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    (a) Radius–height cross section of tangential velocity υ and (b) vertical profiles of domain-averaged zonal velocity in cases with translation speed of 5 m s−1. The line color darkens with decreasing the magnitude of shear. The contour interval in (a) is 2 m s−1.

  • View in gallery

    (a) Time series of maximum azimuthally averaged tangential velocity (black line) at z = 2 km with axisymmetricity of PV γPV averaged from z = 2 to 8 km inside the RMW (blue line). (b) Time series of intensification rate of azimuthally averaged tangential velocity at z = 2 km (thin dashed line). The thick solid line is the result after applying a 1–2–1 average for 20 times. The vertical line indicates the time of t = 102 h, which is defined as the onset of RI.

  • View in gallery

    (top),(middle) Horizontal sections of total condensates qtot (shaded) and sea level pressure (contours) from t = 42 to 106 h. The condensates are vertically averaged from z = 2 to 8 km. The contour interval is 3 hPa, with a highest value of 1010 hPa. Each panel covers a 240 km × 240 km area. The shear vector points eastward (to the right of each panel). (bottom) Radius–height cross sections of azimuthally averaged diabatic heating rate Q (shaded) and tangential velocity υ (contours) at four times. The contour interval is 10 m s−1, and the contours of 20 and 40 m s−1 are highlighted. Each panel covers from the center to 150-km radius and from the surface to 16-km height.

  • View in gallery

    Radius–time section of CAPE (shaded), equivalent potential temperature θe averaged below the 500-m level (thin blue contours), diabatic heating rate Q averaged from z = 2 to 10 km (thick red contours), and RMW (gray dashed line). The surface of 1 K h−1 of Q is drawn to identify the strong heating region. The contours for θe are drawn from 355 to 370 K with an interval of 5 K. The horizontal dash–dotted line indicates the time of RI onset.

  • View in gallery

    Time series of RMC estimated by the symmetric component of radial velocity averaged below z = 2 km. The thick line shows the filtered value, which is the running average in an 8-h window. The gray line shows the eyewall radius, for which the specific definition is described in the text. The radii are normalized by the RMW of tangential velocity at z = 2 km. The vertical line represents the onset of RI.

  • View in gallery

    (a) Time series of difference in vortex center between the surface and at z = 8 km. (b) Time series of difference in temperature T between the center and r = 200 km at z = 5 (black) and 8 km (gray). The vertical lines show the onset of RI.

  • View in gallery

    Time series of vorticity budgets in the upper layer of the low-level center, which are averaged from z = 5 to 8 km and inside the RMW. The gray, black, red, and blue lines represent the horizontal advection, vertical advection, stretching, and tilting terms, respectively. The inset panel shows the time series from t = 78 to 102 h.

  • View in gallery

    Horizontal cross sections of condensates (shaded), vertical velocity (blue contours), and the sum of the vertical advection, tilting, and stretching terms (red contours) from t = 86 to 108 h. The quantities are vertically averaged from z = 5 to 8 km. The contours indicate 3 m s−1 of the vertical velocity and 2 × 10−6 s−2 of the vorticity budget. The dashed circle around the center indicates the RMW at z = 2 km. The centroid of ζ|ζ| at z = 8 km is shown as the white cross with black circle.

  • View in gallery

    (a) Time series of TC intensity defined as the maximum tangential velocity at z = 2 km υm and (b) histogram of lifetime-maximum intensity for the ensemble simulations. The black lines in (a) show members of ensemble simulations that experience RI, whereas those without RI are shown by the gray lines. The black and gray bars in (b) show the RI and non-RI cases, respectively. The bin width is 4 m s−1. The thin black and gray lines in (b) indicate the numbers of each bin.

  • View in gallery

    Time series of (a) TC intensity υm, (b) RMW at z = 2 km rm, (c) difference in rotation center between z = 0 and 8 km Δxc, (d) axisymmetricity averaged inside the RMW and from z = 2 to 8 km γPV, (e) equivalent potential temperature averaged below the 1-km height θe, and (f) RMC for the convergence averaged below 2 km, which is normalized by the RMW, for TCs experiencing RI in the ensemble simulations. The times in the horizontal axes are subtracted by the time of onset of RI. The thick black line indicates the ensemble average, and the gray region is plus and minus one standard deviation from the ensemble mean.

  • View in gallery

    As in Fig. 10, but for the (a) absolute vorticity, (b) vertical advection term, (c) stretching term, (d) tilting term, and (e) horizontal advection term. Note that the range of vertical axis in (e) is 103 times smaller than the other three terms in (b)–(d).

  • View in gallery

    Histograms of (a) intensification rate ∂tυm, (b) υm, (c) RMW at z = 2 km rm, (d) difference in the rotation center between z = 2 and 8 km normalized by the RMW Δxc/rm, (e) axisymmetricity averaged inside the RMW and from z = 2 to 8 km γPV, (f) θe vertically averaged from z = 0 to 1 km, (g) absolute value of background velocity averaged between 2 and 8 km, (h) absolute value of vertical shear (difference from 8 to 2 km), and (i) as in (h), but for TCs when initial υm ≤ 10 m s−1. They are normalized by the bin with the largest number of samples. The bin width is determined as the horizontal range divided by 15. The black and gray bins show the RI and non-RI cases, respectively. The black and gray dashed vertical lines represent the ensemble average for the two groups, respectively.

  • View in gallery

    As in Fig. 12, but for (a) Rossby number (Ro) defined in Eq. (3), (b) dimensionless parameter ϒ representing the tendency of formation of an upright vortex defined in Eq. (4), and (c) Ro times ϒ.

  • View in gallery

    As in Fig. 12, but for (a) axisymmetricity of cloud-top height averaged inside the 200-km radius, (b) bulk dimensionless parameter ϒb defined in Eq. (5), and (c) Rob times ϒb, where Rob is bulk Rossby number using the radius of 100 km rather than RMW.

