## 1. Introduction

*α*is nearly a constant,

*r*is the radius,

_{m}*w*the vertical velocity, and

_{m}*η*the azimuthal vorticity at the radius and height of the maximum tangential wind (

_{m}*r*,

_{m}*z*). In numerical simulations designed to conform closely to the approximations used to derive EPI, Bryan and Rotunno (2009a, their Fig. 12) found good agreement between PI

_{m}^{+}and the numerically simulated maximum tangential wind. Motivated by this finding, the objective of the present study is to develop a better theoretical understanding of

*r*, with emphasis on its variation with

_{m}w_{m}η_{m}*r*.

_{m}Models for the supergradient wind are perforce boundary layer models of momentum transfer from the “free” (frictionless) atmosphere above to the frictional surface below. For rotating-flow boundary layer models, the most problematic aspect is the specification of the imposed radial pressure gradient, or equivalently, the gradient wind *υ _{g}*(

*r*). In contradistinction to parallel-flow boundary layers, the secondary circulation

^{1}of a rotating-flow boundary layer may modify the “free” atmosphere

^{2}turning a relatively straightforward calculation of the boundary layer winds into the far-more-complex problem of solving for the “imposed”

*υ*(

_{g}*r*) together with the boundary layer winds (Rotunno 2014). As demonstrated by Bryan and Rotunno (2009a, their sections 4c and 5),

*υ*(

_{g}*r*) calculated from a numerical simulation that includes a boundary layer should not in general be the same as that from a model that does not include the boundary layer radial-momentum feedback

^{3}on

*υ*(

_{g}*r*) (as in Emanuel 1986; Frisius et al. 2013; Tao et al. 2020a). With this understanding, (1) represents the sum of the gradient wind that would occur in a model absent the boundary layer radial-momentum feedback on

*υ*(

_{g}*r*) plus the supergradient term

*αr*; further elaboration on this point is in section 5.

_{m}w_{m}η_{m}Frisius et al. (2013) carried out the most comprehensive theoretical unification of EPI with a model for supergradient winds. Using a modified version of Emanuel’s (1986) original formulation, Frisius et al. (2013) derive *υ _{g}*(

*r*) for use as an input to a boundary layer model. Frisius et al. (2013) find (their Fig. 3) that

*υ*, the maximum supergradient wind in the boundary layer, increases with the radius of the maximum

_{m}*υ*(

_{g}*r*), as it varies from 5 to 50 km. Analysis of a high-resolution global model (Miyamoto et al. 2014) as well as that of idealized axisymmetric numerical simulations (Tao et al. 2020b, their Fig. 12a) finds that supergradient winds are stronger in larger simulated tropical cyclones.

Tao et al. (2020a) analyzed the quasi-steady states of a series of numerical simulations of idealized axisymmetric tropical cyclones varying physical constants such as the mixing lengths and air–sea transfer coefficients, with all cases having the same thermodynamic initial condition and sea surface temperature. For each set of physical constants, they varied the initial radius *r _{mi}* of the initial tangential-wind maximum

*υ*in the initial profile

_{mi}*υ*(

_{i}*r*,

*z*) as in (1) of Xu and Wang (2018), with the same

*υ*in all cases. Tao et al. (2020a) found that, for each set of physical constants, the steady-state

_{mi}*r*increases as

_{m}*r*increases (consistent with Rotunno and Emanuel 1987, their Table 2) and hypothesized that the contribution from the supergradient wind to

_{mi}*υ*increases with increasing

_{m}*r*(p. 5). In looking at the latter result in the context of (1), the author found that

_{m}*αr*increases relative to EPI

_{m}w_{m}η_{m}^{2}with increasing

*r*. It was not obvious to the author what accounts for this dependence of

_{m}*αr*on

_{m}w_{m}η_{m}*r*, or more generally, what were its determining factors. Montgomery and Smith (2017, p. 567) suggest that the answers lie in the boundary layer equation for the radial momentum. The present study pursues the suggested line of inquiry as it seeks to shed light on the factors determining

_{m}*αr*.

_{m}w_{m}η_{m}After a brief summary of the numerical-model setup in section 2, an analysis of the model data is presented in section 3, including the identification of relevant model-output parameters needed to estimate *αr _{m}w_{m}η_{m}*; methods to relate these model-output parameters to

*r*are developed in this section through a combination of simple scaling and basic theory. A synthesis of these methods is given in section 4 and conclusions summarized in section 5.

_{m}## 2. Model setup

Although the present model setup is described in section 2 of Tao et al. (2020a), some basic information on the simulations is given here for convenience. The numerical model used in the latter study was version 19.7 of Cloud Model 1 (CM1) as described in Bryan and Rotunno (2009b) for axisymmetric tropical cyclone simulations. The domain is 1500 km in the radial direction with grid spacing Δ*r* = 1 km for *r* < 300 km and a linear stretching to Δ*r* = 15 km for *r* ≥ 300 km. The domain is 25 km deep with the lowest grid level at *z* = 25 m, which gradually increases to *dz* = 200 m up to *z* = 5 km above which it remains constant. The sea surface temperature *T _{s}* = 28°C and the simulations are initialized with the Jordan hurricane-season sounding. The simulations are run for 8 days until a quasi-steady state is reached for time

*t*approximately greater than 5 days and the results are averaged over the 1-h output times for the last 24 h of the simulations.

*κ*= 0.4 and

*z*

_{0}is the roughness length; Tao et al. (2020b) specify

*l*

_{∞}= 100 m and

*z*

_{0}= 0.16 m. The control simulations for increasing

*r*have the horizontal mixing length

_{mi}*l*varying linearly from

_{h}*l*

_{h}_{1}(=100 m) to

*l*

_{h}_{2}(=1000 m) as the surface pressure goes from 1015 hPa to below 900 hPa (Stern and Bryan 2018, p. 3902); the control set is therefore designated as RX, where X stands for

*r*in km units. A second set, chosen for detailed comparison with the control set, has the constant value

_{mi}*l*

_{h}_{1}=

*l*

_{h}_{2}= 100 m and is designated as RXL. Thus, one expects horizontal diffusion to have a much bigger effect in the RX compared to the RXL simulations. Detailed analysis is restricted to the cases with the (constant) enthalpy-flux and drag coefficients

*C*=

_{k}*C*= 0.001. Within each set we select from Tao et al. (2020a) the cases with

_{d}*r*= 60, 90, and 120 km with the “skirt” parameter (which controls the radial decay of

_{mi}*υ*(

_{i}*r*,

*z*) for

*r*>

*r*)

_{mi}*B*= 1.0. The initial tangential wind maximum

*υ*= 20 m s

_{mi}^{−1}in all cases. Figure 1 shows the initial velocity profiles at

*z*= 0 and the associated angular-momentum profiles,

*f*= 0.5 × 10

^{−4}s

^{−1}.

