RESEARCH LETTER
10.1002/2013GL058494
Key Points:
• It is a priori not clear if estu-arine convergence enhances exchange flows
• Tidal straining is reversed by strong convergence
• Estuarine convergence tends to increase eddy viscosity during ebb
Correspondence to: H. M. Schuttelaars, H.M.Schuttelaars@tudelft.nl
Citation:
Burchard, H., E. Schulz, and H. M. Schuttelaars (2014), Impact of estuarine convergence on residual circula-tion in tidally energetic estuaries and inlets, Geophys. Res. Lett., 41, doi:10.1002/2013GL058494.
Received 12 NOV 2013 Accepted 27 DEC 2013
Accepted article online 3 JAN 2014
Impact of estuarine convergence on residual circulation
in tidally energetic estuaries and inlets
H. Burchard1, E. Schulz1, and H. M. Schuttelaars2
1Leibniz Institute for Baltic Sea Research Warnemünde , Rostock, Germany,2Delft Institute of Applied Mathematics, Delft University of Technology, Delft, Netherlands
Abstract
Estuarine convergence (landward reduction of width and/or depth) is known to have the potential to significantly enhance estuarine circulation, a result theoretically derived under the assumption of constant eddy viscosity. Recent studies of longitudinally uniform energetic tidal channels indicate that tidal straining, a process driven by tidally varying eddy viscosity, is a major driver of estuarine circulation. The combined effect of estuarine convergence and tidal straining is investigated, for the first time, in this paper. The present idealized numerical study shows that estuarine convergence is reducing or even reversing tidal straining circulation in such a way that estuarine circulation can be weakened. This is a counterintuitive hydrodynamic effect of estuarine convergence, which may reduce (rather than increase) up-estuary particulate matter transport in estuaries and tidal inlets.1. Introduction
Initial studies which assessed estuarine circulation patterns identified the importance of longitudinal buoy-ancy gradients, setting up the so-called gravitational circulation: an up-estuary directed residual flow near the bottom and a down-estuary flow near the surface [see Pritchard, 1956; Hansen and Rattray, 1965;
Chatwin, 1976]. Simpson et al. [1990] recognized that the effects of ebb-flood asymmetries in vertically
dif-ferential advection cause the water column to be periodically stratified: The vertical mixing is enhanced during flood and suppressed during ebb. Jay and Musiak [1994] realized that this vertical mixing asymme-try generates a residual circulation with a vertical structure similar to that of the gravitational circulation. The importance of this tidal straining mechanism was systematically investigated by Burchard and Hetland [2010]: for periodically mixed estuaries residual flows due to tidal straining amount to typically two thirds of the estuarine circulation (with a one-third contribution from gravitational circulation). During the last decade, it has been recognized that lateral variation adds significant complexity to the dynamics of estuar-ies [Lacy et al., 2003]. Later, Lerczak and Geyer [2004] explained how lateral circulation enhances estuarine circulation, while Scully et al. [2009] could demonstrate the significance of this process for the Hudson River estuary. Recently, the interaction of tidal straining with lateral circulation and its impact on estuarine circu-lation has been investigated using cross-sectionally resolving models [Burchard et al., 2011], with the result that tidal straining circulation is strongly enhanced.
In contrast to the extensive discussion of tidal straining as driver of estuarine circulation in the presence of longitudinal buoyancy gradients, its sensitivity to other drivers has never been investigated in detail. In this contribution we will illustrate the importance of tidal straining, even in the absence of longitudinal buoyancy
gradients, by focusing on estuarine circulation patterns generated by cross-sectional convergence.
Follow-ing Burchard and Hetland [2010] and Burchard et al. [2011], to clearly demonstrate its effect on estuarine circulation for different convergence rates and longitudinal density gradients, we concentrate our study on simplified one-dimensional and two-dimensional (cross-sectional) geometries.
