# Characterizing overwater roughness Reynolds number during hurricanes

## Abstract

The Reynolds number, which is the dimensionless ratio of the inertial force to the viscous force, is of great importance in the theory of hydrodynamic stability and the origin of turbulence. To investigate aerodynamically rough flow over a wind sea, pertinent measurements of wind and wave parameters from three data buoys during Hurricanes Kate, Lili, Ivan, Katrina, Rita, and Wilma are analyzed. It is demonstrated that wind seas prevail when the wind speed at 10 m and the wave steepness exceed 9 m s^{−1} and 0.020, respectively. It is found that using a power law the roughness Reynolds number is statistically significantly related to the significant wave height instead of the wind speed as used in the literature. The reason for this characterization is to avoid any self-correlation between Reynolds number and the wind speed. It is found that although most values of \(R_{*}\) were below 500, they could reach to approximately 1000 near the radius of maximum wind. It is shown that, when the significant wave height exceeds approximately 2 m in a wind sea, the air flow over that wind sea is already under the fully rough condition. Further analysis of simultaneous measurements of wind and wave parameters using the logarithmic law indicates that the estimated overwater friction velocity is consistent with other methods including the direct (eddy-covariance flux) measurements, the atmospheric vorticity approach, and the sea-surface current measurements during four slow moving super typhoons with wind speed up to 70 m s^{−1}.

## 1 Introduction

Reynolds number is a key parameter in the theory of hydrodynamic stability and the origin of turbulence, which is of great importance to understand dynamical mechanisms of air–sea interactions. For a history of the Reynolds number, see Rott (1990). This number has been used in atmosphere–ocean interaction (see, e.g. Kraus and Businger 1994) and air–sea interaction (see, e.g. Csanady 2001). For its usage in hurricane physics, see, e.g. Anthes (1982), Davis et al. (2008), Kantha (2008), Smith and Montgomery (2010), Zeng et al. (2010), and Liu et al. (2011). An extensive literature survey related to the wind-stress parametrization at air–sea interface has been conducted by Bryant and Akbar (2016).

Here *U* _{10} is the wind speed at 10 m \(R_{*}\) is the roughness Reynolds number, \(U_{*}\) is the friction velocity, *Z* _{0} is the roughness length and \(\nu\) (= 1.46 × 10^{−5} m^{2} s^{−1}) is the kinematic viscosity of air.

Because there are large scatters in the relation between \(R_{*}\) and \(U_{10}\) (Andreas et al. 2012, Figs. 3, 4), an independent parameterization for the roughness Reynolds number is needed. Since there were no estimations of \(R_{*}\) using independent parameter such as the significant wave height, \(H_{\text{s}}\), based on buoy measurements during a hurricane available in the literature, it is the purpose of this study to find such relation between \(R_{*}\) and \(H_{\text{s}}\) under the fully rough flow over the wind seas during hurricanes.

## 2 Datasets and criteria used in this study

### 2.1 Datasets for this study

*U*

_{10}) and significant wave height (\(H_{\text{s}}\)) and peak or dominant wave period (

*T*

_{p}) from three data buoys during six hurricanes are analyzed. These datasets are listed in Table 1.

Data sources used in this study based on buoy measurements by the National Data Buoy Center (NDBC) (see http://www.ndbc.noaa.gov) during wind seas when \(U_{10}\) ≥ 9 m s^{−1} and \(H_{\text{s}} /L_{\text{p}}\) ≥ 0.020

Hurricane | Year | Month | Date | Buoy | |
---|---|---|---|---|---|

Kate | 1985 | 11 | 19–21 | 42003 | 9–47 |

Lili | 2002 | 10 | 1–3 | 42001 | 9–47 |

Ivan | 2004 | 9 | 12–16 | 42003 | 9–28 |

Katrina | 2005 | 8 | 26–28 | 42003 | 9–29 |

Rita | 2005 | 9 | 21–24 | 42001 | 10–41 |

Wilma | 2005 | 10 | 19–24 | 42056 | 12–32 |

### 2.2 Criterion for atmospheric stability

### 2.3 Criteria for wind speed and wind seas

Here *H* _{s} is the significant wave height in meters, *L* _{p} is the dominant wave length in meters, *T* _{p} is the peak or dominant wave period in seconds, and *g* is the gravitational acceleration (= 9.8 m s^{−2}). Note that the dimensionless parameter \(H_{\text{s}} /L_{\text{p}}\) is called wave steepness, which was proposed by Hsu (1974) and later validated by Taylor and Yelland (2001) for use in the aerodynamic roughness parameterization across the air–sea interface. Using Eq. (4), all datasets during the period as stated in Table 1 are valid under the wind sea conditions.

^{−1}, mixed seas were the general rule. On the other hand, when the wind speed exceeded approximately 9 m s

^{−1}, wind seas prevailed, in support of Eqs. (1) and (4).

## 3 Estimating overwater friction velocity

Since Eqs. (3) and (6) are valid for \(U_{10}\) up to 25 m s^{−1}, it is the purpose of this section to test and extend these linear relations between \(U_{*}\) and \(U_{10}\) to much higher wind speed ranges using other methods.

### 3.1 Using logarithmic wind profile law

Here \(k\) (= 0.4) is the von Karman constant, and \(Z_{ 0}\) is the aerodynamic roughness length.

