## 1. Introduction

*z*is the roughness,

_{o}*u*∗ is the friction velocity, and

*g*is the gravitational constant;

*α*, now known as the Charnock parameter, was assumed initially to be constant. However, it has long been recognized that (1) with a constant

*α*does not adequately describe many datasets. In particular, it was speculated that

*α*was a function of some parameter related to the sea state. Kitaigorodskii and Volkov (1965) proposed that

*α*should depend on the fetch or duration of the wind event. This “wave age” dependence with

*α*as a function of wave age, expressed as either

*c*/

_{p}*u*∗ or

*c*/

_{p}*U*

_{10}

*, found qualitative support in the data of Donelan (1982). Here*

_{N}*c*is the wave phase speed at the peak of the spectrum, and

_{p}*U*

_{10}

*is the 10-m neutral wind speed.*

_{N}*H*/

_{s}*L*, where

*H*is the significant wave height and

_{s}*L*is the wavelength of the dominant waves. Using (1), the roughness can then be expressed in the dimensionless form

*c*= (

*gL*/2

*π*)

^{1/2}, which is valid for deep water. Here we take the “dominant” wave properties to be represented by those at the peak of the amplitude spectrum, denoted with subscript

*p*. Donelan (1990) sought a more general wave-age dependence,

*α*=

*z*

_{o}

*g*/

*u*

^{2}∗ =

*f*(

*u*∗/

*c*

_{p}), where both the dimensionless roughness

*α*and the wave age depend on

*u*∗. This leads to a potential spurious correlation among the variables, one made worse by the fact that for most experiments, variability in

*u*∗/

*c*is dominated by variability in

_{p}*u*∗ (see, e.g., Smith et al. 1992). Lange et al. (2004) showed that the effect of spurious correlation dominated any wave-age signal in the data they examined. In contrast, Drennan et al. (2003; hereinafter DGHQ) sought to reduce this problem by combining data from many field experiments representing a variety of conditions, and grouping the data by

*u*∗, so that the variability in wave age would arise primarily from variations in

*c*. Their resultant relation,

_{p}*L*represents the wavelength at the peak of the wave spectrum. The formula was found to well describe a variety of datasets, both field and laboratory, with the notable exception of very short fetch (or young wave age) field data. This scaling does not suffer from spurious correlation in

_{p}*u*∗.

The two scalings yield rather different estimates of roughness in certain conditions, and our purpose here is to assess how each scaling performs in a range of sea states. To this end, we have assembled eight datasets of momentum flux and wave data. These data were collected from buoys (three datasets), towers (three), and ships (two), in environments ranging from the near-shore to the North Atlantic Ocean. In the following section, we will introduce the various datasets and discuss how the data are analyzed. In section 3, we will test the two models on each individual dataset, and on the data as a whole, with the goal of quantifying where each model does and does not predict the data. In the last section we discuss the successes and limitations of current sea-state parameterization efforts and indicate where planned experimental efforts might yield some exciting results.

## 2. Datasets

Each of the datasets used here includes measurements of wind stress and wave information. The characteristics of the datasets are summarized in Table 1. In seven of the datasets, wind stress is calculated using the “direct” or eddy-correlation (EC) method; in the eighth set, the inertial dissipation (ID) method is used. When the EC measurements are carried out from nonstationary platforms (ships or buoys), all degrees of motion of the platform are measured and used to correct the measured velocity time series. For the ID data, corrections are applied for anemometer response. None of the datasets have been corrected for possible flow distortion effects (Yelland et al. 1998), although these effects are not expected to be significant for these data. The datasets are described briefly below. Further details, including the motion correction and analysis procedures, are provided in the cited references.

*z*is calculated from

_{o}*u*∗ and neutral wind speed at a height of

*z*m,

*U*, after assuming logarithmic profile shapes:

_{zN}*u*∗ = (|

*τ̂*|/

*ρ*)

^{1/2}, where

*τ̂*= −

*ρ*(

*u*′

*w*′

*+*

^{ˆ}_{i}*υ*′

*w*′

*) is the wind stress. We note here that while*

^{ˆ}_{j}*z*is a scalar,

_{o}*τ̂*is a vector. The implications of this are discussed below. The stress

*τ̂*is calculated from the detrended turbulent velocity fluctuations,

*u*′,

*υ*′, and

*w*′, respectively the horizontal inline with the wind, horizontal cross wind, and vertical components. The overbar represents an average over time periods of 17–30 min;

*ρ*is the air density, and

*κ*= 0.4 is the von Kármán constant. The surface drift speed,

*U*, is typically small and is taken here to be zero. Neutral wind speeds are calculated from

_{o}*ψ*, from Donelan (1990);

_{u}*L*, the Obukhov length, is given by

*and*

_{υ}*θ*′

_{υ}represent, respectively, the mean and turbulent components of virtual potential temperature. For the EC data used here, the buoyancy flux

*L*is iterative, and nonconvergent points are removed from the dataset. Also, EC data for which −

*u*′

*w*′

*L*is not well defined in these cases (Drennan et al. 1999). For the ID data,

*L*was calculated using a bulk formula approach to avoid the possibility of bias at low wind speeds (Taylor and Yelland 2000).

