Hurricane Sea Surface Inflow Angle and an ... - AMS Journals

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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 222.68 6 2.28 (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 stormrelative inflow angles (,2508) are found in the fastest-moving storms (.8 m s21) 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 Estimating hurricane surface wind distributions and maxima is an operational requirement of the National Hurricane Center (NHC), as coastal watches and warnings are issued based on storm impacts at landfall, including storm surge. Fairly recent development of reliable instrumentation has resulted in more accurate estimates of tropical cyclone surface wind speed and direction. Currently, remotely sensed surface wind speed observations in tropical cyclones are provided by spaceborne microwave sensors (Katsaros 2010) and airborne stepped-frequency microwave radiometers (SFMR; Uhlhorn and Black 2003; Uhlhorn et al. 2007).

Corresponding author address: Dr. Jun Zhang, NOAA/AOML/ Hurricane Research Division, Universtiy of Miami/CIMAS, 4301 Rickenbacker Causeway, Miami, FL 33149. E-mail: [email protected] DOI: 10.1175/MWR-D-11-00339.1 Ó 2012 American Meteorological Society

In situ near-surface wind vector observations in tropical cyclones are available from global positioning system (GPS) dropwindsondes deployed by research and reconnaissance aircraft (Hock and Franklin 1999). Globally, however, direct measurements of sea surface winds in tropical cyclones are still highly infrequent, so methods have been developed to estimate surface winds from wind data observed at higher altitudes by research aircraft (e.g., Franklin et al. 2003; Powell et al. 2009), from satellite imagery (Velden et al. 2006), and from surface pressure observations (Knaff and Zehr 2007). In comparison to surface wind speed data, wind direction information is exceedingly sparse. Mapping the two-dimensional surface wind vector field in tropical cyclones has several important applications. First, storm surge models are generally forced by surface winds, which not only require the wind magnitude but also the wind direction. It has traditionally been a standard practice to use axisymmetric parametric wind

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models to force storm surge models (e.g., Peng et al. 2006; Rego and Li 2009). These parametric wind models, such as the Sea, Lake and Overland Surges from Hurricanes (SLOSH) wind model (Phadke et al. 2003), Holland’s model (Holland 1980; Holland et al. 2010), and Willoughby’s model (Willoughby et al. 2006) estimate the radial profile of axisymmetric wind speed or tangential wind component without wind direction information. The wind direction is then arrived at by applying a constant inflow angle, and an asymmetry in wind speed is simply assumed due to storm forward motion. Some storm-surge studies (e.g., Westerink et al. 2008) have utilized the National Oceanic and Atmospheric Administration (NOAA)/Hurricane Research Division (HRD) real-time Hurricane WIND analysis system (H*WIND) product (Powell et al. 1998), which estimates surface wind direction applied to SFMR wind speeds as simply a constant angle subtracted from the flight-level wind direction (M. Powell 2005, personal communication). Second, remotely sensed wind direction accuracy in tropical cyclones, particularly in the high-wind innercore region, is often highly degraded as a result of several physical factors. Nadir-viewing passive microwave radiometers (e.g., SFMR) are insensitive to wind direction and spaceborne wide-swath imagers suffer from resolution and rain absorption artifacts (e.g., Connor and Chang 2000). Active microwave sensors such as scatterometers may saturate, are attenuated in heavy precipitation, and are also limited by spatial resolution for tropical cyclone applications, particularly in the inner-core region (Brennan et al. 2009). The resolution limitations can be overcome by synthetic aperture radar (SAR); however, SARs typically provide only a single view and therefore determining the wind direction is an ambiguous problem (Shen et al. 2009). In addition, it is often assumed that the surface roughness elements that provide the radar backscatter mechanism are aligned with the wind direction, which may not always be accurate (e.g., Donelan et al. 1997; Drennan et al. 1999; Grachev et al. 2003; Drennan et al. 2003). Third, predicting tropical cyclone intensity is viewed as a coupled atmosphere–ocean problem, thus understanding the air–sea interaction and ocean feedbacks to hurricanes is of paramount importance (e.g., Black et al. 2007; Shay et al. 1989; Jacob et al. 2000; Shay and Uhlhorn 2008; Jaimes and Shay 2010; Uhlhorn and Shay 2012). Accurately specifying the surface wind direction may benefit wave forecast models, which have been increasingly inserted into the air–sea interface in coupled model applications (e.g., Moon et al. 2007; Zhao and Hong 2011). Accurate representation of the surface wind direction can also help improve our understanding

