The structure and evolution of rapidly intensifying Hurricane Guillermo (1997) is examined using airborne Doppler radar observations. In this first part, the low-azimuthal-wavenumber component of the vortex is presented. Guillermo’s intensification occurred in an environmental flow with 7–8 m s−1 of deep-layer vertical shear. As a consequence of the persistent vertical shear forcing of the vortex, convection was observed primarily in the downshear left quadrant of the storm. The greatest intensification during the ∼6-h Doppler observation period coincided with the formation and cyclonic rotation of several particularly strong convective bursts through the left-of-shear semicircle of the eyewall. Some of the strongest convective bursts were triggered by azimuthally propagating low-wavenumber vorticity asymmetries. Mesoscale budget analyses of axisymmetric angular momentum and relative vorticity within the eyewall are presented to elucidate the mechanisms contributing to Guillermo’s structural evolution during this period. The observations support a developing conceptual model of the rapidly intensifying, vertically sheared hurricane in which shear-forced mesoscale ascent in the downshear eyewall is modulated by internally generated vorticity asymmetries yielding episodes of anomalous intensification.
It is generally well recognized that hurricane intensification results from a combination of internal dynamic and convective processes as well as large-scale environmental forcing and ocean–atmosphere interactions. However, our understanding and ability to forecast these multiscale processes and their interactions remains limited. As a result, current predictions of intensity change exhibit considerably less skill than track predictions (DeMaria et al. 2002). These processes must be better understood before significant advancements in hurricane intensity prediction can be achieved (Marks and Shay 1998).
Rapid intensification (RI) remains one of the greatest challenges to forecasters given its potentially catastrophic impact on unprepared coastal communities. Recently, Kaplan and DeMaria (2003; hereinafter KD03) employed a 12-yr sample of overwater Atlantic Ocean tropical cyclones to statistically define RI (an increase in maximum winds of at least 15.4 m s−1 in 24 h) and to determine the environmental factors most favorable for RI: anomalously warm sea surface temperatures (SSTs), high lower-tropospheric relative humidity, low vertical wind shear, easterly upper-tropospheric flow, and weak forcing from upper-level troughs and cold lows. KD03 summarized that a hurricane far from its maximum potential intensity (MPI) within a favorable environment has the greatest chance of undergoing RI. Their results are qualitatively consistent with numerous observational studies (e.g., Molinari et al. 1995; Bosart et al. 2000; Dunion and Velden 2004) and numerical simulations (e.g., Pfeffer and Challa 1981; Schade and Emanuel 1999; Frank and Ritchie 2001) of hurricane–environment interactions. However, whether such favorable systems ultimately achieve RI may be linked to internal physical processes.
Internally, hurricane intensification proceeds via two conceptually distinct modes: an axisymmetric mode (Ooyama 1969) and an asymmetric mode. In one manifestation of the asymmetric mode the axisymmetrization of potential vorticity (PV) perturbations (e.g., generated by moist convection) leads to intensification of the axisymmetric primary circulation directly through momentum transports by vortex Rossby waves (Montgomery and Kallenbach 1997; Möller and Montgomery 2000; Shapiro 2000). In contrast, Nolan and Grasso (2003) and Nolan et al. (2007) have recently demonstrated that in some cases the baroclinic vortex response to purely asymmetric unbalanced thermal anomalies, or the heating itself, is weakening. In their dry numerical simulations, the impact of the asymmetric mode on intensification was generally subdominant to the axisymmetric mode (i.e., the axisymmetric response to the axisymmetric component of the heating). Another example of the asymmetric mode involves the formation of mesovortices through dynamic instability of the eyewall. It is thought that intensification can be impacted through attendant exchanges of momentum and vorticity between the eye and eyewall (Schubert et al. 1999; Kossin and Schubert 2001; Montgomery et al. 2002; Rozoff et al. 2009) as well as the injection of high-entropy air from the eye into the eyewall (Persing and Montgomery 2003; Cram et al. 2007). Because the axisymmetric and asymmetric modes are necessarily coupled to one another in nature, accurate intensity prediction requires consideration of both (Shapiro 2000; Möller and Shapiro 2005).
The basic internal mechanisms contributing to RI are not expected to be fundamentally different from those responsible for ordinary hurricane intensification. One of the central questions raised in recent studies of intensity change can therefore be restated as: What is the role of the asymmetric mode during periods of extraordinary intensification, and how does its contribution relative to the axisymmetric mode change under different environmental conditions? The problem that motivates the present study is the vertically sheared, yet rapidly intensifying hurricane. When a mature hurricane encounters increasing vertical shear flow, it develops an asymmetric, downshear-oriented convective pattern in response to the shear forcing (e.g., Black et al. 2002; Corbosiero and Molinari 2003). In strongly sheared environments, Frank and Ritchie (2001) proposed that vertical shear negatively impacts hurricane intensification through ventilation of the storm’s upper-level warm core following the development of convective asymmetry and the erosion of the upper-level PV structure. In weak-to-moderate vertical shear RI may still occur if internal mechanisms are able to effectively counter such negative impacts. Reasor et al. (2004) presented a tropical cyclone (TC) resiliency mechanism intrinsic to the dry dynamics by which the tilt asymmetry generated by vertical shearing is effectively damped, thereby maintaining the upright vortex structure. In moist numerical simulations of vertically sheared hurricanes, eyewall mesovortices have been shown to trigger episodes of particularly deep, long-lived convection as they rotate through the left-of-shear semicircle of the storm (Braun 2002; Braun et al. 2006; Braun and Wu 2007). Although these mesoscale regions of convection are forced asymmetrically, their ultimate impact on storm intensification in terms of the asymmetric versus axisymmetric mode remains an unresolved problem.
The goal of this multipart study is to elucidate the relative contributions and essential physical processes associated with the axisymmetric and asymmetric modes in an observed vertically sheared and rapidly intensifying hurricane. We employ a high spatial (∼2 km) and temporal (∼34 min) resolution dual-Doppler dataset collected by two WP-3D research aircraft over an ∼6-h period within eastern Pacific Hurricane Guillermo (1997) during a vortex motion and evolution experiment conducted by the National Oceanic and Atmospheric Administration’s (NOAA) Hurricane Research Division (HRD). In this first part, the basic structure and evolution of Guillermo during RI are reported. A synthesis of aircraft and satellite observations reveals a close relationship between the development of deep asymmetric convection and the storm’s intensification. In section 2 the data and analysis techniques are discussed. Section 3 provides a broad overview of Guillermo’s history, evolution, and inner-core convective pattern. In sections 4 and 5 the axisymmetric and asymmetric components of the wind circulation are documented. Budget contributions to observed changes in angular momentum and relative vorticity are discussed in section 6. Future studies will address the modulation of eyewall convection by asymmetric vorticity structures, mass exchange between the eye and eyewall, and vortex resiliency.