## 3. Analysis

To establish the framework for the present analysis, Fig. 2 shows the steady-state radial and tangential velocities (*u*, *υ*) for R120 with the position of the maximum tangential wind (*r _{m}*,

*z*) shown by the magenta ×. This solution exhibits the familiar pattern of boundary layer radial inflow with a tangential-wind maximum nearly coincident with the

_{m}*u*= 0 contour; the position

*r*

_{min}of the minimum radial velocity

*u*

_{min}(or maximum

*inflow*velocity) is indicated by the blue arrow. Figure 3 shows a closer-in view of the vertical velocity

*w*and azimuthal vorticity

*η*, which play a central role in the present analysis. Table 1 contains the model outputs that the theory developed here seeks to explain with emphasis on the increase of the supergradient wind term

*αr*with

_{m}w_{m}η_{m}*r*.

_{m}Steady-state radial and tangential velocities (*u*, *υ*) for R120. The contour interval (c.i.) for *u* is 10 m s^{−1} (black lines, negative values dashed) and the c.i. = 20 m s^{−1} for *υ* (magenta lines) with the maximum (113.8 m s^{−1}) marked by the magenta × and the position *r*_{min} of *u*_{min} indicated by the blue arrow.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Steady-state radial and tangential velocities (*u*, *υ*) for R120. The contour interval (c.i.) for *u* is 10 m s^{−1} (black lines, negative values dashed) and the c.i. = 20 m s^{−1} for *υ* (magenta lines) with the maximum (113.8 m s^{−1}) marked by the magenta × and the position *r*_{min} of *u*_{min} indicated by the blue arrow.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Steady-state radial and tangential velocities (*u*, *υ*) for R120. The contour interval (c.i.) for *u* is 10 m s^{−1} (black lines, negative values dashed) and the c.i. = 20 m s^{−1} for *υ* (magenta lines) with the maximum (113.8 m s^{−1}) marked by the magenta × and the position *r*_{min} of *u*_{min} indicated by the blue arrow.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Steady-state vertical velocity *w* (green lines; c.i. = 2 m s^{−1}) and azimuthal vorticity *η* (black lines; c.i. = 0.01 s^{−1}); zero lines not plotted. Selected streamlines (gray) bracketing (*r _{m}*,

*z*) marked by the magenta × and the position

_{m}*r*

_{min}of

*u*

_{min}indicated by the blue arrow.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Steady-state vertical velocity *w* (green lines; c.i. = 2 m s^{−1}) and azimuthal vorticity *η* (black lines; c.i. = 0.01 s^{−1}); zero lines not plotted. Selected streamlines (gray) bracketing (*r _{m}*,

*z*) marked by the magenta × and the position

_{m}*r*

_{min}of

*u*

_{min}indicated by the blue arrow.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Steady-state vertical velocity *w* (green lines; c.i. = 2 m s^{−1}) and azimuthal vorticity *η* (black lines; c.i. = 0.01 s^{−1}); zero lines not plotted. Selected streamlines (gray) bracketing (*r _{m}*,

*z*) marked by the magenta × and the position

_{m}*r*

_{min}of

*u*

_{min}indicated by the blue arrow.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Model data at the location (*r _{m}*,

*z*) of the maximum tangential wind

_{m}*υ*;

_{m}*M*is the corresponding angular momentum and

_{m}*M*is the initial angular momentum at the radius

_{mi}*r*of the initial maximum tangential wind.

_{mi}*T*and

_{b}*T*

_{0}are the temperatures at (

*r*,

_{m}*z*) and in the upper-level radial outflow at large radius, respectively, and

_{m}*s*

_{surf}and

*s*

_{0}are the saturation entropy at the sea surface temperature (

*T*) and the entropy at the top of the surface layer (in practice at the first grid level, here at

_{s}*z*= 25 m), respectively; both are evaluated at

*r*=

*r*. The coefficient

_{m}*α*= 1 without dissipation or

*T*/

_{s}*T*

_{0}with dissipation, which is the case in this study.

Tables 1 and 2 list the parameters entering into (3). Table 2 also shows the theoretical maximum potential intensity PI^{+} and the EPI derived from the first term on the rhs of (3). Within cases RX and RXL, one observes a tendency for PI^{+} to increase and E-PI to decrease (slightly) with increasing *r _{mi}*. The difference between PI

^{+}and EPI indicates an increasing contribution from the supergradient-wind term in (3) with increasing

*r*. The last column of Table 2 shows that for the more horizontally diffusive R

_{mi}*X*cases

*υ*(Table 1) tends to be less than PI

_{m}^{+}while the RXL cases tend to exceed slightly PI

^{+}[consistent with Fig. 13 of Bryan (2012)]. Since Table 1 shows that

*r*increases with

_{m}*r*, Table 2 implies that supergradient winds increase with

_{mi}*r*.

_{m}### a. The radius r_{m}

*r*is a linear function of the angular momentum at

_{m}*r*=

*r*,

_{m}*M*(their Figs. 3a–c) and that

_{m}*M*is a linear function of

_{m}*M*(their Figs. 5d–f). It follows, therefore, that

_{mi}*r*is a linear function

_{m}*M*for the same skirt parameter. Analysis of the present cases finds

_{mi}*M*were based on a fixed-height measurement (

_{m}*z*= 1.55 km) in order that they be taken well above the boundary layer. The data shown in Table 1 are instead taken at the position of

_{m}*υ*which occurs at different

_{m}*z*in the different simulations. One may verify from the data in Table 1 that

_{m}*a*′ = 9.177 s km

^{−1}and

*b*′ = 1.1818 km for the RX cases and

*a*′ = 8.161 s km

^{−1}and

*b*′ = −3.964 km for the RXL cases.

