Hurricane Sea Surface Inflow Angle and an Observation-Based

Home , Hurricane Lili (1996), Inflow (meteorology), Radius of maximum wind

NOVEMBER 2012 Z H A N G A N D U H L H O R N 3587

Hurricane Sea Surface Inﬂow Angle and an Observation-Based Parametric Model

JUN A. ZHANG Rosenstiel School of Marine and Atmospheric Science, University of Miami, and NOAA/AOML/Hurricane Research Division, Miami, Florida

ERIC W. UHLHORN NOAA/AOML/Hurricane Research Division, Miami, Florida

(Manuscript received 22 November 2011, in ﬁnal form 2 May 2012)

ABSTRACT

This study presents an analysis of near-surface (10 m) inflow angles using wind vector data from over 1600 quality-controlled global positioning system dropwindsondes deployed by aircraft on 187 flights into 18 hurricanes. The mean inflow angle in hurricanes is found to be 222.6862.28 (95% confidence). Composite analysis results indicate little dependence of storm-relative axisymmetric inflow angle on local surface wind speed, and a weak but statistically significant dependence on the radial distance from the storm center. A small, but statistically significant dependence of the axisymmetric inflow angle on storm intensity is also found, especially well outside the eyewall. By compositing observations according to radial and azimuthal location relative to storm motion direction, significant inflow angle asymmetries are found to depend on storm motion speed, although a large amount of unexplained variability remains. Generally, the largest storm- 2 relative inflow angles (,2508) are found in the fastest-moving storms (.8ms 1) at large radii (.8 times the radius of maximum wind) in the right-front storm quadrant, while the smallest inflow angles (.2108) are found in the fastest-moving storms in the left-rear quadrant. Based on these observations, a parametric model of low-wavenumber inflow angle variability as a function of radius, azimuth, storm intensity, and motion speed is developed. This model can be applied for purposes of ocean surface remote sensing studies when wind direction is either unknown or ambiguous, for forcing storm surge, surface wave, and ocean circulation models that require a parametric surface wind vector field, and evaluating surface wind field structure in numerical models of tropical cyclones.

1. Introduction In situ near-surface wind vector observations in tropical cyclones are available from global positioning system Estimating hurricane surface wind distributions and (GPS) dropwindsondes deployed by research and re- maxima is an operational requirement of the National connaissance aircraft (Hock and Franklin 1999). Glob- Hurricane Center (NHC), as coastal watches and warn- ally, however, direct measurements of sea surface winds ings are issued based on storm impacts at landfall, in- in tropical cyclones are still highly infrequent, so methods cluding storm surge. Fairly recent development of have been developed to estimate surface winds from wind reliable instrumentation has resulted in more accurate data observed at higher altitudes by research aircraft estimates of tropical cyclone surface wind speed and (e.g., Franklin et al. 2003; Powell et al. 2009), from sat- direction. Currently, remotely sensed surface wind ellite imagery (Velden et al. 2006), and from surface speed observations in tropical cyclones are provided pressure observations (Knaff and Zehr 2007). In com- by spaceborne microwave sensors (Katsaros 2010) and parison to surface wind speed data, wind direction in- airborne stepped-frequency microwave radiometers formation is exceedingly sparse. (SFMR; Uhlhorn and Black 2003; Uhlhorn et al. 2007). Mapping the two-dimensional surface wind vector ﬁeld in tropical cyclones has several important applications. First, storm surge models are generally forced by Corresponding author address: Dr. Jun Zhang, NOAA/AOML/ Hurricane Research Division, Universtiy of Miami/CIMAS, 4301 surface winds, which not only require the wind magni- Rickenbacker Causeway, Miami, FL 33149. tude but also the wind direction. It has traditionally been E-mail: [email protected] a standard practice to use axisymmetric parametric wind

