Estimating Overwater Turbulence Intensity from Routine Gust-Factor Measurements

S. A. Hsu Coastal Studies Institute, Louisiana State University, Baton Rouge, Louisiana

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Brian W. Blanchard Coastal Studies Institute, Louisiana State University, Baton Rouge, Louisiana

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Abstract

For overwater diffusion estimates the Offshore and Coastal Dispersion (OCD) model is preferred by the U.S. Environmental Protection Agency. The U.S. Minerals Management Service has recommended that the OCD model be used for emissions located on the outer continental shelf. During southerly winds over the Gulf of Mexico, for example, the pollutants from hundreds of offshore platforms may affect the gulf coasts. In the OCD model, the overwater plume is described by the Gaussian equation, which requires the computation of σy and σz, which are, in turn, related to the turbulence intensity, overwater trajectory, and atmospheric stability. On the basis of several air–sea interaction experiments [the Barbados Oceanographic and Meteorological Experiment (BOMEX), the Air-Mass Transformation Experiment (AMTEX), and, most recently, the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE)] and the extensive datasets from the National Data Buoy Center (NDBC), it is shown that under neutral and stable conditions the overwater turbulence intensities are linearly proportional to the gust factor (G), which is the ratio of the wind gust and mean wind speed at height z (Uz) as reported hourly by the NDBC buoys. Under unstable conditions, it is first shown that the popular formula relating the horizontal turbulence intensity (σu,υ/u∗, where u∗ is the friction velocity) to the ratio of the mixing height (h) and the buoyancy length (L) (i.e., h/L) suffers from a self-correlation problem and cannot be used in the marine environment. Then, alternative formulas to estimate the horizontal turbulence intensities (σu,υ/Uz) using G are proposed for practical applications. Furthermore, formulas to estimate u∗ and z/L are fundamentally needed in air–sea interaction studies, in addition to dispersion meteorology.

Corresponding author address: Dr. S. A. Hsu, Coastal Studies Institute, 308 Howe-Russell Geoscience Bldg., Louisiana State University, Baton Rouge, LA 70803. sahsu@lsu.edu

Abstract

For overwater diffusion estimates the Offshore and Coastal Dispersion (OCD) model is preferred by the U.S. Environmental Protection Agency. The U.S. Minerals Management Service has recommended that the OCD model be used for emissions located on the outer continental shelf. During southerly winds over the Gulf of Mexico, for example, the pollutants from hundreds of offshore platforms may affect the gulf coasts. In the OCD model, the overwater plume is described by the Gaussian equation, which requires the computation of σy and σz, which are, in turn, related to the turbulence intensity, overwater trajectory, and atmospheric stability. On the basis of several air–sea interaction experiments [the Barbados Oceanographic and Meteorological Experiment (BOMEX), the Air-Mass Transformation Experiment (AMTEX), and, most recently, the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE)] and the extensive datasets from the National Data Buoy Center (NDBC), it is shown that under neutral and stable conditions the overwater turbulence intensities are linearly proportional to the gust factor (G), which is the ratio of the wind gust and mean wind speed at height z (Uz) as reported hourly by the NDBC buoys. Under unstable conditions, it is first shown that the popular formula relating the horizontal turbulence intensity (σu,υ/u∗, where u∗ is the friction velocity) to the ratio of the mixing height (h) and the buoyancy length (L) (i.e., h/L) suffers from a self-correlation problem and cannot be used in the marine environment. Then, alternative formulas to estimate the horizontal turbulence intensities (σu,υ/Uz) using G are proposed for practical applications. Furthermore, formulas to estimate u∗ and z/L are fundamentally needed in air–sea interaction studies, in addition to dispersion meteorology.

Corresponding author address: Dr. S. A. Hsu, Coastal Studies Institute, 308 Howe-Russell Geoscience Bldg., Louisiana State University, Baton Rouge, LA 70803. sahsu@lsu.edu

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 σu/U, συ/U, and σw/U, where σu, συ, and σw 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., σy and σz), respectively (see e.g., Panofsky and Dutton 1984; Zannetti 1990; Arya 1999).

