Reexamining the Vertical Structure of Tangential ... - AMS Journals

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Reexamining the Vertical Structure of Tangential Winds in Tropical Cyclones: Observations and Theory DANIEL P. STERN AND DAVID S. NOLAN Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida (Manuscript received 4 August 2008, in final form 28 April 2009) ABSTRACT A few commonly held beliefs regarding the vertical structure of tropical cyclones drawn from prior studies, both observational and theoretical, are examined in this study. One of these beliefs is that the outward slope of the radius of maximum winds (RMW) is a function of the size of the RMW. Another belief is that the outward slope of the RMW is also a function of the intensity of the storm. Specifically, Shea and Gray found that the RMW becomes increasingly vertical with increasing intensity and decreasing radius. The third belief evaluated here is that the RMW is a surface of constant absolute angular momentum M. These three conventional wisdoms of vertical structure are revisited with a dataset of three-dimensional Doppler wind analyses, comprising seven hurricanes on 17 different days. Azimuthal mean tangential winds are calculated for each storm, and the slopes of the RMW and M surfaces are objectively determined. The outward slope of the RMW is shown to increase with radius, which supports prior studies. In contrast to prior results, no relationship is found between the slope of the RMW and intensity. It is shown that the RMW is indeed closely approximated by an M surface for the majority of storms. However, there is a small but systematic tendency for M to decrease upward along the RMW. Utilizing Emanuel’s analytical hurricane model, a new equation is derived for the slope of the RMW in radius–pressure space. This predicts a linear increase of slope with radius and essentially no dependence of slope on intensity. An exactly analogous equation can be derived in log-pressure height coordinates, and a numerical solution yields the same conclusions in geometric height coordinates. These conclusions are further supported by the results of simulations utilizing Emanuel’s simple, timedependent, axisymmetric hurricane model. As both the model and the analytical theory are governed by the dual constraints of thermal wind balance and slantwise moist neutrality, it is demonstrated that it is these two assumptions that require the slope of the RMW to be a function of its size but not of the intensity of the storm. Finally, it is shown that within the context of Emanuel’s theory, the RMW must very closely approximate an M surface through most of the depth of the vortex.

1. Introduction Although the direct impacts of tropical cyclones are generally manifested at and just above the surface, the necessity of understanding tropical cyclone (TC) structure and dynamics is not limited to this region. The importance of the radial structure of the primary circulation has been thoroughly examined in recent years (e.g., Mallen et al. 2005; Reasor et al. 2004; Willoughby and Rahn 2004; Willoughby et al. 2006); the vertical structure has received relatively less attention. This is likely due to the dearth of quality observations of wind fields above the flight level of the aircraft that penetrate

Corresponding author address: Daniel P. Stern, RSMAS/MPO, 4600 Rickenbacker Causeway, Miami, FL 33149. E-mail: [email protected] DOI: 10.1175/2009JAS2916.1 Ó 2009 American Meteorological Society

the storms (typically about 3 km). Although it is expected that the dynamics of TCs are sensitive to vertical structure, few (if any) studies have taken this into account. In most modeling studies, a baroclinic vortex is used as an initial condition, with the particular structure arbitrarily chosen. The baroclinic structure of TCs influences their dynamics in several ways, notably by altering the efficiency with which unbalanced heat energy is converted to balanced mean kinetic energy (Hack and Schubert 1986; Nolan et al. 2007). There remain many unanswered questions regarding the vertical structure of the tangential winds within tropical cyclones, as well as several questionable assumptions made by—and conclusions drawn from—past studies. Furthermore, most of our existing understanding of generalized vertical structure comes from studies that are at least 20 years old. The arrival of copious quantities

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of recent data acquired from airborne Doppler radars, as well as vast improvements in computing power and numerical modeling, allows for the possibility of a comprehensive reexamination of vertical structure similar to that done for radial structure. It is the intent of this study to provide a first step in that direction, by focusing narrowly on three prior beliefs regarding the vertical structure of the radius of maximum winds (RMW). For convenience, we will hereinafter refer to these prior conclusions as the three conventional wisdoms (CWs) of vertical structure and will label them as CW1, CW2, and CW3: 1) CW1 is that the outward slope of the RMW decreases (i.e., becomes more vertical) with increasing intensity. 2) CW2 is that the outward slope of the RMW increases with increasing size of the RMW. 3) CW3 is that the RMW is approximately a surface of constant absolute angular momentum M 5 ry 1 ½fr2, where y is tangential wind speed and f is the Coriolis parameter. The relationships between RMW and slope and between intensity and slope were first explored by Shea and Gray (1973) using flight-level data. Data from all flights from 1957 to 1969 were utilized, consisting of 533 radial legs from 21 hurricanes on 41 days. This was the first study to explicitly compare RMW slopes among storms and attempt to draw general conclusions. Their Fig. 16 is reproduced here as Fig. 1, showing RMW (n mi; 1 n mi [ 1.852 km) versus pressure for all storm days that had data at two or more levels. Each of these missions had data for at the most five levels and usually fewer. Only nine missions with upper-level data were presented in the figure, and no data were present between about 500 and 250 hPa, so the shapes of those curves were presumably an educated guess in the midto-upper troposphere. Storms were classified as ‘‘weak,’’ ‘‘moderate,’’ or ‘‘intense.’’ It was concluded that ‘‘only weak storms exhibit a slope of the RMW with height; more intense storms do not’’; thereby, CW1 was established. A significant fraction of the plotted RMWs apparently had portions that sloped inward with height, a phenomenon that would be currently regarded as atypical. It was further concluded that slope is negligibly small in the lower half of the troposphere, regardless of intensity. Shea and Gray also hypothesized that RMW slope is related to the intensity of cumulus convection, with intense convection leading to vertical RMWs due to vertical advection of angular momentum. This notion is widely believed to be true today. Over the decade following the publication of Shea and Gray (1973), substantial improvements in observational instrumentation were made, including the advent of in-

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ertial navigation, improved radars, and the acquisition of the National Oceanic and Atmospheric Administration (NOAA) P3 aircraft, which are used for research flights to this day. Accordingly, further significant advancements in the understanding of storm structure were made by Jorgensen (1984a), utilizing 141 legs from 10 missions (5 days) into four mature hurricanes. In this study, only Hurricane Allen (1980) was observed at more than two levels, and Anita (1977) and Frederic (1979) were only observed at a single level. Consequently, much of the information on vertical structure was drawn from the radar reflectivity data. This included the assessment of the slope of the radar eyewall (chosen to be the 10-dBZ reflectivity contour) as a proxy for the slope of the RMW. Eyewall slope, defined hereinafter as the ratio of horizontal to vertical displacement, ranged from 1:1 in Anita on 1 September and in Allen on 8 August to 2:1 in Allen on 5 August. Establishing CW2 and reaffirming CW1, Jorgensen concluded that the outward slope of the 10-dBZ surface implied a sloping updraft that ‘‘apparently was related to the eye diameter and storm strength. Steeper slopes appear to be associated with stronger storms that have smaller eyes.’’ It is important to note that his definition of slope (steep meaning closer to vertical) is the opposite of our convention (steep meaning closer to horizontal).1 The ‘‘Dopplerization’’ of the P3 tail radars and the development of techniques for constructing the vector wind field in three dimensions had a tremendous impact on our ability to observe and understand wind structure. One of the earlier studies to make use of this new technology for examining hurricane structure was that of Marks and Houze (1987), who examined Hurricane Alicia (1983). This study was the first to present radius– height cross sections of tangential (and radial) winds derived from Doppler radar, with radial and vertical resolutions of 2 km and 700 m, respectively. The two sets of cross sections represented azimuthal averages over quadrants of the storm, one centered on the northwest and the other on the east-southeast. Importantly, they noted that the conclusion of Shea and Gray that only weak storms have a sloping RMW is apparently inconsistent with the significant outward slope found in Alicia [which had a best-track intensity of 95 kt (1 kt ’ 0.544 m s21) near the time of observation]. This disagreement is implicit in Jorgensen as well, and there are few (if any) subsequent studies that support the 1 Although he then confuses his own terminology in the following sentence, writing that ‘‘Shea and Gray also showed that the slope of the radius of maximum winds decrease for stronger storms’’; by his convention, the slope actually increases for stronger storms.

