## 1. Introduction

The Coupled Ocean–Atmosphere Response Experiment (COARE; Webster and Lukas 1992) triggered a resurgence of interest in measurement and parameterization of air–sea turbulent and radiative fluxes. The program set a 10 W m^{−2} goal for the total uncertainty in long-term (∼1 month) measurement of the surface energy budget for the COARE region. A COARE Flux Working Group (FWG) was formed, which developed a plan for pre-COARE flux cruises, intercomparisons between ships, buoys, and aircraft during COARE, and post-COARE field efforts and calibration studies (Bradley and Weller 1997). Flux estimates originated from ship, buoy, aircraft, satellite, and model data, so to ensure compatibility when comparing results the FWG developed a bulk turbulent flux algorithm as a common standard. The new algorithm needed to accommodate the light wind conditions and large near-surface ocean temperature gradients often encountered in this region. It evolved over several years and was eventually frozen at version 2.5b and published (Fairall et al. 1996a).

The algorithm was developed using COARE measurements, albeit with a relatively limited wind speed range (0–12 m s^{−1}), and has been used by many research groups for COARE analyses (more than 200 citations at the time of this writing). However, it has also been used outside the Tropics in a wide variety of conditions, raising obvious questions about its applicability in colder waters, midlatitudes, and much higher wind speeds. The algorithm was “globalized” in the sense that, wherever possible, physical variables are computed as functions of ambient conditions (e.g., temperature, latitude, solar flux) rather than hardwired to tropical values. Since 1996, work has continued to extend it to higher wind speeds and to verify it against high-quality data outside the Tropics. In 1998 a new version was developed (Bradley et al. 2000), based on six National Oceanic and Atmospheric Administration (NOAA) Environmental Technology Laboratory (ETL) cruises conducted between 1991 and 1993, preliminary results from two other ETL programs, and other published measurements from high wind regions, to extend the applicability to 20 m s^{−1}. It has evolved continuously since then, and is now considered ready for public release as COARE 3.0. Between 1997 and 1999 ETL conducted six more cruises, obtaining direct covariance and inertial-dissipation flux estimates and high-quality bulk meteorological variables, suitable for verifying the new algorithm.

In this paper we describe the new version of the COARE algorithm and its verification. Following this introduction, we give some general background on flux algorithms and the original COARE version (section 2). The new version of the algorithm is described in detail in section 3, and evaluated against the entire ETL database of 7216 1-h values obtained between 1991 and 1999. In section 5 we discuss these results in relation to other algorithms and other measurements. Our conclusions are given in section 6. The measurement system, flux processing methods, and accuracy issues are discussed in the appendix.

## 2. Background on similarity relationships and flux algorithms

### a. Bulk scaling theory

*x,*denoted as

*X,*iswhere the subscript

*h*denotes horizontal components and

*I*

_{x}represents the source term. The quantity of interest is the Reynolds flux enclosed in parentheses. It can be determined by measuring the time or space series of

*w*′ and

*x*′ and computing their mean product; this is referred to as the covariance or eddy-correlation method, and is the only direct flux measurement. There are also many indirect approaches to flux estimation. The inertial-dissipation (ID) method uses characteristics of the inertial subrange of atmospheric velocity and scalar turbulence spectra (Fairall and Larsen 1986; Edson et al. 1991). Details are given in the appendix. Other schemes are based on applications of the Monin–Obukhov similarity theory (MOST) to measured atmospheric properties (these methods yield estimates of the surface flux only, while the covariance method is general). This diversity of flux methods is primarily due to the historical difficulty, expense, and impracticality of making covariance measurements for all variables on the horizontal and temporal scales of interest. Measurements over the ocean involve the additional complication of platform motion, flow distortion over the platform and equipment, and contamination by sea spray and salt particles. More complete information on flux methods can be found in Smith et al. (1996) or Fairall et al. (1997).

*x*can be

*u,*

*υ*wind components, the potential temperature,

*θ,*the water vapor specific humidity,

*q*(see section 3b), or some atmospheric trace species mixing ratio. Here

*c*

_{x}is the bulk transfer coefficient for the variable

*x*(

*d*being used for wind speed) and

*C*

_{x}is the total transfer coefficient. Here Δ

*X*is the air–sea difference in the mean value of

*x,*and

*S*is the mean wind speed (relative to the ocean surface), which is composed of a mean vector part (

*U*and

*V*components) and a gustiness part (

*U*

_{g}) to account for subgrid-scale variability:

*X*

*X*

_{sea}

*X*

*z*

*S*

*U*

^{2}

*V*

^{2}

*U*

^{2}

_{g}

^{1/2}

*n*refers to neutral (

*ζ*= 0) stability,

*z*is the height of measurement of the mean quantity

*X*(

*z*),

*ψ*

_{x}is an empirical function describing the stability dependence of the mean profile,

*κ*is von Karman's constant, and

*z*

_{ox}a parameter called the roughness length that characterizes the neutral transfer properties of the surface for the quantity,

*x.*The MOST stability parameter,

*ζ,*is given bywhere

*T*is temperature,

*g*the acceleration due to gravity, and we are using

*w*′

*u*′

*u*∗ =

*w*′

*u*′

*θ*∗ = −

*w*′

*θ*′

*u*∗, and

*q*∗ = −

*w*′

*q*′

*u*∗ for velocity, temperature and humidity, respectively. It is sometimes convenient (e.g., to reduce variability in MOST relationships) to use a bulk stability parameter, replacing the kinematic fluxes in (5) with

*u*∗,

*θ*∗,

*q*∗, obtained by iteration within the bulk algorithm.

### b. COARE 2.5

*ψ*

_{x}). While there are many algorithms available today, we will restrict the discussion in this section to the Tropical Ocean Global Atmosphere (TOGA) COARE bulk algorithm (Fairall et al. 1996a). Parameterizations in the original COARE model were based on those of the Liu–Katsaros–Businger model (LKB; Liu et al. 1979) but updated. The LKB model allowed for different velocity and scalar von Karman constants, giving them values of 0.35 and 0.45. For the COARE model, the velocity von Karman was set to 0.40. The velocity roughness length was specified as Charnock's (1955) expression plus a smooth flow limit, following Smith (1988):where

*ν*is the kinematic viscosity. The COARE flux data supported Smith's value of the Charnock parameter,

*α*= 0.011. LKB derive the scalar roughness lengths from the principles of surface renewal theory in which, across a thin (∼1 mm) interfacial layer, small eddies transfer heat randomly and intermittently between the bulk fluid and the surface (Liu and Businger 1975). In COARE 2.5 we adopt the LKB representation of the scalar roughness lengths, parameterized in terms of the roughness Reynolds number (

*R*

_{r}=

*z*

_{o}

*u*∗/

*ν*):where scalar roughness Reynolds numbers are

*R*

_{x}=

*u*∗

*z*

_{ox}/

*ν*(see section 3e and Fig. 13). The LKB specification of scalar roughness (7) was retained and the scalar von Karman constant was adjusted to reproduce the average observed latent heat flux over the entire field program; this resulted in a value of 0.40 (i.e., the same as for velocity).

*W*∗,where

*z*

_{i}is the depth of the convective boundary layer,

*θ*

_{υ}is the virtual potential temperature, and

*β*is a parameter presently set to 1.25. If gustiness is not used, then the behavior of the

*ψ*

_{x}functions as |

*ζ*| → ∞ becomes critical in determining fluxes in light winds (Godfrey and Beljaars 1991).

