## 1. Introduction

Theoretical solutions for key aspects of tropical cyclone behavior have been demonstrated to successfully explain observations of real storms in nature. The upper bound on storm peak wind speed is credibly captured by potential intensity (PI) theory (Emanuel 2000; DeMaria and Kaplan 1994). Time-dependent changes in storm intensity are described well by ventilation-modified PI theory (Tang and Emanuel 2012), which provides a solution for the dependence of both peak intensity and intensification rate on the combined effects of wind shear and dry air. The wide variation of overall storm size in nature (Merrill 1984; Chavas and Emanuel 2010; Knaff et al. 2014) is arguably an implicit prediction of PI theory, which is defined in terms of relative changes in angular momentum with radius while remaining agnostic to any specific absolute angular momentum value; the theory further predicts an upper bound on size, given by the ratio of PI to the Coriolis parameter, that is consistent with observations. Finally, storm motion in the tropics is accurately reproduced by beta-and-advection theory (Holland 1984).

Comparably little attention has been given to the physics underlying the radial structure of a tropical cyclone, despite the acknowledged sensitivity of wind and storm surge hazards and the economic damages they inflict to this storm characteristic (Lin and Chavas 2012; Irish and Resio 2010; Czajkowski and Done 2014; Chavas et al. 2013). Indeed, most existing models for radial structure used in practical applications are partially empirical (Holland 1980; Holland et al. 2010; Jelesnianski 1966). Nonetheless, theoretical solutions for radial wind structure do exist. Emanuel (2004) derived models for the outer nonconvecting region where convection is absent, based on the combination of free-tropospheric thermodynamic balance and boundary layer Ekman dynamic balance, and for the inner convecting region, based on boundary layer angular momentum balance and entropy quasi equilibrium; these solutions were then asymptotically merged. More recently, Emanuel and Rotunno (2011) derived an improved solution for the inner region that arises from stratification of the outflow due to Kelvin–Helmholtz turbulence generated by the storm itself, which imposes a radial gradient in convective outflow temperature and, in turn, wind speed at the top of the boundary layer. In particular, this inner-region solution and the outer-region solution of Emanuel (2004) have each been shown to be useful for predicting storm structure in an idealized axisymmetric modeling environment (Emanuel and Rotunno 2011; Chavas and Emanuel 2014).

However, the ability of these theories to explain radial wind structure in observations is currently unexplored. Past observational studies indicate that the inner-core circulation appears to evolve nearly independently of the outer circulation (Merrill 1984; Weatherford and Gray 1988; Chavas and Emanuel 2010; Chan and Chan 2012), which may reflect the contrasting dynamical regimes between the active convection of the inner core (Didlake and Houze 2013a) and the stratiform precipitation or clear skies of the outer region (Didlake and Houze 2013b). This observation is corroborated in idealized axisymmetric modeling work demonstrating a strong sensitivity to the radial turbulent mixing length of the inner, but not outer, radial wind structure (Rotunno and Bryan 2012; Chavas and Emanuel 2014). These findings suggest that the radial structure of a tropical cyclone may, to leading order, be characterized by the juxtaposition of an inner ascending regime and an outer descending regime, yet deeper quantitative comparisons of these solutions to observations is currently lacking.

More broadly, the terms “size” and “structure” are often used interchangeably in the analysis of tropical cyclones. Indeed, in basic research size is often (though not always) taken to be some metric of the outer circulation (Merrill 1984; Liu and Chan 1999; Knaff and Zehr 2007; Dean et al. 2009; Chavas and Emanuel 2010; Knaff et al. 2014), while in risk analysis size typically refers to the radius of maximum wind due to its relevance to damage potential (Irish et al. 2008; Lin et al. 2012). Perspectives aside, determining which of these metrics is more or less “correct” from a physical standpoint depends principally on their covariability. Ultimately, then, the proper interpretation of any particular length scale and its variability requires a holistic analysis of the radial wind structure in order to place it within its appropriate dynamical context. Such an analysis necessitates a model for the complete radial wind structure, preferably one that aligns with known tropical cyclone physics and is capable of reproducing its observed characteristics and variability.

