Uncertainty in TC Maximum Intensity with Fixed Ratio of Surface Exchange Coefficients for Enthalpy and Momentum

洋面焓与动量交换系数比不变条件下的热带气旋最大强度不确定性

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  • Corresponding author: Zhanhong MA, mazhanhong17@nudt.edu.cn
  • Funds:

    Supported by the National Natural Science Foundation of China (42022033 and 41875062) and Natural Science Foundation of Hunan Province, China (2020JJ3040)

  • doi: 10.1007/s13351-022-1120-8

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  • The classical tropical cyclone (TC) maximum intensity theory of Emanuel suggests that the maximum azimuthal wind of TC depends linearly on the ratio of surface exchange coefficients for enthalpy and momentum (Ck and Cd). In this study, a series of sensitivity experiments are conducted with the three-dimensional Cloud Model 1 (CM1), by fixing the ratio of Ck/Cd but varying the specific values of Ck and Cd simultaneously. The results show significant variations in the simulated TC maximum intensity by varying Ck and Cd, even if their ratio is fixed. Overall, the maxi-mum intensity increases steadily with increasing Ck and Cd when their value is smaller than 1.00 × 10−3, and then this increasing trend slows down with further increases in the coefficients. Two previous theoretical frameworks—one based on gradient wind balance and the other incorporating the unbalanced terms—are applied to calculate the maximum potential intensity (PI). The calculated value of the former shows little variation when varying the specific values of Ck and Cd, while the latter shows larger values with increases in both Ck and Cd. Further examination suggests that the unbalanced effect plays a key role in contributing to the increasing intensity with increasing Ck and Cd.
    经典的热带气旋最大强度理论表明热带气旋的最大切向风速正比于洋面焓交换系数(Ck)与动量交换系数(Cd)的比值。本文利用三维的CM1模式进行一系列敏感性实验,各组实验在Ck/Cd不变的基础上采用不同的CkCd。结果表明,在Ck/Cd不变的条件下模拟得到的热带气旋最大强度仍随Ck与Cd而显著的变化。整体而言,当CkCd小于1.00 × 10−3时最大强度的增长趋势最为明显,而后随着CkCd的增大而逐渐放缓。此外本文采用了两种理论框架计算了热带气旋的最大可能强度(PI),一种基于梯度风平衡而另一种考虑了非平衡效应。随CkCd的增大前一种计算的PI的变化较小,而后一种的PI出现了显著的增大。进一步的分析表明非平衡效应对这一现象起到了关键的作用。旋最大强度仍随CkCd而显著的变化。整体而言,当CkCd小于1.00 × 10−3时最大强度的增长趋势最为明显,而后随着CkCd的增大而逐渐放缓。此外本文采用了两种理论框架计算了热带气旋的最大可能强度(PI),一种基于梯度风平衡而另一种考虑了非平衡效应。随CkCd的增大前一种计算的PI的变化较小,而后一种的PI出现了显著的增大。进一步的分析表明非平衡效应对这一现象起到了关键的作用。
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  • Fig. 1.  (a) Time series of the maximum azimuthally averaged tangential wind speed and (b) the steady-state maximum azimuthally averaged tangential wind speed (Vmax, red line) and 10-m wind speed (V10max, blue line) in seven sensitivity simulations with a constant Ck/Cd of 1.3. Cd varies from 0.50 × 10−3 to 2.00 × 10−3 at an interval of 0.25 × 10−3.

    Fig. 2.  Contours of azimuthally averaged angular momentum M (red contours with intervals of 2 × 105 m2 s−1) and saturated moist entropy S* (blue contours with intervals of 10 J kg−1 K−1) in a vertical section of steady-state TCs with varied Cd and Ck/Cd kept constant at 1.3: (a) Cd = 0.50 × 10−3, (b) Cd = 0.75 × 10−3, (c) Cd = 1.00 × 10−3, (d) Cd = 1.25 × 10−3, (e) Cd = 1.50 × 10−3, (f) Cd = 1.75 × 10−3, and (g) Cd = 2.00 × 10−3. The black dashed line denotes the traces of air particles through the location of maximum tangential wind in steady-state TCs.

    Fig. 3.  Variations of (a) simulated azimuthally averaged maximum tangential wind (Vmax) and gradient wind speed (Vgmax) in conjunction with the calculation results of BE98 and BR09c (PI98 and PI09c), and (b) values of VmaxVgmax and PI09c − PI98 with increasing Cd when Ck/Cd remains constant at 1.3 in steady-state TCs.

    Fig. 4.  Variations of azimuthally averaged (a) temperature at the top of the boundary layer (Tb), (b) outflow temperature (Tout), (c) the difference between the outflow temperature and temperature at the top of the boundary layer (Tb Tout), and (d) the enthalpy disequilibrium ($k_{\rm{s}}^* $ka) with the increase of Cd when Ck/Cd remains constant at 1.3 in steady-state TCs.