Unless stated otherwise, the wave information is in the form of directional wave spectra, calculated from surface elevation time series collected with an array of wave gauges, or (for the SWS2 experiment) from buoy accelerations. The spectra are used to determine the relevant wave parameters: *H _{s}*,

*L*,

_{p}*c*and the peak propagation direction. In addition, the directional spectra are partitioned into the components—wind sea and/or swells—making up the wave field. Wind sea is defined as the component of the wave spectrum meeting the criteria |

_{p}*θ*| < 45°, where

_{d}*θ*is the angle between the mean wind-sea direction and that of the wind, and

_{d}*U*

_{10}

*cos*

_{N}*θ*> 0.83

_{d}*c*(Donelan et al. 1985). At most, one wind sea can exist at any one time. All other wave components are classified as swell. Contrary to several other studies (including DGHQ), we have, where possible, avoided the application of a roughness Reynolds number criterion. Lange et al. (2004) have recently shown that such a criterion can result in roughness lengths (and drag coefficients) that are biased high. The reason is that roughness length data taken under rough flow conditions, but which are biased low because of noise, may be falsely identified as smooth flow stress values and eliminated from the dataset. Conversely, data that have little noise or that are biased high are retained, thus causing a high bias in the mean estimate. In their analysis of the RASEX (see below) data, Johnson et al. (1998) applied a roughness Reynolds criterion; therefore, the RASEX roughness data at lower wind speeds are biased high (see below).

_{p}In the following analysis, we distinguish between two broad classes of sea state: wind sea–dominant (WSD) and swell-dominant (SD), depending on whether the majority of the wave energy is in the wind sea or swell. Drennan et al. (2003) considered a subclass of WSD cases, those where the wind-sea energy is much greater than the total swell energy. These very conservative criteria for “pure wind sea” (PWS) were chosen so as to avoid any possible influence of swell on the wave-age relationship. The datasets are described below and are summarized in Table 1.

### a. Agile

For several months in each of the autumns of 1994 and 1995, an air–sea interaction experiment was conducted in the western basin of Lake Ontario (Donelan and Drennan 1995). The 15-m research vessel *Agile* was equipped to measure waves and turbulence on both sides of the air–water interface. Of interest here are stress measurements made using a Gill R2A three-axis sonic anemometer mounted on a bow mast at 7.8 m above mean water level (MWL). Wave data were collected from a bow-mounted wave staff array. The fetch for these data, collected while the vessel steered into the wind, varied from 2 to over 20 km.

The *Agile* dataset includes 101 runs of 20–25 min length. These data represent a wide range of wave ages and steepnesses. Eighty of the *Agile* runs met the WSD criteria, and many of these were strongly underdeveloped (fetch limited). In most of the SD runs the wind was light, and the wind sea was much less than the swell.

### b. AWE

The Adverse Weather Experiment (AWE) was held off the coast of Fort Lauderdale, Florida, during April–May 2000. The goal of the experiment was to investigate the impact of a cold front on mixing, air–sea interaction, and sediment transport in the shallow-water column. The data considered here are from an Air–Sea Interaction Spar (ASIS; Graber et al. 2000) buoy moored several kilometers off the coast in 20 m of water. Wind stress is calculated using measurements from a Gill R2A sonic anemometer mounted on a mast 6.5 m above MWL. An array of wave staffs mounted around the buoy provided directional wave information.

The AWE data used here consists of 565 30-min runs. Most of the AWE data were collected in light–moderate onshore winds, in either swell-dominated or mixed sea conditions. During most of the 335 WSD runs, waves were near full development.

### c. FETCH

The FETCH (Flux, État de la Mer et Télédétection en Condition de Fetch Variable, or flux, sea state, and remote sensing in conditions of variable fetch) experiment took place in the Gulf of Lion in the Mediterranean Sea during March and April 1998. The data considered here are from an ASIS buoy moored 50 km offshore along the 100-m isobath. Estimates of wind stress were made using a Gill R2A sonic anemometer mounted on a mast 7 m above MWL (Drennan et al. 2003).

The FETCH dataset includes a wide variety of sea-state conditions, ranging from strongly forced mistral events to light wind, swell-dominated seas. The dataset consists of 853 28.5-min runs, of which 323 are classified as WSD. (FETCH data were available on the ALBATROS or FETCH databases online at http://dataserv.cetp.ipsl.fr.)

### d. HEXOS

The Humidity Exchange over the Sea (HEXOS) experiment was carried out in 1986 from a tower in the North Sea, at 18-m depth. Here we use the HEXOS data published by Janssen (1997), averaging the stress and mean wind values from the pressure and sonic anemometers. These data, 50 runs, were selected by Janssen for pure wind sea.

### e. RASEX

The Risø Air–Sea Exchange (RASEX) field campaign took place in 1994 at a shallow-water site (4-m depth) off Denmark. We use the RASEX dataset published by Johnson et al. (1998). Wind stress data were collected using a three-axis sonic anemometer mounted on a tower, 3 m above MWL. The mean wind data accompanying the stress data are not from the sonic anemometer, but from a cup anemometer located at 7 m. The tabulated data (80 runs) were selected by Johnson et al. to represent pure wind-sea conditions. However, since only one-dimensional wave spectra are available, the selection criteria are based on spectral width instead of the 2D directional properties used for the other datasets.

### f. SWADE

The Surface Wave Dynamics Experiment (SWADE) took place in 1990/91 off the coast of Virginia. During SWADE, air–sea fluxes, wave spectra, and other parameters were measured from a 20-m swath ship (Donelan et al. 1997). Stress data were obtained from a K-Gill anemometer mounted on a bow tower at 12-m height; wave data were collected from a bow-mounted array. The SWADE dataset consists of 121 17-min runs. The sea state was mostly mixed, often with a strong swell component. The 43 wind sea–dominant runs are either at, or near, full development with a large variation in atmospheric stability.

