1. Introduction

In tropical and subtropical regions, tropical cyclones (TCs; hurricanes or typhoons) represent the major extreme meteorological systems. The strong winds associated with the moving tropical cyclone vortex generate extreme waves which critically impact processes such as: coastal and offshore engineering design, shipping, coastal beach erosion and coastal inundation. The extreme winds (in excess of 40 m s⁻¹), the rapidly turning nature of the vortex, and its translation are a demanding test for our understanding of wind generation physics. The situation is exacerbated by the paucity of quality wind and wave data during intense tropical cyclones. The relatively small geographic size of tropical cyclones means a dense network of observations is required to ensure that extreme conditions will be captured. When they are, the conditions are such that failure of in situ measurement systems is not uncommon. Therefore, the observational database of wave conditions under tropical cyclone forcing is still relatively small, meaning that a comprehensive understanding of the wave fields generated by such systems is still limited.

This paper attempts to build an extensive database of observations of tropical cyclone wave conditions from a number of sources. These sources include a global database of satellite altimeter observations from 13 satellite missions across a period of 33 years together with 37 years of in situ data from the National Data Buoy Center (NDBC) buoy network around the United States. The combined dataset includes altimeter data from 2730 tropical cyclones worldwide and in situ data from 353 tropical cyclones (hurricanes) in the North American region. This combined dataset is analyzed to provide an overview of the spatial distributions of significant wave height and wind speed, as well as details of the directional wave spectrum generated in tropical cyclones.

The arrangement of this paper is as follows. Following this introduction, section 2 provided a brief overview of previous observations and computations of the tropical cyclone wave field, and section 3 describes the data sources used to construct the present tropical cyclone databases. The details of the altimeter observations are provided in section 4, followed by the in situ buoy observations in section 5. In addition to the distribution of significant wave height, the in situ buoy data provide a comprehensive view of the directional wave spectrum. The one-dimensional (1D) energy density spectrum is described in section 6, followed by the directional properties of the spectrum in section 7. Conclusions and discussion appear in section 8.

2. Observations of tropical cyclone wave fields

The earliest attempts to model/describe the tropical cyclone wave field assumed that it would mirror the wind field (Bretschneider 1959, 1972; USACE 1977; Ross 1976). However, the first limited observations from in situ instruments (Patterson 1974; Whalen and Ochi 1978; Black 1979; Ochi and Chiu 1982; Ochi 1993; Young 1998; Young 2006) and remote sensing observations, largely using aircraft-based synthetic aperture radar and orbiting altimeter data (Elachi et al. 1977; King and Shemdin 1978; Gonzalez et al. 1982; McLeish and Ross 1983; Holt and Gonzalez 1986; Beal et al. 1986; Young and Burchell 1996; Wright et al. 2001; Black et al. 2007) showed a more complex structure. These measurements indicated that, ahead of the tropical cyclone vortex, the wave field was composed of locally generated wind seas (aligned with the local wind direction) and remotely generated waves which had been generated in the intense wind regions of the storm and then propagated ahead of the storm. These observations have also been augmented with model simulations of tropical cyclone conditions (SWAMP Group 1985; Young 1988; The WAMDI Group 1988).

These measurements and modeling led to the conclusion that the forward motion of the storm was an important variable in understanding wave generation within tropical cyclones. As a result, the concept of an “extended” or “trapped” fetch was developed (Young 1988; Young and Burchell 1996; Bowyer and MacAfee 2005; Young and Vinoth 2013). In this representation, waves generated to the right (Northern Hemisphere) of the storm center propagate forward with the storm and hence stay in the strong wind region for an extended period of the time (the extended fetch). Hence, both the velocity of forward movement and the maximum wind speed in the storm are important in determining the wave field.

A number of studies have aggregated data from multiple tropical cyclones to provide information on the spatial distribution of tropical cyclone waves. Young (1998, 2006) considered data from directional wave buoys off the northwest coast of Australia during the passage of nine tropical cyclones. To aggregate the data from the multiple storms, a frame of reference moving with the tropical cyclone was adopted. The analysis confirmed the conclusions from the remote sensing data that, ahead of the storm, the wave field was composed of locally generated wind sea and remotely generated waves radiating out from the intense wind regions near the center of the storm. The directional properties obtained from the buoys showed strong directional skew, with remotely generated waves and wind sea often propagating in directions separated by more than 90°.

