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  • View in gallery

    Best track of Hurricane Rita from the NHC, with the central pressures indicated at landfall and at time of maximum storm strength (see http://www.nhc.noaa.gov/pdf/TCR-AL182005_Rita.pdf). The shaded box indicates the RAINEX investigation area on 22 Sep 2005.

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    Central pressure and maximum wind speed throughout the lifetime of Rita during September 2005. The vertical line marks the time of the observations used in this study.

  • View in gallery

    Plan view of Hurricane Rita’s concentric eyewalls at 4-km altitude as observed by the ELDORA radar during 1800–1820 UTC 22 Sep 2005. The flight leg began and ended in the southwestern portion of the storm where there is a gap in the data. (a) Radar reflectivity. (b) Tangential velocity relative to the storm center. Positive values are cyclonic. (c) Vertical velocity perturbations, defined as velocity components from wavenumbers 2 and higher. (d) Vertical vorticity perturbations, defined as vorticity components from wavenumbers 2 and higher. The red boxes show the domains analyzed in Fig. 4.

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    (a) Vertical vorticity perturbations at 2.4-km altitude from the domain labeled A in Fig. 3d (note the different altitudes). (b) As in (a), but for the domain labeled B in Fig. 3d. The red lines show cross sections analyzed in Figs. 8 and 9.

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    (a) Azimuthally averaged field of vertical velocity in the concentric eyewalls. Average reflectivity values are overlaid as black contours (dBZ). (b) As in (a), but for radial velocity. Positive velocities point away from the center.

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    GBR values at 500-m altitude calculated for each dropsonde at a particular radius.

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    (a) Azimuthally averaged field of tangential velocity relative to the storm center. Positive values are cyclonic. Average reflectivity values are overlaid as black contours (dBZ). (b) As in (a), but for vertical vorticity.

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    Vertical cross section of vertical vorticity perturbations along the line AB in Fig. 4b. Vertical velocity contours are overlaid at intervals of 1.5 m s−1. The black contours are positive, white is negative, and gray is the zero contour.

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    (a) Vertical cross section of reflectivity along the line CD in Fig. 4b. Vertical velocity contours are overlaid at intervals of 1.5 m s−1. The black contours are positive, white is negative, and gray is the zero contour. (b) As in (a), but for vertical vorticity perturbations.

  • View in gallery

    Plan view of reflectivity at 2.8-km altitude. Each black dot represents the reflectivity maximum within a 9° sector of the annulus containing the secondary eyewall. The axis coordinates are horizontal distance (km), where the origin is the center of the storm.

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    (a) Composite of vertical velocity perturbations from radially aligned cross sections that are 26 km in length. The centers of the individual cross sections (denoted by the origin of the x axis) are located on the black dots in Fig. 10. (b) As in (a), but for radial velocity perturbations. Positive values point away from the center. (c) As in (a), but for vertical vorticity perturbations. (d) As in (a), but for tangential velocity perturbations. Positive values are cyclonic.

  • View in gallery

    Vertical profiles of the momentum tendency terms from Eq. (2), summed over the secondary eyewall annulus (32–58-km radius) and weighted by mass. (a) Terms from the mean kinematic field (u advection, w advection, and angular momentum conservation). (b) Terms from the perturbation kinematic field (u advection, υ advection, w advection, and angular momentum conservation).

  • View in gallery

    Vertical profiles of the vorticity tendency terms from Eq. (3), summed over the secondary eyewall annulus (32–58-km radius) and mass. (a) Terms from the mean kinematic field (u advection, w advection, tilting, and stretching). (b) Terms from the perturbation kinematic field (u advection, υ advection, w advection, tilting, and stretching).

  • View in gallery

    Summed vertical profiles of the mean vorticity tendency terms, the perturbation vorticity tendency terms, and all vorticity tendency terms.

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    Radial profiles of vorticity tendency terms summed over the altitudes of 0.8–5.2 km and weighted by volume and density. Totals are normalized (separately for each plot) and without units. (a) Terms from the mean kinematic field (u advection, w advection, tilting, and stretching). (b) Terms from the perturbation kinematic field (u advection, υ advection, w advection, tilting, and stretching).

  • View in gallery

    (a) Radial profiles of the mean vorticity tendency terms, the perturbation vorticity tendency terms, and all vorticity tendency terms summed over the altitudes of 0.8–5.2 km. Totals are normalized and without units. The radii of maximum vorticity and tangential velocity within the secondary eyewall are denoted by the solid and dashed vertical lines, respectively. (b) As in (a), but for momentum tendency terms.

  • View in gallery

    (a) Plan view schematic of concentric eyewalls within a mature hurricane. Reflectivity contours are drawn showing a portion of the primary and secondary eyewalls, along with their embedded tangential wind jets and . The clear regions of the moat and eye are labeled. The dotted line through the secondary eyewall represents the cross section conceptualized in (b). The plus sign and counterclockwise arrow indicate a region of increasing vertical vorticity, while the minus sign and clockwise arrow indicate a region of decreasing vertical vorticity. (b) Schematic of the mean kinematics along the cross section of a mature secondary eyewall. Reflectivity contours are drawn. The solid arrows represent the mean secondary circulation. The circled region indicates the tangential jet . The plus signs and minus signs indicate regions of increasing vorticity and decreasing vorticity, respectively.

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Kinematics of the Secondary Eyewall Observed in Hurricane Rita (2005)

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Abstract

Airborne Doppler radar data collected from the concentric eyewalls of Hurricane Rita (2005) provide detailed three-dimensional kinematic observations of the secondary eyewall feature. The secondary eyewall radar echo shows a ring of heavy precipitation containing embedded convective cells, which have no consistent orientation or radial location. The axisymmetric mean structure has a tangential wind maximum within the reflectivity maximum at 2-km altitude and an elevated distribution of its strongest winds on the radially outer edge. The corresponding vertical vorticity field contains a low-level maximum on the inside edge, which is part of a tube of increased vorticity that rises through the center of the reflectivity tower and into the midlevels. The secondary circulation consists of boundary layer inflow that radially overshoots the secondary eyewall. A portion of this inflowing air experiences convergence and supergradient forces that cause the air to rise and flow radially outward back into the center of the reflectivity tower. This mean updraft stretches and tilts the vorticity field to increase vorticity on the radially inner side of the tangential wind maximum. Radially outside this region, perturbation motions decrease the vorticity at a comparable rate. Thus, both mean and perturbation motions actively strengthen the wind maximum of the secondary eyewall. These features combine to give the secondary eyewall a structure different from the primary eyewall as it builds to become the new replacement eyewall.

