Axisymmetric and three-dimensional simulations are used to evaluate the theory of tropical cyclone (TC) intensification proposed by K. A. Emanuel, which is based on gradient wind balance and moist-neutral ascent along angular momentum (M) surfaces. According to the numerical model results, the intensification of the TC can be divided into two periods, phase I and phase II. During phase I, the TC intensifies while the M and saturation entropy (s*) surfaces evolve from nearly orthogonal to almost congruent. During phase II, the M and s* surfaces in the eyewall and outflow are congruent as the TC intensifies, which is consistent with Emanuel’s study. Therefore, the condition of moist slantwise neutrality in Emanuel’s study is sufficiently satisfied throughout the intensification in phase II. It is also found that the sensitivity of the intensification rates to the surface exchange coefficient for entropy Ck matches Emanuel’s theoretical result, which is that the intensification rate is proportional to Ck. However, the intensification rate varies in proportion to the surface exchange coefficient for momentum Cd, while the Emanuel model growth rate is insensitive to Cd. Furthermore, although the tendency diagnosed by Emanuel is qualitatively similar to the numerical model result during phase II, it is not quantitatively similar. The present analysis finds the inclusion of non–gradient wind effects in the theoretical framework of Emanuel’s study produces an intensification rate that is quantitatively similar to the numerical model results. The neglect of non–gradient wind effects in Emanuel’s study may be the reason for the different dependence of its intensification rate on Cd compared to that of the numerical model. Other aspects of Emanuel’s study in the context of recent research on TC intensification are discussed.
In two recent review papers, Montgomery and Smith (2014, 2017) describe the progression of ideas in roughly the past half century concerning the intensification of tropical cyclones (TCs). Restricting attention to the idealized axisymmetric TC, one can see that all theories 1) make a conceptual separation between an interior region and a boundary layer and 2) attempt to reconcile the disparate dynamics of smaller-scale cumulus convection with the larger-scale TC circulation. The focus of the present paper is the recent model for axisymmetric TC intensification developed in Emanuel (2012, hereafter E12), which has the advantage of possessing an approximate analytical solution. Specifically, we use a cloud-resolving axisymmetric nonhydrostatic model, set up to follow closely the physical content of the E12 model, to assess the latter’s predictions and to further clarify its dynamics. Of particular interest here are the effects of the surface exchange coefficients on TC intensification as predicted by the E12 model.
Emanuel (1986, hereafter E86) developed a steady-state axisymmetric TC model in which the interior is assumed to be in hydrostatic and gradient wind balance, and convective heat transport (saturated moist entropy transport ) occurs along angular momentum (M) surfaces; the boundary layer supplies a relation between and M. A basic result of the E86 model is that of the maximum tangential wind, where Ck and Cd are the surface exchange coefficients for entropy and momentum, respectively. The E86 model forms the basis for the time-dependent theoretical model in Emanuel (1997, hereafter E97) and the improved version in E12. This time-dependent system consists of three main equations, formulated in M coordinates. As in E86, hydrostatic and gradient wind balance in the TC interior and moist neutrality along M surfaces are used to relate the tangential wind speed V at the top of boundary layer to the M gradient of and the difference between the boundary layer and outflow temperatures. The second equation is a diagnostic equation for outflow temperature, which is based on the self-stratification theory1 proposed by Emanuel and Rotunno (2011, hereafter ER11). The third equation of the E12 model is the boundary layer equation, which predicts the evolution of in M coordinates. Thus, the time change of TC intensity in E12 results from an imbalance between the effects of sea surface entropy and momentum fluxes. According to the solution of the approximate system of E12 [his (19)], the intensification rate is proportional to and inversely proportional to .2
Craig and Gray (1996), using a nonhydrostatic axisymmetric cloud model, found that the intensification rate increases with increasing values of the exchange coefficients for heat and moisture, which is qualitatively consistent with the E12 model. However, they found that the intensification rate is relatively insensitive to changes in .