Xu and Wang (2018) showed in their Figs. 4–6 that the processes of vortex intensification take place near and radially inward of *r _{mi}* due to Ekman pumping and subsequent convection (see also Peng et al. 2019). As a complement to the analysis in Xu and Wang (2018), Fig. 4 shows

*M*(

*r*,

*z*,

*t*) for the RX cases at

*z*=

*z*indicated in Table 1. Although

_{m}*M*is not conserved, its value at the quasi-steady position

*r*is not much different from

_{m}*M*(see also Table 1) which implies that the air arriving at

_{mi}*r*retains a strong “memory” of the angular momentum at

_{m}*r*. Theories for the gradient wind find that

_{mi}*r*, the radius of maximum gradient wind, should increase with the angular momentum at an outer radius (see, e.g., Fig. 7a of Tao et al. 2020b). Figure 4 suggests that

_{mg}*r*is the effective outer radius and that therefore, for constant

_{mi}*υ*,

_{mi}*M*increases with

_{mi}*r*(Table 1). It follows therefore that

_{mi}*r*increases with

_{mg}*r*(or

_{mi}*M*). Boundary layer theory and simulations predict

_{mi}*r*∝

_{m}*r*so that one can also expect

_{mg}*r*to increase with

_{m}*r*for the same skirt parameter.

_{mi}*M*(*r*, *z*, *t*) at *z* = *z _{m}* in indicated in Table 1 for cases (a) R60, (b) R90, and (c) R120. The 100 m s

^{−1}contour for

*υ*is indicated in black.

_{m}Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

*M*(*r*, *z*, *t*) at *z* = *z _{m}* in indicated in Table 1 for cases (a) R60, (b) R90, and (c) R120. The 100 m s

^{−1}contour for

*υ*is indicated in black.

_{m}Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

*M*(*r*, *z*, *t*) at *z* = *z _{m}* in indicated in Table 1 for cases (a) R60, (b) R90, and (c) R120. The 100 m s

^{−1}contour for

*υ*is indicated in black.

_{m}Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

### b. The product r_{m}w_{m}η_{m}

*r*,

_{m}*z*) is related to the radially inward boundary layer flow through the continuity equation as

_{m}*r*, (5) is simplified by approximating the integrand by its finite-difference approximation across an annulus inside of

_{m}w_{m}η_{m}*r*

_{min}and roughly centered at

*r*to arrive at

_{m}*r*

_{min}is the radius where the radial velocity reaches its minimum value and

*δr*is defined as the interval over which |

*ru*(

*r*,

*z*)| becomes small relative to |

*r*

_{min}

*u*(

*r*

_{min},

*z*)| (see, e.g., Fig. 2); although

*r*

_{min}

*u*(

*r*

_{min},

*z*) and

*δr*can vary with height, they are approximated here by their surface values. Letting

*u*(

*r*

_{min}, 0) ≡

*u*

_{min}with the foregoing approximations, (5) becomes

*δr*at

*z*=

*z*equals the horizontal mass flux in the boundary layer at

_{m}*r*=

*r*

_{min}through the “corner” flow illustrated by the streamlines in Fig. 3. Table 3 contains

*u*

_{min},

*r*

_{min}, and

*δr*; given the crude approximation to the radial derivative in (5),

*δr*is regarded as a tuning parameter to maximize agreement between ∂(

*ru*)/∂

*r*at (

*r*,

*z*) = (

*r*, 0) and its approximation in (6).

_{m}Model outputs *u*_{min}, inflow angle *ϕ*, *r*_{min}, parameters *δr* and *λ _{υ}*, vertical mixing length

*l*(

_{υ}*z*), and approximations to

_{m}*r*from (9) and from replacing

_{m}w_{m}η_{m}*β*= 3.32 and 3.76 for the RX and RXL cases, respectively. The last column is based on the approximations developed in this paper.

*η*= ∂

*u*/∂

*z*– ∂

*w*/∂

*r*, is to an excellent approximation

*η*≃ ∂

*u*/∂

*z*suggesting the scaling

*λ*is a vertical scale chosen to maximize the agreement between (8) and

_{υ}*η*. Further discussion of

_{m}*λ*is given below.

_{υ}*r*shown in Table 1. The theoretical project then is to find the relation of the various parameters on the rhs of (9) to

_{m}w_{m}η_{m}*r*.

_{m}Tables 1 and 3 indicate that *r*_{min} and *δr* increase in more or less constant proportion to the increase of *r _{m}*.

^{4}It follows therefore that the first factor in (9),

*r*

_{min}/

*δr*≡

*β*has no systematic dependence on

*r*. Table 3 shows the results from (9) with

_{m}*β*≃ 3.32 and 3.76 for the RX and RXL cases, respectively; these results indicate the major variation of

*r*with

_{m}w_{m}η_{m}*r*is retained. Therefore almost all the variation with

_{m}*r*in (9) must come from the factors

_{m}*u*

_{min},

*z*, and

_{m}*λ*. A boundary layer theory is required to understand the behavior of these parameters as functions of

_{υ}*r*and is discussed next.

_{m}### c. The velocity u_{min}

The flow in the tropical cyclone boundary layer, as with all boundary layers, is determined by the imposed pressure gradient and the viscous and/or turbulent stress within it. Exact nonlinear results from rotating-flow boundary layer theory have been found and verified experimentally for two canonical cases (reviewed in Rotunno 2014), they are the boundary layer of 1) a vortex in solid-body rotation (*υ* = *ωr*) over a disc of infinite radius (Bödewadt 1940; herein referred to as the SB case) and 2) a potential vortex (*υ* ∝ 1/*r*) over a disc of finite radius (Burggraf et al. 1971; herein referred to as the PV case). Both solutions have no-slip conditions applied to the velocity at the impermeable lower boundary and constant viscosity. Approximate (but quite complex) analytical solutions for the boundary layer of a Rankine-type vortex (*υ* ∝ *r* for *r* ≪ *r _{m}* and

*υ*∝

*r*for

^{−p}*r*≫

*r*, where 0 <

_{m}*p*≤ 1) with semislip (stress ∝ velocity) lower boundary conditions can be found in Kuo (1971). Kuo’s (1971) boundary layer solutions (see his Figs. 6.1–6.4) show features of the solid-body-rotation vortex for

*r*<

*r*and the potential vortex for

_{m}*r*>

*r*. Numerical simulations of the Navier–Stokes equations for a diffusively evolving Rankine vortex with a no-slip lower boundary are reported in Rotunno (2014). For more recent tropical cyclone–specific idealized boundary layer models see Kepert (2001), Kepert and Wang (2001), Kepert (2012, 2017), and Smith and Montgomery (2020).

