Friction dominated exchange in a Florida estuary

Estuarine, Coastal and Shelf Science 113 (2012) 248e258

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Friction dominated exchange in a Florida estuary Kimberly D. Arnott a, *, Arnoldo Valle-Levinson a, Mark Luther b a b

University of Florida, 365 Weil Hall, PO Box 116580, Gainesville, FL 32611, USA University of South Florida, 140 7th Avenue South, St. Petersburg, FL 33701, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 November 2011 Accepted 16 July 2012 Available online 21 August 2012

The typically observed gravitational circulation in estuaries with lateral variations in bathymetry consists of a combined distribution of vertically and horizontally sheared flows. The distribution features inflow at depth and outflow at the surface and along the sides. However, theoretical results of density-driven exchange flows dominated by frictional effects display a laterally sheared distribution with inflow occupying the deepest portion of the cross-section and outflow over the shoals. The main purpose of this investigation was to obtain observational evidence in support of theoretical results. A tidal cycle survey was conducted on February 24, 2009, to collect current velocity and hydrographic profile data along a cross-estuary transect. Observations from Hillsborough Bay were compared qualitatively to numerical model and analytical solution results. The observed residual exchange flow pattern compared favorably with the results from a numerical model and an analytical solution that used a condition controlled by friction. The relative importance of friction was explored at tidal and subtidal timescales. Intratidally, frictional influences were observed in the spatial distribution of tidal current amplitude and phase, as well as in potential energy anomaly (stratification) variations. Subtidally, frictional influences were observed through the spatial distributions of tidally averaged stratification, turbulent kinetic energy dissipation and eddy viscosity values. The main finding of this study was that relatively weak tidal currents (<0.3 m/s) can still produce dominant frictional influences in the dynamics. Results indicated that bathymetry shaped by rather wide shoals and relatively narrow channel mold the frictionally dominated, laterally sheared net flow. Ó 2012 Elsevier Ltd. All rights reserved.

Keywords: exchange flow dissipation spatial variability estuarine circulation stratification friction

1. Introduction The gravitational circulation of a coastal plain estuary is typically regarded as the result of a momentum balance between the seaward barotropic pressure gradient, the landward baroclinic pressure gradient, and friction (Pritchard, 1956). This balance, applied over flat bathymetry, assumes steady state and linear motion characterized by inflow of relatively denser ocean water at depth and outflow of lighter water near the surface. In contrast, over lateral variations in bathymetry the exchange flow pattern consists of inflow in the thalweg and outflow over the shallow flanks (Wong, 1994). This laterally sheared exchange flow is consistent with a highly frictional regime. Typically, the exchange flow observed in estuaries exhibits a combined vertically and laterally sheared distribution with inflow at depth and outflow at

* Corresponding author. E-mail addresses: kimdarnott@ufl.edu (K.D. Arnott), arnoldo@ufl.edu (A. ValleLevinson), [email protected] (M. Luther). 0272-7714/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ecss.2012.07.034

the surface and on the shallow sides (Friedrichs and Hamrick, 1996; Valle-Levinson, 2008). Estuarine circulation can arise from longitudinal density gradients, nonlinear tidal advection (tidal stress), tidal asymmetries in mixing, and river discharge (e.g. Burchard and Hetland, 2010; Cheng et al., 2010). When the motion is nonlinear, the momentum budget is dominated by the pressure gradient, friction, and advective acceleration. As advection becomes prominent, the momentum budget yields a two-layer, vertically sheared residual exchange flow (Lerczak and Geyer, 2004). Asymmetries in mixing associated with periodic density stratification can produce a vertically sheared residual exchange flow (Stacey et al., 2008; Burchard et al., 2011; Cheng et al., 2011). On the other hand, tidal rectification processes can produce a residual exchange flow that features laterally sheared velocities (Li and O’Donnell, 2005; Huijts et al., 2009; Waterhouse and Valle-Levinson, 2010). Presently, there is practically no observational evidence that supports Wong’s (1994) density-driven, exclusively laterally sheared exchange flow (inflow in the entire water column) because of the requirement of high frictional conditions. Such conditions may imply large tidal forcing that can also influence exchange flows

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via advective processes and flood-to-ebb asymmetries in mixing. However, numerical model results (Meyers et al., 2007) of the micro-tidal estuary, Hillsborough Bay, a branch of Tampa Bay, showed a horizontally sheared exchange flow pattern in which inflow occupies the entire water column in the channel and outflow develops over the shoals. These numerical model results also showed that the net exchange flow decreased under reduced river discharge conditions. Such findings of laterally sheared flow, presumably driven by density gradients established by river discharge, motivated an expedition with the purpose of obtaining observational evidence supporting the exchange flow pattern. It was hypothesized that observations of residual exchange flow would be driven by density gradients under marked frictional influences. In order to examine possible frictional influences on exchange flows, tidal and tidally averaged fields were determined. The distribution of tidal current amplitude and phase was calculated at an estuary cross-section with an analytical model. Additionally, estimates of intratidal variations of potential energy anomaly illustrated frictional impacts on stratification. On a tidally averaged time scale, frictional influences were explored with the distributions of density anomaly, buoyancy frequency, and potential energy anomaly. Frictional influences were also studied with the distribution of tidally averaged turbulent kinetic energy dissipation and vertical eddy viscosity values at an estuarine crosssection. Time- and depth-averaged values of advection, Coriolis, and friction terms from the momentum balance were also compared. Finally, two analytical solutions were applied over the observed bathymetry to compare observed to theoretical residual flows under varying frictional conditions and to describe the effects of using constant eddy viscosities in the analytical models. 1.1. Study area Tampa Bay is located on the Gulf of Mexico side of the Florida peninsula in the United States (Fig. 1). Four subsections comprise the entire bay: Old Tampa Bay, Hillsborough Bay, Middle Tampa Bay, and Lower Tampa Bay (Fig. 1b). Extending over a surface area of 1000 km2 and an area weighted mean depth of < 4 m, Tampa Bay is one of the largest shipping ports in Florida (Morisson et al., 2006). A system of shipping channels with depths of w15 m and widths of

