## 1. Introduction

The Gulf of Mexico is a semienclosed basin connected to the Atlantic Ocean by the Straits of Florida and to the Caribbean Sea by the Yucatan Channel. A notable feature of the Gulf of Mexico tides as compared with other places around the world is a dominance of the diurnal over the semidiurnal constituents (Reid and Whitaker 1981). This is in contrast with the east coast of the United States, where the tides are predominantly semidiurnal (Zetler and Hansen 1971). Previous Gulf of Mexico tides studies concluded that the diurnal tide is primarily co-oscillating, entering the Gulf of Mexico through the Straits of Florida and exiting through the Yucatan Channel. (Grace 1932; Zetler and Hansen 1971).

Located on the eastern side of the Gulf of Mexico, the west Florida continental shelf (WFS) is one of North America's broadest continental shelves. Apparently different from most of the Gulf of Mexico basin, semidiurnal tides are appreciable here. Although this tidal structure was discussed in previous studies (Clarke 1991; Reid and Whitaker 1981), descriptions of WFS tides are limited to coastal sea level and a few regional current measurements. Koblinsky (1981) described the *M*_{2} tide over the southern portion of the WFS using velocity measurements from five moorings over a 2-yr duration. The surface wave crests of the *M*_{2} tide, considered to be stationary, linear, and barotropic, were found to parallel the isobaths. Internal tides were not found and the temperature distribution was only slightly distorted by the surface wave. Marmorino (1983) added analyses of velocity data from four moorings deployed in the Florida Big Bend region. Semidiurnal tidal constituent (*M*_{2} and *S*_{2}) energy was found to decrease in the offshore direction, whereas the diurnal tidal constituent (*K*_{1} and *O*_{1}) energy was more uniform across the shelf. As a result, the characterization of the tidal fluctuations changes from predominantly diurnal in deep water to semidiurnal near the coast. Such spatial inhomogeneity exists throughout the WFS. Weisberg et al. (1996) using velocity data from the 47-m isobath, showed that particle displacements in the semidiurnal and diurnal bands are typically about 1 km. However, inertial oscillations, during months when the water column is stratified, can cause an increase in the diurnal band particle excursions to about 5 km.

A deficiency of moored current meter data alone is that realistic arrays are insufficient to map the tides over the entire WFS. To do this we need a combination of in situ data and a high-resolution, area-encompassing model. The present paper combines recent in situ measurements with a three-dimensional, primitive equation numerical model to describe and map the four major tidal constituents (*M*_{2}, *S*_{2}, *K*_{1}, and *O*_{1}) on the shelf from the Florida Keys to the Mississippi River. We concentrate on the barotropic tides since the baroclinic tides are seasonally modulated and an order of magnitude smaller. Exceptions to the baroclinic motions being relatively small are seasonally modulated inertial oscillations whose frequency overlaps with the diurnal tides. Baroclinic tides and inertial oscillations, whose effects on the deterministic barotropic tides are negligible (Clarke 1991), will be the subject of a future correspondence.

We begin in section 2 with the observations of coastal sea level and shelfwide currents, analyzing the barotropic tides by standard methods. Section 3 then introduces the regional model and discusses how it is driven at the open boundary by deep-ocean barotropic tides. The modeled tides and their comparisons with the in situ data for the *M*_{2}, *S*_{2}, *K*_{1}, and *O*_{1} constituents are described in section 4. Based on these comparisons we provide maps of the principal constituent tidal ellipses over the entire domain using the model. We also use the model to estimate the Lagrangian transports induced by the barotropic tides and their attendant nonlinear interactions. Since the amplitude of the tides determines their contribution to turbulent mixing, this is a topic of discussion in section 5. The results are summarized in section 6.

## 2. Observations

The in situ observations are taken from nine tide gauges operated by NOAA National Ocean Service (NOS) and 12 current meter moorings deployed by the University of South Florida (Fig. 1). The nine tide gauges, referred to as South Pass, Waveland, Pensacola, Panama City, Apalachicola, Cedar Key, St. Petersburg, Naples, and Key West span the northeast Gulf of Mexico from the Mississippi River to the Florida Keys. The velocity data are from acoustic Doppler current profilers deployed across the WFS between the 300-m and 10-m isobaths.