### g. SWS2

The Storm Wave Study second experiment (SWS2) took place in autumn 1997 off the coast of Newfoundland (Dobson et al. 1999). Wind measurements were made using a sonic anemometer at 5.75 m on a 6-m Nomad buoy. The friction velocity was calculated from the anemometer data using the inertial dissipation method with correction for anemometer response (Henjes et al. 1999) and bulk stability calculation (Taylor and Yelland 2000). The dataset consists of 1383 runs of 52-s duration, which have been quality controlled for buoy orientation and the spectral slope within the inertial subrange. The buoy was equipped with motion sensors, and these were used to estimate *H _{s}* and

*L*. With

_{p}*H*reaching 9 m, the SWS data include much higher seas than the other studies used here. The sea state during SWS2 was mixed, with wind sea and swell(s) of similar magnitudes. Such conditions are typical of the open ocean at high latitudes. Using wave data from a nearby Directional Wave Rider buoy, only 198 of the SWS2 runs are classified as wind sea–dominant.

_{s}### h. WAVES

The Water–Air Vertical Exchange Study (WAVES) experiment was conducted over 3 yr, 1985–87, in the western basin of Lake Ontario (Donelan et al. 1999). The principal platform was a research tower located 1.1 km offshore, at 12-m depth. Wind stress measurements were made from a bivane anemometer on a mast at 12 m above MWL. Here we use the wind stress dataset from the 1987 WAVES field season (Drennan et al. 1999). The WAVES dataset used here includes 238 30-min runs, covering a wide range of sea states, from strongly forced, and fetch-limited, to swell dominated; 156 of the runs are classified as wind sea–dominated.

## 3. Tale of two models

The dimensionless roughness *z _{o}*/

*H*of the 3391 data points are plotted in Fig. 1a against inverse wave age,

_{s}*u*∗/

*c*. The data are grouped into eight wave-age classes and averaged logarithmically in both roughness and inverse wave age, with the group means and error bars showing two standard errors indicated on the plot. The DGHQ curve (4) is seen to describe well the younger sea states (

_{p}*u*∗/

*c*> 0.05). On the contrary, it is clear that the wave-age scaling will give very poor results with swell-dominated data (low

_{p}*u*∗/

*c*).

_{p}For comparison, we show in Fig. 1b the same data plotted using the wave steepness scaling: *z _{o}*/

*H*versus

_{s}*H*/

_{s}*L*. The curve (5) proposed by TY2001 (solid line in Fig. 1b) predicts the mean behavior of the steeper waves but the data with lower steepness are poorly described. These latter data are mostly collected in the swell-dominated regime. Thus neither scaling is expected to give useful predictions in the swell-dominated regime.

_{p}We now restrict our dataset to WSD conditions (Fig. 2), using the criteria described above. Using the wave-age scaling, the curve of DGHQ is seen to follow the data (1265 points) over most of the wave-age range (Fig. 2a). Also shown are the data averaged (logarithmically) in eight wave-age groups, showing two standard errors in both axes. Sixteen points with *z _{o}*/

*H*< 10

_{s}^{−6}are considered as outliers, and omitted from the averages. There is no significant difference between the pure wind-sea data (darker dots, 659 points) and those where there is additional nondominant swell. Hence we conclude that the DGHQ wave-age relation applies equally well for WSD as for PWS cases. Although the performance of the wave-age model decreases somewhat for fully developed waves (0.025 ≤

*u*∗/

*c*≤ 0.05), it has improved in comparison with the results shown by DGHQ. This is due to the inclusion of the smooth flow data in the present analysis. If these data are excluded, as by DGHQ and others, the roughness in fully developed conditions is biased high.

_{p}In Fig. 2b, the same WSD data are plotted using the wave steepness scaling. The TY2001 curve (5) fits the averages well for most of the steepness range, although it is evident that *z _{o}*/

*H*is significantly underestimated when

_{s}*H*/

_{s}*L*< 0.02. We therefore propose this as a threshold below which the steepness formulation should not be applied. This is discussed further below.

_{p}Comparing either Figs. 1a with 1b or Figs. 2a with 2b, it appears that *z _{o}*/

*H*correlates better with wave age,

_{s}*u*∗/

*c*, than with significant steepness,

_{p}*H*/

_{s}*L*. However, this difference in part reflects the different effects of self correlation for the two scalings. For wave-age scaling, any spurious variations in

_{p}*u*∗ effect both dimensionless variables in a similar sense, improving the apparent correlation (see Lange et al. 2004). For steepness scaling, spurious variations of

*H*have inverse effects on the dimensionless variables, degrading the correlation.

_{s}*C*

_{10}

*has been made for each WSD dataset, with both scalings. The results are shown in Figs. 3 and 4. Here the PWS data are distinguished by black points. When applying the wave-age scaling, an iterative routine was used, starting from*

_{N}*H*,

_{s}*c*, and the wind speed. In Fig. 5, the same comparison is made with

_{p}*C*

_{10}

*calculated from the Smith (1980) bulk relation,*

_{N}*C*

_{10}

*× 1000 = 0.61 + 0.063*

_{N}*U*

_{10}

*. This provides a baseline against which to compare the two sea-state dependent models.*

_{N}In Tables 2–5, we provide the number of samples, mean error, correlation coefficient, linear regression coefficients (maximum likelihood, assuming equal uncertainty in the measured and predicted *C*_{10}* _{N}*s), and the percentage of points falling within the expected range based on sampling variability. Model assessment is based on three statistics: a

*t*test on the hypothesis (H1) that the mean difference between predicted and measured

