# 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

• 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
• 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。
• 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.

•  [1] Bister, M., and K. A. Emanuel, 1998: Dissipative heating and hurricane intensity. Meteor. Atmos. Phys., 65, 233–240.. [2] Braun, S. A., and W.-K. Tao, 2000: Sensitivity of high-resolution simulations of Hurricane Bob (1991) to planetary boundary layer parameterizations. Mon. Wea. Rev., 128, 3941–3961.. [3] Bryan, G. H., 2012: Effects of surface exchange coefficients and turbulence length scales on the intensity and structure of numerically simulated hurricanes. Mon. Wea. Rev., 140, 1125–1143.. [4] Bryan, G. H., 2013: Comments on ‘sensitivity of tropical-cyclone models to the surface drag coefficient’. Quart. J. Roy. Meteor. Soc., 139, 1957–1960. doi: 10.1002/qj.2066. [5] Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic numerical models. Mon. Wea. Rev., 130, 2917–2928.. [6] Bryan, G. H., and R. Rotunno, 2009a: The influence of near-surface, high-entropy air in hurricane eyes on maximum hurricane intensity. J. Atmos. Sci., 66, 148–158.. [7] Bryan, G. H., and R. Rotunno, 2009b: The maximum intensity of tropical cyclones in axisymmetric numerical model simulations. Mon. Wea. Rev., 137, 1770–1789.. [8] Bryan, G. H., and R. Rotunno, 2009c: Evaluation of an analytical model for the maximum intensity of tropical cyclones. J. Atmos. Sci., 66, 3042–3060.. [9] Chen, Y. J., F. Q. Zhang, B. W. Green, et al., 2018: Impacts of ocean cooling and reduced wind drag on Hurricane Katrina (2005) based on numerical simulations. Mon. Wea. Rev., 146, 287–306.. [10] Donelan, M. A., B. K. Haus, N. Reul, et al., 2004: On the limiting aerodynamic roughness of the ocean in very strong winds. Geophys. Res. Lett., 31, L18306.. [11] Emanuel, K., 2018: 100 years of progress in tropical cyclone research. Meteor. Monogr., 59, 15.1–15.68.. [12] Emanuel, K., and R. Rotunno, 2011: Self-stratification of tropical cyclone outflow. Part I: Implications for storm structure. J. Atmos. Sci., 68, 2236–2249.. [13] Emanuel, K. A., 1986: An air-sea interaction theory for tropical cyclones. Part I: Steady-state maintenance. J. Atmos. Sci., 43, 585–605.. [14] Emanuel, K. A., 1988: The maximum intensity of hurricanes. J. Atmos. Sci., 45, 1143–1155.. [15] Emanuel, K. A., 1995: Sensitivity of tropical cyclones to surface exchange coefficients and a revised steady-state model incorporating eye dynamics. J. Atmos. Sci., 52, 3969–3976.. [16] Frisius, T., and D. Schönemann, 2012: An extended model for the potential intensity of tropical cyclones. J. Atmos. Sci., 69, 641–661.. [17] Frisius, T., D. Schönemann, and J. Vigh, 2013: The impact of gradient wind imbalance on potential intensity of tropical cyclones in an unbalanced slab boundary layer model. J. Atmos. Sci., 70, 1874–1890.. [18] Hill, K. A., and G. M. Lackmann, 2009: Analysis of idealized tropical cyclone simulations using the weather research and forecasting model: Sensitivity to turbulence parameterization and grid spacing. Mon. Wea. Rev., 137, 745–765.. [19] Holland, G. J., 1997: The maximum potential intensity of tropical cyclones. J. Atmos. Sci., 54, 2519–2541.. [20] Ma, Z. H., and J. F. Fei, 2022: A comparison between moist and dry tropical cyclones: the low effectiveness of surface sensible heat flux in storm intensification. J. Atmos. Sci., 79, 31–49.. [21] Ma, Z. H., J. F. Fei, Y. L. Lin, et al., 2020: Modulation of clouds and rainfall by tropical cyclone’s cold wakes. Geophys. Res. Lett., 47, e2020GL088873.. [22] Ma, Z. H., Z. L. Zhang, J. F. Fei, et al., 2021: Imprints of tropical cyclones on structural characteristics of mesoscale oceanic eddies over the western North Pacific. Geophys. Res. Lett., 48, e2021GL092601.. [23] Malkus, J. S., and H. Riehl, 1960: On the dynamics and energy transformations in steady-state hurricanes. Tellus, 12, 1–20.. [24] Montgomery, M. T., M. M. Bell, S. D. Aberson, et al., 2006: Hurricane Isabel (2003): New insights into the physics of intense storms. Part I: Mean vortex structure and maximum intensity estimates. Bull. Amer. Meteor. Soc., 87, 1335–1348.. [25] Montgomery, M. T., R. K. Smith, and S. V. Nguyen, 2010: Sensitivity of tropical-cyclone models to the surface drag coefficient. Quart. J. Roy. Meteor. Soc., 136, 1945–1953. doi: 10.1002/qj.702. [26] Nystrom, R. G., R. Rotunno, C. A. Davis, et al., 2020: Consistent impacts of surface enthalpy and drag coefficient uncertainty between an analytical model and simulated tropical cyclone maximum intensity and storm structure. J. Atmos. Sci., 77, 3059–3080.. [27] Ooyama, K., 1969: Numerical simulation of the life cycle of tropical cyclones. J. Atmos. Sci., 26, 3–40.. [28] Peng, K., R. Rotunno, and G. H. Bryan, 2018: Evaluation of a time-dependent model for the intensification of tropical cyclones. J. Atmos. Sci., 75, 2125–2138.. [29] Persing, J., and M. T. Montgomery, 2003: Hurricane superintensity. J. Atmos. Sci., 60, 2349–2371.. [30] Persing, J., M. T. Montgomery, J. C. McWilliams, et al., 2013: Asymmetric and axisymmetric dynamics of tropical cyclones. Atmos. Chem. Phys., 13, 12,299–12,341.. [31] Rosenthal, S. L., 1971: The response of a tropical cyclone model to variations in boundary layer parameters, initial conditions, lateral boundary conditions, and domain size. Mon. Wea. Rev., 99, 767–777.. [32] Rotunno, R., and K. A. Emanuel, 1987: An air–sea interaction theory for tropical cyclones. Part II: Evolutionary study using a nonhydrostatic axisymmetric numerical model. J. Atmos. Sci., 44, 542–561.. [33] Shapiro, L. J., and H. E. Willoughby, 1982: The response of balanced hurricanes to local sources of heat and momentum. J. Atmos. Sci., 39, 378–394.. [34] Smith, R. K., M. T. Montgomery, and G. L. Thomsen, 2014: Sensitivity of tropical-cyclone models to the surface drag coefficient in different boundary-layer schemes. Quart. J. Roy. Meteor. Soc., 140, 792–804. [35] Wang, Y. Q., and J. Xu, 2010: Energy production, frictional dissipation, and maximum intensity of a numerically simulated tropical cyclone. J. Atmos. Sci., 67, 97–116.. [36] Zhang, J. S., Y. L. Lin, and Z. H. Ma, 2021: Footprint of tropical cyclone cold wakes on top-of-atmosphere radiation. Geophys. Res. Lett., 48, e2021GL094705..
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## 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.

Reference (36)

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