## 1. Introduction

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 *M*) surfaces; the boundary layer supplies a relation between *M.* A basic result of the E86 model is that of the maximum tangential wind, *C*_{k} and *C*_{d} 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 ^{1} 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 *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 ^{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

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

## 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 ^{−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

### a. Control simulation

For the control simulation, ^{−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 *V*_{max} = 99 m s^{−1}.

*M*is the absolute angular momentum. To determine

*u*is largest. From Fig. 1, it is found that the diagnosed

^{−1}and then reaches a steady state. The tendency of

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 ^{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 *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

### b. Sensitivity simulations

To verify the generality of our results and to investigate the sensitivity of the numerical solutions to

^{−1}. When the drag coefficient is kept constant, the maximum intensity increases with increasing

^{−1}up to nearly 40 m s

^{−1}. Therefore, according to these observational studies, the values of

Maximum intensity *V*_{max} (m s^{−1}) from simulations that use different values for *C*_{k} (10^{−3}) and *C*_{d} (10^{−3}).

Dependence of *V*_{max} on the ratio of exchange coefficients from E86 (red line) and ER11 (blue line); dots are *V*_{max} for all simulations from Table 1.

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Dependence of *V*_{max} on the ratio of exchange coefficients from E86 (red line) and ER11 (blue line); dots are *V*_{max} for all simulations from Table 1.

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Dependence of *V*_{max} on the ratio of exchange coefficients from E86 (red line) and ER11 (blue line); dots are *V*_{max} for all simulations from Table 1.

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

According to the approximate solution for

Time series of maximum azimuthal velocity *V*_{m} (m s^{−1}) from simulations with different ^{−3}) for the simulations with (a) *C*_{k}/*C*_{d} = 1 and different values of

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Time series of maximum azimuthal velocity *V*_{m} (m s^{−1}) from simulations with different ^{−3}) for the simulations with (a) *C*_{k}/*C*_{d} = 1 and different values of

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Time series of maximum azimuthal velocity *V*_{m} (m s^{−1}) from simulations with different ^{−3}) for the simulations with (a) *C*_{k}/*C*_{d} = 1 and different values of

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Figure 3b shows the sensitivity of the intensification rate to the variation of ^{4} In summary, the numerical model evidence presented here and in past studies strongly suggest that the intensification rate is proportional to

In the following section, we investigate the reasons for the different dependence of intensification rate on *C*_{d} between the E12 theoretical model and the present numerical model.

## 3. Evaluation of E12

To determine why diagnosed

### a. Moist slantwise neutrality

The E12 model assumes that *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

Entropy *s* (contour interval = 10 J kg^{−1} K^{−1}; green lines) and angular momentum *M* (contour interval = 0.2 × 10^{6} m^{2} s^{−1}; black lines) from the experiment with (a) *M* surface that passes through the location of maximum tangential wind.

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Entropy *s* (contour interval = 10 J kg^{−1} K^{−1}; green lines) and angular momentum *M* (contour interval = 0.2 × 10^{6} m^{2} s^{−1}; black lines) from the experiment with (a) *M* surface that passes through the location of maximum tangential wind.

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Entropy *s* (contour interval = 10 J kg^{−1} K^{−1}; green lines) and angular momentum *M* (contour interval = 0.2 × 10^{6} m^{2} s^{−1}; black lines) from the experiment with (a) *M* surface that passes through the location of maximum tangential wind.

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

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 *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).

As in Fig. 4, but for the control experiment at (a) 42 and (b) 60 h.

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

As in Fig. 4, but for the control experiment at (a) 42 and (b) 60 h.

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

As in Fig. 4, but for the control experiment at (a) 42 and (b) 60 h.

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

### b. Gradient wind balance

*b*denotes evaluation at the top of the boundary layer,

*w*is the vertical velocity component, and

Figure 6 compares the differences among ^{−1} less than *M* surfaces in the simulation with large *M* surfaces from the top of boundary layer to the tropopause.

