5. Estuarine Secondary Circulation: Robert J Chant Rutgers University

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1 5. Estuarine Secondary Circulation: Robert J Chant Rutgers University 5.1 Introduction While the majority of theories developed to describe the dynamics of estuarine circulation are devoted to the study of along channel flows at both tidal (Friedrichs and Aubrey 1988) and subtidal frequencies (Pritchard 1956; Hansen and Rattray 1966; Geyer et al. 000; MacCready 004), in recent years numerous studies have concentrated on secondary flows and their importance in the along channel dynamics (Lerczak and Geyer 004) and along-channel dispersion (Smith 1977; Smith 1978; Geyer et al. 008). While detailed theoretical work by Smith (1976) preceded these more recent studies by several decades, the recent modeling and observational studies discussed here have more clearly elucidated the complex interplay between lateral mixing, along channel dynamics and dispersion. These studies have also emphasized the importance of the pioneering work by Ronald Smith (1977,1976) who was awarded the BH Ketchum award in 1996 for this seminal work. Thus, in addition to the recent work described in detail in this chapter, the reader is encouraged to study the work of Smith (1977, 1978). Secondary flows are defined by flow that is normal to the main along channel flow. For natural flows the directionality of this may be defined by flows normal to channel orientation. This is often somewhat ambiguous, so in practice the cross-channel direction is usually defined by either principal component analysis or tidal ellipse analysis of current meter data. Typically the strength of secondary flows is <10% of the strength of along channel flows. However, because cross-channel gradients in velocity, salt, turbidity and other tracers are often larger than their respective along-channel gradients, the magnitude of cross-channel advective terms in the momentum and tracer equations are often as large or larger than their respective along-channel counterparts. Moreover, it is through secondary flows that material is mixed across a channel and thus accurately modeling secondary flows is imperative to make detailed estimates of the dispersive nature of an estuary and to determine the fate and transport of material discharged along an estuarine shore. An early and insightful example of the role of lateral processes driving along channel dispersion is contained in the pioneering work of Okubo (1973) who demonstrated that 1

2 the advection of material into and out of shoreline coves can drive strong along channel dispersion. In addition, the interaction between lateral shear and lateral mixing associated with secondary flows drives an along channel dispersion that can dominate the dispersive nature of many estuaries (Fischer, 197; Smith, 1977; Geyer et al., 008). These processes are discussed in more detail later in the chapter. Finally, lateral circulation shapes channel morphology and often produces channels that are laterally asymmetric around the channel axis. Moreover, cross-channel variations in channel depth play a central role in driving secondary flows through differential advection (Nunes and Simpson 1985; Lerczak and Geyer 004) and thus provides a positive feedback between secondary flow and channel morphology (Huijts et al. 006; Fugate et al. 007). It is noteworthy that secondary flows are typically characterized as baroclinic features with a zero depth-averaged component. While depth-averaged cross-channel flows will develop when along-channel gradients in channel morphology exist (Figure 1a), in this chapter only the case when crosschannel flows are characterized by closed circulation cells with zero cross-section averaged flows (Figures 1bc) is considered. 5. Driving Mechanisms In this section the major mechanisms that drive secondary flows are discussed. One mechanism is Ekman forcing, which represents a dynamical balance between friction and the Coriolis acceleration. Often there is a misconception that Earth s rotation is unimportant in narrow estuaries. This chapter will hopefully convince the reader that this is not the case and that in fact the earth s rotation can be an important contributor to the structure of not only lateral flows but also of along channel flows, even in estuaries that are significantly narrower than an internal Rossby Radius. This radius is defined as the ratio of the internal wave speed (g h) 1/ to the local Coriolis frequency f, where g is reduced gravity, gg is acceleration due to gravity and and are the density difference between the upper and lower layers and the mean density, nominally ~1000 kg/m 3, of the two layers. The second mechanism driving secondary flows is related to flow curvature, which has long been recognized to drive a helical lateral flow normal to the stream-wise flow (Rozovski, 1957). The third mechanism is linked to cross-channel baroclinic pressure gradients that arise from differential advection of the longitudinal density gradient. For completeness, the forcing of

3 secondary flows by diffusive boundary layers is also discussed briefly. While this mechanism appears to be important on continental shelves and slopes (Garrett et al. 1993) it does not appear to be a major contributor to secondary flows in estuaries (Lerczak and Geyer 004). The mechanisms driving secondary flows are clearly identified by analysis of a crossstream momentum equation written in a curvilinear coordinate system: u t n u s u s n s u R fu s 1 P 0, (5.1) n z where u n is the cross-stream (secondary) flow, u s the streamwise flow, R the radius of curvature in the streamwise flow, f the local Coriolis parameter, n the cross-stream direction, s the streamwise direction, z the vertical direction, P is pressure and stress. Equation 5.1 assumes the cross-stream flows u n are much weaker than the streamwise flows. The first term represents the acceleration of the secondary flows, the second term represents the streamwise advection of streamwise gradients in the cross-stream flow. The third term is centrifugal acceleration, the fourth term is the Coriolis acceleration and the fifth and sixth terms represent the pressure gradient and a vertical stress divergence, respectively. The pressure gradient has contributions both from cross-stream sea-level slopes and from cross-stream density gradients: 1 P g g z. (5.) n n n Taking the vertical gradient of 5.1 and using 5. to describe the pressure gradient we find un u s u t z z s A B n u s un u s u f s z R z C D s g 0. (5.3) n z Equation 5.3 is an equation that concisely describes the three major forcing mechanisms. Term A is the local acceleration of the secondary flows; term B represents the straining of secondary flows by vertical shear in the stremwise flow; and term C is the streamwise advection of streamwise gradients in secondary flows. The next three terms represent the main forcing mechanisms that generate estuarine secondary flows. Terms D represent forcing by flow curvature and the earth s rotation. The forcing represented by terms D increases with vertical shear, and the relative importance of flow curvature to the earth s rotation is represented by a Rossby Number (u s /f R), which expresses the relative importance of inertia to rotation. The centrifugal acceleration term is an advective term that takes this form in the curvilinear E F 3