Corresponding author address: Anthony C. Didlake Jr., Department of Atmospheric Sciences, University of Washington, Box 351640, Seattle, WA 98195. E-mail: didlake@washington.edu

Abstract

Airborne Doppler radar data collected from the concentric eyewalls of Hurricane Rita (2005) provide detailed three-dimensional kinematic observations of the secondary eyewall feature. The secondary eyewall radar echo shows a ring of heavy precipitation containing embedded convective cells, which have no consistent orientation or radial location. The axisymmetric mean structure has a tangential wind maximum within the reflectivity maximum at 2-km altitude and an elevated distribution of its strongest winds on the radially outer edge. The corresponding vertical vorticity field contains a low-level maximum on the inside edge, which is part of a tube of increased vorticity that rises through the center of the reflectivity tower and into the midlevels. The secondary circulation consists of boundary layer inflow that radially overshoots the secondary eyewall. A portion of this inflowing air experiences convergence and supergradient forces that cause the air to rise and flow radially outward back into the center of the reflectivity tower. This mean updraft stretches and tilts the vorticity field to increase vorticity on the radially inner side of the tangential wind maximum. Radially outside this region, perturbation motions decrease the vorticity at a comparable rate. Thus, both mean and perturbation motions actively strengthen the wind maximum of the secondary eyewall. These features combine to give the secondary eyewall a structure different from the primary eyewall as it builds to become the new replacement eyewall.

Corresponding author address: Anthony C. Didlake Jr., Department of Atmospheric Sciences, University of Washington, Box 351640, Seattle, WA 98195. E-mail: didlake@washington.edu

1. Introduction

During the lifetime of some intense tropical cyclones, outer rainbands or other convective entities spiral inward and axisymmetrize to form a partial or complete ring of heavy precipitation around the eyewall, and this ring is collocated with a well-defined wind maximum. Observational studies have documented the formation of this feature, known as the secondary (or outer) eyewall, and the characteristic intensity changes of the storm that follow (Willoughby et al. 1982; Willoughby 1990; Black and Willoughby 1992; Franklin et al. 1993). As the secondary eyewall takes shape, it contracts and strengthens with strong subsidence occurring radially inward of the wind maximum (Houze et al. 2007). This subsidence inhibits the secondary circulation of the primary eyewall, which is responsible for the cycling of energy in the storm, and as a result the storm weakens. Over the next 24–48 h, the outer eyewall replaces the inner eyewall and becomes the new primary eyewall, but at a larger radius from the center, and thus contains weaker tangential winds.

Despite understanding the general behavior of secondary eyewalls, there are many aspects of this phenomenon that remain unclear. The primary formation mechanism of secondary eyewalls is not known, although several theories propose that certain factors are responsible, including large-scale environmental influences (Nong and Emanuel 2003; Ortt and Chen 2006), internal wave dynamics (Montgomery and Kallenbach 1997), and the interaction of the vortex with small-scale disturbances (Kuo et al. 2004; Terwey and Montgomery 2008; Judt and Chen 2010). Shapiro and Willoughby (1982) demonstrate how the secondary eyewall may induce subsidence over the inner eyewall and form the clear moat region that separates the eyewalls; however, the exact interaction among these three features is still unclear. The observational data that form the basis of our current understanding are mostly flight-level observations and radar reflectivity data, which are insufficient for a complete kinematic description. More complete observations of the secondary eyewall are needed in order to highlight important processes that govern the overall dynamics of this feature and to ensure that models are accurately simulating eyewall replacement cycles for the correct reasons.

During the Hurricane Rainband and Intensity Change Experiment (RAINEX; Houze et al. 2006, 2007) in 2005, high-resolution aircraft observations of Hurricane Rita’s concentric eyewalls were obtained with the National Center for Atmospheric Research (NCAR) Electra Doppler Radar (ELDORA). While pseudo-dual-Doppler analyses of secondary eyewalls have been done in the past (Franklin et al. 1993; Dodge et al. 1999), RAINEX was the first time that data were collected of concentric eyewalls such that a full dual-Doppler analysis could be performed. As a result, the full three-dimensional kinematic field of the secondary eyewall could be retrieved. Houze et al. (2007) examined a dual-Doppler analysis of this unique dataset, along with dropsonde data, and documented basic features of the concentric eyewall structure. They found that Rita’s secondary eyewall contained an overturning circulation similar to the primary eyewall, with a moat region similar in both thermodynamic and dynamic characteristics to the eye. In this paper, we use the same dual-Doppler analysis as Houze et al. (2007) to examine the kinematic fields of the secondary eyewall in greater detail. The objectives of this study are to characterize the axisymmetric mean and convective-scale perturbation wind fields of the secondary eyewall, and to infer the roles of the kinematic features in the secondary eyewall evolution. Sections 2 and 3 describe the data and methods of analysis. Sections 46 describe the overall structure of the concentric eyewalls. Section 7 investigates composites of the convective-scale features. Section 8 examines tendencies in the tangential momentum and vertical vorticity fields. Finally, section 9 summarizes the key findings from the observations.

2. Data and methodology

On 22 September 2005, the NCAR ELDORA instrument was deployed in RAINEX on board the Naval Research Laboratory (NRL) P3 aircraft to investigate the eyewall replacement of Hurricane Rita. During this mission, the NRL P3 employed curved flight tracks, running parallel to the primary and secondary eyewalls, rather than straight flight-leg segments crossing the eyewalls, to optimally map the eyewall three-dimensional wind field (for details, see Houze et al. 2006, 2007). The data used for this study were observed during 1800–1820 UTC. ELDORA is an X-band dual-Doppler radar noted for its high sampling resolution. It operates with two beams, pointing approximately 16.5° fore and aft, such that the beams intersect at 400-m intervals as the aircraft flies along a track [for more on ELDORA, see Hildebrand et al. (1996)]. For the current study, we assumed that the data at the beam crossings were observed instantaneously, providing two independent observations of the three-dimensional wind vector. The radar data were first corrected for navigation errors (Bosart et al. 2002) and manually edited using the NCAR Solo II software (Oye et al. 1995) to remove noise and radar artifacts. The reflectivity and velocity data were then interpolated to a Cartesian grid with a resolution of 600 m in the horizontal and 400 m in the vertical. The lowest vertical level of data was 800 m, as sea spray contaminates the radar observations below this level. The three-dimensional wind field was obtained using a variational technique that simultaneously solves the radar projection equations and anelastic mass continuity equation, along with vertical velocity boundary conditions. A complete description of this technique and an evaluation of its accuracy are given in Reasor et al. (2009). The storm translation was assumed to be constant and was removed from the wind field. A two-step three-dimensional Leise filter was then applied, yielding a minimum resolvable wavelength of approximately 5 km.