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FIG. 1. Variation of the RMW with elevation for (left) storms with simultaneous lower- and upper-tropospheric data and (right) storms with two or more simultaneous flights in the lower troposphere only. [Reproduced from Fig. 16 of Shea and Gray (1973)].

contention that the RMWs of moderate and intense storms have no slope. However, CW1 itself, that the slope is a function of intensity, has received support in other Doppler winds studies (e.g., Roux and Viltard 1995; Dodge et al. 1999). The third conventional wisdom (CW3) of vertical structure is that the RMW is a surface of constant absolute angular momentum M. Jorgensen (1984b) was the first to make this claim observationally, based on data from Hurricane Allen. Composite cross sections of M were made for both 5 August (north-northwest quadrant only, relative to the RMW of each individual leg) and 8 August (relative to the average RMW at each height). It was stated that, at the RMW, M surfaces ‘‘sloped outward, corresponding to the outward slope of the RMW’’ and that the slope of M surfaces increased with increasing radius. The classification of ‘‘approximate’’ to the constancy of M along RMWs is inevitably subjective, and the magnitude at which deviations from the exact become relevant to structure and dynamics is an important unanswered question. Jorgensen attempted to explain the more vertical RMW with decreasing radius (for Allen on 5 and 8 August) as a consequence of conservation of absolute angular momentum (and thereby using CW3 to

explain both CW1 and CW2), deriving the following equation: tan(f) 5

1 V ›V , RMW ›z

(1.1)

where V is the tangential wind speed and f is the angle of M relative to the horizontal. It is assumed that M is conserved at the RMW.2 From the measured quantities on the rhs, f was calculated and compared with the observed value. Jorgensen found the observed (228 and 748 from horizontal) and calculated (218 and 658 from horizontal) angles to be approximately the same, given the observational limitations. Unfortunately, (1.1) is only a diagnostic for verifying the approximate constancy of angular momentum along the RMW; it does not really explain the variation of slope with intensity and RMW. It indicates that if the vertical shear of the tangential wind along a given m surface were to be held constant while the m surface (which is also the RMW) contracted inward, then the slope of that m surface must 2

As noted by Jorgensen, this equation is actually strictly valid for the slope of surfaces of relative angular momentum m, where m 5 rV.

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decrease (become more vertical), with an additional equal contribution resulting from the required increase in V. There is no prior reason to assume that the shear would remain constant, however, and indeed Jorgensen calculated the shear to be about 4 times as large on 5 August as on 8 August. Furthermore, there is no reason to assume that the angular momentum of the RMW has been conserved between the two days. In other words, even when the RMW is an m (or M) surface, it is not necessarily true that this surface is the same surface on both days. Indeed, it is clearly evident from his Fig. 12 that they are not the same surface. This can happen for several reasons, the most likely of which is that an eyewall replacement cycle occurred. So, in the end, it is clear that an explanation for the intrastorm (and interstorm) variability of RMW slope has not actually been provided, except in the singular case of an RMW that contracts while both maintaining constant m and somehow holding shear along itself constant. This equation cannot be used to claim general relationships between slope, the RMW, and intensity; even if it could, the relationships to intensity and RMW cannot be separated from each other, because they are assumed to be perfectly correlated. We would also argue that the shear values chosen are too high, leading to a calculated slope that is too large. It appears that the shear values given are representative of the total vertical shear (the use of which is incorrect here), not the vertical shear along the RMW surface (which is considerably less), so it is possible that the apparently close agreement between observation and calculation is fortuitous or an artifact. Since the publication of Jorgensen, several other observational studies have found that the RMW is an M surface (e.g., Marks et al. 1992; Franklin et al. 1993). This has led to the acceptance of CW3 as being generally true of all storms. Powell et al. (1996) assumed that the RMW in Hurricane Andrew (1992) was an M surface, used Jorgensen’s equation to estimate the surface RMW from the flight-level RMW, and in turn used this to estimate the surface winds at landfall. This is problematic for at least two reasons: the vertical wind shear was estimated from an approximate thermal wind equation, which gives the shear in the vertical and not along the RMW, and thermal wind balance is severely violated in the boundary layer. Additionally, as noted by Kepert (2006), the RMW slopes outward with height in the boundary layer for entirely different reasons than those that cause its outward slope in the free atmosphere. In a Doppler winds study, Lee et al. (2000) assumed that the RMW of Typhoon Alex (1987) was an M surface and cited Jorgensen in claiming that the slope of that M surface was a function of both intensity and the RMW, thereby supporting all 3 CWs. Additionally, they claimed

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that the slope of the M surface was consistent with the slope of the radar eyewall (in this case, picking the 25-dBZ contour). Finally, Powell et al. (2009) state without reference that ‘‘the RMW above the boundary layer is to good approximation a constant M surface.’’ They then use this assertion as a physical basis for developing a relationship between flight-level winds and surface winds, one that is already used operationally and likely will be used to reassess the intensity of past storms. It is therefore clearly of importance that the validity of CW3 be further investigated. Observationally, our goal is to verify or falsify these conventional wisdoms of vertical structure. This will be accomplished through the analysis of a dataset of threedimensional (3D) Doppler wind fields comprising (at this time) seven hurricanes observed on 17 different days. The use of this dataset will offer at least two critical improvements over those used in prior studies. First, all studies prior to 1984 relied on flight-level wind measurements. The inherently coarse vertical resolution and limited azimuthal sampling contribute to large uncertainty and likely biases, which may have had an impact on the results and conclusions of such studies. The use of Doppler wind fields offers substantial improvement in these respects, with nominal vertical resolution of 500 m and near-complete azimuthal sampling. Second, all known prior studies utilizing Doppler wind fields are case studies of single storms. As is evident from a review of past literature, there is large interstorm variability in structure, and no single hurricane can be representative of a generic cyclone. It is difficult to reconcile (or even find) some of the differing results of these individual case studies, because the papers were written by different authors over a period of two decades, with differing methodologies regarding both the Doppler synthesis and the analysis. Therefore, a single study that examines vertical structure in a consistent manner from a multitude of cases is highly desirable. The body of literature dealing with the theoretical vertical structure of tropical cyclones is much smaller. There is essentially one theory that makes predictions regarding the vertical structure, the maximum potential intensity (MPI) theory of Emanuel (1986, hereinafter E86). MPI theory predicts an exact axisymmetric structure of tangential winds. The tangential wind field above the boundary layer is completely determined by SST and outflow temperature, along with a few other minor contributing parameters. The analytical model is based on the assumptions of thermal wind balance and slantwise moist neutrality everywhere above the boundary layer. This constrains surfaces of absolute angular momentum and saturated equivalent potential temperature u* e to be congruent. In turn, this implies that the lapse rate is moist

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adiabatic along M surfaces, and there is no local buoyancy relative to the vortex. Although thermal wind balance is generally (though not universally) accepted to be a good approximation in the free atmosphere for real storms (e.g., Willoughby 1990), the general validity of slantwise moist neutrality is still debated (e.g., Braun 2002; Camp and Montgomery 2001; Persing and Montgomery 2003). Although a steady state is assumed in order to solve for the MPI itself, the predictions of axisymmetric structure are equally valid for non-steady-state cyclones, as long as the evolution is slow enough that the assumption of thermal wind balance remains valid (Emanuel 2004). For cyclones that are not at their MPI, however, the radial pressure distribution along the top of the boundary layer (which is required to solve for the structure above) cannot be determined a priori. If the pressure (or wind speed) is specified along the lower boundary, the axisymmetric structure can then be determined for an arbitrary slantwise moist neutral tropical cyclone in thermal wind balance. The actual intensity of such a vortex is not constrained by the theoretical steady-state maximum intensity. Note also that although there are other existing theories of MPI (e.g., Holland 1997), they make no predictions of the axisymmetric structure; therefore, Emanuel’s MPI provides the sole existing theoretical basis for quantitative understanding of the vertical structure of tangential winds within tropical cyclones. In section 2, we describe the dataset and methods of analysis. Section 3 presents our observational results regarding the three conventional wisdoms. In section 4, we use Emanuel’s theory to derive an equation for the slope of the RMW (and M surfaces in general) and compare the result with our observational findings. In section 5, we demonstrate that the RMW should closely approximate an M surface within the context of Emanuel’s MPI theory. In section 6, we examine the relationships among RMW, intensity, and slope in a simple time-dependent axisymmetric hurricane model (Emanuel 1995, hereinafter E95). A discussion of our results is presented in section 7.

