## 1. Introduction

Vertical turbulent fluxes of momentum and salt are of fundamental importance to estuarine water motion and mass transport, hence strongly influencing the estuarine morphology, biology, and ecology. There fluxes are usually parameterized by means of a downgradient approach using a vertical eddy viscosity and diffusivity (based on Fick’s law). Due to strong temporal variability of small-scale turbulence, the vertical eddy viscosity can change significantly in time (see, e.g., Peters 1997, 1999). Of particular interest in the present paper are the variations of vertical eddy viscosity and diffusivity at the dominant tidal frequency and their interactions with the water motion and salt transport. The dominant tidal component of these coefficients parameterizes variations of small-scale turbulence during the tidal cycle and is responsible for flood–ebb asymmetry in small-scale turbulence. Such asymmetries were first observed by Simpson et al. (1990) and will be called asymmetric tidal turbulence (ATT) hereafter.

Observations have highlighted the importance of ATT to estuarine circulation in many systems (Jay and Musiak 1994; Stacey et al. 2001; Scully and Friedrichs 2003, 2007). Nevertheless, different flood–ebb asymmetries of turbulence were observed across different marine systems, implying the source of ATT and its impact on water motion and mass transport can also vary significantly. In the lower Hudson River estuary, Liverpool Bay, and the Dee estuary, for example, higher turbulence levels were observed during flood than ebb (Geyer et al. 2000; Simpson et al. 2002; Bolaños et al. 2013). In the Gironde estuary, higher turbulence levels were observed during ebb under neap conditions. However, during spring tides, turbulence was higher during flood (Ross et al. 2019).

These different ATT patterns are associated with many processes. Higher turbulence levels during the flood tide are usually attributed to strain-induced periodic stratification (SIPS; Simpson et al. 1990, 2002, 2005; Stacey and Ralston 2005; Cheng et al. 2010), flood-dominant bottom-shear-generated turbulence (BGT; see, e.g., West and Shiono 1988; Li and Zhong 2009), and convective instabilities (Rippeth et al. 2001; Prandle 2004; Stacey and Ralston 2005). In contrast, stronger turbulence during ebb is attributed to shear instability (Geyer 1995), ebb-dominant BGT (Simons et al. 2010; Ross et al. 2019), and lateral processes (Scully and Geyer 2012; Basdurak et al. 2017). In addition, the tidally varying water depth (TWD) effectively influences the development of turbulence in shallow waters by limiting the maximum eddy size (Uijttewaal and Booij 2000), and may also result in a nonnegligible contribution to the ATT in shallow, meso/macrotidal estuaries.

The contribution of ATT to estuarine circulation originates from the eddy viscosity–shear covariance (ESCO), hence will be called ESCO circulation, as proposed by Dijkstra et al. (2017). Many process-based models have been employed to understand and quantify the contribution of this circulation. Stacey et al. (2008) found the ESCO circulation can be more significant than gravitational circulation (GC) under moderate water depths and stratification, and the phasing of ATT relative to the shear determines whether the ESCO circulation has a similar or opposite vertical structure to GC. These results were confirmed by Cheng et al. (2010) with a width-averaged analytical model as well as a three-dimensional numerical model (considering weakly stratified narrow estuaries with a constant water depth). The relative importance of ESCO circulation to estuarine circulation was first quantified by Burchard and Hetland (2010). Using a one-dimensional water column model coupled to a turbulence closure model, they found that in periodically stratified estuaries ESCO circulation is about twice as important as GC, and the significance of the ESCO circulation decreases with increasing down-estuary wind straining and residual runoff. They also found the ESCO circulation has a vertical structure similar to GC but the structure is “reversed” when considering strong down-estuary wind straining. Extending Burchard and Hetland’s (2010) model to a (two-dimensional) cross-sectional model, Burchard et al. (2011) found strong dependence of the relative importance of ESCO circulation on the Simpson number *S*_{i} (ratio of horizontal buoyancy gradient to bottom frictional velocity scale). For large *S*_{i}, GC is stronger than ESCO circulation; for small *S*_{i}, ESCO circulation dominates the estuarine circulation. Following the same framework, Burchard and Schuttelaars (2012) found that the lateral advection of longitudinal momentum significantly influences ESCO circulation in estuaries with a parabolic cross section. Their results highlight the importance of lateral processes to the along-channel estuarine circulation, as found by Lerczak and Geyer (2004) using a three-dimensional (3D) numerical model. Using the same approach of Burchard and Hetland (2010), Dijkstra et al. (2017) found that ESCO directly influences estuarine circulation through not only the covariance between tidal shear and tidal variations of vertical eddy viscosity, but also the covariance between shear and eddy viscosity at twice the dominant tidal frequency. Moreover, the tidally varying eddy viscosity can interact with GC and indirectly contribute to ESCO circulation. By employing a numerical width-averaged model, they further demonstrated significant variations of the relative importance of GC, and the direct and indirect ESCO circulation components along the Scheldt estuary.

