A set of 327 dropsondes from the NOAA G-IV aircraft was used to create a composite analysis of the azimuthally averaged absolute angular momentum in the outflow layer of major Hurricane Ivan (2004). Inertial instability existed over a narrow layer in the upper troposphere between the 350- and 450-km radii. Isolines of potential and equivalent potential temperature showed that the conditions for both dry and moist symmetric instability were satisfied in the same region, but over a deeper layer from 9 to 12 km. The radial flow maximized at the outer edge of the unstable region. The symmetrically unstable state existed above a region of outward decrease of temperature between the cirrus overcast of the storm and clear air outside. It is hypothesized that the temperature gradient was created as a result of longwave heating within the cirrus overcast and longwave cooling outside the cloudy region. This produced isentropes that sloped upward with radius in the same region that absolute momentum surfaces were flat or sloping downward, thus creating symmetric instability. Although this instability typically follows rather than precedes intensification, limited numerical evidence suggests that the reestablishment of a symmetrically neutral state might influence the length of the intensification period.
The potential importance of outflow-layer inertial instability in tropical cyclone formation and intensification has been addressed for many years [see review by Yanai (1964)]. Sawyer (1947) argued that negative absolute vorticity within the outflow layer, possibly associated with cumulus convection in a precursor disturbance, would produce divergent outflow in the upper troposphere that might further organize core convection and thus lead to intensification. Alaka (1962) and Black and Anthes (1971) provided evidence that local regions of inertial instability exist within tropical cyclone outflow.
Kleinschmidt (1951) made similar arguments to Sawyer (1947) but examined inertial stability along vertically sloping equivalent potential temperature (θe) surfaces, a measure of (moist) symmetric instability. This instability exists in tropical cyclones if the radial slope of the θe surface exceeds that of the absolute angular momentum (M) surface [e.g., Fig. 18 in Yanai (1964)]. Ooyama (1966) noted that such flow is unstable to displacements in both radial directions, but if the mean flow is outward, then the acceleration would be expected to be outward, within the constraints of mass conservation.
The arguments that inertial and/or symmetric instabilities feed back to produce intensification of tropical cyclones were physically appealing, but in early numerical simulations of tropical cyclones, Ooyama (1969) and Anthes (1972) found that inertial instability followed rather than preceded intensification. As a result, interest in this instability and its potential role in tropical cyclones nearly vanished for many years. Möller and Shapiro (2002) and Bui et al. (2009) found small regions of symmetric instability in the mature tropical cyclone outflow layer, expressed by an ellipticity condition for a second-order balanced vortex equation not being met, but the instabilities were examined only in terms of their being an obstacle to the solution of the equations. Rappin et al. (2011) investigated the role of varying background rotation on tropical cyclone intensification but did not address a possible role for symmetric instability.
In this paper the existence of such instability in major Hurricane Ivan (2004) will be described. The means of generation of inertial and symmetric instabilities and their possible roles in the structure and intensity of the storm will be addressed.
2. Data and methods
The data for this study comprise 327 GPS sondes (Hock and Franklin 1999) released in Hurricane Ivan by the National Oceanic and Atmospheric Administration (NOAA) G-IV aircraft. They covered the period 6–16 September, as the storm moved from near 10°N, 50°W to the central Gulf of Mexico. For all but 12 h of the sonde collection, Ivan was a major hurricane with maximum winds exceeding 110 knots (57 m s−1). The history of Hurricane Ivan is described by Franklin et al. (2006).
The sondes were processed following Molinari et al. (2012, 2014). The average sonde release level was near 13 km. The data were interpolated to 100-m vertical resolution within 100-km radial bins. For convenience, the radial bins will be identified by their midpoint; that is, the 100–200-km bin will be labeled as the 150-km radius. All analyses will be shown in radius–height (r–z) sections. Each plot will display an average value of all sondes within the radius bin at each level. Owing to the extremely high vertical resolution, combined with relatively coarse horizontal resolution, a 1–2–1 smoother was applied five times in the vertical in all r–z cross sections.