2. Data The data utilized are the three-dimensional Doppler wind fields that were obtained from the NOAA Hurricane Research Division (HRD; available online at http://www. aoml.noaa.gov/hrd/data_sub/radar.html). These data were collected by the X-band Doppler tail radars in the two NOAA P3 aircraft. The raw data are therefore similar to those used in most prior studies. However, the Doppler wind fields are synthesized using a relatively recent method by Gamache et al. (2004). It is similar to the original variational method of Gamache (1997), with the important exception that quality control is done

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automatically. This allows a vast reduction in the time required to produce an analysis (hours versus months). The wind fields produced by the automatic method have been demonstrated to be of comparable quality to manually produced fields. Furthermore, the most likely errors would occur in the unfolding of aliased velocities, and any such error would be visually obvious in plots of the analyzed data. The automatic analyses can only be performed on flights operating in fore–aft scanning technique (FAST) mode, in which the radar beam is tilted 208 fore and aft of the plane perpendicular to the flight track (Gamache et al. 1995). However, almost all missions in the last 10 years had at least one plane collecting data in FAST mode (J. Gamache 2008, personal communication); therefore, the dataset used for this study could be substantially expanded. The data as acquired are text files, each containing the automatic Doppler analysis from a single flight leg. Each file contains values of the three Cartesian components of the vector wind, along with reflectivity, on a 96 3 96 3 37 grid. The horizontal resolution in both x and y is 2.0 km, whereas that in the vertical is 500 m. The first vertical level is at the surface (0 km), and the top is at 18 km. The storm center lies at the center of the domain (at the point between the four innermost grid points), and the data are all storm centered; the winds are earth relative, however. The center is the operational flight-level center and is independent of height, so any physical tilt of the vortex will be apparent in the wind fields. Based on the domain size, the largest radius that could contain complete azimuthal coverage is 96 km. However, data are rarely available out to that range. Because the focus of this study is on the symmetric vertical structure, the data are azimuthally averaged. Before doing so, however, several steps are required. An important deficiency of several previous studies is the lack of even azimuthal sampling. It is well known that substantial azimuthal asymmetries in the wind field are generally present in tropical cyclones, and, more importantly, these asymmetries vary in amplitude and phase with height (e.g., Marks et al. 1992). Therefore, it is critical to sample as symmetrically as possible when taking an azimuthal average; otherwise, quantitative and even qualitative biases will likely be present. A single analysis generally contains insufficient azimuthal coverage. Therefore, in this study, data from as many legs as possible are composited together in an effort to maximize data coverage. This inevitably results in both a loss of time resolution and a smearing or diffusion of the wind fields (especially when compositing storms that are rapidly changing structure with time). However, this concern is outweighed by the increased azimuthal coverage. Also, the compositing is done prior to any azimuthal

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averaging. In this way, each grid point is represented only once in the azimuthal average. This can potentially introduce aliasing caused by moving asymmetries that are sampled only at certain grid points, because the number of legs that contribute to the average at each grid point is not constant in space. The alternative, in which all grid points at a given radius from all legs are averaged together, is worse, however, because the grid points that are sampled by more legs are improperly weighted more than those that are sampled fewer times. The time-composited three-dimensional wind fields have generally good azimuthal coverage below about 10-km height. However, it is desirable to further increase azimuthal coverage, because data holes remain. Because we have exhausted the actual sampling of the data, to do this it is necessary to somehow ‘‘fill’’ the data holes with pseudodata derived from the surrounding real data (this process is also referred to as ‘‘inpainting’’). This is done with an acquired MATLAB (proprietary software) script that essentially solves a Laplace equation to both fill holes and extrapolate outward. The outward extrapolation beyond the radii containing real data may be entirely unrealistic, but this is of no consequence to our analysis, because these regions are ignored. The filling of the holes appears to look qualitatively realistic, except inside the eye (where few radar scatterers are present), and this region is ignored as well. Note that all existing real data points are entirely unaltered by this process. The resulting azimuthal mean fields are only used for quantitative purposes where there is a large enough proportion (;80%) of real data points. The final step prior to averaging is a cubic spline interpolation in the horizontal to 500-m resolution. This allows a more smoothly height-varying RMW to be found following the calculation of the azimuthal mean tangential winds. It is unlikely to alter quantitative results (positively or negatively), but it substantially aids plotting and interpretation. The use of interpolation is also another reason why the filling in of data holes is desirable; interpolation on fields containing holes will create bigger holes, thereby losing real data. Finally, the azimuthal means are calculated by grouping all data into bins of 500-m radial width and averaging within each bin. To address whether our method of creating azimuthal means produces biases in our results, particularly through the use of the inpainting scheme, we also tried two alternate methods of averaging that do not utilize inpainting. One method is to directly average data binned at a radial resolution of 2 km, without regard to obtaining complete azimuthal coverage. The other method is to linearly interpolate the Cartesian data onto circles at 500-m radial intervals, linearly interpolate in azimuth on each circle to fill all holes, and then take the mean. Our results

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in section 3 are largely insensitive to the method used (not shown); therefore, we will only present the results using our original method. An example of the process just described is illustrated in the following figures. In Fig. 2a are tangential winds at 3-km height from Hurricane Ivan (2004) on 7 September from 1 leg, with data from 1724 to 1751 UTC. These data are unaltered from their original form, other than the trivial calculation of tangential winds from u and y. Shown in Fig. 2b are data from another leg, taken from 1802 to 1826 UTC. On their own, each of these Doppler analyses samples a large portion of the inner core. There are substantial holes in each, however—to the northwest in leg 1 and to the southwest in leg 2. When composited together, most of the holes in one leg are covered by data in the other (not shown). At this level, at least, the coverage of the inner core is now essentially complete outside of the eye, and the RMW is sampled at all azimuths. Next, Fig. 2c demonstrates the results of filling the remaining holes with the inpainting scheme. The winds in the extrapolation region are unrealistic but are not used in our analysis. Figure 2d is a radius–height cross section of azimuthal mean tangential winds from the same data. The vertical axis ranges from 0- to 10-km height, because data coverage generally (but not always) declines rapidly above 10 km. The solid black line is the RMW. In this case, it is readily seen that the RMW is nearly vertical. Based on the conclusions of prior studies, one may be inclined to assert that this near-vertical RMW is the result of Ivan’s intensity, because maximum low-level azimuthal mean tangential winds are greater than 45 m s21. Classification of a storm as intense is relative and arbitrary, however. At what intensity do we expect to find a vertical RMW? This question has never been answered. Previous studies suggest that if Ivan is found to be more (less) intense at another time, then the outward slope of the RMW will be smaller (larger). In single case studies, this hypothesis could not be properly tested. Fortunately, we now have the data to do this.

3. Observational results The tangential winds in Ivan on 14 September are shown in Fig. 3. Ivan at this time is much stronger than it was on 7 September, with low-level winds in excess of 65 m s21. There is now also an appreciable outward slope to the RMW. The slope varies substantially with height, but it is clearly everywhere larger than that of the RMW on 7 September. This variability within a single storm is in opposition to CW1. Another obvious difference between Ivan on 7 and 14 September is that the RMW on 14 September is more than 2 times as large. Unlike the change of slope with intensity, this increase

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FIG. 2. Tangential winds at 3-km height in Hurricane Ivan on 7 September from (a) leg 1, (b) leg 2, and (c) the filled composite after inpainting. (d) Azimuthal mean tangential winds. The solid line in (d) is the RMW.

of slope with increasing RMW is qualitatively consistent with the results of prior studies. It can also be inferred that the RMW on 14 September does not represent the same material surface as the RMW on 7 September, and this poses a significant problem for Jorgensen’s argument. Because the intensity and RMW have both increased, the RMW must have gained a significant quantity of absolute angular momentum. This is indeed the case, as shown in Fig. 4. A likely scenario is that one or more eyewall replacement cycles took place during this interval, and an examination of individual radar legs reveals that a secondary eyewall was present on 12 September (not shown). Inevitably, the outer RMW forms in a region of larger M than the inner eyewall. Once the outer eyewall becomes dominant, it can contract to a radius that is still larger than that of the prior eyewall while achieving comparable or greater intensity.

With the absolute angular momentum data, we can begin to test CW3. On 7 September, the RMW appears to parallel the M contours, at least below 6-km height. It appears that M is approximately constant along the RMW on 14 September as well. Examples in which M is unambiguously not constant along the RMW can be found, as shown in Fig. 5, from Hurricane Dennis (2005) on 7 July. Angular momentum clearly decreases upward along the nearly vertical RMW in this case. Dennis was not particularly intense at the time of these observations, having just reached hurricane status. This would be inconsistent with the idea that the RMWs of weak storms have relatively large slopes. This example also demonstrates that a near-vertical RMW may not necessarily imply a very slow decay of wind speed with height. A final example is of Dennis three days later. Dennis on 10 July is somewhat stronger, much smaller,

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FIG. 3. Azimuthal mean tangential winds of Hurricane Ivan on 14 September. The solid line is the RMW. The shading scale is as in Fig. 2d.

and with an RMW that has a small but perceptible outward slope. In contrast to its earlier behavior, M is now very close to constant along the RMW. These examples serve to illustrate two interrelated facts: that there is large variability in vertical structure among tropical cyclones and that general ‘‘rules’’ of structure cannot be robustly determined from a few case studies. Therefore, we will attempt to examine as many cases as possible in aggregate. The 17 storm days used in this study are listed in Table 1, along with some basic characteristics. All automatic Doppler analyses available from HRD were acquired and examined. Several storm days had only 1 leg of data, so these were unsuitable for this study. Other storm days were subjectively excluded for various reasons, including close proximity to land, insufficient azimuthal coverage following compositing, RMW inside or outside of radial region with sufficient data, and poorly organized or excessively broad RMW. The storms used in this study include Hurricanes Frances (2004), Ivan (2004), Dennis (2005), Katrina (2005), Rita (2005), Wilma (2005), and Helene (2006). A compositing of wind fields from multiple storms would not be meaningful, because of the intrinsic structural variability. This would be the case even if winds were composited relative to individual RMWs [as was done by Shea and Gray (1973)]. Rather, we will examine

specific properties of structure from a multitude of cases, without compositing. In Fig. 6, the RMWs of all 17 cases are plotted versus height. It can be seen that there is a wide variation in RMW size among this sample, ranging from 10 to 70 km at low levels. It also appears that there is a pattern of increasing slope with increasing RMW. To quantify this apparent relationship, we first need to quantify the slope. This is done by calculating the best-fit line to each RMW between 2- and 8-km heights and taking the slope of this line to be the slope of the RMW. These slopes are plotted versus the size of the RMWs (at 2-km height),3 and this is shown in Fig. 7a. There is a bit of scatter, but there appears to be a clear relationship between RMW and slope, which is in qualitative agreement with prior observational studies and thereby supports CW2. This relationship is apparently linear, which we believe to be a novel result. An even better relationship exists between the RMW and the slope of the M surface that originates at the RMW at 2-km height, as shown in Fig. 7d.

