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Revisiting the Balanced and Unbalanced Aspects of Tropical Cyclone Intensification 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 Tropical cyclones (TCs) can be viewed as a quasiaxisymmetric primary circulation superimposed by a thermally and frictionally driven secondary circulation. While the primary circulation remains nearly in gradient wind balance as a ‘‘slowly evolving’’ system, the secondary circulation with radial inflow in the lower troposphere, upward motion in the eyewall, and outflow in
Corresponding author: Prof. Yuqing Wang,
[email protected]
the upper troposphere, can be considered as a response to diabatic heating in the eyewall and the momentum forcing mainly as a result of surface friction (Shapiro and Willoughby 1982). The secondary circulation can bring large absolute angular momentum (AAM) inward in the lower troposphere to spin up the tangential wind and thus lead to the intensification of a TC (Shapiro and Willoughby 1982; Schubert and Hack 1982; Pendergrass and Willoughby 2009). This spinup process was first explored based on the balanced dynamics: namely, the Sawyer–Eliassen (SE) equation (Eliassen 1951). The 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).
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can be well explained by the balanced dynamics (Schubert and Hack 1982; Shapiro and Willoughby 1982; Pendergrass and Willoughby 2009; Fudeyasu and Wang 2011). In addition to the theoretical studies, the SE equation has also been applied to diagnose the processes of TC intensification from the simulations and observations and has been shown to be able to reproduce the secondary circulation during the intensification period, even in the boundary layer, where the balanced assumption is invalid (Molinari and Vollaro 1990; Molinari et al. 1993; Möller and Shapiro 2002; Persing et al. 2002; Hendricks et al. 2004; Montgomery et al. 2006). Therefore, these theoretical and diagnostic studies have demonstrated that the intensification of a TC can be largely explained by the balanced dynamics. Bui et al. (2009) have criticized the classic understanding of TC intensification based on balanced dynamics, as their balanced solution using the SE equation considerably underestimated the boundary layer inflow in their full-physics model simulation. They thus concluded that the balanced dynamics significantly underestimates the boundary layer inflow and thereby the spinup of tangential wind in the inner-core region, and thus the unbalanced dynamics should be largely responsible for TC intensification. Motivated by the results of Bui et al. (2009), Smith et al. (2009) argued that the balanced mechanism spins up the TC vortex above the boundary layer and increases the TC size. The inner-core spinup in the boundary layer is largely attributed to the unbalanced dynamics associated with surface friction. Surface friction can break down gradient balance and enhance the boundary layer inflow. In their viewpoint, the loss of AAM due to surface friction is less than the radial transport of AAM due to frictionally induced inflow. In this case, although the AAM is not conserved, the local tangential wind would still increase in the inner core in the boundary layer, resulting in a low-level jet in tangential wind with supergradient nature near the radius of maximum wind (RMW). They seemed to suggest that the occurrence of unbalanced supergradient wind in the interior of the boundary layer as a result of surface friction plays an important role in spinning up the inner core of a TC. This argument gave the impression that surface friction and its associated unbalanced processes can dominate the balanced dynamics in spinning up the TC in the inner core in the boundary layer. This is in sharp contrast to the traditional view that surface friction is the major energy sink of a TC system, thus contributing negatively to TC intensification and maximum intensity (Emanuel 1989; Raymond et al. 1998; Kepert 2010).
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The argument on the positive contribution by surface friction and the associated unbalanced dynamics to TC intensification seems to be consistent with the positive contribution by the unbalanced flow in the boundary layer to TC maximum intensity (MI) in some previous studies. Smith et al. (2009) speculated that the gradient wind imbalance in the inflow boundary layer could substantially increase TC MI. Bryan and Rotunno (2009) developed a diagnostic model for TC MI by incorporating the effect of unbalanced flow in the boundary layer and found that the unbalanced flow could contribute to TC MI by 5%–30%. Frisius et al. (2013) extended the maximum potential intensity (MPI) theory of Emanuel (1986, 1995) by including the effect of unbalanced flow in a slab boundary layer model and found that the maximum TC intensity can be up to 18% higher than that from the MPI that ignores the unbalanced effect. Note that both studies of Bryan and Rotunno (2009) and Frisius et al. (2013) defined the maximum intensity as the maximum wind speed in the interior of the boundary layer: namely, including the supergradient wind component, which is often about 10%–20% of the gradient wind near the top of the inflow boundary layer in TCs (Kepert and Wang 2001; Kepert 2006; Schwendike and Kepert 2008). Since the supergradient wind is well above the surface, it does not directly contribute to the surface energy production or loss and the TC intensity, which is measured by the nearsurface wind speed. Stern et al. (2015) challenged the hypothesized positive contribution of surface friction to TC intensification proposed by Smith et al. (2009) based on the results from a linearized vortex model, the ThreeDimensional Vortex Perturbation Analysis and Simulation (3DVPAS) of Nolan and Montgomery (2002). They found that the effect of surface friction is only significant in the boundary layer with a strengthened inflow layer and a corresponding shallow outflow layer immediately above. Although surface friction can substantially enhance the boundary layer inflow, the net dynamical effect of surface friction is negative because the positive tangential wind tendency as a result of frictionally induced inflow could not offset the direct spindown by surface friction. Note that Stern et al. (2015) also confirmed the importance of vertical shear of tangential wind in the boundary layer to the frictionally induced inflow in the balanced response, as shown in Bui et al. (2009). Smith and Montgomery (2015) criticized that the linear model used in Stern et al. (2015) has its limitations and could not be used to access the spinup mechanism in the boundary layer because the processes therein are intrinsically nonlinear.
