Wind Speed Analysis of Hurricane Sandy

: The database of the HWind project sponsored by the National Oceanic and Atmospheric Administration (NOAA) for hurricanes between 1994 and 2013 is analysed. This is the ﬁrst objective of the current research. Among these hurricanes, Hurricane Sandy was selected for a detailed study due to the number of ﬁles available and its social relevance, with this being the second objective of this study. Robust wind speed statistics showed a sharp increase in wind speed, around 6 m s − 1 at the initial stage as Category 1, and a linear progression of its interquartile range, which increased at a rate of 0.54 m s − 1 per day. Wind speed distributions were initially right-skewed. However, they evolved to nearly symmetrical or even left-skewed distributions. Robust kurtosis was similar to that of the Gaussian distribution. Due to the noticeable fraction of wind speed intermediate values, the Laplace distribution was used, its scale parameter increasing slightly during the hurricane’s lifecycle. The key features of the current study were the surface and recirculation factor calculation. The surface area with a category equal to, or higher than, a tropical storm was calculated and assumed to be circular. Its radius increased linearly up to 600 km. Finally, parcel trajectories were spirals in the lower atmosphere but loops in the mid-troposphere due to wind translation and rotation. The recirculation factor varied, reaching values close to 0.9 and revealing atmospheric stratiﬁcation.


Introduction
Hurricanes are among the most destructive natural hazards, not only in terms of human life, but also on built infrastructure [1]. Since their consequences are devastating, the effects of these winds on structures such as bridges, transmission lines, offshore wind turbines, skyscrapers, or water supplies are the subject of research [2][3][4][5][6]. However, the most common constructions are usually low-rise buildings, where the direct impact perceived by most of the population is on the building roofs [7]. Moreover, coastal economic activities, such as fisheries, are affected [8]. Last but not least, the longest-lasting consequences are observed on human health [9].
Due to their global impact, hurricanes have been studied in numerous analyses, some of which, such as the relationship between pressure or temperature and wind, are experimentally based [10,11]. Other studies consider the vertical wind profile [12], which is extremely useful for evaluating the structural safety of buildings in extreme wind conditions. The hurricane trajectory analysis is one line of research [13], with other studies focus on gauging their size [14]. Theoretical research has also occasionally been conducted [15].
Due to the close link between ocean surface temperatures and hurricanes, Hosseini et al. [16] obtained a high correlation between sea surface temperature, the increase in which was attributed to climate change, and hurricane frequency over the last century. Another consequence of recent atmospheric warming might be the decrease in translation speed and the increase in the rain rate [17]. However, Rojo-Garibaldi et al. [18] reported a decreasing trend in the number of hurricanes in the Gulf of Mexico and the Caribbean over a wider period, 1749-2010, which they correlated with sunspot activity.
The current paper is divided into two unequal sections. The first analyses a hurricane database spanning a period of nearly twenty years in order to explore its main features and hurricane distributions, with the extended analysis period being one of the prominent aspects in this section. The remainder of the research is devoted to Hurricane Sandy, which hit in October 2012 and gained prominence due to its trajectory, which ran parallel to the US coast, sparking tremendous fear in much of the population. Its origin was traced to a Saharan dust event on 8 October 2012 that moved west until cyclogenesis conditions were reached over the Caribbean [19]. Its evolution was affected by polar and subtropical jet streams [20]. Finally, the "Greenland block", a high-pressure area near Greenland, turned the hurricane towards the northwest coast of the United States [21]. This left turn was unusual [22] and the wind intensified due to a warm seclusion phase [23]. Its effects, such as moderate flooding and sand deposition [24] or soil contamination [25], were noticeable on the coast. However, water clarity and nutrients reached pre-storm conditions in about one year on the Hudson-Raritan estuary, whereas biotic recovery took longer than that in other estuaries [26]. Recent analyses of this hurricane have focused on its reintensification after its extratropical transition [27], its roll vortices, which were a prominent feature due to their large wavelengths [28], its surges [29,30], its effects on infrastructures [31], and its westward turn when it made landfall [32,33]. In the second section of this paper, modelled wind fields are used to explore the progression of wind statistics and the hurricane size. As a specific contribution of this research, this area is determined by plotting isotachs. Finally, since air parcel trajectories are spirals or similar, the evolution of the recirculation factor is studied, with the analysis of this factor being another original contribution of this research.

