## 1. Introduction

The response of tropical cyclone (TC) structure and intensity to vertical shear has been a major area of study and research. It is generally accepted that strong shear acts to weaken TCs and leads to tilting of the vortex with height, but recent studies have documented many other details of the TC vortex and convective response to shear. In this study, we analyze the response of TC eyewall slope (e.g., Stern and Nolan 2009; Stern et al. 2014; Hazelton and Hart 2013, hereafter HH13) to vertical shear, specifically investigating how slope varies around the eyewall in a shear-relative reference frame.

The response of the TC vortex to shear has been a topic of intensive study for years. Details of particular interest have included the tilt of the TC vortex and the asymmetric convective response of a TC due to shear. This has been studied using both numerical models and the latest observational techniques. Jones (1995) was one of the first studies to document the vortex tilt and induced circulation due to shear, using a numerical model on an *f* plane. Reasor et al. (2000), in a case study of Hurricane Olivia (1994) using airborne Doppler radar data, found a larger vortex tilt and an increase in convection in the downshear region of the TC as the magnitude of the shear increased. That study also found a wavenumber-2 asymmetry in the vorticity field below a height of 3 km. Frank and Ritchie (2001) used the National Center for Atmospheric Research (NCAR) Nonhydrostatic Mesoscale Model (NMM) to study the asymmetries and intensity change induced by shear. They found that a pronounced wavenumber-1 asymmetry developed and that the TCs tended to weaken more quickly when shear magnitude increased. This numerical study also found that rising motion tended to be concentrated on the left side of the shear vector. Jones (2004) used an adiabatic simulation from a Boussinesq numerical model to document the variation of TC vortex response to shear: in some cases, the vortex remains resilient in the presence of shear for multiple days, while in other cases, the vortex quickly falls apart because of shear. Reasor and Eastin (2012) estimated vortex tilt in Hurricane Guillermo (1997) using airborne radar data, and found that the TC was fairly resilient in the presence of shear, with a small left-of-shear tilt. Several mechanisms have been proposed for this resiliency, including upshear precession of the vortex (Jones 1995) and vortex Rossby wave damping (e.g., Reasor and Montgomery 2001; Schecter et al. 2002). Riemer et al. (2010) proposed a mechanism for TC weakening due to shear, in which downdrafts due to the shear-induced wavenumber-1 asymmetry cause lower theta-e air to enter the eyewall, reducing the intensity of the TC heat engine.

Corbosiero and Molinari (2002, 2003) provided some of the first comprehensive observational analysis of the TC convective response to shear, using lightning data as a proxy for deep convection, and found that convection occurred preferentially in the downshear-left quadrant of a TC. Black et al. (2002) also performed detailed analysis of the effects of vertical shear on the convective structure of two eastern Pacific hurricanes. Their study suggested that convective bands aid a TC in resisting the effects of shear. They also found that shear led to a wavenumber-1 asymmetry in vertical motion and reflectivity, with updrafts initiated in the downshear region and downdrafts primarily found upshear. With this pattern of vertical motion, the maximum reflectivity was found to the left of shear, as the hydrometeors generated downshear were rotated by the primary circulation of the TC. Eastin et al. (2005) noted a similar pattern in the eyewall region of TCs using aircraft observations, noting that buoyant updrafts tended to be concentrated downshear left and left of shear. This downshear and left-of-shear asymmetry in convection and rainfall has been noted by multiple more recent studies, including Reasor et al. (2009), which studied a period of intensification of Hurricane Guillermo in 7–8 m s^{−1} of shear, and found that the convection associated with intensification occurred primarily in the downshear-left quadrant. This shear-relative convective structure was also shown in composite radar analyses by Reasor et al. (2013). Uhlhorn et al. (2014) investigated the wavenumber-1 asymmetry in low-level wind speed in TCs due to motion and shear by analyzing Stepped Frequency Microwave Radiometer (SFMR) and flight-level data for 128 flights into 35 different hurricanes. This study found that the peak surface wind speed was generally found left of shear, and also found that the impacts of shear became more important than asymmetries due to storm motion as the magnitude of the shear increased, consistent with other prior studies such as Chen et al. (2006).

Eyewall slope is an aspect of TC inner-core structure that is connected to changes in both the horizontal and vertical structure of the vortex and warm core (e.g., Shapiro and Willoughby 1982). Shea and Gray (1973) was the first study to observationally document the outward slope of the radius of maximum wind (RMW) based on wind data at two flight levels. Jorgensen (1984) and Marks (1985) both noted slope in the reflectivity surfaces of the eyewall. Marks and Houze (1987) observed this feature in Hurricane Alicia and also observed the outward slope of the region of maximum winds. Black et al. (1994) analyzed the slope of reflectivity and angular momentum surfaces in a case study of Hurricane Emily (1987). Kepert (2006) invoked frictional processes to explain why the RMW in the boundary layer of a TC slopes even in the absence of a core warm anomaly in this region. More recent studies have attempted to provide a more complete analysis of eyewall slope in TCs using airborne radar data. Stern and Nolan (2009) and Stern et al. (2014) performed a climatology of the slope of the azimuthal mean RMW using airborne Doppler radar data, and found that the RMW slope tended to be related to the size of the RMW but not the intensity of the TC. HH13 performed a climatology of eyewall slope using the 20-dB*Z* contour as a proxy for the edge of the eyewall, and found a relatively strong relationship between slope and eye size, as well as a relationship between 20-dB*Z* slope and TC intensity. HH13 also looked at the difference in the slope of the 20-dB*Z* surface between the downshear and upshear sides of the eyewall. Neither that paper nor any of the other previous slope studies, however, have quantified shear-relative azimuthal asymmetries in eyewall slope for multiple cases.