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To include the nonlinearity, Heng and Wang (2016a) used a nonlinear model, the Tropical Cyclone Model, version 4 (TCM4), to study the effects of surface friction and the associated unbalanced dynamics on TC intensification. They conducted two idealized numerical experiments: one without surface friction and the other with surface friction. To isolate the dynamical effect of surface friction, diabatic heating in the eyewall was prescribed so that the possible feedback of surface friction to diabatic heating was excluded in the experiment with surface friction. The results showed that surface friction has a net negative contribution to TC intensification, in agreement with Stern et al. (2015). They also found that the unbalanced dynamics due to surface friction acts as a process to prevent the further enhancement of agradient flow in the frictional boundary layer by spinning up tangential wind in the surface layer near the RMW, where the flow is strongly subgradient, and by spinning down immediately above where the flow is strongly supergradient. Heng and Wang (2016a) also noticed that, although surface friction has an overall net negative dynamical effect on TC intensification, it plays a critical role in producing the realistic boundary layer structure, such as the enhanced inflow and supergradient wind near the top of the inflow boundary layer near the RMW. Smith and Montgomery (2016) criticized the work of Heng and Wang (2016a) and argued that the experiments with and without surface friction were not adequate to isolate the frictional effect on TC intensification, and the presence of supergradient winds in the simulation with surface friction implies the importance of surface friction to TC intensification. Heng and Wang (2016b) argued that the supergradient wind is well above the surface, results as a fast adjustment (response) to surface friction, exists in all stages of a TC, and thus could not be a justification of the unbalanced contribution to TC intensification. Heng and Wang (2016b), however, also indicated that they did not consider any possible feedback to eyewall convection due to the presence of surface friction since the heating rate in their numerical experiments was prescribed. This study is an extension of the study by Heng and Wang (2016a) and revisits the dynamical effects of surface friction and its associated unbalanced processes on TC intensification in a full-physics model simulation. Different from Heng and Wang (2016a), the balanced and unbalanced contributions to TC intensification in this study are analyzed based on the full-physics model simulation in which diabatic heating contains the feedback of the unbalanced dynamics in the frictional boundary layer. The rest of the paper is organized as follows. Section 2 briefly describes the TCM4 and the
numerical solution of the SE equation. In section 3, the balanced contribution to the intensification of the simulated TC is diagnosed using the SE equation. The sensitivity of the balanced solution to the procedures used to ensure the ellipticity condition of the SE equation is examined in section 4. Major findings and conclusions are given in the last section.
2. Methodology The quadruply nested, fully compressible, nonhydrostatic TCM4 was used to perform an idealized numerical simulation of a TC. A complete description of TCM4 can be found in Wang (2007) and it has been used in studies on TC structure and intensity changes, including Wang (2008a,b), Wang (2009), Xu and Wang (2010a,b), Li et al. (2015), and Heng and Wang (2016a). All model settings, including the initial conditions, were identical to those used in the control experiment in Wang and Heng (2016). To avoid any duplication, the readers are referred to Wang and Heng (2016) for details. The model was run for 168 h, with the model outputs at 6-min intervals for diagnostic analysis. To evaluate the balanced and unbalanced contributions to TC intensification, the SE equation was applied to calculate the balanced response from the model outputs. We used the SE equation in height coordinates as given in Bui et al. (2009). The diagnostic equation for streamfunction in the radius–height plan was given in Eq. (2) in Heng and Wang (2016a), and a full description of every quantity in Eq. (2) can be found therein as well. Given the azimuthal-mean primary circulation, which was assumed in gradient wind and hydrostatic balances (except for in the frictional boundary layer), the SE equation was solved numerically with radial grid spacing of 2.5 km and vertical grid spacing of 250 m in a domain extending from the storm center to a radius of 300 km and from the surface to a height of 18 km. The ellipticity condition for convergent numerical solutions requires the discriminant D . 0, where D 5 2g
2 ›x ›x › jx(§ 1 f ) 1 C 2 (xC) , ›z ›r ›z
(1)
where r and z are radius and height, g is the gravitational acceleration, x 5 1/u, where u is potential temperature, C 5 y 2 /r 1 f y denotes the sum of centrifugal and Coriolis forces, and j 5 2y/r 1 f , with f being the Coriolis parameter. As in previous studies, there were some grid points where the ellipticity condition was not satisfied in our simulation. To assure the ellipticity of the SE equation, we first set h 5 0:01f0 at points where h 5 f0 1 z , 0:01f0 to remove any points where the
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FIG. 1. Time evolution of the maximum azimuthal-mean tangential wind speed (m s21) at the lowest model level (26.5 m above the sea surface) in the TCM4 simulation.
absolute vertical vorticity was negative. If the ellipticity condition was still not satisfied at any grid points, the vertical shear of tangential wind [›(xC)/›z] at those grid points was then reduced by iterations. Namely, the vertical shear of tangential wind was reduced by a factor of 0.8 at those points where D # 0. The ellipticity condition was then rechecked. If the ellipticity condition was still not satisfied at some grid points, the vertical shear of tangential wind was reduced by a factor of 0.8 only at those points where D # 0. Similar iterations were repeated until the ellipticity condition was satisfied at all grid points (but limited not to exceed eight iterations in our case discussion in sections 3 and 4). Similar adjustment procedures were also used in Wang et al. (2016) and in Heng and Wang (2016a). In addition, in our calculations, the radial wind at the surface (z 5 0) was extrapolated based on the vertical shear of radial wind from the model outputs between the 125-m height and the lowest model level at 26.5 m.