DOI: 10.1175/MWR-D-11-00339.1

Ó 2012 American Meteorological Society Unauthenticated | Downloaded 09/24/21 12:31 AM UTC 3588 MONTHLY WEATHER REVIEW VOLUME 140 models to force storm surge models (e.g., Peng et al. of the ocean response to hurricanes when a parametric 2006; Rego and Li 2009). These parametric wind models, wind model is used to force an ocean model (e.g., Price such as the Sea, Lake and Overland Surges from Hur- 1983; Yablonsky and Ginis 2009; Halliwell et al. 2011). ricanes (SLOSH) wind model (Phadke et al. 2003), Because of the ubiquitous cyclonic flow near the sur- Holland’s model (Holland 1980; Holland et al. 2010), face in tropical cyclones, documentation of observed and Willoughby’s model (Willoughby et al. 2006) esti- surface wind directions is typically described in terms of mate the radial profile of axisymmetric wind speed or surface inflow angles, although such studies are rela- tangential wind component without wind direction in- tively sparse. Numerical studies (e.g., Kepert 2010a,b; formation. The wind direction is then arrived at by Bryan 2012) often cite the observational result pre- applying a constant inflow angle, and an asymmetry in sented by Powell (1982, hereafter P82) from data ob- wind speed is simply assumed due to storm forward tained in Hurricane Frederic (1979). Earth-relative motion. Some storm-surge studies (e.g., Westerink et al. inflow angles over the open ocean were found by P82 to 2008) have utilized the National Oceanic and Atmo- vary from outflow of 1128 to inflow of 2558, with greater spheric Administration (NOAA)/Hurricane Research inflow in the right-rear (RR) quadrant and weaker in- Division (HRD) real-time Hurricane WIND analysis flow in the left-front (LF) quadrant, and with a mean value system (H*WIND) product (Powell et al. 1998), which of 2228. Powell et al. (2009) examined a large sample of estimates surface wind direction applied to SFMR wind dropwindsonde data and found a mean inflow angle of speeds as simply a constant angle subtracted from the 2238, although details regarding asymmetric structure flight-level wind direction (M. Powell 2005, personal were not investigated. Note that the original analytical communication). treatment of tropical cyclone inflow was presented by Second, remotely sensed wind direction accuracy in Malkus and Riehl (1960), who suggested an axisymmetric tropical cyclones, particularly in the high-wind inner- average inflow angle of 2208 to 2258 outside of the eye- core region, is often highly degraded as a result of sev- wall, but decreasing to less than 258 at the radius of eral physical factors. Nadir-viewing passive microwave maximum wind (Rmax), was consistent with boundary radiometers (e.g., SFMR) are insensitive to wind di- layer energy constraints. This conclusion also depended on rection and spaceborne wide-swath imagers suffer from knowledge of the surface exchange coefficients of mo- resolution and rain absorption artifacts (e.g., Connor mentum and moist enthalpy, and boundary layer depth, and Chang 2000). Active microwave sensors such as which were not very well known at the time (e.g., French scatterometers may saturate, are attenuated in heavy et al. 2007; Zhang et al. 2008, 2009; Zhang 2010; Haus precipitation, and are also limited by spatial resolution et al. 2010; Kepert 2010a; Smith and Montgomery 2010). for tropical cyclone applications, particularly in the The purpose of this paper is to investigate the mean inner-core region (Brennan et al. 2009). The resolution and asymmetric structure of near-surface inflow angle limitations can be overcome by synthetic aperture radar (at an altitude of 10 m) over a broad range of tropical (SAR); however, SARs typically provide only a single cyclone characteristics, including storm motion, in- view and therefore determining the wind direction is an tensity, and size, utilizing the extensive database of GPS ambiguous problem (Shen et al. 2009). In addition, it is dropwindsonde wind vector observations. Based on the often assumed that the surface roughness elements that data analysis results, a simple parametric model of the provide the radar backscatter mechanism are aligned mean plus wavenumber-1 asymmetric inflow angle field with the wind direction, which may not always be ac- is developed and tested. Section 2 describes the data curate (e.g., Donelan et al. 1997; Drennan et al. 1999; sources, quality control, and analysis methodology. In Grachev et al. 2003; Drennan et al. 2003). section 3, analysis results are presented for both mean Third, predicting tropical cyclone intensity is viewed and asymmetric fields. Section 4 describes the para- as a coupled atmosphere–ocean problem, thus under- metric model development, evaluation, and case-study standing the air–sea interaction and ocean feedbacks comparisons and section 5 summarizes the results and to hurricanes is of paramount importance (e.g., Black discusses the applications of the parametric model. et al. 2007; Shay et al. 1989; Jacob et al. 2000; Shay and Uhlhorn 2008; Jaimes and Shay 2010; Uhlhorn and Shay 2. Data and quality control 2012). Accurately specifying the surface wind direction may benefit wave forecast models, which have been in- GPS dropwindsonde data used in this study were creasingly inserted into the air–sea interface in coupled collected on 187 hurricane research and reconnaissance model applications (e.g., Moon et al. 2007; Zhao and flights in 18 hurricanes (Table 1) between 1998 and 2010. Hong 2011). Accurate representation of the surface Detailed description of dropwindsonde instrumentation wind direction can also help improve our understanding and data accuracies can be found in Hock and Franklin

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TABLE 1. Storm information and number of ﬂights and drop- sondes. Numbers in parentheses are sondes held out of model development for validation.

Storm intensity No. of No. of 2 Storm name Year range (m s 1) flights sondes Bonnie 1998 50–52 5 85 Danielle 1998 33–37 6 107 Georges 1998 35–68 7 87 Bret 1999 52–57 1 18 Floyd 1999 47–63 4 39 Fabian 2003 54–62 9 95 Isabel 2003 67–72 9 51 Frances 2004 45–64 6 (7) 79 (88) Ivan 2004 54–72 10 (17) 101 (172) Jeanne 2004 44–54 2 (8) 13 (66) Dennis 2005 36–62 (14) (110) Katrina 2005 52–77 2 54 Rita 2005 33–77 (17) (187) Dean 2007 38–77 (7) (54) Gustav 2008 33–62 (11) (87) Paloma 2008 34–59 5 43 FIG. 1. Storm-relative two-dimensional distribution of drop- Bill 2009 48–59 12 (4) 144 (15) windsonde surface observation locations between 0 and 12.5r*. Earl 2010 38–64 19 (5) 183 (46) Cross- and along-track positions are normalized by the radius of Totals 97 (90) 1099 (825) maximum wind at the time of observation. The arrow indicates the storm motion direction. Range rings are plotted every 2.5r*, and radials are every 458. (1999). The near-surface fall speed of a dropwindsonde 2 is 12–14 m s 1, while the typical sampling rate is 2 Hz, yielding an approximately 5–7 m vertical sampling. However, in no case are winds extrapolated to the 10-m Note that the 5-s filter, which is typically applied in the level if a sonde terminates above this level. Data loca- postprocessing, effectively reduces the vertical resolu- tions are transformed to a polar coordinate system tion to roughly an order of magnitude lower than the measured radially (r) from the center and azimuthally original sampling. The accuracy of the horizontal wind clockwise from storm motion direction (u). The storm 2 speed measurements is ;0.5 m s 1. The dropwindsonde center positions have been determined using storm data have been postprocessed and quality controlled tracks produced by NOAA/HRD based on the flight- using the HRD Editsonde (Franklin et al. 2003) soft- level wind data (Willoughby and Chelmlow 1982, ware for the data before 2005. Data obtained after 2005 hereafter WC82). Radial distances are normalized by have been postprocessed using the National Center for the estimated radius of maximum wind speed, Rmax, Atmospheric Research (NCAR) Atmospheric Sound- determined from approximately concurrent SFMR ing Processing Environment (ASPEN) software. Recent surface wind observations (r* 5 r/Rmax). The Rmax value studies have indicated little difference exists between represents an average of individually observed wind the Editsonde- and ASPEN- processed wind data (e.g., maxima along each radial leg for a single flight. Data are Barnes 2008). Although there have been several minor reasonably evenly azimuthally distributed, as shown in improvements to the dropwindsonde design and pro- Fig. 1. Figure 2 shows the radial distribution of the cessing since the original documentation (Hock and number of observations. After normalizing the radial