Rosenthal (1971) used an axisymmetric, multilevel primitive equation model with a modified Kuo cumulus parameterization scheme to examine the dependence of the intensification rate on . He found that increases in lead to faster growth rates (but weaker peak intensities). In a recent study, Montgomery et al. (2010) conducted idealized three-dimensional numerical simulations to investigate the sensitivity of the TC intensification rate to changes in the surface drag coefficient. They also found the TC intensification rate increases with increasing until a certain threshold value is reached, beyond which the TC intensification rate is relatively insensitive to further increases in . This latter finding may explain the insensitivity of intensification rate to increases in found in Craig and Gray (1996), as their values of (which included the full increase with wind speed) were relatively large. Given the somewhat contrasting results to be found among numerical studies and the theoretical model of E12, this study uses an axisymmetric, nonhydrostatic cloud model to study the sensitivity of TC behavior to changes in and varied independently.
The paper is organized as follows. In section 2 we describe the numerical simulations used here to evaluate the E12 theoretical model. In section 3 the numerical simulations are compared to the numerical integrations of the E12 model and certain theoretical predictions are tested, especially with reference to the effects of and on TC intensification. In section 4 we seek to clarify aspects of the E12 model by transforming the numerical solutions of E12 from angular momentum to cylindrical coordinates. In section 5 a series of 3D simulations with more-standard physical parameterizations were conducted to check whether the primary results of the idealized axisymmetric simulations remain applicable. Section 6 contains a summary and conclusions.
2. Numerical simulations
Bryan and Rotunno (2009a, hereafter BR09a) compared the maximum intensity of numerically simulated TCs with the E86 theoretical estimate for maximum potential intensity at steady state. Following BR09a, we use here the axisymmetric, nonhydrostatic Cloud Model, version 1 (CM1), as described in Bryan and Rotunno (2009b, hereafter BR09b), to perform a series of numerical experiments examining the intensification rate of the simulated TCs. Most of the model settings used in the present simulations are identical to those used in BR09a. The initial environmental sounding is saturated and was constructed assuming constant pseudoadiabatic equivalent potential temperature as in BR09a (see their Fig. 1). As in BR09a, we employ a small horizontal turbulence length scale and immediately remove condensate in excess of 0.1 g kg−1. These settings are chosen to provide a direct comparison to the idealized framework of E12. The initial vortex is identical to that in Rotunno and Emanuel (1987, hereafter RE87) except here the initial maximum tangential wind speed is 22.5 m s−1. For all the experiments, dissipative heating is not included and the surface exchange coefficients are set to be constant. The model domain is 1500 km in radial direction with a radial grid spacing of 1 km for and a stretched radial resolution for . The vertical grid spacing varies from 50 to 500 m for and becomes 500 m from 5 to 25 km. The stretched grid resolution is believed to be adequate for resolving the boundary layer and inner-core dynamics, which are very important in E12’s theory.
a. Control simulation
For the control simulation, and are set to . Figure 1 shows a time series of the maximum tangential velocity for the control experiment. The TC keeps intensifying for 72 h until it reaches an intensity of 100 m s−1. A slight weakening occurs over the next several hours, and then an approximately steady state is reached after 96 h. We quantify the maximum intensity of tropical cyclones in this article by as the maximum value of a 6-h average of the maximum tangential wind from hourly output. For the control simulation, Vmax = 99 m s−1.
Based on hydrostatic and gradient wind balance and the assumption of slantwise moist neutrality, the approximate diagnostic equation for the wind speed in E12 [his (13)] is
where is the azimuthal velocity at the top of the boundary layer, is the temperature at the top of the boundary layer, is the outflow temperature, is the saturation entropy, and M is the absolute angular momentum. To determine from the numerical solution, we use the rhs of (1), where all terms are evaluated at the radius and height of , . For , we find the temperature at the point in the upper-level outflow where the tangential velocity is zero and the radial wind speed u is largest. From Fig. 1, it is found that the diagnosed is very noisy during the first 32 h; after this time there is a steady increase until 96 h, when a steady state is reached. According to the different behaviors of , the intensification period for the control experiment can be divided into two phases, phase I (0–32 h) and phase II (32–96 h). During phase II, increases from 58 to 70 m s−1 and then reaches a steady state. The tendency of and during phase II is qualitatively similar, as they both increase and reach a steady state at around 96 h. However, the rate of intensification and maximum intensity in E12 is systematically lower than that in the numerical simulation.