3. Diagnostics of the balanced contribution Figure 1 shows the time evolution of the maximum azimuthal-mean tangential wind speed at the lowest model level in the simulation. After about a 26-h initial adjustment period during which the inner-core air column was moistened by the surface fluxes, the TC vortex intensified rapidly from 26 to 91 h with a mean intensification rate of 18.7 m s21 day21. As in Bui et al. (2009), we chose the weak stage of the storm at 50 h of simulation when the storm had the maximum azimuthalmean tangential wind of 26 m s21 and the strong stage at 70 h of simulation when the storm had its maximum azimuthal-mean tangential wind of 42 m s21; both times were in the rapid intensification period. Figure 2 shows the azimuthal-mean tangential wind and temperature anomaly of the simulated storm
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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 dynamical parameters that appeared in the SE equation [see Eq. (1)], including the azimuthal-mean density, potential temperature, absolute vertical vorticity, and inertial stability, were obtained directly from the model outputs. The azimuthal-mean diabatic heating rate and momentum forcing from the model outputs at the two chosen times are shown in Fig. 3. At 50 h, the heating rate covers a broad region between 15 and 100 km from the storm center and has two maxima in the middle troposphere: one is near a radius of 20 km in the eyewall and the other is near a radius of 70 km related to the outer rainbands (Fig. 3a). At 70 h, the maximum heating rate in the eyewall increases to over 20 K h21, more than 3 times that at 50 h. The heating rate in the outer region remains similar to that at 50 h. The momentum forcing is dominated by the sink associated with surface friction within a shallow layer near the surface at the two given times (Figs. 3c,d). The maximum negative value is about 215 m s21 h21 at 50 h and about 230 m s21 h21 at 70 h near the RMW. Some positive values inside the RMW above the boundary layer are due to eddy momentum mixing (both horizontal and vertical). Figure 4 compares the radial wind and vertical motion from the model simulation with those diagnosed from the SE equation at 50 and 70 h. The balanced solution captures the secondary circulation in the model simulation quite well in both the pattern and magnitude throughout the troposphere, even in the boundary layer (Figs. 4a–d). The similarity between the balanced solution and the model simulation can be seen more clearly from the area-averaged radial wind speed within a radius of 160 km shown in Figs. 4e and 4f. This is in sharp contrast to Bui et al. (2009), who showed that the balanced solution could only capture less than half of the boundary layer inflow in the full-physics model simulation in their calculation (this matter will be further discussed in section 4). Note that the maximum boundary layer inflow in the balanced solution occurs at the radius of 40 km, which is about 10 km outside of that from the model simulation at both times, implying that the
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FIG. 2. The radius–height cross sections of the azimuthal-mean (a),(b) tangential wind speed (m s21) and (c),(d) temperature anomaly (8C) averaged at (a),(c) 50 and (b),(d) 70 h of simulation.
balanced dynamics underestimates the inward penetration of the boundary layer inflow into the eye region, or, alternatively, the unbalanced dynamics contributes to inward penetration of boundary layer inflow into the eye and thus the contraction of the RMW, which is consistent with the finding in Heng and Wang (2016a). Note also that the radial wind in the balanced solution near the surface is slightly stronger than that in the TCM4 simulation, in particular at 70 h (Figs. 4d,f). This is most likely a result of the extrapolation of the radial wind at the surface with the same reduction factor as that between 125 and 26.5 m at the corresponding radius in the full-physics TCM4 simulation. This may cause some errors, since the radial distribution of radial wind is slightly shifted outward in the balanced solution, as mentioned above. This also indicates that the interpretation of the results in the lowest 70-m layer should be done with caution. Because of the linearity of the SE equation, contributions to the secondary circulation by the azimuthal mean diabatic heating and momentum forcing can be separately calculated. Namely, the respective balanced solutions can be derived by solving the SE equation twice: once with diabatic heating only and once with
momentum forcing only. This is similar to what was done in Rozoff et al. (2012) in their explanation of the respective roles of diabatic heating and momentum forcing in secondary eyewall formation in a full-physics model simulation using the linearized vortex model 3DVPAS (Nolan and Montgomery 2002), as also used in Stern et al. (2015). Here, we discuss results at 70 h of simulation only, since results from other times are consistent (Figs. 3b,d). Diabatic heating drives a deep inflow layer in the mid- to lower troposphere and a broad outflow layer in the upper troposphere (Fig. 5a). The maximum heating-induced inflow reaches 6 m s21 near the surface at a radius of 40 km. Meanwhile, the balanced upward motion forced by diabatic heating is comparable to that of the model simulation (Figs. 4d, 5c), suggesting that diabatic heating is a major driving force for the deep secondary circulation, including part of the strong inflow in the boundary layer. This is consistent with the full nonlinear solution shown in Fig. 3a in Heng and Wang (2016a). The solution of the SE equation with momentum forcing shows a boundary layer inflow that penetrates more inward into the eye region. The maximum frictionally induced inflow reaches about 5 m s21 near the surface at a radius of 27.5 km (Fig. 5b).
