1. Introduction
The aim of this study is to quantify the contributions of major processes to longitudinal and lateral residual circulation in tidally energetic estuaries. For this purpose, a two-dimensional cross-sectional numerical model is applied, with prescribed longitudinal salinity gradients and barotropic pressure gradients induced by semidiurnal tidal forcing, under the assumption that all other longitudinal gradients are negligible. The residual circulation is decomposed into residual exchange and local runoff contributions from straining, gravitational circulation, lateral advection, and lateral mixing, both in the longitudinal and lateral directions. Zero net river discharge and a rigid lid (neglect of Stokes drift) are assumed, and a symmetric, parabolic bathymetry is used. Effects of the earth’s rotation, wind straining, bathymetric variations, residual river runoff, and Stokes drift will be investigated in follow-up studies. The major tool used in the present study is the estuarine circulation analysis recently introduced by Burchard and Hetland (2010). This analysis provides a semianalytical decomposition of the residual flow profiles into contributions from all processes included in the momentum balance. These contributions are solved numerically, because analytical solutions are available only for strongly simplified situations. This analysis tool is extended here for two-dimensional cross-sectional models, including longitudinal, lateral, and vertical velocity components.
This semianalytical method is based on the same principles as previous approaches that applied a vertically and temporally constant eddy viscosity and calculated contributions to residual currents by tidally averaging the momentum equations and vertically integrating the resulting equations twice (Hansen and Rattray 1965; Chatwin 1976; MacCready 2004; Huijts et al. 2006). Recently, Huijts et al. (2009) demonstrated how analytical solutions for the major contributions to residual currents can be obtained for longitudinal and lateral components, assuming constant eddy viscosity (thus eliminating tidal straining) and diagnostically prescribed lateral density gradients (thus removing parts of the contributions from lateral advection driven by lateral density gradients). In the present study, some simplifications made by Huijts et al. (2009) due to the requirement for analytical solutions are removed by solving the resulting complex analytical expressions numerically. Apart from that, we follow the approach of Huijts et al. (2009).
Classically, estuarine circulation has been attributed to gravitational circulation (see Pritchard 1952, 1954, 1956; Hansen and Rattray 1965; Chatwin 1976). The analytical solutions for gravitational circulation based on constant eddy viscosity originally presented by Hansen and Rattray (1965) have been extended to include wind straining (Wong 1994; Ralston et al. 2008) and parabolic eddy viscosity (McGregor 1972; Ianniello 1977; Burchard and Hetland 2010). All these theories have in common that a tidal modulation of the estuarine circulation other than a modification of the tidal-mean eddy viscosity is not considered.
Effects of ebb–flood asymmetries in vertical differential advection have first been acknowledged by van Aken (1986) and Simpson et al. (1990) as processes leading to a destabilization of the water column during flood (denser offshore waters sheared over less dense onshore waters) and vice versa during ebb. Jay and Musiak (1994) suggested a mechanism how the associated tidal mixing asymmetry (enhancement of vertical mixing during flood and suppression of vertical mixing during ebb) contributes to estuarine circulation: during flood, considerably more momentum is mixed down to the near-bottom region than during ebb, resulting in a near-bottom up-estuary excess velocity. In the present study, this contribution to estuarine circulation is referred to as tidal straining circulation. Burchard and Baumert (1998) showed in a (two-dimensional longitudinal) numerical model study of a partially mixed estuary that this tidal straining circulation contributes more to the generation of estuarine circulation than gravitational circulation. The importance of tidal straining circulation has later been highlighted by Stacey et al. (2001, 2008). Recently, Burchard and Hetland (2010) showed for periodically mixed estuaries that the tidal straining amounts to typically two-thirds of the estuarine circulation (with a one-third contribution from gravitational circulation) for negligible wind stress, no river discharge, and no effects from the earth’s rotation and channel geometry. However, this dominance of the tidal straining was shown to decrease with the stability of the flow.
Another important mechanism for creating estuarine circulation has been proposed by Lerczak and Geyer (2004), who found in an idealized model study of a straight channelized estuary that redistribution of longitudinal momentum by lateral advection generates residual currents in support of estuarine circulation. The underlying mechanism for generating the lateral circulation required for this advectively driven redistribution has already been described by Nunes and Simpson (1985) for weakly stratified estuaries: during flood, lateral differential advection builds up a salinity maximum in the center line of the estuary, which leads to lateral salinity gradients toward both sides of the estuary, generating in turn a lateral estuarine circulation with near-bottom currents directed to the sides and converging near-surface currents, both associated with downwelling in the center of the flow. With this, the advectively driven contribution to estuarine circulation can be explained as follows: During flood, the higher longitudinal momentum is advected near the bottom toward the sides (there increasing momentum). Furthermore, the downwelling advects faster near-surface currents down to the near-bottom region in the center of the flow. As a compensation, near the surface, slower velocities are advected toward the center of the flow, there decreasing the flood momentum. Because the same happens during ebb, but with different signs of velocities and their gradients (such that the advective terms have the same sign during ebb and flood), a residual current with a near-bottom up-estuary component is forced. This mechanism has recently been confirmed for stratified estuaries by means of idealized (Cheng and Valle-Levinson 2009) and realistic numerical model simulations (Scully et al. 2009; Cheng et al. 2009). For scenarios with weak stratification, Lerczak and Geyer (2004) found that this advectively driven contribution to estuarine circulation is of the same order of magnitude as the gravitational circulation. Moreover, the advectively driven contribution to residual longitudinal circulation scales roughly with the relative longitudinal buoyancy gradient (Scully et al. 2009), as well as the gravitational circulation (Geyer et al. 2000) and the tidal straining circulation (Burchard and Hetland 2010).
