## 1. Introduction

### a. Estuarine circulation

The circulation in tidal estuaries has been investigated for more than half a century, but the driving mechanisms are still not fully understood. In the beginning, only the gravitational (also called gravity-, density-, or buoyancy-driven) circulation was considered (e.g., Pritchard 1952; Agnew 1960; Hansen and Rattray 1965): The salinity (and thus density) decrease from the ocean toward the riverine freshwater reach induces an up-estuary baroclinic pressure gradient force that increases from surface to bottom. The down-estuary barotropic pressure gradient force is vertically constant. Combined, these forces drive an up-estuary bottom flow of saline water and a down-estuary surface flow of fresher water. This is the classical estuarine circulation.

Linden and Simpson (1988) and Simpson et al. (1990) discovered the stratifying and destratifying effect of the vertically sheared ebb and flood current in the presence of a longitudinal (along-channel, axial) salinity gradient: since the absolute velocity increases with the distance from the bottom, the ebb current transports fresher over more saline water, resulting in a stably stratified water column, and the flood current reduces or completely erases this stratification. Simpson et al. (1990) called this process strain-induced periodic stratification (SIPS) and described its dependency on the horizontal gradient Richardson number, now called the Simpson number (Burchard et al. 2011).

Even though it was already known that stratification suppresses turbulence (Linden 1979, 1980) and understood how turbulent mixing reduces the intensity of a density-driven flow (Linden and Simpson 1986), it took some time before the feedback of SIPS on the tidal currents was recognized. Jay and Musiak (1994, 1996) explained how stratification during ebb decreases the eddy viscosity and thus allows stronger shear, that is, surface-enhanced ebb currents, and how a homogeneous water column during flood increases the eddy viscosity and thus prohibits strong flood shear, that is, leads to bottom-enhanced flood currents. The residual (tidally averaged, subtidal) circulation contribution resulting from this tidal asymmetry of eddy viscosity and shear has the same orientation as the classical estuarine circulation and is called tidal straining circulation (Burchard et al. 2011; Geyer and MacCready 2014).

One- and two-dimensional model studies (Burchard and Hetland 2010; Burchard et al. 2011) revealed that this is the dominant circulation contribution in periodically stratified estuaries. The first field observations of tidal straining circulation in tidally energetic, weakly stratified channels (Becherer et al. 2011) supported these results. Current measurements (Stacey et al. 2001) and simulations (Burchard et al. 2011) also showed that the tidal straining circulation can reverse, that is, oppose the orientation of the classical estuarine circulation, when stratification is strong during flood (by lateral straining, see below).

A classical longitudinal salinity gradient provided (salinity decreasing in the up-estuary direction), down-estuary shear and stratification enforce each other (see above, SIPS and tidal straining circulation) until the flow becomes unstable, that is, the instantaneous local gradient Richardson number, Ri = *N*^{2}/*S*^{2}, falls below the critical value of 0.25 [*N*^{2} = −*g*/*ρ*_{0} ∂*ρ*/∂*z* Brunt–Väisälä or buoyancy frequency, *S*^{2} = (∂*u*/∂*z*)^{2} + (∂*υ*/∂*z*)^{2} shear squared; Kundu and Cohen 2002].

The existence of transverse (cross-sectional, secondary) circulation was first postulated by Smith (1976) and confirmed by observations of surface convergence during flood (e.g., Nunes and Simpson 1985). It was attributed to bathymetry-induced lateral (across-channel) shear of the longitudinal tidal current (faster above deep channel), leading to lateral salinity gradients that generate a transverse gravitational circulation. Lerczak and Geyer (2004) described the effect of differential advection of along-channel density gradients and the resulting transverse circulation on the longitudinal momentum and circulation. To clearly indicate the mechanism driving these circulations, we refer to it as laterally differential longitudinal advection (of salinity).

During flood, the salinity increases from the shoals to the thalweg (deepest channel) so that the transverse circulation is surface-convergent. This is called flood-oriented transverse circulation (Burchard et al. 2011). It advects high longitudinal momentum from the surface center downward and low momentum from the shoals toward the thalweg, at the surface, which leads to bottom-enhanced flood currents. During ebb, the salinity decreases from the shoals to the thalweg so that the transverse circulation is surface-divergent. This is called ebb-oriented transverse circulation. It advects high momentum from the surface center toward the shoals and low momentum from the shoals toward the thalweg, at the bottom, which leads to surface-enhanced ebb currents. The residual longitudinal circulation contribution resulting from this tidally asymmetric shear also has the same orientation as the classical estuarine circulation and is called advectively driven circulation (Burchard et al. 2011).

Measurements (e.g., Stacey et al. 2001; Lacy et al. 2003; Becherer et al. 2011; Scully and Geyer 2012; Becherer et al. 2015; Purkiani et al. 2015) and numerical studies (Scully et al. 2009; Purkiani et al. 2015) revealed that transverse circulation can also induce stratification and that this lateral straining process is more important during stronger mixing conditions (flood, spring tide). Thus, stratification can be stronger during flood than during ebb, which is contrary to the longitudinal straining (SIPS, see above).

Field observations and model results (Valle-Levinson et al. 2000; Valle-Levinson 2008; Huijts et al. 2011) showed that transverse circulation and lateral convergence can also result from Coriolis deflection of the longitudinal tidal currents, that is, transverse circulation does not require (but may be enhanced by) salinity gradients. Yet another lateral effect is related to turbulent mixing being constantly high above the shoals and periodic above the channel (Scully and Friedrichs 2007). This gradient increases the phase lag between shoals and channel, particularly around slack after ebb, and drives a laterally sheared residual longitudinal circulation with up-estuary flow above the shoals and down-estuary flow above the channel.

