## Abstract

Different measures of wind influence the ocean in different ways. In particular, the time-averaged mixed layer turbulent energy production rate is proportional to 〈*u*^{3}_{*}〉, where *u*_{*} is the “oceanic friction velocity” that is based on wind stress. Estimating 〈*u*^{3}_{*}〉 from monthly averages of wind stress or wind speed may introduce large biases due to the day-to-day variability of the direction and magnitude of the wind. The authors create monthly climatologies of 〈*u*^{3}_{*}〉 from daily wind stress measurements obtained from the Goddard Satellite-based Surface Turbulent Fluxes version 2 (GSSTF2; based on satellite microwave measurements), the Quick Scatterometer (QuikSCAT; based on satellite scatterometry measurements), and the National Centers for Environmental Prediction (NCEP) reanalysis wind. The differences among zonal averages of these climatologies and of a similar climatology based on the da Silva version of the Comprehensive Ocean–Atmosphere Data Set (COADS) have a complex dependence on latitude. These differences are typically 10%–30% of the climatological values. The GSSTF2 data confirm that 〈*u*^{3}_{*}〉 is much larger than estimates from monthly averaged wind stress or wind speed, especially outside the Tropics.

## 1. Introduction

Wind drives the ocean by transferring momentum, and so observed surface wind stress data are needed to force ocean models. There are at least two other ways in which near-surface wind data are important in influencing ocean behavior. One is through the heat flux, because both sensible and evaporative heat fluxes increase with surface wind speed (Pickard and Emery 1990, their section 5.3). The other is through the production of turbulent kinetic energy near the ocean surface (Kraus and Businger 1994, their section 5.2). While the wind influences all three processes (momentum transfer, heat transfer, and turbulent energy production), the *way* in which it influences each one has important, unique features. The purpose of this note is to highlight a careful treatment of wind climatology based on these features.

There is a nonlinear relationship between wind velocity measured at some standard height (generally 10 m) and wind stress ** τ** on the ocean surface. However, the ocean responds linearly to

**. As far as momentum flux is concerned, it is appropriate to drive an ocean with a form of**

*τ***that is linearly filtered in time (such as monthly averaging), at least for studying ocean behavior on time scales appropriate to the given time filter.**

*τ*The link between wind and oceanic turbulence is via the oceanic friction velocity (actually a speed, henceforth referred to simply as the friction velocity), *u*_{*}, defined by the equation

where *τ* is the magnitude of ** τ** and

*ρ*is the ocean density. One may argue that the friction velocity affects oceanic turbulent energy production

_{o}*ɛ*via

where *k* is a nondimensional constant and *z* is the depth below the surface of the ocean (Kraus and Businger 1994, their section 5.2). The appropriate time scale for this equation is one that is long compared to the time scale of small-scale turbulence in the ocean mixed layer but short compared to diurnal or atmospheric synoptic cycles. Substituting Eq. (1) into Eq. (2) and writing *τ* in terms of zonal and meridional vector components (*τ _{x}*,

*τ*), we have

_{y}This nonlinear relationship between *ɛ* and ** τ** means that the average

*ɛ*does not equal

*ɛ*calculated from the average

**. If we want monthly average production, we need the monthly average of a nonlinear function of**

*τ**u*

_{*}or

*τ*.

Nonlinearities are also present in the bulk formulas for freshwater flux and sensible heat flux, which depend on the product of *U* (wind speed at 10-m height) and the differences from the ocean surface to a 10-m height of, respectively, water vapor mixing ratio Δ*q* and temperature Δ*T* (Pickard and Emery 1990; Smith 1988). Similarly, the momentum flux is given by

where *ρ _{a}* is the density of air at the surface and

*C*has a complicated dependence on a number of variables (see, e.g., Fairall et al. 2003). In Eq. (4)

_{d}*U*

^{2}is analogous to

*U*Δ

*T*and

*U*Δ

*q*. We assume here that

*U*≫ surface velocity, which is an excellent approximation except for small regions where ocean surface velocity is large and

*U*is relatively small; in such regions, uncertainty in

*C*is comparable to errors induced by this assumption. Previous studies have shown that there may be significant errors if fluxes are calculated from a monthly average of

_{d}*U*, Δ

*q*, and Δ

*T*rather than the average of the appropriate product (Esbensen and Reynolds 1981; Hanava and Toba 1987; Ledvina et al. 1993; Gulev 1994; Josey et al. 1995; Zhang 1995; Esbensen and McPhaden 1996).

