Revisiting the Balanced and Unbalanced Aspects of Tropical Cyclone Intensiﬁcation

AUGUST 2017 H E N G E T A L . 2575

JUNYAO HENG Paciﬁc Typhoon Research Center, and Key Laboratory of Meteorological Disaster, Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, China

YUQING WANG State Key Laboratory of Severe Weather, Chinese Academy of Meteorological Sciences, China Meteorological Administration, Beijing, China, and International Paciﬁc Research Center, and Department of Atmospheric Sciences, School of Ocean and Earth Science and Technology, University of Hawai’i at Manoa, Honolulu, Hawaii

WEICAN ZHOU Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science and Technology, Nanjing, China

(Manuscript received 16 February 2017, in ﬁnal form 13 May 2017)

ABSTRACT

The balanced and unbalanced aspects of tropical cyclone (TC) intensification are revisited with the balanced contribution diagnosed with the outputs from a full-physics model simulation of a TC using the Sawyer–Eliassen (SE) equation. The results show that the balanced dynamics can well capture the secondary circulation in the full-physics model simulation even in the inner-core region in the boundary layer. The balanced dynamics can largely explain the intensification of the simulated TC. The unbalanced dynamics mainly acts to prevent the boundary layer agradient flow in the inner-core region from further intensification. Although surface friction can enhance the boundary layer inflow and make the inflow penetrate more inward into the eye region, contributing to the eyewall contraction, the net dynamical effect of surface friction on TC intensification is negative. The sensitivity of the balanced solution to the procedure used to ensure the ellipticity condition for the SE equation is also examined. The results show that the boundary layer inflow in the balanced response is very sensitive to the adjustment to inertial stability in the upper troposphere and the calculation of radial wind at the surface with relatively coarse vertical resolution in the balanced solution. Both the use of the so-called global regularization and the one-sided finite-differencing scheme used to calculate the surface radial wind in the balanced solution as utilized in some previous studies can significantly underestimate the boundary layer inflow. This explains why the boundary layer inflow in the balanced response is too weak in some previous studies.

1. Introduction the upper troposphere, can be considered as a response to diabatic heating in the eyewall and the momentum Tropical cyclones (TCs) can be viewed as a quasi- forcing mainly as a result of surface friction (Shapiro and axisymmetric primary circulation superimposed by a Willoughby 1982). The secondary circulation can bring thermally and frictionally driven secondary circulation. large absolute angular momentum (AAM) inward in the While the primary circulation remains nearly in gradient lower troposphere to spin up the tangential wind and wind balance as a ‘‘slowly evolving’’ system, the sec- thus lead to the intensification of a TC (Shapiro and ondary circulation with radial inflow in the lower tro- Willoughby 1982; Schubert and Hack 1982; Pendergrass posphere, upward motion in the eyewall, and outflow in and Willoughby 2009). This spinup process was first explored based on the balanced dynamics: namely, the Sawyer–Eliassen (SE) equation (Eliassen 1951). The Corresponding author: Prof. Yuqing Wang, [email protected] theoretical studies have shown that TC intensification

DOI: 10.1175/JAS-D-17-0046.1 Ó 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 09/30/21 11:42 AM UTC 2576 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 can be well explained by the balanced dynamics The argument on the positive contribution by surface (Schubert and Hack 1982; Shapiro and Willoughby 1982; friction and the associated unbalanced dynamics to TC Pendergrass and Willoughby 2009; Fudeyasu and intensification seems to be consistent with the positive Wang 2011). In addition to the theoretical studies, the contribution by the unbalanced flow in the boundary SE equation has also been applied to diagnose the layer to TC maximum intensity (MI) in some previous processes of TC intensification from the simulations studies. Smith et al. (2009) speculated that the gradient and observations and has been shown to be able to wind imbalance in the inflow boundary layer could reproduce the secondary circulation during the in- substantially increase TC MI. Bryan and Rotunno tensification period, even in the boundary layer, where (2009) developed a diagnostic model for TC MI by in- the balanced assumption is invalid (Molinari and corporating the effect of unbalanced flow in the Vollaro 1990; Molinari et al. 1993; Möller and Shapiro boundary layer and found that the unbalanced flow 2002; Persing et al. 2002; Hendricks et al. 2004; could contribute to TC MI by 5%–30%. Frisius et al. Montgomery et al. 2006). Therefore, these theoretical (2013) extended the maximum potential intensity (MPI) and diagnostic studies have demonstrated that the in- theory of Emanuel (1986, 1995) by including the effect tensification of a TC can be largely explained by the of unbalanced flow in a slab boundary layer model and balanced dynamics. found that the maximum TC intensity can be up to 18% Bui et al. (2009) have criticized the classic un- higher than that from the MPI that ignores the understanding of TC intensification based on balanced balanced effect. Note that both studies of Bryan and dynamics, as their balanced solution using the SE Rotunno (2009) and Frisius et al. (2013) defined the equation considerably underestimated the boundary maximum intensity as the maximum wind speed in layer inflow in their full-physics model simulation. They the interior of the boundary layer: namely, including the thus concluded that the balanced dynamics significantly supergradient wind component, which is often about underestimates the boundary layer inflow and thereby 10%–20% of the gradient wind near the top of the inflow the spinup of tangential wind in the inner-core region, boundary layer in TCs (Kepert and Wang 2001; Kepert and thus the unbalanced dynamics should be largely 2006; Schwendike and Kepert 2008). Since the super- responsible for TC intensification. Motivated by the gradient wind is well above the surface, it does not di- results of Bui et al. (2009), Smith et al. (2009) argued rectly contribute to the surface energy production or loss that the balanced mechanism spins up the TC vortex and the TC intensity, which is measured by the near- above the boundary layer and increases the TC size. The surface wind speed. inner-core spinup in the boundary layer is largely at- Stern et al. (2015) challenged the hypothesized tributed to the unbalanced dynamics associated with positive contribution of surface friction to TC in- surface friction. Surface friction can break down gradi- tensification proposed by Smith et al. (2009) based on ent balance and enhance the boundary layer inflow. In the results from a linearized vortex model, the Three- their viewpoint, the loss of AAM due to surface friction Dimensional Vortex Perturbation Analysis and Sim- is less than the radial transport of AAM due to fric- ulation (3DVPAS) of Nolan and Montgomery (2002). tionally induced inflow. In this case, although the AAM They found that the effect of surface friction is only is not conserved, the local tangential wind would still significant in the boundary layer with a strengthened increase in the inner core in the boundary layer, result- inflow layer and a corresponding shallow outflow layer ing in a low-level jet in tangential wind with super- immediately above. Although surface friction can gradient nature near the radius of maximum wind substantially enhance the boundary layer inflow, the (RMW). They seemed to suggest that the occurrence net dynamical effect of surface friction is negative of unbalanced supergradient wind in the interior of because the positive tangential wind tendency as a the boundary layer as a result of surface friction plays result of frictionally induced inflow could not offset an important role in spinning up the inner core of a the direct spindown by surface friction. Note that TC. This argument gave the impression that surface Stern et al. (2015) also confirmed the importance of friction and its associated unbalanced processes can vertical shear of tangential wind in the boundary layer dominate the balanced dynamics in spinning up the to the frictionally induced inflow in the balanced re- TC in the inner core in the boundary layer. This is in sponse, as shown in Bui et al. (2009). Smith and sharp contrast to the traditional view that surface Montgomery (2015) criticized that the linear model friction is the major energy sink of a TC system, thus used in Stern et al. (2015) has its limitations and could contributing negatively to TC intensification and not be used to access the spinup mechanism in the maximum intensity (Emanuel 1989; Raymond et al. boundary layer because the processes therein are in- 1998; Kepert 2010). trinsically nonlinear.