In the atmospheric surface (or constant flux) layer (i.e., within 100 m above the surface where the variation of vertical turbulent flux with altitude is less than 10% of its magnitude), the turbulence intensities can be estimated (see Arya 1999) for neutral and stable conditions as follows:
i1520-0450-43-12-1911-e1
where Uz 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 σw/ Uz, including its range, is provided in Zannetti (1990, his Table 7-1, p. 148).
In boundary layer meteorology
i1520-0450-43-12-1911-e6
where Cd is the drag coefficient, which is related to the roughness length (Z0). Because Z0 over land is fixed for a given environment, Cd is also known. However, in the marine environment, Z0 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 = Ugust/Uz) 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

At the air–sea interface when the temperature difference between the sea (Tsea) and air (Tair) is zero (i.e., Tsea = Tair), 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 Uz = U10. When Tsea = Tair, the neutral wind is U10 = U10n at 10 m. In Fig. 1, the gust factor is plotted against U10 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
i1520-0450-43-12-1911-e7
Further verification of Eq. (8) is provided in Hsu (2003).
Now, substituting Eq. (8) into Eqs. (1)–(3), one obtains
i1520-0450-43-12-1911-e9
Because hourly values of 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 (σu,υ/u∗) and (h/L)

For unstable conditions when Tsea > Tair, 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.

In the convective boundary layer (see e.g., Kaimal and Finnigan 1994, p. 22),
i1520-0450-43-12-1911-e12
where 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 κ1/3 should be κ−1/3.
Now, by rearranging Eq. (4) we have
i1520-0450-43-12-1911-e13
and from Eq. (12)
i1520-0450-43-12-1911-e14
Substituting Eq. (14) into Eq. (13), one gets
i1520-0450-43-12-1911-e15
From the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) (see Fairall et al. 1996), it is given that
i1520-0450-43-12-1911-e16
where β (=1.25) is the gustiness parameter.
Under unstable conditions, σuσυ (see, e.g., Arya 1999 and our Table 1), Eq. (16) becomes
i1520-0450-43-12-1911-eq1
Setting β = 1.25, we have
wσu,υ
Substituting Eq. (17) into Eq. (15) yields
i1520-0450-43-12-1911-e18
which is near the values under neutral conditions [see Eq. (1)]. Therefore, the self-correlation between (σu,υ/u∗) and (h/L) exists in Eq. (4). An alternative method is provided in the next section.

An alternative relationship between (σu,υ/Uz) and (z/L)

From Eq. (5),
i1520-0450-43-12-1911-e19
and from Table 1,
i1520-0450-43-12-1911-e20
Therefore, σw may be replaced by σu numerically, so that
i1520-0450-43-12-1911-e21
From Pond et al. (1971), Cd = 1.52 × 10−3, or
i1520-0450-43-12-1911-e22
Thus,
i1520-0450-43-12-1911-e23
and from Table 1
i1520-0450-43-12-1911-e24
Note that from Smith (1980), when |z/L| = 0, σu/Uz 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 σu, συ, and σw 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 (σu/Uz), based on Eq. (24), and the horizontal axis represents the measured (σu/Uz). 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 (σu/Uz) and the stability parameter (z/L).

Now, analogous to the estimation of wind maxima, Umax (Panofsky and Dutton 1984),
UmaxUσU
we postulate that, from buoy measurements, Ugust = U ± S + U, or
i1520-0450-43-12-1911-e25
where S represents system accuracy of a buoy, such as instrument accuracy, sensor location, and angles of pitch and roll, and C is a coefficient.
Substituting Eq. (24) into Eq. (25), one gets the following statistical equation:
i1520-0450-43-12-1911-e26
where coefficients α and γ must be determined from field measurements. This is done in Fig. 3, which shows that
i1520-0450-43-12-1911-e27
with the correlation coefficient 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.

