the mouth and river end and is dissipated by the salinity variance dissipation. Vertical variance (stratification) in turn influences the magnitude of salinity variance dissipation. To study how total salinity variance, exchange flow, and salinity variance dissipation are balanced during the spring–neap cycle, the spring–neap variabilities of the individual terms in the salinity variance equation [Eq. (2)] are analyzed. The tidally averaged terms in Eq. (2) are shown in Fig. 5. Not only does the dissipation term vary due to changes in stratification and turbulence intensity through the spring–neap cycle, but the advective input of variance also varies due to changes in the strength of the exchange flow and the salinities of the inflowing and outflowing water. The advective input of variance reaches its peak just after neap tides, and the dissipation term peaks between neap and spring. The competition between the advective input and dissipation results in the temporal variation of total salinity variance inside the estuary, which increases during the transition from spring to neap tides, and decreases during the transition from neap to spring tides.
Spring–neap variations of the three terms in Eq. (2) under a 500 m3 s−1 river discharge condition.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0032.1
The elements of the advective salinity variance input are shown in Fig. 6. River flow and the inflow branch of exchange flow always drive salinity variance into estuary, and the outflow branch of exchange flow removes salinity variance (Fig. 6a). Tidally averaged salinity variance flux driven by river flow is almost constant during the spring-neap cycle (Fig. 6a), so the variation of the net variance input mainly depends on the part that is driven by exchange flow, that is, 
Spring–neap variations of the elements of the advection term (blue line in Fig. 5) under a 500 m3 s−1 river discharge condition, including (a) the salinity variance flux related to exchange flow and river flow. The blue solid line is the same as the blue line in Fig. 5. Note that Qout is negative. (b) Inflow and outflow volume flux and (c) flux-weighted salinity variance.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0032.1
b. Mechanisms for the spring–neap variation of salinity variance dissipation
To study the mechanisms for the spring–neap variation of salinity variance dissipation, the relationship among the variations of salinity variance dissipation, turbulent buoyancy flux, shear production, and stratification is discussed. Figure 7 indicates that the maximum shear production occurs during maximum spring tide, but the maximum turbulent buoyancy flux occurs several days before maximum shear production (Fig. 7b). As Eq. (4) shows, compared to turbulent buoyancy flux, salinity variance dissipation is more sensitive to stratification, so the maximum salinity variance dissipation occurs closer to neap tides (Fig. 7c), when the stratification is maximal, as indicated by the vertical salinity variance (orange dashed lines on Fig. 7). Therefore, although shear production reaches its maximum due to strong tidal current amplitude, neither the turbulent buoyancy flux nor salinity variance dissipation reaches its maximum due to weak background stratification. In fact, the temporal variation of salinity variance dissipation is almost proportional to stratification (Fig. 7c) over the spring–neap cycle. Both the flux Richardson number and the mixing ratio 
Spring–neap variations of the tidally averaged and estuarine volume-integrated mixing variables under a 500 m3 s−1 river discharge condition. The tidally and estuarine volume-averaged vertical salinity variance 
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0032.1
The dependence of salinity variance dissipation on stratification is also clear in the spatial distributions of P, −B, and χs during maximum spring and minimum neap tides (Figs. 8, 9). During the spring tide, both the turbulent buoyancy flux and salinity variance dissipation mainly occur in the region influenced by the bottom boundary layer shear stress (Fig. 8). During the flood tide, turbulent buoyancy flux and salinity variance dissipation mainly occur above the bottom due to the weak stratification near the bottom (Fig. 8a). During the ebb tide, the structures of turbulent buoyancy flux and salinity variance dissipation are similar to shear production, which propagate from the bottom near to the surface (Fig. 8b). Shear production is also found to be enhanced at some sections with an abrupt change of depth along the channel, as discussed in detail in Wang et al. (2017). During the neap tide, bottom shear production is limited under the halocline (Fig. 9). Turbulent buoyancy flux occurs both in the bottom boundary layer and halocline, especially during the flood tide. In contrast, most of the salinity variance dissipation occurs near the halocline, where stratification is the strongest (Fig. 9). Comparing spring to neap tides, even though the shear production is stronger during the spring tide than during the neap tide (Figs. 8, 9), the magnitude of salinity variance dissipation is much smaller owing to weak stratification during spring tides (Fig. 8). The turbulent buoyancy flux is stronger during spring tides, because it is less sensitive to stratification than salinity variance dissipation.
