## 1. Introduction

*χ*

^{s}(Nash and Moum 2002; Burchard and Rennau 2008; Burchard et al. 2009; Wang et al. 2017; Li et al. 2018; MacCready et al. 2018; Burchard et al. 2019; Warner et al. 2020), since it best characterizes the transformation of river water to ocean water as discussed by MacCready et al. (2018) and Burchard et al. (2019). By evaluating the salinity variance budget in terms of the Knudsen relations (Knudsen 1900), MacCready et al. (2018) relate the long-term average, estuary-wide mixing

*M*to the characteristics of the exchange flow and river discharge,

*s*

_{in}and

*s*

_{out}are the bulk values of the salinities of the inflow and outflow, and

*Q*

_{r}is the river discharge. For the computation of these bulk values, the total exchange flow (TEF) analysis framework (MacCready 2011) is used. Burchard et al. (2019) generalized the approximate long-term averaged mixing relation (1) to an exact transient relation. The mixing

*M*can be used to compare the mixing of estuaries with each other independent of the mechanisms responsible for the mixing, e.g., tidal with nontidal estuaries.

While the existing mixing relations, either stationary or transient, can be applied to classical estuaries with river runoff as the only freshwater forcing, they do not cover inverse estuaries since evaporation is neglected (MacCready et al. 2018; Burchard et al. 2019). Furthermore, tropical and subtropical estuaries can be a combination of inverse and classical estuaries (Valle-Levinson 2010). Strong evaporation may form so-called salt plugs, a region of maximum salinities greater than the ocean salinity inside the estuary (Wolanski 1986). For such estuaries, the freshwater forcing may change between seasons, i.e., the estuary may change from a classical estuary type during the wet season to an inverse type in dry season (Valle-Levinson and Bosley 2003).

Simply replacing the incoming river runoff *Q*_{r} in (1) with the outgoing evaporation transport *Q*_{evap}, *Q*_{evap} < 0, would result in *M* < 0. For the exchange flow, this might make sense, as the inflowing water is transformed into more saline outflowing water which requires demixing of the inflowing water since no salt is added in the estuary. But *M* is by definition a strictly positive number. As we will show in this study, this problem is solved by adding a cross-surface salinity square transport proportional to the square of the surface salinity, ensuring *M* ≥ 0. Stern (1968) already states that salt variance added by surface fluxes has to be destructed on microscales due to molecular diffusion to achieve a steady-state solution for the World Ocean.

By this generalization of the Knudsen (1900) relations and the MacCready et al. (2018) mixing relation, not only estuaries may be characterized, but also bays, lagoons, and shallow coastal zones which are influenced by precipitation or evaporation. Furthermore, these relations provide a very good estimate of mixing by evaluating boundary transports without the need for a three-dimensional numerical model.

The paper is structured as follows: first, we derive the exact mixing relation including the cross-surface freshwater transport in section 2, as an extension to the relations presented in MacCready et al. (2018) and Burchard et al. (2019). We further simplify the exact relation by different assumptions that make it easier to apply. In section 3 we present a schematic box model explaining the new, simplified mixing relation for a strictly inverse estuary. In section 4 we present the application of the new mixing relations to idealized, two-dimensional estuary simulations, and the first application to a realistic, three-dimensional simulation, the Persian Gulf. In section 5 we discuss the results of this study and conclude.

## 2. Derivation of the mixing relations

### a. Budget equations

#### 1) Volume

*x*,

*y*,

*z*), the time

*t*and the velocity vector (

*u*,

*υ*,

*w*). We start at the Reynolds-averaged continuity equation in Boussinesq approximation,

*η*and the bottom −

*H*,

*E*is the evaporation rate and

*P*the precipitation rate (both in m s

^{−1}), and

*H*is the water depth. Carrying out the volume integration over the whole volume

*V*of the estuary and considering the kinematic boundary conditions, (3) and (4), the continuity equation (2) reads as

*A*

_{b}describes the open boundary area,

*A*

_{r}the river boundary area for which we assume that the river water is fresh and the salinity along

*A*

_{r}is always zero,

*A*

_{surf}the surface area, and

*u*

_{n}denotes the normal velocity component (positive outwards).

