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\title{Validating Wind Profile Equations during Tropical Storm Debby in 2012}
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\begin{document}

             \author[1]{Prof. S. A.  Hsu}

             \affil[1]{  Louisiana State University}

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\date{\small \em Received: 9 December 2013 Accepted: 4 January 2014 Published: 15 January 2014}

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\begin{abstract}
        


Comparisons of logarithmic and power-law wind profiles are made for offshore conditions during Tropical Storm Debby in 2012 over the Gulf of Mexico. It is found that both laws are validated up to 122m and that the power law is as good as the log law statistically. For practical applications, the exponent of power law can be determined from the gust factor measurement available routinely from National Data Buoy Center (NDBC) buoys.

\end{abstract}


\keywords{logarithmic wind profile, power-law wind profile, gust factor, tropical storm debby.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
n the atmospheric boundary layer, vertical distribution of the wind speed (under strong wind conditions so that the thermal effects may be neglected, see \hyperref[b2]{Hsu, 2003)} can be formulated according to the logarithmic wind profile (e.g. \hyperref[b8]{Panofsky and Dutton, 1984)} as:U z = (U*/k) Ln ((Z-d)/Z 0 )\textbf{(1)}\par
Where U z is the wind speed at height Z, U* is the friction velocity, k (=0.4) is the von Karman constant, d is the displacement height, and Z 0 is the roughness length.\par
Note that when Z is much larger than d, Eq. (1) may be reduced to Uz = (U*/k) Ln (Z/Z 0 )\par
(2)\par
Note also that, for offshore conditions, Z o can vary with wave characteristics since the sea state is mobile depending on the wind speed, duration and fetch \hyperref[b1]{(Hsu, 1988)}.\par
Because the log-law requires several parameters including U*, d, and Z 0 , the power-law wind profile has been widely used instead in the engineering community (e.g. \hyperref[b6]{Irwin, 2006}) that:U 2 /U 1 = (Z 2 /Z 1 ) \textasciicircum  p (3) For Z 2 > Z 1\par
Where U 2 and U 1 are the wind speed at Z 2 and Z 1 , respectively, and p is the exponent, which is related to the gust factor, G, via following formulation (see Hsu, that,G = 1 + 2p\textbf{(4)}\par
The purpose of this study is to validate windprofile equations, whether it is logarithmic or power, for over water applications. 
\section[{II.}]{II.} 
\section[{Methods}]{Methods}\par
In order to compare Equations (  {\ref 2}) and (3) statistically, they are rearranged as follows: From Equation (2), we have Ln Z = Ln Z 0 + (k/U*) U z \hyperref[b4]{(5)} This equation has a least-square linear regression form such thatY = A + B X\textbf{(6)}\par
Where Y = Ln Z, A = Ln Z 0 or Z 0 = e A , X = U z , and B = k/U*.\par
If the anemometer is located at 10 m, one can normalize the wind speed at higher elevation by NDBC measurements so that Equation (3) becomes U z /U 10 = (Z/10) \textasciicircum  p \hyperref[b6]{(7)} Where U 10 is the wind speed at 10 m, which is routinely available from Buoy 42040. 
\section[{III.}]{III.} 
\section[{Validating Wind Profile Equations}]{Validating Wind Profile Equations}\par
When Tropical Storm Debby in 2012 was over the Gulf of Mexico (www.nhc.noaa.gov), there were 3 meteorological stations, which measured wind speed at 3 different heights ranging between 10 and 122m above the sea surface. These data are available online from NDBC (http //www. ndbc.noaa.gov). Since these 3 stations were close-by, their data are listed in Table \hyperref[tab_0]{1}. Using the mean value for each height as listed in the Table, Figures \hyperref[fig_0]{1 and 2} provide the analyses for the logarithmic and the powerlaw wind profiles, respectively. Since the coefficient of determination, R 2 , values are very high, both profiles are verified. For operational applications off shore, it is found that the power-law is as good as the log-law. This finding is very important because over vast ocean, only one level measurement of wind speed from few buoys or ships is available. I    
\section[{Vertical Variation of the Gust Factor}]{Vertical Variation of the Gust Factor}\par
Vertical variation of the gust factor for offshore conditions has been studied by \hyperref[b5]{Hsu (2012)}. Our results during Debby are shown in Figs.  {\ref 3 and 4}. It is found that for operational applications, the gust factor does not decease with height between 10 and 160 m as compared to that over land as shown in Fig.  {\ref 5} near land-falling Hurricanes Frances and Jeanne in 2004 (based on \hyperref[b7]{Merceret, 2009)} and Fig.  {\ref 6} for Typhoon Muifa in 2012 (based on \hyperref[b0]{An et al.,2012)}.\par
