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

Hexin YE, Zhanhong MA, and Jianfang FEI

College of Meteorology and Oceanography, National University of Defense Technology, Changsha 410005

ABSTRACT

Key words: tropical cyclone, maximum intensity, surface exchange coefficients, unbalanced effect

1. Introduction

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 withV2maxvarying proportionally toCk/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 withCk/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 withCkandCd, 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 ofCk/Cdfor the maximum intensity,whereas the PI is proportional to (Ck/Cd)1/2in 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 constantCk/Cdratio in simulations, especially whenCk/Cdis 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 inCkandCdwhen theCk/Cdratio 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 constantCk/Cd.The conclusions and discussion are provided in Section 4.

2. Methods

2.1 Setup of model and parameters

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 threedimensional 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−1at 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, withCdvarying from 0.50 × 10−3to 2.00 × 10−3at an interval of 0.25 × 10−3by fixingCk/Cdto be a constant of 1.3. The value ofCk/Cdwas within the reasonable range ofCk/Cdas 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 averageintensity, similar to the study of N20, and the maximum TC intensity was defined as the average intensity during the steady-state period.

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

2.2 Theoretical framework for PI

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

wherermaxis the radius ofVmax,wmaxis the vertical wind speed at the location ofVmax, andηmaxis the azimuthal vorticity at the location ofVmaxandηmax= ∂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

3.1 Intensity

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 decreasingCdandCk.The storm intensities experience notable fluctuations for the experiments withCdlarger 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 bothCkandCd(Peng et al., 2018). Additionally, it can be seen that the maximum TC intensity also increases whenCdgrows with constantCk/Cd. This is different from the maximum intensity theory of E86, which predicts an invariable value if fixingCk/Cd, regardless of the specific values ofCkandCd.

In order to quantitatively reflect the relationship between the maximum intensity andCkandCd, the variations of azimuthally averaged maximum tangential wind speed (Vmax) and maximum 10-m wind speed (V10max) are presented in Fig. 1b. The variableVmaxof TC varies proportionally with the surface exchange coefficients. WhenCdis relatively large (Cd> 1.00 × 10−3), the increasing trend ofVmaxgradually slows down. The changes ofV10maxshow the same pattern but with relatively smaller values thanVmaxas a result of the surface friction effect.Besides, the values ofV10maxare almost constant withCdlarger than 1.00 × 10−3. This is different from the results of N20, who found an increasing maximum intensity due to increasedk∗s–kaasCddecreased whenCk/Cdwas kept constant, especially withCk/Cd> 1.0.

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.

3.2 Moist slantwise neutrality

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 ofMandS*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 ofMandS*also change with the variation in the surface exchange coefficients. WhenCdincreases with constantCk/Cd, the radial gradients of bothMandS*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, whenCdis greater than 1.0 ×10−3, the variations in the distribution ofMandS*tend to slow down as well.

3.3 Comparison with PI

Figure 3a presents the analysis results of PI for steadystate TCs in seven sensitivity experiments, derived from Eqs. (2) and (3), respectively (hereafter PI98and PI09c).The variableVmaxand azimuthally averaged maximum gradient wind (Vgmax) are also shown for comparison.When the ratio ofCktoCdis kept constant, the result of PI98almost remains unchanged asCdincreases, with just a slight decrease whenCdis larger than 1.00 × 10−3(less than 10%). In contrast to PI98, whenCk/Cdis kept constant at 1.3,Vgmaxvaries proportionally toCdand the value ofVgmaxis relatively less than PI98withCd≤ 1.50× 10−3, as the difference between PI98andVgmaxdecreases whenCdgrows. Thus, PI98may reasonably reflect the balanced effect, especially whenCdis relatively large. Furthermore, PI09c, which incorporates the unbalanced effect, increases firstly and then basically levels off with increasingCd. This trend is consistent with the changes ofVmax. For all the sensitivity experiments, the values of PI09care larger thanVmax. However, with the increment inCd, the discrepancy between PI09candVmaxalso decreases sharply, implying a better performance of PI09cin capturingVmaxwhenCdincreases.

To examine the unbalanced effect with varyingCdandCk, the differences ofVmax−Vgmaxand PI09c− PI98were computed for the sensitivity experiments (Fig. 3b). Overall, both increase rapidly withCd, and then level off whenCdexceeds a certain value (1.00 × 10−3forVmax−Vgmaxand 1.50 × 10−3for PI09c− PI98). Their difference can be neglected whenCdis less than 1.00 × 10−3and gradually increases asCdbecomes larger. These features indicate that the unbalanced effect also increases asCdincreases and then slows down with relatively largeCdwhenCk/Cdremains constant at 1.3. The contribution of the unbalanced effect may be one of the factors causing the increase in the model maximum intensity whenCdincreases with constantCk/Cd.