While the composite analyses of TC eyewall slope for multiple TCs have only considered azimuthal mean slope, there have been a couple of case studies that have briefly mentioned shear-related asymmetries in slope, but no comprehensive analysis of the subject has been performed. Halverson et al. (2006) noted an apparent along-shear asymmetry in the slope of reflectivity surfaces in a case study of Hurricane Erin (2001) based on data from the Fourth Convection and Moisture Experiment (CAMEX-4) field campaign. Rogers and Uhlhorn (2008) looked at azimuthal variation in RMW slope in a case study of Hurricane Rita (in the lower levels, based on surface and flight-level wind data). They found that the change in slope across the eyewall increased as the shear increased over the TC and the vortex became more tilted; however, this study was also limited in scope, as it was only focused on three flights into one TC.

In this study we build on those previous azimuthal mean climatologies and case studies of azimuthal variance by investigating the role of shear in causing azimuthal variation in eyewall slope for a dataset compiled from multiple TCs over the period from 1997 to 2010. Section 2 discusses the radar data used in this study, describes the slope metrics analyzed, and also lists the factors used to select cases suitable for analysis. Section 3 contains the results of the analysis, including overall results on the variation of slope for all cases, as well as a discussion of the results when the data are filtered by shear magnitude and storm intensification. Finally, section 4 discusses physical implications of the results and also lists the conclusions of this study.

## 2. Data and methodology

### a. Radar data used for analysis

The radar data for this project come from the merged Doppler radar analyses from research flights by the National Oceanic and Atmospheric Administration (NOAA) Hurricane Research Division (HRD) P-3 aircraft (Gamache 1997; Reasor et al. 2009). Each merged analysis typically consists of radar data for 3–4 passes across the center of the storm over a period of 3–5 h. The raw data have been processed onto a Cartesian grid with a horizontal resolution of 2 km and a vertical resolution of 0.5 km. The center of the grid is at the center of the storm, and the grid extends 200 km from the center. The vertical extent of the data is 18 km, although most cases only have reliable coverage up to 11–12 km at most, due to the relative scarcity of hydrometeors at upper levels. This dataset has been used extensively in previous composite analyses of TC structure, such as Rogers et al. (2012), Rogers et al. (2013), Reasor et al. (2013), and DeHart et al. (2014).

The vertical shear data come from the developmental dataset for the Statistical Hurricane Intensity Prediction Scheme (SHIPS; DeMaria and Kaplan 1994). The shear vector is obtained from the Global Forecast System (GFS) analyses by removing the TC vortex and then averaging the 850–200-hPa shear over a radius from 0 to 500 km relative to the storm center; thus, it is a measure of the large-scale environmental shear affecting a TC. The shear data are available for the 6-hourly synoptic times (0000, 0600, 1200, and 1800 UTC), and this study used the synoptic time closest to each flight.

To calculate the shear-relative asymmetries in eyewall slope, the radar data were mapped from the Cartesian grid to a polar grid rotated such that the shear vector points to the right (i.e., east), using the methodology of Reasor et al. (2013). This polar grid extends radially outward from the storm center to *r* = 150 km with a 2-km radial resolution and a 5° azimuthal resolution.

### b. Slopes calculated

HH13 analyzed the azimuthal mean slope of the 20-dB*Z* surface in the eyewall. Stern and Nolan (2009) extensively discussed the azimuthal mean slope of the RMW. Stern et al. (2014) recently updated the climatology of the azimuthal mean RMW slopes, including more recent cases, and also looked at the azimuthal mean slope of both momentum surfaces and the 20-dB*Z* surface. In this study, we analyze the azimuthal variance of all three of these metrics. For the slope of an angular momentum (*M*) surface we choose the *M* surface that corresponds with the RMW at *z* = 2 km. By analyzing azimuthal variation in slope of both dB*Z* surfaces and the RMW and *M* surfaces, we can analyze the effects of shear on slope through both convective processes and processes related to wind asymmetries or vortex tilt. HH13 analyzed the slope of the 20-dB*Z* surface over the layer from *z* = 2 to *z* = 11 km; however, in this study, we only look at the slope between *z* = 2 and *z* = 8 km. This is done to maintain consistency with the RMW slope analyses in Stern and Nolan (2009) and Rogers et al. (2012), and is necessary because Doppler wind data are often missing above 8 km. The slope is defined in terms of a ratio (*dr*/*dz*), rather than an angle from the vertical as in HH13, to be consistent with the theoretical analysis of RMW slope by Stern and Nolan (2009). For some of the cases, azimuthal coverage would be sufficient to calculate a slope at each azimuth every 5° around the eyewall; however, to maintain consistency with previous analyses of shear-relative structure, and to account for cases where a small portion of a storm may have had missing data, the slopes are calculated from the data averaged over each of four shear-relative quadrants: downshear left (DSL), upshear left (USL), upshear right (USR), and downshear right (DSR).