In addition, according to Hsu (2012), the mean overwater gust factor between 5 and 160 m during Tropical Storm Lee in 2011 is 1.273 with the standard deviation of 0.11 so that the coefficient of variation, which is the ratio of standard deviation and the mean, is 8.6\% (which is within the 10\% composite error margin in the field measurements for the wind speed). This means that p=0.137 is a good value to use for offshore conditions. Comparison of this p value with the powerlaw analysis shown in Fig.  {\ref 2} (where p=0.142) indicates that P = 0.14 should be useful for practical applications. Furthermore, by substituting this p (=0.142) value from fig.  {\ref 2} into Eq.4, we have G=1+2*0.142=1.284, which is in good agreement with the normally quoted G value of 1.30. In other words, the common use of 30\% gust factor for offshore applications receives further support from this study.   
\section[{Conclusions}]{Conclusions}\par
On the basis of foregoing analyses and discussions, following conclusions can be drawn:\par
? A comparison between logarithmic and power-law wind profiles is made for offshore conditions (up to 122m or 400ft from the sea surface). It is found that, although the log law has more theoretical support, statistical analyses. Therefore, the powerlaw is recommended for operational applications since it involves only one unknown parameter, which is the exponent. ? The exponent of the power-law wind profile can be determined from the gust factor which is routinely available from NDBC measurements. It is also found the gust factor does not vary with the altitude statistically within 160m or 525ft from the sea surface. This finding, which is contrary to the common belief for onshore conditions, is very important for wind loading analyses for offshore structures as well as search and rescue mission planning during storms at sea.\par
the power-law is as good as the log-law based on\begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Figure 1 :}\end{figure}
   \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{}
\end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.1579646017699115\textwidth}P{0.08838495575221238\textwidth}P{0.026327433628318582\textwidth}P{0.08086283185840708\textwidth}P{0.048893805309734514\textwidth}P{0.05077433628318584\textwidth}P{0.05077433628318584\textwidth}P{0.09402654867256638\textwidth}P{0.0018805309734513274\textwidth}P{0.11847345132743362\textwidth}P{0.06769911504424779\textwidth}P{0.06393805309734514\textwidth}}
\tabcellsep Year\tabcellsep \multicolumn{2}{l}{month}\tabcellsep day\tabcellsep hour\tabcellsep min\tabcellsep wind\tabcellsep \tabcellsep U122m\tabcellsep U54.9m\tabcellsep U10m\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep UTC\tabcellsep \tabcellsep direction\tabcellsep \tabcellsep m/s\tabcellsep m/s\tabcellsep m/s\\
\tabcellsep 2012\tabcellsep \tabcellsep 6\tabcellsep 25\tabcellsep 19\tabcellsep 30\tabcellsep 340\tabcellsep \tabcellsep 13\tabcellsep 10\tabcellsep 8\\
\tabcellsep 2012\tabcellsep \tabcellsep 6\tabcellsep 25\tabcellsep 18\tabcellsep 30\tabcellsep 340\tabcellsep \tabcellsep 14\tabcellsep 11\tabcellsep 9\\
\tabcellsep 2012\tabcellsep \tabcellsep 6\tabcellsep 25\tabcellsep 16\tabcellsep 30\tabcellsep 350\tabcellsep \tabcellsep 15\tabcellsep 12\tabcellsep 11\\
\tabcellsep 2012\tabcellsep \tabcellsep 6\tabcellsep 25\tabcellsep 6\tabcellsep 30\tabcellsep 360\tabcellsep \tabcellsep 18\tabcellsep 16\tabcellsep 12\\
\tabcellsep 2012\tabcellsep \tabcellsep 6\tabcellsep 25\tabcellsep 4\tabcellsep 30\tabcellsep 360\tabcellsep \tabcellsep 19\tabcellsep 18\tabcellsep 14\\
Year 2014\tabcellsep 2012 2012 2012\tabcellsep \tabcellsep 6 6 6\tabcellsep 25 25 24\tabcellsep 3 0 23\tabcellsep 30 30 30\tabcellsep 10 360 360\tabcellsep \tabcellsep 19 24 22\tabcellsep 18 22 20\tabcellsep 14 16 16\\
2\tabcellsep 2012 2012\tabcellsep \tabcellsep 6 6\tabcellsep 24 24\tabcellsep 20 19\tabcellsep 30 30\tabcellsep 360 360\tabcellsep \tabcellsep 19 19\tabcellsep 17 16\tabcellsep 14 13\\
XIV Issue I Version I\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep 18.2\tabcellsep 16\tabcellsep 12.7\\
( ) Volume\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
Global Journal of Researches in Engineering\tabcellsep \tabcellsep U z /U 10\tabcellsep 0.4 0.6 1 1.2 1.4 1.6 0.8\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \multicolumn{2}{l}{y = 0.997x 0.141}\\
\tabcellsep \tabcellsep \tabcellsep 0.2\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \multicolumn{2}{l}{R² = 0.998}\\
\tabcellsep \tabcellsep \tabcellsep 0\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
\tabcellsep \tabcellsep \tabcellsep 0\tabcellsep \tabcellsep 2\tabcellsep 4\tabcellsep 6\tabcellsep 8\tabcellsep 10\tabcellsep 12\tabcellsep 14\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep Z/10\tabcellsep \tabcellsep \end{longtable} \par
  {\small\itshape [Note: © 2014 Global Journals Inc. (US) E Figure 2 : Power-law wind profile over the Gulf of Mexico during Tropical Storm Debby in 2012 mean IV.]} 
\caption{\label{tab_1}}\end{figure}
 			\footnote{© 2014 Global Journals Inc. (US)} 		 		\backmatter  			 
\subsection[{This page is intentionally left blank}]{This page is intentionally left blank}			 			  				\begin{bibitemlist}{1}
\bibitem[Hsu and Meteorology ()]{b1}\label{b1} 	 		\textit{},  		 			S A Hsu 		,  		 			Meteorology 		.  		1988. San Diego, CA: Academic Press. p. 260.  	 
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\end{document}