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC AUGUST 2017 H E N G E T A L . 2577

To include the nonlinearity, Heng and Wang (2016a) numerical solution of the SE equation. In section 3, the used a nonlinear model, the Tropical Cyclone Model, balanced contribution to the intensification of the sim- version 4 (TCM4), to study the effects of surface friction ulated TC is diagnosed using the SE equation. The and the associated unbalanced dynamics on TC insensitivity of the balanced solution to the procedures tensification. They conducted two idealized numerical used to ensure the ellipticity condition of the SE equa- experiments: one without surface friction and the other tion is examined in section 4. Major findings and con- with surface friction. To isolate the dynamical effect of clusions are given in the last section. surface friction, diabatic heating in the eyewall was prescribed so that the possible feedback of surface friction to diabatic heating was excluded in the experi- 2. Methodology ment with surface friction. The results showed that The quadruply nested, fully compressible, non- surface friction has a net negative contribution to TC hydrostatic TCM4 was used to perform an idealized intensification, in agreement with Stern et al. (2015). numerical simulation of a TC. A complete description of They also found that the unbalanced dynamics due to TCM4 can be found in Wang (2007) and it has been used surface friction acts as a process to prevent the further in studies on TC structure and intensity changes, in- enhancement of agradient flow in the frictional bound- cluding Wang (2008a,b), Wang (2009), Xu and Wang ary layer by spinning up tangential wind in the surface (2010a,b), Li et al. (2015), and Heng and Wang (2016a). layer near the RMW, where the flow is strongly sub- All model settings, including the initial conditions, were gradient, and by spinning down immediately above identical to those used in the control experiment in where the flow is strongly supergradient. Heng and Wang and Heng (2016). To avoid any duplication, the Wang (2016a) also noticed that, although surface fric- readers are referred to Wang and Heng (2016) for de- tion has an overall net negative dynamical effect on TC tails. The model was run for 168 h, with the model out- intensification, it plays a critical role in producing the puts at 6-min intervals for diagnostic analysis. realistic boundary layer structure, such as the enhanced To evaluate the balanced and unbalanced contribu- inflow and supergradient wind near the top of the inflow tions to TC intensification, the SE equation was applied boundary layer near the RMW. Smith and Montgomery to calculate the balanced response from the model (2016) criticized the work of Heng and Wang (2016a) outputs. We used the SE equation in height coordinates and argued that the experiments with and without sur- as given in Bui et al. (2009). The diagnostic equation for face friction were not adequate to isolate the frictional streamfunction in the radius–height plan was given in effect on TC intensification, and the presence of super- Eq. (2) in Heng and Wang (2016a), and a full description gradient winds in the simulation with surface friction of every quantity in Eq. (2) can be found therein as well. implies the importance of surface friction to TC in- Given the azimuthal-mean primary circulation, which tensification. Heng and Wang (2016b) argued that the was assumed in gradient wind and hydrostatic balances supergradient wind is well above the surface, results as a (except for in the frictional boundary layer), the SE fast adjustment (response) to surface friction, exists in equation was solved numerically with radial grid spacing all stages of a TC, and thus could not be a justification of of 2.5 km and vertical grid spacing of 250 m in a domain the unbalanced contribution to TC intensification. Heng extending from the storm center to a radius of 300 km and Wang (2016b), however, also indicated that they did and from the surface to a height of 18 km. The ellipticity not consider any possible feedback to eyewall convec- condition for convergent numerical solutions requires tion due to the presence of surface friction since the the discriminant D . 0, where heating rate in their numerical experiments was prescribed. ›x ›x › 2 D 52g jx(§ 1 f ) 1 C 2 (xC) , (1) This study is an extension of the study by Heng and ›z ›r ›z Wang (2016a) and revisits the dynamical effects of surface friction and its associated unbalanced processes on where r and z are radius and height, g is the gravitational TC intensification in a full-physics model simulation. acceleration, x 5 1/u, where u is potential temperature, Different from Heng and Wang (2016a), the balanced C 5 y2/r 1 f y denotes the sum of centrifugal and Coriolis and unbalanced contributions to TC intensification in forces, and j 5 2y/r 1 f , with f being the Coriolis pa- this study are analyzed based on the full-physics model rameter. As in previous studies, there were some grid simulation in which diabatic heating contains the feed- points where the ellipticity condition was not satisfied in back of the unbalanced dynamics in the frictional our simulation. To assure the ellipticity of the SE boundary layer. The rest of the paper is organized as equation, we first set h 5 0:01f0 at points where follows. Section 2 briefly describes the TCM4 and the h 5 f0 1 z , 0:01f0 to remove any points where the

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC 2578 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74

averaged in hour 50 and hour 70 of simulation, respectively. As we can see from Figs. 2a and 2b, although the storm intensified rapidly from 50 to 70 h, the RMW of the storm showed little contraction and was located at around 20 km from the storm center. The tangential wind shows a much stronger and deeper primary circulation at 70 h than at 50 h. The corresponding temperature anomaly shows the warm-core structure with the maximum temperature anomaly of 3 K at 50 h and over 5 K at 70 h in the mid- to upper troposphere. The cold anomaly under and outside the eyewall in the lower troposphere (Figs. 2c,d) was due to evaporation of rain and melting of snow and graupel therein. All other dy- FIG. 1. Time evolution of the maximum azimuthal-mean tan- 2 namical parameters that appeared in the SE equation [see gential wind speed (m s 1) at the lowest model level (26.5 m above Eq. (1)], including the azimuthal-mean density, potential the sea surface) in the TCM4 simulation. temperature, absolute vertical vorticity, and inertial stability, were obtained directly from the model outputs. absolute vertical vorticity was negative. If the ellipticity The azimuthal-mean diabatic heating rate and mo- condition was still not satisfied at any grid points, the mentum forcing from the model outputs at the two vertical shear of tangential wind [›(xC)/›z] at those grid chosen times are shown in Fig. 3. At 50 h, the heating points was then reduced by iterations. Namely, the rate covers a broad region between 15 and 100 km from vertical shear of tangential wind was reduced by a factor the storm center and has two maxima in the middle of 0.8 at those points where D # 0. The ellipticity con- troposphere: one is near a radius of 20 km in the eyewall dition was then rechecked. If the ellipticity condition and the other is near a radius of 70 km related to the was still not satisfied at some grid points, the vertical outer rainbands (Fig. 3a). At 70 h, the maximum heating shear of tangential wind was reduced by a factor of 0.8 2 rate in the eyewall increases to over 20 K h 1, more than only at those points where D # 0. Similar iterations were 3 times that at 50 h. The heating rate in the outer region repeated until the ellipticity condition was satisfied at all remains similar to that at 50 h. The momentum forcing is grid points (but limited not to exceed eight iterations in dominated by the sink associated with surface friction our case discussion in sections 3 and 4). Similar adjust- within a shallow layer near the surface at the two given ment procedures were also used in Wang et al. (2016) times (Figs. 3c,d). The maximum negative value is and in Heng and Wang (2016a). In addition, in our cal- 2 2 2 2 about 215 m s 1 h 1 at 50 h and about 230 m s 1 h 1 at culations, the radial wind at the surface (z 5 0) was 70 h near the RMW. Some positive values inside the extrapolated based on the vertical shear of radial wind RMW above the boundary layer are due to eddy mo- from the model outputs between the 125-m height and mentum mixing (both horizontal and vertical). the lowest model level at 26.5 m. Figure 4 compares the radial wind and vertical motion from the model simulation with those diagnosed from 3. Diagnostics of the balanced contribution the SE equation at 50 and 70 h. The balanced solution Figure 1 shows the time evolution of the maximum captures the secondary circulation in the model simu- azimuthal-mean tangential wind speed at the lowest lation quite well in both the pattern and magnitude model level in the simulation. After about a 26-h initial throughout the troposphere, even in the boundary layer adjustment period during which the inner-core air col- (Figs. 4a–d). The similarity between the balanced solu- umn was moistened by the surface fluxes, the TC vortex tion and the model simulation can be seen more clearly intensified rapidly from 26 to 91 h with a mean in- from the area-averaged radial wind speed within a ra- 2 2 tensification rate of 18.7 m s 1 day 1.AsinBui et al. dius of 160 km shown in Figs. 4e and 4f. This is in sharp (2009), we chose the weak stage of the storm at 50 h of contrast to Bui et al. (2009), who showed that the bal- simulation when the storm had the maximum azimuthal- anced solution could only capture less than half of the 2 mean tangential wind of 26 m s 1 and the strong stage at boundary layer inflow in the full-physics model simula- 70 h of simulation when the storm had its maximum tion in their calculation (this matter will be further dis- 2 azimuthal-mean tangential wind of 42 m s 1; both times cussed in section 4). Note that the maximum boundary were in the rapid intensification period. layer inflow in the balanced solution occurs at the radius Figure 2 shows the azimuthal-mean tangential wind of 40 km, which is about 10 km outside of that from the and temperature anomaly of the simulated storm model simulation at both times, implying that the

