## Abstract

Using airborne Doppler radar data from 39 flights into hurricanes from 2004 to 2010, the authors examine the outward slope of the eyewall, revisiting the recent studies of Stern and Nolan. The slope of the radius of maximum winds (RMW) is found to increase nearly linearly with size and is uncorrelated with intensity. The slope of the eyewall absolute angular momentum surface *M* increases with increasing size (strong correlation) and decreases with increasing intensity (weak to moderate correlation). Two other measures of eyewall slope are also investigated: the 20-dB*Z* reflectivity isosurface (dBZ20) and the radius of maximum azimuthal-mean updraft (RWMAX). The slopes of both dBZ20 and RWMAX increase with their size. The slope of dBZ20 decreases with intensity, though the correlation is weak, while the slope of RWMAX is uncorrelated with intensity. The absolute angular momentum decreases on average along the RMW by 9% from 2- to 8-km heights. With this larger dataset, the previous results are generally confirmed: the slope of the eyewall is mostly a function of the size of the RMW.

The vertical decay rate of the maximum tangential winds (Vmax) is also reexamined. On average, Vmax decreases by 20% from 2- to 8-km heights, but this varies from 8% to as large as 42%. This percentage decay rate increases with increasing size and decreases with increasing intensity. Three cases are found where Vmax increases with height from 2 to 4 km, which is likely a consequence of unbalanced flow.

## 1. Introduction

In a previous study, Stern and Nolan (2009, hereafter SN09) constructed azimuthal-mean (axisymmetric) wind fields of Atlantic hurricanes based on pseudo-dual-Doppler radar observations obtained by research aircraft of the National Oceanographic and Atmospheric Administration (NOAA). Through examination of 17 cases (each of which were composites of data taken over a period of several hours), SN09 found that the outward slope of the radius of maximum tangential winds (RMW) as a function of height is strongly and linearly correlated with the RMW at a height of *z* = 2 km. (In other words, the outward slope of the RMW is proportional to the size of the RMW itself). No relationship was found between the slope of the RMW and the current intensity of the storm. Using the same dataset, Stern and Nolan (2011, hereafter SN11) documented another prevailing feature of the inner-core wind fields: between *z* = 2- and 8-km heights, the maximum tangential wind decreased by about the same fraction (20% ± 5%) for all storms, regardless of size or intensity.

Since the publication of SN09, more data have become available from storms after the period previously used (2004–06). The same methods have been applied to these additional datasets, resulting in an increase of the number of azimuthal-mean wind field composites from 17 to 39. This paper provides documentation of this expanded dataset. In addition, two more indicators of the size of the hurricane eyewall are analyzed: the radius of maximum updraft (RWMAX) and the radius of the inner surface of the 20-dB*Z* contour of radar reflectivity (dBZ20). These structures show similar relationships, sloping outward more for larger storms, with little to no relationship to intensity. Very recently, Hazelton and Hart (2013, hereafter HH13) examined the slope of dBZ20, using a similar dataset to ours, but with differences in methodology. They found a weak (but statistically significant) relationship between slope and intensity and a moderate relationship between slope and size. In addition to presenting our results, we will discuss how our results relate to those of HH13.

## 2. Data

The data presented here are processed in exactly the same manner as described in detail in SN09. SN09 and SN11 did not discuss vertical velocities or reflectivities; however, those variables are analyzed, composited, and azimuthally averaged with almost exactly the same procedure as the tangential wind as described in SN09. A brief summary is as follows. Each leg of data acquired from NOAA contains the three-dimensional fields *u*, *υ*, *w*, and reflectivity (dB*Z*) that have been derived through the automatic processing and quality control technique of Gamache et al. (2004). We average the data from multiple flight legs over a period of several hours to maximize azimuthal coverage, and remaining data gaps are filled through an inpainting technique. Each field is horizontally interpolated to 500-m grid spacing (the native vertical grid spacing is already 500 m) using cubic splines, and is azimuthally averaged by calculating the mean within each 500-m bin. Finally, the azimuthal-mean fields are lightly smoothed in the radial direction (10 passes of a 1–2–1 smoother).^{1} An example illustrating this overall process for Hurricane Ivan (2004) is shown in Fig. 2 of SN09.

Some of the datasets for the new cases are different from those of SN09 in one respect: the horizontal resolution of the three-dimensional grids is coarser. Whereas the original 17 cases had grid spacings of 2 km, some of the new cases have grid spacings of either 3 or 5 km.^{2} Details on this and other characteristics of each of the 39 cases presented here are given in Tables 1 and 2. The original cases of SN09 include data from Hurricanes Frances (2004), Ivan (2004), Dennis (2005), Katrina (2005), Rita (2005), Wilma (2005), and Helene (2006). The expanded dataset also includes storm days from Hurricanes Felix (2007), Gustav (2008), Ike (2008), Paloma (2008), Bill (2009), Earl (2010), and Karl (2010). Additionally, the expanded dataset includes some storm days from the original set of storms that were not used in the original dataset.

