1. Introduction
Along-channel transport in many estuaries is caused primarily by estuarine circulation: subtidal currents that bring freshwater seaward in the upper portion of the water column, with an associated near-bed landward flow bringing salty water. Estuarine circulation can interact with plankton or sediments to create a net landward or seaward transport that can be much larger than the mean transport associated with the river flow. Positively buoyant material tends to be transported seaward, whereas negatively buoyant materials, such as suspended sediments, are transported landward.
Early studies of estuarine circulation (Pritchard 1952, 1954, 1956; Hansen and Rattray 1965; Chatwin 1976) assumed that the estuarine circulation was directly driven by longitudinal buoyancy gradients. According to Hansen and Rattray (1965), such gravitational circulation scales with ∂xbH3/(48Aυ) for a flat bottom estuary, with the horizontal buoyancy gradient ∂xb, the mean water depth H, and the mean eddy diffusivity Aυ. The latter quantity is typically used as a tuning parameter to fit observed estuarine circulation to this theory.
The assumption of the dominant contribution of gravitational circulation to estuarine circulation has been challenged by a number of studies during the last two decades. Simpson et al. (1990) first described the process of strain-induced periodic stratification (SIPS), where flood tide straining of the density field destabilizes the water column and vice versa on ebb tide, resulting in substantially enhanced small-scale turbulence and vertical mixing during flood and suppressed vertical mixing during ebb. Jay and Musiak (1994) showed that this internal tidal asymmetry transports more momentum down to the near-bed region during flood than during ebb, thus generating residual transports in support of estuarine circulation and in addition to gravitational circulation. They suggested that this internal tidal asymmetry (which we refer to here as “tidal straining”) is a major mechanism to create near-bed residual flow convergence near the tip of the salt intrusion, such that generation of estuarine turbidity maxima (ETMs) is triggered. In a numerical model study, Burchard and Baumert (1998) underlined the importance of tidal straining for generation of estuarine circulation by demonstrating that switching off tidal straining did reduce estuarine circulation (and thus ETM generation) more substantially than elimination of gravitational circulation (by switching off the internal pressure gradient forcing). This is in accordance with observations by Stacey et al. (2001, 2008), who show that the creation of residual flow profiles is dominated by pulses because of interaction of shear, stratification, and mixing (as opposed to steady gravitational circulation forcing).
The strong vertical mixing asymmetry between ebb and flood resulting from tidal straining has been shown in a number of field studies including turbulence observations (Peters 1999; Peters and Bokhorst 2000; Rippeth et al. 2001; Simpson et al. 2005) and numerical model studies (Simpson et al. 2002; Baumert and Peters 2007; Burchard et al. 2008; Burchard 2009; Verspecht et al. 2009b). Despite this evidence, to better compare to classical theory or to allow for analytical solutions, many model studies of estuarine circulation are based on the assumption of constant eddy viscosity and eddy diffusivity, thus eliminating the effect of tidal straining (Lerczak and Geyer 2004; Hetland and Geyer 2004; Huijts et al. 2006; Valle-Levinson 2008). Recently, Stacey et al. (2009, manuscript submitted to J. Phys. Oceanogr.) have shown that this assumption of a tidal-mean effective eddy viscosity is an oversimplification, because the tidal-mean momentum flux and the tidal-mean shear may have the same sign in large parts of the water column, leading there to a negative effective eddy viscosity.
Various other processes have been identified to generate residual flows in estuaries, such as wind straining (Scully et al. 2005; Ralston et al. 2008; Verspecht et al. 2009b; Chen and Sanford 2009), flow curvature (Chant 2002), rotation of the earth (Lerczak and Geyer 2004; Huijts et al. 2006; Valle-Levinson 2008), and lateral advection (Lerczak and Geyer 2004; Cheng and Valle-Levinson 2009; Scully et al. 2009).
However, independently of wind forcing, bathymetry, and latitude, gravitational circulation and tidal straining are ubiquitous processes in estuaries. So far, the relative contribution of these two processes to estuarine circulation has not been quantified. This is surely due to the fact that both shape residual flow profiles in a similar manner. Direct quantification of these processes is difficult, because it would require accurate observations of time-dependent eddy viscosity profiles, which is difficult for estuarine flows (Lu and Lueck 1999; Simpson et al. 2005). A further problem in determining the relative importance of gravitational circulation and tidal straining to estuarine circulation is the large variety of estuaries and within each estuary the high variability of forcing conditions (spring–neap cycle and variations in runoff and wind forcing). Therefore, we follow here the approach of Burchard (2009), who showed that the flow characteristics of periodic estuarine and coastal circulation with constant wind forcing and horizontal buoyancy gradients depend on only a few nondimensional parameters. These parameters are mainly the horizontal Richardson number, the inverse Strouhal number, the inverse Ekman number, the relative contribution of wind to the velocity scale, and the wind direction. With this, one-dimensional water column models for estuarine circulation can be applied to explore this relevant parameter space with high resolution.
In the present study, we derive analytical expressions for all contributions to residual velocity profiles, including tidal straining, gravitational circulation, wind straining, and residual runoff. These are evaluated by means of a numerical water column model.
We have restricted our attention to periodically stratified estuaries, neglecting the earth’s rotation. This approach, then, ignores the case in which the estuarine bottom boundary layer does not penetrate through the water column, and the outflowing layer may be considered to be inviscid (Geyer et al. 2000; Stacey and Ralston 2005; Ralston et al. 2008). In this paper, we consider only flows that have reached a periodic state, which is in contradiction to the assumption of permanent stratification at constant horizontal buoyancy gradient forcing (Blaise and Deleersnijder 2008).