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC AUGUST 2017 H E N G E T A L . 2579

21 FIG. 2. The radius–height cross sections of the azimuthal-mean (a),(b) tangential wind speed (m s ) and (c),(d) temperature anomaly (8C) averaged at (a),(c) 50 and (b),(d) 70 h of simulation. balanced dynamics underestimates the inward penetra- momentum forcing only. This is similar to what was tion of the boundary layer inflow into the eye region, or, done in Rozoff et al. (2012) in their explanation of the alternatively, the unbalanced dynamics contributes to respective roles of diabatic heating and momentum inward penetration of boundary layer inflow into the eye forcing in secondary eyewall formation in a full-physics and thus the contraction of the RMW, which is consis- model simulation using the linearized vortex model tent with the finding in Heng and Wang (2016a). Note 3DVPAS (Nolan and Montgomery 2002), as also used in also that the radial wind in the balanced solution near Stern et al. (2015). Here, we discuss results at 70 h of the surface is slightly stronger than that in the TCM4 simulation only, since results from other times are con- simulation, in particular at 70 h (Figs. 4d,f). This is most sistent (Figs. 3b,d). Diabatic heating drives a deep inflow likely a result of the extrapolation of the radial wind at layer in the mid- to lower troposphere and a broad the surface with the same reduction factor as that be- outflow layer in the upper troposphere (Fig. 5a). The 2 tween 125 and 26.5 m at the corresponding radius in the maximum heating-induced inflow reaches 6 m s 1 near full-physics TCM4 simulation. This may cause some the surface at a radius of 40 km. Meanwhile, the bal- errors, since the radial distribution of radial wind is anced upward motion forced by diabatic heating is slightly shifted outward in the balanced solution, as comparable to that of the model simulation (Figs. 4d, 5c), mentioned above. This also indicates that the in- suggesting that diabatic heating is a major driving force terpretation of the results in the lowest 70-m layer for the deep secondary circulation, including part of the should be done with caution. strong inflow in the boundary layer. This is consistent Because of the linearity of the SE equation, contri- with the full nonlinear solution shown in Fig. 3a in Heng butions to the secondary circulation by the azimuthal and Wang (2016a). The solution of the SE equation mean diabatic heating and momentum forcing can be with momentum forcing shows a boundary layer inflow separately calculated. Namely, the respective balanced that penetrates more inward into the eye region. The solutions can be derived by solving the SE equation maximum frictionally induced inflow reaches about 2 twice: once with diabatic heating only and once with 5ms 1 near the surface at a radius of 27.5 km (Fig. 5b).

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC 2580 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74

21 FIG. 3. The radius–height cross sections of the azimuthal-mean (a),(b) diabatic heating rate (K h ) and (c),(d) 2 2 momentum forcing (m s 1 h 1) derived from the TCM4 simulation averaged at (a),(c) 50 and (b),(d) 70 h.

Note that the response to the momentum forcing pri- storm, we performed the tangential wind tendency marily due to surface friction only occurs in the boundary budget at 70 h of simulation. The tangential wind ten- layer (Figs. 5b,d) with weak upward motion in the eyewall dency equation in the cylindrical coordinates can be region and only reaches about 2 km where a weak outflow written as (Xu and Wang 2010a,b; Heng and Wang layer associated with supergradient wind occurs. 2016a) The results discussed above demonstrate that the secondary circulation with a deep inflow layer in the ›y ›y ›y0 52u h 2 w 2 u0h0 2 w0 1 F , (2) mid- to lower troposphere, eyewall updraft in the region ›t ›z ›z y of diabatic heating, and the outflow layer in the upper troposphere is mainly driven by diabatic heating in the where t is time and z is height; u, y, and w are radial, eyewall. Although surface friction explains roughly tangential, and vertical winds; and h is vertical absolute 40%–50% of the inflow in the lower part of the vorticity. The overbar denotes the azimuthal mean, and boundary layer, the associated secondary circulation is the prime denotes the deviation from its azimuthal very shallow, with the forced upward motion occurring mean. The azimuthal-mean surface friction of tangential only below 2-km height in the eyewall. Even though the wind together with vertical mixing and horizontal dif- boundary layer was well coupled with the free atmo- fusion of tangential wind is denoted by Fy. The five sphere above with a full nonlinearity in TCM4, the linear terms on the right-hand side of Eq. (2) are, respectively, diagnostics based on the SE equation can still largely the azimuthal-mean radial advection contributed by the recover the diabatic heating and surface-friction- radial flux of absolute vertical vorticity, vertical advec- induced secondary circulation to a large extent. Thus, tion of azimuthal-mean tangential wind by the this strongly suggests that the SE equation is useful for azimuthal-mean vertical velocity, the corresponding understanding TC intensification processes. eddy radial advection and eddy vertical advection, and To investigate the balanced and unbalanced contri- the vertical mixing (including surface friction) and hor- butions to the spinup of tangential wind in the simulated izontal diffusion of tangential wind.

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC AUGUST 2017 H E N G E T A L . 2581

21 FIG. 4. The radius–height cross sections of the azimuthal-mean radial wind (contour interval of 1.0 m s ) and 2 vertical motion (shading; m s 1) (a),(c) derived from the TCM4 output and (b),(d) diagnosed from the SE solution at (a),(b) 50 and (c),(d) 70 h of simulation. The vertical proﬁles of the radial wind averaged within the radius of 160 km in the TCM4 output (blue) and in the SE solution (red) at (e) 50 and (f) 70 h are also shown.