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of the ocean response to hurricanes when a parametric wind model is used to force an ocean model (e.g., Price 1983; Yablonsky and Ginis 2009; Halliwell et al. 2011). Because of the ubiquitous cyclonic flow near the surface in tropical cyclones, documentation of observed surface wind directions is typically described in terms of surface inflow angles, although such studies are relatively sparse. Numerical studies (e.g., Kepert 2010a,b; Bryan 2012) often cite the observational result presented by Powell (1982, hereafter P82) from data obtained in Hurricane Frederic (1979). Earth-relative inflow angles over the open ocean were found by P82 to vary from outflow of 1128 to inflow of 2558, with greater inflow in the right-rear (RR) quadrant and weaker inflow in the left-front (LF) quadrant, and with a mean value of 2228. Powell et al. (2009) examined a large sample of dropwindsonde data and found a mean inflow angle of 2238, although details regarding asymmetric structure were not investigated. Note that the original analytical treatment of tropical cyclone inflow was presented by Malkus and Riehl (1960), who suggested an axisymmetric average inflow angle of 2208 to 2258 outside of the eyewall, but decreasing to less than 258 at the radius of maximum wind (Rmax), was consistent with boundary layer energy constraints. This conclusion also depended on knowledge of the surface exchange coefficients of momentum and moist enthalpy, and boundary layer depth, which were not very well known at the time (e.g., French et al. 2007; Zhang et al. 2008, 2009; Zhang 2010; Haus et al. 2010; Kepert 2010a; Smith and Montgomery 2010). The purpose of this paper is to investigate the mean and asymmetric structure of near-surface inflow angle (at an altitude of 10 m) over a broad range of tropical cyclone characteristics, including storm motion, intensity, and size, utilizing the extensive database of GPS dropwindsonde wind vector observations. Based on the data analysis results, a simple parametric model of the mean plus wavenumber-1 asymmetric inflow angle field is developed and tested. Section 2 describes the data sources, quality control, and analysis methodology. In section 3, analysis results are presented for both mean and asymmetric fields. Section 4 describes the parametric model development, evaluation, and case-study comparisons and section 5 summarizes the results and discusses the applications of the parametric model.

2. Data and quality control GPS dropwindsonde data used in this study were collected on 187 hurricane research and reconnaissance flights in 18 hurricanes (Table 1) between 1998 and 2010. Detailed description of dropwindsonde instrumentation and data accuracies can be found in Hock and Franklin

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

Storm name

Year

Storm intensity range (m s21)

Bonnie Danielle Georges Bret Floyd Fabian Isabel Frances Ivan Jeanne Dennis Katrina Rita Dean Gustav Paloma Bill Earl Totals

1998 1998 1998 1999 1999 2003 2003 2004 2004 2004 2005 2005 2005 2007 2008 2008 2009 2010

50–52 33–37 35–68 52–57 47–63 54–62 67–72 45–64 54–72 44–54 36–62 52–77 33–77 38–77 33–62 34–59 48–59 38–64

No. of flights

No. of sondes

5 6 7 1 4 9 9 6 (7) 10 (17) 2 (8) (14) 2 (17) (7) (11) 5 12 (4) 19 (5) 97 (90)

85 107 87 18 39 95 51 79 (88) 101 (172) 13 (66) (110) 54 (187) (54) (87) 43 144 (15) 183 (46) 1099 (825)

(1999). The near-surface fall speed of a dropwindsonde is 12–14 m s21, while the typical sampling rate is 2 Hz, yielding an approximately 5–7 m vertical sampling. Note that the 5-s filter, which is typically applied in the postprocessing, effectively reduces the vertical resolution to roughly an order of magnitude lower than the original sampling. The accuracy of the horizontal wind speed measurements is ;0.5 m s21. The dropwindsonde data have been postprocessed and quality controlled using the HRD Editsonde (Franklin et al. 2003) software for the data before 2005. Data obtained after 2005 have been postprocessed using the National Center for Atmospheric Research (NCAR) Atmospheric Sounding Processing Environment (ASPEN) software. Recent studies have indicated little difference exists between the Editsonde- and ASPEN- processed wind data (e.g., Barnes 2008). Although there have been several minor improvements to the dropwindsonde design and processing since the original documentation (Hock and Franklin 1999), overall data accuracy has not changed significantly to impact results in this study. To study the near-surface inflow angle, we only use dropwindsonde data with wind vector measurements near the surface (#10 m), totaling 1924 sondes. The horizontal Cartesian wind-vector components (u, y) are linearly interpolated to the 10-m level if not directly observed at that level. All sondes report winds to the surface, although data dropouts over a profile may exist when GPS satellite tracking is temporarily degraded.