After the wind field was retrieved, the data were initially examined using the NCAR Zebra analysis and visualization software (Corbet et al. 1994), which interactively plots overlays of multiple parameters from horizontal and vertical cross sections of the dataset. We then interpolated the data to a cylindrical coordinate system with a radial resolution of 600 m and an azimuthal resolution of 0.5°. Following Reasor et al. (2000), vortex centers at each height were determined by a simplex algorithm (Neldar and Mead 1965) that maximizes the symmetric component of tangential winds within a 15-km-wide annulus centered on the radius of maximum wind. The final center of the coordinate system was calculated by averaging the vortex centers from the 1.2–10-km levels. The wavenumber-1 structure of the kinematic field was estimated by performing a running average around a semicircle centered upon each radius and altitude. Smaller-scale features of wavenumbers 2 and higher (deemed “perturbations”) were determined by subtracting the azimuthal mean and wavenumber-1 structures from the full kinematic field. Since missing values are common in radar data, an individual data point was not included in the field if 30% of its corresponding averaged points were missing. The primary results presented in this paper are not sensitive to the precise values in the center-finding and averaging techniques.

During RAINEX, dropsondes were continually released from three aircraft platforms, providing measurements of the horizontal wind, pressure, temperature, and relative humidity. Thirty dropsondes provided coverage of the storm’s inner core. The dropsondes were quality controlled with either the NCAR Aspen or National Oceanic and Atmospheric Administration (NOAA) Hurricane Research Division (HRD) Editsonde software. We have projected the dropsonde data collected from 1710 to 1910 UTC onto a cylindrical coordinate system centered on the wind center fixes provided by NOAA/HRD.

3. Storm overview

Figure 1 shows the track of Hurricane Rita, and Fig. 2 shows the storm intensity throughout its lifetime. Rita began as a tropical depression just east of the Bahamas (21.3°N, 69.9°W) on 18 September 2005. The storm strengthened as it tracked across the waters north of Cuba and reached hurricane status upon entering the Gulf of Mexico. Over the next 36 h, Rita intensified rapidly into a category 5 (Saffir 2003) storm, with maximum sustained winds of 80 m s−1 and a central pressure of 897 hPa. On 22 September 2005, Rita weakened slightly as it entered the beginning stages of an eyewall replacement cycle. The shaded area in Fig. 1 shows the investigation area for the NRL P3 aircraft mission on this day, and airborne radar imagery confirmed that a secondary eyewall had formed. At 1800 UTC, Rita had a central pressure of 914 hPa and maximum sustained winds of 65 m s−1. By 0600 UTC on 23 September, the eyewall replacement cycle was complete, and the decreasing maximum wind speed leveled off briefly. However, strong wind shear began to disrupt the storm’s circulation as it tracked to the northwest, and Rita continued to decrease in strength until making landfall near the Louisiana–Texas border on 24 September as a category 3 hurricane.

Fig. 1.
Fig. 1.

Best track of Hurricane Rita from the NHC, with the central pressures indicated at landfall and at time of maximum storm strength (see http://www.nhc.noaa.gov/pdf/TCR-AL182005_Rita.pdf). The shaded box indicates the RAINEX investigation area on 22 Sep 2005.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

Fig. 2.
Fig. 2.

Central pressure and maximum wind speed throughout the lifetime of Rita during September 2005. The vertical line marks the time of the observations used in this study.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

4. Horizontal structure

The composite reflectivity structure for the flight time of this study showed a classic concentric eyewall pattern, with mature primary and secondary eyewalls separated by a well-defined moat. Figure 3a shows the reflectivity pattern at 4-km altitude. The secondary eyewall is a continuous ring of heavy rainfall containing embedded convective cells with reflectivities greater than 40 dBZ scattered throughout. These cells do not exhibit any preferred shape, orientation, or radial location within the secondary eyewall. For example, the strong cell located in the south-southeast extends over 14 km and angles across the width of the ring. In contrast, the cells located in the north vary in shape, have no clear orientation, and occur between the center and outer portions of the ring. The moat is clearest in the northeast, with an especially sharp reflectivity gradient defining the inner boundary of the secondary eyewall. The primary eyewall contains the strongest reflectivity values, which form a connected band over roughly half of its circular ring. In its eastern and southern sectors, a rainband originates in the moat region and spirals into the primary eyewall.

Fig. 3.
Fig. 3.

Plan view of Hurricane Rita’s concentric eyewalls at 4-km altitude as observed by the ELDORA radar during 1800–1820 UTC 22 Sep 2005. The flight leg began and ended in the southwestern portion of the storm where there is a gap in the data. (a) Radar reflectivity. (b) Tangential velocity relative to the storm center. Positive values are cyclonic. (c) Vertical velocity perturbations, defined as velocity components from wavenumbers 2 and higher. (d) Vertical vorticity perturbations, defined as vorticity components from wavenumbers 2 and higher. The red boxes show the domains analyzed in Fig. 4.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

Figure 3b shows the tangential wind field at 4 km. A wedge of data is missing in the southwestern section of the domain where the plane completed its circumnavigation. Since this area is only approximately 5% of the nearly circular domain, it is unlikely that the unobserved features from this region have a significant impact on the calculations and qualitative conclusions presented later in this paper. The wind maxima associated with both primary and secondary eyewalls have a clear wavenumber-1 asymmetry that can be partially explained by the 200–850-hPa environmental wind shear, which was weak in magnitude and from the west-southwest throughout the day of these observations. Many studies have shown that in tropical cyclones affected by wind shear, enhanced convection occurs left of the shear while enhanced upward motion occurs downshear-left within the inner core of the storm (e.g., Frank and Ritchie 2001; Corbosiero and Molinari 2002). The enhanced secondary circulation also causes the local tangential circulation to increase. Within Rita’s primary eyewall, upward motion (not shown), reflectivity (Fig. 3a), and tangential winds are enhanced in the expected locations relative to the shear vector. Within the secondary eyewall, the tangential wind field has a clear wavenumber-1 structure with the same alignment as in the primary eyewall. The reflectivity wavenumber-1 asymmetry is much less pronounced, although it is evident as weaker reflectivities in the southwestern portion of the ring. While the primary eyewall contains a mostly smoothed distribution of tangential winds, the secondary eyewall tangential wind field has a more distinctly cellular structure, with numerous cells ranging from 5 to 15 km in diameter. The radial gradient of angular velocity (not shown) within the primary eyewall was 2–3 times greater than that within the secondary eyewall, indicating a contrast in differential advection in the two regions. As a result, convective-scale disturbances within the primary (secondary) eyewall are more (less) likely to be axisymmetrized, and less (more) likely to be long-lived, which could explain the pattern seen in Fig. 3b. Thus, we suspect that convective-scale disturbances within the secondary eyewall were having a significant impact on the evolution of the tangential wind field.