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FIG. 3. The radius–height cross sections of the azimuthal-mean (a),(b) diabatic heating rate (K h21) and (c),(d) momentum forcing (m s21 h21) derived from the TCM4 simulation averaged at (a),(c) 50 and (b),(d) 70 h.
Note that the response to the momentum forcing primarily due to surface friction only occurs in the boundary layer (Figs. 5b,d) with weak upward motion in the eyewall region and only reaches about 2 km where a weak outflow layer associated with supergradient wind occurs. The results discussed above demonstrate that the secondary circulation with a deep inflow layer in the mid- to lower troposphere, eyewall updraft in the region of diabatic heating, and the outflow layer in the upper troposphere is mainly driven by diabatic heating in the eyewall. Although surface friction explains roughly 40%–50% of the inflow in the lower part of the boundary layer, the associated secondary circulation is very shallow, with the forced upward motion occurring only below 2-km height in the eyewall. Even though the boundary layer was well coupled with the free atmosphere above with a full nonlinearity in TCM4, the linear diagnostics based on the SE equation can still largely recover the diabatic heating and surface-frictioninduced secondary circulation to a large extent. Thus, this strongly suggests that the SE equation is useful for understanding TC intensification processes. To investigate the balanced and unbalanced contributions to the spinup of tangential wind in the simulated
storm, we performed the tangential wind tendency budget at 70 h of simulation. The tangential wind tendency equation in the cylindrical coordinates can be written as (Xu and Wang 2010a,b; Heng and Wang 2016a) ›y ›y ›y0 5 2u h 2 w 2 u0 h0 2 w0 1 F y , ›t ›z ›z
(2)
where t is time and z is height; u, y, and w are radial, tangential, and vertical winds; and h is vertical absolute vorticity. The overbar denotes the azimuthal mean, and the prime denotes the deviation from its azimuthal mean. The azimuthal-mean surface friction of tangential wind together with vertical mixing and horizontal diffusion of tangential wind is denoted by F y . The five terms on the right-hand side of Eq. (2) are, respectively, the azimuthal-mean radial advection contributed by the radial flux of absolute vertical vorticity, vertical advection of azimuthal-mean tangential wind by the azimuthal-mean vertical velocity, the corresponding eddy radial advection and eddy vertical advection, and the vertical mixing (including surface friction) and horizontal diffusion of tangential wind.
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FIG. 4. The radius–height cross sections of the azimuthal-mean radial wind (contour interval of 1.0 m s21) and vertical motion (shading; m s21) (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 azimuthalmean tangential wind because of the total (both horizontal and vertical) advection calculated directly from the full-physics model simulation and from the balanced solution at 70 h of simulation. Overall, the balanced solution reproduces the major feature of the tangential wind budget in both the spatial pattern and magnitude (Figs. 6a,b), in agreement with the similarity in the secondary circulation, as shown in Figs. 4c and 4d. Note that some differences between the tendencies calculated from the model output and from the balanced solution appear near and inside the RMW below 2-km height, where the flow is strongly subgradient/supergradient (Fig. 6f). This suggests that the unbalanced contribution
is only important near and inside the RMW below 2-km height. To explore the frictional effect on TC intensification in this full-physics model simulation, the tangential wind tendencies as a result of diabatic heating only and momentum forcing only, respectively, are examined based on the balanced SE solutions (Figs. 6c,d). The total advection due to diabatic heating spins up tangential wind near and outside the RMW while it spins down tangential wind inside the RMW. The frictionally induced secondary circulation contributes to the spinup of tangential wind below 1 km while it spins down tangential wind immediately above (Fig. 6d). Note that the tangential wind tendency due to advection by the
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FIG. 5. The radial–height cross sections of the azimuthal-mean (a),(b) radial wind (m s21) and (c),(d) vertical motion (m s21) 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 that by heating-induced inflow in the boundary layer, although the inflow induced by the former is weaker than that induced by the latter. This is because the frictionally induced inflow penetrates more into the eye region, where the absolute vorticity is much higher than that outside the RMW, leading to larger advective tangential wind tendency. However, the large positive tangential wind tendency induced by the frictionally induced boundary layer inflow (Fig. 6d) is still smaller than the negative tendency due to surface friction itself under and outside the eyewall, leading to a net negative tangential wind tendency near the RMW in the boundary layer (Fig. 6e). The positive tangential wind tendency centered at about 0.5-km height in the eye region results mainly from the inward penetration of the boundary layer inflow into the eye. Since the positive tendency is well inside the RMW, it does not contribute to the intensification of the storm but spins up the cyclonic circulation in the eye region. This demonstrates that the net dynamical effect of surface friction on TC intensification is negative, and the balanced response to diabatic heating in the eyewall (including considerable modifications by the presence of surface friction and its related unbalanced dynamics) is the sole mechanism of TC intensification (Heng and Wang 2016a,b). Figure 6f shows the unbalanced contribution to the tangential wind budget, which is defined as the residual
between the total tendencies calculated from the model output and those calculated from the balanced flow,1 as in Heng and Wang (2016a). We can see that the unbalanced dynamics acts to spin up tangential winds in the lower part of the inflow boundary layer while it spins down tangential winds near the RMW immediately above and outside the RMW in the surface layer. The unbalanced contribution to the tangential wind budget above the boundary layer is negligible. This is consistent with the results in Heng and Wang (2016a), who suggested that the unbalanced dynamics due to the presence of surface friction contributes to a spinup of tangential wind in the surface layer near the RMW where the flow is strongly subgradient and a spindown immediately above where the flow is strongly supergradient. Figure 7 shows the total frictional effect, which is defined as the sum of the unbalanced contribution and the net frictional effect, since the unbalanced
1 Note that the differences plotted in Fig. 6f also include those caused by numerical approximations arising from the transform from Cartesian to cylindrical coordinates, the azimuthal average, the different grid spacing, and different finite-difference operators in TCM4 compared to those in the Sawyer–Eliassen calculation. As a result, Fig. 6f shows not just the effects of the unbalanced flow, but also these numerical issues. Nevertheless, the overall feature is largely contributed by the involved physics rather than by numerical errors, as also shown in Heng and Wang (2016a).