2. Data and methodology
a. Radar observations
Two NOAA WP-3D research aircraft (referred to here as N42RF and N43RF) observed the inner core of Hurricane Guillermo from 1830 UTC 2 August to 0030 UTC 3 August. Both aircraft were equipped with a 5.5-cm wavelength lower fuselage (LF) radar and a 3.2-cm wavelength tail Doppler radar (Jorgensen 1984). N42RF made 10 passes at ∼3-km height across the inner core, while N43RF made 6 passes at ∼6-km height. During the first two passes, the aircraft flew coordinated, near-orthogonal, figure-four patterns to obtain maximum dual-Doppler coverage. The tail radar on N42RF scanned in the track-normal plane, while N43RF employed the fore/aft scanning technique (FAST) in which the antenna scans in a cone 20° from the track-normal plane alternately fore and aft of the aircraft. Details of the two scanning strategies are described by Gamache et al. (1995). During passes 3 and 4, N43RF performed a large-scale survey of the storm, while N42RF remained within the inner core and switched to FAST mode to maintain temporal continuity of the dual-Doppler observations. This pattern of alternating simultaneous aircraft measurements and single aircraft measurements was repeated through pass 10 (see Table 1).
The geometry of both scanning strategies requires a time lag between each radar measurement at a given location within the domain. In the present analysis, the maximum time lag was ∼6 min and occurred along the outer edges of the domain (outside the eyewall) when only FAST observations were available. The addition of simultaneous track-normal measurements increases the dual-Doppler coverage but creates a more complicated spatial distribution of time lag. Because no time weighting of observations was used here, the addition of a third Doppler measurement increases the average time lag at some grid points and decreases it at others. Flow evolution during this short time period can impact the components of the wind vector, although as shown below in section 2d, Doppler-derived and aircraft flight-level tangential wind, and to a lesser extent radial wind, were well correlated.
The duration of the eyewall passes, and hence the compositing period of the inner-core wind field, averaged 18 min. Because the magnitude of the radial wind is typically much less than peak tangential wind above the inflow layer and below the upper-tropospheric outflow layer of a hurricane, it is the azimuthal structure of the wind field that is most likely to be distorted during the compositing period. The average orbital period at the location of maximum axisymmetric tangential wind in Guillermo was approximately 1 h. Thus, features being passively advected would rotate through 108° in the time it took the aircraft to pass from one side of the domain to the other. When the domain is limited to only that portion containing the maximum tangential wind, the rotation angle is reduced to 54°. In their study of Hurricane Olivia (1994), Reasor et al. (2000, hereafter R00) observed that low azimuthal wavenumber wind components in the eyewall sometimes advect around the vortex on time scales greater than the orbital period and are thus better resolved than passively advected features during the compositing period. Nevertheless, some aliasing of the asymmetric structure will occur in the Doppler composite analyses, but its impact on the low-wavenumber structure in the eyewall should be relatively small.
Because of time spent in the downwind leg of the flight pattern, the average time elapsed from the center of one pass to the next was 34 min. Although it was not possible to resolve the structural details of convective cluster evolution on this time scale, aspects of the more slowly evolving mesoscale structure of the vortex were adequately represented. Propagation of specific kinematic features about the eyewall was confirmed through careful comparison of the wind analyses with high temporal resolution LF reflectivity observations.
b. Doppler analysis method
The raw reflectivity (without attenuation correction) and Doppler velocity data were first edited following standard procedures whereby appropriate navigation corrections and spurious echoes (e.g., sea clutter) were removed (e.g., Bosart et al. 2002). Following the procedure outlined in the appendix, the edited fields were then interpolated to a storm-centered 120 km × 120 km Cartesian domain extending from the surface to 20-km height. Little useful data were found below 1 km due to sea clutter contamination or above 12 km due to a lack of hydrometeors. The prevalence of hydrometeors below 3 km within the eye during many passes provides a unique opportunity to document the low-level eye circulation. Although the along-beam resolution was 150 m, the along-track spacing between consecutive scans in either the fore or aft directions was 1.6–1.8 km. The vertical resolution was limited at long range by the 1.9° vertical beamwidth of the tail radar antenna. Thus, uniform horizontal and vertical grid spacing of 2.0 and 1.0 km, respectively, were used in the present analysis.
The Doppler radar analysis methodology of Gamache (1997) was used to simultaneously solve the radar projection equations and anelastic mass continuity equation for the three-dimensional wind field. The solution was subject to a zero vertical velocity constraint at the surface and just above echo top. The reader is referred to the appendix for further details of the Doppler analysis technique. In an application of the technique to idealized wind fields, Gamache (1997) found that the interpolation from radar to Cartesian coordinates generally made a much greater contribution to analysis error than the solution technique. In real-atmosphere situations, time evolution of the wind field, beam filling issues, and inaccuracies in target tracking may also make significant contributions to error.
c. Wind decomposition
The Doppler-derived total wind field was decomposed following the methodology of R00. The storm-relative wind was first obtained by subtracting the time-dependent storm motion vector from the total wind. Storm motion, interpolated from a high-resolution cubic-spline track (Willoughby and Chelmow 1982), was relatively steady with average zonal and meridional components of −4.5 and 0.7 m s−1, respectively, during the observation period. Next, a vortex center-finding algorithm was used to reposition the initial storm-centered analysis. The algorithm specifies a 16-km-wide annulus centered on an initial-guess radius of maximum tangential wind (RMW) at each height. The origin of the annulus within which the average tangential wind is maximized is defined as the center at that height. The new analysis center was defined as a depth average of the 1–10-km height vortex centers. A time average of the centers at each height relative to this depth-averaged center is shown in Fig. 1. During individual passes (not shown) the vortex centers did not depart from the analysis origin by more than 4 km through the depth. This maximum displacement represented 10%–15% of the average RMW. The time-averaged structure evident in Fig. 1 with the low-level center to the north of the upper-level center was a common feature of most passes and did persist in center-finding sensitivity tests. The storm-relative wind was then expressed in a cylindrical coordinate system with the above depth-averaged center as its origin. Unless otherwise indicated, the vortex tilt asymmetry was not removed from the analysis, as done in prior studies (Marks et al. 1992; R00). Finally, the tangential, radial, and vertical wind components were Fourier decomposed in azimuth to facilitate analysis of the low-wavenumber (n = 0–4) structure of the vortex. Hereafter, the n = 0 (n = 1–4) component will be referred to as the axisymmetric (asymmetric) component.