FIG. 1. Storm-relative two-dimensional distribution of dropwindsonde surface observation locations between 0 and 12.5r*. Cross- and along-track positions are normalized by the radius of 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.

However, in no case are winds extrapolated to the 10-m level if a sonde terminates above this level. Data locations are transformed to a polar coordinate system measured radially (r) from the center and azimuthally clockwise from storm motion direction (u). The storm center positions have been determined using storm tracks produced by NOAA/HRD based on the flightlevel wind data (Willoughby and Chelmlow 1982, hereafter WC82). Radial distances are normalized by the estimated radius of maximum wind speed, Rmax, determined from approximately concurrent SFMR surface wind observations (r* 5 r/Rmax). The Rmax value represents an average of individually observed wind maxima along each radial leg for a single flight. Data are reasonably evenly azimuthally distributed, as shown in Fig. 1. Figure 2 shows the radial distribution of the number of observations. After normalizing the radial coordinate by Rmax (Fig. 2b) we find the largest number of sondes is clustered around r* 5 1, corresponding to eyewall deployments, with a secondary peak in the number of sondes deployed near the storm center. Dropwindsonde data are analyzed and grouped in a composite framework. The composite analysis method has been used in previous studies investigating the hurricane inner-core structure (e.g., Frank 1984; Rogers et al. 2012), vertical wind profile structure (e.g., Franklin et al. 2003; Powell et al. 2003), surface layer air–sea

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FIG. 2. Radial distribution of dropwindsonde counts per bin as a function of (a) real distance and (b) normalized distance. Counts are per bin widths of (a) 20 km and (b) 0.5r*.

thermal structure (e.g., Cione et al. 2000; Cione and Uhlhorn 2003), and boundary layer structure (Zhang et al. 2011a). The advantage of the composite analysis method is that it helps to fill data voids and provides a general characterization of the fields under investigation. The most important drawback to compositing is that it tends to smooth the data from a large number of storms that may not be similar. The success of a composite analysis depends on the similarity of the events studied, thus we initially restrict our analysis to data collected in hurricanes (Vmax . 33 m s21, where Vmax is the maximum 1-min wind speed), and radially outward to r* 5 12.5. For each dropwindsonde, Vmax and storm speed (Vs) and direction are obtained from the 6-hourly best-track database (Jarvinen et al. 1984) interpolated to the time of observation. The frequency distributions of Vmax, Rmax, and Vs indicate that observations represent a broad spectrum of storms (Fig. 3). Storm intensities range between 33 , Vmax , 77 m s21, sizes between 10 , Rmax , 72 km, and motion speeds between 0.8 , Vs , 12.3 m s21. The median storm intensity for the whole sample is Vmax 5 56.7 m s21 (Saffir–Simpson category 3), radius of maximum wind is Rmax 5 31.8 km, and storm motion speed is Vs 5 5.5 m s21. The inflow angle1 (a) is defined as the arctangent of the ratio of radial (y r) to tangential (y t) wind components [a 5 tan21(y r/y t)]. Note that storm-relative inflow angle (aSR) is used exclusively throughout this study.

1 Inflow is defined as yr , 0, although we still refer to ‘‘inflow angle’’ when outflow (yr . 0) occurs. Also, a ‘‘larger inflow angle’’ or the like will generally indicate a more negative value throughout this article.