## 3. Slope of the RMW and eyewall *M* surface

*M*

SN09 found that the slope of the RMW increased with increasing size of the RMW at low levels. As an example of this relationship, Fig. 1 shows the radius–height structure of the azimuthal-mean tangential wind for Hurricane Ivan on four different days. On 7 and 9 September, the RMW is nearly vertical (zero slope), consistent with its small size. The RMW more than doubles in size between 9 and 12 September, and the slope correspondingly increases. Finally, on 13 September, the low-level RMW has contracted somewhat, and the slope has become smaller. As discussed in SN09, while case study analyses such as these are illustrative, they cannot by themselves actually demonstrate a robust relationship between size and slope. Further, from Fig. 1 alone, one might get the impression that slope is also an increasing function of intensity, whereas, in fact (as shown in SN09), slope is uncorrelated with intensity. Only by determining the slopes over a large number of cases can the statistical relationships be ascertained. This was done by SN09, who found an approximately linear relationship between slope and size and no relationship between slope and intensity.^{3 }Figures 2a and 2b show scatterplots of the slope of the RMW versus size and intensity, respectively, for the expanded dataset. The linear relationship between slope and size is confirmed by the addition of the 22 new cases (*r* = 0.601),^{4} as is the lack of relationship between slope and intensity (*r* = 0.0036; not significant).

SN09 found a strong^{5} linear relationship between the slope of the absolute angular momentum [*M* = *rV* + (1/2)*fr*^{2}] surface that originates at the 2-km RMW and the size of the RMW and showed that the theory of Emanuel (1986) predicts this same relationship. Figure 2c shows that this relationship is confirmed in the expanded dataset (*r* = 0.724). Unlike for RMW, the slope of the *M* surface is weakly correlated with intensity in terms of the maximum wind (Vmax) at 2 km (Fig. 2d; *r* = −0.345, significant at the 95% level), and moderately correlated with best-track Vmax (not shown; *r* = −0.505) and best-track minimum pressure (Pmin; not shown; *r* = 0.504). As discussed by SN09, a number of observational case studies have found evidence that the RMW is approximately a surface of constant *M* (Jorgensen 1984b; Marks et al. 1992; Franklin et al. 1993). Using their 17 cases, SN09 found that the RMW does on average approximate an *M* surface, but that there is also a systematic decrease of *M* upward along the RMW. This remains true with the expanded dataset, as on average, *M* decreases along the RMW by 9% from 2- to 8-km heights (Fig. 3). The degree of this departure of the RMW from an *M* surface decreases somewhat with increasing intensity (not shown; *r* = 0.444, *p* = 0.005), and has no relationship with storm size. The slopes of *M* and the RMW are moderately correlated with each other (not shown; *r* = 0.519). SN09 did not explicitly show the relationship between the slope of the RMW or *M* surface and the intensity tendency. Using the expanded dataset, Figs. 4a and 4b show scatterplots of slope versus the 12-h change in Pmin. There is no relationship between the slope of either the RMW or the eyewall *M* surface and the intensity tendency (this is also the case when using Vmax; not shown).

## 4. Slope of the eyewall reflectivity and updraft

### a. Previous work

A number of earlier studies examined the slopes of either the eyewall updraft or a reflectivity isosurface for individual cases. Jorgensen (1984a) evaluated the slope of the 10-dB*Z* isosurface from individual cross sections in four hurricanes. He estimated slopes ranging from 1.0 to 2.0, and stated that the slope “…apparently was related to the eye diameter and storm strength.” These reflectivity slopes were also implied to be representative of the slopes of the eyewall updraft and RMW. In a companion study, Jorgensen (1984b) found for a single storm on 2 days that the slope of the RMW was similar to the slope of the 10-dB*Z* isosurface. In one of the first studies to perform Doppler wind analyses for tropical cyclones, Marks and Houze (1987) assessed the slopes of the RMW, the eyewall updraft (maximum vertical velocity), and the maximum reflectivity for Hurricane Alicia (1983). They found that the slope of the RMW was larger than either of the other two slopes. In contrast, Marks et al. (1992) found that in Hurricane Norbert (1984), the slope of the updraft was greater than that of the RMW.^{6 }Lee et al. (2000) analyzed Typhoon Alex (1987), and they found that the slope of the eyewall *M* surface had nearly the same slope as the 25-dB*Z* isosurface. Corbosiero et al. (2005) composited multiple cross sections of vertical-incidence radar data over 6-h periods for Hurricane Elena (1985) and calculated the slope of the 10-dB*Z* isosurface separately from 0–6- and 6–12-km heights. Near peak intensity (category 3), the slopes in the lower and upper 6 km were 30° and 46°, respectively. For a period of weakening intensity (though Elena was still strong), the slopes were 34° and 74°, respectively. From this, Corbosiero et al. (2005) concluded that the increase in upper-level slope was due to the weaker intensity.