Figure 6 compares the tendencies of the azimuthal- is only important near and inside the RMW below mean tangential wind because of the total (both hori- 2-km height. zontal and vertical) advection calculated directly from To explore the frictional effect on TC intensiﬁcation the full-physics model simulation and from the balanced in this full-physics model simulation, the tangential wind solution at 70 h of simulation. Overall, the balanced tendencies as a result of diabatic heating only and mo- solution reproduces the major feature of the tangential mentum forcing only, respectively, are examined based wind budget in both the spatial pattern and magnitude on the balanced SE solutions (Figs. 6c,d). The total ad- (Figs. 6a,b), in agreement with the similarity in the vection due to diabatic heating spins up tangential wind secondary circulation, as shown in Figs. 4c and 4d. Note near and outside the RMW while it spins down tan- that some differences between the tendencies calculated gential wind inside the RMW. The frictionally induced from the model output and from the balanced solution secondary circulation contributes to the spinup of tan- appear near and inside the RMW below 2-km height, gential wind below 1 km while it spins down tangential where the ﬂow is strongly subgradient/supergradient wind immediately above (Fig. 6d). Note that the tan- (Fig. 6f). This suggests that the unbalanced contribution gential wind tendency due to advection by the

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC 2582 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74

21 FIG. 5. The radial–height cross sections of the azimuthal-mean (a),(b) radial wind (m s ) and (c),(d) vertical 2 motion (m s 1) from the balanced SE solution (a),(c) with diabatic heating rate only and (b),(d) with momentum forcing only at 70 h of simulation. frictionally induced inflow is considerably larger than between the total tendencies calculated from the model that by heating-induced inflow in the boundary layer, output and those calculated from the balanced flow,1 as although the inflow induced by the former is weaker in Heng and Wang (2016a). We can see that the un- than that induced by the latter. This is because the balanced dynamics acts to spin up tangential winds in frictionally induced inflow penetrates more into the eye the lower part of the inflow boundary layer while it spins region, where the absolute vorticity is much higher than down tangential winds near the RMW immediately that outside the RMW, leading to larger advective tan- above and outside the RMW in the surface layer. The gential wind tendency. However, the large positive unbalanced contribution to the tangential wind budget tangential wind tendency induced by the frictionally above the boundary layer is negligible. This is consistent induced boundary layer inflow (Fig. 6d) is still smaller with the results in Heng and Wang (2016a), who sug- than the negative tendency due to surface friction itself gested that the unbalanced dynamics due to the pres- under and outside the eyewall, leading to a net negative ence of surface friction contributes to a spinup of tangential wind tendency near the RMW in the bound- tangential wind in the surface layer near the RMW ary layer (Fig. 6e). The positive tangential wind ten- where the flow is strongly subgradient and a spindown dency centered at about 0.5-km height in the eye region immediately above where the flow is strongly super- results mainly from the inward penetration of the gradient. Figure 7 shows the total frictional effect, which boundary layer inflow into the eye. Since the positive is defined as the sum of the unbalanced contribution tendency is well inside the RMW, it does not contribute and the net frictional effect, since the unbalanced to the intensification of the storm but spins up the cy- clonic circulation in the eye region. This demonstrates that the net dynamical effect of surface friction on TC 1 Note that the differences plotted in Fig. 6f also include those intensification is negative, and the balanced response to caused by numerical approximations arising from the transform diabatic heating in the eyewall (including considerable from Cartesian to cylindrical coordinates, the azimuthal average, modifications by the presence of surface friction and its the different grid spacing, and different finite-difference operators related unbalanced dynamics) is the sole mechanism of in TCM4 compared to those in the Sawyer–Eliassen calculation. As a result, Fig. 6f shows not just the effects of the unbalanced flow, TC intensification (Heng and Wang 2016a,b). but also these numerical issues. Nevertheless, the overall feature is Figure 6f shows the unbalanced contribution to the largely contributed by the involved physics rather than by numer- tangential wind budget, which is defined as the residual ical errors, as also shown in Heng and Wang (2016a).

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC AUGUST 2017 H E N G E T A L . 2583

21 21 FIG. 6. The radial–vertical cross sections of the azimuthal-mean tangential wind tendencies (m s h ) due to total mean advection at 70 h of simulation (a) from the TCM4 model output and (b) from the SE solutions with both diabatic heating and momentum forcing, (c) with diabatic heating rate only, and (d) with momentum forcing only. (e) The net frictional effect [defined as the sum of the advective tendency momentum forcing in (d) and surface friction itself from the model output in Fig. 3d]. (f) The residual. The difference between the total tendencies calculated (g) from the model outputs and (h) from the SE balanced solution. contribution resulted primarily from surface friction Heng and Wang (2016a), but in contrast to the hypoth- (Heng and Wang 2016a). The negative tangential wind esis of Bui et al. (2009) and Smith et al. (2009). tendency near and outside the RMW implies that sur- Recently, Smith and Montgomery (2016) criticized face friction and the associated unbalanced dynamics the work of Heng and Wang (2016a) in that the exis- play a role in spinning down the simulated storm. The tence of supergradient winds in Heng and Wang (2016a) positive tangential wind tendency inside the RMW in- reflected the positive contribution of unbalanced dy- dicates that surface friction and the associated un- namics to TC intensification, while Heng and Wang balanced dynamics contribute to the eyewall contraction (2016b) responded that the supergradient wind results of the storm and also drive the tangential wind inside the from a quick adjustment of a balanced vortex to surface RMW from U shaped to V shaped. This is also in friction and did little to the intensification of their sim- agreement with the results of Stern et al. (2015) and ulated storm. To isolate the possible feedback of surface

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC 2584 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74

FIG. 7. The radial–vertical cross section of the sum of the net frictional effect (Fig. 6e) and the unbalanced residual (Fig. 6f), showing the overall unbalanced frictional effect on tangential wind 2 2 tendency (contour interval of 1 m s 1 h 1), with shading showing the FIG. 8. Grid points that are not satisﬁed by the ellipticity condition 2 # azimuthal-mean tangential wind (m s 1) at 70 h of simulation. criterion, namely, with D 0 [gray shading; see Eq. (1)].