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To calculate the storm-relative inflow angle, the storm motion vector is removed from the dropwindsondeobserved 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 (y t , 0). The frequency distribution of aSR for the initial sample is shown in Fig. 4a. The distribution is super-Gaussian (normalized kurtosis k 5 12.7), which is a primarily a result of numerous 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 and Montgomery 2000). By simply eliminating observations where r* , 0.5, the frequency distribution of aSR becomes more normal (Fig. 4b), suggesting an improved representation of the expected inflow angle in tropical cyclones. Because the accuracy of y r and y t, and therefore aSR, depend on the storm center position accuracy, the impact of the storm center position error on the computed inflow angle is briefly examined. WC82 claimed that the storm center based on flight-level wind observations can be determined to around 3-km accuracy, although Kepert (2005) showed that the center position error within the hurricane boundary layer for a translating storm can easily be 5 km or more using the WC82 method. The impact of storm center position error on the uncertainty of inflow angle is simulated by assuming the storm center position is in error (one standard deviation, s) by 5 km, and a normal distribution of inflow angles is generated by Monte Carlo simulation of 1000 realizations. Figure 5 shows the simulated inflow angle error (normalized by the sample s 5 18.38 as indicated in Fig. 4b) versus r*, where the sample median (minimum) Rmax of 32 (10) km is used to normalize the radial distance. For comparison, a 2-km center position errorinduced inflow angle error is shown, representing the estimated accuracy of the translating pressure center tracking method proposed by Kepert (2005). Except for the smallest storms, a 5-km center position error induces an inflow angle error smaller than 18.38 outside of r* 5 1, and would likely be buried in the natural surface wind variability. Some improvement to the accuracy could be made by utilizing the pressure-based 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 specification. Although data are included all the way into the estimated center, inflow angles 62s away from the sample

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FIG. 3. Frequency distribution of dropwindsondes according to the corresponding maximum wind speed (Vmax, m s21), radius of maximum wind speed (Rmax, km), and storm motion speed (Vs, m s21).

mean of 222.68 indicated in Fig. 4b are rejected as highly unrepresentative, which restricts the range of acceptable 10-m level, storm-relative inflow angles between 259.28 and 114.08, resulting in a final working sample of 1613 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 developmental sample; therefore, both datasets may be regarded as random samples of the population.

potentially important detail in ation near the eyewall will be work as our focus here is on variability of the inflow angle

the inflow angle variinvestigated in future the overall structural throughout a tropical

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 (20.538 6 0.238 per r* units; i.e., significantly different from zero slope based on a Student’s t test). This significant dependence of aSR on r* appears especially well pronounced between the eyewall and just outside of the eyewall as indicated by the bin-averaged values. This

FIG. 4. Frequency distribution of storm-relative inflow angle (aSR, 8) for (a) full sample and for (b) sondes radially outward of r* 5 0.5. Sample size (n), mean (m), standard deviation (s), and normalized kurtosis (k) are indicated. Dashed lines represent 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 speed appear to have little relationship with the inflow angle. Although small, the axisymmetric inflow angle has a statistically significant dependence on the storm intensity, particularly at large radii where inflow angles are on average ;58 larger for more intense storms (Fig. 7d). Based on this result, we further stratify the 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– 77 m s21) and six radial band groups (r* 5 0–1, 0.5–1.5, 1–5, 2.5–7.5, 5–10, and 7.5–12.5). Bins are partially overlapped to provide continuity among groups, at the expense of smoothing possibly relevant finescale details. Figures 8a–f show individual observations, binned averages, and 95% confidence intervals of aSR versus Vmax for the six radial bands, along with linear regression fits. An increasing slope of aSR versus Vmax, as radial distance increases, is found outward to r* ’ 10. At larger radii, this increase becomes less apparent as the inflow possibly becomes mixed with the background environmental flow. Linear trends are statistically significant in Figs. 8c–e. Previous studies have also shown that intense storms tend to have more sharply peaked wind profiles

FIG. 6. Storm-relative inflow angle (aSR, 8) as a function of (a) 10-m storm-relative wind speed (U10SR, m s21) and (b) radius normalized by the radius of maximum wind speed (r*, dimensionless). Solid and dashed lines are linear regression fits and 95% confidence intervals, respectively, and points with error bars are bin averages and 95% confidence intervals. Total number of observations (n), correlation coefficient (r2), and trend lines with 95% confidence intervals for parameters are indicated.

