## 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). 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 Franklin1999). 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 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 inner-core 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 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 +12° to inflow of −55°, with greater inflow in the right-rear (RR) quadrant and weaker inflow in the left-front (LF) quadrant, and with a mean value of −22°. Powell et al. (2009) examined a large sample of dropwindsonde data and found a mean inflow angle of −23°, 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 −20° to −25° outside of the eyewall, but decreasing to less than −5° at the radius of maximum wind (*R*_{max}), 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 (1999). The near-surface fall speed of a dropwindsonde is 12–14 m s^{−1}, 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 s^{−1}. 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.

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

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*, *υ*) 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. 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 (*θ*). The storm center positions have been determined using storm tracks produced by NOAA/HRD based on the flight-level wind data (Willoughby and Chelmlow 1982, hereafter WC82). Radial distances are normalized by the estimated radius of maximum wind speed, *R*_{max}, determined from approximately concurrent SFMR surface wind observations (*r** = *r*/*R*_{max})_{.} The *R*_{max} 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 *R*_{max} (Fig. 2b) we find the largest number of sondes is clustered around *r** = 1, corresponding to eyewall deployments, with a secondary peak in the number of sondes deployed near the storm center.

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) 0.5*r**.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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) 0.5*r**.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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) 0.5*r**.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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 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 (*V*_{max} > 33 m s^{−1}, where *V*_{max} is the maximum 1-min wind speed), and radially outward to *r** = 12.5. For each dropwindsonde, *V*_{max} and storm speed (*V _{s}*) 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

*V*

_{max},

*R*

_{max}, and

*V*indicate that observations represent a broad spectrum of storms (Fig. 3). Storm intensities range between 33 <

_{s}*V*

_{max}< 77 m s

^{−1}, sizes between 10 <

*R*

_{max}< 72 km, and motion speeds between 0.8 <

*V*< 12.3 m s

_{s}^{−1}. The median storm intensity for the whole sample is

*V*

_{max}= 56.7 m s

^{−1}(Saffir–Simpson category 3), radius of maximum wind is

*R*

_{max}= 31.8 km, and storm motion speed is

*V*= 5.5 m s

_{s}^{−1}.

Frequency distribution of dropwindsondes according to the corresponding maximum wind speed (*V*_{max}, m s^{−1}), radius of maximum wind speed (*R*_{max}, km), and storm motion speed (*V _{s}*, m s

^{−1}).

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Frequency distribution of dropwindsondes according to the corresponding maximum wind speed (*V*_{max}, m s^{−1}), radius of maximum wind speed (*R*_{max}, km), and storm motion speed (*V _{s}*, m s

^{−1}).

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Frequency distribution of dropwindsondes according to the corresponding maximum wind speed (*V*_{max}, m s^{−1}), radius of maximum wind speed (*R*_{max}, km), and storm motion speed (*V _{s}*, m s

^{−1}).

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

The inflow angle^{1} (*α*) is defined as the arctangent of the ratio of radial (*υ _{r}*) to tangential (

*υ*) wind components [

_{t}*α*= tan

^{−1}(

*υ*/

_{r}*υ*)]. Note that storm-relative inflow angle (

_{t}*α*

_{SR}) is used exclusively throughout this study. 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 ±90° half-plane, eliminating the possibility of anticyclonic flow (

*υ*< 0). The frequency distribution of

_{t}*α*

_{SR}for the initial sample is shown in Fig. 4a. The distribution is super-Gaussian (normalized kurtosis

*k*= +2.7), 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

*α*

_{SR}becomes more normal (Fig. 4b), suggesting an improved representation of the expected inflow angle in tropical cyclones.

Frequency distribution of storm-relative inflow angle (*α*_{SR}, °) for (a) full sample and for (b) sondes radially outward of *r** = 0.5. Sample size (*n*), mean (*μ*), standard deviation (*σ*), and normalized kurtosis (*k*) are indicated. Dashed lines represent normal distributions for the given *μ* and *σ* of each sample.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Frequency distribution of storm-relative inflow angle (*α*_{SR}, °) for (a) full sample and for (b) sondes radially outward of *r** = 0.5. Sample size (*n*), mean (*μ*), standard deviation (*σ*), and normalized kurtosis (*k*) are indicated. Dashed lines represent normal distributions for the given *μ* and *σ* of each sample.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Frequency distribution of storm-relative inflow angle (*α*_{SR}, °) for (a) full sample and for (b) sondes radially outward of *r** = 0.5. Sample size (*n*), mean (*μ*), standard deviation (*σ*), and normalized kurtosis (*k*) are indicated. Dashed lines represent normal distributions for the given *μ* and *σ* of each sample.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Because the accuracy of *υ _{r}* and

*υ*, and therefore

_{t}*α*

_{SR}, 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,

*σ*) 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

*σ*= 18.3° as indicated in Fig. 4b) versus

*r**, where the sample median (minimum)

*R*

_{max}of 32 (10) km is used to normalize the radial distance. For comparison, a 2-km center position error-induced 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 18.3° outside of

*r** = 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.

