1. Introduction
Investigations of atmospheric turbulence over the world's oceans have shown that the interaction of wind with surface waves results in flow characteristics that differ substantially from a horizontally homogeneous terrestrial surface layer. A simple illustration of this is given by consideration of the surface roughness. Over land, the surface roughness can often be treated as constant or slowly varying as a result of vegetative changes. Over the ocean, the surface roughness or drag is determined by the wave field, which is largely determined by the wind—the stronger the winds, the rougher the seas. Therefore, the exchange of momentum and energy is largely governed by the wave field near the ocean surface.
Above this wave-influenced layer lies a layer where the turbulent flow is governed by the generation of turbulence by wind shear and its generation–suppression by buoyancy–stratification. Many turbulent statistics obey Monin–Obukhov similarity (Obukhov 1971; Monin and Obukhov 1954) in this region, which states that these turbulent statistics are a universal function of z/L after normalization by the appropriate scaling parameters. Here, z is the height above the surface, and L is known as the Monin–Obukhov (MO) length, which represents the height at which the generation of turbulence by shear and buoyancy are equal. A number of studies have shown that MO similarity is valid as long as you are in the surface layer above wave influences (e.g., Edson and Fairall 1998; Edson et al. 2004).
As a result, marine meteorologists and physical oceanographers often divide the boundary layer close to the ocean surface into the surface layer where wind shear and buoyancy–stratification govern the turbulent flow (i.e., an MO layer) and a wave boundary layer (WBL) where additional scaling parameters are required for similarity. The search for these scaling parameters, and hypotheses for their use, has been going on for many years (e.g., Charnock 1955; Miles 1957; Hsu 1974; Plant 1982; Geernaert et al. 1986; Donelan 1990; Donelan et al. 1993; Dobson et al. 1994; Hare et al. 1997; Johnson et al. 1998; Bourassa et al. 1999; Drennan et al. 2005), but consensus remains elusive.
This study presents results from several field programs that we specifically designed to investigate the interaction of turbulent flow over surface waves in the marine surface layer. These investigations rely on a set of data collected from the R/P FLIP and an offshore tower during the Marine Boundary Layer (MBL; Hristov et al. 2003), Risø Air–Sea Experiment (RASEX; Mahrt et al. 1996), and Coupled Boundary Layers Air–Sea Transfer at Low Winds (CBLAST-LOW; Edson et al. 2007) programs sponsored by the Office of Naval Research. The study also takes advantage of a dataset collected the National Science Foundation (NSF) sponsored Climate Variability and Predictability (CLIVAR) Mode Water Dynamic Experiment (CLIMODE; Marshall et al. 2009) conducted over two winter seasons in the North Atlantic about the northern wall of the Gulf Stream.
The inclusion of the measurements made during CLIMODE allows an investigation of the transfer coefficients at high wind speeds. The CLIMODE momentum fluxes used in this investigation are provided by the direct covariance (DC) technique from two highly instrumented platforms: a moored 2.7-m-diameter foam-hull buoy and a drifting Air–Sea Interaction Spar (ASIS). The ASIS package included a Direct Covariance Flux Systems (DCFS) with a sonic anemometer, infrared hygrometer, and motion correction system that provides estimates of the momentum, sensible heat, and latent heat fluxes using the DC method. The ASIS was deployed during the January 2006 and February 2007 field programs for 10 and 14 days, respectively. A low-power version of the DCFS (without the infrared hygrometer) was deployed for 15 months on the moored buoy, as described by Weller et al. (2012) and Bigorre et al. (2013). The ASIS and buoy used in CLIMODE are shown in Fig. 1.
(left) The 2.7-m foam-hull buoy and (right) ASIS platform used during the CLIMODE program to provide DC estimates of the momentum and heat fluxes. The moored buoy was successfully deployed for 15 months in the Gulf Stream, while the ASIS was deployed for 14 days.
Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0173.1
The combined MBL, RASEX, CBLAST, and CLIMODE dataset covers a wide range of sea states and wage ages. The wave-age parameter 
The frequency of occurrence of wave ages from (top three rows) different field programs: CLIMODE, CBLAST, and MBL. The solid red line is for a wave age of 1.2 that is commonly associated with fully developed seas. Values <1.2 indicate developing (young) seas, while values >1 indicate decaying (old) seas. (bottom) Composite of all the data.
Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0173.1
2. Parameterizations of momentum exchange
Direct estimates of the momentum flux (surface stress) vs relative wind speed adjusted to 10 m and neutral stability from four field programs and five platforms. No ship data are included in the analysis to reduce the effect of flow distortion. (top) The individual flux estimates from each experiment and (bottom) the data averaged over wind speed bins. The dashed line represents the original COARE 3.0 bulk algorithm and the solid black line is the modified COARE 3.5 algorithm, as described in the text.
Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0173.1
a. Dimensionless shear
Flux–profile measurements were made during the RASEX, MBL and CBLAST programs that utilized two oversea towers and the R/P FLIP as shown in Fig. 4. The setups used on the RASEX and CBLAST towers are described by Vickers and Mahrt (1999) and Edson et al. (2007), respectively. Briefly, the RASEX results are limited to the “long”-fetch (i.e., fetches >15 km) conditions discussed in Mahrt et al. (1996) where the water depth is approximately 3 m at the tower. The cup anemometers used in this study were positioned at 7, 15, 20, 29, and 38 m above mean sea level. A 10-min averaging time was used to compute the fluxes from 3-axis sonic anemometers located at 6, 10, 18, and 32 m.