Data gaps in the analysis can impact the representation of the low-wavenumber wind components. The largest gaps occurred along the outer edges of the dual-Doppler domain or within the center of the eye where radar reflectivity was either absent or below our noise threshold of −7 dBZ. Gaps surrounded by wind data were filled through the mass continuity and filter constraints of the Doppler analysis technique [Eqs. (A3) and (A4)], while gaps along the edge of the domain were filled through linear extrapolation. Excluding aliasing issues arising from flow evolution during the compositing period (see section 2a), the low-wavenumber Fourier components are well resolved if contiguous azimuthal gaps span less than a quarter wavelength. Although the accuracy of the low-wavenumber analysis ultimately depends on the azimuthal distribution of wind observations, data at a given radius and height were not plotted if more than 40% of the wind values around the azimuthal circle at that location were filled values. The primary results presented here are not sensitive to the precise value of this threshold.
d. Data quality
The quality of the Doppler analyses was assessed by comparing aircraft flight-level measurements with the Doppler-derived winds at 3- and 6-km height. Initially, a Bartlett filter was applied to the flight-level observations to effectively remove oscillations smaller than 2 km. Following Gamache et al. (1995), the filtered storm-relative, flight-level data were renavigated and interpolated to the Doppler analysis grid. A gridpoint-by-gridpoint comparison between each Doppler-derived and interpolated flight-level wind component (zonal, meridional, vertical, radial, and tangential) was then performed for all passes. Summary statistics computed for each component include mean difference (or bias), RMS difference, and linear correlation coefficient. Statistics were also computed for combinations of three stratifications: flight altitude (3 and 6 km), region (eye and eyewall), and Doppler-derived wind type (observed or filled). The eye (eyewall) region was defined as any location inside (outside) a 20-km radius, which roughly corresponded to the azimuthal-mean radius of 10-dBZ radar reflectivity in the 3–6-km layer along the eye–eyewall boundary. As previously discussed, the variational scheme fills those regions void of Doppler velocity measurements. Over 90% (270 of 296) of the comparable filled grid points occurred within the eye region. Table 2 summarizes these statistics.
A gridpoint-by-gridpoint comparison between the flight-level and Doppler-derived horizontal wind is shown in Figs. 2a–d. Overall, the summary statistics agree with previously reported values in hurricanes (Marks et al. 1992; Gamache et al. 1995; R00). The largest RMS difference for the horizontal components occurred within the eye where the wind was weakest, and the largest bias and weakest correlation were associated with the eye's interpolated data. The weaker correlation for radial wind was primarily a consequence of the smaller range in values (Fig. 2d). The overall RMS difference of 3.23 m s−1 for the radial wind was comparable to that of the tangential wind (3.38 m s−1), but in terms of percentage error it was larger. Also of note was the significant positive tangential wind bias of 2.89 m s−1 in the eye (compared to 0.41 m s−1 in the eyewall). When no Doppler wind measurements are available, the analysis technique effectively linearly interpolates the zonal and meridional wind components across the eye. Thus, the positive bias is explained by the fact that the tangential wind profile within the eyes of strong, intensifying hurricanes (like Guillermo) is typically U shaped (Kossin and Eastin 2001) rather than a linear function of radius.
The vertical wind showed less correlation than the horizontal components and a positive bias at the two heights and within the two regions. Figures 3a,b show histograms of the frequency of flight-level and Doppler-derived vertical velocity values at each height. The magnitude of the mean bias here (0.3–0.4 m s−1) is smaller than the ∼0.5 m s−1 value reported by Marks et al. (1992) in Hurricane Norbert (1984). The Doppler analysis generally underestimated (overestimated) the frequency of downdrafts (updrafts). In contrast, Marks et al. (1992) found that at 3-km height the Doppler analysis overestimated the frequency of downdrafts compared to the flight-level data. Differences in scanning technique, navigation correction method, and synthesis technique could be responsible for this discrepancy. Overall, the vertical motion distributions are consistent with prior analyses (Marks et al. 1992).
3. Storm overview
a. Storm history and environment
Hurricane Guillermo (1997) developed from an easterly wave disturbance that crossed Central America into the eastern Pacific (Lawrence 1999). Figure 4 shows the best-track position and intensity for Guillermo following hurricane classification at 1800 UTC 1 August. A period of RI spanning the aircraft observation period occurred from approximately 0600 UTC 2 August to 1200 UTC 3 August. During this time, Guillermo moved over 29°–30°C SSTs (Reynolds and Smith 1993) to the west-northwest under the influence of a ridge to its north. By the end of the RI period, Guillermo was a major hurricane within approximately 18 m s−1 of its empirical MPI of 87 m s−1 (DeMaria and Kaplan 1999). The probability method of KD03 was successful in forecasting the RI event. The RI probability increased to ∼50% 24 h prior to the observed start of RI, and it remained at that level until 24 h prior to the observed end of RI when it began to decrease (J. Kaplan 2007, personal communication). The leading contributions to the relatively high probabilities were the potential of the storm (i.e., departure from SST-based MPI) and symmetry of the infrared (IR) satellite representation of the cold cloud tops.
The contribution from the 850–200-hPa environmental vertical shear to RI probabilities was weak, suggesting that the shear magnitude was marginal but conducive for RI. Values from the Statistical Hurricane Intensity Prediction Scheme (SHIPS) database (DeMaria et al. 2005) during the observation period were 7–8 m s−1 from the north-northwest. Using the Doppler analysis, a local vertical shear was also estimated. The hodograph shown in Fig. 5 was constructed by area averaging the horizontal wind at each height within a polar domain centered on the local vortex center and extending out to a maximum radius of 60 km. The deep-layer local shear corresponding to the SHIPS estimate was defined using the 1–3-km (∼850 hPa) and 8–10-km (∼200 hPa) averaged winds. Remarkable agreement between the two shear estimates was found, indicating that the local shear strongly reflected the environmental flow. Additionally, the northwest to southeast tilt of the vortex indicated in Fig. 1 was consistent with this north-northwesterly shear. Vertical shear of 7–8 m s−1 over the vortex depth would not ordinarily be considered detrimental to development, and in fact Guillermo underwent RI despite its presence. The mode of intensification, however, may have been influenced by the persistent asymmetric vertical shear forcing of the vortex. In the following discussion, we therefore reference the southern (northern) side of the vortex as the downshear (upshear) side and the eastern (western) semicircle as the left-of-shear (right of shear) semicircle.
b. Domain-integrated evolution
To demonstrate that the storm intensity evolution during the ∼6-h aircraft observation period not only reflected the longer-term satellite-based RI trend but also exhibited shorter-term variability, additional intensity metrics were considered. In Table 3 Doppler-derived maximum wind speed at 1-km height is recorded for each pass. The maximum wind speed exhibited little discernable trend with large pass-to-pass fluctuations. Because local measures of intensity are more prone to such short-term fluctuations, global metrics were thus used to characterize the basic intensity trend. The domain-integrated, storm-relative absolute angular momentum (AAM) and kinetic energy (KE) are given by
where u, υ, and w are the storm-relative radial, tangential, and vertical velocities, respectively, ρ is the density, and f is the Coriolis parameter. To limit the impact of regions absent of observed Doppler wind on the analysis, an annular domain was used to estimate (1) and (2) with z0 = 1 km, H = 12 km, r0 = 20 km, and R = 60 km. The results for each pass are summarized in Table 3.