Figures 3c and 3d show the fields of vertical velocity and vorticity perturbations, respectively, in which the wavenumber-0 and -1 structures were removed. The secondary eyewall contains cores of positive and negative vertical velocity perturbations. In much of the southern half, the perturbations have a regularly alternating up–down pattern that follows the reflectivity ring. The perturbations in the northeastern portion of the secondary eyewall and throughout most of the primary eyewall are not as strictly organized in the azimuthal direction, which may be due to the stronger differential advection occurring in those locations. In the vorticity field, positive and negative perturbations are arranged in a complicated structure that seems to have no apparent organization. However, Fig. 4 provides a closer look at sections in the secondary eyewall where a rough pattern can be discerned. Vorticity couplets are arranged in narrow bands that are oriented at a slightly oblique angle to the annulus of the outer eyewall. These features are reminiscent of the small-scale vorticity couplet found by Marks and Houze (1984) on the eyewall of Hurricane Debby (1982) in the first airborne Doppler study of a tropical cyclone. This pattern is investigated further in section 6.

Fig. 4.
Fig. 4.

(a) Vertical vorticity perturbations at 2.4-km altitude from the domain labeled A in Fig. 3d (note the different altitudes). (b) As in (a), but for the domain labeled B in Fig. 3d. The red lines show cross sections analyzed in Figs. 8 and 9.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

The reflectivity spiral connected to the eyewall (Fig. 3a) appears to be associated with a positive vorticity anomaly (Fig. 3d) extending from the eyewall. This convectively coupled feature may appear to be contributing to the wavenumber-1 asymmetry in the tangential winds, but its scale length in the reflectivity and vorticity fields is closer to that of a wavenumber-2 structure. It is possibly a vortex Rossby wave emanating from the inner eyewall, as its characteristics are similar to those convectively coupled waves in observations and modeling studies (Reasor et al. 2000; Corbosiero et al. 2006).

5. Axisymmetric structure

Figures 5a and 5b display the azimuthal average of vertical velocity and radial velocity, respectively, with contours of the average reflectivity values overlaid. The primary eyewall (between 10- and 30-km radius) has typical eyewall features: a deep reflectivity tower containing values of 20 dBZ at 12-km altitude, and a structure that slopes outward with height. Below 7 km, the main updraft lies just inside the reflectivity maximum, but the two become collocated in the upper levels. Consistent with previous studies, the airflow reaches its greatest vertical velocity at 10–12-km altitude, where the flow then turns into the characteristic radial outflow of a tropical cyclone (Dodge et al. 1999; Black et al. 1996). Embedded in the low levels of the reflectivity tower is downward motion that is undoubtedly induced by the heavy precipitation (Marks and Houze 1987). The moat contains inward flowing and sinking air, with descent being most prominent at 6–12-km altitude.

Fig. 5.
Fig. 5.

(a) Azimuthally averaged field of vertical velocity in the concentric eyewalls. Average reflectivity values are overlaid as black contours (dBZ). (b) As in (a), but for radial velocity. Positive velocities point away from the center.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

The secondary eyewall is a wider band of heavy precipitation between 30- and 60-km radius. Its reflectivity tower tilts slightly more than the primary eyewall, especially in the mid-to-upper levels. Radially outward, stratiform rain occurs as indicated by the bright band near 5-km altitude in the reflectivity signal. Unlike in the primary eyewall, the secondary eyewall updraft is centered within the reflectivity tower in the low to midlevels. The average updraft becomes wider with height, but its maximum remains collocated with the reflectivity maximum. Below 3 km, the mean updraft is collocated with radial outflow, indicating that low-level air feeding the secondary eyewall convection comes from radially inward of this feature. Radar observations are contaminated in the layer below 800 m; however, radial inflow must have been occurring here as part of the vortex secondary circulation. Dropsonde data verified the existence of low-level radial inflow; furthermore, the depth of the inflow layer varied between 500 and 1000 m in the annulus of the secondary eyewall. The resulting airflow must have been radial inflow in the low levels that turned upward and outward on the inner side of the secondary eyewall.

The reversal in the low-level airflow from the moat region toward the secondary eyewall is like the boundary layer flow into the primary eyewall that Smith et al. (2008) recognized in their variation of the Emanuel (1986) idealized model. In their study, boundary layer air spiraling into the eyewall region gathers enough momentum that its tangential velocity component becomes supergradient (i.e., outward-directed Coriolis and centrifugal forces exceed the inward pressure gradient force acting on the wind, causing deceleration of the radial inflow). As the surface airflow slows down, it underruns the primary eyewall, enters the eye, and then reverses direction in a layer atop the inflow layer as it seeks gradient balance. We hypothesize that air flows into the secondary eyewall in a similar fashion. As boundary layer air rapidly spirals into the secondary eyewall region, it overshoots the secondary eyewall and its tangential component becomes supergradient; thus, the radial inflow at this location slows down. Radial convergence due to this deceleration, as well as convergence with subsiding air in the moat, causes a portion of the low-level flow to rise. If the air does not have enough momentum to overcome this region of supergradient flow, it will reverse its direction in a layer atop the inflow layer from which it originated. This rising outflow eventually finds gradient balance at the radius of the secondary eyewall.

We briefly assess this hypothesis by examining dropsonde observations of the boundary layer flow. We define the gradient balance residual (GBR) as
e1
where υ is the tangential velocity, r is the radius, f is the Coriolis parameter, ρ is the density, and p is the pressure. The terms on the right-hand side of Eq. (1) are the centrifugal, Coriolis, and pressure gradient terms, respectively. When GBR > 0, the flow is supergradient, and when GBR < 0, the flow is subgradient. To calculate GBR, the pressure data was first fit to a sixth-degree polynomial under the assumption that an individual pressure measurement represented an axisymmetric value. The analytical derivative of this polynomial and other appropriate dropsonde measurements were then substituted into Eq. (1) at each dropsonde location. Dropsondes with missing values were omitted from this calculation. Figure 6 shows GBR values at 500-m altitude from all dropsondes. Between 35- and 50-km radius, which is within the secondary eyewall, four dropsondes indicate supergradient flow and have the largest GBR values outside of the primary eyewall region (which exhibited the highest degree of supergradient flow). With the exception of the near-surface flow, this radial range contains positive GBR values throughout a 1-km depth. Although the dropsonde analysis presented here is limited in many ways, these calculations are consistent with our suggestion that the boundary layer airflow is supergradient as it feeds into the secondary eyewall.
Fig. 6.
Fig. 6.