*C*

_{10}

*is zero, a test (H2) that the regression line between measured and predicted values is near unity, and a test (H3) that the observed variability can be explained by the sampling variability. To carry out this last test, we calculate confidence regions for the scatterplots and determine whether the fraction of points falling outside the region is consistent with the value expected based on sampling variability. Most data predictions show more variability than is predicted by H3. This is likely due to our underestimation of the variability in the model prediction. We consider H3 to be met if 80% or more of the points fall within the 90% confidence limits.*

_{N}*ε*, have been calculated for each experiment following Donelan (1990). Here,

*U*is taken as the mean wind speed for an experiment. The coefficient 9.2 in (10) takes into account the variability in −

*u*′

*w*′

*U*

_{10}

*. The variability in −*

_{N}*u*′

*w*′

*C*

_{10}

*due to wind speed. We assume similar errors for the predicted as for the measured*

_{N}*C*

_{10}

*s. The sampling errors in*

_{N}*C*

_{10}

*range from 12% for RASEX (*

_{N}*z*= 3 m, ϒ = 1800 s) to 33% for SWADE (12 m, 1020 s), with a mean of 21%. For the SWS2 data, calculated by ID, we assume an error equal to the mean sampling error for the EC data (21%). In Figs. 3–6, the 90% confidence regions appear as the areas between the dashed lines.

We first consider the *u*∗/*c _{p}* scaling (Fig. 3, Table 2). This scaling was developed using PWS subsets of the data from AGILE, FETCH, HEXOS, SWADE, and WAVES. Qualitatively, the scaling is seen to work well for the AGILE, FETCH, and SWS2 datasets and fairly well for HEXOS and WAVES. The RASEX data are poorly modeled by

*u*∗/

*c*scaling, with much higher variability predicted than was observed. This will be discussed below. Of the eight WSD datasets, H1 (mean difference) is met with AGILE, SWADE, SWS2, and WAVES, and also with the combined WSD dataset; H2 is met, with 90% confidence levels counting both the slope and intercept, for AGILE, FETCH, SWS2, WAVES, and the full WSD dataset; H3 is met at the 90% level with HEXOS, SWADE, and SWS2, and at the 80% level also with AGILE, AWE, FETCH, and the full WSD dataset.

_{p}For the most part there is little difference between the pure wind-sea data (black points) and those with additional (nondominant) swell (gray points). The only significant differences arise with the WAVES dataset: H3 is met at the 80% level for PWS data, but not WSD.

It should be emphasized that wind-sea peak parameters were used when applying the *u*∗/*c _{p}* model. For most WSD data above, these wind-sea parameters were also the overall peak parameters. However, for SWS2, this was frequently not the case. During SWS2, the sea state consisted of wind sea and swell of similar magnitudes. Given that swell fields tend to be more narrow (in frequency and direction) than wind seas, it was often observed that

*E*

_{swell}<

*E*

_{ws}but

*f*=

_{p}*f*

_{p−}_{swell}. If the

*u*∗/

*c*scaling is applied with

_{p}*f*instead of

_{p}*f*

_{p}_{−windsea}, the model is seen to significantly underpredict

*C*

_{10}

*(Fig. 6a).*

_{N}We next apply the steepness scaling to the WSD datasets (Fig. 4, Table 3). This scaling was developed using the HEXOS and RASEX data together with data from Anctil and Donelan (1996), a subset of the WAVES database. Qualitatively, the steepness model is seen to perform well for the HEXOS and SWS2 data, and fairly well for AWE, FETCH, and RASEX. Quantitatively, H1 is met with the AWE, HEXOS, and WAVES datasets; H2 is met with HEXOS, SWS2, and with the full WSD dataset; H3 is met at 90% for two datasets (AWE and HEXOS), and at 80% also for SWADE and SWS2.

The *H _{s}*/

*L*scaling was developed using the peak parameters—not necessarily those of the wind sea. In the case of the bimodal SWS2 data, use of the wind-sea parameters in the scaling results in a significant overprediction of

_{p}*C*

_{10}

*(Fig. 6b). Since the steepness scaling does not rely on wind-sea parameters, there is no a priori need to restrict the scaling to WSD. In Figs. 1b and 2b, the*

_{N}*H*/

_{s}*L*scaling is seen to perform best for

_{p}*H*/

_{s}*L*≥ 0.02. Such a simple criterion is more easily applied than the spectrally derived WSD. The

_{p}*H*/

_{s}*L*scaling predictions for each dataset with

_{p}*H*/

_{s}*L*≥ 0.02 are given in Table 4. For four of the datasets, essentially the same data are selected by the WSD and

_{p}*H*/

_{s}*L*≥ 0.02 criteria. However for FETCH, SWADE, and especially SWS2 the latter criterion has the advantage of selecting a significantly larger number of points than the former. The steepness scaling predictions of this new SWS2 data subset are shown in Fig. 6c. Although the mean errors are somewhat higher with the steepness cutoff criterion, the correlation coefficients, slopes and

_{p}*P*

_{90}statistics remain similar for most datasets. Hence the steepness model can be applied in both WSD and the more general “mixed sea” conditions. For the most part seas with

*H*/

_{s}*L*< 0.02 are dominated by swells (although 6% of the WSD cases fell below the threshold).

_{p}Last, we compare the two sea-state formulas with the simple bulk relation of Smith (1980). The results for the WSD data are shown in Fig. 5, and Table 5. Qualitatively, the bulk relation is seem to perform reasonably well for the RASEX data. However, typical for bulk relations, much less variability is predicted than is present in the data. This is not unexpected, as the sampling variability of mean wind speed (on which bulk estimates are based) is much less than that of momentum flux. Hypothesis H1 is met with two datasets (AWE and SWADE), and nearly met with FETCH. Test H2 (slope) is met only with RASEX. Test H3 is met at the 90% level with AWE and SWADE and at the 80% level with FETCH, SWS2, and the full WSD dataset. The results of the bulk formulation applied to the data with *H _{s}*/

*L*≥ 0.02 are similar, and not shown here.

