1. Introduction
Roughness elements and rough bathymetric features generate resistance to flow through skin friction and pressure anomalies, which are referred to as frictional drag (or frictional skin drag) and pressure drag (or form drag), respectively. Frictional drag is a tangential stress on the surface caused by the molecular diffusion of momentum across a velocity interface (Schlichting 1962) and also applies to heat, salt, and other scalars. This is primarily true for hydrodynamically smooth flows where the bulk of the force on the surface is by viscosity because pressure forces are negligible. However in rough flows, where the roughness elements are larger than the molecular sublayer, the force on the surface is transmitted mainly by the pressure forces on roughness elements, and the force is parameterized as frictional drag (Schlichting 1962; Tennekes and Lumley 1972). The frictional drag depends strongly on the hydrodynamic roughness or the roughness Reynolds number (e.g., Nikuradse 1950). The parameterization of frictional drag is typically derived from inertial or wall boundary layer theory (e.g., Tennekes and Lumley 1972; Stull 1988) and has been applied in both atmosphere and ocean models (Blumberg and Mellor 1987; Hodur 1997).
Form drag is related to the dynamic pressure differences across an obstacle arising from upstream blocking and separation of flow and formation of eddies, hydraulic jumps, and lee-wave disturbances downstream of the obstacle (Gill 1982; Smith 1989; Baines 1995). The nature of the flow depends on the governing parameters such as Froude number Fr, nondimensional obstacle height
In stratified depth-limited flows, the nondimensional height of the bank or nonlinearity parameter
Form drag for airflow over mountains has been studied extensively in both field observations and numerical simulations with special emphasis on improving the parameterization of mountain wave drag (Smith 1978, 1979, 1980; Bougeault et al. 1990; Durran 1990; Clark and Miller 1991; Olafsson and Bougeault 1996; Doyle and Jiang 2006). Several decades of mountain wave studies were beneficial to the estimation of mountain wave drag based mainly on linear internal wave solutions. Such parameterizations have resulted in substantial improvements in current weather prediction models (e.g., Kim and Arakawa 1995; Wood et al. 2001; Kim et al. 2003). Form drag has been computed for both small and large atmospheric mountain ranges. On a smaller scale, Smith (1978) evaluated pressure drag in the Blue Ridge Mountains in the United States using several microbarographs across the ridge, resolving the pressure field over the ridge similar to the oceanic observations made by Warner et al. (2013). However, for most atmospheric observations, measurements of pressure over mountain terrains are quite sparse, as the instruments cannot cover the entire mountain range. Hafner and Smith (1985) used a limited number of microbarographs over the European Alps to evaluate the pressure drag by assuming lateral pressure gradients were constant; even though these observations do not resolve the entire pressure field over the complex terrains, a rough estimate of form drag was obtained. In recent mountain wave experiments such as the Terrain-Induced Rotor Experiment (T-REX; Grubisic et al. 2008) microbarographs measurements were used, but a few microbarographs measurements are not sufficient to compute the pressure drag directly. Instead, pressure measurements have been used to obtain information on internal lee waves.
In the ocean, observational studies of form drag are very limited (e.g., Nash and Moum 2001; Warner et al. 2013). Form drag acts on the entire water column over the topographic feature and leads to the generation of mixing and turbulence, eddies, and internal waves away from the bottom (e.g., Polzin et al. 1997; Jayne and St. Laurent 2001; Pawlak et al. 2003; Garrett and Kunze 2007). Some aspects of topographic–tidal flow interaction and formation of lee waves have been studied analytically and numerically using nonhydrostatic models (Bell 1975; Nakamura et al. 2000; Khatiwala 2003; Legg and Huijts 2006). Several numerical studies have been conducted to examine the dynamics of flows over small banks (e.g., Lamb 1994; Skyllingstad and Wijesekera 2004; Seim et al. 2012). Lamb (1994) and Skyllingstad and Wijesekera (2004) reported qualitatively comparable results to the theoretical and laboratory results of Long (1955) and Baines (1979) for a similar dynamical parameter range (1 < K < 2 and
The total form drag consists of different components that involve isopycnal displacement (referred as “internal”) and surface tilt (referred as “external”; e.g., McCabe et al. 2006). Different components of form drag over coastal banks and ridges were examined by several investigators (Moum and Nash 2000; Nash and Moum 2001; Edwards et al. 2004; McCabe et al. 2006). Warner et al. (2013) were first to measure the total form drag from high-precision pressure sensors (referred to as Ppods; Moum and Nash 2008) deployed over Three Tree Point (TTP), a headland in Puget Sound, Washington. Warner et al. (2013) reported that the form drag coefficient is at least one order of magnitude larger than the frictional drag coefficient at the bottom.
In the following, we compute both form drag and frictional drag from in situ measurements collected over the East Flower Garden Bank (EFGB) on the Texas–Louisiana shelf (Fig. 1) as part of projects sponsored by the Naval Research Laboratory [NRL; Mixing Over Rough Topography (MORT)] and by the Bureau of Ocean Energy Management [BOEM; Currents over Banks (COB)]. As described below, the flow over the EFGB consists of
(left) Habitat characterization map of the EFGB, along with mooring locations. The edge of the habitat map is the boundary of the marine sanctuary. Habitat characterization was provided by NOAA (available at http://www.ncddc.noaa.gov/website/google_maps/FGB/mapsFGB.htm). Red, blue, light green, green, dark blue, and yellow regions denote coral reefs, coralline algal reef, algal nodules, coral community zone, and soft bottom, respectively. Barny moorings (M1–M8, M4a, and M5a) are also marked on the map. (right) Roughness elements associated with corals. A diver was examining the underside of a dislodged coral head after Hurricane Ike in September 2008. Photographer: E. Hickerson, Flower Garden Banks National Marine Sanctuary. This image is part of a larger collection of sanctuary media found online (at https://marinelife.noaa.gov/media_lib).