    Fig. 5.  Variations of the azimuthally averaged (a) unbalanced term (αrmaxwmaxηmax), (b) radius of Vmax (rmax), (c) vertical wind speed at the location of Vmax (wmax), (d) azimuthal vorticity (ηmax), (e) vertical shear of radial wind ($ \text{∂} $u/$ \text{∂} $z), and (f) horizontal shear of vertical wind ($ \text{∂} $w/$ \text{∂} $r) with the increase in Cd when Ck/Cd is kept constant at 1.3 in steady-state TCs.

    Fig. 6.  Vertical sections of the azimuthally averaged ratio of tangential wind (V) to gradient wind (Vg) (shading) and agradient force (contours) with different Cd when Ck/Cd remains constant at 1.3 in steady-state TCs: (a) Cd = 0.50 × 10−3, (b) Cd = 0.75 × 10−3, (c) Cd = 1.00 × 10−3, (d) Cd = 1.25 × 10−3, (e) Cd = 1.50 × 10−3, (f) Cd = 1.75 × 10−3, and (g) Cd = 2.00 × 10−3. The horizontal coordinate is the ratio of r to rmax.

    Fig. 7.  Variations in the (a) maximum and (b) minimum azimuthally averaged ratio of V to Vg from Figs. 6a–g.

    Fig. 8.  Vertical sections of azimuthally averaged vertical wind (red) and radial wind (blue) with different Cd when Ck/Cd remains constant at 1.3 in steady-state TCs: (a) Cd = 1.00 × 10−3 and (b) Cd = 2.00 × 10−3. The negative values are represented by dashed contours. The black dashed line denotes the traces of air particles through the location of maximum tangential wind in steady-state TCs.

    Fig. 9.  Radial distributions of azimuthally averaged (a) enthalpy flux and (b) enthalpy disequilibrium in steady-state TCs with different Cd representations when Ck/Cd remains constant at 1.3. The horizontal coordinate is the ratio of r to rmax.

    Fig. 10.  As in Fig. 3, but for experiments with Ck/Cd held constant at 0.8.

    Table 1.  Values of Ck and Cd in different sensitivity experiments

    Exp. No.1234567
    Ck (10−3)0.65 0.9751.30 1.6251.95 2.2752.60
    Cd (10−3)0.500.751.001.251.501.752.00
    Download: Download as CSV

    Table 2.  Values of Ck and Cd in sensitivity testing for Ck/Cd

    Exp. No.123456
    ${C }_{k }\;\text{(}{\text{10} }^{-\text{3} }\text{)}$0.600.801.001.201.401.60
    ${C} _{d}\; \text{(}{\text{10} }^{-\text{3} }\text{)}$0.751.001.251.501.752.00
    Download: Download as CSV
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Uncertainty in TC Maximum Intensity with Fixed Ratio of Surface Exchange Coefficients for Enthalpy and Momentum

    Corresponding author: Zhanhong MA, mazhanhong17@nudt.edu.cn
  • College of Meteorology and Oceanography, National University of Defense Technology, Changsha 410005
Funds: Supported by the National Natural Science Foundation of China (42022033 and 41875062) and Natural Science Foundation of Hunan Province, China (2020JJ3040)

Abstract: The classical tropical cyclone (TC) maximum intensity theory of Emanuel suggests that the maximum azimuthal wind of TC depends linearly on the ratio of surface exchange coefficients for enthalpy and momentum (Ck and Cd). In this study, a series of sensitivity experiments are conducted with the three-dimensional Cloud Model 1 (CM1), by fixing the ratio of Ck/Cd but varying the specific values of Ck and Cd simultaneously. The results show significant variations in the simulated TC maximum intensity by varying Ck and Cd, even if their ratio is fixed. Overall, the maxi-mum intensity increases steadily with increasing Ck and Cd when their value is smaller than 1.00 × 10−3, and then this increasing trend slows down with further increases in the coefficients. Two previous theoretical frameworks—one based on gradient wind balance and the other incorporating the unbalanced terms—are applied to calculate the maximum potential intensity (PI). The calculated value of the former shows little variation when varying the specific values of Ck and Cd, while the latter shows larger values with increases in both Ck and Cd. Further examination suggests that the unbalanced effect plays a key role in contributing to the increasing intensity with increasing Ck and Cd.