For negligible density gradients and eddy viscosity constant in time, Ianniello [1979] showed that strong enough cross-sectional convergence of an estuary results in residual inflow near the bed and outflow near the surface. To explain this, consider a tidal flow in a converging estuary. During flood, the tidal flow is accelerated due to the convergence of the channel, while during ebb it is decelerated. Therefore, −u𝜕xu
is negative during both flood and ebb and thus causes a seaward acceleration of the flow at any time over the entire water column. This advection term has largest values near the surface where the current speed reaches its maximum. Combined with an acceleration of the water column by a vertically invariable barotropic pressure gradient, the surface currents are accelerated seaward and the near bottom currents
Geophysical Research Letters
10.1002/2013GL058494
are accelerated landward, a process which leads to estuarine circulation without any baroclinic pressure gradient forcing.
We show that time variation in vertical mixing strongly reduces the magnitude of this contribution. If both convergence and longitudinal buoyancy gradients are considered, the tidal straining contribution results in a large part of the parameter space in a reduction of the magnitude of the estuarine circulation, compared to the circulation observed in a nonconverging estuary. This is counterintuitive, as the estuarine convergence itself enhances the estuarine circulation.
2. Dynamic Equations
Estuarine circulation caused by topographic convergence is investigated here in a laterally aver-aged framework. Under that assumption, the underlying longitudinal momentum equation is of the following form: 𝜕t(Du) +𝜕x(Du 2 ) +𝜕𝜎(̄wDu) − 𝜕𝜎 (A v D𝜕𝜎u ) = D2 ∫ 0 𝜎 𝜕 ∗ xb d𝜎 ′− [DP(t)], (1)
where𝜎 = (z−𝜂)∕(𝜂+H) is the nondimensional vertical coordinate, with the sea surface elevation 𝜂(x, t), the mean depth H(x), the depth D =𝜂 + H, the geopotential coordinate z, and the time t; x is the longitudinal coordinate, with𝜕xdenoting the longitudinal derivative with respect to constant𝜎 and 𝜕
∗
xdenoting the
lon-gitudinal derivative with respect to constant z; and ̄w is the vertical velocity with respect to 𝜎 layers, 𝜕∗ xbis
the horizontal buoyancy gradient, Avis the vertical eddy viscosity, and [DP(t)] is the barotropic pressure
gra-dient term (see Burchard and Petersen [1997] for details of the transformation to general vertical coordinates, of which the𝜎 coordinate is a special case). The incompressibility condition reads
W𝜕t𝜂 + 𝜕x(WDu) + W𝜕𝜎(D̄w) = 0, (2)
with W(x) denoting the varying width of the estuary or tidal channel.
2.1. One-Dimensional Idealization
To systematically investigate the dependence of estuarine circulation on various parameter choices, we focus on a setup that results in an along-channel uniform exchange flow, following the approach of, e.g.,
Hansen and Rattray [1965], Simpson et al. [1990], and Burchard and Hetland [2010]. To that end, the
follow-ing assumptions are made: (i) no surface elevation (𝜂 = 0), (ii) no vertical flow through layers of constant 𝜎 (̄w = 0), (iii) constant in time buoyancy gradient, and (iv) variable width but constant depth. Under these assumptions, (2) becomes𝜕x(Au) = 0, such that𝜕xu = −(u∕A)𝜕xA, with the cross-sectional area A = HW, and
(1) can be transformed to 𝜕t(Au) −𝜕𝜎 (A v H2𝜕𝜎(Au) ) = 2(Au)2[1 A2𝜕xA ] −𝜎[AH𝜕∗ xb ] − [AP(t)]. (3)
These conditions contradict the assumption of a zero longitudinal velocity gradient which is often made for estuaries in morphodynamic equilibrium [Friedrichs, 2010]. However, due to coastal engineering measures in tidal estuaries such as deepening (to provide safe fairways) or narrowing (e.g., for coastal protection or land reclamation), estuaries may in many parts be far away from morphodynamic equilibrium conditions, with the consequence of regions along the estuary with significant deviations from𝜕xu = 0. For constant depth,
H, and longitudinally homogeneous eddy viscosity, Av, it can be ensured that all terms in (3) are independent
of x. A is chosen here such that𝜕x
[
𝜕xA∕A
2]= 0, a requirement which is consistent with
A(x) = A0
1 +1 2cx
, (4)
where c = −2A0𝜕xA∕A
2is the convergence of the estuary (positive for an estuary with decreasing width for
increasing x) and A0the area at x = 0. In addition, the longitudinal buoyancy gradient forcing,
[
AH𝜕∗ xb
] , is
Figure 1. Dimensional tidally averaged velocity profiles for a tidal velocity amplitude ofU0= 1m s −1
,H = 10m, M2tidal period, and a
constant eddy viscosity ofAv= 10 −2
m s−2
for three different convergence parametersc. (a) No horizontal buoyancy gradient (𝜕∗ xb = 0).
