A mechanism for the transition of tropical cyclones (TCs) to the spontaneous rapid intensification (RI) phase is proposed based on numerical results of a three-dimensional full-physics model. The intensification phase of the simulated TC is divided into three subphases according to the rate of intensification: 1) a slowly intensifying phase, 2) an RI phase, and 3) an adjustment phase toward the quasi-steady state. The evolution of a TC vortex is diagnosed by the energy budget analysis and the degree of axisymmetric structure of the TC vortex, and the simulated TC is determined to be axisymmetrized 12 h before the onset of RI. It is found that equivalent potential temperature θe in the lowest layer suddenly increases inside the radius of maximum azimuthally averaged horizontal wind rma after the TC becomes nearly axisymmetric. Forward trajectory analyses revealed that the enhanced convective instability in the TC core region where the eyewall subsequently forms results from the increased inertial stability of the TC core after the axisymmetrization. Since fluid parcels remain longer inside rma, owing to the increased inertial stability, the parcels obtain more enthalpy from the underlying ocean. As a result, low-level θe and hence convective available potential energy (CAPE) increase. Under the condition with increased CAPE, the eyewall is intensified and the secondary circulation is enhanced, leading to the increased convergence of low-level inflow; this process is considered to be the trigger of RI. Once the eyewall forms, the simulated TC starts its RI.
The generation and intensification processes of tropical cyclones (TCs) have been investigated for decades. Observations have revealed that strong TCs often experience a rapidly intensifying period after an initial slow development (e.g., Sitkowski and Barnes 2009). In their statistical study, Kaplan and DeMaria (2003) defined a rapid intensification (RI) phase as the rate of change in maximum sustained wind speed being equal to or greater than 15.4 m s−1 (24 h)−1. They showed that all Atlantic basin hurricanes that are either category 4 or 5 undergo an RI phase at least once during their lifetimes. Therefore, the intensification phase of such strong TCs can be divided into two different phases in terms of the rate of intensification. This infers that such TCs mainly intensify through at least two mechanisms during the whole period of TC intensification.
The mechanisms for the intensification as well as the steady-state maintenance of TCs have been studied from observational, analytical, and numerical points of view. Emanuel (1986, 1989) proposed an air–sea interaction theory on TC intensification by assuming moist slantwise neutrality in the eyewall, so-called the wind-induced surface heat exchange (WISHE) theory. This theory describes that the intensification process is spontaneous as follows: given a cyclonic disturbance on the warm ocean, winds at the ocean surface drive sea surface enthalpy fluxes and the increased enthalpy fluxes, whose magnitude approximately depends linearly on the wind speed, enhance the radial gradient of equivalent potential temperature θe in the upper troposphere, which in turn accelerates the tangential wind in the lowest layer of the atmosphere [see Fig. 1 of Montgomery et al. (2009)]. Such a spontaneous intensification is one of the fastest intensifying processes that a TC experiences during its lifetime and therefore is referred to as RI in this study.
However, during an initial intensification before the RI, the vorticity of TCs is originally provided from both large-scale tropical disturbances (e.g., Ritchie and Holland 1999) and mesoscale convective disturbances (e.g., Bister and Emanuel 1997; Ritchie and Holland 1997). Recent studies have pointed out the importance of convective-scale processes, such as vortical hot towers (VHTs), that are characterized by strong convection with a significant cyclonic vorticity (Hendricks et al. 2004), especially during the early stage of TC intensification. Montgomery et al. (2006) conducted a numerical experiment to show that the simulated TC-scale vortex is generated from a mesoscale convective vortex–like disturbance and intensifies through the merger of numerous VHTs. The VHTs are strengthened by stretching of the vertical vorticity through convective updrafts. Once VHTs are generated on the cyclonic disturbance, the VHTs move to the core region where the vorticity of background circulation is large. In general, positive (negative) vortices move up the background gradient of positive (negative) vorticity (Schecter and Dubin 1999, 2001). Therefore, convectively generated VHTs merge with each other around the core region. Van Sang et al. (2008) examined the TC intensification by conducting numerical simulations of a three-dimensional nonhydrostatic model and showed that the mergers of VHT-like vortices gradually enhance the low-level circulation with the increase in the size of each convective cell. From the analysis of satellite data for Tropical Storm Gert (2005), Braun et al. (2010) examined the intensifying mechanism for the vortex merger and emphasized the importance of VHTs for the intensification of the tropical storm.
As vorticity increases through the merger of VHTs and/or mesoscale convective systems (Ritchie and Holland 1997; Simpson et al. 1997), a TC goes into its RI phase. This phase change can be recognized as an increase in the rate of intensification (e.g., Sitkowski and Barnes 2009; Barnes and Fuentes 2010). Although the initial intensification and the RI seem to be relatively well understood, the transition process of TCs to the RI phase is not clear.
