Quarterly Journal of the Royal Meteorological Society Inner-Core Vacillation Cycles during the Rapid Intensification of Hurricane Katrina 1

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1 Page of Quarterly Journal of the Royal Meteorological Society Inner-Core Vacillation Cycles during the Rapid Intensification of Hurricane Katrina 0 Inner-Core Vacillation Cycles during the Rapid Intensification of Hurricane Katrina Nguyen Chi Mai a, Michael J. Reeder a, Noel E. Davidson b, Roger K. Smith c, Michael T. Montgomery d a School of Mathematical Sciences, Monash University, Melbourne, Australia b Centre for Australian Weather and Climate Research, Bureau of Meteorology, Melbourne, Australia c Meteorological Institute, University of Munich, Munich, Germany d Dept. of Meteorology, Naval Postgraduate School, Monterey, CA & NOAA s Hurricane Research Division Correspondence to: School of Mathematical Sciences, Monash University, Melbourne, Australia. Mai.Nguyen@sci.monash.edu.au A simulation of Hurricane Katrina (00) using the Australian Bureau of Meteorology model TCLAPS shows that the simulated vortex vacillates between almost symmetric and highly asymmetric phases. During the symmetric phase, the eyewall comprises elongated convective bands and the low-level potential vorticity (PV) and pseudo-equivalent potential temperature θ e fields exhibit a ring structure with the maximum at some radius from the vortex centre. During this phase, the mean flow intensifies comparatively rapidly as the maximum acceleration of the mean tangential wind occurs near the radius of maximum mean tangential wind (RMW). In contrast, during the asymmetric phase, the eyewall is more polygonal with vortical hot towers (VHTs) located at the vertices. The low-level PV and θ e fields have monopole structures with the maximum at the centre. The intensification rate is lower than during the symmetric phase because the mean tangential wind accelerates most rapidly well within the RMW. The symmetric-to-asymmetric transition is accompanied by the development of VHTs within the eyewall. The VHTs are shown to be initiated by barotropicconvective instability associated with the PV ring structure of the eyewall where the convective instability is large. During the reverse asymmetric-to-symmetric transition, the VHTs weaken as the local vertical wind shear increases and the convective available potential energy is consumed by convection. The weakened VHTs move outwards similarly to vortex Rossby waves and are stretched by the angular shear of the mean vortex. Simultaneously, the rapid filamentation zone outside of the RMW weakens, becoming more favourable for the development of convection. The next symmetric phase emerges as the convection reorganises into a more symmetric eyewall. It is proposed that the vacillation cycles occur in young tropical cyclones and are distinct from the eyewall replacement cycles which tend to occur in strong and mature tropical cyclones. Copyright c 0 Royal Meteorological Society Key Words: Tropical cyclone; Vortex Rossby Waves; barotropic instability; vortical hot towers; eyewall replacement cycles; rapid filamentation zones. Received... ; Revised October, 0 Citation:... Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

2 Quarterly Journal of the Royal Meteorological Society Page of Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0) 0. Introduction et al. 00, Nguyen et al. 00). Although occurring in the same small area of the inner core of the tropical cyclone, The changes in intensity of tropical cyclones and the associated changes in their inner-core structure have been studied for nearly half a century. A recent review of paradigms for tropical-cyclone intensification is given by Montgomery and Smith (0). The key result from the early work is that tropical cyclones intensify as the symmetric overturning circulation these processes have been mostly studied separately and in different contexts. As a first step to understanding inner-core intensity and structure changes, it is pertinant to explore the relationship between these processes and tropical cyclone intensification in a model. The present study examines how these different inner-core mechanisms change the intensity of tropical draw air from outer radii while partially conserving absolute angular momentum. This symmetric intensification mechanism was successfully simulated first by Ooyama (). The contraction of the symmetric eyewall was explained later by Shapiro and Willoughby () in the theoretical framework devised by Eliassen (). The recently, the symmetric spin-up problem has been revisited by Smith et al. (00), who have shown that the inner core intensifies by the radial convergence within the boundary layer, whereas convergence above the boundary layer strengthens principally the winds at outer radii. Tropical cyclone intensification by internal processes has received considerable attention over the last two decades. Several important processes and phenomena have been discovered, including those in which asymmetries play important roles. The inner core processes most relevant to the intensification of the vortex are: (i) Eyewall Replacement Cycles (ERCs) (Willoughby et al. ) ; (ii) Vortex Rossby Waves (VRWs) (Guin and Schubert, Montgomery and Kallenbach, Möller and Montgomery, 000); (iii) Barotropic instability and eyewall mesovortices (Schubert et al., Kossin and Schubert 00); and d) Vortical Hot Towers (VHTs) (Hendricks et al. 00, Montgomery et al. 00, Tory cyclones. For this purpose, a high-resolution (0.0 degrees horizontal resolution) version of the Australian Bureau of Meteorology operational model for tropical-cyclone prediction (TCLAPS) is used to simulate Hurricane Katrina (00). TCLAPS is a hydrostatic, limited-area Numerical Weather Prediction model, which includes a tropical approach was to solve for the response of a symmetric cyclone bogus scheme and assimilation technique specially balanced vortex to imposed local sources of heat and designed for predicting tropical cyclones. Details of this momentum representing the eyewall. Based on aircraft model and the vortex initialisation procedure therein are data, Willoughby (0) showed that tropical-cyclone described by Davidson and Puri (), Puri et al. (), intensification is accompanied normally by the contraction Davidson and Weber (000). of the maxima of the axisymmetric swirling wind. More An ensemble of simulations of Hurricane Katrina has been performed using different initial and boundary conditions, different configurations of the model, and different initial vortex structures, the details of which can be found in Nguyen (0). The simulated track and intensity of most of the members show a good agreement with the observations from Hurricane Katrina. Moreover, during rapid intensification, most of the ensemble members exhibit vacillations between almost symmetric and highly asymmetric states in association with marked changes in the rates of intensification. The vortex structures during these two phases are similar to the two regimes found in the flight level data collected over a 0 year period (-) for the Atlantic hurricanes and reported by Kossin and Eastin (00). In the first regime, which often accompanies rapid intensification, the profiles of vorticity and equivalent potential temperature (θ e ) exhibit a ring structure with elevated values near the eyewall and smaller values in the eye. In contrast, in the second regime a Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