### a. Tidal height

The sea level data are analyzed using the least squares error method of Foreman and Henry (1979). The analysis includes as many as 146 possible tidal constituents, 45 of these being astronomical in origin and the remaining 101 being shallow water constituents (e.g., Godin 1972) that arise by distortions of the astronomical tidal constituents due to the nonlinear effects of shoaling depth. Year-long time series of sea level are used for each of the nine sea level station analyses inclusive of the years 1996–99. Data gaps preclude using the same year for all stations. Table 1 lists the amplitudes and phases (relative to the Greenwich meridian) for the *M*_{2}, *S*_{2}, *K*_{1}, and *O*_{1} constituents, which in sum contain over 90% of the tidal variance. We note that the semidiurnal tides to the south of are larger than those west of Apalachicola Bay. In contrast with the diurnal tides that have similar amplitudes at all nine stations, the semidiurnal tides show large spatial variations. All of the four major constituents have amplitudes that peak near Cedar Key.

*ζ*

_{p}, is constructed by adding the four major tidal constituents. With

*ζ*

_{0}being the observed sea surface height time series after removing the mean, the root-mean-square tidal residual (i.e., the subtidal component),

*σ*

_{r}, and the normalized residual,

*σ*

_{r}/

*σ*

_{0}, where

*σ*

_{0}is the standard deviation of

*ζ*

_{0}, are calculated over the time series record length,

*T,*as

*σ*

_{r}/

*σ*

_{0}, are given as percentages in Table 2. We note that west of Apalachicola Bay, where the tides are generally small, the subtidal variability accounts for more than 70% of total variability. South of Apalachicola Bay, where the tides are larger, the tidal and the subtidal variabilities contribute about equally to the total sea level variability.

### b. Tidal currents

In parallel with sea level the tidal analyses for the moored velocity data are performed using the least squares error method of Foreman (1978). Since the 12 mooring locations were generally not codeployed, the analysis intervals (between 1996 and 1999) differ in record length and time origin. We generally used the entire record length available at each site, and these varied from around three months to one year.

Parameters describing the tidal hodographs are calculated at each depth bin (from 0.5 to 10 m depending on water depth) for each mooring. The amplitudes of the ellipse semimajor axis and the ellipse orientations (measured anticlockwise from east) are shown as a function of depth in Figs. 2 and 3 for the four major constituents: *M*_{2}, *S*_{2}, *O*_{1}, and *K*_{1}. For a purely barotropic tide the amplitude and orientation would be independent of depth and appear as vertical lines. Depth profile deviations from straight lines are related to bottom friction, baroclinicity, contamination by inertial motions, or combinations of these. As shown in Fig. 2 for relatively shallow depths and in Fig. 3 for relatively deeper depths, the semidiurnal tides indeed appear to be barotropic, which agrees with the finding of Koblinsky (1981) for the WFS *M*_{2} tide. In contrast, the diurnal tides, especially in deeper water, tend to show vertical structure. This is most likely due to contamination by inertial motions since in the central WFS the inertial period is about 25.90 h, which is close to the *O*_{1} and *K*_{1} periods of 25.82 and 23.93 h, respectively. WFS inertial oscillations are modulated in time along with the seasonally varying stratification (Weisberg et al. 1996), and they are evident in velocity component plots as baroclinic modes consistent with the orientation reversals seen for the deeper records in Fig. 3.

Overall, the observed tidal current amplitudes on the WFS are weak (on the order of a few centimeters per second). Of the principal constituents, the *M*_{2} current is the strongest; its amplitude tends to increase shoreward, and its ellipse orientation tends to align perpendicular to the isobaths. While weaker, the diurnal constituent amplitudes are more uniformly distributed across the shelf, and these findings agree with those of Marmorino (1983) for the Florida Big Bend region.

### c. Phases of tidal height and tidal currents

One question of interest (particularly for recreational fishermen) is the relative phase between the tidal height at the coast and the tidal currents on the WFS. We address this question relative to the St. Petersburg tide gauge, the reference gauge for much of the WFS, in two different ways. Table 3a shows the phases (and corresponding time lags) between high tide at St. Petersburg and maximum tidal currents (directed toward the coast) at the 12 ADCP mooring sites for each of the major tidal constituents. For the semidiurnal tides, the times of maximum tidal current generally lead the high tide at St. Petersburg by 2–9 h. On the other hand, the maximum tidal currents of the diurnal tide, lag the high tide time at St. Petersburg by 10–20 h.