However, as longitudinal salinity gradients need to be prescribed in water-column and cross-sectional models and lateral processes are neglected in width-averaged models, the 3D interactions between ATT and salinity gradients remain poorly understood, as well as their influence on the gravitational and ESCO circulation. Moreover, most three-dimensional and width-averaged modeling studies have focused on long estuaries, in which the estuarine length and tidal wavelength are within the same order of magnitude. The importance of asymmetric tidal turbulence to estuarine circulation and salt transport in short estuaries (estuarine length ≪ tidal wavelength) is yet to be evaluated.

One major difference between short estuaries and long estuaries lies in the tidal propagation. The tide behaves mostly as a standing wave (with some features of a progressive wave) in short estuaries and a progressive wave in long estuaries (Dronkers 1986). This has direct consequences for estuarine circulation and salt transport. For example, the residual circulation due to tidal return flow (that compensates the Stokes drift) is close to zero in very short tidal basins as a result of weak correlations between the tidal surface elevation and surface velocities. In long estuaries, however, tidal return flow can generate strong seaward residual flow (Dronkers 1986). Moreover, the residual salt flux due to tidal advection of salinity (i.e., tidal pumping) dominates the landward salt transport in long estuaries (Wei et al. 2017), but is close to zero in very short estuaries due to weak correlations between the tidal velocity and salinity (Schettini et al. 2017). Another important distinction is linked to the along-channel salinity gradients, which are usually large in short estuaries (such as the Tamar, Tees, and Wyre estuaries in the United Kingdom) compared to long estuaries (Lewis and Uncles 2003) and can lead to stronger gravitational circulation in shorter estuaries (with large *S _{i}* number). This implies that our knowledge of dominant processes for estuarine circulation and salt transport established for long estuaries may not hold for short estuaries.

Hence, there are three main goals in this study: 1) to develop a semi-analytical model that allows for a three-dimensional investigation of the mutually interacting water motion, salinity distribution, and small-scale turbulence, focusing on ATT in tidally dominated, periodically weakly stratified estuaries; 2) to systematically quantify the individual contributions of asymmetric bottom-shear-generated turbulence, strain-induced periodic stratification, and water depth variations to ATT, estuarine circulation, and salt transport in a short estuary; and 3) to investigate the feedback of water motion and salt transport on ATT. The solution method employed in this study also allows for a systematic interpretation framework for the complex variations of turbulence commonly observed in numerical models and field measurements.

This paper is organized as follows: the research method is described in section 2; in section 3, the contributions of strain-induced turbulence, asymmetric bottom-shear-generated turbulence, and water depth variations to the asymmetric tidal turbulence, and the mutual interactions between asymmetric tidal turbulence, currents, and salinity distribution are investigated; section 4 briefly discusses the model capabilities and limitations; and conclusions are presented in section 5.

## 2. Research method

### a. Model description

The estuarine dynamics considered here is strongly nonlinear due to interactions between turbulence, shear, and stratification: small-scale turbulence can strongly influence the water motion and salinity distribution; meanwhile, the shear acts to generate turbulence, which is inhibited by stable stratification and promoted by unstable stratification. To resolve the estuarine dynamics in a way that allows for a systematic decomposition of the abovementioned interactions, the 3D semi-analytical model of Wei et al. (2017) is extended in this study. In Wei et al. (2017), the nonlinearly coupled water motion and salt dynamics are resolved, decomposing temporal variations into a semidiurnal (M_{2}) tidal constituent and a residual (M_{0}) signal both for the water motion and salinity. In this model, a time-independent vertical eddy viscosity *A*_{υ} and diffusivity *K*_{υ} are prescribed, thus neglecting the influence of temporal variations of *A*_{υ} and *K*_{υ} on the hydro- and salt dynamics. To dynamically include this temporal variability which is directly linked to the asymmetric tidal turbulence (ATT), three main extensions are made. First, *A _{υ}* and

*K*are dynamically coupled to the water motion and stratification, resulting in a tidally varying

_{υ}*A*and

_{υ}*K*. Second, the contribution of this temporal variability of

_{υ}*A*and

_{υ}*K*to estuarine circulation (ESCO) and salt transport are taken into account. Third, the quarter-diurnal (M

_{υ}_{4}) tidal motion [see Eq. (9)] is resolved because it plays a nonnegligible role in the generation of ATT and is partly originated from the covariance of eddy viscosity and shear at the semidiurnal frequency.

#### 1) Extending the 3D semi-analytical model

##### (i) Governing equations and boundary conditions

Here *t* denotes time, *η* is the free surface elevation, and **U** = (*u*, *υ*, *w*) is the velocity vector, with *u*, *υ*, and *w* the velocity components in *x*, *y*, and *z* directions, respectively. The acceleration due to gravity is denoted by *g*. The Coriolis parameter is given by *f*. The estuarine water density is represented by *ρ*, which is assumed to depend only on salinity *S* as *ρ* = *ρ*_{c}(1 + *β*_{s}*S*), with *β*_{s} = 7.6 × 10^{−4} psu^{−1} and a constant background density *ρ*_{c} = 1000 kg m^{−3}. The term *K*_{h} is the horizontal eddy diffusivity.