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Structural Changes Preceding Rapid Intensification in Tropical Cyclones as Shown in a Large Ensemble of Idealized Simulations

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  • 1 Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida, and Advanced Institute for Computational Science, RIKEN, Kobe, Japan
  • | 2 Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida
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Abstract

Structural changes that precede rapid intensification (RI) of tropical cyclones (TCs) are examined in a full-physics model by conducting a large ensemble (270) of idealized TC simulations. The processes leading to RI in a representative case with moderate shear are consistent with previous studies for weakly sheared cases. The most distinct changes are that the vortex tilt and the vortex size begin to decrease more rapidly 6 h before the onset of RI. A vorticity budget analysis for the upper layer around the low-level center reveals that the vertical vorticity is increased by vertical advection, stretching, and tilting terms before RI, whereas the horizontal advection is small. Thus, the upright vortex structure is not achieved through a vortex alignment process but rather is built upward by deep convection.

The ensemble simulations are generated by changing the intensity and size of the initial vortex, the magnitude of vertical wind shear, and the translation speed. The ensemble members that show RI are consistent with the control case and many previous studies: before the onset of RI, the intensity gradually increases, the radius of maximum tangential wind (RMW) decreases, the flow structure becomes more symmetric, the vortex tilt decreases, and the radius of maximum convergence approaches the radius of maximum winds. A dimensionless parameter representing a tendency for the formation of the vertically upright structure is considered. The product of this parameter and the local Rossby number is significantly larger for TCs that exhibit RI in the next 24 h.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yoshiaki Miyamoto, ymiyamoto@rsmas.miami.edu

Abstract

Structural changes that precede rapid intensification (RI) of tropical cyclones (TCs) are examined in a full-physics model by conducting a large ensemble (270) of idealized TC simulations. The processes leading to RI in a representative case with moderate shear are consistent with previous studies for weakly sheared cases. The most distinct changes are that the vortex tilt and the vortex size begin to decrease more rapidly 6 h before the onset of RI. A vorticity budget analysis for the upper layer around the low-level center reveals that the vertical vorticity is increased by vertical advection, stretching, and tilting terms before RI, whereas the horizontal advection is small. Thus, the upright vortex structure is not achieved through a vortex alignment process but rather is built upward by deep convection.

The ensemble simulations are generated by changing the intensity and size of the initial vortex, the magnitude of vertical wind shear, and the translation speed. The ensemble members that show RI are consistent with the control case and many previous studies: before the onset of RI, the intensity gradually increases, the radius of maximum tangential wind (RMW) decreases, the flow structure becomes more symmetric, the vortex tilt decreases, and the radius of maximum convergence approaches the radius of maximum winds. A dimensionless parameter representing a tendency for the formation of the vertically upright structure is considered. The product of this parameter and the local Rossby number is significantly larger for TCs that exhibit RI in the next 24 h.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yoshiaki Miyamoto, ymiyamoto@rsmas.miami.edu

1. Introduction

The statistical analysis of Kaplan and DeMaria (2003) found that strong hurricanes in the Atlantic experience rapid intensification (RI) at least once during their lifetimes. Using a global dataset, Lee et al. (2016) showed that the distribution of lifetime-maximum intensity of tropical cyclones (TCs) is bimodal, with one peak around 30 m s−1 and the other around 60 m s−1. They found that nearly all TCs in the second group experienced RI at some point. Hence, predicting RI is critically important for the intensity forecast of TCs (Kaplan et al. 2010, 2015; Rozoff and Kossin 2011). Observations have shown that many strong TCs have two phases with different intensification rates: a slow intensification and a rapidly intensifying period (e.g., Sitkowski and Barnes 2009).

TCs intensify by diabatic heating in the convection around the circulation center (e.g., Ooyama 1969; Schubert and Hack 1982). The diabatic heating inside the radius of maximum tangential wind (RMW) more effectively accelerates the low-level tangential winds than heating outside the RMW (Shapiro and Willoughby 1982; Pendergrass and Willoughby 2009; Vigh and Schubert 2009). Rogers et al. (2013) found that the deep convection tends to be present inside the RMW in intensifying TCs but shifts to outside the RMW after intensification. Nolan and Grasso (2003) and Nolan et al. (2007) showed that the asymmetric component of diabatic heating has little effect and the intensification is determined by the magnitude of the symmetric-mean heating. An observational study of Shimada et al. (2017) found that TC intensification rate increases with symmetry of the precipitation field.

Hendricks et al. (2010) analyzed the environmental conditions associated with RI and found that the conditions are often not very different for RI and non-RI cases, suggesting that RI is at least controlled partly by internal processes. Composite analyses obtained from satellite and airborne observations have shown that TCs tend to exhibit a symmetric cloud region (eyewall) inside the RMW during RI (Harnos and Nesbitt 2011; Kieper and Jiang 2012; Jiang 2012; Zagrodnik and Jiang 2014; Zawislak et al. 2016; Rogers et al. 2016). Modeling studies also have also found that the flow field is highly axisymmetric in the inner core during RI (Miyamoto and Takemi 2013, hereafter MT13; Wang and Wang 2014; Chang and Wu 2017). Other studies reported that RI can possibly occur under sheared TCs without axisymmetric convective heating (Molinari and Vollaro 2010; Chen et al. 2011; Nguyen and Molinari 2012; Zhang and Chen 2012; Chen and Zhang 2013; Chen and Gopalakrishnan 2015; Miller et al. 2015). Zhang and Chen (2012) and Chen and Zhang (2013) investigated a simulation of RI in Hurricane Wilma (2005) and concluded that the RI starts after the formation of a warm core. The warm core resulted from the development of a convective region, which was advected to the downshear region where convection can be intensified. Nevertheless, as discussed in the composite analyses introduced above, such cases in which convection is highly asymmetric might be less frequent.