*η*and the major component of tidal current

*u*defined as

*R*(

*τ*) =

*E*[

*η*(

*t*)

*u*(

*t*+

*τ*)];

*σ*

_{η}and

*σ*

_{u}are the standard deviation for

*η*and

*u,*respectively; and

*τ*is the time lag. The time lags determined for the composite are similar to the time lags found for

*M*

_{2}constituent alone (Table 3a), reiterating the dominance of the

*M*

_{2}constituent on the WFS.

## 3. Hydrodynamic model

Our work is preceded by several numerical model studies of Gulf of Mexico tides. Reid and Whitaker (1981) applied a finite-difference version of the linearized Laplace tidal equations in a two-dimensional 15′ by 15′ (∼28 km) horizontal grid to portray the barotropic response of the Gulf of Mexico to tidal forcing. While describing the deep water tides well, their coarse grid size was insufficient to describe the tidal structures on the shelf. Westerink et al. (1993) applied a finite-element-based hydrodynamic model (ADCIRC-2DDI), also depth integrated and two-dimensional, to the western North Atlantic, the Gulf of Mexico, and the Caribbean Sea, to develop a tidal constituent database. Computations were presented for varying horizontal resolutions ranging from coarse (1.6° × 1.6°) to fine (6′ × 6′), and discussion was given to the importance of model resolution. Their model comparisons with sea level data show good agreements in the Atlantic Ocean, but persistent errors occurred in the Gulf of Mexico and the Caribbean Sea. These authors suggested that insufficient resolution over the continental shelf, particularly in the shallowest regions, might be responsible for the relatively poor numerical convergence in those regions. More recently, L. Kantha et al. (2002, personal communication; material available online at http://www.ssc.erc.msstate.edu/Tides2D/) developed a data-assimilative, barotropic tidal model for the Gulf of Mexico with similar grid resolution as Reid and Whitaker (1981). Improved results follow from the assimilation of selected coastal tide gauge data. Nevertheless, this model has insufficient resolution to describe the structure of the tidal variations on the WFS.

*z*is the conventional vertical coordinate (positive upward from zero at the mean water level),

*H*is the local water depth, and

*η*is the tidal variation about the mean water depth. Our model domain (Fig. 4) extends from the Mississippi River in the northwest to the Florida Keys in the southeast, with one open boundary that arcs between these two locations. Horizontal resolution varies from less than 2 km near the coast to about 6 km near the open boundary, and the minimum water depth is set at 2 m. This grid allows us to resolve the complicated features of the coastline and the isobath geometries in order to explore the associated structures of the WFS tides. We use 21 sigma layers in the vertical with higher resolution near the bottom to better resolve the frictional boundary dynamics. Bottom stress is obtained by a quadratic drag law in which a nondimensional drag coefficient is calculated on the basis of a specified (0.01 m) bottom roughness length. The model has a total of 121 × 81 × 21 grid points, and the time step for the external mode is 12 s.

Water density is assumed homogeneous in this three-dimensional barotropic tide study. The three-dimensional distribution of the vertical eddy viscosity is computed using the Mellor and Yamada (1982) level-2.5 turbulence closure scheme, and the horizontal eddy viscosity is calculated using the shear-dependent Smagorinsky formulation (Smagorinsky 1963) with a coefficient of 0.2. Dissipation by vertical friction is found to largely exceed that by horizontal friction.

The bathymetry adopted in the model begins with the World Ocean Elevation Data ETOPO5 5′ × 5′ global bathymetry dataset. Since this dataset is inaccurate in the southern portion of WFS we corrected it by incorporating data from NOAA navigation charts and National Geophysical Data Center (NGDC) bathymetry contours. Our modified bathymetry has been used in other WFS modeling studies with satisfactory results (e.g., He and Weisberg 2002; Weisberg et al. 2001; Yang et al. 1999).

Tidal forcing of the model is exclusively at the open boundary. There, the tidal elevations are specified by a linear interpolation of the output from the TOPEX/Poseidon data assimilated global tidal model of Tierney et al. (2000). This barotropic model, with 15′ resolution, provides eight tidal constituents (*M*_{2}, *S*_{2}, *N*_{2}, *K*_{2}, *K*_{1}, *O*_{1}, *P*_{1}, and *Q*_{1}) over a nearly global domain that extends from 80°S to 66°N. The model assimilates both coastal tide gauge data and ocean tides derived from four years of TOPEX/Poseidon satellite altimetry data. Since it uses procedures that tend to preserve the spatial tidal structures in shallow water, it is considered to be a suitable product for forcing higher-resolution regional tidal models. Figure 5 shows the distributions of tidal amplitudes and phases for *M*_{2}, *S*_{2}, *K*_{1}, and *O*_{1} constituents along the open boundary where we have a total of 121 model grids.