For one-dimensional geometries, Burchard and Hetland (2010) had identified the covariance between longitudinal vertical shear and eddy viscosity as the tidal straining contribution to estuarine circulation, as sketched by Jay and Musiak (1994). In the more complex two-dimensional situation investigated here, the modification of longitudinal shear by lateral advection additionally impacts on this shear–viscosity covariance, such that it cannot be identified with classical longitudinal straining alone anymore. In the present theory, it will not be possible to differentiate between the impacts of these two effects on the shear–viscosity covariance. Therefore, we are using the term “tidal straining” here in a more general sense, including all contributions to the shear–viscosity covariance.
In the present study, we focus on the quantification of these three processes (tidal straining, gravitational, and advectively driven circulations) and their lateral variation in such a way that the superposition of these contributions exactly amounts to the estuarine circulation. By doing so, a complete decomposition of the lateral, longitudinal, and vertical components of the estuarine circulation into its contributions from these processes is obtained. The parameter space under consideration covers variations in the relative longitudinal buoyancy gradient (Simpson number) and the relative tidal frequency (unsteadiness parameter). Furthermore, the sensitivity of the composition of estuarine circulation on bed roughness and the width of the estuary is tested.
The use of a constant in time and space longitudinal salinity gradient restricts this study to periodically mixed estuaries, of which a few examples exist (Burchard and Hetland 2010). However, by going toward the limit of almost permanently stratified estuaries, it is shown here how the dynamical regime shifts from tidal straining dominated to gravitationally and advectively dominated, as it is assumed for partially mixed estuaries.
This manuscript is organized as follows: First, major nondimensional numbers are defined (section 2a) and the underlying dynamic equations are given (section 2b). In section 2c, the residual circulation analysis is outlined (see the appendix for details). The numerical model is briefly described in section 3. Various scenarios are discussed in detail in section 4, with emphasis on a reference scenario (section 4a), the dependence on the Simpson number (section 4b), the unsteadiness parameter (section 4c), the bed roughness (section 4d), and the channel width (section 4e). Some conclusions are given in section 5.
2. Theory
a. Nondimensional numbers
It should be noted that, in earlier papers (Monismith et al. 1996; Stacey et al. 2001; Burchard 2009), the Simpson number defined in (1) has generally been denoted as the horizontal Richardson number. However, to acknowledge the introduction of this number by Simpson et al. [1990, see their Eq. (5)], the notation Simpson number is introduced here. The unsteadiness parameter defined in (2) has been denoted in earlier papers (Baumert and Radach 1992; Burchard 2009) as the inverse Strouhal number, but here we suggest using the term unsteadiness parameter to avoid suggesting a relation to eddy shedding processes, which are generally associated with a Strouhal number.
b. Dynamical equations
Dynamic boundary conditions are assuming law of the wall dynamics near the bottom and a prescribed turbulent momentum flux (τxs, τys) at the surface the direction of which is opposed to the near-surface wind direction. Note that the surface momentum flux is considered in the derivations here for future reference only. Buoyancy fluxes are assumed to be zero across all boundaries.
The eddy viscosity Aυ and the eddy diffusivity Kυ are calculated by means of a turbulence closure model (see section 3).
c. Residual circulation analysis
〈u1e〉 and 〈υ1e〉 are exchange profiles due to the covariance between the eddy viscosity and the longitudinal and the lateral vertical shear, because 〈A′υ∂zu′〉 and 〈A′υ∂zυ′〉, of which classical tidal straining (Jay and Musiak 1994) is an important part. Other straining processes, such as the lateral advection of longitudinal shear, may also contribute to this shear–viscosity covariance. This term would be zero for an eddy viscosity constant in time, as assumed in the classical theory of Hansen and Rattray (1965).
〈u2e〉 and 〈υ2e〉 are exchange profiles due to gravitational acceleration, because these are the only terms containing the longitudinal and the lateral buoyancy gradients. Therefore, these terms can be denoted as the gravitational circulation profiles.
〈u3e〉 and 〈υ3e〉 are exchange profiles due to surface stresses and can therefore be denoted as the wind straining circulation terms.