Previous studies have proposed several factors that play an important role in initiating RI: the increase in low-level mass convergence produced by convective downdrafts (Rogers 2010), the presence of strong deep convection inside the RMW (Guimond et al. 2010; Wang and Wang 2014; Rogers et al. 2013, 2015), symmetric convective structure (Kaplan et al. 2010; Zagrodnik and Jiang 2014; Miyamoto and Takemi 2015, hereafter MT15; Rogers et al. 2016), high relative humidity in the middle troposphere (Kieu et al. 2014; Zawislak et al. 2016), increase in the difference in low-level equivalent potential temperature θe between the eye and outside (Barnes and Fuentes 2010; Dolling and Barnes 2012; MT13; Wang and Heng 2016), formation of the warm core (Vigh and Schubert 2009; Chen et al. 2011; Zhang and Chen 2012; Chen and Zhang 2013; Kieu et al. 2014; Chang and Wu 2017), and vertical alignment of the vortex (Stevenson et al. 2014; Hogsett and Zhang 2011; Yamada et al. 2012; Davis and Ahijevych 2012; Rogers et al. 2015). It remains to be determined which of these processes are predominant, coincidental, or even irrelevant.

MT13 and MT15 proposed the following transition mechanism for RI. When a TC first becomes nearly axisymmetric, the inertial stability increases, increasing the residence time of parcels that enter the core. During their longer residence, the parcels obtain enthalpy from the ocean, and the equivalent potential temperature and convective available potential energy (CAPE) increase significantly. The intensity gradually increases in this period because of the presence of convection in the core. Based on the Ekman pumping theory for a circular flow (Eliassen and Lystad 1977; Kepert 2001), the maximum upward velocity at the top of boundary layer is located closer to the vortex center when the vortex is not so strong. This radius of maximum upward velocity, or radius of maximum convergence (RMC), increases during the slow intensification. When the RMC approaches the RMW, longer-lived convective cells located around the RMC (and just inside the RMW) can develop. The axisymmetric component of diabatic heating increases rapidly, and RI begins. Whereas the hypothesis of MT13 and MT15 was validated based on their idealized simulations and one realistic simulation, the vertical shear in the cases they examined was zero or very weak. Hence, further investigation is needed, especially for moderate- and strong-shear cases.

The purpose of this paper is to examine the processes leading to RI and test the hypothesis of MT13 and MT15 under various conditions, including horizontal flow and vertical wind shear. For this purpose, we conducted a large number of numerical simulations using a full-physics model. The experimental settings for the simulations and analysis method are introduced in section 2. We first focus on a representative member of the ensemble to examine structural changes before the onset of RI, including a vorticity budget analysis in section 3. Then we examine the results across all ensemble members in section 4. The results are discussed and parameters describing the likelihood of initiating RI are proposed in section 5. This study is concluded in section 6.

2. Simulation design

We produced an ensemble of idealized TC simulations with 270 members by changing the vertical wind shear, background horizontal velocity, and the intensity and size of each vortex at the initial time. Table 1 shows the specific values of these parameters. The simulations used the Weather Research and Forecasting (WRF) Model, version 3.7 (Skamarock et al. 2008), which is a widely used, fully compressible mesoscale model that solves prognostic equations for mass, momentum, potential temperature, and scalars for water content. A background velocity that is horizontally uniform is added to the initial field with low-level easterly flow and westerly vertical wind shear. The point downscaling method of Nolan (2011) was used to maintain horizontally homogeneous background flow and wind shear in a large, doubly periodic domain without horizontal temperature gradients. Three nested, storm-following domains were used with grid spacings of 18, 6, and 2 km with grid sizes of 480 × 190, 192 × 192, and 180 × 180, respectively. Forty vertical levels were used with the grid spacing expanding vertically with the minimum spacing of 107.5 m at the lowest model layer and the maximum spacing of 1812.1 m at the uppermost layer. The integration period was 144 h for each member with an output interval of 60 min.

Table 1.

List of parameters in the ensemble simulations.

Table 1.

The initial sounding was the moist tropical sounding of Dunion (2011). All simulations used a constant Coriolis parameter, 5 × 10−5 s−1, and a constant sea surface temperature, 301.15 K, over the domain. A 6-class single-moment scheme (Hong and Lim 2006) was used for microphysical processes. Diffusion in the boundary layer was solved by the Yonsei University planetary boundary layer scheme (Noh et al. 2003; Hong et al. 2006). The surface layer was parameterized by Monin–Obukhov similarity theory (Jiménez et al. 2012). The aerodynamic formulas of Davis et al. (2008) and Dudhia et al. (2008) were used for the momentum and heat fluxes at the ocean surface. The radiation scheme was not incorporated. Radiation changes the temperature field in the numerical domain; hence, including radiation causes the environmental sounding to change with time. Since we focus on impacts of each of the experimental parameters, temporal variation of environment should be minimized as much as possible.

We first focus on a single simulation (hereafter control simulation), of which the initial intensity, RMW, background velocity, and vertical shear are 15 m s−1, 150 km, −7.5 m s−1, and 5.0 m s−1, respectively (see Table 1). Figure 1a depicts the radius–height cross section of tangential velocity of the initial vortex for the control simulation. The low-level flow is easterly with westerly wind shear defined by the cosine-shaped profiles seen in Fig. 1b. The surface value corresponds to the translation speed (−5 m s−1 in the figure), and the velocity increases with height depending on the magnitude of the shear. The meridional wind is zero at the initial time. The shear value in Table 1 is the difference in velocity between the 2- and 12-km levels, whereas the shear is estimated as a difference between 2 and 8 km for the main analysis shown below.