_{2}, its first overtide, M

_{4}, and a residual sea surface elevation, M

_{0}, with the width-averaged residual sea surface amplitude equal to zero. A freshwater discharge

*Q*is prescribed at the landward boundary,

*x*=

*L*, where a weir is located. The closed boundaries are impermeable, and the normal component of the depth-integrated water flux is zero at these boundaries. By requiring depth-averaged water flux to vanish at the closed boundaries (instead of a no-slip condition), the dynamics within regions close to these boundaries, where horizontal stresses play an important role in the momentum balance, is not resolved in this model. At the free surface, kinematic and no stress boundary conditions are prescribed. The bottom is impermeable and a partial slip condition is applied, assuming linearized bottom shear stress as first proposed by Lorentz (1926):

*s*the partial slip parameter. The partial slip parameter is chosen such that the energy dissipation per tidal cycle equals the dissipation obtained with a quadratic bottom friction. The partial slip bottom boundary condition is applied at the top of the logarithmic boundary layer, as described in Zitman and Schuttelaars (2012). The linearized bottom friction assumption is essential to directly solving the water motion at each tidal frequency and dissecting the contributions of different processes to residual circulation and salt transport in this model. To close the salinity problem, a spatially uniform tidally averaged salinity

*S*

_{m}is prescribed at the mouth. The salt flux through the free surface and bottom, and the depth-integrated tidally averaged salt transport normal to the closed boundaries are equal to zero.

##### (ii) Semi-analytical approach

A semi-analytical approach is taken to solve the water motion, salinity, and vertical eddy viscosity and diffusivity, combining a perturbation method with a finite element method and assuming all these variables to consist of different tidal constituents. As the first step of the perturbation method, the system of equations is reduced to an ordered system of equations by making the equations dimensionless and comparing each term with a small parameter *ε* ≪ 1, the ratio of the semidiurnal tidal surface amplitude to the undisturbed water depth at the mouth. This results in a system of equations at each order of *ε*, which describes the water motion and salinity at specific tidal frequencies (appendixes A–D). The equations at each order can be solved separately. When using a vertically uniform tidally averaged vertical eddy viscosity and diffusivity, the perturbation method allows the vertical shape functions of all physical variables at each tidal frequency to be obtained analytically from the top of the logarithmic layer to the undisturbed water level. Then, the horizontal distribution of all physical variables is solved using a finite element method.

_{2}tidal constituent, and a first-order M

_{0}and M

_{4}component [neglecting terms of

*O*(

*ε*

^{2}) and higher],

_{2}tidal frequency,

_{4}tidal frequency, and

_{2}tide (i.e., advection,

_{4}tidal flow can be decomposed into contributions by tidal return flow (

_{4}tide (

_{0}component,

_{2}component,

_{0}, M

_{2}, M

_{4}, M

_{6}) due to generation of overtides. Focusing on the ATT, this study considers only the residual component of the vertical eddy viscosity and diffusivity (i.e.,

Note that the temporal correlations between

#### 2) The ESCO flow

_{2}vertical shear:

Here *j* = 0, 4. For *j* = 0, overbars indicate tidal average. For *j* = 4, overbars indicate that only the M_{4} tidal component is selected.

**n**

_{h}is the unit horizontal vector normal to the boundaries. At the undisturbed water level

*z*= 0, the shear stress is equal to zero,

*z*= −

*H*, a linearized partial slip boundary condition is applied,

With known information on the M_{2} tidal motion,

#### 3) Parameterizing small-scale turbulence

*A*

_{υ}and diffusivity

*K*

_{υ}are derived for each tidal frequency using an empirical formula. Basdurak et al. (2013) proposed a vertical eddy viscosity formula including dependencies on both the depth-averaged velocity and the gradient Richardson number

*R*

_{i},

*c*

_{0}= 2.5 × 10

^{−3},

^{2}s

^{−1}, following Davies et al. (1997) and Pacanowski and Philander (1981), respectively. Here

*H*is the undisturbed water depth, and

*β*

_{z}is a function of depth, which is equal to unity above

*z*= −0.1

*H*(following Davies et al. 1997), and is smaller than unity below −0.1

*H*. This formula considers not only the fact that large gradient Richardson numbers dampen turbulence, but also that turbulent flow generates mixing. By fitting

*A*

_{υ}in the James River estuary under weakly stratified conditions except over the northern shoal. This implies that the dominant influencing factors for

*A*

_{υ}, i.e., current and stratification, are reasonably well represented by this formula for weakly stratified conditions. Therefore, this formula is considered appropriate for parameterizing the contributions of these two factors to

*A*

_{υ}in the idealized periodically stratified estuary focused upon in the present study.

Here, *β*_{z} is taken to be unity at all depths for simplicity. This simplification does not account for the tendency that the size of turbulent eddies decreases with distance toward the bottom within the logarithmic boundary layer (see, e.g., Townsend 1961), i.e., Prandtl’s mixing length concept. This is reasonable because the logarithmic boundary layer is not resolved in the idealized model due to the partial slip condition being applied at a distance above the bed as described before. More importantly, including the vertical variations of *A*_{υ} near the bottom does not qualitatively impact the tidal or residual flow patterns (Zitman and Schuttelaars 2012), even though quantitative changes may be found such as an overestimation of *A*_{υ} and velocities near the bed (Davies et al. 1997).