The integrated AAM and KE increased by approximately 6% and 13%, respectively, over the observation period. Both global intensity metrics showed an increase through 1933 UTC followed by a decrease through 2042 UTC. The majority of the storm’s intensification occurred during the period 2042–2225 UTC. Guillermo then weakened slightly and resumed a more gradual intensification through 2404 UTC. Overall, the magnitude of the intensity decreases were relatively small compared to the much larger intensification, suggesting that some aspects of the long-term and short-term RI processes may be captured by the data.
c. Convective structure and evolution
Figure 6 shows the LF radar reflectivity at 3-km height during each pass. In general, the eyewall exhibited a quasi-persistent azimuthal wavenumber-1 pattern with reflectivity maxima consistently located in the left-of-shear semicircle. Figure 7 shows time-azimuth Hovmöller diagrams of Doppler-derived low-wavenumber vertical velocity averaged over 2–6-km height and from 20–30-km radius within the eyewall (Fig. 7a), and peak reflectivity at 2-km height within the 20–30-km radius annulus (Fig. 7b). When viewed in a time-averaged sense, the azimuthal distribution of vertical motion in the 2–6-km layer also exhibited a distinct azimuthal wavenumber-1 signature with maximum ascent (descent) in the downshear left (upshear right) quadrant located azimuthally upwind (downwind) of the reflectivity maximum. Thus, the gross convective pattern during Guillermo’s RI appears to have been partially constrained by the interaction of the vortex with the environmental vertical shear flow (Fig. 5) in a manner broadly consistent with the Black et al. (2002) conceptual model of shear-induced convective asymmetry (see their Fig. 17). The downshear (upshear) eyewall has been argued to be predisposed to rising (descending) motion, even in the absence of moist processes, through the asymmetric secondary circulation forced by vertical shearing of the vortex (Jones 1995; Wang and Holland 1996; Frank and Ritchie 1999; Reasor et al. 2004).
Figure 6 further indicates that the evolution of Guillermo’s convective pattern during RI was much more complex than a simple model of the shear-induced convective asymmetry might predict. The eyewall reflectivity distribution alternated between open eyewall periods (<1855, 2042–2117, and 2258–2333 UTC) and closed eyewall periods (1933–2001, 2154–2225, and >2404 UTC). Superimposed on this structure were multiple transient convective cells. Examination of animated LF reflectivity revealed that the cells (>30 dBZ, with diameters <10 km) regularly developed in the downshear eyewall and cyclonically rotated around to the upshear side before dissipating (Eastin et al. 2005). Multiple cells were also observed to periodically develop in the upshear right quadrant and rotate around to the downshear left quadrant (examples are labeled with an X). Such episodes coincided with (and helped define) the closed eyewall periods. These initially upshear right convective cells would intensify and coalesce into mesoscale clusters during their rotation through the downshear eyewall. At the same time, a weak echo region and opening of the right-of-shear semicircle of the eyewall would develop. Each cluster often embodied the eyewall reflectivity maximum observed during the open eyewall periods.
Figure 7 supports such episodic convective evolution. It was not possible to resolve the structure and evolution of individual convective cells given the 2-km horizontal resolution and ∼34-min temporal resolution of the Doppler dataset. Rather, the low-wavenumber analysis documents the anomalous mesoscale vertical motions associated with the clusters. To facilitate future discussion, prominent downshear left clusters exhibiting 2–6-km-layer average vertical velocity values >5 m s−1 and 2-km reflectivity values >30 dBZ will be referred to as convective bursts. The convective bursts were associated with the previously identified high LF reflectivity clusters and are labeled A–H at the locations of maximum 2–6-km-layer average vertical velocity. In Fig. 7 the azimuthal offset between the labels and locations of peak radially averaged vertical velocity arises due to the radially tilted structure of the updraft maxima. It should also be noted that the animated LF reflectivity suggests that burst pair B–C embodied a single rotating feature. Coincident with each burst in the downshear left quadrant was a prominent mesoscale downdraft in the upshear semicircle of the eyewall in the 2–6-km layer. The downdrafts were generally located downwind of the regions of maximum reflectivity, suggesting they were at least partially driven by precipitation loading and either coincided with or preceded the development of a right-of-shear weak-echo region in the eyewall.
The convective bursts identified in the 2–6-km layer were documented at various stages of development with some exhibiting greater vertical extent than others. The upper-level low-wavenumber vertical motion and horizontal wind averaged from 9–11-km height and 10-km reflectivity >15 dBZ are shown in Fig. 8 for each pass to highlight these differences. In a time-averaged sense, the upper-level updraft pattern was more axisymmetric than at low levels, but it still exhibited a distinct pass-to-pass evolution closely related to the timing of the convective bursts. In particular, each convective burst was associated with distinct updraft and reflectivity maxima that first appeared within, and were initially confined to, the downshear left quadrant. Convective bursts A and C showed the greatest vertical development with upper-level average vertical velocity exceeding 8 m s−1 and values of reflectivity at 10-km height close to 30 dBZ. As each burst intensified and rotated into the upshear quadrants, the upper-level updraft and reflectivity pattern wrapped further around the eye and temporarily evolved into a more axisymmetric distribution. We define a convective burst cycle as the period beginning with the first appearance of convective bursts in the downshear left quadrant and ending with the cyclonic wrapping of upper-level updrafts into the downshear side of the eyewall. Three cycles of such evolution were evident during the observation period: <1855–1933, 2001–2154, and 2225–2404 UTC. It is perhaps no coincidence that the global intensity metrics decreased with the initiation of the upshear weak-echo regions and subsequently increased as the eyewall closed and the upper-tropospheric updrafts began to rotate through the right-of-shear semicircle of the eyewall.
Figure 9 shows the cloud-top brightness temperature evolution during the observation period using Geostationary Operational Environmental Satellite (GOES) IR imagery. Consistent with the convective burst evolution exhibited at lower levels, each IR-indicated convective burst (yellow shading) was initially observed in the left-of-shear semicircle and was followed by the expansion and wrapping of the cold cirrus anvil (red shading) around the upshear quadrants of the eyewall. The most dramatic deep convective event occurred from 2015 to 2215 UTC in association with bursts B–E and coincided with the middle of the three convective burst cycles. The general agreement between the upper-level updraft locations (Fig. 8) and the coldest cloud tops is striking, and it provides independent confirmation that the Doppler analyses effectively resolved the strong, long-lived, convective bursts. A more detailed documentation of the asymmetric convective evolution within the context of the evolving low-wavenumber flow is currently under way.
4. Axisymmetric structure and evolution
a. Primary circulation
Figure 10 shows the axisymmetric tangential wind (), relative vertical vorticity [ = r−1 ∂(r)/∂r], and AAM per unit mass (M = r + fr 2/2) averaged over the observation period. At 1-km height the tangential wind (Fig. 10a) had a maximum value of 54.6 m s−1 at 30-km radius. The RMW sloped outward with height, expanding to 48 km at 12-km height. The vertical vorticity (Fig. 10b) exhibited a deep annular structure that is often observed at low levels within intensifying hurricanes (Kossin and Eastin 2001; Corbosiero et al. 2005). The annulus sloped outward with height and was at a maximum approximately 12–15 km inside the RMW. The relative vorticity was positive out to at least twice the RMW at all levels, and its radial gradient beyond the maximum was both negative and nonnegligible. The presence of a vorticity skirt outside the vortex core was argued by Reasor et al. (2004) to contribute to the resiliency of the hurricane in vertical shear and has been verified for a large number of tropical cyclone cases by Mallen et al. (2005) using flight-level data. The outward slope of the axisymmetric AAM surfaces above 1 km (Fig. 10c) is consistent with previous observations (Black et al. 1994) and numerical simulations (e.g., Zhang et al. 2001) of intensifying storms. In observational studies of the inner core of weakening hurricanes (e.g., Marks et al. 1992; R00), pronounced inward-sloping AAM surfaces below 2 km were argued to be indicative of a net axisymmetric spindown due to friction.