Further possible contributions to the estuarine circulation arise from channel convergence (e.g., Ianniello 1979; Burchard et al. 2014), curvature (Geyer 1993; Chant 2002; Becherer et al. 2015), wind (Scully et al. 2005; Waterhouse et al. 2013), and trapping due to complex geometry (Lacy et al. 2003; Giddings et al. 2012). The reviews by MacCready and Geyer (2010) and Geyer and MacCready (2014) contain comprehensive summaries of estuarine circulation processes.

### b. Motivation

Lerczak and Geyer (2004) suggested that lateral effects would be stronger in narrower estuaries and recommended further studies to investigate the interaction of the transverse and the longitudinal circulation and the salinity field. Simulations with three different widths (Burchard et al. 2011) could not strongly confirm this assumption but revealed a clear dependency of the tidal straining circulation (and thus the total, estuarine circulation) on the width, with the intensity increasing with the width.

The numerical study presented here provides the first systematic investigation of the impact of the aspect ratio (depth-to-width ratio) on the estuarine circulation. It spans four orders of magnitude of the aspect ratio and shows that the dependency described by Burchard et al. (2011) holds only for narrow and medium-wide estuaries [aspect ratio ≳0.004, depending on other parameters; see section 4a(1)]. It reveals that for wider estuaries, the intensity of the dominant, tidal straining circulation decreases with the width, that is, the estuarine circulation is maximum for a certain aspect ratio, a phenomenon we explain here.

Lerczak and Geyer (2004) described how the transverse circulation advects the salinity, together with the longitudinal momentum, and numerous observations revealed its stratifying effect (lateral straining, see above). While the feedback of the momentum advection on the estuarine circulation was recognized (advectively driven circulation), the feedback of the stratification, via the interaction with eddy viscosity and shear (tidal straining circulation), was not considered.

Our study shows that lateral straining (∂_{z}*υ*∂_{y}*S*) can lead to stratification during or at the end of flood, depending on the aspect ratio. This also has far-reaching consequences for the stratification during ebb, the tidal asymmetry, and the estuarine circulation. We suggest calling this process secondary strain-induced periodic stratification (2SIPS).

This paper is structured as follows. The next section introduces the theoretical background, the channel geometry, and nondimensional numbers. Section 3 explains the two-dimensional numerical model, the parameters, and the analysis of the model output. Then the results are presented and discussed: section 4a describes the impact of the aspect ratio on the estuarine circulation, particularly the tidal straining contribution, and on the eddy viscosity and the stratification; section 4b clarifies the role of lateral and vertical straining and advection for the stratification; and section 4c substantiates the results by means of additional experiments and simulations. In the conclusion, the 2SIPS process and its influence on the longitudinal tidal straining circulation in dependency on the aspect ratio are summarized. The appendixes provide further equations and details of the numerical experiments as well as information on a three-dimensional model.

## 2. Theory

### a. Basic equations

_{2}tidal current:where

*A*is the cross-sectional area,

*U*

_{t}is the prescribed cross-sectionally averaged M

_{2}velocity amplitude,

*ω*= 2

*π*/

*T*is the circular frequency, and

*T*= 44 714 s is the periodic time of the M

_{2}tidal current.

Note that river discharge is ignored here. Looking into tidal inlet systems such as those in the Wadden Sea, discharge is typically negligible. A horizontal density gradient can also be established with zero discharge, for example, by differential heating (Burchard et al. 2008).

The lateral barotropic pressure gradient is calculated in such a way that the surface elevation is constant, *η*(*y*, *t*) = 0 (rigid lid). It should be noted that this does not mean that the lateral barotropic pressure gradient is zero, but that it balances the vertically averaged baroclinic pressure gradient so that the vertically integrated lateral velocity vanishes,

The results and sensitivities to parameters obtained with the simplified two-dimensional model (more details in section 3a) are in qualitative agreement with those obtained with a three-dimensional model (appendix B).

*ρ*(

*y*,

*z*,

*t*) is the density,

*ρ*

_{0}= 1025 kg m

^{−3}is the reference density,

*β*= 7.8 × 10

^{−4}psu

^{−1}is the haline contractivity, and

*S*(

*y*,

*z*,

*t*) is the salinity. Thus, the buoyancywhere

*g*is the gravitational acceleration, becomes

### b. Channel geometry

*W*representing the prescribed channel width. The coefficients

*c*

_{1},

*c*

_{2}, and

*c*

_{3}are determined by the following conditions:where

*H*

_{min}and

*H*

_{max}are the prescribed minimum and maximum depth at the shores (

*y*= 0,

*W*) and the thalweg (

*y*=

*W*/2), respectively. The parameter

*λ*is related to the prescribed full width at half maximum depth (FWHM; Fig. 1b):where

*λ*equals the fraction of the channel width with

*H*(

*y*) ≤

*H*

_{max}/2 and thus determines the steepness of the channel slopes, together with

*H*

_{min}and the aspect ratio [(11)].

### c. Nondimensional numbers

The system described above is characterized by the nine dimensional parameters numbered in Table 1. These parameters involve two dimensions (length in meters and time in seconds) so that the system is well defined by seven linearly independent nondimensional numbers (9 − 2 = 7; e.g., Buckingham 1914; Kalagnanam et al. 1994). A possible, and in parts commonly used, set is that given in Table 1.