A limitation of these earlier studies is that they were case studies, and so it is difficult to draw general conclusions regarding the relationships among the slopes of these various surfaces and between the slopes and intensity and size. Further, each study used unique definitions of slope as well as differing methodology, which makes intercomparison difficult. With a larger dataset, and a more uniform and systematic analysis, SN09 determined that while the slope of the RMW increases with its size, the slope is actually unrelated to intensity. SN09 examined only the slopes of the RMW and *M* surfaces, though both reflectivity and vertical velocity data are available in the same dataset. Following upon the work of SN09, HH13 assessed the slope of the 20-dB*Z* isosurface (dBZ20) in 15 different TCs using 121 flight legs.^{7} The slope was calculated between 2- and 11-km heights. They found a weak but statistically significant relationship between the slope and the best-track intensity (both wind and pressure), whereby stronger storms tend to have smaller slopes. They also found a moderate relationship between the slope and the radius of dBZ20 at 2-km height. As in SN09, there was no relationship between size and intensity in their dataset. When they separately evaluated slope in the lower (2–6.5 km) and upper levels (6.5–11 km), they found that the slope was greater at upper levels. Interestingly, HH13 found that while the lower-level slope was correlated with intensity, the upper-level slope was not.

### b. Reflectivity

Here, we examine the slope of dBZ20 within our expanded dataset. As described in section 1, we use the same general methodology to calculate the azimuthal-mean reflectivity as we do for the azimuthal-mean tangential wind. Reflectivity is a nonlinear function of *Z*, which in turn is a nonlinear function of the drop size distribution. It is not obvious whether it is more meaningful to directly calculate the azimuthal mean of dB*Z* or to determine dB*Z* from the azimuthal mean of *Z*. These two averaging methods could potentially yield qualitatively different results, although it is impossible to know this a priori. Therefore, we have performed the analysis using both methods. For both methods, we determine the slope of dBZ20 by finding the best-fit line to the set of points that are directly extracted from the MATLAB contouring algorithm. In many cases, the maximum vertical extent of dBZ20 is less than 8-km height. Because of this, we calculate the slope of dBZ20 between 2- and 6-km heights (for all cases). There are still a number of cases for which this slope cannot be calculated, and so the sample size is less than that for the RMW. For azimuthally averaging dB*Z* (dBZ20 refers to this method), there are 27 cases, whereas for azimuthally averaging *Z* (dBZ20Z), there are 25 cases.

Figure 5a shows the slope of dBZ20 versus its own radius at 2-km height (the radius used by HH13; cf. their Fig. 8). There is a moderate correlation (*r* = 0.609), as the slope increases with radius. A similar relationship exists between the slope of dBZ20 and the RMW at 2-km height (not shown), except that dBZ20 is always inward of the RMW. The median RMW is 13.0 km larger than dBZ20 at *z* = 2 km, and there is a tendency for this difference to increase with the RMW itself (*r* = 0.700; not shown). An apparent outlier in terms of slope versus size is Helene on 17 September, with a slope of 3.93. This large slope is in part due to the fact that dBZ20 barely reaches 6-km height, and so the isosurface becomes close to horizontally oriented near the top of its vertical extent. As with the RMW, in this dataset there is no relationship between the radius of dBZ20 at 2 km and storm intensity, as measured by best-track Vmax and Pmin or the maximum azimuthal-mean tangential wind at 2 km (not shown). There is a weak positive correlation (*r* = 0.330) between the slope of dBZ20 and Pmin that is significant at the 90% level, meaning that slope increases as intensity decreases. It is unclear if this relationship is robust, as the correlation with maximum wind speed is not significant (not shown). Further, the correlation of slope with Pmin is rendered insignificant (*r* = 0.251, *p* = 0.206) when the Helene outlier is removed. The slope of dBZ20 is moderately correlated with the slope of the RMW (*r* = 0.480, significant at 95%) and with the slope of the *M* surface (*r* = 0.584).