adjustment to the inertial instability [related to the first friction to diabatic heating, Heng and Wang (2016a) term on the rhs in Eq. (1)], mostly in the outflow layer, prescribed eyewall heating in their simulations. Here, and the other is the adjustment to vertical shear of we have included full moist processes and gotten similar tangential wind [related to the second term on the rhs in results, indicating that the main conclusions in Heng and Eq. (1)], mostly in the frictional boundary layer (Fig. 8). Wang (2016a,b) remain valid. Namely, surface friction We also evaluated the possible effect of a bug in the code and the associated unbalanced dynamics contribute for vertical shear adjustment used in Bui et al. (2009).In negatively to TC intensification. The unbalanced dy- addition, we examined the effect of the one-sided finite- namics acts to spin up tangential wind in the surface differencing scheme used to calculate the surface radial layer near the RMW where the flow is strongly sub- wind utilized in Bui et al. (2009) and also later in Abarca gradient and to spin down tangential wind immediately and Montgomery (2014). above where the flow is strongly supergradient. This can also be seen from the total budgets of tangential wind a. Adjustment to inertial stability calculated from the TCM outputs (Fig. 6g) and from the Because of the development of the anticyclonic cir- SE solution (Fig. 6h). The above results thus strongly culation in response to the outflow in the upper tropo- suggest that, even though surface friction contributes to sphere, weak symmetry instability often occurs in the 40%–50% of the total boundary layer inflow in the upper troposphere with small negative absolute vertical inner-core region and thus substantially enhances posi- vorticity (Möller and Shapiro 2002; Hendricks et al. tive tangential wind tendency near the RMW in the 2004; Fudeyasu and Wang 2011; Li and Wang 2012). The boundary layer, the positive tendency is not large negative inertial stability leads to a negative discrimi- enough to offset the negative tangential wind tendency nant for the ellipticity condition in some grid points. To directly induced by surface friction near and outside the relax the negative discriminant D in Eq. (1), a regula- RMW. This is also in agreement with the results of Stern rization process is often employed to remove those re- et al. (2015) gions with negative inertial stability. In our calculations discussed in section 3, we set the absolute vertical vor- h 5 : h 5 1 z , : 4. Sensitivity of the balanced solution to various ticity 0 01f0 at grid points where f0 0 01f0 assumptions to remove the inertial instability, the so-called local adjustment. In Bui et al. (2009), the so-called global To understand the difference in the balanced solution adjustment was used. Namely, when the parameter in this study [and also in Heng and Wang (2016a)] from 2 I 5 xj( f0 1 h) 1 C›x/›r, an analog to inertial stability, that in Bui et al. (2009), we examined the sensitivity of was negative at some grid points, a minimum value2 at the balanced response to two assumptions used in solving the SE equation. We noticed that the major difference between the two studies lies in the adjustment to 2 2 2 2 The minimum value in our case is 21.18 3 10 15 K 1 s 2. Al- ensure the ellipticity condition of the SE equation and though the absolute value is quite small compared to the values in the calculation of radial wind at the surface (z 5 0) from the inner-core region, it is comparable to the positive values in the the streamfunction of the SE equation. One is the outer-core region outside a radius of 60 km in our simulated storm.

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC AUGUST 2017 H E N G E T A L . 2585

21 FIG. 9. The radius–height cross sections of the azimuthal-mean (a),(c) radial wind (m s ) and (b),(d) vertical 2 motion (m s 1) in calculations with (a),(b) the local adjustment and (c),(d) the global adjustment, along with the 2 2 differences (global minus local) in the azimuthal-mean (e) radial wind (m s 1) and (f) vertical motion (m s 1) between the two calculations. these grid points was chosen, and 1% of its absolute calculations results mainly from the different adjust- value was then added to all grid points in the whole ments to inertial stability. computational domain to ensure the inertial stability After the global adjustment was applied, the number in the upper troposphere. Here we compare the cal- of the grid points where the ellipticity condition was not culations using the local and global adjustments at 70 h satisfied decreased greatly compared to that after the of simulation. Because there are some grids where the local adjustment was performed (not shown). However ellipticity is still not satisfied because of the large the global adjustment had a much broader effect than vertical shear of tangential wind in the lower bound- the local adjustment, since it was applied to all grid ary layer, an additional adjustment to the vertical points in the computational domain. As a result, the shearisconsideredaswell.Notethatherethesame solution with the global adjustment could considerably adjustment to vertical shear of tangential wind de- underestimate the radial inflow in the boundary layer veloped in this study as described in section 2 was and outflow in the upper troposphere. This can be applied after the adjustment to the inertial stability. clearly seen in Figs. 9a and 9c. Although the maxima of This means that the difference between the two the boundary layer inflow near the RMW are similar in

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC 2586 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 the two balanced SE solutions, both inflow in the underestimated in the lowest 500 m, where the strongest boundary layer and outflow in the upper troposphere inflow often exists (Zhang et al. 2011). were considerably weaker outside the RMW in the To examine the possible contribution of the one-sided calculation with the global adjustment than those with finite-differencing scheme used to calculate the radial the local adjustment (Fig. 9e). This is mainly because the wind at the surface to the underestimation of the global adjustment added a positive value of a small I2 at boundary layer inflow in Bui et al. (2009), we repeated all grids in the computational domain. Although the the calculation of radial wind at the surface using our added value was about three orders smaller than the one-sided finite-differencing scheme but with degraded inertial stability near the eyewall (Bui et al. 2009), it vertical resolution of 500 m. Specifically, we calculated increased the inertial stability in the outer region where the radial wind at the surface using the one-sided finite the inertial stability was much smaller and acted as a differencing of streamfunction between the surface and resistance to the radial inflow/outflow in the SE solution. 500-m height. In this case, for consistency, the radial Overall, the radial inflow/outflow was reduced by as wind at 250 m is taken as the average between the radial 2 much as over 2 m s 1, or about 20%–50% of the solution wind at 500 m and that at the surface calculated using the with the local adjustment. This is in sharp contrast to one-sided finite-differencing scheme. The radius–height that in Bui et al. (2009, p. 1720), who stated that ‘‘this cross section of radial wind below 3-km height thus procedure does not affect the general characteristics of obtained is compared with that from the TCM4 output the solution outside the regions where the regularization and that from our standard calculation with the local is applied.’’ Unlike the large difference in the radial adjustment for inertial stability in Fig. 10. As we already wind, the difference in vertical motion is relatively small mentioned in section 3, the maximum boundary layer (Figs. 9b,d,f). inflow in the balanced solutions (Figs. 10b,c) occurs about 10 km outside of that from the TCM4 simulation b. Calculation of radial wind at the surface (Fig. 10a) because the balanced dynamics considerably Another factor potentially contributing to the un- underestimates the inward penetration of the boundary derestimation of the boundary layer inflow in Bui et al. layer inflow into the eye region. Compared with our (2009) is the one-sided finite-differencing scheme used algorithm, the one-sided finite differencing considerably to calculate the radial wind at the surface (z 5 0) from underestimates the radial wind below 250 m in the inner- the streamfunction of the SE solution.3 We examined core region within a radius of 70 km (Fig. 10e). such a possibility in our calculation and found that this The above results demonstrate that the global ad- effect is marginal, with the vertical resolution of 250 m justment to inertial stability can also lead to the un- used in our balanced solution. This is because, with the derestimation of radial wind in the lower part of the vertical resolution of 250 m, the radial wind at the sur- boundary layer mainly outside the inner core (Fig. 9e) face calculated using the one-sided finite-differencing and the one-sided finite-differencing scheme used to scheme is equivalent to that at 125-m height. Zhang et al. calculate the radial wind at the surface can lead to the (2011) found that the maximum of the radial wind in underestimation of the boundary layer inflow in the observed strong TCs was located at about 150 m above inner-core region (Fig. 10e). To see the combined effect the sea surface. Therefore, the insensitivity in our ver- of the two factors discussed above, we repeated the tical resolution of 250 m is understandable. However, we above calculation using both the one-sided finite- noticed that in Bui et al. (2009) the one-sided finite- differencing scheme and the global adjustment to inertial differencing scheme was used to calculate the radial stability, with the results shown in Fig. 10d. Now it is wind at the surface in the balanced solution with the clearly seen that the combined effect of the global ad- vertical grid spacing of 500 m. This means that the radial justment to inertial stability and the one-sided finite- wind at the surface (z 5 0) was equivalent to that at differencing scheme used to calculate the radial wind at 250-m height. Therefore, in the calculations of the surface with the vertical resolution of 500 m in Bui Bui et al. (2009), the radial wind could be considerably et al. (2009) is an underestimation of radial wind in the 2 boundary layer by as much as over 3 m s 1.Thismeans that the two factors can explain 30%–40% near the 3 The one-sided finite-differencing scheme was also used to cal- surface across the RMW (Fig. 10f). culate the radial wind at the surface (z 5 0) from the balanced solution in Abarca and Montgomery (2014) with the vertical c. Other possible effects resolution of 500 m. We thus consider that the results discussed in this subsection are also applicable to explain part of the In addition to the two factors discussed above, Bui underestimation of the boundary layer inflow in Abarca and et al. (2009) constructed the balanced vortex in thermal Montgomery (2014). wind balance for the balanced solution, including in the

