## Abstract

Multiyear in situ Eulerian acoustic Doppler current profiler measurements were obtained at 5-, 10-, and 19-m depths off the Big Bend coast, and in 19 m off the Florida Peninsula to the south. Analysis on subinertial time scales, dominated by weatherband frequencies, led to the following conclusions. At the 19-m Big Bend site (K-Tower), consistent with coastally trapped wave (CTW) theory, the along-isobath flow is not proportional to the local along-isobath wind stress, but rather to the alongshore wind stress to the south along the west Florida shelf (WFS). At the southern 19-m site, consistent with previous work, the along-isobath flow is driven by , but is weakened by an alongshore pressure gradient brake. Via CTW dynamics this brake is due to the abrupt “end” of the WFS at the Florida Keys. By contrast, along-isobath flow at the shallow 5-m site is driven by the local wind in a constant stress turbulent frictional layer. Because of the freshwater flux near the coast, density usually increases seaward. This leads to a strong asymmetry in the cross-isobath frictional bottom boundary layer (BBL) flow when the subinertial along-isobath flow direction changes. In one case the BBL flow is shoreward and gravitationally stable while, in the other, the BBL flow is gravitationally unstable as less dense water is forced under more dense water. Seasonal changes in the seaward horizontal density gradient also shear the seasonal along-isobath flow via thermal wind dynamics.

## 1. Introduction

The wide west Florida shelf (WFS) is unusual in that it is cut off at its southern end by the Florida Keys, and in the north the orientation of the coast changes by 90° at the junction of the Florida Peninsula and the Florida Panhandle. While currents on the central and southern parts of the WFS have been measured for more than a decade and much has been learned [see the summary by Weisberg et al. (2005), (2009)], the only published in situ Eulerian records in the Florida Big Bend are the comparatively short records at K-Tower, an Air Force navigational tower in 19 m of water about 33 km south of the Big Bend coast (Fig. 1). Summer currents there (31 days, 15 August–15 September 1978) were reported at 4- and 8-m above the bottom by Marmorino (1983a); winter currents (1-m above the bottom in 1989/90 and 1990/91, and 12-m above the bottom in 1991/92) were reported by Weatherly and Thistle (1997). Both papers indicate that the substantial bend in the coastline makes the area of particular dynamical interest since the subinertial along-shelf wind stress is not in a simple local frictional balance with the subinertial along-shelf current, and no clear-cut relationships exist (Marmorino 1983a).

Utilizing new multiyear acoustic Doppler current profiler (ADCP) high-resolution current measurements, we investigate the dynamics of the subinertial flow at four locations near the Big Bend. Section 2 gives an overview of observations and data processing, including a brief description of the low-pass filter used to obtain the “subinertial” data to be analyzed. Most of the subinertial (in our data periodicity longer than a few days) energy is in the weatherband (periodicity from a few days to a few weeks), so our analysis of the subinertial data in sections 3–7 is essentially an analysis of the weatherband variability. Seasonal flows, typically with a variance of about 15% of the total subinertial variance, are treated separately in section 8. Coordinate systems for each site are analyzed in section 3. Vertical structure and flow strength for both along- and cross-isobath subinertial flow are described in section 4. The dynamics of the along-isobath flow at 19-m sites S and K-Tower and near-shore sites B and A are analyzed in sections 5 and 6, respectively. Section 7 contains a discussion of the subinertial across-isobath flow, section 8 the seasonal flow, and the concluding remarks are given in section 9.

## 2. Data

Observations used to investigate the dynamics of the Big Bend region's subinertial circulation include current, wind stress, sea level, and hydrographic data. These are described below.

### a. Currents

As part of a state-funded north Florida red tide monitoring program, current measurements using a bottom-mounted, upward-looking Teledyne RD Instruments (RDI) Workhorse 300-kHz ADCP were begun on 19 January 2007 near K-Tower, an Air Force navigation tower in 19 m of water about 33 km south of the north Florida Big Bend coast (see map in Fig. 1, location as latitude and longitude coordinates in Table 1, and timeline of measurements in Fig. 2). Current velocity measurements are ongoing and from 3- to 15-m above the bottom with 1-m vertical resolution.

The red tide monitoring program ended in 2009, but in 2008 funding from the Northern Gulf of Mexico Institute (NGI) became available so that not only were the K-Tower measurements continued, but additional measurements were taken at two sites A and B inshore of K-Tower along a line approximately perpendicular to the coast (Fig. 1). At both site A (5-m depth, 6 km from the Big Bend coast, 11 March 2008–present) and site B (10-m depth, 15 km from the Big Bend coast, 19 June 2008–16 June 2009 and 6 July 2010–present) bottom-mounted, upward-looking Nortek 1-MHz acoustic wave and current (AWAC) profilers were deployed. The current measurements were made with a 0.5-m vertical resolution at depths from 1- to 3-m above the bottom at site A and from 1- to 7-m above the bottom at site B.

Finally, under funding from the red tide monitoring project, another bottom-mounted, upward-looking AWAC profiler was deployed at site S, about 45 km south of K-Tower and 70 km from the Big Bend coast in a water depth of 19 m, the same depth as at K-Tower. The first deployment (23 April–17 November 2009) measured currents from 1- to 15-m the bottom with 1-m vertical resolution, while the second deployment (17 November 2009–9 July 2010) had a 0.5-m vertical resolution.

Separate velocity measurements in the bottom log-layer were made by two Falmouth Scientific, Inc., two-dimensional acoustic current meters (FSI 2D-ACM) 25- and 85-cm above the bottom at K-Tower for several time intervals in 2007/08 and at site S for both deployment periods. However, at site S the FSI instruments failed on the second deployment, so no log-layer results are available then.