Database Description
Files considered in the current research correspond to the HWind Project [34,35], which are freely available [36]. This database extended mainly from 1994 to 2013, covering over 200 hurricanes. The information for each hurricane includes real-time observations and maps with the wind field and wind speed components in a grid, where the origin lies in the hurricane centre. These latter files were used in this paper.
One section of this study is devoted to analysing air parcel trajectories. These trajectories were calculated with the METeorological data EXplorer (METEX) model [37], which provides, among other variables, the hourly longitude and latitude of the air parcel one day before it reaches the desired point.

Statistics
Wind fields are described using robust statistics. The median, Q 0.5 , is considered as the location. Spread is quantified by the interquartile range, IQR = Q 0.75 − Q 0.25 , where Q 0.25 and Q 0.75 are the first and third quartiles, respectively. Symmetry is determined by the Yule-Kendall index, γ YK [38] γ YK = (Q 0.25 − 2Q 0.5 + Q 0.75 )/IQR (1) Finally, the distribution flatness is given by the robust kurtosis, RK [39] where D 0.1 , and D 0.9 are the first and ninth deciles, respectively. RK is 0.263 for a Gaussian distribution.

The Laplace Distribution
Its probability density function, f, is defined as where the median is the location parameter, and b is the scale parameter, and where n is the number of x i , which are the wind speed values of each wind field.

Recirculation Factor
This factor was introduced by Allwine and Whiteman [40] to indicate the presence of recirculation on a given timescale, usually one day. Figure 1a illustrates its calculation. The air parcel moves from point B, with longitude λ 1 and latitude ϕ 1 , to point A(λ 0 , ϕ 0 ). S i is the distance travelled by the air parcel each hour, and the wind run, S, is its sum.
where the median is the location parameter, and b is the scale parameter, and where n is the number of xi, which are the wind speed values of each wind field.

Recirculation Factor
This factor was introduced by Allwine and Whiteman [40] to indicate the presence of recirculation on a given timescale, usually one day. Figure 1a illustrates its calculation. The air parcel moves from point B, with longitude λ1 and latitude , to point A(λ0, ). Si is the distance travelled by the air parcel each hour, and the wind run, S, is its sum.
Transport distance, L, is the straight distance between the beginning, B, and end of trajectory, A, and the recirculation factor is which lies between 0 and 1. Since Si and L are arcs on the Earth's surface, these distances are calculated by the Sinnott equation [41], which for arc L is R equal to 0 means no recirculation, since the trajectory is a straight line. However, when R is equal to 1, the air parcel has returned to its origin. Figure 1b presents different recirculation factors assuming that the corresponding trajectories are circumference arcs.   Transport distance, L, is the straight distance between the beginning, B, and end of trajectory, A, and the recirculation factor is

Database Analysis
which lies between 0 and 1. Since S i and L are arcs on the Earth's surface, these distances are calculated by the Sinnott equation [41], which for arc L is Atmosphere 2021, 12, 1480 4 of 13 R equal to 0 means no recirculation, since the trajectory is a straight line. However, when R is equal to 1, the air parcel has returned to its origin. Figure 1b presents different recirculation factors assuming that the corresponding trajectories are circumference arcs.  Figure 2b shows that hurricanes in this database took place from May to December. This number gradually increased until the marked growth observed in August, with 67 hurricanes, followed by September, which had slightly fewer. A sharp decrease was noted in November, with nine hurricanes, the minimum being recorded in December with only two hurricanes. From a practical point of view, only hurricanes with the so-called GriddedData files may be useful for detailed analyses. These files are not available for the oldest hurricanes, and have only been present since 1998 for most hurricanes. However, a varying number of files may be found for each hurricane, reaching up to 60 files at different hours and days. Figure 2c presents the average number of files per hurricane. Although the mean is 9.7 and the number of files is close to this mean, this number is very noticeable in 2004, 2005, and 2008, with between 15 and 20 files per hurricane. with the least number of hurricanes recorded, only seven, followed by the greatest number of hurricanes, 18, recorded in 2005. Another possible temporal analysis may be made by considering monthly distribution. Figure 2b shows that hurricanes in this database took place from May to December. This number gradually increased until the marked growth observed in August, with 67 hurricanes, followed by September, which had slightly fewer. A sharp decrease was noted in November, with nine hurricanes, the minimum being recorded in December with only two hurricanes. From a practical point of view, only hurricanes with the so-called GriddedData files may be useful for detailed analyses. These files are not available for the oldest hurricanes, and have only been present since 1998 for most hurricanes. However, a varying number of files may be found for each hurricane, reaching up to 60 files at different hours and days. Figure 2c presents the average number of files per hurricane. Although the mean is 9.7 and the number of files is close to this mean, this number is very noticeable in 2004, 2005, and 2008, with between 15 and 20 files per hurricane.