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 (*σ* = 18.43°) indicated in Fig. 4b. Solid lines are for 5-km error and dashed lines are for 2-km error. Black lines are for *R*_{max} = 32 km, representing the sample median, and gray lines are for *R*_{max} = 10 km, representing the sample minimum.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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 (*σ* = 18.43°) indicated in Fig. 4b. Solid lines are for 5-km error and dashed lines are for 2-km error. Black lines are for *R*_{max} = 32 km, representing the sample median, and gray lines are for *R*_{max} = 10 km, representing the sample minimum.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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 (*σ* = 18.43°) indicated in Fig. 4b. Solid lines are for 5-km error and dashed lines are for 2-km error. Black lines are for *R*_{max} = 32 km, representing the sample median, and gray lines are for *R*_{max} = 10 km, representing the sample minimum.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Although data are included all the way into the estimated center, inflow angles ±2*σ* away from the sample mean of −22.6° indicated in Fig. 4b are rejected as highly unrepresentative, which restricts the range of acceptable 10-m level, storm-relative inflow angles between −59.2° and +14.0°, 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.

## 3. Data analysis results

### a. Axisymmetric distribution

Figures 6a,b show the storm-relative inflow angle, *α*_{SR}, as functions of local storm-relative wind speed (*U*_{10SR}) and *r**, respectively. Linear regression of *α*_{SR} on *U*_{10SR} (Fig. 6a) indicates little dependence of the angle on wind speed. In contrast to wind speed independence, a significant increase in *α*_{SR} 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 (−0.53° ± 0.23° per *r** units; i.e., significantly different from zero slope based on a Student’s *t* test). This significant dependence of *α*_{SR} on *r** appears especially well pronounced between the eyewall and just outside of the eyewall as indicated by the bin-averaged values. This potentially important detail in the inflow angle variation near the eyewall will be investigated in future work as our focus here is on the overall structural variability of the inflow angle throughout a tropical cyclone. With reasonably even azimuthal sampling, this result (Fig. 6b) describes the axisymmetric mean inflow angle radial profile.

Storm-relative inflow angle (*α*_{SR}, °) as a function of (a) 10-m storm-relative wind speed (*U*_{10SR}, m s^{−1}) 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 (*r*^{2}), and trend lines with 95% confidence intervals for parameters are indicated.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Storm-relative inflow angle (*α*_{SR}, °) as a function of (a) 10-m storm-relative wind speed (*U*_{10SR}, m s^{−1}) 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 (*r*^{2}), and trend lines with 95% confidence intervals for parameters are indicated.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Storm-relative inflow angle (*α*_{SR}, °) as a function of (a) 10-m storm-relative wind speed (*U*_{10SR}, m s^{−1}) 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 (*r*^{2}), and trend lines with 95% confidence intervals for parameters are indicated.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Further stratification of *α*_{SR} among weak/strong, small/large, and fast/slow-moving storms is shown in Figs. 7a–f. The data are divided into *V*_{max}, *R*_{max}, and *V _{s}* 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 ~5° larger for more intense storms (Fig. 7d). Based on this result, we further stratify the inflow angle according to five intensity groups (

*V*

_{max}= 33–42.5, 37.5–52.5, 47.5–62.5, 57.5–72.5, and 67.5–77 m s

^{−1}) and six radial band groups (

*r** = 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

*α*

_{SR}versus

*V*

_{max}for the six radial bands, along with linear regression fits. An increasing slope of

*α*

_{SR}versus

*V*

_{max}, 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 (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.

Storm-relative inflow angle (°) as a function of 10-m wind speed (m s^{−1}) stratified according to (a) *V*_{max}, (b) *R*_{max}, and (c) *V _{s}*. (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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Storm-relative inflow angle (°) as a function of 10-m wind speed (m s^{−1}) stratified according to (a) *V*_{max}, (b) *R*_{max}, and (c) *V _{s}*. (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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Storm-relative inflow angle (°) as a function of 10-m wind speed (m s^{−1}) stratified according to (a) *V*_{max}, (b) *R*_{max}, and (c) *V _{s}*. (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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Storm-relative inflow angle (*α*_{SR}, °) as a function of storm intensity (*V*_{max}, m s^{−1}) 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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Storm-relative inflow angle (*α*_{SR}, °) as a function of storm intensity (*V*_{max}, m s^{−1}) 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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Storm-relative inflow angle (*α*_{SR}, °) as a function of storm intensity (*V*_{max}, m s^{−1}) 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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

### b. Asymmetric distribution

Although the axisymmetric mean storm-relative inflow angle (*α*_{SR}) 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 turn our attention to resolving the asymmetric structure. Figure 9 shows the storm motion direction-relative 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 *V _{s}* = 5.5 m s

^{−1}. 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

*α*

_{SR}on the storm motion speed. Relatively larger

*α*

_{SR}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.