Time series of *V*_{m} (m s^{−1}) from the control simulation (black line), *V _{e}* calculated from (1) (red line), and

*V*

_{a}from (3), which includes

*V*

_{e}and the non–gradient wind balance term (blue line), for (a)

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Time series of *V*_{m} (m s^{−1}) from the control simulation (black line), *V _{e}* calculated from (1) (red line), and

*V*

_{a}from (3), which includes

*V*

_{e}and the non–gradient wind balance term (blue line), for (a)

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Time series of *V*_{m} (m s^{−1}) from the control simulation (black line), *V _{e}* calculated from (1) (red line), and

*V*

_{a}from (3), which includes

*V*

_{e}and the non–gradient wind balance term (blue line), for (a)

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

As discussed in section 2b, the dependence of the intensification rate on *C*_{d} 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

### c. Self-stratification of TC outflow

_{c}is the critical Richardson number and

The relationship between ^{−2} s) and

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

The relationship between ^{−2} s) and

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

The relationship between ^{−2} s) and

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

## 4. Other aspects of the E12 model

The foregoing results show the E12 model can qualitatively describe the evolution of

Figure 8 shows *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 *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

^{−6} m^{2} s^{−1}; black lines) from the E12 time-dependent model for (a) 18 and (b) 36 h. The thick arrows in (a) represent the derived inflow and updraft for E12 model. The red lines in (a) and (b) are the *M* surfaces that pass through *V*_{m}. (c) The tendency of *M* (m^{2} s^{−2}; color shading) averaged from 18 to 36 h. The lines in (c) are *M* surfaces passing through *V*_{m} at 18 (solid) and 36 h (dashed).

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

^{−6} m^{2} s^{−1}; black lines) from the E12 time-dependent model for (a) 18 and (b) 36 h. The thick arrows in (a) represent the derived inflow and updraft for E12 model. The red lines in (a) and (b) are the *M* surfaces that pass through *V*_{m}. (c) The tendency of *M* (m^{2} s^{−2}; color shading) averaged from 18 to 36 h. The lines in (c) are *M* surfaces passing through *V*_{m} at 18 (solid) and 36 h (dashed).

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

^{−6} m^{2} s^{−1}; black lines) from the E12 time-dependent model for (a) 18 and (b) 36 h. The thick arrows in (a) represent the derived inflow and updraft for E12 model. The red lines in (a) and (b) are the *M* surfaces that pass through *V*_{m}. (c) The tendency of *M* (m^{2} s^{−2}; color shading) averaged from 18 to 36 h. The lines in (c) are *M* surfaces passing through *V*_{m} at 18 (solid) and 36 h (dashed).

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

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 *M* from the boundary layer (Figs. 9b,c; Schmidt and Smith 2016; Kilroy et al. 2016).

Analysis of the inward transport of the angular momentum surfaces through (a) the time change (m^{2} s^{−2}, averaged from 48 to 60 h), (b) selected *M* surfaces (contour interval = 0.2 × 10^{−6} m^{2} s^{−1}) and the radial–vertical wind vectors in the box shown in (a), and (c) the profiles of vertical advection, horizontal advection, and diffusion of *M* (m^{2} s^{−2}) at the location of its maximum time change. The lines in (a) are *M* surfaces passing through *V*_{m} at 48 (solid) and 60 h (dashed), and the position of *V*_{m} is indicated by the dot in (b).

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Analysis of the inward transport of the angular momentum surfaces through (a) the time change (m^{2} s^{−2}, averaged from 48 to 60 h), (b) selected *M* surfaces (contour interval = 0.2 × 10^{−6} m^{2} s^{−1}) and the radial–vertical wind vectors in the box shown in (a), and (c) the profiles of vertical advection, horizontal advection, and diffusion of *M* (m^{2} s^{−2}) at the location of its maximum time change. The lines in (a) are *M* surfaces passing through *V*_{m} at 48 (solid) and 60 h (dashed), and the position of *V*_{m} is indicated by the dot in (b).

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Analysis of the inward transport of the angular momentum surfaces through (a) the time change (m^{2} s^{−2}, averaged from 48 to 60 h), (b) selected *M* surfaces (contour interval = 0.2 × 10^{−6} m^{2} s^{−1}) and the radial–vertical wind vectors in the box shown in (a), and (c) the profiles of vertical advection, horizontal advection, and diffusion of *M* (m^{2} s^{−2}) at the location of its maximum time change. The lines in (a) are *M* surfaces passing through *V*_{m} at 48 (solid) and 60 h (dashed), and the position of *V*_{m} is indicated by the dot in (b).