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w300 m has been dredged from the entrance to the bay toward all four subsections, as shown in Fig. 1b with bathymetry and topography data obtained from the United States Geological Survey. These channels are the main conduit of ocean water into the bay and allow development of water column stratification. The water column in Tampa Bay is considered partially stratified to well mixed, with maximum tidally averaged vertical density differences of 0.6 kg/m3 in this study. The tidal regime of the central Gulf of Mexico is mixed with semidiurnal dominance, featuring a form factor (F ¼ (K1 þ O1)/(M2 þ S2)) of 1.18 and an average tidal range of 0.67 m (Brooks and Doyle, 1998). The maximum variance of flows in Tampa Bay is produced by the tides, accounting for 95% of current energy (Weisberg and Zheng, 2006). Wind and freshwater input also contribute to the variability (Brooks and Doyle, 1998). Tidal current amplitudes show maximum values of 1.2e 1.8 m/s at the bay entrance and decay to values of w0.20 m/s in Hillsborough Bay (Brooks and Doyle, 1998). Tampa Bay has a subtropical climate, characterized by warm, humid summers and mild winters (Goodwin, 1987). The maximum seasonal rainfall (50 cm) occurs during the summer and the minimum occurs in the fall (15 cm) with the total annual rainfall averaging 140 cm/yr (Schmidt and Luther, 2002). Precipitation accounts for 43% of the freshwater input (Meyers et al., 2007), 41% is contributed by river discharge, and the remaining input is provided by domestic and industrial point sources, groundwater, and springs (Zarbock et al., 1995). Precipitation and groundwater discharge combine for annual average freshwater inputs of w63 m3/s (Schmidt and Luther, 2002). The wind regime in the area has annual mean speeds of 3.8 m/s that originate from the northeast. Seasonal wind patterns vary from 6 to 7 m/s (northeasterly) during winter, 4.5e6 m/s (easterly) during spring, 4.5e5.5 m/s (southeasterly) during summer, and 4.5e6.5 m/s (easterly) during fall months (Virmani, 2005). Variations are observed due to tropical storms and cold fronts, which periodically pass every six to ten days during winter, when winds can exceed 10 m/s (Brooks and Doyle, 1998). A diurnal sea breeze often develops throughout the year. This study was conducted in Hillsborough Bay, the northeastern section of Tampa Bay, with dimensions of 7 km wide and 15 km long (Brooks and Doyle, 1998). The bathymetry at the entrance to

Fig. 1. (a) Coastline of Florida and (b) bathymetry of Tampa Bay, red line signifies study transect line.

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Hillsborough Bay is mostly shallow (<6 m). Looking into the bay, a cross-section reveals two shoals separated by a 14 m deep channel that is shifted toward the north (left) shoal, which is narrower than the south shoal. The paper is organized as follows. Section 2 describes data collection and processing techniques, followed by Results. Section 3.1 displays tidal variations of stratification indicated by the potential energy anomaly. Section 3.2 describes the variability of tidal flow through depth-averaged flows, flood-averaged and ebbaveraged velocity profiles, and observed and theoretical results for tidal current amplitude and phase. Together, these results highlight frictional influences on the system at tidal timescales. Tidally averaged fields of density anomaly, buoyancy frequency and potential energy anomaly are depicted in Section 3.3 to show the effects of stratification on subtidal timescales. The observed residual exchange flow is described in Section 3.4 and tidally averaged TKE dissipation values, as well as eddy viscosity estimates, are presented in Section 3.5 to explore frictional influences. Theoretical solutions for mean flow are compared to observations in Section 3.6. Lastly, the results are discussed and conclusions are proposed in Sections 4 and 5. 2. Methodology With the purpose of determining the residual exchange flow structure at the Hillsborough Bay entrance, a hydrographic

(temperature and salinity) and current velocity data collection expedition was conducted on February 24, 2009 over one semidiurnal tidal period. An Acoustic Doppler Current Profiler (ADCP) was used to collect velocity data, and a Self Contained Autonomous Microstructure Profiler (SCAMP) for hydrography. The transect, shown as a red line (in the web version) in Fig. 1, included five hydrographic stations, four located over depths <6 m (the shoals) and one in the channel (w14 m). Measurements were collected for approximately 11.5 h, which allowed a total of 12 transect repetitions with 6 of those repetitions including hydrographic profiles. The wind remained below 5 m/s from the northeast and northwest during the sampling period. A 1200 kHz ADCP was mounted on a 1.2 m-long catamaran and towed off the starboard side of a 280 boat, at typical speeds of 1.5e 2 m/s. Velocity measurements were recorded at 2 Hz, with a bin size of 0.5 m that ranged from 1.7 m to 14.7 m. Each ensemble analyzed consisted of 40 profiles averaged over 20 s, giving a spatial resolution of 30e40 m. These raw data were first corrected by calibrating the ADCP compass taking into account the ship’s velocity (Joyce, 1989) obtained with a Garmin GPS Map 182C. The transect origin was defined to separate the continuous data set into transect repetitions, after which data were interpolated onto a regular grid. At the end, the data set consisted of a regular matrix of 90 columns and 28 rows with a horizontal resolution of 50 m and a vertical resolution of 0.5 m, repeated 12 times.