3 This height was chosen because it is the lowest level that is both above the boundary layer (where RMW and slope are strongly influenced by frictional processes) and where the winds from the Doppler analyses are believed to be reliable (J. Gamache 2008, personal communication).

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FIG. 4. Azimuthal mean absolute angular momentum of Hurricane Ivan on (a) 7 and (b) 14 September. The solid line is the RMW.

Figure 7b shows that there appears to be little or no relationship with intensity. This result disagrees with CW1. The lack of relationship between intensity and slope is robust to different measures of intensity, including maximum azimuthal mean tangential wind from

the radar (shown in Fig. 7b) and best-track maximum sustained 10-m winds (not shown). In this sample, at least, there is also almost no relationship between intensity and RMW. This latter result is contrary to prior studies and conventional wisdom,

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FIG. 5. Azimuthal mean tangential winds of Dennis on (a) 7 and (b) 10 July, with the RMW plotted (solid line). RMW (solid) and the M surface (dashed) that originates at the RMW at 2-km height for Dennis on (c) 7 and (d) 10 July. The dotted curves are the M contours corresponding to values of 90% and 110% of the value of the dashed contour.

because intensity and RMW are generally thought to be inversely related. This question was reexamined recently in Kimball and Mulekar (2004). Using the extended best-track dataset, they found that median RMW does indeed decrease with Saffir–Simpson category, from 55.5 km for tropical storms to 27.8 km for category-5 storms. However, the changes only occur between categories 2 and 3 and between categories 4 and 5. There are also far fewer records of category 5 in the dataset. Nevertheless, there is a physical expectation that intensity should have some correlation with RMW, because storms generally contract during intensification. The correlation is apparently not that strong, however, given the nearly identical distribution of RMWs between categories 3 and 4, so it is perhaps not terribly anomalous that our limited dataset here displays no

relationship. The weak correlation can also be explained by the fact that a physical relationship is only to be expected within an individual storm; even then, it is only prior to any eyewall replacement cycles. The range of tropical cyclone initial sizes is huge, and therefore the sizes of hurricanes of similar intensity also have a substantial (though much smaller) variability. Eyewall replacement cycles also act to confound any general relationship and can remove much of the correlation between intensity and RMW, even across the history of a single storm. We can also examine the degree to which M is constant along the RMW among the cases in this dataset. This is done by comparing the slope of the RMW to the slope of the M surface that passes through the RMW at a height of 2 km. The results of the comparison are

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STERN AND NOLAN TABLE 1. List of storm days used in the study.

Storm: date

Time range (UTC)a (No. of legs)

Best-track MSLPb (hPa)

Best-track VMAXc (kt)

RMWd (nm)

Slope of RMW (ratio/8)e

Frances: 30 Aug 2004 (1) Frances: 31 Aug 2004 (2) Frances: 1 Sep 2004 (3) Frances: 4 Sep 2004 (4) Ivan: 7 Sep 2004 (5) Ivan: 9 Sep 2004 (6) Ivan: 12 Sep 2004 (7) Ivan: 13 Sep 2004 (8) Ivan: 14 Sep 2004f (9) Dennis: 7 Jul 2005g (10) Dennis: 10 Jul 2005h (11) Katrina: 28 Aug 2005 (12) Rita: 20 Sep 2005 (13) Rita: 22 Sep 2005 (14) Rita: 23 Sep 2005 (15) Wilma: 20 Oct 2005 (16) Helene: 17 Sep 2006 (17)

1753–2132 (3) 1707–1940 (2) 1651–2016 (3) 1842–2310 (3) 1724–1826 (2) 1628–1859 (3) 1630–2051 (4) 1758–2406 (6) 1902–2725 (11) 2445–2740 (4) 1608–1952 (6) 1725–2404 (5) 1541–1836 (3) 1435–2100 (7) 1640–2319 (7) 1824–2328 (4) 1547–1906 (6)

956/948/946 949/942/941 937/941/939 962/962/958 963/956/950 919/921/923 919/920/916 915/912/914 931/928/935 989/982/972 930/942/970 909/902/905 985/975/967 908/914/915 927/930/931 910/917/924 976/970/962

100/110/110 120/125/120 120/120/120 90/90/95 100/105/115 140/130/130 135/130/140 140/140/140 120/120/120 60/70/80 120/110/45 145/150/140 70/85/95 140/125/120 115/110/105 130/130/130 80/90/100

15/15/15 20/20/15 15/15/15 35/35/40 15/15/10 10/10/10 15/15/15 15/15/15 30/25/25 20/20/20 10/10/20 20/20/20 35/35/20 10/10/15 20/20/20 25/25/20 20/20/25

2.19/658 0.91/428 0.30/178 2.70/708 0.24/148 0.10/68 1.32/538 20.38/2218 0.95/448 20.27/2158 0.25/148 0.81/398 1.40/548 0.50/278 0.69/358 0.91/428 1.60/588

a

Time is in hours after 0000 UTC on the day on which the mission started, and therefore time may be 2400 UTC or greater. MSLP is minimum sea-level pressure. These are generally the values at the best-track time closest to the radar data, along with the values at the best-track times 6 h earlier and later. For most cases, the order is 1200/1800/0000 UTC. c VMAX is the maximum 1-min sustained wind speed valid at 10-m height. d RMW is from the extended best track and is only given to a precision of 5 n mi. e As defined in the text, slope is the ratio of horizontal to vertical displacement of the RMW. Here, we also provide slope in terms of degrees from the vertical axis. In both cases, larger numbers indicate an increasingly outward slope. f Best-track times are 1800/0000/0600 UTC. g This mission started on 6 Jul, and therefore times reflect hours after 0000 UTC 6 Jul. Because all radar data was gathered after 0000 UTC 7 Jul, this case is labeled as 7 Jul. Best-track times are 1800/0000/0600 UTC. h Dennis made landfall at 1930 UTC 10 Jul, and this is reflected in the 0000 UTC best-track data. b

shown in Fig. 7c. If the slopes were equal, then the points would lie along the 1:1 line. The distance away from the line indicates the amount of deviation from constant M along the RMW. It can be seen that a majority of the points lie relatively close to the 1:1 line. However, there is a small but systematic tendency for M to decrease upward along the RMW, because only two points lie below and to the right of the line. Figure 8a shows the profile of M along the RMW for each case, normalized by the value of M at 2-km height. Although M does not always monotonically decrease with height along the RMW, every case exhibits a net loss of M between 2- and 8-km heights, with an average loss of 8.3%. The tangential winds along the RMW as a function of height for all cases are shown in Fig. 8b. Obviously, this dataset is biased toward strong storms. This is partially due to the fact that storms tend to be mature by the time they are within range of the P3s, but it also reflects the difficulty in finding weak storms whose structure is organized enough to be straightforwardly analyzed. For instance, data for Hurricane Ophelia were available, but the wind field was broad and ill defined and was often too large for the inner core to be contained within the range of the Doppler analyses. Another weak hur-

ricane, Katrina on 25 August, had excellent data coverage but had to be excluded, because a portion of its circulation was over land and there was a significant tilt of the center with height. Nevertheless, there are still category-1 and category-2 storms represented among the dataset, including Dennis on 7 and 10 July and Rita on 20 September. Although the winds decrease monotonically with height for almost all cases, the rate of decay is never particularly large; hurricanes apparently maintain much of their strength through large depths of the troposphere. This is quantified in Fig. 8c, showing tangential wind along the RMW for each storm normalized by its value at 2-km height. The majority of storms retain 75%–85% of their 2-km tangential wind speed at 8-km height. Because the tangential winds must asymptotically approach zero at some height, most of the decay must be occurring between 8 km and the tropopause. The shape of the wind profile in the upper troposphere is difficult to determine however, because data coverage is increasingly sparse with height. We have shown that the slope of the RMW (and the slope of its initial M surface) increases approximately linearly with increasing RMW. This is in general

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FIG. 6. RMW for all 17 cases vs height. The numbers correspond to the storms listed in Table 1.

agreement with prior studies, and confirms CW2. To a certain extent, CW3 is also supported by our data, although there is substantial evidence for a systematic decrease of M upward along the RMW. On the other hand, our results are contrary to several prior studies regarding CW1, in that we find no relationship between slope and intensity. Before addressing the disparity between our results and those of Shea and Gray (and others), we will first investigate whether there is any theoretical reason to expect that the size of the RMW and the intensity of a hurricane should control the slope of the RMW.