_{0}, M

_{2}) can then be derived by extracting the tidally averaged and semidiurnal components of

*A*

_{υ}(ignoring the other frequencies) using a Fourier transform (

_{2}vertical eddy viscosity can be also written as

_{2}tidal frequency. Higher-frequency terms are neglected in the present study to focus on the ATT associated with the flood–ebb turbulence asymmetries. As the residual and quarter-diurnal tidal currents are usually one order of magnitude smaller than the semidiurnal currents, and

#### 4) Dissecting the asymmetric tidal turbulence

#### 5) Iterative method

An iterative approach is used to solve the coupled system of equations. For a given distribution of _{2} tidal water motion can be calculated following the same procedure of Kumar et al. (2016). However, due to the dependence of _{2} flow itself [see Eq. (25)], the leading-order M_{2} water motion needs to be calculated iteratively together with _{2} tide. Then, the information on the M_{2} tide is used in Eq. (25) to derive a new _{2} water motion, which is then fed back into Eq. (25) to derive a new _{2} water motion are obtained when the domain-averaged relative difference between the old _{2} tide and

The gravitational circulation, GC, is coupled with salinity, and needs to be solved together with salinity (Wei et al. 2017). The ESCO circulation is strongly determined by _{0}, M_{2}, and M_{4} water motion including GC and ESCO [see Eq. (20) and Eq. (23)], and contributes to the salinity distribution. Therefore, salinity,

### b. Experiments design

Parameters representative of the Blackwater estuary (in Essex, United Kingdom), which are typical for short estuaries, are used as a template in this study. Four experiments are conducted. In experiment I, Eq. (23) is used to calculate

#### 1) Default experiment

Parameters characteristic for the Blackwater estuary are considered in the default experiment. The estuary is 28 km long (with a weir located at the estuarine head, see Fig. 2a), and has a tidal amplitude of ~2 m, representative of short, mesotidal estuaries.

*B*

_{0}the width at the mouth, and

*L*

_{b}the estuarine convergence length. The water depth in the idealized estuary is described by

*H*

_{1},

*H*

_{2}, and

*H*

_{3}prescribed depth parameters,

*C*

_{s}the shoal parameter, and

*L*the estuarine length; see Table 1 for their values. Comparing Figs. 2a and 2b, the idealized bathymetry is featured with a deeper channel in the middle and shallower shoals on each side, the main bathymetric feature of the real Blackwater estuary. The minimum water depth in the idealized case is 3.2 m (larger than the tidal amplitude), hence effects of intertidal flats are not considered.

Definitions of acronyms.

The dominant tidal constituent here is the M_{2} tidal constituent, with a tidal amplitude of 2 m at the mouth. The river discharge in the Blackwater estuary fluctuates from ~1 m^{3} s^{−1} to up to 30 m^{3} s^{−1}. During low river flow conditions, the estuary is mostly well mixed throughout the tidal cycle. During high river flows, the water column is well mixed during flood and weakly stratified during ebb. To focus on periodically stratified conditions, a constant river discharge *Q* = 20 m^{3} s^{−1} is used. Previous observations in this estuary show the water column can exhibit slightly unstable stratification during flood tides (Talbot 1967; Fox et al. 1999). To parameterize the enhanced turbulence due to convective instability in case of unstable stratification, the minimum depth-mean gradient Richardson number (

*K*

_{h}. This coefficient is related to the local cross-sectional area and the along-channel salinity gradient, according to Savenije (2015). For simplicity, it is assumed to linearly decrease with the estuarine width following Wei et al. (2017),

*C*

_{h}= 10 m

^{2}s

^{−1}. All other parameter values are given in Table 2.

Parameters for the idealized Blackwater estuary.

#### 2) Sensitivity experiments

Equations (27)–(29) are respectively used in experiments II–IV (see Table 3), to investigate the individual contribution of BGT, SIPS, and TWD to the M_{2} vertical eddy viscosity and estuarine circulation (see section 3b).

Parameterizing formula for *A*_{υ} and *K*_{υ} in each experiment.

## 3. Results

The patterns of the ATT, residual circulation, and salt transport in the idealized short estuary are shown in section 3a. The individual contributions of SIPS, BGT, and TWD to ATT are quantified in section 3b. The interactions between ATT, water motion, and salinity distribution are unraveled in section 3c. The semidiurnal and quarter-diurnal tidal properties are shown in appendix E.

### a. Varying patterns of vertical eddy viscosity, water motion, and salinity distribution

#### 1) Vertical eddy viscosity

The tidally averaged vertical eddy viscosity *x* ≈ 10 km), however,

The depth-mean phase of *x* < 12 km and is almost constant across the channel further upstream. This is probably related to the water depth, which has strong lateral variations in the downstream and is almost laterally uniform in the upstream (Fig. 2b). In the central estuary (*x* ≈ 10 km), the depth-mean phase of

#### 2) Residual circulation

The depth-mean total residual circulation is up to 0.08 m s^{−1} and consists of a landward flow in the channel and a seaward flow on the shoals (Fig. 4a). This total circulation is then decomposed into different components following Eq. (8). The contributions of the dominant residual circulation components are shown in Fig. 4. The depth-averaged gravitational circulation (GC), with a maximum velocity of 0.10 m s^{−1}, is the largest estuarine circulation component, followed by ESCO circulation (up to 0.047 m s^{−1}) and advection driven circulation (AC, up to 0.040 m s^{−1}). The depth-averaged ESCO circulation shows an outflow in the channel and inflow on the shoals, contrasting the patterns of GC and AC. The differences in depth-averaged patterns between ESCO circulation and GC agree with findings of Scully and Friedrichs (2007) in the York River estuary. The residual circulation due to the no-shear surface condition, tidal return flow and river discharge are small, hence are not shown.