When azimuthally averaging *Z* instead of dB*Z*, the results are qualitatively similar, but the correlations are reduced and are sometimes rendered insignificant owing to the increase in the number of large outliers (not shown). The radius of dBZ20Z is also systematically smaller than that of dBZ20 at *z* = 2 km, with a median difference of 4.9 km. This difference is because for dBZ20Z, taking the antilog of dB*Z* (and dividing by 10) prior to averaging increases the relative weight of larger values of *Z*, and so the mean dB*Z* so calculated is increased from that which comes from simply averaging in dB*Z*. Therefore the minimum radius at which this contour is found in dBZ20Z is decreased from that in dBZ20.

While there are differences of opinion regarding which method of averaging is more appropriate, discussions with radar experts have convinced us that averaging in dB*Z* may be most meaningful (P. Reasor 2013, personal communication). Additionally, Lakshmanan (2012) show statistically that in image processing, it is generally better to use dB*Z* than *Z*. Based on this and the fact that the relationships between dBZ20 and slope in our study are more robust, we use only dBZ20.

### c. Eyewall updraft

The slope of the maximum azimuthal-mean eyewall updraft (RWMAX) is calculated in the same way as is the slope of the RMW. RWMAX in our dataset is in most cases inwards of the RMW, which is generally consistent with previous studies (e.g., Marks and Houze 1987). Therefore, it occurs where there are fewer radar scatterers, and so the azimuthal data coverage is worse. Furthermore, vertical velocity has relatively more energy at high wavenumbers than does tangential wind speed, and so data gaps are more problematic. In an observing-system simulation experiment using the analysis software of Gamache et al. (2004) on output from the Hurricane Weather Research and Forecasting Model (HWRF), Lorsolo et al. (2013) have recently shown that the relative errors in *w* are larger than for tangential and radial winds, and in particular, that the azimuthal-mean vertical velocity field inside the eye is poorly retrieved. They found that the location of the eyewall updraft was accurate at midlevels, but that it did not extend to the lowest levels as in the “truth,” and that the magnitude of the mean updraft is too strong by 1–3 m s^{−1} at upper levels. Because of these uncertainties in data quality, we carefully examined each case and made a subjective decision on whether to include it in the dataset. In the end, we included 23 cases for the slope of the eyewall updraft.

Figure 5b shows the slope of RWMAX versus its own radius at 2-km height. There is a moderate correlation (*r* = 0.442, *p* = 0.035), with the slope of the updraft increasing with size. The same relationship is seen between the slope of RWMAX and the 2-km RMW (not shown). There is no relationship between the slope of RWMAX and intensity, by any measure (not shown). The median RMW is 8 km larger than RWMAX at 2-km height. Unlike with dBZ20, there is not a significant relationship between the RMW and the distance from the RMW to RWMAX. Note that because we are assessing the slope of the eyewall, we subjectively excluded cases where the strongest updraft was a large distance away from the RMW. Figure 6 compares the slope of the RMW to the slopes of *M*, dBZ20, and RWMAX. The slope of the *M* surface originating at the 2-km RMW is almost always greater than the slope of the RMW, consistent with the fact that *M* decreases upward along the RMW. In most cases, the slope of dBZ20 is also greater than the slope of the RMW, although there are a number of cases close to the 1:1 line. For RWMAX, there is a more even distribution about the 1:1 line, indicating that the slope is almost as likely to be smaller than that of the RMW than it is to be larger. The median over all cases (nonhomogeneous comparison) of slope is 0.813, 1.460, 1.300, and 0.786 for the RMW, *M*, dBZ20, and RWMAX, respectively. That the slope of dBZ20 is on average 50% larger than the slope of the RMW implies that dBZ20 approaches the RMW with increasing height.

To better illustrate the results from Fig. 6, the azimuthal-mean tangential wind field for four different cases are shown in Fig. 7, with the RMW, RWMAX, and *M* surface shown from 2 to 8 km, and dBZ20 shown from 2 to 6 km. These examples are chosen because they represent storms of very different intensities and sizes. Ivan on 9 September (Fig. 7a) is small and intense, Wilma on 20 October (Fig. 7b) is of moderate size and intense, Frances on 30 August (Fig. 7c) is large and of moderate intensity, and Helene on 17 September (Fig. 7d) is large and relatively weak. In each case, RWMAX at 2 km is inward of the RMW, and dBZ20 is in turn inwards of RWMAX. The slope of *M* is larger than that of the RMW in each case as well, though by varying degrees. In both Wilma and Frances, the RMW appears to be nearly an *M* surface up to about 6-km height, with *M* sloping more outward above. On the other hand, *M* diverges from the RMW by 4-km height in Ivan and Helene. Note that Frances has two distinct eyewalls, with the outer eyewall clearly more intense, and Ivan has a secondary wind maximum that is evident from 1- to 5-km heights, with the inner eyewall more intense. For each case with multiple eyewalls or wind maxima, we analyze only the eyewall that is associated with the stronger wind maximum. In Frances and Helene, the RWMAX approximately parallels the RMW. In Wilma, the nearly vertical RWMAX diverges from the sloping RMW, while in Ivan, the sloping RWMAX approaches the nearly vertical RMW. For all but Helene, dBZ20 approximately parallels the RMW. As discussed previously, Helene is an outlier in terms of the slope of dBZ20, which is a consequence of the fact that dBZ20 barely reaches 6-km height.

