The Spatial Distribution of Ocean Waves in Tropical Cyclones

AUGUST 2020 T A M I Z I A N D Y O U N G 2123

ALI TAMIZI AND IAN R. YOUNG Department of Infrastructure Engineering, University of Melbourne, Melbourne, Australia

(Manuscript received 27 January 2020, in ﬁnal form 28 May 2020)

ABSTRACT

The spatial structure of both the wind and wave fields within tropical cyclones is investigated using two large databases. The first of these was compiled from global overpasses of tropical cyclones by satellite altimeters. The second dataset consists of an extensive collection of North American buoy measurements during the passage of tropical cyclones (hurricanes). The combined datasets confirm the vortex structure of the tropical cyclone wind field with the strongest winds to the right (Northern Hemisphere) of the storm. The wave field largely mirrors the wind field but with greater right–left asymmetry that results from the extended fetch to the right of the translating tropical cyclone. The extensive in situ buoy database confirms previous studies indicating that the one-dimensional spectra are generally unimodal. The directional spectra are, however, directionally skewed, consisting of remotely generated waves radiating out from the center of the storm and locally generated wind sea. The one-dimensional wave spectra have many similarities to fetch-limited cases, although for a given peak frequency the spectra contain less energy than for a fetch-limited case. This result is consistent with the fact that much of the wave field is dominated by remotely generated waves.

1. Introduction This paper attempts to build an extensive database of observations of tropical cyclone wave conditions In tropical and subtropical regions, tropical cyclones from a number of sources. These sources include a (TCs; hurricanes or typhoons) represent the major global database of satellite altimeter observations extreme meteorological systems. The strong winds as- from 13 satellite missions across a period of 33 years sociated with the moving tropical cyclone vortex gen- together with 37 years of in situ data from the National erate extreme waves which critically impact processes Data Buoy Center (NDBC) buoy network around the such as: coastal and offshore engineering design, ship- United States. The combined dataset includes altim- ping, coastal beach erosion and coastal inundation. The 2 eter data from 2730 tropical cyclones worldwide and extreme winds (in excess of 40 m s 1), the rapidly in situ data from 353 tropical cyclones (hurricanes) in turning nature of the vortex, and its translation are a the North American region. This combined dataset is demanding test for our understanding of wind gener- analyzed to provide an overview of the spatial distri- ation physics. The situation is exacerbated by the butions of significant wave height and wind speed, as paucity of quality wind and wave data during intense well as details of the directional wave spectrum gen- tropical cyclones. The relatively small geographic size erated in tropical cyclones. of tropical cyclones means a dense network of obser- The arrangement of this paper is as follows. Following vations is required to ensure that extreme conditions this introduction, section 2 provided a brief overview of will be captured. When they are, the conditions are previous observations and computations of the tropical such that failure of in situ measurement systems is not cyclone wave field, and section 3 describes the data uncommon. Therefore, the observational database of sources used to construct the present tropical cyclone wave conditions under tropical cyclone forcing is still databases. The details of the altimeter observations are relatively small, meaning that a comprehensive un- provided in section 4, followed by the in situ buoy ob- derstanding of the wave fields generated by such sys- servations in section 5. In addition to the distribution of tems is still limited. significant wave height, the in situ buoy data provide a comprehensive view of the directional wave spectrum. Corresponding author: Ian R. Young, ian.young@unimelb. The one-dimensional (1D) energy density spectrum is edu.au described in section 6, followed by the directional

DOI: 10.1175/JPO-D-20-0020.1 Ó 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 09/29/21 05:32 AM UTC 2124 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 50 properties of the spectrum in section 7. Conclusions and the center of the storm. The directional properties ob- discussion appear in section 8. tained from the buoys showed strong directional skew, with remotely generated waves and wind sea often 8 2. Observations of tropical cyclone wave fields propagating in directions separated by more than 90 . The directional properties indicated that the wave The earliest attempts to model/describe the tropical systems (remotely generated waves and wind sea) did cyclone wave field assumed that it would mirror the not completely decouple and there was a continuum of wind field (Bretschneider 1959, 1972; USACE 1977; energy between the systems. As a result, the 1D spectra Ross 1976). However, the first limited observations from generally remained unimodal. In addition, these 1D in situ instruments (Patterson 1974; Whalen and Ochi spectra had many of the attributes previously observed 1978; Black 1979; Ochi and Chiu 1982; Ochi 1993; Young for fetch-limited wind seas. The nondimensional en- 5 2 4 1998; Young 2006) and remote sensing observations, ergy g Etot/U10 and nondimensional peak frequency largely using aircraft-based synthetic aperture radar and n 5 fpU10/g were approximately related by the same orbiting altimeter data (Elachi et al. 1977; King and power law as observed for fetch-limited wind seas. In

Shemdin 1978; Gonzalez et al. 1982; McLeish and Ross these relations, Etot is the total energy of the waves, fp is 1983; Holt and Gonzalez 1986; Beal et al. 1986; Young the spectral peak frequency, U10 is the wind speed, and and Burchell 1996; Wright et al. 2001; Black et al. 2007) g is gravitational acceleration. The high-frequency face 2 showed a more complex structure. These measurements of the spectrum decayed at ;f 4,wheref is frequency, indicated that, ahead of the tropical cyclone vortex, the and the energy level of this region of the spectrum was wave ﬁeld was composed of locally generated wind seas similar to fetch-limited spectra. The peak frequency (aligned with the local wind direction) and remotely values were, however, such that these waves were generated waves which had been generated in the in- propagating faster than the local wind. This is consis- tense wind regions of the storm and then propagated tent with these waves being generated in other parts of ahead of the storm. These observations have also been the storm and then propagating to the observation site. augmented with model simulations of tropical cyclone Young (2006) concluded that these observed spectral conditions (SWAMP Group 1985; Young 1988; The shapes were consistent with an energy balance in which WAMDI Group 1988). nonlinear wave–wave interactions played a dominant These measurements and modeling led to the con- role (Young and van Vledder 1993). clusion that the forward motion of the storm was an Hu and Chen (2011) considered directional spectra important variable in understanding wave generation recorded at 12 buoys in the Gulf of Mexico during within tropical cyclones. As a result, the concept of an the passage of seven tropical cyclones (hurricanes). ‘‘extended’’ or ‘‘trapped’’ fetch was developed (Young Consistent with Young (2006), they found the 1D 1988; Young and Burchell 1996; Bowyer and MacAfee spectra were generally unimodal with the exception of