### c. Selection of cases

The database of merged radar analyses currently contains data for 75 different flights into 19 different TCs (18 Atlantic, 1 Pacific) from 1997 to 2010. To get cases that were suitable for this analysis, we filtered out some cases based on several criteria. We removed all cases where the storm was below hurricane strength based on the “best track” (National Hurricane Center 2014) data from the National Hurricane Center (Vmax < 33 m s^{−1}) in order to ensure that all cases had a developing inner-core structure. We also filtered out cases where the center of the TC was less than 50 km from land in order to ensure that the inner-core processes observed were not being driven by land interaction, as well as two cases that were slightly more than 50 km from land but were moving toward land and made landfall less than 12 h after the flight. Almost all of the cases had enough data to calculate azimuthal mean slope, but a few had more than 50% of the data in a quadrant missing (especially wind data). In these situations, the entire case was removed in order to maintain a homogeneous dataset for comparison between the shear-relative quadrants. Finally, subjective analysis identified several cases where a TC was undergoing an eyewall replacement cycle (e.g., Willoughby et al. 1982), and the 20-dB*Z* surface was associated with the inner eyewall while the RMW (and hence momentum surface) was associated with the outer eyewall. Other observational studies have also noted cases where some convection remains in the inner eyewall even as the outer eyewall grows (e.g., Bell et al. 2012). To maintain consistency and allow for a comparison between the different slope metrics, we also removed these cases from the dataset. After applying these filters, the dataset consists of 34 different flights into 14 different TCs (13 Atlantic, 1 Pacific) from 1997 to 2010. All but one of the TCs (Guillermo in 1997) were from the 2003 to 2010 time period. Figure 1 shows the locations of the TCs during the flight legs of interest, based on the best track data.

Table 1 provides a summary of basic information about the cases used, including the dates and times of the flights, the maximum wind speed, maximum potential intensity (MPI) of the TC based on SST (Emanuel 1986, 1988), the shear magnitude and direction from the SHIPS dataset, the intensity change in the 12 h after the flight, and the radar-derived RMW of the TC (at *z* = 2 km). The median intensity of the TCs used in the study was 115 kt (1 kt = 0.5144 m s^{−1}), so the dataset focuses on strong hurricanes with well-developed inner-core regions. The median MPI was 141 kt, and the median difference between MPI and intensity was 24 kt, so that a majority of the TCs were in an environment that was thermodynamically favorable for intensification. The median shear magnitude was 12.9 kt, and the median shear heading was 137° (slightly south of east). The median 12-h intensity change was 0 kt, such that the TCs were spread approximately equally between those that intensified after the period of interest, and those that weakened.

Summary of statistics of the TCs used in the dataset.

## 3. Results

### a. Mean slope composite results

Table 2 summarizes all of the slopes in each quadrant for each of the metrics analyzed, as well as the downshear and upshear mean of each metric. The table also lists the quadrant differences in slope that are statistically significant at the 95% level. Discussion of each of the slope metrics follows.

Mean slopes defined by all three metrics in each shear-relative quadrant. Statistical significance was determined based on one-tailed *t* tests.

Figure 2 (black line) shows the 2–8-km RMW slopes in each shear-relative quadrant. The azimuthal mean RMW slope found by averaging the slopes from all four quadrants was 0.80, very similar to the mean RMW slope values of 0.84 found by Stern and Nolan (2009) and 0.86 found by Stern et al. (2014). Based on a one-tailed *t* test (all statistical tests performed, unless otherwise noted, were one-tailed *t* tests) of the difference in the means, each of the two downshear quadrants has a statistically significant (*p* < 0.05) higher slope than each of the two upshear quadrants. In addition, when the overall downshear versus upshear slopes were compared by averaging the two downshear and two upshear quadrants, the mean upshear slope (0.45) was statistically significantly less (*p* < 0.01) than the downshear mean (1.15). This composite result is consistent with Rogers and Uhlhorn (2008), which found that as the vortex of Hurricane Rita became more tilted due to shear, there was an along-shear asymmetry in RMW slope at lower levels.