There are many tidally energetic estuaries and coastal areas for which periodic stratification, including full mixing during a certain phase of the tide, is typical. Several examples for such estuaries are the Conwy Estuary in North Wales (Nunes and Simpson 1985; Simpson et al. 2001), the Chesapeake Bay tributary York River during spring tides (Simpson et al. 2005), and the Skagit River in Washington State (Yang and Khangaonkar 2009). Examples for coastal areas are tidal gulleys in the Wadden Sea of the German Bight (Buijsman and Ridderinkhof 2008; Burchard et al. 2008) or in lagoon systems such as Willapa Bay in Washington State (Banas and Hickey 2005). Also in coastal embayments, periodic stratification is typical, such as in Liverpool Bay (Simpson et al. 1990; Sharples and Simpson 1995; Verspecht et al. 2009a), for which SIPS has been defined.
The paper is organized as follows: First, the theory is presented in section 2, including the presentation of the basic equations (section 2a) and their nondimensional formulation (section 2b) and the derivation of residual profiles (section 2c) and their idealized analytical solutions (section 2d). In section 3, the model simulations are described, including the model setup (section 3a) and the results for basic simulations and sensitivity studies without wind straining and residual runoff (section 3b), as well as with wind straining or residual runoff (section 3c). The results are then discussed in section 4, and major conclusions are summarized in section 5.
2. Theory
a. Dynamical equations
The term Aυ is eddy viscosity and Kυ is eddy diffusivity, both calculated by means of a two-equation turbulence closure model (for details, see the appendix). Momentum boundary conditions at the bottom are no slip with u(−H, t) = 0. At the surface a constant momentum flux τxs = −Aυ∂zu for z = 0 is prescribed. The resulting bed momentum flux is denoted as τxb = −Aυ∂zu for z = −H.
b. Nondimensional form of equations
Here, the superscript wt denotes that the velocity scale is composed of contributions from wind w and tide t and that, therefore, the inverse Strouhal number and the horizontal Richardson number are modified in the sense that they are based on the wind–tide velocity scale Uwt instead of only the tidal velocity scale Umax. It should be further noted that here the current velocity scale and not the friction velocity scale is used to allow for easier comparison of this theory to field observations (see also Burchard 2009).
c. Residual profiles
For clarity, we denote 〈ut〉 = 〈u1〉 (tidal straining), 〈ug〉 = 〈u2〉 (gravitational circulation), 〈uw〉 = 〈u3〉 (wind straining), and 〈ur〉 = γ(z)Ur (residual runoff profile). With (18), the residual longitudinal velocity profile is expressed as the sum of four tidal-mean velocity profiles with the following properties:
The first term on the right-hand side 〈ut〉 contains the correlation between eddy viscosity and vertical shear and thus represents the tidal straining part of 〈u〉. This term vanishes when the correlation between eddy viscosity and vertical shear is zero: for example, for the case of constant eddy viscosity.
The second term on the right-hand side 〈ug〉 is proportional to the horizontal buoyancy gradient and therefore represents the gravitational circulation part of 〈u〉.
The third term on the right-hand side 〈uw〉 represents wind-induced straining.
The last term on the right-hand side 〈ur〉 is proportional to the vertically averaged tidal-mean velocity and therefore represents the contribution to 〈u〉 from residual runoff.
At the bottom (z = −H), all four residual profiles individually fulfill a no-slip boundary condition with 〈ut〉 = 〈ug〉 = 〈uw〉 = 〈ur〉 = 0.
The vertical mean of the first three profiles is individually zero, whereas the vertical mean of the last profile equals the vertically averaged tidal-mean velocity.
d. Idealized analytical solutions
3. Model experiments
a. Model setup
Experiments are carried out with the numerical model described in section 2a and the turbulence closure model presented in the appendix. To cover a wide range of possible scenarios, the following parameter space is explored here: Rixwt = 0, … ,1 × 10−3 and Stiwt = 1 × 10−4, … , 1.1 × 10−3, with a resolution of 41 × 41 simulations. Variations of Rixwt and Stiwt are achieved by systematic combinations of Uwt and ∂xb (for details, see Burchard 2009). The dimensionless bed roughness length and surface roughness length are held at constant values of z̃0b = 5 × 10−5 and z̃0s = 1 × 10−5, respectively. The upper limit for Rixwt is given here by the transition to permanently stratified regimes that cannot be treated with the assumption of constant ∂xb. It should be noted that for simulations including the earth’s rotation, the limit for permanent stratification is generally much higher (for details, see Burchard 2009). For an M2 tide with T = 44 714 s, the upper limit for Stiwt is equivalent to Uwt > H/(50 s) (e.g., 0.2 m s−1 for H = 10 m), which means that we do not consider weak tidal flow here (which would most likely lead to permanent stratification). All simulations were carried out with a time step of Δt = T/1000 and 100 equidistant vertical layers, thus providing a resolution high enough to avoid significant discretization errors [see Burchard et al. (2005) and Umlauf et al. (2005) for details of the discretization schemes].
b. Simulations without wind straining and residual runoff
Figure 2 shows various parameters during a full tidal cycle for a reference case with Rixwt = 5 × 10−4 and Siwt = 5 × 10−4. Although this is a case with weak stratification, the classical SIPS dynamics can be clearly identified by stable stratification after ebb flow (t̃ = 1) and fully mixed conditions after flood (t̃ = 0.5), by stronger eddy viscosity Ãυ during flood than during ebb and largely positive buoyancy production b̃ during flood. For increasing buoyancy gradient forcing, the duration of the fully mixed phase (nonnegative b̃) will be shorter (see, e.g., Fig. 2 of Burchard 2009) until the buoyancy gradient forcing exceeds a threshold value beyond which stable stratification persist permanently, thus not allowing for periodic states within the current model framework.