The directional properties indicated that the wave systems (remotely generated waves and wind sea) did not completely decouple and there was a continuum of energy between the systems. As a result, the 1D spectra generally remained unimodal. In addition, these 1D spectra had many of the attributes previously observed for fetch-limited wind seas. The nondimensional energy $\mathit{ϵ}=g^2{E}_{\mathrm{tot}}/U_{10}^4$ and nondimensional peak frequency $\nu =f_p{U}_{10}/g$ were approximately related by the same power law as observed for fetch-limited wind seas. In these relations, E_tot is the total energy of the waves, f_p is the spectral peak frequency, U₁₀ is the wind speed, and g is gravitational acceleration. The high-frequency face of the spectrum decayed at ~f⁻⁴, where f is frequency, and the energy level of this region of the spectrum was similar to fetch-limited spectra. The peak frequency values were, however, such that these waves were propagating faster than the local wind. This is consistent with these waves being generated in other parts of the storm and then propagating to the observation site. Young (2006) concluded that these observed spectral shapes were consistent with an energy balance in which nonlinear wave–wave interactions played a dominant role (Young and van Vledder 1993).

Hu and Chen (2011) considered directional spectra recorded at 12 buoys in the Gulf of Mexico during the passage of seven tropical cyclones (hurricanes). Consistent with Young (2006), they found the 1D spectra were generally unimodal with the exception of spectra far from the tropical cyclone center (r > 6R_m, where R_m is the radius to maximum winds) and spectra to the left of the storm center. Esquivel-Trava et al. (2015) considered spectra from four buoys in the Gulf of Mexico and Caribbean during the passage of 14 tropical cyclones (hurricanes). Again, they found that the 1D spectra were unimodal. However, their results indicated that the energy in the high-frequency tail of the spectrum was lower than proposed by Young (2006).

Collins et al. (2018) reported on observations from two Taiwanese buoys during the passage of three tropical cyclones (typhoons) plus a tropical storm. They confirmed that the spectra were commonly directionally skewed. However, in contrast to the previous measurements, they found that a significant percentage of the 1D spectra were bimodal.

In a more recent series of papers (Hwang 2016; Hwang and Fan 2017; Hwang et al. 2017; Hwang and Walsh 2016, 2018a,b), scanning radar altimeter data were used to examine directional properties of waves from six North American hurricanes. These data confirmed a relationship between ϵ and ν that is similar to that proposed by Young (2006) and that the spectra were composed of a mix of remotely generated waves and wind sea, which varied spatially within the tropical cyclone.

The analyses described above indicate that one of the defining features of the tropical cyclone wave field is that the spectrum consists of a combination of remotely generated wave energy and locally generated wind sea. The remotely generated wave systems have often been termed “swell” in previous studies. As will be shown clearly below, although they may not be locally generated, these remotely generated waves remain coupled to the local wind sea through nonlinear interactions. Therefore, to avoid confusion we have termed such components of the spectrum “remotely generated waves” rather than swell.

The observations described above provide some insight into the characteristics of tropical cyclone waves. However, even the combined dataset, represents observations in only a small number of tropical cyclones. To cover the full range of possible conditions, a much larger dataset is required. The analysis below describes such a dataset.

3. Tropical cyclone database

Our aim is to compile a large database of tropical cyclone track, wind field parameters, significant wave height, wind speed and spectral information. To do this, it has been necessary to obtain data from a number of different sources. The data used are outlined below.

d. Tropical cyclone databases

The above information was used to form two tropical cyclone databases consisting of: global analysis of altimeter tracks over tropical cyclones (U₁₀ and H_s) and North American analysis of tropical cyclone passes in the vicinity of buoys [U₁₀, H_s, and E(f, θ)].

The global altimeter database was constructed using the following steps:

• The altimeter data of Ribal and Young (2019) are available in 1° × 1° regions.

• The IBTrACS TC tracks are defined at 6-hourly values.

• For each IBTrACS TC position location, altimeter data in a region of 6.4° × 6.4° around the location were extracted.

• These data were then searched for altimeter passes with ±3 h of the TC position time.

• These passes were extracted and stored.

• Because there will be a time mismatch between the altimeter pass and the IBTrACS position location, the TC position was then interpolated to the same time as the altimeter pass (i.e., moved along the IBTrACS path by an amount equal to V_fmΔt, where Δt is the time mismatch (because IBTrACS data are typically every 6 h, Δt is a maximum of 3 h) and V_fm is the velocity of forward movement of the storm, calculated from successive location positions in IBTrACS.