Fig. 2. (a) Potential energy anomaly F (J/m3), (b) depth-averaged along-estuary tidal flow velocity (cm/s), (c) flood and ebb-averaged channel and shoal profiles (m/s).

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To resolve hydrography and turbulent kinetic energy (TKE) dissipation, the SCAMP was deployed in descending mode at five stations along the transect. The instrument descended at speeds near 10 cm/s, sampling at rates of 100 Hz. The SCAMP processing software (produced by Precision Measurement Engineering, Inc.) was used to extract profiles of temperature and electrical conductivity, from which parameters such as salinity, density, buoyancy frequency and TKE dissipation were derived. The latter variable was used to quantify frictional effects throughout the water column and the transect. To estimate the TKE dissipation, the SCAMP processing software takes each temperature profile, calculates the temperature vertical gradient, divides the profile of temperature gradient into stationary segments, and calculates the TKE dissipation, 3 , for each segment by fitting the temperature vertical gradient to the Batchelor spectrum. For a detailed discussion of the Batchelor spectrum fitting technique, refer to the Appendix. Profiles of density were arranged by transect and interpolated using Delaunay Triangulation onto a uniform grid. The TKE dissipation and buoyancy frequency profiles were vertically interpolated to 0.25 m resolution. The profiles were organized by transect repetition and then were tidally averaged. Lastly, the tidally averaged profiles were interpolated onto a uniform grid. The same technique was used to calculate vertical eddy viscosity values. Density, TKE dissipation, buoyancy frequency and vertical eddy viscosity distributions are presented across the transect. 3. Results The results of this investigation are presented as tidal and tidally averaged distributions of stratification along the sampling transect. Tidal distributions of stratification are described in conjunction with tidal flows. Subsequently, the observed residual exchange flow structure is explained in terms of the influence by turbulent kinetic energy dissipation and vertical eddy viscosity distributions along the transect. These distributions are used to justify the application of analytical solutions that help explain the observed residual flow structure and to propose alternative reasons for the flows observed. 3.1. Tidal variation of stratification Early in the measurement period, the largest density anomaly (st) range was observed in the first two repetitions of the hydrographic transect (not shown). The largest top to bottom density difference in the w14 m deep channel was w1 kg/m3 (from 22.1 to 23.2 kg/m3), indicating a weakly stratified water column in general. More uniform conditions were observed over the shoal regions and during other tidal phases. Tidal variations of stratification are represented by one value: the potential energy anomaly, F, which is a proxy for vertical stratification and the deficit of potential energy in the water column (displacement of the water column’s center of mass from mid-depth). The potential energy anomaly is represented by (Simpson et al., 1990)

f¼

g h

Z

0 h

ðrm  rÞz dz:

(1)

In Eq. (1), g is gravity acceleration (9.81 m/s2), h is the depth of the water column, and r is the water’s density. One value of F was calculated from the depth mean density, rm, for each density profile across the sampling transect. This resulted in a sequence of F values for each transect repetition. Values of F obtained during every transect repetition allowed construction of Hovmöller diagrams that described spatial variations of stratification as a function of time throughout the tidal

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cycle. Values of F across the transect showed the greatest stratification in the channel (located at 1 km on the horizontal axis) during the beginning and the end of the sampling period, representing the end of ebb and beginning of a subsequent ebb tide, respectively (Fig. 2a). The potential energy anomaly varied from 5 J/m3 in the channel to nearly 0 J/m3 over the shoals, displaying the variation from stratified to mixed water column conditions throughout the cross-section. These variations in F followed the changes in tidal flow as indicated next. 3.2. Tidal flow variability Another Hovmöller diagram (distance vs. time) showed the progression of the depth-averaged tidal flow across the transect (Fig. 2b). Positive tidal flow values (flood) dominated the period of observation. Rather than being influenced by meteorological forcing, this bias to flood flow was a result of the mixed nature of the tide in the area. The ebb period sampled was the weaker of that particular day, which resulted in stronger floods than ebbs. Ebb periods were sampled only at the beginning of the survey (end of ebb and maximum stratification) and toward the end. The largest depth-averaged flood velocities (w0.30 m/s) were observed over the narrow channel, while the largest depth-averaged ebb velocities (w0.05 m/s) occurred over the north shoal. Consistent with tidal straining of the density field (Simpson et al., 1990), end of ebb coincided with the period of maximum stratification. Profiles of along-channel tidal velocities, averaged throughout flood and ebb phases, were compared in the channel and over shoal locations (Fig. 2c). Flood flows in the channel featured a parabolic shape of all positive values with minima at the surface and nearbottom. In contrast, the mean ebb profile in the channel displayed two layers, with ebb velocities near the surface and flood velocities beneath. The flow profile over the shoals showed similarities from mean flood to mean ebb, with magnitudes decaying with depth. On the other hand, asymmetries between flood and ebb profiles suggested dense water intrusion at depths > 5 m, which enhanced the velocities at those depths during flood. During ebb, the intrusion overcame tidal velocities at those depths, which resulted in a two-layer flow. Tidal variability was further explored to ascertain the strength of semidiurnal tidal currents and their influence on frictional effects in estuaries. The along-estuary semidiurnal tidal current amplitude contours showed isopleths that resembled the bathymetry (Fig. 3a). This feature was a clear indication of the effects of bottom friction on tidal currents, causing the flow to decrease toward the bottom (e.g. Valle-Levinson and Lwiza, 1995). The largest amplitudes were observed at the surface in the channel with values of w30 cm/s, in comparison to w10 cm/s values at the surface of the right shoal. Relatively lower absolute values of phase in the transect represent areas where tidal currents initially reversed with the transition between tides. The along-estuary tidal current phase distribution showed that tidal currents led by w70e80 near the bottom of the shoals and right slope of the channel (looking into the estuary) relative the surface of the channel (Fig. 3b). Momentum was overcome earlier in these areas, allowing for the flow to reverse earlier than at the surface (e.g. Valle-Levinson and Lwiza, 1995). Results of an analytical tidal model (Friedrichs and Hamrick, 1996; Huijts et al., 2006) were compared to observations to verify the effects of friction on tidal dynamics. The momentum balance of the analytical model is given by:

vut vh v2 ut ¼ g t þ Az 2 vt vx vz

(2)

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Fig. 3. Observed along-estuary (a) tidal current amplitude (cm/s) and (b) tidal current phase (deg) and Huijts et al.’s (2006) analytical model results for along-estuary. (c) Tidal current amplitude (cm/s) and (d) tidal current phase (degrees).

fut ¼ g

vht v2 vt þ Az 2 vy vz

vvt vwt þ ¼ 0 vy vz

(3) (4)

Eqs. (2) and (3) present the along and across-estuary momentum balance, where ut is the tidal velocity, g is gravity, ht is sea-level, f is the Coriolis parameter, and Az is the vertical eddy viscosity. The left hand side (l.h.s.) term of Eq. (2) represents local accelerations, the first right hand side (r.h.s.) term represents pressure gradient forcing from water level slope, and the second term on the r.h.s. represents stress divergence or turbulent friction. The across-estuary momentum balance shown in Eq. (3) is composed of Coriolis forcing (l.h.s.), surface gradient forcing (first term r.h.s.), and turbulent friction (second term r.h.s.). Eq. (4) is continuity, which assumes along-channel uniformity. These equations represent a damped (tidal) wave. Inputs to the analytical model consisted of the observed bathymetry h(y), a sectionally averaged tidal flow amplitude Ut of 10 cm/s, and an eddy viscosity value Az of 0.0025 m2/s. Values for Ut and Az are derived from observations (e.g. Fig. 3a and Section 3.5, respectively for Ut and Az). Results for tidal current amplitude (Fig. 3c) ranged from 0 to w30 cm/s and displayed the largest tidal flows at the surface and in the channel. Amplitudes also decreased with depth and were generally comparable to observations. The tidal current phase (Fig. 3d) showed values between 80 and 60 near the bottom and over the far left and right shoals. Phase values of w130 were located from 2 to 6 m depths in the channel. This distribution indicated that the largest phase lags were observed in the channel, where the tidal currents were greatest and therefore took longer to change between tidal stages. Analytical model results for tidal current phase overall compared favorably with observations, although the analytical solution yielded slight

differences in the surface waters of the channel. The channel area of large phase values was much narrower in the observed phase distribution than in the analytical solution. In order to determine the sensitivity of the analytical model results to the prescribed value of Az, the eddy viscosity value was increased and decreased relative to that used in Fig. 3 (2.5  103 m2/s). A near-order-of-magnitude increase in Az to 1  102 m2/s accentuated lateral gradients while reducing vertical gradients in flow and phase (Fig. 4a, b). Increased friction caused the strongest flows and largest phase lags to be concentrated in the channel. Conversely, a near-order-of-magnitude decrease in the prescribed Az to 1  104 m2/s caused more laterally uniform but more vertically variable distributions of current phase and amplitude throughout the transect (Fig. 4c, d) than with Az of O(103 m2/ s). These model results indicate then that enhanced frictional effects will reduce vertical gradients and augment lateral gradients. Overall similarities between observed and predicted tidal current phase and amplitude distributions strongly suggested that the along-channel momentum consisted of local accelerations caused by pressure gradient and damped by friction. Slight inconsistencies between the analytical model and the observations could be attributed mainly to three causes: a) influences from advection that were neglected in the momentum balance; b) effects from Coriolis acceleration that were neglected in the along-estuary dynamics of the analytical model; and c) prescription of a constant vertical eddy viscosity value. This last cause of the discrepancy between tidal model and observations became evident from the eddy viscosity distributions derived from SCAMP measurements, which showed marked spatial variability in the transect (Section 3.5). Enhanced friction (higher eddy viscosities) over the right shoal, relative to the channel, should account for the lower observed tidal current amplitude values than those predicted by the model. Furthermore, temporal (tidal) variability of eddy

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Fig. 4. Huijts et al.’s (2006) analytical solution results with Az ¼ 0.01 m2/s (a) amplitude (cm/s) and (b) phase (degrees); analytical solution results with Az ¼ 0.0001 m2/s (c) amplitude (cm/s) and (d) phase (degrees).