## 5. The decay of maximum winds with height

SN11 used the same dataset as SN09 to investigate the rate at which the maximum azimuthal-mean tangential winds decay with height. This was done by normalizing each profile of maximum tangential winds by the respective value at 2-km height. They found that storms exhibit a common profile of normalized maximum winds, decreasing by approximately the same percentage rate between 2- and 8-km heights. Though there was some dependence of this rate on both size and intensity, they showed that this sensitivity was small enough such that the mean decay profile could be used to accurately “predict” the profiles of most individual storms with an accuracy of ±3 m s^{−1}. Figure 8a shows the vertical profiles of Vmax, while Fig. 8b shows the respective normalized profiles (Vmaxnorm) for the expanded dataset. The new profiles are generally similar to the original profiles. The median Vmaxnorm at *z* = 8 km is 0.796, and the 25th and 75 percentiles are 0.734 and 0.843, respectively. Compared to the original dataset, there is a greater spread in Vmaxnorm, and there are more outliers.

Figure 9 shows scatterplots of the normalized maximum tangential winds evaluated at 8-km height versus the maximum tangential winds and RMW at 2-km height, respectively. Vmaxnorm increases with increasing maximum wind speed (*r* = 0.479, *p* = 0.002) and decreases with increasing RMW (*r* = −0.324, *p* = 0.044). This systematic relationship between the normalized vertical decay rate of Vmax and intensity and size confirms what was shown with the original dataset in SN11. Also shown in Fig. 9 are the absolute decay rates (m s^{−1} km^{−1}) from 2- to 8-km heights. The absolute decay rate ranges from −0.686 to −2.98 m s^{−1} km^{−1}. The magnitude of the decay rate increases with increasing size, and excluding the obvious outliers, it appears that the absolute decay rate approximately doubles as the low-level RMW increases from 10 to 50 km. While there is not a statistically significant relationship between Vmax at 2 km and the absolute decay rate, the decay rate does appear to increase somewhat with increasing intensity, excluding the outliers. Figure 10 shows the difference between the “predicted” Vmax using the mean profile of Vmaxnorm and the actual Vmax. The addition of the new cases in the expanded dataset substantially expands the range of this difference, with values as large as ±8 m s^{−1} (cf. Fig. 4b in SN11). It does remain true that in most cases, the error caused by using the mean decay profile is relatively small: less than ±4 m s^{−1}. Nevertheless, it seems that the observed vertical profile of maximum tangential wind is further from “universal” than was apparent in SN11.

For Vmaxnorm at 8 km, there are three obvious positive outliers, where the decay rate is much less than average. These are Dennis on 10 July 2005 (Vmaxnorm = 0.896), Rita on 21 September 2005 (Vmaxnorm = 0.916), and Felix on 3 September 2007 (Vmaxnorm = 0.895). From the vertical profiles (Fig. 8b), it is clear that the deviation in the decay rate actually occurs between 2 and 5 km, where Vmax is nearly constant or even increases with height. Radius–height plots of the azimuthal-mean tangential wind for these three cases are shown in Fig. 11. It can be seen that in each case, there is a secondary maximum in tangential wind at 4–4.5 km. This double maximum is also evident in individual legs and prior to any inpainting/interpolation (not shown), and so we are confident that these structures are robust. SN11 hypothesized that this structure in Dennis was a consequence of unbalanced flow, and showed evidence from an idealized simulation (shown here in Fig. 11d) that was consistent with this hypothesis. It seems likely that the anomalous profiles in Rita and Felix have the same cause.