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FIG. 6. The radial–vertical cross sections of the azimuthal-mean tangential wind tendencies (m s21 h21) 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). The negative tangential wind tendency near and outside the RMW implies that surface friction and the associated unbalanced dynamics play a role in spinning down the simulated storm. The positive tangential wind tendency inside the RMW indicates that surface friction and the associated unbalanced dynamics contribute to the eyewall contraction of the storm and also drive the tangential wind inside the RMW from U shaped to V shaped. This is also in agreement with the results of Stern et al. (2015) and
Heng and Wang (2016a), but in contrast to the hypothesis of Bui et al. (2009) and Smith et al. (2009). Recently, Smith and Montgomery (2016) criticized the work of Heng and Wang (2016a) in that the existence of supergradient winds in Heng and Wang (2016a) reflected the positive contribution of unbalanced dynamics to TC intensification, while Heng and Wang (2016b) responded that the supergradient wind results from a quick adjustment of a balanced vortex to surface friction and did little to the intensification of their simulated storm. To isolate the possible feedback of surface
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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 tendency (contour interval of 1 m s21 h21), with shading showing the azimuthal-mean tangential wind (m s21) at 70 h of simulation.
friction to diabatic heating, Heng and Wang (2016a) prescribed eyewall heating in their simulations. Here, we have included full moist processes and gotten similar results, indicating that the main conclusions in Heng and Wang (2016a,b) remain valid. Namely, surface friction and the associated unbalanced dynamics contribute negatively to TC intensification. The unbalanced dynamics acts to spin up tangential wind in the surface layer near the RMW where the flow is strongly subgradient and to spin down tangential wind immediately above where the flow is strongly supergradient. This can also be seen from the total budgets of tangential wind calculated from the TCM outputs (Fig. 6g) and from the SE solution (Fig. 6h). The above results thus strongly suggest that, even though surface friction contributes to 40%–50% of the total boundary layer inflow in the inner-core region and thus substantially enhances positive tangential wind tendency near the RMW in the boundary layer, the positive tendency is not large enough to offset the negative tangential wind tendency directly induced by surface friction near and outside the RMW. This is also in agreement with the results of Stern et al. (2015)
4. Sensitivity of the balanced solution to various assumptions To understand the difference in the balanced solution in this study [and also in Heng and Wang (2016a)] from that in Bui et al. (2009), we examined the sensitivity of 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 ensure the ellipticity condition of the SE equation and the calculation of radial wind at the surface (z 5 0) from the streamfunction of the SE equation. One is the
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FIG. 8. Grid points that are not satisfied by the ellipticity condition criterion, namely, with D # 0 [gray shading; see Eq. (1)].
adjustment to the inertial instability [related to the first term on the rhs in Eq. (1)], mostly in the outflow layer, and the other is the adjustment to vertical shear of tangential wind [related to the second term on the rhs in Eq. (1)], mostly in the frictional boundary layer (Fig. 8). We also evaluated the possible effect of a bug in the code for vertical shear adjustment used in Bui et al. (2009). In addition, we examined the effect of the one-sided finitedifferencing scheme used to calculate the surface radial wind utilized in Bui et al. (2009) and also later in Abarca and Montgomery (2014).
a. Adjustment to inertial stability Because of the development of the anticyclonic circulation in response to the outflow in the upper troposphere, weak symmetry instability often occurs in the upper troposphere with small negative absolute vertical vorticity (Möller and Shapiro 2002; Hendricks et al. 2004; Fudeyasu and Wang 2011; Li and Wang 2012). The negative inertial stability leads to a negative discriminant for the ellipticity condition in some grid points. To relax the negative discriminant D in Eq. (1), a regularization process is often employed to remove those regions with negative inertial stability. In our calculations discussed in section 3, we set the absolute vertical vorticity h 5 0:01f0 at grid points where h 5 f0 1 z , 0:01f0 to remove the inertial instability, the so-called local adjustment. In Bui et al. (2009), the so-called global adjustment was used. Namely, when the parameter I 2 5 xj( f0 1 h) 1 C›x/›r, an analog to inertial stability, was negative at some grid points, a minimum value2 at
2 The minimum value in our case is 21.18 3 10215 K21 s22. Although the absolute value is quite small compared to the values in the inner-core region, it is comparable to the positive values in the outer-core region outside a radius of 60 km in our simulated storm.