(e.g., Mallen et al. 2005; Willoughby et al. 2006), and model simulations show that more peaked storms have stronger inflow outside the radius of maximum wind speed (e.g., Kepert and Wang 2001; Kepert 2006a,b), suggesting that the increased inflow angle at large radius is consistent with the dynamics.

b. Asymmetric distribution Although the axisymmetric mean storm-relative inflow angle (aSR) appears largely independent of the local wind speed and weakly dependent on radial distance from the storm center and storm intensity, there remains a large amount of residual variability that is not explained by the measurement error. Therefore, we next

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FIG. 7. Storm-relative inflow angle (8) as a function of 10-m wind speed (m s21) 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 structure. Figure 9 shows the storm motion directionrelative azimuthal distribution of the inflow angle at two radial bands, grouped according to the storm motion speed greater/less than the observed median speed of Vs 5 5.5 m s21. By fitting a harmonic function consisting of a mean plus wavenumber-1 component to the data, a clear asymmetry emerges, which possibly indicates both an amplitude and phase dependence of aSR on the storm motion speed. Relatively larger aSR are found to the right of the storm motion direction, and smaller angles to the left of the motion direction. Furthermore, as the asymmetry amplitude increases, and phase shifts downwind (i.e., counterclockwise in the Northern Hemisphere) for storms moving faster than the sample median motion speed. At each radial band, the mean (i.e., azimuthally averaged) inflow angle is statistically equivalent for the two storm motion speed groups.

At this point, it is worth noting the rather large amount of inflow angle variability over and above the wavenumber-1 asymmetry. The dropwindsonde-observed winds have been only mildly low-pass filtered (5 s) during the postprocessing from their raw, relatively instantaneous values. Thus, the observations are expected to contain a significantly greater level of natural turbulence, for example, as compared to previously reported buoy observations which are averaged over an extended time period (typically .5 min). It is expected that applying additional averaging to the individual sonde profiles as is done for operational purposes (Franklin et al. 2003) would reduce the overall variance in inflow angles, at the expense of capturing the variability in surface wind data. Since there is apparently a dependence of inflow angle asymmetry on the storm motion speed, aSR versus azimuthal direction, u, is grouped according to the storm motion speed (Vs 5 0–3.6, 3.6–5.5, 5.5–7.2,

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FIG. 8. Storm-relative inflow angle (aSR, 8) as a function of storm intensity (Vmax, m s21) 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.

and .7.2 m s21), based on the sample distribution quartiles, and the radial band groups as earlier defined. Harmonic functions are fit to the observed aSR versus u data to estimate the asymmetry amplitude (Aa1) and phase (Pa1) for each group. The resulting amplitude is normalized by the mean, Aa0, for each subsample. The inflow angle asymmetry is presented in Figs. 10 and 11, for the normalized amplitudes and phases, respectively. As shown in Figs. 10a–f, the asymmetry amplitude typically ranges from 0.25 to 1.0 times the symmetric mean of aSR, increasing as the storm motion speed increases, especially inward of r* 5 5. At larger radii (r* . 5), the amplitude dependence on the storm motion speed becomes somewhat less clear, although the asymmetry itself remains evident. Similar to the amplitude, the phase of the aSR asymmetry (Pa1), defined as the azimuthal direction of the maximum inflow, is plotted as a function

of Vs and r* (Figs. 11a–f). At all radii, the peak aSR is found to the right and right rear of the storm (between 1908 and 11358 azimuth) for slower storms, and rotates downwind toward the front of the storm (08 to 1458) as Vs increases. At all radii, linear trends in the asymmetry amplitude and phase with increasing Vs are statistically significant. There is some hint of a quadratic dependence of the phase on Vs, as the phase shift appears to reverse when a critical speed of Vs ’ 6 m s21 is reached; however, the sample statistics are not currently satisfactory to confidently resolve whether this is significant.

4. Parametric inflow angle model a. Model development With fairly clear dependencies of inflow angle asymmetry 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 according the storm speed: Vs , 5.5 m s21 (blue) and Vs . 5.5 m s21 (red). Phase is measured clockwise from the front of the storm.

model of the two-dimensional (2D) surface stormrelative inflow angle aSR(r*, u) in a tropical cyclone can be constructed provided the storm intensity (Vmax), and storm motion speed (Vs) as parameters. The proposed model is developed based on a subset (;64%) of the full observation sample as previously mentioned (see Table 1). The parametric model estimates the stormrelative inflow angle, aSR, according to the following relationship: aSR (r*, u, Vmax , Vs ) 5 Aa0 (r*, Vmax ) 1 Aa1 (r*, Vs ) 3 cos[u 2 Pa1 (r*, Vs )] 1 «,

(1)

where « is the model error. The axisymmetric inflow angle, Aa0, was found to depend primarily on r* and Vmax (Fig. 7d). Based on this result, a linear function is fit to the Aa0 (r*, Vmax) binned observations. The function assumes the linear form: Aa0 5 aA0 r* 1 bA0 Vmax 1 cA0 ,