The three platforms used to directly measure flux–profile relationships during the CBLAST, MBL, and RASEX programs. (left) The ASIT tower used in CBLAST where the profiling mast is at far left and the mast holding the fixed sensors is nearer the platform. (middle) The R/P FLIP used in MBL and (right) the tower used in the RASEX program.
Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0173.1
The CBLAST results are restricted to wind directions between 190° and 245° to provide “infinite” fetch and minimize the flow distortion by the tower, which faces the open ocean to its south. The water depth is approximately 15 m at the tower. A profiling mast (Fig. 1) supporting a moving sensor package holding a 2-axis sonic anemometer was used to measure the wind speed at approximately 3, 5, 7, 10, 13.5, and 15.5 m after adjustment for tides. The array was used to calibrate 3-axis sonic anemometers deployed at fixed locations of approximately 4, 6.5, 10, 15, 18, and 20 m. A 20-min average was used to compute the fluxes in unstable conditions, while the average of two 10-min-averaged fluxes was used in stable conditions.
The MBL results are limited to a 7-day period with optimal winds from the northwest as described by Friehe et al. (2001). The setup shown in Fig. 4 for the R/P FLIP consisted of a vertical array of five 3-axis sonic anemometers to measure momentum and buoyancy fluxes at approximately 4, 5, 9, 14, and 18 m above mean sea level. Eleven cup/vane anemometers were used to measure the mean wind speed between 3 and 17 m above the ocean surface. The cup–vane pairs located on either side of the R/P FLIP's port boom were excluded from the analysis because of flow distortion. The remaining nine cup anemometers were locate at approximately 2.9, 3.8, 4.7, 5.6, 6.8, 7.8, 12.7, 14.7, 15.7, and 16.2 m above mean sea level. These remaining cups were corrected for over speeding and the sonics were motion corrected as described by Edson et al. (1998) and Miller et al. (2008). A 15-min average is used to compute the fluxes. The 10–20-min averaging times in these experiments are chosen to maximize the correlation between mean wind speed and wind stress (Mahrt et al. 1996), but are short enough to limit the impact of nonstationarity on the fluxes.
As shown in Fig. 2, the CBLAST data are often characterized by old seas with low wind conditions over swell. A number of studies (e.g., Smedman et al. 1994, 1999) have shown that swell can have a significant impact on wind profiles under these conditions. For example, the large-eddy simulations (LES) conducted by Sullivan et al. (2008) show that fast moving swell can impact the wind profiles throughout the surface layer under light wind conditions. Therefore, the data used in the following investigation of the dimensionless shear is limited to wave ages 
In all experiments, the wind shear was calculated from a least squares fit to U versus ln(z) using a second-order polynomial. The dimensionless shear was computed using the local values of the momentum flux and MO length as described by Vickers and Mahrt (1999). The dimension shear was computed at the sonic anemometer heights of 6 and 10 m for RASEX; 4, 6.5, and 10 m for CBLAST; and 5 m for FLIP. Although the depth of the WBL for momentum exchange is not universally defined—for example, see the discussions in Chalikov (1986, 1995), Belcher and Hunt (1993), Mastenbroek et al. (1996), Kudryavtsev et al. (2001), Moon et al. (2004), and Chalikov and Rainchik (2011)—these heights are expected to be within the surface layer and generally above the WBL for 
Measurements of the dimensionless shear from the RASEX, MBL, and CBLAST experiments are shown in Fig. 5. The bin-averaged data agree very well with the current formulations used in the COARE 3.0 algorithm (Fairall et al. 2003), which is based on the Kansas (Businger et al. 1971) and over ice Surface Heat Budget of the Arctic Ocean (SHEBA) (Persson et al. 2002) experiments. The agreement with the commonly used Businger–Dyer formulations (Businger 1988) in unstable conditions is not surprising since this form of the dimensionless shear has been successfully used to compute bulk fluxes over the oceans for decades. A fit to the data between 
(top) Individual estimates of the dimensionless shear vs the stability parameter z/L from the RASEX, MBL, and CBLAST programs for 
Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0173.1
There is more uncertainty in the dimensionless shear under stable conditions, but the same can be said for surface layers over land. The average data follow the Businger–Dyer function out to z/L ~ 0.5 but then increase less rapidly. The COARE 3.0 algorithm relies on the formulation presented by Beljaars and Holtslag (1991) for stable conditions, which models the roll off under highly stable conditions using several tunable parameters. The values used in the COARE 3.0 function agree well with the bin-averaged data as shown in Fig. 5. It should be noted that the data do not compare well with the RASEX parameterization under stable conditions reported by Vickers and Mahrt (1999). However, this discrepancy is effectively removed by limiting the data to wind directions that provide long fetch. This restriction is believed to remove many of the complications that arise because of surface-layer adjustment from land to sea over short fetch as described in Mahrt et al. (1998, 2001).