## 4. Modeled elevations and currents

### a. General features

The model is forced at the open boundary by specifying the composite *M*_{2}, *S*_{2}, *K*_{1}, and *O*_{1} constituent elevations there. Elevations, currents, and turbulence quantities are computed over the model interior by integrating forward in time beginning from a state of rest. An initial spinup period of five inertial cycles (about five days) is used to suppress transients. The subsequent 30 days are then analyzed using the same techniques as for the in situ data to retrieve individual constituent amplitude and phase distributions over the model domain. Table 1 provides amplitude and phase comparisons between the modeled and observed elevations of *M*_{2}, *S*_{2}, *K*_{1}, and *O*_{1} at the nine coastal tide gauge stations spanning the analysis domain. Amplitude differences are generally less than 2 cm; the singular exception being at Cedar Key where it is 2.5 cm for *M*_{2}. Phase differences are generally less than 10°; with Naples at 10.1° for *O*_{1} being the largest. Both amplitude and phase differences show equally likely plus or minus values demonstrating that the modeled tides are not biased. Root-mean-square (rms) values of the phase differences, when converted to time, amount to less than 10 min (20 min) for the semidiurnal (diurnal) species. Considering the model tide gauge sampling relative to the actual tide gauge positions and the nearshore masking (2 m being the shallowest model depth), these phase agreements are both good and as good as can be expected. Therefore, no model tuning was considered.

The alongcoast variations of the amplitudes and phases of the four major tidal constituents are graphically shown in Fig. 6, where circles denote the observations and crosses indicate the model results. Allowing for scale changes in the abscissa we see a relative spatial homogeneity in the diurnal species compared with a more spatial inhomogeneous distribution for the semidiurnal species. We also note several bull's-eyes in the model–observation comparison.

The agreements of Table 1 and Fig. 6 justify using the model to produce coamplitude and cophase charts for the *M*_{2}, *S*_{2}, *O*_{1}, and *K*_{1} constituents (Fig. 7). In general agreement with the findings of the previously cited Gulf of Mexico tide studies, these maps provide further details of the tidal structures on the WFS. The semidiurnal and diurnal species are distinctly different. The phase of the *M*_{2} tide advances toward the northwest. Apalachicola Bay separates the *M*_{2} tidal regime into two parts. To the west the *M*_{2} tide is weak. To the east it is strong. Both the Florida Big Bend and the Florida Bay regions show relative maxima with a relative minimum in between just to the south of Tampa Bay. The *S*_{2} constituent shows features similar to the *M*_{2} constituent, but with a much smaller amplitude. The phase patterns for the semidiurnal species appear to result by diffraction around the Florida Keys and onto Cape San Blas.

Amplification of the semidiurnal tides in the Florida Big Bend has been discussed by previous investigators. Reid and Whitaker (1981) considered a resonance deriving from the counterclockwise speed of propagation matching the group speed for a gravest mode edge wave (*gS*/*f* where *S* is the bottom slope—Kajiura 1958). With these speeds in close alignment and with the *M*_{2} tide wrapping around the basin approximately once per tidal cycle, near resonance may be achieved (these authors estimated an average *M*_{2} propagation speed of 98 m s^{−1} and an edge wave speed of 120 m s^{−1}). While this type of resonance is possible, we note that bottom topography and edge wave speeds vary around the shelf as does the *M*_{2} tide amplitude and phase gradient.

*ru*/

*h*), the linear Laplace tidal equations are

*α*and width

*a,*that is,

*h*(

*x*) =

*αx,*0 ≤

*x*≤

*a,*they expressed sea level in terms of a zero order Bessel function as

*ω*

^{2}−

*f*

^{2}> 0) on a wide shelf (

*a*/

*α*is large),

*μa*→ 1 and

*η*(

*a*)/

*η*(0) → 0, implying amplification of semidiurnal tides across a wide shelf. To account for alongshore variations in the tides, Lentz et al. (2001) developed an analytical, flat-bottom, two-dimensional model with varying shelf width. These analytical models are both consistent with the observed semidiurnal tide behaviors on the WFS. They suggest that the coamplitude distributions may be the result of local geometry, as contrasted with the previous basin-scale resonance arguments.