(2)

where the coefficients, (a, b, c), are determined by weighted least squares multiple regression. The observations are weighted inversely by the 95% confidence interval on bin averages, such that values with higher statistical confidence (i.e., smaller variance and/or more observations) are given more weight. In particular, data closer to the storm center (r* , 2) carry a greater weight. The fitted function is shown in Fig. 12, indicating that the smallest azimuthally averaged inflow angle is typically

found in the weakest hurricanes near Rmax, while the largest inflow angle is found in the most intense storms well outside of the eyewall. Similarly, the estimated normalized amplitude (2Aa1 / Aa0 ) and phase (Pa1 ) of the inflow angle asymmetry are computed based on observations shown in Figs. 10 and 11, with resulting model fits shown in Fig. 13: A 2 a1 5 aA1 r* 1 bA1 Vs 1 cA1 , Aa0

(3a)

Pa1 5 aP1 r* 1 bP1 Vs 1 cP1 .

(3b)

The model fits reflect that the smallest asymmetry amplitude is generally found in slow-moving, weak hurricanes, and the largest asymmetry amplitude is found in intense, fast-moving hurricanes. The azimuthal location of the maximum inflow angle is typically in the right-rear quadrant for slow-moving storms and rotates downwind toward the front of storms as the storm motion speed increases. Coefficients and associated statistics for Eqs. (2) and (3) are listed in Table 2. Based on the linear correlation coefficient of variation, the symmetric mean inflow angle is the most accurately estimated quantity (r2 5 0.77), while the asymmetry amplitude is the least (r2 5 0.17).

b. Estimated 2D fields As an example of the model’s application, the 2D inflow angle distribution is constructed (Fig. 14) for a range of storm motion speeds (2 , Vs , 8 m s21) 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 (Vs, m s21) 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.

intensities (35 , Vmax , 65 m s21). Both the increase in the aSR asymmetry amplitude as well as downwind rotation of the maximum inflow angle with increased storm motion speed found in the observations are captured. As storms become more intense, the increase in aSR well outside of the inner core suggests that on average, angles of aSR , 2508 are likely to be found to the right of track for fast-moving storms. Since outflow is found for ,15% of the whole sample (Fig. 4b), the model does not estimate aSR . 08 in any case. Since the inflow angle represents the local trajectory of mass transport toward the storm center, it may be inferred from the above analyses that slower-moving storms tend to import relatively more near-surface air from the right-rear quadrant, while faster-moving storms import more air from the right-front quadrant. Since sea surface cooling is well known to typically be maximized in the right-rear quadrant of Northern

Hemisphere tropical cyclones, slower-moving storms may be more susceptible to the negative storm intensity feedback by the storm-generated cool wake than fastermoving storms. This is in addition to the fact that slowermoving storms tend to generate a more intense cold wake response than faster-moving storms (e.g., Bender and Ginis 2000). In developing the Statistical Hurricane Intensity Prediction Scheme (SHIPS), DeMaria and Kaplan (1994) found that fast-moving storms tended to intensify more than slow-moving storms, which is attributed to greater ocean cooling typically found in slow storms.

c. Parametric model evaluation An independent sample of inflow angle observations is gathered from the aircraft missions as listed in Table 1, which was not used for model development, but used 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 as for the dependent sample results in an independent dataset of n 5 621 observations. To ensure that both samples are drawn from the same aSR population, the cumulative distributions are plotted in Fig. 15. Both a Student’s t test of the means and a Kolmogorov– Smirnov test (Massey 1951) of the distributions indicate no significant differences at the 95% confidence level. For each independent inflow angle observation, the parent storm’s parameters (i.e., Vmax, Vs, and Rmax), are obtained from the best-track and SFMR surface wind data as for the dependent sample, which are input to the parametric model to compute the inflow angle. A scatterplot of observed versus model-predicted inflow angles is shown in Fig. 16a. Regression statistics indicate the model explains 24% of the overall inflow angle variance, with a root-mean-square (RMS) residual of 14.68, or an improvement of around 3.78 (;20%) over simply using a mean value. Considering observations

within 61s of the mean value, as shown by the cumulative distribution function (Fig. 16b), the model’s accuracy increases to within 11.98 (RMS). Some of this unexplained residual error is likely due to high wavenumber variability not captured by the model (e.g., turbulence, local convective downdrafts, etc.). However, the inherent smoothing introduced by compositing observations over many cases will tend to damp variability that might be resolved in any individual case. If smallscale fluctuations are not considered to be important for a particular application, then this smoothing may be a desirable result.

d. Case studies The proposed parametric model of near surface inflow angle is compared with individual, semisynoptic cases to better understand its accuracy and limitations. Four cases from two hurricanes are examined: Hurricane Frederic (1979), previously documented by P82; and recent multiaircraft observations in Hurricane Earl (2010).