#### 2) Salinity

*K*

_{h}and

*K*

_{υ}denote the horizontal and vertical eddy diffusivities, respectively, and

*s*the salinity. Volume integration of the salinity equation (6) requires boundary conditions for the diffusive salt flux. We demand that there is no net salt flux through the surface and the bottom. The surface boundary condition for salt flux reads therefore as

_{n}denoting the derivative normal to the transect. Furthermore, we define the tracer flux

*c*being a generic tracer, in Eq. (9)

*c*=

*s*. Equation (9) only allows exchanging salt through the open ocean boundary.

#### 3) Salinity square

*s*leads to an equation for

*s*

^{2},

*s*, one finds corresponding boundary conditions for

*s*

^{2},

#### 4) Salinity variance

*s*′ one finds

*χ*

^{s}describes the destruction of salinity variance (and salinity square) per unit volume by turbulent eddies. For steady-state, homogeneous turbulence,

*χ*

^{s}is equal to the destruction of salinity variance on molecular scale (Nash and Moum 2002). The respective variance boundary conditions at the surface and the bottom can be derived by multiplying (7) and (8) with 2

*s*′:

*χ*

^{s}denotes a sink in both the salinity squared equation (15) and the salinity variance equation (19). The formalism of the salinity squared equation (15) has the advantage that it only depends on boundary fluxes and not on any internal information of the estuary. The variance on the other hand depends on the mean salinity

### b. Time averaging and reformulation in terms of TEF bulk values

*c*above a certain salinity

*S*through any cross section is defined as

*A*(

*S*) being the fraction of the transect with

*s*>

*S*(MacCready 2011) and ⟨⟩ denotes temporal averaging. The inflowing and outflowing bulk properties are defined as the fraction of inflowing/outflowing tracer transport divided by the respective volume transport:

*Q*

^{c}(

*S*) with considerations of the extreme values (see Lorenz et al. 2019). With

*c*= 1, the volume transport bulk values,

*Q*

_{in},

*Q*

_{out}, their respective bulk salinities (

*c*=

*s*),

*s*

_{in},

*s*

_{out}, bulk salinities squared (

*c*=

*s*

^{2}), (

*s*

^{2})

_{in}, (

*s*

^{2})

_{out}, bulk salinity variances (

*c*=

_{in}, (

_{out}, and with the definitions of river discharge

*Q*

_{evap}≤ 0,

*Q*

_{precip}≥ 0, surface transport of salinity squared

*M*≥ 0, one can rewrite the time-averaged budget equations (5), (9), (15), and (19):

### c. Exact and approximated mixing relations

*Q*

_{surf}and an additional

*s*

^{2}transport through the surface,

*P*=

*E*= 0 (32) reduces to Eq. (36) of Burchard et al. (2019). Assuming constancy (subscript

*c*), i.e., the temporal average of the squared property is the same as the time averaged quantity squared, e.g., (

*s*

^{2})

_{in}= (

*s*

_{in})

^{2}, (

*s*

^{2})

_{out}= (

*s*

_{out})

^{2}, (32) reduces to

*p*), i.e., a tidally periodic estuary, or a long-term average of the estuary, the storage terms are approximately zero. The averaging period that is needed to justify this assumption, may strongly differ for each estuary, depending on its size and driving forces. Under these assumptions (32) yields

*cp*), (32) reads as

*Q*

_{evap}< 0 a nonnegative

*M*

_{cp}is obtained due the additional squared surface salinity term with

*s*

_{surf}≥ (

*s*

_{in}

*s*

_{out})

^{1/2}. If the surface salinity of an inverse estuary does not fulfill this condition, the assumptions of stationarity and constancy are invalid.