In typical execution of a bulk algorithm, the atmospheric variables (*U,* *V,* *T,* *q*) at reference height *z* are provided through measurement or model output; the surface properties (current vector, water temperature) are also provided. Strictly speaking, (3) requires the true interface temperature, *T*_{s}, but usually only the temperature at some depth, *T*_{w}(*D*), is available so that a method of estimating *T*_{s} from *T*_{w} is needed. The COARE algorithm incorporates submodels that represent the millimeter-scale cool skin near the interface and the diurnal (solar) warm layer in the upper few meters of the ocean (Fairall et al. 1996b; Webster et al. 1996). The cool skin implies that the true interface temperature is several tenths (°C) cooler than the bulk water near the surface. In light wind and sunny conditions the sun may warm the upper few meters of the ocean by 1°–3°C. The surface value for specific humidity is computed from the surface temperature and the vapor pressure of seawater (0.98 times the vapor pressure of pure water; Kraus and Businger 1994).

### c. Discussion of recent bulk models

Almost two decades have passed since Blanc (1985) published his careful and rather alarming review of the large differences between more than 10 bulk flux algorithms. At that time, many of the physical processes involved in air–sea transfer of heat or momentum were embedded within the transfer coefficient for a particular entity. While transfer coefficients were known to be a function of wind speed, height, and atmospheric stability, more often than not the accuracy of available test data did not justify anything other than a constant value for the transfer coefficient of scalars, or a simple wind speed–dependent value for stress. Such simple expressions have become inadequate for several reasons, mostly connected with the burgeoning requirements of climate research, and recognition of the sensitivity of numerical climate models to small changes in air–sea flux values, particularly when attempting to couple oceanic and atmospheric GCMs. We will briefly describe some other bulk flux algorithms that have emerged over the period of development of the COARE algorithm, and give their main features in Table 1. The list is not exhaustive or even possibly up to date.

The algorithm described by Zeng et al. (1998, hereafter ZZD) was developed to suit the needs of numerical model codes. As with COARE 2.5, ZZD use the Kansas profile functions for near-neutral atmospheric stability, with the convective forms of Kader and Yaglom (1990) and the relations of Holtslag et al. (1990) in very stable conditions. Equation (6), with *α* = 0.013, is used for *z*_{o} and the Brutsaert (1982) formulation [see Eq. (29)] is used for the scalar roughness lengths, with constant coefficients obtained from the R/V *Moana Wave* COARE data (Fairall et al. 1996a). ZZD include a “gustiness” correction, with *β* = 1.0, and allowance for the 2% reduction in saturated specific humidity over seawater, which they find decreases latent heat flux by 20% at 14 m s^{−1}. The algorithm is tuned to the same dataset (the R/V *Moana Wave* observations by Fairall et al. 1996a) as COARE version 2.5 for low to moderate wind speeds, and the Humidity Exchange Over the Sea (HEXOS) data (De Cosmo et al. 1996) up to 18 m s^{−1}. However, they use the HEXOS data as presented by the authors, rather than modified as described in section 3e. Various empirical constants are given values that ensure reasonable agreement with observations over the wind speed range 0–18 m s^{−1}.

The Bourassa et al. (1999, hereafter BVW) bulk model is notable for its attempt to relate surface roughness lengths, and hence the exchange coefficients, to various aspects of sea state, swell, gravity waves, and capillary waves. They draw on, and extend, published representations of wave structure in their analyses, but adequate validating data are not yet available to test the potential of this algorithm. BVW point out that, particularly under low wind conditions where swell, wind velocity, stress, and current directions are not necessarily parallel, their algorithm allows the cross-wind component of stress to be calculated. They note that, with the coexistence of these different wave types and interactions between them, it is unclear what the proper coordinate frame of reference should be for the wind and water velocities. For *z*_{o}, BVW use a version of Eq. (6) in which the “Charnock” term is replaced by one involving wave parameters. For the scalar roughness lengths, they use the Reynolds parameters, *R*_{t} = 0.4 and *R*_{q} = 0.62, given by Brutsaert (1982; section 5). BVW adopt the profile stability functions of Beljaars and Holtslag (1991) for stable and of Benoit (1977) for unstable conditions, and gustiness with *β* = 1.25.

The Clayson et al. (1996, hereafter CFC) algorithm is based on the surface renewal theory of Brutsaert (1975), with an alternative timescale parameterization. CFC use a simplified form of the BVW sea state surface roughness model, and the same profile stability functions as BVW, which lead to Brutsaert-like expressions for the scalar roughness lengths. Unlike BVW, they do not include a gustiness term, relying on their capillary wave parameterization and surface renewal theory to obtain correct fluxes in low wind, convective conditions. Because this algorithm was developed for the assessment of fluxes from satellite data, they incorporate a model of the ocean cool skin (Wick et al. 1996). They also adjust surface humidity for the 2% reduction of vapor pressure over saline water. However, they do not correct SST for the diurnal thermocline.

Zhang and McPhaden (1995, hereafter ZM95) studied the relationship between SST and latent heat flux in the equatorial Pacific Ocean, using data from the TOGA Tropical Atmosphere Ocean (TAO) moored array. Their bulk flux algorithm takes the simplest possible route consistent with the need to take account of low wind and highly convective conditions, and is computationally economical. They adopt the standard Kansas stability profile functions across the entire unstable range from near-neutral to free convection, and Eq. (6) with *α* = 0.011 for the momentum roughness length. However, following Geernaert (1987) they set the temperature and moisture roughness lengths equal and constant at 2 × 10^{−5} m.

Beljaars (1995) approaches the problem from the perspective of the numerical modeler seeking economical solutions without violating physical reality. He shows that normal MOST can be used in the surface layer without modification for free convection (i.e., no ⅓ power law), as long as a convective scaling velocity [Eq. (8)] is included in the bulk equation. He adopts the standard Kansas expressions, arguing that the behavior of the stability functions for large (−*ζ*) is not too critical because vertical gradients are small in the well-mixed regime, and contribute little to the air–surface velocity, temperature, or moisture difference. His analysis is general for the surface layer over land and ocean, rough and smooth surfaces, and suggests a typical value *β* = 1.2 for the gustiness parameter. For air–sea transfer over the ocean, Beljaars (1995) uses Eq. (6) for *z*_{o}, with *α* = 0.018, and the same Brutsaert (1982) expressions for the scalar roughness lengths as BVW, applied to both smooth and rough flow. He justifies this cautiously, on the grounds that empirical evidence points to an almost constant moisture transfer coefficient over the entire low to high wind speed regime (Smith 1989). Approaching zero wind speed, his transfer coefficients for both heat and moisture follow quite well the increasing trend shown by the low wind data of Bradley et al. (1991).

Using COARE and TAO mooring data, ZZD have compared momentum and scalar fluxes from several of the algorithms described above and show that, up to around 6 m s^{−1}, their performance is very similar. Differences appear at higher wind speeds, but uncertainty about the quality of the test data in this regime makes judgement difficult. ZZD also consider the parameterizations employed in several numerical models, and conclude that these are seriously defective. It is probably fair to comment that almost any of the modern algorithms, with reasonable roughness parameterizations and stability correction, would considerably improve the performance of the models.

## 3. Advances for COARE 3.0

For the improved algorithm three major issues were addressed. First, the COARE model was fit to the average flux observed but was not a perfect fit to the observed wind speed dependence (e.g., the peak in the 10-m neutral transfer coefficient *C*_{e10n} in the LKB model at wind speeds around 6 m s^{−1} was not apparent in the data). Second, the model was published as being valid over the COARE wind range from 0 to 10 m s^{−1}, and extension to higher wind speeds was needed. Third, it needed to be generalized for more global applications, and tested against a much broader dataset. Preliminary discussion of these and other minor improvements appear in Bradley et al. (2000) and Fairall et al. (2001); they are summarized as follows:

- The empirical constant in the convective portion of the scalar profile function has been changed for improved matching to direct profile observations (Grachev et al. 2000).
- The Kansas stable profile functions (Businger et al. 1971) have been replaced by those from Beljaars and Holtslag (1991) which, based on new profile data taken over the Arctic ice cap (Persson et al. 2002), appear to be a better fit at extreme stability.
- The stability iteration loop has been reduced from 20 to 3 by taking advantage of a bulk Richardson number parameterization for an improved first guess (Grachev and Fairall 1997).
- The latent heat flux has been reformulated in terms of mixing ratio instead of water vapor density.
- Above 10 m s
^{−1}the Charnock parameter takes a simple wind speed dependence based on data from various sources (e.g., Hare et al. 1999). - An option has been added to allow the velocity roughness to be affected by wave parameters.
- The LKB scalar roughness relationship [
*f*_{x}(*R*_{r})] has been replaced with a much simpler one that fits both the COARE and Humidity Exchange Over the Sea Main Experiment (HEXMAX) databases.