Fig. 1.
Fig. 1.

(a) Radius–height cross section of tangential velocity υ and (b) vertical profiles of domain-averaged zonal velocity in cases with translation speed of 5 m s−1. The line color darkens with decreasing the magnitude of shear. The contour interval in (a) is 2 m s−1.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

The TC center was defined as the centroid of surface pressure (Braun 2002). To evaluate the magnitude of vortex tilt due to the vertical wind shear, we examined the distance of the rotation center between the surface and z = 8 km. After applying a 1–2–1 smoothing 20 times on the horizontal field of vertical relative vorticity ζ at z = 8 km, the upper-level center was defined as the centroid of ζ|ζ| within the radius of 180 km from the surface center.

TC intensity is defined in this study as the maximum azimuthally averaged tangential velocity at z = 2 km, hereafter denoted by υm. In the ensemble simulations, the onset of RI was defined as the time when the intensification rate exceeds a threshold, 0.54 m s−1 over 1 h [corresponding to 25.2 kt (1 kt = 0.51 m s−1) over 24 h], for the subsequent 6 h, and the rate is less than the threshold for the previous 2 h. Note that the intensification rate was calculated as the temporal difference in TC intensity in a 2-h window centered at each output time. Out of 270 ensemble members, 119 simulations met the criteria, whereas the other 151 members were diagnosed as the non-RI cases.

3. TC evolution in the control simulation

a. Overview and definition of RI period

Figure 2a shows a time series of TC intensity υm and the axisymmetricity of Ertel’s potential vorticity (PV) averaged from z = 2 to 8 km inside the RMW. The axisymmetricity γ represents the degree of symmetry of a given field as defined by MT13:
e1
where r, λ, z, and t, respectively, represent the radius, azimuth, height, and time, respectively; the bar shows the azimuthal average; and the prime denotes the deviation from the azimuthal average. The value of γPV ranges from 0 to 1, and the distribution of PV is perfectly axisymmetric when γPV is 1. The intensification rate in Fig. 2b shows that, overall, the simulated TC has two different intensification phases: a slowly intensifying phase up to t = 102 h followed by RI. The TC intensity is approximately 30 m s−1 when RI begins and reaches about 70 m s−1 at the end of simulation. The axisymmetricity is initially large since the initial vortex is axisymmetric. It decreases rapidly to 0.2 then starts increasing after t = 90 h. At the onset of RI (t = 102 h), γPV is 0.45, indicating that the structure of the simulated TC is very symmetric. The axisymmetricity is large especially in the lower levels. Note that the results shown here use data from the innermost domain (Δx = 2 km).
Fig. 2.
Fig. 2.

(a) Time series of maximum azimuthally averaged tangential velocity (black line) at z = 2 km with axisymmetricity of PV γPV averaged from z = 2 to 8 km inside the RMW (blue line). (b) Time series of intensification rate of azimuthally averaged tangential velocity at z = 2 km (thin dashed line). The thick solid line is the result after applying a 1–2–1 average for 20 times. The vertical line indicates the time of t = 102 h, which is defined as the onset of RI.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

Figure 3 depicts horizontal sections of vertically averaged total condensates qtot from z = 2 to 8 km beginning at t = 42 h. A strong convective region is present in the downshear-left quadrant (t = 42 h), and it rotates counterclockwise to the upshear region with time (t = 94 h). The azimuthal motion of the strong convective region is possible because the wind shear is not so strong (5 m s−1) in this case. Note that the cyclone translates right to left in each panel and the vector of vertical wind shear is opposite to the motion vector. The horizontal distribution of condensates becomes fairly axisymmetric after the onset of RI.

Fig. 3.
Fig. 3.

(top),(middle) Horizontal sections of total condensates qtot (shaded) and sea level pressure (contours) from t = 42 to 106 h. The condensates are vertically averaged from z = 2 to 8 km. The contour interval is 3 hPa, with a highest value of 1010 hPa. Each panel covers a 240 km × 240 km area. The shear vector points eastward (to the right of each panel). (bottom) Radius–height cross sections of azimuthally averaged diabatic heating rate Q (shaded) and tangential velocity υ (contours) at four times. The contour interval is 10 m s−1, and the contours of 20 and 40 m s−1 are highlighted. Each panel covers from the center to 150-km radius and from the surface to 16-km height.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

The bottom panels in Fig. 3 depict the radius–height cross section of azimuthally averaged diabatic heating rate by phase change of water and tangential velocity every 4 h from t = 94 h. Prior to the RI, the tangential velocity has a peak in the lower troposphere, and the symmetric component of diabatic heating appears around the RMW but with weak amplitude. Once RI begins, the tangential velocity increases substantially in the lower troposphere, and a strong peak of symmetric diabatic heating appears around the RMW, consistent with the horizontal cross section.

Figure 4 shows the radius–time cross section of θe averaged from z = 0 to 0.5 km, CAPE, and diabatic heating Q averaged from z = 2 to 8 km. Note that CAPE was calculated by lifting air parcels in the vertical direction, not along a sloped and slanted trajectory. The parcels were defined by quantities vertically averaged in the lowest 500 m. Before the onset of RI, θe and CAPE increase inside the RMW, which is consistent with the idealized nonsheared experiment of MT13. The RMW is defined as the radius at which the υm occurs. A strong diabatic heating region forms around the RMW, and CAPE rapidly decreases inside the RMW after RI begins. This suggests that the enhanced CAPE is partially consumed to form the developing eyewall. Another likely reason for decreasing CAPE is the development of the warm core. The RMW contracts with time and becomes close to its lifetime minimum (~40 km) a few hours after the onset of RI, similar to many observed and simulated TCs (Stern et al. 2016).

Fig. 4.
Fig. 4.