• TC tracks and altimeter passes were rotated such that the TC was propagating toward the north, and Southern Hemisphere storms were “flipped” left–right. In this manner a common reference frame was formed with all storms propagating in the same direction and in the same hemisphere.

A similar approach was used for the buoy data:

• A region with a 5° radius was defined around each buoy.

• IBTrACS TC paths that transected this region were extracted.

• Data from the buoy covering the period of the TC transect were extracted and associated with the IBTrACS pass.

• Again, the TC track and each directional band of the directional spectra [Eq. (1)] were rotated, such that the TC is propagating toward the north. The data were then placed in a frame of reference moving with the TC. Note that this was not necessary with the altimeter passes, because the altimeter pass is “almost instantaneous” as compared with the V_fm of the TC.

Figure 1 shows the distribution of the buoys used in the database. Because there are a large number of buoys (see section 5), only buoys with data from 10 or more tropical cyclones are shown.

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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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The TC altimeter database consists of a total of 36 509 altimeter passes through 2730 TCs. Figure 2 shows histograms of the key parameters within this extensive database. Figure 2a shows the histogram of the maximum value of H_s for each altimeter pass, showing values up to H_s ≈ 15 m. Similarly, the histogram of the maximum values of U₁₀ for each altimeter pass in Fig. 2b shows values up to U₁₀ ≈ 50 m s⁻¹. These panels also show many low values of H_s and U₁₀ associated with passes far from the centers of tropical cyclones. Note that throughout this study we consider values of H_s and U₁₀ out to a spatial distance of 10R_m (10 radii to maximum winds). To better highlight the extreme data near the centers of tropical cyclones, Figs. 2a and 2b also include “breakout” plots of the tails of the histograms. This shows that, because of the size of the databases, there are many hundreds of altimeter passes with H_s > 10 m and U₁₀ > 30 m s⁻¹.

Summary of the altimeter TC database: (a) maximum significant wave height H_s 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 V_fm 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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Summary of the altimeter TC database: (a) maximum significant wave height H_s 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 V_fm 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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Summary of the altimeter TC database: (a) maximum significant wave height H_s 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 V_fm 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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Values of the central pressure p₀ range from 990 hPa to as low as p₀ ≈ 870 hPa in Fig. 2c. The velocity of forward movement V_fm was determined from successive positions of the TC center in the IBTrACS archive. The majority of the values of V_fm are between the values of 5 and 15 m s⁻¹ (Fig. 2d). The distribution of radius to gales R₃₄ is shown in Fig. 2e, and the distance from the TC center to the location of the maximum H_s measured by the altimeter pass is shown in Fig. 2f (expressed nondimensionally in terms of the radius to maximum winds R_m). As can be seen in Fig. 2f, the amount of usable data decreases near the tropical cyclone centers (i.e., distance/R_m < 2). This occurs because of the degradation of the altimeter radar returns in heavy rain. Despite this data loss, because of the very large database, there are still approximately 2500 altimeter passes within 2R_m of storm centers.

The corresponding set of distributions for the TC NDBC buoy database is shown in Fig. 3. This database includes a total of 2902 buoy records (TC passage past the buoy) from a total of 353 TCs. As noted above, the spatial distribution of these buoys is shown in Fig. 1. Although this is a much smaller dataset than the TC altimeter database, it is significantly larger than previous in situ buoy datasets and importantly includes directional spectral estimates. Only NDBC buoys with directional wave sensors were used for this analysis. The distributions of maximum U₁₀ and H_s (Figs. 3a,b) are remarkably similar to the TC altimeter database (Figs. 2a,b). Note, however, that the geographic distributions of the two databases are different (global, TC altimeter database; North America, TC buoy database). This becomes clear in the distributions of the TC parameters p₀ (Fig. 3c), V_fm (Fig. 3d), and R₃₄ (Fig. 3e). In particular, it is clear that the global database has consistently larger values of R₃₄, presumably because of the large number of Asian typhoons in the database, which are known to be spatially larger than North American hurricanes (Mok et al. 2018). It is also clear that the distribution of the distance to the recorded maximum H_s (Fig. 3f) does not decrease near the center of storms as for the TC altimeter database. This lends support to the assumption that this feature of the altimeter database was due to data loss in the altimeters due to high rain rates near the centers of TCs.