Dimensional parameters and nondimensional numbers with their reference values or variation ranges. The numerals in the first column indicate the system-characterizing nine parameters and seven linearly independent nondimensional numbers. The circles mark model input. The stars mark varied parameters and numbers; others are invariant. Parameters and numbers without an equation are directly prescribed; others are calculated. Sigma is the bottom-following coordinate, *σ*(*y*, *z*) = *z*/*H*(*y*); *κ* = 0.4 is the von Kármán constant. Note that the quasi-lateral diffusivity of salinity here equals that of momentum,

The Simpson number is also known as the horizontal Richardson number; the unsteadiness number is also known as the Stokes number (Souza 2013) or inverse Strouhal number (Burchard et al. 2011, and references therein). Their dynamic impact and that of the nondimensional bottom roughness length were systematically investigated in previous studies (Burchard and Hetland 2010; Burchard et al. 2011). These studies used the average water depth as length scale, *H* = *H*_{mean}, but here we choose the maximum water depth, *H* = *H*_{max}, which is independent of the channel shape.

*W*and the root-mean-square (RMS) friction velocity scale

*α*on the estuarine circulation and its contributions.

## 3. Methods

### a. Numerical model, experiments, and setup

The dimensional equations are solved numerically by means of the three-dimensional General Estuarine Transport Model (GETM; http://www.getm.eu) incorporating a two-equation *k*–*ε* turbulence closure model [General Ocean Turbulence Model (GOTM; http://www.gotm.net)]. See Burchard et al. (2011) for details on the two-dimensional mode of GETM, that is, the *y*–*z* slice model with vanishing longitudinal gradients except for the baroclinic and the barotropic pressure gradients.

To investigate the importance of transverse advection of momentum and salinity for the longitudinal circulation, we carry out further experiments with reduced physics, in addition to the reference experiment [experiment A (full physics); equations in appendix A]: experiment C (no momentum advection), experiment D (no lateral internal pressure gradient), and experiment F (no transverse salinity advection). Experiments C and D have already been carried out by Burchard and Schuttelaars (2012); experiment F is new. [It should be noted that we do not call it B or E in order to avoid confusion with Burchard and Schuttelaars (2012).]

The domain of the cross-sectional slice model is shown in Fig. 1a. It lies in the *y*–*z* plane and has a lateral resolution of 200 cells and a vertical resolution of 100 sigma layers with zooming toward the bottom. The surface elevation is constant, *η*(*y*, *t*) = 0 (rigid lid; see Burchard et al. 2011).

The model is forced by a constant baroclinic pressure gradient and a periodic barotropic pressure gradient [(1)]. The computational time step is Δ*t* = *T*/20 000 ≈ 2.2 s except for very narrow channels, for which numerical stability requires Δ*t* = *T*/40 000 ≈ 1.1 s (*W* ≤ 300 m) or even Δ*t* = *T*/80 000 ≈ 0.6 s (*W* ≤ 100 m). The model is started from rest at slack after ebb and run for 10 tidal cycles to ensure periodicity.

It should be noted that we tested the periodicity as well as the influence of the starting point. If started at slack after flood, the model gives the same results, except for an expectable offset in salinity and thus density and buoyancy.

### b. Prescribed parameters

The invariant parameters are prescribed with the reference values listed in Table 1 unless otherwise noted. They yield the nondimensional numbers also listed in Table 1, for example, the Simpson number Si = 0.48 and the unsteadiness number Un = 0.064.

For comparison, applying the average water depth and the amplitude of the friction velocity (

In this contribution, we focus on variation of the width with 100 m ≤ *W* ≤ 10^{6} m, yielding the aspect ratio 0.15 ≥ *α* ≥ 1.5 × 10^{−5}, that is, spanning four orders of magnitude. This parameter range is investigated with about 60 simulations.

Please note that not the entire *α* range can be found in reality, but it is used here to also cover marginal phenomena, for example, to show the (transition to) quasi-one-dimensional behavior of wide estuaries. Aspect ratios of several real estuaries are listed in Table 2 for comparison. It should be noted that the model results presented here are not applicable to all of these estuaries since some of them are governed by permanent stratification, Coriolis forcing, or curvature.

Aspect ratios of several real estuaries, calculated from depth and width as found in the given literature and divided into three size classes [see section 4a(1), Fig. 2]. It should be noted that *H* = *H*_{mean} in the first three examples.

To vary *α* but no other nondimensional number, the full width at half maximum depth and the dimensional quasi-lateral diffusivities *λ*) and the nondimensional diffusivities remain invariant (see Table 1). Instead of varying *α* by means of *W* (requiring the variation of FWHM and *α* could be varied by means of *H*_{max} (requiring the variation of *H*_{min}, *U*_{t}, and

As a side note, if the water depth is varied but *α*), there is an additional impact on the dynamics. However, this impact is small compared to the impact of *α* (to be shown): varying

It should be noted that preliminary tests confirmed self-similarity when the dimensional parameters are varied but the nondimensional numbers are invariant. For example, if *H*_{max} is doubled, invariance of the nondimensional numbers requires doubling of *H*_{min}, *W*, FWHM, and *U*_{t}, as well as quadrupling of

Please also note that, for the default parameter values (Table 1), Coriolis force has no significant effect on the residual circulation [section 4c(2)].