Looking at a transect at *x* = 10 km, the total residual circulation is apparent with a strong near-bottom inflow in the channel and near-surface outflow on the shoals (Fig. 4e). There is also a weak inflow in the upper layers of the channel, which does not agree with the “classical” estuarine circulation (with near-bottom inflow and near-surface outflow; see, e.g., Lerczak and Geyer 2004). This discrepancy is probably related to the overestimated vertical eddy viscosity near the surface, as a result of taking *β*_{z} = 1 at all depths. In reality, turbulence has to vanish at the free surface, potentially increasing vertical shear (by reducing the vertical exchange of momentum) and enhancing the near-surface outflow due to tidal return flow, ESCO, and river discharge.

The cross-sectional distribution of GC is similar to the total circulation (Fig. 4f), highlighting the dominant role of GC in determining the structure of the total residual circulation in this idealized estuary. The patterns of GC are in agreement with existing analytical studies for large Ekman number (Kasai et al. 2000; Valle-Levinson et al. 2003; Huijts et al. 2009), where vertical eddy viscosity was assumed to be vertically uniform. ESCO circulation shows an outflow in the deep channel and inflow on the shoals (Fig. 4g), in agreement with model results of the Delaware estuary (Geyer et al. 2020). The patterns of GC and ESCO circulation are also consistent with the numerical results of Burchard et al. (2011) for small Simpson number. Advectively driven circulation shows inflow on the right side of the channel (looking landward) and outflow on the left side, consistent Huijts et al. (2009).

#### 3) Salinity distribution and transport

The tidally averaged salinity _{2} tidal salinity has a small spatial phase difference (Fig. 5d), lagging behind the M_{2} velocity by ~90°.

The relative importance of different processes to the residual salt transport in the idealized estuary is evaluated by calculating the cross-sectionally integrated residual salt transport due to each process, following Wei et al. (2017). Figure 6 shows the salt transport contributions due to tidal advection of salinity (TASF, red), ESCO circulation (ESCO, blue), gravitational circulation (GC, green), advection (AC, orange), shear-free surface (NS, yellow), tidal return flow (TRFSD, brown), horizontal diffusion (DIFF, pink), and river flow (RD, gray). In the idealized estuary, the residual salt balance is dominated by the landward salt transport due to horizontal diffusion and the seaward transport due to river discharge. TASF and GC play an almost equally important role in transporting salt into the estuary, followed by AC. ESCO circulation, due to its reversed structure, tends to flush salt out of the estuary. The salt transport contributions due to all residual flow components, except river flow, vanish after the first 15 km because

### b. Dissecting drivers of asymmetric tidal turbulence

To understand the response of the vertical eddy viscosity *A _{υ}* to the flow and salinity changes, the tidal variations of vertical profiles of the along-channel velocity, salinity, gradient Richardson number

*R*

_{i}, and

*A*

_{υ}are plotted in Fig. 7 for two locations, representing the channel and shoal, respectively. The longitudinal velocity is flood dominant in the channel and ebb dominant on the shoal. At both locations, unstable stratification (with higher salinities at the top than the bottom) arises during the flood tide while stable stratification occurs during the ebb, consistent with previous observations in the Blackwater estuary (Talbot 1967). The unstable stratification results in negative values of gradient Richardson number

*R*

_{i}during flood and positive values during ebb. Negative values of gradient Richardson number (

*R*

_{i}≪ −0.01) are found in most of the water column under flood conditions. Since the vertical variations of

*A*

_{υ}can only originate from

*β*

_{z}= 1), which is equal to

*A*

_{υ}is vertically homogeneous in most of the water column (Fig. 7g). During ebb, large positive values of

*R*

_{i}occur due to small vertical shear. As the vertical eddy viscosity is reversely related to

*R*

_{i}when it is positive,

*A*

_{υ}approaches zero at depths with

*R*

_{i}> 0.25. This is consistent with the laboratory findings of Rohr et al. (1988) that turbulence decays for

*A*

_{υ}is larger during flood than ebb in the channel but is larger during ebb on the shoal. This implies that the periodic stratification, which tends to increase

*A*

_{υ}during flood and reduce it during ebb at both locations [following Eq. (28)], is not always the dominant driver of asymmetric tidal turbulence (ATT), for example, on the shoal.

To identify the dominant driver(s) of ATT in this idealized estuary, the individual contributions of strain-induced periodic stratification (SIPS), asymmetric bottom-shear-generated turbulence (BGT) and tidal variation of water depth (TWD) to the semidiurnal variations of vertical eddy viscosity are calculated using Eqs. (27)–(29), respectively. Both SIPS and BGT are dominant drivers of ATT in this idealized estuary, whereas the contribution of TWD to ATT is small (Figs. 8a–c). The depth-mean amplitude of SIPS induced M_{2} vertical eddy viscosity, _{2} vertical eddy viscosity, _{2} vertical eddy viscosity due to all processes shows stronger channel-to-shoal than along-channel variability (Figs. 8d–f). The phase of

### c. Interaction mechanisms between asymmetric tidal turbulence, water motion and salinity distribution

The decomposition of ATT into different contributions allows a systematic investigation of the interacting mechanisms between ATT, water motion and salinity distribution. In this section, the interactions are shown by illustrating the influence of ATT on residual circulation [section 3c(1)] and salinity [section 3c(2)], and the impact of tidal straining and flow asymmetry on ATT [section 3c(3)].