Rogers (2010) carried out a numerical simulation of Hurricane Dennis (2005) and examined the convective processes before and after the onset of RI of the simulated TC. It was found that the RI is triggered by the enhanced low-level convergence (i.e., secondary circulation). In addition, Rogers suggested that the inertial stability of a TC vortex should be a suitable factor for predicting whether a TC goes into its RI phase, because the large inertial stability in the core region results effectively in condensation heating for accelerating primary circulation (Schubert and Hack 1982; Shapiro and Willoughby 1982; Nolan et al. 2007). The importance of the inertial stability for the structural evolution of TCs is suggested by the theoretical study of Vigh and Schubert (2009). Based on analytical solutions of a balanced, frictionless fluid system (Sawyer–Eliassen equation), they found that high inertial stability plays an important role in rapidly intensifying a warm core. Guimond et al. (2010) examined the airborne Doppler radar and satellite observations of Hurricane Dennis (2005) and showed that during the initial development of Dennis (2005), convective downdrafts increase the temperature in the eye and the inward motions toward the eye. Barnes and Fuentes (2010) examined the observational data obtained from integrated aircraft reconnaissance and suggested that an eye excess energy, defined as the vertically integrated difference in θe between the eye and eyewall, can diagnose well the onset of RI. Once the eye excess energy is accumulated enough, the energy is released in the eyewall region; this condition is favorable for RI. In other words, the increase in θe in the eye is necessary for the transition to the RI phase.
Although prior studies suggest that the thermodynamic and kinematic properties of the TC core play an important role in initiating RI, it is still not clear how and when the TC vortex changes its phase to the RI. Therefore, the purpose of this study is to examine a mechanism for the transition of TCs to the RI phase. One control and two sensitivity experiments under idealized conditions are conducted with the use of a three-dimensional, nonhydrostatic numerical model. The model settings are described in section 2. In section 3, the results of the experiments are shown. Based on the numerical results, a mechanism for the transition to RI is proposed in section 4. We conclude this study in section 5.
2. Numerical model and experimental design
The Advanced Research Weather Research and Forecasting Model (ARW-WRF, version 3.0; Skamarock et al. 2008) is used to simulate the development of a TC. The WRF solves a fully compressible, nonhydrostatic equation set using the third-order Runge–Kutta time-integration scheme (Wicker and Skamarock 2002) and the fifth-order upwind-biased scheme for advection terms. The prognostic equations are for momentum in three dimensions—perturbation geopotential, perturbation potential temperature, perturbation surface pressure of dry air, and the mixing ratios of water contents [see details in Skamarock et al. (2008)].
Our approach to determine the experimental setups for the present numerical simulations is to set the model physics and atmospheric conditions as simple as possible in order to focus on the dynamics of TCs. With this thinking, a Kessler (1969)-type scheme is used for cloud microphysics, and the effects of radiation are not included. The subgrid turbulence is represented by the eddy viscosity calculated using stress tensor and length scale (Smagorinsky 1963; Lilly 1962). The surface exchange coefficients for momentum and enthalpy, which are critical parameters for the TC intensity during the intensification as well as the steady state (Emanuel 1995, 1997; Gray and Craig 1998; Montgomery et al. 2010; Wang and Xu 2010; Xu and Wang 2010; Miyamoto and Takemi 2010; Ito et al. 2011), are formulated based on recent experimental and observational studies (Donelan et al. 2004; Zhang et al. 2008).
The initial state of the model atmosphere is a horizontally homogeneous, resting atmosphere whose vertical profiles of temperature and water vapor mixing ratio are determined by a tropical mean sounding of Jordan (1958). An axisymmetric vortex formulated by Rotunno and Emanuel (1987) with the maximum wind speed of 12 m s−1 at the radius of 80 km is inserted at the center of the computational domain at the initial time. The experiment is carried out using a vortex-following two-way nesting method on an f plane with the constant f value of 5 × 10−5 s−1. The outer numerical domain covers a 3000 km × 3000 km area with 6-km horizontal grid spacing, while the inner domain covers a 600 km × 600 km area at 2-km resolution. Both domains use 44 vertical levels stretched with height up to the 50-hPa level. In the upper part of the domain, a damping layer of the 2846-m depth is applied to minimize unwanted gravity waves reflecting at the upper boundary. No orthogonal fluxes are permitted at the lateral and top boundaries. The bottom boundary is assumed to be sea surface whose temperature is fixed uniformly at 300 K during the integration. Under these experimental settings, the governing equations are integrated numerically for 200 h, with model output saved every hour except for the period from t = 75 to 120 h, during which the output is saved every 10 min. Analyses are performed primarily for this control experiment.
To confirm that the simulated results from the control experiment do not depend on the model parameterization and/or the initial condition, we carry out two experiments that investigate sensitivities to the cloud-microphysics parameterization and to the initial environment. The first experiment is done by changing the microphysics parameterization to a scheme including ice-phase processes (Goddard microphysics; Tao et al. 1989). The second experiment uses a mean tropical sounding obtained during the observational campaign in 2008 by the research vessel (R/V) Mirai of the Japan Agency for Marine-Earth Science and Technology (JAMSTEC; Geng et al. 2011). Note that the experimental settings are identical to the control experiment except for the microphysics or the initial sounding.
a. Intensification and evolution of the simulated vortex
1) Definition of intensification phase
To diagnose the TC intensity, the maximum horizontal wind speed at the 1-km height is used in this study. Figure 1a shows the time series of the maximum wind speed and the radius of maximum horizontal wind rm. The maximum wind speed of the simulated TC starts increasing from t = 70 h and achieves the maximum around t = 150 h, while rm decreases from t = 70 h and becomes nearly constant after t = 113 h. After the intensification finishes (t = 150 h), the simulated TC reaches a quasi-steady state at which both the maximum wind speed and rm remain almost constant.