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FIG. 9. The radius–height cross sections of the azimuthal-mean (a),(c) radial wind (m s21) and (b),(d) vertical motion (m s21) in calculations with (a),(b) the local adjustment and (c),(d) the global adjustment, along with the differences (global minus local) in the azimuthal-mean (e) radial wind (m s21) and (f) vertical motion (m s21) between the two calculations.
these grid points was chosen, and 1% of its absolute value was then added to all grid points in the whole computational domain to ensure the inertial stability in the upper troposphere. Here we compare the calculations using the local and global adjustments at 70 h of simulation. Because there are some grids where the ellipticity is still not satisfied because of the large vertical shear of tangential wind in the lower boundary layer, an additional adjustment to the vertical shear is considered as well. Note that here the same adjustment to vertical shear of tangential wind developed in this study as described in section 2 was applied after the adjustment to the inertial stability. This means that the difference between the two
calculations results mainly from the different adjustments to inertial stability. After the global adjustment was applied, the number of the grid points where the ellipticity condition was not satisfied decreased greatly compared to that after the local adjustment was performed (not shown). However the global adjustment had a much broader effect than the local adjustment, since it was applied to all grid points in the computational domain. As a result, the solution with the global adjustment could considerably underestimate the radial inflow in the boundary layer and outflow in the upper troposphere. This can be clearly seen in Figs. 9a and 9c. Although the maxima of the boundary layer inflow near the RMW are similar in
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the two balanced SE solutions, both inflow in the boundary layer and outflow in the upper troposphere were considerably weaker outside the RMW in the calculation with the global adjustment than those with the local adjustment (Fig. 9e). This is mainly because the global adjustment added a positive value of a small I2 at all grids in the computational domain. Although the added value was about three orders smaller than the inertial stability near the eyewall (Bui et al. 2009), it increased the inertial stability in the outer region where the inertial stability was much smaller and acted as a resistance to the radial inflow/outflow in the SE solution. Overall, the radial inflow/outflow was reduced by as much as over 2 m s21, or about 20%–50% of the solution with the local adjustment. This is in sharp contrast to that in Bui et al. (2009, p. 1720), who stated that ‘‘this procedure does not affect the general characteristics of the solution outside the regions where the regularization is applied.’’ Unlike the large difference in the radial wind, the difference in vertical motion is relatively small (Figs. 9b,d,f).
b. Calculation of radial wind at the surface Another factor potentially contributing to the underestimation of the boundary layer inflow in Bui et al. (2009) is the one-sided finite-differencing scheme used to calculate the radial wind at the surface (z 5 0) from the streamfunction of the SE solution.3 We examined such a possibility in our calculation and found that this effect is marginal, with the vertical resolution of 250 m used in our balanced solution. This is because, with the vertical resolution of 250 m, the radial wind at the surface calculated using the one-sided finite-differencing scheme is equivalent to that at 125-m height. Zhang et al. (2011) found that the maximum of the radial wind in observed strong TCs was located at about 150 m above the sea surface. Therefore, the insensitivity in our vertical resolution of 250 m is understandable. However, we noticed that in Bui et al. (2009) the one-sided finitedifferencing scheme was used to calculate the radial wind at the surface in the balanced solution with the vertical grid spacing of 500 m. This means that the radial wind at the surface (z 5 0) was equivalent to that at 250-m height. Therefore, in the calculations of Bui et al. (2009), the radial wind could be considerably
3 The one-sided finite-differencing scheme was also used to calculate the radial wind at the surface (z 5 0) from the balanced solution in Abarca and Montgomery (2014) with the vertical resolution of 500 m. We thus consider that the results discussed in this subsection are also applicable to explain part of the underestimation of the boundary layer inflow in Abarca and Montgomery (2014).
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underestimated in the lowest 500 m, where the strongest inflow often exists (Zhang et al. 2011). To examine the possible contribution of the one-sided finite-differencing scheme used to calculate the radial wind at the surface to the underestimation of the boundary layer inflow in Bui et al. (2009), we repeated the calculation of radial wind at the surface using our one-sided finite-differencing scheme but with degraded vertical resolution of 500 m. Specifically, we calculated the radial wind at the surface using the one-sided finite differencing of streamfunction between the surface and 500-m height. In this case, for consistency, the radial wind at 250 m is taken as the average between the radial wind at 500 m and that at the surface calculated using the one-sided finite-differencing scheme. The radius–height cross section of radial wind below 3-km height thus obtained is compared with that from the TCM4 output and that from our standard calculation with the local adjustment for inertial stability in Fig. 10. As we already mentioned in section 3, the maximum boundary layer inflow in the balanced solutions (Figs. 10b,c) occurs about 10 km outside of that from the TCM4 simulation (Fig. 10a) because the balanced dynamics considerably underestimates the inward penetration of the boundary layer inflow into the eye region. Compared with our algorithm, the one-sided finite differencing considerably underestimates the radial wind below 250 m in the innercore region within a radius of 70 km (Fig. 10e). The above results demonstrate that the global adjustment to inertial stability can also lead to the underestimation of radial wind in the lower part of the boundary layer mainly outside the inner core (Fig. 9e) and the one-sided finite-differencing scheme used to calculate the radial wind at the surface can lead to the underestimation of the boundary layer inflow in the inner-core region (Fig. 10e). To see the combined effect of the two factors discussed above, we repeated the above calculation using both the one-sided finitedifferencing scheme and the global adjustment to inertial stability, with the results shown in Fig. 10d. Now it is clearly seen that the combined effect of the global adjustment to inertial stability and the one-sided finitedifferencing scheme used to calculate the radial wind at the surface with the vertical resolution of 500 m in Bui et al. (2009) is an underestimation of radial wind in the boundary layer by as much as over 3 m s21. This means that the two factors can explain 30%–40% near the surface across the RMW (Fig. 10f).