Summary of the in situ buoy TC database: (a) H_s for each transect of a TC, (b) U₁₀ for each transect, (c) p₀ of each storm in the database, (d) V_fm 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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Summary of the in situ buoy TC database: (a) H_s for each transect of a TC, (b) U₁₀ for each transect, (c) p₀ of each storm in the database, (d) V_fm 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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Summary of the in situ buoy TC database: (a) H_s for each transect of a TC, (b) U₁₀ for each transect, (c) p₀ of each storm in the database, (d) V_fm 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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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As noted above, our interest is in defining the spatial distribution of the wave field in tropical cyclones. Consistent with previous studies (Young 2006; Hu and Chen 2011; Collins et al. 2018) we have defined this region as being ±10R_m from the center of the storm. At distances greater than this definition, the background synoptic wind field often starts to dominate over the tropical cyclone vortex.

4. Altimeter observations

With all TCs rotated such that they propagate toward the north and by normalizing the spatial scale in terms of R_m, it is possible to pool the TC altimeter data. Figure 4 shows the altimeter tracks pooled in this manner with the color proportional to the observed H_s. The data have been partitioned based on p₀ [p₀ < 960 hPa (Fig. 4a); 960 < p₀ < 980 hPa (Fig. 4b)]. Figure 4a contains a total of 1993 passes, and Fig. 4b has 4562 passes. The total number of passes summed over these two figures (6555) is less than the total number of altimeter passes in the database (36 509), because only cases in which the IBTrACS archive also recorded values of R_m are included. A 2R_m × 2R_m grid was placed over the data shown in Fig. 4, and the median of the values within each grid region was determined. Note that the mean was also determined and produced very similar results. The median is preferred here, because it is less impacted by outliers in the large dataset. Contour plots of the median H_s and U₁₀, normalized by the maximum median value, are shown for each grid square in Fig. 5. Thus contour values range between 0 and 1. To give an indication of the magnitude of the values in these normalized plots, the maximum values of H_s and U₁₀ for each subplot are given in the figure caption.

Values of H_s 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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Values of H_s 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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Values of H_s 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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Contour plots of the spatial distributions of (left) H_s 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 $H_s^m$ = 14.5 m for (a), $U_{10}^m$ 47.3 m s⁻¹ for (b), $H_s^m$ = 15. 6 m for (c), $U_{10}^m$ = 52.7 m s⁻¹ for (d), $H_s^m$ = 15. 6 m for (e), and $U_{10}^m$ = 52.7 m s⁻¹ for (f).

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Contour plots of the spatial distributions of (left) H_s 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 $H_s^m$ = 14.5 m for (a), $U_{10}^m$ 47.3 m s⁻¹ for (b), $H_s^m$ = 15. 6 m for (c), $U_{10}^m$ = 52.7 m s⁻¹ for (d), $H_s^m$ = 15. 6 m for (e), and $U_{10}^m$ = 52.7 m s⁻¹ for (f).

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Contour plots of the spatial distributions of (left) H_s 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 $H_s^m$ = 14.5 m for (a), $U_{10}^m$ 47.3 m s⁻¹ for (b), $H_s^m$ = 15. 6 m for (c), $U_{10}^m$ = 52.7 m s⁻¹ for (d), $H_s^m$ = 15. 6 m for (e), and $U_{10}^m$ = 52.7 m s⁻¹ for (f).

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Spatial distributions of TC H_s and U₁₀ have previously been produced from model data (Tolman and Alves 2005) but we believe that these are the first such results obtained from actual measurements.

Vortex models of TC wind fields have been proposed by a number of authors (e.g., Holland 1980; Willoughby and Rahn 2004; Willoughby et al. 2006; Holland et al. 2010). These models define the characteristics of the TC wind field consisting of a calm eye, an asymmetric wind field, with the strongest winds to the right (Northern Hemisphere) of the eye and a wind field which decays exponentially from the area of maximum winds. The wind vectors spiral around the storm center in an anticlockwise direction with a slight inflow toward the center.

The results in Fig. 5 are partitioned by central pressure p₀. Figures 5a and 5b show moderately intense storms (960 < p₀ < 980 hPa), Figs. 5c and 5d show intense storms (p₀ < 960 hPa), and Figs. 5e and 5f show all data. The wind fields (Figs. 5b,d,f) show a number of the characteristics defined by the vortex models. The wind fields are asymmetric, with the peak wind occurring to the right of the storm center. The overall wind field is asymmetric. For instance, U₁₀ is larger for positive values of X/R_m (right of storm center) than for negative values (left of storm center). The wind speed increases as p₀ decreases (more intense storms). The clear vortex decay extends out to approximately 10R_m. The results do not show a calm eye, but the data were gridded at 2R_m, so this is not unexpected. Individual passes of the altimeters certainly show such structure.