*s*

_{surf}≤ (

*s*

_{in}

*s*

_{out})

^{1/2}. This condition is fulfilled, as for this kind of estuary the water column would be stratified and therefore

*s*

_{surf}≤

*s*

_{out}≤

*s*

_{in}.

## 3. Simplistic box model

To exemplify the derived mixing relation (35) for an inverse estuary, we assume a box model as shown in Fig. 1 with *Q*_{r} = *δV* = *δS* = 0. The steady-state estuary consisting of a surface layer (volume *V*_{surf} with salinity *s*_{surf}) and a bottom layer (volume *V*_{b} with salinity *s*_{out}) (see Fig. 1a) is maintained by the coaction of evaporation and inverse estuarine circulation during an infinitesimal time interval Δ*t*. We describe the process of evaporation via *Q*_{surf} < 0 as spatially constant removal of freshwater from the saline surface volume. This freshwater loss increases the surface salinity from *s*_{surf} to *Q*_{in}, *s*_{in}) and an outflow of estuarine bottom water (*Q*_{out}, *s*_{out}) keeping the salt in the estuary constant (Fig. 1c).

*Q*

_{surf}< 0) can be calculated as

*a*and

*b*,

*c*back to state

*a*is given by (Σ

_{c}− Σ

_{a})/Δ

*t*and thus

_{b}− Σ

_{a})/Δ

*t*in (43) is equivalent to the additional

*s*

^{2}transport

*M*

_{cp}are well illustrated: the salinity square (variance) loss due to the exchange flow and the salinity square (variance) gain due to evaporation.

## 4. Numerical model setups

To test the derived relations (32)–(35) in a numerical model, we employ a 2D model similar to the setup of Warner et al. (2005) and Burchard et al. (2019), and a realistic three-dimensional model of the Persian Gulf, also known as Arabian Gulf. The model of our choice is the General Estuarine Transport Model (GETM; Burchard and Bolding 2002), a coastal ocean model, solving the hydrostatic Boussinesq equations (Klingbeil et al. 2018). For turbulence closure, the turbulence module of the General Ocean Turbulence Model (GOTM; Burchard and Bolding 2001) is coupled to the model. Here, we use the *κ*–*ϵ* model in all simulations as well as the total variation diminishing (TVD) Superbee scheme (Pietrzak 1998) for the advection discretization. Further details on the different setups are described in the respective sections.

### a. Calculation of mixing and ${F}_{\text{surf}}^{{s}^{2}}$ in the model

In GETM all forms of variance changes are quantified in every grid cell according to the discrete variance decay analysis method of Klingbeil et al. (2014). This also includes the application of (43) to the single surface grid cells. The corresponding variance change defines the local contribution to

### b. Idealized two-dimensional simulations

First, we applied the relations (32)–(35) to an idealized setup with different forcing. Furthermore, we investigated approximations of the surface salinity square transport, since the exact value is difficult to observe in high temporal and spatial resolution. The idealized estuary is 100 km long with a resolution of *dx* = *dy* = 200 m. The topography is linearly increasing from 15-m depth at the ocean boundary to 5-m depth where the river is entering the domain, see Fig. 2. In the vertical 60 equally distributed *σ*-coordinates are used. The open boundary is prescribed with a constant salinity of 30 g kg^{−1}, and an M_{2} tidal amplitude of 0.5 m, if tides are applied. Freshwater transports that are river discharge as well as spatially integrated precipitation or evaporation, are all 5.0 m^{3} s^{−1}. The TEF analysis is done during the model run for each baroclinic time step using 250 equidistant salinity bins from 0 to 40.0 g kg^{−1} as proposed by Lorenz et al. (2019). Depending on the experiment, see Table 1, different combinations of forcing are applied. The temperature is kept constant in these simulations. The different experiment simulations are integrated into a quasi-periodic state (500 tidal cycles), which is not a real periodic state due to waves forced by the surface flux gradients which propagate through the domain, flushing the high salinity water from the shallow part out of the estuary. Therefore, a long-term average of 100 tidal cycles is carried out to minimize the storage terms in the budget equations, see Fig. 2.

