An Analytic Model for Tropical Cyclone Outer Winds

The variation of Tropical cyclone azimuthal wind speed (V) with distance from storm center (r) is a fundamental aspect of storm structure with important implications for risk and damages. The theoretical model of Emanuel (2004, https://doi.org/10.1017/CBO9780511735035.010), which applies outside the rainy core of the storm, matches radiatively‐driven subsidence and Ekman suction rates just above the boundary layer to obtain a nonlinear differential equation for dV/dr. This model is appealing because of its strong physical foundation, but lacks a known analytic solution for V(r). In this paper, I obtain an analytic solution to V(r) for the Emanuel (2004, https://doi.org/10.1017/CBO9780511735035.010) outer wind model. Following previous work, I then use this solution to explore properties of merged wind models that combine the outer model with an inner model, which applies to the rainy core of a storm.

• Analytic solutions are derived for the previously unsolved outer wind model of Emanuel (2004,  2 of 10 model of Emanuel (2004) is a major theoretical accomplishment that remains under-appreciated, likely due to the lack of known closed-form solutions. The code provided as part of this work (Cronin, 2023) may be useful to researchers who model hurricane risk, as it accelerates merged wind profile calculations by a factor of ∼50 relative to an existing method (Chavas, 2022).
The second goal is to leverage analytic solutions to consider constraints on V(r) in present or future climates. I find that in the relevant region of parameter space, merged profiles follow a scaling close to 2 −1 ∼ 2 0 , where f is the Coriolis parameter, c D the drag coefficient, and w r the radiative-subsidence speed (Section 4). This scaling arises from equating the total ascent and descent of the overturning circulation, and indicates that in a future climate, storms with the same outer size will likely have a smaller radius of maximum winds due to increases in V m and decreases in w r . Although my findings here do not rely on the analytic outer wind solution, this section is facilitated by faster merged wind profile calculations and by prior discussion of both inner and outer wind profiles. Finally, I close with a summary of findings, and some thoughts about limitations and future directions (Section 5). Emanuel (2004) derives an expression for the radial gradient of the azimuthal wind (dV/dr) outside the rainy core of a Tropical cyclone, based on the budget of absolute angular momentum in the boundary-layer inflow. In steady state at a given radius, the absolute angular momentum averaged over the boundary layer depth, = + 1 2 2 , is increased by inward radial advection of air with higher M, and decreased by torque due to surface stress, c D V 2 . Taking ψ as the cyclone's overturning circulation streamfunction in the radius-height plane at the top of the boundary layer (vertical velocity = 1 ), this balance is:

Derivation
In the outer regions of the storm, where there are no convective updrafts, ψ must increase with decreasing radius to accommodate sinking air at the top of the boundary layer. This air is thermodynamically constrained to descend at the radiative-subsidence speed = −̇∕ , where ̇ is the radiative heating rate of air just above the boundary layer, and θ is the potential temperature (w r is defined to be positive for subsiding flow). Over Tropical oceans, w r is typically several millimeters per second, and the drag coefficient c D ∼ 10 −3 . If the circulation of the storm vanishes at some outer radius, r 0 , and w r is taken as a constant, then the streamfunction at r < r 0 can be directly obtained by integrating w r over the annulus between r and r 0 : ( ) = ( 2 0 − 2 ) ∕2 (e.g., Figure 1). This balance can equivalently be viewed as matching radiative subsidence and Ekman suction just above the boundary layer, because the absolute vorticity f + ζ in the denominator of the Ekman suction can be written as 1 . Either view leads to the same conclusion: the absolute angular momentum in the outer region of the storm increases with radius according to: which gives the following equation for V: Figure 1. (a) Radial profile of azimuthal or swirling winds, V(r), for a Tropical cyclone. General features include the radius of maximum wind, r m , the maximum wind speed V m , and the radius of vanishing wind, r 0 . The specific profile drawn in black merges the Emanuel (2004) outer wind model (cyan dashed line) and the Emanuel and Rotunno (2011) inner wind model (red dashed line), following Chavas et al. (2015). The theoretical absolute-angular-momentum-conserving wind profile (green), and the merge radius r a are also drawn. (b) The overturning circulation in the radius-height plane generally includes ascent at small radii, and sinking at large radii. Merged wind profiles of Chavas et al. (2015) have a continuous overturning streamfunction (ψ) at r a , and assume a constant radiative-subsidence speed w r for r > r a .