4 coordinate system (in a Cartesian coordinate system the centrifugal acceleration is contained in the horizontal advective terms). Note that because velocity is squared, the centrifugal acceleration has the same sign on flood and ebb, while the Coriolis acceleration changes sign between flood and ebb. Thus, for purely harmonic tidal motion the tidal average of the Coriolis term will be zero. In contrast, the effects of flow curvature acting on an oscillatory flow result in a strong rectified motion due to the non-linear centrifugal acceleration. Furthermore, the sum of the Coriolis and centrifugal accelerations will augment during one phase of the tide and compete on the opposite phase, thus producing tidal asymmetries in secondary flows in regions of flow curvature. Term E represents a forcing associated with cross-stream density gradients and is often a dominant term driving secondary circulations. Cross-stream density gradients, however, can also shut down secondary flows, similar to the shelf dynamics associated with the arrested Ekman Layer (MacCready and Rhines 1991). Finally, term F in equation 5.3 represents friction, which typically balances one of the forcing terms that drive secondary flows. 5.3 Development of lateral Density Gradients It is instructive to discuss mechanisms that produce cross-channel baroclinic forcing before discussing the details of the dynamics of secondary flows because lateral buoyancy forcing plays a critical role in both the generation and the arresting of lateral flows. This discussion begins with the salt balance equation: s s s s s u v w Kv, (5.4) t x y z z z where K v is the vertical diffusivity of salt, u is the along-channel (x) flow, v is the cross-channel (y) flow and w is the flow in the vertical direction z. Taking the cross channel gradient of equation 5.4 yields:. s s u s s v s s w s s s u v w K v y t t y y x y x y y y z y z y z z y A A B B' C C' Term A in equation 5.5 is the tendency term and represents the time rate of change of crosschannel salinity gradients. Note that the primed terms cannot generate cross-channel density D D' E (5.5) 4

5 gradients but rather can only produce a local change in the cross-channel gradient by the advection of existing cross-channel gradients. Term C modifies existing cross-channel salinity gradients by the compression of isohalines (Figure a). In contrast terms B and D actually generate cross-channel density (salinity) gradients. Term B generates cross-channel density u gradients due to lateral shears in the along-channel flow y acting on the along-channel s salinity gradient (Figure b), also known as differential advection. Later it will be shown x that this is often a dominant mechanism driving secondary flows, as described in Nunes and Simpson (1985). Term D acts to tilt the vertical stratification by cross-channel variability in the vertical motion (Figure c) and requires the existence of secondary flow. This term has been shown in a number of studies to arrest secondary flows (Seim and Gregg, 1997; Chant and Wilson 1997; Chant 00; Lerczak and Geyer, 004). Finally, term E represents cross-channel variations in mixing that would generate cross-channel salinity gradients and thus drive secondary flows. However, while this mechanism has been suggested to be important in shelf/slope dynamics (Phillips et al. 1986; Garrett et al. 1993) it appears to only play a minor role in generating estuarine secondary flows (Lerczak and Geyer, 004). Even though there may be some estuarine environments where cross-channel gradients in mixing are important contributors to secondary flows, the importance of this term in driving estuarine secondary flows has yet to be demonstrated. In summary, there are three major mechanisms that drive estuarine secondary flows: differential advection, Coriolis acceleration and flow curvature. In addition, cross-channel salinity gradients can only be generated by three mechanisms (although only two are believed to be dominant): a) differential advection, which is the development of cross channel gradients set up by lateral shears in the along-channel flow acting on the along channel salinity gradient; b) the tilting of vertical stratification by cross-channel variability in vertical motion associated with the secondary flows themselves; and c) cross-channel variations in mixing. Thus, from the point of view of the mass field, it is only differential advection that can initiate secondary flows because the second mechanism requires their existence and typically acts to shut down secondary flows. Considering only these two terms, the tendency equation 5.5 can be simplified to: 5

6 s u s w s. (5.6) t y y x y z The first term on the right hand side is typically associated with the generation of secondary flows due to differential advection, while the second term on the right hand side tends to shut down existing secondary flows. 5.4 Differential Advection Depth-averaged along channel tidal currents tend to be strongest over deeper parts of the channel. Consequently an isohaline in the middle of the channel will advance further upstream (downstream) on the flood (ebb) tide than on the flanks and produce tidal period variations in cross-channel density gradients (Figure 3). A classic example of differential advection was identified by Nunes and Simpson (1985) who showed that during the flood tide a pair of counter rotating secondary flow cells develop in the vertical plane. The flow cells result in surface flows that converge over the deep channel becoming visible as a line of flotsam oriented in the alongchannel direction over the deep channel (Figure 3a). This secondary flow is driven by a crosschannel baroclinic pressure gradient that is characterized by heavy, more saline waters in the deep channel relative to the flanks. The deep water in mid-channel then sinks under the influence of gravity and is replaced by fresher fluid from the flanks. During ebb tide the opposite may occur with differential advection producing fresher water in the main channel relative to the flanks. Here, the saline water on the flanks sinks towards mid channel while the fresh water in mid channel rises and spreads to the flanks producing an opposite lateral flow pattern than the flow that occurs during flood (Figure 3b). Smith (1977) and Nunes and Simpson (1985) assume that the two dominant terms in the cross-stream momentum balance for secondary flows driven by differential advection are pressure gradient and friction. This is analogous to the along channel momentum balance assumed in Hansen and Rattary (1965), MacCready (004) and summarized in Lerczak and Geyer (004). Moreover, similar to the Hansen and Rattray (1966) theory, a constant vertical eddy viscosity is assumed and a steady-state assumption is made. With these simplifying assumptions the cross-channel momentum balance can be written as: g u z K y z (5.7) z 6