(b) Positive horizontal buoyancy gradient (𝜕∗
xb = 5⋅ 10 −6
s−2
). Note that the velocity profile in Figure 1b forc = 0is equivalent to the classical Hansen and Rattray [1965] solution.
chosen as a constant and the external pressure gradient forcing is chosen such that the depth-mean veloc-ity is a harmonic function with the frequency of the semidiurnal M2tide,𝜔, and an amplitude of U0A0∕A,
where U0is the velocity amplitude at x = 0. Despite these restricting conditions, entire estuaries can be
described by a one-dimensional water column model, including the dynamics of limited estuarine reaches if their length is comparable to the tidal excursion.
Nondimensionalization (using H as length scale,𝜔−1as time scale, and the root mean square friction velocity
scale U∗as velocity scale, see Burchard et al. [2013]) for x = 0 leads to
Un𝜕̃t̃u − 𝜕𝜎(̃Av𝜕𝜎̃u
)
= −Cõu2
− Si𝜎 −[̃P(̃t)], (5)
with the unsteadiness number Un = H𝜔∕U∗, the Simpson number Si = 𝜕 ∗ xbH
2∕U2
∗, and the convergence
number Co = cH. A further nondimensional parameter of the study is the relative bottom roughness̃zb
0 =
zb
0∕H with the dimensional bottom roughness z b
0. All nondimensional variables X in (5) are denoted as ̃X.
For display, velocity profiles are nondimensionalized by the tidal velocity amplitude (̂u = u∕U0). Tidal
resid-ual velocity profiles (denoted as⟨̂u⟩) are decomposed into contributions from various processes such as convergence (⟨̂uc⟩), tidal straining (⟨̂us⟩), and gravitational circulation (⟨̂ug⟩) using a method proposed by
Burchard and Hetland [2010]:
⟨̂u⟩ = ⟨̂uc⟩ + ⟨̂us⟩ + ⟨̂ug⟩ (6)
3. Numerical Experiments
The numerical simulations carried out here are discretizations of (5) using the General Ocean Turbulence Model (www.gotm.net) for sections 3.1 and 3.2 and the General Estuarine Transport Model (www.getm.eu) for section 3.3 (including additional terms for lateral momentum advection).
3.1. Constant Eddy Viscosity
To compare the convergence-driven estuarine circulation with the classical theory of gravitational circula-tion as developed by Hansen and Rattray [1965], numerical experiments with constant eddy viscosity and zero river runoff are carried out (see the caption of Figure 1 for the dimensional parameters). For the case of no horizontal buoyancy gradient (as it may occur in tidal estuaries upstream of the salt intrusion; see Figure 1a), a convergence of c = 5⋅ 10−5m−1(equivalent to a 50% reduction in width over 20 km) already
results in an upstream residual current near the bed of 1% of the tidal velocity amplitude, which more than doubles for c = 1⋅ 10−4m−1(50% reduction in width over 10 km). Compared to the classical Hansen and
Rattray [1965] solution, estuarine circulation is more than doubled for c = 5⋅ 10−5m−1and substantially
increased by more than a factor of 3 for strong convergence (see Figure 1b). These results suggest that strong estuarine convergence can substantially increase estuarine circulation, as argued by Ianniello [1979].
Geophysical Research Letters
10.1002/2013GL058494
Figure 2. Decomposition of nondimensional residual velocity profiles (red) into contributions from tidal straining (green), gravitational
circulation (blue), and estuarine convergence (black). Simulations have been carried out with a two-equation turbulence closure model atUn = 0.025(U0= 1.77m s −1 ,H= 10 m, M2tidal frequency,̃z b 0= 5⋅ 10 −5
). (a) No horizontal buoyancy gradient, strong convergence. (b) Horizontal buoyancy gradient (Si = 0.6, strong convergence). (c) Horizontal buoyancy gradient (Si = 0.6, no convergence).