Franklin 1999), overall data accuracy has not changed coordinate by Rmax (Fig. 2b) we find the largest number significantly to impact results in this study. of sondes is clustered around r* 5 1, corresponding to To study the near-surface inflow angle, we only use eyewall deployments, with a secondary peak in the dropwindsonde data with wind vector measurements number of sondes deployed near the storm center. near the surface (#10 m), totaling 1924 sondes. The Dropwindsonde data are analyzed and grouped in horizontal Cartesian wind-vector components (u, y) are a composite framework. The composite analysis method linearly interpolated to the 10-m level if not directly has been used in previous studies investigating the hur- observed at that level. All sondes report winds to the ricane inner-core structure (e.g., Frank 1984; Rogers surface, although data dropouts over a profile may exist et al. 2012), vertical wind profile structure (e.g., Franklin when GPS satellite tracking is temporarily degraded. et al. 2003; Powell et al. 2003), surface layer air–sea

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To calculate the storm-relative inﬂow angle, the storm motion vector is removed from the dropwindsonde- observed Cartesian wind vector before transforming to radial and tangential components relative to the storm center location. The angle calculation is restricted to the standard arctangent 6908 half-plane, eliminating the

possibility of anticyclonic ﬂow (yt , 0). The frequency distribution of aSR for the initial sample is shown in Fig. 4a. The distribution is super-Gaussian (normalized kurtosis k 512.7), which is a primarily a result of nu- merous outliers exhibiting unrealistically large outﬂow. These measurements are mostly found very close to the estimated storm center, and are likely due to errors in the wind-determined storm center location, along with the possibility of multiple wind minima existing (Nolan

FIG. 2. Radial distribution of dropwindsonde counts per bin as and Montgomery 2000). By simply eliminating obser- , a a function of (a) real distance and (b) normalized distance. Counts vations where r* 0.5, the frequency distribution of SR are per bin widths of (a) 20 km and (b) 0.5r*. becomes more normal (Fig. 4b), suggesting an improved representation of the expected inflow angle in tropical thermal structure (e.g., Cione et al. 2000; Cione and cyclones. y y a Uhlhorn 2003), and boundary layer structure (Zhang Because the accuracy of r and t, and therefore SR, et al. 2011a). The advantage of the composite analysis depend on the storm center position accuracy, the im- method is that it helps to fill data voids and provides pact of the storm center position error on the computed a general characterization of the fields under investiga- inflow angle is briefly examined. WC82 claimed that the tion. The most important drawback to compositing is storm center based on flight-level wind observations that it tends to smooth the data from a large number of can be determined to around 3-km accuracy, although storms that may not be similar. The success of a com- Kepert (2005) showed that the center position error posite analysis depends on the similarity of the events within the hurricane boundary layer for a translating studied, thus we initially restrict our analysis to data storm can easily be 5 km or more using the WC82 21 method. The impact of storm center position error on collected in hurricanes (Vmax . 33 m s , where Vmax is the maximum 1-min wind speed), and radially outward the uncertainty of inflow angle is simulated by assuming the storm center position is in error (one standard de- to r* 5 12.5. For each dropwindsonde, Vmax and storm viation, s) by 5 km, and a normal distribution of inflow speed (Vs) and direction are obtained from the 6-hourly best-track database (Jarvinen et al. 1984) interpolated to angles is generated by Monte Carlo simulation of 1000 the time of observation. The frequency distributions of realizations. Figure 5 shows the simulated inflow angle error (normalized by the sample s 5 18.38 as indicated Vmax, Rmax, and Vs indicate that observations represent a broad spectrum of storms (Fig. 3). Storm intensities in Fig. 4b) versus r*, where the sample median (mini- 21 mum) Rmax of 32 (10) km is used to normalize the radial range between 33 , Vmax , 77 m s , sizes between 10 , distance. For comparison, a 2-km center position error- Rmax , 72 km, and motion speeds between 0.8 , Vs , 2 12.3 m s 1. The median storm intensity for the whole induced inflow angle error is shown, representing the 21 estimated accuracy of the translating pressure center sample is Vmax 5 56.7 m s (Saffir–Simpson category 3), tracking method proposed by Kepert (2005). Except radius of maximum wind is Rmax 5 31.8 km, and storm 2 motion speed is V 5 5.5 m s 1. for the smallest storms, a 5-km center position error s 8 The inflow angle1 (a) is defined as the arctangent of induces an inflow angle error smaller than 18.3 outside of r* 5 1, and would likely be buried in the natural the ratio of radial (yr) to tangential (yt) wind compo- 21 surface wind variability. Some improvement to the nents [a 5 tan (yr/yt)]. Note that storm-relative inflow accuracy could be made by utilizing the pressure-based angle (aSR) is used exclusively throughout this study. method, especially close to the center in small storms, but it appears that the vast majority of data would not be strongly impacted by the error in storm center 1 Inflow is defined as y , 0, although we still refer to ‘‘inflow r specification. angle’’ when outflow (yr . 0) occurs. Also, a ‘‘larger inflow angle’’ or the like will generally indicate a more negative value throughout Although data are included all the way into the esti- this article. mated center, inflow angles 62s away from the sample