洋面焓与动量交换系数比不变条件下的热带气旋最大强度不确定性

经典的热带气旋最大强度理论表明热带气旋的最大切向风速正比于洋面焓交换系数(Ck)与动量交换系数(Cd)的比值。本文利用三维的CM1模式进行一系列敏感性实验,各组实验在Ck/Cd不变的基础上采用不同的CkCd。结果表明,在Ck/Cd不变的条件下模拟得到的热带气旋最大强度仍随Ck与Cd而显著的变化。整体而言,当CkCd小于1.00 × 10−3时最大强度的增长趋势最为明显,而后随着CkCd的增大而逐渐放缓。此外本文采用了两种理论框架计算了热带气旋的最大可能强度(PI),一种基于梯度风平衡而另一种考虑了非平衡效应。随CkCd的增大前一种计算的PI的变化较小,而后一种的PI出现了显著的增大。进一步的分析表明非平衡效应对这一现象起到了关键的作用。旋最大强度仍随CkCd而显著的变化。整体而言,当CkCd小于1.00 × 10−3时最大强度的增长趋势最为明显,而后随着CkCd的增大而逐渐放缓。此外本文采用了两种理论框架计算了热带气旋的最大可能强度(PI),一种基于梯度风平衡而另一种考虑了非平衡效应。随CkCd的增大前一种计算的PI的变化较小,而后一种的PI出现了显著的增大。进一步的分析表明非平衡效应对这一现象起到了关键的作用。
    • Tropical cyclones (TCs) are one of the most influential weather systems resulted from atmosphere–ocean interactions (Ma et al., 2020, 2021; Zhang et al., 2021). The surface exchange coefficients for enthalpy (Ck) and momentum (Cd) largely determine the energy source and sink of a TC system (Malkus and Riehl, 1960; Ooyama, 1969; Rosenthal, 1971). In the theoretical framework of Emanuel (1986, hereafter E86), a conceptual model of a Carnot engine, based on the assumptions of hydrostatic, axisymmetric, and gradient wind balance, was put forward to describe the mature TC; an equation for the relationship between the maximum potential intensity (PI) and surface exchange coefficients was deduced with V2max varying proportionally to Ck/Cd. After that, more details were appended to the Carnot model to evaluate the PI of TCs (Rotunno and Emanuel, 1987; Emanuel, 1988). After including the impacts of eye dynamics, Emanuel (1995) modified previous equations and estimated the sensitivity of TCs to the exchange coefficients. Furthermore, Bister and Emanuel (1998, hereafter BE98) found that the maximum intensity of both numerical and analytical models can be approximately 20% greater compared with previous results when accounting for the dissipative heating. Emanuel and Rotunno (2011) examined the stratification of outflow and revised the equations for PI, which was proven to be more accurate with Ck/Cd < 1.0 (Peng et al., 2018).

      However, the theory of PI cannot explain the superintensity phenomenon observed or simulated by previous studies (Holland, 1997; Persing and Montgomery, 2003; Montgomery et al., 2006; Bryan and Rotunno, 2009a, b; Ma and Fei, 2022). In order to reveal the causes of superintensity, many efforts have been made in theoretical derivation and numerical simulations. Bryan and Rotunno (2009c, hereafter BR09c) found that the superintensity mainly results from the unsatisfied assumption of gradient balance in the simulation; thus, a new framework was then proposed to account for the unbalanced flow that was omitted in the former deduction, and the analytical results became more accurate than previous theoretical models. Additionally, the effects on the superintensity of other factors, including the convective available potential energy (CAPE) and the inclusion of horizontal momentum diffusion, have also been examined in recent papers (Frisius and Schönemann, 2012; Frisius et al., 2013). Through analyzing the distribution of energy production and dissipation in simulated TCs, Wang and Xu (2010) pointed out that the dissipation is greater than the production outside the eyewall and that the opposite is true internally, indicating that superintensity is mainly caused by the energy inflow from the outside of eyewall.

      In recent years, many studies have investigated the relationship of maximum intensity with Ck and Cd, and found some phenomena that are distinct from classical theories (Emanuel, 2018). Montgomery et al. (2010) applied a three-dimensional model and found that there is an optimal value of Ck/Cd for the maximum intensity, whereas the PI is proportional to (Ck/Cd)1/2 in classical theory. Nevertheless, Bryan (2013) suggested that the existence of thresholds in the maximum intensity is just caused by the relatively short integration time. Besides, Nystrom et al. (2020, hereafter N20) found that, different from the classical theories, the maximum intensity increases as the surface coefficients decrease with a constant Ck/Cd ratio in simulations, especially when Ck/Cd is larger than 1.0, owing to the changes in enthalpy disequilibrium near the radius of maximum intensity (rmax).

      In this paper, a three-dimensional, nonhydrostatic model was used to investigate the variation in the maximum intensity with changes in Ck and Cd when the Ck/Cd ratio remains constant—similar to the setting in N20 but with the numerical model extended to three dimensions, which can simulate the development of a TC with greater accuracy than an axisymmetric model. Meanwhile, the sensitivity of the TC structure to the surface exchange coefficients was also examined.

      The rest of the paper is organized as follows: Section 2 describes the numerical model and basic setup of parameters and two theoretical frameworks of PI. Section 3 shows the results from sensitivity simulation experiments and theoretical calculations with constant Ck/Cd. The conclusions and discussion are provided in Section 4.

    2.   Methods
    • The numerical model selected in this study is the nonhydrostatic numerical model called Cloud Model 1 (CM1; Bryan and Fritsch, 2002), version 20.1. A three-dimensional framework is selected on the basis that it produces more reliable simulations than an axisymmetric framework. The domain size is 3000 km × 3000 km × 20 km, with stretched grids in the vertical and horizontal directions. The vertical grid spacing is set to 50 m near the surface and stretched to 500 m above 5500 m. The horizontal grid spacing is set to 3 km within the inner 1200 km × 1200 km mesh grid, with the spacing gradually stretched to 15 km outside the distance of 1200 km from the domain center. The initial maximum wind speed of the vortex is 15 m s−1 at a radius of 75 km. As for the environmental parameters, the sea surface temperature (SST) is constant at 28°C, and the input sounding used here is the approximately moist-neutral sounding (Rotunno and Emanuel, 1987). We set the horizontal turbulence length to 1000 m and the vertical turbulence length to 50 m, which is recommended in the CM1 model for the most reasonable match to the observations (Bryan, 2012). This is different from several recent studies (Peng et al., 2018; N20). Dissipative heating is included in all the simulations because of its significant effects on the maximum intensity put forth in BE98. Other unmentioned parameters remain at their default values.