Radius–time section of CAPE (shaded), equivalent potential temperature θe averaged below the 500-m level (thin blue contours), diabatic heating rate Q averaged from z = 2 to 10 km (thick red contours), and RMW (gray dashed line). The surface of 1 K h−1 of Q is drawn to identify the strong heating region. The contours for θe are drawn from 355 to 370 K with an interval of 5 K. The horizontal dash–dotted line indicates the time of RI onset.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

As noted in the introduction, MT13 and MT15 proposed that RI begins when the RMC reaches the radius of developing eyewall. Figure 5 shows the time series of RMC and eyewall radius rew. We define rew as RMW minus Δrew, where Δrew is the distance between the RMW and inner radius of strong heating region (qtot > 1.5 g kg−1) at t = 102 h, and Δrew does not vary in time. The RMC is initially located at a small radius relative to the RMW, increases with time, approaches the eyewall radius rew around the onset of RI, and approximately corresponds to the eyewall radius afterward. The results obtained here for the control simulation (a moderate-shear case) are quite consistent with those of the zero-shear case in MT13 and MT15.

Fig. 5.
Fig. 5.

Time series of RMC estimated by the symmetric component of radial velocity averaged below z = 2 km. The thick line shows the filtered value, which is the running average in an 8-h window. The gray line shows the eyewall radius, for which the specific definition is described in the text. The radii are normalized by the RMW of tangential velocity at z = 2 km. The vertical line represents the onset of RI.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

Figure 6a shows a time series of the distance between the vortex center at the surface and at z = 8 km (i.e., the magnitude of vortex tilt Δxc). The value of Δxc decreases from 38.5 km at t = 78 h to less than 10 km by t = 96 h, which is 6 h before the onset of RI, and it becomes nearly constant afterward. Note that the large majority of the reduction of tilt occurs before RI begins. Figure 6b shows a time series of the horizontal differences in azimuthally averaged temperature from the vortex center to r = 200 km as a proxy of warm-core strength at z = 5 and 8 km. The difference in temperature at both levels starts increasing around t = 96 h, which is 6 h before the onset of RI and corresponds to when the upright vortex forms. Nevertheless, no clear signal was observed in the temperature difference immediately before the onset of RI.

Fig. 6.
Fig. 6.

(a) Time series of difference in vortex center between the surface and at z = 8 km. (b) Time series of difference in temperature T between the center and r = 200 km at z = 5 (black) and 8 km (gray). The vertical lines show the onset of RI.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

b. Vorticity budget analysis for the upper layer before RI

Figure 6a shows that the simulated TC becomes vertically aligned before the onset of RI. The processes leading to the vertically upright structure are investigated by conducting a vorticity budget analysis in the upper troposphere around the center of the low-level vortex. The equation for vertical vorticity in Cartesian coordinates can be written as
e2
where (u, υ, w) are the velocities in the (x, y, z) directions, ζ = ∂xυ − ∂yu is the vertical component of relative vorticity, t is the time, and f is the Coriolis parameter. The first and second terms on the right-hand side represent the horizontal advection, the third term represents the vertical advection, the fourth term is the stretching, the fifth and sixth terms are the tilting, and Dζ includes solenoidal effects and diffusion.

Figure 7 shows a time series of vorticity budget terms: the horizontal advection, vertical advection, stretching, and tilting terms. The terms are first calculated in Cartesian coordinates and then averaged inside the RMW and vertically from z = 5 to 8 km. The solenoidal term and the diffusion term are not shown because the former is small and the latter generally reduces the magnitude of vorticity tendency. Before the onset of RI, the stretching term plays a major role, and both the vertical advection and tilting terms contribute positively, whereas the horizontal advection term is negligible. After t = 106 h (4 h after the start of RI), the tilting term is dominant, whereas both the vertical advection and stretching terms contribute negatively. The horizontal advection term is small compared with the other terms. Hence, the formation of the upright structure of this simulated TC was not caused by the alignment of a tilted vortex. Rather, the vortex was strengthened in the upper layer through the vertical advection and stretching of vorticity from the lower troposphere. The sign of the stretching term changes from positive to negative in the 5–8-km layer after the onset of RI. This is due to the change in the maximum level of diabatic heating from upper to lower levels (figure not shown).

Fig. 7.
Fig. 7.

Time series of vorticity budgets in the upper layer of the low-level center, which are averaged from z = 5 to 8 km and inside the RMW. The gray, black, red, and blue lines represent the horizontal advection, vertical advection, stretching, and tilting terms, respectively. The inset panel shows the time series from t = 78 to 102 h.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

Figure 8 displays the structural evolution of condensates, vertical velocity, and the sum of stretching and vertical advection terms that play an important role in increasing vorticity in the upper layer before and after the onset of RI. The quantities are averaged vertically from z = 5 to 8 km. The large condensate region is highly asymmetric and located in the upshear to upshear-left side before t = 100 h. A more symmetric condensate region in the 5–8-km layer then develops around the RMW. The vorticity production terms are large in the regions with large vertical velocity and condensates. This indicates that the budget terms are intensified by the strong vertical velocity that is produced by the convection. The upper-level center, defined as the centroid of ζ|ζ| at z = 8 km, moves toward the surface center, and their difference is approximately less than 10 km after t = 102 h (cf. Fig. 6a).

Fig. 8.
Fig. 8.