2005; Young and Vinoth 2013). In this representation, spectra far from the tropical cyclone center (r . 6Rm, waves generated to the right (Northern Hemisphere) of where Rm is the radius to maximum winds) and spec- the storm center propagate forward with the storm and tra to the left of the storm center. Esquivel-Trava et al. hence stay in the strong wind region for an extended (2015) considered spectra from four buoys in the period of the time (the extended fetch). Hence, both the Gulf of Mexico and Caribbean during the passage of velocity of forward movement and the maximum wind 14 tropical cyclones (hurricanes). Again, they found speed in the storm are important in determining the that the 1D spectra were unimodal. However, their wave field. results indicated that the energy in the high-frequency A number of studies have aggregated data from tail of the spectrum was lower than proposed by multiple tropical cyclones to provide information on the Young (2006). spatial distribution of tropical cyclone waves. Young Collins et al. (2018) reported on observations from (1998, 2006) considered data from directional wave two Taiwanese buoys during the passage of three trop- buoys off the northwest coast of Australia during the ical cyclones (typhoons) plus a tropical storm. They passage of nine tropical cyclones. To aggregate the data confirmed that the spectra were commonly directionally from the multiple storms, a frame of reference moving skewed. However, in contrast to the previous measure- with the tropical cyclone was adopted. The analysis ments, they found that a significant percentage of the 1D confirmed the conclusions from the remote sensing data spectra were bimodal. that, ahead of the storm, the wave field was composed of In a more recent series of papers (Hwang 2016; locally generated wind sea and remotely generated Hwang and Fan 2017; Hwang et al. 2017; Hwang and waves radiating out from the intense wind regions near Walsh 2016, 2018a,b), scanning radar altimeter data

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC AUGUST 2020 T A M I Z I A N D Y O U N G 2125 were used to examine directional properties of waves In the present analysis, data after 1980 have been from six North American hurricanes. These data considered. confirmed a relationship between and n that is sim- b. Altimeter data ilar to that proposed by Young (2006) and that the spectra were composed of a mix of remotely gener- There are a number of satellite systems that can po- ated waves and wind sea, which varied spatially within tentially provide information on tropical cyclone wind the tropical cyclone. speed U10 and significant wave height Hs. These include The analyses described above indicate that one of scatterometer (U10), radiometer (U10), and altimeter the defining features of the tropical cyclone wave field (U10 and Hs). We had access to long-term databases is that the spectrum consists of a combination of re- (30 years) for each of these systems [radiometer (Young motely generated wave energy and locally generated et al. 2017), altimeter (Ribal and Young 2019), and wind sea. The remotely generated wave systems have scatterometer (Ribal and Young 2020)]. All instruments often been termed ‘‘swell’’ in previous studies. As will were tested as potential data sources. All of these sys- be shown clearly below, although they may not be tems are, to varying degrees, impacted by heavy rain locally generated, these remotely generated waves near the center of tropical cyclones. This was particu- remain coupled to the local wind sea through nonlin- larly the case for radiometer, where there were very few ear interactions. Therefore, to avoid confusion we tropical cyclone passes without significant data loss near have termed such components of the spectrum ‘‘re- the storm centers. Noting this, and the fact that altimeter motely generated waves’’ rather than swell. measures both U10 and Hs, the altimeter database of The observations described above provide some in- Ribal and Young (2019) was used in this analysis. This sight into the characteristics of tropical cyclone waves. database includes data from all 13 altimeters that were However, even the combined dataset, represents ob- in operation between 1985 and 2018. Each of the al- servations in only a small number of tropical cyclones. timeters was calibrated against buoy data, including

To cover the full range of possible conditions, a much consideration of performance for extreme values of U10 larger dataset is required. The analysis below describes and Hs (important for this tropical cyclone application). such a dataset. Altimeters are typically placed in near-polar orbits such that they orbit south to north (ascending pass) and north 3. Tropical cyclone database to south (descending pass). Along track, altimeters have relatively high spatial resolution, providing a measure-

Our aim is to compile a large database of tropical ment of both U10 and Hs approximately every 10 km. cyclone track, wind field parameters, significant wave However, because they measure only over a narrow height, wind speed and spectral information. To do beam below satellite nadir, there are typically hundreds this, it has been necessary to obtain data from a of kilometers between tracks. As such, the altimeter number of different sources. The data used are out- may not always provide data for tropical cyclones be- lined below. cause of their relatively small geographic scale. a. IBTrACS track data c. NDBC buoy data The International Best Track Archive for Climate The NDBC operates the most extensive wave buoy Stewardship (IBTrACS) dataset (Knapp et al. 2010, network in the world (Evans et al. 2003). Of particular 2018) was developed by the NOAA National Climatic relevance for the present application, this network Data Center. The archive synthesizes and merges covers the Atlantic, Pacific, and Gulf of Mexico regions best-track data from all official Tropical Cyclone for North American hurricanes. The NDBC buoy data

Warning Centers and the WMO Regional Specialized typically includes hourly measurements of Hs,the1D Meteorological Centers. The dataset contains data energy density spectrum E(f), where including time, position, maximum sustained winds, sðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi minimum central pressure, p and storm nature 0 H 5 4 E(f ) df , (i.e., tropical cyclone, tropical storm, etc.). In addi- s tion, information such as the radius to maximum winds Rm and the radius to gales R34 is provided and the cross-spectral moments a1, b1, a2,andb2. Following forsomestorms.Thedataareprovidedgloballyat Longuet-Higgins et al. (1963) and Young (1994), the di- 6-h intervals. Although the archive contains data rectional spectrum was determined as E(f, u) 5 E(f)D(f, beginning from 1848, the quality of data before u), whereÐ D(f, u) is a directional spreading function such the satellite period is obviously of lower quality. that D(f , u) df 5 1 and is constrained to the form

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC 2126 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 50 u 2 u (f ) A similar approach was used for the buoy data: D(f , u) 5 A(f ) cos2s( f ) m . (1) 2 d A region with a 58 radius was deﬁned around each buoy. d IBTrACS TC paths that transected this region were The mean direction u and the spreading parameter s m extracted. are determined from the ﬁrst two spectral moments a1 d Data from the buoy covering the period of the TC and b1 as ÂÃ transect were extracted and associated with the 5 21 IBTrACS pass. um(f ) tan b1(f )/a1(f ) and (2) d Again, the TC track and each directional band of the r (f ) directional spectra [Eq. (1)] were rotated, such that s(f ) 5 1 , (3) 2 the TC is propagating toward the north. The data 1 r1(f ) were then placed in a frame of reference moving 2 5 2 1 2 with r1 a1 b1. The normalization factor A(f)in with the TC. Note that this was not necessary with Eq. (1) is to ensure the integral of D(f, u) is equal to 1. the altimeter passes, because the altimeter pass is