While there are no data^{8} that will allow us to quantitatively determine whether unbalanced flow is responsible for these structures, these storms share other characteristics that might be conducive for large unbalanced flow. Dennis and Felix are both very small storms, and while closer to average, Rita is still relatively small. All three were major hurricanes, and Rita and Felix were category 5. The linear analytical theory of Kepert (2001) predicts that the strength of the supergradient jet at the top of the boundary layer should increase with the maximum gradient winds (Vgmax), as does the nonlinear numerical boundary layer model of Kepert and Wang (2001). In their full-physics idealized simulations, SN11 also found that the supergradient flow increases strongly with Vgmax (their Fig. 11). SN11 showed that just above the supergradient jet, there is a region of subgradient flow. In these simulations, it is this subgradient “jet,” combined with the existence of yet another weakly supergradient jet above, that causes the secondary maximum in tangential wind speed. Figure 11d shows the azimuthal-mean wind field from a simulation of SN11 (this is similar to Fig. 17a of SN11), with the unbalanced flow overlaid. The subgradient jet sandwiched between the two supergradient jets is evident, and the vertical structure of the tangential wind field in this idealized simulation is strikingly similar to the observations of Rita at this time. This general configuration of unbalanced flow actually existed in all of SN11’s simulations at almost all times once hurricane intensity was achieved. Whether an elevated maximum in tangential winds was seen depended on the strength of the unbalanced jets and their location relative to the radius of maximum gradient wind (RMVg).

Returning to our three observed cases, they are all very intense, which should lead to stronger unbalanced flows, and is therefore conducive for producing elevated maxima in tangential wind. On the other hand, their small size should decrease the magnitude of unbalanced flows and therefore should be less conducive for producing elevated maxima in tangential wind. Kepert’s theory predicts that the supergradient jet should weaken with decreasing RMW, and the simulations of SN11 support this (their Fig. 16). There are a few possibilities for why the elevated wind maximum might occur preferentially in smaller storms, despite the fact that size itself should lead to weaker unbalanced flows. First, all else being equal, the strength of the supergradient jet should increase with decreasing inertial stability (Kepert 2001). As shown in SN11, a vortex that initially has weaker inertial stability outside of the RMW (i.e., a more rapid decrease of tangential winds with increasing radius) will contract to smaller steady-state size. Therefore, it might be the case that small, strong storms have weaker inertial stability just outside the RMW than large, strong storms, and so perhaps they are more likely to exhibit elevated wind maxima. Second, the unbalanced jets are generally inwards of the RMVg, and therefore the RMW is displaced inwards of the RMVg within and near the boundary layer. SN11 showed that the amount by which the RMW is displaced increases with increasing RMVg, so that for very small storms, the RMW is nearly collocated with the RMVg. It is possible that this allows for larger unbalanced flows at the RMW itself, despite the jet maximum being weaker.

## 6. Comparison to previous studies

This study is an extension of SN09 and SN11, and so it is useful to examine to what degree our results here remain consistent with these previous studies. The main observational result of SN09 is that the slope of the RMW increases approximately linearly with size and has no relationship with intensity. This is confirmed with our larger dataset. SN09 also found that the RMW is approximately a surface of constant *M*, but that *M* systematically decreases upward along the RMW. This too is confirmed in the expanded dataset, as is the fact that the slope of the *M* surface that is coincident with the RMW at 2-km height has a strong linear relationship with size. SN09 did not examine the relationship between the slope of *M* and intensity. Here, we have found that there is a moderate relationship with the best-track intensity, with the strongest storms having the smallest (most vertical) slopes of *M*. This relationship is not predicted by theory (Emanuel 1986; SN09). The departure of the RMW from an *M* surface is also not predicted by theory, and we have found here that the degree of departure becomes less with increasing intensity.

SN11 showed (using the same dataset as SN09) that there was a common profile for the fractional decay rate of the maximum tangential winds in tropical cyclones and that this result was generally consistent with both theory and numerical simulations. With the expanded dataset, we find that the average decay rate is the same as in SN11, but that there is more variability in the new dataset. In the original dataset, SN11 found that Vmaxnorm did increase with increasing intensity and decreasing size, and this result is supported by the expanded dataset. In the context of Emanuel’s theory (as shown in SN11), Vmaxnorm should be nearly independent of both size and intensity, and the absolute decay rate should therefore increase linearly with intensity and be independent of size. SN11 found that the observed systematic relationship between normalized decay rate and intensity is at least in part due to the fact that *M* decreases with height along the RMW, and this remains the case with the expanded dataset (not shown). Why there is such a deviation from constant *M*, and why the magnitude of the deviation decreases with increasing intensity is unclear. We speculate that perhaps stronger storms more closely satisfy one of the two main assumptions of Emanuel’s theory: slantwise moist neutrality.^{9} On the other hand, Stern (2010) found that there were substantial departures from neutrality in simulations, and that that structure and magnitude of these deviations remained similar at all intensities.