Thus, this work seeks insight into the complete radial structure of the tropical cyclone wind field in nature for the dual purposes of testing its theory and improving our understanding of its characteristics (Part I) and variability (Chavas and Lin 2015, manuscript submitted to *J. Atmos. Sci.*, hereafter Part II). Here we first develop a simple new model for the complete tropical cyclone radial wind structure at the boundary layer top by mathematically merging the inner region solution of Emanuel and Rotunno (2011, hereafter ER11) and the outer region solution of Emanuel (2004, hereafter E04). In essence, this is a “first guess” physical model: inner and outer solutions adjoined directly with a vanishingly small transition region. Can this simple model reproduce the radial wind structure of real storms in nature? To answer this question, we employ two observational databases of radial wind structure to assess the model and its component parts. First, the outer solution is compared with a QuikSCAT-based global dataset of outer wind structure (1999–2009). Second, the inner solution and, subsequently, the complete model are compared with an HWind-based dataset of radial wind structure in the Atlantic and eastern Pacific basins (2004–12). Section 2 develops the theoretical model. Section 3 describes the observational datasets and methodology for comparing model and data, and the results are presented in sections 4 and 5. Finally, section 6 summarizes key conclusions and explores limitations of this analysis and avenues for future work. Part II will characterize the modes of wind field variability that emerge under an alternative application of this model and their relationship to those found in observations.

## 2. Theory

### a. Review of existing theory

*r*is the radius from the axis of rotation,

*V*is the azimuthal wind, and

*f*is the Coriolis parameter. The low-level circulation of a tropical cyclone can be understood qualitatively as the partial conservation of absolute angular momentum by inflowing boundary layer air. Beginning at some outer radius

Quantifying the precise rate at which absolute angular momentum is lost with decreasing radius, though, requires an accounting of the broader dynamics and thermodynamics of the system. Existing theory achieves such a goal, albeit in distinct thermodynamic regimes: the inner ascending region characterized by persistent strong convection (ER11) and the outer descending region characterized by quiescent, convection-free conditions (E04). Here we seek to mathematically merge these two solutions for the purpose of creating a complete solution for the radial distribution of absolute angular momentum, and in turn the radial profile of the rotating wind, in a tropical cyclone. We begin with a review of the theory and solutions for each region.

#### 1) Outer-region structure model: E04

*V*is the azimuthal wind in the boundary layer and

*χ*is given bywhere

*u*is the radial velocity, and

*h*is the boundary layer depth, with the steady-state slab boundary layer angular momentum budgetwhere the near-surface wind velocity

**V**is approximated by its azimuthal component

*V*. In short, this model directly connects the thermodynamics of the quiescent free troposphere to the local relative vorticity of the boundary layer flow.

Mathematically, Eq. (2) is a Riccati equation that lacks a known analytical solution but can be solved numerically. Equation (2) has two parameters: *χ* and *χ*, is a multiplicative factor on the rhs of Eq. (2) and therefore directly modulates the variation of angular momentum (and wind speed) with radius. The latter,

As noted earlier, *χ* is an environmental parameter that ought to vary relatively slowly across space, as *χ* in observations in section 3.

#### 2) Inner-region structure model: ER11

^{1}and

This inner-region model results from linking the radial distribution of angular momentum at the top of the boundary layer to the stratification of the outflow aloft. Small-scale shear-induced turbulence stratifies the outflow, which translates to an increase in outflow temperature

Equation (6) has three parameters:

### b. Complete radial structure model

*M*and its radial derivative each be continuous at a merge point, denoted

This system has three equations [Eqs. (8)–(10)] and seven parameters: six storm-specific unknowns (*f*) and one environmental parameter *χ* that can be estimated from known environmental conditions. Naturally, we seek to predict the merge point *χ* and a latitude value at the storm center, this leaves two free parameters for external specification from among

This construction offers two possible routes. First, the model may take as input *r*–*V* space and then appends the outer solution [Eq. (2)] to its tail. Second, the model may take either *r*–*V* space and then finds the inner solution that matches. In either case, Eqs. (8) and (9) lead to a quartic function in

In practice, though, a geometric approach offers useful conceptual insight while achieving the same result. Geometrically, a merge point equates to a tangent point between the two curves. Given the oppositely signed curvatures of the inner and outer solution, the tangent point solution is unique over a wide range of values of all parameters. This fact extends beyond the above case to include deviations of *χ*, for which analytical solutions lose their tractability. Thus, for our purposes we obtain the merged solution iteratively by converging toward this geometric constraint, the integration for which is fast

In essence, this is the simplest possible model for the complete radial wind structure that is rooted in existing structural theory—one that assumes an infinitesimal transition between the strongly convecting inner core and the nonconvecting outer region. Note, however, that in the neighborhood of the merge point the two models are mathematically similar by construction. Thus, for the purpose of predicting the radial wind structure, the transition region between strong convection and convection free need not be a single point but rather may be of finite width so long as it is sufficiently narrow.