7 The cross-channel density gradient, which produces the lateral baroclinic forcing, is u t y y x (5.8) u The lateral shear ( u y ) associated with the oscillatory tide is defined as sin t, where B is the tidal frequency, u is the amplitude of the velocity difference between mid channel (y = B/) and the channel edges (y = 0, B) and B is the channel width. Thus, the cross-channel density u gradient scales as, which corresponds to a density difference between the main channel B x u and the channel edges of. Integrating equation 5.6 twice vertically one obtains a x scale for the cross-channel flow V DA in an expression identical to the along channel scale (Hansen and Ratray, 1966; Lerczak and Geyer, 004) gh 1 gh ( y) 1 gh u( y) VDA ~. (5.9) 48 Av o y 48 Av B o 4 o Av x B Equation 5.9 indicates that when lateral friction balances the cross-channel pressure gradient (set up by differential advection) the strength of the secondary flows is linearly proportional to the lateral shear, inversely proportional to the vertical eddy viscosity A v and varies with the cube of the channel depth H. Equation 5.9 also suggests that if the lateral shear shows no tidal period variability, then the secondary flows should be of the same magnitude on flood as on ebb. The flood tide causes convergence in the middle of the channel and divergence at depth, while the ebb tide causes divergence at the surface and convergence at depth. As will be shown later, however, lateral circulation in fact varies in strength with the tidal period because of tidal asymmetries in vertical stratification of density (Lerczak and Geyer, 004). The tendency for stratification to suppress secondary flows and thus the physics missing in equation 5.7 becomes apparent by comparing the strength of secondary flows predicted by equation 5.9 with field observations. For example, during neap tides in the Hudson River estuary, estimates of vertical eddy viscosity A v in the halocline are between 10-5 to 10-4 m /s (Peters 000; Chant et al; 007; Geyer et al 008), lateral shear is ~ m/s and the along channel density gradient is ~ kg/m 4. Equation 5.9 would predict lateral flows of over 1 m/s, which is at least an order of magnitude larger than observed lateral flows in this system 7

8 (Chant and Wilson 1997; Lerczak and Geyer 004). Lateral flows are reduced because, as will be discussed later, they are shut down by density stratification and this mechanism is not included in the underlying physics contained in equations 5.7 and Flow Curvature It long has been recognized that a vertically sheared flow with curvature will develop a secondary flow with the lower layer flow directed towards the inside of the bend and the upper layer flow directed away from the bend (Rozovski, 1957). The dynamics that drive this lateral flow are easily conceptualized by first considering how flow rounds a bend. Consider a channel with a 90º bend as depicted in Figure 4. Flow must accelerate from a north-westward flow to a south-eastward flow in order for the depth-averaged flow to round the bend. This acceleration is driven by a pressure gradient that sets up by the inertia in the flow. If the pressure gradient is too weak the depth averaged flow will not make it around the bend and will cause water to pile up at the outer bend, eventually generating exactly the correct pressure gradient that will steer the flow around the bend. If the flow is vertically sheared, as we would expect a boundary layer flow to be, with swifter flows near the surface, the fluid at the surface has too much momentum and will head towards the outside of the bend. Upon reaching the outer boundary it flows downward. In contrast, flow in the lower layer is moving slower and thus the acceleration will drive the flow to the inside of the bend where it will upwell at the wall. This results in the classic helical flow pattern shown in Figure 4b. A second way to conceptualize the effects of flow curvature is by using a vorticity argument. In the upstream flow depicted in Figure 4a, the flow is vertically sheared and the relative vorticity vector is pointed to the south-east. This vorticity is depicted as a cylinder in the flow that rotates clockwise (looking from the south-east) with the vertical shear. As this fluid rounds the bend, this south-easterly pointing vorticity vector corresponds to a secondary flow towards the inside of the bend at depth and towards the outside of the bend at the surface (Figure 4b). Eventually, bottom friction will generate vertical shear and a vorticity vector that points towards the west. (Figure 4a). These dynamics are more precisely described, of course, with the use of a momentum equation written in streamwise coordinates (analogous to equation 5.1) for a homogenous fluid and neglecting the earth s rotation: 8

9 u t n u s u s n s u R g A n z v u z n 0. (5.10) Taking the depth-average of this equation and neglecting stress at the sea surface results in: u s u s n s u R g b n H 0, (5.11) where b is the bottom stress. Note that the first term in equation 5.10 vanishes upon vertical integration because the depth average of the cross-stream flow (u n ) is exactly zero. Taking the difference between equations 5.10 and 5.11 and neglecting time dependence and the first tem in 5.11, which is generally small, yields: u s u s n A z v un b u s z H u R s. (5.1) The first term in equation 5.1 is the streamwise advection of lateral flow, the second and third terms are frictional forces. The term on the right hand side is the shear forcing and drives the secondary flows. Clearly, equation (5.1) indicates that the forcing increases quadratically with the shear. Kalkwijk and Booij (1985) provided analytical solutions for equation 5.11 for a case with a logarithmic velocity profile and a parabolic eddy viscosity profile. Given a nondimensional quadratic drag coefficient, C D = 3x10-3, the maximum strength of secondary flows as in Kalwijk and Booij (1985) is 6 u s H / R (Geyer, 1993) (Figure 5). However, observations of secondary flows around a headland by Geyer (1993) clearly demonstrated that lateral flows were twice as large as predicted by Kalkwijk and Booij (1985). Geyer (1993) attributed the discrepancy between the theory and observations to the effects of stratification and assessed the role of stratification by numerically solving equation 5.1, neglecting the first term. Geyer (1993) noted that stratification allowed the vertical shear in the streamwise flow to be stronger than it would be in a logarithmic layer and thus the forcing term to the secondary flows, us u s / R is stronger. In addition, because this term is balanced by friction, A z v u z n, reduced eddy viscosity in the presence of stratification requires stronger shear in the secondary flow for frictional forces to balance the (increased) shear forcing. Thus, Geyer (1993) argued that the 9