### b. Wind

Meteorological measurements are available at K-Tower and two nearby National Data Buoy Center (NDBC) (http://www.ndbc.noaa.gov) buoys: 42036 northwest of Tampa, Florida, and 42039 southeast of Pensacola, Florida (Fig. 1).

### c. Hydrography

Since 14 November 2006, approximately monthly (see the timeline in Fig. 2) hydrographic measurements have been made at five locations along a section from the Florida State University Coastal Marine Laboratory (FSUCML) to K-Tower. These measurements were typically taken at 0.5-m resolution and to 1 m from the bottom and 1 m from the surface. Also, hourly temperature and conductivity are available at K-Tower at 3- and 9-m above bottom since 19 December 2007.

### d. Coastal sea level and atmospheric pressure

National Ocean Service/Center for Operational Oceanographic Products and Services (NOS/CO-OPS) (http://tidesandcurrents.noaa.gov/) provides 6-min coastal sea level and atmospheric pressure measurements at various locations along the WFS. Near the current mooring locations, they are at Apalachicola on the Big Bend coast (station 728690) and at Cedar Key on the Florida Peninsula coast (station 8727520). Similar data were also available from the closest station to the current sites, the NDBC station Shell Point, Florida (SHPF1), owned and maintained by the University of South Florida.

### e. Data processing

All current velocity time series were checked for gaps and bad data, and long-period tidal variability (lunar fortnightly and lunar monthly) was removed via a least squares fit. To isolate the subinertial variability, the time series were low-pass filtered using a cosine–Lanczos filter (Emery and Thomson 2001) that passes 50% power at frequency 2*π*/40 h and 10% power at 2*π*/30 h. The time series filtered in this way are here called subinertial or low frequency interchangeably. Since near-surface contaminated (side lobe) current data were removed and not available to us, for depth-averaged calculations, which require an integral to the surface, we approximated the near-surface flow using the topmost good current value. All other analyses were based on a nonextrapolated record.

The sea level time series were adjusted for atmospheric pressure and filtered in the same way as those for the currents. The wind stress components were calculated from hourly wind speed and wind direction time series following Large and Pond (1982) and then low-pass filtered in the same manner as the currents. Since the wind stress components at the offshore NDBC buoys have high correlation and nearly unit regression coefficients (on average, *r* = 0.92, *r*_{95%} = 0.06, and the regression coefficient is 0.97), we use the NDBC buoy 42039 measurements to represent the regional large-scale winds. The near-shore winds at K-Tower are, in general, weaker than the offshore winds, so we use the K-Tower wind record to illustrate the local wind effects near the Big Bend.

Whenever mentioned, correlations at the 95% confidence level were calculated following Ebisuzaki (1997), and all the presented correlations are significant at the 95% confidence level. With regard to regression coefficients, when two variables *X* and *Y* in a linear regression both contain “noise,” the regression of *Y* on *X* is not the reciprocal of the regression of *X* on *Y*. When the relative size of that noise is not known, the appropriate unbiased regression coefficient is the ratio of their standard deviations (Clarke and Van Gorder 2013). Since the measured physical quantities here all are contaminated by noise, often of uncertain amplitude, we will use this regression coefficient.

## 3. Coordinate system for the subinertial flow

The shelf bathymetry in the region is complex, and the isobaths consequently vary irregularly on short spatial scales (see, e.g., Fig. 3). Theoretically, in the absence of friction, by the conservation of potential vorticity in shallow water, flow should approximately follow the isobaths at subinertial frequencies and at “large” spatial scales. But does that mean that the low-frequency flow should follow the highly irregular isobaths in Fig. 3 or some larger-scale topographic trend? At what scales, dynamically, should we expect the flow to follow the isobaths?

For the flow to approximately follow the isobaths, the relative vorticity *ζ* should be small compared to the local Coriolis parameter *f*, for then, by the conservation of potential vorticity , the low-frequency flow will follow *f*/*h* and, hence, locally the depth *h*. For a flow following the isobaths, the relative vorticity depends on horizontal velocity shear and curvature vorticity , where is the depth-averaged along-isobath flow and *R* is the local radius of curvature of the isobath. Since the velocity shear is small compared to *f*, the flow will tend to follow the isobaths provided is small, that is, for isobaths with radius of curvature *R* ≫ . Mean flows are weak in the Big Bend region, and the subinertial flows have an amplitude of about 10 cm s^{−1}. Since in the Big Bend *f* ≈ 7 × 10^{−5} s^{−1}, *R* must have a scale much greater than km for low-frequency flows to remain approximately on the isobaths.

The above theoretical ideas were checked by first calculating the principal axes of the depth-averaged subinertial flow at the four measurement sites A, B, K-Tower, and S (Table 1). At each of the four measurement sites (see Fig. 4) the *y* axis is defined to lie along the principal axis such that its eastward component is positive. Consequently, the *x* axis in this right-handed system points offshore.

To see whether the *y* axis as defined above is really approximately “along isobath” at the appropriate topographic scales, we analyzed the 3-arcsec (~90-m resolution) bathymetric data provided by the National Geophysical Data Center (http://www.ngdc.noaa.gov/mgg/coastal/crm.html). At each of the four measurement sites the topographic data in a latitude–longitude square box, centered on the particular site, was approximated by a least squares fit to the linearly sloped topography:

where *x _{*}* (

*y*) are eastward (northward) coordinates relative to the box center. The resulting constant topographic gradient

_{*}*α*

**e**

_{1}+

*β*

**e**

_{2}(

**e**

_{1}= eastward unit vector,

**e**

_{2}= northward unit vector) is perpendicular to the unit vector (

*β*

**e**

_{1}–

*α*

**e**

_{2})/(

*α*

^{2}+

*β*

^{2})

^{1/2}in the along-isobath direction. Squares with sides of lengths

*L*ranging from 1 to 35 km were used to estimate “along isobath ” directions at various scales.