_{p}Among the wind-sea datasets, RASEX is particularly poorly modeled by the wave-age scaling. Also, the low wind RASEX data are not predicted by steepness scaling. The RASEX data of Johnson et al. (1998) have been the subject of some controversy for several reasons. One relates to the fluxes and mean winds having been measured at different elevations. TY2001 exclude all RASEX data for wind speeds below 10 m s^{−1} because of the possibility of noise contamination, and also because only the higher wind data agree with the results of Vickers and Mahrt (1997) using data from a different anemometer during the same experiment. Recently Lange et al. (2004) have suggested that the use of the roughness Reynolds number criterion caused a spurious trend in the Johnson et al. (1998) data. Thus for wind speeds above about 10 m s^{−1} the Vickers and Mahrt (1997) mean values were at or above the Reynolds number cutoff, but for lower wind speeds data around their mean values would have been eliminated by Johnson et al. Thus the Johnson et al. data below 10 m s^{−1} are biased significantly high. These data points are those appearing below the scatter limits in the RASEX plots of Figs. 3–5. Excluding these data the steepness scaling adequately describes the RASEX data. In contrast fully 90% of the high-wind RASEX data fall *outside* the 90% confidence range of the wave-age scaling: the *u*∗/*c _{p}* scaling predicts much higher values of

*C*

_{10}

*than were measured. Unfortunately, no directional wave spectra are available for these data, and the classification of wind sea was made using a wave spectral bandwidth parameter (a second point of controversy). Johnson et al. (1998) point out the relatively high wave spectral width in the RASEX data, and also that clear two-peaked behavior was seen in some cases. It is not clear that the criteria used by Johnson et al. to select the data were sufficient to exclude swell contaminated (or dominated) cases. Whether swell, and effects of shallow water, could significantly alter the roughness from the expected wind-sea values remains a point of debate among the present authors.*

_{N}We next consider the swell-dominated data (Fig. 7). Here we do not distinguish between the various datasets, because the results are similar for all six (RASEX and HEXOS have no swell data). As could be predicted from Fig. 1, both the wave-age and steepness formulations significantly underpredict the roughness (and *C*_{10}* _{N}*) in swell-dominated conditions where

*H*/

_{s}*L*< 0.02. The bulk formulation, on the other hand, predicts the mean

_{p}*C*

_{10}

*for these data quite well, although the bias noted above is very evident: high/low measured*

_{N}*C*

_{10}

*are under/over estimated. The mean behavior of the data is fairly well recovered, but the variability is missed.*

_{N}For swell-dominated conditions where *H _{s}*/

*L*> 0.02, the steepness formulation does well in comparison with the other two (Fig. 7, gray points). Indeed the distinction between swell and wind sea is less relevant with the steepness formulation than the cutoff based on steepness alone.

_{p}## 4. Discussion and conclusions

We have tested the wave-age model of Drennan et al. (2003) and the wave steepness model of Taylor and Yelland (2001) against eight datasets collected in a variety of conditions. To summarize, no formulation is able to model all WSD datasets. However, both sea-state parameterizations were seen to do significantly better than the Smith (or other) bulk formula for most WSD datasets. Comparing the two sea-state parameterizations, the wave-age formula performed better with the more strongly forced field data (AGILE, FETCH, and to a lesser degree, WAVES). This is also evident in Fig. 2: roughness estimated using the *u*∗/*c _{p}* formula agree well with the data for the youngest waves (

*u*∗/

*c*> 0.05), whereas roughness predicted by the steepness formula is below the median value for the steepest waves.

_{p}On the other hand, the steepness formula better predicts the RASEX and HEXOS data, and also the near fully developed AWE data. In addition, the steepness formula was seen to be well suited to mixed sea conditions, with dominant swell (*H _{s}*/

*L*> 0.02). It should be noted that both sea-state parameterizations perform better on the datasets that were used in their development (AGILE, FETCH, HEXOS, SWADE, and WAVES for the wave-age formula and HEXOS and RASEX for the steepness formula). Presently, we do not fully understand why the performance of the different formula varies in cases such as AGILE, RASEX, and HEXOS. Hence the user should be aware that significant errors in roughness length estimation are possible in limited fetch conditions.

_{p}Up to this point, we have evaluated the roughness parameterizations on their ability to predict individual data points. Often, however, one is more interested in the average momentum flux over time scales of weeks or months. Hence, we investigate the ability of the parameterizations to predict the mean observed *C*_{10}* _{N}* versus wind speed relation for each experiment. In Fig. 8, the mean curves are shown for WSD. For the SWS2 panel alone, the steepness criterion (

*H*/

_{s}*L*> 0.02) is used to select the data for all but the wave-age curve.

_{p}Using this method of comparison the Smith (1980) formula generally appears better than in the previous results. In addition to AWE, the data for FETCH follow the Smith relationship for most wind speeds above 8 m s^{−1}. If it is accepted that the RASEX data below 10 m s^{−1} are biased, then the Smith relationship holds for this experiment too (as confirmed by the Vickers and Mahrt 1997 analysis). Much of the SWADE (*U*_{10} < 10 m s^{−1}) and SWS2 data (*U*_{10} < 12 m s^{−1}) also follow the Smith relationship. However it remains the case that, for several datasets [AGILE, HEXOS, RASEX, SWS2 (*U*_{10} > 12 m s^{−1}), and WAVES] the Smith relationship significantly under predicts the observed roughness.