The radius of each sonde was determined by the great circle distance from the storm center. The latter was defined using the best-track data from the National Hurricane Center, linearly interpolated to 1-min positions. This high time resolution of the track allowed the sonde radius to be determined at each level; that is, the drift of the sonde was taken into account. In reality, of course, the storm does not move linearly between the 6-h best-track positions. Rather, the 1-min interpolation prevented artificial jumps in sonde positions during their descent.
The distribution of sonde release locations with respect to the center of the storm is given in Fig. 1. The distribution is reasonably uniform in radius and azimuth except in three regions. Only a single sonde was released within the 100-km radius, and that region will be omitted. Few sondes were released north of the storm between the storm center and the 250-km radius, and south of the center outside of the 600-km radius. To investigate the impact of these data-sparse regions, azimuthally averaged absolute angular momentum was calculated by
where υλ is storm-relative tangential velocity. Figure 2 shows an r–z composite of this field with 100-km radial spacing and 100-m vertical spacing, as noted above. The stippling shows regions in which M decreased outward on geopotential height surfaces. The M fields exhibited a smaller outward slope in the lower and middle troposphere than in the upper troposphere, consistent with smaller vertical shear of the tangential wind in this region. The M surfaces bowed outward in the upper troposphere, reflecting the development of anticyclonic flow at larger radii. To the authors’ knowledge, no examples of the observed M distribution on the scale of Fig. 2 are available in the published literature. We have obtained M cross sections that extend beyond the 400-km radius from real-data simulations carried out by Nolan et al. (2013), Chen and Zhang (2013), and Huang et al. (2012). The M distribution in Fig. 2 resembles those provided by these authors within 550 km of the center, including a region of outward decrease of M in the first two of those papers. The focus of this paper will be on the decrease of M with radius between 350 and 450 km in Fig. 2 and related inertial and symmetric instabilities. Figure 1 indicated that the data coverage and uniformity in that radial region were reasonably good. The missing data north of the storm inside r = 300 km is not within this region of interest and thus should not have a major influence on the conclusions of this paper.
Figure 2 also reveals an inertial wall (a strong outward increase in M) from 7- to 11-km heights at r = 600 km. Over the next 100 km, M was nearly constant with radius in the upper troposphere. At these large radii, neither of these characteristics seems consistent with any physical process. The M values calculated using gridded analyses from the European Centre for Medium-Range Weather Forecasts (ECMWF; not shown) indicated minima in the upper troposphere south of the center outside the 600-km radius, a region with a dearth of data (Fig. 1). It is possible that the inertial wall was an artifact that resulted from the discontinuity in data density. As a result, subsequent analyses will show only from the 150–550-km radii. The 550-km radius encompassed most of the region of cyclonic flow within the storm, as well as the full structure of the central overcast (see Figs. 3 and 4). To focus on the outflow layer, only the levels from 7 to 13 km are displayed.
A second potential source of error could arise from an uneven distribution of sondes with time. The mean sonde release time and its standard deviation are given in Table 1 for each radial bin. Although the data were collected over 10 days, the mean times at adjacent radii fell in the middle of the period at all radii of interest, with no more than 18-h time difference. In addition, the standard deviation was virtually identical at all radii, indicating a relatively uniform spread of observations in time. The number of sondes peaked at the radii of greatest interest. Figure 1 and Table 1 suggest that the spatial and temporal distributions of sondes should not significantly bias the results. Subsequent r–z sections are interpreted as azimuthal-mean fields over the life of Hurricane Ivan. No steady-state assumption is required. This issue will be addressed further in section 4.
