^{*}

For wind-wave interaction studies, from the literature, two criteria for the wind sea are used: one is the wave age and the other wave steepness. Analysis of pertinent datasets from a data buoy near the passage of Hurricane Matthew in 2016 indicates that C_{p}/U_{10} = -15 H_{s}/L_{p} + 1.3. Here C_{p}/U_{10} is the wave age and H_{s}/L_{p} is the wave steepness, where C_{p} is the phase speed of the peak wave, U_{10} is the wind speed at 10 m, H_{s} is the significant wave height, and L_{p} is the peak or dominant wave length. It is found that 87% of the variation between wave age and wave steepness can be explained by this relation. Application of this relation to estimate U_{10} and friction velocity from wave parameters are also presented for practical environmental and engineering use.

In the realm of air-sea interaction, particularly in wind-wave interaction research, parameters of wave age and its steepness have been employed extensively (for literature reviews, see [

On October 2 and 3, 2016, Hurricane Matthew impacted the NDBC Buoy 42058 located in the deep central Caribbean Sea. During the growing wind seas, simultaneous meteorological and oceanographic (met-ocean) measurements of the wind speed at 5 m, U_{5}, wind gust, U_{gust}, significant wave height, H_{s}, dominant wave period, T_{p}, and other parameters were made by the NDBC (see www.ndbc.noaa.gov) as provided in ^{-1}, wind gust from 20 to 41 m s^{-1}, H_{s} from 4 to 10 m, and the barometric pressure dropped from 1002 to 958 hPa. This 44 hPa drop in 23 hour constitutes that Matthew was a meteorological bomb.

Day | Hour UTC | Min | U_{5} m s^{-1} |
U_{gust} m s^{-1} |
H_{s} m |
T_{p} sec |
Baro hPa | H_{s}/L_{p} |
C_{p}/U_{10} |

2 | 6 | 50 | 16.1 | 20.4 | 4.16 | 10.81 | 1002 | 0.02 | 1.0 |

2 | 7 | 50 | 16.6 | 20.3 | 4.64 | 10 | 1001.2 | 0.03 | 0.9 |

2 | 8 | 50 | 15.9 | 20.2 | 5.7 | 10 | 1001 | 0.04 | 0.9 |

2 | 9 | 50 | 16.8 | 21.3 | 5.51 | 11.43 | 1001.2 | 0.03 | 1.0 |

2 | 10 | 50 | 18.2 | 22.4 | 6.14 | 11.43 | 1001.4 | 0.03 | 0.9 |

2 | 11 | 50 | 18 | 22.5 | 6.82 | 11.43 | 1001.1 | 0.03 | 0.9 |

2 | 12 | 50 | 18 | 23.8 | 7.22 | 10.81 | 1001.3 | 0.04 | 0.9 |

2 | 13 | 50 | 21 | 26.9 | 6.09 | 10.81 | 1000.9 | 0.03 | 0.7 |

2 | 14 | 50 | 20.5 | 24.9 | 7.07 | 11.43 | 1000.4 | 0.03 | 0.8 |

2 | 15 | 50 | 19.3 | 23.8 | 6.74 | 10.81 | 999.4 | 0.04 | 0.8 |

2 | 16 | 50 | 19.8 | 25.8 | 6.23 | 10 | 998.1 | 0.04 | 0.7 |

2 | 17 | 50 | 21.9 | 26.2 | 6.77 | 10 | 995.6 | 0.04 | 0.6 |

2 | 18 | 50 | 22.5 | 29.7 | 7.23 | 10 | 993.9 | 0.05 | 0.6 |

2 | 19 | 50 | 23.7 | 29.3 | 8.4 | 10 | 992.1 | 0.05 | 0.6 |

2 | 20 | 50 | 24.2 | 29.6 | 7.98 | 10.81 | 991 | 0.04 | 0.6 |

2 | 21 | 50 | 24.3 | 29.9 | 7.9 | 10 | 989.5 | 0.05 | 0.6 |

2 | 22 | 50 | 23.2 | 28.5 | 7.32 | 10 | 990.2 | 0.05 | 0.6 |

2 | 23 | 50 | 22.6 | 29.2 | 6.97 | 10 | 990.1 | 0.04 | 0.6 |

3 | 0 | 50 | 24 | 29 | 6.22 | 9.09 | 989.6 | 0.05 | 0.5 |

3 | 1 | 50 | 26.1 | 33.8 | 7.18 | 9.09 | 987.2 | 0.06 | 0.5 |

3 | 2 | 50 | 27.8 | 35 | 7.71 | 10 | 983.5 | 0.05 | 0.5 |

3 | 3 | 50 | 31.6 | 38.4 | 10.09 | 10 | 972.6 | 0.06 | 0.4 |

3 | 4 | 50 | 33 | 40.9 | 10.35 | 10.81 | 958.1 | 0.06 | 0.5 |

In order to minimize the effects of swell, conditions under the wind sea are investigated and acquired. According to [