Azimuthal variation of inflow angle (left) near the eyewall and (right) outside the core (2.5 < *r** < 7.5). Observations (×, o), bin averages (solid squares with error bars), and least squares fits (solid lines) are grouped according the storm speed: *V _{s}* < 5.5 m s

^{−1}(blue) and

*V*> 5.5 m s

_{s}^{−1}(red). Phase is measured clockwise from the front of the storm.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Azimuthal variation of inflow angle (left) near the eyewall and (right) outside the core (2.5 < *r** < 7.5). Observations (×, o), bin averages (solid squares with error bars), and least squares fits (solid lines) are grouped according the storm speed: *V _{s}* < 5.5 m s

^{−1}(blue) and

*V*> 5.5 m s

_{s}^{−1}(red). Phase is measured clockwise from the front of the storm.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Azimuthal variation of inflow angle (left) near the eyewall and (right) outside the core (2.5 < *r** < 7.5). Observations (×, o), bin averages (solid squares with error bars), and least squares fits (solid lines) are grouped according the storm speed: *V _{s}* < 5.5 m s

^{−1}(blue) and

*V*> 5.5 m s

_{s}^{−1}(red). Phase is measured clockwise from the front of the storm.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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, *α*_{SR} versus azimuthal direction, *θ*, is grouped according to the storm motion speed (*V _{s}* = 0–3.6, 3.6–5.5, 5.5–7.2, and >7.2 m s

^{−1}), based on the sample distribution quartiles, and the radial band groups as earlier defined. Harmonic functions are fit to the observed

*α*

_{SR}versus

*θ*data to estimate the asymmetry amplitude (

*A*

_{α}_{1}) and phase (

*P*

_{α}_{1}) for each group. The resulting amplitude is normalized by the mean,

*A*

_{α}_{0}, 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

*α*

_{SR}, increasing as the storm motion speed increases, especially inward of

*r** = 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

*α*

_{SR}asymmetry (

*P*

_{α}_{1}), defined as the azimuthal direction of the maximum inflow, is plotted as a function of

*V*and

_{s}*r** (Figs. 11a–f). At all radii, the peak

*α*

_{SR}is found to the right and right rear of the storm (between +90° and +135° azimuth) for slower storms, and rotates downwind toward the front of the storm (0° to +45°) as

*V*increases. At all radii, linear trends in the asymmetry amplitude and phase with increasing

_{s}*V*are statistically significant. There is some hint of a quadratic dependence of the phase on

_{s}*V*, as the phase shift appears to reverse when a critical speed of

_{s}*V*≈ 6 m s

_{s}^{−1}is reached; however, the sample statistics are not currently satisfactory to confidently resolve whether this is significant.

Inflow angle asymmetry amplitude (*A _{α}*

_{1}, °), normalized by the symmetric mean (−

*A*

_{α}_{o}, °), as a function of storm motion speed (

*V*, m s

_{s}^{−1}) 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**=10.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Inflow angle asymmetry amplitude (*A _{α}*

_{1}, °), normalized by the symmetric mean (−

*A*

_{α}_{o}, °), as a function of storm motion speed (

*V*, m s

_{s}^{−1}) 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**=10.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Inflow angle asymmetry amplitude (*A _{α}*

_{1}, °), normalized by the symmetric mean (−

*A*

_{α}_{o}, °), as a function of storm motion speed (

*V*, m s

_{s}^{−1}) 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**=10.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

As in Fig. 10, but for asymmetry phase (*P _{α}*

_{1}, °). Linear trends are significant at the 95% confidence level at all radii. Phase is measured clockwise from the front of the storm.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

As in Fig. 10, but for asymmetry phase (*P _{α}*

_{1}, °). Linear trends are significant at the 95% confidence level at all radii. Phase is measured clockwise from the front of the storm.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

As in Fig. 10, but for asymmetry phase (*P _{α}*

_{1}, °). Linear trends are significant at the 95% confidence level at all radii. Phase is measured clockwise from the front of the storm.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

## 4. Parametric inflow angle model

### a. Model development

*α*

_{SR}(

*r**,

*θ*) in a tropical cyclone can be constructed provided the storm intensity (

*V*

_{max}), and storm motion speed (

*V*) 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 storm-relative inflow angle,

_{s}*α*

_{SR}, according to the following relationship:

*A*

_{α}_{0}, was found to depend primarily on

*r** and

*V*

_{max}(Fig. 7d). Based on this result, a linear function is fit to the

*r**,

*V*

_{max}) binned observations. The function assumes the linear form:

*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

*R*

_{max}, while the largest inflow angle is found in the most intense storms well outside of the eyewall.