Figure 11 shows a time-radius Hovmöller diagram of axisymmetric tangential wind and relative vorticity averaged over 1–3-km height. Changes in the tangential wind maximum (Fig. 11a) were qualitatively consistent with the global intensity metrics. A decrease of ∼2 m s−1 between 1855 and 2001 UTC was followed by a 4.5 m s−1 increase over a subsequent 2-h period. The strengthening of the low-level wind began with the initiation of downshear convective burst B, and it continued through convective bursts C–E. After a brief period of weakening, the maximum wind increased by ∼2 m s−1 from 2225 to 2404 UTC following convective bursts F–H.
The net increase in maximum tangential wind coincided with an increase in eyewall relative vorticity (Fig. 11b) as expected from the Stokes theorem. The radial distribution of low-level vorticity remained annular, but its width and peak location fluctuated. In Fig. 12, mean relative vorticity computed from the flight-level data at 3-km height for each pass is compared to the Doppler-derived axisymmetric vorticity at the same level (using a relaxed coverage threshold in the eye). Overall, the two estimates of the mean vorticity structure agreed remarkably well, even within the eye (less than 10–15 km). Aside from the anomalous peak in vorticity near 25-km radius at 1855 UTC, the vorticity profile remained relatively broad through 2001 UTC. A pronounced annular profile then developed between 2001 and 2154 UTC during the period of greatest convective burst activity (see Figs. 6 –9). Then, from 2154 to 2404 UTC the vorticity profile broadened into a distribution resembling that between 1933 and 2001 UTC, but with larger vorticity values inside the RMW. R00 found a similar broadening of the radial vorticity profile in weakening Hurricane Olivia (1994) and suggested that the evolution resulted in part from a vortex-scale instability process (e.g., Schubert et al. 1999). Kossin and Eastin (2001) noted such evolution was common after maximum intensity, marking the end of intensification, but they did not find robust evidence of episodic broadening during intensification. Guillermo’s axisymmetric vorticity evolution suggests the presence of a vortex-scale instability process during an RI period.
b. Secondary circulation
Guillermo’s axisymmetric time-averaged secondary circulation is shown in Fig. 13. The radial component (Fig. 13a) was dominated by an upper-level outflow maximum resulting from ascent approximately along the outward-sloping AAM surfaces. The top of the inflow layer was resolved outside the RMW, but because of sea clutter contamination the strongest inflow below 1 km could not be observed in the Doppler analyses. The tongue of weak axisymmetric inflow near 4-km height has also been noted in observed (Eastin et al. 2005) and numerically simulated (Liu et al. 1999; Braun 2002) intensifying hurricanes. Eastin et al. (2005) attributed such midlevel inflow to an ensemble of locally buoyant convective updrafts in the eyewall. The region of relatively strong axisymmetric outflow (>1.5 m s−1) inside the RMW and below 3 km has previously been observed in Doppler analyses of weakening hurricanes (Marks et al. 1992; R00). Liu et al. (1999) described this flow in simulations of intensifying Hurricane Andrew (1992) as a low-level return outflow resulting from AAM transports in the marine boundary layer. They argued that it might aid intensification through the transport of high equivalent potential temperature air from the eye into the eyewall, supporting or enhancing convection.
The axisymmetric vertical velocity (Fig. 13b) was dominated by the outward-sloping updraft maximum in the eyewall, located approximately 4–6 km outside the vorticity maximum and approximately 5–8 km inside the RMW. A zone of weak (<1 m s−1) outward-sloping axisymmetric descent was evident along the inner eyewall edge, although its vertical coherency cannot be established due to the lack of Doppler observations at midlevels. Such structure is consistent with previous flight-level observations (Jorgensen 1984; Corbosiero et al. 2005) and numerical simulations (e.g., Liu et al. 1999) of intensifying hurricanes. The increase in eyewall vertical velocity with height is also consistent with the preponderance of locally buoyant (i.e., upward accelerating) convective updrafts documented by Eastin et al. (2005) using the flight-level data from the same period.
Figure 14 shows the time evolution of the axisymmetric secondary circulation. The vertical velocity in the 5–7-km layer (Fig. 14a) was generally positive throughout the domain and largely followed the convective burst evolution. Despite the prominent azimuthal wavenumber-1 vertical velocity structure (Fig. 7), the average updraft value in the eyewall exceeded that of the average downdraft, resulting in an enhanced axisymmetric component of upward motion coincident with the enhanced asymmetric convective activity. Three periods of enhanced axisymmetric upward motion (>2.5 m s−1) were observed: ∼1855, 2001–2154, and 2258–2333 UTC. Recall that the greatest increase in axisymmetric tangential wind occurred during or shortly after the latter two periods. The axisymmetric radial velocity in the 1–3-km layer (Fig. 14b) was dominated by outflow associated with the sloping eyewall updraft and the aforementioned return flow. The two prominent outflow maxima preceded the midlevel vertical velocity maxima, and thus may be partially responsible for the enhanced axisymmetric updraft as envisioned by Liu et al. (1999). Outside the RMW, the radial flow exhibited an oscillatory pattern in time with anomalous axisymmetric inflow during the periods of enhanced upward motion.
Finally, it should be noted that the radius of maximum updraft exhibited a modest ∼4-km contraction over the observation period (Fig. 14a). A similar contraction was apparent in the axisymmetric tangential wind (Fig. 11) and radar reflectivity (Fig. 6) fields. Such evolution is broadly consistent with the Shapiro and Willoughby (1982) conceptual model of axisymmetric eyewall contraction and vortex intensification in response to point sources of heat and momentum (i.e., convection). The evolution is also consistent with the Schubert et al. (1999) model of asymmetric contraction in response to vortex-scale instability processes. The evolution of the axisymmetric vorticity profile (Fig. 12) provides indirect evidence for the asymmetric mechanism.
5. Asymmetric structure and evolution
According to global intensity metrics, the greatest intensification occurred from 2042 to 2225 UTC and coincided with a period of convective burst activity in the downshear left quadrant of the eyewall. Other downshear left convective bursts that occurred just before and at the end of the observation period also appeared to correlate well with storm intensification. Although the existence of such a relationship between deep eyewall convection and intensity change is not surprising, the observational documentation of the relationship in mesoscale kinematic fields with relatively high spatial and temporal resolutions has heretofore been elusive.
The asymmetric forcing of Guillermo by 7–8 m s−1 deep-layer vertical shear resulted in periods of significant low-wavenumber asymmetry in the vertical motion field. Figure 15a shows the time-averaged axisymmetric and low-wavenumber contributions to the low-level (1–3 km) vertical motion. The axisymmetric and wavenumber-1 components were dominant within the eyewall (both peaked at 26-km radius) and comparable in magnitude. In the dry numerical simulations of Reasor et al. (2004), vertically sheared hurricane-like vortices were found to develop a steady-state vertical tilt to the left of downshear. The associated dry adiabatic dynamics requires a steady wavenumber-1 vertical motion asymmetry with maximum ascent downshear. Even with the addition of moist ascent, the mechanisms identified in the dry context may still be relevant to the evolution of the hurricane in vertical shear (Reasor et al. 2004; Braun et al. 2006; Schecter and Montgomery 2007). However, an effect not explicitly represented in prior idealized modeling is the seemingly perpetual episodic generation of deep convection in the downshear eyewall, its rotation through the left-of-shear semicircle (with possible enhancement), and dissipation on the upshear side. Such convective evolution likely made a significant contribution to Guillermo’s wavenumber-1vertical velocity amplitude. At the same time, because the magnitude of the downshear-left ascent was generally larger than the upshear descent, a large positive axisymmetric component of vertical motion resulted.