Horizontal cross sections of condensates (shaded), vertical velocity (blue contours), and the sum of the vertical advection, tilting, and stretching terms (red contours) from t = 86 to 108 h. The quantities are vertically averaged from z = 5 to 8 km. The contours indicate 3 m s−1 of the vertical velocity and 2 × 10−6 s−2 of the vorticity budget. The dashed circle around the center indicates the RMW at z = 2 km. The centroid of ζ|ζ| at z = 8 km is shown as the white cross with black circle.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

4. Ensemble results

a. Time series of quantities in RI cases

Figure 9a shows time series of TC intensity for the ensemble simulations. Using the criteria shown in section 2, 119 simulations out of 270 were diagnosed to have RI within the integration period (144 h). Almost all the TCs that experience RI achieve strong intensity consistent with previous statistical studies (Kaplan and DeMaria 2003; Lee et al. 2016). Only six members reached strong intensity (greater than 30 m s−1) without RI.

Fig. 9.
Fig. 9.

(a) Time series of TC intensity defined as the maximum tangential velocity at z = 2 km υm and (b) histogram of lifetime-maximum intensity for the ensemble simulations. The black lines in (a) show members of ensemble simulations that experience RI, whereas those without RI are shown by the gray lines. The black and gray bars in (b) show the RI and non-RI cases, respectively. The bin width is 4 m s−1. The thin black and gray lines in (b) indicate the numbers of each bin.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

A histogram for the lifetime-maximum TC intensity in the ensemble simulations is shown in Fig. 9b. The intensity distribution is bimodal, and the stronger and weaker peaks correspond to the members with RI and without RI. The distributions for the RI and non-RI groups are clearly different: TCs with RI attain stronger intensity in their lifetime than those without RI except for some cases that do not satisfy the RI condition even though they reach strong intensity. This is consistent with the statistical analysis of Lee et al. (2016).

The ensemble-average time series of TC intensity of members with RI are shown in Fig. 10a, in which the time is subtracted by the time of RI onset for each member. The ensemble shows that the intensification rate greatly increases after the onset of RI, as usually seen in a strong TC (e.g., Barnes and Fuentes 2010). Before RI, the RMW is decreasing and begins to shrink more rapidly before RI. Notably, the intensity at the start of RI does not vary much around the mean value of 29.5 m s−1. After RI, the RMW continues to shrink for another 12 h then remains constant.

Fig. 10.
Fig. 10.

Time series of (a) TC intensity υm, (b) RMW at z = 2 km rm, (c) difference in rotation center between z = 0 and 8 km Δxc, (d) axisymmetricity averaged inside the RMW and from z = 2 to 8 km γPV, (e) equivalent potential temperature averaged below the 1-km height θe, and (f) RMC for the convergence averaged below 2 km, which is normalized by the RMW, for TCs experiencing RI in the ensemble simulations. The times in the horizontal axes are subtracted by the time of onset of RI. The thick black line indicates the ensemble average, and the gray region is plus and minus one standard deviation from the ensemble mean.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

Figure 10c shows the ensemble-average time series of vortex tilt Δxc estimated in the same way as in Fig. 6a. The tilt decreases with time before the onset of RI, and it becomes quite small at RI onset, whereas the variance is large. The large variance is because of various values of vortex size and vertical shear in the ensemble. Figure 10d shows the time series of axisymmetricity averaged inside the RMW and from z = 2 to 8 km, showing that the flow structure becomes more symmetric before the onset of RI. The equivalent potential temperature θe below the 1-km level averaged inside the RMW steadily increases with time (Fig. 10e). Figure 10f shows the RMC for the convergence vertically averaged below 2 km. The RMC approaches the RMW at the 2-km level and becomes close to it around the onset of RI. These behaviors shown in Fig. 10 are qualitatively the same as in the control simulation and previous studies.

Figure 11a shows a time series of vertical vorticity averaged inside the RMW and from z = 5 to 8 km. The vorticity steadily increases until RI and then more rapidly increases after RI. Figures 11b–e show the vorticity budget terms for the upper layer around the surface center, also averaged inside the RMW and from z = 5 to 8 km. Prior to the onset of RI, the stretching, tilting, and vertical advection terms play important roles in enhancing vorticity. Once RI starts, the tilting term is the dominant term that increases the vorticity in the upper layer. The stretching term and vertical advection contribute negatively, whereas the magnitude of vertical advection is one order smaller. The horizontal advection is negligible. The positive contribution of stretching and vertical advection to vorticity tendency before RI onset is consistent with the control simulation.

Fig. 11.
Fig. 11.

As in Fig. 10, but for the (a) absolute vorticity, (b) vertical advection term, (c) stretching term, (d) tilting term, and (e) horizontal advection term. Note that the range of vertical axis in (e) is 103 times smaller than the other three terms in (b)–(d).

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

b. Likelihood of RI in subsequent 24 h

Using the ensemble, we seek to identify external and internal factors that are favorable for RI. For this purpose, we defined the intensification rate as the change in TC intensity in each 12-h period and then computed that maximum 12-h intensification rate within the time window of the next 24 h. Applying a threshold of 9.6 m s−1 of change in υm over 12 h for RI, we obtain two groups for the RI and non-RI cases. The total number of samples for the RI and non-RI cases are 1019 and 11 487, respectively. Note that here we are taking many samples from the hourly output of each simulation, some of which have periods of RI and some which do not. Since the number of samples with RI is much less than non-RI cases, the histograms in Figs. 1214 are normalized by the bin of category that has the largest number of samples in order to compare the statistical features of distributions between the RI and non-RI groups. The dark bars show data for TC periods that achieved RI in the next 24 h, while the gray bars are data for TC periods that were not followed by RI.

Fig. 12.
Fig. 12.

Histograms of (a) intensification rate ∂tυm, (b) υm, (c) RMW at z = 2 km rm, (d) difference in the rotation center between z = 2 and 8 km normalized by the RMW Δxc/rm, (e) axisymmetricity averaged inside the RMW and from z = 2 to 8 km γPV, (f) θe vertically averaged from z = 0 to 1 km, (g) absolute value of background velocity averaged between 2 and 8 km, (h) absolute value of vertical shear (difference from 8 to 2 km), and (i) as in (h), but for TCs when initial υm ≤ 10 m s−1. They are normalized by the bin with the largest number of samples. The bin width is determined as the horizontal range divided by 15. The black and gray bins show the RI and non-RI cases, respectively. The black and gray dashed vertical lines represent the ensemble average for the two groups, respectively.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

Fig. 13.
Fig. 13.