Based on this method, the two sea-state–based formulas perform with similar success for AWE and WAVES. The wave-age formula also performs well for AGILE, FETCH, SWADE, and SWS2. The two failures for this scaling are HEXOS, for which the roughness at the lower wind speeds is underpredicted, and RASEX, where much higher roughness is predicted than that observed. In these two experiments very different roughness values occurred at similar wave ages. It is a strength of the steepness formula that it can predict the roughness variations found during both RASEX and HEXOS. However, this formula overpredicts the roughness observed during FETCH, SWADE, and SWS2 (for *U*_{10} < 15 m s^{−1}) and underpredicts the AGILE data. The poor performance for SWS2 contrasts with the apparently good predictions shown by Taylor and Yelland (2001). The difference is caused by the application of the *H _{s}*/

*L*> 0.02 criterion in the present study.

_{p}In Fig. 9, all data are combined. To test the scalings on the same data, Fig. 9a shows WSD data with *H _{s}*/

*L*> 0.02; Fig. 9b shows swell-dominant data with

_{p}*H*/

_{s}*L*< 0.02. For WSD data, the two sea-state models give similar mean curves for most of the wind speed range, except for

_{p}*U*< 8 ms

^{−1}, where the steepness model is closer to the data. Either sea-state model is preferred to Smith for WSD data. For the swell-dominated data, neither sea-state model does well, and Smith (1980) provides the best fit to the mean

*C*

_{10}

*curve.*

_{N}The poor performance of both scalings on swell data would indicate that sea-state parameters (steepness or wave age) based on swell components do not offer useful first-order information on air–sea fluxes. The large-scale swells may dominate the sea surface, but to first-order it is the smaller-scale wind-driven waves which determine the air–sea fluxes. That said, swell waves have been observed to significantly modify the air–sea transfers (e.g., Volkov 1970). Donelan et al. (1997) reported a doubling of *C*_{DN} over pure wind-sea values with strong swells running against the wind, while Drennan et al. (1999), Grachev and Fairall (2001) and others have observed a similarly strong reduction of *C*_{DN} in strong following swells. Clearly any extension of the sea-state models to include swell must account for not only the magnitude, but also the direction of the swell waves.

As pointed out above, the wind stress is a vector quantity. It is generally assumed that the wind stress vector is aligned with the mean wind, in which case −*υ*′*w*′

In addition to steering the stress direction, dominant swell has also been reported to modify the logarithmic wind profile shape (Donelan 1990). This would invalidate our use of profile functions (6) and (7) in calculating the roughness in these conditions. In contrast, Edson and Fairall (1998) found support for the logarithmic profiles in typical mixed sea conditions. Hence our approach should be valid in pure wind-sea and mixed sea conditions where it is claimed to apply.

As existing results on swell effects on stress are largely qualitative, it is not yet possible to account for them in roughness models such as those described here. Recently there has been a renewed interest in quantifying for the effects of swell on drag coefficient parameterizations. Several new experimental campaigns with these goals in mind are underway. Although we cannot yet claim complete success in efforts to model roughness over the sea, much progress has been made in recent years. It is hoped that the coming experiments will yield sufficient data to address the outstanding issues.

A further point of investigation should be to extend the study of flow distortion effects on flux measurements. These have been quantified for flux data obtained using the ID method on ships (Yelland et al. 1998, 2002; Dupuis et al. 2003), and Oost et al. (1994) quantified the effects of the flow around the instrumented boom on the EC fluxes obtained during HEXOS. In contrast, the effect of turbulent flow distortion around ships or other sizable platforms on EC fluxes has not been quantified.

In the meantime, what formula would we recommend? Much depends on what wave data are available. If no wave data are available, then one is forced to use a bulk formula such as Smith (1980), but one should be aware that the roughness is likely to be underestimated under strongly forced conditions. In passing, we note that this underestimation would be worse for the relationship proposed by Smith (1988), and hence we favor the earlier paper. If spectral wave data are available then the wave-age–based formula was both developed from and better fitted a larger number of the datasets. But this success is dependent on having knowledge of wind-sea parameters (or otherwise that the sea state is wind sea–dominated). If only bulk sea-state properties are available (such as *H _{s}* and

*L*) then the steepness formula was more robust for cases of mixed wind sea and swell, and for swell dominant conditions with

_{p}*H*/

_{s}*L*> 0.02. However, where the swell is dominant (with

_{p}*H*/

_{s}*L*< 0.02) then the Smith (1980) formula should be used.

_{p}## Acknowledgments

This paper arose out of a comment and reply between the authors regarding Taylor and Yelland (2001). We thank editor Peter Müller for suggesting this cooperative effort. We also appreciate the comments of Fred Dobson and two anonymous reviewers. Many of the experimental data used here were collected in collaboration with colleagues. In particular, we acknowledge M. A. Donelan, H. C. Graber, D. Hauser, and E. A. Terray, along with the technical support of J. Gabriele, R. Pascal, and M. Rebozo. Support to WMD from ONR (N00014-00-1-0594) and NSF (OCE-9871855 and OCE-0220459) is gratefully acknowledged.

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Dimensionless roughness *z _{o}*/

*H*vs (a) inverse wave age

_{s}*u*∗/

*c*and (b) significant steepness

_{p}*H*/

_{s}*L*for the combined WSD dataset (1265 points). The 659 darker points represent pure wind-sea conditions. In each panel, the data (excluding 16 points with

_{p}*z*/

_{o}*H*< 10

_{s}^{−6}) are bin-averaged, with the bin means (logarithmic) indicated by circles. The error bars represent two standard errors about the mean. Also shown (solid lines) are the curves of Drennan et al. (2003) and Taylor and Yelland (2001) in (a) and (b), respectively, and the curve

*H*/

_{s}*L*= 0.02 (dashed).