3 Page of Quarterly Journal of the Royal Meteorological Society Inner-Core Vacillation Cycles during the Rapid Intensification of Hurricane Katrina 0 monopole structure with the maximum at the centre is observed for both vorticity and θ e. The present work focuses on the evolution of this pattern of the vortex structure. This paper presents a detailed analysis of the vortex structure during the two phases for a particular member of the ensemble. The ensemble member chosen is the one that most closely matches the observations. It uses the boundary conditions from the analyses of the Australian global model GASP, and sea surface temperature analysis of the same week (which was normally unavailable in real-time). The physical mechanisms involved in the transitions between the two phases are identified and related to the inner-core processes described above. This paper is organised as follows. Section describes the evolution of the simulated Hurricane Katrina, paying particular attention to the development of asymmetries in the PV. In Section, the structure of the simulated hurricane during the two phases is analysed and compared with satellite images of Hurricane Katrina. The physical processes that occur during transitions between the two phases are discussed in Section and the conclusions are presented in Section.. The evolution of the hurricane and development of asymmetries The flow asymmetries are characterised here by the amplitudes of different azimuthal wavenumbers at hpa, obtained by Fourier decomposition, and the maximum standard deviation of potential vorticity (PV) from its azimuthal mean, SDPVmax. Figure shows the evolution of the PV asymmetries at hpa, with the SDPVmax shown in panel (a) and the magnitudes of wavenumbers from 0 to at km radius in panel (b). In this simulation, which starts from 00 UTC August 00, there are three phases when the vortex is prominently asymmetric: these phases are denoted by A, A and A and run from - h, - h, and - h, respectively. The maximum amplitude of asymmetries occur at times t A = h, t A = h and t A = h (see panel a). Increases in the amplitude of the asymmetries during these phases are associated with the amplification of wavenumbers to (panel b). These asymmetric phases are separated by two prominent phases of relatively high symmetry (i.e. with a minimum in SDPVmax) between - h and -. We refer to these as the symmetric phases S and S, respectively. The maximum symmetry occurs at times t S = h and t S = h, respectively. During the symmetric phases, the amplitude of the symmetric component, wavenumber 0, is relatively large while those of the asymmetric components, wavenumbers greater than 0, are small (panel b). Thus, the symmetric component is largest during the symmetric phase, but rapidly decreases in amplitude as the asymmetric structures grow. Figure. Evolution of maximum mean tangential wind at hpa (dotted line with crosses), maximum total wind (solid line with circles) and minimum surface pressure (solid line). It has been shown by several authors (Black et al. 00, Rogers et al. 00, Braun et al. 00) that wavenumber asymmetries tend to be associated with the environmental vertical wind shear. In the present simulation the shear is relatively small (less than m s ) and throughout the integration period the amplitude of wavenumber is, indeed, small (Figure b). Consequently, the evolution of asymmetries in the model is unlikely to be associated with the environmental vertical wind shear and seems more likely to be connected to the internal processes in the inner core. The variation of the asymmetries is highly correlated with the rate-of-change of the maximum azimuthal mean tangential wind v max / t. (Note especially the anti Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

4 Quarterly Journal of the Royal Meteorological Society Nguyen Chi Mai, Michael J. Reeder, Noel E. Davidson, Roger K. Smith, Michael T. Montgomery Page of 0 Figure. Evolution of PV asymmetries at hpa: (a) SDPVmax (dotted line with circles, [PV Unit = m K kg ]), maximum mean tangential wind v max (solid line, [m s ]), and its tendency v max/ t (dashed line with triangles, [m s hour ]). (b) Amplitudes of azimuthal wavenumbers from 0 to of PV [PVU] at the km radius. The horizontal axis shows hours elapsed from 00 UTC August 00. correlation between the dashed line with circles and the solid line with triangles in Figure a.) The correlation coefficient between SDPVmax and v max / t is relatively high (-0.) during the period from h (t A ) to h (t A ), and even higher over shorter time periods (e.g., - 0. from h to h). This correlation is consistent with the observational study of Willoughby (0), which showed that the more symmetric tropical cyclones tend to have higher intensification rates than those that are more asymmetric. Figure shows the evolution of the maximum azimuthal mean tangential wind v max at hpa, the maximum total wind speed at any level, and the minimum surface pressure. The minimum surface pressure falls most rapidly during the asymmetric phases, the reasons of which will be discussed in Section.. In contrast, both maximum azimuthal mean and total winds increase rapidly during the symmetric phases but stall during the asymmetric The maximum occurs typically near 00 hpa. phases. The maximum total wind speed peaks during the asymmetric phases before decreasing ahead of the next symmetric phase. The occurence of strong winds during the asymmetric phase is associated with the VHTs, which develop within the eyewall (shown in the next section), and is a result of the increased vertical vorticity produced by the vorticity stretching in the updraught of the VHTs. Thus, the simulated tropical cyclone vacillates between states of relatively high and low symmetry while intensifying. These vacillation cycles are the focus of the remainder of this study.. The evolution of the vortex.. Symmetric and asymmetric phases The structure of the vortex at hpa at times t S and t A is examined now (Figure ). The vertical motion fields at these two times (panels (a) and (b)) show distinctly different features. Although there are still noticeable asymmetries, Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