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
2. Observations
Sampling was conducted at the EFGB, located approximately 190 km southeast of Galveston, Texas, in the northwestern Gulf of Mexico (GoM). The EFGB is part of the National Marine Sanctuary System and is one of many banks formed by salt domes on the continental shelf. The EFGB is a biologically diverse coral reef consisting of brightly colored sponges, plants, and other marine life. The habitat map and an example of the size of a typical coral head are shown in Fig. 1. The EFGB is a pear-shaped bathymetric feature rising from below 100-m water depth to 18 m below the surface (Figs. 1, 2). The bank is about 10 km long and about 6 km wide, and its projected area to the horizontal plane encompassed by the 100-m bathymetry contour is approximately 40 km2. The EFGB has steep lateral slopes (~50-m height change within 500-m distance) at the eastern and southern boundaries. The top, located just off the southern peak, is nearly flat with more gentle slopes westward and northward, but contains rough bathymetric elements such as corals (Figs. 1, 2). The top of the bank is dominated by relatively large, head-forming corals, with Montastrea annularis, known as boulder star corals, composing 30% of the total coverage between 18- and 36-m depth, and the species Diploria strigosa, Colpophyllia natans, Montastrea cavernosa, Millepora alcicornis, and Porites astreoides account for an additional 20% coverage (Fig. 1). The total coral coverage on the upper reef averages approximately 55% (Bright et al. 1984). Between 30- and 40-m depth, algal ridges are common, and below 50-m, algal nodules (rhodoliths) are dominant. In these deeper zones nodules can reach 1–2-m height due to lower current velocities (Minnery 1990; Fig. 1).
(a) Bathymetric map of the bank with contours every 10 m. The red bullets are the long-term Barny mooring locations, and the blue bullets are the short-term mooring locations. The locations of Ppods are marked in green circles. Note that the string moorings (S1–S5 and S8) were located about 100–200 m from the Barny locations. The green lines are June transects, and blue lines are August transects. The black dashed lines shows the coordinate system (x, y), where the x axis is parallel to the M2–M5 mooring line, and the y axis is perpendicular to the mooring line. Note that U is positive in the x direction, and V is positive in the y direction. (b) Cross section of bathymetry along M2–M5–M4 line. Distance (positive eastward) is from M2 and the height of bank (positive upward) is from 106-m bathymetry contour. (c) Longitude–depth cross sections of bathymetry along T1 and T2 transects.
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
The observational program over the EFGB consisted of two 6-month mooring deployments (from December 2010 to June 2011 and from June 2011 to December 2011), a 2-week intensive observational period (IOP) from 31 May to 14 June 2011, and a week-long microstructure survey during a coral spawning event in August 2011 as a part of the MORT and COB projects. Two ships [Research Vessel (R/V) Pelican and R/V Manta] took measurements over the bank during the IOP. Microstructure surveys were conducted from the R/V Manta, while moorings were deployed from the R/V Pelican. Detailed discussions of instrumentation, data collection, sampling methods, and data processing for the long-term and short-term deployments are in Teague et al. (2013) and in Jarosz et al. (2014, manuscript submitted to J. Geophys. Res.). Here we describe the subset of the data relevant to this study, that is, measurements made during the IOP and in August 2011 as well as from the moorings deployed during the second half of the COB experiment (June–December 2011). Bathymetry and locations of these short- and long-term moorings at the EFGB are shown in Figs. 1a and 2a.
a. Moorings: Currents, pressure, and hydrography
A total of 10 Teledyne RD Instruments’ (RDI) acoustic Doppler current profilers (ADCPs) were deployed in bottom-mounted, trawl-resistant housings referred to as “Barnys” (Perkins et al. 2000) at stations M1–M8, measuring east, north, and vertical velocities (Figs. 1, 2a). The ADCP at M5 operated at 600 kHz and all others operated at 300 kHz. Transducer heads were approximately 0.5 m above the sea floor. Yearlong currents and hydrographic fields were collected from five ADCP moorings (M1–M5) and four subsurface string moorings (S1–S4) equipped with temperature T, conductivity C, and pressure P (TCP) sensors. ADCPs recorded nearly full water column current profiles every 15 min (an ensemble average of 120 realizations) during long-term deployments. During the IOP high temporal resolution, current and hydrographic data were collected from another five bottom-mounted ADCPs at stations M4a, M5a, M6, M7, and M8 and two string moorings at S5 and S8. Samples were averaged every minute at M4a, M5a, and M6 and every 2 min at M7 and M8. Vertical bin sizes were 1 m at M5a, 2 m at M4a, and 4 m at M6, M7, and M8. Accuracy for the ADCP measurements is 0.5% of the observed water velocity ± 0.5 cm s−1.