Figure 1b shows the time series of the rate of change in the maximum wind speed, a proxy for the rate of intensification of the simulated TC. The intensification rate fluctuates around the zero value from t = 20 to 70 h and linearly increases after t = 70 h. Then, the rate sharply increases from t = 105 h and reaches the maximum value of 3.38 m s−1 h−1 at t = 113 h. Since the rate of the intensification rate between t = 105 h and 113 h rapidly increases with time, this period can be regarded as a spontaneous intensification phase, that is, an RI phase. After t = 113 h, the intensification rate sharply drops until t = 122 h and gradually decreases. Subsequent to t = 150 h, the rate varies around the zero value. Therefore, the intensification of the simulated TC occurs during the period from t = 70 to 150 h. We define this period as an intensification phase. According to the difference in the rate of the intensification rate, this phase is further divided into three subphases: 1) a slowly intensifying phase between t = 70 and 105 h, 2) an RI phase in which the rate of intensification sharply increases from t = 105 to 113 h, and 3) an adjustment phase toward the maximum intensity from t = 113 h to the end of intensification (t = 150 h). Since the definition of RI in this study is the spontaneous intensification during which a TC enhances the intensification rate by itself, the RI phase is defined here as the period when the intensification rate is large and increases with time.
Since the purpose of this study is to examine a transition mechanism for the RI, we mainly focus on the physical processes prior to phase 2 (i.e., a later period of phase 1). The processes in phase 3 are not discussed in the current paper.
2) Intensifying processes
To investigate transition processes to the RI, the evolution of the simulated TC structure is at first examined. Figures 2 depicts the spatial distributions of the vertically averaged Ertel’s potential vorticity (PV) at t = 85, 95, 105, and 115 h. The averages are indicated by the bar in this paper and hence the vertically averaged PV is written as , where z represents the vertical direction. In Fig. 2, PV is averaged vertically between the levels of 1.5 and 12.0 km. Note here that in this study, the centers of the simulated TC are determined by centroids estimated by pressure distributions at the sea level (Braun 2002). A lot of local peaks characterized by a couple of positive and negative contours are seen in the early period of the simulation (t = 85, 95 h, and earlier; figures not shown). This is consistent with the previous numerical study of Van Sang et al. (2008). However, the horizontal scale of the dipole structures is smaller than that shown in Van Sang et al. (2008), whose experiment simulated dipole vortices with the 20-km horizontal scale. Meanwhile, the spatial distributions of at t = 105 and 115 h show that vortices with only positive peaks survive around the center (Figs. 2c and 2d). A circular area of positive values with about a 15-km radius becomes dominant after t = 105 h. The magnitude of the positive peaks increases with time [29.1, 46.5, 95.0, and 147.8 PV units (PVU; 1 PVU = 1.0 × 10−6 m2 s−1 K kg−1) at the four times shown in Fig. 2], while the magnitude of the negative peaks decreases with time (−22.8, −19.7, −15.5, and −13.1 PVU). In this way, Fig. 2 indicates that once the single-peak structure becomes significant (between t = 95 and 105 h), the TC vortex starts spinning up fast.
To demonstrate physical processes before and during the RI (phase 2), temporal evolutions of thermodynamic variables are examined here. Since the overall evolution of TCs can be clearly indicated by θe in the upper troposphere, temporal evolutions of θe at the 12-km height at three different radii are diagnosed (Fig. 3a). The θe values shown in Fig. 3a are the areal average within the eye region (r ≤ 5 km), and the azimuthal averages at the 40-km radius and at the 200-km radius. Prior to the later period of phase 1, θe gently increases in every region. In particular, in the eye and at the 40-km radius increase at the same rate. However, the rate of change in θe suddenly becomes large only in the eye region after the later stage of phase 1 until the end of phase 2. The sudden increase in in the eye during this period corresponds well to the sudden increase in the rate of intensification (Fig. 1b). Meanwhile, at the 40- and 200-km radii no longer increases during phase 2. Thus, the radial gradient of θe in the upper troposphere rapidly increases during the RI (phase 2).
The time series of the surface enthalpy flux and the vertically averaged θe in the lowest 1 km at rm are shown in Fig. 3b. It is indicated that both quantities rapidly increase after t = 105 h, again coinciding with the rapid increase in the rate of intensification. Therefore, it is considered that the intensified maximum wind speed results in the enhancement of the surface enthalpy flux, which in turn increases in the lowest 1 km. In addition, examining angular momentum fields indicated that the constant angular momentum line passing through rm approximately coincided with a constant surface of θe during phase 2 (figures not shown). The results shown in Fig. 3 as well as the angular momentum field strongly suggest that during phase 2, the simulated TC vortex intensifies through the WISHE mechanism that is a spontaneous intensifying process, as described in the introduction.
b. Axisymmetric and asymmetric evolution of the TC
The results described in section 3a showed that the horizontal distribution of of the simulated TC has a single strong peak around the TC center after the onset of RI. This indicates that during RI, the TC intensifies through axisymmetric processes. However, it was shown that before the RI there are numerous PV couplets, indicating that the TC circulation involves asymmetric processes before t = 105 h. Hence, it is expected that the simulated TC evolves from an asymmetric to an axisymmetric structure before the end of phase 1. To diagnose the time when the vortex becomes axisymmetric, this subsection focuses on phases 1and 2 in terms of the asymmetric and axisymmetric structures of the simulated TC.