At the suggestion of a reviewer, we investigated the possibility that the wavenumber-1 structure that is presumed to be associated with shear and/or vortex tilt might be responsible for the deviations of the observations from theory. The hypothesis was that storms with a greater wavenumber-1 amplitude of tangential winds at the RMW would have a lesser azimuthal-mean tangential wind and that this could manifest as a more rapid decay of azimuthal-mean tangential winds with height, as well as a larger deviation of the RMW from an *M* surface. This is plausible, as Marks et al. (1992) and Roux and Marks (1996) showed the wavenumber-1 amplitude is maximized near the RMW. We examined this hypothesis by calculating the wavenumber-1 tangential wind amplitude as a function of height along the RMW for each case for both the Earth-relative and storm-relative winds (not shown). We found that the correlations of the wavenumber-1 fields with the decay rates of azimuthal-mean tangential winds and *M* are generally insignificant and that where they are very weakly significant, the sign of the relationship is opposite to that of the hypothesis. Therefore, it appears that the wavenumber-1 tangential winds do not explain the spread in the observations or the deviations from theory.

HH13 investigated the slope of the 20-dB*Z* isosurface using a dataset that has a large overlap with ours. They found that the slope of dBZ20 has a statistically significant but weak (*r* = 0.33) relationship with Pmin and a moderate relationship with Vmax (*r* = −0.41). This is somewhat consistent with our results, although in our case the correlation with Vmax is too weak to be statistically significant, and the correlation with Pmin is very dependent on a single outlier. HH13 found a moderate relationship between the size of dBZ20 and its slope (*r* = 0.56), and this is consistent with our current results and is similar to the relationships found here and in SN09 between the RMW and its slope and RMWAX and its slope. Although HH13 emphasize the relationship between slope (of dBZ20) and intensity, their results actually indicate a substantially stronger relationship between slope and size.

There are a number of differences in methodology between HH13 and our study, and we discuss some of them here. Most importantly, they assessed slope from 2- to 11-km heights, whereas we use 2–6 km owing to the large number of cases for which dBZ20 does not extend above 6 km. When dBZ20 did not extend to 11 km, HH13 calculated the slope using data up until the maximum height of the isosurface. It is not clear how many such cases there are in HH13, but it is presumably a substantial portion of their dataset. Using a range of heights that varies among the cases could potentially lead to biases, as there is likely a relationship between the vertical extent of dBZ20 and intensity. HH13 found that the peak of the distribution of slope angles for dBZ20 was 70° (from a histogram with 5° bins), though the mean was 58°; here we find that they are 55° and 48°, respectively (not shown). This is very likely due to the differences in vertical range used (as also speculated by HH13). Indeed, HH13 found that the mean over the 2–6.5-km layer was 46°, nearly identical to what we find from 2 to 6 km.

Another important difference from SN09 and our current study is that HH13 calculated slope in terms of the angle from the vertical, whereas we use the ratio of horizontal to vertical displacement. As angle is a nonlinear function of the ratio (e.g., an increase in angle from 70° to 80° is actually a much larger increase in slope than is an increase in angle from 40° to 50°), this may lead to differences in the relationships between slope and intensity and size. For example, the mean slope of the RMW in our expanded dataset is 0.863, but the tangent of the mean angles is 0.610, 30% less, and this underestimate of the mean using angles is similar for dBZ20, RWMAX, and the *M* surface as well (not shown). HH13 stated that the correlations are not very sensitive to which definition of slope is used, and we also find that this is generally true (not shown), though the correlation between the slope of the RMW and its size is substantially reduced (from *r* = 0.601 to 0.466) when using the angle. The theoretical linear relationship between the slope of *M* and radius is for slope defined as a ratio and not as an angle. Indeed, a distinct departure from linearity can be seen in the observed relationship between the angle slope of *M* and size (not shown). Therefore, we suggest that future studies should define the eyewall slope as a ratio.