At steady state, (1), together with the steady-state boundary layer entropy equation (section 2 of ER11), is essentially the maximum potential intensity (MPI) derived from the E86 analytical model. BR09a investigated the reasons for the underestimation of the model-predicted maximum intensity by the MPI and found that the neglect of unbalanced dynamics can account for the difference. We will return to this point later.
Figure 1 also shows the time series of maximum tangential wind directly integrated by the time-dependent theoretical model of E12, denoted ; the details of the numerical model can be found in E12. By adjusting the initial distribution ,3 the vortex is initialized with a maximum tangential wind of 20 m s−1 [(20) in E12]. The initial size of the vortex is the same as the RE87 vortex with the tangential wind going to zero at radius . The reference value of angular momentum . The temperatures and are set to 200 and 300 K, respectively; and . The depth of the boundary layer is 5000 m, as used in E12. In sympathy with the sentiment expressed in “The mathematical elegance of the formulation in E97 is achieved at the cost of making the physical processes of spin-up less transparent” (Montgomery and Smith 2014, p. 48),” in section 4 we transform the E12 theoretical model results from M coordinates to r–z coordinates to help clarify the intensification process embodied in the E12 model. For now, we note that the evolution of is very similar to during phase II (Fig. 1).
b. Sensitivity simulations
To verify the generality of our results and to investigate the sensitivity of the numerical solutions to and , we conduct several sensitivity simulations. The values of and are set as , , , and .
Table 1 shows the maximum intensity for all the experiments. We first observe that in the numerical experiments is a function of , as there are only slight variations along the diagonals in the table. Furthermore, we see that when the ratio is reduced from 8 to 1/8, reduces from 211 to 42 m s−1. When the drag coefficient is kept constant, the maximum intensity increases with increasing . In the experiments in which is held constant, the maximum intensity decreases with increasing . These results are in basic agreement with the predictions of MPI. To dig a little deeper, we consider that in E86 the MPI is proportional to , while in ER11 in which the self-stratification of TC outflow is taken into account, the MPI is modified such that
where is the MPI defined in E86. Figure 2 shows the dependence of wind speed on the ratio of exchange coefficients calculated from E86 and ER11. The results show that when is less than 1, the numerical model maximum intensity is somewhat closer to the profile (2) proposed by ER11. However, when is larger than 1, the relationship between maximum intensity and the ratio of over in the numerical simulations is closer to . Observations suggest should be between 0.6 and 0.7 (Black et al. 2007; Zhang et al. 2008) and the recent laboratory work by Haus et al. (2010) suggests a value of of approximately 0.5 for winds greater than 30 m s−1 up to nearly 40 m s−1. Therefore, according to these observational studies, the values of in nature are recognized as <1. Thus, at least within the observed range of variation, (2) is an improvement over .
According to the approximate solution for in E12 [his (19)], the rate of intensification is proportional to . As discussed above in regard to Table 1, is shown to be function of in both numerical and theoretical results. This means that with held fixed, the intensification rate should be proportional to . Conversely, with held fixed, the E12 approximate solution intensification rate should vary inversely with (however, solutions to the unapproximated E12 model show that the intensification rate is nearly insensitive to ; see footnote 2). Figure 3a shows from simulations with and varying from to . For the same , the steady-state values of maximum intensity in the simulations are similar; however, it takes only 24 h to reach the steady state for the simulation with and more than 144 h for . These results support the idea that the TC intensification rate is proportional to , which is consistent with E12’s theory and other previous studies (e.g., Craig and Gray 1996).
Figure 3b shows the sensitivity of the intensification rate to the variation of when is equal to (the results for simulations with other values of are qualitatively similar). Compared to the changes in intensification rate with changes in , the intensification rate is less sensitive to changes in the drag coefficient, which is consistent with the results in Craig and Gray (1996). However, the results still suggest a proportional relationship between the intensification rate and for small , and then no significant difference in intensification rate is found as increases beyond approximately . These results are similar to the sensitivity of the intensification rate to changes in found in Montgomery et al. (2010, their Figs. 1 and 3), which indicate the intensification rate of the vortex increased with increasing until a certain threshold value is attained and then it remained relatively constant. It should be noted that, in contrast with Montgomery et al. (2010), the maximum intensity at steady state decreases with increasing in our simulations.4 In summary, the numerical model evidence presented here and in past studies strongly suggest that the intensification rate is proportional to for small values, while the prediction from the E12 model is for an intensification rate that is insensitive to (Emanuel 2018).