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21 FIG. 3. Frequency distribution of dropwindsondes according to the corresponding maximum wind speed (Vmax,ms ), radius of maximum 21 wind speed (Rmax, km), and storm motion speed (Vs,ms ). mean of 222.68 indicated in Fig. 4b are rejected as highly potentially important detail in the inflow angle vari- unrepresentative, which restricts the range of acceptable ation near the eyewall will be investigated in future 10-m level, storm-relative inflow angles between 259.28 work as our focus here is on the overall structural and 114.08, resulting in a final working sample of 1613 variability of the inflow angle throughout a tropical independent, quality-controlled, observations between 0 , r* , 12.5. Note that the sonde count of 1538 indicated in Fig. 4b does not contain data where r* , 0.5, which were excluded for quality-control purposes as mentioned earlier. Hereafter, analysis of inflow angles will consider this full sample; however, the sample will be split prior to developing the proposed parametric inflow angle model, such that 621 observations (;38%) from various storms are held out for model evaluation purposes. This validation sample will be shown not to be statistically significantly different from the devel- opmental sample; therefore, both datasets may be re- garded as random samples of the population.

3. Data analysis results a. Axisymmetric distribution Figures 6a,b show the storm-relative inflow angle, aSR, as functions of local storm-relative wind speed (U10SR) and r*, respectively. Linear regression of aSR on U10SR (Fig. 6a) indicates little dependence of the angle on wind speed. In contrast to wind speed independence, a significant increase in aSR with the radial distance from the center is indicated (Fig. 6b). Between 0 , r* , 12, the slope of the inflow angle dependence on radial distance is significant at the 95% confidence level (20.53860.238 per r* units; i.e., significantly different FIG. 4. Frequency distribution of storm-relative inflow angle from zero slope based on a Student’s t test). This sig- a 8 a ( SR, ) for (a) full sample and for (b) sondes radially outward of nificant dependence of SR on r* appears especially well r* 5 0.5. Sample size (n), mean (m), standard deviation (s), and pronounced between the eyewall and just outside of the normalized kurtosis (k) are indicated. Dashed lines represent eyewall as indicated by the bin-averaged values. This normal distributions for the given m and s of each sample.

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FIG. 5. Normalized inflow angle error due to inaccurate storm center location as a function of radial distance from center. Error is standard deviation normalized by the sample standard deviation (s 5 18.438) indicated in Fig. 4b. Solid lines are for 5-km error and dashed lines are for 2-km error. Black lines are for Rmax 5 32 km, representing the sample median, and gray lines are for Rmax 5 10 km, representing the sample minimum. cyclone. With reasonably even azimuthal sampling, this result (Fig. 6b) describes the axisymmetric mean inflow angle radial profile.

Further stratiﬁcation of aSR among weak/strong, small/large, and fast/slow-moving storms is shown in

Figs. 7a–f. The data are divided into Vmax, Rmax, and Vs groups according to their respective sample median values, as previously stated. Both storm size and motion FIG. 6. Storm-relative inflow angle (aSR, 8) as a function of 21 speed appear to have little relationship with the inflow (a) 10-m storm-relative wind speed (U10SR,ms ) and (b) radius angle. Although small, the axisymmetric inflow angle normalized by the radius of maximum wind speed (r*, dimen- has a statistically significant dependence on the storm sionless). Solid and dashed lines are linear regression fits and 95% confidence intervals, respectively, and points with error bars are intensity, particularly at large radii where inflow angles bin averages and 95% confidence intervals. Total number of ob- are on average ;58 larger for more intense storms servations (n), correlation coefficient (r2), and trend lines with 95% (Fig. 7d). Based on this result, we further stratify the confidence intervals for parameters are indicated. inflow angle according to five intensity groups (Vmax 5 33–42.5, 37.5–52.5, 47.5–62.5, 57.5–72.5, and 67.5– 2 77 m s 1) and six radial band groups (r* 5 0–1, 0.5–1.5, (e.g., Mallen et al. 2005; Willoughby et al. 2006), and 1–5, 2.5–7.5, 5–10, and 7.5–12.5). Bins are partially model simulations show that more peaked storms have overlapped to provide continuity among groups, at the stronger inflow outside the radius of maximum wind expense of smoothing possibly relevant finescale details. speed (e.g., Kepert and Wang 2001; Kepert 2006a,b), Figures 8a–f show individual observations, binned av- suggesting that the increased inflow angle at large radius erages, and 95% confidence intervals of aSR versus Vmax is consistent with the dynamics. for the six radial bands, along with linear regression fits. b. Asymmetric distribution An increasing slope of aSR versus Vmax, as radial distance increases, is found outward to r* ’ 10. At larger Although the axisymmetric mean storm-relative in- radii, this increase becomes less apparent as the inflow flow angle (aSR) appears largely independent of the lo- possibly becomes mixed with the background environ- cal wind speed and weakly dependent on radial distance mental flow. Linear trends are statistically significant in from the storm center and storm intensity, there remains Figs. 8c–e. Previous studies have also shown that intense a large amount of residual variability that is not ex- storms tend to have more sharply peaked wind profiles plained by the measurement error. Therefore, we next

