## Introduction

*U*

_{Z}is the wind speed at height

*Z, u*∗ is the friction velocity,

*κ*is the von Kármán constant,

*Z*

_{0}is the aerodynamic roughness length, and

*L*is the Monin–Obukhov stability length.

_{m}has the form (Paulson 1970)

*x*= [1 − 16(

*Z*/

*L*)]

^{1/4}and under stable conditions (Panofsky and Dutton 1984, 136),

In the offshore environment, such as over the Gulf of Mexico, daily operations occurring on many tall platforms (>50 m) require wind-loading estimations. If the platform does not have in situ wind measurements, the wind information is generally extrapolated from buoy measurements at 5–10 m above the sea surface. The purpose of this research note is to provide offshore technicians with a quick and accurate alternative to Eq. (2) for estimating the wind shear. A comparison between shear values derived from Eq. (2) and those from our simplified formula will also be made.

## The simplified formula

*a*and

*b*are coefficients to be obtained from the values of (−

*Z*/

*L*) versus Ψ

_{m}(Businger), as provided by Panofsky and Dutton (1984, 135–136). Our results (see Fig. 1) show that

^{2}= 98% of the variability in Ψ

_{m}(

*Z*/

*L*). Because the total system accuracy for wind speed from data buoys is about ±10% (National Data Buoy Center 1990), our Eq. (5) is well within the measurement accuracy and thus should be useful since it is much easier to compute with a calculator by field operators.

## A comparison between Eqs. (2) and (5)

*U*

_{Z}is measured at 12 m above the mean sea surface, we set

*Z*= 12 m. The von Kármán constant

*κ*is taken to be 0.4 (Donelan et al. 1997).

Figure 2 shows our results. Note that *Z*/*L* ranges between −0.01 and −9.23 and that the rmse is only 0.09 for the range of ln(*Z*/*Z*_{0}) between approximately 7 and 17. Thus the difference between Eqs. (2) and (5) is almost negligible so that for offshore applications, Eq. (5) should be useful.

## Further comparison between Eqs. (2) and (5) for operational use

*u*∗ is needed. For example, the wind-drift sea surface velocity,

*u*

_{s}≈ 0.55

*u*∗ (Wu 1975; Garratt 1992, 98). To estimate

*u*∗ one may “bypass”

*Z*

_{0}by using the drag coefficient formulation

*C*

_{DN}is the near-neutral drag coefficient and

*U*

_{10N}the near-neutral wind speed at 10 m. Since under near-neutral conditions, −

*L*→ ∞ so that

*Z*/

*L*→ 0. Thus the limit of Eq. (1) when Ψ

_{m}(

*Z*/

*L*) → 0 is

*C*

_{DN}is obtained from Garratt (1992, 101) as follows:

*C*

_{DN}

*U*

_{10N}

^{−3}

Since values of *U*_{10N} are also provided by Donelan et al. (1997), Eq. (10) is used to compare the difference between Eqs. (2) and (5) (setting *Z* = 10 m and *U*_{Z} = *U*_{10N}). Our results are shown in Fig. 3. The rmse here is only 0.23 cm s^{−1} for the *u*∗ range between about 10 and 60 cm s^{−1}, therefore we can say that Eq. (5) is nearly identical to Eq. (2) and thus it is applicable for operational use.

## Conclusions

The mathematical representation of Ψ_{m}(*Z*/*L*) used for the wind speed profile in the unstable atmospheric surface layer originally developed by Paulson in 1970 has been simplified. Comparisons between Paulson’s [shown as Eq. (2)] and our simplified formula [shown as Eq. (5)] indicate that, for overwater aerodynamic roughness length and shear velocity determination, their difference is negligible for offshore applications.

## REFERENCES

Donelan, M. A., W. M. Drennan, and K. B. Katsaros, 1997: The air–sea momentum flux in conditions of wind sea and swell.

*J. Phys. Oceanogr.,***27,**2087–2099.Garratt, J. R., 1992:

*The Atmospheric Boundary Layer.*Cambridge University Press, 316 pp.National Data Buoy Center, 1990: Climatic summaries for NDBC buoys and stations, Update 1, 454 pp.

Panofsky, H. A., and J. A., Dutton, 1984:

*Atmospheric Turbulence.*Wiley, 397 pp.Paulson, C. A., 1970: The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer.

*J. Appl. Meteor.,***9,**857–861.Wu, J., 1975: Wind-induced drift currents.

*J. Fluid Mech.,***68,**49–70.