viscosities, as they covary with tidal vertical shears, will generate higher order terms (higher than tidal flows) that will influence residual flows through ebb-to-flood asymmetric mixing (e.g., Cheng et al., 2011). Analytical model results then underscored the limitations of using a constant eddy viscosity. However, the main features of the model and observations were represented reasonably well and were similar to those observed in the Chesapeake Bay (ValleLevinson and Lwiza, 1995) and obtained with a similar analytical solution (Winant, 2007). Such distributions of tidal current amplitude and phase highlighted the importance of friction to tidal dynamics, which also translated in the relevance of friction to the residual, or tidally averaged, exchange flow. 3.3. Tidally averaged density fields Tidally averaged contours of density anomaly and buoyancy frequency were compared to tidally averaged values of potential energy anomaly (Fig. 5). The buoyancy frequency, N, is a measure of the frequency that a particle of water will oscillate about an equilibrium depth (Apel, 1987) and is given by ðg=rvr=vzÞ1=2 with units of s1. Spatial variations in density anomaly st identified regions of weakly stratified and mixed conditions. Transect contours of tidally averaged density anomaly showed weak stratification in the channel, varying only by 0.5 kg/m3 (Fig. 5a). The right (south) shoal depicted nearly mixed conditions with a minor density variation of < 0.2 kg/m3. The spatial distribution of tidally averaged N2 showed the lowest values near the bottom of the left (north) shoal and at depth in the channel, while the largest values were at the surface (Fig. 5b). The largest values of N2 indicated greatest

stratification in the channel, while the smallest values were located over the left (North) shoal. Tidally averaged values of F also showed the greatest stratification in the channel (Fig. 5c). The salient point of these three fields (st, N2, and F) was that density stratification developed only in the channel, albeit with low values. These values of net stratification were linked to the structure of the net exchange flow. 3.4. Tidally averaged exchange flow Consistent with numerical model results in the study area (Meyers et al., 2007), the spatial variation of the observed alongestuary residual exchange flow showed volume inflow in the channel and outflow of less-dense water over the shoals (Fig. 6). The inflow strength increased with depth in the channel, portraying nearly zero values at the surface and values up to 25 cm/s near the bottom. Outflow appeared over the shoals and was weaker than inflow. The isotachs over the shoals roughly followed the bathymetry, indicating frictional influences from bottom drag. Results from Stacey et al. (2008), Cheng et al. (2010), and Burchard et al. (2011) displayed a vertically sheared residual exchange flow dominated by tidal asymmetries in mixing. Alternatively, observations by Waterhouse and Valle-Levinson (2010) and analytical results by Huijts et al. (2009) showed horizontally sheared exchange flow within the channel, i.e., reversal of flow within the channel, driven by tidal rectification. Both Meyers et al. (2007) numerical results and the observed residual exchange flow in Hillsborough Bay also depicted a horizontally sheared exchange structure. In contrast to being laterally sheared with reversal of flow occurring inside the channel, reversals of net flow in Hillsborough Bay occurred at the transitions from shoal to channel. This

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Fig. 5. Tidally averaged (a) density anomaly sT (kg/m3), (b) squared buoyancy frequency log10[N2 (s1)] and (c) potential energy anomaly F (J/m3).

observed horizontal exchange flow pattern was consistent with analytical solutions for density-driven exchange flow under high frictional conditions (Huijts et al., 2006; Valle-Levinson, 2008). Such detail on where the net flow reverses direction with respect to bathymetry indicated that the residual exchange flow in Hillsborough Bay was most likely density-driven. This density-driven residual exchange flow pattern occurred because the dense ocean water followed the path of least resistance, thereby intruding in the deep channel. The overall shape of this distribution resembled theoretical results that were obtained for a net zero volume transport condition. However, the observations suggested that net transport of water was directed into the estuary because there was more inflow than outflow. This is explained further in the Discussion.

the surface over the right shoal. The vertical structure of 3 showed increased values with depth across the transect because of bottom friction effects. Values of 3 were combined with buoyancy frequency values, N, to determine vertical eddy viscosity values, Az, following (Osborn, 1980):

3.5. TKE dissipation Additional evidence that frictional influences in the water column shaped the exchange flows observed was provided by the spatial distribution of TKE dissipation, 3 . The lateral distribution of tidally averaged 3 showed the largest values in the near-bottom region over the shoals, with values ranging from 106 m2/s3 (or Watts/kg) to 105.5 m2/s3 (Fig. 7a). The lowest values, ranging from 107.5 to 106.5 m2/s3, were located at depth in the channel and near

Fig. 6. Observed residual exchange flow (cm/s).

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exchange, leading to the disregard of advective terms. Scully et al. (2009) discussed that the advective terms are cancelled out by asymmetries in tidal mixing. Qualitatively, advective terms should cause vertically sheared exchange flows (Cheng and Valle-Levinson, 2009), or laterally sheared flows inside the channel (Huijts et al., 2011), which was not the case in Hillsborough Bay. Because of the likely counterbalance to advective accelerations by other dynamical agents, analytical solutions were applied under the assumption of linear dynamics (Huijts et al., 2006; ValleLevinson, 2008). This approach allowed a comparison of results obtained with simple dynamics to flows observed, and thus to draw inferences on nonlinear effects. Thus, neglecting advection terms, Eqs. (6) and (7) become

f v ¼ g

fu ¼ g Fig. 7. Tidally averaged (a) TKE dissipation (m2/s).