Three chords of length *R*_{curv} can approximately describe the perimeter of a semicircle of radius *R*_{curv} and length *πR*_{curv} ≈ 3*R*_{curv}, but fewer than three are inadequate. This indicates that the straight isobaths in the squares with sides of length *L* above can approximately describe curves with radius of curvature *L*, but no smaller, that is, *R*_{curv} ≥ *L*. Figure 5a shows the along-isobath direction as a function of *L* at site B. For *L* from 1 to 10 km this direction, measured as degrees north of due east, changes considerably owing to the irregular nature of the topography. In this range of *L*, the along-isobath direction at each *L* is indicative of isobaths with radius of curvature approximately equal to that *L*. For *L* from about 10 to 35 km the along-isobath direction does not change, suggesting that the isobaths in this direction have a radius of curvature of at least 35 km.

From the small relative vorticity constraint discussed earlier, we expect the depth-averaged flow to follow the isobath for radius of curvature *R*_{curv} ≫ 1.4 km. The result in Fig. 5a, therefore, suggests that the angular difference *δθ* between the along-isobath direction and the major axis of the low-frequency flow (see Fig. 3) should be small for *L* ≫ 1.4 km.

Figure 5b shows plots of *δθ* as a function of *L* at each of the sites, negative *δθ* corresponding to the current major axis being rotated to the left of the isobath direction. Plots at A, B, and K-Tower all show small slightly positive for *L* between 4 and 8 km, qualitatively consistent with the expected small *δθ* for *L* ≫ 1.4 km. For larger *L*, *δθ* remains small at K-Tower and site B, but at site A *δθ* increases to about 40°. This may be due to larger-scale topography not being indicative of more local variations that are large enough scale to cause along-isobath flow. More likely, at least at times of strong wind forcing, is that frictional effects are very strong in the very shallow (5-m depth) water at site A and play a fundamental role in the dynamics (see section 6). When frictional effects are dominant we do not expect the low-frequency flow to follow the isobaths. Unlike the results at the other three sites, at site S *δθ* is comparatively large in magnitude until box sizes increase to *L ≈* 30 km. This may be due to the almost constant depth topography near site S that makes the estimation of accurate topographic gradients difficult.

## 4. Spatial structure and strength of the subinertial flow

The low-frequency wind-driven along-isobath flow on this wide shelf is expected to be barotropic (Clarke and Brink 1985; Mitchum and Clarke 1986b). The vertical profiles of the first mode empirical orthogonal function (EOF) for the along-isobath flow at K-Tower in all four seasons are shown in Fig. 6b. The first mode EOF explains about 90% of the variance of the along-isobath flow in each case. The profiles are approximately independent of depth except that the strength of the flow decreases toward the bottom over a depth of about 8–9 m or so. Such a decrease because of the bottom friction over a vertical scale of about this size is expected based on previous estimates of the vertical turbulent frictional scale in this region (Marmorino 1983b).

The winter profile also shows a slight decrease of the flow near the surface, consistent with the vertical scale of the surface turbulent frictional layer. This decrease can be associated with the local surface Ekman flow in the direction opposite to the main barotropic along-isobath flow. This effect (see Fig. 7) is most noticeable in the stormy El Niño winter 2009/10. In that winter, the much stronger local cross-isobath wind stress drove an along-isobath transport that weakened the flow amplitude in the surface frictional layer. Figure 7 shows that the removal of the Ekman part of the flow near the surface makes the along-isobath flow approximately independent of depth above the frictional bottom layer. The along-isobath flow in all four seasons is of similar strength, about 8 cm s^{−1} (Fig. 6b). Note that for all the EOFs the principal components have been normalized so that their variance is 0.5, enabling the EOF vertical profiles to be representative of the flow amplitudes.

The first EOF of the cross-isobath profiles at K-Tower in Fig. 6a explains most of the variance in each season, but the variance explained is smaller than for the along-isobath case. This may be due to a greater percentage of noise expected in the smaller cross-isobath signal. It may also have to do with the nonlinearity of the cross-isobath flow response, which will be discussed later in section 7a. The vertical structure of the across-isobath flow is such that the depth-averaged across-isobath current is much smaller than the depth-averaged along-isobath current, in keeping with the depth-averaged flow being approximately along the isobath (see section 3).

Figure 8b shows first mode EOF along-isobath vertical structures for all four current measurement sites during all the times that the measurements were taken at each site. Results for the along-isobath flow in the same 19 m of water depth at sites S and K-Tower are similar, but at the shallower sites A and B the velocity is more sheared. Such shear is expected in shallower water where the flow is almost entirely in the turbulent bottom boundary layer. This is discussed further in section 6.

The cross-isobath first mode EOF vertical structures at all four sites are shown in Fig. 8a. Analogously to Fig. 6a, the vertical structure of the across-isobath flow at each site is such that the depth-averaged across-isobath current is much smaller than the depth-averaged along-isobath current. The variance explained by the across-isobath vertical EOF structures is only about 60% and, as noted above, may not be an adequate representation of the across-isobath flow.

At each mooring site the along-isobath vertical profile of the first mode EOF does not change sign with depth and explains about 90% of the variance. While the cross-isobath flow *u* at some sites is not of opposite sign near the surface and bottom, possibly because of the strongly varying topography alongshore or to the direction of our axes being slightly in error, it is still true that is comparatively small to . Based on these results, in the next section we will use the depth-averaged along-isobath subinertial flow as a representative of the along-isobath subinertial flow.