_{p}Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Dimensionless roughness *z _{o}*/

*H*vs (a) inverse wave age

_{s}*u*∗/

*c*and (b) significant steepness

_{p}*H*/

_{s}*L*for the combined WSD dataset (1265 points). The 659 darker points represent pure wind-sea conditions. In each panel, the data (excluding 16 points with

_{p}*z*/

_{o}*H*< 10

_{s}^{−6}) are bin-averaged, with the bin means (logarithmic) indicated by circles. The error bars represent two standard errors about the mean. Also shown (solid lines) are the curves of Drennan et al. (2003) and Taylor and Yelland (2001) in (a) and (b), respectively, and the curve

*H*/

_{s}*L*= 0.02 (dashed).

_{p}Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Dimensionless roughness *z _{o}*/

*H*vs (a) inverse wave age

_{s}*u*∗/

*c*and (b) significant steepness

_{p}*H*/

_{s}*L*for the combined WSD dataset (1265 points). The 659 darker points represent pure wind-sea conditions. In each panel, the data (excluding 16 points with

_{p}*z*/

_{o}*H*< 10

_{s}^{−6}) are bin-averaged, with the bin means (logarithmic) indicated by circles. The error bars represent two standard errors about the mean. Also shown (solid lines) are the curves of Drennan et al. (2003) and Taylor and Yelland (2001) in (a) and (b), respectively, and the curve

*H*/

_{s}*L*= 0.02 (dashed).

_{p}Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured 10-m neutral drag coefficients (×1000) vs those predicted using the Drennan et al. (2003) wave-age scaling (using wind-sea peak parameters) for dominant wind-sea conditions with eight datasets. The black points represent pure wind-sea conditions. The thick (thin) solid lines show the best linear fit (1:1 agreement). Dashed lines represent 90% confidence limits based on the sampling variability of each dataset.

Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured 10-m neutral drag coefficients (×1000) vs those predicted using the Drennan et al. (2003) wave-age scaling (using wind-sea peak parameters) for dominant wind-sea conditions with eight datasets. The black points represent pure wind-sea conditions. The thick (thin) solid lines show the best linear fit (1:1 agreement). Dashed lines represent 90% confidence limits based on the sampling variability of each dataset.

Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured 10-m neutral drag coefficients (×1000) vs those predicted using the Drennan et al. (2003) wave-age scaling (using wind-sea peak parameters) for dominant wind-sea conditions with eight datasets. The black points represent pure wind-sea conditions. The thick (thin) solid lines show the best linear fit (1:1 agreement). Dashed lines represent 90% confidence limits based on the sampling variability of each dataset.

Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured 10-m neutral drag coefficients (×1000) vs those predicted using the Taylor and Yelland (2001) significant steepness scaling for dominant wind-sea conditions with eight datasets. The black points represent pure wind-sea conditions. The thick (thin) solid lines show the best linear fit (1:1 agreement). Dashed lines represent 90% confidence limits based on the sampling variability of each dataset.

Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured 10-m neutral drag coefficients (×1000) vs those predicted using the Taylor and Yelland (2001) significant steepness scaling for dominant wind-sea conditions with eight datasets. The black points represent pure wind-sea conditions. The thick (thin) solid lines show the best linear fit (1:1 agreement). Dashed lines represent 90% confidence limits based on the sampling variability of each dataset.

Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured 10-m neutral drag coefficients (×1000) vs those predicted using the Taylor and Yelland (2001) significant steepness scaling for dominant wind-sea conditions with eight datasets. The black points represent pure wind-sea conditions. The thick (thin) solid lines show the best linear fit (1:1 agreement). Dashed lines represent 90% confidence limits based on the sampling variability of each dataset.

Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured 10-m neutral drag coefficients (×1000) vs those predicted using the Smith (1980) bulk relation for dominant wind-sea conditions with eight datasets. The black points represent pure wind-sea conditions. The thick (thin) solid lines show the best linear fit (1:1 agreement). Dashed lines represent 90% confidence limits based on the sampling variability of each dataset.

Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured 10-m neutral drag coefficients (×1000) vs those predicted using the Smith (1980) bulk relation for dominant wind-sea conditions with eight datasets. The black points represent pure wind-sea conditions. The thick (thin) solid lines show the best linear fit (1:1 agreement). Dashed lines represent 90% confidence limits based on the sampling variability of each dataset.

Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured 10-m neutral drag coefficients (×1000) vs those predicted using the Smith (1980) bulk relation for dominant wind-sea conditions with eight datasets. The black points represent pure wind-sea conditions. The thick (thin) solid lines show the best linear fit (1:1 agreement). Dashed lines represent 90% confidence limits based on the sampling variability of each dataset.

Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured vs predicted 10-m neutral drag coefficients (×1000) for the SWS2 WSD dataset. (a) Prediction is via the Drennan et al. (2003) wave-age formulation using the peak frequency (not the peak wind-sea frequency). (b) Prediction is via the Taylor and Yelland (2001) significant steepness scaling using the peak wind-sea frequency (not the peak frequency). These two panels demonstrate the importance of using the correct peak parameters in applying the models. (c) Steepness formulation is again used, this time with the data selected using the criterion *H _{s}*/

*L*> 0.02, instead of the spectral WSD.