### c. Analysis and notation

*u*,

*υ*,

*w*), vertical eddy viscosity

*A*

_{z}, and salinity

*S*are analyzed based on the last (tenth) tidal cycle of the model output, when a periodic state is reached. The time and spatial coordinates are nondimensionalized by their characteristic scales:The nondimensional time is defined such that we have full flood at

*U*

_{t}(Burchard et al. 2011) and the eddy viscosity by

*H*

_{max}and

It should be noted that these contributions are not independent of each other but that they interact nonlinearly. For example, the tidal straining circulation is associated with stratification and shear, both of which are influenced by the total residual circulation. Still, the direct forcing mechanisms are identified with this method and can then be further investigated, as we will show for the tidal straining circulation [sections 4a(3) and 4b].

For visualization of the transverse circulation, a particle tracking model is applied. The utilized integration scheme is fourth-order Runge–Kutta with a time step of Δ*t* = *T*/500 ≈ 89.4 s. The interpolation is linear in time and cubic in space. The model is implemented in MATLAB.

About 8000 particles are released in one-half of the cross section (symmetric, Fig. 1b). They are initially (at slack after ebb) distributed on a rectangular grid with a nondimensional spacing of ^{−3} in the vertical and ^{−3} in the lateral direction. (Note that the model is dimensional.) The particles are moved solely by transverse advection (

It should be noted that the transverse advection is not independent of the longitudinal advection (

## 4. Results and discussion

### a. Aspect ratio and estuarine circulation

#### 1) Impact of the aspect ratio on the estuarine circulation contributions

The impact of a channel’s aspect ratio on the intensity of the estuarine circulation and its contributions is shown in Fig. 2. For convenience, we define the following three size classes: wide/shallow channels with *α* ≤ *α*_{1} = 0.002, narrow/deep channels with *α* ≥ *α*_{2} = 0.015, and medium channels in between. These limits (solid vertical lines in Fig. 2) coincide with distinct changes of the residual longitudinal circulation, particularly the tidal straining and the advectively driven contribution.

It should be noted that the values of *α*_{1} and *α*_{2}, as well as the course of the functions *α*, depend on the prescribed parameters (Table 1), which vary along real estuaries, just as *α*.

In wide channels, the estuarine circulation is relatively weak, with the tidal straining circulation being the dominant contribution (73%–79%), followed by the gravitational circulation (21%–16%).

At *α*_{1} = 0.002, the intensity of the tidal straining circulation starts to increase sharply, the advectively driven circulation starts to increase, and the gravitational circulation increases slightly. In medium channels, the total circulation reaches a maximum, which is caused primarily by a maximum of the tidal straining circulation at *α*_{m} = 0.004. The advectively driven circulation continues to increase while the gravitational circulation remains about constant.

At *α*_{2} = 0.015, the advectively driven circulation becomes stronger than the tidal straining circulation so that, in narrow channels, the advectively driven circulation is the dominant contribution. The intensity of the tidal straining circulation falls below zero at *α* = 0.03, that is, the tidal straining circulation reverses and opposes the classical estuarine circulation in very narrow channels. The gravitational circulation remains about constant.

These results raise the main question of this paper: Why is the tidal straining circulation maximum for a certain aspect ratio? Furthermore, why is it reverse in very narrow channels? Before we answer this [sections 4a(3) and 4b(1)], let us consider the residual velocity profiles establishing the circulation intensities.

#### 2) Comparison of residual profiles for different aspect ratios

Cross-sectional views of the residual circulation contributions are shown in Fig. 3 for a wide (*α*_{w} = 0.0004), a medium (*α*_{m} = 0.004), and a narrow (*α*_{n} = 0.04) channel. In agreement with the measure of their intensity (Fig. 2, *α* values marked at lower abscissa), the medium channel exhibits the strongest down-estuary residual flow at the surface and up-estuary flow above the bottom (Fig. 3a2), which is mainly caused by the strong tidal straining circulation (Fig. 3b2). The gravitational (Fig. 3c2) and the advectively driven contribution (Fig. 3d2) are much weaker.

In the wide channel, all circulation contributions are weaker than in the medium channel (Figs. 3a1,b1,c1,d1 versus Figs. 3a2,b2,c2,d2). The advectively driven circulation (Fig. 3d1) has almost ceased, so that the wide channel resembles a one-dimensional situation, that is, horizontal gradients are negligible (except for the prescribed longitudinal pressure gradients).

In the narrow channel, the advectively driven circulation is stronger than in the medium channel (Fig. 3d2 versus Fig. 3d3) and the tidal straining circulation exhibits up-estuary flow below the surface and down-estuary flow at the slopes and in the center of the channel (Fig. 3b3), which opposes the classical estuarine circulation [

Also shown in Fig. 3 are the residual eddy viscosity and the salinity. In the wide channel, salinity decreases from the thalweg to the shoals and only very slightly from the bottom to the surface (Fig. 3f1). Consequently, stratification is very weak. Eddy viscosity, on the other hand, is strong in the center of the channel (Fig. 3e1).

In the medium channel, salinity stratification is strong, with highest values above the center of the channel (Fig. 3e2). Eddy viscosity is relatively small and confined to the deep parts of the channel (Fig. 3f2).

In the narrow channel, salinity varies only very little, with strongest stratification occurring below the surface (Fig. 3f3). There, eddy viscosity is comparable to that in the medium channel; in the lower part, it is slightly higher (Fig. 3e3).

A detailed explanation of the differences in circulation, eddy viscosity, and stratification for different channel widths follows in sections 4a(3) and 4b.