The agreement between the individual datasets and previously used parameterizations strongly suggests that the use of flux–profile relationships based on MO similarity is valid in the marine surface layer for 
This suggests that the data may still be influenced by waves, which violates the assumptions made for MO similarity. For example, upon close examination of the individual datasets, the RASEX data taken over shallow water with generally younger sea conditions fall slightly below the CBLAST and FLIP taken under more mature sea conditions. However, these differences are subtle, and an investigation on the impact of surface waves on shear production is ongoing. Therefore, for the remainder of this investigation, it is assumed that the measurements are generally made above the WBL (i.e., for z ≥ 4 m) and that MO similarity is valid. Stability corrections are made using the COARE 3.0 algorithm.
b. Neutral drag coefficient
The investigation will focus on the parameterization of the rough-flow component through the Charnock coefficient. This coefficient was originally referred to as the Charnock constant but is now known to vary as a function of, for example, wind speed, wave age, and sea state. The behavior of the Charnock coefficient as a function of wind speed is investigated in section 2c. This is followed by investigations of the wage-age and sea-state dependence of the Charnock coefficient in sections 2d and 2e; where wave age quantifies the stage of wave development, while sea state characterizes the current conditions in term of, for example, wave height, wave period, and wave steepness. The investigation then provides a means to reconcile the wind speed– and wave age–dependent formulation over the open ocean in section 3, and discusses their behavior at high and low winds in sections 3a and 3b. The investigation concludes with a summary that includes a comparison of the DC momentum fluxes versus the parameterizations developed in this study in section 3c.
c. Wind speed–dependent formulation
(top),(middle) Direct estimates of the drag coefficient plotted vs relative wind speed. The values have been adjusted to 10 m and corrected for atmospheric stability using MO similarity theory (Fairall et al. 2003). (top) The individual data from each platform. (middle) The bin-averaged drag coefficients vs wind speed where the error bars represent the standard deviation about the mean. The dashed line represents the COARE 3.0 algorithm, while the solid line is a modification to this algorithm designated as COARE 3.5. The dashed–dotted line is the function provided by Large and Pond (1981). (bottom) Estimates of the Charnock coefficient averaged over wind speed bins. The error bars represent the standard error about the mean.
Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0173.1
The neutral drag coefficients are in good agreement with COARE 3.0 over moderate wind conditions. However, there are differences at the lowest and highest wind speeds where COARE 3.0 over- and underestimates the drag, respectively. Therefore, the combined dataset is used to refine the dependence of the Charnock coefficient as a function of wind speed. This is accomplished through the following steps.
Individual estimates of the neutral drag coefficients at 10 m are computed from measurements following (5) as shown by the upper panel of Fig. 6.
The measured
Likewise, the measurements of
The bin-averaged values of
The bin-averaged drag coefficients and friction velocities are used to compute the roughness length for rough flow from
The roughness length and friction velocities are used to compute the Charnock coefficient from
d. Wave age–dependent formulation
This analysis relies on the wide range of wage ages captured during CBLAST and CLIMODE deployments and, to a lesser extent, the wind event captured during the MBL experiment to reduce the problem of self-correlation (Donelan et al. 1992; Lange et al. 2004; Drennan et al. 2005). The Charnock coefficient is computed using individual estimates of the drag coefficient and friction velocity using steps 5) and 6) as in section 2c. The natural log of the Charnock coefficient 
(top) Individual and bin-averaged estimates of the Charnock coefficient vs inverse wave age where the error bars represent the standard deviation about the mean. (middle) The bin-averaged data on a linear scale. The error bars represent the standard error about the mean. The solid line is a fit to the data using (15), while the other lines present previously reported relationships as labeled. The dashed vertical line represents the fully developed value of inverse wave age. (bottom) All of the observations found over the narrow range of phase speeds (m s−1) vs inverse wave age. The lines representing the COARE 3.5 function are then generated by fixing the phase speed in the middle of each range and allowing the friction velocity to vary.
Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0173.1
As in previous studies, the Charnock coefficient is seen to increase with inverse wave age reflecting the commonly held view that the younger the waves, the rougher the surface. Research has shown that a fully developed sea occurs at 
These open-ocean values provide significantly less variability in the Charnock coefficient than the coefficients reported in shallow water and fetch-limited environments by Smith et al. (1992), Johnson et al. (1998), and Oost et al. (2002) as shown by the middle panel in Fig. 7. These coefficients have generally been tuned to data over a narrow range of wave ages. However, Charnock coefficients determined experimentally over the ocean generally range from 0.011 to 0.018 from fully developed seas (Kraus and Businger 1994). As shown in Fig. 7, the COARE 3.5 parameterization spans that same range for 
Obviously, the value of the wave age and therefore the value of the Charnock coefficient can be driven by variability in 
The RASEX data were not included in the bin-averaged results because they did not exhibit similar drag coefficients at the same inverse wave ages of the other datasets. The RASEX experiments were conducted in shallow water where bottom friction drives much of the variability in 
e. Sea state– and wave age–dependent formulation
The surface roughness scaled by significant wave height averaged over inverse wave age bins. The error bars represent (top) the standard deviation and (bottom) the standard error about the mean. The solid line is from (19) with D = 0.09, while the other lines present previously reported relationships as labeled. These parameterizations are normalized by the significant wave height using σH = 4σ as appropriate where σ is the RMS value used in some of the previous studies. The dashed vertical line represents the fully developed value of inverse wave age.
Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0173.1
3. Reconciling wind speed– and wave age–dependent formulations
Inverse wave age plotted vs relative wind speed. (top) The individual data from each experiment, and (middle) the data averaged over wind speed bins. The RASEX data are not included in this average. The dashed black line represents the inverse wave age commonly associated with fully developed seas. The dashed–dotted line is a linear fit to the averaged data, while the solid line is a third-order fit. (bottom) As in Fig. 6, but with the addition of the green line representing the function derived by ECMWF as given by (20), and the red line that combines the third-order fit with (15).
Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0173.1
Perhaps more surprising is the agreement between ECMWF and the COARE 3.5 algorithm given the nature of the two parameterizations. Specifically, it has been argued that some of the discrepancy between bulk estimates and direct measurements of the fluxes reflect the variability in wave properties at any given wind speed. However, the COARE algorithm matches the observations well without any wave information. Furthermore it is nearly identical to the function representing the globally averaged drag coefficient from the wave age–based model. This begs the following question: why do wave age– and wind speed–dependent formulation give such similar results?
The answer is found by looking at the relationship between inverse wave age and wind speed shown in Fig. 9. The measurements indicate that fully developed seas (
The third-order fit also highlights the observation that the wage age goes to a finite positive value under very low wind conditions because of the ubiquitous nature of swell (Hanley et al. 2010) and the role of gustiness in maintaining momentum exchange under these conditions (Fairall et al. 1996). Perhaps more importantly, the fit also supports the idea (e.g., Hsu 1974) that the wave field saturates at high wind conditions. The behavior of momentum exchange at extremely high and very low winds is explored further in the following sections.
a. High wind speeds
The measured drag coefficients are larger than a number of previous open-ocean formulas such as Liu et al. (1979), Large and Pond (1981), and Smith (1988) at high winds. However, there is increasing evidence from shipboard observations (e.g., those that were used to develop COARE 3.0) that measured drag coefficients are significantly larger than these parameterizations over the open ocean. Direct covariance and mean wind measurements from ship-based observations suffer from flow distortion and imperfect motion correction (Edson et al. 1998), which is why the investigation described here has focused on data from fixed towers and low-profile platforms designed to minimize flow distortion.
Nonetheless, recent observational studies and numerical model predictions indicate that the drag coefficient should level off and even decrease at extreme wind conditions in order for hurricanes to develop. A Charnock coefficient that continues to increase with increasing winds does not support those observations and predictions. However, although the data are sparse above 22 m s−1 in this study, the values of the Charnock coefficient shown in Fig. 6 indicate that they level off at 
These results are consistent with the recent investigations by Foreman and Emeis (2010) and Andreas et al. (2012), which provide insight into the asymptotic behavior of the drag coefficient at extreme winds. In their approach, the drag coefficient is determined by a fit of the friction velocity to the wind speed for wind speeds that correspond to fully rough seas. This approach is used to produce the result shown in Fig. 10, where the friction velocity is plotted against 
(top) Individual estimates of friction velocity vs relative wind speed and (middle) their wind speed bin averages. The dashed line is equal to 
Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0173.1
b. Low wind speeds
The bin-averaged drag coefficients fall below both the COARE 3.0 and 3.5 parameterization at the lowest wind speed (e.g., 
This upward exchange of momentum as a result of wave-driven winds (Hanley and Belcher 2008) is expected to reduce the total momentum flux, which would act to reduce the drag under these conditions. As a result, the roughness appears smoother than the smooth-flow conditions measured under laboratory conditions. The impact of the upward momentum flux has been investigated in a number of studies including those by Drennan et al. (1999), Smedman et al. (1994, 1999, 2003), Grachev and Fairall (2001), Grachev et al. (2003), and Hanley and Belcher (2008). Therefore, there is a growing consensus that wind–swell interaction is the leading cause for the reduction of the drag on low winds (Hanley et al. 2010).
c. Flux comparison
This investigation is concluded with a comparison of the DC estimates of the friction velocity versus bulk estimates using wind speed–, wave age–, and wave slope–dependent parameterization of the Charnock coefficient using (13), (15), and (18), respectively, as shown in Fig. 11. The DC and bulk estimates from RASEX are also plotted to provide independent comparisons (i.e., using data that are not used to develop the parameterizations) and to test COARE 3.5 in a fetch-limited shallow water environment. The RASEX dataset is that used in Vickers and Mahrt (1999). However, the data for all values of the fetch are used in this comparison.
Scatterplots of DC estimate of the surface stress vs (top) (left) the COARE 3.0 algorithm and (right) wind speed; and (bottom) (left) wave age, and (right) wave slope based parameterization developed in this study for COARE 3.5. The red points are from the CLIMODE buoy, the blue from MBL, the green from CBLAST, and the black from RASEX.
Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0173.1
The performance of each parameterization is determined from the RMS difference between the direct and bulk estimates of the flux. The overall RMS would be overwhelmed by the CLIMODE dataset because of its large size. Therefore, the RMS is computed from the individual datasets and then averaged. These results are summarized in Table 1, where we have added the Smith et al. (1992) and Donelan (1990) parameterizations for comparison.
The RMS difference between the directly measured stress and estimates using COARE 3.0 and the wind speed–, wave age–, and wave slope–dependent parameterizations developed in this study for inclusion in COARE 3.5. The average is the mean of the values from the three experiments used to develop the parameterizations and the RASEX program is included as an independent test. The percent uncertainty represents the average RMS divided by the average mean (× 100) Results based on the parameterizations reported by Smith et al. (1992) and Donelan (1990) are provided for comparison.
The results show that the wind speed–dependent formulation is most accurate for the open-ocean datasets. This is true for both the average and individual datasets suggesting that the wind speed–dependent formulation in COARE 3.5 is an improvement over COARE 3.0 even at the lower wind speeds experienced in CBLAST and MBL. The wind speed–dependent formulation also gives the best agreement with the RASEX data. The wave slope–dependent formulation is slightly more accurate than the wave age–dependent formulation. However, all of the formulations developed in this investigation give similar agreement over the wide range of wind speed, wave age, and wave slopes found in the combined datasets.
4. Summary
The combination of data collected over a wide range of wind speed, sea state, and atmospheric stability conditions during the RASEX, MBL, CBLAST, and CLIMODE programs is used to improve parameterization of the drag coefficient over the ocean. All of these programs measured the momentum, heat, and mass fluxes directly using the DC method. The RASEX, MBL, and CBLAST programs also measured wind profiles to estimate the dimensionless shear over the ocean. The combined dataset is in good agreement with the dimensionless shear formulation used in COARE 3.0 that is based on over-land and over-ice experiments. The dimensionless shear shows little influence of the waves on the wind profiles above 4 m indicating that the measurements are above the WBL for momentum—at least for 
The study then investigates the behavior of the surface roughness and drag coefficients at high wind speed using data collected during the CLIMODE program. The new data resulted in minor changes to the wind speed–dependent Charnock coefficient used in the COARE 3.0 algorithm at wind speeds over 13 m s−1. These modifications are included in the COARE 3.5 algorithm, which gives good agreement with the stress estimates collected under a wide range of wind and wave conditions during the CBLAST, MBL, RASEX, and CLIMODE programs.
Numerous investigations have shown that the Charnock coefficient is also dependent on the state of the surface wave field. Therefore, the combined dataset is used to develop wave age– and wave slope–dependent parameterizations of the surface roughness, which are included in COARE 3.5. These parameterizations also give good agreement with the directly measured momentum fluxes over a wide range of sea states and wave ages. However, the COARE 3.5 wind speed–dependent formulation is shown to provide better agreement with the DC stress measurements without any wave information. Furthermore, it is nearly identical to the function representing the globally averaged drag coefficient from a wave age–based model run at the ECMWF.
These findings are easily reconciled using the observed linear relationship between wind speed and inverse wave age over the open ocean. The reason for this is found in the wind and wave data; namely, in storm passage after storm passage, the wave age varies nearly linearly with wind speed. The composite of all these storms shows that young waves are almost always found under high wind conditions, and old waves are found in their wake after winds have calmed down. Therefore, there is not a pronounced functional difference between drag coefficients based on wind speed and on wave age over the open ocean up to approximately 25 m s−1.
It is fair to ask if the observations used in this analysis are representative of the entire ocean, since they were mainly taken in midlatitudes. However, the good agreement between the observations and the ECMWF globally averaged fields suggest that the COARE 3.5 parameterization can be used to give accurate momentum fluxes over the open ocean, with the greatest uncertainty at low wind speeds in the presence of swell. However, the ubiquitous nature of swell (e.g., Hanley et al. 2010) and its overall tendency to reduce the total momentum flux argues for a parameterization that reduces the drag compared to COARE 3.0 under light wind conditions.
It is also evident that the nearly linear relationship between wind speed and inverse wave age breaks down in the fetch-limited and shallow-water environment that characterized the RASEX program as shown in Fig. 9. However, the wave age– and wave slope–dependent parameterizations of the Charnock coefficient give good agreement with the directly measured fluxes for all of the field programs including RASEX. Although these functions are tuned to data with infinite fetch used in this analysis, this implies that the parameterizations are applicable to a wide range of marine environments.
Lastly, the results argue that it is difficult to improve upon a wind speed–dependent parameterization under any conditions. This may simply be due to the fact that wind-driven waves support the majority of the surface stress, and the modulation of the surface stress by longer waves is a second-order effect under most conditions. Furthermore, the inclusion of additional dependent variables with their own measurement uncertainties in the bulk flux algorithm tends to increase the uncertainties in the fluxes. Therefore, the potential improvements from the wave age– and wave slope–dependent parameterizations may be better utilized in applications where higher quality wave measurements are available.