Analysis of Hurricane Sandy
Two criteria were considered when selecting a hurricane for detailed analysis: the number of GiddedData files and the social impact. Only 24 hurricanes occurring between

Analysis of Hurricane Sandy
Two criteria were considered when selecting a hurricane for detailed analysis: the number of GiddedData files and the social impact. Only 24 hurricanes occurring between 2005 and 2012 were described with at least 20 GriddedData files. Of these, six reached the highest category, Category 5, and only Hurricane Katrina, in 2005, left a marked social impact. However, the current research investigates Hurricane Sandy, which occurred in 2012. Although it was only Category 3 [42], it is described by 48 GriddedData files accompanied by a noticeable social impact reflected by the extremely high number of web search results to emerge. The files available cover from 23 to 30 October 2012. Moreover, each file contains over 20,000 wind speed values distributed in networks centred on the hurricane between about 1000 km × 1000 km and 2000 km × 2000 km.     Figure 4b. The correlation coefficient of this fit is statistically significant at a 0.1% level. In a similar period, the Yule-Kendall index evolved linearly from positive skewness, around 0.4, to nearly symmetrical or even slightly left-skewed distributions, seen in Figure 4c. The correlation coefficient is also statistically significant at a 0.1% level for this variable. Finally, the robust kurtosis was similar to that of a Gaussian distribution, seen in Figure 4d.    Figure 5a shows an example of the wind speed histogram. Its shape describes extremely high frequencies in few central wind speed values bounded by low frequencies in the remaining wind speeds, with decreasing frequencies when the wind speed moves away from the central values. This shape suggests that the wind speed is similar in most of the region covered by the hurricane, and the Laplace distribution may suitably describe wind speed. One parameter of this distribution is the median wind speed, whose evolution was previously presented. The other parameter is the scale parameter, which may be calculated with Equation (4). Figure 5b shows its progression, where a slight linear trend with a rate of around 0.21 m s −1 each day was observed from 24 to 29 October 2012, revealing that the interval of frequent wind speeds grew wider over time, in agreement with the interquartile range. In any case, progression is somewhat irregular, with certain abrupt changes. However, the correlation coefficient is statistically significant at a 0.1% level.

Surface Analysis
Isotachs were calculated each twelve hours. By way of an ex the wind speed on 27 October 2012 at 1.30 UTC. From plots suc sponding to the boundaries of the Saffir-Simpson scale were s where the hurricane reaches each category was calculated. More the hurricane resembles a circle, the radius corresponding to the s than, a tropical storm was calculated and is presented in Figure  linearly at a rate of around 86 km per day, with a correlation coef significant at a 0.1% level.

