Mariners Weather Log Vol. 59, No. 3, December 2015

Estimating Wave Height Using Wind Speed during a Tropical Cyclone

Professor S. A. Hsu
Louisiana State University
Email: shihahsu@outlook.com

Abstract

Analysis of wind and wave measurements from five NDBC data buoys during three hurricanes and one typhoon indicates that the significant wave height is approximately 30 percent of the wind speed at 10 meters. This 30% rule between wave height and wind speed should be useful for mariners as a safety guide during tropical cyclones.

1. Introduction

The wave height used in this study is the significant wave height, H_s, which is defined as the average height of the highest one-third of the waves observed at a specific point (see, e.g., Hsu, 1988). This parameter is very useful because it is approximately equal to the wave height that a trained observer would visually estimate for a given sea state (e.g. Bishop, 1984). During wind seas when the effect of swell is negligible, Carter (1982) developed the following equations for duration and/or fetch limited seas (see also, Taylor and Yelland, 2001) that for fetch limited seas,

H_s = 0.0163X^0.5 U₁₀, (1)

T_p = 0.566X^0.3 U₁₀^0.4, (2)

And for duration limited seas,

H_s = 0.0146D^5/7 U₁₀^9/7, (3)

T_p = 0.540D^3/7 U₁₀^4/7, (4)

The sea is considered fetch limited if

D > 1.167X^0.7U₁₀^-0.4(5)

Where H_s is the significant wave height in meters, T_p is the dominant wave period in seconds, U₁₀ is the wind speed at 10 meter in meters per second, X is the fetch in kilometer, and D is the duration in hours.

Because both fetch and duration parameters are not normally available, it is the purpose of this study to simplify aforementioned equations during tropical cyclones. Furthermore, operational formulas will be provided so that rapid estimation of H_s from U₁₀ may be made.

2. Data Analysis and Results

Because the parameter of fetch is not available routinely, we first eliminate it from Equations (1) and (2), and get, For fetch limited seas,

U₁₀ = 13413H_s³/ T_p⁵, (6)

Similarly, for duration limited seas by eliminating the duration parameter from Equations (3) and (4),

U₁₀ = 14754H_s³/ T_p⁵, (7)

According to Drennan et al (2005), a wind sea is defined when

H_s / L_p = 0.020, (8)

L_p / (g/2^pi)^T_p² = 1.56T_p²,(9)

Here L_p

is the dominant wave length and g is the gravitational acceleration. Note that the parameter H_s / L_p is called wave steepness.

Now, we can investigate the relation between H_s and T_p. During Hurricane Katrina over Buoy 42040 (from 0200UTC on 27 thru 1400UTC on 30 August 2005), an extreme wave height of 16.91m (55ft) was recorded by the National Data Buoy Center (NDBC) (www.ndbc.noaa.gov). Using the criteria as set in Equations (8) and (9) during the passage of Katrina near 42040, one can plot H_s against T_p as shown in Figure 1.

It is found that 86 percent of the relation between H_s and T_p can be explained by a power law (since the coefficient of determination, R² = 0.86). Because the exponent of this power law relation, which is 0.37, is very close to 0.40, we may replace it by 0.40. Our analysis is provided in Figure 2, which shows that,

T_p = 5.3H_s^0.4, (10)

Equation (10) should be useful, if the correlation coefficient, R = 0.88, is acceptable.

Substituting Equations (10) into (6) and (7), respectively, we have, For fetch limited seas,

H_s = 0.33^U10, (11)

And for duration limited seas,

H_s = 0.30^U10, (12)

In this study, four datasets (including three hurricanes and one typhoon) are analyzed. They are Hurricane Kate at Buoy 42003 (from 1600UTC on 18 thru 2300UTC on 21 in November 1985), Hurricane Ivan at 42003 (from 0100UTC on 13 thru 1100UTC on 16 in September 2004), Hurricane Katrina at 42003 (from 0400UTC on 26 thru 0500UTC on 28 in August 2005), and Typhoon Russ at 52009 near Guam (from 0100UTC on 16 thru 1400UTC on 20 in December 1990). Note that, while the anemometer at Buoy 42003 was located at 10m, the anemometer at Buoy 52009 during Typhoon Russ was located at 5m. Therefore, the wind data from 52009 were corrected from 5 to 10m using the logarithmic wind profile law (Hsu, 1988) and the roughness length parameterization based on Taylor and Yelland (2001). Our results are presented in Figure 3.

3. Conclusions

Since our result shown in Figure 3 is identical to Equation (12), it is concluded that the significant wave height can be estimated linearly from the wind speed. Furthermore, although the difference between Equations (11) and (12) is about 9%, which is within the 10% composite margin of error in wind measurements at sea according to NDBC, Equation (11) may also be employed for safety reasons.

Acknowledgements

Buoy measurements provided by NDBC are greatly appreciated.

Figure 1. Relation between T_p and H_s at NDBC Buoy 42040 during Hurricane Katrina.

Figure 2. Relation between T_p and H_s^0.4 at 42040 during Katrina.

Figure 3. Relation between H_s and U₁₀ during Hurricanes Kate, Ivan, Katrina and Typhoon Russ.

References

Bishop, J. M., 1984: Applied Oceanography, Wiley, New York.

Drennan, W. M., P. K. Taylor and M. J. Yelland, 2005: Parameterizing the sea surface roughness, Journal of Physical Oceanography, 35, 835-848.

Hsu, S. A., 1988: Coastal Meteorology, Academic Press, San Diego, CA.

Carter, D. J. T., 1982: Prediction of wave height and period for a constant wind velocity using the JONSWAP formula, Ocean Engineering, 9, 17-33.

Taylor, P. K., and M. J. Yelland, 2001: The dependence of sea surface roughness on the height and steepness of the waves, Journal of Physical Oceanography, 31, 572-590.