From Eq. (27), we have
i1520-0450-43-12-1911-e28
Substituting Eq. (28) into Eq. (24), one gets
i1520-0450-43-12-1911-e29
Furthermore, from Eqs. (5) and (28) and Fig. 1, we have
i1520-0450-43-12-1911-e30
Equations (29) and (30) are our proposed horizontal and vertical turbulence intensities for unstable conditions.
In addition, from Eqs. (17) and (29), we have
wUzG
which means that the convective velocity parameter may be computed directly by the surface wind and gust measurements. Thus, Eq. (31) bypasses the utilization of mixing height h that was needed in Eq. (12).

Conclusions

Several conclusions may be drawn from this study:

  1. 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).

  2. 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.

  3. 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).

  4. For practical applications the gust factor is found to be related to (z/L), which was shown statistically in Eq. (27).

  5. 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.

  6. In order to further verify Eqs. (9), (10), (11), (29), and (30), turbulent intensity measurements on an NDBC buoy are necessary. Because U10, 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.

REFERENCES

  • Arya, S. P. 1999. Air Pollution Meteorology and Dispersion. Oxford University Press, 310 pp.

  • Fairall, C. W., E. F. Bradley, D. P. Rogers, J. B. Edson, and G. S. Young. 1996. Bulk parameterization of air–sea fluxes for Tropical Ocean–Global Atmosphere Coupled Ocean–Atmosphere Response Experiment. J. Geophys. Res 101 C2:37473764.

    • Search Google Scholar
    • Export Citation
  • Fujitani, T. and T. Hayashi. 1975. Direct measurement of turbulent fluxes on RV Keifu-Maru. Disaster Prevention Research Institute AMTEX '75 Data Rep., Vol. 4, 125–128.

    • Search Google Scholar
    • Export Citation
  • Glickman, T. S. Ed.,. 2000. Glossary of Meteorology. 2d ed. Amer. Meteor. Soc., 797 pp.

  • Hsu, S. A. 1988. Coastal Meteorology. Academic Press, 260 pp.

  • Hsu, S. A. 2003. Estimating overwater friction velocity and exponent of power-law wind profile from gust factor during storms. J. Waterw. Port Coastal Ocean Eng 129:174177.

    • Search Google Scholar
    • Export Citation
  • Hsu, S. A. and B. W. Blanchard. 2003. Recent advances in air–sea interaction studies applied to overwater air quality modeling: A review. Pure Appl. Geophys 160:297316.

    • Search Google Scholar
    • Export Citation
  • Kaimal, J. C. and J. J. Finnigan. 1994. Atmospheric Boundary Layer Flows. Oxford University Press, 289 pp.

  • Panofsky, H. A. and J. A. Dutton. 1984. Atmospheric Turbulence. Wiley-Interscience, 397 pp.

  • Pond, S., G. T. Phelps, J. E. Paquin, G. McBean, and R. W. Stewart. 1971. Measurements of turbulent fluxes of momentum, moisture, and sensible heat over the ocean. J. Atmos. Sci 28:901917.

    • Search Google Scholar
    • Export Citation
  • Smith, S. D. 1980. Wind stress and heat flux over the ocean in gale force winds. J. Phys. Oceanogr 10:709726.

  • Yelland, M. and P. K. Taylor. 1996. Wind stress measurements from the open ocean. J. Phys. Oceanogr 26:541558.

  • Yelland, M., B. I. Moat, P. K. Taylor, R. W. Pascal, J. Hutchings, and V. C. Cornell. 1998. Wind stress measurements from the open ocean corrected for airflow distortion by the ship. J. Phys. Oceanogr 28:15111526.

    • Search Google Scholar
    • Export Citation
  • Zannetti, P. 1990. Air Pollution Modeling. Van Nostrand Reinhold, 444 pp.

Fig. 1.
Fig. 1.

A relationship between the gust factor and the drag coefficient under neutral conditions in the Gulf of Mexico

Citation: Journal of Applied Meteorology 43, 12; 10.1175/JAM2174.1

Fig. 2.
Fig. 2.

An analysis of the rmse between Eq. (24) and the AMTEX '75 dataset. The anemometer was located at 18 m above the sea surface

Citation: Journal of Applied Meteorology 43, 12; 10.1175/JAM2174.1

Fig. 3.
Fig. 3.