Acknowledgments
This work was funded by the National Science Foundation Grant OCE04-24536 as part of the CLIVAR Mode Water Dynamics Experiment (CLIMODE) and the Office of Naval Research Grant N00014-05-1-0139 as part of the CBLAST-LOW program. We thank Jon Ware and Steve Faluotico (WHOI-AOP&E) and the personnel of the Upper Ocean Processes group (WHOI-PO), who designed, calibrated, maintained, and deployed the components of the ASIT, ASIS, and surface mooring. This paper is dedicated to the memory of Prof. Carl Friehe at the University of California, Irvine, who conceived and led many of these investigations. His scientific insight, curiosity, and humor remain an inspiration for all of us.
APPENDIX
Relative Wind Speed
The CLIMODE dataset taken in the vicinity of the Gulf Stream is a good dataset to demonstrate the importance of including 
Drag coefficient vs wind speed for the COARE 3.5 algorithm computed using (left) relative winds and (right) absolute winds. (top) Individual direct-covariance stress estimates and (bottom) bin-averaged values. Colors indicate relatively small (blue) and large (red) sea surface current speeds that correspond to the buoys being located outside and within the Gulf Stream, respectively.
Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0173.1
REFERENCES
Andreas, E. L, L. Mahrt, and D. Vickers, 2012: A new drag relation for aerodynamically rough flow over the ocean. J. Atmos. Sci., 69, 2520–2537.
Banner, M. L., and W. L. Peirson, 1998: Tangential stress beneath wind-driven air-water interfaces. J. Fluid Mech., 364, 115–145.
Belcher, S. E., and J. C. R. Hunt, 1993: Turbulent shear flow over slowly moving waves. J. Fluid Mech.,251, 109–148.
Beljaars, A. C. M., and A. A. M. Holtslag, 1991: Flux parameterization over land surfaces for atmospheric models. J. Appl. Meteor., 30, 327–341.
Bigorre, S., R. A. Weller, J. Lord, J. B. Edson, and J. D. Ware, 2013: A surface mooring for air–sea interaction research in the Gulf Stream. Part II: Analysis of the observations and their accuracies. J. Atmos. Oceanic Technol.,30, 450–469.
Black, P. G., and Coauthors, 2007: Air–sea exchange in hurricanes: Synthesis of observations from the coupled boundary layer air–sea transfer experiment. Bull. Amer. Meteor. Soc., 88, 357–374.
Bourassa, M. A., D. G. Vincent, and W. L. Wood, 1999: A flux parameterization including the effects of capillary waves and sea state. J. Atmos. Sci., 56, 1123–1139.
Businger, J. A., 1988: A note on the Businger–Dyer profiles. Bound.-Layer Meteor., 42, 145–151.
Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Flux profile relationships in the atmospheric surface layer. J. Atmos. Sci., 28, 181–189.
Chalikov, D. V., 1986: Numerical simulation of the boundary layer above waves. Bound.-Layer Meteor., 34, 63–98.
Chalikov, D. V., 1995: The parameterization of the wave boundary layer. J. Phys. Oceanogr., 25, 1335–1349.
Chalikov, D. V., and S. Rainchik, 2011: Coupled numerical modeling of wind and waves and the theory of the wave boundary layer. Bound.-Layer Meteor., 138, 1–41.
Charnock, H., 1955: Wind stress on a water surface. Quart. J. Roy. Meteor. Soc., 81, 639–640.
Dobson, F. W., S. D. Smith, and R. J. Anderson, 1994: Measuring the relationship between wind stress and sea state in the open ocean in the presence of swell. Atmos.–Ocean, 32, 237–256.
Dobson, F. W., R. J. Anderson, P. K. Taylor, and M. J. Yelland, 1999: Storm Wind Study II: Open ocean wind and sea state measurements. Proceedings of Symposium on the Wind-Driven Air–Sea Interface: Electromagnetic and Acoustic Sensing, Wave Dynamics and Turbulent Fluxes, M. L. Banner, Ed., University of New South Wales, 295–296.
Donelan, M. A., 1990: Air–sea interaction. The Sea, B. LeMehaute and D. M. Hanes, Eds., Ocean Engineering Science, Vol. 9, Wiley and Sons, 239–292.
Donelan, M. A., M. G. Skafel, H. Graber, P. Liu, D. Schwab, and S. Venkatesh, 1992: On the growth rate of wind-generated waves. Atmos.–Ocean, 30, 457–478.
Donelan, M. A., F. W. Dobson, S. D. Smith, and R. J. Anderson, 1993: On the dependence of sea surface roughness on wave development. J. Phys. Oceanogr., 23, 2143–2149.
Drennan, W. M., H. C. Graber, and M. A. Donelan, 1999: Evidence for the effects of swell and unsteady winds on marine wind stress. J. Phys. Oceanogr.,29, 1583–1864.
Drennan, W. M., P. K. Taylor, and M. J. Yelland, 2005: Parameterizing the sea surface roughness. J. Phys. Oceanogr.,35, 835–848.
Edson, J. B., and C. W. Fairall, 1998: Similarity relationships in the marine atmospheric surface layer for terms in the TKE and scalar variance budgets. J. Atmos. Sci., 55, 2311–2328.
Edson, J. B., A. A. Hinton, K. E. Prada, J. E. Hare, and C. W. Fairall, 1998: Direct covariance flux estimates from mobile platforms at sea. J. Atmos. Oceanic Technol., 15, 547–562.