## 5. Along-isobath flow dynamics in 19-m water at sites K-Tower and S

According to coastally trapped wave (CTW) theory (see, e.g., Gill and Schumann 1974; Gill and Clarke 1974; Clarke 1977), seaward of an “inner shelf” region where surface and bottom Ekman layers overlap (Mitchum and Clarke 1986a), subinertial along-isobath flow is found by integrating the alongshore component of the wind stress along characteristics corresponding to CTW modes (Mitchum and Clarke 1986b). Since the dominant first CTW mode propagates from the Florida Keys to the Big Bend in a day or so, a time much shorter than subinertial periodicity, to a first approximation for the subinertial time series of interest, the wave propagation speed is effectively infinite. This implies that the dominant first mode characteristic integral reduces to a simple integral of the alongshore component of the wind stress from the Florida Keys to the Big Bend. Since the low-frequency wind stress is of large spatial scale, dynamically we would expect the along-isobath wind-driven flow to be more highly correlated with wind stress in the along-shelf direction of the WFS (see the arrow in Fig. 4), rather than the local along-isobath direction. Thus, even though the along-isobath direction at K-Tower is close to the direction of (see Fig. 4), we expect that , the depth-averaged along-isobath flow at K-Tower, should be more highly correlated with than . Figures 9a and 9b illustrate that this, indeed, is the case for the winter 2009/10, and Table 2 shows that is more highly correlated with than with for all seasons of the year. Correlations of with are highest in winter and lowest in summer, the season when the subinertial is weakest. The correlation is the highest when lags by about 16 h, a result expected from previous analyses (Mitchum and Clarke 1986a; Liu and Weisberg 2005a) and discussed later below.

A referee wondered whether the depth-averaged along-isobath flow at K-Tower could be driven by the local cross-isobath wind stress there since this is in the same direction as . But, while subinertial cross-isobath wind stress can affect coastal sea level and cross-isobath flow (see, e.g., Tilburg 2003), its effect on depth-averaged along-isobath flow is small compared to the same strength along-isobath wind [e.g., compare the term proportional to *υ* in Figs. 7a,b and 8a,b in Tilburg (2003)]. Furthermore, as we have seen earlier in the second paragraph of section 4, the flow generated by the cross-shelf forcing is opposite to the observed along-isobath flow.

Site S is approximately on the same 19-m isobath as K-Tower and, compared to the length of the Florida Peninsula, is only a short distance from it (see Fig. 1). Consequently the flow at K-Tower and site S should be forced by essentially the same integral of the wind stress along the coast from the Florida Keys. According to CTW theory, the depth-averaged alongshore flow at site S () and K-Tower () should therefore be very similar and they are (Fig. 9c). The maximum winter 2009/10 correlation coefficient *r*_{max} = 0.95 is when lags by 1 h, but the correlation is nearly the same (*r* = 0.94) when there is no lag. Figure 9c also shows that the flow amplitudes are essentially the same, at least for winter 2009/10. Correlation and regression results for the site S record indicate (Table 3) that and closely approximate each other for the other seasons of the year, the agreement again being lower in the summer when is weakest.

Earlier in this section, we found that lags by about 16 h. Since ≈ , we can establish the reason for this lag if we can establish why lags by about 16 h. In the remainder of this section, we investigate why such a lag might arise.

Consider the linearized depth-averaged along-isobath momentum equation

where the overbar denotes depth average, *t* is time, *ρ*_{*} the mean water density, and are, respectively, the wind stress and the bottom stress in the along-isobath direction. Based on a theoretical analysis, Mitchum and Clarke (1986a) suggested that (2) can be approximated, at 19-m depth, by

In deriving (3) was assumed to be negligible and the pressure barotropic, so that with as the sea level corrected for the effect of atmospheric pressure. In addition, the along-isobath bottom stress was represented as

In 19-m depth

with *δ* being the *e*-folding decay scale for the surface and bottom Ekman layers.

At the southern “end” of the WFS the normal and tangential coordinate system that we have chosen is severely contorted, isobaths heading westward along the northern side of the Keys and then back eastward again on the southern side of the Keys to join with isobaths on the eastern coast of the United States. Mitchum and Clarke (1986b) recognized that it would be difficult for energy to propagate from the southern side of the Keys, around their end, and enter the WFS. This is especially so given the existence of the very narrow shelf and strong northward Florida Current south of the Keys. Consistent with this difficulty, sea level at Key West is much smaller than coastal sea levels further to the north [see, e.g., Fig. 2 of Marmorino (1982)], and Mitchum and Clarke (1986b) found that the essential large-scale dynamics of the low-frequency variability on the WFS could be described by putting *η* = 0 at the WFS southern end. Thus the WFS can be regarded, approximately, as a broad shelf with northward/northwestward isobaths “ending” at the Keys where *η* = 0.

Note that, by geostrophy, *η* = 0 across the southern WFS “boundary” marked by the Keys corresponds, as far as the WFS is concerned, to = 0 there. Consequently, we expect from (3) that near the Keys

This is consistent with Mitchum and Clarke (1986b), their (4.2) and (4.5a,b), for small *y*. Farther to the north, CTW theory (Mitchum and Clarke 1986b) shows that these terms no longer exactly balance, and the imbalance results in a nonzero according to (3). The alongshore sea level gradient, resulting from the “cut off” shelf at the Florida Keys, thus acts as a “brake” on the wind stress forcing and weakens . Cancellation of by *gη _{y}* is consistent with the results of Liu and Weisberg (2005a) and the observed and calculated increasing coastal sea level amplitudes northward of the Florida Keys by CTW theory (Mitchum and Clarke 1986b).

The solution of (3) is

so the initial condition is unimportant after times more than about twice the frictional spin down time *h*/*R*. Note that (7) predicts that lags the forcing *F*, which, as we have seen, is proportional to a reduced . For low enough frequency forcing the lag is approximately *h*/*R*. Since lags by about 16 h (see Table 2), we expect *h*/*R* ~16 h.