      A total of seven sensitivity experiments were carried out, with Cd varying from 0.50 × 10−3 to 2.00 × 10−3 at an interval of 0.25 × 10−3 by fixing Ck/Cd to be a constant of 1.3. The value of Ck/Cd was within the reasonable range of Ck/Cd as proposed by Emanuel (1995), and one that has also been used in previous TC modeling studies (e.g., Braun and Tao, 2000; Montgomery et al., 2010; Bryan, 2013). Table 1 lists the specific values for all experiments. All the simulations were integrated for 24 days, except the Cd0.0005 experiment (Cd = 0.50 × 10−3), which was integrated for 35 days. This ensured all storms had reached their steady-state maximum intensity during the simulation (Bryan, 2013). The TC intensity at any given time was defined as the maximum azimuthally averaged tangential wind speed (Vmax) found by traversing all the points in the mesh grid. The steady-state period was defined as the 24-h period with maximum average intensity, similar to the study of N20, and the maximum TC intensity was defined as the average intensity during the steady-state period.

      Exp. No.1234567
      Ck (10−3)0.65 0.9751.30 1.6251.95 2.2752.60
      Cd (10−3)0.500.751.001.251.501.752.00

      Table 1.  Values of Ck and Cd in different sensitivity experiments

    • In the classical Carnot engine theory proposed in E86, the PI is estimated by the equation

      $$ {V}^{{2}}=\frac{{C}_{k}}{C_{d}}\left({{T}}_{\rm{b}} - {{T}}_{\rm{out}}\right)\frac{{{k}}_{{\rm{s}}}^{{*}}-{{k}}_{{\rm{a}}}}{{{T}}_{\rm{s}}}, $$ (1)

      where Tb and Tout represent temperatures at the top of the boundary layer and the outflow region, respectively. The former is defined as the temperature at the location of Vmax following BR09c, and the latter is defined as the area-averaged temperature within radii of 250–300 km from the TC center along the trajectories of air parcels originating from Vmax. Ts is the SST, and $k_{\rm{s}}^* $ka is the enthalpy disequilibrium between the air at the lowest model layer and the sea surface at the radius of maximum tangential wind speed. The enthalpy is defined as k = cpT(1 – q) + (clT + Lv)q, where cp denotes the heat capacity of dry air at constant pressure, q is the specific humidity of air, cl denotes the heat capacity of liquid water, and Lv is the latent heat of vaporization and takes the value of 2.5 × 106 J kg−1 (Emanuel, 1995). With dissipative heating, the equation can be modified to the form in BE98:

      $$ {V}^{{2}}=\frac{{C}_{k}}{C_{d}}\left({{T}}_{\rm{b}} - {{T}}_{\rm{out}}\right)\frac{{{k}}_{{\rm{s}}}^{{*}}-{{k}}_{{\rm{a}}}}{{{T}}_{\rm{out}}}. $$ (2)

      In addition, as shown in BR09c, by considering the contribution of unbalanced flow to the maximum intensity in the boundary layer, the equation can be expressed as

      $$ {V}^{\rm{2}}=\frac{{{C}}_{{k}}}{{{C}}_{{d}}}\left({{T}}_{{\rm{b}}}-{{T}}_{\rm{out}}\right)\frac{{{k}}_{{\rm{s}}}^{\rm{*}}-{{k}}_{{\rm{a}}}}{{{T}}_{\rm{out}}} + \frac{{{T}}_{{\rm{s}}}}{{{T}}_{\rm{out}}}{{r}}_{\rm{max}}{{w}}_{\rm{max}}{\eta }_{\rm{max}}, $$ (3)

      where rmax is the radius of Vmax, wmax is the vertical wind speed at the location of Vmax, and ηmax is the azimuthal vorticity at the location of Vmax and ηmax = ∂u/∂z – ∂w/∂r. These three different equations are used to estimate the maximum intensity of the stimulated TCs and all the variations in the equations are the average values during the steady-state period.

    3.   Results
    • Figure 1a presents the time series of maximum wind in different simulations. All the storms tend to intensify at first and then basically reach their steady states. The time period required for a storm to reach its maximum intensity generally extends with decreasing Cd and Ck. The storm intensities experience notable fluctuations for the experiments with Cd larger than 1 × 10−3, suggesting that an integration longer than 10 days may be required for a storm to reach its maximum intensity. The TC intensification rates increase as the surface exchange coefficients increase, which is consistent with the notion that intensification rate is proportional to both Ck and Cd (Peng et al., 2018). Additionally, it can be seen that the maximum TC intensity also increases when Cd grows with constant Ck/Cd. This is different from the maximum intensity theory of E86, which predicts an invariable value if fixing Ck/Cd, regardless of the specific values of Ck and Cd.