5 Page of Quarterly Journal of the Royal Meteorological Society Inner-Core Vacillation Cycles during the Rapid Intensification of Hurricane Katrina 0 Symmetric phase S at h (a) (c) (e) Asymmetric phase A at h (b) (d) (f) Vertical motion ω Potential vorticity θ e Figure. Vortex structure at hpa during symmetric phase S (top panels) and asymmetric phase A (bottom panels). Left panels (a,b): vertical motion [Pa s ] (updraughts are shaded) and horizontal wind magnitude (contours). Middle panels (c,d): PV [PVU](shaded). Right panels (e,f): equivalent potential temperature θ e [K] (shaded). The contours in panels c-f are vertical motion. the eyewall at t S has a quasi ring-like structure comprising elongated bands of moderately strong updraft. In contrast, at t A, the eyewall has a triangular shape with three intense updrafts at the vertices. These updrafts have enhanced rotation and are the model representation of VHTs. The structure of PV at times t S and t A are shown in Figures c and d, respectively, and the corresponding structures of θ e are shown in Figures e and f. At time t S, both fields have a ring-like structure with maxima at radii of about km, while at t A they have a monopole structure with maxima located near the centre. The foregoing structures are representative of the phases S and A, respectively, and are similar also during the other phases of strong symmetry (S) and asymmetry (A). For this reason, we use the times t S The eyewall is defined arbitrarily as the regions having an the minimum upward velocity of - Pa s, which is equivalent to cm s at hpa. and t A subsequently to illustrate the characteristics of the symmetric and asymmetric phases, respectively. During the symmetric and asymmetric phases, the vortex structure in the simulation is similar to the observed structure during Regimes and reported by Kossin and Eastin (00), respectively, which are described in Section. There are other aspects of the simulation that are similar to Kossin and Eastin s regimes also. First, the rate of intensification is higher during their Regime than during Regime. Second, the transition from regime to occurs over a short time interval (typically to h). The time scale for the reverse transition is not reported by Kossin and Eastin (00), whereas it is about to h in the simulation. The simulation allows one to investigate a range of fields that are not readily available from observations such as local wind speed, the surface heat fluxes and the rate-ofstrain. These fields are of interest because of their relevance to understanding the role of VHTs during the asymmetric phase and are shown in Figure at times t S and t A. Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

6 Quarterly Journal of the Royal Meteorological Society Nguyen Chi Mai, Michael J. Reeder, Noel E. Davidson, Roger K. Smith, Michael T. Montgomery Page of 0 Symmetric phase S at h (a) (c) (e) Asymmetric phase A at h (b) (d) (f) Local wind maxima Surface laten heat flux Rate-of-strain Figure. As Figure but for other fields. (a, b): The Laplacian of the horizontal wind magnitude V mag [ (ms) ] (shaded), gray (black in the color version) contours show horizontal wind speed. (c, d) Surface latent heat flux (shaded, [ W m ]. Dark (blue) contours in panels a-d show vertical motion. Right panels (e, f): Horizontal rate-of-strain S (shaded, [ s ]). Contours show relative vorticity [ s ] The regions of locally-high wind speed at hpa in panels (a) and (b) are highlighted by the negative Laplacian of the wind speed ( V mag ). Panels (c) and (d) show the surface latent heat flux and panels (e) and (f) show the horizontal rate-of-strain at hpa. The horizontal rate-of-strain, S, is defined as S = ( U/ x V/ y) + ( V/ x + U/ y), where U and V are wind components in zonal and meridional directions x and y, respectively. During the asymmetric phase, the VHTs are clearly associated with strong local wind speeds (Figure b), high θ e values (Figure f) at hpa, and enhanced surface fluxes (Figure d). The colocation of these dynamical and thermodynamical features may be understood as follows. Bursts of convection increase the surface wind speed in the vicinity as air is drawn into updraught. In turn, the increased wind speeds enhance the surface latent heat fluxes, increasing θ e at low levels. As a result, the buoyancy of the air within the updraughts of the VHTs is enhanced, leading to stronger convection. In contrast, during the symmetric phase (Figures a, e, and c), these variables have comparatively lower magnitudes and more axisymmetric structures, consistent with the ring-like eyewall. There is a region of large rate-of-strain just outside of the eyewall during both the symmetric and asymmetric phases (Figures e and f, respectively). This region is associated in part with the strong angular shear of the azimuthal mean tangential flow and in part with the strong horizontal shear associated with the VHTs during the asymmetric phase. In this area, there is a large negative radial gradient of relative vorticity compared with the weak gradient in the core region, where the relative vorticity is comparatively large. For this reason, the flow changes from one of relatively large vorticity and low rate-of-strain in the core to one of low vorticity and large rate-of-strain outside the core. Rozoff et al. (00) proposed that the rapid filamentation zones, where the square of rate-of-strain (S ) is larger than the enstrophy (ζ ), are unfavourable Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