Mitchum and Clarke (1986b) tested (3) using a short current meter record off Cedar Key (see Fig. 1) in 21 m of water with measured wind stress and estimated from coastal sea levels at Cedar Key and St. Petersburg. Good agreement was obtained with *δ* = 8.5 m and, hence, by (5) with Using a much more extensive dataset off Sarasota (see Fig. 1), Liu and Weisberg (2005a) verified that the along-shelf momentum balance (2) could be approximated by (3) with The balance (3) has also been checked on other continental shelves in similar water depths with similar *R* [e.g., off California by Lentz and Winant (1986) and Hickey et al. (2003)].

We tested whether the above physics also applies at site S. As was done for the WFS by Mitchum and Clarke (1986a) and Liu and Weisberg (2005a), we used coastal sea level to estimate *η _{y}* at the shallow water site, in the present case writing

with Δ*η* the sea level at Cedar Key minus that at Shell Point and Δ*y* = 170 km, the alongshore distance between these stations (see Fig. 1). Liu and Weisberg accounted for the possible difference between *η _{y}* at the coast and

*η*in 15 m of water by multiplying Δ

_{y}*η*/Δ

*y*by a factor

*μ*= 0.7, and here we do the same.

The bottom friction parameter *R* needed for the calculation at site S can be obtained from (4) by regressing onto at site S, the bottom stress being calculated from the two log-layer bottom boundary layer FSI current measurements (see section 2a) at heights *ζ*_{1} = 25- and *ζ*_{2} = 85-cm above the bottom. This calculation proceeds in standard fashion as follows. In the logarithmic layer the direction of the stress is the same as the direction of the flow, and the speed of the flow *U* is a function of the height *ζ* above the bottom according to

where , *κ* is von Kármán's constant, and *ζ*_{0} is the roughness length. From (9) it follows that

and, hence, that |** τ**| can be estimated as

Since |** τ**| and the direction of

**are known for the subinertial time series, can be calculated. Based on (4), regression of on at site S gave for the period 23 April–17 November 2009. A similar value () was obtained at K-Tower based on several measurement periods in 2007/08. From (5) and , for ,**

*τ**δ*≈ 17 m, about double a previous estimate of 8.5 m by Marmorino (1983b).

We compared the theoretical estimate for from (7) with the observed using the log-layer *R* estimate and *μη _{y}*(

*t*) based on (8) with

*μ*= 0.7 as explained above. The theoretical and observed time series were well correlated (

*r*

_{max }

*=*0.6 when the observed flow lags the model by 6 h) and had similar amplitude (regression coefficient = 0.89). The correlation was improved and the lag reduced if we chose smaller

*μ*= 0.3–0.5 for

*R*= 4.5–6 (× 10

^{−4}m s

^{−1}) (Fig. 10a). This value of

*R*corresponds to

*δ*= 12.8–17 m. Figure 10b suggests that, even as far distant from the Keys as site S, the along-isobath pressure gradient term is acting as an

*O*(1) brake on the flow. In reporting this result we recognize that the estimate of

*η*at S in 19 m is crude since we use a finite difference of coastal sea level, one of the sea level stations (Shell Point) being in the topographically complex region near the Big Bend (Fig. 1). Nevertheless, the

_{y}*O*(1) contribution made by the alongshore pressure gradient estimate still permits good agreement between the measured and modeled

*υ*, suggesting that the pressure gradient estimate has some validity.

Another source of model error is the neglect of the terms in (2) to get (3). The term may not be small. However, inclusion of this term into the forcing in (3) did not improve the modeled result, probably due to the difficulty in estimating when the instruments do not measure a part of the water column.

## 6. Along-isobath inner-shelf flow at sites A and B

Since an average estimated *e*-folding scale for the Ekman depth is ~8.5 m or more in the region [see section 5 and Marmorino (1983b)], it is likely that at site A (water depth 5 m) and site B (water depth 10 m) the surface and bottom Ekman layers will overlap. Equation (2) suggests that, as the coast is approached and the water depth , the surface and the bottom stress must balance; if they did not, the stress divergence force would approach infinity, and the flow would become very large. The idea that the turbulent stress becomes constant and equal to the wind stress as is consistent with the overlapping Ekman layer shallow-water theory of Mitchum and Clarke (1986a). As pointed out by Mitchum and Clarke, this result means that the currents in very shallow water are driven by the local wind stress. This is quite different from the currents at K-Tower and site S, which are driven remotely by an integration of the alongshore wind stress along the WFS from the Florida Keys.

There is some evidence that the currents at the 5-m site A are driven by local rather than remote forcing, especially in the winter when the forcing is strongest. At site A the major axis of the depth-averaged flow is at −30° (Fig. 4), that is, 30° south of east. The largest correlation of with different wind stress components in all directions occurs when the wind stress angle is about −20° (Table 4), which is much closer to the major axis of the flow than the −60° orientation of , appropriate for remotely forced flow.

Since the flow over the whole depth is in a constant stress layer as , in that limit for an eddy viscosity formulation for the stress we have

where *A* is the constant eddy viscosity. Integrating (12) from the bottom, where *υ* = 0, to some general *z* gives the linear profile

This is consistent with the Mitchum and Clarke (1986a) results when .

We tested (13), and Fig. 11 summarizes these results by showing the depth-averaged correlations between *υ* and the local along-isobath wind stress at site A, as well as the *e*-folding scale *δ* for all calendar months when the correlations are significant. The average δ = 8–9 m is comparable to the 8.5 m value found by Marmorino (1983b) using a different analysis and data. The Ekman *e*-folding scale *δ* is smaller in the summer, consistent with higher stratification and the weaker wind forcing resulting in lower turbulence then. The small depth theory (13) is invalid in the summer because it relies on the water depth to be less than about half of *δ,* and in the summer *h* ≈ *δ* (Fig. 11).

At site B the water depth is 10 m, comparable to, or larger than *δ*. Therefore, the constant stress dynamics of site A is not expected to be valid. More telling, however, is that the along-isobath component of the subinertial low-frequency depth-averaged flow responds best to the wind stress in the direction −60° (Table 2), or , rather than to the local along-isobath wind stress, indicating dynamics more like that at K-Tower than at site A.