As in Fig. 12, but for (a) Rossby number (Ro) defined in Eq. (3), (b) dimensionless parameter ϒ representing the tendency of formation of an upright vortex defined in Eq. (4), and (c) Ro times ϒ.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

Fig. 14.
Fig. 14.

As in Fig. 12, but for (a) axisymmetricity of cloud-top height averaged inside the 200-km radius, (b) bulk dimensionless parameter ϒb defined in Eq. (5), and (c) Rob times ϒb, where Rob is bulk Rossby number using the radius of 100 km rather than RMW.

Citation: Journal of the Atmospheric Sciences 75, 2; 10.1175/JAS-D-17-0177.1

Figure 12a shows the histogram of TC intensification rate in the ensemble experiments. The distribution for RI samples has an ensemble mean of 1.0 m s−1 h−1, whereas that for non-RI samples has a peak at slightly positive value ~0.2 m s−1 h−1. Figure 12b shows the histogram of TC intensity. The RI cases clearly have much stronger intensity (~35.7 m s−1) than that for the non-RI cases (~16.9 m s−1). The two groups also have a clear difference in RMW: the RI cases tend to have small values compared with the non-RI cases (Fig. 12c).

Figure 12d shows the vortex tilt normalized by the RMW, such that values larger than 1 indicate that the magnitude of vortex tilt Δxc is larger than the RMW. The RI cases generally have small magnitude of tilt (0.16 on average) compared with the non-RI cases (0.36). It is worth noting that no RI cases exceed 0.5, indicating that RI does not occur in the present ensemble when the vortex tilt is greater than 0.5 × RMW. The axisymmetricity is larger in the RI cases than in the non-RI cases (Fig. 12e). The low-level equivalent potential temperature θe averaged below the 1-km level also shows a clear difference between the two groups.

Figures 12g and 12h show histograms of mean flow speed and vertical wind shear for the RI and non-RI cases. In each case, the mean wind is computed in a 900 km × 900 km square around the vortex center; the mean flow is absolute value of the vector mean wind from 2 to 8 km, while the wind shear is the absolute value of the vector difference between 2 and 8 km.

The histogram shows little difference for the likelihood of RI for varying values of mean flow. In contrast, the vertical wind shear is significantly larger in the non-RI cases. Particularly, few TCs experience RI when the vertical shear is higher than 7.5 m s−1 in this ensemble. This feature is more clearly observed in TCs with weaker initial intensity. Figure 12i shows a histogram of wind shear for RI events for members initialized with vortices with tangential wind of 10 m s−1. For these simulations, RI only occurs when the surrounding wind shear is less than 3 m s−1. In contrast, a histogram for only simulations initialized with tangential wind of 15 m s−1 or greater shows no differentiation of RI by wind shear (not shown).

5. Parameters for RI prediction

The present results imply that the theory for nonshear cases proposed by MT13 and MT15 can possibly be applied to sheared cyclones, in which Rossby number is a parameter that controls the onset time of RI (cf. Fig. 20 of MT15). The Rossby number is defined by
e3
where υm is the maximum tangential velocity at z = 2 km, f is the Coriolis parameter, and rm is the radius of υm. Figure 13a shows the histogram of Ro. The ensemble mean for the RI group is 14.0, which is approximately 4 times larger than the non-RI group. This is consistent with MT15 and can be simply interpreted by the results shown above: stronger TC intensity and smaller RMW are features of the RI group (cf. Figs. 12b and 12c).
Previous studies and the present results suggest that the formation of an upright vortex structure is important for initiating RI (cf. Figs. 6a, 10d, and 12d). The vorticity budget analysis indicates the formation of this upright structure is related to the strong vertical velocity associated with the convection. A parameter to represent the favorability of upright vortex formation can be represented in this way:
eq1
The parameter ϒ qualitatively represents that at a given magnitude of vortex tilt, higher symmetric structure and weaker tilt tendency are more favorable for the formation of the upright vortex. The present results indicate that factors to reduce tilt may be related to stronger convective intensity and low-level vorticity, whereas increase of tilt is simply caused by vertical wind shear. Precession and realignment are not found to play a significant role here as the contribution of horizontal vorticity advection is negligibly small. Using the parameters shown in the present study, ϒ can be defined as
e4
where is the normalized difference in vortex center between the surface and 8 km, is the normalized radial difference in equivalent potential temperature in the boundary layer, ζ is the vertical absolute vorticity at 2-km altitude, and is the vertical shear of background velocity between z = 2 and 8 km. Note that the radial difference in θe is included as an indicator of convective instability, because we have observed in this study and also other cases that θe increases inside the RMW in the lower layer, while θe in the middle troposphere is not significantly changed before RI (e.g., MT13; MT15). In such a case, it is suggested that an increase in θe inside the RMW from the environment at the same altitude simply represents an increase in vertical gradient of θe in the core region and hence convective instability. The parameter ϒ qualitatively indicates the tendency for formation of an upright vortex and hence indicates that larger is more favorable for RI. The histogram of ϒ is shown in Fig. 13b, in which most of the quantities were calculated in the same way as those in Fig. 12, while ζ was approximated by υm/rm + f. The difference in ensemble mean between the RI and non-RI groups is remarkable: the mean is 157.7 for the RI group, while it is 5.1 for the non-RI group.