It may seem that the monthly average of *u*^{3}_{*} can be calculated directly from the monthly average of *U*, since both are scalar (as opposed to vector) averages. However, even if *u*_{*} and *U* are proportional to each other, the nonlinearity of *u*^{3}_{*} causes problems once again. Denoting an average with 〈 〉, and letting *x*′ = *x* − 〈*x*〉, we have

so that the greater the variability within a month (〈*x*′^{2}〉 and 〈*x*′^{3}〉), the greater the discrepancy between 〈*u*^{3}_{*}〉 and 〈*u*_{*}〉^{3}. Therefore, we should construct monthly averages and a monthly climatology of *u*^{3}_{*} to supplement the monthly climatologies of the other fields.

Observations from the Special Sensor Microwave Imager (SSM/I) flown on satellites from 1987 to 2000 have been used to construct a near-global wind dataset known as GSSTF2 (Goddard Satellite-based Surface Turbulent Fluxes version 2; Chou et al. 2003). Mears et al. (2001) compare wind speed measurements from this dataset to in situ measurements from buoys at various latitudes (mostly in the tropical and North Pacific) and find that daily measurements have a standard deviation difference of 1.3 m s^{−1} and a long-term average difference of less than 0.5 m s^{−1}. Chou et al. (2003) compare GSSTF2 to 10 ship-based experiments at widely dispersed locations and seasons and find similar agreement for speed; for stress, they find a standard deviation of 0.07 N m^{−2} m^{2} for daily measurements and a monthly average difference of roughly 0.01 N m^{−2}. Chou et al. (2003) show discrepancies between GSSTF2 and other datasets such as the National Centers for Environmental Prediction (NCEP) reanalysis (Kalnay et al. 1996) and a collection of marine data (da Silva et al. 1994). Note that the surface marine data of da Silva et al. (1994) are based on the Comprehensive Ocean–Atmosphere Data Set (COADS) of Woodruff et al. (1987); henceforth we refer to the da Silva et al. (1994) version of these data simply as COADS, though it differs from the original COADS release. Chou et al. (2003) argue that COADS overestimates *U* in windy regions because of underestimated anemometer heights, and that the difference from NCEP is due to NCEP *U* being too weak at low latitudes (Wang and McPhaden 2001; Smith et al. 2001) and too strong at high latitudes (Renfrew et al. 2002).

Chou et al. (2003) and Mears et al. (2001) make a strong case for GSSTF2 as a near-global wind dataset that can provide monthly statistics for the 1990s. These wind data are available via the Internet (see online at http://daac.gsfc.nasa.gov/precipitation/gsstf2.0.shtml). This dataset includes estimates of *U* and ** τ** on a 1° latitude–longitude grid for daily values, monthly averages, and a monthly 1988–2000 climatology.

We also construct *u*^{3}_{*} climatologies from other datasets. One dataset is the NCEP reanalysis, from which we use daily average data from the period 1980–2004. The data are available online (see ftp://ftp.cdc.noaa.gov/Datasets/ncep.reanalysis.dailyavgs/surface_gauss/). The other is the Quick Scatterometer (QuikSCAT) from the period 2000–04. QuikSCAT is a satellite scatterometer that beams microwaves at the sea surface and infers the wind speed and wind stress from the characteristics of the reflected signal. It has been in operation since July 1999. We use the “level 3” dataset, which consists of twice-daily measurements interpolated to a 1/4° latitude–longitude grid (Physical Oceanography DAAC 2003; available online at http://podaac.jpl.nasa. gov/ovw/). Ebuchi et al. (2002) compared QuikSCAT winds to ocean buoy measurements and found a root-mean-square difference of 1.0 m s^{−1}. Bourassa et al. (2003) compared QuikSCAT winds to in situ measurements from research vessels and estimated an average uncertainty of 0.3 m s^{−1} for QuikSCAT wind speed. We also compare climatologies derived from these datasets to the *u*^{3}_{*} climatology of COADS, which contains data from 1945 to 1989.

## 2. u^{3}_{*} climatologies

We compile statistics based on “daily” data, which for both satellite climatologies (GSSTF2 and QuikSCAT) contain missing data due to ice, gaps in the area covered by satellite on a given day, and grid points in which rain occurred during the satellite overpass. Geographical coverage is good, with the QuikSCAT satellite providing 90% daily coverage and GSSTF2 coverage comparable for most its record because of the presence of two satellites. In QuikSCAT, the daily data often consist of two measurements at a given grid point, corresponding to “ascending” and “descending” satellite passes. Excluding rainy days may introduce a bias, if at a given grid point rainy days tend to be more or less windy than nonrainy days. We did not attempt to estimate the difference in wind strength between rainy and nonrainy days. However, the more rainy days at a given grid point, the greater the bias induced by a given correlation between wind and rain. For GSSTF2, much of the ocean has coverage density of at least 7/8 at each grid point, with somewhat lower values in the vicinity of the subtropics (Fig. 1). There are bands of lower sampling density (typically around half of the days missing) just north of the equator, in the western tropical Pacific, and in the Bay of Bengal. The large fraction of missing days in these areas seems to be primarily due to rain.