For the basic situation without wind forcing and residual runoff, Fig. 3 shows nondimensional residual velocity profiles 〈ũ〉 and their contributions from tidal straining 〈ũt〉 and gravitational circulation 〈ũg〉 for nine different combinations of the inverse Strouhal number Stiwt and the horizontal Richardson number Rixwt. As expected, all profiles are directed up estuary in the lower half of the water column and down estuary in the upper half of the water column; that is, tidal straining and gravitational circulation enhance each other and add up to the total estuarine circulation. Estuarine circulation intensifies with increasing horizontal Richardson number (because of the increased buoyancy gradient forcing) and is weakly reduced with increasing inverse Strouhal number. The latter can be explained in the following way: An increased Stiwt can be obtained with an increased tidal frequency (decreased tidal period T), which in turn will not provide enough time for the tidal asymmetry to sufficiently develop, with the result that the residual circulation is weakened. All profiles in Fig. 3 clearly show that the residual circulation resulting from tidal straining is about a factor of 2 stronger than the gravitational circulation, thus contributing about two-thirds of the estuarine circulation.
This is accurately quantified in the top-right panel of Fig. 4, where the relative contribution of tidal straining to the residual profile is shown for the whole parameter space. For large parts of the space, this contribution lies between 0.6 and 0.7 (i.e., around two-thirds). Only near the transition to permanent stratification (i.e., for high Rixwt), gravitational circulation is more important. The top-left panel shows that the estuarine circulation is indeed more strongly and about linearly depending on Rixwt and weakly decreasing with Stiwt. The two bottom panels in Fig. 4 show the contributions of tidal straining and gravitational circulation separately, suggesting that the gravitational circulation is almost invariant with Stiwt. In contrast to this, the tidal straining contribution does significantly decrease with increasing Stiwt, supporting the argument that increased tidal frequency reduces tidal asymmetry.
A quantitative scaling analysis (see Fig. 5) shows that, in the limit of weak buoyancy gradient forcing, estuarine circulation and its contributions from tidal straining circulation and gravitational circulation all scale linearly with Rixwt. For larger buoyancy gradients, dependence on Rixwt becomes stronger and nonlinear, as suggested by Fig. 4. Gravitational circulation is largely independent of Stiwt, which is in good agreement with the analytical solution presented in Eq. (26) for constant-in-time parabolic eddy viscosity profiles. This analytical solution for gravitational circulation is a good approximation for weak buoyancy gradient forcing because for strong buoyancy gradient forcing nonlinearities are expected because of the dependence of the eddy viscosity on stratification. Figure 5 shows further that the dependence of estuarine circulation and tidal straining circulation on Stiwt is weakly negative, with best-fit exponents of −0.2 and −0.28, respectively.
In the following, results from a number of model experiments are discussed. These results shed more light into the generation mechanism of estuarine circulation and also give an idea about the sensitivity of the model results to the turbulence closure model.
To estimate the importance of tidal straining circulation and gravitational circulation for estuarine circulation (Burchard and Baumert 1998) using a two-dimensional along-estuary vertically resolving model for exploring generation mechanisms of estuarine turbidity maxima switched off tidal straining in one experiment by setting N 2 to zero for all turbulence closure calculations and switched off gravitational circulation in another experiment by setting the internal pressure gradient to zero in the momentum equation. The same is done here, and the resulting estuarine circulation intensity is shown in Fig. 6. When tidal straining is technically removed (left panel), estuarine circulation is substantially decreased and varies linearly with Rixwt. The remaining contribution of tidal straining falls below 0.3 but does not vanish completely because turbulence still contains some remaining tidal asymmetry resulting from asymmetric shear (and thus shear production) caused by gravitational circulation. Compared to the full model (bottom-right panel in Fig. 4), gravitational circulation is weaker here, especially for high Rixwt, because tidally averaged eddy viscosity (which occurs in the denominator of 〈ũg〉) is not damped by stable stratification generated by tidal straining. This is also the reason why permanent stratification does not occur in the parameter space shown. These results compare well with the analytical gravitational circulation solution (26). Looking up the relative intensity of residual circulation for the same roughness length, z̃0b = 5 × 10−5, results in a higher circulation intensity for the analytical solution (see Fig. 1). This can be explained by the constant-in-time eddy viscosity used in the analytical problem, which is consistent with fully developed currents but overestimates the low mixing during slack tides resulting for the eddy viscosity calculated by the turbulence closure model. Stacey et al. (2008) show that during these slack tide periods of low vertical mixing, pulses of residual circulation are triggered.
The dependence of estuarine circulation and the contribution from tidal straining on specifications in the turbulence closure model is shown in Fig. 7. A decreased steady-state Richardson number, resulting in decreased mixing efficiency (see appendix for details), slightly increases the relative importance of tidal straining and vice versa. This means that stronger turbulence damping by stable stratification (smaller steady-state Richardson number and smaller mixing efficiency) increases tidal asymmetry and thus tidal straining.