10 effects of stratification were two-fold in augmenting secondary circulation for it a) increased the shear in the streamwise flow that forces the secondary flows, and b) stratification suppresses vertical eddy viscosity thus requiring stronger lateral shears (i.e. secondary flows) to balance the shear forcing. Similar conclusions were reached by Chant (00) based on a long-term mooring deployment in Newark Bay, New Jersey, near a region of strong flow curvature. During times of low river flow a strong secondary flow was observed as linearly proportional to tidal current amplitude (Fig 6ab). Chant (00) argued that the shear forcing would increase quadratically with tidal current speed and this would be balanced by a quadratic increase in the cross-channel frictional term. The quadratic increase in the frictional term occurs because eddy viscosity increases linearly with tidal current speed, as suggested by Bowden and Fairbairn (195a, b). Thus, a linear increase in vertical eddy viscosity and in the vertical shear (secondary flows) will cause the stream-normal stress term to increase quadratically to match the quadratic increase in the forcing to the secondary flow. Therefore, secondary flows increase linearly with tidal current speed, despite the quadratic increase in forcing. However, in the same record, during times of high river discharge Chant (00) noted that secondary flows were shut down due to buoyancy effects, and this is discussed in more detail later (Figure 6c). Finally, note that the balance between centrifugal forcing and friction can only occur in the region of flow curvature. Downstream, where the flow may encounter a straight channel, the forcing would be shut down resulting in a momentum balance between downstream advection of the secondary flows ( u s u s n ) and friction, resulting ultimately in a frictional spin-down of the secondary flows downstream of the region of flow curvature. Geyer (1993) suggested that this spin down time-scale is dominated by the effects of bottom friction and can be estimated as H/C D u s. Given typical estuarine values of these parameters (H = 10m, C D = 3x10-3 and u s = 0.5 m/s) yields a spindown time of a few hours. In contrast, Fong et al (in press), suggest a spindown time of H/u*, which is (C d ) 1/ shorter than the time suggested by Geyer (1993) and corresponds to 400 seconds. Note that these two decay time scales correspond to downstream distances, for the above velocity scale, of 4000 meters for the former and 00 meters for the latter. Observations by Fong (008), based on multiple ADCP s in a sinuous channel suggest that the spin-down length scale is even smaller and indicate that internal friction appears to dominate over bottom 10

11 friction. Nevertheless, the downstream effect of secondary flows driven by flow curvature in weakly stratified cases appears to be limited. The case of the downstream effect in highly stratified environments is discussed next. 5.6 Effects of Coriolis. A common misconception regarding estuarine circulation is that the effect of the earth s rotation is unimportant when the estuarine channel s width is smaller than the internal Rossby radius of deformation. This is indeed false! In reality, the first order momentum balance in the cross-channel direction is often geostrophic. Consider, for example the depth-averaged momentum equation in the cross-channel direction for a homogenous fluid and (momentarily) neglecting the advective term: v P y fu 1 x z. (5.13) t An along channel flow (u) will accelerate the cross-channel velocity, which tends to tilt the sealevel upwards to the right side of the channel (looking downstream in the northern hemisphere) and produces a lateral pressure gradient. If the along-channel flow has no shear, then no crosschannel flows will develop and the flow will be geostrophic. However, as is always the case, there will be shear in the along channel flow due to frictional and density effects and thus there will be an imbalance between the depth-independent cross-channel barotropic pressure gradient and the depth-dependent Coriolis acceleration. This produces an ageostrophic cross-channel flow. In a similar approach to that used to investigate the effects of curvature, the effects of Earth s rotation on driving secondary flows are studied by subtracting the vertical average of equation 5.13 from 5.13, itself. This analysis, for now, assumes a homogeneous fluid and continues to neglect the advective terms. With these assumptions, the vertical average of equation 5.13 is: v by f u g t y H Subtracting equation 5.13 from 5.14 results in:. (5.14) 11

12 v v t by y f ( u u) H z. (5.15) For a steady-state case the shear forcing associated with Coriolis accelerations is balanced by friction, consistent with Ekman dynamics. Thus, in the case of a river flowing seaward the bottom layer flow will be directed to the left of the depth averaged flow, as expected in a bottom Ekman layer. Using analytical solutions of Kalkwijk and Booij (1985), Geyer (1993) finds that for a quadratic bottom drag coefficient of 3x10-3 the magnitude of the lateral circulation forced by Coriols accelerations is 3f H. Interestingly, the magnitude of the secondary flow is not proportional to the flow speed. This is in contrast to curvature-induced secondary flows, which are proportional to flow speed. A second difference between curvature-forced secondary flows and those associated with Coriolis is that the direction of lateral circulation associated with Coriolis forcing changes sign with the sign of the along-channel flow whereas in the case of curvature-forced secondary flows the sign of the secondary flow is independent of the sign of the along-channel or streamwise flow. The strength of the Coriolis forcing, however, does not have a simple relationship with A boundary layer thickness as given by v (where is the tidal frequency, Lerczak and Geyer, 004). Yet, given the range of vertical eddy viscosities A v considered in Lerczak and Geyer (3- x 10-4 m /s), the strength of secondary flows is independent of and has a tidally varying amplitude of 1 f U 8 o where U o is the tidal current amplitude. Thus, for a mid-latitude estuary with tidal currents of 1 m/s Coriolis-forced secondary flows will have an amplitude of 9 cm/s. The effects of rotation can be important, even in narrow estuaries, because of the lateral flows produced. The vertical structure of secondary flows that are forced by the earth s rotation varies with the Ekman number (H). When the boundary layer occupies the entire water column secondary flows are characterized by a single cell, while for small Ekman Numbers (thin boundary layers) a more complex lateral flow structure develops that is comprised of multiple cells that vary over the tidal cycle (Lerczak and Geyer, 004). A major impact of Coriolis-forced motion on estuarine circulation is that it generates a cross-channel asymmetry in the structure of along-channel exchange flow. This asymmetry is 1