〈u4e〉 and 〈υ4e〉 are exchange profiles due to lateral and vertical advection and will therefore be denoted as advectively driven residual circulation profiles. Contributions from horizontal and vertical advection are combined here to present advection along sloping coordinates (such as the bottom) as one process.
〈u5e〉 and 〈υ5e〉 are exchange profiles due to lateral (along coordinate) mixing, a process parameterizing eddy-related lateral transport of momentum caused by processes not resolved in a two-dimensional lateral model such as lateral shear instabilities.
〈u6e〉 and 〈υ6e〉 are exchange profiles due to the Coriolis rotation.
3. Numerical model
As a numerical modeling tool, the General Estuarine Transport Model (GETM; see online at http://www.getm.eu and see Burchard and Bolding 2002) has been used in a two-dimensional vertical mode covering the cross-sectional y–z plane. Such lateral GETM scenarios have already been applied by Umlauf et al. (2010) for studying channelized dense bottom currents and by Hofmeister et al. (2010) to quantify numerical mixing in a coastal upwelling scenario. Both horizontal velocity components are calculated such that the interaction between longitudinal and lateral components through advection and Coriolis rotation can be included. The advantage of this two-dimensional approach over a full three-dimensional model with approximately homogeneous conditions in the longitudinal direction (see, e.g., Lerczak and Geyer 2004; Chen and Sanford 2009; Cheng and Valle-Levinson 2009) is the high spatial resolution at comparably low computational effort. In the lateral model, gradients in the longitudinal direction are either neglected (as for the velocity gradients) or prescribed (as for the salinity gradient). The barotropic forcing is applied in a way that in each time step the longitudinal velocity is corrected by a constant value such that the prescribed transport according to (7) is achieved (see Burchard 1999). The rigid-lid condition is fulfilled by correcting the lateral velocity in each water column by a constant value such that the vertical mean vanishes [see (8)].
As turbulence closure model, a two-equation k–ε model with transport equations for the turbulent kinetic energy (TKE) k and the dissipation rate of the TKE ε is used. These equations are coupled to a second-moment closure suggested by Cheng et al. (2002). Details of the closure model are given by Umlauf and Burchard (2005) and Burchard and Hetland (2010). This type of closure model has proven to quantitatively reproduce marine and limnic turbulence observations (see, e.g., Simpson et al. 2002; Burchard et al. 2002; Stips et al. 2002; Arneborg et al. 2007).
To minimize numerical dissipation and mixing, the advection terms for momentum and salinity are calculated by means of nonlinear monotone [total variation diminishing (TVD)] one-dimensional methods in a directional split mode. The scheme used here is a third-order monotone Universal Limiter for Transport Interpolation Modeling of the Advective Transport Equation Quadratic Upstream Interpolation for Convective Kinematics with Estimated Upstream Terms (ULTIMATE QUICKEST) method (see Leonard 1991; Pietrzak 1998; for numerical tests, see Burchard and Bolding 2002).
As the lateral depth distribution, a symmetric parabolic function has been chosen, with a maximum depth of Hmax = 15 m in the center and a minimum depth of Hmin = 5 m at the sides (see Fig. 2) as in Lerczak and Geyer (2004). Thus, the average depth is
The residual circulation analysis is carried out inside the numerical model, averaging the relevant terms using all time steps during the last of 10 tidal periods. To calculate A′υ, u′, and υ′ during each time step of the 10th tidal period, the tidal-mean values 〈Aυ〉, 〈u〉, and 〈υ〉 are calculated already during the 9th tidal period.
The momentum advection terms are calculated exactly as they occur in the discretized momentum equations, using the TVD alternating direction schemes presented in section 3, without direct rediscretization. This procedure reduces the discretization error of the residual circulation analysis.
4. Scenario studies
A reference scenario and a number of sensitivity scenarios with variation of the Simpson number Si, the unsteadiness parameter Un, the bed roughness length z0b, and the width of the estuary W are discussed here.
Practically, the cross-sectionally averaged velocity amplitude Ut and the longitudinal salinity gradient ∂xS are prescribed. Furthermore, the bed roughness and the bathymetry [including the depth H(y) and the width W] are directly prescribed, as well as the tidal frequency ω = 2π/T with T = 44 714 s.