In the following section, we investigate the reasons for the different dependence of intensification rate on Cd between the E12 theoretical model and the present numerical model.
3. Evaluation of E12
To determine why diagnosed in (1) is not quantitatively similar to the numerical model results in phase II, we evaluate three primary components of the E12 model in the next several subsections. In the following subsections, we do not address the realism of the numerical simulations or whether the E12 model components are appropriate for natural tropical cyclones. Indeed, as discussed in BR09a and BR09b, tropical cyclones simulated with the control setup have some obviously unnatural features, such as the pseudoadiabatic assumption and small horizontal mixing length. Rather, we seek to ascertain which approximations in the E12 model are responsible for the qualitative differences in TC intensification rate between E12 and the present numerical simulation model.
a. Moist slantwise neutrality
The E12 model assumes that is a function of M alone in the free atmosphere during the intensification period. In Fig. 4 we show the distribution of entropy s as formulated for pseudoadiabatic processes (Bryan 2008) and the distribution of M during the intensification in phase II for three different simulations. To a good approximation, the s and M surfaces are congruent in the eyewall in the interior free atmosphere and TC outflow. Consistent with the E12 assumption of moist slantwise neutrality, there is approximately one value of s for a given value of M in the eyewall region. During phase I, the TC intensifies while the M and s surfaces evolve from nearly orthogonal to almost congruent (not shown). The details of the intensification progress in phase I will be discussed in a follow-on paper. Based on these analyses, we conclude that an approximate condition of moist slantwise neutrality is achievable in the eyewall and outflow of a numerical simulation during the intensification in phase II. It follows that this component of E12 is probably not the source of the discrepancy between and in the simulations.
Figure 5 shows the s and M surfaces for the control experiment at different times during phase II. At 42 h the M surface passing through the location of is at a radius of approximately 40 km; 18 h later, is found at a radius of approximately 30 km. Figure 5 shows that during phase II, the s and M surfaces in the eye region are moving inward with an approximately self-similar pattern during the intensification in phase II. The M and s surfaces in the eyewall are contracting as the TC intensifies, which is similar to the eyewall frontogenesis illustrated in E97 (the predecessor of E12).
b. Gradient wind balance
BR09a showed that a major difference between the steady-state MPI of E86 and the steady states found in their numerical simulation was the presence in the model of strong non–gradient wind effects. As the E12 model is also based on the assumption of gradient wind balance, it is reasonable to look at the influence of non–gradient wind effects on the TC intensification rate. To determine the effect of the unbalanced term, we use a derivation with similar approximations to those used in BR09a and express a modified maximum wind speed containing the unbalanced terms,
[see also BR09a, their (20)], where the subscript b denotes evaluation at the top of the boundary layer, w is the vertical velocity component, and is the azimuthal vorticity.
where all terms are evaluated at the location of maximum tangential velocity.
Figure 6 compares the differences among , , and in several simulations. It is clear that the evolution of is in considerably better agreement with , which means the neglect of non–gradient wind effects in E12 may be the reason for the quantitative difference between and . BR09a found that including unbalanced flow effects [through the second term on the rhs of (3)] in the E86 analytic model of maximum intensity considerably improves the agreement between theory and the numerical estimates of the maximum tangential wind speed at steady state. In our study, is 9 m s−1 less than for the mature stage in the control experiment. A difference of this magnitude is seen in all the simulations (less than 10% of ). Figures 6b and 6c show the results for the experiments with the same and different ( in Fig. 6b and in Fig. 6c). When is small, the non–gradient wind balance term is smaller (Bryan 2012) and the E12 model can better describe the evolution of the TC in the numerical simulations. Comparing the shape of the M surfaces in the simulation with large (; Fig. 4c) to the simulation with small (; Fig. 4b), the wave associated with the unbalanced flow is seen by the undulation in the M surfaces from the top of boundary layer to the tropopause.