The NDBC buoys were also used to measure wind ‘‘almost instantaneous’’ as compared with the Vfm speed. Because anemometers on these buoys are at a range of the TC. of heights, all measurements were converted to a standard Figure 1 shows the distribution of the buoys used in the reference height of 10 m, assuming a neutral stability log- database. Because there are a large number of buoys arithmic boundary layer (Ribal and Young 2019). (see section 5), only buoys with data from 10 or more d. Tropical cyclone databases tropical cyclones are shown. The TC altimeter database consists of a total of 36 509 The above information was used to form two tropical altimeter passes through 2730 TCs. Figure 2 shows his- cyclone databases consisting of: global analysis of al- tograms of the key parameters within this extensive timeter tracks over tropical cyclones (U and H ) and 10 s database. Figure 2a shows the histogram of the maxi- North American analysis of tropical cyclone passes in mum value of H for each altimeter pass, showing values the vicinity of buoys [U , H , and E(f, u)]. s 10 s up to H ’ 15 m. Similarly, the histogram of the maxi- The global altimeter database was constructed using s mum values of U10 for each altimeter pass in Fig. 2b the following steps: 21 shows values up to U10 ’ 50 m s . These panels also d The altimeter data of Ribal and Young (2019) are show many low values of Hs and U10 associated with available in 18318 regions. passes far from the centers of tropical cyclones. Note d The IBTrACS TC tracks are deﬁned at 6-hourly values. that throughout this study we consider values of Hs and d For each IBTrACS TC position location, altimeter U10 out to a spatial distance of 10Rm (10 radii to maxi- data in a region of 6.4836.48 around the location mum winds). To better highlight the extreme data near were extracted. the centers of tropical cyclones, Figs. 2a and 2b also in- d These data were then searched for altimeter passes clude ‘‘breakout’’ plots of the tails of the histograms. with 63 h of the TC position time. This shows that, because of the size of the databases, d These passes were extracted and stored. there are many hundreds of altimeter passes with Hs . 21 d Because there will be a time mismatch between 10 m and U10 . 30 m s . the altimeter pass and the IBTrACS position loca- Values of the central pressure p0 range from 990 hPa tion, the TC position was then interpolated to the to as low as p0 ’ 870 hPa in Fig. 2c. The velocity of same time as the altimeter pass (i.e., moved along forward movement Vfm was determined from successive the IBTrACS path by an amount equal to VfmDt, positions of the TC center in the IBTrACS archive. The where Dt isthetimemismatch(becauseIBTrACS majority of the values of Vfm are between the values of 5 2 data are typically every 6 h, Dt is a maximum of 3 h) and 15 m s 1 (Fig. 2d). The distribution of radius to gales

and Vfm is the velocity of forward movement of the R34 is shown in Fig. 2e, and the distance from the TC storm, calculated from successive location positions center to the location of the maximum Hs measured by in IBTrACS. the altimeter pass is shown in Fig. 2f (expressed non- d TC tracks and altimeter passes were rotated such that dimensionally in terms of the radius to maximum winds

the TC was propagating toward the north, and Southern Rm). As can be seen in Fig. 2f, the amount of usable Hemisphere storms were ‘‘ﬂipped’’ left–right. In this data decreases near the tropical cyclone centers (i.e.,

manner a common reference frame was formed with all distance/Rm , 2). This occurs because of the degrada- storms propagating in the same direction and in the same tion of the altimeter radar returns in heavy rain. Despite hemisphere. this data loss, because of the very large database, there

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC AUGUST 2020 T A M I Z I A N D Y O U N G 2127

FIG. 1. Locations of NDBC buoys used for the in situ database of wind speed and directional wave properties. Only buoys for which data from more than 10 tropical cyclones were available are shown.

are still approximately 2500 altimeter passes within 2Rm region as being 610Rm from the center of the storm. At of storm centers. distances greater than this definition, the background The corresponding set of distributions for the TC NDBC synoptic wind field often starts to dominate over the buoy database is shown in Fig. 3. This database includes a tropical cyclone vortex. total of 2902 buoy records (TC passage past the buoy) from a total of 353 TCs. As noted above, the spatial dis- 4. Altimeter observations tribution of these buoys is shown in Fig. 1. Although this is a much smaller dataset than the TC altimeter database, it is With all TCs rotated such that they propagate toward significantly larger than previous in situ buoy datasets and the north and by normalizing the spatial scale in terms of importantly includes directional spectral estimates. Only Rm, it is possible to pool the TC altimeter data. Figure 4 NDBC buoys with directional wave sensors were used for showsthealtimetertrackspooledinthismannerwiththe this analysis. The distributions of maximum U10 and Hs color proportional to the observed Hs. The data have (Figs. 3a,b) are remarkably similar to the TC altimeter been partitioned based on p0 [p0 , 960 hPa (Fig. 4a); database (Figs. 2a,b). Note, however, that the geographic 960 , p0 , 980 hPa (Fig. 4b)]. Figure 4a contains a total distributions of the two databases are different (global, TC of 1993 passes, and Fig. 4b has4562passes.Thetotal altimeter database; North America, TC buoy database). number of passes summed over these two figures (6555) This becomes clear in the distributions of the TC parame- is less than the total number of altimeter passes in the da- ters p0 (Fig. 3c), Vfm (Fig. 3d), and R34 (Fig. 3e). In par- tabase (36 509), because only cases in which the IBTrACS ticular, it is clear that the global database has consistently archive also recorded values of Rm are included. A larger values of R34, presumably because of the large 2Rm 3 2Rm grid was placed over the data shown in Fig. 4, number of Asian typhoons in the database, which are and the median of the values within each grid region was known to be spatially larger than North American hurri- determined. Note that the mean was also determined and canes (Mok et al. 2018). It is also clear that the distribution produced very similar results. The median is preferred here, of the distance to the recorded maximum Hs (Fig. 3f) does because it is less impacted by outliers in the large dataset. not decrease near the center of storms as for the TC al- Contour plots of the median Hs and U10, normalized by timeter database. This lends support to the assumption that the maximum median value, are shown for each grid square this feature of the altimeter database was due to data loss in in Fig. 5. Thus contour values range between 0 and 1. the altimeters due to high rain rates near the centers of TCs. To give an indication of the magnitude of the values in

As noted above, our interest is in defining the spatial these normalized plots, the maximum values of Hs and distribution of the wave field in tropical cyclones. U10 for each subplot are given in the figure caption. Consistent with previous studies (Young 2006; Hu and Spatial distributions of TC Hs and U10 have previously Chen 2011; Collins et al. 2018) we have defined this been produced from model data (Tolman and Alves 2005)

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC 2128 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 50

FIG. 2. Summary of the altimeter TC database: (a) maximum signiﬁcant wave height Hs for each pass of an altimeter, (b) maximum wind speed U10 for each pass of an altimeter, (c) central pressure p0 of each storm in the database, (d) velocity of forward movement Vfm of each storm in the database, (e) radius to gales R34 of each storm in the database, and (f) minimum distance between altimeter and storm eye for each altimeter pass.