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21 FIG. 7. Storm-relative inflow angle (8) as a function of 10-m wind speed (m s ) stratified according to (a) Vmax, (b) Rmax, and (c) Vs. (d)–(f) The inflow angle as a function of normalized radius stratified as for (a)–(c). Data are grouped according to the sample median values shown, with blue representing groups less than the median, and red the groups greater than the median. Solid and dashed lines are linear regression fits and 95% confidence intervals, respectively. Solid squares and error bars are bin averages and 95% confidence intervals, respectively. turn our attention to resolving the asymmetric struc- At this point, it is worth noting the rather large ture. Figure 9 shows the storm motion direction- amount of inflow angle variability over and above the relative azimuthal distribution of the inflow angle at wavenumber-1 asymmetry. The dropwindsonde-observed two radial bands, grouped according to the storm mo- winds have been only mildly low-pass filtered (5 s) tion speed greater/less than the observed median speed during the postprocessing from their raw, relatively in- 21 of Vs 5 5.5 m s . By fitting a harmonic function con- stantaneous values. Thus, the observations are expected sisting of a mean plus wavenumber-1 component to the to contain a significantly greater level of natural turbu- data, a clear asymmetry emerges, which possibly in- lence, for example, as compared to previously reported dicates both an amplitude and phase dependence of buoy observations which are averaged over an extended aSR on the storm motion speed. Relatively larger aSR time period (typically .5 min). It is expected that ap- are found to the right of the storm motion direction, plying additional averaging to the individual sonde and smaller angles to the left of the motion direction. profiles as is done for operational purposes (Franklin Furthermore, as the asymmetry amplitude increases, et al. 2003) would reduce the overall variance in inflow and phase shifts downwind (i.e., counterclockwise in angles, at the expense of capturing the variability in the Northern Hemisphere) for storms moving faster surface wind data. than the sample median motion speed. At each radial Since there is apparently a dependence of inflow band, the mean (i.e., azimuthally averaged) inflow an- angle asymmetry on the storm motion speed, aSR gle is statistically equivalent for the two storm motion versus azimuthal direction, u, is grouped according to speed groups. the storm motion speed (Vs 5 0–3.6, 3.6–5.5, 5.5–7.2,

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21 FIG. 8. Storm-relative inflow angle (aSR, 8) as a function of storm intensity (Vmax,ms ) for six radial bands: (a) 0 , r* , 1, (b) 0.5 , r* , 1.5, (c) 1 , r* , 5, (d) 2.5 , r* , 7.5, (e) 5 , r* , 10, and (f) 7.5 , r* , 12.5. Solid lines are linear regression best fits to individual observations, dashed lines are 95% confidence intervals on regression lines. Bin averages and 95% confidence intervals are plotted as squares and error bars, respectively. Linear trends are significant at the 95% confidence level in (c),(d),(e), but not significant elsewhere.

21 and .7.2 m s ), based on the sample distribution of Vs and r* (Figs. 11a–f). At all radii, the peak aSR is quartiles, and the radial band groups as earlier deﬁned. found to the right and right rear of the storm (between

Harmonic functions are fit to the observed aSR versus u 1908 and 11358 azimuth) for slower storms, and rotates data to estimate the asymmetry amplitude (Aa1) and downwind toward the front of the storm (08 to 1458)as phase (Pa1) for each group. The resulting amplitude is Vs increases. At all radii, linear trends in the asymmetry normalized by the mean, Aa0, for each subsample. The amplitude and phase with increasing Vs are statistically inflow angle asymmetry is presented in Figs. 10 and 11, significant. There is some hint of a quadratic dependence for the normalized amplitudes and phases, respectively. of the phase on Vs, as the phase shift appears to reverse 21 As shown in Figs. 10a–f, the asymmetry amplitude typ- when a critical speed of Vs ’ 6ms is reached; how- ically ranges from 0.25 to 1.0 times the symmetric mean ever, the sample statistics are not currently satisfactory to of aSR, increasing as the storm motion speed increases, confidently resolve whether this is significant. especially inward of r* 5 5. At larger radii (r* . 5), the amplitude dependence on the storm motion speed be- 4. Parametric inflow angle model comes somewhat less clear, although the asymmetry it- a. Model development self remains evident. Similar to the amplitude, the phase of the aSR asymmetry (Pa1), defined as the azimuthal With fairly clear dependencies of inflow angle asym- direction of the maximum inflow, is plotted as a function metry on the storm motion speed, a simple analytical

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FIG. 9. Azimuthal variation of inﬂow angle (left) near the eyewall and (right) outside the core (2.5 , r* , 7.5). Observations (3, o), bin averages (solid squares with error bars), and least squares ﬁts (solid lines) are grouped 21 21 according the storm speed: Vs , 5.5 m s (blue) and Vs . 5.5 m s (red). Phase is measured clockwise from the front of the storm.

model of the two-dimensional (2D) surface storm- found in the weakest hurricanes near Rmax, while the relative inflow angle aSR(r*, u) in a tropical cyclone can largest inflow angle is found in the most intense storms be constructed provided the storm intensity (Vmax), and well outside of the eyewall. storm motion speed (Vs) as parameters. The proposed Similarly, the estimated normalized amplitude (2Aa1/ model is developed based on a subset (;64%) of the Aa0) and phase (Pa1) of the inflow angle asymmetry are full observation sample as previously mentioned (see computed based on observations shown in Figs. 10 and 11, Table 1). The parametric model estimates the storm- with resulting model fits shown in Fig. 13: relative inflow angle, aSR, according to the following A relationship: 2 a1 5 1 1 aA1r* bA1Vs cA1 , (3a) Aa0 a u 5 1 SR(r*, , Vmax, Vs) Aa0(r*, Vmax) Aa1(r*, Vs) 5 1 1 Pa a r* b Vs c . (3b) 3 u 2 1 « 1 P1 P1 P1 cos[ Pa1(r*, Vs)] , (1) The model fits reflect that the smallest asymmetry am- where « is the model error. plitude is generally found in slow-moving, weak hurri-