_{m}The steady-state angular momentum *M*(*r*, *z*) in Fig. 5 suggests that the simulated flow is characterized by a Rankine-type vortex above the boundary layer. Closer examination of the velocity profile *υ*(*r*) above the boundary layer (not shown) for *r* > *r _{m}* shows, however, that

*υ*(

*r*) ∝

*r*with

^{−n}*n*≈ 0.8 (observations show

*n*< 0.7; Mallen et al. 2005). Although qualitatively reasonable solutions for the boundary layer of this type of vortex can be found using linear theory (e.g., Kepert 2017), they are technically limited by the requirement that the secondary-circulation velocities be much smaller that the tangential velocity maximum

*υ*. Tables 1 and 3 show that

_{m}*u*

_{min}/

*υ*≈ 0.35–0.38 which is not ≪1 and important features of the (nonlinear) dynamics are not accounted for. In what follows, the case is made that, while not strictly applicable, the nonlinear theory for the boundary layer of potential vortex (

_{m}*n*= 1.0) provides a useful interpretive tool for the present results.

*M*(*r*, *z*) (green; c.i. = 0.5 km^{2} s^{−1}) and the dominant terms in the radial momentum equation [black, *τ _{rz}*/∂

*z*; c.i. =0.03 m s

^{−2}, zero line not plotted]. Position of

*υ*indicated by the magenta × and the position

_{m}*r*

_{min}of

*u*

_{min}is indicated by the blue arrow.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

*M*(*r*, *z*) (green; c.i. = 0.5 km^{2} s^{−1}) and the dominant terms in the radial momentum equation [black, *τ _{rz}*/∂

*z*; c.i. =0.03 m s

^{−2}, zero line not plotted]. Position of

*υ*indicated by the magenta × and the position

_{m}*r*

_{min}of

*u*

_{min}is indicated by the blue arrow.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

*M*(*r*, *z*) (green; c.i. = 0.5 km^{2} s^{−1}) and the dominant terms in the radial momentum equation [black, *τ _{rz}*/∂

*z*; c.i. =0.03 m s

^{−2}, zero line not plotted]. Position of

*υ*indicated by the magenta × and the position

_{m}*r*

_{min}of

*u*

_{min}is indicated by the blue arrow.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

*u*(

*r*,

*z*) in the outer PV-like boundary layer and its inward decrease to zero in the inner SB-like boundary layer, as exemplified in Fig. 2. Figure 5 also displays the dominant terms

^{5}in the radial momentum equation, which can be written in the form

*D*stands for the radial diffusion terms [Eq. (13) of Rotunno and Emanuel 1987], which are dominated by the vertical diffusion of

_{u}*u*(

*r*,

*z*) to the ground (or “friction”). The first term on the rhs of (10) is the difference between the outward-directed centrifugal plus Coriolis accelerations and the inward-directed pressure-gradient acceleration.

The radial momentum budget in Fig. 5 shows that the pressure-gradient term overcomes the centrifugal/Coriolis term for the outer PV-like boundary layer producing an inward radial acceleration (Fig. 2); this term changes sign for the inner SB-like boundary layer and thus produces an outward radial acceleration (cf. Fig. 6d of Smith et al. 2009). This latter outward acceleration is entirely due to *r _{m}*,

*z*) illustrating that the supergradient wind is an essential feature of the boundary layer near

_{m}*r*=

*r*. The frictional term opposes the inward radial velocity at and near

_{m}*z*= 0 for both the inner-core and outer portions of the boundary layer; of special interest here is the radius at which the inward radial acceleration of the PV-like boundary layer is first cancelled by the outward frictional acceleration, indicated by the blue arrow. Not coincidentally, this radius is

*r*

_{min}. Figure 5 shows that

*r*

_{min}is also approximately the radius above the boundary layer marking the transition between the inner (SB-like) and the outer (PV-like) regions of

*M*(

*r*,

*z*).

The approach here is to use results from the literature on SB-like and PV-like boundary layers to interpret the dependencies of the factors in (9) on *r _{m}* and thus to the external parameters of the problem through (4). The biggest difference between the SB and PV boundary layers is the existence of a local similarity solution for the former and its nonexistence in the latter case (Rotunno 2014). In other words, for the SB case, the local radius

*r*, viscosity

*K*, and rotation rate

*ω*of the flow above the boundary layer are sufficient to determine the local solution.

^{6}In contrast, in the PV case the solution at any

*r*depends on the upstream evolution of the boundary layer, i.e., nonlinear advection is indispensable. Figure 4 suggests one can consider the simulated tropical cyclone vortex as having three regions: an inner region of nearly solid-body rotation, a middle region of nearly constant

*M*, and an outer region of gradually increasing

*M*. In this latter region one can ignore advection and consider, for example, the utility of “column” models (e.g., Bryan et al. 2017). Based on the latter three-zone vortex model, the working hypothesis here is that the middle, PV-like boundary layer controls

*u*

_{min}independently of the outer zone, while the SB-like boundary layer controls the vertical scales

*z*and

_{m}*λ*since both are defined at

_{υ}*r*which is clearly within the SB-like region.

_{m}*u*

_{min}would consist of (10) and

*D*stands for the diffusion terms in the azimuthal direction [Eq. (14) of Rotunno and Emanuel 1987] and the continuity equation. This model would require using a flow-dependent

_{υ}*K*and horizontal viscosity. Lacking such an analytical model, one can however use the numerical model results and (10) to demonstrate the importance of nonlinear advection. Figure 6 shows the major terms in (10) at

_{υ}*z*= 0, viz.,

*z*= 0 from

*r*

_{min}to

*r*gives

_{mi}*u*

^{2}(

*r*) is small and can been neglected. Equation (13) yields the numerically simulated

_{mi}*u*

_{min}(Table 3) to within ±1 m s

^{−1}(not shown) and confirms that

*u*

_{min}is controlled by the middle region (

*r*

_{min}<

*r*<

*r*) of nearly constant

_{mi}*M*. This result is in broad agreement with Smith and Montgomery (2020, p. 3443) on the importance of nonlinear advection in the radial-momentum budget. Note incidentally that using the mixed-layer form of (11) to deduce

*u*(

*r*) in the boundary layer would fail in the limit (∂

*M*/∂

_{g}*r*) → 0 [see, e.g., Eq. (A13) of Peng et al. 2018] for a potential vortex as would any linear theory.