Model-estimated axisymmetric inflow angle (*A*_{a0}) as functions of storm intensity (*V*_{max}, m s^{−1}) 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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Model-estimated axisymmetric inflow angle (*A*_{a0}) as functions of storm intensity (*V*_{max}, m s^{−1}) 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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Model-estimated axisymmetric inflow angle (*A*_{a0}) as functions of storm intensity (*V*_{max}, m s^{−1}) 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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

*r*

^{2}= 0.77), while the asymmetry amplitude is the least (

*r*

^{2}= 0.17).

(a) Asymmetric storm-relative inflow angle model normalized amplitude (−*A _{α}*

_{1}/

*A*

_{α}_{0}) and (b) phase (

*P*

_{α}_{1}, °) as functions of storm motion speed (

*V*

_{s}, m s

^{−1}), 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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

(a) Asymmetric storm-relative inflow angle model normalized amplitude (−*A _{α}*

_{1}/

*A*

_{α}_{0}) and (b) phase (

*P*

_{α}_{1}, °) as functions of storm motion speed (

*V*

_{s}, m s

^{−1}), 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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

(a) Asymmetric storm-relative inflow angle model normalized amplitude (−*A _{α}*

_{1}/

*A*

_{α}_{0}) and (b) phase (

*P*

_{α}_{1}, °) as functions of storm motion speed (

*V*

_{s}, m s

^{−1}), 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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

### 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 < *V _{s}*

*<*8 m s

^{−1}) and intensities (35 <

*V*

_{max}< 65 m s

^{−1}). Both the increase in the

*α*

_{SR}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

*α*

_{SR}well outside of the inner core suggests that on average, angles of

*α*

_{SR}< −50° 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

*α*

_{SR}> 0° in any case.

Storm-relative inflow angle field computed by the parametric model for storm motion speeds of *V _{s}* = 2, 4, 6, and 8 m s

^{−1}(columns) and intensities of

*V*

_{max}= 35, 45, 55, and 65 m s

^{−1}(rows). In all panels storm direction is toward the top.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Storm-relative inflow angle field computed by the parametric model for storm motion speeds of *V _{s}* = 2, 4, 6, and 8 m s

^{−1}(columns) and intensities of

*V*

_{max}= 35, 45, 55, and 65 m s

^{−1}(rows). In all panels storm direction is toward the top.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Storm-relative inflow angle field computed by the parametric model for storm motion speeds of *V _{s}* = 2, 4, 6, and 8 m s

^{−1}(columns) and intensities of

*V*

_{max}= 35, 45, 55, and 65 m s

^{−1}(rows). In all panels storm direction is toward the top.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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 faster-moving storms. This is in addition to the fact that slower-moving 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 validation. Applying the same quality-control criteria as for the dependent sample results in an independent dataset of *n* = 621 observations. To ensure that both samples are drawn from the same *α*_{SR} 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.

Cumulative probability distributions of storm-relative inflow angle (*α*_{SR}, °) for model development dependent sample (dashed line) and independent evaluation sample (solid line).

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Cumulative probability distributions of storm-relative inflow angle (*α*_{SR}, °) for model development dependent sample (dashed line) and independent evaluation sample (solid line).

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Cumulative probability distributions of storm-relative inflow angle (*α*_{SR}, °) for model development dependent sample (dashed line) and independent evaluation sample (solid line).

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

For each independent inflow angle observation, the parent storm’s parameters (i.e., *V*_{max}, *V*_{s}, and *R*_{max}), 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 14.6°, or an improvement of around 3.7° (~20%) over simply using a mean value. Considering observations within ±1*σ* of the mean value, as shown by the cumulative distribution function (Fig. 16b), the model’s accuracy increases to within 11.9° (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 small-scale fluctuations are not considered to be important for a particular application, then this smoothing may be a desirable result.

Observed vs model-predicted storm-relative inflow angle (*α*_{SR}, °). (left) Paired samples with linear regression (thick black line) statistics: number of observations (*n*), RMSE of the residual, and correlation coefficient (*r*^{2}) are indicated. (right) Cumulative probability distributions for observations (solid line) and model (dashed line).