1) Definition of axisymmetricity
To diagnose the axisymmetricity of the simulated TC, we defined a parameter to quantitatively measure the degree to which a TC vortex is axisymmetric. At first, a physical variable is divided into the azimuthally averaged (axisymmetric) component and the deviation from the azimuthal mean (asymmetric component), written as
where φ is a variable; r and λ are the radial and the tangential direction, respectively; and the prime stands for the deviation from the azimuthal average. Then, a parameter representing the degree of axisymmetricity of a vortex is defined as the ratio of the squared azimuthally averaged component to the sum of squared axisymmetric and asymmetric components, as shown:
This parameter is referred to as axisymmetricity in this study. When there are exactly no asymmetries (φ′ = 0), γ equals 1. Because Fig. 2 clearly demonstrates the characteristic evolution of the simulated TC, PV is used as φ in Eq. (2).
The axisymmetricity using PV (γPV) averaged within the 100-km radius and between z = 1.5 and 12.0 km (i.e., ), is also indicated in Fig. 2. The term is small (0.39) at t = 85 h, when the horizontal distribution of is dominated by individual convection (Fig. 2a), while becomes larger after the circular area with positive values forms (Figs. 2c and 2d). These results indicate that is able to well represent the degree to which the distribution is axisymmetric.
Figure 4a shows the time series of as well as the TC intensity. Since an axisymmetric vortex is embedded in the domain at the initial time, is 1.0 at t = 0 h. Several hours later suddenly decreases, and after that remains small up to t ≈ 90 h. Then, rapidly increases to a high value (=0.590) at t = 93 h and subsequently keeps constant with fluctuation to t = 122 h. Afterward, reaches its maximum. The temporal change shown in Fig. 4a suggests that the simulated TC vortex becomes nearly axisymmetric at t = 93 h, which is 12 h before the onset of RI (t = 105 h). In contrast, the gentle slope of from t = 70 to 93 h indicates that the intensifying process at the early stage of phase 1 (t < 93 h) corresponds to the axisymmetrizing process.
The axisymmetricities for other variables related to TC intensity—that is, kinetic energy γKE, tangential velocity γυ, and vertical velocity γw—are also calculated (Fig. 4b). The values are also averaged both in the radial and the vertical directions as in Fig. 2. When kinetic energy or tangential velocity is used as φ in Eq. (2), the increase in axisymmetricity (i.e., or ) is much smaller than that for PV. Especially, approximately holds the value of 1 during the integration period. However, increases after the RI starts and keeps small values (does not exceed 0.5). In this way, γPV is a better parameter to distinguish the axisymmetricities than γKE, γυ, and γw.
2) Energetic viewpoint of TC evolution
To understand how the axisymmetric and asymmetric processes contribute to the intensification of the simulated TC, we examine the processes before and after the RI starts (t = 105 h) from an energetic point of view by carrying out an energy budget analysis based on Kwon and Frank (2008).
Kinetic energy density (KE) and available potential energy density (PE) are defined as
where u and υ are the radial and tangential velocities, respectively; h ≡ R(p/pr)κ/p; p is the pressure and pr is the reference pressure set as 1000 hPa; κ = R/cp; R is the gas constant for dry air; cp is the specific heat at constant pressure; s2 ≡ −h(∂θ0/∂p); the zero subscripts represent the initial state; and is the deviation of potential temperature from the initial state.
Figure 5 shows the time series of KE and PE integrated in the volume within the 100-km radius and from the surface to the 16-km height. It should be noted that the scales of the vertical axes for the variables in the two panels of Fig. 5 are different. According to the decomposition defined in Eq. (1), KE and PE are divided into axisymmetric components that correspond to zero wavenumber component (WN = 0) and asymmetric ones (WN > 0). The axisymmetric part of kinetic energy density (KEWN=0) starts increasing after t = 30 h, and the rate of change is enhanced at t = 93 h. KEWN=0 achieves the maximum value at around t = 165 h. In contrast, the axisymmetric part of available potential energy density (PEWN=0) continuously increases throughout the integration. The increasing rate of PEWN=0 saturates at around t = 80 h and constantly increases during the rest of the simulation time.
The asymmetric components of both kinetic energy density and available potential energy density (KEWN>0 and PEWN>0) are further separated into higher wavenumber components (WN > 5) and the total asymmetric ones (WN > 0). Figure 5 shows that the asymmetric components (i.e., WN > 5 and WN > 0) of both KE and PE increase at least until t = 93 h. Subsequently, KEWN>0 decreases sharply until t = 113 h and then remains approximately constant until the end of the intensification (t = 150 h), whereas PEWN>0 steadily increases until the end of the intensification phase.