The role of convective mixing during parts of the flood (for details, see Burchard 2009) is highlighted by two experiments with modified parameterizations for convective mixing. In the first experiment, the generic choice of c3+ = 1 (see appendix for details) is replaced by c3+ = c3−, reducing the dissipation rate for convective mixing by converting the buoyancy forcing in the dissipation rate Eq. (A2) to a sink term and thus enhancing convective mixing. This has the effect that the tidal straining is more pronounced (see Fig. 8, top left). In the second experiment, convective mixing is reduced by setting all instable values of N 2 to zero for the turbulence closure model, thus assuming that convection does not enhance vertical mixing. With this, the tidal straining contribution to estuarine circulation is significantly reduced (see Fig. 8, top right), but it is still larger than 0.5 in most parts of the considered parameter space. This means that the intensity of convective mixing strongly influences the tidal straining component of the residual estuarine circulation. However, the strength of the estuarine circulation itself is not strongly affected by the different compositions of estuarine circulation. This can be explained by a negative feedback process: stronger convective mixing during flood increases the correlation between vertical shear and eddy viscosity and thus tidal straining, but it also decreases gravitational circulation by increasing tidal-mean viscosity [cf.
It can be seen from the two bottom panels in Figs. 7 and 8 that the changes in tidal straining circulation are very close to the changes in residual circulation. Thus, the gravitational circulation is almost unchanged by changes in the turbulence closure model (not shown).
c. Simulations with wind straining or residual runoff
Effects of wind straining and residual runoff on estuarine circulation can easily be assessed with the analytical method developed here. The effect of residual runoff on shaping residual velocity profiles is shown in Fig. 9 for a horizontal Richardson number of Rixwt = 5.0 × 10−4 and an inverse Strouhal number of Stiwt = 5.0 × 10−4. With the depth-mean residual velocity Ur ranging between 1% (ũr = Ur/Uwt = −0.01) and 10% (ũr = −0.1) of the velocity scale, significant variations of the contributions from tidal straining ũt and gravitational circulation ũg are seen. For a small residual runoff of 1% and 2%, near-bed residual flow is directed up estuary and 〈ũt〉 is still typically a factor of 2 larger than 〈ũg〉. However, with increasing runoff, the contribution from tidal straining decreases. This can be explained by the fact that the runoff itself adds extra turbulence production because of the increased bottom stress, which in turn decreases the tidal asymmetry. This increasing importance of gravitational circulation is quantified in Fig. 10, which shows that the relative contribution of tidal straining decreases with increasing runoff for large part of the considered parameter space. A near-linear residual velocity profiles is given for example in the bottom-left panel of Fig. 9 for ũr = −0.05, in a similar way as predicted by the analytical solution for gravitational circulation with a balance between runoff and horizontal Richardson number [see Eq. (28)].
An interesting effect can be studied when wind straining is applied instead of residual runoff (see Fig. 11), where the impact of weak (ru = 0.1) and moderate (ru = 0.2) up-estuary and down-estuary winds on residual profiles is shown for a horizontal Richardson number of Rixwt = 2.0 × 10−4 and an inverse Strouhal number of Stiwt = 5.0 × 10−4. As expected, estuarine circulation is enhanced by down-estuary winds and reduced by up-estuary winds. Down-estuary wind stabilizes the water column such that, also during flood, convective overturning is suppressed, with the consequence that tidal straining is significantly reduced for scenarios with moderate down-estuary wind; thus, gravitational circulation becomes generally more important than tidal straining. Near-bed tidal straining may even be reversed by down-estuary wind, as clearly seen in the top-right panel of Fig. 11. This effect will be strongest for small Rixwt and can be explained as follows: During ebb, down-estuary wind straining increases vertical shear and thus increases turbulence production, whereas during flood vertical shear and thus turbulence are reduced. This generates an inverse tidal asymmetry also with zero or small horizontal buoyancy gradients, slightly weakening the joint effect of wind straining and gravitational circulation.
The extent of inverse tidal straining is highlighted in Fig. 12. For relatively strong down-estuary wind straining (ru = 0.2), for almost the entire parameter space inverse straining acts such that the near-bed residual velocity resulting from tidal straining ũt is negative and thus reversed. In this situation, for most of the parameter space, gravitational circulation is stronger than tidal straining circulation. The opposite happens for up-estuary wind: for the entire parameter space under investigation, tidal straining is significantly stronger than gravitational circulation, because tidal mixing asymmetry resulting from wind straining is supporting tidal mixing asymmetry resulting from tidal straining.
4. Discussion
For a periodically stratified situation with negligible residual runoff and no wind forcing, the contribution of tidal straining to estuarine circulation is significantly stronger than gravitational circulation. This has been shown for a variety of parameters in the turbulence closure model and is therefore a robust result.
Also, for various wind forcing and runoff situations, tidal straining is generally more important than gravitational circulation, with a few exceptions for down-estuary winds, when convective overturning during flood tide and thus tidal asymmetry is decreased (section 3c). The remaining question is, for which suite of realistic estuarine and coastal situations is this result representative?
In some estuaries, the water column is capped by a permanently stratified layer such that the present result is not directly transferable. However, it may be hypothesized that, in the periodically stratified lower part of the water column in such estuaries, the situation is similar because also there the stable stratification is eroded during flood and restratified during ebb. This would be in accordance with the simple linear stress model developed by Stacey and Ralston (2005) and needs to be investigated in a further study in which the assumption of a constant horizontal buoyancy gradient is relaxed. Such a study could be done in a similar way as suggested by Blaise and Deleersnijder (2008); that is, expressing ∂xb as a function of b, with the consequence that the constant ∂xb would need to be replaced by the tidally averaged vertical profile of the horizontal buoyancy gradient in Eq. (15).