a. Reference scenario
For the reference scenario, a channel of width W = 1000 m with Si = 0.0911 and Un = 0.0343 was chosen. The bed roughness length was set to z0b = 10−3 m. This results in a classical strain-induced periodic stratification (SIPS; see Simpson et al. 1990) situation with full vertical mixing at the end of the flood and marginally stable stratification during ebb. Figure 3 shows results for velocity and salinity during full flood (t/T = 0.25; Figs. 3a,b), slack after flood (t/T = 0.5; Figs. 3c,d), full ebb (t/T = 0.75; Figs. 3e,f), and slack after ebb (t/T = 1.0; Figs. 3g,h). During full flood, salinity is almost vertically homogeneous because of classical flood straining, but, in the center of the channel, salinity is about 0.5 psu higher than at the sides. This is due to longitudinal differential advection of salinity: that is, because the longitudinal velocity has a distinct local maximum in the center of the estuary, where the depth is 3 times as high as at the sides. In turn, the horizontal salinity gradients lead to a weak lateral circulation, with divergent flow at the bed and convergent flow at the surface, as already observed by Nunes and Simpson (1985) (see also Fig. 4). This lateral circulation is strongly enhanced to a lateral velocity of up to υ̃ = 0.07 by the end of the flood, when vertical mixing is reduced because of decreasing bed shear such that the dense, saline water masses in the center of the flow sink down. Furthermore, the longitudinal buoyancy gradient acts in such a way that the near-surface longitudinal flow reverses toward ebb already before slack tide, whereas the near-bottom flow is still flood oriented. At full ebb, horizontal differential advection has already generated a salinity minimum near the surface in the center of the channel, and by the end of the ebb also the bottom salinity in the center of the channel has reached a lateral minimum such that high salinities have accumulated near the sides of the estuary. However, the ebb lateral circulation is very weak. By the end of the ebb, when vertical mixing is reducing, the potential energy of this lateral stratification is released in such a way that strong lateral flow convergence (negative values of ∂yυ) is generated near the bed (Fig. 4f), causing significant upwelling in the center of the flow (with a maximum vertical velocity of w̃ = 0.007) and flow divergence near the surface.
In Fig. 4, the tidal dynamics are shown in high temporal resolution for the center of the flow. The classical tidal straining as sketched by Jay and Musiak (1994) is clearly visible. During flood (t/T = 0.25), denser downstream waters are sheared over less dense upstream waters, which results into marginally unstable stratification (surface salinity exceeds bottom salinity by up to 0.03 psu, which is not visible from Fig. 4c), strong vertical mixing, and velocity profiles that are almost homogeneous in the upper half of the water column. During ebb (t/T = 0.75), vertical differential advection forces stabilization of the water column, resulting in marginally stable salinity profiles (Fig. 4c), suppressed vertical mixing (Fig. 4d), and velocity profiles that are sheared up to the surface (Fig. 4a). Because of more intense mixing during flood, the vertical momentum flux is larger during flood than during ebb (Fig. 4b). Interestingly, the vertical velocity (Fig. 4e) has maximum values that are shifted in time with respect to the instances of slack tides. The strongest negative vertical velocities after the flood occur at t/T ≈ 0.42, just when the vertical mixing ceases abruptly (Fig. 4d). The strongest upward velocities occur at t/T ≈ 0.08, which coincides with minimum vertical mixing after the ebb current. This asymmetry in transition to low vertical mixing is caused by the fact that the estuarine circulation causes a background shear in the upper part of the water column, which increases ebb shear and decreases flood shear. With this, the phase of active shear production in the upper half of the water column is significantly shorter during flood than during ebb. It is furthermore clearly seen in Fig. 4f that surface convergence directly coincides with downward vertical velocities in the center of the flow and vice versa.
The tidally resolved dynamics discussed for the reference scenario result in the expected estuarine circulation pattern shown in Fig. 5c. The longitudinal exchange flow pattern shows a remarkable similarity to the flow pattern simulated by Lerczak and Geyer (2004) for their weakly stratified case (their Fig. 8). The residual vertical momentum flux 〈Aυ∂zu〉 (Fig. 5d) shows the expected pattern, with negative values in the lower part of the center of the channel because of the observed flood shear dominance in that region.
Figure 6 shows the residual circulation decomposition into major contributions. The left column (Figs. 6a,c,e,g) presents the longitudinal total contributions 〈ũia〉 to the residual circulation according to (11), together adding up to the residual circulation 〈ũ〉. Both the gravitational and the advectively driven component provide strong up-estuary contributions in the center of the flow, compensated by down-estuary contributions near the sides. For the gravitational component, this can be explained such that the baroclinic pressure gradient (forcing up-estuary flow), which increases with depth, is balanced by a depth-independent barotropic pressure gradient (forcing down-estuary flow). Because both have to balance across the estuary, the baroclinic pressure gradient forcing dominates in deeper water and the barotropic pressure gradient dominates in shallower water.
The contribution from the advectively driven total residual circulation 〈ũ4a〉 is even stronger (see Fig. 6e). During flood, the negative vertical velocity advects positive momentum downward in the center of the channel (Fig. 3a); during ebb, the opposite happens in the center of the channel. Near the sides, momentum advection leads to decrease of longitudinal momentum, both during flood and ebb. The contribution from horizontal mixing is opposite to the advectively driven forcing: high positive momentum during flood is decreased in the center of the channel by horizontal mixing and vice versa during ebb.