GBR values at 500-m altitude calculated for each dropsonde at a particular radius.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

Returning to Fig. 5b, we see that the radial inflow layer extends to 5-km altitude just outside of the secondary eyewall. Since this radial wind slows down with decreasing radius, convergence occurs. This air likely joins the prevailing rising air. Franklin et al. (1993) observed a similarly deep inflow layer, which they also suggested was feeding into the secondary eyewall. Within the reflectivity tower, radial inflow occurs at 4-km altitude as part of a slight inward jog taken by the rapidly rising air. This pattern of low-level radial outflow followed by a radially inward jog is similar to the rising airflow in the primary eyewall simulated by Bryan and Rotunno (2009). In their study, the eyewall flow oscillated between supergradient and subgradient as it sought gradient balance, which is possibly the case here in the secondary eyewall. Above 5 km, the airflow returns to radial outflow and joins outflow from the inner eyewall in the upper levels.

Figures 7a and 7b display the azimuthal average of tangential wind and vertical vorticity, respectively. The tangential wind field of the inner eyewall consists of a maximum embedded in the reflectivity tower at 2-km altitude, and decreasing winds with radius and height. The corresponding vorticity field is a tall tower of large values positioned along the inner edge of the eyewall. These fields of tangential wind and vorticity are consistent with past eyewall observations (e.g., Marks and Houze 1987; Bell and Montgomery 2008). The secondary eyewall, however, takes on an axisymmetric structure that is different from that of the primary eyewall. In the low levels of the secondary eyewall, a wind maximum occurs near the center of the reflectivity tower at 2-km altitude, as in the primary eyewall. However, above this level, the vertical distribution of tangential winds differs considerably. On the inner side, the tangential winds of the secondary eyewall drop off rapidly with height, reaching speeds of 50 m s−1 at 2.4-km altitude. On the outer side, the vertical gradient of tangential wind is much weaker, and wind speeds do not drop to 50 m s−1 until 5-km altitude. The wind minimum that is often identified as a moat feature occurs on the inner side of the secondary eyewall reflectivity tower. This structure of the secondary eyewall wind maximum is unlike that seen in either the primary eyewall or the principal rainband (Hence and Houze 2008; Didlake and Houze 2009). Flight-level observations of Hurricane Gilbert (1988) indicated a secondary wind maximum on the inner edge of the reflectivity tower (Black and Willoughby 1992; Dodge et al. 1999); however, that data came from single radial passes and not total azimuthal averages.

Fig. 7.
Fig. 7.

(a) Azimuthally averaged field of tangential velocity relative to the storm center. Positive values are cyclonic. Average reflectivity values are overlaid as black contours (dBZ). (b) As in (a), but for vertical vorticity.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

The vorticity field (Fig. 7b) shows a horizontal layer of elevated vorticity extending from the moat to the center of the secondary eyewall, reaching its maximum on the boundary between the two. The elevated vorticity then bends upward through the center of the reflectivity tower and points slightly inward at 4-km altitude. This vorticity structure appears to follow the dominant secondary circulation flow, where air rising slightly in the low levels travels radially outward into the reflectivity tower, then rises rapidly with some slight inward motion. Also, like the dominant updraft, the vorticity maximum of the outer eyewall is in a different location within the reflectivity tower than that of the inner eyewall.

6. Vorticity perturbation couplets

We now examine the vertical structure of the vorticity perturbation features seen in Fig. 4. Figure 8 shows a near-tangential cross section of the secondary eyewall. Alternating vorticity perturbations extend from the lowest level to near 9-km altitude and they are tilted upwind. This tilt is consistent with the vertical wind shear of the tangential wind, as low-level features are advected faster than upper-level features. Between 2- and 6-km altitude, vertical velocity perturbations alternate with nearly the same length scale as the vorticity perturbations. A radial cross section of the same feature shows that the vorticity perturbation features are leaning radially outward with height (Fig. 9b). This tilt is also consistent with the vertical wind shear of the radial wind, where air flows inward in the low levels and outward in the upper levels. The positive vorticity perturbation appears to be convectively coupled as it is collocated with a strong reflectivity signal that similarly tilts outward (Fig. 9a). As seen in the tangential cross section, the vertical velocity and vorticity perturbations have nearly the same radial length scale. From these cross sections, we see vorticity perturbation couplets arranged such that they are descending in altitude as they spiral inward within the secondary eyewall. The similar length scales also suggest a close association between the vertical velocity and vorticity perturbations.

Fig. 8.
Fig. 8.

Vertical cross section of vertical vorticity perturbations along the line AB in Fig. 4b. Vertical velocity contours are overlaid at intervals of 1.5 m s−1. The black contours are positive, white is negative, and gray is the zero contour.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

Fig. 9.
Fig. 9.

(a) Vertical cross section of reflectivity along the line CD in Fig. 4b. Vertical velocity contours are overlaid at intervals of 1.5 m s−1. The black contours are positive, white is negative, and gray is the zero contour. (b) As in (a), but for vertical vorticity perturbations.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

7. Maximum reflectivity composites

We have shown from the azimuthal averages that the secondary eyewall has a unique overall kinematic structure. Cross sections of the embedded cells suggest that there may also be some organized kinematic features on the convective scale. Organized convective-scale motions occur in other prominent tropical cyclone features. Black et al. (2002) documented recurring intense updraft cores embedded in the primary eyewall. Braun et al. (2006) and Marks et al. (2008) have associated such updraft cores with eyewall mesovortices interacting with the environmental wind shear. Hence and Houze (2008) and Didlake and Houze (2009) identified overturning convective-scale updrafts and two types of convective-scale downdrafts that occur regularly in the principal rainbands of tropical cyclones. Here we examine the organization of the convective-scale kinematics within the secondary eyewall to obtain a better understanding of how this feature evolves.