_{p}Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured vs predicted 10-m neutral drag coefficients (×1000) for the SWS2 WSD dataset. (a) Prediction is via the Drennan et al. (2003) wave-age formulation using the peak frequency (not the peak wind-sea frequency). (b) Prediction is via the Taylor and Yelland (2001) significant steepness scaling using the peak wind-sea frequency (not the peak frequency). These two panels demonstrate the importance of using the correct peak parameters in applying the models. (c) Steepness formulation is again used, this time with the data selected using the criterion *H _{s}*/

*L*> 0.02, instead of the spectral WSD.

_{p}Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured vs predicted 10-m neutral drag coefficients (×1000) for the SWS2 WSD dataset. (a) Prediction is via the Drennan et al. (2003) wave-age formulation using the peak frequency (not the peak wind-sea frequency). (b) Prediction is via the Taylor and Yelland (2001) significant steepness scaling using the peak wind-sea frequency (not the peak frequency). These two panels demonstrate the importance of using the correct peak parameters in applying the models. (c) Steepness formulation is again used, this time with the data selected using the criterion *H _{s}*/

*L*> 0.02, instead of the spectral WSD.

_{p}Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured 10-m neutral drag coefficients (×1000) vs those predicted for swell-dominated conditions. The models used are (a) wave age, (b) steepness, and (c) the Smith (1980) bulk relation. The dark points represent the subset of points with *H _{s}*/

*L*< 0.02. Note that many of the lighter points lie behind the dark ones.

_{p}Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured 10-m neutral drag coefficients (×1000) vs those predicted for swell-dominated conditions. The models used are (a) wave age, (b) steepness, and (c) the Smith (1980) bulk relation. The dark points represent the subset of points with *H _{s}*/

*L*< 0.02. Note that many of the lighter points lie behind the dark ones.

_{p}Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Measured 10-m neutral drag coefficients (×1000) vs those predicted for swell-dominated conditions. The models used are (a) wave age, (b) steepness, and (c) the Smith (1980) bulk relation. The dark points represent the subset of points with *H _{s}*/

*L*< 0.02. Note that many of the lighter points lie behind the dark ones.

_{p}Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Mean 10-m neutral drag coefficients vs wind speed for dominant wind-sea conditions with eight datasets. The curves represent data (dotted) and predictions using Drennan et al. (2003) wave-age formula (thick), Taylor and Yelland (2001) wave steepness formula (dashed), and Smith (1980) (thin).

Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Mean 10-m neutral drag coefficients vs wind speed for dominant wind-sea conditions with eight datasets. The curves represent data (dotted) and predictions using Drennan et al. (2003) wave-age formula (thick), Taylor and Yelland (2001) wave steepness formula (dashed), and Smith (1980) (thin).

Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Mean 10-m neutral drag coefficients vs wind speed for dominant wind-sea conditions with eight datasets. The curves represent data (dotted) and predictions using Drennan et al. (2003) wave-age formula (thick), Taylor and Yelland (2001) wave steepness formula (dashed), and Smith (1980) (thin).

Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Mean 10-m neutral drag coefficients (×1000) vs wind speed for (a) WSD conditions with *H _{s}*/

*L*> 0.02 and (b) swell dominant with

_{p}*H*/

_{s}*L*< 0.02. The curves represent data (dotted) and predictions using Drennan et al. (2003) wave-age formula (thick), Taylor and Yelland (2001) wave steepness formula (dashed), and Smith (1980) (thin). Error bars show two standard errors.

_{p}Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Mean 10-m neutral drag coefficients (×1000) vs wind speed for (a) WSD conditions with *H _{s}*/

*L*> 0.02 and (b) swell dominant with

_{p}*H*/

_{s}*L*< 0.02. The curves represent data (dotted) and predictions using Drennan et al. (2003) wave-age formula (thick), Taylor and Yelland (2001) wave steepness formula (dashed), and Smith (1980) (thin). Error bars show two standard errors.

_{p}Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Mean 10-m neutral drag coefficients (×1000) vs wind speed for (a) WSD conditions with *H _{s}*/

*L*> 0.02 and (b) swell dominant with

_{p}*H*/

_{s}*L*< 0.02. The curves represent data (dotted) and predictions using Drennan et al. (2003) wave-age formula (thick), Taylor and Yelland (2001) wave steepness formula (dashed), and Smith (1980) (thin). Error bars show two standard errors.

_{p}Citation: Journal of Physical Oceanography 35, 5; 10.1175/JPO2704.1

Summary of the eight experiments showing, from top to bottom, which data were used in the development of the wave-age (A) and steepness (S) models (S* refers to the use of a data subset), instrument platform (MPN is the Meetpost Noordwijk platform, and NWRI is the National Water Research Institute tower), experiment location, water depth, the flux measurement method (EC is eddy correlation and ID is inertial dissipation), whether 1D or 2D wave spectra were used, fetch range, anemometer type, anemometer height, sampling period, and number of data.

Measured drag coefficients *C*_{10}* _{N}* of WSD data compared with estimates from the wave-age model (Drennan et al. 2003). Error indicates two standard errors about the mean;

*P*

_{90}indicates actual percent of points within 90% confidence bands.

Same as in Table 2, but compared with estimates from the wave steepness model (Taylor and Yelland 2001).

Measured drag coefficients *C*_{10}* _{N}* of data with

*H*0.02 compared with estimates from the wave steepness model (Taylor and Yelland 2001). Error indicates two standard errors about the mean;

_{s}/L_{p}>*P*

_{90}indicates actual percent of points within 90% confidence bands.

Measured drag coefficients *C*_{10}* _{N}* of WSD data compared with estimates from the Smith (1980) bulk model. Error indicates two standard errors about the mean;

*P*

_{90}indicates actual percent of points within 90% confidence bands.