Overview of four idealized model experiments of different combinations of freshwater forcing; *Q*_{evap} and *Q*_{precip} are the area integrated evaporation and precipitation rates of *E* and *P*.

Experiment A (Table 1, Fig. 2a) demonstrates the bottom water formation solely due to evaporation, without any tides. Over the inner half of the domain, constant evaporation is prescribed. Due to the slope of the bathymetry, the salinity increases toward the shallow end and creates a baroclinic pressure gradient. Opposing a barotropic pressure gradient due to the lower surface at the coast is formed. The water of high salinities flows down the slope at the bottom, whereas the water of less salinity enters the evaporation zone, indicated by the arrows in Fig. 2a. As shown in Table 2, *M*_{p} shows a small discrepancy to *M*_{e}, whereas *M*_{c} shows a large deviation, although (*s*^{2})_{in} ≈ (*s*_{in})^{2} and (*s*^{2})_{out} ≈ (*s*_{out})^{2}. Therefore, *M*_{cp} also shows a large deviation. This is due to the overall low mixing in this simulation.

Results of the mixing analysis for the idealized model experiments listing the variables needed to compute the mixing with relations (32)–(35). In the model mixing is directly diagnosed (*M*_{e} equal to machine accuracy.

Experiment B adds river discharge, *Q*_{r} = 5 m^{3} s^{−1}, to experiment A, while also shifting the evaporation area further offshore and reducing the area from half the domain to one-third of the domain, keeping the freshwater loss to −5 m^{3} s^{−1}. Furthermore, a semidiurnal M_{2} tide of 0.5 m is prescribed. The resulting salinity distribution, see Fig. 2b, shows a salinity maximum near the left border of the evaporation zone of ~37 g kg^{−1}, a so-called salt plug which can be often found in tropical estuaries during dry season (Wolanski 1986; Valle-Levinson and Bosley 2003). In the salt plug, the mean velocity is downward. The circulation along the analyzed transect follows an inverse circulation. In addition, due to the river discharge, a classical estuarine circulation is created near the coast, extending into the evaporation zone. There the strong evaporation is able to transform the brackish water to higher salinities than the prescribed 30 g kg^{−1} at the open boundary. Compared to the first experiment the mixing is larger by a factor of ~20, although

Experiment C replaces the river discharge with a precipitation region in the coastal area, see Fig. 2c. The results show a similar salinity distribution to experiment B and the mixing is comparable as well. The major difference to experiment B is the numerical mixing that has more than doubled which can be explained by stronger salinity gradients since a thin surface layer of fresher water is maintained, see Fig. 2c. This strong salinity gradient is prone to be mixed due to numerical mixing.

Experiment D adds river discharge, yielding a net positive freshwater budget adding 5 m^{3} s^{−1} to the setup. The overall circulation follows a classical estuarine circulation with brackish water leaving the estuary at the surface, while saline water enters at the bottom, see Fig. 2d. The evaporation is not enough to form a salt plug as shown in experiments B and C. The surface salinity square transport

*s*

_{surf}, Eq. (36), to model mean surface salinities. The best approximation for these experiments of the exact

Comparison of sea surface salinity and *s*_{surf,model,evap}⟩ and ⟨*s*_{surf,model,precip}⟩ are the respective mean surface salinities of the evaporation/precipitation zones, whereas ⟨*s*_{surf,model}⟩ is the mean surface salinity of the whole estuary.

### c. Realistic three-dimensional simulation

In this section we apply the mixing relations to a realistic model setup of a large, inverse estuary, the Persian Gulf, see Fig. 3.