Terms in (12) for the (a) RX and (b) RXL cases with −*udu*/*dr* in green, *τ _{rz}*/∂

*z*in black; magnitudes in m s

^{−2}. The blue stars indicate the position of

*r*

_{min}in each case.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Terms in (12) for the (a) RX and (b) RXL cases with −*udu*/*dr* in green, *τ _{rz}*/∂

*z*in black; magnitudes in m s

^{−2}. The blue stars indicate the position of

*r*

_{min}in each case.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Terms in (12) for the (a) RX and (b) RXL cases with −*udu*/*dr* in green, *τ _{rz}*/∂

*z*in black; magnitudes in m s

^{−2}. The blue stars indicate the position of

*r*

_{min}in each case.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

*ϕ*≡ tan

^{−1}(

*u*

_{min}/

*υ*), where

_{ms}*υ*= 0.8

_{ms}*υ*

_{m}^{7}is the maximum tangential wind at the surface, displayed in Table 3. These values are consistent with the observationally estimated value of −22.6° ± 2.2° (Zhang and Uhlhorn 2012). Using

*ϕ*one can recast (9) as

*r*in cases with larger

_{m}w_{m}η_{m}*υ*, other things being equal. As there is little or no systematic dependence of

_{m}*β*,

*ϕ*, or

*υ*with

_{m}*r*within the RX and RXL cases individually, (14) implies that the dependence of

_{m}*r*on

_{m}w_{m}η_{m}*r*is embodied in

_{m}*z*/

_{m}*λ*.

_{υ}### d. The height z_{m}

*z*(

_{m}*r*). Looking for guidance from theory, both linear (Kepert 2001) and nonlinear (Rotunno and Bryan 2012) theory for the SB-like inner region of the simulated tropical cyclone vortex predict the depth scale

_{m}*I*= 2

*ω*for the SB vortex;

*K*is the vertical eddy viscosity. Taking into account the semislip lower boundary condition, the height of the tangential-wind maximum is given by the first of Kepert’s (2001) Eq. (27) as

_{υ}*V*a characteristic tangential velocity. The arctangent function in (15) varies from

*π*/2 to 3

*π*/4 as

*χ*ranges from 0 to ∞ representing the lower boundary condition ranging from free-slip to no-slip.

^{8}For the SB vortex

*ω*is constant; however, the simulated tropical cyclone vortices are Rankine-like with the inner core

*ω*varying from a constant in the inner SB vortex and diminishing as the inverse square of radius in the outer PV vortex. Figure 7 shows

*ω*(

*r*) at

*z*= 3 km, where

*υ*≃

*υ*. For the horizontally diffusive RX cases,

_{g}*ω*is relatively constant for

*r*<

*r*(denoted by the placement of the crosses in the radial direction), while for the RXL cases,

_{m}*ω*is peaked just within

*r*=

*r*, indicative of vorticity concentration in the eyewall. The author’s best estimate of the effective value of

_{m}*ω*to use in (15) is given by the placement of the crosses in the

*ω*direction in Fig. 7. To discover the relation

*z*(

_{m}*r*), one must find a relation between

_{m}*ω*and

*r*. Figure 8 shows of

_{m}*υ*/

_{m}*r*versus the

_{m}*ω*estimated from Fig. 7; this analysis suggests

*υ*/

_{m}*r*≃

_{m}*μω*with

*μ*varying between 1.3 and 1.4. Letting

*ω*≃

*μ*

^{−1}

*υ*/

_{m}*r*and

_{m}*V*=

*υ*, (15) can be written as

_{m}*υ*is relatively constant within the RX and RXL cases individually. It follows therefore that the dependence

_{m}*z*(

_{m}*r*) by (16) depends directly on

_{m}*r*and indirectly on the variation of

_{m}*K*with

_{υ}*r*.

_{m}Estimated rotation rate *ω* of the gradient wind at *z* = 3 km for the (a) RX and (b) RXL cases with *X* = 60 (magenta), 90 (green), and 120 (blue). The placement of the crosses in the *ω* direction indicates the estimated rotation rate of the inner core and their placement in the radial direction indicates *r _{m}*.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Estimated rotation rate *ω* of the gradient wind at *z* = 3 km for the (a) RX and (b) RXL cases with *X* = 60 (magenta), 90 (green), and 120 (blue). The placement of the crosses in the *ω* direction indicates the estimated rotation rate of the inner core and their placement in the radial direction indicates *r _{m}*.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Estimated rotation rate *ω* of the gradient wind at *z* = 3 km for the (a) RX and (b) RXL cases with *X* = 60 (magenta), 90 (green), and 120 (blue). The placement of the crosses in the *ω* direction indicates the estimated rotation rate of the inner core and their placement in the radial direction indicates *r _{m}*.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

The rotation rate *υ _{m}*/

*r*(s

_{m}^{−1}) (Table 1) vs the estimates of

*ω*(s

^{−1}) from Fig. 7. The line

*υ*/

_{m}*r*=

_{m}*μω*with

*μ*= 1.3 and 1.4 are plotted for reference.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

The rotation rate *υ _{m}*/

*r*(s

_{m}^{−1}) (Table 1) vs the estimates of

*ω*(s

^{−1}) from Fig. 7. The line

*υ*/

_{m}*r*=

_{m}*μω*with

*μ*= 1.3 and 1.4 are plotted for reference.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

The rotation rate *υ _{m}*/

*r*(s

_{m}^{−1}) (Table 1) vs the estimates of

*ω*(s

^{−1}) from Fig. 7. The line

*υ*/

_{m}*r*=

_{m}*μω*with

*μ*= 1.3 and 1.4 are plotted for reference.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

*K*; however, Figs. 9a and 9b show that it is highly variable, especially near the level

_{υ}*z*(marked by the crosses) in the RXL cases. Figures 9c and 9d show that to a very good approximation

_{m}*z*and a secondary maximum aloft, Stern et al. (2020) computed

_{m}*S*is the vertical deformation [Eq. (16) of Bryan and Rotunno 2009b] and found reasonably good agreement between the formula and the model results across simulations with different values of

_{υ}*l*

_{∞}and

*C*(their Fig. 19). However, in the present problem, at and below

_{d}*z*,

_{m}*K*has significant variation, especially in the RXL cases. For

_{υ}*K*and

_{υ}*υ*approximately constant, (16) implies

_{m}*r*, more like

_{m}*z*∝

_{m}*r*.