The axisymmetric inflow angle, Aa0, was found to canes, and the largest asymmetry amplitude is found in depend primarily on r* and Vmax (Fig. 7d). Based on this intense, fast-moving hurricanes. The azimuthal location result, a linear function is fit to the Aa0 (r*, Vmax) binned of the maximum inflow angle is typically in the right-rear observations. The function assumes the linear form: quadrant for slow-moving storms and rotates downwind toward the front of storms as the storm motion speed increases. Coefficients and associated statistics Aa 5 a r* 1 b V 1 c , (2) 0 A0 A0 max A0 for Eqs. (2) and (3) are listed in Table 2. Based on the linear correlation coefficient of variation, the symmetric where the coefficients, (a, b, c), are determined by mean inflow angle is the most accurately estimated weighted least squares multiple regression. The obser- quantity (r2 5 0.77), while the asymmetry amplitude is vations are weighted inversely by the 95% confidence the least (r2 5 0.17). interval on bin averages, such that values with higher statistical confidence (i.e., smaller variance and/or more b. Estimated 2D fields observations) are given more weight. In particular, data closer to the storm center (r* , 2) carry a greater weight. As an example of the model’s application, the 2D The fitted function is shown in Fig. 12, indicating that the inflow angle distribution is constructed (Fig. 14) for 21 smallest azimuthally averaged inflow angle is typically a range of storm motion speeds (2 , Vs , 8ms ) and

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FIG. 10. Inflow angle asymmetry amplitude (Aa1, 8), normalized by the symmetric mean (2Aao, 8), as a function of storm motion speed 21 (Vs,ms ) at various radii bins: (a) 0 , r* , 1, (b) 0.5 , r* , 1.5, (c) 1 , r* , 5, (d) 2.5 , r* , 7.5, (e) 5 , r* , 10, and (f) 7.5 , r* , 12.5. Solid squares and error bars indicate bin averages and 95% confidence intervals, and lines are linear least squares best fits to averages. Linear trends are significant at the 95% confidence level at radii up to r*510.

21 intensities (35 , Vmax , 65 m s ). Both the increase in Hemisphere tropical cyclones, slower-moving storms the aSR asymmetry amplitude as well as downwind ro- may be more susceptible to the negative storm intensity tation of the maximum inflow angle with increased feedback by the storm-generated cool wake than faster- storm motion speed found in the observations are cap- moving storms. This is in addition to the fact that slower- tured. As storms become more intense, the increase moving storms tend to generate a more intense cold in aSR well outside of the inner core suggests that on wake response than faster-moving storms (e.g., Bender average, angles of aSR ,2508 are likely to be found to and Ginis 2000). In developing the Statistical Hurricane the right of track for fast-moving storms. Since outflow is Intensity Prediction Scheme (SHIPS), DeMaria and found for ,15% of the whole sample (Fig. 4b), the Kaplan (1994) found that fast-moving storms tended to model does not estimate aSR . 08 in any case. intensify more than slow-moving storms, which is at- Since the inflow angle represents the local trajectory tributed to greater ocean cooling typically found in slow of mass transport toward the storm center, it may be storms. inferred from the above analyses that slower-moving c. Parametric model evaluation storms tend to import relatively more near-surface air from the right-rear quadrant, while faster-moving An independent sample of inflow angle observations storms import more air from the right-front quadrant. is gathered from the aircraft missions as listed in Table 1, Since sea surface cooling is well known to typically which was not used for model development, but used be maximized in the right-rear quadrant of Northern for testing the model’s accuracy by performing a cross

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FIG. 11. As in Fig. 10, but for asymmetry phase (Pa1, 8). Linear trends are signiﬁcant at the 95% conﬁdence level at all radii. Phase is measured clockwise from the front of the storm.

validation. Applying the same quality-control criteria within 61s of the mean value, as shown by the cumu- as for the dependent sample results in an independent lative distribution function (Fig. 16b), the model’s ac- dataset of n 5 621 observations. To ensure that both curacy increases to within 11.98 (RMS). Some of this samples are drawn from the same aSR population, the unexplained residual error is likely due to high wave- cumulative distributions are plotted in Fig. 15. Both number variability not captured by the model (e.g., a Student’s t test of the means and a Kolmogorov– turbulence, local convective downdrafts, etc.). However, Smirnov test (Massey 1951) of the distributions in- the inherent smoothing introduced by compositing ob- dicate no significant differences at the 95% confidence servations over many cases will tend to damp variability level. that might be resolved in any individual case. If small- For each independent inflow angle observation, the scale fluctuations are not considered to be important for parent storm’s parameters (i.e., Vmax, Vs, and Rmax), are a particular application, then this smoothing may be obtained from the best-track and SFMR surface wind a desirable result. data as for the dependent sample, which are input to the d. Case studies parametric model to compute the inflow angle. A scat- terplot of observed versus model-predicted inflow an- The proposed parametric model of near surface inflow gles is shown in Fig. 16a. Regression statistics indicate angle is compared with individual, semisynoptic cases the model explains 24% of the overall inflow angle to better understand its accuracy and limitations. Four variance, with a root-mean-square (RMS) residual of cases from two hurricanes are examined: Hurricane 14.68, or an improvement of around 3.78 (;20%) over Frederic (1979), previously documented by P82; and re- simply using a mean value. Considering observations cent multiaircraft observations in Hurricane Earl (2010).

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hurricane inflow angles. Given its uniqueness, it represents a high-quality basis for evaluating the parametric inflow angle model developed herein. Earth-relative inflow angles computed from ship, buoy, and aircraft wind observations were composited over a 24-h period relative to Frederic as it traveled across the central Gulf of Mexico (see Fig. 6 in P82). For direct comparison with the parametric model, storm-relative inflow angles are computed based on the observed storm motion (heading 2 3338 at 5 m s 1) of Hurricane Frederic at the time of interest (0400 UTC 12 September 1979). The corresponding mean and asymmetric fields are then estimated using the method employed for the dropwindsonde observations. Figure 17 compares storm-relative inflow angles observed in Hurricane Frederic with model-estimated FIG. 12. Model-estimated axisymmetric inflow angle (Aa0)as 21 angles computed from Frederic’s input parameters: functions of storm intensity (V ,ms ) and normalized radial max 5 21 5 5 21 distance from the storm center (r*). The numbers on the plots are Vs 5ms , Rmax 33 km, and Vmax 45 m s . the mean and standard deviation of inflow angle at each intensity/ Qualitatively, the model inflow compares well with the radius bin. individual observations (Fig. 17a), with the largest storm-relative inflow found to the right of storm motion direction, and the smallest inflow to the left. 1) HURRICANE FREDERIC (1979) Direct quantitative comparison of observed versus Surface inflow angles observed around Hurricane model estimated inflow angle values (Fig. 17b) shows Frederic were documented by P82, and to date this good correlation (r2 5 0.80), although the model un- analysis remains one of the few semisynoptic analyses of derrepresents the dynamic range of observed angles.