4. Thermodynamic control of the slope of the RMW As far as we are aware, there exists no adequate theoretical explanation for what controls the outward slope of the RMW or even why it slopes outward at all. Some have argued (Holland 1997; Kepert 2006; Wang 2002) that because hurricanes are baroclinic vortices the RMW must slope outward. This argument is also implicit in Powell et al. (2009), who state that ‘‘the magnitude of RMW tilt is greatest in the upper troposphere, where the warm core is strongest.’’ But these arguments are incomplete, because it is trivial to create a balanced baroclinic vortex with a vertical RMW (e.g., using the iterative solution of Nolan and Montgomery 2002). The M surfaces must slope outward in such a vortex, but there is no need for the RMW to be an M surface in

a steady-state, balanced baroclinic vortex. Camp and Montgomery (2001) argue that ‘‘in nature, to conserve angular momentum, the eyewall slopes outward with height.’’ This line of reasoning, which has appeared in several papers (e.g., Holland 1997), is circular: it fails to explain the existence of the particular baroclinic structure (which is implicitly assumed to exist) that determines the degree of outward slope. Arguments such as these also fail to explain why the RMW, which is not a material surface, should conserve angular momentum.4 We find from observations that the RMW is indeed relatively close to an M surface in most cases. It turns out that an explanation for this phenomenon as well as our other observational findings can be found within the framework of Emanuel’s MPI theory. MPI theory appears to contain within it predictions for the relationship between slope and RMW and between slope and intensity, although to our knowledge this has yet to be made explicit. In his seminal paper,

4 It is not clear to us whether Camp and Montgomery intend to refer to the slope of the RMW, or to some other surface, which they define to be the eyewall. ‘‘Eyewall’’ is an ambiguous term within the literature, often taken to be the inner edge of the column of maximum radar reflectivity but also sometimes taken to be the interface between cloud and clear air, the momentum surface that goes through the location of maximum updraft, or the RMW itself (e.g., Wang 2002).

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FIG. 7. Scatterplots of the slope of the RMW vs (a) the RMW at 2-km height, (b) the maximum azimuthal mean tangential wind at 2-km height, and (c) the slope of the M surface that originates at the RMW at 2-km height. (d) Scatterplot of the slope of the M surface vs the RMW. The best-fit line to the data is plotted in (a),(b), and (d), and the 1:1 line is plotted in (c).

Emanuel (1986) presents an equation for the slope of M surfaces in radius–pressure space [his Eq. (10)]: ›r r3 ds* ›T 5 . ›p M 2M dM ›p s*

(4.1)

In section 5, we will offer an explanation as to why the RMW may itself be approximately an M surface in MPI theory; in section 6, we will confirm numerically that it is, so this equation also approximately gives the slope of the RMW.5 The slope depends on radius, absolute angular momentum, the moist adiabatic lapse rate along the RMW, and the rate of change of saturated moist 5 For mathematical accuracy, we will continue to refer to M instead of RMW in the equations themselves.

entropy s* with absolute angular momentum. It is difficult to further interpret the sensitivity of slope to radius and intensity from the equation in this form. Among other problems, radius and intensity are both implicitly present in absolute angular momentum in the denominator, whereas radius is also explicit in the numerator. However, the relationships can be made clearer through the use of Emanuel’s Eq. (11), which is an equation for radius along an M surface found by integrating Eq. (4.1) upward along M surfaces: r 2 M 5

M . ds* (T T out ) 2 dM

(4.2)

Rearranging Eq. (4.2), we can solve for ds*/dM and substitute this into Eq. (4.1) to find

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›r r 5 ›p 2(T T M

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›T . out ) ›p M

(4.3)

Derivatives are evaluated along M surfaces, because s* is constant with M. The slope is now only dependent on radius, temperature and its derivative with respect to pressure along the RMW, and outflow temperature Tout. The negative sign is present because pressure increases downward and therefore an outward slope in pressure coordinates will be negative. By the slantwise moist neutral assumption, the lapse rate is the moist adiabatic rate, which depends on pressure and temperature and will asymptotically approach the dry adiabatic lapse rate with increasing height. This lapse rate should only be weakly dependent on intensity and radius. Generally, this equation predicts a linear increase in the slope of M surfaces with radius at constant pressure. So, if we were to compare the slopes of different M surfaces within an individual storm, the slopes of the same M surface between different storms, or the slopes of different M surfaces between different storms, we would (in theory) find that the slope varies nearly linearly with radius. Note, however, that slope also varies along any given M surface and that this relationship is nonlinear. As we move upward along an M surface, we are also moving outward and slope will be increasing. More importantly, temperature is decreasing toward the outflow temperature, and it can be seen from Eq. (4.2) that 1/(T 2 Tout) varies with r2 along an M surface. Therefore, the combined dependence of slope on radius along an M surface is actually r3, as can be seen directly from Eq. (4.1). Therefore, a particular M surface will flare outward as it approaches the tropopause. Nevertheless, over a large portion of the lower and middle troposphere, the shape of the RMW should be approximately linear in radius–height space. That this is true is still a bit difficult to see from Eq. (4.3), which is in pressure coordinates. This assertion could be made more convincingly by deriving an equivalent expression to Eq. (4.3) in height coordinates, but unfortunately this cannot be done analytically. An exactly analogous form can, however, be derived (see the appendix) in log-pressure coordinates, which closely approximate geometric height coordinates. In log-pressure coordinates, Eq. (4.3) becomes ›r r ›T 5 . ›Z M 2(T T out ) ›Z M FIG. 8. (a) Normalized M, (b) tangential wind, and (c) normalized tangential wind along the RMW vs height for all cases. For each case in (a) and (c), respectively, the value of M and the tangential winds at each level are divided by their respective values at 2-km height. The numbers in (b) correspond to the storms listed in Table 1. The dashed line in (a) is the mean.

(4.4)

It is now clear that the dependence of slope (at constant height) on radius is indeed linear, to the extent that logpressure height is a valid approximation of actual height.

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We can also now see that the increase of the moist adiabatic lapse rate6 with height will lead to an additional doubling (approximately) of the slope of the RMW from the lower to upper troposphere. Strikingly, intensity appears to be absent as a contributor to the slope of the RMW. The moist adiabatic lapse rate along the RMW will go down slightly with increasing intensity, but this is negligibly small relative to the other factors, particularly radius. The strength of the balanced temperature perturbation associated with a storm will increase with increasing intensity, but most of the warming occurs radially inward of the RMW. This implicit effect of intensity is for slope to decrease with increasing intensity, but this should be—and is (as we will show in section 6)—very small. Therefore, it appears that MPI theory supports our observational finding that the slope of the RMW (at a given height) depends on its radius (and approximately linearly) but not on intensity. It does appear that MPI theory predicts a dependence of slope on potential intensity though, through both SST (which controls the maximum value of T) and Tout. As either SST increases or outflow temperature decreases, MPI increases and slope decreases. How can slope depend on MPI but not intensity itself? In MPI theory, this apparent paradox is resolved by the fact that the model is steady state and intensity is always at MPI. There is no freedom to vary intensity independently of potential intensity. It is important to emphasize, however, that the free-atmosphere component of the Emanuel model remains valid for non-steady-state storms at arbitrary intensities7 (Emanuel 2004) and that actual intensity is not present in Eq. (4.4). So MPI theory predicts that for a given potential intensity the slope of the RMW will not vary appreciably with actual intensity. A category-1 storm should have about the same slope as a category-5 storm in the same environment if the radius of the low-level RMW is the same. In reality, most storms never come close to reaching their MPI, and those that do only exist in that state for a small portion of their lifetime. Therefore, over all storms, the correlation between actual and potential intensity is relatively weak, so if E86 is valid we would not expect to see the apparent relationship be6 Although we refer to the ‘‘lapse rate’’, the decay rate of temperature with height along a sloping surface of constant saturation moist entropy is not actually equivalent to the local value of the moist adiabatic lapse rate. The former will also depend on the radial gradient of the actual (absolutely stable) vertical lapse rate. This effect is very small, however. 7 The free-atmosphere wind structure is determined by the freeatmosphere component of the model, along with appropriate boundary conditions. These boundary conditions could come from the steady-state MPI boundary layer model, but they also can be arbitrarily specified with no additional restrictions.

FIG. 9. Schematic diagram illustrating the momentum surface that originates at the low-level RMW and arbitrary momentum surfaces on either side of the RMW.

tween slope and intensity produced by the relationship between slope and MPI.