#### 3) Origin of the maximum and the reverse tidal straining circulation

To investigate the maximum of the tidal straining circulation in the medium channel, we now take a closer look at tidal cycles of the longitudinal velocity, its vertical shear, the eddy viscosity, the salinity, the Brunt–Väisälä frequency, and the gradient Richardson number at the thalweg (Fig. 4). Recall that the covariance of viscosity and vertical shear of longitudinal velocity divided by residual viscosity is the driving term for the tidal straining circulation [

In the wide channel, the expected behavior of weak, strain-induced periodic stratification is observed (SIPS; see section 1a and references therein): the shear is weakly asymmetric, with negative shear being slightly stronger and longer lasting than positive shear (*N*^{2} and Ri) during full and late flood (Figs. 4d1,e1,f1).

The tidal asymmetries grow drastically when going to medium channels. (Recall that only the aspect ratio is changed, in terms of nondimensional numbers.) In the medium channel, negative shear is much stronger than the positive shear (−1.99 versus 1.24), particularly in the upper half of the thalweg water column (−1.14 versus 0.31; Figs. 4a2,b2). This leads to very high shear fluctuations with

In the narrow channel, negative shear is still stronger and longer lasting than positive shear (−1.14 versus 1.03) (Figs. 4a3,b3). Close to the surface, this negative shear even persists over the entire tidal cycle. Eddy viscosity is now stronger during ebb than during flood (Fig. 4c3), but its maximum is smaller than that in wide or medium channels. This reverse eddy viscosity asymmetry is due to a reverse stratification pattern with strongest (though still weak) stratification during and shortly after flood and homogeneous water columns in the second half of the ebb phase (Figs. 4d3,e3,f3). This behavior cannot be explained by the longitudinal tidal currents either.

Three characteristics lead to the observed maximum of the tidal straining circulation in the medium channel (Table 3): (i) positive shear fluctuation (

Overview of the characteristics leading to maximum tidal straining circulation as well as their fulfilment (☑) for the three size classes. [An empty check box, ☐, corresponds to the notation (¬*i*), etc., in the text.]

In the wide channel, only characteristic (i) is clearly fulfilled; positive shear fluctuation coincides with strong viscosity fluctuation during flood. Negative shear fluctuation coincides with moderate viscosity, that is, positive viscosity fluctuation (not shown), around full ebb [(ii) not fulfilled, indicated as (¬ii)]; however, negative shear fluctuation coincides with very weak viscosity, that is, negative viscosity fluctuation, at the beginning of ebb and around slack after ebb (ii). The residual tidal straining circulation in the wide channel is classically oriented, but the opposing contribution during ebb (¬ii) and the high residual viscosity (¬iii) yield a weak intensity.

In the narrow channel, the eddy viscosity asymmetry is reverse, but the shear asymmetry is not. (The surface shear is negative during flood, but its fluctuation is positive like in wide and medium channels; not shown.) Positive shear fluctuation coincides with moderate viscosity, that is, positive viscosity fluctuation, around full flood (i) but with very weak viscosity, that is, negative viscosity fluctuation, at the beginning and end of flood (¬i). Negative shear fluctuation coincides with very weak viscosity, that is, negative viscosity fluctuation, at the beginning and end of ebb (ii) but with strong viscosity, that is, positive viscosity fluctuation, around full ebb (¬ii). The classical-opposing contributions are slightly larger than the classical so that the residual tidal straining circulation opposes the classical orientation, that is, the tidal straining circulation is reverse in narrow channels. The fact that these contributions almost balance yields a weak intensity even though the residual viscosity is low (iii).

### b. Transverse stratifying and destratifying processes

#### 1) Stratification by transverse salinity advection

As explained above, the vertical shear of the longitudinal velocity is affected by the eddy viscosity, which itself is affected by the stratification. The observed change of the stratification pattern with only the aspect ratio (Figs. 4d1,d2,d3,e1,e2,e3) suggests the importance of transverse, stratifying as well as destratifying processes. To understand these processes, we now consider the lateral and vertical advection of salinity over one tidal cycle. Figure 5 shows snapshots of the lateral velocity and the salinity.

We begin with the medium channel and the flood phase. The slope ∂_{y}*H*(*y*) causes laterally differential longitudinal advection of salinity, which builds a lateral salinity gradient with highest salinity at the thalweg (Figs. 5b2,c2,d2,e2). This generates flood-oriented transverse circulation (surface convergence) (Figs. 5c2,d2,e2), which strains the salinity field (Figs. 5d2,e2) and induces strong stratification already shortly before slack after flood (Fig. 5e2).

The strong stratification in the center of the channel persists throughout the ebb and into the flood phase (Figs. 5e2,f2,g2,h2,a2,b2,c2). Laterally differential longitudinal advection during ebb builds a salinity gradient with lowest salinity at the thalweg, in the upper half of the cross section (Figs. 5g2,h2,a2). This generates ebb-oriented transverse circulation (Figs. 5h2,a2,b2), which also enhances stratification (Figs. 5a2,b2).

Considering the effect of stratification on eddy viscosity and thus vertical shear of horizontal velocities, the key leading to the maximum of the tidal straining circulation in the medium channel appears to be the matching length scales of half the channel width and the lateral salinity advection during the second half of flood (Figs. 5c2,d2,e2), which leads to strong stratification already at the very beginning of the ebb phase (Fig. 5e2). Because of the stratification-shear feedback during ebb (enforcing each other until the gradient Richardson number Ri falls below 0.25, at the surface around full ebb; Fig. 4f2), the ebb viscosity is very low and thus the tidal asymmetry very strong.