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This is a Riccati equation with no known closed-form solution, but it can be transformed into a second-order ordinary differential equation (ODE) by a change of variables. I show below that this transformed equation is amenable to a quickly-converging power series solution when expanded in a coordinate x ≡ 1 − r/r 0 that varies from 0 at the outer edge of the storm to 1 at storm center.
Using primes to denote derivatives of a function q with respect to r, a Riccati equation of the form: can be rewritten as a second-order homogeneous ODE in a transformed function y, where qA(r) = −y′/y: Applying this result to Equation 4 with q = rV and simplifying slightly gives: If a solution for y(r) can be found, then V is given by where the first term, labeled V AMC (r), is the absolute-angular-momentum-conserving azimuthal wind speed for inflow from a quiescent state at radius r 0 inward to radius r. The second term, labeled G(r), is the fractional reduction of wind speed relative to V AMC due to loss of absolute angular momentum by surface friction. Physical must be bounded on [0,1], and satisfy the boundary condition of G(r 0 ) = 1. Note that y′/y has dimensions of inverse distance, w r distance per time, f inverse time, and c D is dimensionless, so G(r) is indeed dimensionless.
Equation 7 can be solved with a power series in r, but this series converges slowly and has an undetermined free parameter that does not clearly relate to the outer boundary condition (G(r 0 ) = 1). However, a change of variables in Equation 7, to: gives a power series solution that both converges comparatively quickly and naturally matches G(r 0 ) = 1. Since dx = −dr/r 0 , Equation 7 expressed in terms of x (with an (x) subscript on a primed term denoting a derivative with respect to x) becomes: where ≡ 0 −1 is identical to the nondimensional outer wind parameter found in . Note that the solution for G is expressed in terms of The power series solution to Equation 10, given by = ∑ ∞

=0
, can be taken generally to have a 0 = 1 (the choice of a 0 does not affect G since it does not alter the ratio ′ ( ) ∕ ), leading to the first few terms and recurrence relation for coefficients as follows: 4 of 10 (Here, terms outside braces that are factored out show that one can write a n as 1/(n!) 2 multiplied by a degree-n polynomial in γ with integer coefficients-a fact used further in Text S1 in Supporting Information S1.) The power series of the derivative ′ ( ) , is given by ′ ( ) = ∑ ∞ =0 ( + 1) +1 , so: The last line here also shows that this solution satisfies the outer boundary condition of G(r 0 ) = 1 because x = 0 at r = r 0 .
The wind speed relative to the absolute-angular-momentum-conserving limit, G(r), is a function of γ ≡ c D fr 0 / w r . G(r) decreases slowly with decreasing radius for small γ, and strongly with decreasing radius, particularly near r = r 0 , for larger γ (Figure 2a). A larger outer radius, drag coefficient, or Coriolis parameter all correspond to a greater torque on the inflow and a greater reduction in absolute angular momentum, whereas a larger radiative-subsidence speed leads to stronger radial advection of absolute angular momentum by a stronger overturning circulation, and thus a weaker dependence of G on r. For real-world storms with typical values of γ ∼ 10-100, a few dozen terms (for both numerator and denominator) in G are sufficient to give errors less than 0.01% ( Figure 2b). Series truncation errors are calculated relative to a solution with 100 terms, and for a given error tolerance, the number of required terms increases with γ. Series solutions are thus relatively efficient at calculating outer wind profiles, though more computationally efficient methods may exist. Further details of results including numerical implementation of vectorized calculation of G(r) and approximate solutions to G(r) are presented in Texts S1 and S2 in Supporting Information S1, respectively. Chavas et al. (2015) merge solutions for the outer wind profile of Emanuel (2004) and the convective core wind profile of Emanuel and Rotunno (2011). I follow the same procedure, whereby V and dV/dr are matched for inner and outer profiles, but using analytic outer wind profiles.