      Figure 1.  (a) Time series of the maximum azimuthally averaged tangential wind speed and (b) the steady-state maximum azimuthally averaged tangential wind speed (Vmax, red line) and 10-m wind speed (V10max, blue line) in seven sensitivity simulations with a constant Ck/Cd of 1.3. Cd varies from 0.50 × 10−3 to 2.00 × 10−3 at an interval of 0.25 × 10−3.

      In order to quantitatively reflect the relationship between the maximum intensity and Ck and Cd, the variations of azimuthally averaged maximum tangential wind speed (Vmax) and maximum 10-m wind speed (V10max) are presented in Fig. 1b. The variable Vmax of TC varies proportionally with the surface exchange coefficients. When Cd is relatively large (Cd > 1.00 × 10−3), the increasing trend of Vmax gradually slows down. The changes of V10max show the same pattern but with relatively smaller values than Vmax as a result of the surface friction effect. Besides, the values of V10max are almost constant with Cd larger than 1.00 × 10−3. This is different from the results of N20, who found an increasing maximum intensity due to increased $k_{\rm{s}}^* $ka as Cd decreased when Ck/Cd was kept constant, especially with Ck/Cd > 1.0.

    • As one of the primary assumptions in theoretical models, moist slantwise neutrality is the foundation of equations for maximum PI. To examine whether this assumption is satisfied in all the simulations, we drew the distribution of azimuthally averaged angular momentum (M) and saturated moist entropy (S*) in a vertical section of steady-state TCs (Fig. 2). The contour lines of M and S* are approximately congruent along the traces of particles through the location of maximum tangential wind, which indicates that moist slantwise neutrality was basically met above the boundary layer for all the sensitivity experiments. Therefore, Eqs. (2) and (3) can be applied to calculate the maximum PI of TCs. However, the moist slantwise neutrality is not met well in Fig. 2a, probably because the TC in the Cd0.0005 experiment had not fully reached its steady-state structures due to the relatively low intensification rate. It is noteworthy that the structures of M and S* also change with the variation in the surface exchange coefficients. When Cd increases with constant Ck/Cd, the radial gradients of both M and S* become larger. This is related to the fact that both stronger diabatic heating and frictional forcing cause stronger radial inflow in the boundary layer (Shapiro and Willoughby, 1982). Besides, resembling the variation trend of maximum intensity, when Cd is greater than 1.0 × 10−3, the variations in the distribution of M and S* tend to slow down as well.

      Figure 2.  Contours of azimuthally averaged angular momentum M (red contours with intervals of 2 × 105 m2 s−1) and saturated moist entropy S* (blue contours with intervals of 10 J kg−1 K−1) in a vertical section of steady-state TCs with varied Cd and Ck/Cd kept constant at 1.3: (a) Cd = 0.50 × 10−3, (b) Cd = 0.75 × 10−3, (c) Cd = 1.00 × 10−3, (d) Cd = 1.25 × 10−3, (e) Cd = 1.50 × 10−3, (f) Cd = 1.75 × 10−3, and (g) Cd = 2.00 × 10−3. The black dashed line denotes the traces of air particles through the location of maximum tangential wind in steady-state TCs.

    • Figure 3a presents the analysis results of PI for steady-state TCs in seven sensitivity experiments, derived from Eqs. (2) and (3), respectively (hereafter PI98 and PI09c). The variable Vmax and azimuthally averaged maximum gradient wind (Vgmax) are also shown for comparison. When the ratio of Ck to Cd is kept constant, the result of PI98 almost remains unchanged as Cd increases, with just a slight decrease when Cd is larger than 1.00 × 10−3 (less than 10%). In contrast to PI98, when Ck/Cd is kept constant at 1.3, Vgmax varies proportionally to Cd and the value of Vgmax is relatively less than PI98 with Cd ≤ 1.50 × 10−3, as the difference between PI98 and Vgmax decreases when Cd grows. Thus, PI98 may reasonably reflect the balanced effect, especially when Cd is relatively large. Furthermore, PI09c, which incorporates the unbalanced effect, increases firstly and then basically levels off with increasing Cd. This trend is consistent with the changes of Vmax. For all the sensitivity experiments, the values of PI09c are larger than Vmax. However, with the increment in Cd, the discrepancy between PI09c and Vmax also decreases sharply, implying a better performance of PI09c in capturing Vmax when Cd increases.

      Figure 3.  Variations of (a) simulated azimuthally averaged maximum tangential wind (Vmax) and gradient wind speed (Vgmax) in conjunction with the calculation results of BE98 and BR09c (PI98 and PI09c), and (b) values of VmaxVgmax and PI09c − PI98 with increasing Cd when Ck/Cd remains constant at 1.3 in steady-state TCs.

      To examine the unbalanced effect with varying Cd and Ck, the differences of VmaxVgmax and PI09c − PI98 were computed for the sensitivity experiments (Fig. 3b). Overall, both increase rapidly with Cd, and then level off when Cd exceeds a certain value (1.00 × 10−3 for VmaxVgmax and 1.50 × 10−3 for PI09c − PI98). Their difference can be neglected when Cd is less than 1.00 × 10−3 and gradually increases as Cd becomes larger. These features indicate that the unbalanced effect also increases as Cd increases and then slows down with relatively large Cd when Ck/Cd remains constant at 1.3. The contribution of the unbalanced effect may be one of the factors causing the increase in the model maximum intensity when Cd increases with constant Ck/Cd.