The spatial distributions of H_s (Figs. 5a,c,e) largely mirror those of U₁₀. There is, however, a suggestion of slightly greater right–left asymmetry for significant wave height than for wind speed. This is consistent with the assumptions of an extended or trapped fetch model of TC wave generation (e.g., Young 2017). These extended fetch models suggest that the velocity of forward movement V_fm should play a role in the generation and distribution of H_s within the TC. We attempted to partition the data by both p₀ and V_fm, but, even with this extensive dataset, the data became too sparse and noisy to obtain reliable spatial distributions.

Note that in compiling these results only altimeter data with a quality flag of “1” (Ribal and Young 2019) were retained. That is, only data regarded as of “good” quality were used. However, visual inspection of altimeter passes showed that probably the largest quality issue associated with the results in Fig. 5 was due to IBTrACS TC location information. On a number of occasions, altimeter passes clearly passed through the calm eye of the TC, accurately positioning the eye. In a number of these cases, the IBTrACS position was in error by up to 100 km. The IBTrACS values of p₀ may also be in error, but we have no way of determining the magnitude of such errors. Also, we expect this to have less impact on the results in Fig. 5 than do TC location errors. Random errors in the position of the eye will tend to “smear” the wind speed and significant wave height median values shown in Fig. 5, this effect being more pronounced close to the centers of the storm, where the spatial gradients of wind speed and significant wave height are greatest.

5. In situ buoy observations

As in Fig. 4, Fig. 6 shows recorded buoy data during the passage of TCs near buoys. Note that all data are now in a frame of reference moving with the TC. Figure 6 contains a total of 2081 TC passes near buoys, a significantly small dataset than for the altimeters. Also, note that this total (2081) is less than the total number of in situ buoy records (2902) because, again, only data with recorded IBTrACS values of R_m are retained. However, the dataset is still much larger than any previous in situ datasets. Measurements of both H_s and U₁₀ were again binned into a regular grid of 3R_m × 3R_m and the median values found for each grid square. The coarser 3R_m binning was used for the buoy data, because tests with a 2R_m grid (as used for the altimeter data) were unacceptably noisy.

Values of H_s at in situ buoys during the passage of tropical cyclones. All storms were rotated to be propagating toward the north, and data were placed into a frame of reference moving with the storm.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Values of H_s at in situ buoys during the passage of tropical cyclones. All storms were rotated to be propagating toward the north, and data were placed into a frame of reference moving with the storm.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Values of H_s at in situ buoys during the passage of tropical cyclones. All storms were rotated to be propagating toward the north, and data were placed into a frame of reference moving with the storm.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Figure 7 shows color contour plots for the median values of both H_s (Fig. 7a) and U₁₀ (Fig. 7b). As with the altimeter data in Fig. 5, the contoured values were normalized by the maximum median value and the maximum values are provided in the figure caption. Because there are significantly fewer data than for the altimeters, the data were not partitioned by p₀. Although the distributions of median H_s and U₁₀ are noisier than the corresponding altimeter results (Fig. 5), they show the same general attributes. That is, the maximum values of H_s and U₁₀ are to the right of the storm center, the wind and wave height fields decay with radius (X/R_m and Y/R_m), and there is greater right–left asymmetry for H_s than U₁₀.

Contour plots of the spatial distributions of (a) H_s 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. Maximum values are $H_s^m$ = 16.9 m for (a) and $U_{10}^m$ = 47.2 m s⁻¹ for (b).

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Contour plots of the spatial distributions of (a) H_s 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. Maximum values are $H_s^m$ = 16.9 m for (a) and $U_{10}^m$ = 47.2 m s⁻¹ for (b).

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Contour plots of the spatial distributions of (a) H_s 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. Maximum values are $H_s^m$ = 16.9 m for (a) and $U_{10}^m$ = 47.2 m s⁻¹ for (b).

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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The peak wave direction ${\overline{\theta }}_w$ (mean direction at the spectral peak frequency) was determined from the NDBC data for each of the buoy observations, and the values were again binned into 2R_m × 2R_m bins. The mean values for each bin are shown in Fig. 8a, with the corresponding mean wind speeds $\overline{{\theta }_U}$ given in Fig. 8b. As expected, $\overline{{\theta }_U}$ 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 ${\overline{\theta }}_w$ (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, $\overline{{\theta }_U}$ and ${\overline{\theta }}_w$ 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, ${\overline{\theta }}_w$ 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 (presumably aligned with the local wind direction $\overline{{\theta }_U}$) and remotely generated waves, which were generated in the intense wind regions near the center of the TC, have “outrun” the TC and now appear ahead of the storm as remotely generated waves. The left-rear quadrant shows a confused situation for ${\overline{\theta }}_w$, indicating a complex spectral form with possible multiple peaks.