Barny moorings were also equipped with Ppods (Moum and Nash 2008). Ppods are modified Paroscientfic pressure sensors and have a precision of about 0.14 mm or 1 Pa (1 Pa = 1 N m−2). A total of five Ppods were deployed during MORT (Fig. 2a) at M2, M4, M5, M7, and M8, and their pressure is referred to as PM2, PM4, PM5, PM7, and PM8. Three of the Ppods (PM2, PM4, and PM5) were deployed along a line over the bank for a period of nearly 6 months (between June and December 2011), and the other two (PM7 and PM8) were deployed for 2 weeks in June 2011 during the IOP. We noted mismatches in time stamps and nonuniform sampling rates in the long-term deployed Ppods. The first step was to obtain correct time stamps and sampling rates. The time stamps of PM2 and PM4 were accurate, but their sampling rates deviated from 1 Hz, indicating drifts in the sampling clocks. Note that the average sampling rate was 0.97 Hz. To fix sampling rates caused by clock drifts, we resampled (i.e., interpolated) PM2 and PM4 data at 1-Hz sampling rates between accurate time stamps of starting and ending times. We noted that the accurate clock in PM5 was broken in August 2011, but the sampling clock had an accurate time stamp with a sampling rate of 1 Hz; when compared with the rest of the Ppods, the sampling clock in PM5 was consistent with the other accurate clocks in PM2 and PM4. Therefore, we used the time stamp of the sampling clock as the accurate time stamp for PM5.
Four TCP string moorings (S1–S4) were deployed close to M1–M4 during the first half of the deployment and only three (S2–S4) were redeployed near M2–M4 during the second half (Fig. 2a). Each string mooring contained 8 to 12 TCP sensors that were approximately equally spaced between 7 and 12 m below the surface and 1 m above the bottom (Teague et al. 2013). We do not address processing and data quality issues here, since those topics were discussed in Teague et al. (2013).
For the current data, we used local coordinates with the U component oriented toward the axis of the M2–M4 mooring line and the V component oriented perpendicular to the mooring line (Fig. 2). We rotated ADCP currents by 19° counterclockwise to the horizontal direction to match with the direction of the form drag estimated in section 4. Low-frequency east–west flow showed a two-layer structure around the bank (Teague et al. 2013). A similar flow pattern was found in U. Figure 3 shows the depth-averaged low-frequency [<0.02 cycles per hour (cph)] currents (UU, VU) in the upper 50-m layer. The flow UU was nearly eastward during the first half of the record and changed direction after yearday 263 (Figs. 3a,b). Currents in the lower layer (UL, VL) were smaller than in the upper layer and were highly influenced by the bank (Figs. 3c,d; Teague et al. 2013). Mean currents approximately 20 m above the bottom followed the bottom topography (Teague et al. 2013). Shipboard and moored velocity measurements made during the IOP (2–13 June 2011) showed generation of submesoscale and mesoscale motions and reversals of bottom currents on the lee side of the bank (Jarosz et al. 2014, manuscript submitted to J. Geophys. Res.). Based on all the ADCP velocity records, we noted that the currents in the upper 50-m layer at M1 were least affected by the bank. M1 was located at the northwest corner of the bank where the bottom slope was less steep compared to slopes south and east side of the bank (Fig. 2). Near-surface intensified high-frequency currents (>0.2 cph) were observed over the top of the bank and were closely related with wind and surface waves (Wijesekera et al. 2013).
Time series of 48-h low-pass filtered upper-layer (above 50 m) and lower-layer (below 50 m) velocities (cm s−1). (a) Upper-layer velocity along the direction of the mooring axis M2–M4 (UU). (b) Upper-layer velocity perpendicular to the mooring axis (VU). (c) Lower-layer velocity along the direction of mooring axis M2–M4 (UL). (d) Lower-layer velocity perpendicular to the mooring axis (VL). (e) Color image of depth–time series of 15-min averaged temperature (°C) at S2.
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
The spectra of UU and VU show several distinctive frequency bands associated with tidal/inertial waves and subtidal flows (Fig. 4). The near-inertial period (25.52 h) lies between the K1 (23.93 h) and O1 (25. 82 h) tidal periods. Teague et al. (2013) reported that near-inertial currents were as large as 20 cm s−1, especially during summer months, and that barotropic diurnal tides were less than about 4 cm s−1. The tidal–inertial components of east–west and north–south directions had similar magnitudes, while the semidiurnal components in the north–south direction were stronger than the east–west components (Fig. 4). The subtidal band contains 4–6-, 10–12-, and 16–17-day fluctuations. There was no clear spring–neap tidal signal (period ~14 day) in the subtidal band (Fig. 4). In the northern GoM, the 4–6-day signal may relate to synoptic-scale weather systems (e.g., Donohue et al. 2006). Donohue et al. (2006, 2008) observed 16- and 14-day oscillations in bottom pressure in the north central GoM, but these signals were unexplained. However, Donohue et al. (2008) further noted that the 16-day signal is in phase with water level variations measured by coastal tide gauges on the west Florida shelf.
Depth-averaged frequency-weighted velocity spectra in the upper 50 m based on time series of velocity shown in Figs. 3a and 3b. (a) UU and (b) VU. Spectra at M1, M4, and M5 are plotted. The vertical lines denote frequencies of M2, K1, O1, f (inertial), 4-day, and 17-day motions.