We carry out this examination in cross sections averaged such that a composite pattern is centered on a local maximum in reflectivity, thus allowing for the radius of the cell to vary within the precipitation ring. We assume that the greatest effects on the vortex will be from the most intense cells. On this basis, we construct composite patterns for only the most intense reflectivity cells. The exact procedure is to divide the cylindrical domain into a specified number of sectors. Within each sector, the maximum reflectivity at a specified altitude is located within the annulus containing the secondary eyewall (32–58-km radius). This point becomes the fixed center of a radial cross section composite spanning a radial distance of 26 km, the approximate width of the secondary eyewall. For this study, the data are divided into 40 sectors and the reflectivity maxima used to center the composite are found at the 2.8-km level. The composite center locations are shown in Fig. 10.

Fig. 10.
Fig. 10.

Plan view of reflectivity at 2.8-km altitude. Each black dot represents the reflectivity maximum within a 9° sector of the annulus containing the secondary eyewall. The axis coordinates are horizontal distance (km), where the origin is the center of the storm.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

Figure 11a presents the composite of vertical velocity perturbations with reflectivity contours overlaid. A perturbation updraft is centered within the reflectivity tower. Its maximum occurs at 5-km altitude and it extends from the lowest level to approximately 10 km. As with the azimuthal average updraft (Fig. 5a), upward perturbations in the low levels extend to the radially inner side of the reflectivity tower, where overshooting boundary layer air rises as it returns back to the secondary eyewall cell. Perturbation downdrafts occur radially outside of the maximum-reflectivity tower, extending from the lowest levels to nearly 8 km. The upper-level portion of this downdraft column may be caused by pressure perturbations associated with the adjacent buoyant updraft (as in regions adjacent to the principal rainband; see Didlake and Houze 2009) and/or they may be induced by cooling from sublimation of the abundant ice particles that precipitate from both eyewall outflows. Stratiform precipitation, which is indicated in this region by the bright band, may induce downdrafts in the low levels as the air attains negative buoyancy from evaporative and melting cooling (Houze 1997). Similar to the perturbation composite, the azimuthal average contains a bright band and downward motion on the radially outer side of the secondary eyewall reflectivity tower (Fig. 5a). Within the heavy rain of the reflectivity tower, downdraft perturbations occur; these downdrafts are most likely due to precipitation loading.

Fig. 11.
Fig. 11.

(a) Composite of vertical velocity perturbations from radially aligned cross sections that are 26 km in length. The centers of the individual cross sections (denoted by the origin of the x axis) are located on the black dots in Fig. 10. (b) As in (a), but for radial velocity perturbations. Positive values point away from the center. (c) As in (a), but for vertical vorticity perturbations. (d) As in (a), but for tangential velocity perturbations. Positive values are cyclonic.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

The radial velocity perturbations have a distinct pattern associated with local reflectivity maxima (Fig. 11b). Below 5-km altitude, outflow perturbations occur on the inner side of the maximum reflectivity tower while inflow perturbations occur on the outer side. These perturbations meet within the reflectivity tower, indicating converging airflow. An exception exists below 1.2 km where inflow perturbations persist throughout the reflectivity tower. Above 5 km, the pattern is reversed. Inflow perturbations occur on the inner side of the reflectivity tower while outflow perturbations occur on the outer side. Comparing to Fig. 5b, the perturbation composite and the azimuthal average both contain low-to-midlevel radial outflow on the inner side of the reflectivity tower and radial inflow on the outer side. They also both show radial outflow in the mid- to upper levels on the outer side.

The vorticity perturbation composite in Fig. 11c resembles the vorticity perturbation features seen in Fig. 9b. A tilted structure of positive vorticity begins in the low levels inward of the reflectivity tower, then extends outward and up to 8-km altitude. The maximum of this positive vorticity is centered within the reflectivity tower near 3-km altitude. This tilted structure does not exactly follow the azimuthally averaged vorticity maximum (Fig. 7b), which has a sharp bend in the low levels within the reflectivity tower. However, local maxima in positive vorticity perturbations and the averaged vorticity maximum are collocated at 3-km altitude within the reflectivity tower and in the lowest levels radially inside the reflectivity tower. The perturbation composite also contains tilted negative perturbations on both sides of the central positive perturbation, and a final tilted positive perturbation 5–10 km outside of the center. This oscillating, wavelike structure was coherent in numerous variations of the composite specifications (reflectivity altitude and sector count).

One possible explanation for this peculiar pattern is obtained from vortex Rossby wave theory, which is detailed in Montgomery and Kallenbach (1997). As seen in Fig. 7b, the secondary eyewall contains a significant vorticity maximum through a depth of 6 km, with corresponding radial vorticity gradients on either side. Any vorticity disturbance that may occur on either gradient would propagate vortex Rossby waves in the radial direction away from the vorticity maximum. The composite perturbation vorticity pattern could be a footprint of vertically structured, sheared vortex Rossby waves generated when the intense convective cells disturb the wind field of the secondary eyewall. Since the vorticity gradient changes sign in the moat, waves cannot propagate far from the secondary eyewall on the radially inward side. As the vorticity gradient extends farther on the radially outward side, the wave features are found at a greater distance from the vorticity maximum on that side.

The composite tangential wind perturbations (Fig. 11d) comply with the vorticity perturbations. Between 2- and 6-km altitude, a tangential wind minimum occurs on the inner side of the reflectivity tower. A perturbation maximum occurs in the lowest levels within the reflectivity tower and connects to a broader region of positive perturbations radially outside the tower, extending into the upper levels. The pattern of these perturbations closely follows the azimuthally averaged tangential wind (Fig. 7a). Negative tangential wind perturbations are located in the same region as the local minimum of the azimuthal average. Positive perturbations are collocated with the low-level wind maximum and the elevated strong winds on the radially outer side of the reflectivity tower.

The composite perturbation structures from Fig. 11 correspond well with the mean kinematic structures from Figs. 5 and 7. For each parameter, positive (negative) perturbations mostly coincide with higher (lower) values in the azimuthal average. A brief analysis of the full kinematic field showed that azimuthal averages of radii not chosen for the maximum reflectivity composite were qualitatively identical to those in Figs. 5 and 7. From these correlations between the perturbation and mean structures, we suggest that the convective-scale kinematics associated with locally intense cells are axisymmetrizing to build the mean kinematic structure of the secondary eyewall. This upscale process shapes the vortex-scale circulation of the secondary eyewall as it matures and evolves into the primary eyewall.