13 characterized by the inflow (when looking towards the ocean in the Northern Hemisphere) tending to the right side of the channel and the outflow to the left (Valle-Levinson et al. 000; Lerczak and Geyer 004; Valle-Levinson et al. 007; Valle-Levinson 008). Lerczak and Geyer (004) suggest this asymmetry would tend to accumulate sediment on the right side of Northern hemisphere channels and produce a morphologically laterally asymmetric channel, such as that found in the Hudson River and other estuarine channels. This asymmetry could feed back by driving lateral flows associated with differential advection, which would tend to augment the trapping of sediment on the right flank. The feedback between geomorphology and lateral flow processes has been observed to occur in estuarine systems (Geyer et al. 001; Fugate et al. 007) and treated analytically by Huijts et al. (006). 5.7 Effects of stratification While Geyer (1993) suggested that stratification tends to strengthen secondary flows, others (Chant and Wilson 1997; Seim and Gregg 1997; Chant 00; Lerczak and Geyer 004) show clear evidence that secondary flows are shut down by the buoyancy effects of vertical density stratification. This discrepancy occurs because while under weak stratification the dynamics are consistent with Geyer s analysis (discussed earlier), under stronger stratification the tilting of the halocline by secondary flows produces a baroclinic pressure gradient that opposes the shear forcing. This tilting occurs via the advective term w s, which appears as y z the second term on the right hand side of equation 5.6. The tendency for vertical stratification to suppress secondary flows is analogous to the shutting down of Ekman transport during upwelling conditions on sloping continental shelves as described by MacCready and Rhines (1993) and Garrett et al (1993). Chant and Wilson (1997) presented data from the Hudson River estuary around a region of strong flow curvature, where secondary flows were relatively weak (similar to the situations depicted in figure 6c). Chant and Wilson added baroclinic pressure gradient to equation 5.1 and compared it to estimates of the shear forcing. The cross-stream baroclinic pressure gradient was defined as g n z 0 h g ( z') dz' ( z') dz', (5.16) n 0 13

14 where the overbar represents depth average. The second term represents then the depth-averaged value of the baroclinic pressure gradient. Chant and Wilson (1997) computed equations 5.1 and 5.16 using CTD data and with the forcing term us u s / R based on shipboard ADCP data. They found them to be nearly in balance. As with weakly stratified flows, the momentum balance between pressure gradient and centrifugal accelerations must change away from the region of flow curvature as the shearforcing term goes to zero. However, rather than simply spinning down, the cross-stream density gradient (set up in the region of flow curvature) will adjust and result in a lateral sloshing of the secondary flows (Figure 7). On one hand this could potentially extend the downstream influence of curvature-induced secondary flows because of the reduced internal friction associated with stratification allowing the lateral sloshing to potentially extend far downstream. On the other hand, the oscillatory nature of a lateral sloshing will limit the cross-stream excursion of a parcel of fluid and thus limit the cross-stream mixing. Chant (00) provided a scaling of the lateral mixing associated with a cross-stream seiche and suggested that it was generally a weak mechanism driving cross-stream mixing. As alluded to earlier, stratification also constrains lateral flows generated by differential advection. If the channel is sufficiently stratified, the tendency for lateral flows to tilt isopycnals will produce a baroclinic pressure gradient that can shut down secondary flows. Lerczak and Geyer (004) introduced a parameter that characterizes the relative importance of stratification on suppressing secondary flows to the generation of secondary flows by differential advection. The parameter, is the ratio of the forcing associated with the tilting of the isopcynals to the forcing that drives secondary flows from differential advection, i.e. zw y xu y, which is the ratio of the two right hand terms in equation 5.6. Using expression 5.9 to relate the lateral shear in the along channel flow to the scale of the secondary flows and scaling the cross-channel gradient in vertical velocity as w y Hu B v, Lerczak and Geyer (004) obtained: 1 N H H 1 1 T frtt ~, (5.17) 4 B A 4 T Iw 14

15 where N is the buoyancy frequency and T T, T fr and T Iw are the timescales associated with the tide, friction and internal waves respectively. The internal wave time scale is set by the cross-channel travel time for an internal wave, and is equal to the ratio of the internal wave speed squared (N H ) to the width of the channel squared (B ). The frictional time scale (H /A v ) is the time scale required to mix momentum completely in the vertical. The tidal time scale is, of course, the tidal period. Lerczak and Geyer (004) argue that when «1, stratification is unable to suppress secondary circulation. However, as approaches unity, tilting of the pycnocline by the secondary flow significantly suppresses the lateral motion. As will be discussed later, the variability in stratification at both tidal and fortnightly time scales has been shown to play a central role in the modulation of the dynamics that govern estuarine circulation over the spring/neap cycle. 5.8 Diffusive Boundary Layer The final process that can drive lateral circulation consists of mixing along the sloping boundaries. The no-flux boundary condition requires that the vertical density gradient vanish near the bottom and results in a cross-channel baroclinic pressure gradient that draws fluid up from the lower layer toward the shoaling flank (Figure 4d). For a fluid with a constant eddy viscosity and a Prandtl number (ratio of eddy viscosity to eddy diffusivity of salt) of 1, Garrett et al (1993) find that the thickness of the diffusive boundary layer,, is ( f N A v sin ) (5.18) where is the slope of the bottom relative to the horizontal. The maximum cross-channel flow that develops from diffusive boundary layers v BL is given by (Garrett et al, 1993, Lerczak and Geyer, 004): Av v cot BL. (5.19) In a series of idealized numerical simulations inspired by conditions in the Hudson River estuary, lateral flows associated with diffusive boundary layers as scaled by equation 5.19 were typically cm/s and nearly an order of magnitude smaller than lateral circulation driven by other processes (Lerczak and Geyer, 004). Subsequently, diffusive boundary layer processes appear to play only a minor role in the dynamics of estuarine secondary circulation. Note, however, that the Lerczak and Geyer (004) numerical simulations only covered a range of estuarine parameter 15