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
Near-surface temperature showed seasonal warming in summer and cooling in late fall (Fig. 3e). The advection of cold water, especially near the bottom, appeared after yearday 240 [lasting close to the end of the observational period (Fig. 3e)] and changed the vertical structure of temperature and density fields, which in turn altered bottom pressure across the bank on subseasonal to seasonal time scales. Teague et al. (2013) suggested a plausibility of advection of cold water by eddies in the region. In the following analysis, we focus on motions that can be well resolved by our data, and therefore we limit the computation of the form drag to tidal (12–25 h) and subtidal (4–17 days) motions.
b. Ship surveys: Currents, hydrography, and microstructure
Ship surveys were conducted from the National Oceanic and Atmospheric Administration (NOAA) R/V Manta, a 25-m catamaran vessel out of Galveston, Texas. A 300-kHz RDI ADCP was mounted on the side of the boat and recorded currents with a vertical bin size of 4 m. Velocities were processed using the University of Hawai‘i’s Common Oceanographic Data Access System (CODAS; Firing et al. 1995) to correct for ship motion and were averaged over 4-min intervals. A BioSonics DT-X digital scientific echosounder was used to acquire bathymetric data based on acoustic backscatter. The echosounder sampled at a rate of 3 Hz with a frequency of 123 Hz. The average horizontal sampling distance was approximately 25 cm.
Additionally, vertical profiles of small-scale velocity and temperature along with finescale T, C, and P fields were collected from a free-falling Rockland Scientific vertical microstructure profiler (VMP), equipped with a pressure sensor, shear probes, a fast thermistor, microconductivity, and Sea-Bird Electronics temperature and conductivity sensors (e.g., Wolk et al. 2002). These profiles were collected while the R/V Manta was drifting or steaming slowly along quasi-straight lines over the bank at speeds of 0.5 to 1 m s−1 (blue and green lines in Fig. 2a). The velocity shear accuracy was 5%, and the VMP probe had an average descent speed of 0.8 m s−1, producing shear measurements approximately every 0.2 cm. Shear measurements were utilized to estimate turbulent kinetic energy (TKE) dissipation rates using
3. Form and frictional drags
a. Form drag

















b. Frictional drag















4. Evaluation of form drag
a. Bottom pressure anomaly from Ppods
Figure 5 shows bottom pressure fluctuations at five mooring sites (Figs. 1, 2) between 2 June (yearday 152) and 13 June (yearday 163) 2011. The pressure fluctuations were constructed by removing the temporal mean for a given record length of 12 days. The demeaned pressure fluctuations show diurnal tides and a segment of the spring–neap cycle. Tides around and over the bank had similar phases (Fig. 5a) since the spatial scales of tidal motions are much larger than the lateral spacings between moorings. Unlike pressure, the depth-averaged velocities at M1–M5 were not always in phase, especially at the beginning of the records (Figs. 5b,c). While pressure fluctuations were dominated by diurnal tides, velocity fluctuations contained combinations of tides and near-inertial waves.
(a) Demeaned bottom pressure (P − 〈P〉) at M2, M4, M5, M7, and M8 between 2 Jun (yearday 152) and 13 Jun (yearday 163) 2011, where 〈P〉 is the time-averaged pressure. Pressure at M7 (south of the bank; Fig. 2a), marked in cyan differs from the rest of the observations. (b) Depth-averaged velocity in the direction of the mooring axis (UD) and (c) velocity perpendicular to the mooring axis (VD) at M1–M5.
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
Pressure spectra calculated at M2, M4, and M5 (Fig. 6) show several distinctive features including the surface–gravity wave band between 0.08 and 0.2 Hz, the infragravity wave band between 0.002 and 0.08 Hz, the semidiurnal M2 tide (period = 12.42 h), and the diurnal tidal bands K1 (period = 23.93 h) and O1 (period = 25.82 h). Our pressure spectra also indicate variability in the synoptic time scale (4–6 days) and in 10–17-day fluctuations, similar to velocity spectra shown in Fig. 4. The amplitude of a given tidal frequency was estimated by integrating the pressure spectrum for a specified spectral band, and the amplitudes of O1, K1, and M2 tides are 0.17, 0.17, and 0.06 m, respectively. The pressure spectrum (Fig. 6) has a slope of about
(a) Frequency spectra of bottom pressure at M2, M4, and M5. The spectra were based on the entire pressure record with a sampling rate of 1 Hz. Spectra were averaged into 210 frequency bins. (b) Enlarged plot showing diurnal tides O1 and K1 and semidiurnal tide M2.