Merging With the Inner Wind Profile
I consider the maximum azimuthal wind speed V m and the radius of maximum winds r m as known variables, and the merge radius between inner and outer profiles r a and the outer radius r 0 as unknowns (r a and r 0 are generally shown normalized by r m ). For a ratio of enthalpy exchange to drag coefficients c k /c D = 1, the inner wind profile from Emanuel and Rotunno (2011) (their Equation 36) becomes: where V x and r x are nominally the maximum azimuthal wind speed and radius of maximum winds (Emanuel & Rotunno, 2011). A subtle distinction must be drawn here: these nominal values V x and r x are roughly but not exactly equal to V m ≡ max(V in ) and the radius r m where this occurs. Although V x converges to V m (as does r x to r m ) for V x /(fr x ) ≫ 1, this convergence is slow: r m lies about 5% inward of r x for V x /(fr x ) = 10 and about 0.5% inward of r x for V x /(fr x ) = 100. Such differences are negligible for scaling arguments below, but they lead to unacceptably large mismatches when comparing to previous code if left uncorrected (i.e., if V x = V m and r x = r m are used in Equation 13). Thus, similarly to Chavas (2022), I use several iterations to solve for values of V x and r x in Equation 13 that give a wind profile consistent with the user-input values of V m = max(V in ) at radius r m .
Taking V m and r m as known parameters, two dimensionless variables that govern merged solutions are: where ̃ is a normalized radiative-subsidence speed (following Emanuel, 2004;Chavas & Emanuel, 2014, it represents a ratio of the outer descent rate to the Ekman pumping ascent in the center of the storm), and Ro is the inner-core Rossby number. Although the outer wind profile is now analytic (Equation 12), the merge radius r a and outer radius r 0 must still be solved for numerically as a function of Ro and ̃ . For a given (Ro,̃) pair, the inner wind profile is specified and the outer wind profile depends on the to-be-determined value of r 0 . An iterative loop scans through several choices of r 0 to find a value that gives an outer wind profile tangent to the inner wind profile at a single point: the merge radius r a . This follows a similar approach to Chavas and Lin (2016), but they search through slightly different variables.
The normalized outer radius r 0 /r m increases with decreasing ̃ and increasing Ro, while the normalized merge radius r a /r m increases with increasing ̃ and increasing Ro (Figure 3). The outer wind parameter, = 0 −1 = ( 0∕ )̃− 1 Ro −1 , thus increases with decreasing ̃ and Ro-unsurprising from its definition-but indicating that r 0 /r m increases sub-linearly with Ro in this parameter range. For sufficiently large ̃ , particularly at small Ro, there is no merge point and no outer wind regime at all: the inner wind profile of Emanuel and Rotunno (2011) extends to the edge of the storm (sections shaded gray in Figure 3). This matches the finding of Cronin and Chavas (2019) that wind profiles for dry hurricanes have little contribution from the outer wind regime, due to greater radiative-subsidence speeds and a more symmetric distribution of vertical velocities in dry radiative-convective equilibrium. In Text S3 in Supporting Information S1, I use analytic outer wind solutions to derive an approximate bound on this subset of parameter space, and find that it corresponds roughly to the inequality:̃≥̃ * The dotted line in Figure 3 shows that this approximation generally succeeds in delimiting the part of parameter space without an outer-wind component to the merged profiles, particularly at lower Ro.