^{−1}. The wave parameterizations have been added primarily as a response to requests from various users. We have taken two models from the recent literature (Taylor and Yelland 2001; Oost et al. 2002) that are wave age and/or wave slope based.

MATLAB and FORTRAN versions of both COARE 2.5b and 3.0 have been made publicly available online at ftp://ftp.etl.noaa.gov/et7/anonymous/cfairall/bulkalg/. Included is a description of the codes and a test dataset file. The programs are set up to read the test file and output the results; output files and graphs of results are also provided.

### a. Stability function considerations

The unstable profile stability functions used in the COARE algorithm are a blend of Kansas forms (valid for −1 < *ζ* < 0) and forms that scale as (*a*_{x}*ζ*)^{−1/3} in convective conditions (*a*_{x} is an empirical constant and the −1/3 power is the asymptotic limit of MOST; see Grachev et al. 2000). In COARE 2.5 *a*_{u} and *a*_{s} were set to 12.87; Grachev et al. (2000) showed that the values *a*_{u} = 10.15 and *a*_{s} = 34.15 gave the smoothest blend of Kansas and convective forms so these values have been adopted for COARE 3.0. The modification of the profile function on the stable side is based on more extensive observations. The original Kansas observations were limited to 0 < *ζ* < 1. The form of the stable side functions as *ζ* ≫ 1 has implications for the numerical characteristics of the stability iteration.

*C*

_{b}=

*κc*

^{1/2}

_{tn}

*c*

_{dn}≈

*κ*/

*c*

^{1/2}

_{dn}

*R*

_{ib}, isThe behavior of (9) in unstable versus stable conditions becomes the basis of the improved first guess of

*ζ*for the iteration. The stable functions from the Kansas experiment (which were used in COARE 2.5b) take the form

*ψ*

_{u}

*ψ*

_{T}

*Bζ,*

*B*is about 5.0. Substituting (11) into (9) and ignoring the small difference in velocity and temperature transfer coefficients in the stability terms, then

*ζ*

*C*

_{b}

*R*

_{ib}

*BR*

_{ib}

^{−1}

*R*

_{ib}

*B*

^{−1}

This result is consistent with the original Kansas analysis that suggested a critical gradient Richardson number on the order of 0.2. The physical interpretation is debatable, but numerically the iteration based on (9) does not converge if *R*_{ib} exceeds the threshold described by (13). If one views this condition as complete suppression of turbulent transport, then the fluxes should simply be set to zero. Using extensive measurements from a 100-m tower over land, Beljaars and Holtslag (1991) found finite, but highly intermittent, values for fluxes in very stable conditions. They produced empirical functions that fit their data (and the Kansas data) and do not result in a critical Richardson number, but lead to rapidly decreasing fluxes as stability increases. A preliminary analysis of tower data over sea ice in the Surface Heat Budget of the Arctic (SHEBA) experiment (Persson et al. 2002) has also found small but finite fluxes in very stable conditions. These small fluxes may be caused by breaking atmospheric gravity waves or some other process of nonshear-driven turbulence for which Monin–Obukov scaling is inappropriate. Pending more information, we adopt the Bejaars and Holtslag functions, which have eliminated occasional pathological results obtained with version 2.5.

### b. Conservative moisture variables and the Webb et al. (1980) correction

*r*

_{x}, of some scalar quantity with mass concentration,

*x,*Webb et al. (1980) showed that the true interfacial flux can be represented aswhere

*ρ*

_{a}is the total density of air,

*m*

_{a}is the molecular weight of air and

*m*

_{υ}the molecular weight of water vapor (

*m*

_{a}/

*m*

_{υ}= 1.61),

*Q*is the water vapor density, and

*W*is the very small mean vertical velocity required to conserve mass because temperature and moisture affect the density of air:

*w*′

*Q*′

*q*=

*Q*/

*ρ*

_{a}. However, we can apply (14) to moisture flux (i.e., set

*x*=

*Q*) and then the actual latent heat cooling realized at the interface is derived from the conservative flux. Note that

*q*=

*r*

_{Q}/(1 +

*r*

_{Q}) and, because

*r*

_{Q}is typically on the order of 0.01, water vapor mixing ratio,

*r*

_{Q}, and specific humidity are essentially interchangeable. The COARE 2.5 algorithm represents this aswhere

*C*

_{Q}was obtained by fits to the COARE measured flux data via (16) and the correction term is returned by the model. For version 3.0, the data were reanalyzed using the proper conservative quantity

*w*′

*r*

^{′}

_{x}

*C*

_{Q}and

*C*

_{e}are simply related

*C*

_{e}parameterizes the fundamentally conserved flux, we argue that it is the fundamental transfer coefficient and

*C*

_{Q}would, thus, depend not only on the ocean surface properties (i.e., the roughness lengths) but also on mean conditions as described by (19). The ratio

*C*

_{e}/

*C*

_{Q}will typically vary from 1.05 in the Tropics to 1.02 at high latitudes. Because we incorporate the Webb correction into our parameterization of

*C*

_{e}, our values are 2%–5% larger than conventional representations based on analysis of water vapor mass flux alone. The new version of the model computes

*w*′

*r*

^{′}

_{Q}

*W*is no longer needed for the moisture flux, but it can be used to compute Webb corrections for other trace gases where the basic measured quantity is mass concentration.

### c. The Charnock parameter

*α,*which appears in (6). For version 2.5 of the COARE algorithm we took

*α*= 0.011. This was the value suggested by Smith (1988), which also agreed with that obtained by direct measurement during COARE. By rearranging (4) and (6), and calculating neutral values at 10-m standard height,

*z*

_{o10n}

*κ*

*c*

^{1/2}

_{d10n}

^{−1}; subsequent measurements in higher wind speed regions found larger values for

*α*as wind speeds increased beyond 10 m s

^{−1}. One key result that appeared in the literature (Yelland and Taylor 1996) was a substantial analysis of inertial-dissipation (ID; see the appendix) stress measurements for wind speeds between 6 and 26 m s

^{−1}, which showed

*α*increasing monotonically from 0.011 to 0.017. Covariance measurements by ETL and the Woods Hole Oceanographic Institution (WHOI) on R/V

*Wecoma*off California during the Marine Boundary Layers (MBL) Experiment gave preliminary results with similar or even greater increases in

*α.*Figure 1 shows measurements of

*α*from several field campaigns that were available to us for the initial development of version 3.0 of the COARE algorithm. Based on this evidence, we adopted the form shown in Fig. 1, where

*α*increases linearly from 0.011 at

*U*

_{10n}= 10 m s

^{−1}to 0.018 at

*U*

_{10n}= 18 m s

^{−1}, and remains constant for lack of better information beyond this wind speed.

Yelland and Taylor's (1996) results were a major factor in our decision to allow *α* to increase with wind speed above 10 m s^{−1}. Ironically, in more recent papers Yelland and Taylor have reanalyzed their ID-based flux estimates so that they no longer support the increase in *α* specified in COARE 3.0. Yelland et al. (1998) revised their 1996 results using corrections to mean winds and measurement heights from computational fluid dynamics (CFD) calculations on models of their ships. Another revision (Taylor and Yelland 2000) was made after adopting an improved dimensionless dissipation function to calculate their ID stress values. These changes have reduced their values for *α* to about 0.011 for all wind speeds. Thus, the behavior of Charnock's parameter at wind speeds above 10 m s^{−1} remains controversial, and we will look to analysis of the large number of direct covariance and ID stress measurements that are accumulating in the ETL database to help resolve the matter. It is important because of the increasing application of bulk flux algorithms to severe storm situations.