We repeated the model simulation of Lorenz et al. (2020) for the year 2011 which is a representative year of the exchange flow of the Persian Gulf. The difference between this simulation and the results of Lorenz et al. (2020) is an online TEF analysis as proposed by Lorenz et al. (2019) rather than an offline analysis of hourly output. We used 250 salinity bins from 0 to 50 g kg^{−1}. The setup has a horizontal resolution of one nautical mile (1 n mi = 1.852 km) based on ETOPO1 (Amante and Eakins 2009), a vertical resolution of 40 vertically adaptive coordinates (Hofmeister et al. 2010, 2011; Gräwe et al. 2015). It is forced with hourly meteorological data from the NCEP Climate Forecast System version 2 (CFSv2; Saha et al. 2011), and uses 3-hourly HYCOM boundary conditions (Chassignet et al. 2007) with tides from the Oregon State University Tidal Prediction Software (OTPS; Egbert and Erofeeva 2002) (30-min resolution). For further details of the setup and the validation of the model, we refer to Lorenz et al. (2020).

The Persian Gulf is a semi-enclosed marginal sea with net freshwater loss due to evaporation. Its exchange flow is, therefore, following an inverse estuarine circulation. Although the evaporation is dominating the freshwater budget, there is an annual mean river discharge of ≈1600 m^{3} s^{−1} with the Shatt Al Arab as the major contributor with a mean discharge of 1400 m^{3} s^{−1}. The circulation of the Gulf is strongly dependent on the seasonal cycle. In winter, the Gulf is well mixed in most parts due to strong surface cooling, evaporation, winds, and its shallowness (mean depth ≈ 40 m). In summer, the Gulf is stratified and a general counterclockwise circulation with large eddies (~100-km diameter) can be observed (Reynolds 1993; Kämpf and Sadrinasab 2006; Thoppil and Hogan 2010a; Yao and Johns 2010a; Pous et al. 2015). In contrast, in winter, the large eddies have dissipated and a lot of much smaller eddies (~10-km diameter) are occurring (Kämpf and Sadrinasab 2006; Yao and Johns 2010a; Pous et al. 2015).

The seasonality of the forcing translates into the salt mixing, see in Fig. 4, where the vertically integrated, monthly mean *χ*^{s} is shown (*χ*^{s} includes both physical and numerical contributions). Four persistent salt mixing hotspots can be identified: the western shelf of the Gulf of Oman, the Strait of Hormuz, the river plume of the Shatt Al Arab, and the shallow zone north of Bahrain and west of Qatar.

The western shelf of Oman is the region where the Persian Gulf water stratifies into the Indian Ocean in around 250-m depth. The Gulf water interacts with the steep topography and forms eddies of mesoscale to submesoscale size (L’Hégaret et al. 2015; Vic et al. 2015; Morvan et al. 2019). This process is not sufficiently resolved in the hydrostatic model (Klingbeil and Burchard 2013) and the mixing in our simulation is explained by numerical mixing due to internal pressure gradient errors within the terrain-following coordinates (Shchepetkin and McWilliams 2003).

The hotspot located in the Strait of Hormuz is where the exchange flow is occurring, see Fig. 5 for the mean salinity and mean velocity structure in summer and winter, respectively. The high mixing there is related to topographic features in the channel, strong tidal currents, and eddy activity (Swift and Bower 2003), which mix the saline outflowing water with the inflowing water. Along the Iranian coast in the Strait, there is only Indian Ocean water of almost constant salinity (see Fig. 5), which explains the low salt mixing values as there are only small salinity gradients. West of the Strait there is more mixing occurring in fall when the buoyancy loss due to the atmospheric forcing occurs. Almost all year there is a line of high mixing which is the front between the Iranian Coastal Jet of Indian Ocean Surface Water and the more saline Persian Gulf water (Kämpf and Sadrinasab 2006).