5. The relationship between the RMW and M surfaces We now offer a qualitative argument for why the RMW should be approximately an M surface within the context of MPI theory. The results will be supported by numerical simulations presented in the next section. Let us consider three surfaces of absolute angular momentum, as idealized in Fig. 9. The middle surface M0 originates at the RMW at an arbitrary lower boundary Z0, whereas M1 and M21 are located outside and inside M0, respectively. The tangential wind speed at any point along M0 is V0 5 M0/R0 2 (½)fR0, with analogous expressions for V1 and V21. We know that, at all heights, M1 . M0 and that R1 . R0. So, at any height along M0 and M1, V0 . V1 only if 1 2 R1 M0 2 fR0 .1 1 R0 M1 fR21 2

or

1 M1 fR21 R1 2 . . 1 R0 M0 fR20 2

(5.1)

(5.2)

This can be rewritten as R1 m . 1, R0 m0

(5.3)

where m 5 ry is the relative angular momentum. In Eq. (5.3), the relative angular momentum m1 and m0 are evaluated along the surfaces of constant absolute angular momentum M1 and M0, so the rhs is not constant.

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However, although the radii of initially congruent surfaces of relative and absolute angular momentum diverge from each other above ;8-km height (not shown), the variation of relative angular momentum along an absolute angular momentum surface is small (for the MPI vortex of E86, the value of m along the M surface originating at the surface RMW decreases to 98% and 94% of its surface value at 10- and 12-km heights, respectively), and the contribution of this variation to the variability of the ratio m1/m0 is negligible up to great heights (;12 km; not shown). We can therefore approximate the rhs of Eq. (5.3) as the constant M1/M0. We can ensure that the RMW will not move outward with height across M surfaces if R1/R0 can be guaranteed to be everywhere larger than that constant (M1/M0). Similarly, we can ensure that the RMW will not move inward with height across M surfaces if R21/R0 can be guaranteed to be everywhere larger than M21/M0 (which is a constant). When both of these inequalities are satisfied, then the RMW will remain along M0. We already know that these relationships must hold at the initial height, because M0 was defined to be located at the RMW there. To understand what happens above the initial height, we recall Eq. (4.4), which showed an approximately linear dependence of the slope of M surfaces on radius. If the slope were (at constant height) an exact linear function of radius (i.e., neglecting the radial dependence of T 2 Tout and the moist adiabatic lapse rate), then it can be shown that derivatives (of every order) of slope with respect to height along M surfaces are also exact linear functions of radius. This means that, at any given height, the slope has changed from its initial value (at Z0) by the same factor at all radii. In other words, the ratio of the slopes of any two M surfaces is constant with height. Now, because the slope itself is a linear function of radius, then the ratio of the radii of any two M surfaces is r1/r0 5 slope1/slope0. Because the ratio of the slopes is constant with height, the ratio of the radii must also be constant with height. Therefore, the ratios of the radii are exactly constant with height if the slopes of M surfaces are locally an exact linear function of radius. If these ratios were exactly constant with height, then it would be impossible for the RMW to move across momentum surfaces, because R1/R0 would always be greater than M1/M0 and R21/R0 would always be greater than M21/M0. There are two reasons why the ratio of the radii is not exactly constant with height. This can most easily be seen by referring to Emanuel’s Eq. (49), which we now rederive and write as Eq. (5.6) below, which relates the radius of an M surface to its initial radius rB at the top of the boundary layer. Applying Eq. (4.2) to an M surface at rB gives

r2B M 5

M , ds* 2 (T B T out ) dM

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(5.4)

and therefore M 5 r2B M (T B T out ). ds*/dM

(5.5)

Because ds*/dM is constant along an M surface, Eq. (5.5) can be substituted into Eq. (5.4) to give T B T out , (5.6) r2 M 5 r2B M T T out which is Eq. (49) from E86. Taking the ratio of Eq. (5.6) applied to two arbitrary M surfaces, ! ! r2 M1 r2B M1 T B T out1 T 0 T out0 5 . r2 M0 r2B M0 T B T out0 T 1 T out1

(5.7)

This equation provides a complementary explanation of why the ratio of the radii of any two M surfaces is nearly (but not quite) constant with height. It is clear that the variation with height of the ratio of the radii of any two M surfaces depends only on T and Tout, because TB (the temperature at the top of the boundary layer) is assumed to be constant in MPI theory. If we further approximate Tout as a constant (as is done within some parts of MPI theory, as well as the simple axisymmetric model of E95), then, when evaluated with temperature as the vertical coordinate (instead of height), the ratio of the radii of any two M surfaces is exactly constant. When the radii are (more appropriately) evaluated with height as the vertical coordinate, their ratio is not constant, because temperature decreases with increasing radius. Additionally, if we allow Tout to be variable (as is done in section 2d of E86), then the ratio of the radii of two M surfaces will no longer be constant, even when evaluated at constant temperature. It is important to point out however that the lack of exact constancy of the ratio with height does not necessarily imply that the RMW is not exactly an M surface, only that we cannot be sure that it is. It can be seen that both of these effects act in the same direction (this is true to the extent that the temperature difference between two M surfaces increases with height). Outside of M0, R1/R0 will increase slightly with height, and it will remain impossible for the RMW to move outward across momentum surfaces. Inside of M0, R21/R0 will decrease with height, so it is possible for the RMW to move inward across momentum surfaces at some finite height. However, we expect both of these effects to be small (although we cannot

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show this a priori) and that the RMW will remain very close to (if not exactly) an M surface throughout the lower to midtroposphere and perhaps throughout the depth of the vortex, depending on the specific parameters. The degree to which the nonconstancy of the ratios allows the RMW to depart from an M surface is difficult to evaluate analytically, because it depends on the radial profile of tangential wind on the lower boundary; that is, for a given slope of the M surfaces, a more sharply defined RMW will remain closer to its initial M surface than would a broader RMW. We will show numerically that the RMW is indeed extremely close to an M surface for the steady-state vortices predicted by MPI theory and for time-evolving vortices that are governed by the same constraints as in MPI theory.

6. Verification in geometric height coordinates a. RMW slope versus low-level RMW Although we cannot solve for analytical relationships in geometric height coordinates for the variables that control the slope of the RMW, we can do so numerically, following the iterative approach of E86 to solve for the structure of a storm at MPI. The RMW as a function of height can be found, and the slope (evaluated from 2- to 8-km height) can be calculated in the same way as for the observational data. RMW is not actually a parameter in the Emanuel model, nor is it truly predicted. Rather, the RMW is a function of an outer radius r0, which is specified as an input parameter. This is the radius at which the radial pressure gradient at the top of the boundary layer vanishes, and it acts as a strong control on the RMW of the steady-state MPI vortex. The RMW is also a function of latitude, SST, and outflow temperature, although E86 stresses ‘‘that neither the core radius nor the outer radius are directly related to intensity . . . .’’ Because the outer radius is not really known a priori, we cannot speak of the RMW as a predicted variable. Nevertheless, Emanuel’s Eq. (46) (not shown) provides a direct relationship between the RMW and the outer radius, along with the other aforementioned variables. We can explore the relationship between the RMW and slope by systematically varying r0 while holding other parameters constant, and this is shown in Fig. 10a. As expected, slope increases nearly linearly with RMW. The rate of increase with RMW depends on SST; slope is less sensitive to RMW when SST is higher. This figure can also be interpreted as showing the relationship between slope and SST; at constant RMW, slope increases with decreasing SST. At small RMW, the dependence on SST is very small, but it becomes increasingly strong with increasing RMW. One would expect to be able to detect this effect observationally only in storms with

FIG. 10. Slope of RMW vs the RMW at 2-km height, as predicted by Emanuel’s MPI theory (E86). This relationship is shown (a) for varying values of SST and (b) for varying values of Tout. (c) Slope of RMW vs the slope of the M surface originating at the RMW at 2-km height for varying values of SST. The 1:1 line is plotted in (c).

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large RMWs, perhaps larger than 50 km. We can now also confirm that the RMW is to a very close approximation an M surface within MPI theory, at least within the 2–8-km height layer. The slope of the RMW versus the slope of its initial M surface is shown in Fig. 10c, and it is clear that their respective slopes are essentially equal. Slope also increases with increasing outflow temperature, as shown in Fig. 10b, although this sensitivity is substantially less than for SST. The differing sensitivities between SST and Tout are explained by the fact that the SST parameter is also implicitly contained within the lapse rate term, whereas Tout is not. We can therefore infer that it is the effect of SST on the moist adiabatic lapse rate that provides most of the control of SST on slope, rather than the direct effect of changing SST itself. As changes in SST and Tout lead to changes in potential intensity, it is evident that the slope of the RMW is a function of potential intensity. It is difficult to systematically vary potential intensity itself from the MPI parameters in this framework. Furthermore, it is impossible to explore any possible sensitivity of structure to actual intensity here. We can make further progress by moving a step forward in complexity to a timedependent model.

b. Further results from a time-dependent model Emanuel (1989) and E95 developed a highly simplified axisymmetric time-dependent hurricane model that allows a weak initial vortex to intensify to MPI. In this model, there are only two layers, the subcloud layer and the rest of the troposphere. The free troposphere is always in thermal wind balance and slantwise moist neutral, which are the same constraints imposed by the MPI steady-state model. Convection is crudely parameterized through a quasi-equilibrium assumption on boundary layer entropy. The horizontal coordinate of the model is potential radius R, given by f 2 f R 5 ry 1 r2 . 2 2