Lateral advection is another process that had to be neglected in this one-dimensional parameter space study. It has already been shown by Nunes and Simpson (1985) that tidally energetic estuaries may develop a characteristic transverse circulation with surface convergence during flood and bottom convergence during ebb. Lerczak and Geyer (2004) were the first to show that this process leads to enhancement of classical estuarine circulation. Scully et al. (2009) could show that also this process scales with the horizontal Richardson number and may be of the same order as gravitational circulation. We hypothesize here that tidal straining, which has been neglected in the studies by Lerczak and Geyer (2004) and Scully et al. (2009), is important also in estuaries of limited width and must not be neglected as a process contributing to estuarine circulation.
The effect of the earth’s rotation may be relatively small for narrow estuaries. In wide estuaries and coastal regions, however, rotation has the effect of destabilizing the water column, such that periodic stratification may be seen for much larger horizontal Richardson numbers as compared to the case without rotation. Some of these dynamics are also reproduced by one-dimensional models, as discussed in detail by Burchard (2009). The present theory can easily be extended for rotational one-dimensional flows. In principle, extensions to two-dimensional (laterally homogeneous or longitudinally homogeneous) estuaries should also be possible, as well as extensions to fully three-dimensional estuaries, as long as strictly periodic solutions are analyzed. For this, the analysis would have to be extended to all terms in the longitudinal momentum equation, including the advective terms, which may play an important role (Li and O’Donnell 2005; Scully et al. 2009). Such an analysis would distinguish between the contributions of the relevant processes to estuarine circulation much more accurately than the method applied by Burchard and Baumert (1998), who switched off single processes in different numerical experiments, thus providing an invasive numerical analysis.
The question for the relative importance of tidal straining and gravitational circulation to estuarine circulation is of more interest than theoretical alone. Specifically for coastal flows with small horizontal buoyancy gradients, estimates only based on scalings for gravitational circulation may lead to a strong underestimation of estuarine circulation. Postma (1954) and Zimmerman (1976), for example, conclude that buoyancy gradients in the Marsdiep in the Dutch Wadden Sea are not sufficiently strong to drive net transport of suspended particulate matter. This is in contrast to recent model results by Burchard et al. (2008) showing that even weak buoyancy gradients can drive net SPM transport into the Wadden Sea.
The reason why the role of tidal straining for estuarine circulation is generally underestimated is that its contribution to residual profiles has a shape similar to that of gravitational circulation and that it even obeys a similar scaling. As shown in Fig. 5, estuarine circulation, tidal straining, and gravitational circulation scale linearly with the horizontal Richardson number Rix for the limit of weak buoyancy gradient forcing, a result that has been suggested for gravitational circulation already by Scully et al. (2009); see also Geyer et al. (2000). For estuarine circulation and tidal straining, additional weak negative dependence on the inverse Strouhal number is given. It would be a great (maybe impossible) challenge for future studies to directly observe these subtle differences in the field. A more promising strategy for supporting the present model results by field observations would be to obtain a close agreement between model results and field observations for a variety of parameters including turbulence and stress measurements and then to analyze the individual contributions to residual circulation from the model results.
Although in the past turbulence closure models have been assessed extensively for their performance in stably stratified shear flow, the level of predictability of these models for sheared convective flow is low. This may be due to the fact that strong convective mixing would approximately homogenize all tracers in the convective boundary layer, such that even crude methods such as convective adjustment lead to acceptable results. However, shown here, the rate of convective mixing of momentum is decisive for shaping residual circulation (see Fig. 8). Specifically, the relative importance of tidal straining increases with increased convective mixing during flood tide. This is because the no-slip bottom boundary condition for velocity prescribes zero-bed velocity and increased convective mixing increases near-bed shear and thus residual circulation resulting from tidal straining. It is a future challenge to turbulence closure modeling to refine parameterizations of sheared convective flow. Such an improved parameterization should fit into the general framework of two-equation turbulence closure models, and a reliable fit for the buoyancy parameter c3+ in the dissipation rate Eq. (A2) or any other length scale equation would already be a great progress.
5. Conclusions
This study provides mathematical definitions of tidal straining, gravitational circulation, wind straining, and residual runoff contributions to estuarine circulation for periodic tidal flows. These are derived analytically from the tidally averaged one-dimensional momentum equations.
The major result of this study is that tidal straining is the dominant process in generating estuarine circulation for periodically stratified estuaries. As a rule of thumb, for situations without wind straining and residual runoff, tidal straining contributes two-thirds and gravitational circulation contributes one-third of the estuarine circulation.
It is shown that tidal straining may not only be driven by tidal asymmetries resulting from horizontal buoyancy gradients but also from other asymmetric processes such as wind straining. For weak horizontal buoyancy gradients and down-estuary wind straining, tidal straining may be reversed; that is, it opposes estuarine circulation.
The intensity of estuarine circulation scales linearly with the horizontal Richardson number and weakly with the Strouhal number, where the latter contribution results from tidal straining. Extensions of the present theory to partially mixed and strongly stratified estuaries as well as to two and three dimensions, including additional terms such as advection and the earth’s rotation, are possible.
Acknowledgments
The work of Hans Burchard has been supported by the EU-funded project ECOOP (European COastal-shelf sea OPerational monitoring and forecasting system Contract 36355). Robert Hetland thanks the Fulbright Commission for funding a sabbatical at the Leibniz Institute for Baltic Sea Research, which made this collaborative work possible. The authors are indebted to Elisabeth Schulz (Warnemünde) for carefully correcting the manuscript.
REFERENCES
Banas, N. S., and B. M. Hickey, 2005: Mapping exchange and residence time in a model of Willapa Bay, Washington, a branching, microtidal estuary. J. Geophys. Res., 110 , C11011. doi:10.1029/2005JC002950.
Baumert, H., and H. Peters, 2000: Second-moment closures and length scales for weakly stratified turbulent shear flows. J. Geophys. Res., 105 , 6453–6468.