Figure 2 (dark gray line) also shows the slopes of the angular momentum surface corresponding to the 2-km RMW. The USR quadrant is the most upright in the mean, and is statistically significantly different from all three other quadrants. The DSL quadrant is most sloped, and is statistically significantly greater in slope than the USR and DSR quadrants (*p* < 0.05). The overall downshear versus upshear contrast is not as clear as in the RMW slope, with the USL quadrant being almost as sloped as the DSL quadrant and the mean downshear slope (1.86) not showing statistical difference from the mean upshear slope (1.49). It should be noted that the *M* slopes tended to be greater than the slopes of the RMW (i.e., *M* decreased with height along the RMW), which is consistent with the azimuthal mean results from Stern and Nolan (2009). The mean *M* slope after averaging over all four quadrants and all 34 cases was 1.68, again very similar to the mean value of 1.75 found in Stern et al. (2014) and much larger than the mean RMW slope. The *M* slope tends to be larger than RMW slope in all four quadrants, as well, as seen in Fig. 2.

Finally, Fig. 2 (light gray line) also shows the azimuthal variation of the slope of the 20-dB*Z* contour. The mean value of 1.89 is higher than that found in Stern et al. (2014), likely because their dataset stopped at *z* = 6 km for the dB*Z* slope, while the slopes extended up to *z* = 8 km in this study. While individual cases showed large variance, overall there was less difference between the four quadrants for the dB*Z* slope than for the other metrics, and there were no statistically significant differences between the quadrants. This seems to indicate that the dB*Z* slope is driven by a combination of vortex tilt and other processes, such as convective generation downshear and subsequent rotation of frozen hydrometeors around a significant portion of the vortex (Marks and Houze 1987; Houze et al. 1992; Rogers et al. 2009). This latter effect was invoked by Rogers et al. (2013) as a possible explanation for a lack of difference in reflectivity structure above 5 km between intensifying and steady-state cases and by DeHart et al. (2014) to explain the shear-relative distribution of reflectivity at lower and upper levels using a similar set of Doppler analyses. In our analysis, it may be that the radar data are observing the TCs in different phases of the rotation of hydrometeors and preventing a significant signal in the dB*Z* slope (in the mean) from either convective generation downshear or vortex tilt. Conversely, as will be discussed later, the dB*Z* slope and its relationship with the *M* slope is an important factor in distinguishing between steady-state and intensifying TCs.

Figures 3–5 show composites of the data used for each slope metric, and further illustrate the variation seen in Fig. 2, although the statistics above are based on calculations from each case individually. Since the TCs had eyewalls of various sizes, the compositing was done by first converting the data from each case into a coordinate system with the radial coordinate defined by distance relative to the RMW at *z* = 2 km (*r** = *r*/RMW). This technique was also used in Rogers et al. (2013) and Reasor et al. (2013). However, since angular momentum is dependent on the physical radius, the angular momentum was calculated for each individual case before compositing. In all three of these figures, example profiles such as those used to calculate slope for the individual cases are shown, in order to allow for a comparison between the different metrics and quadrants.

The results from the composited figures are generally consistent with those found by considering all 34 cases separately. In particular, the downshear side of the eyewall shows a greater slope of the RMW and *M* surfaces, especially at upper levels, and the 20-dB*Z* contour shows the greatest slope in the USR quadrant. This quadrant also has generally lower reflectivity than the others, consistent with the findings of Reasor et al. (2013).

### b. Lower slope versus upper slope composite results

Next, the vertical variation of eyewall slope is examined in a shear-relative sense for each metric. Corbosiero et al. (2005) discussed this vertical difference in slope based on the 10-dB*Z* contour in a study of Hurricane Elena (1985), and HH13 quantified the difference between lower and upper azimuthal mean slope of the 20-dB*Z* surface. Here, we investigate the vertical variation of slope in a shear-relative framework by separating the slopes into a “lower slope” from *z* = 2 to 5 km and an “upper slope” from *z* = 5 to 8 km. The results are shown in Figs. 6a–c. Based on the difference in the mean between the upper slopes (which are hypothesized to be greater) and the lower slopes, we find that for RMW slope two quadrants (DSR and USR) have a greater slope in the upper troposphere (*p* < 0.05), for *M* slope all four quadrants have a greater slope in the upper layer (*p* < 0.05), and for dB*Z* slope three quadrants (DSL, USL, and DSR) have a greater slope in the upper layer (*p* < 0.05). These results are consistent with the case study by Corbosiero et al. (2005) and also with HH13. It should also be noted that the layers used for lower and upper slope (2–5 and 5–8 km) are slightly different than those used in HH13, since the current study only used data up to 8 km. It is possible that the differences in lower and upper slope (especially for the dB*Z* slope) would be greater with the higher altitude used in HH13, but to allow for a consistent comparison between the three metrics, we limited the data to 2–8 km.

### c. Wavenumber-1 analysis

*S*

_{0}represents an “azimuthal mean” slope,

*S*

_{1}is the amplitude of the slope asymmetry (proportional to variation in slope between the quadrants), and

*ϕ*is the phase shift of the maximum slope away from

*α*(

*α*= 45° DSL, 135° USL, −45° DSR, −135° USR). The errors from this technique were small throughout the dataset, indicating that the wavenumber-1 fit was generally a good description of the slope asymmetries despite the small number of points that went into each fit. The errors were very consistent between quadrants, with a median slope error of 0.21 for RMW slope in each quadrant, a median error of 0.08 for the

*M*slope in each quadrant, and a median error of 0.20 for the dB

*Z*slope in each quadrant.