(6.1)

The physical radii of R surfaces are predicted quantities in the E95 model, so the RMW is now explicitly predicted. Most importantly, actual intensity is predicted and varies in time. Our focus here is not on the details and realism of hurricane genesis and intensification, but rather on the axisymmetric structure of hurricanes and the factors that control the structure at any given time. Therefore, this freely available model is appropriate for further investigating the sensitivity of structure to RMW and intensity. An exploration of the parameter space in the E95 model was performed by using differing initial conditions and analyzing the structure at various stages of

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vortex evolution. The relationship between slope and RMW is shown in Fig. 11a, and the relationship between slope and intensity is shown in Fig. 11b. The slope of the RMW was determined in a manner similar to that done with observations, by finding the slope of the best-fit line to the RMW from 2- to 8-km height.8 For the simulation of a single storm, the slope of the RMW is linearly related to its size, as in the MPI model; as the RMW contracts with time, the slope decreases. The intensity also appears to be very much related to slope, although the relationship is nonlinear. In fact, for a given simulation, the slope is approximately inversely proportional to intensity. This would appear at first to contradict our observational results and the predictions of MPI theory. However, this correlation occurs because we are only looking at a single simulation. The storm intensifies while contracting and therefore RMW and intensity will be related for a single intensification process. When we vary the initial conditions (in particular, the outer radius, which contributes most to determining the time evolution of the RMW), the specific relationship between RMW and intensity changes. Now, when we examine the relationships between slope and RMW and between slope and intensity over multiple simulations, a more consistent picture emerges. The RMW remains very well correlated with its slope, whereas the correlation between slope and intensity begins to decrease after combining the results of only a few simulations. There is a tremendous range of intensities associated with a given slope. In contrast, each slope is associated with a very narrow range of RMWs. This is illustrated in Figs. 11a and 11b, in which the ovals enclose the same seven points that have approximately the same slope. The intensity for these points ranges from 20 to 55 m s21, whereas the RMWs are all located within a few kilometers of one another. Therefore, although RMW is an excellent predictor of the slope, intensity is a very poor predictor, and the results of this simple model are in general agreement with both analytical theory and our observational findings. Finally, Fig. 11c shows the relationship between the slope of the RMW and the slope of its initial M surface. The points cluster tightly around the 1:1 line, so M is nearly constant along the RMW. At large slope (corresponding to large RMW), there is a small but systematic deviation of the points to the right

8 Although the prognostic equations of the E95 model are solved for in just two layers and in potential radius space, the tangential wind field is diagnostically solved and provided in geometric radius–height coordinates with horizontal and vertical resolutions of about 1 km and 200 m, respectively. This can be done because of the assumptions of slantwise moist neutrality and thermal wind balance.

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of the 1:1 line, indicating an increase of M with height along the RMW. The reason for this is currently unclear, but we note that these slope differences correspond to the RMW being found only a single grid point (1 km) outside of its initial M surface.

7. Discussion and conclusions Three conventional wisdoms regarding the vertical structure of tropical cyclones have been reexamined from both an observational and theoretical perspective. Using a dataset of 3D Doppler wind fields acquired from NOAA/HRD, we examined the azimuthal mean structure of tangential winds in seven hurricanes on 17 days. We found that the outward slope of the RMW increased with increasing size of the low-level RMW. This relationship between slope and RMW size is in agreement with prior studies and confirms the validity of CW2. This is the first study to quantify this relationship, which we found to be linear. Unlike for RMW, we found no apparent relationship between slope and intensity. This is in contrast to several previous observational studies— most notably that of Shea and Gray (1973), who found that slope decreased with increasing intensity. This discrepancy can likely be explained by the differences in sampling and data quality. In calculating the slope of the RMW, we utilize data between 2- and 8-km heights, whereas the flight-level records used by Shea and Gray contained no data between ;5- and 12-km heights. The Doppler winds we use have a vertical resolution of 500 m, whereas the data of Shea and Gray were at only 2–3 lower-tropospheric levels, sometimes supplemented with an upper-tropospheric flight between 180 and 260 hPa. Furthermore, our azimuthal sampling is much more dense and symmetric. As numerous studies have found (e.g., Marks et al. 1992), stationary asymmetries in the tangential wind field are often significant, even in strong tropical cyclones, so it is critical to maximize the density and symmetry of azimuthal sampling in order to derive a representative azimuthal mean structure. We also note that the three storms categorized by Shea and Gray as intense all had RMWs of less than 10 n mi (;18.5 km). Because no large intense storms were sampled, it would not have been possible for their study to have found an intense storm with a large slope, and this is likely the most important contributing factor to their conclusions. In our dataset, there are several large intense storms, and we have

FIG. 11. For the E95 model, slope of RMW vs (a) the RMW, (b) VMAX (the maximum azimuthal mean tangential wind speed), and (c) the slope of the M surface that originates at the RMW at 2-km height. Where not specified otherwise, simulations used the default parameters SST 5 278C, r0 5 350 km, RMW0 5 80 km, and

VMAX0 5 15 m s21; RMW0 and VMAX0 are the initial values of RMW and VMAX, and r0 is the initial radius at which the tangential wind vanishes. The ovals are drawn for purposes of comparison of (a) with (b), and they enclose the same seven data points.

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much more precise and accurate information on the intensity and the RMW of storms than was available to Shea and Gray. Therefore, although our sample size is still somewhat smaller than desired, we are confident that our observational results are robust and that CW1 is false. Lacking a preexisting theoretical explanation for the slope of the RMW, we derived an alternate (and simpler to interpret) form of Emanuel’s original equation for the slope of M surfaces in radius–pressure space. Depending only on the assumptions of thermal wind balance and slantwise moist neutrality, we found that the slope of each M surface (at constant height) increases linearly with radius and has no explicit (and only weakly implicit) dependence on intensity. This is in broad agreement with our observational results, as long as the RMW is itself an M surface. Using a simple geometrical argument, we showed that, within MPI theory, the RMW must closely approximate an M surface through most of the depth of the vortex. Therefore, the tendency of well-developed tropical cyclones to be near a state of slantwise moist neutrality is the primary reason that the RMW slopes outward with height. The imposition of thermal wind balance alone provides no constraint on the slope of M surfaces, does not require the RMW to lie along an M surface, and therefore does not on its own provide any control on the degree (or occurrence at all) of outward slope of the RMW. The only additional assumption within Emanuel’s model is slantwise moist neutrality, so this must be the cause of the outward slope of the RMW. To maintain slantwise moist neutrality, at a given height a momentum surface must have a greater outward slope if it is located at a larger radius, regardless of the particular quantity of momentum that characterizes the surface; therefore, it is the size of the RMW that determines the degree of outward slope and not the intensity of the cyclone. We further demonstrated (in the appendix) that this result is equally true in log-pressure height coordinates. We then showed numerically that this result remains essentially the same in geometric height coordinates.9 Finally, using the simple time-dependent model of E95, we verified the predictions of the steadystate analytical theory and demonstrated that (in this model) slope only has a relation to intensity for an individual storm that contracts while maintaining nearly constant angular momentum along the RMW. This correlation is completely due to the near-perfect inverse relationship between intensity and RMW for a single case, so intensity is a very poor predictor of slope over all storms.

9 Although this was shown for MPI vortices, it is true of any balanced, slantwise moist neutral vortex.

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That the RMW is an M surface within MPI theory appears to provide strong evidence for the validity of CW3, given that the predictions made by the theory regarding CW1 and CW2 are in accord with the observations. We also examined CW3 within our observational dataset and found that, for a majority of storms, the RMW was fairly well approximated as an M surface. On the other hand, M decreased upward along the RMW in all cases; although this decrease was relatively small on average, it was substantial in several cases. It therefore appears that, although the RMW is approximately an M surface for most hurricanes at most times, there are occasions where this does not hold true. The reason for this phenomenon and its significance are currently unclear. It has been suggested that nearly vertical RMWs occur because of intense convection and consequent vertical advection of angular momentum (Shea and Gray 1973). The implication of this is either that the RMWs of rapidly intensifying storms have a tendency to depart from M surfaces or that the maximum tangential winds do not decay with height. It seems at least plausible that the former could be true, because slantwise moist neutrality is more likely to be violated in rapidly intensifying storms, and we have shown that it is the assumption of such neutrality that leads to the theoretical near constancy of M along the RMW. On the other hand, we are hesitant to adopt this belief for several reasons. The first reason is that we [and others, such as Franklin et al. (1993)] have shown that a nearly vertical RMW may still remain approximately an M surface, and in these cases the lack of outward slope can be well explained by the size of the RMW. Another reason that we remain skeptical of the rapid intensification theory is that it is not borne out in our dataset. There are two cases (Dennis on 7 July and Ivan on 13 September) in our dataset in which the calculated slope of the RMW is slightly negative (inward with height), but with substantial outward slopes of the M surface at the RMW (1.15 and 1.35, respectively). At the time of our observations, Dennis was rapidly intensifying, although it was also the weakest storm in our sample. Ivan, on the other hand, was a nearly steady state category-5 storm. What they have in common is that their inner cores must not be slantwise moist neutral. The reason why this occurs is a subject of ongoing investigation. Acknowledgments. The authors thank John Gamache for helpful discussions and for allowing us to make use of his data and Kerry Emanuel for helpful discussions and for making his simple time-dependent hurricane model readily available. We also appreciate the helpful comments provided by Gary Barnes and one anonymous reviewer. This research was supported by the NSF under

DECEMBER 2009

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STERN AND NOLAN

Grant ATM-0432551. Additional support for Daniel Stern was provided by a University of Miami graduate fellowship.