## 7. Summary and conclusions

In this study, we have revisited the work of SN09 using a dataset of Doppler radar data from 39 flights into Atlantic hurricanes to evaluate the slopes of the eyewalls and their relationships to storm size and intensity. With this larger dataset (as compared to 17 flights used by SN09), we find that the slope of the RMW (from 2- to 8-km heights) increases linearly with the RMW at 2-km height and is not a function of intensity. The *M* surface that originates at the 2-km RMW also increases linearly with the RMW itself, as do the slopes of the azimuthal-mean 20-dB*Z* isosurface and the azimuthal-mean eyewall updraft. Although the slopes of the RMW and updraft are uncorrelated with intensity, the slope of the *M* surface has a weak to moderate correlation with intensity. The slope of the 20-dB*Z* isosurface also appears to have a weak relationship with intensity, although it is unclear if this is robust. No measure of eyewall slope appears to have any relationship with intensity change. The slopes of the RMW and eyewall updraft are on average smaller than those of the eyewall *M* surface and the 20-dB*Z* isosurface. Consistent with the fact that the slope of *M* is larger than that of the RMW, *M* on average decreases with height along the RMW by about 9% from 2 to 8 km. The RMW becomes closer to being a surface of constant *M* as intensity increases. Overall, this expanded dataset confirms the primary result of SN09: the slope of the eyewall is largely a function of the RMW itself.

We have also used the expanded dataset to revisit some of the results of SN11, which utilized the same dataset as SN09. The average percentage rate of decay with height of the maximum tangential winds is about the same in the expanded dataset, with about a 20% decrease from 2- to 8-km heights. The majority of the decay of the balanced wind field therefore occurs above 8 km in all hurricane-strength vortices. The normalized maximum winds decrease more slowly with height in stronger storms and decrease more rapidly with height in larger storms. A few cases exhibit regions of increasing maximum winds with height from about 2 to 4 km. We believe that this structure is a manifestation of unbalanced flow, consistent with the idealized simulations shown in SN11.

With this comprehensive dataset of airborne Doppler radar data, it is now possible to draw general conclusions about the vertical structure of the eyewall region of tropical cyclones, and it is evident that in a number of ways, this structure is predictable. Beyond basic knowledge itself, this should prove useful for model verification. Until very recently, most forecasts and simulations were verified only by the “intensity,” as represented by maximum 1-min sustained 10-m wind speed or minimum sea level pressure. It is becoming increasingly evident that these basic metrics are limited in their ability to encompass the many other relevant characteristics of tropical cyclones (e.g., Vukicevic et al. 2014). Our results regarding the slope of the eyewall and vertical decay rate of maximum wind speed provide guidance for evaluating the realism and accuracy of numerical simulations. In most storms, there will not be adequate observations to determine the actual slope of the eyewall. However, based on the values of slopes presented here, and the systematic relationship between slope and size, it is possible to evaluate the simulated vertical structure based only on knowledge of the low-level size. Similarly, it is possible to assess whether the simulated vertical decay rate of the maximum winds falls within the known observed range. Deficiencies in simulated eyewall slopes have indeed been noted in previous studies of high-resolution hurricane simulations by Nolan et al. (2009, 2013), with the more recent study performing comparisons to the dataset presented here. Ultimately, this should lead to improvements in forecasts of both structure and intensity, as well as improved understanding of the dynamics of the tropical cyclone eyewall.

## Acknowledgments

The authors thank three anonymous reviewers for their helpful comments and suggestions. D. Stern was supported by an NSF-AGS Postdoctoral Research Fellowship (AGS-1231193). D. Nolan was supported by NSF Grant AGS-1132626 and by NOAA through the Hurricane Forecast Improvement Program.

## REFERENCES

*26th Conf. on Hurricanes and Tropical Meteorology,*Miami, FL, Amer. Meteor. Soc., 5D.4. [Available online at http://ams.confex.com/ams/pdfpapers/75806.pdf.]

**71,**1292–1304, doi:.

## Footnotes

^{1}

Note that SN09 neglected to mention the radial smoothing.

^{2}

The choice of grid spacing is made by the analyst who runs the automatic processing software and is a compromise between domain size and computational speed and storage space. Note that the intrinsic resolution of the radar is the same among all cases.

^{3}

A slightly more complete version of the analysis in SN09, demonstrating the statistical significance of those results, can be found in the dissertation of Stern (2010).

^{4}

Except where noted otherwise, all correlations are significant at the 99% level.

^{5}

We somewhat arbitrarily define relationships as strong, moderate, and weak when the correlations exceed 0.6, are between 0.4 and 0.6, and are less than 0.4, respectively.

^{6}

Though this can be seen from the data presented, Marks et al. (1992) actually state that the slopes are similar.

^{7}

HH13 state that there are 124 legs, but the list in their Table 3 sums to 121.

^{8}

We did examine radial profiles of flight-level temperature for Rita and found the mean profile exhibited a negative radial gradient, consistent with a decrease in gradient wind with height and further supporting the hypothesis that the increase in tangential winds with height is due to unbalanced flow.

^{9}

As the boundary layer supergradient jet increases systematically with increasing intensity, it is unlikely that the other primary constraint of Emanuel’s theory, thermal wind balance, is more closely satisfied in stronger storms.