8. Momentum and vorticity tendencies

The qualitative nature of the mean and convective-scale patterns of the kinematic fields described above suggest structures that may help explain how the secondary eyewall is developing. We quantify this thinking by calculating instantaneous tendencies of the kinematic fields. The tendencies of the azimuthally averaged tangential momentum and vertical vorticity are determined in cylindrical coordinates (r, θ, z) and decomposed into mean and perturbation terms. The Coriolis, friction, and pressure perturbation terms are ignored for simplicity. The tendency equations are given by
e2
e3
where u, υ, and w are the radial, tangential, and vertical velocities, ζ is the vertical vorticity, the overbar represents the azimuthal mean, and the prime represents perturbations from the mean. In Eq. (2), the first three terms represent advection of mean momentum by the mean wind components, and the fourth term conserves angular momentum of the mean tangential wind. The corresponding terms for the perturbation wind components are in brackets. In Eq. (3), the first three terms represent advection of mean vorticity by the mean wind components, the fourth and fifth terms represent tilting of the mean horizontal vorticity, and the sixth term represents stretching of the mean vertical vorticity. The corresponding terms for the perturbation components are again in brackets. The second term in each equation is equal to zero.

To isolate the impact of smaller-scale features within the secondary eyewall, the following analysis does not include the contributions of the wavenumber-1 structure in the perturbation terms in Eqs. (2) and (3). Since the tangential wind and vorticity fields are most active below 6-km altitude, we pay particular attention to how the dynamical field is changing in the low to midlevels. Figures 12a and 12b display the vertical profiles of momentum tendency for the mean and perturbation terms in the secondary eyewall. The most active terms in the mean momentum tendency are the vertical advection and angular momentum terms, where the vertical advection term is mostly positive in the low to midlevels, and the angular momentum term changes sign around 2 km. The most significant perturbation term is the vertical advection term, which indicates that the vertical velocity perturbations are decreasing momentum above 4 km and increasing momentum below this level.

Fig. 12.
Fig. 12.

Vertical profiles of the momentum tendency terms from Eq. (2), summed over the secondary eyewall annulus (32–58-km radius) and weighted by mass. (a) Terms from the mean kinematic field (u advection, w advection, and angular momentum conservation). (b) Terms from the perturbation kinematic field (u advection, υ advection, w advection, and angular momentum conservation).

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

The vertical profiles of vorticity tendency are shown in Fig. 13. The largest mean term in the low levels is stretching, which is increasing vorticity below 5-km altitude. The vorticity perturbation terms have mixed results, as two terms are increasing vorticity and three terms are decreasing vorticity. The magnitudes of the perturbation terms are slightly greater than those of the mean terms. In a vorticity analysis of a primary eyewall, Reasor et al. (2009) also found comparable magnitudes of the mean and perturbation contributions, consistent with the importance of smaller-scale processes in both the primary and secondary eyewalls. The total of the perturbation terms, shown in Fig. 14, is decreasing vorticity in the low levels, while the total of the mean terms is increasing vorticity. The dominant term in the perturbation total is the vertical advection term, as the sum of the remaining four terms is nearly zero (Fig. 13b). Between 2 and 5 km, the decrease in vorticity by the perturbations outweighs the increase in vorticity by the mean motions.

Fig. 13.
Fig. 13.

Vertical profiles of the vorticity tendency terms from Eq. (3), summed over the secondary eyewall annulus (32–58-km radius) and mass. (a) Terms from the mean kinematic field (u advection, w advection, tilting, and stretching). (b) Terms from the perturbation kinematic field (u advection, υ advection, w advection, tilting, and stretching).

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

Fig. 14.
Fig. 14.

Summed vertical profiles of the mean vorticity tendency terms, the perturbation vorticity tendency terms, and all vorticity tendency terms.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

Figures 15a and 15b show the mean and perturbation radial profiles for the vorticity tendency below 5.2-km altitude. The largest mean terms are tilting and stretching. The tilting term changes sign around 45-km radius, which is roughly the center of the mean updraft (Fig. 5a). This tendency pattern is consistent with updraft-induced tilting of the horizontal vorticity constituted by the decreasing mean winds with height over the analysis depth. Also at 45-km radius, the maximum vertical gradient of updraft speed accounts for the peak in stretching of the mean vorticity.

Fig. 15.
Fig. 15.

Radial profiles of vorticity tendency terms summed over the altitudes of 0.8–5.2 km and weighted by volume and density. Totals are normalized (separately for each plot) and without units. (a) Terms from the mean kinematic field (u advection, w advection, tilting, and stretching). (b) Terms from the perturbation kinematic field (u advection, υ advection, w advection, tilting, and stretching).

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

The radial profile of perturbation vorticity tendency, like its vertical profile (Fig. 13b), shows two terms increasing the vorticity and three terms decreasing the vorticity throughout the secondary eyewall. Vertical advection is the largest in magnitude and it maximizes its vorticity decrease at 48-km radius, which is the radially outer portion of the reflectivity tower.

While the mean kinematic fields can readily explain the distributions of the mean tendency terms (e.g., tilting and stretching), the distributions of the perturbation tendency terms cannot be so readily explained by the perturbation composites. Although coherent patterns emerge in the perturbation composites, vorticity tendency calculations from these composites do not agree with the total perturbation tendency profiles shown in Figs. 13b and 15b. This disagreement is a result of the highly variable distributions of kinematic perturbations in individual cross sections. Individual cross sections of vorticity tendency terms thus are also highly variable; however, the summation of their total impact around storm provides insight into the synergistic role that convective-scale features have in the evolution of the secondary eyewall.

The total vorticity tendencies (Fig. 16a) show that the mean kinematic field predominantly increases vorticity throughout the secondary eyewall while the perturbations decrease vorticity. They combine to increase vorticity most at 44-km radius and decrease vorticity most at 50-km radius. In relation to the tangential wind distribution, the total tendency is increasing counterclockwise rotation (ζ > 0) radially inward and clockwise rotation (ζ < 0) radially outward of the maximum tangential wind. This juxtaposition implies that the mean and perturbation motions are acting together, but playing different roles, to increase the low-level wind maximum of the secondary eyewall. The radial profile of total momentum tendency (Fig. 16b) supports this implication, as the mean and perturbation contributions combine to yield the greatest momentum increase at the radius of the maximum tangential wind. Although the perturbation momentum tendency (unlike the perturbation vorticity tendency) has a smaller magnitude than that of the mean, it still has a nonnegligible contribution to the total.

Fig. 16.
Fig. 16.