The wavenumber decomposition of low-level vertical vorticity (Fig. 15b) shows a significantly different relationship, with the asymmetric components ∼20% of the axisymmetric peak amplitude in the eyewall. Wavenumber 2 made the greatest contribution to the vorticity asymmetry and exhibited two maxima located ∼8 km inside and outside the axisymmetric peak (or collocated with the maximum radial gradients in the axisymmetric component). Note that when the vortex tilt (Fig. 1) was effectively removed from the analysis following Marks et al. (1992) and R00, the wavenumber-1contribution was reduced with little impact on the amplitude of the wavenumber-2 component (not shown). Such relative wavenumber contributions to the vorticity asymmetry are consistent with R00’s analysis of vertically sheared Hurricane Olivia (1994). A time-radius Hovmöller diagram of the low-level wavenumber-2 amplitude shown in Fig. 16 confirms the double peak in the radial distribution. The pronounced outer peak persisted over a 3-h period from 1855 to 2154 UTC and then reemerged at the end of the observation period. The radially outward shifting and reduction of the wavenumber-2 amplitude between 2154 and 2225 UTC coincided with a broadening of the axisymmetric vorticity profile (Figs. 11b, 12) and outward shifting of the axisymmetric updraft maximum (Fig. 14a).
The structure and evolution of the low-level, low-wavenumber vertical motion and vorticity fields are shown in Fig. 17. The prominence of wavenumber 2 in the vorticity field was evident between 1855 and 2154 UTC, despite some masking by the wavenumber-1 tilt asymmetry. The wavenumber-2 component appeared to complete more than one rotation around the eyewall during this period. The rotation was confirmed during those passes when the eyewall reflectivity took on an elliptical structure (see Fig. 6) and the major axis could be clearly tracked by LF radar. The estimated period of rotation was ∼140 min, which is 2.2–2.4 times the local orbital period at the RMW. Similar propagation characteristics of the wavenumber-2 component in the eyewall have been reported in both observational studies (Kuo et al. 1999; R00) and numerical simulations (Wang 2002b; Braun 2002).
The role of propagating vorticity asymmetry in organizing eyewall convection within vertically sheared hurricanes was recently examined numerically by Braun et al. (2006) and Braun and Wu (2007). In weak-to-moderate vertical shear, they found that convective bursts were initiated in the downtilt eyewall (downshear to downshear left direction) when a strong cyclonic vorticity anomaly, or mesovortex, rotated into the downtilt side and the low-level inflow associated with the vertical shear flow converged with the cyclonic circulation of the mesovortex. It was speculated that the flow of high entropy air from the eye into the eyewall might further support the deep convective bursts. As the mesovortices rotated into the uptilt (or upshear) side of the eyewall, the associated updrafts weakened.
A similar modulation of convection was observed in the low-wavenumber fields of Guillermo’s eyewall between 2001 and 2117 UTC (see Fig. 17). At 2001 UTC, a prominent cyclonic vorticity asymmetry was observed in the downshear left quadrant of the eyewall. Where the asymmetric outflow associated with this feature met the low-level southeasterly environmental flow, enhanced convergence occurred and convective burst B was triggered. The dominant wavenumber-2 component of this vortex-scale asymmetry rotated through ∼90° between 2001 and 2042 UTC. Enhanced outflow from the eye continued to converge with the environmental flow, sustaining the deep convection and ultimately convective burst C. Amplification and growth in size of the vorticity asymmetry was accompanied by the development of a closed cyclonic mesocirculation in the perturbation wind field during this period. When the orientation between the asymmetric cyclonic flow and the environmental flow ceased to favor enhanced convergence, the triggering of updrafts diminished. However, other features continued to enhance convection in the downshear left quadrant. At 2117 UTC, a wavenumber-4 vorticity asymmetry along the inner eyewall edge appeared to promote an exchange between the eye and the eyewall, leading to enhanced convection. The mixing of high entropy air into the eyewall via the low-wavenumber vorticity asymmetries may have served to counter the negative impacts of vertical shear. Such a role for the asymmetric mode has observational support (Eastin et al. 2005) and has been discussed recently by Cram et al. (2007) in their trajectory analyses of numerically simulated hurricanes.
In addition to the more compact eyewall vorticity anomalies, trailing bands of vorticity were observed outside the RMW. The most prominent bands were found at 2001 and 2117 UTC, coincident with bands of enhanced upward motion. Not all trailing updraft bands, however, were associated with strong cyclonic vorticity (e.g., at 2333 and 2402 UTC), but the majority were associated with enhanced reflectivity (see Fig. 6). Corbosiero et al. (2006) demonstrated a relationship between observed trailing-spiral reflectivity bands in the hurricane inner core and sheared vortex-Rossby waves through an analysis of their propagation characteristics. Numerical studies of the hurricane eyewall region have also confirmed the presence of sheared vortex-Rossby waves (Chen and Yau 2001; Wang 2002a). They may arise due to the axisymmetrization of convectively generated vorticity anomalies or vortex-scale asymmetry, like the prominent wavenumber-2 vorticity observed from 1855 to 2154 UTC. Recent work by Reasor et al. (2004) cites vertical shearing of the hurricane as another source of sheared-vortex Rossby waves in the eyewall region.
6. Kinematic budget analyses
a. Absolute angular momentum
Budget analyses of axisymmetric AAM and relative vertical vorticity were performed to elucidate mechanisms contributing to structural changes and the RI of Guillermo during the observation period. Previous observational studies of the hurricane AAM budget have focused on the large-scale vortex (e.g., Palmen and Riehl 1957; Holland 1983). Here, we focus on the vortex inner core above the boundary layer using mesoscale-resolving Doppler observations. The axisymmetric Lagrangian (storm following) AAM budget equation neglecting friction and the departure of the Coriolis parameter from a constant reference value, f0, is given by
where the bars and primes denote axisymmetric and asymmetric (eddy) quantities, respectively, and all velocities are storm relative. The local time tendency, ∂L/∂t, was evaluated in a coordinate system translating with the storm motion. The rhs contributions to local tendency in (3) are the axisymmetric radial and vertical advections and the corresponding eddy advection, respectively. Figure 18 shows the axisymmetric (Figs. 18a,b) and eddy (Figs. 18c,d) contributions to the total AAM budget tendency averaged from 1855 to 2404 UTC. Consistent with numerically based budget studies of the hurricane inner core (e.g., Zhang et al. 2001), the radial and vertical advections of AAM largely canceled one another so that the total tendency above the frictional inflow layer was a small residual between the terms. The radial advection of relatively low AAM in the outward-sloping eyewall was approximately balanced by upward advection of high AAM in the same region, with peak values occurring above 8-km height where both the radial and vertical velocities were largest. A secondary maximum in the negative radial advective tendency was observed at 2-km height just inside the RMW at that level. A similar feature was observed by Zhang et al. (2001) resulting from the strong outflow there and it was largely offset in their simulated storm by a positive vertical advective tendency. Here, the vertical advective tendency at low levels was much weaker than the positive tendency, resulting in a net spindown tendency where the observed change in AAM was clearly positive over the observation period. Such inconsistencies highlight the possibility of noncancelling errors (e.g., due to interpolation of raw radar data, the mass continuity constraint, and fall speed removal) present in each of the budget terms, and thus we focus here on interpretation of the contribution of each individual term rather than their sum.