Here L_{p} is the dominant wave length in meters and g is the gravitational acceleration (= 9.8 m s^{-2}). Note that the parameter H_{s}/L_{p} is called wave steepness, which is available from routine buoy measurements. On the other hand, a wind sea exits when the wave age, C_{p}/U_{10}≤1.2 (see, e.g., [_{p} is the phase speed of the peak wave so that

Because the wind speeds were recorded at 5-m instead of 10-m at Buoy 42058 during Matthew in 2016, one needs to adjust U_{5} to U_{10} using the power-law wind profile (see, e.g., [

Here p = (U_{gust}/U_{5} -1)/2 = (G-1)/2, where G is the gust factor and U_{gust} is the wind gust measured at the buoy [

Using Eq. (5), we can now adjust the wind speed from 5 to 10 m.

Gust factor, G = U_{gust}/U_{5}, measured at Buoy 42058 during Hurricane Matthew in 2016.

On the other hand, the logrithmic wind profile [

And from [

Here U_{z} is the wind speed at height Z, U* is the friction velocity, k (= 0.4) is the von Karman constant, and Z_{o} is the roughness length.

Since U_{z} is available from the NDBC buoy measurements and by eliminating (U*/k) from Eq. (6), we have

A comparison between the power-law wind profile based on Eq. (4) and the logrithmic wind profile from Eq. (8) is demonstrated as follows: During Hurricane Delta in 2020, NDBC Buoy 42002 (located in the western Gulf of Mexico) was impacted. Similar to Eq. (4), we have

Here p = (U_{gust}/U4.1 -1)/2 = (1.32-1)/2 = 0.16, where G (=U_{gust}/U4.1) is the gust factor and U_{gust} is the wind gust measured at Buoy 42002 during Delta as provided in

On the other hand, based on Eq. (8) and set Z = 4.1m, one gets

Here Z_{o} is based on Eq. (7) by using the simultanueous wave measurements made at Buoy 42002 during Delta. Our result is presented in ^{2} are near unity, we are confident to say that the power-law and log-law wind profiles are compatible. However, since the log-law requires both wind and wave datasets so that the wave age can be computed which may suffer the self correlation, we are using only the wind gust factor as employed in

Measurements of the gust factor at NDBC Buoy 42002 during Hurricane Delta in 2020.

A comparison of U_{10} estimates based on the power-law (in vertical axis) and logrithmic wind profile law (in horizontal axis) at Buoy 42002 during Hurricane Delta in 2020.

On the basis of aforementioned methods and datasets from

Relation between wave age (C_{p}/U_{10}) and wave steepness (H_{s}/L_{p}) based on the last 2 columns in

As shown in ^{2} = 0.87, meaning that 87% of the variation between wave age and wave steepness can be explained by Eq. (12).

Rearranging Eq. (12), we have

Because NDBC Buoys 42003 and 42056 recorded all three parameters, i.e., the wind speed at 10 m, U_{10}, H_{s}, and T_{p}, during 4 hurricanes: Ivan (Sep. 13-16, 2004) and Katrina (Aug. 26-28, 2005) at 42003, Emily (July 17-18, 2005) and Wilma (Oct. 18-23, 2005) at 42056 (see www.ndbc.noaa.gov), these datasets are employed to compute U_{10} according to Eq. (13). Our results are presented in _{10} from wave parameters during wind seas. Using 2 other independent datasets [_{10} and H_{s} [

A comparison of Eq. (13) and measurements during 4 hurricanes (see text).

Further verification of Eq. (13) using 2 independent datasets.

In addition, using the estimated U_{10} and measured wave parameters, the shear or friction velocity, U*, a basic parameter for air-sea interaction, can now be determined based on Eqs. (6, 7, and 13) so that

On the basis of aforementioned analysis and discussions, it is concluded that for the relation between wave age and wave steepness, Eq. (12) may be employed. Therefore, this research note reconciles the long-standing difference or question whether the wave age or the wave steepness is superior to use in wind-wave interaction investigations. However, since wave steepness is measured routinely by the data buoys, it may be used to estimate the wave age using Eq. (12). Applications of this relation to estimate the wind speed at 10 m and friction velocity from wave parameters during the growing wind seas are also provided in Eqs. (13) and (14), respectively.

Appreciation goes to the NDBC for providing the datasets used in this research.

The author did all the research work of this study.

The author has declared that no competing interests exist.