The results of applying this fit to all cases are shown in Figs. 7–8. Figure 7 shows the phase of the fit (i.e., the azimuthal direction where the fit to the slopes was a maximum) for all cases, a possible proxy for the tilt of the vortex due to shear. Consistent with the earlier results, the RMW and *M* slopes showed a general downshear phase (especially the RMW slope), with 26 of the 34 RMW slope phases downshear and 22 of the 34 *M* slope phases downshear. The slope for dB*Z* showed more variance, with only 17 of 34 cases showing downshear asymmetry phase. The median RMW slope phase was 11.7° (11.7° left of shear), and the median *M* slope phase was 40.3° (40.3° left of shear). Both of these results (especially the RMW slope) are relatively consistent with the average vortex tilt of approximately 10° left of shear found by Reasor et al. (2013), although the *M* slope tended to be rotated a little more left of shear than the RMW slope. The slope of the 20-dB*Z* surface did show a slight peak phase in the DSL quadrant, but the distribution was more flat than for the other two metrics, once again possibly due to the rotation of hydrometeors by the TC primary circulation.

Figure 8 shows the distributions of the amplitudes of the wavenumber-1 fits for each slope metric. In general, although the *M* slope is the only metric with no amplitudes greater than 3, the RMW slope has more low-amplitude fits than either the *M* slope or the dB*Z* slope. The median RMW amplitude was 0.69, lower than that of the *M* slope (1.04) and the dB*Z* slope (1.03). Because of the highly nonnormal nature of this distribution (Fig. 8), a Wilcoxon rank-sum test was used to analyze this difference. The RMW slope median amplitude was marginally statistically significantly different from both the *M* and dB*Z* slope (*p* = 0.056, 0.063). This difference, as well as the greater tendency for downshear slope in RMW (Fig. 7) potentially indicates that the asymmetry in the RMW slope is on the vortex scale and is more consistently described by the wavenumber-1 asymmetry maximized downshear, while the *M* and dB*Z* slope are connected to shorter-scale convective or other higher-order asymmetries that cannot be assessed with this dataset.

### d. Stratification by shear magnitude

Next, the study investigated whether there was any apparent relationship between the magnitude of the shear encountered by the TC and the degree of slope asymmetry.

#### 1) Phase/amplitude by shear magnitude

First, we analyzed the wavenumber-1 asymmetry discussed earlier in terms of stratification by shear magnitude. The cases were separated into high-shear cases with shear >7 m s^{−1} (~14 kt) and low-shear cases with shear <4 m s^{−1} (~8 kt), similar to the thresholds of Reasor et al. (2013). The low-shear cases were a subset of 10 TCs from the dataset, while there were 14 high-shear TCs. Once again, because of the nonnormal nature of the amplitude distributions, a Wilcoxon rank-sum test was used to compare the distributions. The results are summarized in Table 3. For RMW, the median of the low-shear cases was statistically significantly lower than the median of the high-shear cases (*p* = 0.04). This result is generally consistent with the case study of Hurricane Rita by Rogers and Uhlhorn (2008), although their study only considered the RMW below flight level (generally ~3 km). For the *M* slope, however, the low-shear set was only marginally significantly different (based on the rank-sum test) from the high-shear cases (*p* = 0.08). The dB*Z* slope also showed less of a relationship between amplitude and shear, once again likely due to the higher variance in that metric. The low-shear group was again only marginally significantly different from the high-shear set (*p* = 0.07).

Comparison of high-shear (greater than 7 m s^{−1}) and low-shear (less than 4 m s^{−1}) median azimuthal standard deviations and amplitude of wavenumber-1 asymmetries of eyewall slope for each metric. For the standard deviations, differences that are statistically significant are set in italics. For the wavenumber-1 amplitude, differences that are statistically significant are set in boldface.

The above results indicate that amplitude of the wavenumber-1 asymmetry showed some connection to shear magnitude, mainly for the RMW slope. The phase of the asymmetry for these metrics also showed some relationship to shear magnitude for the RMW slope. Figure 9 shows scatterplots of phase versus shear magnitude. The absolute value of phase was plotted since values from −90° to 90° represent the downshear quadrants, and values less than −90° or greater than 90° represent the uphsear quadrants. Once again, because of the nonnormal distributions, a Wilcoxon rank-sum test was used to compare the medians of the high-shear and low-shear sets. Although there is considerable spread, both the RMW slope (*p* = 0.057) and the *M* slope (*p* = 0.054) show a marginally significant difference between the high-shear and low-shear median phases, while the dB*Z* slope shows no difference. The result seems to indicate an overall tendency for the eyewall to tilt more downshear when the shear magnitude is greater, although there is significant spread. The downshear-left tilt appears particularly favored for the RMW slope as shear increases (not shown), as 9 of the 14 RMW slope phases were in the DSL quadrant when shear was greater than 7 m s^{−1}, compared with 14 of the 34 cases overall.