The QuikSCAT level-3 data include two wind stress estimates based on two formulas. We use wind stress based on the Liu and Tang (1996) formula.

We calculate *u*_{*} at each grid point from ** τ** for each day for which there was a measurement, form an average of

*u*

^{3}

_{*}for each month in the 13-yr time series, and create a climatology by averaging the monthly values for each of the 12 months of the annual cycle. The four climatologies all show similar spatial structures for June and December (Fig. 2). For instance, all the climatologies show a winter intensification of

*u*

^{3}

_{*}at high latitudes, some signs of local maxima in the trade winds belts, and relatively weak friction velocity in the eastern and western tropical Pacific. There are numerous differences, both in the magnitude of

*u*

^{3}

_{*}and in details of the patterns. For instance, in December, the high-latitude North Pacific friction velocity maximum occurs in the eastern half of the basin in both satellite data climatologies, while it is more centrally located in NCEP.

To get a better quantitative picture of *u*^{3}_{*} dependence on latitude, we plot the zonal average for each climatology for June and December (Figs. 3a,b). Once again, all climatologies show similar patterns, with values ranging from around 0.5 cm^{3} s^{−3} at the equator to 3–5 cm^{3} s^{−3} at high latitudes. The seasonal cycle is more pronounced at high northern latitudes than at high southern latitudes. To compare the climatologies to each other, we examine the ratio *u*^{3}_{*}/*u*^{3}_{*a}, where *u*^{3}_{*a} is the average of all four climatologies. This ratio shows that discrepancies among the datasets have a complicated structure (Figs. 3c,d). NCEP wind is generally weaker than the other climatologies in the Tropics, reaching down to less than 0.6 of the *u*^{3}_{*a} at the equator. The low tropical values for NCEP are consistent with previous studies (Wang and McPhaden 2001; Smith et al. 2001). COADS is generally weak (compared to the others) in the Southern Hemisphere, also going lower than 0.6 of the average near 60° S. GSSTF2 is relatively small at high latitudes, especially in the summer hemisphere where it is roughly 0.7 of the average. QuikSCAT tends to have the largest values, rising to nearly 1.4 of the average at the equator in June.

The different climatologies are based on different observational methods as well as different time periods. To see the influence of the time period, we compared the NCEP friction velocity for three periods: 1980–2004 (the default value), 1988–2000 (as in GSSTF2), and 2000–04 (as in QuikSCAT). For June and December (not shown), 1988–2000 zonal averages are generally less than 5% different from the full record, while 2000–04 are generally less than 10% different from the full record. Therefore, most of the discrepancies between datasets shown in Figs. 3c,d do not appear to be due to the different sampling periods. The pointwise difference fields also have complicated latitude–longitude structure, with both positive and negative values at a given latitude.

As discussed above, the strength of the different climatologies (relative to each other) varies as a complicated function of latitude. The annual cycle at a given latitude shows a somewhat simpler structure. For the entire year, at a latitude of 44.5°N (Fig. 4a), NCEP and QuikSCAT are very close to each other while GSSTF2 is smaller; at 44.5°S (Fig. 4a), QuikSCAT is the strongest. At the equator (Fig. 4c), NCEP is the weakest and QuikSCAT is slightly larger than GSSTF2 for all months. At 14.5°S (Fig. 4d), NCEP is again the weakest but GSSTF2 is larger than QuikSCAT, especially from September–October. In contrast, at 14.5°N (Fig. 4b), the strongest winds switch between QuikSCAT and GSSTF2 in different months. At several latitudes, COADS data do not maintain a consistent strength relative to the other climatologies. (The *u*^{3}_{*} climatologies can be downloaded via anonymous ftp to ftp://ftp.grads.iges.org,subdirectorypub/klinger/GSSTF2.)

## 3. Comparison to other estimates of friction velocity

As discussed in section 1, calculating climatological *u*^{3}_{*} from climatological values of wind stress ** τ** or wind speed

*U*will introduce errors if the wind stress varies in time. Here we compare

*u*

^{3}

_{*}climatologies calculated from GSSTF2 〈

**〉 and 〈**

*τ**U*〉 to the GSSTF2 〈

*u*

^{3}

_{*}〉 discussed above (where 〈〉 represents the averaging needed to form a climatology).

As expected, the calculation based on 〈** τ**〉 seriously underestimates 〈

*u*

^{3}

_{*}〉, with zonal average values ranging from less than 0.2 to less than 0.9 of 〈

*u*

^{3}

_{*}〉 (Fig. 6a). This ratio is generally lower at midlatitudes than in the Tropics, presumably because of the more vigorous time variability at midlatitudes associated with synoptic weather systems.