The rough position of real Tropical cyclones in this joint (̃, Ro) parameter space in Figure 3 is indicated by colored dots for representative median storms of different intensity categories, using data from Figure 10 of Chavas et al. (2015). Colors of light gray, dark gray, green, yellow, orange, and red, respectively, indicate low-intensity Tropical Storms, high-intensity Tropical Storms, Category 1 Hurricanes, Category 2 Hurricanes, Category 3 Hurricanes, and Category 4/5 Hurricanes. Fixed values of c D = 0.001 and w r = 0.002 m s −1 are used in plotting these points. As in Chavas et al. (2015), the ratio r 0 /r m -of outer size to the radius of maximum winds-increases strongly with intensity, the normalized merge radius r a /r m increases weakly with intensity, and (not discussed previously) γ ≈ 15-20 is strikingly similar across representative storms from different intensity CRONIN 10.1029/2023GL103942 6 of 10 7 of 10 classes. Because γ = c D fr 0 /w r -and f, c D , and w r all vary comparatively little with storm intensity-the relative constancy of γ with storm intensity is consistent with the known weak correlation between intensity and storm outer radius (e.g., Chavas & Emanuel, 2010).
Further details of methods and results for how merged wind profile calculations are performed and benchmarked against previous code ( Figure S1 in Supporting Information S1) are presented in Text S4 in Supporting Information S1. By using the analytic outer wind profiles described above, together with vectorized calculations of multiple wind profiles at once and use of lookup tables for key variables (Texts S1 and S4 in Supporting Information S1), acceleration by about a factor of ∼50 is obtained relative to the code of Chavas (2022), with comparable or greater accuracy. This corresponds to a computation time of about 10 −4 -10 −3 s per wind profile on a single core of a laptop computer when many (>100) profiles are computed at a time.

Discussion and Scaling of Merged Profiles
In the parameter space near present-day Tropical cyclones (5 < Ro < 50 and 0.02 <̃< 0.2 ; see Figure 3), the outer radius in merged solutions is approximated well by a power law 0∕ ∼ Ro 0.5̃−0.5 . These powers are approximate (e.g., the best-fit exponent for Ro is less than 1/2), but this form is used because a clean approximate scaling relationship results from it among V m , r m , and r 0 : How to consider this relationship depends on which storm properties one views as externally constrained, and which others one thus seeks to predict. In a diagnostic sense, this scaling seems promising in its capacity to explain and in some cases reconcile seemingly disparate dependences of r 0 on sea-surface temperature, rotation rate, and surface moisture availability (Cronin & Chavas, 2019;Khairoutdinov & Emanuel, 2013;Zhou et al., 2014). Recent work on cyclone outer size, however, suggests taking the perspective that r 0 , V m , c D , f, and w r may all be viewed as externally constrained under future climate change (e.g., Chavas & Reed, 2019). Rearranging this expression to solve for the radius of maximum winds suggests that r m will likely decrease with warming for storms of the same outer size, the same or greater intensity, and in similar latitude bands. Before discussing this implication, however, I try to provide some physical basis for Equation 17.
The wind merger condition of continuity in V and dV/dr also implies that inner and outer streamfunctions must match at the merge radius r a . Equation 17 can be rearranged to a relationship between upward mass transport in the inner region (left-hand side) and downward mass transport in the outer region (right-hand side): Note that I use "mass transport" here as a stand-in for the more accurate term "volume transport"-reasonable if imperfect when referring to transport just above a cyclone's boundary layer at different radii where density may vary by ∼10% (the two are also implicitly equated in Emanuel (2004)). The downward mass transport can be approximated as 2 0 , because constant subsidence has been assumed over the annulus between r a and r 0 , and ( 2 0 − 2 ) ≈ 2 0 if r 0 ≫ r a . But why does the upward mass transport scale as 2 −1 ? If r a /r m were constant, then the inner part of the storm would have upward mass transport that scaled with inner-core Ekman pumping rate, or 2 (e.g., Khairoutdinov & Emanuel, 2013), yet this scaling differs slightly. Rearranging Equation 1 shows that the overturning streamfunction can be calculated if V and M are known: In Text S5 in Supporting Information S1 I find that this allows the integrated mass transport for the inner wind profile (Equation 13) to be approximated as: If r a /r m depends primarily on Ro, as seen near the colored dots in Figure 3, then this may be subject to further simplification. In the special case that r a /r m ∼ Ro 1/3 , the approximate form in Equation 18 is recovered exactly.