### d. Surface wave influence on roughness parameters

For wind speeds greater than about 5 m s^{−1}, surface waves are a dominant factor in the surface roughness of the ocean. A simple description of surface roughness such as (6) represents the average surface wave climate of an ensemble of measurements. If this is reasonably similar to that of the open ocean, then the results are useful in many applications. However, it is sometimes desirable to know the stress appropriate for the actual wave conditions, for example, in coastal regions where the wave climate is different from that of the open ocean (Gulev et al. 1998). Furthermore, if we knew more about the linkage between wave properties and surface roughness, measurements from diverse regions could be associated more rationally than by simple averaging. The literature abounds with analyses and models that address this issue, ranging from crude parameterizations based on the simplest wave properties (e.g., significant wave height or phase speed of the dominant waves) to complicated integrations based on the two-dimensional wave spectrum.

The original COARE bulk flux model did not consider wave conditions, primarily because no detailed wave measurements were made but also partly because the wave stress community lacked consensus on how to handle waves. Subsequently some progress has been made in this regard, although it is fair to say that consensus is still lacking. However, we have incorporated into COARE 3.0 two recent parameterizations, each of which allows the Charnock parameter or velocity roughness length to be calculated from specified wave properties.

*z*

_{o}

*h*

_{s}

*h*

_{s}

*L*

_{p}

^{4.5}

*h*

_{s}is the significant wave height (mean height of the upper third of the height distribution that is approximately 4 times the rms surface elevation at a point) and

*L*

_{p}the wavelength associated with the peak of the wave frequency–size spectrum (the so-called dominant wave period,

*T*

_{p}). Therefore

*h*

_{s}/

*L*

_{p}is approximately the slope of the dominant waves. The coefficients are purely empirical, and are based on fits to extensive observations by Taylor, Yelland, and colleagues plus several sets of data from other experiments (including shallow water). The second relationship is from Oost et al. (2002):

*α*

*C*

_{p}

*u*

^{−2.5}

*C*

_{p}is the phase speed of the dominant wave and

*C*

_{p}/

*u*∗ is a measure of “wave age” (wave age is sometimes also defined as

*C*

_{p}/

*U*

_{10}). This result is based entirely on data obtained in several campaigns at the Dutch research platform in the North Sea.

*α*= 0.011. COARE 3.0 specifies

*α*to increase slightly with wind speed in the 12–18-m s

^{−1}range as described above, and is provided with a switch so that the user can choose between the average Charnock relation or either of these two wave parameterizations. With (25a) one must provide significant wave height and wave period; with (25b) only wave period is required. The relevant parameters are computed in the model using standard deep-water gravity wave relationships:These options may enable users to assess potential wave-induced uncertainties in the flux results. They are offered in the model without critical comment except to note that they were produced by two different experimental groups with well-deserved reputations for careful and thoughtful measurements and analysis. Figure 2 shows a comparison of the neutral momentum transfer coefficient for fully developed seas using the three COARE 3.0 options and the measurements of Smith (1980). We use the formulas for fully developed sea given by Taylor and Yelland (2001):

*h*

_{s}

*U*

^{2}

_{10n}

*T*

_{p}

*U*

_{10n}

*C*

_{p}/

*U*

_{10n}= 1.14; some wave practitioners prefer to specify fully developed conditions in terms of

*C*

_{p}/

*u*∗ ≈ 28.

We cannot claim that the wave climate of the dataset used to develop COARE 3.0 is, on average, well developed at all wind speeds, so Fig. 2 does not imply that one model is better than the other. The duration/fetch required to reach the fully developed state increases with mean wind speed, so it seems likely that well-developed conditions are more commonly observed at moderate winds. However, this issue is confused by possible contributions to the stress of swell waves that are not associated with the local winds; this becomes increasingly significant at low wind speeds and/or in high wind regions where larger swell is more common.

### e. Scalar roughness length parameterization

LKB (and COARE 2.5) parameterized the scalar roughness Reynolds numbers in terms of *R*_{r} [Eq. (7)], based on sublayer transfer and surface renewal theory, supported with a limited amount of field data. To produce the new scalar roughness parameterization for COARE 3.0, we combined some ETL datasets (see section 4) and added a reanalysis of data from HEXOS conducted on the Dutch tower in the North Sea (Smith et al. 1996). The COARE measurements on R/V *Moana Wave* produced about 850 h of usable humidity flux data. We focus on the behavior of moisture parameters, because the range of latent heat flux is much greater than that for sensible heat in our datasets, and historically it has received the greater attention. Data from two field programs prior to COARE [the Tropical Instability Wave Experiment (TIWE) and the Atlantic Stratocumulus Transition Experiment (ASTEX)] and one subsequent to COARE [the San Clemente Ocean Probing Experiment (SCOPE)] provided an additional 450 usable values. All measurements in this combined set, which we call COARE-plus, were made with the same system and processing methods. The SCOPE data were obtained from R/P *FLIP* which, because of minimal motion and flow distortion, are of unusually high quality and consistency (Grachev and Fairall 1997). The fluxes and bulk variables were averaged in 10-m neutral wind speed bins, and transfer coefficients were computed. This analysis gave a reasonably clean depiction of the wind speed dependence of *C*_{e10n} for *U* between 0 and 10 m s^{−1}.

For the higher wind speed regime, we used published results (DeCosmo et al. 1996) from HEXMAX. To make the HEXMAX transfer coefficients consistent with our COARE-plus data, they were modified to account for the cool skin (the cooler interface implies lower water vapor pressure, so that a larger transfer coefficient is required to produce the observed flux), the 2% reduction in water vapor pressure over seawater, and the Webb et al. (1980) correction. These changes increased their transfer coefficients by about 8%; the DeCosmo et al. (1996) *median* values were used to reduce the sensitivity to outliers and a skewed distribution. The maximum wind speed for usable humidity data from HEXMAX was about 18 m s^{−1}. The two datasets are shown in Fig. 3 as a function of wind speed. Because the HEXMAX conditions were much rougher than typical open ocean regions at the same wind speed, we have converted the results to roughness Reynolds number, *R*_{r}, which we take to be a more fundamental representation of air–sea surface interaction properties than the wind speed; these results are shown in Fig. 4. The compatibility of the ship-based, primarily tropical data with the platform-based, North Sea data is striking and encouraging.

*R*

_{t}and

*R*

_{q}, so that the scalar roughness lengths and therefore the transfer coefficients were different. The behavior of

*z*

_{oq}with

*R*

_{r}shown in Fig. 4 offers a much simpler mathematical relationship than LKB to represent (7). We, therefore, adopt the following empirical fit to the COARE-plus and HEXOS data:

*z*

_{oq}

^{−4}

^{−5}

*R*

^{−0.6}

_{r}

*z*

_{ol}. Also shown in Fig. 4 are the original LKB (=COARE 2.5) parameterization and a parameterization from Garratt (1992), based on the surface renewal approach of Brutsaert (1975), where

*z*

_{oq}is given as

*z*

_{oq}

*z*

_{o}

*R*

^{1/4}

_{r}

*C*

_{e10n}, the differences between the three models would be difficult to distinguish in a few 1-month field campaigns. The decrease of

*z*

_{oq}with

*R*

_{r}(or wind speed) is usually attributed to the relatively increasing importance of the molecular transfer bottleneck near the interface; momentum transfer has an additional transfer mechanism due to pressure and does not exhibit this behavior. Figure 5 shows version 2.5 and 3.0 neutral exchange coefficients as a function of wind speed. Beyond 10 m s

^{−1}COARE 2.5 is a pure extrapolation, with no validation.