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FIG. 12. Model-estimated axisymmetric inflow angle (Aa0) as functions of storm intensity (Vmax, m s21) and normalized radial distance from the storm center (r*). The numbers on the plots are the mean and standard deviation of inflow angle at each intensity/ radius bin.

1) HURRICANE FREDERIC (1979) Surface inflow angles observed around Hurricane Frederic were documented by P82, and to date this analysis remains one of the few semisynoptic analyses of

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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 3338 at 5 m s21) 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 angles computed from Frederic’s input parameters: Vs 5 5 m s21, Rmax 5 33 km, and Vmax 5 45 m s21. Qualitatively, the model inflow compares well with the 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. Direct quantitative comparison of observed versus model estimated inflow angle values (Fig. 17b) shows good correlation (r2 5 0.80), although the model underrepresents the dynamic range of observed angles.

FIG. 13. (a) Asymmetric storm-relative inflow angle model normalized amplitude (2Aa1/Aa0) and (b) phase (Pa1, 8) as functions of storm motion speed (Vs, m s21), 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 Eqs. (2) and (3). Means and standard deviations (italicized) are tabulated. The correlation coefficient (r2) and RMSE for the fits are also shown. Eq.

Variables

(2)

Aa0 (8)

(3a)

2 Aa1 /Aa0

(3b)

Pa1 (8)

a

b

c

r2

RMSE

20.90 0.29 0.04 0.04 6.88 5.80

20.90 0.07 0.05 0.06 -9.60 9.42

214.33 4.22 0.14 0.32 85.31 56.86

0.77

2.00

0.17

0.18

0.31

41.40

The axisymmetric storm-relative inflow angle (Fig. 17c) is not significantly different from the dropwindonsdeobserved average value. There is a small linear increase in the inflow angle with radial distance, although with relatively small sample of observations (n 5 24) in the Frederic analysis, the trend is not statistically significant.

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In contrast to the well-estimated mean, the asymmetry amplitude (Fig. 17d) is around 1.5 times as large as the symmetric average, which is approximately twice as large as predicted by the model for the given storm speed and intensity. Typically, the model estimated aSR ranges from 2108 to 2408, while inflow angles in Hurricane Frederic were found to vary approximately between 1158 and 2608. To explain this large asymmetry, P82 noted a large difference in veering of wind 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 due to boundary layer stability associated with the storm’s cool ocean wake. However, another plausible explanation rests on the impact of environmental vertical wind shear on modulation of inflow angle (Thompson 1974), which is beyond the scope of this study and is left for future work. We also note that inflow angles derived from ships, buoys, and aircraft

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 s21 (columns) and intensities of Vmax 5 35, 45, 55, and 65 m s21 (rows). In all panels storm direction is toward the top.

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FIG. 15. Cumulative probability distributions of storm-relative inflow angle (aSR, 8) for model development dependent sample (dashed line) and independent evaluation sample (solid line).

observations (and thus using a longer averaging period) might differ from the estimates derived from the dropsonde-based model [which uses a very short (5 s) averaging period]. The observed versus model-predicted asymmetry phase (Fig. 17e) compares very well considering the model accuracy, as the peak storm-relative inflow for Hurricane Frederic was found around 908 to the right of the storm motion direction, as would be expected.