7 Page of Quarterly Journal of the Royal Meteorological Society Inner-Core Vacillation Cycles during the Rapid Intensification of Hurricane Katrina 0 for convection if the filamentation time scale (τ fil = (S described in the previous sections. Specifically, the structure ζ ) / ) is less than convective time scale. Thus, during the at UTC on August (panel a) and 0 UTC on asymmetric phase, (Figure f), the large rate-of-strain and August (panel f) resemble the symmetric phase in the hence strong filamentation that occurs just outside of the model simulation, whereas UTC and 00 UTC on VHTs is unfavourable for the development of convection. August (panels b and c, respectively) correspond to the Another significant difference between the symmetric asymmetric phase. To complete the sequence, Figures d and the asymmetric phases is the area of large rate-ofstrain inside the eyewall during the asymmetric phase to the symmetric phase. From this perspective, hurricane and e resemble the transition period from the asymmetric (Figure f). This region has comparatively high values Katrina goes through one vacillation cycle (symmetric to of vorticity (see contours), which are associated with the VHTs. Consequently, the vorticity is rapidly stirred in the region between the eye and the eyewall during asymmetric phase. was to h, it is still the same order of magnitude. Observational evidence for the two phases in Katrina.. Evolution of the mean fields On August, the National Hurricane Center reported an ERC in Hurricane Katrina (Knabb et al. 00). However, we show here evidence that the evolution of the structure of Hurricane Katrina resembles a vacillation cycle during this period. Figure shows a sequence of IR satellite images from UTC August to 0 UTC August 00. These images are at irregular time intervals that depend on the availability of data from polar-orbitting satellites. During this period, Hurricane Katrina changes from having a relatively symmetric ring structure (panel a) to a highly asymmetric structure with a seemingly broken eyewall, comprising more than one strong overshooting embedded convective region (marked by the bright shade in panels b and c). Subsequent figures (d and e) show some weakening of the convection near the core while the vortex becomes more symmetric. The large area of cold cloud tops (bright shading) to the south-east of the hurricane centre in (d) and (e) mark cirrus clouds, which are most likely the remnants from deep convection. At 0 UTC on the August (panel f), Hurricane Katrina became symmetric again without embedded strong deep convection. The observed evolution of Hurricane Katrina is consistent with the symmetric and asymmetric phases asymmetric to symmetric) during a period of h from UTC on August (panel a) to 0 UTC August (panel e). While this is longer than the simulation, which The relationship between the rates of intensification (discussed in Section ) and the azimuthal mean fields are examined now. Tangential and radial winds Figure a shows radius-time plots of the isotachs of azimuthal mean tangential wind at hpa and its tendency. The region of maximum v/ t occurs near the radius of azimuthal-mean tangential wind (RMW) during the symmetric phases moves inward during the transition towards the following asymmetric phase, which is consistent with the variation of intensification rate v max / t described in Section. The rate at which v max intensifies during the asymmetric phase is slow because the largest acceleration occurs at inner radii instead of at the RMW as during the symmetric phase. The acceleration of v at inner radii and deceleration near the RMW during the asymmetric phase can be understood by stirring effects of the mesoscale circulations of the VHTs which are strongest during this phase. Fast moving air near the RMW is mixed with the slower-moving air at inner radii, thereby leading to a large acceleration at inner radii and a smaller acceleration near the RMW. Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

8 Quarterly Journal of the Royal Meteorological Society Nguyen Chi Mai, Michael J. Reeder, Noel E. Davidson, Roger K. Smith, Michael T. Montgomery Page of 0 a) UTC August b) UTC August c) 00 UTC August d) UTC August e) 0 UTC August f) 0 UTC August Figure. Infrared satellite images at km-resolution for hurricane Katrina during and August 00. (a) (b) (c) v/ t at hpa u/ t at 00 hpa ω at hpa Figure. Radius-time plots of (a) the tendency of mean tangential wind v/ t [m s hour ] (shaded) at hpa; b) the tendency of mean radial wind u/ t [m s hour ] (shaded) at 00 hpa; and c) the mean vertical motion ω [Pa s ] at hpa. In a and b, solid contours show the mean tangential wind v [m s ] at the corresponding levels, dashed tilted lines indicate the axes of maximum v/ t at hpa. Bold solid lines in a,b,c show the RMW at the corresponding levels. Two curved solid lines in c mark the axes of maximum ω. Figure a shows also regions of weak acceleration of v in a large area at outer radii during the asymmetric phases (e.g. from 0. to. degree during A). The increase of v at these radii can be explained by the effects of Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

9 Page of Quarterly Journal of the Royal Meteorological Society Inner-Core Vacillation Cycles during the Rapid Intensification of Hurricane Katrina 0 (a) (b) (c) PV/ r θ e CAPE Figure. Radius-time plots for: (a) radial gradient of mean PV P V / r [ PVU m ] at hpa, negative values are shaded; (b) θ e [K] at hpa; (c) Convective Available Potential Energy CAPE [ J kg K ]. Solid contours and dashed tilted lines show the same features as those in Figures a,b. Bold solid lines show the RMW at hpa. Two curved solid lines in c mark the axes of maximum ω. VHTs in producing near-surface, system-scale convergence within the boundary layer. In an azimuthally-averaged sense, this air partially conserves its absolute angular momentum and spins faster. Indeed, Figure b, which shows the acceleration of the mean radial wind u/ t at 00 hpa (which is in the boundary layer ), has corresponding regions of accelerating radial inlow ( u/ t < 0) at outer radii during the asymmmetric phases (see e.g. the shaded region from 0. to. degree radius during A). Likewise, the acceleration of v near the RMW during the symmetric phases coincides with the acceleration of u in the boundary layer. Thus, Figures a and b indicate that there are two processes increasing v at hpa (which is just above the boundary layer). In the first process, the local VHTcircultions stir the air, leading to an acceleration of v at radii well inside the RMW and a deceleration near In this study, the boundary layer is defined as the layer adjacent to the surface where frictional drag reduces the absolute angular momentum and leads to strong radial inflow. The boundary layer is seen in vertical crosssections as the layer where the absolute angular momentum increases with height. With this definition, the 00 hpa level lies in the boundary layer and the hpa level is just above the boundary layer. the RMW, during the asymmetric phase. In the second process, the acceleration of the radial inflow in the boundary layer leads to the strengthening of v near the RMW during the symmetric phase and at outer radii during the asymmetric phase. The second process is consistent with the intensification process identified by Smith et al. (00) for the spin up of the vortex inner core (see Section ). Vertical motion The vacillation of the azimuthal-mean vertical motion ω (shown in Figure c) is similar to that found in ERCs. For example, prior to the symmetric phase S, the mean eyewall moves inwards (solid lines mark this inward movement). During the asymmetric phase (e.g. A), the inner eyewall weakens while the strongest convection develops at some outer radius, similar to the formation of the outer eyewall during ERCs. Subsequently, this region of strong convection contracts, again reminiscent of a typical ERC. The intensity changes during the vacillation cycles are analogous to those of ERCs also. Specifically, the smaller intensification rates of the vacillation cycles during the asymmetric Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