The total straining contribution to the residual currents is predominantly promoting an estuarine circulation exchange flow in the vertical (Fig. 6a). Additionally, in the lateral direction, up-estuary flows dominate at the sides and down-estuary flow in the center of the channel, thus opposing the lateral contributions from gravitational and advectively driven circulation. Assuming that the total contribution from straining results from a tidal-mean area-integrated balance between vertical momentum flux near the bed (which is negative in the center of the channel and positive at the sides; see Fig. 3d) divided by the local water depth and the negative tidal-mean pressure gradient, it is clear that vertically integrated total straining circulation is down estuary in the center and up estuary at the sides. This can be verified by checking that the tidal-mean vertical momentum flux at the bed, divided by the local depth, vanishes when averaged across the width of the estuary (see Fig. 3d).
The right column of Fig. 6 shows a decomposition of the longitudinal residual velocity 〈ũ〉 into residual exchange circulation profiles (Figs. 6b,d,f) and local residual runoff (Fig. 6h) according to (10a). Furthermore, Table 2 gives quantitative measures for the intensities of the exchange flows
The up-estuary residual current in the lower half of the center of the flow is furthermore enhanced by a local residual runoff circulation (Fig. 6h), which has a pronounced maximum in the center of the channel. In real, long estuaries with variable water depth and tidal waves progressing up estuary, the Stokes drift (higher water levels during flood than during ebb) and river discharge would counteract against this longitudinal exchange flow (Li et al. 1998; Li and O’Donnell 2005).
For the central water column, Fig. 7 shows in detail all contributions to the longitudinal residual profile: from the total (exchange + local runoff) residual profiles (Fig. 7a) calculated according to (11), the residual exchange circulation profiles (Fig. 7b) calculated according to (10a), and the local runoff contributions (Fig. 7c) calculated according to the difference between (10a) and (11). The total residual profiles due to straining, gravitation, advection, and horizontal mixing are exactly the sum of the respective exchange and local runoff contributions. In all three panels, the black curve (residual in Figs. 7a,b and local residual in Fig. 7c) is the sum of all the other curves displayed. Figure 7b shows that the residual exchange circulation comes largely from the straining contribution, although its total contribution has almost no up-estuary component (Fig. 7a) because of the down-estuary straining contribution to the local runoff profile (Fig. 7c). The local runoff contributions as well as the local residual runoff profile show shapes like classical logarithmic velocity profiles. Contributions from advection and horizontal mixing residual exchange circulation are negligible in the center of the flow. The former, however, shows two pronounced contributions to estuarine circulation at y/W = 0.25 and y/W = 0.75 (see Fig. 6). This is in agreement with the study by Lerczak and Geyer (2004) (see also the discussion in section 1). The horizontal mixing residual exchange contribution to estuarine circulation is small throughout the cross section for all scenarios tested (see Table 2). In the following, we emphasize on the analysis of longitudinal and lateral residual exchange circulation, 〈ũie〉 and 〈υ̃ie〉.
The lateral residual circulation is fairly weak for the reference scenario (see Fig. 5c). Near the surface, a two-cell circulation with surface convergence is detectable (consistent with stronger lateral circulation during flood than during ebb), and, near the bed, two cells with weak bottom convergence and positive vertical velocity in the center of the flow can be seen, as characteristic for ebb flow.
Lateral gravitational circulation (Fig. 6d) is directly related to the mean salinity distribution (Fig. 5a) and shows the strongest contribution to lateral residual circulation. Near the bottom, these gravitational currents are directed upslope, driven by the pronounced lateral salinity gradients. Lateral straining circulation is largely opposite to the gravitational circulation, with significant near-bed downslope currents. However, it will be seen that this feature is not persistent throughout most of the sensitivity studies, because small changes in Si and Un may reverse the strain-induced lateral circulation. It seems that the orientation of these circulation cells depends sensitively on the timing of the lateral potential energy releases and the vertical mixing (cf. with Figs. 4e,f). Both, gravitational and strain-induced lateral circulation support upward residual vertical velocity in the center of the flow. However, this is completely compensated by a strong downward advection-induced vertical velocity and can clearly be seen in all of the following sensitivity studies. This process induces a tendency of the residual vertical velocity to be directed downward in the center of the channel.
b. Sensitivity to Si
The Simpson number Si can easily be modified by changing the longitudinal salinity gradient. Setting the Simpson number to a value of Si = 0.0456 (i.e., taking half of the value of the reference case) results in a strong weakening of the tidal asymmetry, as can be seen in a 50% reduction in residual vertical momentum flux (not shown). Residual circulation is also reduced by about 50% (see Table 2), which is consistent to the finding that the intensity of estuarine circulation scales with Si (see Geyer et al. 2000; Scully et al. 2009; Burchard and Hetland 2010). Also gravitational circulation, advectively driven circulation, and residual runoff circulation are substantially reduced for this low Si case (see Table 2).