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC AUGUST 2017 H E N G E T A L . 2587

21 FIG. 10. The radius–height cross sections of the azimuthal-mean radial wind (m s ) from (a) TCM4 outputs, (b) the SE solution of this study, (c) the SE solution with the local adjustment for inertial stability, but with one- sided finite differencing for the calculation of radial wind at the surface (z 5 0) using degraded vertical resolution of 500 m, as used in Bui et al. (2009), and (d) the SE solution with the global adjustment for inertial stability, but with one-sided finite differencing for the calculation of radial wind at the surface (z 5 0) using degraded vertical resolution of 500 m, as used in Bui et al. (2009). (e) The difference between (c) and (b); (f) the difference between (d) and (b). boundary layer. In this case, a cold-core structure ap- Abarca and Montgomery (2014).4 This may introduce pears in the boundary layer. This indeed can largely imbalance in the basic vortex as well and possibly reduce strengthen the vertical temperature gradient and thus the boundary layer inflow in the balanced response. static stability in the boundary layer, contributing to suppression of vertical motion and also the radial wind in the boundary layer. This has been demonstrated by Bui et al. (2009) in their appendix and also in Stern et al. 4 We found that the vertical shear of tangential wind in the (2015). We noticed that the cold temperature anomaly boundary layer at grid points that are not satisfied with the ellip- in the boundary layer was not adjusted to satisfy the ticity condition (mainly in the core region as seen in our Fig. 8) was reduced by a factor of 0.8 in Bui et al. (2009) and by 0.4 in Abarca thermal wind balance after the adjustment to vertical and Montgomery (2014) (M. T. Montgomery 2017, personal com- shear of the tangential wind in the boundary layer to munication), but both studies left the originally balanced temper- satisfy the ellipticity condition in Bui et al. (2009) and ature anomaly unchanged.

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC 2588 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74

Our additional calculations (not shown) indicate that and an outflow layer in the upper troposphere. The reducing the vertical shear of tangential wind in the balanced response to momentum forcing due to surface boundary layer can also lead to a decrease of the boundary friction is a shallow transverse circulation mainly in the layer inflow. This is consistent with the calculations using lower 3 km of the atmosphere with a strengthened inflow the solver of Bui et al. (2009) and the latest version in in the boundary layer and a weaker outflow layer im- Abarca and Montgomery (2014) (M. T. Montgomery 2017, mediately above the inflow boundary layer. Different personal communication). Therefore, we believe that, in from diabatic heating, the frictionally induced boundary addition to the two factors discussed above, both the re- layer inflow shows an inward penetration into the eye duced vertical shear of tangential wind in the boundary region inside the RMW, contributing to the contraction layer and the unbalanced large cold temperature core that of the eyewall. is not adjusted according to the corresponding reduced The azimuthal-mean tangential wind tendency due to vertical shear of tangential wind in the boundary layer may both horizontal and vertical advections calculated using account for an additional part of the underestimation of the balanced solution compares well with that using the the boundary layer inflow in Bui et al. (2009) and Abarca output of the model simulation except for the small re- and Montgomery (2014). gion across the RMW in the lowest 2 km of the model We thus conclude that the balanced dynamics can atmosphere, where the flow is strongly unbalanced and indeed reproduce much of the secondary circulation in agradient. We showed that the unbalanced contribution full-physics model simulations, including in the bound- to the tangential wind budget of the simulated storm is ary layer. The extremely weak boundary layer inflow in mainly restricted within a small region near the RMW the balanced solutions given in Bui et al. (2009) and also below about 2-km height, with a spinup of tangential in Abarca and Montgomery (2014) results primarily wind in the lower part of the boundary layer where the from unphysical reasons. Therefore, in agreement with flow is subgradient and a spindown immediately above Stern et al. (2015) and Heng and Wang (2016a), but in where the flow is strongly supergradient. We also found contrast to Bui et al. (2009) and Smith et al. (2009), re- that although the frictionally induced boundary layer in- sults from our study strongly suggest that the spinup of flow induces a large positive tendency of tangential wind tangential wind or the intensification of a TC is driven in the boundary layer, the tendency is not large enough to largely by diabatic heating in the eyewall and not by offset the negative tendency directly because of surface surface friction. This spinup/intensification can be well frictional drag. As a result, the net dynamical effect of explained by the balanced dynamics. surface friction is negative to TC intensification. We also showed that the combined contribution of surface friction and the associated unbalanced dynamics to tangential 5. Conclusions and discussion wind budget is negative tendencies near and outside the This study is an extension of the previous work by RMW and positive tendencies in the eye region, thus Heng and Wang (2016a,b) and revisited the balanced slowing down the intensification but contributing to the and unbalanced aspects of TC intensification and ex- contraction of the eyewall of the simulated storm. amined the sensitivity of the balanced solution to the These results are, in general, consistent with those procedure used to ensure the ellipticity condition of discussed in Heng and Wang (2016a) based on simula- the SE equation and the calculation of radial wind at the tions with prescribed eyewall heating. However, our surface based on the outputs from an idealized full- results do not support the inner-core spinup mechanism physics simulation using the nonhydrostatic, fully com- proposed by Smith et al. (2009), who stated that the pressible nonlinear model TCM4. An hourly mean unbalanced dynamics contributed to the spinup of the vortex structure together with both diabatic heating and inner core in the boundary layer. We show that the un- momentum forcing directly from the full-physics model balanced dynamics due to the presence of surface fric- simulation during the rapid intensification period of the tion acts to prevent the unbalanced flow in the boundary simulated storm was used to analyze the balanced and layer from further intensification. Results from this unbalanced contributions to the TC intensification. The study together with those from Heng and Wang (2016a) results show that the balanced solution of the SE equa- and Stern et al. (2015) strongly suggest that TC in- tion can reproduce most of the azimuthal-mean sec- tensification can be primarily explained by the balanced ondary circulation in the full-physics model simulation dynamics in response to diabatic heating in the eyewall, even in the boundary layer where the flow is agradient. and the unbalanced dynamics due to surface friction The balanced response to diabatic heating in the eyewall prevents the agradient wind in the boundary layer from is a deep transverse circulation with an inflow layer in further intensification. Our findings are in disagreement the mid- to lower troposphere, updrafts in the eyewall, with those shown in Bui et al. (2009), who found that in