3.4 Physical quantities

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 Vmax − Vgmax 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 ( ks∗ – 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 asCdincreases, resembling the pattern of difference between PI09cand 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 factorαis the ratio ofTstoTout. AsTsis constant in all the simulations,αchanges inversely toToutand thus has positive impacts on the unbalanced effect due to the decreasingTout(Fig. 4b). The second part in the unbalanced term,rmax, is inversely proportional toCdwith constantCk/Cdand then gradually levels off whenCdis comparatively large (Fig. 5b). This is consistent with the consensus that surface friction leads to a contraction of the eyewall. Additionally,wmaxchanges proportionally toCdand then plateaus withCdlarger than 1.25 × 10−3(Fig. 5c). The same pattern has also been found in the variation ofηmaxbut with a notable increasing trend even ifCdis relatively large (Fig. 5d). The variableηmaxis formed by two parts, including the vertical shear of radial wind (∂u/∂z) and horizontal shear of vertical wind (∂w/∂r) at the location ofVmax(Figs. 5e, f).These two components are separately examined in Figs.5e, f. The vertical shear of radial wind, ∂u/∂z, also increases withCd. In contrast with ∂u/∂z, the magnitude of∂w/∂ris relatively small and can be omitted inηmax(Fig.5f), which implies that the variation ofηmaxcan be represented by the changes in ∂u/∂z. In conclusion, althoughrmaxdecreases withCd,wmaxandηmaxas well asαall have positive impacts on the unbalanced term. Therefore,the value of the unbalanced term in Eq. (3) increases with the growth inCdwhen the ratio ofCktoCdis kept to 1.3,indicating the enhancement of the unbalanced effect eventually causes the increase in maximum intensity in steady-state TCs. In particular,wmaxandηmaxincrease significantly withCdwhenCk/Cdis kept the same,thereby dominating the increasing trend of the unbalanced effect.

3.5 Structures of steady-state TCs

In order to further examine the variation of unbalanced flow asCkandCdincrease, 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:

wherevis the tangential wind speed,rdenotes the radius,θvis the virtual potential temperature of air, and π′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 ofFaand subgradient flow occur just outsidermaxnear the surface. Furthermore, it can be qualitatively seen that the intensity of both subgradient and supergradient wind increases asCdincreases with constantCk/Cd, implying an increasing contribution of the unbalanced effect.

To quantitatively evaluate the variation of supergradient and subgradient wind, Fig. 7 presents the changes in maximum and minimum values ofV/Vgin Fig. 6. With the growth ofCd, the maximum ratio increases from approximately 1.10 whenCd= 0.50 × 10−3to over 1.30 whenCd= 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 ofV/Vgindicate intensification of the unbalanced flow whenCdincreases asCk/Cdremains constant.

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 ( ∂u/ ∂z), and (f) horizontal shear of vertical wind ( ∂w/ ∂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.

Figure 8 presents the steady-state distributions of azimuthally averaged vertical and radial wind in a vertical section withCd= 1.00 × 10−3or 2.00 × 10−3. AsCdincreases, the strength of vertical motion in the eyewall grows. In addition, the height of the ascending area also expands asCdchanges from 1.00 × 10−3to 2.00 × 10−3.Therefore, the air particles in the eyewall can reach higher heights with increasingCd, which leads to the decrease inTout(Fig. 4b). This can increase the efficiency of the Carnot engine, as indicated by Eq. (2). The increases in surface windV10andCkcan also lead to an increase in surface enthalpy flux (Fig. 9a), although the enthalpy disequilibrium ofks∗–kadecreases with increasingCdwhenCk/Cdremains 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 thermaxin the boundary layer (Smith et al., 2014). Another nonnegligible feature is the relatively larger radial inflow and vertical shear of radial wind speed withCd= 2.00 × 10−3than that withCd= 1.00 × 10−3in the boundary layer, as a result of the increasing friction effect from the surface, which also implies enhancement of the unbalanced effect.

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.

3.6 Sensitivity test

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

Figure 10a presents the variation ofVmaxandVgmaxtogether with PI98and PI09cin these experiments. Overall,Vmaxincreases asCdincreases whenCk/Cdis set to 0.8,consistent with Fig. 3a. The discrepancy between PI98andVgmaxalso decreases with increasingCd, in conjunction with the discrepancy between PI09candVmax. Furthermore, the increasing trend ofVmax−Vgmaxand PI09c−PI98remains remarkable in Fig. 10b, implying enhanced unbalanced effects with increasingCdwhenCk/Cdis set to 0.8—the same as the results whenCk/Cdis kept at 1.3.These results with differentCk/Cdindicate that the main results are insensitive to the specific value ofCk/Cd.

4. Conclusions

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

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.

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

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 constantCk/Cd. Seven sensitivity experiments withCk/Cdfixed 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 inCdwhenCk/Cdremains constant and the increasing trend gradually slows down whenCdis larger than 1.00 × 10−3, with a similar pattern found in the variation ofV10max. This phenomenon is different from the conclusions of classical theories and N20, which predict an unvarying and decreasing maximum intensity with increasingCdbut fixedCk/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 whenCdis larger than 1.00 × 10−3. Furthermore, whenCk/Cdremains constant, the PI98remains 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. AsCdincreases, the difference between PI09cand PI98also grows, consistent with the variation trend of the difference betweenVmaxandVgmax, 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∗s–kagradually decreases to an extent up to about 25%, in agreement with the trend in N20. This trend largely eliminates the positive impacts ofToutandTb−Toutand eventually results in the negligible changes in the balanced component. On the contrary, owing to the significant increment ofwmaxandηmax, the term representing the unbalanced effect increases notably whenCdgrows with constantCk/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 whenCkandCdgrow, corresponding to the changes in agradient force. These features also reflect that the unbalanced effect increases asCdgrows whenCk/Cdis 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 increasingCd, which corresponds to the development of subgradient flow near the surface. This might mainly result from intensification of the surface friction effect reflected by increasingCd. 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 inCk. 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 constantCk/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 constantCk/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.