The differences of both KE and PE between the higher (WN > 5) and the total (WN > 0) asymmetric components are generally small, indicating that the effects of the low-wavenumber components (1 ≤ WN ≤ 5) are small. In other words, low-wavenumber motions such as vortex Rossby waves (e.g., Guinn and Schubert 1993; Montgomery and Kallenbach 1997) have little impact on the intensification during the period and thus high-wavenumber components that are considered to include convective-scale processes play a role in maintaining the asymmetric motion. Figure 6 shows the time series of the ratio of KEWN>0 to KEWN=0 within the volume defined above. The ratio before t = 93 h is larger than that after t = 93 h. This temporal change in the ratio indicates that the contribution of convective-scale processes should be significant in the TC core (within the 100-km radius) up to t = 93 h. After t = 93 h when the rate of change in KEWN=0 suddenly increases, the contributions of asymmetric components gradually become smaller than those of the axisymmetric components. This is consistent with the results that the simulated TC becomes well axisymmetric at t = 93 h as diagnosed by γPV (Fig. 4a).
The result that the high-wavenumber component of KE is dominant compared to the low-wavenumber component seems to be inconsistent with the results of Wang (2008) in his idealized numerical simulation. His results indicate that the low-wavenumber component of KE is significant in the core region, while the high-wavenumber component mainly presents only outside the TC core (see his Fig. 14). However, in the current experiment, it is also shown that the distributions of both high- and low-wavenumber components in a radius–time cross section are qualitatively similar to those of Wang (2008) (figures not shown). Since the components of KEWN>0 and KEWN>5 in the present experiment (Fig. 5) are evaluated in the volume including both the core and parts of the outer region, the high-wavenumber component, which is more dominant in the outer radii but less dominant in the core region, becomes more significant than the low-wavenumber one.
As a next step, the energy budgets are calculated with the use of the conservation equations of KE and PE as follows (Kwon and Frank 2008):
where and are KE of axisymmetric and asymmetric motions, respectively; and are PE of axisymmetric and asymmetric motions, respectively; w is the vertical velocity in a pressure coordinate; and Q is the heating rate through diabatic processes. The first terms on the right-hand side of the above four equations consist of the radial and vertical advection terms (subscript “adv”). The second terms in Eqs. (5) and (6) stand for the barotropic energy conversion between the axisymmetric and asymmetric components of KE, whereas the second terms in Eqs. (7) and (8) represent the baroclinic energy conversion between the axisymmetric and asymmetric components of PE (conWN=0→WN>0). Note that the components at the origin (terminate) of the arrow are the losses (gains) of the energy. Positive values of the energy conversion terms mean that the energy is converted from the axisymmetric to the asymmetric components. The third terms represent the baroclinic energy conversion of axisymmetric and asymmetric components between KE and PE (conPE→KE and conPE←KE). The positive sign of both components represents the energy conversion from PE to KE. The fourth terms of the PE equations, Eqs. (7) and (8), are the source terms proportional to diabatic heating (DH) and to the deviation of potential temperature from the initial state. The last terms of the four equations stand for the diffusion due to subgrid-scale turbulent processes. In these equations, the bar (superscript λ omitted here) represents the azimuthal mean.
Figure 7 shows the time series of the budgets of Eq. (5). The energy conversion term between PE and KE in Eq. (5) is always positive, which means that PE is baroclinically converted to KE. This term generally increases throughout the three phases. However, after t = 93 h the barotropic energy conversion of KE between the WN = 0 and the WN > 0 components is mostly negative, indicating that KEWN=0 is converted to KEWN>0. The magnitude of the advection terms is significantly smaller than that of the other terms.
The budgets of Eq. (7) and the asymmetric heating term (DHWN>0) in Eq. (8) are shown in Fig. 8. Prior to t = 93 h, both the axisymmetric and asymmetric components of the energy source due to the diabatic heating are equally present (Fig. 8b). After t = 93 h (Fig. 4), the axisymmetric heating (DHWN=0) significantly increases, which continues during phase 2. This is consistent with the results diagnosed by the axisymmetricity [Eq. (2)]. After t = 120 h, the increase in the heating rate becomes smaller until the heating approaches the maximum value. This temporal change in DHWN=0 seems to be well correlated with that of the energy conversion term between KE and PE. In contrast, DHWN>0 no longer increases after t = 93 h. Corresponding to the decrease in DHWN>0, the energy conversion of PE from the WN = 0 to the WN > 0 component is small. As in the budgets of KE, the contribution of the advection terms is negligible.
The intensification phase of the simulated TC from the energy budget analysis is summarized as follows. As the TC vortex becomes nearly axisymmetric (Fig. 4), the axisymmetric component of diabatic heating increases (Fig. 8b). The heating enhances PEWN=0, which is in turn converted to the axisymmetric component of KE (Figs. 7 and 8a). As a result, KEWN=0 increases and then the simulated TC intensifies.