## 7. Across-isobath flow

In section 7a, we consider the across-isobath flow in the frictional bottom boundary layer and in section 7b the frictional surface boundary layer.

### a. Asymmetric near-bottom across-isobath flow

As we saw earlier in section 5, the subinertial along-isobath flow at K-Tower is, to a first approximation, remotely driven by . When is positive (southward), *υ* is positive (eastward) at K-Tower, and, when is negative, *υ* is negative. This can be seen in Fig. 12b, showing the typical along-isobath flows at K-Tower, site B, and site A when (thick lines) and (thin lines). The K-Tower profiles correspond to the longest lines, the site B profiles to the next longest, and site A profiles to the shortest as the depth decreases toward the coast. As noted in sections 5 and 6, the K-Tower profile is quasi barotropic, but the site A profile is strongly sheared because of bottom friction. Since is approximately parallel to the *y* axis at site A (see Fig. 4), we expect that the locally driven flow at site A would also have the same sign as , as Fig. 12b shows. Site B, being dynamically between K-Tower and site A but closer to K-Tower (see the end of section 6), also should have the same sign response as , which is so (Fig. 12b). Figure 12b indicates that *υ* for positive is slightly greater than *υ* for negative, and this is at least partly explained by being slightly larger in magnitude on average when it is positive than when it is negative (about 14 cPa compared to about −12 cPa).

Because of bottom friction, in standard fashion we expect the along-isobath flows shown in Fig. 12b to induce cross-isobath bottom boundary layer flow. Specifically, since *υ* is quasigeostrophic above the bottom boundary layer (Gill and Schumann 1974; Mitchum and Clarke 1986b; Liu and Weisberg 2005a), and since *υ* and, consequently, the Coriolis force weaken much more than the cross-isobath pressure gradient force in the bottom boundary layer, cross-isobath flow is induced in the bottom boundary layer. This flow should be shoreward (negative) when *υ* is positive and seaward (positive) when *υ* is negative. Figure 12 shows that this is observed for the two deeper sites where there is a definite bottom boundary layer; near the bottom of the water column the thick and thin *υ* curves in the lower panel correspond to thick and thin *u* curves of opposite sign in the upper panel. However, note that, while the *υ* fields in the lower panel are comparable in size, the *u* fields near the bottom differ by a factor of 4 or more and have less vertical shear.

Why should there be such a strong asymmetry in the *u* field? A similar flow asymmetry has been previously observed and explained by Weisberg et al. (2001) (see also Liu and Weisberg 2005b, 2007; Liu et al. 2006) for the WFS off central Florida. The Weisberg et al. explanation involved vertical stratification and no vertical mixing, but, in the Big Bend over the relevant lower half of the water column, the vertical density gradient essentially vanishes, and it is the horizontal density gradient that generally seems more important to the dynamics. The strength of the horizontal density gradients can be seen from hydrographic sections showing temperature, salinity, and density profiles (Fig. 13). Especially in the lower part of the water column in the deeper water, these profiles are strongly dependent on *x* and nearly independent of *z*. Seasonal horizontal density gradients are very large, mainly attributed to the terrestrial freshwater flux of lower density water (see the discussion in section 8b).

But why does this horizontal density gradient cause an asymmetry in the cross-shelf near-bottom flow? Consider an idealized case where the mean density is lower near the coast and independent of depth. By the standard boundary layer dynamics discussed earlier, a quasigeostrophic along-isobath flow induces a bottom boundary layer transport to the left of the flow in the Northern Hemisphere (Fig. 14). Since there is no mean vertical stratification initially, when *υ* < 0 and the bottom boundary layer flow is driven seaward by the cross-isobath pressure gradient (Fig. 14a), this immediately leads to less dense water under more dense water, a gravitational instability, and thus mixing. This spreads the seaward flow over an increased depth and weakens it. By contrast, in Fig. 14b, when the along-isobath flow is reversed, the induced shoreward flow leads to more dense water underlying less dense water and gravitational stability. Consequently, for a given alongshore flow speed, the induced near-bottom *u* field should be much stronger when the bottom boundary layer flow is toward rather than away from the coast. This is consistent with Fig. 12. When is positive the flow is favorable for upwelling and southward over the Florida Peninsula, the along-isobath flow is eastward (positive) in the Big Bend (Fig. 12a), and the bottom boundary layer velocity is large and shoreward (Fig. 12b). This shoreward transport is much larger than the seaward transport when is negative and the Big Bend near-bottom along-isobath flow is westward. A similar mixing asymmetry occurs over the northern California shelf (Lentz and Trowbridge 1991), although in that case the mechanism is associated with vertical stratification and a sloping bottom.

Sometimes, as in the bottom three panels in Fig. 13, the “mean” density gradient can be reversed. In this case, probably because of the 2007/08 La Niña, terrestrial conditions were drier, the salinity seaward gradient was greatly reduced, and from 13-m depth to K-Tower the colder near-shore temperatures reversed the horizontal density gradient. By the bottom boundary layer asymmetry mechanism, when the “mean” horizontal density gradient is reversed, the vertical structure of the cross-isobath boundary layer flow should change. Evidence for this is provided by the extreme upwelling case at K-Tower. For the usual case when density increases from the coast, the shoreward bottom boundary layer transport is gravitationally stable and strong (see Fig. 12; repeated in Fig. 15 as the Δ*ρ* > 0 case). But when the “mean” horizontal density gradient reverses (Δ*ρ* < 0 in Fig. 15), shoreward bottom boundary layer flow is gravitationally unstable, and the resultant observed flow is weaker and is not concentrated in a bottom boundary layer.