The wide channel has a gentler slope so that the lateral gradients of the longitudinal velocity and thus of the salinity are smaller than in the medium channel. (Note that the salinity range over

The narrow channel has a steeper slope so that the lateral gradients of the longitudinal velocity and thus of the salinity are larger than in the medium channel. (Note that the salinity range over

#### 2) Distinction of the stratifying processes: Particle tracking and mathematical decomposition

Figure 5 clearly suggests but cannot prove that transverse salinity advection is the dominant stratifying process in the medium channel. By means of particle tracking, the contribution of transverse advection to the salinity redistribution can be visualized.

The results of the transverse particle tracking are presented in Fig. 6 for the second half of flood. Shown are the instantaneous positions of the particles, marked by dots, and their color-coded salinity (see below and figure caption). The initially equidistant particle distribution (at slack after ebb, not shown) becomes inhomogeneous because of interpolation and truncation errors of the integration scheme. (Note that there is no convergence or divergence in the transverse plane,

- Fig. 6a - instantaneous, that is, at present position and time (full flood, contours in Fig. 5c);
- Fig. 6b - memorized, that is, at previous position and time (full flood, particles as information carrier); and
- Fig. 6c - instantaneous (slack after flood, Fig. 5e).

The wide channel is vertically homogeneous at full flood with salinity increasing toward the thalweg (Fig. 6a1). The generated transverse circulation induces very weak stratification above the slopes at slack after flood (Fig. 6b1). Comparison of Fig. 6b1 with Fig. 6c1 indicates that this is the dominant stratifying process in this part of the tidal cycle.

The medium channel is also vertically homogeneous at full flood with salinity increasing toward the thalweg, except for subsurface stratification at the thalweg (Fig. 6a2), which is a remainder from strong stratification during ebb (Figs. 4d2,e2). The transverse circulation induces strong stable stratification at slack after flood, except for the thalweg and the bottom at the slopes (Fig. 6b2). Comparison of Fig. 6b2 with Fig. 6c2 indicates that this is the dominant stratifying process here, too. The actual salinity field (Fig. 6c2) is vertically homogeneous above the bottom at the slopes because of turbulent diffusion and more saline toward the thalweg because of further laterally differential longitudinal advection.

The narrow channel is already weakly stratified in the upper center of the channel at full flood (Fig. 6a3). The transverse circulation stirs the salinity field and induces unstable stratification at slack after flood (Fig. 6b3). The actual salinity field (Fig. 6c3) is not unstable, though, but stably stratified because of the negative shear close to the surface even during flood [section 4a(3), Fig. 4b3].

Without the residual, (23) is similar to the evolution equation of stratification in Giddings et al. [2011, their (7)] and provides an alternative to the evolution equation of potential density anomaly (Burchard and Hofmeister 2008; de Boer et al. 2008).

Since longitudinal SIPS is a universal process in all regions with longitudinal density gradients and tidal currents, we call it primary strain-induced periodic stratification (1SIPS) in the following. Accordingly, we suggest calling periodic stratification by lateral/vertical straining secondary/tertiary strain-induced periodic stratification (2/3SIPS).

Laterally/vertically advected periodic stratification is abbreviated 2/3APS. The combination of APS and SIPS is already known as ASIPS (de Boer et al. 2008; Howlett et al. 2011), in our case 2ASIPS for laterally and 3ASIPS for vertically advected and strain-induced periodic stratification.

All terms in (23) are nondimensionalized by multiplication with

In wide channels (without Coriolis forcing), 1SIPS is more important than 2- and 3ASIPS. In very wide channels, 2- and 3ASIPS are negligible.

In medium channels, 1SIPS is about 4 times as strong as in wide channels (feedback of transverse processes), but 2SIPS increases and dominates considerably, though it is partly compensated by negative 2APS, that is, lateral advection of weaker stratification (from shoals to thalweg, not shown). Negative 3SIPS and positive 3APS approximately balance each other.

In narrow channels, 2SIPS is still the dominant stratifying term, but it decreases. Positive 3SIPS contributes about half as strong as 1SIPS. Close to the limit between medium and narrow channels (*α* ≈ 0.02), 3SIPS and positive 3APS, that is, vertical advection of stronger stratification (from surface to bottom, not shown), concur so that 3ASIPS is about 3 times as important as 1SIPS.

#### 3) Secondary strain-induced periodic stratification

Since 2SIPS is the dominant stratifying term except in wide channels (Fig. 7), it is now elucidated a bit further. It has a periodicity of *T*/2: in the medium channel, transverse circulation is generated (or rather turned into the orientation corresponding to the respective tidal phase) around full tides (Figs. 5c2,h2), when laterally differential longitudinal advection has already built a considerable lateral salinity gradient (schematic Fig. 8a; cf. Figs. 6a1,a2). As soon as lateral straining has flattened this gradient, stratification is maximum (Fig. 8b2). In the medium channel, this occurs around slack tides (Figs. 5b2,e2, 6b2).

The further development depends on the tidal phase. After slack after ebb, during the first half of flood, stratification is decreased (Figs. 5b2,c2,d2); after flood, during ebb, it is increased (Figs. 5e2,f2,g2) because of 1SIPS (periodicity *T*).

In addition to multidirectional SIPS and APS, shear-induced turbulence (periodicity *T*/2) interferes with the stratification (highest shear, turbulent diffusion, and destratification at full tides; cf. Fig. 4). Furthermore, all these processes interact nonlinearly so that it is difficult to distinguish the effect of 2SIPS.

In wider channels, 2SIPS is weaker and limited to the slopes (Figs. 5a1,b1,f1, 6b1) because the length scale of the lateral velocity is smaller than half the channel width (Fig. 8b1). Lateral straining and stratification are maximum only after slack tides (Figs. 5b1,f1).