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Observed vs model-predicted storm-relative inflow angle (*α*_{SR}, °). (left) Paired samples with linear regression (thick black line) statistics: number of observations (*n*), RMSE of the residual, and correlation coefficient (*r*^{2}) are indicated. (right) Cumulative probability distributions for observations (solid line) and model (dashed line).

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

Observed vs model-predicted storm-relative inflow angle (*α*_{SR}, °). (left) Paired samples with linear regression (thick black line) statistics: number of observations (*n*), RMSE of the residual, and correlation coefficient (*r*^{2}) are indicated. (right) Cumulative probability distributions for observations (solid line) and model (dashed line).

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

### 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).

#### 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 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 333° 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 angles computed from Frederic’s input parameters: *V _{s}* = 5 m s

^{−1},

*R*

_{max}= 33 km, and

*V*

_{max}= 45 m s

^{−1}. 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 (

*r*

^{2}= 0.80), although the model underrepresents the dynamic range of observed angles. The axisymmetric storm-relative inflow angle (Fig. 17c) is not significantly different from the dropwindonsde-observed average value. There is a small linear increase in the inflow angle with radial distance, although with relatively small sample of observations (

*n*= 24) in the Frederic analysis, the trend is not statistically significant.

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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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 *α*_{SR} ranges from −10° to −40°, while inflow angles in Hurricane Frederic were found to vary approximately between +15° and −60°. 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 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 90° 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 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 (<−25°) 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.

As in Figs. 17a,b, but for three sample periods (columns) in Hurricane Earl.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

As in Figs. 17a,b, but for three sample periods (columns) in Hurricane Earl.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

As in Figs. 17a,b, but for three sample periods (columns) in Hurricane Earl.

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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** = 1, where the inflow is maximized around 40° 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 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.

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

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

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

Citation: Monthly Weather Review 140, 11; 10.1175/MWR-D-11-00339.1

## 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 −22.6° ± 2.2°, with 95% statistical confidence, which agrees well with previous results (P82; Powell et al. 2009). There, is however, a large amount of variability (*σ* = 18.3°) 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 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 de-alias 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.

## 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 Sim Aberson for constructive comments on the early version of this paper. Finally, we also thank the two anonymous reviewers for their constructive comments, which substantially improved our paper.

## REFERENCES

Barnes, G. M., 2008: Atypical thermodynamic profiles in hurricanes.

,*Mon. Wea. Rev.***136**, 631–643.Bender, M. A., and I. Ginis, 2000: Real-case simulations of hurricane–ocean interaction using a high-resolution coupled model: Effects on hurricane intensity.

,*Mon. Wea. Rev.***128**, 917–946.Black, P. G., and Coauthors, 2007: Air–sea exchange in hurricanes: Synthesis of observations from the Coupled Boundary Layer Air–Sea Transfer experiment.

,*Bull. Amer. Meteor. Soc.***88**, 357–374.Brennan, M. J., C. C. Hennon, and R. D. Knabb, 2009: The operational use of QuikSCAT ocean surface vector winds at the National Hurricane Center.

,*Wea. Forecasting***24**, 621–645.Bryan, G. H., 2012: Effects of surface exchange coefficients and turbulence length scales on the intensity and structure of numerically simulated hurricanes.

,*Mon. Wea. Rev.***140**, 1125–1143.Bryan, G. H., and R. Rotunno, 2009: The maximum intensity of tropical cyclones in axisymmetry numerical model simulations.

,*Mon. Wea. Rev.***137**, 1770–1789.Cangialosi, J. P., 2011: Tropical cyclone report: Hurricane Earl (AL072010) 25 August–4 August 2010. National Hurricane Center, 29 pp. [Available online at http://www.nhc.noaa.gov/pdf/TCR-AL072010_Earl.pdf.]

Cione, J. J., and E. W. Uhlhorn, 2003: Sea surface temperature variability in hurricanes: Implications with respect to intensity change.

,*Mon. Wea. Rev.***131**, 1783–1796.Cione, J. J., P. G. Black, and S. H. Houston, 2000: Surface observation in hurricane environment.

,*Mon. Wea. Rev.***128**, 1550–1561.Connor, L. N., and P. F. Chang, 2000: Ocean surface wind retrievals using the TRMM Microwave Imager.

,*IEEE Trans. Geosci. Remote Sens.***38**(4), 2009–2016.DeMaria, M., and J. Kaplan, 1994: A Statistical Hurricane Intensity Prediction Scheme (SHIPS) for the Atlantic basin.

,*Wea. Forecasting***9**, 209–220.Donelan, M. A., W. M. Drennan, and K. B. Katsaros, 1997: The air–sea momentum flux in mixed wind sea and swell conditions.