Two more mixing hotspots are located further inside the Gulf: the Shatt Al Arab river plume area and the area north of Bahrain and west of Qatar. The high mixing values are explained by the strong vertical salinity gradients in those regions. The fresh, thus buoyant water of the Shatt Al Arab stratifies over the saline Gulf water forming large vertical salinity gradients which are mixed, for example, by tidal currents. The vertical salinity gradients north of Bahrain are created by a dense gravity current flowing into the Gulf’s interior. Hypersaline water of salinities greater than 50 g kg^{−1} is formed in the shallow south of Bahrain, which stratifies below the adjacent Gulf water. Although dense water is formed in this region, it is negligible as a source of outflowing Persian Gulf Water (Lorenz et al. 2020).

Besides these persistent hotspots, seasonality can be observed also in other parts of the Gulf. The vertically integrated salt mixing is smaller in winter (JFM) and summer (JAS) than in spring (AMJ) and fall (OND). The explanation is found in the vertical structure of the Gulf: in winter most regions of the Gulf are well mixed, i.e., there are no salinity gradients to be mixed. In spring, stratification and the basinwide circulation start to form, which creates shear and therefore salt mixing. In summer, stratification is strongest, inhibiting vertical mixing. In fall the stratification is mixed away due to surface cooling, strong evaporation, and winds, reflected in the high local mixing values.

The surface salinity square transport is strongest in November and weakest in March, see contours in Fig. 4. From March onward the transport grows from the shallow coasts offshore until September, before it decreases again starting from the shallows. The shallow areas have higher surface salinities (Reynolds 1993; Kämpf and Sadrinasab 2006; Yao and Johns 2010a; Pous et al. 2015) which combined with the dry easterly to southeasterly winds cause this spatial distribution.

It is well known that the outflowing water is formed in late fall and winter in the northwest and southern shallows, which then propagates as a bottom current to the Strait of Hormuz (Reynolds 1993; Kämpf and Sadrinasab 2006; Yao and Johns 2010b; Pous et al. 2015; Lorenz et al. 2020). Lorenz et al. (2020) show that in late July/early August a regime shift in the outflowing water occurs. From February to August, most outflowing water in the Strait of Hormuz, except entrained surface waters, originated in the southern shallows in fall and winter. The propagation time is 2–3 months. From August onward most outflowing water was formed in the northwest during fall and winter, more than 6 months prior. During this time period, water from the southern shallows is not dense enough to become part of the bottom water (Yao and Johns 2010b). This shift in water masses is reflected in the increase of the salinity of the outflow in early August.

A time series for 2011 of the exchange flow properties across the transect in the Strait of Hormuz, see Fig. 3 for the location and Fig. 5 for the spatial structure of the exchange flow, reflects the described seasonality (see Fig. 6). The volume exchange is highly variable at time scale of a few days. On a weekly time scale, a spring–neap cycle can be observed, especially in the outflow. The inflow is near the surface and therefore affected more by winds. On seasonal time scales, we find that the volume exchange is stronger in the first half of the year, when the well-mixed state transitions into the stratified state, than the second half where increased vertical mixing weakens the exchange flow (Yao and Johns 2010b; Pous et al. 2015; Lorenz et al. 2020). The salinity of the inflow *s*_{in} is ~37 g kg^{−1} whereas the salinity of the outflow follows the seasonal cycle described before (see Fig. 6b). The salinity of the outflow *s*_{out} decreases from March to July since the salinity of the surface waters inside the Persian Gulf decreases during that time period (increasing salinity stratification inside the Gulf due to more inflow). On the dense water’s path from its origin to the transect, it is entrained with the less saline surface water. The increase of *s*_{out} afterward is due to the arrival of dense winter water formed in the north (Lorenz et al. 2020). For this simulation, (*s*^{2})_{in,out} ≈ (*s*_{in,out})^{2}, and both are therefore not shown. The basin-averaged model surface salinity, Fig. 6b, shows a seasonal cycle with the lowest salinities in summer and highest in winter (Kämpf and Sadrinasab 2006). The surface salinity bulk value, *s*_{surf} defined in Eq. (36), is more variable than the model surface salinity and is highest in summer and lowest in winter, contrary to the model surface salinity. This is due to the weighting of the model surface salinity with the evaporation rate which is most of the year higher near the shallow coasts where the highest surface salinities are found.