As seen in Fig. 12, both the tilt and shear in Eq. (4) are significantly smaller in the RI group. It is indicated that the tilt is forced by the shear and RI likely occurs as long as the tilt is weak. Whereas the vertical wind shear is fixed throughout the simulation in the present ensemble experiments, the temporal change of vertical shear might be important as a predictor since it changes the tilt.

We combine the two dimensionless parameters, Ro and ϒ. The former represents the likelihood of RI based on the cyclone size and intensity, whereas the latter indicates the tendency for the formation of an upright vortex. Figure 13c shows the histograms of Roϒ. The RI group has an ensemble mean of 3.6 × 103 that is more than 50 times larger than the mean of non-RI group (0.04 × 103). The 99% quantile of ϒ is 2.1 × 103 in the non-RI group. It suggests that the proposed dimensionless parameter Roϒ can predict the occurrence of RI in the next 24 h.

Unfortunately, many of the parameters in Eqs. (3) and (4) are difficult to observe in real time. For practical use, we modify the parameters by replacing the RMW by a fixed radius of 100 km and the axisymmetricity of PV by that of cloud-top height . In the statistical RI index of Kaplan and DeMaria (2003) and Kaplan et al. (2010), the parameter cloud-top standard deviation (CTSD) is computed from the standard deviation of brightness temperature as routinely observed by satellites. Here, we use an axisymmetricity of cloud-top height rather than CTSD; hence, larger values represent more symmetric cloud structure. We define the cloud-top height as the highest altitude at which the mixing ratio of condensates exceeds 0.05 g kg−1. The tilt can possibly be estimated from differences in the center of condensate field as seen from 85- and 37-GHz channels in microwave satellites. Figure 14a shows a histogram of averaged inside the 200-km radius and that the RI group has larger values than the non-RI group. Equation (4) may be rewritten as
e5
The histogram for ϒb is shown in Fig. 14b. Despite the fact that ϒb is an approximated version of ϒ, it does differentiate the RI and non-RI cases. Combining with Ro but using the radius of 100 km instead of RMW (so notation is changed to Rob), the parameter Robϒb is much larger in the RI group, 1.7 × 103 as compared to 0.01 × 103 in the non-RI group (Fig. 14c).

6. Conclusions

We examined structural changes preceding RI in TCs simulated by a three-dimensional full-physics model with various intensities and sizes of the initial vortex, vertical wind shear, and background velocity. The analyses for the control simulation (a member of the ensemble) with moderate vertical shear revealed that the onset of RI corresponds to the formation of strong axisymmetric heating region around the RMW. The formation of the symmetric heating region was well correlated with the approach of the RMC toward the RMW, which indicates that the symmetric component of low-level convergence is strong immediately inside the RMW where the symmetric heating region forms. These processes for the moderate-shear case are consistent with the weakly sheared cases examined in MT15.

The vortex tilt decreased until about 6 h before the onset of RI. RI almost always started after the TC attained an upright vortex structure. A vorticity budget analysis revealed that the stretching term played an important role in increasing vorticity in the upper layer around the low-level center. The vertical advection and tilting terms also had nonnegligible magnitude. The stretching, tilting, and vertical advection terms were accompanied by strong vertical velocity in the convective region. The horizontal advection term was negligibly small compared with the other three terms, indicating that the upright structure of the simulated TC was achieved not by vortex alignment but by the intensification of vorticity associated with strong convective updrafts in the upper level.

In the ensemble simulations, 119 members out of 270 simulations experienced RI. The results across the ensemble were qualitatively similar to those of the control simulation. The ensemble average of RI cases had two distinct intensification phases, consistent with observed strong TCs. The RMW and the vortex tilt decreased with time, while the equivalent potential temperature θe in the lowest 1 km and the RMC increased before the onset of RI. The vorticity budget in the ensemble mean showed that the formation of an upright vortex before RI was a result of the stretching, the vertical advection, and the tilting term. As in the control simulation, the horizontal advection term was negligible.

We proposed a dimensionless parameter ϒ, Eq. (4), that represents the tendency to form an upright vortex. In particular, the parameter ϒ consists of the axisymmetricity (representing the extent to which the vortex is axisymmetric), the inverse of vortex tilt, and the ratio of radial difference in θe times the low-level vertical vorticity to the vertical wind shear that tends to increase tilt. Consistent with MT15, the Rossby number was large in the samples experiencing the RI in the subsequent 24 h compared with the samples without the RI. In addition, ϒ is much larger in the RI group than in the non-RI group. The proposed dimensionless parameter ϒ multiplied by Ro was more than 50 times larger when the simulated TCs experienced RI in the subsequent 24 h. The proposed parameter was approximated by replacing the quantities that are difficult to observe in practice. Using the standard deviation of cloud-top height from the azimuthal mean and neglecting the RMW and the radial difference in θe, the parameter combined with Rossby number is also significantly larger in the RI group than in the non-RI group.

Whereas the ensemble simulations cover a wide range of parameters, they do not consider some processes that might play a role in initiating RI. Incorporating the processes such as radiation and assessing their effect remains for future work. The impact of moisture in the environment should be examined in a future study, because the present study utilized Dunion (2011) tropical sounding for all the simulations and moisture can strongly affect the persistence of deep convection by entrainment (Zawislak et al. 2016; Rogers et al. 2016; Nguyen et al. 2017). The parameter developed should be tested by using data from real-case simulations or operational forecasts.

Acknowledgments

Y. Miyamoto was partly supported by JSPS Scientific Research 26-358 for the JSPS Fellowship program for overseas researchers. D. Nolan was supported by NASA through Grant NNX16AP19G. The numerical simulations were performed at the Center for Computational Sciences at the University of Miami. The authors are grateful for fruitful discussions with Drs. Rob Rogers, John Kaplan, Hua Chen, and Leon Nguyen and also thank two anonymous reviewers for providing comments and suggestions.

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