_{m}Vertical eddy viscosity *K _{υ}* for the (a) RX (c.i. = 50 m

^{2}s

^{−1}) and (b) RXL (c.i. = 100 m

^{2}s

^{−1}) cases, with

*X*= 60, 90, and 120 (magenta, green, and blue, respectively). The major contributor to

*K*, (17), for the (c) RX and (d) RXL cases with the same c.i. as in (a) and (b). The × marks the location of (

_{υ}*r*,

_{m}*z*) for each case.

_{m}Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Vertical eddy viscosity *K _{υ}* for the (a) RX (c.i. = 50 m

^{2}s

^{−1}) and (b) RXL (c.i. = 100 m

^{2}s

^{−1}) cases, with

*X*= 60, 90, and 120 (magenta, green, and blue, respectively). The major contributor to

*K*, (17), for the (c) RX and (d) RXL cases with the same c.i. as in (a) and (b). The × marks the location of (

_{υ}*r*,

_{m}*z*) for each case.

_{m}Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Vertical eddy viscosity *K _{υ}* for the (a) RX (c.i. = 50 m

^{2}s

^{−1}) and (b) RXL (c.i. = 100 m

^{2}s

^{−1}) cases, with

*X*= 60, 90, and 120 (magenta, green, and blue, respectively). The major contributor to

*K*, (17), for the (c) RX and (d) RXL cases with the same c.i. as in (a) and (b). The × marks the location of (

_{υ}*r*,

_{m}*z*) for each case.

_{m}Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

*u*/∂

*z*≃ −

*u*

_{min}/

*λ*, but further explanation of the origin of

_{υ}*λ*was deferred. It may be verified by the data in Tables 1 and 3 that

_{υ}*γ*≃ 13 ± 1 and

*l*(

_{υ}*z*) is given by (2). A rationale for (18) is given in the last part of this section, but for now, its consequences w.r.t. to (16) are developed. Substitution of (17) and (18) into (16) gives

_{m}*z*(

_{m}*r*), but first consider (19) with the tan

_{m}^{−1}factor replaced by

*σ*. Neglecting

*z*

_{0}in (2) and substituting the result into (19) gives

*z*(

_{m}*r*). Equation (2) suggests the nondimensionalization

_{m}*u*

_{min}/

*υ*= 0.8 tan

_{m}*ϕ*, leads to the solution

The data in Tables 1 and 3 have an average *ϕ* = −24.1°; with *γ* = 13.1, *C* = 0.0112*μ*. Figure 10 shows in dimensional form the exact solution of (19) and the approximate solution (21) with *σ* = *π*/2 and *μ* = 1.35. Evaluation of the tan^{−1} factor in (19) from the exact solution for *μ* = 1.35, or from substituting the approximate solution for *z _{m}* into the formula for

*χ*, both give

*σ*(

*r*) ranging from 1.59 to 1.62 between the limits of Fig. 10. Figure 10 also plots the data points

_{m}*z*(

_{m}*r*) from Table 1. Based on the values of

_{m}*μ*reported above, all the data points should straddle the

*μ*= 1.35 line which is approximately the case for the RXL cases. However, the theoretical curve underestimates

*z*(

_{m}*r*) for the RX cases; in these cases there appears to be a systematic offset from the theoretical curve. In the RX cases, horizontal diffusion is as large or larger than the vertical diffusion terms in the angular momentum budget (not shown); to take the larger total diffusion into account (at least superficially), the solution with

_{m}*μ*= 2.7 (2 × 1.35) appears to account for

*z*(

_{m}*r*) in the RX cases.

_{m}Solution for *z _{m}*(

*r*) from the exact (19) (solid blue line) and the approximate (21) (dashed line) for

_{m}*μ*= 1.35 along with the RX and RXL data. Also shown is the exact solution curve for

*μ*= 2.7.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Solution for *z _{m}*(

*r*) from the exact (19) (solid blue line) and the approximate (21) (dashed line) for

_{m}*μ*= 1.35 along with the RX and RXL data. Also shown is the exact solution curve for

*μ*= 2.7.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Solution for *z _{m}*(

*r*) from the exact (19) (solid blue line) and the approximate (21) (dashed line) for

_{m}*μ*= 1.35 along with the RX and RXL data. Also shown is the exact solution curve for

*μ*= 2.7.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

### e. The scale λ_{υ}

*τ*can be approximated by

_{rz}*z*= 0 to

*z*=

*z*at

_{m}*r*=

*r*with

_{m}*D*≃ ∂

_{u}*τ*/∂

_{rz}*z*and using (22) gives

*F*represents horizontal diffusion and the (small) influence of filters. From (23) therefore

^{9}

*U*

^{2},

*γ*, and

*λ*=

_{υ}*γl*(

_{υ}*z*), shown in Table 4. The values of

_{m}*γl*are generally consistent with the original estimate of

_{υ}*λ*shown in Table 3 but are on the low side, which can be traced back to the fact that (17) slightly underestimates the actual

_{υ}*K*. The author has been unable to find useful estimates of the rhs of (23); however, the main virtue of (23) and (24) is that they provide a theoretical rationale for the estimate of

_{υ}*η*from (8) with (18), viz.,

_{m}## 4. Synthesis

*z*(

_{m}*r*;

_{m}*μ*) can be estimated from (21) with

*μ*= 2.7 and 1.35 for the RX and RXL cases, respectively, and

*l*(

_{υ}*z*) from (2). Recalling that

_{m}*β*= 3.32 and 3.76 for the RX and RXL cases, respectively, and

*γ*= 13, the final estimate of

*r*based on (27) is given in the last column of Table 3. Although the numerical values differ considerably from the values of

_{m}w_{m}η_{m}*r*given in Table 1, the general increase with

_{m}w_{m}η_{m}*r*is a robust result upon which a physical interpretation can be based.