Az ¼ 0:2

3

2

3

vh g vr v2 u  z þ Az 2 vx ro vx vz

vh g vr v2 v  z þ Az 2 vy ro vy vz

(8)

(9)

(m /s ) and (b) vertical eddy viscosity Az

3

N2

(5)

Values of TKE dissipation and vertical eddy viscosity both provide assessments on frictional influences in the system. Tidally averaged values of vertical eddy viscosity obtained with Eq. (5) ranged from 104.5 m2/s to 102 m2/s and followed the same general trend as 3 (Fig. 7b). The largest vertical eddy viscosity values (w102 to 102.5 m2/s) were near the bottom of the shoals and at 2 m depth in the channel, while the smallest values (w104.5 m2/s) were observed at depth in the channel and upper portion of the water column above the right shoal. 3.6. Theoretical solution for mean flow The distributions of 3 and Az indicated that frictional effects must influence the dynamics despite the relatively moderate tidal currents in this part of the estuary. A theoretical solution that compares Earth’s rotation to friction in terms of the Ekman number (Huijts et al., 2006; Valle-Levinson, 2008) is used to assess frictional influences. For this solution, the momentum balance is given by

u

vu vu vu vh g vr v2 u  z þ Az 2 þv þw  f v ¼ g vx ro vx vx vy vz vz

(6)

u

vv vv vv vh g vr v2 v þ v þ w þ fu ¼ g  z þ Az 2 vx vy vz vy ro vy vz

(7)

Eqs. (6) and (7) represent the along- and across-channel momentum balances, where u is the mean along-channel velocity and h is the sea surface height. The terms on the left hand side of Eqs. (6) and (7) are the advective and Coriolis terms, which are balanced by the barotropic and baroclinic pressure gradient (first two terms on the right hand side) and stress divergence or friction (last term), which assumes constant Az. Even though the spatial distributions of tidally averaged Az show a complicated, non-constant structure, the assumption of constant Az allows a comparison of net flows observed with those derived from simplified dynamics. According to Geyer et al. (2000), the advective terms in Eqs. (6) and (7) are considered to be at least one order of magnitude smaller than the other terms. In addition, Huijts et al. (2006) found that advection has a negligible impact on density-driven residual

The solution to these equations, which also considers the width of the estuary as determined by the Kelvin number Ke (width of basin over internal Rossby radius) has been obtained for density-driven flows by Valle-Levinson (2008). A conceptually similar set of equations was also solved by Huijts et al. (2006, 2009). Both solutions require prescription of water level slopes or density gradients as inputs. Following Huijts et al. (2009) the along-estuary water level slope vh=vx is related to the along-estuary density gradient vr=vx, over uniform bathymetry, byvh=vx ¼ ð3=8Þðhmax =ro Þ ðvr=vxÞ. In turn, the horizontal density gradient (required by Huijts et al.’s model) and the lateral slope in water level (required by Valle-Levinson’ model) are related by

vr 240ro Ek vh ¼  hmax vx vy

(10)

where Ek is the Ekman number and equals Az/(fh2max) and hmax is the maximum depth of the cross-section. The solution of Huijts et al. (2006) was obtained by prescribing the bathymetry h(y), an Az of 0.0007 m2/s, and a horizontal density gradient of 1  104 kg/m4 (Fig. 8a). Prescribed inputs produced net flows of similar magnitude to those observed. The analytical solution of Valle-Levinson (2008) was also obtained for comparison over the same bathymetry h(y), with values of vh=vy of 1  107 and Ke ¼ 1, under varying values of vertical eddy viscosity, Az. Four values of Ek representing weak (0.001), moderate (0.07 and 0.14), and high frictional influences (0.3) were used to investigate variations in the distribution of the exchange flows (Fig. 8bee). Along-estuary density gradients associated with each Ek value, according to Eq. (10), were 2  106, 1  104, 2  104, and 5  104 kg/m4, respectively. The theoretical solution obtained with Huijts et al.’s (2006) approach produced a horizontally sheared exchange flow (Fig. 8a) that roughly resembled the observed flow (Fig. 5). It also was similar to Valle-Levinson’s (2008) approach, which basically differs in the expression for conservation of mass, under high Ek (Fig. 8e). Solutions with the smallest Ek showed a vertically sheared structure (Fig. 8b). The theoretical exchange flow most consistent with the observed exchange flow was obtained using Ek of 0.07 and 0.14 (Fig. 8c, d). Results with Ek of 0.14 resulted in the flow structure that was most consistent, overall throughout the transect, with the observed exchange flow. However, the solution with Ek of 0.07 featured isotachs in the channel region of the transect that greatly resembled those observed. The inconsistencies between the linear solutions of Fig. 8 and observations are most likely a result of both the analytical solution’s

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Fig. 8. (a) Huijts et al.’s (2006) analytical solution showing along-estuary residual exchange flow, Valle-Levinson’s (2008) analytical solution showing along-estuary residual exchange flow in Ekman and Kelvin numbers (b) Ek ¼ 0.001 (weak friction), (c) Ek ¼ 0.07 (moderate friction), (d) Ek ¼ 0.14 (above moderate friction), and (e) Ek ¼ 0.3 (high friction).