Quantitatively, the average tendencies observed in Guillermo agreed with those documented in prior numerically simulated intensifying hurricanes. The total (axisymmetric plus eddy) radial advective tendency exhibited a peak value of −14 × 105 m2 s−1 h−1 at 9–10-km height and near 30-km radius. The total vertical advective tendency peaked at a similar location with a value of 17 × 105 m2 s−1 h−1. Zhang et al. (2001), in a simulated storm of slightly larger size (35–40-km RMW at 1–2-km height), greater intensity (peak tangential winds of 65–70 m s−1 at 1–2-km height), and peak axisymmetric updraft of comparable magnitude but at slightly lower altitude than in the present case, documented 1-h average radial (vertical) advective tendencies close to −30 × 105 (+30 × 105) m2 s−1 h−1 at 10–11-km height. Thus, the Doppler analysis seems to yield a reasonable AAM budget for an intensifying storm. However, due to aforementioned noncancelling errors in the observational data, the rhs of (3) could not be used to reliably predict the magnitude of the residual.
In terms of sign, the eddy terms contributed to the total tendency in much the same way as their axisymmetric counterparts. The eddy radial advective tendency exhibited lower- and upper-tropospheric negative maxima, reinforcing the axisymmetric tendency. The eddy vertical advective tendency also reinforced the positive axisymmetric tendency, but it was maximized at low to middle levels of the eyewall just inside the RMW. Although the partitioning between axisymmetric and asymmetric components cannot be represented exactly in an airborne Doppler analysis due to nonstationarity of the flow during the sampling period, the eddy component should be represented to the extent that its impact on the budget tendency can be meaningfully compared to the axisymmetric contribution. Near the locations of peak eddy contribution, the eddy values accounted for up to 30%–40% of the radial and vertical tendencies, supporting a nonnegligible role for the eddies in the evolution of axisymmetric AAM within the eyewall. In their examination of a 120-h average AAM budget during the mature stage of a simulated hurricane, Yang et al. (2007) found that resolved eddies contributed to a positive tendency (approximately 0.7 × 105 m2 s−1 h−1) just inside the eyewall in the lower to middle troposphere. They argued that while small, the eddy advection played an important role in reducing the outward slope of AAM surfaces in their simulated hurricane during this period. Wang (2002b) found instantaneous eddy fluxes associated with spiral vortex Rossby waves in a simulated hurricane to contribute to an increase in AAM radially inside the lower- to middle-tropospheric eyewall. More detailed analyses of local AAM budgets using airborne Doppler analyses are ongoing.
b. Relative vorticity
Mechanisms contributing to the increase in axisymmetric vorticity in the eyewall during the convectively active period from 2001 to 2154 UTC were investigated through a budget analysis of relative vertical vorticity. The axisymmetric Lagrangian relative vorticity budget equation neglecting frictional and solenoidal effects, and the departure of the Coriolis parameter from a constant reference value, f0, is given by
where the terms on the rhs of (4) are the axisymmetric horizontal and vertical advection of vorticity, axisymmetric stretching of absolute vorticity [with δ = r−1 ∂(ru)/∂r + r−1 ∂υ/∂λ], axisymmetric tilting of horizontal vorticity, and the corresponding eddy contributions, respectively. Figure 19 shows the budget terms averaged from 2001 to 2154 UTC. The largest observed low-level increase in axisymmetric vorticity during the period occurred between 20- and 25-km radius in association with the development of the pronounced peaked annular profile (Figs. 11b, 12). In terms of axisymmetric contributions to positive vorticity tendency near this location, vertical advection and tilting of radial vorticity contributed most at the inner edge of the annulus, while radial advection and stretching of vertical vorticity contributed most at the outer edge. In terms of magnitude, the stretching and tilting tendencies were greatest. The largest eddy contribution to positive vorticity tendency at low levels was associated with tilting of vorticity. Because the eddy tilting tendency was collocated with the axisymmetric tilting tendency, and of comparable magnitude, in total the tilting term provided the greatest source of low-level positive vorticity. The magnitude of the average tilting tendency was approximately 3–5 × 10−3 s−1 h−1, which was adequate to explain the 2–3 × 10−3 s−1 increase in axisymmetric vorticity within the eyewall between 2001 and 2154 UTC. Thus, according to the observationally based budget analysis, tilting of symmetric and eddy horizontal vorticities combined with the axisymmetric stretching of vorticity was the most likely source for the development of the peaked annular profile of vorticity.
An investigation of hurricane RI in a vertically sheared environment was carried out using airborne Doppler radar data collected from 1845 to 0012 UTC 2–3 August 1997 in the inner core of Hurricane Guillermo. The dual-aircraft experiment permitted 10 consecutive Doppler analyses of the mesoscale wind structure. In this first part, we examined the analysis quality and then presented an overview of Guillermo’s basic structure and evolution.
The interaction of 7–8 m s−1 deep-layer environmental vertical shear with the vortex organized the eyewall convection into a wavenumber-1 distribution with updrafts dominant in the downshear left semicircle and downdrafts predominant upshear. On shorter time scales, such structure was composed of deep convective bursts that were episodically initiated downshear. With the initiation of the convective bursts, the reflectivity would temporarily evolve into an open eyewall pattern with maximum reflectivity downshear left and a weak-echo band wrapping around the right-of-shear semicircle. Periods of brief weakening coincided with the development of the weak-echo bands. As the bursts rotated around the left-of-shear semicircle, intensification of the storm occurred. Occasionally, convective cells would develop or rotate through the right-of-shear semicircle, leading to a closed appearance to the eyewall reflectivity. These cells were often the seedlings of a subsequent convective burst. The greatest intensification of the storm coincided with a 2-h period of deep convective bursts in the downshear left quadrant of the eyewall. These convective bursts were closely related to the rotation of low-wavenumber vorticity asymmetries around the eyewall. The observed triggering of deep convection in regions where outflow enhanced by the vorticity asymmetries converged with low-level environmental flow supports recent high-resolution numerical simulations of vertically sheared hurricanes (Braun 2002; Braun et al. 2006; Braun and Wu 2007).