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

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).

Schematic diagram of the angular momentum (*M*) budget of (a) the present numerical model (cf. Fig. 9) and (b) the E12 model (cf. Fig. 8). In the boundary layer, both models exhibit the inward transport of *M* and the loss of *M* to the lower surface. At a fixed radius in the numerical model in (a), *M* has a local maximum at a level lower than the level where *u* = 0, so there is both inward and upward transport between these two levels. In the E12 model in (b), the upward transport at the top of its boundary layer is implicit. Above the boundary layer, the numerical model has a radial–vertical velocity such that there is a component in the opposite direction of

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Schematic diagram of the angular momentum (*M*) budget of (a) the present numerical model (cf. Fig. 9) and (b) the E12 model (cf. Fig. 8). In the boundary layer, both models exhibit the inward transport of *M* and the loss of *M* to the lower surface. At a fixed radius in the numerical model in (a), *M* has a local maximum at a level lower than the level where *u* = 0, so there is both inward and upward transport between these two levels. In the E12 model in (b), the upward transport at the top of its boundary layer is implicit. Above the boundary layer, the numerical model has a radial–vertical velocity such that there is a component in the opposite direction of

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

Schematic diagram of the angular momentum (*M*) budget of (a) the present numerical model (cf. Fig. 9) and (b) the E12 model (cf. Fig. 8). In the boundary layer, both models exhibit the inward transport of *M* and the loss of *M* to the lower surface. At a fixed radius in the numerical model in (a), *M* has a local maximum at a level lower than the level where *u* = 0, so there is both inward and upward transport between these two levels. In the E12 model in (b), the upward transport at the top of its boundary layer is implicit. Above the boundary layer, the numerical model has a radial–vertical velocity such that there is a component in the opposite direction of

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

## 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 (*l*_{h} = 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

Maximum wind speed (m s^{−1}) at 10 m MSL (averaged over 1 day, using hourly output) from 3D full-physics simulations.

As in Fig. 3, but for 3D full-physics simulations.

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

As in Fig. 3, but for 3D full-physics simulations.

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

As in Fig. 3, but for 3D full-physics simulations.

Citation: Journal of the Atmospheric Sciences 75, 6; 10.1175/JAS-D-17-0382.1

## 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

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 *C*_{d}, the neglect of non–gradient wind effects in E12 seems likely to be the reason for the different dependence on *C*_{d} 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

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.

## Acknowledgments

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.

## APPENDIX

### Transformation from *M* Coordinates to *r**–**z* Coordinates

*M*

*r*

*z*

*M*surfaces. The equation for

*r–T*coordinates to obtain

*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

*M*surfaces implies that the saturated moist static energy, defined as

*z*is obtained from (A2):

*L*is the latent heat of vaporization. Since

*a*denotes the ambient variable. The gradient wind relationship can be written as

*r*and then the boundary layer pressure

*π*can be expressed as

Substituting *z* and *T*. Finally, we interpolate *M* coordinates to *r–z* coordinates by the same method.

*u*in the boundary layer, we consider the

*M*equation in cylindrical coordinates,

*M*, which is expressed using the standard aerodynamic formula,

*ρ*is the density and

*w*is the vertical velocity. A mass streamfunction

*ψ*may be defined by virtue of (A10), such that

*z*to yield

*h*is the depth of boundary layer, and the boundary condition

*w*.

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^{1}

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.

^{2}

Figure 1 of E12 suggests that both the E12 approximate analytical and full numerical solutions support a growth rate that is inversely proportional to *C*_{d}; however, an error in the caption of Fig. 1 in E12 (*C*_{k} was varied, not *C*_{d}) does not support any conclusion on how the E12 full numerical solution growth rate depends on *C*_{d}. Although the E12 approximate equation [his (19)] suggests a growth rate inversely proportional to *C*_{d}, solutions of the full E12 numerical model, holding *C*_{k} constant and varying *C*_{d}, show that the growth rate is roughly insensitive to *C*_{d} (Emanuel 2018).

^{3}

This is done by setting a constant that determines the size of the initial vortex, the maximum value of saturation entropy

^{4}

Bryan (2013) explains this contrast by showing a longer period of integration is needed for several of the Montgomery et al. (2010) experiments to reach a steady state.