## 4. Flux and transfer coefficient evaluation

Table 2 lists the series of 12 deployments of the ETL seagoing flux system, beginning before the COARE field program and ending in late 1999. The basic flux measuring system and software (see the appendix) remained essentially the same but sensor models, data acquisition structure, computers, etc., were upgraded over the period. As described in section 3e, data from the first six cruises formed the COARE-plus database. Because this contained only 67 h of data at wind speed greater than 10 m s^{−1}, it was augmented with 94 h of HEXMAX data and used to formulate COARE 3.0. Data from the six later cruises have been combined with the first six to form a larger database (ETL1999) containing 7216 h of data, including about 800 h with wind speeds exceeding 10 m s^{−1} and 2200 h at high latitudes. This allows us to test the algorithm over the wind speed range from 0 to 20 m s^{−1}.

We first scan the database to select a subset of covariance and ID flux estimates that satisfy criteria designed to reject invalid or unreliable points. The criteria include experimental aspects (e.g., relative wind direction within a certain sector to avoid interference by the ship's structure), instrument performance indicators, avoidance of ship maneuvers, and requirements that certain variables (e.g., variances) fall within physically reasonable limits. We might reject a specific flux value if the standard deviation of wind speed normalized by the mean wind speed exceeded some limit, but never reject a flux value based on its comparison with the bulk model. At sea, the shipboard system records continuously, irrespective of weather or operational conditions, so that such quality controls are needed to ensure a dataset that is clean, coherent, and relevant to the geophysical problem under investigation. In the present case, 4946 h for stress and 4276 h for latent heat flux were accepted (covariance, ID, and bulk).

The next step is to compare the values of fluxes obtained from the bulk algorithm with the measurements in some rational fashion. Usually we are interested in the average performance of the bulk algorithm, with information on its statistical scatter about the observations. In this analysis we show comparisons of quantities averaged in bins of 10-m neutral wind speed with the additional condition that the air–sea specific humidity difference exceeds 2.0 g kg^{−1}. Figure 6 shows such a comparison for latent heat flux. The bin width increases slightly at the data-sparse higher wind speeds; bins with fewer than five values are not shown. The turbulence values are the average of covariance and ID values; the bulk values are COARE 3.0. We plot both medians and means to reveal skewed distributions or effects of outliers. Figure 7 shows these same values plotted on an *x*–*y* linear scale. Figure 8 shows a similar comparison for stress, using only medians and averaging the covariance and ID values separately.

*G*=

*S*/

*U*is the gustiness factor and, for simplicity, we assume

*V*= 0. The 10-m neutral wind is computed from the wind measured at height

*z*asand the 10-m air–sea humidity difference isTo reduce sampling noise and avoid artificial correlations, we use bulk-derived values of MOST scaling parameters to compute the neutral 10-m values.

The bulk algorithm is used to compute bulk fluxes and transfer coefficients that are then averaged in each bin. The required transfer coefficient is the mean bulk coefficient multiplied by the ratio of mean measured to mean computed (bulk) fluxes. If the bulk model is reasonably accurate, (33) yields the transfer coefficient associated with the *measured* average flux.

In Fig. 9 we show the comparison between measured and modeled *C*_{e10p} using (33) and the average of mean and median values (we also show results of the SM using medians). The rms deviation of these values from the model is 4.0%. The statistical uncertainty in the mean 10-m neutral transfer coefficients, shown as error bars, was estimated by dividing the standard deviation of points within each wind speed bin, *σ*_{Ce}

Corresponding results for the momentum transfer coefficient are shown in Fig. 10, where again the error bars correspond to the uncertainty in the estimates. Figure 11 shows the Charnock parameter computed from the transfer coefficients given in Fig. 10 using (21) and (22). The large amount of new data at higher wind speeds strongly support the increasing *α* built into COARE 3.0 on the evidence of Fig. 1, and suggest that it continues to increase beyond 18 m s^{−1}. The comparison is not valid below 5 m s^{−1} because *z*_{o} approaches the smooth flow limit 0.11 *ν*/*u*∗.

*z*

_{oq}and

*R*

_{r}are quite sensitive to the method used to obtain

*z*

_{o}and

*u*∗, and confused by the contributions of swell, which are not accounted for in MOST. Apart from the highest and lowest wind speeds, agreement between COARE 3.0 and the measurements is excellent, and noticeably better than that shown in Fig. 9. This reflects the compensating effect of deviations of

*C*

_{d10n}from the model, as shown in Fig. 10, on measured

*C*

_{e10n}, since

*R*

_{q}as a function of

*R*

_{r}. Again, the model fits the ETL1999 measurements well. The original Garratt–Brutsaert model, referred to in section 3e and Fig. 4, fits poorly, but becomes a good fit with slight change of parameters:

*R*

_{q}

*R*

_{r}

*R*

^{1/4}

_{r}

Since the total variation of *R*_{q} is relatively small over the range of *R*_{r} from 0.1 to 100, a two-parameter fit is adequate.

## 5. Discussion

Here we compare the results presented above with previous measurements of air–sea exchange coefficients. There have been several recent comparisons of flux algorithms (CFC; ZZD; Chang and Grossman 1999) that showed that for *U* < 10 m s^{−1} results agreed quite closely, but diverged at higher wind speeds. However, few quality observations for *U* > 10 m s^{−1} existed to validate these comparisons. Shallow-water observations (e.g., HEXMAX) require some physical model to relate to open-ocean conditions. For stress, a consensus model is presently lacking; for scalar transfers we use roughness Reynolds number similarity, as described in the previous section. The new results presented in this paper, for air–sea fluxes and exchange coefficients, and on which the revised COARE bulk algorithm is based, derive from a very large database (ETL1999). The total number of hours of observation used in this study significantly exceeds the total number considered in previous reviews. They also benefit from great advances in sensor and computational technology as described in the appendix.

In Fig. 9, the increase in *C*_{e10n} toward very low wind speeds confirms, with about 700 h of data below 2.5 m s^{−1}, the behavior observed by Bradley et al. (1991) and embodied in LKB and both versions 2.5 and 3.0 of the COARE algorithm. For wind speeds above 4 m s^{−1}, previous estimates of *C*_{e10n} are usually given as a constant value because the accuracy ascribed to the measurements does not warrant more detail. The review by Smith (1989) found *C*_{e10n} = 1.2 (±0.1) × 10^{−3} for winds from 4 to 14 m s^{−1}; Garratt (1992) and Smith et al. (1996) quote 1.1 × 10^{−3} ±15% for winds between 3 and 20 m s^{−1}; for the HEXMAX data DeCosmo et al. (1996) give 1.1 × 10^{−3} and find no significant variation with wind speed up to 18 m s^{−1}. To compare with these, we calculate the constant value for *C*_{e10n} which fits the data above 5 m s^{−1}, and find 1.15 × 10^{−3} within 5.3%. This value may not compare directly because it includes the following: 1) reduction in seawater vapor pressure by 2% due to salinity, 2) a true air–water interface temperature (i.e., cool skin corrected), and 3) the Webb et al. correction. These factors combine to increase the mean moisture transfer coefficient by more than 6%.

However, as shown in Fig. 9, both the early data used to tune COARE 3.0 and the large ETL1999 database clearly demonstrate that *C*_{e10n} increases steadily toward higher winds. Previously, the HEXMAX data were the most significant high wind speed results for the moisture transfer coefficient. After adjustment as described in section 3e, HEXMAX is consistent with our measurements, whether graphed as *C*_{e10n} versus wind speed or *z*_{oq} versus *R*_{r}.