### b. Near-surface across-isobath flow

We analyzed the near-surface across-isobath flow at K-Tower by considering the linearized along-isobath momentum equation averaged over a surface turbulent layer of depth *D*:

In (14) *τ ^{y}*(−

*D*) is the turbulent stress in the

*y*direction at depth

*D*, and, as noted earlier, is the wind stress in the

*y*direction. If were due to Ekman transport and if

*D*were large enough so that

*τ*(−

^{y}*D*) were negligible compared to , then (14) would simplify to

In that case would be entirely due to the local along-isobath wind stress.

We tested the simple local balance (15) using K-Tower wind stress and calculated from the current observations there. Figure 16 shows the winter 2009/10 results with *D* = 9 m, an appropriate depth given the structure of *u* in Fig. 6. The correlation between measured and predicted is *r* = 0.62, but Fig. 16 shows that the measured and predicted amplitudes do not agree, and, consistently, the regression coefficient is not unity.

Figure 11 shows that in winter *δ* ≈ 9–10 m, so *D* ≈ *δ*. When *D* = *δ*, standard Ekman layer theory shows that *τ ^{y}*(−

*D*) is not completely negligible in (14), and

Thus, the cross-isobath wind stress can contribute significantly to cross-isobath flow even when the surface flow is averaged over the *e*-folding depth *δ*. We modified the formula for in (15) using (16), but the results in Fig. 16 changed negligibly.

We also used the measured current data to calculate and, from (14), tested whether could be calculated from

Again, there was negligible improvement compared to the results shown in Fig. 16.

According to (14), the only other term contributing to is the along-isobath pressure gradient contribution, −*gη _{y}*/

*f*. We had no along-isobath pressure or sea level measurements to estimate this term, but we tried to estimate it using a finite difference between the coastal sea levels at Shell Point and Apalachicola (see Fig. 4) and an adjustable scale factor to take into account the change in pressure gradient from the coast. Again, there was a negligible improvement in the Fig. 16 results. But we cannot rule out

*−gη*/

_{y}*f*as a major contributor to since, in the topographically complex Big Bend region, it is likely that

*η*at K-Tower is not accurately estimated from a finite difference of the coastal Apalachicola and Shell Point sea levels.

_{y}Other possible contributors to the measured Eulerian flow are the nonlinear terms omitted from (14) and subinertial contributions driven by subinertial changes in the size and direction of propagating surface gravity waves (see, e.g., Xu and Bowen 1994). Based on the coherence and similar size of the velocities at the B, K, and S sites in Fig. 4, the nonlinear terms are negligible. Calculations based on the surface wave data from the ADCP instrument at K-Tower suggest that wave-generated subinertial near-surface flows are also much smaller than . So, it seems that the alongshore pressure gradient forcing, which we could not estimate accurately, is the likely reason for the discrepancy in Fig. 16.

## 8. Seasonal and mean flow

As mentioned in the introduction, subinertial variability is dominated by the weatherband frequencies (periodicity from a few days to a few weeks). Although they are weaker, the lower-frequency seasonal flows are in the same direction for longer time intervals and so can often transport particles farther than the stronger higher-frequency flows. Because of this, seasonal and mean flows are of importance to the life cycles of some of Florida's commercially valuable fisheries like the gag grouper (see, e.g., Fitzhugh et al. 2005), red tide blooms (Carlson and Clarke 2009), and the transport of pollutants like oil spills.

Notwithstanding the importance of seasonal and mean flows, there have been few estimates of such flows in the Big Bend region. DiMarco et al. (2005) and Ohlmann and Niiler (2005) analyzed Lagrangian surface drifter current estimates for the entire northern Gulf of Mexico continental shelves and suggested that the mean and surface flows are comparatively small in the Big Bend region. On the other hand, Carlson and Clarke (2009) calculated geostrophic seasonal surface flow using TOPEX/Poseidon/*Jason-1* along-track shelf-averaged seasonal surface flow near the Big Bend and estimated that seasonal shelf currents could change by ~20 cm s^{−1} over the calendar year. Numerical model results published for the northern Gulf of Mexico and WFS (Morey et al. 2005; Weisberg et al. 2005) also suggest strong seasonal variations in the flow, with the wind being the major driving force on the inner continental shelf. Long in situ current records in the central and southern WFS have enabled documentation and discussion of the seasonal flows there (Liu and Weisberg 2012), but these measurements are all south of the Big Bend.

Based on the above, there is a need to document and understand in situ estimates of the mean and seasonal flow in the Florida Big Bend. In this section, we provide such estimates from the multiyear in situ measurements now available to us.

### a. Depth-averaged along-isobath flow

Our longest records are in 19-m depth at K-Tower, and we focus most of our attention on the results there. As expected from the subinertial results in section 5, and consistent with CTW dynamics, the depth-averaged monthly along-isobath flow at K-Tower is more highly correlated with than with , and depth-averaged monthly along-isobath flow at site S is very similar to that at K-Tower (see Fig. 17).

Based on the above monthly data, the depth-averaged along-isobath flow at K-Tower, averaged for each of the 12 calendar months, is as shown in Fig. 18. The flow is eastward along the Big Bend coast during June, July, and from October to March, and westward during April, May, August, and September. The maximum eastward flow is 4 cm s^{−1} in November and maximum westward in May is 3 cm s^{−1}. Consistent with the correlations in Fig. 17a, the stronger eastward flows in January, February, and November correspond to being positive (southward) then, and the stronger westward flows in May and September are consistent with negative (northward) then. Note that these results are based on coincident wind stress and current records for the 40-month period upon which Fig. 18 was constructed.