_{m}As already noted *β* and *υ _{m}* have little variation with

*r*within the RX and RXL cases individually. Thus the major variation with

_{m}*r*on the rhs of (27) comes from the ratio

_{m}*z*(

_{m}*r*)/

_{m}*l*(

_{υ}*z*) which indicates, by (7) and (8), that

_{m}*r*and

_{m}w_{m}*η*have inverse variations with

_{m}*z*and, since

_{m}*z*increases with

_{m}*r*, they have inverse variations with

_{m}*r*. Using the above analysis, the relation (7) can be written as

_{m}*r*≃

_{m}w_{m}*β*tan

*ϕz*(

_{m}*r*)

_{m}*υ*and relation (26) as

_{m}*η*≃ tan

_{m}*ϕυ*/[

_{m}*γl*(

_{υ}*z*)]. Although

_{m}*z*increases with

_{m}*r*(Fig. 10) and

_{m}*l*(

_{υ}*z*) increases with

_{m}*z*by (2), the latter reaches the asymptotic limit

_{m}*l*

_{∞}and thus the increase of

*r*with

_{m}w_{m}*z*dominates the decrease of

_{m}*η*with increasing

_{m}*l*.

_{υ}In physical terms the near constancy of *ϕ* within the RX and RXL cases individually, and increasing *z _{m}* with

*r*in all cases, imply increasing horizontal mass flux through the boundary layer (−

_{m}*r*

_{min}

*u*

_{min}

*z*at

_{m}*r*=

*r*

_{min}) and, by continuity, increasing vertical mass flux (

*r*at

_{m}w_{m}δr*z*=

*z*). As established in section 3b,

_{m}*r*

_{min}and

*δr*increase with

*r*such that their ratio is approximately fixed and therefore they do not contribute to increasing

_{m}*r*with

_{m}w_{m}*r*. The form of the stress (22) implies that

_{m}*η*≃ (∂

_{m}*u*/∂

*z*)

*∝ [*

_{m}*l*(

_{υ}*z*)]

_{m}^{−1}and that therefore the offsetting decrease of

*η*with

_{m}*l*(

_{υ}*z*) in

_{m}*r*is limited by the approach of [

_{m}w_{m}η_{m}*l*(

_{υ}*z*)]

_{m}^{−1}to a constant with increasing

*z*(

_{m}*r*).

_{m}*z*

_{0}= 0) to obtain

*κ*,

*γ*, and

*l*

_{∞}are regarded as “fixed.” Substitution of (28) into (1) with

*υ*=PI

_{m}^{+}in (28) gives

*ϕ*(≃−23°) and

*z*(≃500 m). However,

_{m}*β*(=

*r*

_{min}/

*δr*) is harder to determine from these analyses. All the data show that

*r*

_{min}tracks with

*r*; however,

_{m}*δr*, which is a measure of the eyewall thickness, can vary from

*δr*≃

*r*

_{min}in the composites (Fig. 5 of Zhang et al. 2011) to something much smaller in case studies (Fig. 4 of Bell and Montgomery 2008). With

*α*= 1.5,

*γ*= 13.1,

*l*

_{∞}= 100 m,

*ϕ*= −24.1°, and

*z*= 500 m, (29) gives PI

_{m}^{+}/EPI = (1.04, 1.09, 1.15) for

*β*= (1, 2, 3). Increasing PI

^{+}/EPI with

*β*is consistent with the less horizontally diffusive simulations (with thinner eye walls) having larger supergradient winds (Fig. 1 of Rotunno and Bryan 2012).

A caveat to the foregoing is that the current observational estimates of *C _{d}* ≃ 0.0025 and

*C*≃ 0.001 (Bell et al. 2012) imply

_{k}*C*/

_{k}*C*= 0.4 instead of

_{d}*C*/

_{k}*C*= 1.0 used in this study. Other things being equal, a smaller

_{d}*C*/

_{k}*C*would decrease EPI (3) and thus increase PI

_{d}^{+}/EPI. Numerical simulations reported in (Bryan 2012, his Fig. 12) indicate a significant increase of PI

^{+}/EPI for

*C*/

_{k}*C*< 1.0 and small

_{d}*l*.

_{h}## 5. Summary and conclusions

Bryan and Rotunno (2009a) derive the formula (1) in which the supergradient winds are represented by the term *αr _{m}w_{m}η_{m}*, where

*α*is nearly a constant for a given thermodynamic environment. It is argued herein that understanding

*r*requires an understanding of the simulated tropical cyclone boundary layer. Based on the continuity equation and the stress model used in the numerical simulations, it is found that

_{m}w_{m}η_{m}*u*

_{min}is the maximum radial inflow velocity,

*l*(

_{υ}*z*) is the vertical mixing length, (2), and (

*r*,

_{m}*z*) is the location of maximum tangential wind

_{m}*υ*. The increase of the supergradient wind term with

_{m}*r*, as expressed by

_{m}By examining a set of simulations in which the radius of maximum tangential wind of the initial vortex *r _{mi}* increases, it is found that

*r*increases with

_{m}*r*. Based on theories for the maximum gradient wind

_{mi}*υ*(Emanuel 1986; Frisius et al. 2013; Tao et al. 2020b), the radius

_{mg}*r*of the maximum gradient wind

_{mg}*υ*should increase with increasing angular momentum at an outer radius. The present numerical simulations suggest the radius of the initial tangential wind maximum

_{mg}*r*is the effective outer radius. At

_{mi}*r*the angular momentum

_{mi}*M*increases for fixed initial maximum tangential velocity; consistent with the gradient-wind theories and boundary layer theory,

_{mi}*r*∝

_{m}*r*so that

_{mg}*r*increases with

_{m}*r*(or

_{mi}*M*).

_{mi}To understand the dependence of *r _{m}*, appeal is made to an approximate version of the simulated tropical cyclone vortex. Herein the case is made that a useful compromise between fidelity and simplicity is the Rankine vortex in which the inner core is in solid-body rotation (SB) and the outer core is characterized by a potential vortex (PV). The working hypothesis in this paper is that the mechanism(s) controlling the merger of PV outer-core and SB inner-core boundary layers is at the crux of the understanding of supergradient winds. Lacking an analytical solution for the Rankine-vortex boundary layer, the present approach is to examine the model data in light of the known properties of the SB and PV boundary layers individually.