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
We adopted the following procedure to compute pressure anomalies. First high-frequency pressure fluctuations and spikes were removed by filtering raw pressure to remove frequencies higher than 10−3 Hz. Second, seasonal and subseasonal motions with time scales greater than 30 days were filtered out, since these motions are not well resolved, and our focus is on tidal and subtidal (4–17 day) oscillations. These bandpass-filtered pressure fluctuations at M2, M4, and M5 are referred as
Bottom pressure anomalies (thin gray lines) at (a) M4,
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
As described above, the bottom pressure anomalies west, east, and in the middle of the bank (M2, M4, and M5, respectively) were obtained by (i) bandpass filtering of pressure between 1000 s and 30 days, (ii) subtracting a selected reference pressure outside the bank (i.e., pressure at M2), and (iii) subtracting the pressure due to rotational effects. Bottom pressure anomalies at M2, M4, and M5 are, respectively,
b. Form drag
We used three pressure sensors at M2, M5, and M4 (Fig. 2a) to compute the form drag over the bank. The drag was computed along the mooring line that was oriented about 19° counterclockwise from the east–west direction (Fig. 2). The topographic section along the mooring line is close to “top hat”–shaped bathymetry. In general, the EFGB has steeper slopes along the east and south boundaries and a significant portion of the bank top is relatively flat, except near the peak (Fig. 2). The major challenge is to evaluate the form drag [(1)] from only three pressure measurements. Hafner and Smith (1985) estimated the form drag over the European Alps using a limited number of pressure measurements. They transformed the surface integral [(1)] to the flux integral and evaluated the form drag as the product of the horizontal, locally constant pressure gradient and the volume of the mountain terrain. This is the equivalent of assuming that the local pressure gradient is linear between two nearby observation sites. Hafner and Smith (1985) discussed the uncertainty of pressure that was determined by a linear formula based on their surface stations.
The form drag is large in regions where both bathymetric slopes and pressure anomalies are significant. Warner et al. (2013) examined the sensitivity of a number of pressure measurements by evaluating the integral [(1)] over a sloping headland, TTP in Puget Sound, Washington, and found that four sensors (two on either slope of the ridge; Fig. A1) can provide an accurate estimate. We also examined sensitivity for our three-point measurement setup (one on the top and one on either side of the ridge; Fig. A1) using Warner’s (2012) numerically generated pressure fields over the TTP ridge. As described by Warner (2012), at TTP, the large bottom pressure anomalies were found where the topography was the steep. The model was able to capture some, but not all of these pressure anomalies. The spatial resolution of the model and the required smoothing of the topography for the model to run were major factors for simulating an accurate pressure field over the bank (Warner 2012). Since the Ppods in MORT were not located on the sloping sections of the topography, some of the largest pressure anomalies could have been missed.
A sensitivity analysis described in the appendix allows the examination of uncertainties in our computation based on three pressure measurements, although the geometrical and dynamical settings of this study and Warner et al.’s (2013) are not identical and the model results may not perfectly capture the distribution of bottom pressure anomalies due to numerical limitations. We compared form drag estimates based on multiple pressure measurements versus three pressure measurements as described in the appendix. We used three methods to compute the form drag over the bank. Method I computes the form drag by approximating bottom pressure for a given slope as the mean pressure between the top and the outer edge of the ridge and then multiplying it by the mean slope. Method II computes the form drag as a product of the observed slope and a linearly interpolated pressure between the outer edges and the top of the ridge. Method III computes the form drag as a product of the mean pressure for a given slope and the maximum magnitude of that slope. The estimates from methods I and II were comparable, but their maximum magnitudes were a factor of 2 smaller than those obtained from the method III and the fully resolved solution. Here we applied all three methods to compute the form drag along the M2–M4 mooring line, and results are illustrated in Fig. 8. Topographic gradients of the bank are small or near zero over the top and outside the 100-m bathymetry contour compared to the two major slopes of the bank (Figs. 1a, 2). Therefore, we can expect major contributions to the form drag from sloping boundaries. The EFGB is asymmetric, especially toward the southern end, where the lateral slopes over the bank are not negligible. We may not compute the form drag completely since our computation is limited to the east–west direction. Drags estimated from linearly interpolated pressure and from mean pressure are comparable. The form drag estimated from method III is twice large as those that calculated from methods I and II, similar to the findings in the appendix, indicating the sensitivity to the slope.
Form drag estimated from the three methods described in the appendix.
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
In the following analyses, we used the form drag computed from method II, similar to Hafner and Smith (1985), even though we may underestimate the expected value by a factor of 2. To examine the time variability of the total form drag, we examine tidal and subtidal components (time scales greater than 2 days) separately. The corresponding drag estimates based on method II are shown in Figs. 9a and 9b. The tidal band contains diurnal and semidiurnal tides, and the subtidal band contains multiple time scales (4–17 days). Here it is assumed that form drag can be expressed as a linear summation of several components. However, this assumption may have limitations when the processes that generate form drag become highly nonlinear. Frequency spectra of Dform were also computed to examine multiple time-scale processes related to the total form drag (Fig. 10). The semidiurnal and diurnal tidal components, 4–6-day synoptic time-scale variability, and 10–17-day low-frequency variability are major contributors to the total form drag, similar to the velocity spectrum shown in Fig. 3. The most dominant components of Dform are the diurnal and 4–17-day band with drag magnitudes in excess of 2000 N m−1 (Table 1; Figs. 9). Note that the mean form drag for the tidal band was computed by averaging maximum tidal magnitudes, and the mean for the 4–17-day band was computed by averaging maximum magnitudes in every 4-day time interval.
Total form drag Dform based on interpolated pressure fluctuations (method II). (a) Tidal band and (b) subtidal bands (period > 2 days). The blue line in (a) and (b) is the inertial component of the form drag Diner due to acceleration/deceleration of the flow.
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
Variance-preserving power spectra of the form drag [(N m−1)2 × 10−7]. Thin lines with crosses denote 95% confidence limits. Vertical lines denote periods of K1, O1, 4-day, and 17-day motions.