Surface Analysis
Isotachs were calculated each twelve hours. By way of an example, Figure 6a shows the wind speed on 27 October 2012 at 1.30 UTC. From plots such as this, isotachs corresponding to the boundaries of the Saffir-Simpson scale were selected, and the surface where the hurricane reaches each category was calculated. Moreover, since the shape of the hurricane resembles a circle, the radius corresponding to the surface equal to, or higher than, a tropical storm was calculated and is presented in Figure 6b. This radius increases linearly at a rate of around 86 km per day, with a correlation coefficient that is statistically significant at a 0.1% level. the hurricane resembles a circle, the radius corresponding to the s than, a tropical storm was calculated and is presented in Figure  linearly at a rate of around 86 km per day, with a correlation coef significant at a 0.1% level.  (b) Evolution of the hurricane's radius, assuming a circular shape for wind speed equal to, or higher than, that of a tropical storm. The linear increase was noticeable until 29 October 2012. Figure 7a presents the air parcel trajectories ending at the hurricane centre at 10, 1000, and 5000 m in height on 27 October 2012 at 13.30 GMT. Although recirculation is noticeable for these trajectories, the main difference is that trajectories at 10 and 1000 m are simpler, like spirals, than the 5000 m trajectory, which presents loops. This behaviour may be due to surface friction, which is more accentuated near the surface, causing lower wind speeds than those recorded in the mid troposphere. These high wind speeds and the composition of rotation and translation would be responsible for complex trajectories observed at this level. Moreover, Figure 7b,c presents the evolution of the recirculation factor in the whole troposphere, calculated from trajectories obtained with the METEX model, ending at the hurricane's centre at selected times. From a practical point of view, the layer below 1000 m is the most interesting due to its impact on human life. The recirculation factor is relatively uniform in this layer and its values are relatively high (around 0.6) most of the time. In order to investigate the influence of the end point of the air parcel trajectories on the recirculation factor, these factors were also calculated for trajectories ending at a point whose latitude was 2 • lower than the hurricane's centre latitude, but with the same longitude. Although the resulting recirculation factors varied enormously, they were, on average, around 0.1 lower for these latter trajectories in the low atmosphere, from 10 to 500 m. of the time. In order to investigate the influence of the end point of the air parcel trajectories on the recirculation factor, these factors were also calculated for trajectories ending at a point whose latitude was 2° lower than the hurricane's centre latitude, but with the same longitude. Although the resulting recirculation factors varied enormously, they were, on average, around 0.1 lower for these latter trajectories in the low atmosphere, from 10 to 500 m.

Database Analysis
Chavas et al. [43] presented the spatial distribution of hurricane tracks on the Earth in the period 1999-2009 where the lowest number was observed over the Southern Pacific Ocean and Southern Indian Ocean. In the remaining basins, these trajectories were similar for the Atlantic and East Pacific basins. In both regions, hurricanes approached the continent from low latitudes, then veered and moved away from the continent to higher latitudes, with the greatest latitudes being reached in the Atlantic basin. However, hurricanes in the West Pacific basin were confined to low latitudes.
Delgado et al. [44] reanalysed the North Atlantic hurricane database for the period 1954 to 1963. They obtained an average of around 11 storms (tropical storms and hurricanes) per year, with the number varying between 7 and 16. This average was around six for hurricanes each year, ranging between three and nine, and around three per year for major hurricanes, ranging between none and five.
Most hurricanes described in Section 3.1 occur in the second part of the year. This result is in agreement with the study presented by Corporal-Lodangco and Leslie [45], who investigated tropical cyclones in the Philippine region during the period 1945-2011. They reported two seasons: the less active season, from January to May, with a median seasonal frequency of two, and the more active season, from June to December, with a median seasonal frequency of 15. However, although hurricane seasons are usually felt to exist, their start is not fixed since this may be influenced by atmospheric processes. For example, a significant delay in the start of the hurricane season in the western North Pacific was observed after a strong El Niño in the preceding winter [46].
Anomalies in the sea surface are responsible for changes in hurricane frequency. Wang et al. [47] investigated Atlantic hurricanes from 1951 to 2010 and concluded that warm anomalies in the sea surface temperature in wintertime in the main Atlantic development region are frequently followed by unusually active hurricane seasons. Moreover, the hurricane frequency may be affected by certain trends, since analyses of cyclone genesis frequency from October to December over the western North Pacific revealed a decreasing trend with two periods, the first from 1980 to 1995, with about ten hurricanes per year, seven of which could be assigned to the eastern region, and the second from 1996 to 2011, with about six hurricanes per year, where around four may be assigned to the western region [48]. Another factor that may impact on this frequency could be climate change, since analyses of the strength and distribution of hurricanes during the 2016 North Atlantic hurricane season revealed that said year was noticeable for a series of events never before observed, such as the observation site for a high category or the number of high category hurricanes in the same month [49].