Interestingly, although a linear correlation did show a significant inverse relationship between shear magnitude and intensity (*r* = −0.40, *p* = 0.02) within this dataset, there was an even stronger correlation between wavenumber-1 RMW slope asymmetry and intensity (*r* = −0.56, *p* < 0.01). This indicates that although there was a significant variation in the response of the vortex to shear, consistent with prior studies, cases where the shear caused significant vortex tilt had a more difficult time reaching extreme intensities.

#### 2) Overall slopes by shear magnitude

Next, we analyzed the variation in slope asymmetry with shear using the slope values themselves, rather than the wavenumber-1 least squares fits, to examine the consistency of the results. This was done by examining the variance of the slopes in each shear-relative quadrant for all cases. For the RMW slope, the median variation of the high-shear cases was indeed statistically higher than the low-shear set (*p* = 0.04) based on a Wilcoxon rank-sum test, consistent with the wavenumber-1 amplitude. Similar to the results of the wavenumber-1 analysis, this test also showed that there was only a marginally statistically significant difference between the sets for the *M* slope (*p* = 0.08) and the dB*Z* slope (*p* = 0.07).

As an illustration of the difference in eyewall slope asymmetry (specifically the RMW slope) between the high-shear and low-shear cases, the composite tangential wind in each quadrant for each of the sets is shown in Figs. 10–11. The figures also show the profiles of eyewall slope for each metric. A comparison of these figures shows that the slopes (especially the RMW slope) indeed seem to vary more around the eyewall for the high-shear cases. For the low-shear cases, there appears to be a gentler slope in all four quadrants, and it is relatively symmetric around the eyewall. The difference between the sets appears to be particularly pronounced when looking at the slopes above 5 km. Some of this difference in the tangential wind fields may be due to the fact that the low-shear cases had a higher median intensity (61.7 m s^{−1}) than the high-shear cases (56.6 m s^{−1}). This is also reflected in the lower inertial stability (not shown) that was observed, especially above 5 km, in the high-shear cases, making the vortex more susceptible to tilt. For the low-shear cases, inertial stability in excess of 2 × 10^{−5} s^{−2} was observed all the way up about 10 km, while for the high-shear cases this value was only 1–1.5 × 10^{−5} s^{−2}. These results are consistent with the statistical results discussed above, and also consistent with the findings of idealized simulations by Jones (2000), which found reduced vortex tilt due to shear in a baroclinic vortex as the magnitude of the upper-level cyclonic tangential wind increased. Once again, this result is indicative of the fact that vortex tilt due to shear is reflected in the slope of the eyewall, and the degree of asymmetry across and along the eyewall appears to be related to shear magnitude.

### e. Stratification by intensity change

Finally, the study investigated whether there were any systematic differences in eyewall slope in any of the quadrants between intensifying TCs and weakening/steady TCs. Other structural differences between weakening and steady-state TCs were discussed extensively in Rogers et al. (2013). The threshold used to define an intensifying TC was a 12-h increase in intensity of 10 kt or more after the period of interest, similar to the definition used in Rogers et al. (2013). Weakening or steady TCs were defined as those TCs where the 12-h intensity change was 0 or less (i.e., no intensity change or a decrease in wind speed). In addition, since we were interested only in storms that had a thermodynamic potential to intensify, to explore potential dynamical factors for intensification or weakening, we removed steady/weakening cases where the TC was 20 kt or less from its thermodynamic MPI, as defined by Emanuel (1986, 1988). The intensifying cases consisted of a subset of 10 flights, and the weakening/steady cases (not close to MPI) were a subset of 10 other flights. The median best track intensity for the two sets was very similar (54.0 m s^{−1} for intensifying TCs, and 56.6 m s^{−1} for weakening/steady TCs). The median shear was also very similar (6.91 m s^{−1} for intensifying TCs, 7.09 m s^{−1} for weakening/steady TCs).