The mean storm-relative radial velocity field in Hurricane Ivan (Fig. 3a) reveals maximum outflow exceeding 8 m s−1 at the 12-km level and the 550-km radius. Outflow increased with radius from r = 250 km to the edge of the figure over a layer from 10 to 13 km. The variation was even stronger with respect to mass flux (not shown); the outward flux at the 12-km level at 550-km radius was 8 times that at 250-km radius. Outflow increased with height outside the 250-km radius above the 9-km level. Points A and A′ in Fig. 3a will be discussed with subsequent figures. Storm-relative tangential velocity (Fig. 3b) revealed deep cyclonic flow. Anticyclonic flow existed in the upper troposphere outside of the 450-km radius.
The same M cross section as in Fig. 2 is plotted with the mean relative humidity in Hurricane Ivan in Fig. 4. Relative humidity was calculated with respect to ice at temperatures below 0°C and with respect to water otherwise. The edge of the cirrus overcast in the composite was identified by the 80% contour of relative humidity. The central overcast extended outward to beyond the 350-km radius between 10.2- and 11.6-km heights. This relative humidity contour provides only an approximation of the cloud edge. In particular, the known dry bias in these dropsondes at low temperatures [see the appendix in Molinari et al. (2014)] makes the upper limit of the overcast uncertain. Cairo et al. (2008) found the cirrus overcast in a tropical cyclone extended all the way to the tropopause, and it seems likely this would hold in a tropical cyclone as intense as Hurricane Ivan. As a result, the relative humidity below 80% at upper levels in Fig. 4 does not preclude the central overcast having reached those levels.
The almost-constant M field with radius above the 12.5-km level from 250- to 350-km radii in Fig. 4 represented near-neutral inertial stability. The M surfaces sloped downward from 350- to 450-km radii between the 10.7- and 12.2-km levels, indicating inertial instability (e.g., Kepert 2010). The unstable layer was present near the outer edges of the central cloud cover.
The same M contours are shown in Fig. 5a, this time superimposed over equivalent potential temperature θe. Although the M surfaces sloped downward with radius near the edge of the central cloud cover, the θe surfaces retained an upward slope. This distribution of M and θe represents moist symmetric instability (e.g., Emanuel 1983). When θe increases upward and M decreases upward, this condition can be expressed by
indicating that the azimuthally averaged θe surfaces slope upward with radius more than the M surfaces (Yanai 1964). Moist symmetric instability (shading in Fig. 5a) existed at the same radius as inertial instability, but through a deeper layer. The consequences of this deep layer of instability can be seen by tracing flow from point A in Fig. 5a. An axisymmetric tube of air moving outward while conserving θe (i.e., moving along the θe surface passing through A) would possess M larger than its environment and accelerate outward [see similar arguments by Emanuel (1983) and Thorpe et al. (1989)]. It is apparent from the figure that this acceleration would continue beyond the edge of the plot near the 11-km level, because the parcel would not yet have returned to its original M surface. The impact of the unstable region thus extends beyond the radius where the condition in Eq. (2) is met. Consistent with these arguments, the radial velocity (Fig. 3a) increased outward within the unstable region to a maximum at the 550-km radius. Figures 4 and 5 suggest that the magnitude and structure of the outflow were connected to the presence of this instability.
An additional symmetrically unstable area in Fig. 5a was located above the 12-km level in the storm core where the flow was close to inertially neutral (nearly flat M surfaces). The instability arose because the θe surfaces sloped upward in the presence of a radial temperature gradient associated with the warm core. The radial velocity also accelerated outward in this region (Fig. 3a), supporting the hypothesis that symmetric instability was influencing the outflow structure of the storm over a fairly deep layer.