(a) Radial profiles of the mean vorticity tendency terms, the perturbation vorticity tendency terms, and all vorticity tendency terms summed over the altitudes of 0.8–5.2 km. Totals are normalized and without units. The radii of maximum vorticity and tangential velocity within the secondary eyewall are denoted by the solid and dashed vertical lines, respectively. (b) As in (a), but for momentum tendency terms.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

9. Conclusions

The ELDORA radar system’s high-resolution reflectivity and velocity data collected from the secondary eyewall of Hurricane Rita (2005) provide a uniquely detailed kinematic analysis of this tropical cyclone feature. We have analyzed both the axisymmetric and convective-scale structures of the mature secondary eyewall, and the main conclusions are summarized in the conceptual model shown in Fig. 17. The plan view in Fig. 17a shows the secondary eyewall as a band of heavy precipitation encircling the primary eyewall with an intervening annulus that is relatively precipitation-free. Intense reflectivity cells embedded in the secondary eyewall have no specific shape or orientation, and they are not located at a specific radius from the storm center. The cross-sectional view in Fig. 17b shows that the secondary eyewall consists of a reflectivity tower that has a slight tilt radially outward in the low to midlevels and an increased tilt in the upper levels. Outside of the main reflectivity tower, a bright band indicates the presence of stratiform precipitation.

Fig. 17.
Fig. 17.

(a) Plan view schematic of concentric eyewalls within a mature hurricane. Reflectivity contours are drawn showing a portion of the primary and secondary eyewalls, along with their embedded tangential wind jets and . The clear regions of the moat and eye are labeled. The dotted line through the secondary eyewall represents the cross section conceptualized in (b). The plus sign and counterclockwise arrow indicate a region of increasing vertical vorticity, while the minus sign and clockwise arrow indicate a region of decreasing vertical vorticity. (b) Schematic of the mean kinematics along the cross section of a mature secondary eyewall. Reflectivity contours are drawn. The solid arrows represent the mean secondary circulation. The circled region indicates the tangential jet . The plus signs and minus signs indicate regions of increasing vorticity and decreasing vorticity, respectively.

Citation: Journal of the Atmospheric Sciences 68, 8; 10.1175/2011JAS3715.1

Tangential wind jets, denoted by and , occur near the centers of both the primary and secondary eyewalls. Note that is divided into many arrows to emphasize the persistence of convective-scale features in the tangential wind structure of the secondary eyewall. Convective-scale features are less persistent in the primary eyewall tangential wind structure because of stronger differential advection, and thus is drawn as a single arrow. Note that is embedded in the secondary eyewall reflectivity tower near 2-km altitude (Fig. 17b). This wind maximum is actively being strengthened by both the mean kinematics on the vortex scale and the perturbation kinematics and the convective scale.

The mean secondary circulation consists of radial inflow in the boundary layer. As the boundary layer air flows past the secondary eyewall, its tangential velocity component becomes supergradient and outward-directed forces slow down the airflow. The radial inflow converges with itself and with air that descends in the moat, denoted by the arrow radially inside the secondary eyewall. The air flowing into the secondary eyewall then rises and reverses into radial outflow, similar to the way boundary layer flow into the primary eyewall behaves in the model of Smith et al. (2008). As the air rises within the secondary eyewall, the mean flow jogs slightly radially inward between 4 and 6 km and then returns to radial outflow throughout the upper levels, joining the radial outflow from the primary eyewall.

Calculations with the momentum and vorticity equations show that the mean kinematic field in the low to midlevels is increasing the mean vorticity, primarily through stretching and tilting by the updraft. This tendency of increasing vorticity , indicated by plus signs in the conceptual model, creates a local enhancement of counterclockwise circulation radially inward of the tangential wind maximum. As a result, the mean kinematic field is strengthening and the entire tangential wind field in the radially outer portion of the reflectivity tower.

Perturbations from the mean kinematic field are highly variable in individual cross sections of the secondary eyewall; however, composites of those perturbations associated with intense convective cells show that they have coherent structures. One notable structure is an alternating pattern of outward-leaning vorticity perturbations, which are possibly attributed to vortex Rossby wave dynamics. All of the perturbation composites seen in Fig. 11 strongly resemble the mean kinematic field, which suggests that locally intense cells significantly contribute to building the mean structure of the secondary eyewall in an upscale process. Thus, the axisymmetrization of convective-scale kinematics allows for vortex-scale dynamics to influence the secondary eyewall evolution. These convective-scale dynamics have a direct role in modifying the wind field. Calculations from section 8 show that perturbation motions create a region of negative vorticity tendency, largely through vertical advection of vorticity, located radially outside of . This negative vorticity tendency is denoted by minus signs in the conceptual model. The resulting enhancement of clockwise circulation radially outside of acts to strengthen this jet at 2 km and the entire midlevel tangential wind field above the jet.

Thus, both the mean and perturbation motions strengthen the wind maximum in the secondary eyewall, but they have different roles in the strengthening process. Radially inside of the tangential jet, the mean motions increase vorticity, while radially outside of the tangential jet, the perturbation motions decrease vorticity. Both vorticity tendencies have similar magnitudes, demonstrating the importance of processes from all scales in the evolution of the secondary eyewall.

Although secondary eyewalls have been observed in tropical cyclones for several decades, it has been a significant challenge for models to simulate them. In full-physics, three-dimensional simulations of Hurricane Rita, Houze et al. (2007) varied the grid spacing in multiple model runs and found that a secondary eyewall formed only in the highest-resolution simulation (1.67 km horizontal grid spacing). Only recently have full-physics models been well resolved enough to consistently produce concentric eyewalls, and as a result only a few published modeling studies focus on them (e.g., Terwey and Montgomery 2008; Qiu et al. 2010; Judt and Chen 2010). Unfortunately, these models still have difficulty producing the level of detail necessary to resolve features of the type identified in this paper. With the future advancement of computer technology, finer-resolution modeling will become available, which will allow these details to be identified; thus, a better understanding of secondary eyewall formation and eyewall replacement cycles should result.

In the future, we plan to analyze the structure of inner core rainbands that occurred prior to the formation of Rita’s secondary eyewall. Further investigation of the thermodynamic structure and boundary layer processes is also needed, as both aspects were limitations in the current study.

Acknowledgments

We gratefully appreciate the help and comments from Stacy Brodzik, Shuyi Chen, Deanna Hence, Bradley Smull, and Jian Yuan. Beth Tully assisted with graphics and editing. Finally, we thank Michael Bell for graciously providing the dataset used in this study. This research was supported by the National Science Foundation under Grants ATM-0432623 and ATM-0743180, and the Department of Defense through the National Defense Science and Engineering Graduate Fellowship Program.

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