Unlike the semidiurnal tides, the diurnal tides show little coamplitude variations across the analysis domain. Both amplitudes and phases for the *O*_{1} and *K*_{1} constituents are nearly spatially uniform. This is consistent with the Zetler and Hansen (1971) finding that the diurnal tide is essentially in phase throughout the Gulf of Mexico. It is also consistent with Eq. (4) (Clarke 1991), since for *ω* ≈ *f* on the WFS, *μa* → 0, so the diurnal tides do not amplify toward the coast. The *O*_{1} and *K*_{1} constituents also have comparable amplitudes with mean ratio of 0.95. Basin-scale resonance for these constituents is not expected because their propagation speeds are about half those of the semidiurnal constituents, so they do not circumscribe the basin within a tidal period as do the semidiurnal species (R. Reid 2001, personal communication).

The ratio between the amplitudes of the principal semidiurnal and diurnal tide constituents, (*M*_{2} + *S*_{2})/(*K*_{1} + *O*_{1}), is shown in Fig. 8. While the WFS tidal regime is generally characterized as mixed and mainly of diurnal type, this is not true of the inner half of the WFS where the ratio exceeds 0.5 and especially of the Florida Big Bend and Florida Bay regions where the semidiurnal tides dominate. It is only over the outer portion of the WFS and to the west of Apalachicola Bay where the diurnal species dominate.

Given the model sea level descriptions and the model and in situ data agreements for sea level at the coast, we now examine the tidal currents in a similar manner. Figure 9 compares the depth-averaged tidal current hodographs (calculated for all of the in situ data locations of Fig. 1) with the depth-averaged tidal current hodographs calculated from the model, sampled at all of these in situ data sites. Qualitatively, the comparisons are very good for the semidiurnal constituents with ellipse semimajor amplitudes, eccentricities, and orientations all in agreement (see Table 4 for listings of the hodograph ellipse semimajor axes, semiminor axes, and orientations). Particularly notable is how the ellipse orientations diverge away from the region of relative minimum semidiurnal tide elevation just south of Tampa Bay. Eccentricities at the stations close to the coast show the largest departures between the data and the model, which may be a consequence of the model turbulence parameterization. Deviations from an inviscid fluid semiminor to semimajor axis ratio of *ωf*^{−1} are due to the friction, which is consistent with the largest departures being in the shallowest water. Nevertheless, the agreements are still very good. The overall agreements between the observed and modeled diurnal constituents, while also good, are degraded somewhat from those of their semidiurnal brethren, and there are additional reasons for this behavior.

As pointed out by Reid and Whitaker (1981), modeled currents are more sensitive to factors such as bottom friction, horizontal eddy diffusivity, and topography than are modeled water levels. A particular problem in the Gulf of Mexico for diurnal constituents is the closeness in frequency between these and the local inertial period. Large inertial oscillations exist in the WFS observations when the water column is stratified. While not a subject of this paper, we did investigate several different techniques in attempting to minimize inertial oscillation interference in the data. For instance, we broke the data records into month-long pieces to distinguish barotropic tides during months without stratification versus mixed barotropic tides and inertial oscillations during months with stratification. Nonuniform modulation made this too subjective, however. Recognizing that the inertial oscillations when present are predominantly of first baroclinic mode structure we opted to perform the data analyses on depth-averaged currents, effectively averaging out the inertial oscillations. Nevertheless, some nontidal inertial oscillatory behavior may still be contaminating the diurnal constituent hodographs of Fig. 9. Similar problems do not exist for the semidiurnal constituents since they are removed in frequency from the inertial oscillations. Some amplification is observed in the data during months when the WFS is stratified, but this is small relative to the barotropic tides. Whether such amplification is the result of internal tide generation or a reduction in the effects of bottom friction by the stratification remains to be determined.

The agreements between the observed and modeled tidal current ellipses are sufficiently good to warrant the mapping of these ellipses over the entire model domain. These are shown in Fig. 10 for the *M*_{2}, *S*_{2}, *O*_{1}, and *K*_{1} constituents. The general patterns seen in the coamplitude contours are reflected here. The *M*_{2} currents amplify from very small values in deep water to largest values (approaching 20 cm s^{−1}) near shore in the Florida Big Bend and Florida Bay. The relative minimum in between to the south of Tampa Bay is evident with the hodograph orientations diverging away from this point. To the west of Apalachicola Bay we see very little *M*_{2} tidal current except just to the east of the Mississippi River where the shelf again widens away from the DeSoto Canyon. The *S*_{2} tidal currents behave very similarly to the *M*_{2} tidal current except that they are weaker. The *O*_{1}, and *K*_{1} ellipses appear very similar to one another, being weaker than the *M*_{2} and stronger than the *S*_{2} ellipses over the WFS, but stronger than either of these to the west of Apalachicola Bay. Eccentricities in general agree with the tendencies from inviscid theory that the tidal ellipses should be more circular (*ωf*^{−1}) for the diurnal species than for the semidiurnal species, and the polarizations are clockwise.