The neutral drag coefficient is well known to increase with wind speed and is often given an empirical linear form above about 5 m s^{−1} (e.g., Garratt 1992; Smith 1980; Yelland et al. 1998). The ETL1999 data in Fig. 10 confirm that a linear representation is not unreasonable, but is less enlightening than a model based on the physical concepts of air–sea exchange described in section 2. The Smith (1988) model embodied in COARE 3.0, Eq. (6), predicts an increase in *C*_{d10n} toward low wind speeds. As shown in both the covariance and ID measurements in Fig. 10, only for wind speeds less than about 1 m s^{−1}, does *C*_{d10n} increase above its minimum value of about 1.0 × 10^{−3}, which appears constant to about 5 m s^{−1}. Between about 6 and 14 m s^{−1}, the ID values are on average 3.0% lower than the covariance values, but the difference increases above 15 m s^{−1}. Agreement between COARE 3.0 and combined ID and covariance 10-m neutral drag coefficients is about 4%. In fact, the ID values for stress and drag coefficient agree closely with the model, while the covariance values tend to be higher at high wind speed and slightly lower at low wind speed. Covariance could be overestimated by about 10% because of ship flow distortion; the ID results could be underestimated by a similar amount because of loss of pressure transport production of turbulent kinetic energy (TKE) to the growing waves (Janssen 1999).

Most of our measurements for *U* > 12 m s^{−1} were acquired in the Fronts and Atlantic Storms Experiment (FASTEX) (North Atlantic) and Moorings (North Pacific) field studies and the number of useable observations for *U* > 15 m s^{−1} is fairly small (133 h for stress; 85 h for moisture). In the high wind regime, our drag coefficient values are somewhat higher than those of Smith (1980) and Taylor and Yelland (2000). For simplicity, consider values of 10^{3}*C*_{d10n} at *U* = 20 m s^{−1}: Smith, 1.93; Taylor and Yelland, 1.92; this dataset, covariance, 2.30 and ID, 2.07. The average of these four values is 2.06 and the spread is from +12% to −7%. Part of the difference between our ID result and that of Taylor and Yelland lies in the choice of dimensionless dissipation function. A change in Kolmogorov constant from 0.55 to 0.53 [the value consistent with Edson and Fairall (1998, hereafter referred to as EF)] would increase the Taylor and Yelland neutral transfer coefficient by 3.8% to 1.99. The COARE 3.0 value at 20 m s^{−1} is 2.06. Overall, we estimate that the COARE 3.0 transfer coefficients when applied to 1-h bulk measurements over the open ocean are accurate to about 5% for wind speeds 0–10 m s^{−1} and better than 10% for 10–20 m s^{−1}.

In the wind speed range 0–20 m s^{−1} the major remaining surface physics issue is the influence of surface waves on the fluxes. With present techniques, a huge number of observations will be required to obtain definitive results because of the addition of one or two independent variables. High-quality, routine measurements of wave properties is an important technical challenge, so we must hope for a breakthrough in theory or modeling. A second major issue is the application of our measurement-based algorithms in numerical models. One aspect is the difference between one-dimensional and two-dimensional representations (Vickers and Esbensen 1998) but most of the problems are associated with resolved versus unresolved (subgrid scale) processes and variability. The COARE algorithm accounts for velocity variability caused by boundary layer–scale eddies, but that is the only unresolved process explicitly in the model. Of course, when the model is applied to point measurements of the appropriate timescale, other sources of variability are explicitly resolved by the input data. In many numerical models moist convective processes are at least partly unresolved and this is a large source of variability. Some investigators (Jabouille et al. 1996; Zulauf and Krueger 1997; Redelsperger et al. 2000; Zeng et al. 2002) have tried adding a second gustiness velocity term, based on convective mass flux or precipitation rate, to the boundary layer convection-driven gustiness velocity (8). These studies used high-resolution models to simulate convective variability and examined the effect of spatial–temporal averages on the bulk flux relationships. The results have been encouraging but observational verification with conventional flux measurements is not straightforward. Rainfall studies encounter sampling problems and a point measurement of rainfall is a poor indicator of convective activity (at sea one can be surrounded by rainstorms for days and not collect any rain at the ship). Aircraft and ship-based scanning radars may be well suited to attack this problem.

The above analysis has ignored the possible effects of sea spray droplets on the moisture transfer coefficient. Our measurements of water vapor flux do not differentiate between evaporation from the sea surface and subsequent evaporation from sea spray droplets below (or above) the sensors. Such effects are expected to become significant at winds exceeding 15–20 m s^{−1}, but production of spray droplets as a function of wind speed are still uncertain (Andreas et al. 1995) and the threshold for measurable effects is not known. Recent model studies (Pattison and Belcher 1999; Andreas 2001) suggest droplet effects on the order of 10 W m^{−2} for HEXOS-type conditions where direct evaporation is about 250 W m^{−2}. In our opinion, such models are not yet sufficiently accurate to justify their inclusion in COARE 3.0.

## 6. Conclusions

The COARE bulk flux algorithm has been updated and its range of wind speed validity is extended to 0–20 m s^{−1}; we designate this version COARE 3.0. The updates include improvements to the stability functions, shortening the stability iteration, and eliminating the need for a Webb correction to latent heat flux. The wind speed dependence of both velocity and scalar transfer coefficients is changed slightly, particularly above 10 m s^{−1}. The modifications were based on nearly 2800 h of direct flux measurements during six ETL cruises in the COARE era (referred to as COARE-plus), augmented with about 100 h of data at wind speeds above 10 m s^{−1} from the HEXMAX experiment. Neutral moisture exchange coefficients from the two data sources merged extremely well after the published HEXMAX results were adjusted for three established correction factors.

In LKB and COARE 2.5, the scalar exchange coefficients were based on a relationship between scalar and velocity surface roughness Reynolds numbers. Analysis of the combined dataset suggested a much simpler mathematical relationship for COARE 3.0, directly relating the scalar roughness length to roughness Reynolds number. The drag coefficient was changed via a revised Charnock constant, which increases above 10 m s^{−1} on the evidence of observations from several sources, including some of the COARE-plus cruises.

Flux measurements from six later cruises (1997–99) have been added to COARE-plus to form a very large air–sea interaction database (ETL1999). ETL1999 contains about 7200 h of direct covariance and inertial-dissipation flux observations, 800 h for wind speeds greater than 10 m s^{−1} and 2200 h outside the Tropics, as well as concurrent measurements of bulk meteorological, radiation, and ocean variables. When sorted into wind speed bins of about 1-m s^{−1} width, the average latent heat flux ranges from 40 to 250 W m^{−2} and average stress from 0.001 to 1.0 N m^{−2}. The large number of observations in the database removes most of the statistical uncertainty in determining the mean fluxes and transfer coefficients. There is much more data than previously existed for both high (>10 m s^{−1}) and low winds (<2 m s^{−1}). When subjected to quality control filters, there remain about 4500 direct flux observations of very high quality for research and validation purposes, including here the evaluation of COARE 3.0.

The observations in ETL1999 strongly support new relationships built into the revised algorithm, for the scalar roughness length dependence on roughness Reynolds number (Fig. 12) and the Charnock parameter, *α* (Fig. 11). Our combined covariance and ID measurements indicate *α* constant at 0.011 up to about 10 m s^{−1}, increasing thereafter to about 0.020 at 19 m s^{−1}.

On the basis that *C*_{e10n} is often represented as a constant value between 5 and 20 m s^{−1}, our analysis indicates a value 1.15 × 10^{−3}, close to several other recent determinations. However, our measurements clearly indicate that *C*_{e10n} increases steadily with wind speed from about 1.08 × 10^{−3} at 5 m s^{−1} to 1.2 × 10^{−3} at 18 m s^{−1}. COARE 3.0 fits these bin-averaged measurements with an rms deviation of 4.0%. The measured *C*_{d10n} values increase from about 1.0 × 10^{−3} at 3 m s^{−1} to 2.30 × 10^{−3} at 20 m s^{−1} if covariance fluxes are used, or 2.07 × 10^{−3} for ID fluxes. The difference may be partly due to flow distortion effects on the covariance values, and above 15 m s^{−1} a possible indication of the wave-pressure effects on TKE dissipation and hence ID estimates (Janssen 1999). The average of 2.18 × 10^{−3} is significantly larger (i.e., separated by several standard deviations) than the classic covariance measurements of Smith (1980) and the large ID database of Taylor and Yelland (2000). The COARE 3.0 value at 20 m s^{−1} is 2.06. For wind speed greater than 2 m s^{−1} it fits the combined covariance and ID measurements of *C*_{d10n} within 4.2%. At low wind speeds both *C*_{e10n} and *C*_{d10n} increase toward lower wind speeds as has been observed previously.