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
Averaged form drag, velocity, power, and bulk drag coefficient [(4)]. Estimates of Pform were limited to low-frequency motions (Fig. 11). The magnitudes of K1 and O1 tidal velocities estimated by Teague et al. (2013) are also included in the table. The values inside the brackets are 95% confidence limits, and the values inside the curly brackets are minimum and maximum of the record. The term CD for the diurnal tides was computed from velocity estimates given in Teague et al. (2013). The estimated bulk drag coefficient for semidiurnal tides by Warner et al. (2013) is given in the bottom row.
The total form drag [(1), (8)] may contain inertial pressure exerted by the oscillatory nature of the flow (e.g., Warner and MacCready 2009; Warner et al. 2013). Because of tidal acceleration, such as during slack tides, the barotropic pressure gradient (i.e., sea surface slope
c. Power: Pform
Both currents and the estimated form drag consist of nonstationary processes, and conventional Fourier cospectrum analysis does not provide information about power variations within data records. Therefore, the power per unit length Pform is estimated as a product of Dform and Um [(2)], where Um is the free-stream velocity in the direction of the form drag at an upstream location where the flow is undisturbed by the topographic feature. Unfortunately, our observations outside the bank were close to the walls of the bank. The flow below the height of the bank was impacted by the topography, and among those observations, the flow in the upper 50 m at M1 was the least affected. Therefore, we used the upper 50-m averaged velocity UU at M1 as the undisturbed mean flow Um. A time series of Pform for the subtidal band at M1–M5 is shown in Fig. 11. For most occasions, −UU and Dform were in phase, indicating kinetic energy loss in mean currents due to the form drag over the bank with negative Pform. Positive values of Pform may reflect phase differences between the drag and the velocity, but some differences may be attributed to uncertainties in Dform. The energy loss (Pform < 0) has a wide range of variability with values as large as 800 W m−1 (Fig. 11). The 6-month averaged energy loss (Pform < 0) at M1 was about 85 W m−1 with 95% confidence limits between 76 and 93 W m−1 (Table 1). If the energy loss is computed from the depth-averaged velocity, the resulting 6-month averaged energy loss at M1 is 73 W m−1, about 14% less than the estimate based on UU (Table 1). The power associated with the diurnal tidal components was not computed, since tidal velocities were heavily contaminated by inertial motions. The magnitude of average energy loss per unit area, Pform/l0, at subtidal (4 to 17 days) bands is about 10−2 W m−2. Energy loss over the bank can be expressed as (Pform/l0)A0, where A0 is the projection of the bank to the horizontal plane. The long-term average energy losses for the 4–17-day band is about 0.32 MW, where l0 = 9 km, and A0 (=40.76 km2) is the projected bank area encompassed by the 100-m bathymetric contour.
Time series of (a) upper 50-m layer velocity UU at M1–M5, (b) Dform for the subtidal band, and (c) Pform at M1–M5. Thick black lines in (a) and (c) are UU and Pform at M1.
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
d. Drag coefficient
The bluff–body drag coefficient CDF [(3)] and the bulk drag coefficient CD [(4)] were computed, and the estimates of CD are presented here since CDF/CD = l0/h0. The bulk drag coefficient was calculated from the observed relation between
Form drag per unit area Dform/l0 for the subtidal band plotted against
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
e. Wave drag from linear theory






Dform for subtidal motions vs the estimated wave drag from the linear theory [(16)] at M1 along with 95% confidence levels (vertical lines).
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
5. Estimates of frictional drag
We compute the frictional drag coefficient Cd [(13)] and the roughness length z0 [(15)] over the bank by combining measured TKE dissipation rates and currents near the bottom. The spacing between VMP profiles is approximately 200 m on average, much larger than individual coral features. Hence, our measurements can be best utilized to estimate the range of roughness over the top of the bank rather than a precise value at an individual location. The measured values of velocity and TKE dissipation rate ε were interpolated to a grid with 0.5-m vertical spacing. The bottom boundary layer depth where density changed by 0.01 kg m−3 from the near-bottom density was about 4 m from the bottom for nearly 75% of the VMP profiles collected over the bank. The inertial boundary layer is typically less than 20% of the boundary layer depth and is expected to be less than 1 m. Dissipation measurements were averaged over the first 4 m from the bottom (or approximately over the bottom boundary layer), and five sequential profiles were combined, resulting in spacing on the order of 1 km. The maximum likelihood estimator method (Baker and Gibson 1987) was used to calculate the expected value and 95th percentile confidence limits. Velocities were also averaged in time using five adjacent profiles, and any measurements more than 7.5 m above the bottom were discarded. The frictional velocity
Habitat composition maps show algal nodules covering most of the bank, with the coral reef zone (55% coverage) over the shallower regions (Fig. 1). The expectation is that the bank will contain a range of values, including both smooth areas and rough (coral coverage) regions. Prior estimates indicate that the roughness lengths over corals are expected to be several centimeters (Rosman and Hench 2011), much larger than what was seen at the moorings located just off and on top of the bank. Calculated roughness lengths [(15)], measured dissipation rates, and bottom velocities along the two transects shown in Fig. 14 contain significant variability, partly due to limited sampling in both space and time. Dissipation, velocity magnitude, and frictional velocity (multiplied by 10) are shown in the first two panels of Fig. 14 for T1 and T2 (the transects highlighted in Figs. 2a,c). The third panel shows two estimates of the roughness length
Near-bottom TKE dissipation rate ε, near-bottom velocity
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
The calculated roughness lengths [(15)] and the associated drag coefficients [(13)] over the EFGB vary spatially (Fig. 15). Given the transect range of the ship ADCP, estimates of
All values of (a) z0 (cm) and log10 values of (b) bottom drag coefficient (Cd) over the bank.