Figure 4 indicates that the coefficient of determination \(R^{2}\) = 0.95, meaning that 95% of the variation between \(U_{*}\) and \(U_{10}\) can be explained by Eq. (9). In other words, if one accepts the high correlation coefficient *R* = 0.97, Eq. (9) is useful in air–sea interaction. It is very surprising that Eq. (9) is almost identical to Eq. (6), indicating that the results obtained from the use of logarithmic wind-profile approach is as effective as the direct method using the eddy-covariance flux measurements as employed by Andreas et al. (2012) and Edson et al. (2013).

### 3.2 Using atmospheric vorticity method

*h*is the height of hurricane boundary layer.

Based on the datasets of \(U_{10}\) (used here as a surrogate) and \(U_{*}\) as provided, Fig. 7 shows the results. Since the slope is unity and the correlation coefficient is 0.95, one may say that Eq. (9) is further verified.

### 3.3 Using sea-surface current method

Here \(U_{\text{sea}}\) is the sea-surface drift velocity.

On the basis of aforementioned analysis and discussions, it is demonstrated that, when the wind speed exceeds 9 m s^{−1} during wind seas, the linear relation between overwater friction velocity and the wind sped at 10 m indeed exists as shown in Eq. (9). This formula is almost identical to that using the direct eddy-covariance flux measurements as provided in Eq. (6). Equation (9) is further verified independently using the atmospheric vorticity method during a hurricane and using the sea-surface current velocity measurements by drifters during four super typhoons. Since Eqs. (6) and (9) are nearly identical and Eq. (6) is based on direct eddy-covariance method, Eq. (6) can now be extended up to the wind speed of 70 m s^{−1} and used in the following analysis.

## 4 Estimating the roughness length

Now, using the measured \(U_{10} ,\,U_{*}\) and \(Z_{0}\) can be computed from Eqs. (6) and (12), respectively.

## 5 Characterizing the roughness Reynolds number

On the basis of aforementioned analyses, we can now continue our search for the relation between \(R_{*}\) and \(H_{\text{s}}\) as follows.

### 5.1 Relation between \(R_{*}\) and \(H_{\text{s}}\) during Kate

*R*

^{2}= 0.86, meaning that 86% of the variation can be explained by the significant wave height or the high correlation coefficient,

*R*= 0.93, are acceptable, we have:

For 9 ≤ \(U_{10}\) ≤ 47 m s^{−1},

Equation (13) indicates that when \(H_{\text{s}}\) ≥ 1.5 m in a wind sea, \(R_{*}\) ≥ 2.6. According to Eq. (2) the air flow over this wind sea is fully rough, implying that the viscous effect on the wind sea is negligible and that the familiar logarithmic wind profile law is valid over the wind sea.

### 5.2 Relation between \(R_{*}\) and \(H_{\text{s}}\) during other five hurricanes

### 5.3 Characterizing overwater Reynolds number during all six hurricanes

Now, if we substitute *R* _{*} = 2.5 from Eq. (2) into Eq. (19), *H* _{s} = 1.6 m. In other words, when *H* _{s} are approximately higher than 2 m or during fresh breeze in Beaufort scale 5 for moderate waves, the airflow over the wind sea is already aerodynamically fully rough. Note that the three ‘outlier’ dots with *R* _{*} > 500 in Fig. 9 are all associated with time when hurricane passed directly through the buoys, future study is needed to investigate whether the wind-wave dynamics within the hurricane eyewall lead to the different behavior of these dots. Sapsis and Haller (2009) shows the extreme vorticity structure along the hurricane eyewall, which could explain the very strong Reynolds number *R* _{*} in Fig. 9.

## 6 Conclusions

On the basis of aforementioned analyses and discussions, it is concluded that, during six hurricanes as listed in Table 1, when both neutral-stability wind speed at 10 m, \(U_{10} ,\) and wave steepness, \(H_{\text{s}} /L_{\text{p}} ,\) exceed 9 m s^{−1} and 0.020, respectively, the roughness Reynolds number can be characterized by a power law using significant wave height, which has a high correlation coefficient of 0.92. It is found that although most values of \(R_{*}\) were below 500, they could reach to approximately 1000 near the radius of maximum wind. Furthermore, it is found that, when the significant wave height exceeds approximately 2 m in a wind sea, the air flow over this wind sea is under fully rough conditions, implying that the viscous effect on the wind sea is negligible and that the familiar logarithmic wind profile law is valid over the wind sea. In other words, when the significant wave heights are approximately higher than 2 m or during fresh breeze in Beaufort scale 5 for moderate waves, the airflow over the wind sea is already aerodynamically fully rough. Additional analyses of simultaneous measurements of wind and wave parameters using the logarithmic wind-profile law indicates that the estimated overwater friction velocity is consistent with other methods including the direct (eddy-covariance flux) measurements, the atmospheric vorticity approach, and the sea-surface current measurements during four slow moving super typhoons with wind speed up to 70 m s^{−1}.

## Notes

### Acknowledgements

This research was supported in part by the Chinese National Program on Global Change and Air–Sea Interaction (no. GASI–IPOVAI-04) and the international cooperation project of National Natural Science Foundation of China (no. 41620104003).

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