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC AUGUST 2020 T A M I Z I A N D Y O U N G 2129

FIG. 3. Summary of the in situ buoy TC database: (a) Hs for each transect of a TC, (b) U10 for each transect, (c) p0 of each storm in the database, (d) Vfm of each storm in the database, (e) R34 of each storm in the database, and (f) minimum distance between TC eye and buoy for each case in the database.

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC 2130 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 50

FIG. 4. Values of Hs along altimeter passes over tropical cyclones. All storms are rotated to be propagating toward the north. Southern Hemisphere storms are left–right ‘‘flipped’’ to ensure consistency with Northern Hemisphere storms. Shown are (a) cases with p0 , 960 hPa and (b) cases with 960 , p0 , 980 hPa. but we believe that these are the first such results obtained assumptions of an extended or trapped fetch model of TC from actual measurements. wave generation (e.g., Young 2017). These extended Vortex models of TC wind fields have been proposed fetch models suggest that the velocity of forward move- by a number of authors (e.g., Holland 1980; Willoughby ment Vfm should play a role in the generation and dis- and Rahn 2004; Willoughby et al. 2006; Holland et al. tribution of Hs within the TC. We attempted to partition 2010). These models define the characteristics of the TC the data by both p0 and Vfm, but, even with this extensive wind field consisting of a calm eye, an asymmetric wind dataset, the data became too sparse and noisy to obtain field, with the strongest winds to the right (Northern reliable spatial distributions. Hemisphere) of the eye and a wind field which decays Note that in compiling these results only altimeter data exponentially from the area of maximum winds. The wind with a quality flag of ‘‘1’’ (Ribal and Young 2019)were vectors spiral around the storm center in an anticlockwise retained. That is, only data regarded as of ‘‘good’’ quality direction with a slight inflow toward the center. were used. However, visual inspection of altimeter passes The results in Fig. 5 are partitioned by central pressure showed that probably the largest quality issue associated p0. Figures 5a and 5b show moderately intense storms with the results in Fig. 5 was due to IBTrACS TC location (960 , p0 , 980 hPa), Figs. 5c and 5d show intense information. On a number of occasions, altimeter passes storms (p0 , 960 hPa), and Figs. 5e and 5f show all data. clearly passed through the calm eye of the TC, accurately The wind fields (Figs. 5b,d,f) show a number of the positioning the eye. In a number of these cases, the characteristics defined by the vortex models. The wind IBTrACS position was in error by up to 100 km. The

fields are asymmetric, with the peak wind occurring to IBTrACS values of p0 may also be in error, but we have the right of the storm center. The overall wind field is no way of determining the magnitude of such errors. asymmetric. For instance, U10 is larger for positive Also, we expect this to have less impact on the results in values of X/Rm (right of storm center) than for negative Fig. 5 than do TC location errors. Random errors in the values (left of storm center). The wind speed increases position of the eye will tend to ‘‘smear’’ the wind speed as p0 decreases (more intense storms). The clear vortex and significant wave height median values shown in Fig. 5, decay extends out to approximately 10Rm. The results this effect being more pronounced close to the centers of do not show a calm eye, but the data were gridded at the storm, where the spatial gradients of wind speed and

2Rm, so this is not unexpected. Individual passes of the signiﬁcant wave height are greatest. altimeters certainly show such structure. The spatial distributions of Hs (Figs. 5a,c,e)largely 5. In situ buoy observations mirror those of U10. There is, however, a suggestion of slightly greater right–left asymmetry for signiﬁcant wave As in Fig. 4, Fig. 6 shows recorded buoy data during height than for wind speed. This is consistent with the the passage of TCs near buoys. Note that all data are now

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC AUGUST 2020 T A M I Z I A N D Y O U N G 2131

FIG. 5. Contour plots of the spatial distributions of (left) Hs and (right) U10 for altimeter passes over tropical cyclones, showing (a),(b) data for 960 , p0 , 980 hPa; (c),(d) data for p0 , 960 hPa; and (e),(f) all data. All values have been normalized by the maximum value of the median quantities over the spatial ﬁeld. Maximum values are m 5 m 21 m 5 m 5 21 m 5 Hs 14.5 m for (a), U10 47.3 m s for (b), Hs 15. 6 m for (c), U10 52.7 m s for (d), Hs 15. 6 m for (e), m 5 21 and U10 52.7 m s for (f). in a frame of reference moving with the TC. Figure 6 datasets. Measurements of both Hs and U10 were again contains a total of 2081 TC passes near buoys, a signiﬁ- binned into a regular grid of 3Rm 3 3Rm and the median cantly small dataset than for the altimeters. Also, note values found for each grid square. The coarser 3Rm binning that this total (2081) is less than the total number of in situ was used for the buoy data, because tests with a 2Rm grid buoy records (2902) because, again, only data with recor- (as used for the altimeter data) were unacceptably noisy. ded IBTrACS values of Rm are retained. However, the Figure 7 shows color contour plots for the median dataset is still much larger than any previous in situ values of both Hs (Fig. 7a) and U10 (Fig. 7b). As with the

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC 2132 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 50

As expected, uU shows the structure of a vortex, with the wind spiraling around the TC center in an anticlockwise direction (Northern Hemisphere). The corresponding mean wave directions uw (Fig. 8a) show a more complex spatial structure, consistent with the limited previous measurements (Young 2006; Hu and Chen 2011; Esquivel- Trava et al. 2015; Collins et al. 2018). In the right-rear quadrant, uU and uw are well aligned. In the right-front quadrant, the waves tend to propagate more in the direction of forward movement of the storm. In the left-front quadrant, uw appears to propagate from a region near the center of the storm. This behavior in the left- and right- front quadrants is consistent with the wave conditions being made up of a combination of local wind sea (pre-

FIG. 6. Values of Hs at in situ buoys during the passage of tropical sumably aligned with the local wind direction uU )and cyclones. All storms were rotated to be propagating toward the remotely generated waves, which were generated in the north, and data were placed into a frame of reference moving with intense wind regions near the center of the TC, have the storm. ‘‘outrun’’ the TC and now appear ahead of the storm as remotely generated waves. The left-rear quadrant shows altimeter data in Fig. 5, the contoured values were a confused situation for u , indicating a complex spectral normalized by the maximum median value and the w form with possible multiple peaks. maximum values are provided in the figure caption. As noted above, these spatial distributions are con- Because there are significantly fewer data than for the sistent with previous in situ and satellite data but pro- altimeters, the data were not partitioned by p . Although 0 vide many more data than were previously available and the distributions of median H and U are noisier than s 10 hence provide a more comprehensive distribution of the corresponding altimeter results (Fig. 5), they show the mean wind and peak wave direction vectors, covering same general attributes. That is, the maximum values of the full spatial field out to 610R . Figure 8a, shows the H and U are to the right of the storm center, the wind m s 10 occasional vector of u that appears to be inconsistent and wave height fields decay with radius (X/R and w m with its neighbors. In all of these cases, they are associ- Y/R ), and there is greater right–left asymmetry for H m s ated with 2R grid squares where there are few in situ than U . m 10 observations, despite the overall size of the dataset. The peak wave direction uw (mean direction at the spectral peak frequency) was determined from the 6. 1D spectral form NDBC data for each of the buoy observations, and the values were again binned into 2Rm 3 2Rm bins. The Visual inspection of the recorded 1D spectra in the TC mean values for each bin are shown in Fig. 8a, with the in situ buoy database, indicated that the vast majority corresponding mean wind speeds uU given in Fig. 8b. were unimodal. Therefore, following Young (2006), the

FIG. 7. Contour plots of the spatial distributions of (a) Hs and (b) U10 for transects of tropical cyclones near in situ buoys. All values have been normalized by the maximum value of the median quantities over the spatial ﬁeld. m 5 m 5 21 Maximum values are Hs 16.9 m for (a) and U10 47.2 m s for (b).