The situation profoundly changes when the Simpson number is doubled with respect to the reference scenario: that is, to Si = 0.182 (see Figs. 8 –10). This value of Si is at the margin of periodic stratification, and a slight increase would result in runaway stratification, because a longitudinal salinity gradient constant in time and space is not consistent anymore (see the discussion in section 5). For such high values of Si, Burchard and Hetland (2010) have shown that residual circulation no longer scales proportionally to Si. As a consequence, the residual circulation is now more than doubled compared to the reference case and amounts to a maximum value of
The total contributions to residual circulation from gravitational circulation and advectively driven circulation (Figs. 9c,e) are strongly increased (and only partially compensated by an increased down-estuary contribution from tidal straining and horizontal mixing; Figs. 9a,g) such that now the local runoff circulation in the center of the flow amounts to a vertical-mean value of Ur(W/2)/Ut > 0.07 (Fig. 9h).
An explanation for the collapse of the tidal straining circulation in the center of the channel can be found in Fig. 10. In the near-surface region low eddy viscosity (dark blue color in Fig. 10d, indicating negative values of A′υ) coincides with negative shear (negative value of ∂zu′), such that 〈A′υ∂zu′〉 > 0 in that area, just as in the case of classical straining circulation in the lower half of the water column (see, e.g., Fig. 4). Therefore, the longitudinal straining circulation shows a strong up-estuary component near the surface. Because the shear–viscosity covariance is scaled by the tidal-mean viscosity 〈Aυ(z)〉 [see the term 
The dependence of the longitudinal estuarine circulation and its composition on the Simpson number is shown in Fig. 11a, where Si varies over more than one order of magnitude from Si = 0.01 to Si = 0.2. Between Si = 0.05 and Si = 0.12 estuarine circulation as well its contributors scales about linearly with Si. Gravitational circulation scales linearly with Si even for the entire range Si < 0.12, whereas estuarine circulation and its contributions from tidal straining and lateral advection increase significantly more than linearly with Si for Si < 0.05. For Si > 0.12, the contributions from gravitational circulation and lateral advection increase dramatically to compensate for the increasingly negative contribution from tidal straining. For the whole range Si < 0.12 (i.e., for tidally energetic flows), the relative contribution of the tidal straining circulation to longitudinal estuarine circulation amounts to about two-thirds, with a maximum of 73% for Si = 0.05.
The lateral circulation and its components are weak for Si < 0.05 (Fig. 11c). For even larger values of Si (Si ≈ 0.1), flood-oriented lateral gravitational circulation (up slope near the bed) and ebb-oriented lateral straining circulation (down slope near the bed) compensate each others (see also Figs. 6b,d), such that lateral residual circulation is weak (Fig. 5c). For larger values of Si, the advective contribution increases such that, for these situations with relative strong horizontal density gradients, lateral residual circulation is strongly flood oriented (see also Fig. 8c).
c. Sensitivity to Un
Although, for fixed Si, changes in Un are in reality (and in the model used here) obtained by adjusting the tidal velocity amplitude (e.g., shift between spring and neap tide) and the longitudinal salinity gradient (variations in river runoff) [see (12)], the effect of changed unsteadiness can most easily be explained by variations in tidal frequency ω. Low tidal frequencies (i.e., relatively long tidal periods) provide more time for the salinity gradient and the currents to interact. Thus, for a smaller value of Un, more pronounced ebb–flood asymmetries would be expected, resulting in stronger estuarine circulation.
By means of a one-dimensional model study, Burchard and Hetland (2010) found a dependence of estuarine circulation on the unsteadiness parameter Un (denoted as inverse Strouhal number in their study), which is proportional to Un−0.2. This dependence is reassessed here for the parabolic channel bathymetry by means of varying Un over more than one order of magnitude between Un = 0.009 and Un = 0.14. As clearly seen in Fig. 11b, estuarine circulation depends on the unsteadiness parameter Un (i.e., on the relative tidal frequency). A relatively large value of Un (i.e., a relatively high tidal frequency) implies that the time for the tidal flow to establish tidal asymmetries is relatively short, with the result that the residual circulation intensity is small. For small values of Un (Un < 0.02), the sensitivity of the estuarine circulation and its composition is weak, because they scale with about Un−0.2 … Un−0.4. This weakly negative but significant scaling is in agreement with the simplified one-dimensional situations studied by Burchard and Hetland (2010). For small values of Un, the advective contribution is comparable to the straining contribution. However, for increasing values of Un, the advection contribution collapses (see also Table 2). This is because it takes quite some time for the lateral circulation to evolve, which is the driver of the advective contribution to longitudinal circulation.