It was also shown in the analyses of the axisymmetricity and the energy budget that the simulated TC vortex becomes axisymmetric at t = 93 h, which is 12 h before the onset of RI. In the observational study of Sitkowski and Barnes (2009), the inner-core region of Hurricane Guillermo (1997) clearly indicated a circular-shaped structure during the RI. The structure of the simulated TC core is consistent with this observed feature.
c. Dynamic and thermodynamic properties around the onset of RI
From the previous results, it was indicated that the simulated TC goes into the RI phase at t = 105 h, which is 12 h after the TC becomes axisymmetric. In this subsection, the dynamic and thermodynamic properties before and after the RI starts are focused on in order to demonstrate the processes of the transition to the RI phase.
Figure 9 depicts the horizontal distributions of and water content, which are averaged vertically between the levels of 1.5 and 12.0 km. The thick positive contours of with a 50-km horizontal scale indicate that the overall structure of the simulated TC is almost axisymmetric. The contours are concentrated around the TC center at the four times. Meanwhile, the local peaks of the vertically averaged water content, which are characterized by a spiral shape, are significant at earlier times (Figs. 9a–c). This spiral structure gradually organized into a ring-shaped structure. At t = 106 h, the ring-shaped distributions of the vertical averages of both PV and water content indicate that the eyewall forms at r ≈ 10 km, when the TC goes into the RI phase (t = 105 h). In other words, the eyewall formation is considered to be closely related to the onset of RI of the simulated TC.
Figure 10 exhibits the horizontal distributions of vertical velocity and potential temperature deviation from the horizontal average , averaged vertically between the 1.5- and 12.0-km heights. At the four times, small peaks of both and are seen in the whole region, while is high inside the 25-km radius. As time passes, around the TC center increases, meaning that a warm-core structure becomes significant. The spiral peaks of appear at t = 100 and 103 h, which corresponds to those of the averaged water content as shown in Fig. 9. After the onset of RI, circular ring-shaped updrafts are clearly seen around the rotation center.
Figure 11 depicts the radius–height sections of the tangentially averaged diabatic heating rate , radial velocity , and absolute angular momentum at the same four times as in Figs. 9 and 10. At all of the given times, the positive and negative components of are present in the upper troposphere (z ≈ 15 km) and below the 1.5-km height, respectively. Although the outflow in the upper troposphere exists soon after the numerical integration starts (figure not shown), it is after t = 97 h that the outflow becomes significant, as shown by Fig. 11a.
As time passes, the contours of and proceed inward, which is consistent with the gradual decrease in rm as shown in Fig. 1. The contours of align in the vertical and the distance between the contour lines becomes narrow around the eyewall at t = 106 h. Since a fluid particle moves with conserving its absolute angular momentum in an inviscid rotating fluid system, the concentration of contours of in the free atmosphere means the increase in upward motion. As shown in Fig. 10d, the vertical velocity is significant in the core region at t = 106 h. Correspondingly, becomes significant and is also confined in the narrow region (4 ≤ r ≤ 12 km). This confined diabatic heating region (Fig. 11d) is where the water content is high (Fig. 9d) and corresponds to the eyewall of the simulated TC.
As Rogers (2010) and Guimond et al. (2010) stated, the enhancement of low-level convergence is a triggering mechanism for RI of Hurricane Dennis (2005). As shown in Fig. 11, the enhancement of convergence of low-level inflow is linked to the intensification of updrafts. Figures 10 and 11 show that the strong upward motion confined in the TC core is closely related to the initiation of RI. Hence, temporal changes in the intensity and location of vertical mass fluxes are examined. The time series of the local and azimuthally averaged maximum vertical mass fluxes at z = 12.5 km (Fm and ) and their radii ( and ) as well as the radius of maximum azimuthally averaged horizontal wind rma are plotted in Fig. 12. The temporal changes in Fm and are generally similar to each other, although before t = 105 h the fluctuation of Fm is larger than that of . It is seen that both Fm and rapidly increase immediately before t = 105 h. However, highly fluctuates before t = 105 h and then the radius becomes smaller than rma after the RI starts; also locates around rma during and after the RI phase.
The vertical mass flux is well correlated with the diabatic heating after the onset of RI as shown in Figs. 9–11, and so is the confinement of as well as around rma. Thus, the onset of RI corresponds to the time of the eyewall formation (around t = 105 h). In other words, the TC structure, including the eyewall, the secondary circulation (updraft in the eyewall and low-level convergence), and the vigorous peak of the vertically averaged PV at the center of circulation, seems to be established at t = 105 h, although the TC vortex becomes axisymmetric at t = 93 h (Fig. 4).