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC AUGUST 2020 T A M I Z I A N D Y O U N G 2133

FIG. 8. Spatial distributions of (a) peak wave direction and (b) mean wind direction obtained from the TC in situ buoy database. All storms were rotated such that storms propagate toward the north.

1D spectra were modeled by the generalized Joint remotely generated waves in the present context. It is North SeaWave Project (JONSWAP) form proposed by clear in Fig. 9 that a significant proportion of the Young and Verhagen (1996): spectra is in this category and that remotely generated wave energy is an important component of many ob- 5 2 24 2(51n) n E(f ) bg (2p) fp f served spectra. The importance of remotely generated 2 ! 3 24 waves in defining the wave field is also clear in the mean 4n f 5 exp[2( f 2f )2=(2s2f 2)] wave directions shown in Fig. 8a. 3 exp g p p . (4) 4 fp Fetch-limited wind-sea spectra typically have n from approximately 24to25(Hasselmann et al. 1973; In the spectral form in Eq. (4), g is gravity and the Donelan et al. 1985). The data here scatter around these 52 spectral parameters are a ‘‘Phillips’’ scale parameter values, with a mean value of n 4.68, shown in Fig. 9a. The values of peak enhancement factor g are shown in b, the spectral peak frequency fp, the high-frequency spectral decay exponent n, and the spectral width pa- Fig. 9c, along with the relationship proposed by Donelan rameter s. This form differs from the classic JONSWAP et al. (1985) for fetch-limited data. Although the values form in that the exponent n is not set to a constant (e.g., of g are comparable to fetch-limited results, they are n 525 for JONSWAP). The nondimensional energy consistently larger. This result indicates the spectra and nondimensional frequency n were then deter- are more peaked than in fetch-limited cases. Whereas mined as the present data for n, g,ands (not shown) show no clear trend as a function of U10/Cp, consistent with 5 2 = 4 g Etot U10 and (5) fetch-limited wind seas, there is a clear relationship between b and U10/Cp in Fig. 9b. Donelan et al. (1985) 5 = n fpU10 g, (6) also found such a functional dependence for fetch- Ð limited conditions, which they approximated by b 5 5 0.55 where Etot E(f ) df . 0.006(U10/Cp) . Young (2006) reported TC data Figure 9 shows plots of the spectral parameters n that are generally consistent with this relationship. (Fig. 9a), b (Fig. 9b), and g (Fig. 9c), as a function of the The present extensive dataset shows a relationship inverse wave age U10/Cp, where Cp is the phase speed of that increases more rapidly as a function of U10/Cp, components at the spectral peak frequency fp. The ver- which can be approximated by tical line shown in each of the panels of Fig. 9 is drawn 1:21 5 5 : = at U10/Cp 0.83. This is the swell–wind-sea limit pro- b 0 0045 U10 Cp . (7) posed by Donelan et al. (1985). Values of U10/Cp , 0.83 will not be receiving active local wind forcing at This relationship (and the present data) yield values the spectra peak and are usually regarded as swell, or of b that are generally consistent with the fetch-limited

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC 2134 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 50

FIG. 9. One-dimensional spectrum parameters [Eq. (4)] for in situ buoy data for tropical cyclones: (a) spectral decay parameter n as a function of inverse wave

age U10/Cp (the horizontal solid line is the mean: n 524.68, and the dashed line is drawn at n 524); (b) as in (a), but for spectral level parameter b {the solid line through the data is the best ﬁt [Eq. (7)], and the dashed line is the fetch-limited result of Donelan et al. (1985)}; and (c) as in (a), but for peak enhancement parameter g [the dashed line is the fetch-limited relationship of Donelan et al. (1985)]. The vertical solid lines in (a)–(c) mark the demarcation between swell and wind sea:

U10/Cp 5 0.83.

results for U10/Cp ’ 1 but yield values of b that are less That is, there is less energy in the tail of the spectrum than the fetch-limited results for values of U10/Cp , 1. (lower b) and therefore lower total energy (Etot and ). That is, the high-frequency tail of the TC wave spectrum has less energy than is typically seen in fetch-limited 7. Directional spectra wind seas. A close examination of the limited data points of Young (2006) (Fig. 10 of Young 2006), shows that the The four Fourier directional moments recorded by the Young (2006) data are actually also consistent with (7). in situ wave buoys are used to estimate the directional Figure 10 shows the nondimensional energy as a spectrum E(f, u). This is often described by the 1D function of the nondimensional peak frequency n, to- spectrum, E(f) and the directional spreading function gether with the fetch-limited relationship of Donelan D(f, u) [Eq. (1)]. Figures 11 and 12, show the spa- et al. (1985) and the wind-sea–swell demarcation of n 5 tial wave ﬁeld divided into octants. For reference, we 0.13. Again, many of the data are for n , 0.13 (i.e., re- number these octants in an anticlockwise direction from motely generated waves) and are in reasonable agree- the x axis. Therefore, octant 1 is ENE, octant 2 is NNE, ment with the fetch-limited functional dependence. octant 3 is NNW, ..., octant 8 is ESE. For each of these Closer examination of the data does, however, show that octants, representative values of mean wave direction, for TC conditions is slightly smaller than the fetch- uw (dashed arrow) and mean wind direction, uU (solid limited result for n , 0.13. This is consistent with the arrow) are shown in Figs. 11 and 12. Figure 11 also shows smaller values of b also seen in Fig. 9c for TC conditions. representative 1D spectra E(f) at each of these points.

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC AUGUST 2020 T A M I Z I A N D Y O U N G 2135

the wind and spectral peak wave directions gradually increases until it is in excess of 1208 in octant 5 (WSW). The large directional skew in the octants to the left of the storm center (octants 4, 5, and 6) accounts for the bi-/trimodal spectra observed in these octants (Fig. 11) and the confused mean wave directions (Fig. 8a). As already noted by Young (2006), even though there is signiﬁcant directional skew, the spectra never degrade into separate wave systems. That is, there is a continuum

of energy from fp to 5fp. Young (2006) interprets this as indicating that there is a continual cascade of energy throughout the skewed spectrum due to nonlinear wave–wave interactions (Hasselmann et al. 1973, Young and van Vledder 1993).