10 Quarterly Journal of the Royal Meteorological Society Page of Nguyen Chi Mai, Michael J. Reeder, Noel E. Davidson, Roger K. Smith, Michael T. Montgomery 0 phase correspond to the slow intensification rates of hurricances during the weakening of the inner eyewall in the ERCs (Willoughby et al. ). On the other hand, the symmetric phase of the vacillation cycles corresponds to the contraction phase of an ERC. Nevertheless, it is important to note that the vacillation cycles described here are essentially different processes to ERCs. Specifically, the break-down of the eyewall during the asymmetric phase is not due to the formation of the outer eyewall as in ERCs. Rather, the eyewall breaks down as asymmetries develop within the eyewall itself (see e.g. Figure a,d), suggesting that these structure change cycles are an alternative means for the inner core to rapidly intensify. Moreover, unlike ERCs, the vacillations are associated with vortex-scale episodic stirring. Potential vorticity The evolution of the radial gradient of the azimuthal mean PV at hpa shown in Figure a illustrates the vacillation between the ring and monopole structure, which characterize the symmetric and asymmetric phases, respectively. The monopole structure has negative radial gradient at all radii from the vortex centre, whereas the ring structure has positive radial gradient at inner radii and negative radial gradient at outer radii. Thus, regions with shaded values from the centre indicate a monopole structure, and the regions with blank near the centre changing to shaded at some radius indicate a ring structure. Monopole structures occur during the periods - h, - h, and - h, whereas the ring structures are found during the intervening periods - h, - h, and after h. These periods of monopole and ring structures (except for the times after h) indeed correspond to the symmetric and asymmetric phases described earlier in Section. A change in the sign of the mean PV gradient of the ring structure is a necessary condition for barotropic instability (Rayleigh 0, Schubert et al. ) suggesting that the periods when the vortex is in the symmetric phase are susceptible to barotropic instability. Thus, the growth of VHTs, which herald the end of the symmetric phase and the transition to the asymmetric phase may be initiated and organised by this barotropic instability. The effects of the latter will be examined in the next section. However, during the mature stage (after h of integration), the simulated vortex reaches a quasi-steady configuration in which the mean PV maintains a ring-like structure without vacillating between ring and monopole structures. The reasons why the vacillation ceases will be addressed in the next section. Equivalent potential temperature Figure b shows radius-time plots of azimuthal mean θ e at hpa. Here again, the vacillation pattern is evident with a ring structure during the symmetric phase and the monopole structure during the asymmetric phase. During the symmetric phase, there is a ring of high θ e at some radius (centered at km during S and km during S). During the asymmetric phase, a θ e maximum occurs at the vortex centre and decreases significantly near the RMW. An analysis of the terms in the tendency equation for θ e, which was carried out in Nguyen (0) (not shown), indicates that the high θ e ring structure during the symmetric phase is due primarily to vertical advection of θ e. On the other hand, the decrease of θ e near the RMW during the asymmetric phase is accompanied by the horizontal advection, which is consistent with the stirring effects of the VHTs during this phase (as discussed earlier in Figure f). CAPE Convective Available Potential Energy (CAPE) and convection are highly interdependent. Increases in CAPE increase the intensity of convection, while, in turn, the convection consumes the the CAPE and stabilizes the atmosphere. This negative feedback is demonstrated clearly in the evolution of CAPE shown in Figure c. CAPE is computed as p LF C p LNB R d (T v T venv)dlnp for an air parcel lifted from the surface. R d is specific gas constant for dry air, LNB and LFC are level of neutral buoyancy and level of free convection, Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

11 Page of Quarterly Journal of the Royal Meteorological Society Inner-Core Vacillation Cycles during the Rapid Intensification of Hurricane Katrina 0 During the transitions to the asymmetric phases, the CAPE decreases substantially at the outer side of the axes of maximum v/ t (dashed tilted straight lines), which is where convection is most active. Just prior to the times of maximum asymmetries, regions of minimum CAPE coincide with the maximum updraught (two solid lines), indicating the convection consumes the CAPE. Conversely, during the transition from the asymmetric to the symmetric phase, the CAPE near the RMW is restored to higher values (see regions near the RMW from A to S and A to S). This increase in CAPE may be explained in part by fewer and weaker VHTs during this period, and in part by the increase of θ e at low levels (see the above explanation of the ring structure in θ e ). While CAPE is reduced by deep convection, it is replenished by surface latent heat fluxes. However, the evolution of the surface latent heat fluxes exhibits a steady increase as the mean circulation strengthens (not shown here). Thus, it appears that the VHTs play a dominant role in consuming CAPE, thereby reducing the potential for the further development of convection.. Transition mechanisms.. Symmetric to Asymmetric transition This section examines the physical mechanisms that produce asymmetric structures and are responsible for the change from a symmetric to an asymmetric vortex. The role of barotropic instability Kossin and Eastin (00) suggest that the transition from Regime to Regime (equivalent to the symmetric to asymmetric transition) is a consequence of barotropic instability. Their hypothesis is that the instability leads to the breakdown of the PV ring with the formation of respectively, p is pressure, T v and T venv are virtutal temperatures of the air parcel and of the environment, respectively. The calculation is based on stcript plotskew.gs, which is available from Grid Analysis and Display System (GrADS) users community. Figure. Evolution of the amplitude of azimuthal wavenumber calculated from PV (lines) at hpa along the km radius. E-folding time [h] of perturbations imposed on the symmetric vorticity profiles of the simulated vortex is shown by columns. The growth rates slower than h are not plotted. Short columns indicate faster growth rates. mesovortices and that the accompanying mixing results in the monopole structure (Regime or the asymmetric phase). A difficulty with the this argument is that the growth rate of barotropic disturbances appears to be too small to account for this regime transition. In the calculations presented by Schubert et al. (), the whole process, including the breaking down of the ring-like structure and the formation of the monopole, takes about h. This time is much longer than both that observed by Kossin and Eastin (00) (within to h) and that found in the present simulation (about to h). The growth rates of disturbances in the analytical model and numerical analogues thereof (Schubert et al., sections and ) are dependent on the amplitude and width of the annulus and, therefore, these growth rates and corresponding life-cycles may be faster. On the other hand, Schecter and Montgomery (00) indicate that when the effects of cloudiness are accounted for, the predicted growth rates may be reduced, depending on the extent of cloudiness. In the light of these results, the barotropic instability calculations should give an upper bound on the linear growth rate, provided that they are carried out for the simulated profiles. As a means of estimating the time-scale for barotropic instability in the present flow, we apply the linear stability analysis worked out by Weber and Smith (). This analysis determines the normal modes of the mean tangential wind profiles from the simulations using the nondivergent barotropic vorticity equation. Figure Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