As discussed in section 2b, the dependence of the intensification rate on Cd in the E12 model is different from the present numerical results and many other studies (e.g., Rosenthal 1971; Montgomery et al. 2010). The non–gradient wind balance term is strongly related to surface drag, which in turn is related to . The present analysis suggests that the neglect of non–gradient wind effects in the diagnostic equation for wind speed of E12 [(1)] may be the reason for the different dependence of the intensification rate on between the E12 model and the present numerical simulations.
c. Self-stratification of TC outflow
The self-stratification of the outflow temperature used in the E12 model is determined by small-scale turbulence that limits the Richardson number to a critical value for the onset of turbulence, which is proposed by ER11 [their (31)]:
where Ric is the critical Richardson number and is the physical radius at which the Richardson number first attains its critical value. In our simulations the region with near-critical Ri is found close to the eyewall region (not shown), which is consistent with ER11. Figure 7 shows the relationship between and from hourly output in the control simulation during phase II. The quantity is calculated at the location of , . Referring to ER11, is evaluated in the region for small Ri (<1). Fitting a straight line to the data between and gives a slope of , which is close to the value of when and . Therefore, the self-stratification hypothesis of outflow in the E12 model is consistent with our numerical simulations during the growth stage in phase II.
4. Other aspects of the E12 model
The foregoing results show the E12 model can qualitatively describe the evolution of in the numerical simulations during phase II; however, the physical character of the evolution in the commonly used cylindrical coordinate system is not immediately obvious.
Figure 8 shows from the E12 theoretical model (see the appendix) in a window extending to 80 km (the full domain is 412.5 km). This figure shows that during the intensification period, the M surfaces are moving radially inward in an approximately self-similar pattern and display a type of frontogenesis, similar to the intensification features during phase II of the numerical simulations shown in Fig. 5. The derived radial and vertical velocities in the boundary layer for the E12 model are shown in Fig. 8a. Although described in detail in the appendix, a brief explanation of how the interior distribution of evolves is given here. The E12 model updates in the boundary layer; (1) therefore gives , which allows the calculation of . As indicated in Fig. 8, shows the inward movement of large M in the boundary layer that is instantaneously communicated to the interior through the requirements of balance and moist neutrality (see the appendix). Figure 8c displays the change of M over the period of evolution shown in Fig. 8b. In the region above the boundary layer, M is conserved and therefore which implies that the perpendicular component of the implicit secondary circulation vector, , is in the opposite direction of .
To obtain a more complete picture of the advective and turbulent processes in more realistic conditions, we return to the axisymmetric numerical simulations. Figure 9a shows the time change of M in a 12-h period, which can be compared to that of the E12 model shown in Fig. 8c. We observe a qualitative similarity, in that the maximum tendency is concentrated in the eyewall in both models over a deep layer. Major differences include the width of the region of updraft as the positive vertical motion in the E12 model (Fig. 8a) changes to negative for radii beyond roughly 200 km (not shown). Also, the inward–outward excursions of the M surfaces, and the corresponding local extrema of evident in Fig. 9a, are absent in Fig. 8c. These local extrema are associated with non–gradient wind effects as documented by Bryan and Rotunno (2009a) for the steady-state version of the present model. The strong local positive values of (Fig. 9a) at the top of the boundary layer support the idea that this part of the eyewall region is spun up by the vertical advection of high M from the boundary layer (Figs. 9b,c; Schmidt and Smith 2016; Kilroy et al. 2016).
Figure 10 summarizes and illustrates the qualitative similarities of the M budget in E12 and the present numerical simulation. This figure shows that while the process of inward transport of M in the boundary layer and vertical transport to the interior is explicit in the numerical simulation (cf. Fig. 9c), it is implicit in the E12 model (cf. Fig. 8a).