### b. Tidal residual current and the Lagrangian transport

*U*

_{res},

*V*

_{res}) defined as the difference between the model responses (

*U,*

*V*) and the four constituent tidal harmonic currents analyzed from the model output. Thus, the Eulerian residual current is calculated as

*u*

_{i},

*υ*

_{i}) and (

*α*

_{i},

*β*

_{i}) are the analyzed harmonic constants for the

*M*

_{2},

*S*

_{2},

*K*

_{1}and

*O*

_{1}tidal constituents. Figure 11 presents the residual currents in both Eulerian and Lagrangian frameworks. The Eulerian representation (upper panel) is the residual surface current field after subtracting from each grid point the linear least squares fit of the four major tidal constituents. The Eulerian residual field is weak (generally less than 0.01 m s

^{−1}) and generally directed toward the north or northwest consistent with the propagation direction for the semidiurnal tides. Vectors appear largest in the Florida Big Bend where they also form an anticyclonic gyre. The Lagrangian representation (lower panel) is calculated by tracking surface drifters originating at every fourth grid point over a 30-day period; that is, they are Lagrangian trajectories for dynamically passive particles released at the surface on day 6 (after the five day spinup period) and tracked through day 35. The patterns appear very similar to the Eulerian representation. Particle displacements are small (order 10 km month

^{−1}) suggesting that barotropic tidal current rectification is not a mechanism of major importance in transporting materials on the WFS.

## 5. Bottom stress and turbulence mixing

Turbulence mixing in this POM-based barotropic tide model derives from bottom stress calculated using a quadratic drag law. Here we examine the bottom stress distribution, its mixing ramifications for the water column, and the implications of these findings for simpler models that employ a linear drag law. For simplicity, only the *M*_{2} current is considered since it is the dominant contributor to the WFS tidal currents.

*τ*

_{b}

*ρ*

*u*

^{2}

_{∗}

*ρ*

*C*

_{D}

*u*

*u*

*ρr*

*u*

*ρ*is the water density and for the first two (depth-dependent) cases,

*u*∗ is the friction velocity,

*C*

_{D}is a bottom drag coefficient, and

*u*is the near-bottom current. For the third (depth averaged) case

*u*

*r*is a resistance coefficient. For our application, we consider the rms value of

*τ*

_{b}calculated over 10

*M*

_{2}tidal cycles from which the friction velocity

*u*∗ is obtained as (

*τ*

_{b}/

*ρ*)

^{1/2}. A map of this

*u*∗ for the WFS

*M*

_{2}tide is given in the upper panel of Fig. 12. The spatial variations of

*u*∗ or the magnitude of turbulence mixing reflects the spatial variations in the tidal current magnitude. Regions of the largest tidal mixing are associated with regions of the strongest tidal currents found in relatively shallow water. In particular the maximum values of

*u*∗ (greater than 0.36 cm s

^{−1}) are found in the Florida Big Bend and in Florida Bay.

*C*

_{D}. The adjustable parameter in

*z*

_{0}. The default value for

*z*

_{0}is 0.01 m. For sensitivity studies we ran a series of model experiments in which we varied

*z*

_{0}between 0.002 and 0.05 m and found no significant differences in model performance. We then replaced Eq. (7) with a smaller spatially uniform drag coefficient (

*C*

_{D}= 0.001). This substantial change in drag coefficient, especially in shallow water, resulted in a disproportionately small change in the model response. In analogy to a damped harmonic oscillator, where proportionately small (large) increases in response occur away from (near)resonance, our sensitivity findings suggest that tidal amplifications in the Big Bend and Florida Bay regions are nonresonant, supporting the earlier assertion that geometry is the cause of the observed amplification.