The latter can be clearly seen in Fig. 11d. For Un > 0.04, lateral residual circulation collapses, whereas, for lower values of Un, the lateral residual circulation is mainly resulting from an imbalance between gravitational and straining residual lateral circulation.
d. Variation of z0b
Changes in bottom roughness z0b primarily have two effects. First, the near-bottom vertical scaling in the water column is changed. Second, the conversion between current velocity scaling by Ut and friction velocity scaling by U* is changed. Thus, dependence of the estuarine circulation and its composition on relative bed roughness, z0b/
e. Variation of width
Table 2 shows a strong impact of estuary width on the intensity of estuarine circulation, mostly on the account of variations in tidal straining circulation. For a narrow estuary (W = 500 m), estuarine circulation [
Concerning the influence of estuary width to the other two important processes for generating estuarine circulation, the effect of gravitational circulation is slightly reduced for the narrow estuary (see Table 2) because of increased tidally averaged eddy viscosity (see Fig. 12). In contrast to that, the lateral advection contribution is reduced for the wide estuary, which is in accordance to the arguments by Lerczak and Geyer (2004), who discuss that for wider estuaries the time to significantly move fluid parcels across the estuary becomes longer than the vertical mixing time scale. It should be noted that, for the limit of an infinitely wide estuary, this two-dimensional case should collapse into the one-dimensional case investigated by Burchard and Hetland (2010), including vanishing contributions from lateral advection. However, for the present case of relatively weak stratification, both contributions are small compared to the tidal straining contribution.
5. Discussion and conclusions
The present study has shed light on a number of processes relevant for longitudinal and lateral estuarine circulation that could not be quantitatively assessed in other studies because of the use of constant viscosity or the assumption of lateral homogeneity. Specifically, the estuarine circulation analysis tool developed by Burchard and Hetland (2010) helped to understand the composition of estuarine circulation in tidally energetic estuaries.
It could be shown that in channelized estuaries tidal straining is the dominant process for the generation of the longitudinal estuarine circulation for relatively weak stratification (Si ≤ 0.15). The value of an approximate ⅔ contribution found by Burchard and Hetland (2010) in a one-dimensional water column study is even exceeded here, most likely because of the additional contribution from lateral advection of longitudinal shear.
For stronger stratification, because it typically occurs in partially mixed estuaries, the contribution from the tidal straining collapses and becomes negative for Si > 0.15. The reasons are found in the near-surface covariance between vertical shear and eddy viscosity (see section 4b). Because the dynamics near the surface strongly depend on surface wind stress, surface heat fluxes, and surface waves, processes which have been ignored here, it is not clear at this point how much stake to put in the exact value of 0.15.
The significant contribution of the lateral advection to longitudinal estuarine circulation, which has been discussed by several authors (e.g., Lerczak and Geyer 2004; Scully et al. 2009), can, for the first time, be quantified here. For a large range of parameters, this advectively driven circulation is comparable in intensity to gravitational circulation, which has been thought until recently to be the dominant process for estuarine circulation. The simulations with strong stratification carried out in the present study show that gravitational circulation may significantly dominate over the advectively driven contribution, probably because of the suppression of lateral circulation by stable stratification, as suggested by Lerczak and Geyer (2004).
Most major contributors to longitudinal estuarine circulation seem to roughly scale with the Simpson number, as shown in Fig. 11a. The estuarine circulation itself is roughly proportional with Si over the entire parameter range investigated here. For large Si, the gravitational and the advectively driven contributions grow overproportionally with Si to compensate for the collapse of the tidal straining contribution.
As discussed in section 4c, estuarine circulation depends on the unsteadiness parameter Un (i.e., on the relative tidal frequency). A relatively large value of Un (i.e., a relatively high tidal frequency) implies that the time for the tidal flow to establish tidal asymmetries is relatively short, with the result that the residual circulation intensity is small. For smaller values of Un (Un < 0.02), the sensitivity of the estuarine circulation and its composition is weak. This result is in agreement with the simplified one-dimensional model study by Burchard and Hetland (2010).
The advective contribution to longitudinal estuarine circulation is generated by the lateral residual circulation. It can be clearly seen in Fig. 11 that the collapse of lateral circulation due to small Si or large Un leads to a substantial decrease of the advectively driven longitudinal residual exchange circulation.
One further important aspect of estuarine circulation has been found in the present study that has not been discussed previously: tidal straining and thus estuarine circulation is strongly dependent on the width of the estuary. This effect is mainly due to a lateral density gradient forcing, which is increased for narrow estuaries (similar lateral density differences due to similar horizontal differential advection), thus promoting increased vertical mixing and substantial reduction of flood to ebb stratification.
These limitations discussed above can only be overcome by applying the residual circulation analysis introduced by Burchard and Hetland (2010) for fully three-dimensional estuary models. Because this method requires a high resolution in space to minimize numerical truncation errors and periodic solutions to compute tidal residuals, the investigation of complex bathymetries or strong density gradient forcing is quite demanding on present-day computers. Furthermore, the inclusion of Stokes drift as a driver for estuarine circulation (which has been switched off in the present study by means of a rigid-lid condition) requires the definition of a relevant time-averaging procedure in a water column with significant depth variations with time.
Acknowledgments
The work of Hans Burchard has been supported by the EU-funded project European Coastal-Shelf Sea Operational Monitoring and Forecasting System (ECOOP; Contract 36355). Elisabeth Schulz is a grant holder of the Interdisciplinary Faculty of the University of Rostock, Germany. We thank Malcolm Scully (Old Dominion University, Norfolk, Virginia) for pointing out that more processes than classical tidal straining contribute to the covariance between longitudinal vertical shear and eddy viscosity. We are furthermore grateful to Richard Hofmeister (Warnemünde, Germany) for supporting us with his python skills for the graphics and to Karsten Bolding (Asperup, Denmark) for keeping GETM up and running. Finally, we thank Parker MacCready (University of Washington, Seattle) and an anonymous referee for their constructive comments on earlier versions of the manuscript.