5. Three-dimensional simulations
Up to this point a simplified axisymmetric numerical model with simple, but unrealistic, initial conditions has been used to facilitate a comparison with the E12 theoretical model. To check whether the primary results of this study remain applicable with more complex dynamics and more realistic conditions, we have conducted an additional set of simulations using three spatial dimensions and the default physical parameterizations for CM1. Specifically, we use the microphysics scheme of Morrison et al. (2009) and a more realistic length scale in the horizontal turbulence code (lh = 750 m). Dissipative heating is included. The vertical turbulence scheme and the distribution of vertical levels are the same as before. The horizontal grid spacing is 2 km over a 400 km × 400 km inner mesh, with increasingly stretched grid spacing beyond, with a maximum grid spacing of 16 km. The entire domain is 3000 km × 3000 km. The initial vortex is the same as before, but random small-amplitude temperature perturbations are added, and the initial sounding is the “moist tropical” composite from Dunion (2011). The sea surface temperature is 28°C. All simulations are integrated for 12 days.
For intensity we use the maximum value of wind speed at 10 m MSL from hourly output and then find the maximum 1-day average. The results (Table 2) have the same qualitative trends as before (Table 1), in that peak intensity increases as increases or as decreases. The maximum wind speeds are generally lower than before, as expected, because of the increased horizontal length scale, although perhaps in part because of a smaller air–sea temperature difference. Time series of maximum wind speed (Fig. 11) show that qualitative trends in intensification rate are also the same as before (Fig. 3); that is, the intensification rate increases as increases when (Fig. 11a) and the intensification rate increases as increases when is held fixed (especially for small values of ). These three-dimensional “full physics” simulations add confidence to the results of the more idealized simulations concerning the effects of and on TC intensity and intensification.
6. Summary and conclusions
In this paper we use a cloud-resolving numerical model to evaluate the time-dependent theoretical model of tropical cyclone intensification proposed by E12. Applying the diagnostic relation [(1)] from E12 to the present numerical simulations indicates a very noisy during the initial adjustment of the vortex to a state of moist neutrality along angular momentum surfaces (phase I). After this adjustment is complete, shows a more regular increase (phase II) that is qualitatively similar to the numerical simulations but not quantitatively. The results of the sensitivity experiments for different drag and entropy surface exchange coefficients show that the maximum tangential wind speed in the numerical experiments is a function of and with a functional dependence that is closer to that found by ER11 for the observationally relevant range in which . The intensification rate in numerical simulations is proportional to , which is consistent with E12’s theory. However, the intensification rate is also proportional to within a certain range in our simulations, while the E12 model is essentially insensitive to . These results hold for both axisymmetric and three-dimensional numerical simulations, using both simplified and realistic physical parameterizations and initial conditions.
We analyze the present numerical simulations to evaluate the assumptions used in the E12 model to find the reason for the difference in results between the E12 model and our numerical model. It is found that during phase I, the TC intensifies while the angular momentum (M) and entropy (s) surfaces evolve from nearly orthogonal to almost congruent, which is not consistent with slantwise moist neutrality. The details of the intensification progress in phase I will be discussed in another paper. During phase II, the M and s surfaces are congruent as the TC intensifies, which means the assumption of slantwise moist neutrality in E12 is valid in the intensification process of phase II.
The effect of non–gradient wind balance on TC intensification and the mature stage in eyewall region has been recognized by several studies (e.g., Smith et al. 2008; BR09a). In the present analysis, we find that the inclusion of non–gradient wind effects in the theoretical framework of E12 produces an intensification rate that is quantitatively similar to the numerical model results. As the non–gradient wind term is closely related to the drag efficient Cd, the neglect of non–gradient wind effects in E12 seems likely to be the reason for the different dependence on Cd of the intensification rate between E12 and the present numerical model.
The self-stratification of the outflow temperature used in the E12 model is also evaluated in our numerical simulation. We found that the region with near-critical Ri is close to the eyewall region; fitting a straight line to the data between and gives a slope of , which is close to the value of when and . These results indicate that the hypothesis of the outflow self-stratification determined by small-scale turbulence that limits the Richardson number to a critical value for the onset of turbulence is appropriate in our numerical simulation during the intensification process of phase II.
Other aspects of the E12 model in the context of recent research on TC intensification are discussed. When transformed to cylindrical coordinates, the qualitative similarity of the E12 model to the present model becomes apparent. Noted differences are that the width of the updraft/eyewall region is much smaller and non–gradient wind effects are apparent in the cloud model. The present analysis of the cloud model angular momentum budget supports the idea that vertical advection from the boundary layer plays a role in eyewall spinup at the top of the boundary layer (e.g., Schmidt and Smith 2016; Kilroy et al. 2016).