    • In this subsection, the components comprising Eq. (3) are examined in order to reveal how the balanced and unbalanced effects change with the increase in the surface enthalpy and momentum exchange coefficients when Ck/Cd remains constant. Referring to the deduction process in BR09c, the right-hand side (rhs) of Eq. (3) is composed of two main parts, representing the impact of the balanced and unbalanced effect on TCs, respectively. The first part on the rhs of Eq. (3) generally reflects the balanced effect on TCs and is identical to the term on the rhs of Eq. (2). Figure 4 presents the changes in physical quantities in this term. When Cd grows with constant Ck/Cd, the variation in Tb is small and can be neglected (Fig. 4a). The outflow temperature decreases monotonically with Cd, from roughly 216 to 204 K (Fig. 4b). Meanwhile, the value of TbTout increases with increasing Cd when Ck/Cd remains constant; the range of variation is relatively large, increasing by 18% with Cd varying from 0.50 × 10−3 to 2.00 × 10−3 (Fig. 4c). Overall, the changes of both Tb − Tout and Tout have positive impacts on the increase in PI98. Nevertheless, $k_{\rm{s}}^* $ka decreases rapidly as Cd increases, with the decreasing amplitude attaining ~25% when Cd increases from 0.50 × 10−3 to 2.00 × 10−3 (Fig. 4d), which is in agreement with the trend in N20. The decreasing trend of $k_{\rm{s}}^* $ka eliminates the positive effects of Tb − Tout and Tout and eventually results in the slight reduction in PI98 when Cd > 1.00 × 10−3 (Fig. 3a).

      Figure 4.  Variations of azimuthally averaged (a) temperature at the top of the boundary layer (Tb), (b) outflow temperature (Tout), (c) the difference between the outflow temperature and temperature at the top of the boundary layer (Tb Tout), and (d) the enthalpy disequilibrium ($k_{\rm{s}}^* $ka) with the increase of Cd when Ck/Cd remains constant at 1.3 in steady-state TCs.

      Additionally, as shown in Fig. 5a, the value of the second term in Eq. (3), which represents the unbalanced effect in TCs, rises steadily from almost zero to values over 6000 as Cd increases, resembling the pattern of difference between PI09c and PI98 (Fig. 3b). To better understand the mechanism of the variation of the unbalanced effect, all the quantities in this term were investigated. The first factor α is the ratio of Ts to Tout. As Ts is constant in all the simulations, α changes inversely to Tout and thus has positive impacts on the unbalanced effect due to the decreasing Tout (Fig. 4b). The second part in the unbalanced term, rmax, is inversely proportional to Cd with constant Ck/Cd and then gradually levels off when Cd is comparatively large (Fig. 5b). This is consistent with the consensus that surface friction leads to a contraction of the eyewall. Additionally, wmax changes proportionally to Cd and then plateaus with Cd larger than 1.25 × 10−3 (Fig. 5c). The same pattern has also been found in the variation of ηmax but with a notable increasing trend even if Cd is relatively large (Fig. 5d). The variable ηmax is formed by two parts, including the vertical shear of radial wind ($ \text{∂} $u/$ \text{∂} $z) and horizontal shear of vertical wind ($ \text{∂} $w/$ \text{∂} $r) at the location of Vmax (Figs. 5e, f). These two components are separately examined in Figs. 5e, f. The vertical shear of radial wind, $ \text{∂} $u/$ \text{∂} $z, also increases with Cd. In contrast with $ \text{∂} $u/$ \text{∂} $z, the magnitude of $ \text{∂} $w/$ \text{∂} $r is relatively small and can be omitted in ηmax (Fig. 5f), which implies that the variation of ηmax can be represented by the changes in $ \text{∂} $u/$ \text{∂} $z. In conclusion, although rmax decreases with Cd, wmax and ηmax as well as α all have positive impacts on the unbalanced term. Therefore, the value of the unbalanced term in Eq. (3) increases with the growth in Cd when the ratio of Ck to Cd is kept to 1.3, indicating the enhancement of the unbalanced effect eventually causes the increase in maximum intensity in steady-state TCs. In particular, wmax and ηmax increase significantly with Cd when Ck/Cd is kept the same, thereby dominating the increasing trend of the unbalanced effect.

      Figure 5.  Variations of the azimuthally averaged (a) unbalanced term (αrmaxwmaxηmax), (b) radius of Vmax (rmax), (c) vertical wind speed at the location of Vmax (wmax), (d) azimuthal vorticity (ηmax), (e) vertical shear of radial wind ($ \text{∂} $u/$ \text{∂} $z), and (f) horizontal shear of vertical wind ($ \text{∂} $w/$ \text{∂} $r) with the increase in Cd when Ck/Cd is kept constant at 1.3 in steady-state TCs.