16 space occurring in the Hudson River, and thus diffusive boundary layers may be more important in systems that fall outside of this regime. Moreover, Lerczak and Geyer (004) ran their simulations with a constant eddy visocity. It is likely that more complex flows that develop with more realistic turbulent closures schemes will show that diffusive boundary layers have a greater impact than suggested by Lerczak and Geyer (004). 5.9 Role of secondary flows on streamwise processes. Thus far this chapter has discussed only the mechanisms that drive lateral circulation and provided field and modeling examples of particular cases. While these studies have advanced our understanding of the dynamics of secondary circulation they may leave the reader wondering what is the significance of lateral circulation on estuarine processes? For example, are secondary flows (as the name may imply) of secondary importance to estuarine dynamics? The answer appears to be a resounding NO! Indeed both analytical (Winant 004), numerical (Lerczak and Geyer, 004), and field observations (Geyer et al 008) highlight the important role that secondary flows have on along channel dispersion and on the very dynamics that drive estuarine exchange, which themselves play a central role in along channel dispersive processes Role in Along-channel Dynamics. Recent numerical model studies have clearly demonstrated that secondary circulation plays an important role in driving the estuarine exchange flow (Lerczak and Geyer, 004; Scully et al. 008). The classic theory of estuarine exchange flow neglects the role of advection and balances the along channel pressure gradient with frictional forces. In this model, friction is assumed to be equal to the tidally averaged vertical shear (i.e. the exchange flow) times a tidally averaged vertical eddy viscosity. Based on this balance the estuarine exchange flow Ue is scaled by (Hansen and Rattray 1966; MacCready 004): 3 1 gh Ue ~. (5.0) 48 A y v Equation 5.0 predicts that the estuarine exchange flow will be inversely proportional to the vertical eddy viscosity A v. Microstructure data from the Hudson River estuary indicate that vertical eddy viscosity varies by an order of magnitude over the spring/neap cycle (Peters 001), while hydrographic sections indicate that during low to moderate river discharge the along channel density gradient is relatively constant over the spring neap cycle (Lerczak et al. 006). o 16

17 Thus, equation 5.0 predicts that the exchange flow should vary by an order of magnitude over the spring neap cycle. However observations clearly show that the exchange flow varies by a factor of -3 over the spring/neap cycle (Geyer et al. 000), suggesting that that important dynamics are missing in the development of classical estuarine theory. Numerical results from Lerczak and Geyer (004) describe in detail how secondary flows accelerate exchange flows during spring tides and thus buffer the 10-fold effect predicted by equation 5.0 over the spring/neap cycle. The key to this buffering effect is the interaction between forces that drive the secondary flows due to differential advection, and the time-varying vertical density stratification at both tidal and spring/neap time scales. On the tidal time scale, this asymmetry is characterized by enhanced stratification on the ebb tide and reduced stratification on flood as a result of tidal straining (Simpson et al; 1990). On the neap/spring time scale, modulations in vertical mixing weaken stratification during spring tide and enhance stratification during neap tide. Thus secondary flows are often suppressed during ebb tides, due to the effects of stratification and, in the case of the modeling by Lerczak and Geyer (004), throughout the tidal cycle during neap tides when the water column remains highly stratified on both flood and ebb. In contrast, during flood phases of the spring tide the model produces the classic two-cell secondary flow field as described by Nunes and Simpson (1985). Lerczak and Geyer (004) model results (Figure 8a-d) show that during flood, lateral circulation advects lowmomentum fluid from the flanks into surface waters in mid-channel, which acts to decelerate the flood or equivalently accelerate the upper layer seaward. Meanwhile the strong downward vertical motion in the center of the channel advects the strong landward currents at the surface to the lower layer, which accelerates the lower layer landward. In contrast, during the ebb tide (Figure 8ab), stratification suppresses the secondary flows and these advective tendencies are not at play. Similarly during the neap tide, strong stratification throughout the tidal cycle suppresses secondary flows and the advective momentum exchange is minimal (Figure 8e-h). Consequently, the tidally averaged effects of the secondary flow on the along-channel momentum balance during spring tides resemble the flood tide conditions. This lateral circulation effect, because it accelerates the lower layer landward and the upper layer seaward, tends to augment the exchange flow. In contrast, during neap tides secondary flows are suppressed due to stratification, on both flood and ebb, and thus the tendency to accelerate the exchange flow is reduced. Thus the tendency for the tidally asymmetric secondary flows to augment exchange flow intensifies with 17

18 increased mixing. This tendency competes with the frictional effect that is included in the classic model whereby the exchange flow decreases with increased mixing. These results are summarized in Figure 9 that plots the exchange flow for a series of simulations that varied A v from to m /s, characteristic of neap tide and spring tide conditions. Results demonstrate that the exchange flow is less sensitive to mixing than predicted by equation 5.0. Indeed, rather than falling as A -1 v the exchange flow in these simulations fell at only A v. This suggests that the reduced sensitivity of estuarine exchange flow to variations in vertical mixing, relative to that predicted by equation 5.0, is caused by the non-linear effects of secondary flows. While the results of Lerczak and Geyer (004) are quite compelling, there are several cautionary notes. First, the results discussed above were run with a constant eddy viscosity, while it is well known that the eddy viscosity varies by orders of magnitude in both space and time. Lerczak and Geyer (004) did run simulations with the k turbulent closure scheme and found that secondary flows did develop during neap tide on flood, but were confined to the bottom boundary layer. While these secondary flows may play an important role in the momentum balance, they do not effectively exchange momentum between the upper and lower layers and thus do not appear to be as effective in augmenting the exchange flow as during weakly stratified conditions. A second cautionary note is that while observations of secondary flows in the Hudson River, and in other estuarine systems, do generally show a reduction in secondary flows during stratified conditions, their structure becomes more complex and characterized by multiple circulation cells in the vertical. Lerczak and Geyer (004) did present results using the kturbulent closure scheme, which exhibited some of this complexity, most notably the circulation in the lower layer. The model failed to capture the upper layer cell apparent in observations during neap tide floods. Finally, recent numerical results by Scully et al. (008), suggest that the augmentation of the exchange flow during spring tide is largely compensated by increased interfacial stresses. This balance between the exchange flow and interfacial stresses appears to explain why the simple scaling argument proposed by Geyer et al (000), which neglects both interfacial mixing and secondary flows, accurately predict the estuarine exchange flow. While it is clear why these two terms (advective effects associated with secondary flows and interfacial stresses) have opposite tendencies, it is unclear why their temporal variability should perfectly compensate for each other over the spring/neap cycle. Is 18