Baumert, H., and H. Peters, 2007: Validating a turbulence closure against estuarine microstructure measurements. Ocean Modell., 19 , 183–203.
Blaise, S., and E. Deleersnijder, 2008: A new parameterisation of salinity advection to prevent stratification from running away in a simple estuarine model. Ocean Sci., 4 , 239–246.
Buijsman, M. C., and H. Ridderinkhof, 2008: Variability of secondary currents in a weakly stratified tidal inlet with low curvature. Cont. Shelf Res., 28 , 1711–1723.
Burchard, H., 1999: Recalculation of surface slopes as forcing for numerical water column models of tidal flow. Appl. Math. Modell., 23 , 737–755.
Burchard, H., 2009: Combined effects of wind, tide, and horizontal density gradients on stratification in estuaries and coastal seas. J. Phys. Oceanogr., 39 , 2117–2136.
Burchard, H., and H. Baumert, 1995: On the performance of a mixed-layer model based on the k-ε turbulence closure. J. Geophys. Res., 100 , 8523–8540.
Burchard, H., and H. Baumert, 1998: The formation of estuarine turbidity maxima due to density effects in the salt wedge. A hydrodynamic process study. J. Phys. Oceanogr., 28 , 309–321.
Burchard, H., and K. Bolding, 2001: Comparative analysis of four second-moment turbulence closure models for the oceanic mixed layer. J. Phys. Oceanogr., 31 , 1943–1968.
Burchard, H., E. Deleersnijder, and G. Stoyan, 2005: Some numerical aspects of turbulence-closure models. Marine Turbulence: Theories, Observations and Models, H. Z. Baumert, J. H. Simpson, and J. Sündermann, Eds., Cambridge University Press, 197–206.
Burchard, H., G. Flöser, J. V. Staneva, R. Riethmüller, and T. Badewien, 2008: Impact of density gradients on net sediment transport into the Wadden Sea. J. Phys. Oceanogr., 38 , 566–587.
Chant, R. J., 2002: Secondary circulation in a region of flow curvature: Relationship with tidal forcing and river discharge. J. Geophys. Res., 107 , 3131. doi:10.1029/2001JC001082.
Chatwin, P. C., 1976: Some remarks on the maintenance of the salinity distribution in estuaries. Estuarine Coastal Shelf Sci., 4 , 555–566.
Chen, S-N., and L. P. Sanford, 2009: Axial wind effects on stratification and longitudinal salt transport in an idealized, partially mixed estuary. J. Phys. Oceanogr., 39 , 1905–1920.
Cheng, P., and A. Valle-Levinson, 2009: Influence of lateral advection on residual currents in microtidal estuaries. J. Phys. Oceanogr., 39 , 3177–3190.
Cheng, Y., V. M. Canuto, and A. M. Howard, 2002: An improved model for the turbulent PBL. J. Atmos. Sci., 59 , 1550–1565.
Geyer, W. R., J. H. Trowbridge, and M. M. Bowen, 2000: The dynamics of a partially mixed estuary. J. Phys. Oceanogr., 30 , 2035–2048.
Hansen, D. V., and M. Rattray, 1965: Gravitational circulation in straits and estuaries. J. Mar. Res., 23 , 104–122.
Hetland, R. D., and W. R. Geyer, 2004: An idealized study of the structure of long, partially mixed estuaries. J. Phys. Oceanogr., 34 , 2677–2691.
Huijts, K. M. H., H. M. Schuttelaars, H. E. de Swart, and A. Valle-Levinson, 2006: Lateral entrapment of sediment in tidal estuaries: An idealized model study. J. Geophys. Res., 111 , C12016. doi:10.1029/2006JC003615.
Jay, D. A., and J. D. Musiak, 1994: Particle trapping in estuarine tidal flows. J. Geophys. Res., 99 , 445–461.
Lerczak, J. A., and W. R. Geyer, 2004: Modeling the lateral circulation in straight, stratified estuaries. J. Phys. Oceanogr., 34 , 1410–1428.
Li, C., and J. O’Donnell, 2005: The effect of channel length on the residual circulation in tidally dominated channels. J. Phys. Oceanogr., 35 , 1826–1840.
Lu, Y., and R. G. Lueck, 1999: Using a broadband ADCP in a tidal channel. Part II: Turbulence. J. Atmos. Oceanic Technol., 16 , 1568–1579.
MacCready, P., 2004: Toward a unified theory of tidally-averaged estuarine salinity structure. Estuaries, 27 , 561–570.
Nunes, R. A., and J. H. Simpson, 1985: Axial convergence in a well-mixed estuary. Estuarine Coastal Shelf Sci., 20 , 637–649.
Osborn, T. R., 1980: Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr., 10 , 83–89.
Peters, H., 1999: Spatial and temporal variability of turbulent mixing in an estuary. J. Mar. Res., 57 , 805–845.
Peters, H., and R. Bokhorst, 2000: Microstructure observations of turbulent mixing in a partially mixed estuary. Part I: Dissipation rate. J. Phys. Oceanogr., 30 , 1232–1244.
Postma, H., 1954: Hydrography of the Dutch Wadden Sea. Arch. Neerl. Zool., 10 , 405–511.
Pritchard, D. W., 1952: Salinity distribution and circulation in the Chesapeake Bay estuarine system. J. Mar. Res., 11 , 106–123.
Pritchard, D. W., 1954: A study of the salt balance in a coastal plain estuary. J. Mar. Res., 13 , 133–144.
Pritchard, D. W., 1956: The dynamic structure of a coastal plane estuary. J. Mar. Res., 15 , 33–42.