The mean and standard errors for each slope metric for each quadrant are shown for both intensifying and steady-state/weakening cases in Table 4. Little statistical difference was found in each individual slope metric category between the sets. However, the dB*Z* slope in the USL quadrant was more upright for intensifying cases than weakening/steady cases (*p* = 0.04), based on a *t* test. This suggests that intensifying cases potentially have more shear-induced convection (rotated by the primary circulation) in the eyewall. This is consistent with Rogers et al. (2013), which found higher eyewall reflectivities overall for intensifying cases as compared to steady-state cases. This point is further illustrated by comparing the slopes of the dB*Z* and *M* surfaces for both sets (Fig. 12). In the DSL and USL quadrants, the 20-dB*Z* surface is, on average, more upright than the *M* surface for intensifying cases, while in the USR quadrant the dB*Z* surface is more sloped and in the DSR quadrant they are approximately the same. For weakening cases, the dB*Z* surface is more sloped in both upshear quadrants and has approximately the same slope as the *M* surface in the downshear quadrants. The difference between intensifying cases and weakening/steady cases is significant USL (*p* = 0.01) and marginally significant DSL (*p* = 0.06) using this metric of the *M*–dB*Z* slope difference. The results indicate that convective generation downshear is manifested by more upright reflectivity surfaces (and left of shear, due to rotation of hydrometeors), and this convective activity in the inner-core region plays a major role in the intensification of TCs. Two case studies illustrating these composite results are shown in Fig. 13. The first (Fig. 13a) is Hurricane Paloma at approximately 1800 UTC 7 November 2008. At this time, Paloma was a category 1 hurricane with maximum sustained winds of 41 m s^{−1} (80 kt). It was over 29°C SST and had an MPI of 154 kt. The shear impacting the storm was from the southwest at 10.8 kt. The second TC (Fig. 13b) is Hurricane Ophelia on 11 September 2005. Ophelia was also a category 1 hurricane with maximum sustained winds of 33.4 m s^{−1} (65 kt). It was also over warm SST (28.5°C), and although the MPI was lower than for Ophelia (119 kt), both TCs were more than 50 kt below their MPI. The shear for Ophelia was not significantly higher (12.1 kt). Figure 13 shows the structural comparison in the DSL quadrant of these two TCs. For Paloma, although both the 20-dB*Z* contour and the *M* surface are sharply sloped, the 20-dB*Z* surface (*dr*/*dz* = 2.33) is much more upright than the *M* surface (*dr*/*dz* = 4.33), indicating rising air and diabatic heating inside the low-level RMW. However, for Ophelia, the *M* surface is more upright (*dr*/*dz* = 1.00) while the 20-dB*Z* surface is sharply sloped (*dr*/*dz* = 2.67), indicating that the updraft and diabatic heating extends outside the RMW. These results are similar to those found by Rogers et al. (2014) for a case study of Hurricanes Earl and Gustav by comparing the updraft core (based on *W*) and the *M* surface. Paloma, with the favorable configuration for intensification (discussed more below), rapidly intensified by 30 kt in the 12 h after this observation. Ophelia, however, remained quasi steady, dropping in intensity by 5 kt over the next 12 h.

Mean and standard errors of slope in each quadrant for intensifying cases vs weakening/steady-state cases.

## 4. Summary and discussion

The results presented above show that the eyewall slope is a metric that can be related to variations in vortex and convective structure due to shear. The tendency for a greater RMW slope downshear is consistent with the well-studied notion of downshear vortex tilt due to shear. It is possible, based on these results, that a comparison of the RMW slope across the eyewall could be used as a proxy for vortex tilt. It is somewhat surprising that this signal is not reflected as strongly in the slope of an *M* surface, as *M* is also related to the tangential winds. It is possible, however, that dependence of *M* on the radius in addition to *V*, and the fact that *M* surfaces tend to tilt more sharply with height as altitude increases (even in the absence of vertical shear), blur some of the signal from this metric. In the comparison of overall average slopes in each quadrant (and for much of the analysis in this study), the slope of the 20-dB*Z* contour showed little statistically significant variation due to shear. A possible explanation for this result, as discussed previously, is the tendency for the hydrometeors generated by convective activity in one quadrant to be “spread around” the TC by the primary circulation. This would tend to obscure any difference in slopes between quadrants. One area where the dB*Z* slope did show a significant signal (as did the other two slope metrics) was in the difference in slope between the lower and upper troposphere, as the dB*Z* slope showed a greater slope in the upper troposphere. This may be partially due to the outward radial advection of the smaller precipitation particles above the freezing layer, as well as the tendency for updrafts to follow the *M* surfaces, which also generally slope more with height. It would likely be insightful to compare the 20-dB*Z* slope in cases with strong eyewall convection versus cases with mainly stratiform precipitation. It seems possible that areas of very deep convection would be less susceptible to tilt due to shear, and this is a possible topic for further investigation.

The isolation of the wavenumber-1 azimuthal variation in eyewall slope using the fit of a cosine function further illustrated the shear-induced asymmetry. As discussed in the results, the tendency for the phase of the maximum slope to occur in one of the downshear quadrants (for both the RMW slope and *M* slope) is consistent with the idea that the downshear side of the eyewall tends to slope more due to shear. The similarity between the median phase of the RMW and *M* slope asymmetries (11.7° and 40.3° left of shear, respectively) and the mean vortex tilt found in observational and modeling studies such as Reasor et al. (2004, 2013) once again suggest that asymmetry in eyewall slope may serve as a proxy for the tilt of the vortex, which multiple prior studies have shown is a key component of TC response to and resilience against vertical shear. The slope of the 20-dB*Z* contour did have a peak in the phase distribution downshear, but the distribution was more spread out overall, consistent with the idea of rotation of hydrometeors by the TC primary circulation. Additionally, it is possible that some of the spread in slopes (of all three metrics) and cases where the maximum slope was not downshear were partially due to precession of the vortex in shear (Reasor and Eastin 2012).