2) HURRICANE EARL (2010) As part of a coordinated NOAA Intensity Forecast Experiment (IFEX, Rogers et al. 2006) and the National

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Aeronautics and Space Administration (NASA) Genesis and Rapid Intensification Processes (GRIP) experiment, a series of multiaircraft missions were conducted to observe 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 a rather poor correlation, as large inflow (,2258) is found to the left of storm motion direction, which is atypical for the given storm motion speed, and therefore the model does not capture. Observed axisymmetric means, and asymmetric amplitudes and phases are fairly well approximated for each case (Fig. 19), with the exception of the asymmetry on 31 August. On this day, an especially large amplitude of the inflow angle asymmetry is found to be approximately 2.3 times the mean at r* 5 1, where the inflow is maximized around 408 downwind (left) of the storm motion direction. In contrast, the asymmetry is almost nonexistent at r* ’ 8. The asymmetry phase at r* ’ 8 is found to be within the expected range; however, the small asymmetry amplitude renders the azimuthal location of the peak inflow relatively uncertain. 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). Twodimensional 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 2011), which is believed to be the main reason that the distribution of the observed inflow angles is very different from that based on our parametric model. Moreover, on this day, the environmental vertical shear also increased continuously, which induced asymmetric convection in the hurricane core, and may be another factor for the discrepancy in inflow angle distribution compared to other days.

5. Discussion and conclusions This study analyzes data from 1613 GPS dropwindsondes deployed by 187 research aircraft in 18 hurricanes to document the distribution of inflow angle near

the sea surface (10 m). The results show that there is essentially no linear dependence of azimuthally averaged storm-relative inflow angle on the local surface wind speed. A small dependence of the axisymmetric inflow angle on radial distance from the storm center and storm intensity is found. The mean inflow angle is estimated to be 222.68 6 2.28, with 95% statistical 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. Approximately 24% of the inflow angle variance is explained by the axisymmetric mean plus wavenumber-1 sinusoidal asymmetry, whose amplitude and phase depend 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 parametric model of the 2D surface inflow angle is proposed, which only requires as input the storm motion speed, maximum wind speed, and radius of maximum wind. Both the observed mean and asymmetric near-surface inflow structure are found to generally agree with the theoretical description suggested by Kepert (2001) and Kepert and Wang (2001). The asymmetry of the inflow angle with strong dependence on the storm motion speed generally agrees with the finding of Shapiro (1983). However, our results indicate that Malkus and Riehl (1960) significantly underestimated the inflow angle near the eyewall, likely a result of their model overestimating the boundary layer depth (;2.2 km), which is now believed to be much shallower (Zhang et al. 2011a). For practical applications, our parametric inflow angle model may be combined with remotely sensed observations of near-surface winds in tropical cyclones to better define the 2D wind vector field, for example, from SFMR surface wind speed measurements. Also, the model-estimated wind direction field can be used to dealias retrieved multivector solutions with added confidence. Storm surge, surface wave, and upper-ocean

simulations in tropical cyclones may benefit from a more accurate representation of the surface wind vector field by implementing the parametric inflow angle model developed in this study. Recently, Kwon and Cheong (2010) indicated accurately initializing the surface-wind vector field was important for hurricane forecasts, and that the inflow angle is a key parameter for proper specification of the wind field. Simulated tropical cyclone intensity has been shown to be sensitive to the representation of boundary layer structure in previous numerical studies (e.g., Nolan et al. 2009a,b). The simulated boundary layer structure in turn depends on the drag coefficient (Montgomery et al. 2010), and horizontal (Bryan and Rotunno 2009; Zhang and Montgomery 2012) and vertical eddy diffusivities (Foster 2009; Zhang et al. 2011b). A model’s accuracy in representing boundary layer structure may also be evaluated by properly computing the inflow angle distribution around the tropical cyclone (e.g., Kepert 2010a; Bryan 2012). As part of NOAA’s Hurricane Forecast Improvement Project (HFIP), the observational data presented in this work will be used to evaluate the representation of boundary layer and/or surface layer 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 NOAA Hurricane Forecast Improvement Project (HFIP). We gratefully acknowledge all the scientists and crews who were involved in the Hurricane Research Division’s field program collecting the data used in this work. We appreciate the efforts of all the scientists and students who helped postprocessing the (pre 2004) dropwindsonde data used in this work. Without their efforts, this work would not have been possible. In particular, we are very grateful to Kathryn Sellwood and Sim Aberson for organizing and maintaining the dropwindsonde data base at HRD and making both the raw and postprocessed data available. We thank Beth Oswald for postprocessing the dropwindsonde collected using the Coupled Boundary Layer Air–Sea Transfer (CBLAST) experiment (2002–04) while working with Peter Black and the author Jun Zhang in 2006 as a summer student at HRD. We thank Robert Rogers and Frank Marks for helpful discussions. We acknowledge Mark Powell and

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