#### 1) Influence of ATT on estuarine circulation

Asymmetric tidal turbulence, ATT, is essential to the generation of ESCO circulation. Because the ESCO circulation modulates the horizontal salinity gradients, it also indirectly influences GC. Figure 9 shows the cross-sectional distribution at *x* = 10 km of the along-channel GC (Figs. 9a–c) and ESCO circulation (Figs. 9d–f) in experiments II–IV as a result of the individual ATT contributions of strain-induced periodic stratification (SIPS, left panels), asymmetric bottom-shear-generated turbulence (BGT, middle panels), and tidal variations of water depth (TWD, right panels). In all experiments, GC is stronger than the ESCO circulation. The spatial pattern of GC is consistent across the experiments, but its strength changes remarkably. When considering SIPS induced ATT only, the along-channel GC is up to 0.12 m s^{−1}. It increases to 0.15 m s^{−1} when considering the BGT effects on ATT, and is up to 0.13 m s^{−1} (same as that in the default experiment) when considering the TWD effects only.

The ESCO circulation changes even more across the experiments, compared to GC. The ESCO induced by SIPS, BGT, and TWD results in a maximal velocity of 0.05, 0.1, and 0.004 m s^{−1}, respectively (Figs. 9d–f). Under the influence of BGT and TWD, the ESCO circulation yields an outflow in the channel and inflow on the shoals. The ESCO circulation caused by SIPS, however, generates an inflow in the channel and outflow on the shoals. That means, in this idealized estuary, the ESCO circulation caused by BGT and TWD act against GC to weaken the total estuarine circulation, while that driven by SIPS is in support of GC to strengthen the total estuarine circulation.

As illustrated by Burchard and Hetland (2010), the ESCO circulation at a certain position is reversely related to the ESCO-induced residual shear stress divided by the residual eddy viscosity (*ϑ* when considering each of the three processes. Concerning the SIPS mechanism, positive values of *ϑ* are found across the section (Fig. 9g) which tend to drive seaward currents. Since the cross-sectionally integrated residual water transport has to vanish and *ϑ* is larger on the shoals than in the channel, the SIPS-induced ESCO consists of an outflow on the shoals and inflow in the channel. Concerning the BGT effects, however, *ϑ* is positive in the channel and negative on the shoals (Fig. 9h). Hence, this yields a strong outflow in the channel and inflow on the sides. ATT also contributes to a small quarter-diurnal tidal flow, as shown in Fig. E1a.

The contribution of SIPS to *ϑ*. This will then contribute to an enhanced ESCO circulation and a weaker total residual circulation (results not shown).

#### 2) Influence of ATT on salinity distribution

The residual salinity distribution is strongly influenced by the residual eddy viscosity

The influence of individual drivers of ATT (i.e., SIPS, BGT, and TWD) on salt transport can also be evaluated by recalculating the cross-sectionally integrated residual salt transport considering each process in separation (experiments II–IV). The SIPS induced ATT tends to import salt in the central estuary and export salt near the mouth through ESCO circulation (not shown). To balance this contribution, the GC induced salt transport is increased near the mouth, and decreased in the central estuary, compared to the default experiment. The BGT- and TWD-induced ATT exports salt throughout the estuary, and effectively enhances the GC induced landward salt transport.

The salinity stratification is related to the vertical structure and amplitude of _{2} sea surface elevation and residual salinity, hence is affected by ATT in the same way as shown above.

#### 3) Impact of water motion and salinity on asymmetric tidal turbulence

In this section, the feedback of water motion and salinity distribution on ATT is explored. The driving mechanism of salinity stratification associated with the semidiurnal tide, which is key for the SIPS induced ATT, is investigated in section 3c(3)(i). Influence of the residual circulation and quarter-diurnal tide on tidal asymmetric BGT, essential for

##### (i) Influence of the semidiurnal tide on asymmetric tidal turbulence

Both _{2}) tide because this tidal constituent controls the vertical shear and stratification. In the idealized estuary, the relative phase of stratification compared to the semidiurnal vertical shear, is almost constant across the estuary as the semidiurnal tide propagates at a pumping mode. Hence,

Equation (33) indicates that, in periodically weakly stratified estuaries dominated by the semidiurnal tide, the temporal variations of stratification are caused by interactions between along-channel salinity gradients and the vertical shear due to along-channel semidiurnal tidal velocities, i.e., along-channel tidal straining (ACS), and interactions between cross-channel salinity gradients and the vertical shear due to cross-channel semidiurnal tidal velocities, i.e., cross-channel tidal straining (CCS). Note that in more strongly stratified conditions (not considered in this study), horizontal advection of vertical salinity gradients can also contribute to tidal variations of stratification (Scully and Geyer 2012).