Figure 11 is a schematic diagram of the present conceptual model^{10} with the key features extracted from R120L. In the PV outer core (where *M _{g}* ≃ const.), the boundary layer inflow velocity −

*u*

_{min}is fundamentally the result of the unbalanced inward radial pressure-gradient force and nonlinear advection that accelerates the boundary layer flow inflow (cf. Burggraf et al. 1971). Analysis of the present simulations (Fig. 6) in terms of (6) and (13) support for this conclusion. A fluid parcel traversing from the PV outer-core to the SB inner-core (where

*dM*/

_{g}*dr*> 0) boundary layer (i.e., from

*r*

_{min}to

*r*), by virtue of its inertia continues to flow inward while partially conserving its angular momentum, which in turn, leads to an increase of the outward-directed centrifugal acceleration w.r.t. to the inward-directed pressure-gradient acceleration in the SB vortex (cf. Bödewadt 1940). The force imbalance decelerates the inward radial flow to zero as the parcels rise up to the level of maximum tangential wind

_{m}*z*(Fig. 3). Although without rigorous justification, it was found that using the formulas for

_{m}*z*from linear and nonlinear theories of the boundary layer of a vortex in solid-body rotation (based on a constant viscosity) together with a close approximation to the model’s flow-dependent eddy viscosity (17), gives a reasonably good estimate of

_{m}*z*(

_{m}*r*) (Fig. 10). The stress model (22) introduces the factor [

_{m}*l*(

_{υ}*z*)]

_{m}^{−1}in

Schematic diagram of the key features of the boundary layer extracted from the R120L simulation. Shown are the angular momentum contours 2.5 and 3.0 × 10^{6} m^{2} s^{−1}, the position of maximum tangential wind (*r _{m}*,

*z*) (magenta cross) and the location

_{m}*r*

_{min}of the minimum (maximum inward) radial velocity.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Schematic diagram of the key features of the boundary layer extracted from the R120L simulation. Shown are the angular momentum contours 2.5 and 3.0 × 10^{6} m^{2} s^{−1}, the position of maximum tangential wind (*r _{m}*,

*z*) (magenta cross) and the location

_{m}*r*

_{min}of the minimum (maximum inward) radial velocity.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Schematic diagram of the key features of the boundary layer extracted from the R120L simulation. Shown are the angular momentum contours 2.5 and 3.0 × 10^{6} m^{2} s^{−1}, the position of maximum tangential wind (*r _{m}*,

*z*) (magenta cross) and the location

_{m}*r*

_{min}of the minimum (maximum inward) radial velocity.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

The physical content in *z _{m}* with

*r*with

_{m}*u*

_{min}∼ ct. (within the individual sets of simulations) implies increasing horizontal mass flux through

*r*

_{min}and therefore, by continuity, increasing vertical mass flux at

*z*near

_{m}*r*=

*r*(

_{m}*r*increases with

_{m}w_{m}*r*). Although the stress model implies

_{m}*η*∝ 1/

_{m}*l*(

_{υ}*z*), the decrease of

_{m}*η*with

_{m}*l*(

_{υ}*z*) is muted by the asymptotic limit

_{m}*l*(

_{υ}*z*) →

*l*

_{∞}for large

*z*(≫

*l*

_{∞}/

*κ*).

*αr*is diagnosed from the fully interactive numerical simulations. A more complete theory yielding consistent interior and boundary layer winds at steady state would presumably come from a solution to the nonlinear partial differential equation (Tao et al. 2020b) from which (1) originates. More specifically, (1) of Tao et al. (2020b) can be written schematically as a nonlinear Helmholtz equation valid for the inviscid, adiabatic flow above the boundary layer,

_{m}w_{m}η*ψ*are on the interior domain (i.e., the lower boundary is at the top of the boundary layer). The nonlinear functional

*F*depends on knowledge of flow within the boundary layer [e.g.,

*M*(

*ψ*)] since the validity of (30) depends on the flow emanating upward from the boundary layer; however, the boundary layer model requires knowledge of the radial pressure gradient above the boundary layer [i.e., the solution of (30)]; hence an iteration would be required. Numerical models, such as the one analyzed here, do this iteration automatically. The author has not yet succeeded in discovering an analytical/theoretical way of unifying the interior and boundary layer solutions.

Second, a more theoretically justifiable theory for *z _{m}*(

*r*) with a flow-dependent eddy diffusion in a Rankine-type vortex would be a welcome addition to the theory of tropical cyclone boundary layers. Third, the form of

_{m}*z*(

_{m}*r*) derived here depends on the assumed functional dependence of

_{m}*l*(

_{υ}*z*) in (2); it remains to be seen what effect other equally reasonable functions

*l*(

_{υ}*z*) has on the simulated

*z*(

_{m}*r*). Finally the variation of the present estimates of the supergradient wind to changes in

_{m}*C*,

_{d}*C*,

_{k}*l*

_{∞}, and initial-vortex structure and amplitude was not addressed.

This is the flow in the radial-vertical plane; the primary circulation is the flow in the azimuthal direction.

By advecting upward an angular momentum distribution *M*(*r*) incompatible with that assumed for *υ _{g}*(

*r*).

Figure 10 of Bryan and Rotunno (2009a) illustrates that this effect moves the maximum-tangential-wind angular-momentum surface inward with respect that implied in the EPI formulation.

A detailed analysis of how these parameters depend on *r _{m}* would entail an analysis of how the flow-dependent diffusion and/or filters behave in cylindrical coordinates; such an analysis is not necessary to achieve the goals of the present study. For additional information on this issue, see the modeling study of Kepert (2017, his Figs. 6b–7b) which suggests eyewall width and radius increase together.

The horizontal diffusion terms in the radial-momentum equation are negligible in the RXL cases and are small (but nonnegligible inward of *r*_{min}) in the RX cases; horizontal diffusion affects *u*(*r*, *z*) in the RX cases mainly through its effect on *M*(*r*, *z*) in and above the boundary layer.

Solutions based on linear theory are all of this type.

The model-derived factor 0.8 is consistent with observations (Powell 1980).

The nonlinear solution (Rotunno and Bryan 2012, their Table 1) shows *z _{m}*(

*K*), the nonlinear equivalent of the tan

^{−1}function with

*K*= 1/

*χ*.

Kepert’s (2012) Eq. (12) is the analogous result for a constant-stress layer (*U*^{2} = 0) and constant *l _{υ}*.

This schematic is similar in concept to Fig. 1b in Montgomery and Smith (2017).

## Acknowledgments.

Informal reviews of the first draft of this paper by Drs. George H. Bryan, Robert G. Nystrom, and Dandan Tao are gratefully acknowledged. Formal reviews by Jeffrey D. Kepert and Michael T. Montgomery greatly improved the clarity and accuracy of the revised manuscript. Richard Rotunno is supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement 1852977.

## Data availability statement.

The data used in this paper are available from the author on request.

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