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
Bottom roughness z0 and frictional drag coefficient Cd estimated over the EFGB. Maximum, minimum, arithmetic mean, and median were estimated from 243 observations. The maximum likelihood estimator (MLE) was computed by assuming lognormal distributions for z0 and Cd, where
The range of frictional regimes over the bank can be seen in Fig. 16, which shows
The term Cd as a function of frictional Reynolds number Re*. The horizontal line is at 0.002, the approximate value of Cd at the mooring locations.
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
6. Discussion
a. Form drag, power, and bulk drag coefficient
Analysis of a 6-month-long time series of pressure over the EFGB revealed that Dform was generated by multiple time-scale processes. The magnitude of Dform at the diurnal tidal band is about 1500 N m−1 for tidal velocities of about 3–5 cm s−1 (Table 1), and the inertial drag was a significant component of the total form drag for this band. Apart from Dform at tidal oscillations, we found large form drags at subtidal bands with time scales of 4–6-, 10–11-, and 16–17-day periods (Figs. 9, 10) for depth-averaged currents less than 10 cm s−1. For these low-frequency bands, the inertial drag was negligible. Values of Dform for diurnal tides and 4–17-day oscillations were similar. Form drags resulting from semidiurnal tides are well documented in previous studies (Nash and Moum 2001; Warner et al. 2013), but there are no quantitative estimates of the form drag for subtidal motions such as 4–17-day oscillations observed in the GoM. Our analysis demonstrates that the form drag resulting from low-frequency currents is an important flow retardation mechanism even in the presence of the large frictional drag associated with coral reefs.
The 6-month averaged power removed from the low-frequency motions by the form drag was 85 W m−1 (Table 1). Jarosz et al. (2014, manuscript submitted to J. Geophys. Res.) computed the along-transect and depth-integrated turbulent kinetic energy dissipation rate E over the bank (transects shown in Fig. 2a), where
b. Parameter dependence and linear theory
We computed the nondimensional parameters
Nondimensional bank height
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
Garrett and Kunze (2007) have suggested extra parameters including a steepness parameter,
Formation of hydrostatic lee waves and the associated form drag for
Jarosz et al. (2014, manuscript submitted to J. Geophys. Res.) reported upstream stagnation flow, flow around the bank, vortex formation, and bottom flow reversals on the lee side of the EFGB. They noted that their findings were qualitatively similar to the laboratory experiments (e.g., Baines 1979; Hunt and Snyder 1980; Vosper et al. 1999). The dynamical parameters (
c. Frictional drag coefficients associated with corals
The frictional drag coefficient over the bank is large due to high roughness elements associated with coral reefs. The observations, which were utilized to estimate this coefficient, were made when winds were weak and surface waves were small. Consequently, surface wave–induced bottom currents could not influence the bottom TKE dissipation rate. The estimated roughness lengths reflect the true roughness lengths of small-scale bathymetric features over the EFGB. The coral coverage is not continuous over the bank, and thus z0 had a range of values from those representing a smooth bottom with a roughness length of 0.001 cm to those for relatively rough regimes with z0 as high as 68 cm. Nearly ⅓ of our estimates were greater than 1 cm, and the average drag was about 6 × 10−3. These higher roughness lengths correspond to high drag coefficients and large frictional Reynolds numbers. The impact of high bottom frictional forces can be important because the strength of the bottom boundary layer stress can change lee-wave formation, decrease the form drag, and reduce the overall momentum loss to bottom obstacles/banks compared to a typically assumed free-slip boundary (Skyllingstad and Wijesekera 2004). Pratt (1986) reported that hydraulic controlled flows over straits and sills with friction can alter the dynamics of the flow such as shifting of control points from a sill to downstream locations. Although these modeling studies do not mimic the EFGB observations, the observed large bottom stress over the bank is a factor to consider in the parameterization of the form drag.
7. Summary and conclusions
Hydrographic, velocity, microstructure, and bottom pressure measurements were used to quantify the magnitude and temporal variability of the form drag, roughness lengths, and associated frictional drag coefficients over the EFGB bank. The EFGB is a rough topographic feature located on the Louisiana–Texas shelf approximately 190 km southeast of Galveston, Texas. The EFGB, about 6 km wide and 10 km long, is located at the shelf edge in 100 m of water depth with a peak rising to about 18 m below the sea surface. The area encompassed by the 100-m bathymetric contour of the bank is about 40 km2. The bank is one of the northernmost tropical coral reefs. The bank includes both coral coverage and sandy bottoms.
Nearly 6-month-long bottom pressure records (June–December 2011) showed significant semidiurnal (M2) tide, diurnal (K1 and O1) tides, and 4–17-day variability over and around the bank. Note that tidal analysis conducted by Teague et al. (2013) showed that the barotropic tidal currents were small (K1, O1 speeds < 5 cm s−1) compared to inertial currents that were over 15 cm s−1. The M2 tidal currents were less than 2 cm s−1, and the depth-averaged subtidal currents were about 4–9 cm s−1.