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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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As noted above, these spatial distributions are consistent with previous in situ and satellite data but provide many more data than were previously available and hence provide a more comprehensive distribution of mean wind and peak wave direction vectors, covering the full spatial field out to ±10R_m. Figure 8a, shows the occasional vector of ${\overline{\theta }}_w$ that appears to be inconsistent with its neighbors. In all of these cases, they are associated with 2R_m grid squares where there are few in situ observations, despite the overall size of the dataset.

6. 1D spectral form

Visual inspection of the recorded 1D spectra in the TC in situ buoy database, indicated that the vast majority were unimodal. Therefore, following Young (2006), the 1D spectra were modeled by the generalized Joint North SeaWave Project (JONSWAP) form proposed by Young and Verhagen (1996):

In the spectral form in Eq. (4), g is gravity and the spectral parameters are a “Phillips” scale parameter β, the spectral peak frequency f_p, the high-frequency spectral decay exponent n, and the spectral width parameter σ. This form differs from the classic JONSWAP form in that the exponent n is not set to a constant (e.g., n = −5 for JONSWAP). The nondimensional energy ϵ and nondimensional frequency ν were then determined as

where $E_{\mathrm{tot}}={\displaystyle \int E\left(f\right)\hspace{0.17em}df}$.

Figure 9 shows plots of the spectral parameters n (Fig. 9a), β (Fig. 9b), and γ (Fig. 9c), as a function of the inverse wave age U₁₀/C_p, where C_p is the phase speed of components at the spectral peak frequency f_p. The vertical line shown in each of the panels of Fig. 9 is drawn at U₁₀/C_p = 0.83. This is the swell–wind-sea limit proposed by Donelan et al. (1985). Values of U₁₀/C_p < 0.83 will not be receiving active local wind forcing at the spectra peak and are usually regarded as swell, or remotely generated waves in the present context. It is clear in Fig. 9 that a significant proportion of the spectra is in this category and that remotely generated wave energy is an important component of many observed spectra. The importance of remotely generated waves in defining the wave field is also clear in the mean wave directions shown in Fig. 8a.

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/C_p (the horizontal solid line is the mean: n = −4.68, and the dashed line is drawn at n = −4); (b) as in (a), but for spectral level parameter β {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 γ [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/C_p = 0.83.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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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/C_p (the horizontal solid line is the mean: n = −4.68, and the dashed line is drawn at n = −4); (b) as in (a), but for spectral level parameter β {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 γ [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/C_p = 0.83.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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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/C_p (the horizontal solid line is the mean: n = −4.68, and the dashed line is drawn at n = −4); (b) as in (a), but for spectral level parameter β {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 γ [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/C_p = 0.83.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Fetch-limited wind-sea spectra typically have n from approximately −4 to −5 (Hasselmann et al. 1973; Donelan et al. 1985). The data here scatter around these values, with a mean value of n = −4.68, shown in Fig. 9a. The values of peak enhancement factor γ are shown in Fig. 9c, along with the relationship proposed by Donelan et al. (1985) for fetch-limited data. Although the values of γ are comparable to fetch-limited results, they are consistently larger. This result indicates the spectra are more peaked than in fetch-limited cases. Whereas the present data for n, γ, and σ (not shown) show no clear trend as a function of U₁₀/C_p, consistent with fetch-limited wind seas, there is a clear relationship between β and U₁₀/C_p in Fig. 9b. Donelan et al. (1985) also found such a functional dependence for fetch-limited conditions, which they approximated by β = 0.006(U₁₀/C_p)^0.55. Young (2006) reported TC data that are generally consistent with this relationship. The present extensive dataset shows a relationship that increases more rapidly as a function of U₁₀/C_p, which can be approximated by

This relationship (and the present data) yield values of β that are generally consistent with the fetch-limited results for U₁₀/C_p ≈ 1 but yield values of β that are less than the fetch-limited results for values of U₁₀/C_p < 1. That is, the high-frequency tail of the TC wave spectrum has less energy than is typically seen in fetch-limited wind seas. A close examination of the limited data points of Young (2006) (Fig. 10 of Young 2006), shows that the Young (2006) data are actually also consistent with (7).