To demonstrate the transition phase from the establishment of axisymmetricity (t = 93 h) to the initiation of RI (t = 105 h), the thermodynamic aspects of the simulated TC are examined here. Figure 13 indicates the temporal and radial changes of convective available potential energy (CAPE), rma, θe averaged in the lowest 500-m layer , and the diabatic heating averaged from the 1.5- to 12-km height . In addition, all quantities are azimuthally averaged. Note that CAPE is calculated by lifting a parcel whose properties are vertically averaged in the lowest 500 m. The rma is located around the radius of 60 km at t = 75 h (prior to the onset of RI) and it shrinks to the minimum value (rma ≈ 10 km) after RI starts (t = 105 h). The increase of occurs preferentially inside rma and correspondingly, the increase of occurs within rma. It is interesting to note that suddenly increases within rma at t = 93 h and remains larger than 2500 J kg−1 until t = 105 h. After the RI starts, is strengthened and confined inside rma (≈8 km) (Figs. 11–13). Once the high appears, the enhanced suddenly decreases.
d. Sensitivity experiments
In this subsection, the sensitivities of the results of the current idealized experiment, especially the enhancement of CAPE within rma, to the physics parameterization and initial condition are examined and the results of the sensitivity experiments are briefly shown. Figures 14a and 14b depict the time–radius cross sections of , rma, , and of the experiments with the ice microphysics and the different initial sounding, respectively. In both figures, it is shown that rma shrinks as time passes, and that both and increase, especially inside rma. Furthermore, once the high forms, the enhanced suddenly decreases and no longer increases. In conclusion, the results shown in these figures are qualitatively consistent with the results of Fig. 13, and therefore it is confirmed that the enhancement of in the core region before the formation of the high is robust.
4. A proposed mechanism for the transition to RI
a. Linkage between CAPE accumulation and axisymmetricity
Based on the results obtained in section 3, a transition mechanism to the RI is investigated in this section. In Fig. 13, it was shown that CAPE rapidly increases after the TC core becomes axisymmetric and the high CAPE condition continues until RI starts, suggesting that the axisymmetrization is a key factor in increasing the CAPE value. Furthermore, from a dynamical aspect, the axisymmetrization will increase the inertial stability of the TC core. Therefore, we hypothesize that the rapid increase in CAPE within rma is closely related to the enhancement of inertial stability.
First, the inertial stability (Schubert and Hack 1982)
is examined. As argued by Rogers (2010), I2 can be useful as an indicator to diagnose whether a TC is at the RI stage: if I2 is sufficiently high (although the critical value was not determined by Rogers), then the RI of the TC is able to start. The time series of , the vertically averaged inertial stability, , and , which are averaged inside rma, are shown in Fig. 15. As in Figs. 4 and 13, Fig. 15 shows that enhances after about t = 93 h and increases from t = 93 to 105 h along with the enhancement of . After the onset of RI (t = 105 h), suddenly decreases, whereas rapidly increases. It is obvious from Fig. 15 that is positively correlated with during the period between the establishment of axisymmetricity (t = 93 h) and the onset of RI (t = 105 h).
It is noteworthy that stops increasing around t = 100 h, although steadily increases. This is considered to be due to the enhanced θ″ around the rotation center (Fig. 10): when the atmosphere becomes statically more stable (i.e., greater θ″), CAPE will be smaller.
As a next step, a Lagrangian trajectory analysis for air parcels originating in the core region is carried out in order to examine the relationship between I2 and air motion within the core region. The movements of the air parcels are calculated by the two-step method of Doty and Perkey (1993). First, 81 parcels are seeded at the grid points at the lowest model layer (z = 30.0 m) within the radius of 20 km at a given time. Then, the trajectories of the parcels are calculated by the simulated wind field at every 10 min until the parcels pass over either the 30-km radius, which approximately corresponds to the outer edge of the high CAPE region (Fig. 13), or the 500-m height below, which the thermodynamic variables are vertically averaged for the calculation of CAPE. The wind outputs at 10-min intervals are linearly interpolated at 1-min resolution. From given position and time, the location of a parcel as a first guess is calculated with the use of the horizontally and vertically interpolated winds at the given time step and then winds at the point after the displacement are obtained by interpolating three-dimensional wind data at the next time step. The winds at the two different locations at two consecutive times are averaged and used to determine the movement of the parcel after a single time step (Takemi and Satomura 2000).
Figure 16a shows the time series of the eye residence time, which represents how long the seeded parcels stay within a control volume inside the 30-km radius and below the 500-m height. Note that the residence time is estimated by the averages for the 81 parcels. It is shown that at t = 93 h, the residence time sharply increases from about 1 to 2–3.5 h and becomes shorter than 2 h after t = 105 h. Figure 16b shows the time series of whether the parcels pass through the control volume at the lateral or the top boundary. If all parcels pass through the lateral boundary, then the value is plotted at “lateral 100%” in the vertical axes. Before t = 93 h all the parcels pass at the lateral boundary, whereas subsequent to t = 93 h, the percentage of the parcels that pass through the top boundary becomes significant. After t = 99 h, few parcels pass through the lateral boundary.
b. Discussion: A proposed mechanism for the transition to RI
The increase in surface enthalpy flux as well as the residence time occurs possibly due to the axisymmetrization of the simulated TC (Fig. 4a). Since the enhanced secondary circulation results in strong vertical velocity below the 500-m height and the increased I2 prevent parcels from passing through the lateral boundary, parcels tend to move upward rather than outward. Inside rma, the vertical motion is an order of magnitude weaker than the horizontal motion (figure not shown). As a result, parcels can stay longer within rma after the TC becomes nearly axisymmetric.