Ralston, D. A., W. R. Geyer, and J. A. Lerczak, 2008: Subtidal salinity and velocity in the Hudson River estuary: Observations and modeling. J. Phys. Oceanogr., 38 , 753–770.
Rippeth, T. P., N. Fisher, and J. H. Simpson, 2001: The cycle of turbulent dissipation in the presence of tidal straining. J. Phys. Oceanogr., 31 , 2458–2471.
Scully, M. E., C. T. Friedrichs, and J. M. Brubaker, 2005: Control of estuarine stratification and mixing by wind-induced straining of the estuarine density field. Estuaries, 28 , 321–326.
Scully, M. E., W. R. Geyer, and J. A. Lerczak, 2009: The influence of lateral advection on the residual estuarine circulation: A numerical modeling study of the Hudson River estuary. J. Phys. Oceanogr., 39 , 107–124.
Sharples, J., and J. H. Simpson, 1995: Semi-diurnal and longer period stability cycles in the Liverpool Bay region of freshwater influence. Cont. Shelf Res., 15 , 295–313.
Shih, L. H., J. R. Koseff, J. H. Ferziger, and C. R. Rehmann, 2000: Scaling and parameterization of stratified homogeneous turbulent shear flow. J. Fluid Mech., 412 , 1–20.
Simpson, J. H., J. Brown, J. Matthews, and G. Allen, 1990: Tidal straining, density currents, and stirring in the control of estuarine stratification. Estuaries, 26 , 1579–1590.
Simpson, J. H., R. Vennell, and A. Souza, 2001: The salt fluxes in a tidally energetic estuary. Estuarine Coastal Shelf Sci., 52 , 131–142.
Simpson, J. H., H. Burchard, N. R. Fisher, and T. P. Rippeth, 2002: The semi-diurnal cycle of dissipation in a ROFI: Model-measurement comparisons. Cont. Shelf Res., 22 , 1615–1628.
Simpson, J. H., E. Williams, L. H. Brasseur, and J. M. Brubaker, 2005: The impact of tidal straining on the cycle of turbulence in a partially stratified estuary. Cont. Shelf Res., 25 , 51–64.
Stacey, M. T., and D. K. Ralston, 2005: The scaling and structure of the estuarine bottom boundary layer. J. Phys. Oceanogr., 35 , 55–71.
Stacey, M. T., J. R. Burau, and S. G. Monismith, 2001: Creation of residual flows in a partially stratified estuary. J. Geophys. Res., 106 , 17013–17037.
Stacey, M. T., J. P. Fram, and F. K. Chow, 2008: Role of tidally periodic density stratification in the creation of estuarine subtidal circulation. J. Geophys. Res., 113 , C08016. doi:10.1029/2007JC004581.
Umlauf, L., 2009: The description of mixing in stratified layers without shear in large-scale ocean models. J. Phys. Oceanogr., 39 , 3032–3039.
Umlauf, L., and H. Burchard, 2005: Second-order turbulence models for geophysical boundary layers. A review of recent work. Cont. Shelf Res., 25 , 795–827.
Umlauf, L., H. Burchard, and K. Bolding, 2005: General Ocean Turbulence Model: Source code documentation. Baltic Sea Research Institute Warnemünde Tech. Rep. 63, 346 pp.
Valle-Levinson, A., 2008: Density-driven exchange flow in terms of the Kelvin and Ekman numbers. J. Geophys. Res., 113 , C04001. doi:10.1029/2007JC004144.
Verspecht, F. I., H. Burchard, T. P. Rippeth, M. J. Howarth, and J. H. Simpson, 2009a: Processes impacting on stratification in a region of freshwater influence: Application to Liverpool Bay. J. Geophys. Res., 114 , C11022. doi:10.1029/2009JC005475.
Verspecht, F. I., T. P. Rippeth, A. J. Souza, H. Burchard, and M. J. Howarth, 2009b: Residual circulation and stratification in the Liverpool Bay region of freshwater influence. Ocean Dyn., 59 , 765–779.
Wong, K-C., 1994: On the nature of transverse variability in a coastal plain estuary. J. Geophys. Res., 99 , 14209–14222.
Yang, Z., and T. Khangaonkar, 2009: Modeling tidal circulation and stratification in Skagit River estuary using an unstructured grid ocean model. Ocean Modell., 28 , 34–49.
Zimmerman, J. T. F., 1976: Mixing and flushing of tidal embayments in the western Dutch Wadden Sea. Part I: Distribution of salinity and calculation of mixing times scales. Neth. J. Sea Res., 10 , 149–191.
APPENDIX
Turbulence Closure Model
For stable stratification, the parameter c3 = c3− regulates the balance between mixing and stratification and defines the steady-state Richardson number Rist at which turbulence in homogeneous shear layers may come to a steady state. According to Shih et al. (2000), the steady-state Richardson number should be Rist = 0.25 for fully developed turbulence. For the second-moment closure by Cheng et al. (2002), this is obtained by setting c3− = −0.74 where Rist is an increasing function of c3−. A steady-state Richardson number Rist > 0.25 would lead to weaker suppression of turbulence by stable stratification and vice versa (for details, see Umlauf and Burchard 2005).