,*J. Phys. Oceanogr.***27**, 2087–2099.Drennan, W. M., K. K. Kahma, and M. A. Donelan, 1999: On momentum flux and velocity spectra over waves.

,*Bound.-Layer Meteor.***92**, 489–515.Drennan, W. M., H. C. Graber, D. Hauser, and C. Quentin, 2003: On the wave age dependence of wind stress over pure wind seas.

,*J. Geophys. Res.***108**, 8062, doi:10.1029/2000JC00715.Foster, R. C., 2009: Boundary-layer similarity under an axisymmetric, gradient wind vortex.

,*Bound.-Layer Meteor.***131**, 321–344.Frank, W. M., 1984: A composite analysis of the core of a mature hurricane.

,*Mon. Wea. Rev.***112**, 2401–2420.Franklin, J. L., M. L. Black, and K. Valde, 2003: GPS dropwindsonde wind profiles in hurricanes and their operational implications.

,*Wea. Forecasting***18**, 32–44.French, J. R., W. M. Drennan, J. A. Zhang, and P. G. Black, 2007: Turbulent fluxes in the hurricane boundary layer. Part I: Momentum flux.

,*J. Atmos. Sci.***64**, 1089–1102.Grachev, A. A., C. W. Fairall, J. E. Hare, J. B. Edson, and S. D. Miller, 2003: Wind stress vector over ocean waves.

,*J. Phys. Oceanogr.***33**, 2408–2429.Halliwell, G. A., Jr., L. K. Shay, J. K. Brewser, and W. J. Teague, 2011: Evaluation and sensitivity analysis of an ocean model response to Hurricane Ivan.

,*Mon. Wea. Rev.***139**, 921–945.Haus, B., D. Jeong, M. A. Donelan, J. A. Zhang, and I. Savelyev, 2010: Relative rates of air-sea heat transfer and frictional drag in very high winds.

,*Geophys. Res. Lett.***37**, L07802, doi:10.1029/2009GL042206.Hock, T. F., and J. L. Franklin, 1999: The NCAR GPS dropwindsonde.

,*Bull. Amer. Meteor. Soc.***80**, 407–420.Holland, G. J., 1980: An analytic model of the wind and pressure profiles in hurricanes.

,*Mon. Wea. Rev.***108**, 1212–1218.Holland, G. J., J. I. Belanger, and A. Fritz, 2010: A revised model for radial profiles of hurricane winds.

,*Mon. Wea. Rev.***138**, 4393–4401.Jacob, S. D., L. K. Shay, A. J. Mariano, and P. G. Black, 2000: The 3D oceanic mixed layer response to Hurricane Gilbert.

,*J. Phys. Oceanogr.***30**, 1407–1429.Jaimes, B., and L. K. Shay, 2010: Near-inertial wave wake of Hurricanes Katrina and Rita over mesoscale oceanic eddies.

,*J. Phys. Oceanogr.***40**, 1320–1337.Jarvinen, B. R., C. J. Newmann, and M. A. S. Davis, 1984: A tropical cyclone data tape for the North Atlantic Basin, 1886-1983: Contents, limitations, and uses. Tech. Rep. 22, NOAA Tech Memo., NWS/NHC, Miami, FL, 21 pp.

Katsaros, K. B., 2010: Discoveries about tropical cyclones provided by microwave remote sensing.

*Oceanography from Space: Revisited,*V. Barale, J. F. R. Gower, and L. Alberotanza, Eds., Springer, 59–71.Kepert, J. D., 2001: The dynamics of boundary layer jets within the tropical cyclone core. Part I: Linear theory.

,*J. Atmos. Sci.***58**, 2469–2484.Kepert, J. D., 2005: Objective analysis of tropical cyclone location and motion from high density observations.

,*Mon. Wea. Rev.***133**, 2406–2421.Kepert, J. D., 2006a: Observed boundary layer wind structure and balance in the Hurricane core. Part I: Hurricane Georges.

,*J. Atmos. Sci.***63**, 2169–2193.Kepert, J. D., 2006b: Observed boundary layer wind structure and balance in the Hurricane core. Part II: Hurricane Mitch.

,*J. Atmos. Sci.***63**, 2194–2211.Kepert, J. D., 2010a: Slab- and height-resolving models of the tropical cyclone boundary layer. Part I: Comparing the simulations.

,*Quart. J. Roy. Meteor. Soc.***136**, 1686–1699, doi:10.1002/qj.667.Kepert, J. D., 2010b: Slab- and height-resolving models of the tropical cyclone boundary layer. Part II: Why the simulations differ.