The E12 model for TC intensification is the only theoretical model with an approximate analytical solution at present. Compared to a nonhydrostatic cloud model, the theoretical model possesses the chief virtue of transparency. On the other hand, transparency usually comes at the expense of accuracy. The purpose of this study is to examine the trade-off between transparency and accuracy in the E12 model of TC intensification. It is hoped that identified strengths and weaknesses can guide future attempts at the construction of a more accurate, yet sufficiently transparent, theoretical model.
The authors are grateful for the constructive comments provided by the reviewers and also thank Fuqing Zhang and Juan Fang for the fruitful discussions. This research was conducted during the first author’s visit to NCAR, which was supported by NCAR’s Advanced Study Program Graduate Student Program and the Short Term Visit Program of Nanjing University. The first author was supported in part by the National Key Research and Development Program of China under Grant 2017YFC1501601 and the National Natural Science Foundation of China Grant 41475046.
Transformation from M Coordinates to r–z Coordinates
According to E12’s theoretical model, we can obtain values of , , , and radius at the top of the boundary layer. The complete two-dimensional structure of the theoretical TC can be obtained by evaluating quantities along M surfaces. The equation for is given by (4) of E12. For convenience, can be calculated as follows:
We then obtain the tangential velocity from the absolute angular momentum—that is, —and finally we interpolate to r–T coordinates to obtain .
To transform all the variables from r–T to r–z coordinates, the relation between z and T needs to be found. As a matter of convenience, the height of the M surfaces may be obtained by noting that the constancy of saturated equivalent potential temperature along M surfaces implies that the saturated moist static energy, defined as
is approximately constant along such surfaces. The height z is obtained from (A2):
where is nondimensional pressure; is a reference pressure; and are the specific heat and gas constants of dry air, respectively; is the saturation mixing ratio; and L is the latent heat of vaporization. Since is a function of temperature and pressure, which is unknown, we derive it in order to infer the height of the surfaces from (A3).
The saturation mixing ratio at the top of the boundary layer at constant temperature may be expressed as
where the subscript a denotes the ambient variable. The gradient wind relationship can be written as
This may be directly integrated once in r and then the boundary layer pressure can be obtained as follows:
The saturation mixing ratio is found from the Clausius–Clapeyron equation:
where is the gas constant for water vapor. According to the definition of saturation entropy, the value of π can be expressed as
Substituting and into (A3), we obtain the desired relation between z and T. Finally, we interpolate to . Other variables can be transformed from M coordinates to r–z coordinates by the same method.
To derive the radial velocity u in the boundary layer, we consider the M equation in cylindrical coordinates,
where is the vertical flux of M, which is expressed using the standard aerodynamic formula, .
The anelastic form of the mass continuity equation can be written as
where ρ is the density and w is the vertical velocity. A mass streamfunction ψ may be defined by virtue of (A10), such that
Since density varies only slightly within the boundary layer, (A12) may be integrated in z to yield
where is a mean density in the boundary layer, h is the depth of boundary layer, and the boundary condition at was used.
The vertical mean radial velocity in the boundary layer may be found by integrating (A11) across the depth of the layer so that
The vertical velocity is related to the streamfunction and can be expressed as
The theory establishes the functional dependence of the outflow temperature as a function of angular momentum such that the local Richardson number is not reduced below a critical minimum.
Figure 1 of E12 suggests that both the E12 approximate analytical and full numerical solutions support a growth rate that is inversely proportional to Cd; however, an error in the caption of Fig. 1 in E12 (Ck was varied, not Cd) does not support any conclusion on how the E12 full numerical solution growth rate depends on Cd. Although the E12 approximate equation [his (19)] suggests a growth rate inversely proportional to Cd, solutions of the full E12 numerical model, holding Ck constant and varying Cd, show that the growth rate is roughly insensitive to Cd (Emanuel 2018).
This is done by setting a constant that determines the size of the initial vortex, the maximum value of saturation entropy , the saturation entropy of the unperturbed environment , and the environment saturation entropy of air at the sea surface .