The comparison of high-shear and low-shear cases showed some differences in structure of the eyewall slope. From the wavenumber-1 analysis, the RMW slope showed a tendency for greater amplitude of the wavenumber-1 asymmetry as shear increased. In addition, the phase tended to be more directly downshear. A similar result was found for the RMW slope by a simple comparison of azimuthal slope variance between the high-shear and low-shear sets. A composite of the data for the high-shear and low-shear sets also strengthened this conclusion, and showed that the variation in slope for high-shear cases was particularly pronounced at upper levels (above 6 km), possibly due to weaker tangential winds and lower inertial stability aloft for the high-shear cases. The results are also consistent with the findings of Rogers and Uhlhorn’s (2008) case study of Hurricane Rita. The schematic from their study (their Fig. 4b) is reproduced in a slightly modified form here (Fig. 14). While their study only focused on data at the surface and flight level (typically around 3 km), this study has confirmed the validity of this model for the azimuthal variation of the RMW slope over a deep layer of the troposphere, further highlighting the notion that this variation of the RMW slope seems to be connected to the tilt of the vortex due to shear. All of the analysis in this study has considered eyewall slope to be dependent on the tilt of the vortex. However, it would also be potentially interesting to quantify eyewall slope in a tilt-relative sense, and to see if there is (as would be expected) less overall variance. This could potentially highlight some of the higher-wavenumber asymmetries that affect slope (such as convective asymmetries), and this is a potential topic for future analysis.

Another aspect of the analysis where the dB*Z* slope did show signal was in the comparison of intensifying versus weakening/steady cases. Specifically, the dB*Z* surfaces on the left-of-shear side of TCs (especially USL) tended to be more upright than the *M* surfaces for intensifying cases, while they tended to be similarly or more sloped than the *M* surfaces in weakening/steady cases. Since the *M* surface begins at the 2-km RMW, this result indicates that the dB*Z* surface is more likely to remain radially inside the low-level RMW in cases with a more upright eyewall (intensifying cases). This highlights the notion that intensifying cases tend to have more convective activity inside the RMW, where diabatic heating can lead to intensification through a Sawyer–Eliassen-like response (e.g., Pendergrass and Willoughby 2009). For example, Schubert and Hack (1982) showed that cumulus convection leads to more heating in regions of high inertial stability (such as the region inside the RMW), and Hack and Schubert (1986) similarly showed that the heating efficiency of a diabatic heating source increased as the vortex intensity and inertial stability increased. In addition, a recent study by Hendricks et al. (2014) used a shallow-water model to show that heating very close to the center of a cyclone is important for intensification. These studies all provide support to the idea that convective heating inside the RMW (as observed by the proxy of reflectivity surfaces) is a favorable configuration for intensification of TCs. The results from this study are also consistent with the findings of recent observational and modeling studies, specifically Rogers et al. (2013), which used Doppler radar composites to connect intensification to heating through convective bursts inside the RMW, and Chen and Zhang (2013), who used a high-resolution model to find that convective bursts inside the RMW were associated with the RI of Hurricane Wilma. Rogers et al. (2014) also observed a similar structure (updraft more upright than the *M* surface) for Hurricane Earl during a period of intensification. A schematic summarizing the slope differences for intensifying and weakening/steady cases is shown in Fig. 15.

Overall, these results indicate that the eyewall slope is a useful metric for assessing the structure of sheared TCs, especially in cases with good azimuthal data coverage. This composite analysis shows that the RMW and *M* slope especially seem to agree well with the results of previous studies noting a tendency for downshear tilt. While the other metrics (in particular the dB*Z* slope) showed less signal in parts of the analysis, the difference between the two metrics (the 20-dB*Z* slope and *M* slope) in intensifying TCs is significant and related to the concept that heating inside the RMW is connected to intensification of tropical cyclones. This topic merits further study to explore the details of how eyewall slope relates to intensification in sheared TCs. In particular, other high-resolution observational or numerical model datasets will be used to further explore the idea that rising air inside the RMW (as illustrated by the 20-dB*Z* and *M* slopes) is a favorable configuration for intensification. Investigating the source of these more upright updrafts, whether it be boundary layer convergence, eyewall instability, or other factors, would likely prove useful for diagnosing and predicting short-term structure and intensity changes. This is the subject of ongoing and future work.

The authors thank the scientists and flight crews of *NOAA-42* and *NOAA-43* for their efforts to collect valuable data, including the radar data used in this study. John Gamache’s three-dimensional Doppler analysis technique was invaluable for constructing the merged analyses used in this study. The authors thank Eric Uhlhorn, Paul Reasor, and Frank Marks for helpful discussions. The comments of two anonymous reviewers led to significant improvements in the analysis and discussions from an earlier version of the manuscript. The lead author was partially supported by the FSU Legacy Fellowship.

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