12 Quarterly Journal of the Royal Meteorological Society Page of Nguyen Chi Mai, Michael J. Reeder, Noel E. Davidson, Roger K. Smith, Michael T. Montgomery 0 illustrates the amplitude evolution of the azimuthalwavenumber component of PV at hpa along the km radius circle (lines) and e-folding times (columns) of the corresponding wavenumber calculated by the Weber and Smith s method. Episodes in which e-folding time is short (i.e. fast growth rates) occur just prior to the subsequent asymmetric phase, at which times wave amplitudes peak (e.g. during the S from to h, and around the symmetric phase S from to h). During the mature phase the amplitudes of wavenumber do not grow even though the necessary conditions for instability are satisfied (e.g. to h). This non-development is consistent with the results of Wang (00a), who found that wavenumbers and higher are damped effectively by rapid filamentation, which is typically strong during the mature stage. The calculated growth rates are slower and the e- folding times are longer than the time taken for the vortex to change from the symmetric phase to the asymmetric phase in the simulation. However, these theoretical calculations are for the linear instabilities of an unforced barotropic vortex, whereas the simulation is baroclinic and includes a relatively sophisticated representation of the physical processes. Nonetheless, the agreement between the timing of the calculated fast growth rates and the simulated peaks in amplitudes of the corresponding wavenumbers suggests that barotropic instability plays a part in promoting the asymmetries during the initial stage of the transition towards the asymmetric phase. A complimentary approach is to examine the budgets of kinetic and potential energies for the mean flow and eddies (i.e. asymmetries). The analysis method is adopted from the work of Kwon and Frank (00), wherein the full derivation of the equations can be found. The barotropic conversion rate, which comprises all the terms that are present in the tendency equations for both the mean and eddy parts of the kinetic energy (but with opposite signs), has the form: [ u u u r + ru v r ( ) v + u r ω u p + v ω v p + u r v v Primes denote the deviations of the variables from their azimuthal means at each level. The radial, tangential and vertical wind components are u, v and ω, respectively, and, as before, p is the pressure. Positive conversion rates indicate that the kinetic energy of eddies is transferred to the kinetic energy of the mean flow. ] () Figure a shows the evolution of the vertical profile of the barotropic conversion rate averaged over radii from to 0 km. After the symmetric phases S and S, the barotropic conversion rate is negative (shaded with dark colour) indicating that the kinetic energy of the mean flow is converted into eddy kinetic energy, which is consistent with the idea that barotropic instability plays a role in the transition. Note that negative barotropic conversion rates occur also during the periods other than the symmetricto-asymmetric transition, especially noticeable after the asymmetric phase A. Thus, barotropic energy conversion does not occur only during the symmetric-to-asymmetric transition, nor it is the only process that occurs during this transition. The role of convective instability We present evidence now suggesting that the development of VHTs plays an important role in the transition to the asymmetric phase. Figure shows the evolution of the vertical structure of θ e near the eyewall. Low-level θ e, and hence convective instability, increases as the vortex becomes more symmetric, reaching its maximum during the symmetric phase. The transfer of eddy potential energy to eddy kinetic energy represents the effects of convection and is defined by hθ ω (Kwon and Frank 00). The evolution of this Wavenumber is analysed here because the structure of this component is the most prominent in the vertical motion field during the symmetric phase S (see Figure a). Here h = (R/p)(p/p ref ) (R/Cp), p ref = Pa, R= J kg K is specific gas constant, and C p = 0 J kg K is specific heat capacity of dry air. Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

13 Page of Quarterly Journal of the Royal Meteorological Society Inner-Core Vacillation Cycles during the Rapid Intensification of Hurricane Katrina 0 a) b) c) Figure. Evolution of the energy conversion rates. a) Barotropic conversion [ m s ] as in Eq. (). b) Convective conversion [ m s ]. c) Baroclinic conversion [ m s ] as in Eq. (). Figure. Evolution of the azimuthally-averaged θ e [K] for the radii between and km, which encompasses the eyewall. term is shown in Figure b. The term is positive throughout the integration indicating that convective processes are (not surprisingly) active in the core region. Further, during the transition from the symmetric to asymmetric phase (e.g. from S to A, and from S to A), the convective conversion term increases in magnitude, reaching maxima at the times of maximum asymmetry (t A, t A ). The role of baroclinic processes For completeness, Figure c shows the baroclinic energy conversion rate, which comprises all the terms that are present in the tendency equations for both the mean and eddy available potential energy: [ (h ) u s θ θ r + ( h s ) ] ω θ θ p where s = h( θ 0 / p), and θ 0 is the mean θ at each level (Kwon and Frank 00). Positive values of this term indicate the conversion from eddy potential energy to mean potential energy. () In Figure c, the baroclinic energy conversion term is positive throughout the troposphere and has a larger magnitude during the symmetric-to-asymmetric transition, indicating that the potential energy of the eddies is transferred to mean potential energy. Thus, this process reduces the amplitudes of the eddies (i.e. the asymmetries), especially during the symmetric-to-asymmetric transition, while increasing the potential energy of the mean flow. Note that the magnitudes of the barotropic and baroclinic conversion terms (Figures a, c) are smaller than the convective conversion term (Figure b) indicating the dominant role of convective processes. The formation of the monopole structure The effects of stirring the PV during the asymmetric phase is examined now. Figure shows the evolution of a local PV maximum identified by the Laplacian of PV ( P V ). Initially, the cyclonic PV anomaly (marked by a filled circle inside of a triangle) is associated with a deep convective area (panel a), but over the next minutes it gradually detaches from the convective area (panels b,c,d) and moves towards the vortex centre (panels e and f). The PV monopole Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