Wind Speed
These measurements have occasionally been taken if the experimental device is near the hurricane route [50]. In these situations, the wind speed gradually increases to the maximum, and then decreases once the hurricane centre moves away from the experimental site. Although measurements are taken at a single point, they may be considered similar at sites that are an equal distance from the centre, since radial symmetry is assumed. Moreover, studies normally present the horizontal wind speed profile, where one maximum is reached at a certain distance [51], and this wind speed shape has occasionally been modelled [52,53].
Although analyses of wind speed distribution are not common, the current study considers the Laplace distribution of the wind speed due to the shape of the wind speed histogram. However, Cui and Caracoglia [54] used the Weibull distribution for annual wind speed maxima.

Radius
For US tropical cyclone forecast centres, estimating the maximum extent of the 17.5, 25.7, and 32.9 m s −1 (34, 50 and 64 kt, respectively) winds in compass quadrants (northeast, southeast, southwest, and northwest) surrounding the hurricane centre is critical [55,56]. These wind thresholds are known as gale-force, destructive, and hurricane-force, and the corresponding distances are called the "wind radii". The average 17.5 m s −1 wind radius calculated in 2014-2015 in the western North Pacific basin was around 248 km, larger than those for the Atlantic and eastern North Pacific basins, which were around 176 and 152 km, respectively [57].
Chavas and Emanuel [58] analysed the azimuthally-averaged radius of 12 m s −1 wind, r 12 , and the radius of vanishing winds, r 0 , for a dataset covering the period 1999-2008. Their global median values were 197 and 423 km, respectively. Moreover, they presented these values in each basin. The largest radii were reached in the West Pacific basin, around 250 and 500 km, whereas the smallest values were observed in the East Pacific basin, slightly below 150 and 350 km.

Air Parcel Trajectories
Wind speed vertical profiles in the boundary layer modelled by Snaiki and Wu [59] revealed the noticeable influence of the surface below 300 m, where the change is around Myers and Malkin [61] presented the spiral trajectory of air parcels in a hurricane and Niu et al. [62] considered its mathematical form by a logarithmic spiral. Spirals observed in trajectories at low altitudes may be attributed to friction with the surface, whereas loops in the mid troposphere may be due to a composition of the wind rotation and the hurricane displacement.

Conclusions
Most of the hurricanes in the HWind project were formed in the Atlantic basin, mainly between August and September, with the southern hemisphere being excluded. Only 24 of these hurricanes between 2005 and 2012 presented more than 20 files per hurricane, thus making detailed analyses possible. Most of these hurricanes occurred in 2005 and 2008. Among these, Hurricane Sandy was the most recent and had the greatest social impact due to its trajectory, which ran parallel to the US coast. For over half of its life cycle, it was Category 1 or even 2. The calculation of different statistics indicated that its median wind speed increased sharply, to around 6 m s −1 , on 25 October 2012. However, the interquartile range increased linearly for most of its life cycle at a rate of 0.54 m s −1 per day and the Yule-Kendall index decreased slightly, revealing a symmetry evolution from right-skewed to nearly symmetrical or even left-skewed distributions. The robust kurtosis remained similar to that of the Gaussian distribution. The Laplace distribution was used for the wind speed, and its scale parameter increased linearly at a rate of 0.21 m s −1 per day. Assuming a circular shape, its radius for wind speed as at least a tropical storm was calculated, and a linear increase of around 86 km per day was observed.
Moreover, the recirculation factor remained around 0.6 close to the ground, although it reached even greater values at heights free from the surface influence. However, this factor should not replace the trajectory representation, since it may not be representative enough of trajectory shape, which is a spiral close to the ground but a loop in the mid-troposphere.
Finally, the database used has proved powerful enough to allow a detailed study of hurricanes, with the analysis presented in the current paper being extensible to other hurricanes in order to gain insights into evolution patterns, since it would be interesting to ascertain whether trends observed in certain statistics reflect general behaviours. Moreover, since all the files can be downloaded, their systematic analyses will improve the time resolution of calculations so as to achieve a full description of the hurricanes in this database. Additionally, specific regions could be selected in the files for particular spatial analyses.