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APPENDIX
Derivation of Residual Circulation Equations
One serious accuracy issue is given by the structure of the integrands 
Conversion factor CD between friction velocity squared U*2 and vertically averaged velocity squared Ut2 as function of the nondimensional roughness length z̃0b = z0b/H.
Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4453.1
Sketch of the orientation of the coordinate system and the velocity components with respect to the cross-sectional model domain.
Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4453.1
Scenario 01 (reference) for (a),(b) full flood (t/T = 0.25); (c),(d) slack after flood (t/T = 0.5); (e),(f) full ebb (t/T = 0.75); (g),(h) and slack after ebb (t/T = 1.0). (a),(c),(e),(g) Velocity distribution with contours and color scale for longitudinal velocity and arrows for lateral and vertical velocity. The vertical velocity is scaled by the aspect ratio between vertical and horizontal scales. (b),(d),(f),(h) Salinity distribution as contours and color scale.
Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4453.1
Scenario 01 (reference): Time series of profiles of (a) longitudinal velocity, (b) vertical momentum flux, (c) salinity, (d) eddy viscosity, (e) vertical velocity, and (f) lateral convergence in the center of the channel.
Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4453.1
Scenario 01 (reference): Nondimensional tidal residual results. Velocities are nondimensionalized with the tidal velocity amplitude Ut, and vertical momentum flux and eddy viscosity are nondimensionalized with the friction velocity amplitude U* and the average depth
Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4453.1
Scenario 01 (reference): Nondimensional tidal residual velocity profiles. Velocities are nondimensionalized with the tidal velocity amplitude Ut. Vertical velocity components are scaled by the aspect ratio between vertical and horizontal scales. The profiles show (left) total and (right) exchange residual longitudinal velocity contributions: (a) total straining residual circulation 〈u1a〉; (c) total gravitational residual circulation 〈u2a〉; (e) total advectively driven residual circulation 〈u4a〉; (g) total horizontal mixing residual circulation, 〈u5a〉; (b) baroclinic straining residual circulation, (〈u1e〉, 〈υ1e〉, 〈w1e〉); (d) baroclinic gravitational residual circulation (〈u2e〉, 〈υ2e〉, 〈w2e〉); (f) baroclinic advectively driven residual circulation (〈u4e〉, 〈υ4e〉, 〈w4e〉); and (h) residual runoff circulation γUr. For simplicity, lateral circulation is shown only in the residual exchange circulation panels. The external parameters of this scenario 01 are given in Table 1.
Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4453.1
Residual profiles for the reference scenario 01 at the central location y = W/2. (a) Total (exchange + runoff) residual profiles 〈ũia〉 calculated according to the first equation of (11). The black curve is residual profile 〈ũ〉. (b) Baroclinic residual profiles 〈ũie〉 calculated according to the first equation of (10). The light blue curve is local residual profile γUr, and the black curve is residual profile 〈ũ〉. (c) Barotropic residual profiles 〈ũia〉 − 〈ũie〉 calculated according to the difference between the first equations of (11) and (10). The black curve is local residual profile γUr.
Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4453.1
Scenario 03 (large Si): Nondimensional tidal residual results; see caption in Fig. 5 for details.
Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4453.1
Scenario 03 (large Si): Nondimensional tidal residual results; see caption in Fig. 6 for details.
Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4453.1
Scenario 03 (large Si): Time series of profiles of (a) longitudinal velocity, (b) vertical momentum flux, (c) salinity, and (d) eddy viscosity in the center of the channel.
Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4453.1
Sensitivity of (a),(b) longitudinal (logarithmic scale) and (b),(d) lateral (linear scale) residual exchange profiles to (a),(c) the Simpson number Si and (b),(d) the unsteadiness number Un as variation of the reference scenario 01. For the lateral circulation, positive values denote flood-oriented circulation (upslope near the bed). Values for the reference scenario are marked with a circle. For Si > 0.17, the truncation error of the lateral circulation analysis became too large for obtaining significant results.
Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4453.1
Comparison of (a),(b) longitudinal velocity and (c),(d) salinity for full flood and (e),(f) eddy viscosity in the central location over a tidal period for scenarios (a),(c),(e) 08 (narrow channel with W = 500 m) and (b),(d),(f) 09 (wide channel with W = 2000 m).
Citation: Journal of Physical Oceanography 41, 3; 10.1175/2010JPO4453.1
Specifications of the scenarios simulated for the present study. Scenario 01 is the reference scenario in relation to which external parameters have been changed for the other scenarios (marked in italics).
Measure of the nondimensional longitudinal residual exchange flow intensity