14 Quarterly Journal of the Royal Meteorological Society Page of Nguyen Chi Mai, Michael J. Reeder, Noel E. Davidson, Roger K. Smith, Michael T. Montgomery 0 (a) : h (b) : h (c) : h (d) :00 h (e) : h (f) :0 h Figure. Local PV maxima at hpa represented by (PV), [ PVU m ] (shaded regions represent positive anomalies). Contours are vertical motion [- Pa s ]. Dotted lines ending with filled circles inside triangles mark the positions of a tracked local PV maximum. The figures show the fields at minutes intervals after : h of integration, during the transition from the symmetric to the asymmetric phase. structure with the maximum at the vortex centre develops through a sequence of similar episodic mixing processes. The movement of the local maxima of cyclonic PV towards the vortex centre can be explained by a process similar to the non-linear β effect leading to north-westward drift of tropical cyclones in the Northern Hemisphere and a south-westward drift in the Southern Hemisphere (see e.g. Chapter in Elsberry et al. ). In the Northern Hemisphere, the advection of planetary vorticity by the vortex circulation induces an anticyclonic tendency on the eastern side of the vortex and a cyclonic tendency on the western side. The ensuing vorticity anomaly pattern is rotated by the cyclonic circulation of the vortex and leads to an induced flow across the vortex centre towards the northwest. This flow accounts for north-westward drift of the vortex. In more complex flows, vortices tend to move with a component towards higher PV. In a barotropic context, the tendency for small-scale cyclonic vortices to migrate to the centre of larger ones was discussed by Ulrich and Smith (). During the symmetric-to-asymmetric transition, a similar process occurs with the VHTs, which are embedded in a negative radial gradient of azimuthal-mean PV (not shown here). Then, analogous to a vortex on a β plane, the VHTs move towards higher PV. During this inward movement, the VHTs stir high PV near the eyewall and low PV within the eye, thereby increasing the PV near the vortex centre. For this reason, the central pressure falls most rapidly during the asymmetric phases (Figure ) as the PV accummulated at the vortex center. As the VHTs move inwards into a region with warm air aloft, their convection decays due to the decreased static instability. However, the low-level PV anomalies Although the physical size of VHTs appear to be small (of the order of -0 km) the advection by their circulations (as determined in budget calculations which are not shown here for brevity) affect much larger surrounding areas (of the order of km). Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

15 Page of Quarterly Journal of the Royal Meteorological Society Inner-Core Vacillation Cycles during the Rapid Intensification of Hurricane Katrina 0 they produce outlive the updraughts and continue moving inwards. These PV anomalies, which are not accompanied by deep convection, appear to be what have been described previously in the literature as eyewall mesovortices (Kossin et al. 00), which have been interpreted as originating from barotropic instability (Kossin et al. 000, Kossin and Schubert 00). We have shown here that mesovortices may originate from VHTs in this case. In summary, the symmetric-to-asymmetric transistion is accompanied by the development of the VHTs within the eyewall. The VHTs are formed as a result of a combination of barotropic and convective instabilities, which can be termed as barotropic-convective instability. Subsequently, the low-level cyclonic PV anomalies associated with the VHTs move towards the centre, leading to the formation of the monopole during the asymmetric phase... Asymmetric to Symmetric transition In contrast to the symmetric-to-asymmetric transition, during which VHTs grow within the eyewall, the VHTs weaken during the asymmetric-to-symmetric transition. After the times of maximum asymmetry, they become elongated convective bands moving outwards and more slowly than the mean tangential wind component. This outward and retrogressive movement is similar to that of VRWs and is to be expected since the radial gradient of azimuthal mean PV is negative. At the same time, the associated large rate-of-strain at their peripheries (see Figures c,d) weaken also thereby producing an environment more favourable for the development of convection at outer radii. The section below examines the reasons why the VHTs weaken, the similarity of the convective bands to VRWs, and the relationship between the rate-of-strain and convection at outer radii. The role of the local vertical shear The reduction of the CAPE (Figure c) is not the only processes weakening the VHTs after the asymmetric phase. The evolution of the local vertical wind shear in the eyewall region plays a role also. Figure. Evolution of the vertical gradient of the mean tangential wind v/ p [ m s Pa ] averaged across the eyewall region (from to km). The decrease of the cyclonic tangential wind with height (which is the signature of warm-cored systems) may create an environment unfavourable for the development of deep convection: while the lower part of a VHT is embedded in the rapidly rotating flow in the lower troposphere, its upper part may encounter much slower cyclonic rotation. As a result, VHTs in such an environment may be tilted backwards (i.e. upstream) with height, leading to a ventilation and weakening of the convection. To study this effect, the evolution of the vertical gradient of mean tangential wind is examined. Figure shows the evolution of the vertical gradient of the mean tangential wind near the eyewall (averaged over - km radii), where the VHTs are located. The vertical shear of mean tangential wind in the upper part of the troposphere (shaded regions above the 00 hpa level) increases noticeably towards the asymmetric phase, as the vortex spins up at low levels. This increase of the mean tangential wind at low levels is a result of both the horizontal stirring by the VHTs and vertical advection by the mean circulation of the vortex (as described in Section. on the evolution of mean radial and tangential wind at low levels). Thus, the resultant large vertical shear of the mean tangential wind that develops after the asymmetric phase is a less favourable environment for convection. This increase in the shear is consistent also with a warmer core during the asymmetric phase. Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