For the case of linear, depth-averaged models bottom stress is set by the resistance coefficient *r.* As examples, Clarke (1991) used *r* = 2 × 10^{−4} m s^{−1}, whereas Lentz et al. (2001) found *r* = 5 × 10^{−4} m s^{−1} to be appropriate for the North Carolina shelf. Using rms bottom stress and depth-averaged current distributions we arrive at *r* estimations here [Eq. (6)] that range across the shelf from 0.6 to 6 (×10^{−4} m s^{−1}) (middle panel of Fig. 12). Thus, the use of constant values for *r,* while appealing for linear model calculations, may either overestimate or underestimate the bottom stress on continental shelves. Value judgements about quadratic versus linear drag laws, aside, we simply emphasize that bottom friction parameterization for models remains a complex, empirical issue.

*ν*controls the dynamics. Within this layer (of a few centimeters thickness) we have

*k*≈ 0.4 is the von Kármán constant.

Above the logarithmic layer is the “outer layer” for which the velocity and turbulence profiles depend on the nature of flow and hence obviates universality.

*A*(e.g., Wimbush and Munk 1970) as

*A*≫

*ν*where the constant stress and logarithmic regions overlap and using Eqs. (8) and (10), we arrive at

*A*

*cu*

*z.*

*A*to estimate the Ekman layer thickness

*δ*for steady-state flow regimes; that is,

*f*is the local Coriolis parameter and

*c*= 0.1–0.4 (e.g., Loder and Greenberg 1986; Weatherly and Martin 1978).

*W*=

*u*+

*iυ,*into the momentum equations,

*W*∝

*e*

^{iσt}, where

*σ*is the frequency of oscillation, we arrive at [in analogy to Eq. (12)]

For the case of the *M*_{2} tide-induced bottom boundary layer on the WFS, with *σ*_{M2}*f*, the bottom boundary layer thickness may be estimated as *δ* ≈ *cu*∗/(3*f*). Using a midrange value of *c* = 0.2, the bottom boundary thickness induced by the *M*_{2} tidal current over most of the WFS is estimated at about 2–5 m (except near Cedar Key and in Florida Bay where it is comparable to the water depth). Therefore, tidal mixing alone is insufficient to mix the water column on the WFS.

The height of the logarithmic layer is taken as some fraction of the Ekman depth. Soulsby (1983), for instance, defines the logarithmic layer height as 0.1*δ* and with this definition the lower panel of Fig. 12 shows the *M*_{2} tidal-current-induced logarithmic layer height for the WFS. Values are generally less than 1 m. Empirical determinations of the log layer based on the in situ bottom boundary layer measurement on the WFS are presently in progress and these will be reported on separately.

## 6. Summary

Using in situ data and a numerical circulation model we examine the structure of the WFS tides and the effects of the tides in distributing water properties and in mixing the water column vertically. Attention is limited to barotropic tides and to the four major tidal constituents (*M*_{2}, *S*_{2}, *O*_{1}, and *K*_{1}) that account for the bulk of the tidal variance (∼90%) of the region. The data are from coastal sea level stations ranging from the Mississippi River delta to the Florida Keys, and from current meter moorings deployed across the shelf between the 10-m and 300-m isobaths. The model is a regional adaptation of the POM extending from west of the Mississippi River to the Florida Keys with a single open boundary arcing between these land termini. Regional tides are produced by forcing the model at its open boundary using the composite *M*_{2}, *S*_{2}, *O*_{1}, and *K*_{1} sea level variability from the global (TOPEX/Poseidon assimilated) tide model of Tierney et al. (2000). With standard tidal analysis tools, we produce coamplitude and cophase maps for each constituent and compare both sea level and currents against actual observations. Based on the fidelity of comparison we then use the model products to discuss the WFS barotropic tides.

Apalachicola Bay, in the Florida Panhandle, is a dividing point between appreciable semidiurnal tides to the east over the WFS and minimal semidiurnal tides from there to the Mississippi River delta. In contrast with the semidiurnal tides, the diurnal tides are spatially more uniform. With respect to overall rms sea level variability, tides account for about 30% west of, versus about 50% southeast of Apalachicola Bay, respectively. The balance of the rms variability is due to synoptic scale weather-induced sea level change. Coamplitude maps show largest tides in the regions where the shelf is widest, that is, in the Florida Big Bend and in Florida Bay. The region just south of Tampa Bay shows a relative minimum. Both the amplification of the semidiurnal constituents and the spatial variations with shelf width are consistent with linear theory (e.g., Clarke 1991; Lentz et al. 2001). Such theory also accounts for the relative spatial uniformity of the diurnal constituents. We conclude that the spatial distributions of the tides on the WFS are the result of local geometry, as contrasted with previous ideas about basinwide tidal resonance.