Two alternative wave parameterizations (Taylor and Yelland 2001; Oost et al. 2002) have been incorporated into COARE 3.0, to enable wave conditions to be used in the calculation of surface roughness. The purpose is to enable the algorithm to be applied in regions, such as the coastal zone, where the wave climate is different from the open ocean. This option has not been evaluated by the authors, for lack of detailed wave data. However, we hope that this capability will encourage users to apply the arcane field of wind–wave relationships to practical situations where wave measurements are available, and thereby advance this area of study.

Twenty years ago, a survey of bulk flux schemes by Blanc (1985) found 30% differences in moisture and momentum transfer coefficients at moderate wind speeds with major problems noted at low and high wind speeds. With advancements in technology, air–sea flux measurements from ships are being made almost routinely by several research groups around the world and the progress has been impressive. Extensive low wind speed measurements in the COARE program and the adoption of gustiness have significantly improved the situation for light winds. We submit that, when applied to 1-h bulk measurements over the open ocean, the COARE 3.0 transfer coefficients are accurate to about 5% for wind speeds 0–10 m s^{−1}. There is still need to resolve differences in the 10–20 m s^{−1} regime, but since Blanc's review these differences have reduced to around 10%. At the highest wind speeds the effects of spray have yet to be satisfactorily quantified. The issue of the effects of flow distortion on fluxes, particularly stress, still needs to be addressed through a combination of numerical, laboratory, and field studies. The great advancement in ship-based covariance measurements is highlighted by recent results on air–sea gas transfer where direct measurements of CO_{2} fluxes and transfer velocity have been made (Fairall et al. 2000; McGillis et al. 2001). Until recently, such measurements were considered to range from the unacceptable to the impossible (Csanady 2001).

This work was supported by the ONR Marine Meteorology program, the DOE Atmospheric Radiation Measurement program, the NOAA Office of Global Programs, the National Science Foundation Climate Dynamics program, and Grant ATM-9727054 from NSF's Mesoscale Dynamics Meteorology program. The authors especially thank Scott Abbott, Dave Costa, Jesse Leach, Cat Russell, Dan Gottas, Jim Jordan, and Dave Gregg for their work, dedication, and personal sacrifice beyond all reason. Numerous discussions with Drs. Peter K. Taylor (SOC), Ed Andreas (CRREL), and Meghan Cronin (NOAA/PMEL) are gratefully acknowledged.

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# APPENDIX

## The ETL Shipboard Flux Measuring System

The ETL seagoing flux and bulk meteorology measurement system was fully described in Fairall et al. (1997). The following deals with specific aspects relevant to computing bulk transfer coefficients.

### Instruments

The basic measurements used in this paper are covariance and inertial-dissipation turbulent flux estimates, combined with measurements of the basic bulk variables as described in section 2. A sonic anemometer (Gill/INUSA RS-2 or RS-2A) is used to obtain the three components of the wind vector (*u*′, *υ*′, *w*′) and the sonic temperature (*T*′). A high-speed infrared hygrometer (Ophir Corporation IR-2000) is used to obtain *Q*′. Velocity fluctuations in fixed-earth coordinates are obtained from the raw anemometer output by applying rotations to account for pitch, roll, and yaw plus corrections for the ship's velocity vector. High-frequency (i.e., surface wave–induced) motions are measured with an integrated package of angular rate sensors and accelerometers (Systron Donner Motionpak) which forms the mounting base of the sonic anemometer. Lower-frequency motions are obtained from GPS, a gyrocompass, and the ship's Doppler speed log. Details of the motion correction are given in Edson et al. (1998) and discussion on higher-frequency data and other covariance processing issues appear in EF. Sonic temperature is corrected for velocity cross talk and the humidity contribution as in Fairall et al. (1997). The ID flux estimates are computed from the variance spectral density of *u*′, *T*′, and *Q*′ in the inertial subrange of locally isotropic turbulence, which is usually at frequencies sufficiently above the wave-induced platform motions so corrections are not needed.

The optics of the high-speed hygrometer can be contaminated by salt (Fairall and Young 1991; Fairall et al. 1997) and require daily washing using specially installed water jets. Data obtained with water on the optics (e.g., during rainfall) are unreliable. In some conditions, sunlight also invalidates the data. The condition of the optics is monitored in the data stream and a threshold is set to reject such data. Because of these three sources of error, usable data for latent heat flux are significantly less than for stress. The hygrometer is located as near to the anemometer as flow distortion considerations allow (usually about 1 m), and below rather than alongside or above the anemometer. To account for the loss of correlation caused by the physical separation of the *w*′ and *Q*′ sensors, a correction (typically 2%–4%) is applied to *w*′*Q*′

Mean wind speed and mean vector wind magnitude are obtained from the sonic anemometer after transformation to fixed-earth coordinates. A floating thermistor is used to obtain a near-surface value for the ocean temperature (the depth is about 5 cm). The COARE cool skin algorithm is used to obtain the interface value, typically 0.3°C cooler than the bulk. Mean air temperature and humidity are obtained with a combined temperature–relative humidity sensor in an aspirated radiation shield. In the early 1990s a Vaisala HMP35 sensor (0.1°C, 3% RH quoted accuracy) was used, later replaced with a Vaisala HMP-235 (0.1°C, 2% RH quoted accuracy).

### Flux processing methods

Covariance and ID fluxes and mean variables are computed in 10-min chunks from a nominally 1-h time section and then averaged to 1-h. A coordinate rotation of the high-speed time series is performed on the mean fixed-earth velocity vector, following Tanner and Thurtell (1969) to produce streamwise coordinates for the 1-h period. Thus, we compute fluxes normal to the mean flow vector, which is subject to a mean tilt of about 5° due to distortion by the ship's structure. Initially the 10-min covariance blocks were averaged and the ID fluxes were computed from the 1-h-averaged spectra. After 1993 we changed to spectral processing for the covariances, too. The time series is time tapered with a Hamming window, the cospectra computed, and the covariances obtained as the integral of the windowed cospectrum. This reduces the sampling noise caused by leakage of low-frequency variations associated with the rectangular window.

*s*

_{x}

*k*

*C*

^{2}

_{x}

*k*

^{−5/3}

*γ*

_{x}

*N*

_{x}

^{1/3}

*k*is the wavenumber,

*C*

^{2}

_{x}

*γ*

_{x}is the Kolmogorov constant,

*N*

_{x}is the rate of dissipation for the variable

*x,*and ε is the rate of dissipation of TKE. In the first relationship the factor 0.25 is a mathematical constant that follows from the Fourier transform relationship between the variance spectrum and the autocorrelation coefficient while, in the second, the Kolmogorov constants are empirically determined by measurements. The structure function parameter specifies small-scale turbulent fluctuations as the mean square difference in the variable

*x*at locations separated by a small distance

*d*:This notation is standard in spectral analysis, and should not be confused with the transfer coefficient as used in (2) and subsequently in the body of this paper.

*C*

^{2}

_{x}

*U*

^{−2/3}

_{r}

*ω*

^{5/3}

*s*

_{x}

*ω*

*f*

_{tay}

*U*

_{r}is the wind speed relative to the ship,

*ω*is the angular frequency, and

*f*

_{tay}is a correction for deviations from Taylor's hypothesis associated with wind variations. To verify the

*k*

^{−5/3}dependence, the spectral slope is computed as a function of frequency from a number of values in the inertial subrange (usually about 1–2 Hz) and a median

*C*

^{2}

_{x}

*C*

^{2}

_{x}

*x*

^{2}

_{∗}

*z*

^{−2/3}

*F*

_{x}

*ζ*

*F*

_{x}are the MOST dimensionless function parameters) are used to compute the MOST scaling parametersThe square root creates a sign ambiguity that we resolve by giving

*T*∗ and

*q*∗ the same sign as their air–sea mean differences. We have used the

*F*

_{x}forms from EF; we use bulk estimates of

*ζ*in (A5) to avoid convergence problems in the ID algorithm.