Snapshots of the longitudinal distributions of shear production, turbulent buoyancy flux, and salinity variance dissipation at (a)–(c) flood and (d)–(f) ebb tides during the maximum spring tide. In (a) and (d), colors indicate the log values of shear production, and gray contours indicate the isohalines with a 2 g kg−1 interval. Colors indicate the log values of turbulent buoyancy flux in (b) and (c) and salinity variance dissipation in (e) and (f). The dashed black line in each panel indicates the line separating the bottom boundary layer and internal shear layer, which is used for the decomposition in Fig. 10.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0032.1
As in Fig. 8, but during the minimum neap tide.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0032.1
The effects of wind and waves are not included in the Hudson estuary model, so the amount of turbulent buoyancy flux and salinity variance dissipation can be divided into two parts caused by two different mechanisms, respectively: the part that is generated at the bottom boundary layer and the other part that is caused by mixing within an internal shear layer. To quantitatively divide the salinity variance dissipation caused by the two mechanisms, we follow the method in Ralston et al. (2010), requiring a local minimum in shear stress to distinguish an internal shear layer from the bottom boundary layer. The turbulent buoyancy flux and salinity variance dissipation below the depth of stress minimum are attributed to bottom boundary layer shear and above the stress minimum are attributed to internal shear. As shown in Fig. 10, during spring tides, a larger part of the turbulent buoyancy flux is induced by the bottom boundary layer stress, and during neap tides, the turbulent buoyancy flux induced by the two mechanisms is comparable. In contrast, because of more sensitivity to stratification, salinity variance dissipation shows a much greater role in the internal shear stress during neap tides and has comparable contributions during spring tides.
Spring–neap and tidal variations of (top) turbulent buoyancy flux and (bottom) salinity variance dissipation induced by bottom boundary layer shear stress and internal shear stress, respectively. The letters A, B, C, and D indicate the time of flood and ebb during maximum spring and minimum neap tides, respectively, i.e., corresponding to the time of the snapshots in Figs. 8 and 9.
Citation: Journal of Physical Oceanography 48, 12; 10.1175/JPO-D-18-0032.1
5. Discussion and conclusions
We have studied the relationship among the variations of turbulence production, turbulent buoyancy flux, salinity variance dissipation, and stratification with river flow and the spring–neap cycle and how they are related to exchange flow in the Hudson estuary. As river flow increases, estuarine volume-integrated turbulence production decreases because salt intrusion becomes shorter, but salinity variance dissipation inside the estuary increases due to more net input of salinity variance, particularly from the increased river flow. The increased input of salinity variance during high river flow results in increased vertical variance, that is, increased stratification, which leads to more efficient conversion of turbulent energy to salinity variance dissipation, as quantified by the mixing ratio 
Under unsteady-state conditions, as illustrated by the spring–neap cycle, the competition between the source, that is, advective input of variance driven by exchange flow and river flow, and the sink, that is, dissipation due to mixing, results in the temporal variation of total variance inside the estuary. Most of that variation in the Hudson is accounted for by the variation in vertical variance, with the horizontal variance remaining nearly constant. The vertical variance is uniquely associated with stratification, which strongly influences the magnitude of salinity variance dissipation, as quantified by the mixing ratio. The mixing ratio increases by more than an order of magnitude between spring and neap tides, leading to the dominance of salinity variance dissipation during neap tides. During neap tides, most of the salinity variance dissipation occurs in the halocline as a result of internal layer shear stress. During most of the transition time from spring to neap tides, the advective input of the variance is larger than the dissipation, resulting in the net increase of the total variance, as well as increasing the vertical and horizontal variance. Therefore, stratification is intensified, and salinity variance dissipation induced by internal shear stress increases near the halocline. During most of the transition time from neap to spring tides, when dissipation becomes larger than the advective input of variance from the open boundaries, variance inside the estuary decreases and stratification is destroyed.
While this paper only considers the variance balance in the context with a single estuary, this approach has promise for comparing mixing processes between estuaries of different types. To remove the influence of different estuarine volumes, the volume-average estimates of shear production, turbulent buoyancy flux, and salinity variance dissipation shown in Table 1 would provide useful comparisons. For instance, MacDonald and Geyer (2004) reported turbulent buoyancy fluxes in the lift-off plume of the Fraser River of more than 10−4 m2 s−3, more than two orders of magnitude higher than reported here. In the salt wedge of the Connecticut River, Holleman et al. (2016) reported local values of χs of more than 0.1 (g kg−1)2 s−1, more than three orders of magnitude larger than the average values of the Hudson River. These are not fair comparisons, because neither the Fraser River nor Connecticut River values are estuarine averages, which would be expected to be considerably lower than the local values in regions of strong mixing. However it is likely that highly stratified estuaries will exhibit much higher values of χs and also of the mixing ratio 
Acknowledgments
TW was supported by the National Key R&D Program of China (Grant 2017YFA0604104), National Natural Science Foundation of China (Grant 41706002), Natural Science Foundation of Jiangsu Province (Grant BK20170864), and MEL Visiting Fellowship (MELRS1617). WRG was supported by NSF Grant OCE 1736539. Part of this work is finished during TW’s visit in MEL and WHOI. We would like to acknowledge John Warner for providing the codes of the Hudson estuary model, and Parker MacCready, the editor, and two reviewers for their insightful suggestions on improving the manuscript.
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