16 Quarterly Journal of the Royal Meteorological Society Page of Nguyen Chi Mai, Michael J. Reeder, Noel E. Davidson, Roger K. Smith, Michael T. Montgomery 0 Thus, the weakening of VHTs after the asymmetric phase can be explained by the collective effects of the reduced CAPE and a high-shear local environment. Vortex Rossby Waves VRWs have been proposed as an asymmetric mode of intensification (Guin and Schubert, Montgomery and Kallenbach, Möller and Montgomery, 000) and are associated with PV asymmetries. These asymmetries are typically created by convection and are commonly associated with inner spiral rain bands. In the environment of the vortex s mean negative radial PV gradient, positive PV anomalies tend to move radially outwards and while they move cyclonically, they retrogress relative to the mean tangential wind. As they move outwards, these PV anomalies experience strong straining by the differential rotation of the mean tangential flow and become thinner and their radial wavenumber increases. Thus, according to the dispersion relation derived by Montgomery and Kallenbach (), the radially-outward group velocity decreases and there may exist a critical radius where the group velocity becomes zero. At this radius, the disturbance gives its energy to the mean flow (Montgomery and Kallenbach, Möller and Montgomery, 000). In the simulation, the weakened VHTs become stretched bands of moderate convection moving outwards, thus, resembling the VRWs. The phase and group propagation speeds of the theoretical VRWs are computed using the formulae of Möller and Montgomery (000) (for constant radial wavenumbers). Figure shows an azimuthal Hovmöller diagram of vertical motion (shaded) overlayed with the PV along the km radius circle. The VHTs, as marked by strong upward motion and positive PV anomalies, are tracked and their azimuthal speeds are compared with the theoretical phase speeds of VRWs. These calculations use azimuthal-mean profiles of tangential wind speed, PV and static stability at hpa of the simulated vortex for disturbances having azimuthal wavenumbers of, or. The radial and vertical Figure. Hovmöller diagram of vertical motion (shaded) and PV (contour) at hpa along the km radius circle. On the horizontal axis, the direction from left to right is the cyclonic displacement. Solid straight lines are the direction of movement with the mean tangential flow. Longdashed lines indicate the movement directions of the tracked VHTs. Longshort-dashed lines show the movement with the theoretical VRW phase speeds for the azimuthal-wavenumber. Note that the PV contours are not the main focus of this plot but are shown to illustrate the typical colocation of updraughts and positive PV anomalies within VHTs. wavenumbers are chosen arbitrarily for the waves having the length scales of km and km in the respective directions. Whether or not the entities tracked can be considered VRWs can be judged by the degree to which the measured and theoretical speeds agree. In Figure, the larger the angle between the trajectory with the solid line (representing the movement with the mean tangential wind), the slower the movement of the VHT compared with the mean flow. The tracked VHTs (marked by dashed lines) move more slowly than both the mean flow (solid lines) and the calculated phase speeds of VRWs (dashed-dotted lines). This retrogression of VHTs is pronounced during the symmetric-to-asymmetric transition (e.g. from S at h to A at h) while the VHTs are strengthening. In contrast, after the asymmetric phase A, the weakening VHTs move nearly with the theoretical VRW Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)

17 Page of Quarterly Journal of the Royal Meteorological Society Inner-Core Vacillation Cycles during the Rapid Intensification of Hurricane Katrina 0 speeds which predicts small retrogression relative to the mean flow (see below). Figure. Retrogression calculated by (v c)/v 0% (circles) of tracked VHTs and the calculated VRW phase speed for wave number (solid thin line). The dashed line represents the smoothed average retrogression of the tracked convective entities. c is the tangential speed of the VHTs. The retrogression of the VHTs is illustrated more clearly in Figure. During the symmetric phases (e.g. S at h and S at h) the average retrogression of the tracked VHTs is of the order of %, whereas it drops to around %, close to the theoretical estimates, just after the asymmetric phase A near - h. To the extent that this retrogression is a signature of VRWs, the tracked convective entities behave more like VRWs during the asymmetric-to-symmetric transition than during the symmetric-to-asymmetric transition. Figure shows the theoretical radial phase and group speeds of the VRWs. The propagation of these waves is superimposed on the tendencies of the mean tangential wind (as in Figure a), the purpose of which is to identify any connection between the VRWs and changes in the mean tangential flow. According to Montgomery and Kallenbach (), Montgomery and Enagonio (), the VRWs interact with the mean tangential flow where the group speed vanishes. In the simulation, the critical radius is near km, at which traces of the radial group propagation (thin solid curves in Figure ) converge. However, there does not seem to be any significant increase in the mean tangential wind at the critical radius, indicating that the wave-mean flow interaction is weak at best. Figure. Radial phase (Cp r) and group (Cg r) speeds of the theoretical VRWs for the azimuthal wavenumber. The shaded areas show positive tendencies of v at hpa. Contours show v. The bold (blue) line represents the RMW. Solid thin curves show the radial propagation with the radial group speed Cg r, dashed straight lines are for radial phase speed Cp r, and solid straight lines illustrate mean radial wind at each point of interest. The reduced rate-of-strain at outer radii Figure. The evolution of the mean vertical motion ω [Pa s ] (dashed line) and the mean filamentation time scale τ fil [minutes] (solid line) at the 0 km radius on hpa. As shown in Figures c and f, the rate-of-strain, and thus the strain-dominated region just outside of the RMW, is more pronounced during the asymmetric phase than during the symmetric phase. As pointed out by Rozoff et al. (00), Copyright c 0 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: (0)