The described seasonality of the local salt mixing is visualized more clearly in the time series of the mixing *M*_{e}, see Fig. 6d. For this simulation the constancy assumption (33) shows only a small error, since the graphs of the bulk salinities and bulk salinity squares show the same temporal evolution, proving that for this exchange flow *δV*, see Fig. 6c, as it is one order of magnitude larger than the surface freshwater forcing *Q*_{surf}, see Fig. 6e, which in turn is one order of magnitude larger than the river discharge *Q*_{r}. The salinity storage *δS* and salinity square storage *δS*^{2} are proportionally larger than *δV* (not shown here). Due to the dominant storage terms, we do not show the mixing relations (34) and (35) in Fig. 6d as these assumptions are clearly not valid for a time series of daily values. The frequent changes are due to tides and wind events (Thoppil and Hogan 2009, 2010b). The wind events are called Shamal which brings dry air to the Gulf leading especially in fall to a great heat loss and high evaporation rates, as seen in November and December Fig. 6e and is investigated in detail by Thoppil and Hogan (2010b). On a weekly time scale, the mixing is a result of surface transports, e.g., the higher mixing in June due to the high *M* are shown by Wang et al. (2017) for the Hudson River and by MacCready et al. (2018) for an idealized estuary. The strongest signal is found to be time scales between 14 and 17 days, which is the time scale of the spring–neap cycle, but also the length of high/low evaporation over the Gulf (see Fig. 6e).

As a measure of stratification and to show its relation to mixing, we used the potential energy anomaly (PEA) Φ(Simpson et al. 1978) (see Fig. 6f). For a well-mixed state Φ = 0, whereas Φ > 0 indicates stratification. The high values of PEA (averaged over the Gulf) in Fig. 6f in summer and low values in winter indicate seasonal stratification. PEA further changes significantly on a biweekly time scale, which is visible more clearly for _{t}Φ, is well (negatively) correlated to the mixing *M*, see Fig. 6g where the anomalies of ∂_{t}Φ, *M*_{e}, and _{t}Φ is negative and vice versa, indicating that mixing reduces stratification and the absence of mixing allows stratification to form or increase. Furthermore, Fig. 6g suggests that most of the variation in *M*_{e} is due to

On seasonal time scales the storage terms minimize and the seasonality of the exchange flow and the forcing is visible. The volume storage is orders of magnitude smaller than the freshwater forcing, see Table 4. The mixing is low from January to April, increases from May to July, decreases until October before reaching its maximum in November and December. This seasonal mixing cycle is modulated with the seasonal surface salinity square transport

Results of the mixing analysis for the whole Persian Gulf in 2011: all annual mean variables needed to compute the mixing with relations (32)–(35) are listed below.

*M*≈ 4.7 × 10

^{6}(g kg

^{−1})

^{2}m

^{3}s

^{−1}; see Table 4 for the exact value. For this simulation the relative errors of the different assumptions are

*s*

_{surf,model}is the mean surface salinity of the model. The errors of all mixing relations and also the approximation of the exact

*s*

_{surf}can be the long-term average surface salinity, instead of the more complex definition (36).

^{−1}; Ibrahim et al. 2020), and following the isohaline theory by Burchard (2020), the long-term averaged mixing due to river runoff integrated over the volume bounded by the river and the 40 g kg

^{−1}isohaline has to be

*M*= (40 g kg

^{−1})

^{2}

*Q*

_{r}. This means that even though river discharge is almost an order of magnitude smaller than evaporation, the riverine mixing is high due to the large salinity range the freshwater has to be entrained through.