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC AUGUST 2017 H E N G E T A L . 2589 their diagnostic analysis using the SE equation the bal- mechanism of a TC proposed therein should not be a anced dynamics could explain one-third of the boundary primary mechanism of TC intensification. The implica- layer inflow and substantially underestimated the tan- tion of the results from Bui et al. (2009) to the secondary gential wind tendency calculated from the full-physics eyewall formation in Abarca and Montgomery (2014) model simulation. also becomes questionable, since Wang et al. (2016) also To understand why the balanced solution in this study demonstrated that the balanced dynamics can capture and that in Bui et al. (2009) are so different, the sensi- well the secondary circulation during the secondary tivity of the balanced solution to the procedure used to eyewall formation in a full-physics model simulation. remove any negative discriminant for the ellipticity Nevertheless, two caveats exist in this study. First, the condition of the SE equation to ensure the convergent axisymmetric vortex structure used in the SE equation in solution was examined. A two-step approach was used this study included the effect of surface friction already, both in our study and in Bui et al. (2009). The first step is which could not be explicitly isolated from the effect of an adjustment to remove any inertial instability at any diabatic heating. Second, although the SE equation as- grid point, often in the upper troposphere, and a second sumes the thermal wind balance of the basic vortex, the step is to reduce the vertical shear of tangential wind, temperature field of the basic vortex in this study was often large in the boundary layer. We found that the use not adjusted to satisfy the thermal wind balance implied of the so-called global regularization to remove inertial by strong vertical shear in the boundary layer, as done in instability used in Bui et al. (2009) resulted in 15%–30% Bui et al. (2009). Note that Bui et al. (2009) showed in underestimations of both inflow in the boundary layer their appendix that the imbalance in the basic vortex and outflow in the upper troposphere in the SE solution could result in almost doubled strength in the boundary compared to the solution with the local adjustment used layer inflow. However, close inspection of their Figs. 6 in this study. We also found that the one-sided finite- and 12 indicates that the boundary layer inflow was still differencing scheme used to calculate the radial wind at about 50% weaker in their balanced solution than in the the surface with the vertical resolution of 500 m for the full-physics model simulation. Furthermore, as in- balanced solution as used in Bui et al. (2009) and Abarca dicated in section 4c, in Bui et al. (2009) the temperature and Montgomery (2014) could lead to a 30%–40% re- anomalies were not adjusted to ensure the thermal wind duction of the boundary layer inflow in the inner-core balance after the reduction of vertical shear of tangen- region. Therefore, we believe that the use of the global tial wind in their second step adjustment for the ellip- regularization to inertial stability together with the one- ticity condition. This means that the basic vortex in the sided finite-differencing scheme used to calculate the boundary layer was not in thermal wind balance either radial wind at the surface with the vertical resolution of in Bui et al. (2009). Finally, the possible feedback of 500 m in the balanced solutions can explain over the surface friction to diabatic heating in the eyewall, as 30%–40% underestimation of the boundary layer inflow mentioned in Heng and Wang (2016b), has not been in Bui et al. (2009). Other possible factors contributing discussed in this study. This indirect effect of surface to the underestimation of the boundary layer inflow in friction on diabatic heating and thus on TC in- Bui et al. (2009) include, but are not limited to, the tensification could be important but could not be iso- massively reduced vertical shear of tangential wind in lated from other effects in one simulation analyzed in the boundary layer and the unchanged strong cold this study. This needs a more deliberate experimental temperature core in the boundary layer. design and is reserved for a future study. As a result, implications from the results of Bui et al. (2009) could be problematic. Smith et al. (2009) pro- Acknowledgments. This study has been supported by posed that the inner-core spinup in the boundary layer the National Natural Science Foundation of China under of a TC is largely the result of frictionally induced inflow Grants 41130964 and 41475091, NSF Grant AGS-1326524, in the boundary layer while the spinup of the tangential Postgraduate Research and Practice Innovation Program wind above the boundary layer is through the balance of Jiangsu Province under Grant CXZZ12-0494, Special response to eyewall heating. They stated that ‘‘although Fund for Meteorological Scientific Research in the Public absolute angular momentum is not materially conserved Interest Grant GYHY201406006, and Priority Academic in the boundary layer, large wind speeds can be achieved Program Development of Jiangsu Higher Education In- if the radial inflow is sufficiently large to bring the air stitutions (PAPD). Y. Wang acknowledges the helpful parcels to small radii with minimal loss of angular mo- discussions with Prof. Chun-Chieh Wu and Dr. Hui Wang mentum’’ (Smith et al. 2009, p. 1332). Since our results in other collaborative research, which motivated the cur- and also those of Stern et al. (2015) have demonstrated rent sensitivity calculations to address some issues that that this is not the case, the boundary layer spinup previously had not been addressed in Bui et al. (2009).

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC 2590 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74