8. Discussion and conclusions

This study has compiled a large database of both satellite altimeter overpasses of tropical cyclones and FIG. 10. Nondimensional energy versus nondimensional peak frequency n from in situ buoy data during the passage of tropical directional buoy measurements during the passage of cyclones. The dashed line through the data is the fetch-limited tropical cyclones. The satellite data confirm that both result of Donelan et al. (1985). The vertical solid line is the de- the wind and wave fields in tropical cyclones are asym- 5 marcation between swell and wind sea: n 0.13. metric, with stronger winds/larger waves to the right of the eye (Northern Hemisphere) of the storm. The wave Figure 12 shows representative directional spreading field has a greater degree of asymmetry than the wind functions D(f, u) at each point. That is, in both Figs. 11 field. This occurs because the generation of large waves and 12 we show typical examples of the spectra rather requires both strong winds and a substantial fetch over than averaging all of the spectra. Such an averaging which the wind must blow. In the case of a translating process would tend to smear the spectra, resulting in tropical cyclone, an extended fetch develops to the right excessive spreading in both frequency and direction. of the storm where the direction of propagation of the Examination of the 1D spectra in Fig. 11 shows that, storm is approximately aligned with the wind direction. with the exception of octant 4 (WNW) and octant 5 Hence, waves generated in this region move forward (WSW), the spectra are unimodal and very similar in with the translating storm and waves can remain in the shape to fetch-limited wind seas. However, the values of intense wind region for an extended duration. The op- peak frequency are such that most of these spectra have posite happens to the left of the storm where the wind waves at the spectra peak that are propagating faster direction is in the opposite direction to the direction than the local wind (remotely generated waves). Octants of translation of the storm. Therefore, waves quickly 4 and 5 (left of storm center) correspond to the region of propagate out of the intense wind region. As a result, confused uw in Fig. 8a. the asymmetry of the wave field is greater than the Although the 1D spectra look very simple (unimodal), wind field. the directional spreading functions D(f, u) are very It is believed that these are the first spatial distribu- complex, with most showing significant directional tions showing the full distribution of significant wave skewing. In each octant, the high-frequency compo- height out to 10Rm. Previous studies addressing such nents of the spectrum (f . 2.5fp)arealignedwiththe distributions have relied on model data (e.g., Young and local wind (solid vertical lines in Fig. 12). However, the Vinoth 2013). Such studies are naturally limited by the components at the spectral peak frequency represent reliability of the model physics under the complex remotely generated waves that have been generated forcing within tropical cyclones. earlier in the intense wind regions near the center of Consistent with a range of previous studies, the ex- the TC and have now propagated to the site (dashed tensive directional buoy data in this study show that vertical lines in Fig. 12). In octant 8 (ESE) the wind and much of the tropical cyclone wave field is dominated by spectral peak wave direction are most aligned and remotely generated waves originating near the center of hence D(f, u) has little directional skew. As one rotates the tropical cyclone. The circulating vortex generates anticlockwise from octant 8 (ESE) the angle between waves in this region which radiate out from the center of

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC 2136 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 50

FIG. 11. Spatial distributions of the one-dimensional spectrum E(f) within tropical cyclones obtained from in situ buoy data. The solid arrows show the mean wind direction; the dashed arrows show the peak wave direction. Representative one-dimensional spectra are shown for each octant.

the storm in all directions. As a result, in most regions of the full tropical cyclone waves field (610Rm). This the tropical cyclone wave field, the spectrum consists of contrasts with previous studies (Young 2006; Hu and remotely generated waves radiating out from the center Chen 2011; Collins et al. 2018) that have only been able of the storm, together with locally generated wind sea. to obtain data from parts of the spatial field. These wave systems can be directionally separated by We typically assume the wave spectral energy balance more than 908. The phase speeds of the remotely gener- is represented by the processes of atmospheric input Sin, ated waves are such that, in most cases, they propagate nonlinear wave–wave interactions Snl, and whitecap faster than the local wind speed. This means the remotely dissipation Sds (Stot 5 Sin 1 Snl 1 Sds; Komen et al. 1984). generated waves are not actively forced by the local wind. At the spectral peak (remotely generated waves peak),

Despite this, the remotely generated waves and wind sea Sin ’ Sds ’ 0, because the waves are propagating faster do not exist as separate systems. Rather there is a con- than the local wind and because the mean squared slope tinuum of spectral energy from the high-frequency wind of the remotely generated waves is small. Therefore, the sea to the low-frequency remotely generated waves. This spectral balance in the vicinity of the spectral peak will continuum of energy is, however, directional skewed. be Stot ’ Snl. The nonlinear term Snl is typically char- Because of the size of the in situ buoy dataset, it has acterized by a transfer of energy from higher frequency been possible to examine the directional spectrum over to lower frequency (Young and van Vledder 1993).

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC AUGUST 2020 T A M I Z I A N D Y O U N G 2137

FIG. 12. Spatial distributions of the directional spreading function D(f, u) within tropical cyclones obtained from in situ buoy data. The solid arrows show the mean wind direction; the dashed arrows show the peak wave direction. Representative values of D(f, u) [Eq. (1)] are shown for each octant. The colored contours show D(f, u) with contours drawn at 0.9, 0.8, ..., 0.1. Consistent with the arrows, the vertical solid lines on the spectra insets show the mean wind direction and the vertical dashed lines show the peak wave direction.

In the present case, this continual feed of energy to Although the datasets used in this analysis are very lower frequencies appears to result in a cascade of en- large, there are still clear limitations resulting from the ergy from the locally generated wind sea to the remotely ﬁnite amounts of data. It was necessary to bin the al- generated waves peak. This results in a smooth variation timeter data at a resolution of 2Rm 3 2Rm and the buoy of the spectrum with a unimodal one-dimensional form. data at an even coarser resolution of 3Rm 3 3Rm.Asa The nonlinear terms result in a one-dimensional result, much of the detailed structure close to the eye of spectral form very similar to fetch-limited waves. The the storm is lost. We have assumed that the spatial high-frequency face decays } f n,wheren ranges structure can be normalized by the radius to maximum between 24and25. Because of the lack of signiﬁcant winds Rm. This is a spatial scaling commonly used. wind input near the spectral peak, the energy in the However, there are other choices which may also be high-frequency face of the spectrum is lower than for appropriate, including R34. The present data do not al- fetch-limited cases and the total energy is also lower low one to explore these differences in detail. There than one would normally expect for waves with the was a strong desire to partition the data by both tropical observed (low) peak frequencies. cyclone intensity (p0 or Vmax) and translation speed Vfm,