The use of θe in Fig. 5a implies saturation during some part of the process. The mean relative humidity (Fig. 4) decreased with radius, and most likely a mix of saturated and unsaturated air existed outside the 350-km radius. Figure 5b substitutes virtual potential temperature θυ for θe in order to identify dry symmetric instability. The layer of such instability narrowed in the vertical compared to Fig. 5a. For instance, air carried outward on an isentropic surface through point B reveals only weak instability compared with Fig. 5a. Below point B was a nearly symmetrically neutral layer at a location that was strongly unstable in Fig. 5a. Figure 3a showed that radial velocity increased outward only above the 10.5-km height. As a result, the dry symmetric instability field, which indicated only small or no instability below 10 km, might have been most relevant in shaping the outflow distribution at these larger radii, where saturation was more intermittent than in the storm core.
b. Numerical modeling example
The greatest uncertainty in Figs. 4 and 5 is the assumption that the composite structure of 327 sondes over 10 days represents a realizable time- and azimuthally averaged state. Evidence was provided in Fig. 1 and Table 1 that the spatial and temporal distributions of data did not bias the analysis over the 350–550-km radii of primary interest. In this section, a similar symmetrically unstable structure will be shown from the high-resolution numerical modeling study of Nolan et al. (2013). This model contained 1-km inner-grid spacing, double-moment microphysics, and cloud radiative interaction with the radiation parameterization called every 6 min. It contained 60 vertical levels, including 8 levels between 9- and 13-km heights. This model thus provided physical parameterizations that effectively simulated the diabatic physics of the central cloud mass of the tropical cyclone. Additional details are provided by Nolan et al. (2013).
Cross sections of M (black contours) and θe (shaded) from the Nolan et al. (2013) simulation are shown in Fig. 6. The region of inertial instability in the simulation closely overlapped that of symmetric instability, and only the former is shaded in Fig. 6. The plots extend 3 km higher than in Fig. 5. Figure 6a represents a single time during the simulation. Instability existed from 200- to 500-km radii between 12.5- and 14-km elevations. Nolan et al. (2013, their Figs. 4 and 17a) showed that Fig. 6a was valid during, rather than prior to, a time of rapid intensification. This is consistent with the arguments of Anthes (1972) and Ooyama (1969) that instability followed, rather than led, significant intensification. This instability relaxed back to a nearly neutral state after 6–8 h (not shown). Rapid intensification ended in the model at about the same time, suggesting that the creation and removal of symmetric instability might have played a role in the length of the rapid intensification period.
The same fields are displayed in Fig. 6b, but averaged over a 48-h period that included the times of rapid intensification. This time average still contained an inertially and symmetrically unstable state. The instability existed outside the model central dense overcast as defined by the relative humidity (not shown), consistent with the structures shown earlier for Hurricane Ivan. Figure 6 provides numerical modeling support for the observed instability shown in Figs. 4 and 5.
Symmetric instability in Hurricane Ivan (2004) occurred over a wider region than inertial instability. One symmetrically unstable region existed above the 12-km level inside the 450-km radius. This area was characterized by nearly flat M surfaces with radius in the presence of sloping θυ and θe surfaces. This unstable region could be produced by near-zero–potential vorticity air from central convection in the presence of a radial temperature gradient associated with the warm core. One might expect such an unstable region to develop with every substantial burst of convection in the storm. The observed outflow in Hurricane Ivan increased outward in this region, suggesting that such instability was frequently active.
A second unstable region existed near the edge of the central cirrus overcast of the storm at 350–450-km radii. Moist symmetric instability existed in this region over a deep layer from 7.5- to 12-km height. Dry symmetric instability was restricted more to the upper troposphere. Rings of air following isentropic surfaces would be accelerated outward beyond the 550-km radius by the symmetric instability. The mean outflow showed outward acceleration in this region.
It seems counterintuitive that a mean state should be unstable. In Hurricane Ivan, one-half of the 6-h G-IV observation periods (7 of 14) experienced at least 5 hPa (6 h)−1 deepening, and these were distributed throughout the 10 days of observation. At least one eyewall replacement cycle occurred during the 10-day period of data collection (Sitkowski et al. 2011). If unstable states developed during these deepening periods, the subsequent response would produce a return to nearly neutral stability, but no further. The average state would be weakly unstable, consistent with Figs. 4, 5, and 6b. For this reason, the use of composite fields in Figs. 4 and 5 does not indicate an assumption of a steady-state storm. Rather, the mean state shown in these figures represents the sum of unstable and neutral states.