Since the trajectory analysis suggests that the residence time is closely related to I2, the relationship between the residence time and the enthalpy that an air parcel (along a trajectory) obtains from the underlying ocean is estimated and shown in Fig. 17. Note that the enthalpy is an integrated value during the period when an air parcel is located at the lowest level (i.e., when a parcel is at 15 ≤ z ≤ 52 m). It is obvious that the enthalpy obtained through the sea surface fluxes before t = 93 h (black circles) is smaller than after t = 93 h (dark gray rectangles and light gray circles). Once the TC is axisymmetric (i.e., t = 93 h), the residence time substantially becomes long and the accumulated enthalpy in turn enhances. After the onset of RI (after t = 105 h), the obtained enthalpy is still large (light gray circles), although the residence time becomes shorter. This seems to result from the enhanced secondary circulation.
It is confirmed that parcels obtain more enthalpy from the ocean as the residence time becomes longer. As a result, θe and hence CAPE increase inside rma after t = 93 h. Once CAPE becomes sufficiently large for continuous convective activity, the TC becomes ready for the formation of the eyewall.
In contrast, the axisymmetrized structure of the TC vortex enhances the axisymmetric component of diabatic heating as well as the secondary circulation. It is shown that as I2 increases, the diabatic heating around the eyewall region effectively works to enhance the TC intensity and the warm core rapidly develops. These processes are consistent with previous studies (e.g., Schubert and Hack 1982; Shapiro and Willoughby 1982; Nolan 2007; Vigh and Schubert 2009). Therefore, the axisymmetrized vortex accelerates the spinup process. Consequently, the axisymmetrization prior to the RI is necessary for both dynamic and thermodynamic evolutions of TCs.
To examine the mechanism for the transition of TCs to the RI phase, numerical experiments are carried out by using a three-dimensional, fully compressible numerical model (WRF). The intensification phase of the simulated TC is divided into three subphases according to the rate of the intensification: 1) a slowly intensifying phase (t = 70–105 h), 2) an RI phase (t = 105–113 h), and 3) an adjustment phase toward the quasi-steady state (t = 113–150 h). Specific analysis focusing on the intensification phase, including the RI phase of the simulated TC, reveals the following points. The WISHE mechanism seems to play a role in RI (phase 2) of the simulated TC (Fig. 3). In contrast, the energy budget analysis shows that PE generated by the axisymmetric diabatic heating is converted to the axisymmetric component of KE, which corresponds to the rotational motion of TC. The contribution of PE to the asymmetric component of KE is small compared to that of the axisymmetric one (Figs. 5–8). Since the intensifying process corresponds to the forming process of a single, vigorous peak of the vertically averaged PV at the center of circulation, a parameter, axisymmetricity γ [Eq. (2)] is defined in order to diagnose the degree to which the vortex is axisymmetric. Using γ, it is found that 12 h before the onset of RI, the simulated TC vortex becomes axisymmetric (Fig. 4). Axisymmetrization of the simulated TC vortex makes the TC core inertially stable. Both θe in the lowest 500 m and CAPE increase—especially inside rma—after the TC becomes nearly axisymmetric (Fig. 13). Subsequent to the onset of RI, the enhanced CAPE suddenly decreases and the convective activity is confined to the eyewall region (Figs. 9–13). In addition, the vertical mass flux is confined to r ≈ rma and becomes significant before the onset of RI (Fig. 12).
The Lagrangian trajectory analysis shows that axisymmetrization of the simulated TC enables air parcels to stay longer within the core region and to obtain a larger amount of enthalpy from the underlying ocean (Figs. 16 and 17). Intensifying surface wind speeds enhance sea surface fluxes and hence provide much enthalpy. Consequently, θe and CAPE increase within rma, where the eyewall subsequently forms and the maximum value of CAPE achieves 3500 J kg−1 (Fig. 13). The sudden increase in the vertical mass fluxes in the core region (or evolution of the secondary circulation) results in forming the eyewall by using the enhanced CAPE. The RI processes of the simulated TC are driven by the formation of the eyewall.
Given a cyclonic disturbance strong enough to obtain energy from the ocean, the transition mechanism to the RI phase of TCs proposed in this study is summarized below. The new findings of this study are items 1, 3, and 4:
A TC vortex becomes axisymmetric or at least starts axisymmetrization processes sufficiently before the start of RI.
As the TC vortex becomes axisymmetric, its I2 increases substantially and the axisymmetric component of diabatic heating effectively accelerates the primary and secondary circulations.
The large I2 increases the residence time of fluid parcels in the TC core, which results in the enhancement of boundary layer θe and hence CAPE within the rma.
An eyewall forms by utilizing the enhanced CAPE in the core region and then the RI processes are driven.
Therefore, the results of this study indicate that a TC needs to be well axisymmetrized before the onset of RI (about 10 h in this case). This time period should be sufficiently long for the increase of CAPE in the core region before the onset of RI. The time needed would be different for each TC.
We would like to express our gratitude to Prof. H. Ishikawa, who provided a lot of fruitful discussion. We also thank Profs. C.-C. Wu and S. Yoden, Drs. G. H. Bryan and K. Ito, and Mr. R. Yoshida for giving us valuable suggestions. The comments by anonymous reviewers are greatly acknowledged for improving the original manuscript. This work is partly supported by JSPS Scientific Research Grant 20-776 for JSPS Research Fellows.
Current affiliation: RIKEN, Hyogo, Japan.