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC 2138 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 50 as model data (Young and Vinoth 2013) indicate these National Data Buoy Center. NOAA/National Data Buoy Center Tech. Doc. 03-02, 44 pp. combined parameters impact both the magnitude of Hs and the spatial structure. Despite the size of the datasets, Gonzalez, F. I., T. E. Thompson, W. E. Brown, and D. E. Weissman, 1982: Seasat wind and wave observations of this was not possible, with such data segmenting result- northeast Pacific Hurricane Iva, 13 August 1978. J. Geophys. ing in unacceptably noisy spatial distributions. In Fig. 5 Res., 87, 3431–3438, https://doi.org/10.1029/JC087iC05p03431. some attempt was made to examine the impact of p0; Hasselmann, K., and Coauthors, 1973: Measurements of wind- however, this is not a comprehensive analysis. Last, the wave growth and swell decay during the Joint North SeaWave IBTrACS data are often limited in their accuracy. This Project (JONSWAP). Deutches Hydrographisches Institut Hydraulic Engineering Rep., 95 pp. clearly results in errors in accurate determination of TC Holland, G. J., 1980: An analytical model of the wind and pres- tracks and tropical cyclone wind field parameters. These sure profiles in hurricanes. Mon. Wea. Rev., 108, 1212–1218, are, however, the best available data for such large https://doi.org/10.1175/1520-0493(1980)108,1212:AAMOTW. datasets. 2.0.CO;2. ——, J. I. Belanger, and A. Fritz, 2010: A revised model for radial profiles of hurricane winds. Mon. Wea. Rev., 138, 4393–4401, Acknowledgments. The authors acknowledge the sup- https://doi.org/10.1175/2010MWR3317.1. port provided to author Tamizi by the University of Holt, B., and F. I. Gonzalez, 1986: SIR-B observations of dominant Melbourne through a Ph.D. scholarship. The development ocean waves near Hurricane Josephine. J. Geophys. Res., 91, of the altimeter database was supported by a grant from 8595–8598, https://doi.org/10.1029/JC091iC07p08595. the Integrated Marine Observing System. Hu, K., and Q. Chen, 2011: Directional spectra of hurricane- generated waves in the Gulf of Mexico. Geophys. Rev. Lett., 38, L19608, https://doi.org/10.1029/2011GL049145. REFERENCES Hwang, P. A., 2016: Fetch- and duration-limited nature of surface wave growth inside tropical cyclones: With applications to Beal, R. C., T. W. Gerling, D. E. Irvine, F. M. Monaldo, and D. G. air–sea exchange and remote sensing. J. Phys. Oceanogr., 46, Tilley, 1986: Spatial variations of ocean wave directional 41–56, https://doi.org/10.1175/JPO-D-15-0173.1. spectra from the Seasat synthetic aperture radar. J. Geophys. ——, and E. J. Walsh, 2016: Azimuthal and radial variation of Res., 91, 2433–2449, https://doi.org/10.1029/JC091iC02p02433. wind-generated surface waves inside tropical cyclones. Black, J. L., 1979: Hurricane Eloise directional wave energy J. Phys. Oceanogr., 46, 2605–2621, https://doi.org/10.1175/ spectra. Proc. 11th Offshore Technology Conf., Houston, TX, JPO-D-16-0051.1. Offshore Technology Conference, OTC-3594-MS, https:// ——, and Y. Fan, 2017: Effective fetch and duration of tropical doi.org/10.4043/3594-MS. cyclone wind fields estimated from simultaneous wind and Black, P. G., and Coauthors, 2007: Air–sea exchange in hurricanes: wave measurements: Surface wave and air–sea exchange Synthesis of observations from the Coupled Boundary Layer computation. J. Phys. Oceanogr., 47, 447–470, https://doi.org/ Air–Sea Transfer Experiment. Bull. Amer. Meteor. Soc., 88, 10.1175/JPO-D-16-0180.1. 357–374, https://doi.org/10.1175/BAMS-88-3-357. ——, and E. J. Walsh, 2018a: Propagation directions of ocean Bowyer, P. J., and A. W. MacAfee, 2005: The theory of trapped-fetch surface waves inside tropical cyclones. J. Phys. Oceanogr., 48, waves within tropical cyclones—An operational perspective. 1495–1511, https://doi.org/10.1175/JPO-D-18-0015.1. Wea. Forecasting, 20, 229–244, https://doi.org/10.1175/WAF849.1. ——, and ——, 2018b: Estimating maximum significant wave Bretschneider, C. L., 1959: Hurricane design—Wave practices. height and dominant wave period inside tropical cyclones. Trans. Amer. Soc. Civ. Eng., 124, 39–62. Wea. Forecasting, 33, 955–966, https://doi.org/10.1175/WAF- ——, 1972: A non-dimensional stationary hurricane wave model. D-17-0186.1. Proc. Fourth Offshore Technology Conf., Houston, TX, ——, Y. Fan, F. J. Ocampo-Torres, and H. García-Nava, 2017: Offshore Technology Conference, OTC-1517-MS, https:// Ocean surface wave spectra inside tropical cyclones. J. Phys. doi.org/10.4043/1517-MS. Oceanogr., 47, 2393–2417, https://doi.org/10.1175/JPO-D- Collins, C. O., H. Potter, B. Lund, H. Tamura, and H. C. Graber, 17-0066.1. 2018: Directional wave spectra observed during intense trop- King, D. B., and O. H. Shemdin, 1978: Radar observations of ical cyclones. J. Geophys. Res. Oceans, 123, 773–793, https:// hurricane wave directions. 16th Int. Conf. Coastal Engineering, doi.org/10.1002/2017JC012943. Hamburg, Germany, ASCE, 209–226, https://doi.org/10.1061/ Donelan, M., M. Hamilton, and W. Hui, 1985: Directional spectra 9780872621909.012. of wind-generated waves. Philos. Trans. Roy. Soc. London, Knapp, K. R., M. C. Kruk, D. H. Levinson, H. J. Diamond, and C. J. A315, 509–562, https://doi.org/10.1098/rsta.1985.0054. Neumann, 2010: The International Best Track Archive for Elachi, C., T. W. Thompson, and D. B. King, 1977: Observations of Climate Stewardship (IBTrACS): Unifying tropical cyclone the ocean wave pattern under Hurricane Gloria with synthetic best track data. Bull. Amer. Meteor. Soc., 91, 363–376, https:// aperture radar. Science, 198, 609–610, https://doi.org/10.1126/ doi.org/10.1175/2009BAMS2755.1. science.198.4317.609. ——, H. J. Diamond, J. P. Kossin, M. C. Kruk, and C. J. Schreck, Esquivel-Trava, B., F. J. Ocampo-Torres, and P. Osuna, 2015: 2018: International Best Track Archive for Climate Stewardship Spatial structure of directional wave spectra in hurricanes. (IBTrACS) Project, version 4. NOAA/National Centers for Ocean Dyn., 65, 65–76, https://doi.org/10.1007/s10236- Environmental Information, accessed 27 January 2020, https:// 014-0791-9. doi.org/10.25921/82ty-9e16. Evans, D., C. L. Conrad, and F. M. Paul, 2003: Handbook of au- Komen, G. J., S. Hasselmann, and K. Hasselmann, 1984: On the tomated data quality control checks and procedures of the existence of a fully developed wind-sea spectrum. J. Phys.