,*Quart. J. Roy. Meteor. Soc.***136**, 1700–1711, doi:10.1002/qj.685.Kepert, J. D., and Y. Wang, 2001: The dynamics of boundary layer jets within the tropical cyclone core. Part II: Nonlinear enhancement.

,*J. Atmos. Sci.***58**, 2485–2501.Knaff, J. A., and R. M. Zehr, 2007: Reexamination of tropical cyclone wind–pressure relationships.

,*Wea. Forecasting***22**, 71–88.Kwon, I., and H. Cheong, 2010: Tropical cyclone initialization with a spherical high-order filter and an idealized three-dimensional bogus vortex.

,*Mon. Wea. Rev.***138**, 1344–1367.Malkus, J. S., and H. Riehl, 1960: On the dynamics and energy transformations in steady-state hurricanes.

,*Tellus***12**, 1–20.Mallen, K. J., M. T. Montgomery, and B. Wang, 2005: Reexamining the near-core radial structure of the tropical cyclone primary circulation: Implications for vortex resiliency.

,*J. Atmos. Sci.***62**, 408–425.Massey, F. J., 1951: The Kolmogorov–Smirnov test for goodness of fit.

,*J. Amer. Stat. Assoc.***46**, 68–78.Montgomery, M. T., R. K. Smith, and S. V. Nguyen, 2010: Sensitivity of tropical cyclone models to the surface drag coefficient.

,*Quart. J. Roy. Meteor. Soc.***136**, 1945–1953.Moon, I.-J., I. Ginis, T. Hara, and B. Thomas, 2007: A physics-based parameterization of air–sea momentum flux at high wind speeds and its impact on hurricane intensity predictions.

,*Mon. Wea. Rev.***135**, 2869–2878.Nolan, D. S., and M. T. Montgomery, 2000: The algebraic growth of wavenumber 1 disturbances in hurricane-like vortices.

,*J. Atmos. Sci.***57**, 3514–3538.Nolan, D. S., J. A. Zhang, and D. P. Stern, 2009a: Evaluation of planetary boundary layer parameterizations in tropical cyclones by comparison of in situ data and high-resolution simulations of Hurricane Isabel (2003). Part I: Initialization, maximum winds, and outer core boundary layer structure.

,*Mon. Wea. Rev.***137**, 3651–3674.Nolan, D. S., J. A. Zhang, and D. P. Stern, 2009b: Evaluation of planetary boundary layer parameterizations in tropical cyclones by comparison of in situ data and high-resolution simulations of Hurricane Isabel (2003). Part II: Inner core boundary layer and eyewall structure.

,*Mon. Wea. Rev.***137**, 3675–3698.Peng, M., L. Xie, and L. J. Pietrafesa, 2006: Tropical cyclone induced asymmetry of sea level surge and fall and its presentation in a storm surge model with parametric wind fields.

,*Ocean Modell.***14**, 81–101, doi:10.1016/j.ocemod.2006.03.004.Phadke, A., C. Martino, K. F. Cheung, and S. H. Houston, 2003: Modeling of tropical cyclone winds and waves for emergency management.

,*Ocean Eng.***30**, 553–578.Powell, M. D., 1982: The transition of the Hurricane Frederic boundary-layer wind field from the open Gulf of Mexico to landfall.

,*Mon. Wea. Rev.***110**, 1912–1932.Powell, M. D., S. H. Houston, L. R. Amat, and N. Morisseau-Leroy, 1998: The HRD real-time hurricane wind analysis system.

,*J. Wind Eng. Ind. Aerodyn.***77**, 53–64.Powell, M. D., P. J. Vickery, and T. A. Reinhold, 2003: Reduced drag coefficient for high wind speeds in tropical cyclones.

,*Nature***422**, 279–283.Powell, M. D., E. W. Uhlhorn, and J. D. Kepert, 2009: Estimating maximum surface winds from hurricane reconnaissance measurements.

,*Wea. Forecasting***24**, 868–883.Price, J. F., 1983: Internal wave wake of a moving storm. Part I: Scales, energy budget, and observations.

,*J. Phys. Oceanogr.***13**, 949–965.Rego, J. L., and C. Li, 2009: On the importance of the forward speed of hurricanes in storm surge forecasting: A numerical study.

,*Geophys. Res. Lett.***36**, L07609, doi:10.1029/2008GL036953.Rogers, R., and Coauthors, 2006: The Intensity Forecasting Experiment: A NOAA multiyear field program for improving tropical cyclone intensity forecasts.

,*Bull. Amer. Meteor. Soc.***87**, 1523–1537.Rogers, R., S. Lorsolo, P. Reasor, J. Gamache, and F. Marks, 2012: Multiscale analysis of tropical cyclone kinematic structure from airborne Doppler radar composites.