Edson, J. B., C. J. Zappa, J. A. Ware, W. R. McGillis, and J. E. Hare, 2004: Scalar flux profile relationships over the open ocean. J. Geophys. Res., 109, C08S09, doi:10.1029/2003JC001960.
Edson, J. B., and Coauthors, 2007: The Coupled Boundary Layers and Air–Sea Transfer Experiment in low winds. Bull. Amer. Meteor. Soc., 88, 341–356.
Fairall, C. W., E. F. Bradley, D. P. Rogers, J. B. Edson, and G. S. Young, 1996: Bulk parameterization of air–sea fluxes for TOGA COARE. J. Geophys. Res., 101, 3747–3764.
Fairall, C. W., E. F. Bradley, J. E. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk parameterization of air–sea fluxes: Updates and verification for the COARE algorithm. J. Climate, 16, 571–591.
Foreman, R. J., and S. Emeis, 2010: Revisiting the definition of the drag coefficient in the marine atmospheric boundary layer. J. Phys. Oceanogr., 40, 2325–2332.
Friehe, C. A., J. A. Smith, K. F. Rieder, N. E. Huang, J.-P. Giovanangeli, and G. L. Geernaert, 2001: Wind, stress and wave directions. Wind Stress over the Ocean, I. S. F. Jones and Y. Toba, Eds., Cambridge University Press, 232–241.
Geernaert, G. L., K. B. Katsaros, and K. Richter, 1986: Variation of the drag coefficient and its dependence on sea state. J. Geophys. Res., 91 (C6), 7667–7679.
Grachev, A. A., and C. W. Fairall, 2001: Upward momentum transfer in the marine boundary layer. J. Phys. Oceanogr., 31, 1698–1711.
Grachev, A. A., C. W. Fairall, J. E. Hare, J. B. Edson, and S. D. Miller, 2003: Wind stress vector over ocean waves. J. Phys. Oceanogr., 33, 2408–2429.
Hanley, K. E., and S. E. Belcher, 2008: Wave-driven wind jets in the marine atmospheric boundary layer. J. Atmos. Sci., 65, 2646–2660.
Hanley, K. E., S. E. Belcher, and P. P. Sullivan, 2010: A global climatology of wind–wave interaction. J. Phys. Oceanogr., 40, 1263–1282.
Hare, J. E., T. Hara, J. B. Edson, and J. M. Wilczak, 1997: A similarity analysis of the structure of air flow over surface waves. J. Phys. Oceanogr., 27, 1018–1037.
Hersbach, H., 2011: Sea surface roughness and drag coefficient as functions of neutral wind speed. J. Phys. Oceanogr., 41, 247–251.
Hicks, B. B., 1978: Some limitations of dimensional analysis and power laws. Bound.-Layer Meteor., 14, 567–569.
Hristov, T. S., S. D. Miller, and C. A. Friehe, 2003: Dynamical coupling of wind and ocean waves through wave-induced air flow. Nature, 422, 55–58.
Hsu, S. A., 1974: A dynamic roughness equation and its application to wind stress determination at the air–sea interface. J. Phys. Oceanogr., 4, 116–120.
Johnson, H. K., and H. Kofoed-Hansen, 2000: Influence of bottom friction on sea surface roughness and its impact on shallow water wind wave modeling. J. Phys. Oceanogr., 30, 1743–1756.
Johnson, H. K., and H. J. Vested, 1992: Effects of water waves on wind shear stress for current modeling. J. Atmos. Oceanic Technol., 9, 850–861.
Johnson, H. K., J. Højstrup, H. J. Vested, and S. E. Larsen, 1998: On the dependence of sea surface roughness on wind waves. J. Phys. Oceanogr., 28, 1702–1716.
Kitaigorodskii, S. A., 1973: The Physics of Air–Sea Interaction. Cambridge University Press, 273 pp.
Kraus, E. B., and J. A. Businger, 1994: Atmosphere-Ocean Interaction. Oxford University Press, 362 pp.
Kudryavtsev, V. N., V. K. Makin, and J. F. Meirink, 2001: Simplified model of the air flow above the waves. Bound.-Layer Meteor., 100, 63–90.
Lange, B., H. K. Johnson, S. Larsen, J. Højstrup, H. Kofoed-Hansen, and M. J. Yelland, 2004: On detection of a wave age dependency for the sea surface roughness. J. Phys. Oceanogr., 34, 1441–1458.
Large, W. G., and S. Pond, 1981: Open ocean momentum flux measurements in moderate to strong winds. J. Phys. Oceanogr., 11, 324–336.
Liu, W. T., K. B. Katsaros, and J. A. Businger, 1979: Bulk parameterization of air–sea exchanges of heat and water vapor including the molecular constraints at the interface. J. Atmos. Sci., 36, 1722–1735.
Mahrt, L., D. Vickers, J. Howell, J. Højstrup, J. M. Wilczak, J. B. Edson, and J. E. Hare, 1996: Sea surface drag coefficients in RASEX. J. Geophys. Res., 101, 14 327–14 335.
Mahrt, L., D. Vickers, J. Edson, J. Sun, J. Højstrup, J. Hare, and J. Wilczak, 1998: Heat flux in the coastal zone. Bound.-Layer Meteor., 86, 421–446.