›a ›a ›s* 5 . ›r Z ›s* Z ›r Z

(A.8)

In E86, it was shown that

APPENDIX Derivation in Log-Pressure Coordinates

(A.1)

where the constant scale height H is H5

RT 0 . g

f1

y r

y5

›f ›r

(A.2)

and

›f gT 5 , ›Z T0

(A.4)

(A.5)

Using the equation of state and the exponential dependence of p on Z,

1 › gH ›s* ›T 2 M 5 . r3 ›Z T 0 R ›r Z ›Z s* r

(A.11)

As in E86, saturated moist entropy is constant along absolute angular momentum surfaces, and Eq. (A.11) can be written as 1 › gH ›T ds* ›M 2 M 5 . r3 ›Z T 0 R ›Z s* dM ›r Z r

(A.12)

›r r3 ds* ›T 5 . ›Z M 2M dM ›Z s*

(A.13)

Equation (A.13) is exactly analogous to Eq. (4.1), with derivatives with respect to p replaced by derivatives with respect to Z. Therefore, in log-pressure coordinates, Eq. (4.3) becomes ›r r ›T 5 . ›Z M 2(T T out ) ›Z M

(A.14)

(A.6) REFERENCES

where a is the inverse density. This is analogous to Eq. (6) of E86: 1 › 2 ›a M 5 . 3 r ›p ›r p r

(A.10)

By dividing by (›M/›r)jZ and recalling the definition of H, Eq. (A.12) becomes

1 › g ›a eZ/H 2 M , 5 p0 3 r ›Z T 0 ›r Z R r

›a H Z/H ›T 5 e . ›s* Z p0 ›Z s*

(A.3)

where f is the geopotential. From Eqs. (A.3) and (A.4), a thermal wind equation can be derived: 1 › g ›T 2 M 5 . r 3 ›Z T 0 ›r Z r

Using Eq. (A.10) in Eq. (A.8) and substituting into Eq. (A.6),

In Eq. (A.1), p is pressure and p0 is a constant reference pressure, usually taken to be 1000 hPa. In Eq. (A.2), R is the gas constant for dry air, T0 is a constant reference temperature, and g is the constant gravitational acceleration. Following Schubert et al. (2007), gradient and hydrostatic balances are given by

(A.9)

The analogous relationship in log-pressure coordinates is

In log-pressure coordinates, we define Z to be p Z 5 H ln 0 , p

›a ›T 5 ›s* p ›p s*

(A.7)

Equation (7) of E86 will be identical in log-pressure coordinates, with derivatives now evaluated at constant Z, which is a function of pressure alone,

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——, 1989: The finite-amplitude nature of tropical cyclogenesis. J. Atmos. Sci., 46, 3431–3456. ——, 1995: The behavior of a simple hurricane model using a convective scheme based on subcloud-layer entropy equilibrium. J. Atmos. Sci., 52, 3960–3968. ——, 2004: Tropical cyclone energetics and structure. Atmospheric Turbulence and Mesoscale Meteorology, E. Fedorovich, R. Rotunno, and B. Stevens, Eds., Cambridge University Press, 165–191. Franklin, J. L., S. J. Lord, S. E. Feuer, and F. D. Marks Jr., 1993: The kinematic structure of Hurricane Gloria (1985) determined from nested analyses of dropwindsonde and Doppler radar data. Mon. Wea. Rev., 121, 2433–2451. Gamache, J. F., 1997: Evaluation of a fully three-dimensional variational Doppler analysis technique. Preprints, 28th Conf. on Radar Meteorology, Austin, TX, Amer. Meteor. Soc., 422–423. ——, F. D. Marks Jr., and F. Roux, 1995: Comparison of three airborne Doppler sampling techniques with airborne in situ wind observations in Hurricane Gustav (1990). J. Atmos. Oceanic Technol., 12, 171–191. ——, J. S. Griffin Jr., P. P. Dodge, and N. F. Griffin, 2004: Automatic Doppler analysis of three-dimensional wind fields in hurricane eyewalls. Preprints, 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.] Hack, J. J., and W. H. Schubert, 1986: Nonlinear response of atmospheric vortices to heating by organized cumulus convection. J. Atmos. Sci., 43, 1559–1573. Holland, G. J., 1997: The maximum potential intensity of tropical cyclones. J. Atmos. Sci., 54, 2519–2541. Jorgensen, D. P., 1984a: Mesoscale and convective-scale characteristics of mature hurricanes. Part I: General observations by research aircraft. J. Atmos. Sci., 41, 1268–1285. ——, 1984b: Mesoscale and convective-scale characteristics of mature hurricanes. Part II: Inner core structure of Hurricane Allen (1980). J. Atmos. Sci., 41, 1287–1311. Kepert, J. D., 2006: Observed boundary layer wind structure and balance in the hurricane core. Part II: Hurricane Mitch. J. Atmos. Sci., 63, 2194–2211. Kimball, S. K., and M. S. Mulekar, 2004: A 15-year climatology of North Atlantic tropical cyclones. Part I: Size parameters. J. Climate, 17, 3555–3575. Lee, W.-C., B. J.-D. Jou, P.-L. Chang, and F. D. Marks Jr., 2000: Tropical cyclone kinematic structure retrieved from singleDoppler radar observations. Part III: Evolution and structures of Typhoon Alex (1987). Mon. Wea. Rev., 128, 3982–4001. Mallen, K. J., M. T. Montgomery, and B. Wang, 2005: Reexamining the near-core radial structure of the tropical cyclone primary

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circulation: Implications for vortex resiliency. J. Atmos. Sci., 62, 408–425. Marks, F. D., Jr., and R. A. Houze Jr., 1987: Inner core structure of Hurricane Alicia from airborne Doppler radar observations. J. Atmos. Sci., 44, 1296–1317. ——, ——, and J. F. Gamache, 1992: Dual-aircraft investigation of the inner core of Hurricane Norbert. Part I: Kinematic structure. J. Atmos. Sci., 49, 919–942. Nolan, D. S., and M. T. Montgomery, 2002: Nonhydrostatic, threedimensional perturbations to balanced, hurricane-like vortices. Part I: Linearized formulation, stability, and evolution. J. Atmos. Sci., 59, 2989–3020. ——, Y. Moon, and D. P. Stern, 2007: Tropical cyclone intensification from asymmetric convection: Energetics and efficiency. J. Atmos. Sci., 64, 3377–3405. Persing, J., and M. T. Montgomery, 2003: Hurricane superintensity. J. Atmos. Sci., 60, 2349–2371. Powell, M. D., S. H. Houston, and T. A. Reinhold, 1996: Hurricane Andrew’s landfall in south Florida. Part I: Standardizing measurements for documentation of surface wind fields. Wea. Forecasting, 11, 304–328. ——, E. W. Uhlhorn, and J. D. Kepert, 2009: Estimating maximum surface winds from hurricane reconnaissance measurements. Wea. Forecasting, 24, 868–883. Reasor, P. D., M. T. Montgomery, and L. D. Grasso, 2004: A new look at the problem of tropical cyclones in vertical shear flow: Vortex resiliency. J. Atmos. Sci., 61, 3–22. Roux, F., and N. Viltard, 1995: Structure and evolution of Hurricane Claudette on 7 September 1991 from airborne Doppler radar observations. Part I: Kinematics. Mon. Wea. Rev., 123, 2611–2639. Schubert, W. H., C. M. Rozoff, J. L. Vigh, B. D. McNoldy, and J. P. Kossin, 2007: On the distribution of subsidence in the hurricane eye. Quart. J. Roy. Meteor. Soc., 133, 595–605. Shea, D. J., and W. M. Gray, 1973: The hurricane’s inner core region. I. Symmetric and asymmetric structure. J. Atmos. Sci., 30, 1544–1564. Wang, Y., 2002: Vortex Rossby waves in a numerically simulated tropical cyclone. Part I: Overall structure, potential vorticity, and kinetic energy budgets. J. Atmos. Sci., 59, 1213–1238. Willoughby, H. E., 1990: Gradient balance in tropical cyclones. J. Atmos. Sci., 47, 265–274. ——, and M. E. Rahn, 2004: Parametric representation of the primary hurricane vortex. Part I: Observations and evaluation of the Holland (1980) model. Mon. Wea. Rev., 132, 3033–3048. ——, 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.