REFERENCES tropical cyclone. J. Atmos. Sci., 69, 997–1020, doi:10.1175/ 2011JAS3690.1. Abarca, S. F., and M. T. Montgomery, 2014: Departures from ——, ——, and Y. Duan, 2015: Impacts of evaporation of rainwater axisymmetric balance dynamics during secondary eyewall on tropical cyclone structure and intensity—A revisit. formation. J. Atmos. Sci., 71, 3723–3738, doi:10.1175/ J. Atmos. Sci., 72, 1323–1345, doi:10.1175/JAS-D-14-0224.1. JAS-D-14-0018.1. Molinari, J., and D. Vollaro, 1990: External influences on hurricane Bryan, G. H., and R. Rotunno, 2009: The influence of near-surface, intensity. Part II: Vertical structure and response of the hur- high-entropy air in hurricane eyes on maximum hurricane ricane vortex. J. Atmos. Sci., 47, 1902–1918, doi:10.1175/ intensity. J. Atmos. Sci., 66, 148–158, doi:10.1175/ 1520-0469(1990)047,1902:EIOHIP.2.0.CO;2. 2008JAS2707.1. ——, ——, and S. Skubis, 1993: Application of the Eliassen bal- Bui, H. H., R. K. Smith, M. T. Montgomery, and J. Peng, 2009: anced model to real-data tropical cyclones. Mon. Wea. Rev., Balanced and unbalanced aspects of tropical cyclone in- 121, 2409–2419, doi:10.1175/1520-0493(1993)121,2409: tensification. Quart. J. Roy. Meteor. Soc., 135, 1715–1731, AOTEBM.2.0.CO;2. doi:10.1002/qj.502. Möller, J. D., and L. J. Shapiro, 2002: Balanced contributions to the Eliassen, A., 1951: Slow thermally or frictionally controlled me- intensification of Hurricane Opal as diagnosed from a GFDL ridional circulation in a circular vortex. Astrophys. Norv., 5, model forecast. Mon. Wea. Rev., 130, 1866–1881, doi:10.1175/ 19–60. 1520-0493(2002)130,1866:BCTTIO.2.0.CO;2. Emanuel, K. A., 1986: An air–sea interaction theory of tropical Montgomery, M. T., M. E. Nicholls, T. A. Cram, and A. B. cyclones. Part I: Steady-state maintenance. J. Atmos. Sci., Saunders, 2006: A vortical hot tower route to tropical cyclo- , 43, 585–604, doi:10.1175/1520-0469(1986)043 0585: genesis. J. Atmos. Sci., 63, 355–386, doi:10.1175/JAS3604.1. AASITF.2.0.CO;2. Nolan, D. S., and M. T. Montgomery, 2002: Nonhydrostatic, three- ——, 1989: The finite-amplitude nature of tropical cyclogenesis. J. At- dimensional perturbations to balanced, hurricane-like vortices. mos. Sci., 46, 3431–3456, doi:10.1175/1520-0469(1989)046,3431: Part I: Linearized formulation, stability, and evolution. J. Atmos. TFANOT.2.0.CO;2. Sci., 59, 2989–3020, doi:10.1175/1520-0469(2002)059,2989: ——, 1995: Sensitivity of tropical cyclones to surface exchange NTDPTB.2.0.CO;2. coefficients and a revised steady-state model incorporating Pendergrass, A. G., and H. E. Willoughby, 2009: Diabatically in- eye dynamics. J. Atmos. Sci., 52, 3969–3976, doi:10.1175/ duced secondary flows in tropical cyclones. Part I: Quasi- 1520-0469(1995)052,3969:SOTCTS.2.0.CO;2. steady forcing. Mon. Wea. Rev., 137, 805–821, doi:10.1175/ Frisius, T., D. Schönemann, and J. Vigh, 2013: The impact of gra- 2008MWR2657.1. dient wind imbalance on potential intensity of tropical cy- Persing, J., M. T. Montgomery, and R. E. Tuleya, 2002: Environ- clones in an unbalanced slab boundary layer model. J. Atmos. mental interactions in the GFDL hurricane model for Hurri- Sci., 70, 1874–1890, doi:10.1175/JAS-D-12-0160.1. cane Opal. Mon. Wea. Rev., 130, 298–317, doi:10.1175/ Fudeyasu, H., and Y. Wang, 2011: Balanced contribution to the 1520-0493(2002)130,0298:EIITGH.2.0.CO;2. intensification of a tropical cyclone simulated in TCM4: Raymond, D. D., C. López-Carrillo, and L. L. Cavazos, 1998: Case Outer-core spinup process. J. Atmos. Sci., 68, 430–449, studies of developing east Pacific easterly waves. Quart. J. Roy. doi:10.1175/2010JAS3523.1. Meteor. Soc., 124, 2005–2034, doi:10.1002/qj.49712455011. Hendricks, E. A., M. T. Montgomery, and C. A. Davis, 2004: The Rozoff, C. M., D. S. Nolan, J. P. Kossin, F. Zhang, and J. Fang, role of ‘‘vortical’’ hot towers in the formation of Tropical 2012: The roles of an expanding wind field and inertial stability Cyclone Diana (1984). J. Atmos. Sci., 61, 1209–1232, in tropical cyclone secondary eyewall formation. J. Atmos. doi:10.1175/1520-0469(2004)061,1209:TROVHT.2.0.CO;2. Sci., 69, 2621–2643, doi:10.1175/JAS-D-11-0326.1. Heng, J., and Y. Wang, 2016a: Nonlinear response of a tropical Schubert, W. H., and J. J. Hack, 1982: Inertial stability and tropical cyclone vortex to prescribed eyewall heating with and without cyclone development. J. Atmos. Sci., 39, 1687–1697, surface friction in TCM4: Implications for tropical cyclone doi:10.1175/1520-0469(1982)039,1687:ISATCD.2.0.CO;2. intensification. J. Atmos. Sci., 73, 1315–1333, doi:10.1175/ Schwendike, J., and J. D. Kepert, 2008: The boundary layer winds JAS-D-15-0164.1. in Hurricanes Danielle (1998) and Isabel (2003). Mon. Wea. ——, and ——, 2016b: Reply to ‘‘Comments on ‘Nonlinear re- Rev., 136, 3168–3192, doi:10.1175/2007MWR2296.1. sponse of a tropical cyclone vortex to prescribed eyewall Shapiro, L. J., and H. E. Willoughby, 1982: The response of bal- heating with and without surface friction in TCM4: Implica- anced hurricanes to local sources of heat and momentum. J. tions for tropical cyclone intensification.’’’ J. Atmos. Sci., 73, Atmos. Sci., 39, 378–394, doi:10.1175/1520-0469(1982)039,0378: 5105–5109, doi:10.1175/JAS-D-16-0262.1. TROBHT.2.0.CO;2. Kepert, J. D., 2006: Observed boundary layer wind structure and Smith, R. K., and M. T. Montgomery, 2015: Towards clarity on balance in the hurricane core. Part II: Hurricane Mitch. understanding tropical cyclone intensification. J. Atmos. Sci., J. Atmos. Sci., 63, 2194–2211, doi:10.1175/JAS3746.1. 72, 3020–3031, doi:10.1175/JAS-D-15-0017.1. ——, 2010: Tropical cyclone structure and dynamics. Global Per- ——, and ——, 2016: Comments on ‘‘Nonlinear response of a spectives on Tropical Cyclones, J. C. L. Chan and J. D. Kepert, tropical cyclone vortex to prescribed eyewall heating with and Eds., World Scientific Series on Asia-Pacific Weather and without surface friction in TCM4: Implications for tropical Climate, Vol. 4, World Scientific, 3–54. cyclone intensification.’’ J. Atmos. Sci., 73, 5101–5103, ——, and Y. Wang, 2001: The dynamics of boundary layer jets within doi:10.1175/JAS-D-16-0163.1. the tropical cyclone core. Part II: Nonlinear enhancement. J. At- ——, ——, and N. V. Sang, 2009: Tropical cyclone spin-up re- mos. Sci., 58, 2485–2501, doi:10.1175/1520-0469(2001)058,2485: visited. Quart. J. Roy. Meteor. Soc., 135, 1321–1335, TDOBLJ.2.0.CO;2. doi:10.1002/qj.428. Li, Q.-Q., and Y. Wang, 2012: Formation and quasi-periodic be- Stern, D. P., J. L. Vigh, D. S. Nolan, and F. Zhang, 2015: Re- havior of outer spiral rainbands in a numerically simulated visiting the relationship between eyewall contraction and

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC AUGUST 2017 H E N G E T A L . 2591

intensification. J. Atmos. Sci., 72, 1283–1306, doi:10.1175/ ——, 2009: How do outer spiral rainbands affect tropical cyclone JAS-D-14-0261.1. structure and intensity? J. Atmos. Sci., 66, 1250–1273, Wang, H., C.-C. Wu, and Y. Wang, 2016: Secondary eyewall for- doi:10.1175/2008JAS2737.1. mation in an idealized tropical cyclone simulation: Balanced ——, and J.-Y. Heng, 2016: Contribution of eye excess energy to and unbalanced dynamics. J. Atmos. Sci., 73, 3911–3930, the intensification rate of tropical cyclones: A numerical study. doi:10.1175/JAS-D-15-0146.1. J. Adv. Model. Earth Syst., 8, 1953–1968, doi:10.1002/ Wang, Y., 2007: A multiply nested, movable mesh, fully com- 2016MS000709. pressible, nonhydrostatic tropical cyclone model—TCM4: Xu, J., and Y. Wang, 2010a: Sensitivity of tropical cyclone inner- Model description and development of asymmetries without core size and intensity to the radial distribution of surface explicit asymmetric forcing. Meteor. Atmos. Phys., 97, 93–116, entropy flux. J. Atmos. Sci., 67, 1831–1852, doi:10.1175/ doi:10.1007/s00703-006-0246-z. 2010JAS3387.1. ——, 2008a: Structure and formation of an annular hurricane ——, and ——, 2010b: Sensitivity of the simulated tropical cyclone simulated in a fully compressible, nonhydrostatic model— inner-core size to the initial vortex size. Mon. Wea. Rev., 138, TCM4. J. Atmos. Sci., 65, 1505–1527, doi:10.1175/ 4135–4157, doi:10.1175/2010MWR3335.1. 2007JAS2528.1. Zhang, J. A., R. F. Rogers, D. S. Nolan, J. Marks Jr., and D. Frank, ——, 2008b: Rapid filamentation zone in a numerically simulated 2011: On the characteristic height scales of the hurricane tropical cyclone. J. Atmos. Sci., 65, 1158–1181, doi:10.1175/ boundary layer. Mon. Wea. Rev., 139, 2523–2535, doi:10.1175/ 2007JAS2426.1. MWR-D-10-05017.1.

Unauthenticated | Downloaded 09/30/21 11:42 AM UTC