The tidal currents on the WFS, especially the semidiurnal constituents, are primarily barotropic. Very good agreements are obtained between the in situ data derived and the modeled tidal hodograph ellipses. Semimajor axes are generally less than 0.1 m s^{−1}, and the ellipses are generally oriented normal to the shoreline. Ellipse orientation, however, does reflect the coamplitude distributions with tidal currents tending to diverge (converge) on regions of relative minima (maxima). A relatively small seasonal modulation is observed in the semidiurnal tidal currents due to either internal tides or reductions in friction by stratification. Larger seasonal modulation is observed in the diurnal constituents due to the generation of inertial oscillations under stratified conditions. Inertial oscillations, when present, are generally organized as first baroclinic modes, so they tend to cancel out when performing vertical averages. This helps to facilitate the barotropic tidal analyses. Baroclinic tides and inertial oscillations will be reported on separately.

The nonlinearity of the model allows it to generate mean currents despite its linear harmonic forcing. We present these residual currents in both Eulerian and Lagrangian frameworks to assess their potential effects on tidal residual material transports. While a coherent Stokes-drift pattern toward the northwest emerges, the magnitudes are small (one to two orders of magnitude less than the semimajor axis speeds) amounting to net particle trajectories of generally less than 10 km per month. Albeit persistent, this is smaller than the potential particle trajectories for either seasonal or synoptic scale forcing (e.g., Weisberg et al. 1996).

With the *M*_{2} tide having the largest tidal current magnitude we use this constituent to examine the spatial distribution of bottom stress by tides on the WFS and the role that this may play in water column mixing. Except for the shallow regions of the Florida Big Bend and Florida Bay, the potential for mixing by tides is weak. Attempting to parameterize friction in a linear model using a resistance coefficient is limited by the spatial variability in bottom stress; nevertheless, the values estimated are within the range of values used in other studies. Estimates of logarithmic layer thickness by the *M*_{2} tide are generally less than 1 m and empirical studies are under way to quantify this better.

In closing, with quantifiable metrics we show that regional tides on the WFS may be adequately described by forcing a regional, primitive equation, three-dimensional model with a global tide model at the open boundary, and that the properties of the tidal variations result primarily from the local geometry.

## Acknowledgments

The in situ data collection and numerical model developments benefited from several sponsors. The measurements were initiated under a cooperative agreement between the U.S. Geological Survey, Center for Coastal Geology and the USF College of Marine Science. Subsequent support by the Minerals Management Service, Cooperative Agreement 14-35-0001-30787, is noted. Our work continues under support by the Office of Naval Research Grant N00014-98-1-0158 and the National Oceanic and Atmospheric Administration Grant NA76RG0463. Special thanks are offered to the family of Elsie and William Knight for the research fellowship endowment that helped to support R. He. Messrs. R. Cole, J. Donovan, and C. Merz are largely responsible for the success of the field program. Dr. S. Meyers contributed analyses and discussions, and we are particularly grateful for the helpful e-mail discussions provided by Prof. Robert Reid. The sensitivity test on the drag coefficient in section 5 was at the suggestion of an anonymous reviewer.

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Comparison of observed and computed tidal elevation at reference sites

Normalized rms tidal residuals showing the contributions of nontidal fluctuations to the sea level variability at the nine tide gauges spanning the study domain

Table 3a. Greenwich phases of the major tidal constituents at the St. Petersburg tide gauge and at the 12 ADCP stations (left-hand columns), plus the relative times between high water at St. Petersburg and maximum (shoreward directed) semimajor axis tidal currents at the 12 ADCP stations (right-hand columns). Phase differences are converted to time differences using speeds of *M*_{2}, *S*_{2}, *O*_{1} and *K*_{1} of 29.98 deg h^{−1}, 30.00 deg h^{−1}, 13.94 deg h^{−1} and 15.04 deg h^{−1}, respectively, where “+” and “−” indicate time lags and leads, respectively

Table 3b. The time lags between high tide at St. Petersburg and the maximum (shoreward directed) semimajor axis tidal currents at the 12 ADCP stations determined by the maximum lag-correlation coefficient between the composites of the *M*_{2}, *S*_{2}, *O*_{1}, and *K*_{1} constituents

A comparison of computed and observed tidal ellipse parameters