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Thus, Equations 17 and 18 emerge from a combination of mass continuity and the dependence of r a /r m on ̃ and Ro-particularly the gradual increase of r a /r m with Ro. I know of no theoretical basis for any specific dependence of r a /r m on Ro, so this result highlights the importance of examining total cyclone upward mass transport in both real and simulated storms in future study. With this physical interpretation established, I consider application of Equation 18 to the question of how storm structure may change with climate warming.
Specifically, I consider how r m varies with warming at fixed r 0 . A bit of explanation is warranted regarding the null hypothesis of constant r 0 with warming, which may surprise some readers (this hypothesis is described and substantiated further by Schenkel et al. (2023)). Past studies differ in their predictions for changes in outer size with climate warming, partly due to use of different metrics of size, and partly due to different idealizations across simulations. Simulations of cyclones on an f-plane often (though not universally) show an outer size that is bounded above by V p /f (e.g., Chavas & Emanuel, 2014, where V p is the potential intensity)-a length scale that increases with climate warming due to increasing V p . The recent work of Wang et al. (2022) has also established an upper limiting "potential size" with similar behavior. The outer size of real-world cyclones, however, increases with latitude, directly counter to a 1/f scaling . Chavas and Reed (2019) hypothesized that a crucial feature missing from f−plane simulations is the latitudinal dependence of f, or beta effect. They used numerical simulations with varied rotation rate and planetary size to show that a vortex Rhines scale ∼( ∕( ∕ )) 1∕2 , where a is the planetary radius and V β an outer circulation wind speed, likely limits cyclone size in Earth's Tropics, while a V p /f bound may apply at higher latitudes. Critically, the vortex Rhines scale is essentially invariant with climate warming. Taken together, these results suggest that cyclones in Tropical latitudes may change little in outer size with climate warming-even if size may increase somewhat with warming at higher latitudes (e.g., Stansfield & Reed, 2021).
Thus, rearranging Equation 17 and taking r 0 and f as constants, the dependence of r m on warming is mediated by changes in V m and w r : The radiative-subsidence speed w r is expected to decrease modestly with surface warming due to increases in lower-tropospheric static stability: along a moist adiabat at 900 hPa, d/dT S [log(dθ/dz)] ≈ 2% K −1 at T S = 290 K and ≈1% K −1 at T S = 300 K. Absent clear changes in lower-tropospheric radiative cooling, this would lead to corresponding decreases in w r of ∼1%-2% K −1 . Potential intensity is expected to increase modestly by ∼1%-2% K −1 with surface warming (e.g., Khairoutdinov & Emanuel, 2013;Zhou et al., 2014), with changes in mean actual intensity somewhat more uncertain. Thus, expected changes in V m and w r combine to predict a d log r m /dT ∼ −5% K −1 decrease in radius of maximum winds (at fixed f, r 0 , and c D ), although some of this decrease could be offset by a poleward expansion of Tropical cyclone tracks. This leads me to hypothesize that more intense storms may have considerably smaller radii of maximum winds in a warmer climate-a result seen in some modeling studies (Chen et al., 2020;Xi et al., 2023) but worthy of deeper future investigation.

Conclusions
The outer wind model of Emanuel (2004) has finally been analytically solved. Solutions are a ratio of two power series in a normalized radius variable x = (1 − r/r 0 ); the power series converge relatively quickly and depend on one nondimensional parameter γ = c D fr 0 /w r (as in . The new solution is used to speed up calculations of complete wind models (merging the outer wind model of Emanuel (2004) and the inner wind model of Emanuel and Rotunno (2011) as in Chavas et al. (2015)). For merged solutions, I find an approximate scaling relationship 0 ∼ 0.5 −0.5 0.5 −0.5 over the relevant range of parameter space for real Tropical cyclones. This scaling is physically consistent with constraints posed by the overturning circulation of a cyclone, together with a dependence of the size of the ascent region on the inner-core Rossby number V m /(fr m ) that is an emergent result of matching wind profiles from the two regions. If future storms have greater maximum wind speeds and a similar distribution of outer sizes (r 0 ), then this scaling predicts decreases in maximum wind radii with climate warming. Although a decrease in r m is good news for wind damage, it may pose forecasting challenges too: Carrasco et al. (2014) found that observed storms with smaller r m are more likely to intensify rapidly.