3.2. Variable Eddy Viscosity
To assess the role of estuarine convergence on estuarine circulation driven by tidal straining, an oscillating tidal flow with a realistic turbulence closure model (two-equation k-𝜀 model with second-moment clo-sure; see Umlauf and Burchard [2005] for details) is analyzed. Such a type of model has proven to reproduce dynamics of energetic tidal flow under the influence of horizontal density gradients [Simpson et al., 2002]. Nondimensional results are shown in Figure 2 for flows with and without a horizontal buoyancy gradient and with and without estuarine convergence. For the simulation without a horizontal buoyancy gradient (Figure 2a), the convergence-driven component of the estuarine circulation behaves as expected like a
Figure 3. Intensityas defined in (7) of (a) residual circulation and contributions from (b) convergence, (c) tidal straining, (d) and gravitational circulation as function of the Simpson numberSi, and the convergence numberCo. The unsteadiness number (Un = 0.025) and the relative bottom roughness (̃zb
0 = 5⋅ 10
−5
) have been kept constant for all simulations. Colored dots mark the points in parameter space for the results shown in Figure 2a (blue), Figure 2b (green), and Figure 2c (red). Gray fields denote nonperiodic solutions.
Figure 4. Intensity of estuarine circulation due to several dynamic processes as function of the convergence number,Co, for a cross-sectionally resolved estuary withSi = 0.18andUn = 0.049, which is the reference case by Burchard et al. [2011]. Note that their
Sivalues have to be multiplied by 2 and theirUnvalues by√2in order to include the root mean square bed friction velocity scale instead of its amplitude as scaling.
classical estuarine circulation profile, which is however opposed by the tidal straining residual flow compo-nent. This counterintuitive result can be explained as follows: During ebb flow the shear in a large part of the water column is enhanced by the estuarine circulation, and during flood it is weakened. This enhanced shear results in a stronger eddy viscosity during ebb than during flood, which is opposite to the result found in the classical tidal straining case described by Jay and Musiak [1994]. Therefore, the covariance between eddy viscosity and shear results in an inverse tidal straining circulation. By comparing Figures 2b and 2c, which show a decomposition of the residual circulation under the presence of a horizontal buoyancy gra-dient, it becomes clear that when longitudinal buoyancy gradients are prescribed, estuarine convergence even decreases the intensity of the estuarine circulation due to enhanced weakening of the tidal straining circulation: The positive contribution from width convergence is negated by a stronger reduction of the tidal straining contribution. This is caused by the fact that the enhancement of the shear depends nonlinearly on the circulation resulting from convergence and the buoyancy gradient.
To explore the parameter space spanned by the Simpson number and the convergence number, 21 × 21 simulations have been carried out to calculate the residual velocity profiles as well as their components for each simulation. The intensity of the residual circulation is quantified by
(⟨̂u⟩) = −H42∫ 0 −H ⟨̂u⟩(z +H 2 ) dz (7)
[Burchard et al., 2011], with(⟨̂u⟩) > 0 denoting classical estuarine circulation (landward residual flow near the bed) and(⟨̂u⟩) < 0 denoting inverse estuarine circulation.
Figure 3 shows that the residual circulation (which is positive over the whole parameter space) is largely resulting from tidal straining and convergence, whereas gravitational circulation is relatively weak in this tidally energetic regime. The strength of the convergence circulation mainly depends on Co and increases slightly with Si. Tidal straining circulation increases with Si as shown by Burchard and Hetland [2010] and decreases with Co as argued above such that it reverses for large values of Co. Consequently, the intensity of estuarine circulation is weakened by estuarine convergence for relatively large Si and small to moderate
Co. Additional numerical experiments for fully propagating tidal waves with elevation amplitudes up to 40% of the water depth showed for situations with high Si and low Co similar reductions of estuarine circulation (not shown).