In narrower channels, 2SIPS sets in very early (Figs. 5b3,f3, 6a3) and is then subject to longitudinal shear (Figs. 4a3,b3) and turbulent diffusion (Figs. 6b3,c3). The length scale of the lateral velocity is larger than half the channel width so that transverse stirring occurs (Fig. 8b3; cf. Fig. 6b3).

### c. Further results and discussion

#### 1) Importance of transverse advection of salinity and momentum

Further evidence of the importance of transverse salinity advection for the maximum of the tidal straining circulation is provided by comparison of the full-physics experiment (A) with reduced experiments ignoring transverse advection of momentum (C), of salinity (F), and of both (D; no lateral internal pressure gradient, i.e., no transverse circulation). The maximum of the tidal straining circulation still exists in experiment C but not in experiments D and F (Fig. 9a). This implies that salinity advection is more important for the phenomenon than momentum advection.

In experiment D, the intensity of the tidal straining circulation is constant over *α*, as expected: without lateral internal pressure gradients, transverse processes are completely switched off so that the lateral dimension does not play a role. The intensities of all circulation contributions in experiment A (Fig. 2) remain constant at the values of experiment D (Fig. 9a) for *α* ≲ 0.0003 because then the slope ∂_{y}*H* is too small to cause significant lateral gradients of salinity and velocity. Thus, transverse processes are negligible (quasi one-dimensional).

The maximum of the tidal straining circulation is due to a maximum of the viscosity–shear covariance in medium channels [*α* ≈ *α*_{m} = 0.004; Fig. 9b, experiments A and C; characteristics (i) and (ii) in section 4a(3)] as well as to a decrease of the residual eddy viscosity around the limit between wide and medium channels [*α*_{1} = 0.002; Fig. 9c; characteristic (iii); also see (18)].

#### 2) Generality of the results

To substantiate the generality of our findings, we carry out further simulations (experiment A) with different prescribed parameters, namely, a nonzero Coriolis parameter, *f* = 1 × 10^{−4} s^{−1} (inverse Ekman number *λ* = 0.75 (dashed line in Fig. 1b; Fig. 10b); and half the longitudinal salinity gradient, ∂_{x}*S* = −1.5 psu m^{−1} (Si = 0.24; Fig. 10c).

The intensities of the circulation contributions in these simulations are weaker (note the different

*α*range. The rotationally driven contribution

*δ*≈ 1.2 (0.9) for the wide (medium, narrow) channel, that is, large friction and negligible Coriolis force.

Furthermore, we cover the two-dimensional parameter space spanned by medium aspect ratios and low-to-moderate Simpson numbers (about 90 simulations; Fig. 11). Note that runaway stratification occurs for higher Si, a common problem of one- and two-dimensional simulations (Geyer and MacCready 2014).

For decreasing Si, the intensity of the tidal straining circulation decreases (Fig. 11; cf. Burchard et al. 2011, 2014) and the maximum is shifted to narrower channels. The aspect ratio of the channel with maximum tidal straining circulation appears to depend exponentially on the Simpson number: *α*_{m} ∝ exp(−Si) (white dotted line; note the logarithmic *α* scale in all figures). Assuming constant *H*, this can be rewritten as *W*_{m} ∝ exp(Si).

A lower Si is equivalent to a smaller longitudinal salinity gradient, which via laterally differential longitudinal advection implies a smaller lateral salinity gradient. This leads to a weaker transverse circulation so that the secondary strain-induced periodic stratification, and thus the intensity of the tidal straining circulation, is maximum in a narrower channel. This functional chain is highly nonlinear, though, for example, because Si also affects the longitudinal gravitational circulation and the background stratification.

In a final step, we carry out three-dimensional simulations with nonzero river discharge and free surface elevation (appendix B). These simulations qualitatively confirm the results obtained with the simplified two-dimensional model (cf. Figs. 2 and A1b).

Please note that the qualitative impact of the aspect ratio on lateral gradients (of *H*, *u*, *S*), on 2SIPS, and thus on the intensity of the tidal straining circulation, is of general nature, though we (can) cover only a small part of the parameter space to prove this. It should be noted again that the results presented in this paper apply to periodically, weakly stratified estuaries. In a permanently stratified estuary, for example, lateral straining probably has a much weaker additional effect on the stratification.

#### 3) A note on stratification and mixing

Stratification decreases the eddy viscosity *A*_{z} and thus the mixing of momentum (see section 1a). It also decreases the eddy diffusivity *K*_{z}, but not the mixing of salinity; this increases with stratification [(27)]. It should be noted that *K*_{z} is decreased more than *A*_{z} in a stably stratified water column, that is, the turbulent Prandtl number is larger than one, Pr = *A*_{z}/*K*_{z} > 1 (Kundu and Cohen 2002).

*ρ*

_{0}is the reference density (see section 2a).

The residual and cross-sectionally averaged nondimensional potential density anomaly and vertical mixing are shown in Fig. 12. They behave very similarly, that is, both stratification and vertical mixing are weak in wide, strong in medium, and moderate in narrow channels. They are maximum at the same aspect ratio as the intensity of the tidal straining circulation (Fig. 2; *α*_{m} = 0.004).