19 this fortuitous? Does it suggest some dynamical link between secondary flows and mixing? Or is it an artifact of the model? 5.10 Role of secondary flows in dispersion. Secondary circulation is the dominant process driving lateral mixing in estuaries. The interaction between lateral mixing and lateral shears drives along channel dispersion via a shear dispersion mechanism. This is analogous to the vertical shear dispersion produced by the interaction between vertical shear and vertical mixing. Like vertical shear dispersion, lateral shear dispersion can occur due to both the mean shear and the tidally oscillatory shear (Wilson and Okubo 1978). In both cases the shear dispersion is orders of magnitude larger than the dispersion driven by small scale turbulence (Fischer et al. 1979). The mechanism of lateral mixing is itself a vertical shear dispersion whereby the vertical shear in the cross-channel flow, associated with the secondary flow interacts with vertical mixing. The rate of vertical shear dispersion in the cross-stream direction (D y ) is Dy v H Kz, (5.1) where v, H and K z are the vertical shear in the cross channel flow (i.e. the secondary circulation), the channel depth and the vertical eddy diffusivity, respectively. The coefficient depends on the vertical structure of the velocity and diffusivity and in estuarine flows it is ~1 to (Geyer et al, 008). Thus for typical estuarine flows (K z = m /s, = 5 x 10-3, v = 0.1 m/s and H = 10m) lateral dispersion associated with vertical shear dispersion will be of order 10 m /s. The along-channel dispersion associated with lateral shear dispersion is identical to equation 5.1 with the denominator replaced by K y and the shear by the cross-channel shear in the along-channel flow, i.e. D x u H. (5.) K Whether lateral shear dispersion is driven by tidally oscillating lateral shear or by the mean shear depends on the lateral mixing time. Unlike steady shear dispersion, which continues to increase with decreasing vertical eddy diffusivity, shear dispersion associated with tidal motion will be maximum when the mixing time is on the order of a tidal cycle. Note that the lateral mixing time y 19

20 T x is given by W /10 K y, where W is the oscillatory lateral-shear dispersion associated with tides. This lateral mixing may be an important mechanism driving along channel dispersion in channels that are 100 to 1000 m wide. Observations of the lateral spread of a dye patch confined to the lower layer in the Hudson River estuary indicated a K y of ~1 m /s during the flood tide. This value is consistent with equation 5.1 using the patch thickness of 3-5m for H of 10 m and vertical eddy diffusivity of m /s (Geyer et al. 008). During the ebb tide, weak lateral flows were suppressed by stratification and caused estimates of K x ~0. m /s, which were consistent with equation 5.1. During spring tides, lateral shear dispersion was the dominant process driving dispersion, because vertical shear dispersion was suppressed by strong mixing. However, during neap tides vertical shear dispersion dominated and was significantly larger than the spring-tide dispersion rates. Nevertheless, in many narrow estuarine systems, lateral shear dispersion is often the dominant process driving along channel dispersion (Fischer 197; Smith 1977). Even in estuaries where along-channel dispersion is dominated by vertical shear dispersion, the fact that secondary flows act to augment the estuarine shear suggests that even in these systems it indirectly adds to the dispersive nature of the estuary. Summary Secondary circulation has a first order impact on the along channel dynamics of an estuary, estuarine dispersion and estuarine geomorphology. There are a number of processes that drive secondary flows but it appears that in many systems differential advection dominates, with the exception in the vicinity of channel bends where flow curvature dominates. Coriolis acceleration also plays a subtle but significant role in the dynamics of secondary flows, even in very narrow estuaries. 0

21 References Chant, R. J. (00). "Secondary flows in a region of flow curvature: relationship with tidal forcing and river discharge." J. Geophys. Res. (C Oceans) /001JC00108, 1 September. Chant, R. J. and R. E. Wilson (1997). "Secondary circulation in a highly stratified channel." Journal of Geophysical Research 10: Fischer, H. B. (197). "Mass transport mechanisms in partially stratified estuaries." Journal of Fluid Mechanics 53( ). Fong, D. A., et al. (In Press). "Turbulent stresses and secondary currents in a tidally-forced channel with significant curvature and asymmetric bed forms." Journal of Hydraulic Eng. Friedrichs, C. T. and D. G. Aubrey (1988). "Non-linear tidal distortion in shallow well-mixed estuaries: a synthesis." Estuarine,Coastal and Shelf Science 7: Fugate, D. C., et al. (007). "Lateral dynamics and associated transport of sediment in the upper reaches of a partially mixed estuary, Chesapeake Bay USA." Continental Shelf Research 7( ). Garrett, C., et al. (1993). "Boundary mixing and arrested Ekman layers: Rotating stratified flow near a sloping boundary." Annual Review of Fluid Mechanics 5: Geyer, W. R., et al. (008). Tidal and spring-neap variations in horizontal dispersion in a partially mixed estuary. Journal of Geophysical Research, 113. Geyer, W. R., et al. (000). "The dynamics of a partially mixed estuary." Journal of Physical Oceanography 30: Geyer, W. R., et al. (001). "Sediment transport and trapping in the Hudson River Estuary." Estuaries 4(5): Hansen, D. V. and M. Rattray (1966). "New dimensions in estuary classification." Limnology and Oceanography 11, Huijts, K. M. H., et al. (006). "Lateral Entrapement of sediment in tidal estuaries: An idealized model study." Journal of Geophysical Research 111, C1016, doi:10.109/006jc003615, 006. Lerczak, J. A. and W. R. Geyer (004). "Modeling the lateral circulation in straight, stratified estuaries." Journal of Physical Oceanography 34:

22 Lerczak, J. A., et al. (006). "Mechanisms driving the time-dependent salt flux in partially stratified estuary." Journal of Physical Oceanography 36(1): g MacCready, P. (004). "Toward a unified theory of tidally-averaged estuarine salinity structure." Estuaries 7(4): MacCready, P. and P. B. Rhines (1991). "Buoyant inhibition of Ekman transport on a slope and its effect on stratified spinp-up." Journal of Fluid Mechanics 3. Nunes, R. A. and J. H. Simpson (1985). "Axial Convergence in a well mixed estuary." Estuar. Coast. Mar. Sci. 0: Okubo, A. (1973). "Effects of shoreline irregularities on streamwise dispersion in estuarine and other embayments." Netherlands Journal of Sea Research 8(13-4). Peters, H. a. R. B. (001). "Microstructure observations of turbulent mixing in a partially mixed estuary, II: Salt flux and stress." J. Phys. Oceanogr. 31: Phillips, O. M., et al. (1986). "An experiment on boundary mixing: Mean circulation and transport rates." Journal of Fluid Mechanics 173. Pritchard, D. W. (1956). "The dynamic structure of a coastal plain estuary." Journal of Marine Research 17: Seim, H. E. and M. C. Gregg (1997). "The importance of aspiration and channel curvature in producing strong vertical mixing over a sill." Journal of Geophysical Research 10: Smith, R. (1977). "Long term dispersion of contaminants in small estuaries " Journal of Fluid Mechanics 8: Smith, R. (1978). "Longitudinal dispersion of a buoyant contaminant in a shallow channel." Journal of Fluid Mechanics 78: 677:688. Valle-Levinson, A. (008). "Density-Driven exchange flows in terms of the Kelvin andn Ekman Numbe." Journal of Geophysical Research 113: C04001 doi:10.109/007jc Valle-Levinson, A., K. Holderied, C. Li and R. J. Chant (007). "Subtidal flow structure at the turning region of a wide outflow plume." Journal of Geophysical Research. 11, C04004, doi:10.109/006jc Valle-Levinson, A., et al. (000). "Convergence of latreal flow along a coastal plain estuary." Journal of Geophysical Research 17:

23 Wilson, R. E. and A. Okubo (1978). "Longitudinal dispersion in a partially mixed estuary." Journal of Marine Research 36(3): Winant, C. D. (004). "Three-dimensional wind-driven flow inn and elongated, rotating basin." Journal of Physical Oceanography 34: Figure Legends. Figure 1) Left panels show channel configuration that can produce depth averaged cross-channel flows. Right panels depict the type of secondary flows that are discussed in this chapter. These flows have zero depth-averaged component and are characterized by closed cells of cross-channel re-circulation. Figure ) Tendency equation for secondary flows. In all figures the dashed lines depict the tendency for the flow field to modify salt field that are depicted by solid lines. a) Plan view depicting compaction/compression of isohalines by term C in equation 5.5. b) Generation of cross-channel density gradients by differential advection as described by term B in equation 5.5. c) Generation of cross-channel density gradient by term D in equation 5.5. d) Generation of cross-channel density gradients associated with diffusive boundary layer (term E in 5.5). Figure 3a) Salinity (dashed line) and sense of lateral flows (arrows) associated with differential advection during flood tide (right panel) and ebb tide (left panel) Figure 4) Upper Panel. Schematic showing the set-up of secondary flows associated with flow curvature. Flow entering the channel is vertically sheared and has a relative vorticity vector pointed to the south-east. Lines at channel bends depict sea-level, with solid lines elevated levels and dashed lines lower levels. L corresponds to the length-scale of the spin-down of secondary flows. Lower panel shows structure of secondary flows in and downstream of channel bend. The channel cross-section in lower panel is drawn asymmetrically to depict the tendency for secondary flows in channel bends to deepen the channel on the outside of the bend. Figure 5) Composite from Geyer 1993 showing secondary flows from shipboard surveys off Martha s Vineyard. In left panel filled arrows are near surface vectors and while arrows are near-bottom vector. Veering angle for transect B is over 30 degrees. Right panel shows stream-wise flows (contours) and secondary flows (arrows) for transect B. 3

24 Figure 6) Composite from Chant 00 showing response of secondary flow to variations in tidal range and stratification from a mooring in Newark Bay. Upper panel shows linear relationship between along channel shear and cross channel shear (i.e. secondary flows). Middle panel shows veering of current vector during low river discharge during ebb tide in a region of counter-clockwise flow curvature. Flow towards the bottom of the page is towards the outside of the bend, while flow towards the upper side of the page represents flow towards the inside of the bend. Data is binned as a function of tidal range. Bottom panel shows lack of veering during times of high river discharge indicating that secondary flows are shut down by stratification. Figure 7) Schematic from Chant and Wilson (1997) showing set-up of cross-channel baroclinicity, down stream adjustment and oscillatory rebound down stream. Figure 8) Cartoon from Chant and Wilson 1997 Figure 8) Results from Lerczak and Geyer (004) showing along channel velocity (contours in left panel), stratification (contours in right panels) and secondary flows (vectors in left panels). Panels ab shows spring tide conditions during ebb tide. Panels cd show springtide conditions during the flood. Panels ef show neap-tide conditions during ebb and panels gh shows neap tide conditions during the flood. Figure 9) Exchange flow from model run (dots) and from the Hansen and Rattray (1966) theory -1 (crosses). The solid line shows the A v relationship prediected by HR66, while the dashed line shows the best fit to the numerical results and emphasized the weakened sensitivity of the exchange flows to vertical mixing due to non-linear processes associated with secondary circulation. 4

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28 5.6 8

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