c. Other possible effects In addition to the two factors discussed above, Bui et al. (2009) constructed the balanced vortex in thermal wind balance for the balanced solution, including in the
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FIG. 10. The radius–height cross sections of the azimuthal-mean radial wind (m s21) from (a) TCM4 outputs, (b) the SE solution of this study, (c) the SE solution with the local adjustment for inertial stability, but with onesided 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 appears in the boundary layer. This indeed can largely strengthen the vertical temperature gradient and thus 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. (2015). We noticed that the cold temperature anomaly in the boundary layer was not adjusted to satisfy the thermal wind balance after the adjustment to vertical shear of the tangential wind in the boundary layer to satisfy the ellipticity condition in Bui et al. (2009) and
Abarca and Montgomery (2014).4 This may introduce imbalance in the basic vortex as well and possibly reduce the boundary layer inflow in the balanced response.
4 We found that the vertical shear of tangential wind in the boundary layer at grid points that are not satisfied with the ellipticity 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 and Montgomery (2014) (M. T. Montgomery 2017, personal communication), but both studies left the originally balanced temperature anomaly unchanged.
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Our additional calculations (not shown) indicate that reducing the vertical shear of tangential wind in the boundary layer can also lead to a decrease of the boundary layer inflow. This is consistent with the calculations using the solver of Bui et al. (2009) and the latest version in Abarca and Montgomery (2014) (M. T. Montgomery 2017, personal communication). Therefore, we believe that, in addition to the two factors discussed above, both the reduced vertical shear of tangential wind in the boundary layer and the unbalanced large cold temperature core that is not adjusted according to the corresponding reduced vertical shear of tangential wind in the boundary layer may account for an additional part of the underestimation of the boundary layer inflow in Bui et al. (2009) and Abarca and Montgomery (2014). We thus conclude that the balanced dynamics can indeed reproduce much of the secondary circulation in full-physics model simulations, including in the boundary layer. The extremely weak boundary layer inflow in the balanced solutions given in Bui et al. (2009) and also in Abarca and Montgomery (2014) results primarily from unphysical reasons. Therefore, in agreement with Stern et al. (2015) and Heng and Wang (2016a), but in contrast to Bui et al. (2009) and Smith et al. (2009), results from our study strongly suggest that the spinup of tangential wind or the intensification of a TC is driven largely by diabatic heating in the eyewall and not by surface friction. This spinup/intensification can be well explained by the balanced dynamics.
5. Conclusions and discussion This study is an extension of the previous work by Heng and Wang (2016a,b) and revisited the balanced and unbalanced aspects of TC intensification and examined the sensitivity of the balanced solution to the procedure used to ensure the ellipticity condition of the SE equation and the calculation of radial wind at the surface based on the outputs from an idealized fullphysics simulation using the nonhydrostatic, fully compressible nonlinear model TCM4. An hourly mean vortex structure together with both diabatic heating and momentum forcing directly from the full-physics model simulation during the rapid intensification period of the simulated storm was used to analyze the balanced and unbalanced contributions to the TC intensification. The results show that the balanced solution of the SE equation can reproduce most of the azimuthal-mean secondary circulation in the full-physics model simulation even in the boundary layer where the flow is agradient. The balanced response to diabatic heating in the eyewall is a deep transverse circulation with an inflow layer in the mid- to lower troposphere, updrafts in the eyewall,
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and an outflow layer in the upper troposphere. The balanced response to momentum forcing due to surface friction is a shallow transverse circulation mainly in the lower 3 km of the atmosphere with a strengthened inflow in the boundary layer and a weaker outflow layer immediately above the inflow boundary layer. Different from diabatic heating, the frictionally induced boundary layer inflow shows an inward penetration into the eye region inside the RMW, contributing to the contraction of the eyewall. The azimuthal-mean tangential wind tendency due to both horizontal and vertical advections calculated using the balanced solution compares well with that using the output of the model simulation except for the small region across the RMW in the lowest 2 km of the model atmosphere, where the flow is strongly unbalanced and agradient. We showed that the unbalanced contribution to the tangential wind budget of the simulated storm is mainly restricted within a small region near the RMW below about 2-km height, with a spinup of tangential wind in the lower part of the boundary layer where the flow is subgradient and a spindown immediately above where the flow is strongly supergradient. We also found that although the frictionally induced boundary layer inflow induces a large positive tendency of tangential wind in the boundary layer, the tendency is not large enough to offset the negative tendency directly because of surface frictional drag. As a result, the net dynamical effect of surface friction is negative to TC intensification. We also showed that the combined contribution of surface friction and the associated unbalanced dynamics to tangential wind budget is negative tendencies near and outside the RMW and positive tendencies in the eye region, thus slowing down the intensification but contributing to the contraction of the eyewall of the simulated storm. These results are, in general, consistent with those discussed in Heng and Wang (2016a) based on simulations with prescribed eyewall heating. However, our results do not support the inner-core spinup mechanism proposed by Smith et al. (2009), who stated that the unbalanced dynamics contributed to the spinup of the inner core in the boundary layer. We show that the unbalanced dynamics due to the presence of surface friction acts to prevent the unbalanced flow in the boundary layer from further intensification. Results from this study together with those from Heng and Wang (2016a) and Stern et al. (2015) strongly suggest that TC intensification can be primarily explained by the balanced dynamics in response to diabatic heating in the eyewall, and the unbalanced dynamics due to surface friction prevents the agradient wind in the boundary layer from further intensification. Our findings are in disagreement with those shown in Bui et al. (2009), who found that in