assumption of uniform eddy viscosity and the possible relevance of nonlinear advection. However, theoretical results suggest moderate frictional influences in the cross-section, yielding a horizontally sheared exchange flow pattern with velocity magnitudes that were comparable to those observed. Inflow occurred throughout the water column in the channel, which increased to 0.2 m/s at depth. Weaker outflow developed over the shoals which decreased with depth and revealed values of 0.05 m/s. 4. Discussion An investigation was undertaken to obtain observational evidence that supports a seldom observed density-driven horizontally sheared exchange flow, which was predicted by numerical

model results in Hillsborough Bay. As predicted by numerical model results, the observed exchange flow displayed a laterally sheared structure, with inflow of dense ocean water in the channel bounded by outflow of less-dense water over the shallow shoal regions (Fig. 6). The residual exchange flow was likely driven by density gradients. Inflow in the channel should be greatest at depth, because the denser ocean water follows the path of least resistance. The lessdense water outflowing over the shoals had the opposite flow distribution, which decreased with depth. This flow structure is typical of theoretical solutions dominated by friction (Wong, 1994). Once it was determined that the observed exchange flow was consistent with numerical model results (Meyers et al., 2007), it was essential to investigate the frictional conditions of the system. One way to quantify frictional influences was through spatial variations

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in stratification. The uniform st distribution over the right shoal was indicative of frictional influences mixing the water column. Top to bottom density differences of 22.6 kg/m3 to 23.2 kg/m3 developed in the channel, showing only weak stratification. Fig. 2b highlighted the channel-to-shoal variation in F of only 5 J/m3, which is rather small when compared to other areas (e.g. Rippeth et al., 2001). Another way of assessing the effect of friction was through tidal flow amplitude. Bathymetric influences had a clear effect on the observed tidal flow. The results of the tidal current amplitude distribution showed the largest flow velocities at the surface in the channel and decreased with depth because of bottom friction (Fig. 3a). The amplitude of tidal currents reached a maximum of w30 cm/s; a value that is rather weak in comparison to other estuaries. Even though tidal currents were relatively weak, the isotachs resembled the shape of the bottom throughout the water column, a consequence of bottom friction. Analytical results of tidal flows also confirmed that the dominant tidal momentum balance was local accelerations and surface gradients balanced by friction (Fig. 3c). More evidence of frictional effects in the cross-section was given by turbulent kinetic energy dissipation and vertical eddy viscosity distributions. The tidally averaged turbulent kinetic energy dissipation contours showed the greatest values near the bottom of the shoals and in the surface waters of the channel, adjacent to the left shoal. The range of magnitude of the tidally averaged TKE dissipation was 107.5 m2/s3 to 105.5 m2/s3. Peters (1997) found a TKE dissipation range of 107 m2/s3 to 104 m2/s3 in the Hudson River and considered the upper limit of values relatively high. The high TKE dissipation values in Hillsborough Bay were an order of magnitude lower than at the Hudson, but still influential. Having a similar distribution as TKE dissipation, the tidally averaged vertical eddy viscosity distribution showed the greatest values near the bottom of the shoals and in the surface waters of the channel. The small vertical eddy viscosity values at depth in the channel indicated that turbulence could have been suppressed by weak density stratification in this region and by the relatively larger depth. This was verified by the tidally averaged potential energy anomaly (Fig. 5c) that yielded larger values in the channel, where the water column was weakly stratified and turbulence dissipation was lowest toward the lower portion of the water column. Even though eddy viscosities were spatially variable and despite possible relevance of nonlinear advection, an analytical solution that assumes linear dynamics and constant eddy viscosity was applied to the transect. The apparently restrictive assumptions of negligible nonlinear advection and constant eddy viscosity were used to compare theoretical results to actual observations and explore the limitation of such assumptions. Another reason for applying this linear model is that its solution is readily available in two forms (Huijts et al., 2006; Valle-Levinson, 2008) that address essentially the same dynamics but with slightly different expressions for continuity. Both solutions (Fig. 8), which depict densitydriven flows, showed similar flow structures. Frictional influences were recognized by increasing the Ekman number, which caused the exchange flow distribution to transition from purely vertically sheared to purely horizontally sheared. The analytical solution results that were most consistent with the observed flows required two different Ekman numbers (0.07 and 0.14) and represented moderate to above moderate frictional conditions (Fig. 8c, d). The key feature of the flow distribution obtained with an Ekman number of 0.07, representing moderate friction, was the alignment and positioning of the isopleths in the channel. Yet, the magnitude of the modeled flow was weaker than that observed. On the other hand, the result obtained with an Ekman number 0.14 generated comparable flow magnitudes to those observed but yielded isopleth shapes that varied from the observed contours. The slight inconsistencies in the channel region could be attributed to