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC AUGUST 2020 T A M I Z I A N D Y O U N G 2139

Oceanogr., 14, 1271–1285, https://doi.org/10.1175/1520- Conf., Houston, TX, Offshore Technology Conference, OTC- 0485(1984)014,1271:OTEOAF.2.0.CO;2. 3228-MS, https://doi.org/10.4043/3228-MS. Longuet-Higgins, M. S., D. E. Cartwright, and N. D. Smith, 1963: Willoughby, H. E., and M. E. Rahn, 2004: Parametric representa- Observations of the directional spectrum of sea waves using tion of the primary hurricane vortex. Part I: Observations and the motion of a floating buoy. Ocean Wave Spectra, Prentice evaluation of the Holland (1980) model. Mon. Wea. Rev., 132, Hall, 111–136. 3033–3048, https://doi.org/10.1175/MWR2831.1. McLeish, W., and D. B. Ross, 1983: Imaging radar observations of ——, R. W. R. Darling, and M. E. Rahn, 2006: Parametric repre- directional properties of ocean waves. J. Geophys. Res., 88, sentation of the primary hurricane vortex. Part II: A new 4407–4419, https://doi.org/10.1029/JC088iC07p04407. family of sectionally continuous profiles. Mon. Wea. Rev., 134, Mok, D. K. H., J. C. L. Chan, and K. T. F. Chan, 2018: A 31-year 1102–1120, https://doi.org/10.1175/MWR3106.1. climatology of tropical cyclone size from the NCEP Climate Wright, C. W., and Coauthors, 2001: Hurricane directional wave Forecast System Reanalysis. Int. J. Climatol., 38, e796–e806, spectrum spatial variation in the open ocean. J. Phys. Oceanogr., https://doi.org/10.1002/joc.5407. 31, 2472–2488, https://doi.org/10.1175/1520-0485(2001)031,2472: Ochi, M. K., 1993: On hurricane-generated seas. Proc. Second Int. HDWSSV.2.0.CO;2. Symp. on Ocean Wave Measurement and Analysis, New Young, I. R., 1988: A parametric hurricane wave prediction model. Orleans, LA, ASCE, 374–387. J. Waterw. Port Coastal Ocean Eng., 114, 637–652, https:// ——, and M. H. Chiu, 1982: Nearshore wave spectra measured doi.org/10.1061/(ASCE)0733-950X(1988)114:5(637). during Hurricane David. Proc. 18th Int. Conf. on Coastal ——, 1994: On the measurement of directional wave spectra. Appl. Engineering, Cape Town, South Africa, ASCE,77–86, https:// Ocean Res., 16, 283–294, https://doi.org/10.1016/0141-1187(94) doi.org/10.1061/9780872623736.005. 90017-5. Patterson, M. M., 1974: Oceanographic data from Hurricane ——, 1998: Observations of the spectra of hurricane generated Camille. Proc. Offshore Technology Conf., Houston, TX, waves. Ocean Eng., 25, 261–276, https://doi.org/10.1016/S0029- Offshore Technology Conference, OTC-2109-MS, https:// 8018(97)00011-5. doi.org/10.4043/2109-MS. ——, 2006: Directional spectra of hurricane wind-waves. J. Geophys. Ribal, A., and I. R. Young, 2019: 33 years of globally calibrated wave Res., 111,C08020,https://doi.org/10.1029/2006JC003540. height and wind speed data based on altimeter observations. Sci. ——, 2017: A review of parametric descriptions of tropical cyclone Data, 6,77,https://doi.org/10.1038/S41597-019-0108-4. wind-wave generation. Atmosphere, 8, 194, https://doi.org/ ——, and ——, 2020: Calibration and cross-validation of global 10.3390/atmos8100194. ocean wind speed based on scatterometer observations. ——, and G. P. Burchell, 1996: Hurricane generated waves as ob- J. Atmos. Oceanic Technol., 37, 279–297, https://doi.org/ served by satellite. Ocean Eng., 23, 761–776, https://doi.org/ 10.1175/JTECH-D-19-0119.1. 10.1016/0029-8018(96)00001-7. Ross, D. B., 1976: A simplified model for forecasting hurricane ——, and G. Ph. van Vledder, 1993: A review of the central role of generated waves. Bull. Amer. Meteor. Soc., 57, 113–115, nonlinear interactions in wind-wave evolution. Philos. Trans. https://doi.org/10.1175/1520-0477-57.1.95. Roy. Soc. London, 342A, 505–524, https://doi.org/10.1098/ SWAMP Group, 1985: Ocean Wave Modelling. Plenum Press, 256 pp. rsta.1993.0030. The WAMDI Group, 1988: The WAM model—A third gener- ——, and L. A. Verhagen, 1996: The growth of fetch limited waves in ation ocean wave prediction model. J. Phys. Oceanogr., 18, water of finite depth. Part II: Spectral evolution. Coast. Eng., 29, 1775–1810, https://doi.org/10.1175/1520-0485(1988)018,1775: 79–99, https://doi.org/10.1016/S0378-3839(96)00007-5. TWMTGO.2.0.CO;2. ——, and J. Vinoth, 2013: An ‘extended fetch’ model for the Tolman, H. L., and J.-H. G. M. Alves, 2005: Numerical modeling spatial distribution of tropical cyclone wind-waves as ob- of wind waves generated by tropical cyclones using moving served by altimeter. Ocean Eng., 70, 14–24, https://doi.org/ grids. Ocean Modell., 9, 305–323, https://doi.org/10.1016/ 10.1016/j.oceaneng.2013.05.015. j.ocemod.2004.09.003. ——, E. Sanina, and A. V. Babanin, 2017: Calibration and cross U.S. Army Corps of Engineers, 1977: Shore Protection Manual.Vols. validation of a global wind and wave database of altimeter, I–III, U.S. Army Coastal Engineering Research Center, 1223 pp. radiometer, and scatterometer measurements. J. Atmos. Whalen, J. E., and M. K. Ochi, 1978: Variability of wave spectral Oceanic Technol., 34, 1285–1306, https://doi.org/10.1175/ shapes associated with hurricanes. Proc. Offshore Technology JTECH-D-16-0145.1.

Unauthenticated | Downloaded 09/29/21 05:32 AM UTC