## 5. Conclusions

Generally, stratification in an estuary is induced by the vertical shear of the longitudinal ebb currents (faster and thus fresher at the surface) and increased during the ebb phase (longitudinally, primary strain-induced periodic stratification, 1SIPS). However, the surface-convergent transverse circulation during flood can induce stratification too (mainly laterally, secondary, 2SIPS): it transports fresher water from the shoals (fresher there because of laterally differential longitudinal advection) to the thalweg, at the surface, and spreads the saltier water from the thalweg in the deep channel and along the bottom of the slopes. For a certain, medium aspect ratio (*α* ≈ 0.004 for our set of nondimensional numbers), the two bodies of fresher water just merge at the thalweg (at the surface) at slack after flood, so that the stratification is very strong already at the beginning of the ebb phase.

It should be noted that the dominance of 2SIPS over 1SIPS in medium and narrow channels implies that the consideration of only longitudinal forcing parameters, for example,

Strong stratification during ebb suppresses the eddy viscosity and gives rise to strong shear. Thus, the tidal asymmetry of stratification, eddy viscosity, and shear is high and drives an intense longitudinal tidal straining circulation in medium channels. Since this is the dominant contribution to the total residual longitudinal circulation in tidally energetic, weakly (primarily) stratified estuaries, the estuarine circulation is maximum in these medium channels.

In wide channels, 2SIPS and other transverse processes are very weak, that is, hardly any additional stratification is induced, so that the tidal straining circulation is comparably weak. Very wide channels are laterally homogeneous and resemble a one-dimensional situation.

In narrow channels, transverse processes (mainly 2SIPS and vertically, tertiary advected and strain-induced periodic stratification, 3ASIPS) are very strong, and stable stratification is induced already before full flood and persists throughout ebb. 1SIPS further increases the stratification, but this also gives rise to the shear, which eventually produces turbulence and thus decreased stratification and increased eddy viscosity during ebb. This reverse tidal asymmetry of eddy viscosity leads to a reverse tidal straining circulation.

The total, estuarine circulation does not reverse because the advectively driven contribution is the dominant one in narrow channels. As suggested by Lerczak and Geyer (2004), the intensity of the advectively driven circulation increases with the aspect ratio.

## Acknowledgments

The studies of Elisabeth Schulz have been funded by a scholarship from the University of Rostock, Interdisciplinary Faculty, Department Maritime Systems. The work of Henk M. Schuttelaars has been supported by a visiting scientist grant through the Leibniz Institute for Baltic Sea Research Warnemünde. Ulf Gräwe has been funded by the German Federal Ministry of Research and Education (BMBF) in the framework of the project PACE (The future of the Wadden Sea sediment fluxes: Still keeping pace with sea level rise?, FKZ 03F0634A). The work of Hans Burchard has been carried out in the framework of the project ECOWS (Role of Estuarine Circulation for Transport of Suspended Particulate Matter in the Wadden Sea) funded by the German Research Foundation (DFG, project BU1199/11). We thank Lars Umlauf (Leibniz Institute for Baltic Sea Research Warnemünde, Rostock, Germany) for valuable discussions and hints. The authors are furthermore grateful to the two anonymous reviewers, whose remarks and suggestions helped to greatly improve this paper.

## APPENDIX A

### Equations and Reduced Experiments

*x*,

*u*) and across (

*y*,

*υ*) the estuary can be written as follows (assumptions as in section 2a):where

*σ*layers and

*P*is the barotropic pressure gradient in the given direction. Here, square brackets, [ ], are used to denote prescribed variables.

*K*

_{z}(

*y*,

*z*,

*t*) is the vertical eddy diffusivity.

The reference experiment A (full physics) uses the full Eqs. (A1)–(A3). The reduced experiments (section 3a) are implemented as follows:

## APPENDIX B

### Three-Dimensional Model

Three-dimensional simulations are carried out by means of the General Estuarine Transport Model (GETM). The model domain has a horizontal resolution of 200 × 60 cells and a vertical resolution of 20 sigma layers with zooming toward the bottom.

The topography is shown in Fig. B1a. The initial salinity decreases linearly from *S*_{a} at the open boundary, *x* = −*L*, to 0 at *x* = −*L*/2 and is 0 between *x* = −*L*/2 and the river mouth, *x* = 0. The prescribed parameters are listed in Table B1.

Prescribed parameters for the 3D simulations. The stars mark varied parameters; others are invariant.

_{2}tidal surface elevation at the open boundary (

*ω*and

*T*as in Table 1),Simulations are started from rest and run for 20 tidal cycles to ensure periodicity.

The aspect ratio is varied by means of the width (about 30 simulations); the river discharge is scaled accordingly (see Table B1) in order to yield comparable residual velocities. The intensity of the estuarine circulation is analyzed at *x*_{α} = −60 km, which lies at the up-estuary end of the convergence zone.

It should be noted that the convergence increases with the width, but it is very weak at *x*_{α} [convergence number Co = 2.9 × 10^{−4}–3.2 × 10^{−4}; cf. Burchard et al. (2014); that is, increasing by a factor of 1.1 over the covered *α* range]. Here, the amplitude of the cross-sectionally averaged longitudinal velocity is about 1.3 m s^{−1} and the residual cross-sectionally averaged longitudinal salinity gradient is about −4 × 10^{−4} psu m^{−1}. The salinity field exhibits weak primary SIPS (not shown).

The impact of the aspect ratio on the intensity of the estuarine circulation is shown in Fig. B1b. The intensity and thus the absolute impact of the aspect ratio (i.e., the intensity range) are considerably weaker in the three-dimensional model than in the two-dimensional model. This might be related to the spatial and temporal variability of the longitudinal salinity gradient,

In the three-dimensional model, the intensity has a maximum at *α* ≈ 0.0067. As in the two-dimensional model (also see Fig. 2), this is due to a maximum of the tidal straining circulation, which is again related to secondary SIPS (not shown).

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