The estimated form drag from bottom pressure anomalies across the bank showed variability on multiple time scales. Here Dform was estimated from three pressure measurements over and around the bank while assuming that the local horizontal pressure gradient was a constant between two observational sites (e.g., Hafner and Smith 1985). It is likely that Dform was underestimated by a factor of 2. Therefore, our estimate of Dform could be treated as a lower bound of the expected value. The drag resulting from diurnal tidal motions and 4–17-day oscillations had magnitudes of about 2000 N m−1. Coefficients of the time-averaged bulk drag
Physical parameters describing the dynamics of flows over the EFGB have a range of values representing a highly nonlinear, hydrostatic flow regime with upstream blocking, where
Using measurements of the velocity and estimated TKE dissipation rates along multiple transects crossing the bank, the range of roughness lengths
Large roughness elements can impact the hydraulic flow control as well as coral reef ecology. Locations with coral coverage have increased drag, leading to more turbulence and mixing at these locations relative to regions without corals. This turbulence in turn increases the delivery of nutrients and particulates to the corals. Additionally, it helps to disperse larvae during spawning events, generating a positive feedback cycle that sustains the health of the coral community. The magnitude of the bottom stress over the bank can modify the separation of flow, lee-wave structure and strength of the form drag; hence, it is a factor to consider in the parameterization of the form drag in numerical models. Consequently, our observations and analysis of the hydrographic, velocity, microstructure, and pressure measurements suggest that the EFGB is a “hot spot” of mixing on the shelf of GoM. The analysis further demonstrates that the form drag resulting from low-frequency currents over an isolated bank on the continental shelf is an important physical process that must be parameterized to represent a wide range of flow states.
Acknowledgments
This work was sponsored by the Office of Naval Research in an NRL project referred to as Mixing Over Rough Topography (MORT) and by the BOEM in the project referred to as Currents Over Banks (COB) through the Interagency Agreement M10PG00038. Support for James Moum was provided through ONR Grant N00014-09-1-0280. The measurements were made in cooperation with the Flower Garden Banks National Marine Sanctuary, administered by the National Oceanic and Atmospheric Administration (NOAA). Assistance provided by Alexis Lugo-Fernandez of BOEM and Emma Hickerson of NOAA and the crew of the R/V Pelican and R/V Manta was greatly appreciated.
APPENDIX
Estimation of the Form Drag Based on Numerical Simulations: Sensitivity Study
The form drag over an underwater topographic feature is estimated from bottom pressure fluctuations and a horizontal slope of the bathymetry, as indicated in (1). The accuracy of the estimated drag depends on the number of pressure measurements and the location of the pressure measurements. A sensitivity study was conducted to examine the minimum number of pressure measurements required to calculate a reasonable estimate of the form drag. Here we used numerically simulated pressure fields over a sloping headland, TTP in Puget Sound, Washington, developed by Warner (2012) as part of her Ph.D. dissertation. The TTP model was based on the Regional Ocean Modeling System (ROMS), and details of the model setup is given by Warner (2012). The study helps us to quantify some uncertainties in an estimation of the form drag over the EFGB based on three pressure measurements. The ROMS results are an approximation to the real observations due to numerical limitations given by Warner (2012). It is also noted that the shape and size of TTP are different from the EFGB and that the locations of Ppods during MORT might have missed large pressure fluctuations. Therefore, the results discussed below have more uncertainties than the model results alone.
A time series of modeled bottom pressure anomalies along the TTP ridge was used to compute the form drag based on (1). The spatial resolution of the model was 50.5 m. The ridge is about 2 km wide and is nearly symmetrical (Fig. A1). Gradients of the ridge, perpendicular to the ridge axis, are strongest on the midsections of the ridge with the maximum on the right side and the minimum on the left side. Total pressure anomalies [(8)] including external, internal, and inertial components at three different locations (near bottom and top of the ridge) are shown in Fig. A2.
(a) Cross section of the ridge transect based on the model bathymetry. Spatial resolution is 50.5 m. The crosses are the data points used in the analysis. The bullets represent locations of pressure time series plotted in Fig. A2. (b) Horizontal gradient of ridge topography, dh/dx across the ridge (from Warner 2012).
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
Dynamic bottom pressure fluctuations at three different locations: X1, X2, and X3. Locations of X1, X2, and X3 are marked in bullets in Fig. A1 (from Warner 2012).
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
a. Method I

b. Method II

c. Method III

The estimated drag over the ridge from the three different methods is illustrated in Fig. A3. The drag resulting from all three methods have the same sign, even though the magnitudes of the complete solution (D0) and method III (D3) are larger than D1 [(A2)] and D2 [(A3)]. The terms D1 and D2 have similar magnitudes, but their maximum magnitudes are about 2 times smaller than the maximum of D0 and D3 (Fig. A3). However, when averaged over the tidal cycle, both D1 and D2 are about 70% of the complete solution D0. There are large variations of the slope across the ridge, and it appears that the slope is an important factor in the computations.
Estimated pressure drag from four different methods: the complete integration D0 [(A1)], mean pressure and mean lateral gradient D1 [(A2)], linearly interpolated solution D2 [(A3)], and mean pressure and maximum amplitude of lateral gradient, D3 [(A4)].
Citation: Journal of Physical Oceanography 44, 9; 10.1175/JPO-D-13-0230.1
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