Here _{2} tidal currents and salinity, see details in appendix D. Second, stratification is linearly dependent on the horizontal gradients of the semidiurnal tidal elevation and

The amplitudes of the depth-mean ^{−1}, is larger than ^{−1}. It implies that the impact of ACS on the intratidal variations of

##### (ii) Influence of residual circulation and quarter-diurnal tide on asymmetric tidal turbulence

The residual circulation plays an important role in the residual salt transport, hence affecting the horizontal salinity gradients and stratification, and modulating ATT through

The residual circulation and quarter-diurnal tidal velocity are of the same order of magnitude in most of estuary (see Figs. 4a,e and 11e), suggesting that both components can play an important role in generating asymmetries in BGT and contribute to *ϕ* there.

## 4. Discussion

To facilitate a comparison of our semi-analytic model results with existing numerical studies that focus on long estuaries, the model was also applied to a long estuary with a length of 100 km (results not shown). The estuarine convergence length is reduced to keep the width at the mouth and the river boundary the same as those in the short estuary. In this case, the ESCO circulation is stronger than gravitational circulation, GC. This opposes the results for the short estuary, where GC is stronger than the ESCO circulation (see Figs. 4b,c). This difference between the short and long estuary seems to agree with the cross-sectional model results of Burchard et al. (2011) for large and small Simpson number (*S*_{i}), respectively. They found that the residual circulation is dominated by GC for large *S*_{i} but dominated by ESCO for small *S*_{i}. This agreement is probably related to the larger *S*_{i} in the short estuary due to relatively large along-channel salinity gradients when compared to the long estuary.

The different asymmetric tidal turbulence and ESCO circulation patterns caused by strain-induced periodic stratification (SIPS) and bottom-shear-generated turbulence (BGT) shown in this study indicate that the variable ESCO circulation patterns found in previous studies may be related to different processes being dominant in different systems. In case ATT is dominated by SIPS, ESCO circulation and GC follow similar patterns. In case ATT is dominated by BGT, the structure of ESCO circulation depends on the tidal asymmetries in BGT. The tidal asymmetries in BGT, due to their dependence on the residual and quarter-diurnal tidal currents, are significantly influenced by estuarine topography, geometry, length, and forcing (Jay 1991; Friedrichs and Aubrey 1994; Ridderinkhof et al. 2014).

Although the semi-analytical model has some limitations with regard to complex estuarine environments due to the adoption of several assumptions [e.g., weak quarter-diurnal variations of turbulence, linearized bottom friction, vertically homogeneous residual vertical eddy viscosity, unity Prandtl–Schmidt number (Pr)], it does provide valuable insights into the dynamic interacting barotropic and baroclinic processes. Nevertheless, these assumptions can result in quantitatively different results when compared to reality.

By focusing on weakly stratified estuaries in this study, it is reasonable to assume that the semidiurnal vertical eddy viscosity is an order of magnitude smaller than the residual viscosity [Eq. (11)]. By making these assumptions, the tidal variations of vertical eddy viscosity and diffusivity (asymmetric tidal turbulence, ATT) do not directly influence the semidiurnal tide, GC, or the residual salinity. Nevertheless, ATT can indirectly influence GC and the residual salinity through the ESCO-induced salt transport. In partially stratified estuaries, where the residual and semidiurnal salinities are within the same order of magnitude, ATT can significantly influence the water motion and salinity at all tidal frequencies. In those systems, the influence of ATT on estuarine circulation and salt transport can be even stronger (see, e.g., Dijkstra et al. 2017).

By tuning the friction parameters to obtain the best fit of the observed sea level and velocities, the linearized bed friction assumption allows good reproduction of the residual and dominant tidal bed stresses (Geyer et al. 2000; Li et al. 2004), but the overtide and its induced tidal asymmetry are not well reproduced (Friedrichs and Aubrey 1994). Therefore, the relative importance of the bottom-shear generated ATT, ESCO circulation, GC, and salt transport can differ quantitatively from those when considering a more realistic quadratic bottom friction.

The tidally averaged vertical eddy viscosity and diffusivity are assumed to be vertically uniform in our model (by taking

*β*_{z}= 1). In reality, these coefficients can change significantly in the vertical (Peters 1997; Huguenard et al. 2015; Ross et al. 2019). Considering vanishing turbulence at the free surface and the bottom, for example, will probably increase near-surface seaward currents and reduce them near the bottom, increase stratification, reduce lateral exchange, and strengthen the along-channel GC. Consequently, the individual contributions of SIPS and BGT to the generation of ATT and ESCO circulation will probably be increased.A unity Pr, which is observed in salt wedge estuaries (Geyer and Smith 1987; Holleman et al. 2016), was considered in this study for simplicity. In reality, Pr is related to the gradient Richardson number, according to Stacey et al. (1999). They found that Pr reduces to 0.7 when the gradient Richardson number approaches zero. Therefore, by assuming Pr = 1, we overestimated the vertical eddy diffusivity particularly during the flood tide. This probably has led to underestimated unstable stratification and underestimated contribution of strain-induced periodic stratification to the asymmetric tidal turbulence and ESCO circulation.