,*Mon. Wea. Rev.***140**, 77–99.Shapiro, L. J., 1983: The asymmetric boundary layer flow under a translating hurricane.

,*J. Atmos. Sci.***40**, 1984–1998.Shay, L. K., and E. W. Uhlhorn, 2008: Loop current response to Hurricanes Isidore and Lili.

,*Mon. Wea. Rev.***136**, 3248–3274.Shay, L. K., R. L. Elsberry, and P. G. Black, 1989: Vertical structure of the ocean current response to a hurricane.

,*J. Phys. Oceanogr.***19**, 649–669.Shen, H., Y. He, and W. Perrie, 2009: Speed ambiguity in hurricane wind retrieval from SAR imagery.

,*Int. J. Remote Sens.***30**(11), 2827–2836.Smith, R. K., and M. T. Montgomery, 2010: Hurricane boundary-layer theory.

,*Quart. J. Roy. Meteor. Soc.***136**, 1665–1670.Thompson, R. O. R. Y., 1974: The influence of geostrophic shear on the cross-isobar angle of the surface wind.

,*Bound.-Layer Meteor.***6**, 515–518.Uhlhorn, E. W., and P. G. Black, 2003: Verification of remotely sensed sea surface winds in hurricanes.

,*J. Atmos. Oceanic Technol.***20**, 99–116.Uhlhorn, E. W., and L. K. Shay, 2012: Loop current mixed layer energy response to Hurricane Lili (2002). Part I: Observations.

,*J. Phys. Oceanogr.***42**, 400–419.Uhlhorn, E. W., P. G. Black, J. L. Franklin, M. Goodberlet, J. Carswell, and A. S. Goldstein, 2007: Hurricane surface wind measurements from an operational stepped frequency microwave radiometer.

,*Mon. Wea. Rev.***135**, 3070–3085.Velden, C., and Coauthors, 2006: The Dvorak tropical cyclone intensity estimation technique: A satellite-based method that has endured for over 30 years.

,*Bull. Amer. Meteor. Soc.***87**, S6–S9.Westerink, J., and Coauthors, 2008: A basin to channel scale unstructured grid hurricane storm surge model applied to southern Louisiana.

,*Mon. Wea. Rev.***136**, 833–864.Willoughby, H. E., and M. B. Chelmlow, 1982: Objective determination of hurricane tracks from aircraft observations.

,*Mon. Wea. Rev.***110**, 1298–1305.Willoughby, H. E., R. W. R. Darling, and M. E. Rahn, 2006: Parametric representation of the primary hurricane vortex. Part II: A new family of sectionally continuous profiles.

,*Mon. Wea. Rev.***134**, 1102–1120.Yablonsky, R. M., and I. Ginis, 2009: Limitation of one-dimensional ocean models for coupled hurricane-ocean model forecasts.

,*Mon. Wea. Rev.***137**, 4410–4419.Zhang, J. A., 2010: Estimation of dissipative heating using low-level in situ aircraft observations in the hurricane boundary layer.

,*J. Atmos. Sci.***67**, 1853–1862.Zhang, J. A., and M. T. Montgomery, 2012: Observational estimates of the horizontal eddy diffusivity and mixing length in the low-level region of intense hurricanes.

,*J. Atmos. Sci.***69**, 1306–1316.Zhang, J. A., P. G. Black, J. R. French, and W. M. Drennan, 2008: First direct measurements of enthalpy flux in the hurricane boundary layer: The CBLAST results.

,*Geophys. Res. Lett.***35**, L14813, doi:10.1029/2008GL034374.Zhang, J. A., W. M. Drennan, P. G. Black, and J. R. French, 2009: Turbulence structure of the hurricane boundary layer between the outer rainbands.

,*J. Atmos. Sci.***66**, 2455–2467.Zhang, J. A., R. F. Rogers, D. S. Nolan, and F. D. Marks, 2011a: On the characteristic height scales of the hurricane boundary layer.

,*Mon. Wea. Rev.***139**, 2523–2535.Zhang, J. A., F. D. Marks, M. T. Montgomery, and S. Lorsolo, 2011b: Estimation of turbulence characteristics of the eyewall boundary layer of Hurricane Hugo (1989).

,*Mon. Wea. Rev.***139**, 1447–1462.Zhao, W., and X. Hong, 2011: Impacts of tropical cyclone inflow angle on ocean surface waves.

,*Chin. J. Oceanol. Limnol.***29**, 460–469.

^{1}

Inflow is defined as *υ _{r}* < 0, although we still refer to “inflow angle” when outflow (

*υ*> 0) occurs. Also, a “larger inflow angle” or the like will generally indicate a more negative value throughout this article.

_{r}