Geophysical Research Letters
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3.3. Cross-Sectional Variation
To make a first assessment of the effects of convergence on tidal straining and estuarine circulation in cross-sectionally varying weakly stratified estuaries, we have extended the reference scenario by Burchard
et al. [2011] by this process. There, a 1000 m wide estuarine cross section with parabolic depth distribution
(15 m in the center and 5 m at the sides) is simulated with a forcing as described above (resulting in Si = 0.18 and Un = 0.049; see caption of Figure 4 for details). Convergence numbers have been varied between
Co = 0 and Co = 0.003, and the results are surprisingly consistent with those of the one-dimensional
simulations (Figure 4): Generally, convergence reduces or even reverses tidal straining circulation such that for low values of Co, estuarine convergence has a reducing net effect on estuarine circulation. However, as tidal straining circulation reverses at relatively high convergence rates, lateral circulation increases such that advectively driven estuarine circulation is significantly increased, with the effect that also the total estuarine circulation is enhanced.
4. Discussion
The model simulations presented here confirm the result by Ianniello [1979] that landward convergence in estuaries and tidal inlets adds a positive contribution to classical estuarine circulation. This contribution scales linearly with the dimensionless convergence number as can be seen in Figure 3b. For the classical test scenario with constant in time and space eddy viscosity as introduced by Hansen and Rattray [1965], estu-arine convergence increases the total estuestu-arine circulation considerably, without and with consideration of the effect of horizontal buoyancy gradients. However, for situations including higher-order turbulence clo-sures, the process of tidal straining which may explain a large part of estuarine circulation in tidally energetic estuaries [Burchard and Hetland, 2010; Burchard et al., 2011] is reversed in such a way that the total estuar-ine circulation is reduced. This counterintuitive result is explaestuar-ined as follows: For no horizontal buoyancy gradient, ebb stress is enhanced and flood stress is weakened due to convergence, resulting in higher eddy viscosity during ebb than during flood. This leads to an effect similar to the tidal straining effect discussed by Jay and Musiak [1994]: The tidal phase with higher eddy viscosity leads to bottom-enhanced velocity, with the effect that the residual near bottom velocity points to the direction of the tidal phase with higher eddy viscosity. This is the flood direction for classical tidal straining and the ebb direction for estuarine con-vergence. For a fixed buoyancy gradient, increasing estuarine convergence reduces the estuarine circulation contribution by tidal straining (see Figure 3c) but enhances the contribution related to convergence itself (Figure 3b). Compared to the circulation in a nonconverging tidal inlet, these competing effects result in a weakening of the estuarine circulation for relatively large Si and small to moderate Co (see Figure 3a) and in strengthening for other combinations of Si and Co.
5. Conclusions
Based on the concept of constant eddy viscosity, estuarine convergence has been believed to generally enhance estuarine circulation. The pivotal role of tidal straining due to eddy viscosity systematically varying with the tidal phase has only been understood during the last two decades, specifically for tidally energetic estuaries and inlets. The present study shows that the combined effect of tidal straining and estuarine con-vergence may have a weakening effect on estuarine circulation, an effect which can have significant impact on transport of suspended particulate matter (SPM). As recently shown by Burchard et al. [2013], especially the process of tidal pumping of SPM (covariance between current velocity and SPM concentration) is sensi-tive to the tidal cycle of bed stress (and thus eddy viscosity), which in turn (as shown in section 3.2) is highly correlated to the impact of convergence on estuarine circulation.
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The present study has been carried out in the framework of the project ECOWS (Role of Estuarine Circulation for Transport of Suspended Particu-late Matter in the Wadden Sea) funded by the German Research Founda-tion (DFG) as project BU1199/11 and by the German Federal Ministry of Research and Education in the frame-work of the project PACE (The future of the Wadden Sea sediment fluxes: Still keeping pace with sea level rise?, FKZ 03F0634A). The work of Henk M. Schuttelaars has been supported by a visiting scientist grant through IOW. The studies of Elisabeth Schulz have been funded by a scholarship by the University of Rostock, Interdisciplinary Faculty, Department Maritime Systems. The Editor thanks Charitha Pattiaratchi, John Simpson, and an anonymous reviewer for their assistance in evaluating this paper.
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