FIG. 13. (a) Asymmetric storm-relative inﬂow angle model normalized amplitude (2Aa1/Aa0) and (b) phase 21 (Pa1, 8) as functions of storm motion speed (Vs,ms ), and normalized radius (r*). The numbers on the plots are the mean and standard deviation of amplitude and phase at each speed/radius bin. Phase is measured clockwise from the front of the storm.

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TABLE 2. Coefficients for the parametric inflow angle model as in In contrast to the well-estimated mean, the asymme- Eqs. (2) and (3). Means and standard deviations (italicized) are 2 try amplitude (Fig. 17d) is around 1.5 times as large as tabulated. The correlation coefficient (r ) and RMSE for the fits the symmetric average, which is approximately twice are also shown. as large as predicted by the model for the given storm Eq. Variables ab cr2 RMSE speed and intensity. Typically, the model estimated a 2 8 2 8 (2) Aa0 (8) 20.90 20.90 214.33 0.77 2.00 SR ranges from 10 to 40 , while inflow angles in 0.29 0.07 4.22 Hurricane Frederic were found to vary approximately 2 (3a) Aa1/Aa0 0.04 0.05 0.14 0.17 0.18 between 1158 and 2608. To explain this large asym- 0.04 0.06 0.32 metry, P82 noted a large difference in veering of wind (3b) Pa1 (8) 6.88 -9.60 85.31 0.31 41.40 5.80 9.42 56.86 direction with height among quadrants, with greater directional wind shear found in the southeast (right rear) quadrant where the largest earth-relative inflow angles were observed, and suggested that this could be The axisymmetric storm-relative inflow angle (Fig. 17c) due to boundary layer stability associated with the is not significantly different from the dropwindonsde- storm’s cool ocean wake. However, another plausible observed average value. There is a small linear in- explanation rests on the impact of environmental crease in the inflow angle with radial distance, although vertical wind shear on modulation of inflow angle with relatively small sample of observations (n 5 24) (Thompson 1974), which is beyond the scope of this in the Frederic analysis, the trend is not statistically study and is left for future work. We also note that significant. inflow angles derived from ships, buoys, and aircraft

21 FIG. 14. Storm-relative inﬂow angle ﬁeld computed by the parametric model for storm motion speeds of Vs 5 2, 4, 6, and 8 m s 21 (columns) and intensities of Vmax 5 35, 45, 55, and 65 m s (rows). In all panels storm direction is toward the top.

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Aeronautics and Space Administration (NASA) Genesis and Rapid Intensification Processes (GRIP) experiment, a series of multiaircraft missions were conducted to ob- serve the evolution of Hurricane Earl in the western Atlantic Ocean. Three semisynoptic 24-h compositing periods are examined centered at 0000 UTC 30 August, 31 August, and 2 September. Dropwindsonde-observed 10-m inflow angles were computed for sondes deployed by the NOAA WP-3D and G-IV, Air Force Reserve Command (AFRC) WC-130J, and NASA DC-8 aircraft. Comparison of observed versus modeled inflow angles for each of the three periods (Fig. 18) generally reflects the variability found in the overall composite analyses, as the range of observed inflow angles is larger than predicted by the model, which is to be expected. In particular, the 0000 UTC 31 August analysis indicates FIG. 15. Cumulative probability distributions of storm-relative ,2 8 inflow angle (aSR, 8) for model development dependent sample a rather poor correlation, as large inflow ( 25 )is (dashed line) and independent evaluation sample (solid line). found to the left of storm motion direction, which is atypical for the given storm motion speed, and therefore the model does not capture. observations (and thus using a longer averaging pe- Observed axisymmetric means, and asymmetric riod) might differ from the estimates derived from the amplitudes and phases are fairly well approximated dropsonde-based model [which uses a very short (5 s) for each case (Fig. 19), with the exception of the averaging period]. The observed versus model-predicted asymmetry on 31 August. On this day, an especially asymmetry phase (Fig. 17e) compares very well con- large amplitude of the inflow angle asymmetry is sidering the model accuracy, as the peak storm-relative found to be approximately 2.3 times the mean at r* 5 1, inflow for Hurricane Frederic was found around 908 to where the inflow is maximized around 408 downwind the right of the storm motion direction, as would be (left) of the storm motion direction. In contrast, the expected. asymmetry is almost nonexistent at r* ’ 8. The asymmetry phase at r* ’ 8 is found to be within the expected 2) HURRICANE EARL (2010) range; however, the small asymmetry amplitude renders As part of a coordinated NOAA Intensity Forecast the azimuthal location of the peak inflow relatively un- Experiment (IFEX, Rogers et al. 2006) and the National certain. It is noteworthy that Hurricane Earl experienced

FIG. 16. Observed vs model-predicted storm-relative inﬂow angle (aSR, 8). (left) Paired samples with linear regression (thick black line) statistics: number of observations (n), RMSE of the residual, and correlation coefﬁcient (r2) are indicated. (right) Cumulative probability distributions for observations (solid line) and model (dashed line).