Limitations of the dataset, foremost of which is the ∼34 min time separation between analyses, complicate a quantitative assessment of the relative contributions of axisymmetric and asymmetric processes to the observed intensity change. However, by any measure, Guillermo’s inner core was asymmetric, and asymmetric processes were involved in the modulation of eyewall convection. The axisymmetric AAM and vertical relative vorticity budgets estimated from the Doppler-derived wind fields indicated that the eddy contributions to structure change within the eyewall were nonnegligible. In dry numerical simulations, Möller and Montgomery (2000) found that intensification from tropical storm to hurricane strength could be accomplished through the axisymmetrization of PV anomalies representing the balanced response to pulsed convective heating, the so-called asymmetric mode. Möller and Shapiro (2005) found that while dry numerical studies have been instrumental in elucidating the basic mechanisms of the asymmetric mode, in their moist simulations of the axisymmetrization process the axisymmetric secondary circulation does not play a passive role in intensity change. During the periods of pronounced convective asymmetry in Guillermo, the axisymmetric component of vertical motion was also enhanced. To a large extent, this enhancement, and the attendant intensification suggested by the axisymmetric AAM budget analysis, may have been a direct consequence of the axisymmetric projection of the episodic convective bursts and their associated heating. In the context of prescribed rotating heat sources in dry, hurricane-like vortices, Nolan et al. (2007) found that by far the greatest contribution to intensification was from the axisymmetric projection of the heating (i.e., the axisymmetric mode).
Finally, it should be noted that Sitkowski and Barnes (2009) recently completed a detailed synthesis of the GPS dropwindsondes deployed outside Guillermo’s eyewall (i.e., outside our Doppler domain) during the same period studied here. Their analysis of the thermodynamic and kinematic structure of the inflow layer revealed minimal evidence that RI resulted from a significant inward advection of momentum or high entropy air from the environment. Rather, their results support the notion that RI was largely controlled by physical processes occurring in and near the eyewall.
The Guillermo dataset analyzed here has given us a unique first look at the detailed mesoscale structure and evolution of the hurricane inner core during RI. Still, many questions remain unanswered. Presently, this unique dataset is being used to document how interactions between the various low-wavenumber structures modulate the frequency and spatial distribution of the convective bursts, as well as the distribution and impact of asymmetric mass exchange between eye and eyewall. Finally, more fundamental research addressing the structure, distribution, forcing, and evolution of inner-core convection is needed throughout the tropical cyclone life cycle. As we gain a better understanding of the various paths by which convection can influence subsequent convection and storm evolution, we will undoubtedly improve our ability to forecast intensity changes.
The authors are grateful to the scientists and flight crews from the NOAA HRD and Aircraft Operations Center for their efforts in collecting the data used in this study. We thank Frank Marks, John Kaplan, Scott Braun, and Dave Nolan for helpful discussions. The first author also thanks Jessica Fieux for her assistance with the budget analyses. Comments by Michael Bell and two anonymous reviewers were very beneficial and improved this paper. A portion of this work was completed while the second author was a National Research Council postdoctoral associate at the NOAA Hurricane Research Division. This work was supported by NSF Grants ATM-0514199 and ATM-0652264.
Radar Analysis Methodology
The Doppler radar analysis methodology used in this paper is similar to the variational approach used in Bousquet and Chong (1998) and Gao et al. (1999). First, the reflectivity and Doppler observations are interpolated to a regularly spaced Cartesian grid. The values at each grid point are determined through a Gaussian weighting of the nearby data. For the 2-km horizontal and 1-km vertical grid spacing used in the present analysis, the maximum distance permitted between a grid point and observation was 4 km (2 km) in the horizontal (vertical), and the e-folding distance for the Gaussian weights was 2 km (1 km) in the horizontal (vertical). The winds are then matched to the Doppler observations while satisfying several weighted constraints, including the anelastic mass continuity equation. The variational method (numerical details are discussed below) solves the problem in the full oblong domain of grid points, which includes the subset of grid points actually containing Doppler observations. The continuity constraint is applied everywhere in the domain, as are constraints that minimize the second derivatives of the wind quantities. The Doppler-radar observations are used where there are data. The second derivative constraints are given greater weight in those regions where there are no Doppler observations. Boundary conditions are applied to the vertical wind but are given the weight of a single Doppler observation. The boundary conditions assume the vertical wind to be zero at the surface and at one grid point above the echo top, as well as at the top of the oblong. Relatively weak boundary constraints permit an analysis that is not dominated by errors in the integration of the continuity equation from the boundaries, which otherwise can create an analysis dominated by such somewhat random errors. This approach was discussed in greater detail by Gao et al. (1999), who chose to use a background wind field rather than use any boundary conditions.
The method solves a system of equations in a variational manner by minimizing the error in the constraint equations used to produce a cost function, F, which consists of a weighted sum of functionals derived from a least squares application of the constraint equations:
where λi represents the Lagrange multiplier associated with functionals Fi. The first N functionals correspond to the Doppler observations and represent the difference between the analysis and the observations:
where VR represents the magnitude of the radial velocity from observation (m, n), or the mth observation from a particular radar n; α, β, and γ represent the direction cosines for the east–west, north–south, and vertical directions, respectively; u, υ, and w represent the analysis values of the east–west, north–south, and vertical wind components, respectively; VT represents the terminal fall speed of precipitation; i, j, and k represent the integral position in the analysis grid; and δ represents the weight of the radar observation (m, n) at grid point (i, j, k). The Mn is the total number of observations from radar n and N is the total number of radars. The fore and aft sweeps of an airborne Doppler radar are considered as being observed from two independent radars. The next functional represents the anelastic three-dimensional mass divergence, which should be zero:
where ρ is air density, and x, y, and z are the positions in the east–west, north–south, and vertical dimensions, respectively. The next functional is the sum of the various partial second derivatives of the wind field. This functional operates as a filter, and its Lagrange multiplier should be kept small:
Finally, the bottom and top boundary conditions are given respectively, by
where δT is zero everywhere except at grid points that are one grid cell higher than the last good Doppler observation and at the upper boundary of the oblong. The value of k for the vertical wind subscript in (A5) is unity, which represents the surface.
Equations (A2) through (A6) give all the values for functionals Fi. The value of unity is given to Lagrange multipliers λ1 through λN, which are multipliers of the functionals representing Doppler observations, the value of 0.01 is given to λN+2, the multiplier of the second-derivative functional, and the value of unity is given to λN+3 and λN+4, the multipliers of the boundary-condition functionals. The value for λN+1 should be equal to Δx2, where Δx is the horizontal grid spacing. Such a value is dictated by the error in the Doppler observations and the space between those observations; however, when this value of λN+1 is used, there are often regions where the three-dimensional mass divergence is systematically one sign or the other over volumes of several grid points and order of magnitude 10−3 s−1. This large value is the approximate size of the error expected from the observations; however, because this error is not randomly distributed, the errors at a grid point represent the integrated errors from a boundary. By increasing the value of λN+1 to 1000 Δx2, the integrated errors in mass continuity reach an acceptable value. With this weight given to the mass continuity functional, the random error in the analysis, when compared to the Doppler observations, is still only of order 1 m s−1, which is the estimated error of the airborne Doppler radial velocities.
Corresponding author address: Paul Reasor, Dept. of Meteorology, The Florida State University, 404 Love Building, Tallahassee, FL 23206. Email: firstname.lastname@example.org