### b. Vertical structure of the along-isobath monthly flow at K-Tower

Figure 19 illustrates the annual cycle of depth-averaged temperature, salinity, and density at sites A, B, and K-Tower (see Fig. 1). The annual cycle of these quantities at a given depth is similar to the depth-averaged values at these locations. The annual cycle in density at each location is mainly governed by temperature (note the similar annual cycles in Figs. 19a,c), but the large seaward density gradient *ρ _{x}* is mainly due to salinity (see Fig. 20). The seaward salinity gradient is largely because of the influence of the annually and interannually varying Apalachicola River, Florida's largest river [see Morey et al. (2009) for a discussion of the variability and influence of this river on the WFS]. The large seaward density gradient should lead to a vertical shear in the alongshore flow via the thermal wind relationship

Integrating (18) vertically from *z* = −11 to −4 m gives

The 4-m depth is the depth nearest the surface where we have reliable velocity data, and the 11-m depth is the deepest available to calculate *ρ _{x}* from hydrographic data at K-Tower and the station next closest to the coast.

Time series of Δ*υ* from the current meter data at K-Tower and the rhs of (19) from the hydrographic data are compared in Fig. 21. The two time series have a similar amplitude and are correlated at *r* = 0.67 (*r*_{95%} = 0.49), suggesting that *ρ _{x}* and the thermal wind balance explains much of the vertical shear. For most of the record Δ

*υ*< 0, corresponding to a velocity that becomes increasingly westward with height and consistent with seaward density increase. An exception is the late fall of 2007 and winter of 2008/09 when the density gradient reverses, and, consistently, Δ

*υ*> 0, as the velocity is increasingly eastward with height.

### c. Cross-isobath seasonal flow at K-Tower

Figure 22 shows the average calendar-month cross-isobath flow at K-Tower based on approximately 4 years of measurements. In the lower part of the water column the flow is toward the shore but nearer the surface, except for September–November, the flow is of opposite sign, particularly in June and July. The shoreward near-bottom flow is likely influenced by the BBL mechanism discussed in section 7a. By that mechanism, in the usual case when density increases seaward, the zero-mean energetic weatherband along-isobath flow will give rise to a stronger shoreward than seaward BBL flow, that is, a rectified near-bottom flow toward the coast.

## 9. Concluding remarks

The preceding analysis of the long time series of currents and hydrography in the Big Bend region has led to some key results summarized below.

The subinertial flow in the Big Bend region in 19 and 10 m of water is largely remotely driven by wind stress parallel to the axis of the Florida Peninsula to the south in accordance with coastally trapped wave dynamics and previous work. The Florida Keys “cut off” the WFS in the south and essentially exert a “brake” on the generation of the along-isobath flow, making it smaller compared to that which would have been generated by the same wind on a shelf without the Keys. The subinertial flow in 5-m water depth, especially in winter, is locally wind driven.

The huge Big Bend cross-shelf density gradients, due largely to the freshwater flux near the coast, have a profound influence on the cross-isobath flow. By standard bottom boundary layer dynamics, the quasigeostrophic along-isobath flow will induce bottom boundary layer transport toward and away from the coast depending on the along-isobath flow direction. However, because density usually increases strongly offshore in the Big Bend region, seaward bottom boundary layer flows result in less dense water being forced under more dense water. Gravitational instability and mixing disrupt the bottom boundary layer flow and severely reduce it. By contrast, shoreward bottom boundary layer flow results in denser water underlying less dense water, and there is no gravitational instability and reduction of the cross-isobath flow by the mixing mechanism.

When there is a winter drought, as often occurs during La Niña (see, e.g., Morey et al. 2009), the freshwater flow is decreased, the seaward salinity gradient decreases, and the cooler winter temperature in the shallower water nearer the coast can reverse the density gradient. In this rarer case it is the shoreward bottom boundary layer flow that moves lighter water under heavier water and, so, causes gravitational instability and mixing and decreased cross-shore flow.

Analysis showed that, if numerical models are to model the flow accurately, they must resolve the bottom topography on scales less than 5 km. Also, they must be able to model correctly the mixing cross-isobath mechanism summarized under (ii) and (iii) above.

The results under (ii) imply that the subinertial across-isobath flow is rectified. Specifically, if the subinertial along-isobath flow has a zero mean, the frictionally induced cross-isobath bottom boundary layer flows will not have a zero mean—the flow will be rectified so that there is a net onshore bottom boundary layer transport, about 10 m thick, toward the coast. This may at least partly explain the mean shoreward bottom boundary transport for all calendar months of the year.

Like the subinertial flow, seasonal along-isobath depth-averaged flow is remotely forced by . It varies from a maximum westward flow of 3 cm s

^{−1}in May to a maximum eastward flow of 4 cm s^{−1}in November, with an overall mean of about 1 cm s^{−1}eastward. Because of the cross-shelf density gradient, the monthly along-isobath flow is vertically sheared in accordance with the thermal wind relationship.

## Acknowledgments

We gratefully acknowledge grant support of the Florida Institute of Oceanography (Grant 4710-1101-05-A), the Florida Fish and Wildlife Research Institute (DO364787, DO591065 and DO1191327), the Northern Gulf Institute (Grant 000013122), BP/the Gulf of Mexico Research Initiative to the Deep-C Consortium, and the National Science Foundation (Grants OCE-0850749 and OCE-1155257). We thank Stephanie Fahrny White and Peter Lazarevich of the FSU Marine Field Group, and the FSU Marine Lab, especially Captain Rosanne Weglinski, for dedicated collection of the data.

## REFERENCES

*Circulation in the Gulf of Mexico: Observations and Models, Geophys. Monogr.,*Vol. 161, Amer. Geophys. Union, 101–110.

*Data Analysis Methods in Physical Oceanography*. 2nd Elsevier Science, 638 pp.

*Circulation of the Gulf of Mexico: Observations and Models, Geophys. Monogr.,*Vol. 161, Amer. Geophys. Union, 203–218.

**29,**

*Circulation of the Gulf of Mexico: Observations and Models, Geophys. Monogr.,*Vol. 161, Amer. Geophys. Union, 325–347.