257

the uniform eddy viscosity and negligible advection assumptions. Nonetheless, the overall resemblance between observations and modeled density-driven flow may be related to the cancellation of two potentially large dynamic terms: nonlinear advection and tidal asymmetries in mixing (e.g. Scully et al., 2009; Huijts et al., 2011). Moreover, it has been shown that the effect of advective accelerations is to enhance vertically sheared exchange flows (Lerczak and Geyer, 2004; Cheng and Valle-Levinson, 2009) whereas observations showed laterally sheared flows. It still remains to be determined whether the mean advection term vvu=vy þ wvu=vz could affect the laterally sheared structure observed. Flows related to this dynamic term can be influential in systems where the mean flow is as large as the tidal flow (Huijts et al., 2011), as in Hillsborough Bay. Unavailability of w from observations hinders reliable estimates of that dynamic term in this study. The analytical solution results of Fig. 8 in conjunction with Fig. 7d suggested that frictional effects in this area were prominent. For this particular condition at Hillsborough Bay, independent of the width of the basin, the dynamic depth was the main cause of the flow distribution (Valle-Levinson, 2008). Therefore, bathymetric variations in the relatively shallow cross-section were responsible for the horizontally sheared exchange flow pattern. According to the theoretical solution, the relatively steep, deep, and narrow channel allowed denser ocean water to intrude along the full range of the water column. 5. Conclusions The exchange flow in Hillsborough Bay consisted of a horizontally sheared structure, with net volume inflow in the channel and outflow over the left shoal. The rarely observed, purely horizontally sheared, residual exchange flow compared favorably with both numerical and theoretical model results. Assessment of the observed hydrography, potential energy anomaly, TKE dissipation, and tidal flow amplitudes indicated that frictional effects were influential in the study area. The laterally sheared exchange flow structure was shaped by variations in bathymetry, i.e., relative shallowness of the shoals combined with the relative deep and narrow channel. Observations in Hillsborough Bay suggested that even in an estuary that features weak tidal currents, frictional influences can still impact the flow. Acknowledgments This project was funded with the support of NSF project OCE0825876. A. Waterhouse, C. Winant, J. Lee, and N. Basdurak are appreciated for their assistance in conducting the field work. The comments by H. de Swart and an anonymous reviewer are gratefully acknowledged. Appendix The SCAMP collects profiles of high resolution temperature gradients. The observed temperature gradient spectrum of each segment is fitted to the Batchelor spectrum (Luketina and Imberger, 2001), a theoretical temperature gradient wavenumber spectrum that describes the behavior of temperature in a turbulent flow (Batchelor, 1959). The Batchelor spectrum is derived assuming that turbulence in the observed fluid is isotropic (statistical properties are independent of axis rotation) and homogeneous (statistically stationary in time). Following Luketina and Imberger (2001), a temperature gradient spectrum in general includes five wavenumber partitions (from low to high wavenumber): fine structure, inertial waves, inertial convective subrange (energy cascade), Batchelor spectrum, and noise. Gibson and Schwartz (1963)

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developed the following one-dimensional Batchelor spectrum SB expression:

3 2 pffiffiffiffiffiffi Z N 2 x 2qcT 4 a2 a e 2 a SB ðkÞ ¼ e 2 dx5 2kB DT a

(1)

where pffiffiffiffiffixffi is distance, kB is the Batchelor wavenumber, a ¼ 2qðk=kB Þ is a dimensionless wavenumber, q (¼3.4) is a constant related to least rate of strain, DT is diffusivity of heat, and XT is the dissipation of temperature variance. The relationship between the Batchelor wavenumber kB and dissipation (Luketina and Imberger, 2001) is:

!1 kB ¼

4

3

(2)

vD2T

where n represents kinematic molecular viscosity of water (106 m2/s). The dissipation of temperature variance is estimated using Steinbuck et al.’s (2009) algorithm, an approach that modified Ruddick et al. (2000) Maximum Likelihood Estimation spectral fitting. The key feature of this modified approach is the inclusion of the noise spectrum in the integration of the temperature gradient, improving the values of TKE dissipation rate. The dissipation of temperature variance XT is obtained by combining the integrals of the best fit Batchelor spectrum at unresolved low wave numbers, the noise-removed observed temperature gradient spectrum, and the best fit Batchelor spectrum at unresolved high wave numbers (Steinbuck et al., 2009), respectively given by:

2 Z cT ¼ 6DT 6 4 Z þ

kL 0

Z SB ðkÞdk þ

kn kL

ðSobs ðkÞ  Sn ðkÞÞdk

3 N kB

7 SB ðkÞdk5

(3)

In Eq. (3) SB, represents the Batchelor spectrum, Sobs is the observed temperature gradient spectrum, Sn is the noise spectrum, and kL, kn, and kB are the lower, noise and Batchelor wave numbers, respectively. The algorithm fits the observed temperature gradient to the Batchelor spectrum using the dissipation of temperature variance (Eq. (3)) as a constraint to the fit (Steinbuck et al., 2009). The goodness of fit is evaluated using the Maximum Likelihood Estimation (MLE) (Ruddick et al., 2000). Finally, the algorithm obtains the Batchelor wavenumber from the best fit, from which the TKE dissipation is calculated following Eq. (2) (within a range of 9  1011 m2/s3 to 1.5  105 m2/s3). References Apel, J.R., 1987. Principles of ocean physics. International Geophysical Series 38, 166e167. Batchelor, G.K., 1959. Small-scale variation of convected quantities like temperature in turbulent fluid. Journal of Fluid Mechanics 5, 113e133. Brooks, G.R., Doyle, L.J., 1998. Recent sedimentary development of Tampa Bay: a microtidal estuary, incised into tertiary platform carbonates. Estuaries 21, 391e406. Burchard, H., Hetland, R.D., 2010. Quantifying the contributions of tidal straining and gravitiation circulation to residual circulation in periodically stratified tidal estuaries. Journal of Physical Oceanography 40, 1243e1262. Burchard, H., Hetland, R.D., Schulz, E., Schuttelaars, H.M., 2011. Drivers of residual estuarine circulation in tidally energetic estuaries: straight and irrotational channels with parabolic cross section. Journal of Physical Oceanography 41, 548e570. Cheng, P., Valle-Levinson, A., 2009. Influence of lateral advection on residual currents in microtidal estuaries. Journal of Physical Oceanography 39, 3177e3190.

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