Figure 10 shows the nondimensional energy ϵ as a function of the nondimensional peak frequency ν, together with the fetch-limited relationship of Donelan et al. (1985) and the wind-sea–swell demarcation of ν = 0.13. Again, many of the data are for ν < 0.13 (i.e., remotely generated waves) and are in reasonable agreement with the fetch-limited functional dependence. Closer examination of the data does, however, show that for TC conditions ϵ is slightly smaller than the fetch-limited result for ν < 0.13. This is consistent with the smaller values of β also seen in Fig. 9c for TC conditions. That is, there is less energy in the tail of the spectrum (lower β) and therefore lower total energy (E_tot and ϵ).

Nondimensional energy ϵ versus nondimensional peak frequency ν from in situ buoy data during the passage of tropical cyclones. The dashed line through the data is the fetch-limited result of Donelan et al. (1985). The vertical solid line is the demarcation between swell and wind sea: ν = 0.13.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Nondimensional energy ϵ versus nondimensional peak frequency ν from in situ buoy data during the passage of tropical cyclones. The dashed line through the data is the fetch-limited result of Donelan et al. (1985). The vertical solid line is the demarcation between swell and wind sea: ν = 0.13.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Nondimensional energy ϵ versus nondimensional peak frequency ν from in situ buoy data during the passage of tropical cyclones. The dashed line through the data is the fetch-limited result of Donelan et al. (1985). The vertical solid line is the demarcation between swell and wind sea: ν = 0.13.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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7. Directional spectra

The four Fourier directional moments recorded by the in situ wave buoys are used to estimate the directional spectrum E(f, θ). This is often described by the 1D spectrum, E(f) and the directional spreading function D(f, θ) [Eq. (1)]. Figures 11 and 12, show the spatial wave field divided into octants. For reference, we number these octants in an anticlockwise direction from the x axis. Therefore, octant 1 is ENE, octant 2 is NNE, octant 3 is NNW, …, octant 8 is ESE. For each of these octants, representative values of mean wave direction, ${\overline{\theta }}_w$ (dashed arrow) and mean wind direction, $\overline{{\theta }_U}$ (solid arrow) are shown in Figs. 11 and 12. Figure 11 also shows representative 1D spectra E(f) at each of these points. Figure 12 shows representative directional spreading functions D(f, θ) at each point. That is, in both Figs. 11 and 12 we show typical examples of the spectra rather than averaging all of the spectra. Such an averaging process would tend to smear the spectra, resulting in excessive spreading in both frequency and direction. Examination of the 1D spectra in Fig. 11 shows that, with the exception of octant 4 (WNW) and octant 5 (WSW), the spectra are unimodal and very similar in shape to fetch-limited wind seas. However, the values of peak frequency are such that most of these spectra have waves at the spectra peak that are propagating faster than the local wind (remotely generated waves). Octants 4 and 5 (left of storm center) correspond to the region of confused ${\overline{\theta }}_w$ in Fig. 8a.

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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Spatial distributions of the directional spreading function D(f, θ) 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, θ) [Eq. (1)] are shown for each octant. The colored contours show D(f, θ) 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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Spatial distributions of the directional spreading function D(f, θ) 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, θ) [Eq. (1)] are shown for each octant. The colored contours show D(f, θ) 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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Spatial distributions of the directional spreading function D(f, θ) 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, θ) [Eq. (1)] are shown for each octant. The colored contours show D(f, θ) 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.

Citation: Journal of Physical Oceanography 50, 8; 10.1175/JPO-D-20-0020.1

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Although the 1D spectra look very simple (unimodal), the directional spreading functions D(f, θ) are very complex, with most showing significant directional skewing. In each octant, the high-frequency components of the spectrum (f > 2.5f_p) are aligned with the local wind (solid vertical lines in Fig. 12). However, the components at the spectral peak frequency represent remotely generated waves that have been generated earlier in the intense wind regions near the center of the TC and have now propagated to the site (dashed vertical lines in Fig. 12). In octant 8 (ESE) the wind and spectral peak wave direction are most aligned and hence D(f, θ) has little directional skew. As one rotates anticlockwise from octant 8 (ESE) the angle between the wind and spectral peak wave directions gradually increases until it is in excess of 120° 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 f_p to 5f_p. 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 directional buoy measurements during the passage of tropical cyclones. The satellite data confirm that both the wind and wave fields in tropical cyclones are asymmetric, with stronger winds/larger waves to the right of the eye (Northern Hemisphere) of the storm. The wave field has a greater degree of asymmetry than the wind field. This occurs because the generation of large waves requires both strong winds and a substantial fetch over which the wind must blow. In the case of a translating tropical cyclone, an extended fetch develops to the right of the storm where the direction of propagation of the storm is approximately aligned with the wind direction. Hence, waves generated in this region move forward with the translating storm and waves can remain in the intense wind region for an extended duration. The opposite happens to the left of the storm where the wind direction is in the opposite direction to the direction of translation of the storm. Therefore, waves quickly propagate out of the intense wind region. As a result, the asymmetry of the wave field is greater than the wind field.