Mahrt, L., D. Vickers, J. Edson, J. Wilczak, J. Hare, and J. Højstrup, 2001: Vertical structure of offshore flow during RASEX. Bound.-Layer Meteor., 100, 47–61.
Marshall J., and Coauthors, 2009: Observing the cycle of convection and restratification over the Gulf Stream system and the subtropical gyre of the North Atlantic Ocean: Preliminary results from the CLIMODE field campaign. Bull. Amer. Meteor., Soc., 90, 1337–1350.
Martin, M. J., 1998: An investigation of momentum exchange parameterizations and atmospheric forcing for the coastal mixing and optics program. M.S. thesis, Massachusetts Institute of Technology, Boston, MA, and the Woods Hole Oceanographic Institution, Woods Hole, MA, 83 pp.
Mastenbroek, C. V., V. K. Makin, M. H. Garrat, and J. P. Giovanangeli, 1996: Experimental evidence of the rapid distortion of the turbulence in the air flow over water waves. J. Fluid Mech., 318, 273–302.
Miles, J. W., 1957: On the generation of surface waves by shear flows. J. Fluid Mech., 3, 185–204.
Miller, S. D., T. S. Hristov, J. B. Edson, and C. A. Friehe, 2008: Platform motion effects on measurements of turbulence and air–sea exchange over the open ocean. J. Atmos. Oceanic Technol., 25, 1683–1694.
Monin, A. S., and A. M. Obukhov, 1954: Basic laws of turbulent mixing in the surface layer of the atmosphere. Tr. Geofiz. Inst., Akad. Nauk SSSR,151, 163–187.
Moon, I.-J., T. Hara, I. Ginis, S. E. Belcher, and H. L. Tolman, 2004: Effect of surface waves on air–sea momentum exchange. Part I: Effect of mature and growing seas. J. Atmos. Sci., 61, 2321–2333.
Nordeng, T. E., 1991: On the wave age dependent drag coefficient and roughness length at sea. J. Geophys. Res., 96, 7167–7174.
Obukhov, A. M., 1971: Turbulence in an atmosphere with a non-uniform temperature. Bound.-Layer Meteor.,2, 7–29.
Oost, W. A., G. J. Komen, C. M. J. Jacobs, and C. van Oort, 2002: New evidence for a relation between wind stress and wave age from measurements during ASGAMAGE. Bound.-Layer Meteor., 103, 409–438.
Persson, O. G. P., C. W. Fairall, E. L Andreas, P. S. Guest, and D. K. Perovich, 2002: Measurements near the atmospheric surface flux group tower at SHEBA: Near-surface conditions and surface energy budget. J. Geophys. Res., 107, 8045, doi:10.1029/2000JC000705.
Plant, W. J., 1982: A relationship between wind stress and wave slope. J. Geophys. Res., 87, 1961–1967.
Powell, M. D., P. J. Vickery, and T. A. Reinhold, 2003: Reduced drag coefficient for high wind speeds in tropical cyclones. Nature, 422, 279–283.
Smedman, A. S., M. Tjernström, and U. Högström, 1994: The near-neutral marine atmospheric boundary layer with no surface shearing stress: a case study. J. Atmos. Sci., 51, 3399–3411.
Smedman, A. S., U. Högström, H. Bergström, A. Rutgersson, K. K. Kahma, and H. Pettersson, 1999: A case study of air–sea interaction during swell conditions. J. Geophys. Res., 104 (C11), 25 833–25 852.
Smedman, A. S., X. G. Larsén, U. Högström, K. K. Kahma, and H. Pettersson, 2003: Effect of sea state on the momentum exchange over the sea during neutral conditions. J. Geophys. Res., 108, 3367, doi:10.1029/2002JC001526.
Smith, S. D., 1988: Coefficients for sea surface wind stress, heat flux, and wind profiles as a function of wind speed and temperature. J. Geophys. Res., 93 (C12), 15 467–15 472.
Smith, S. D., and Coauthors, 1992: Sea surface wind stress and drag coefficients: The HEXOS results. Bound.-Layer Meteor., 60, 109–142.
Sullivan, P. P., J. B. Edson, T. Hristov, and J. C. McWilliams, 2008: Large-eddy simulations and observations of atmospheric marine boundary layers above non-equilibrium surface waves. J. Atmos. Sci., 65, 1225–1245.
Taylor, P. K., and M. J. Yelland, 2001: The dependence of sea surface roughness on the height and steepness of the waves. J. Phys. Oceanogr., 31, 572–590.
Vickers, D., and L. Mahrt, 1999: Observations of non-dimensional wind shear in the coastal zone. Quart. J. Roy. Meteor. Soc., 125, 2685–2702.
Webster, P. J., and R. Lukas, 1992: TOGA COARE: The coupled ocean–atmosphere response experiment. Bull. Amer. Meteor. Soc.,73, 1377–1416.
Weller, R. A., S. P. Bigorre, J. Lord, J. D. Ware, and J. B. Edson, 2012: A surface mooring for air–sea interaction research in the Gulf Stream. Part I: Mooring design and instrumentation. J. Atmos. Oceanic Technol., 29, 1363–1376.
