## Introduction

According to Glickman (2000), the turbulence intensity is defined as the ratio of the root-mean-square of the eddy velocity to the mean wind speed; in general, it is a quantity that characterizes the intensity of gusts in the airflow. Mathematically, the turbulent intensities in horizontal and vertical directions are *σ*_{u}/*U,* *σ*_{υ}/*U,* and *σ*_{w}/*U,* where *σ*_{u}, *σ*_{υ}, and *σ*_{w} are the standard deviations of velocity fluctuations in the *x,* *y,* and *z* directions, respectively, and *U* is the mean wind speed. Note that the mean wind speed, rather than the particular component mean velocity, is used in the definition of turbulence intensities (Arya 1999). In air pollution meteorology and dispersion, these turbulence intensities are related to the particle dispersion parameters in the *y* and *z* directions (i.e., *σ*_{y} and *σ*_{z}), respectively (see e.g., Panofsky and Dutton 1984; Zannetti 1990; Arya 1999).

*U*

_{z}is the wind speed at height

*z,*

*u*∗ is the friction velocity,

*h*is the boundary layer height, and

*L*is the Obukhov (buoyancy) length. Note that the atmospheric stability classification by Pasquill categories using

*σ*

_{w}/

*U*

_{z}, including its range, is provided in Zannetti (1990, his Table 7-1, p. 148).

*C*

_{d}is the drag coefficient, which is related to the roughness length (

*Z*

_{0}). Because

*Z*

_{0}over land is fixed for a given environment,

*C*

_{d}is also known. However, in the marine environment,

*Z*

_{0}varies with the wind, sea, and swell characteristics, in addition to the atmospheric stability parameter (

*z*/

*L*) (see e.g., Hsu 1988). As stated previously, the turbulence intensity is related to the wind gust. Because the gust factor (

*G*=

*U*

_{gust}/

*U*

_{z}) is measured routinely by the National Data Buoy Center (NDBC) buoys, it is the purpose of this study to find practical formulas to estimate turbulence intensities using

*G.*For more detail about NDBC's measurement program, see their Web site (online at seaboard.ndbc.noaa.gov).

## For neutral and stable conditions

*T*

_{sea}) and air (

*T*

_{air}) is zero (i.e.,

*T*

_{sea}=

*T*

_{air}), the stability parameter

*z*/

*L*also is zero (see e.g., Hsu and Blanchard 2003). Under these conditions, neutral stability prevails. With this criterion, we select NDBC buoy 42001 in the central Gulf of Mexico for our analysis. The anemometer was located at the standard 10 m above the sea surface so that

*U*

_{z}=

*U*

_{10}. When

*T*

_{sea}=

*T*

_{air}, the neutral wind is

*U*

_{10}=

*U*

_{10n}at 10 m. In Fig. 1, the gust factor is plotted against

*U*

_{10}and the recent drag coefficient formulations based on extensive datasets in the open ocean by Yelland and Taylor (1996) and Yelland et al. (1998) are superimposed. Excellent agreement between the gust factor and the drag coefficient is reached, such that

*G*are provided by NDBC buoys, Eqs. (9)–(11) are recommended for practical applications for neutral and stable conditions.

## For unstable conditions

### The self-correlation problem between (σ_{u,υ}/u∗) and (h/L)

For unstable conditions when *T*_{sea} > *T*_{air}, Eq. (4) has been used extensively in air pollution meteorology (see, e.g., Panofsky and Dutton 1984; Zannetti 1990; Arya 1999). However, this equation suffers from self-correlation as discussed below.

*w*∗ is the convective velocity and

*κ*(=0.4) is the von Kármán constant. Notice that the same equation is provided in Arya [1999, p. 101, Eq. (4.64)], but that there is an error such that

*κ*

^{1/3}should be

*κ*

^{−1/3}.

*β*(=1.25) is the gustiness parameter.

*σ*

_{u}≃

*σ*

_{υ}(see, e.g., Arya 1999 and our Table 1), Eq. (16) becomes

*β*= 1.25, we have

*w*

*σ*

_{u,υ}

*σ*

_{u,υ}/

*u*∗) and (

*h*/

*L*) exists in Eq. (4). An alternative method is provided in the next section.

### An alternative relationship between (σ_{u,υ}/U_{z}) and (z/L)

*C*

_{d}= 1.52 × 10

^{−3}, or

*z*/

*L*| = 0,

*σ*

_{u}/

*U*

_{z}is also 0.10.

Equation (24) is further verified by another independent dataset. In order to obtain as large a variance of |*z*/*L*| as possible, data from the 1975 Air-Mass Transformation Experiment (AMTEX '75) (see Fujitani and Hayashi 1975) are employed. This dataset includes overwater measurements of *σ*_{u}, *σ*_{υ}, and *σ*_{w} in addition to wind speed and air and sea temperatures. Note that AMTEX '75 was conducted over the open East China Sea in February 1975 and that values of |*z*/*L*| extended to nearly 7. [For the computation of |*z*/*L*|, see Hsu and Blanchard (2003).] Figure 2 is our result. Note that the vertical axis is the estimated (*σ*_{u}/*U*_{z}), based on Eq. (24), and the horizontal axis represents the measured (*σ*_{u}/*U*_{z}). Because the rmse is small in comparison with the data range, we conclude that Eq. (24) is a useful approximation between the longitudinal turbulence intensity (*σ*_{u}/*U*_{z}) and the stability parameter (*z*/*L*).

*U*

_{max}(Panofsky and Dutton 1984),

*U*

_{max}

*U*

*σ*

_{U}

*U*

_{gust}=

*U*±

*S*+

*Cσ*

_{U}, or

*S*represents system accuracy of a buoy, such as instrument accuracy, sensor location, and angles of pitch and roll, and

*C*is a coefficient.

*α*and

*γ*must be determined from field measurements. This is done in Fig. 3, which shows that

*R*= 0.84.

Note that during the first half of January 2002, cold air moved over the eastern Gulf of Mexico, including the freezing temperature line, which eventually draped over the northern Gulf Coast. To further verify Eq. (27), Fig. 4 is provided. If one accepts these smaller rmses as compared with the gust factor analyzed, one can say that Eq. (27) is useful operationally.

## Conclusions

Several conclusions may be drawn from this study:

Under neutral and stable conditions, the turbulence intensities in the horizontal and vertical directions are linearly related to the gust factor as derived in Eqs. (9)–(11).

Under unstable conditions, the ratio of the mixing height and buoyancy length cannot be related to the horizontal turbulence intensity because of the self-correlation problem. Therefore, the popular formula shown in Eq. (4) should not be used for overwater applications.

Because of the self-correlation problem stated above, an alternative formula is proposed in Eq. (24), which relates the horizontal turbulence intensity to the surface-based stability parameter (

*z*/*L*).For practical applications the gust factor is found to be related to (

*z*/*L*), which was shown statistically in Eq. (27).Using Eq. (27), variations in horizontal and vertical turbulence intensities with the gust factor are proposed for practical applications in Eqs. (29) and (30), respectively.

In order to further verify Eqs. (9), (10), (11), (29), and (30), turbulent intensity measurements on an NDBC buoy are necessary. Because

*U*_{10},*G,*and*z*/*L*are available at buoys 42001, 42002, and 42003, it is recommended that further experiments be conducted at these stations if logistically practical.

## Acknowledgments

This study was partially supported by the Minerals Management Service (MMS), U.S. Department of the Interior, through the Coastal Marine Institute of Louisiana State University under a cooperative agreement with Louisiana State University. The contents of this paper do not necessarily reflect the views or policies of the MMS.

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