    • In order to further examine the variation of unbalanced flow as Ck and Cd increase, the low-level structures of steady-state TCs in all the simulations were analyzed and the results are reported in this subsection. The agradient force can be written as:

      $$ {F}_{\rm{a}}=fv+\frac{{v}^{\text{2}}}{r}-{c}_{p}{\theta} _{\rm{v}}\frac{\text{∂}{\text{π}}{'}}{\text{∂}r}, $$ (4)

      where v is the tangential wind speed, r denotes the radius,θv is the virtual potential temperature of air, and $ {\text{π}}^{\text{′}} $ is the nondimensional pressure. Figure 6 presents the distribution of the azimuthally averaged agradient force and the ratio of tangential wind (V) to gradient wind (Vg) in the vertical section of steady-state TCs in the sensitivity experiments. Overall, there are some similar features for all the experiments: a high positive value of the agradient force appears near the location of maximum azimuthal wind, in correspondence with the large supergradient wind in this area. Besides, owing to the friction effect from the surface, negative values of Fa and subgradient flow occur just outside rmax near the surface. Furthermore, it can be qualitatively seen that the intensity of both subgradient and supergradient wind increases as Cd increases with constant Ck/Cd, implying an increasing contribution of the unbalanced effect.

      Figure 6.  Vertical sections of the azimuthally averaged ratio of tangential wind (V) to gradient wind (Vg) (shading) and agradient force (contours) with different Cd when Ck/Cd remains constant at 1.3 in steady-state TCs: (a) Cd = 0.50 × 10−3, (b) Cd = 0.75 × 10−3, (c) Cd = 1.00 × 10−3, (d) Cd = 1.25 × 10−3, (e) Cd = 1.50 × 10−3, (f) Cd = 1.75 × 10−3, and (g) Cd = 2.00 × 10−3. The horizontal coordinate is the ratio of r to rmax.

      To quantitatively evaluate the variation of supergradient and subgradient wind, Fig. 7 presents the changes in maximum and minimum values of V/Vg in Fig. 6. With the growth of Cd, the maximum ratio increases from approximately 1.10 when Cd = 0.50 × 10−3 to over 1.30 when Cd = 2.00 × 10−3 (Fig. 7a). Meanwhile, the minimum value reduces monotonically as well, from larger than 0.80 to nearly 0.60 (Fig. 7b). Both the increase in maximum values and decrease in minimum values of V/Vg indicate intensification of the unbalanced flow when Cd increases as Ck/Cd remains constant.

      Figure 7.  Variations in the (a) maximum and (b) minimum azimuthally averaged ratio of V to Vg from Figs. 6a–g.

      Figure 8 presents the steady-state distributions of azimuthally averaged vertical and radial wind in a vertical section with Cd = 1.00 × 10−3 or 2.00 × 10−3. As Cd increases, the strength of vertical motion in the eyewall grows. In addition, the height of the ascending area also expands as Cd changes from 1.00 × 10−3 to 2.00 × 10−3. Therefore, the air particles in the eyewall can reach higher heights with increasing Cd, which leads to the decrease in Tout (Fig. 4b). This can increase the efficiency of the Carnot engine, as indicated by Eq. (2). The increases in surface wind V10 and Ck can also lead to an increase in surface enthalpy flux (Fig. 9a), although the enthalpy disequilibrium of $k_{\rm{s}}^* $ka decreases with increasing Cd when Ck/Cd remains constant (Fig. 9b). This result is largely consistent with the hypothesis proposed in N20. Besides, the maximum values of vertical wind at low levels of the boundary layer can be observed in Figs. 8a, b, which is caused by the large gradient of radial inflow and the eruption of ascending flow near the rmax in the boundary layer (Smith et al., 2014). Another nonnegligible feature is the relatively larger radial inflow and vertical shear of radial wind speed with Cd = 2.00 × 10−3 than that with Cd = 1.00 × 10−3 in the boundary layer, as a result of the increasing friction effect from the surface, which also implies enhancement of the unbalanced effect.

      Figure 8.  Vertical sections of azimuthally averaged vertical wind (red) and radial wind (blue) with different Cd when Ck/Cd remains constant at 1.3 in steady-state TCs: (a) Cd = 1.00 × 10−3 and (b) Cd = 2.00 × 10−3. The negative values are represented by dashed contours. The black dashed line denotes the traces of air particles through the location of maximum tangential wind in steady-state TCs.

      Figure 9.  Radial distributions of azimuthally averaged (a) enthalpy flux and (b) enthalpy disequilibrium in steady-state TCs with different Cd representations when Ck/Cd remains constant at 1.3. The horizontal coordinate is the ratio of r to rmax.

    • To examine the sensitivity of the above results to the specific value of Ck/Cd, a set of additional experiments were conducted with the value of Ck/Cd being changed to 0.8, as listed in Table 2. The Cd0.0005 experiment was removed because of its very small Ck and Cd, which will result in intensification that is too slow to reach maximum intensity within 24 days (not shown).