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their diagnostic analysis using the SE equation the balanced dynamics could explain one-third of the boundary layer inflow and substantially underestimated the tangential wind tendency calculated from the full-physics model simulation. To understand why the balanced solution in this study and that in Bui et al. (2009) are so different, the sensitivity of the balanced solution to the procedure used to remove any negative discriminant for the ellipticity condition of the SE equation to ensure the convergent solution was examined. A two-step approach was used both in our study and in Bui et al. (2009). The first step is an adjustment to remove any inertial instability at any grid point, often in the upper troposphere, and a second step is to reduce the vertical shear of tangential wind, often large in the boundary layer. We found that the use of the so-called global regularization to remove inertial instability used in Bui et al. (2009) resulted in 15%–30% underestimations of both inflow in the boundary layer and outflow in the upper troposphere in the SE solution compared to the solution with the local adjustment used in this study. We also found that the one-sided finitedifferencing scheme used to calculate the radial wind at the surface with the vertical resolution of 500 m for the balanced solution as used in Bui et al. (2009) and Abarca and Montgomery (2014) could lead to a 30%–40% reduction of the boundary layer inflow in the inner-core region. Therefore, we believe that the use of the global regularization to inertial stability together with the onesided finite-differencing scheme used to calculate the radial wind at the surface with the vertical resolution of 500 m in the balanced solutions can explain over the 30%–40% underestimation of the boundary layer inflow in Bui et al. (2009). Other possible factors contributing to the underestimation of the boundary layer inflow in Bui et al. (2009) include, but are not limited to, the massively reduced vertical shear of tangential wind in the boundary layer and the unchanged strong cold temperature core in the boundary layer. As a result, implications from the results of Bui et al. (2009) could be problematic. Smith et al. (2009) proposed that the inner-core spinup in the boundary layer of a TC is largely the result of frictionally induced inflow in the boundary layer while the spinup of the tangential wind above the boundary layer is through the balance response to eyewall heating. They stated that ‘‘although absolute angular momentum is not materially conserved in the boundary layer, large wind speeds can be achieved if the radial inflow is sufficiently large to bring the air parcels to small radii with minimal loss of angular momentum’’ (Smith et al. 2009, p. 1332). Since our results and also those of Stern et al. (2015) have demonstrated that this is not the case, the boundary layer spinup
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mechanism of a TC proposed therein should not be a primary mechanism of TC intensification. The implication of the results from Bui et al. (2009) to the secondary eyewall formation in Abarca and Montgomery (2014) also becomes questionable, since Wang et al. (2016) also demonstrated that the balanced dynamics can capture well the secondary circulation during the secondary eyewall formation in a full-physics model simulation. Nevertheless, two caveats exist in this study. First, the axisymmetric vortex structure used in the SE equation in this study included the effect of surface friction already, which could not be explicitly isolated from the effect of diabatic heating. Second, although the SE equation assumes the thermal wind balance of the basic vortex, the temperature field of the basic vortex in this study was not adjusted to satisfy the thermal wind balance implied by strong vertical shear in the boundary layer, as done in Bui et al. (2009). Note that Bui et al. (2009) showed in their appendix that the imbalance in the basic vortex could result in almost doubled strength in the boundary layer inflow. However, close inspection of their Figs. 6 and 12 indicates that the boundary layer inflow was still about 50% weaker in their balanced solution than in the full-physics model simulation. Furthermore, as indicated in section 4c, in Bui et al. (2009) the temperature anomalies were not adjusted to ensure the thermal wind balance after the reduction of vertical shear of tangential wind in their second step adjustment for the ellipticity condition. This means that the basic vortex in the boundary layer was not in thermal wind balance either in Bui et al. (2009). Finally, the possible feedback of surface friction to diabatic heating in the eyewall, as mentioned in Heng and Wang (2016b), has not been discussed in this study. This indirect effect of surface friction on diabatic heating and thus on TC intensification could be important but could not be isolated from other effects in one simulation analyzed in this study. This needs a more deliberate experimental design and is reserved for a future study. Acknowledgments. This study has been supported by the National Natural Science Foundation of China under Grants 41130964 and 41475091, NSF Grant AGS-1326524, Postgraduate Research and Practice Innovation Program of Jiangsu Province under Grant CXZZ12-0494, Special Fund for Meteorological Scientific Research in the Public Interest Grant GYHY201406006, and Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). Y. Wang acknowledges the helpful discussions with Prof. Chun-Chieh Wu and Dr. Hui Wang in other collaborative research, which motivated the current sensitivity calculations to address some issues that previously had not been addressed in Bui et al. (2009).
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