With this, the mixing efficiency in steady state is independent of the second-moment closure used (see also Umlauf 2009). Therefore, Rist = 0.25 [equivalent to c3− = −0.74 when using the Cheng et al. (2002) second-moment closure] results in Γ = 0.22, a value close to the estimate by Osborn (1980). A higher value of Rist = 0.28 (equivalent to c3− = −0.61) would result in Γ = 0.234, and a lower value of Rist = 0.22 (equivalent to c3− = −0.92) would result in Γ = 0.2. Thus, higher steady-state Richardson numbers are consistent with higher mixing efficiencies and thus with stronger mixing and vice versa. This actually is a new result, which is relevant here and supports the theories on two-equation turbulence closure modeling developed by Burchard and Baumert (1995), Baumert and Peters (2000), Burchard and Bolding (2001), and Umlauf and Burchard (2005).
For instable stratification, c3 = c3+ = 1 is generally used (see, e.g., Burchard and Bolding 2001), a choice that is made to provide a source term for the dissipation rate equation in the case of free convection with P = 0 and B > 0. However, Umlauf and Burchard (2005) pointed out that c3+ > 1 is not necessary and that, for a free convection scenario, the resulting buoyancy fluxes are more realistic for c3+ = c3−. Therefore, this alternative option is also tested in one of the simulations discussed in section 3b.
Analytical solution for gravitational circulation according to Eq. (26): (left) nondimensional residual velocity profiles for four different values of the bottom roughness length, calculated with Rixt = 5 × 10−4, and (right) variation of the intensity of gravitational circulation with bottom roughness length and horizontal Richardson number, calculated by using the measure introduced in (29).
Citation: Journal of Physical Oceanography 40, 6; 10.1175/2010JPO4270.1
Nondimensional dynamics of strain-induced periodic stratification for Rxwt = 5 × 10−4 and Stwt = 5 × 10−4. Shown are (top) (left) the current velocity ũ and (right) the buoyancy b̃; (middle) (left) the eddy viscosity Ãυ and (right) the vertical momentum flux τ̃ = −Ãυ∂z̃ũ; and (bottom) (left) the shear production P̃ = Ãυ(∂z̃ũ)2 and (right) the buoyancy production b̃ = −k̃υ∂z̃b̃. All variables are shown as functions of nondimensional height z̃ and nondimensional time t̃.
Citation: Journal of Physical Oceanography 40, 6; 10.1175/2010JPO4270.1
Nondimensional residual velocity profiles (red lines), their contributions from tidal straining (green lines), and gravitational circulation (blue lines) for nine different combinations of the inverse Strouhal number and the horizontal Richardson number.
Citation: Journal of Physical Oceanography 40, 6; 10.1175/2010JPO4270.1
(top left) Nondimensional intensity of residual circulation, (top right) relative contribution of tidal straining, (bottom left) nondimensional intensity of the tidal straining contribution to residual circulation, and (bottom right) nondimensional intensity of the gravitational circulation contribution to residual circulation. Intensities of residual circulation are quantified with the measure introduced in (29). Regions marked in gray did not reach a periodic state after 20 tidal periods.
Citation: Journal of Physical Oceanography 40, 6; 10.1175/2010JPO4270.1
Scalings for (left) the intensity of estuarine circulation, (middle) the contribution from tidal straining, and (right) the contribution from gravitational circulation.
Citation: Journal of Physical Oceanography 40, 6; 10.1175/2010JPO4270.1
Nondimensional intensity of residual circulation (left) for switching off of tidal straining by setting N 2 to 0 for the turbulence closure scheme and (right) for switching off of gravitational circulation by setting ∂xb to 0 in the momentum equation.
Citation: Journal of Physical Oceanography 40, 6; 10.1175/2010JPO4270.1
(left) Nondimensional intensity of residual circulation and (right) relative contribution of tidal straining to estuarine circulation (top) for a reduced steady-state Richardson number and (middle) an increased steady-state Richardson number. (bottom) Residual circulation and tidal straining velocity profiles for Rixwt = Stiwt = 5 × 10−4 as function of the steady-state Richardson number.
Citation: Journal of Physical Oceanography 40, 6; 10.1175/2010JPO4270.1
As in Fig. 7, but for (top) the alternative choice of c3+ = c3− and (middle) for switching off of convective mixing by setting N 2 to 0 for unstable stratification for the turbulence closure scheme; and (bottom) for the generic case with c3+ = 1, the case with c3+ = c3−, and the case where convective mixing was switched off.
Citation: Journal of Physical Oceanography 40, 6; 10.1175/2010JPO4270.1
Nondimensional residual velocity profiles (red lines) and their contributions from tidal straining (green lines), gravitational circulation (blue lines), and residual velocity profiles (purple lines) for four different runoff velocities. The calculations are carried out for Rixt = Stiwt = 5 × 10−4. Note the different scalings of the velocity axis.
Citation: Journal of Physical Oceanography 40, 6; 10.1175/2010JPO4270.1
Relative contribution of tidal straining to the joint effect of tidal straining and gravitational circulation for four different residual runoff velocities: (top left) −0.01, (top right) −0.02, (bottom left) −0.05, and (bottom right) −0.1 m s−1.
Citation: Journal of Physical Oceanography 40, 6; 10.1175/2010JPO4270.1
As in Fig. 9, but in addition with wind straining (light blue lines) for down estuary (α = 90°) and up estuary (α = 270°) and for weak wind (ru = 0.1) and strong wind (ru = 0.2); and with Rixt = 2.0 × 10−4. Residual runoff velocity was set to zero.
Citation: Journal of Physical Oceanography 40, 6; 10.1175/2010JPO4270.1
As in Fig. 10, but for (left) moderate and (right) strong wind forcing for (top) down-estuary and (bottom) up-estuary winds. For these scenarios, the residual runoff velocity was set to zero. White dots indicate reversed tidal straining (negative bed value of tidal straining term).
Citation: Journal of Physical Oceanography 40, 6; 10.1175/2010JPO4270.1