      Exp. No.123456
      ${C }_{k }\;\text{(}{\text{10} }^{-\text{3} }\text{)}$0.600.801.001.201.401.60
      ${C} _{d}\; \text{(}{\text{10} }^{-\text{3} }\text{)}$0.751.001.251.501.752.00

      Table 2.  Values of Ck and Cd in sensitivity testing for Ck/Cd

      Figure 10a presents the variation of Vmax and Vgmax together with PI98 and PI09c in these experiments. Overall, Vmax increases as Cd increases when Ck/Cd is set to 0.8, consistent with Fig. 3a. The discrepancy between PI98 and Vgmax also decreases with increasing Cd, in conjunction with the discrepancy between PI09c and Vmax. Furthermore, the increasing trend of VmaxVgmax and PI09c − PI98 remains remarkable in Fig. 10b, implying enhanced unbalanced effects with increasing Cd when Ck/Cd is set to 0.8—the same as the results when Ck/Cd is kept at 1.3. These results with different Ck/Cd indicate that the main results are insensitive to the specific value of Ck/Cd.

      Figure 10.  As in Fig. 3, but for experiments with Ck/Cd held constant at 0.8.

    4.   Conclusions
    • In this study, we utilized an idealized Cloud Model (CM1, version 20.1), similar to that of N20 but extended to a three-dimensional framework, to investigate the sensitivity of maximum intensity to the changes in surfaces exchange coefficients for enthalpy (Ck) and momentum (Cd) with constant Ck/Cd. Seven sensitivity experiments with Ck/Cd fixed at a constant of 1.3, according to the reliable range proposed in previous papers (Emanuel, 1995; Bryan, 2012), were designed and all the TCs were found to reach a steady state after a long enough integration time of 24 days (extended to 35 days for the Cd0.0005 experiment).

      The simulated results showed that the maximum intensity increases with the growth in Cd when Ck/Cd remains constant and the increasing trend gradually slows down when Cd is larger than 1.00 × 10−3, with a similar pattern found in the variation of V10max. This phenomenon is different from the conclusions of classical theories and N20, which predict an unvarying and decreasing maximum intensity with increasing Cd but fixed Ck/Cd, respectively. Subsequent to the numerical simulations, the theoretical frameworks of BE98 and BR09c were applied to calculate the PI of the steady-state TCs in seven sensitivity experiments. Compared with the results of simulations, the outcomes of BE98 (PI98) and BR09c (PI09c) can reasonably reflect the variation trend of simulated maximum gradient wind (Vgmax) and the maximum tangential wind (Vmax) with the growth in surface exchange coefficients, especially when Cd is larger than 1.00 × 10−3. Furthermore, when Ck/Cd remains constant, the PI98 remains roughly invariant despite the changes of both exchange coefficients, in contrast with the increment in PI09c, the latter of which includes the impact of the unbalanced effect. As Cd increases, the difference between PI09c and PI98 also grows, consistent with the variation trend of the difference between Vmax and Vgmax, indicating that enhancement of the unbalanced effect is the factor causing the increase in maximum intensity.

      Through an examination of different components in the equations from BR09c, $k_{\rm{s}}^* $ka gradually decreases to an extent up to about 25%, in agreement with the trend in N20. This trend largely eliminates the positive impacts of Tout and TbTout and eventually results in the negligible changes in the balanced component. On the contrary, owing to the significant increment of wmax and ηmax, the term representing the unbalanced effect increases notably when Cd grows with constant Ck/Cd. An examination of the thermodynamic structure shows that the supergradient flow near the top of boundary layer increases while the subgradient flow near the surface decreases when Ck and Cd grow, corresponding to the changes in agradient force. These features also reflect that the unbalanced effect increases as Cd grows when Ck/Cd is kept unchanged. After analyzing the variation of secondary circulation in steady-state TCs, we found that the radial inflow and vertical shear of radial wind speed are both enhanced with increasing Cd, which corresponds to the development of subgradient flow near the surface. This might mainly result from intensification of the surface friction effect reflected by increasing Cd. At the same time, the convection in the eyewall was also found to strengthen, due to the increasing energy flux from the surface induced by the growth in Ck. According to the second term on the rhs of Eq. (3), the enhancement of the unbalanced effect is mainly caused by the intensification in both convection and the vertical shear of radial wind speed, as a result of increasing surface exchange coefficients with constant Ck/Cd.

      Overall, the results of this study demonstrate the significant impact of surface exchange coefficients on the enhancement of the unbalanced effect, and provide a different aspect to investigate the sensitivity of maximum intensity to the surface exchange coefficients. We speculate that the different conclusion from N20 could be related to the intrinsic discrepancy between axisymmetric and three-dimensional numerical models, as a result of the simplified physical processes in axisymmetric models compared with three-dimensional ones (Persing et al., 2013), or to the discrepancies of horizontal and vertical turbulence lengths, which have a significant impact on the intensity of unbalanced flow even with a constant Ck/Cd (Bryan, 2012, their Fig. 11). The inclusion of dissipative heating may also be a factor influencing the conclusions. Additionally, the dependence of surface exchange coefficients on the wind speed, compared with the constant values in this paper, could impact the results as well (Donelan et al., 2004; Hill and Lackmann, 2009; Chen et al., 2018). To further examine these conjectures, more numerical simulations and investigations are required in future work.

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