An important result of the paper is that analytic solutions can be used to calculate merged wind profiles with considerably less computational cost than the numerical integration of Equation 3 by Chavas (2022). This may 9 of 10 make the code developed here (Cronin, 2023) immediately useful for risk modeling and assessment. A limitation of the analytic approach for the outer wind model, however, is that the drag coefficient, c D , cannot be allowed to vary with wind speed as in existing numerical solutions (Chavas, 2022). Regarding the inner wind model, the code developed here (Cronin, 2023) has been limited thus far to the case of equal exchange coefficients c k / c D = 1-extension to unequal but constant exchange coefficients is straightforward with further work. As far as I am aware, no analytic (or pre-coded numerical) solutions exist for the inner wind model when c k /c D varies with wind speed.
The Emanuel (2004) outer wind model is a major theoretical accomplishment, yet it has not been widely adopted by the Tropical cyclone research community-likely due in part to the lack of a closed-form solution. I hope that the solutions provided here (and the code to implement them) spurs further adoption and testing of the validity of the outer wind model, and perhaps useful approximations of it that are simpler still to implement. A limitation of the outer wind model, especially near r 0 , is that its derivation from Equation 1 has assumed a surface torque that scales as c D V 2 , where V is the swirling wind of the cyclone. For values of V much smaller than a background wind speed V 0 , an azimuthal-mean torque ∼ c D V 0 V would be more appropriate; both limits (V ≫ V 0 and V ≪ V 0 ) can be captured by a torque √ 2 0 + 2 . I have not attempted analytic solution of Equation 1 using such a functional form, and the problem does not seem tractable by the Riccati equation solution method used above.
An extension of this work that is more analytically tractable, and possibly more useful, is the reduction in bias of the merged wind profiles by adding a third region between ascending inner and descending outer regions. Chavas et al. (2015) find that real storms deviate most from the profile of the merged model at radii somewhat greater than the merge radius. In this region, observed winds decrease less rapidly with radius than the merged model predicts, and precipitation extends well beyond r a , violating the assumptions of the outer wind model. Analysis of the overturning circulation above suggests that the jump in assumed behavior at r a is perhaps even more troubling than realized by Chavas et al. (2015): vertical velocities w in within the inner ascending region are often maximal at r a ; this can be seen by plotting:  Chavas et al. (2015) suggest that a natural assumption for an intermediate region would be to take w = 0; as a consequence ψ would be constant in the join region between inner ascending and outer descending wind profiles. This assumption replaces ( 2 0 − 2 ) in the denominator of Equation 4 with a constant. The resulting equation for V is solvable by the same methods I used above, and the intermediate function y is a solution to the Airy equation (y″ − ry = 0). Questions about the utility, uniqueness, and interpretation of such a three-region merged solution for the wind profile are left for future work.
Finally, this study has focused on a steady-state wind profile, in which advection of absolute angular momentum by the mean overturning circulation balances surface friction. Such a framework does not directly provide any information about how the wind profile behaves in time-evolving situations, including what might drive gradual expansion of the outer radius (e.g., Chavas & Emanuel, 2010;Cocks & Gray, 2002), more rapid changes in inner structure where r m and V m vary together, or the important problem of eyewall replacement cycles and secondary eyewall formation. The wind profile model will also fail in regions where other terms are important in the steady budget of absolute angular momentum, including vertical advection by the mean circulation, or convergences of eddy fluxes in the vertical or horizontal. Nevertheless, particularly given the hypothesis that secondary eyewall formation results from a mismatch or adjustment of the inner core to the outer structure of the storm (Shivamoggi, 2022), a solid understanding of a physics-based steady wind profile seems an important foundation for building further insight into the behavior of Tropical cyclones.

Data Availability Statement
MATLAB code to reproduce figures in the paper and make general wind profile calculations is archived on Zenodo (Cronin, 2023, https://doi.org/10.5281/zenodo.7783251). The code version used in this paper is v20230329.