## 5. Discussion and conclusions

In this study, we derived a mixing relation that is valid for all estuaries since it includes surface freshwater fluxes. Crucial to this relation is the newly introduced integrated surface salinity square transport due to precipitation and evaporation,

*s*

^{2})

_{in}= (

*s*

_{in})

^{2}and (

*s*

^{2})

_{out}= (

*s*

_{out})

^{2}, relation (33); (ii) assuming periodicity/long-term averaging, i.e., vanishing storage terms, relation (34); (iii) assuming both at the same time, relation (35). We performed four two-dimensional idealized simulations with different freshwater forcing to exemplify the mixing relations (see section 4b). From the performed idealized simulations we cannot generalize the errors of the assumptions as these seem to be setup specific. For example, the constancy assumption fails for experiment A (only evaporation) but is a good approximation for experiments B (evaporation + river discharge) and C (evaporation + precipitation). It is desirable to approximate the newly introduced term for the integrated surface salinity square transport

*P*≫

*E*or

*E*≫

*P*, as it is in experiment A (evaporation, no precipitation), and also the Persian Gulf simulation (

*E*≫

*P*) (see section 4c). When both are equal,

*E*≈

*P*, or occur in different areas, evaporation and precipitation have to be evaluated separately with the respective surface salinities, see Eq. (47). We showed this in experiments C (evaporation + precipitation) and D (evaporation + precipitation + river discharge), where the net surface freshwater transport is zero. Based on these findings, we conclude that if either

*E*or

*P*dominate the surface freshwater budget and if the averaging period is long enough, the mixing can be sufficiently approximated with Eq. (37),

*s*

_{surf,model}being the simulated long-term mean, estuary-wide averaged surface salinity instead of the surface salinity bulk value we defined in Eq. (36) which is defined using the exact

*s*

_{surf,model}may also be replaced by direct observations of the surface salinity, if available. Relation (52) further allows decomposing the mixing contributions from river runoff and cross-surface transports. For the Persian Gulf, with a total mixing of ~4.7 × 10

^{6}(g kg

^{−1})

^{2}m

^{3}s

^{−1}, the two contributions are almost equal, despite the order of magnitude difference in freshwater forcing, |

*Q*

_{surf}| ≈ 10|

*Q*

_{r}|.

In general, the separate quantification of mixing due to river runoff and mixing due to cross-surface freshwater transports may help to advance the understanding of estuarine dynamics in future studies, e.g., when working with tropical estuaries, where the freshwater forcing may shift from wet to dry season (Wolanski 1986; Valle-Levinson and Bosley 2003) and with it the mixing contributions.

We have shown that for the Persian Gulf changes in freshwater forcing immediately modify the mixing *M*, i.e., evaporation events in Fig. 6g. Other sources of immediate changes in mixing could be discharge events (Lemagie and Lerczak 2020) and wind events (Lange et al. 2020). Some remaining open questions not only for the Persian Gulf but also for any estuary are the following: how do these spatial and temporal changes in mixing interact and change the circulation and salinity fields? When are these changes detectable in the properties of the exchange flow, e.g., *s*_{in} and *s*_{out}? We think that the presented mixing relations as well as tools like the direct analysis of salinity variance (Li et al. 2018; Warner et al. 2020), but also an isohaline mixing analysis (Burchard 2020; Burchard et al. 2021) are suitable tools to answer these questions.

## Acknowledgments

The authors want to thank the two anonymous reviewers for their constructive feedback. This study was conducted within the framework of the Research Training Group “Baltic TRANSCOAST” funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)-GRK 2000 (www.baltic-transcoast.uni-rostock.de). This is Baltic TRANSCOAST publication GRK2000/0036. H.B. and K.K. were further supported by the Collaborative Research Centre TRR 181 on Energy Transfer in Atmosphere and Ocean funded by the German Research Foundation (project 274762653). The numerical simulations were performed with resources provided by the North-German Supercomputing Alliance (HLRN). Most of the analysis work was performed by computers financed by PROSO (FKZ: 03F0779A).

## Data availability statement

The data of the annual Persian Gulf simulation can be found here: https://www.io-warnemuende.de/marvin-lorenz-publications.html. Please contact one of the authors if you are interested in the analysis scripts or the idealized simulations.

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