## 5. Conclusions

Focusing on asymmetric tidal turbulence (ATT) at the semidiurnal (M_{2}) tidal frequency, a semi-analytical model was developed to study the dynamic interactions between asymmetric tidal turbulence (parameterized by the M_{2} variations of vertical eddy viscosity and diffusivity), water motion and salinity in periodically, weakly stratified estuaries. This model was applied to an idealized short estuary. Results show that, gravitational circulation (GC) is the most important estuarine circulation component, followed by the residual circulation related to ATT and eddy viscosity shear covariance (ESCO), and the advectively driven circulation (AC). The depth-mean ESCO circulation exhibits a reversed pattern compared to GC and AC, resulting in a significant residual seaward salt transport. This contrasts the significant landward salt transport induced by GC. Tidal pumping (due to advection of salinity by semidiurnal tidal currents) also plays an important role in transporting salt landward in this short estuary.

Asymmetric tidal turbulence influences estuarine circulation not only by being essential to the generation of the ESCO circulation, but also by modulating the horizontal salinity gradients (due to the ESCO generated salt transport), hence affecting GC. It is found that the impact of ATT on residual circulation and salt transport strongly depends on the dominant mechanisms generating ATT, namely, strain-induced periodic stratification (SIPS) and asymmetric bottom-shear-generated turbulence (BGT). SIPS increases vertical eddy viscosity during flood and reduce it during ebb, and this effect is more significant on the shoals than in the channel. This results in an ESCO circulation with almost the same depth-averaged structure as GC, and tends to transport salt landward. This transport tends to reduce longitudinal salinity gradients and hence weakens GC. BGT yields a higher eddy viscosity during flood in the channel and during ebb over the shoals. This results in a reversed ESCO circulation with depth-averaged patterns opposite to GC. This ESCO circulation acts to increase longitudinal salinity gradients and hence enhances GC.

The feedback of water motion and salinity distribution on ATT is also revealed. The semidiurnal tide predominantly controls the ATT generated by SIPS because this tidal constituent controls the vertical shear and stratification. Due to the dependence of stratification on horizontal salinity gradients, the semidiurnal tide also indirectly affects ATT by modulating these gradients due to the important landward salt transport contribution induced by tidal pumping. Residual circulation and the quarter-diurnal tide both contribute significantly to tidal asymmetries in velocities, hence strongly influencing the ATT induced by BGT. Due to strong tidal velocity asymmetries and weak stratification in this idealized short estuary, the BGT effects dominate over the SIPS effects in the generation of ATT which explains the reversed structure of the total ESCO circulation. These findings highlight the influence of different turbulence sources on ATT, hydro- and salinity dynamics can differ significantly. Since the relative importance of these sources are strongly dependent on estuarine bathymetry, geometry and forcing, the techniques developed here can be used to gain a general understanding of turbulence variations, and their influence and dependence on the estuarine circulation and stratification in realistic estuaries.

## Acknowledgments

We acknowledge funding from the Natural Environment Research Council through the BLUEcoast project (NE/N015894/1). HMS and LOA also acknowledge funding to the SUPREME project from NWO in the Netherlands and the Newton Fund via the Engineering and Physical Sciences Research Council (Grant ALWSD.2016.015 in the Netherlands, and Grant EP/R024480/1 in the United Kingdom). MEW was partially supported by Chilean Fondecyt project 11191077. This work used the NOC cluster MOBIUS and the ARCHER U.K. National Supercomputing Service (http://www.archer.ac.uk). We appreciate the constructive comments from two anonymous reviewers, which significantly improved our paper. All data used in this paper are available on Zenodo (https://doi.org/10.5281/zenodo.4666030).

## APPENDIX A

### Leading-Order Water Motion

_{2}components of the shallow water equations:

For a given _{2} tidal flow and

## APPENDIX B

### First-Order Water Motion

#### a. Governing equations

_{0}and M

_{4}. The governing equations for residual flows at these two frequencies are derived by considering the tidal average or the M

_{4}component of the shallow water equations:

*j*= 0, 4, and the underbrace denotes various mechanisms that force the residual flow. Equations (B1)–(B4) show that the residual flow is forced by advective contributions of the leading order M

_{2}tide (tidal rectification, denoted by AD), density driven gravitational circulation (GC), the stress-free surface condition (NS), river discharge (RD), a return flow (TRFSD), and ESCO circulation. Since the first-order water motion equations (B1) and (B4) are linear, the residual flow components due to these forcing mechanisms can be solved separately. Hence, the solution of the residual water motion can be written as

_{4}forcing for the gravitational circulation.

## APPENDIX C

### Leading-Order Salinity

## APPENDIX D

### First-Order Salinity

## APPENDIX E

### Tidal Water Motion

The semidiurnal (M_{2}) tidal surface amplitude is almost uniform in the estuary, slightly increasing from 1.95 m at the estuarine mouth to 2.05 m at the head of the estuary (Fig. 11a). The phase relative to that at the mouth is less than 5° everywhere, indicating

_{4}) tidal velocities,

*ϕ*) between the semidiurnal and quarter-diurnal tidal velocities,

*ϕ*< 270°) and flood dominance on the shoals (0° < Δ

*ϕ*< 90°). A cross-sectional view of

*ϕ*(at

*x*= 10 km) shows that

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