To examine how well the monthly average *U* climatology predicts the monthly average *u*^{3}_{*}, we must have an expression for *C _{d}* in Eq. (4) so that we can relate

*U*to

*τ*. We will then use Eq. (1) to calculate 〈

*u*

_{*}〉

^{3}from the monthly 〈

*U*〉 climatology included in GSSTF2.

The actual value of *C _{d}* depends on several parameters; here we will attempt to approximate the expression used by Chou et al. (2003) with a simple expression

*C*(

_{d}*U*). To derive such an expression, we insert daily values of

*τ*and

*U*from GSSTF2 into Eq. (4) to calculate

*C*for each grid point on four representative days: the first day of March, June, September, and December 1989. For this calculation, we assume

_{d}*ρ*= 1.2 kg m

_{a}^{−3}. For each of these days, all of the pairs of

*C*and

_{d}*U*are sorted into 1 m s

^{−1 }

*U*bins, and we find the value of the 10th, 50th, and 90th percentile

*C*for each bin (Fig. 5a). The median values do not vary much from day to day (Fig. 5a). The 10th percentile and 90th percentile values are within 20% of the median value for virtually all

_{d}*U*, and within 10% of the median for most

*U*(Fig. 5b). If we take

*C*(

_{d}*U*) to be the average of the 50th percentile curves (heavy curve in Fig. 5a), we can estimate

*τ*for each grid point given only

*U*and the curve

*C*(

_{d}*U*). The resulting errors in

*τ*are less than 0.02 N/m

^{2}for most grid points (Fig. 5c), with the smallest errors in the Tropics (Fig. 5d).

The zonal average of this estimate of *u*^{3}_{*} divided by 〈*u*^{3}_{*}〉 also tends to be less than 1 (Fig. 6b), but not as small as the ratio based on *τ* (Fig. 6a). Again, values were smaller at midlatitudes than in the Tropics, with values ranging from 0.5 to 0.9 over most of the domain. Monthly averaged *U*, like monthly averaged *τ*, cannot be used to give a good measure of the turbulent production in the ocean mixed layer.

## 4. Summary and conclusions

Numerical ocean models need measurements of wind stress ** τ** for both momentum transfer and mechanical energy transfer to turbulence at the sea surface. This mechanical energy transfer is given by

*u*

^{3}

_{*}, where

*u*

_{*}is the oceanic friction velocity. Models often use monthly average values of wind stress, either for specific years or for a climatology derived from several years. However, daily variations in the magnitude and direction of

**can cause large discrepancies between estimates of**

*τ**u*

^{3}

_{*}based on monthly average

**and the true monthly average**

*τ**u*

^{3}

_{*}.

We have used GSSTF2, a recent satellite-based passive microwave measure of surface wind stress, to produce a climatology of monthly average *u*^{3}_{*}. We compared this to estimates based on monthly average wind stress and wind speed; both of these estimates, particularly that based on wind stress, grossly underestimate *u*^{3}_{*}. The underestimate is not as severe in the Tropics, where daily variations are not as large. The comparison indicates that outside the Tropics, it is important to use the proper measure of *u*^{3}_{*} in order to transfer mechanical energy into the ocean properly.

We also created climatologies based on 2000–04 QuikSCAT daily wind stress and 1980–2004 NCEP reanalysis daily wind stress. We did a side-by-side comparison of the three climatologies along with the da Silva et al. (1994) version of COADS, which calculates 〈*u*^{3}_{*}〉 from individual measurements of *u*^{3}_{*}. The four climatologies all have qualitatively similar features, but there are significant differences as well. Zonal average values show differences that have a complex dependence on latitude. In the Tropics, GSSTF2 and QuikSCAT estimates tend to be higher than NCEP (as much as about twice as high at the equator). Poleward of about 40° latitude, GSSTF2 can be more than 30% smaller than QuikSCAT. Elsewhere, GSSTF2 and QuikSCAT are typically 10%–20% different. NCEP reanalysis winds are known to be too weak in the Tropics, but the cause of the other discrepancies between different climatologies is not clear. An analysis of the time variation, as measured in the NCEP reanalysis, suggests that the difference in measurement time period for each climatology is not the primary cause of the differences.

Models may also be run with daily *τ* measurements, which eliminates the averaging problems discussed here. However, the same cannot be said for using a climatology based on many years’ measurements because averaging the data from the same date over many years greatly reduces the daily variability on that date, producing a similar bias to the one discussed here.

## Acknowledgments

Barry A. Klinger was supported by NSF Grant ATM-9910853. The authors thank Roger Lukas, an anonymous reviewer at COLA, and the two journal reviewers for helpful comments.

## REFERENCES

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## Footnotes

*Corresponding author address:* Dr. Barry Klinger, Center for Ocean–Land–Atmosphere Studies, 4041 Powder Mill Road, Suite 302, Calverton, MD 20705-3106. Email: klinger@cola.iges.org