The inertial instability near the edge of the central overcast arose because the M surfaces sloped downward with radius. The radial–vertical circulation of the storm, with upward motion in the core and compensating subsidence outside the central overcast, could have produced such an M distribution. But the θe surfaces should have turned downward by the same process. The fact that they do not is what created the symmetric instability. The reasons for this behavior might resemble those in the Hadley cell circulation (T. Dunkerton 2013, personal communication). Near-zero–potential vorticity upper-tropospheric air emitted from tropical convection produces nearly flat M surfaces on the poleward side. As this air continues poleward in the return branch of the Hadley cell, it eventually subsides and the M surfaces turn downward, creating inertial instability [see discussion by Sato and Dunkerton (2002)]. Wirth and Dunkerton (2006) introduced a soft inertial adjustment (and thus an implied momentum source) to allow parcels to cross M surfaces. This simulated the removal of such inertial instability by small-scale motions. In Hadley cell theory, radiative cooling prevents the isentropes from turning downward as much as the M surfaces, thus creating symmetric instability (Held and Hou 1980). Held and Hou noted that this instability can coexist with a steady Hadley circulation.
Diabatic processes might also help to create the symmetric instability in tropical cyclones. Examination of Fig. 5a shows that radially inside point A, the slope of the θe surfaces with radius was small, whereas outside point A, the slope increased. Figure 4 shows that point A lies near the edge of the cloudy region. The smaller radial potential temperature gradient is consistent with longwave warming that would be expected within the cirrus overcast (Bu et al. 2014; Melhauser and Zhang 2014), which would slow the general decrease of temperature with radius in the warm-core disturbance. Longwave cooling outside the cirrus overcast at the same levels would then locally enhance the radial temperature gradient and turn the isentropes upward with radius. In terms of thermal wind reasoning, the radial temperature gradient created an upward decrease in tangential wind near the edge of the clouds. The weaker tangential flow with radius in that layer would locally reduce M, consistent with the structure shown in Figs. 4 and 5. The net impact of the radial gradient in longwave heating would be to turn the isentropes upward in the same region where the M surfaces are flat or sloping downward, thus creating symmetric instability. It is proposed that diabatic physics might play a significant role in helping to create these dynamical instabilities.
Previous studies by Ooyama (1969) and Anthes (1972), as well as the simulation of Nolan et al. (2013), show that symmetric instability does not appear to initiate rapid intensification in tropical cyclones. The Nolan et al. (2013) results, however, suggested that the process by which symmetric instability is removed might play a key role in the nature and even the length of periods of rapid intensification. As a result, study of both the generation and the removal of symmetric instability in high-resolution numerical models of tropical cyclones is recommended to better understand these processes.
This paper has dealt with composite (and thus essentially azimuthally averaged) fields. The tropical cyclone outflow layer tends to be organized into individual outflow channels (Rappin et al. 2011; Merrill and Velden 1996). These are analogous to maxima in poleward flow of low–potential vorticity air at localized longitudes in the Hadley cell studied by Sato and Dunkerton (2002). Future work will address such asymmetric features in the tropical cyclone.
We thank Drs. Tim Dunkerton of NorthWest Research Associates and David Nolan of the University of Miami for helpful discussions of this work. Dr. Nolan also provided the data for Fig. 6 of this paper. We appreciate the careful reading of the manuscript by three anonymous reviewers. G-IV dropsonde data were obtained from the Hurricane Research Division of NOAA. ERA-Interim data were obtained from the National Center for Atmospheric Research, which is supported by the National Science Foundation (NSF). This paper was supported by NSF Grant AGS1132576 and Office of Naval Research Grant N000141410162.