*ϕ*

_{ε}is the MOST dimensionless TKE dissipation rate function and

*ϕ*

_{Nx}is the scalar dimensionless dissipation rate. Thus, scaling parameters computed similar to (A5) can be used based on inferred dissipation rates. Note, the dissipation rates are

*inferred*rather than measured [i.e., through the empirical relationship (A1)]. Direct measurement of dissipation rates requires integration of the derivative spectrum into the kHz range, which is not feasible over the ocean. Equivalently, (A1) can be used to determine relationships between the dimensionless dissipation functions and the dimensionless structure functions:

*F*

_{x}

*ζ*

*γ*

_{x}

*κ*

^{−2/3}

*ϕ*

_{Nx}

*ζ*

*ϕ*

_{ε}

*ζ*

^{−1/3}

*ϕ*

_{ε}

*ζ*

*ϕ*

_{m}

*ζ*

*ζ.*

The structure function–based approach that we are using employs *F*_{x} functions from EF that are fits of spectral *C*^{2}_{x}*z*^{2/3} and normalized by covariance values of *x*^{2}_{∗}*FLIP.* Here directly measured fluxes are related to directly measured structure functions through a pair of empirically determined coefficients. The TKE approach relies on determination of the terms of the TKE budget equation, the value selected for the Kolmogorov constants, and other empirical functions such as *ϕ*_{m}. The EF approach makes no assumptions about TKE balances or Kolmogorov constants; it is a simple semiempirical combination of measured inertial subrange structure function parameters, covariance fluxes, and stability.

### Accuracy considerations

Covariance flux estimates are subject to random sampling errors associated with atmospheric variability (Wyngaard 1973; Finkelstein and Sims 2001) and other random errors caused by imperfect motion corrections or sensor noise and drift. Systematic errors are caused by incorrect sensor calibration, imperfect motion correction, and flow distortion. Of all the turbulent quantities over the ocean, the wind components (needed for the stress measurement) have the strongest signals, and the most dependable sensor. However, because of the cross talk between velocity components, stress is most susceptible to motion correction and flow distortion. Correlations between *w*′ and sonic-derived *T*′ are also subject to cross-talk errors because both are determined from the same time-of-flight measurements. Even humidity flux is subject to cross talk because of acceleration effects on the hygrometer's rotating light beam chopper. These effects are discussed in detail in Fairall et al. (2000) and McGillis et al. (2001). Near-zero flux measurements when Δ*T* and Δ*q* are very small indicates that cross-talk errors in our scalar fluxes are small, but in the case of sensible *heat* flux they may not be negligible. The accuracy of humidity fluxes is mainly constrained by uncertainty in the response of the humidity sensor to moisture fluctuations in the flux-containing frequency range. Comparisons between sensors of similar and different types suggests this is about 5% with the sensor used here.

For well-placed sensors on ships, flow distortion is a serious concern only for stress. Stress measurements from two research vessels were found to be 10%–15% greater than those obtained on R/P *FLIP* (considered to be largely distortion and motion free) during side-by-side intercomparisons (Edson et al. 1998; EF). The distortion effect will depend on the specific arrangement of sensors relative to the ship structure, so the above results may not translate exactly to the present measurements. Therefore we have applied no empirical distortion correction to our covariance stress data but note a possible systematic uncertainty of about 10%.

The covariance stress vector is composed of streamwise (*τ*_{x}) and cross-stream (*τ*_{y}) horizontal components, which combine to give a magnitude of the stress vector, *τ* = *τ*^{2}_{x} + *τ*^{2}_{y}*τ*_{y} is negligible, then *τ* is equal to the absolute value of *τ*_{x}. It is clear that taking the absolute value of the points in Fig. A1 and then performing an average will not yield the correct average stress on the ocean. For wind speeds above 3 m s^{−1}, *τ*_{y} makes a negligible contribution to the magnitude of the average stress. At very low winds there is a tendency for the streamwise component to average to a negative value that is interpreted as the return of momentum to the atmospheric by waves (Grachev and Fairall 2001). This effect represents another difficulty in MOST; nonzero cross-stream stress is also not accounted for by MOST. The bulk algorithm does not deal with these effects.

The ID flux estimates do not require motion corrections, and variance estimates (i.e., variance spectra) have smaller sampling variability than covariances (Wyngaard 1973). The statistical uncertainty of covariance and ID stress measurements is contrasted in Fig. A2 where variations about the bin medians are plotted as a function of wind speed. Here *σ*_{U∗} is computed as one-half the difference in the stress values corresponding to 84% and 16% cumulative probability within each wind speed bin. One important difference between covariance and ID stress estimates is the positive-definite nature of ID algorithms for stress. In covariance measurements we obtain a distribution of stress values (including negatives) that we average to obtain a mean estimate. In ID measurements we obtain a distribution of *C*^{2}_{x}*C*^{2}_{T}*C*^{2}_{u}*C*^{2}_{u}

The ID estimates are subject to another major error source: uncertainty in the dimensionless structure function parameter. We are using *F*_{x} functions from EF that are fits of spectral *C*^{2}_{x}*z*^{2/3} and normalized by covariance values of *x*^{2}_{∗}*FLIP.* The TKE approach is also subject to errors in the form of the dimensionless dissipation functions and in the value selected for the Kolmogorov constants (see discussion in EF).

The absolute accuracy of transfer coefficient measurements is subject to uncertainties in the mean measurements, the fluxes, and in the case of neutral transfer coefficients (or roughness length), the MOST stability functions (see discussion in Fairall et al. 1996a). Table A1 is a reproduction of accuracies claimed by Fairall et al. (1996a). These accuracies are not derived from factory calibrations for most instruments, but from comparisons with multiple instruments and platforms such as were done in the COARE, FASTEX, and the 1999 cruises on R/V *Ronald H. Brown.* In most cruises, mean and fast humidity sensors are scaled to match a high-quality handheld ventilated psychrometer; the adjustments are typically ±2%. The near-surface water temperature sensor is checked against the ship's thermosalinograph at night when vertical gradients are expected to be low. Flow distortion corrections have been applied for mean wind speed and for the height of measurements based on wind tunnel measurements (R/V *Moana Wave*) and wind flow patterns obtained by computational fluid dynamics (R/V *Knorr* and R/V *Ronald H. Brown*). Height adjustments to mean observations and ID fluxes are made as described by Yelland et al. (1998). The corrections are typically 2%–4% and the height changes on the order of 1 m. Of the 7216 1-h observations in the database, only 390 h from the R/V *Malcolm Baldridge* have no independent distortion estimates. When possible, the wind vector is referenced to the ocean surface to remove the effects of currents. For COARE the currents at a nearby buoy were used to correct the measured earth-frame wind vectors; for the 1999 cruises the ship's Doppler speed log was used as the reference in the motion corrections.

Comparative features of various modern bulk flux algorithms

Summary of ETL air–sea flux and bulk meteorological data cruises used in the analysis: Tropical Instability Wave Experiment (TIWE); Atlantic Stratocumulus Transition Experiment (ASTEX); Coupled Ocean–Atmosphere Response Experiment (COARE); San Clemente Ocean Probing Experiment (SCOPE); Fronts and Atlantic Storms (FASTEX); Joint Air–Sea Monsoon Experiment (JASMINE); Nauru ’99 Experiment (NAURU99); Tropical Rainfall Measuring Mission (TRMM)/Kwajalein Experiment (KWAJEX); Pan American Climate Studies fall 1999 study (PACSF99)

Table A1. Measurement uncertainty estimates from Fairall et al. (1996a)