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FIG. 17. Comparison of model vs observed storm-relative inflow angles in Hurricane Frederic (1979). Two- dimensional field predicted by (a) parametric model and observed values, (b) pair samples of observed vs model angles and linear regression statistics, (c) observed inflow angle and axisymmetric mean, (d) asymmetry amplitude, and (e) phase radial profiles. Observed values and regressions are in black and model-predicted values are in red. Data reproduced by permission of M. Powell. Black arrow in (a) represents the storm motion direction. an eyewall replacement cycle on 31 August (Cangialosi the sea surface (10 m). The results show that there is 2011), which is believed to be the main reason that the essentially no linear dependence of azimuthally aver- distribution of the observed inflow angles is very different aged storm-relative inflow angle on the local surface from that based on our parametric model. Moreover, on wind speed. A small dependence of the axisymmetric this day, the environmental vertical shear also increased inflow angle on radial distance from the storm center continuously, which induced asymmetric convection in the and storm intensity is found. The mean inflow angle is hurricane core, and may be another factor for the discrep- estimated to be 222.6862.28, with 95% statistical ancy in inflow angle distribution compared to other days. confidence, which agrees well with previous results (P82; Powell et al. 2009). There, is however, a large amount of variability (s 5 18.38) around this mean value. Ap- 5. Discussion and conclusions proximately 24% of the inflow angle variance is ex- This study analyzes data from 1613 GPS dropwind- plained by the axisymmetric mean plus wavenumber-1 sondes deployed by 187 research aircraft in 18 hurri- sinusoidal asymmetry, whose amplitude and phase de- canes to document the distribution of inflow angle near pend on the storm motion speed and radial distance from

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FIG. 18. As in Figs. 17a,b, but for three sample periods (columns) in Hurricane Earl. the center of the storm. Based on these results, a para- simulations in tropical cyclones may benefit from a more metric model of the 2D surface inflow angle is proposed, accurate representation of the surface wind vector field which only requires as input the storm motion speed, by implementing the parametric inflow angle model maximum wind speed, and radius of maximum wind. developed in this study. Both the observed mean and asymmetric near-surface Recently, Kwon and Cheong (2010) indicated accu- inflow structure are found to generally agree with the rately initializing the surface-wind vector field was im- theoretical description suggested by Kepert (2001) and portant for hurricane forecasts, and that the inflow angle Kepert and Wang (2001). The asymmetry of the inflow is a key parameter for proper specification of the wind angle with strong dependence on the storm motion field. Simulated tropical cyclone intensity has been speed generally agrees with the finding of Shapiro shown to be sensitive to the representation of boundary (1983). However, our results indicate that Malkus and layer structure in previous numerical studies (e.g., Nolan Riehl (1960) significantly underestimated the inflow et al. 2009a,b). The simulated boundary layer structure angle near the eyewall, likely a result of their model in turn depends on the drag coefficient (Montgomery overestimating the boundary layer depth (;2.2 km), et al. 2010), and horizontal (Bryan and Rotunno 2009; which is now believed to be much shallower (Zhang Zhang and Montgomery 2012) and vertical eddy diffu- et al. 2011a). sivities (Foster 2009; Zhang et al. 2011b). A model’s For practical applications, our parametric inflow angle accuracy in representing boundary layer structure may model may be combined with remotely sensed obser- also be evaluated by properly computing the inflow angle vations of near-surface winds in tropical cyclones to distribution around the tropical cyclone (e.g., Kepert better define the 2D wind vector field, for example, from 2010a; Bryan 2012). As part of NOAA’s Hurricane SFMR surface wind speed measurements. Also, the Forecast Improvement Project (HFIP), the observational model-estimated wind direction field can be used to de- data presented in this work will be used to evaluate the alias retrieved multivector solutions with added confi- representation of boundary layer and/or surface layer dence. Storm surge, surface wave, and upper-ocean structure in tropical cyclone model simulations.

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FIG. 19. As in Figs. 17c–e, except for three sample periods (columns) in Hurricane Earl.

Acknowledgments. This work was supported by the Sim Aberson for constructive comments on the early NOAA Hurricane Forecast Improvement Project version of this paper. Finally, we also thank the two (HFIP). We gratefully acknowledge all the scientists and anonymous reviewers for their constructive comments, crews who were involved in the Hurricane Research which substantially improved our paper. Division’s ﬁeld program collecting the data used in this work. We appreciate the efforts of all the scientists and REFERENCES students who helped postprocessing the (pre 2004) dropwindsonde data used in this work. Without their Barnes, G. M., 2008: Atypical thermodynamic proﬁles in hurri- efforts, this work would not have been possible. In canes. Mon. Wea. Rev., 136, 631–643. Bender, M. A., and I. Ginis, 2000: Real-case simulations of particular, we are very grateful to Kathryn Sellwood and hurricane–ocean interaction using a high-resolution coupled Sim Aberson for organizing and maintaining the drop- model: Effects on hurricane intensity. Mon. Wea. Rev., 128, windsonde data base at HRD and making both the raw 917–946. and postprocessed data available. We thank Beth Oswald Black, P. G., and Coauthors, 2007: Air–sea exchange in hurricanes: for postprocessing the dropwindsonde collected using the Synthesis of observations from the Coupled Boundary Layer Coupled Boundary Layer Air–Sea Transfer (CBLAST) Air–Sea Transfer experiment. Bull. Amer. Meteor. Soc., 88, 357–374. experiment (2002–04) while working with Peter Black Brennan, M. J., C. C. Hennon, and R. D. Knabb, 2009: The op- and the author Jun Zhang in 2006 as a summer student at erational use of QuikSCAT ocean surface vector winds at HRD. We thank Robert Rogers and Frank Marks for the National Hurricane Center. Wea. Forecasting, 24, 621– helpful discussions. We acknowledge Mark Powell and 645.

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