It is believed that these are the first spatial distributions showing the full distribution of significant wave height out to 10R_m. Previous studies addressing such distributions have relied on model data (e.g., Young and Vinoth 2013). Such studies are naturally limited by the reliability of the model physics under the complex forcing within tropical cyclones.

Consistent with a range of previous studies, the extensive directional buoy data in this study show that much of the tropical cyclone wave field is dominated by remotely generated waves originating near the center of the tropical cyclone. The circulating vortex generates waves in this region which radiate out from the center of the storm in all directions. As a result, in most regions of the tropical cyclone wave field, the spectrum consists of remotely generated waves radiating out from the center of the storm, together with locally generated wind sea. These wave systems can be directionally separated by more than 90°. The phase speeds of the remotely generated waves are such that, in most cases, they propagate faster than the local wind speed. This means the remotely generated waves are not actively forced by the local wind. Despite this, the remotely generated waves and wind sea do not exist as separate systems. Rather there is a continuum of spectral energy from the high-frequency wind sea to the low-frequency remotely generated waves. This continuum of energy is, however, directional skewed.

Because of the size of the in situ buoy dataset, it has been possible to examine the directional spectrum over the full tropical cyclone waves field (±10R_m). This contrasts with previous studies (Young 2006; Hu and Chen 2011; Collins et al. 2018) that have only been able to obtain data from parts of the spatial field.

We typically assume the wave spectral energy balance is represented by the processes of atmospheric input S_in, nonlinear wave–wave interactions S_nl, and whitecap dissipation S_ds (S_tot = S_in + S_nl + S_ds; Komen et al. 1984). At the spectral peak (remotely generated waves peak), S_in ≈ S_ds ≈ 0, because the waves are propagating faster than the local wind and because the mean squared slope of the remotely generated waves is small. Therefore, the spectral balance in the vicinity of the spectral peak will be S_tot ≈ S_nl. The nonlinear term S_nl is typically characterized by a transfer of energy from higher frequency to lower frequency (Young and van Vledder 1993). In the present case, this continual feed of energy to lower frequencies appears to result in a cascade of energy from the locally generated wind sea to the remotely generated waves peak. This results in a smooth variation of the spectrum with a unimodal one-dimensional form.

The nonlinear terms result in a one-dimensional spectral form very similar to fetch-limited waves. The high-frequency face decays ∝ fⁿ, where n ranges between −4 and −5. Because of the lack of significant wind input near the spectral peak, the energy in the high-frequency face of the spectrum is lower than for fetch-limited cases and the total energy is also lower than one would normally expect for waves with the observed (low) peak frequencies.

Although the datasets used in this analysis are very large, there are still clear limitations resulting from the finite amounts of data. It was necessary to bin the altimeter data at a resolution of 2R_m × 2R_m and the buoy data at an even coarser resolution of 3R_m × 3R_m. As a result, much of the detailed structure close to the eye of the storm is lost. We have assumed that the spatial structure can be normalized by the radius to maximum winds R_m. This is a spatial scaling commonly used. However, there are other choices which may also be appropriate, including R₃₄. The present data do not allow one to explore these differences in detail. There was a strong desire to partition the data by both tropical cyclone intensity (p₀ or V_max) and translation speed V_fm, as model data (Young and Vinoth 2013) indicate these combined parameters impact both the magnitude of H_s and the spatial structure. Despite the size of the datasets, this was not possible, with such data segmenting resulting in unacceptably noisy spatial distributions. In Fig. 5 some attempt was made to examine the impact of p₀; however, this is not a comprehensive analysis. Last, the IBTrACS data are often limited in their accuracy. This clearly results in errors in accurate determination of TC tracks and tropical cyclone wind field parameters. These are, however, the best available data for such large datasets.