A relationship between the gust factor and stability parameter over the eastern Gulf of Mexico in Jan 2002

Citation: Journal of Applied Meteorology 43, 12; 10.1175/JAM2174.1

Fig. 4.
Fig. 4.

Verifications of Eq. (27) over the central Gulf of Mexico in Feb 2003; (top) U10 > 1 m s−1 and (bottom) U5 > 1 m s−1

Citation: Journal of Applied Meteorology 43, 12; 10.1175/JAM2174.1

Table 1.

Measured and derived boundary layer parameters during the Barbados Oceanographic and Meteorological Experiment (BOMEX) and pre-BOMEX cruises (Pond et al. 1971); z = 8 m and

Table 1.
Save
  • Arya, S. P. 1999. Air Pollution Meteorology and Dispersion. Oxford University Press, 310 pp.

  • Fairall, C. W., E. F. Bradley, D. P. Rogers, J. B. Edson, and G. S. Young. 1996. Bulk parameterization of air–sea fluxes for Tropical Ocean–Global Atmosphere Coupled Ocean–Atmosphere Response Experiment. J. Geophys. Res 101 C2:37473764.

    • Search Google Scholar
    • Export Citation
  • Fujitani, T. and T. Hayashi. 1975. Direct measurement of turbulent fluxes on RV Keifu-Maru. Disaster Prevention Research Institute AMTEX '75 Data Rep., Vol. 4, 125–128.

    • Search Google Scholar
    • Export Citation
  • Glickman, T. S. Ed.,. 2000. Glossary of Meteorology. 2d ed. Amer. Meteor. Soc., 797 pp.

  • Hsu, S. A. 1988. Coastal Meteorology. Academic Press, 260 pp.

  • Hsu, S. A. 2003. Estimating overwater friction velocity and exponent of power-law wind profile from gust factor during storms. J. Waterw. Port Coastal Ocean Eng 129:174177.

    • Search Google Scholar
    • Export Citation
  • Hsu, S. A. and B. W. Blanchard. 2003. Recent advances in air–sea interaction studies applied to overwater air quality modeling: A review. Pure Appl. Geophys 160:297316.

    • Search Google Scholar
    • Export Citation
  • Kaimal, J. C. and J. J. Finnigan. 1994. Atmospheric Boundary Layer Flows. Oxford University Press, 289 pp.

  • Panofsky, H. A. and J. A. Dutton. 1984. Atmospheric Turbulence. Wiley-Interscience, 397 pp.

  • Pond, S., G. T. Phelps, J. E. Paquin, G. McBean, and R. W. Stewart. 1971. Measurements of turbulent fluxes of momentum, moisture, and sensible heat over the ocean. J. Atmos. Sci 28:901917.

    • Search Google Scholar
    • Export Citation
  • Smith, S. D. 1980. Wind stress and heat flux over the ocean in gale force winds. J. Phys. Oceanogr 10:709726.

  • Yelland, M. and P. K. Taylor. 1996. Wind stress measurements from the open ocean. J. Phys. Oceanogr 26:541558.

  • Yelland, M., B. I. Moat, P. K. Taylor, R. W. Pascal, J. Hutchings, and V. C. Cornell. 1998. Wind stress measurements from the open ocean corrected for airflow distortion by the ship. J. Phys. Oceanogr 28:15111526.

    • Search Google Scholar
    • Export Citation
  • Zannetti, P. 1990. Air Pollution Modeling. Van Nostrand Reinhold, 444 pp.

  • Fig. 1.

    A relationship between the gust factor and the drag coefficient under neutral conditions in the Gulf of Mexico

  • Fig. 2.

    An analysis of the rmse between Eq. (24) and the AMTEX '75 dataset. The anemometer was located at 18 m above the sea surface

  • Fig. 3.

    A relationship between the gust factor and stability parameter over the eastern Gulf of Mexico in Jan 2002

  • Fig. 4.

    Verifications of Eq. (27) over the central Gulf of Mexico in Feb 2003; (top) U10 > 1 m s−1 and (bottom) U5 > 1 m s−1

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