Next, we test the extent to which this simple model and its component parts can reproduce the radial wind structure of tropical cyclones in nature. For our analysis, we take as input

## 3. Observational data and methodology

### a. Data

Two databases of radial profiles of the near-surface (*z* = 10 m) azimuthal wind are analyzed: the QuikSCAT-based QSCAT-R database (Chavas and Vigh 2014) for the outer region (1999–2009, global) and an identically constructed HWind-based dataset (Powell et al. 1998) at small and intermediate radii (2004–12, North Atlantic and eastern Pacific). QuikSCAT and HWind data have approximate horizontal resolutions of 12.5 and 6 km, respectively. For both datasets, an estimate of the background flow is removed for each case prior to calculating the radial profile (see supplementary information). Additionally, QuikSCAT rain-rate data are used to explore the radial structure of convection.

Because of the common occurrence of azimuthally periodic asymmetries (Uhlhorn et al. 2014; Reasor et al. 2000), azimuthal data coverage asymmetry is a principal source of uncertainty in radial profile estimation. To quantify this uncertainty, for each dataset we define a data asymmetry parameter *ξ* as the magnitude of the vector mean of all gridpoint distance vectors from center as a function of radius. For small *ξ* implies lower uncertainty;

Storm-center latitude and longitude position and local storm translation vector are interpolated from HURDAT and JTWC best-track data. Full details of all data products are provided in the supplementary information.

### b. Methodology: Comparing model and observations

*V*whose parameters are optimally estimated directly from the data of Donelan et al. (2004). This function is given byUnlike

*χ*as the environmental free parameter in the outer wind model. Here we take

*f*is taken as its value at the latitude of the storm center. We note that the qualitative behavior of the model solution presented in the previous section is not sensitive to the choice of constant or variable

Second, the inner model given by Eq. (6) is compared with the HWind database, where we retain only profiles whose peak azimuthal-mean wind speed exceeds 15 m s^{−1} to remove very weak cases. As noted above, the value of the environmental free parameter

Finally, given optimal estimates of

Note that these datasets are valid near the surface, whereas ideally one would test the model against observations at the top of the boundary layer. However, datasets of comparable quality or quantity at the gradient level are not currently available. Moreover, the nature of the hurricane boundary layer and its role in mediating wind speeds from gradient level to the surface, including the effects of gradient wind imbalance, are not easily accounted for^{2} and remain a subject of active research (Powell et al. 2003; Kepert 2010; Gopalakrishnan et al. 2013; Kepert 2013; Sanger et al. 2014). As a result, here we do not explicitly account for the boundary layer, instead exploring the extent to which a model theoretically valid at the boundary layer top can reproduce near-surface wind structure; further discussion is provided in section 6.

## 4. Radial structure in observations

We begin simply with an exploration of the characteristic radial wind structure in a tropical cyclone from our observational databases. Figure 3 displays a subset of cases from the North Atlantic basin for which HWind and QuikSCAT data are available nearly concurrently. These cases (^{−1}, and (iii) difference in wind speed at the transition point less than 3 m s^{−1}.

From this subset emerges a common overall structure: significant variability in the inner core (^{−1} spans a range of

## 5. Model versus observations

We now quantify the model fit to observations. We start with a simple demonstration example for a storm snapshot. Subsequently, we perform a comprehensive comparison of each model component and the complete model with the observational databases, including optimal estimation of the environmental parameters and discussion of relationships of the estimated values to their underlying physics.

### a. Example: Ivan on 14 September 2004

Figure 4 displays the fit of the inner model alone as well as the complete model to data for Hurricane Ivan on 14 September 2004, when both HWind (1330 UTC) and QuikSCAT (1134 UTC) data are available at comparable times with highly symmetric data coverage out to large radii (^{−1} and HWind data are used radially inward. The inner and complete models are fit to HWind

First and foremost, the complete model significantly improves upon the inner model alone in its representation of the wind structure beyond the inner core. Indeed, as components of the complete model, both the inner and outer models are largely capable of reproducing their respective regions. In the vicinity of the radius of maximum wind, the inner model reproduces the local radial wind structure for given ^{−1} over the wide annulus spanning *r* > 850 km, the model approaches zero more rapidly than observed, though data coverage asymmetry becomes large at such large radii, indicating greater uncertainty. The dominant region of misfit lies just beyond the merge point (^{−1} at *r* = 130.5 km. Notably, in this region the descending outer model is applied, yet the rain-rate data shown in Fig. 4 indicate that at least some intermittent convection is likely occurring. Nevertheless, the model is capable of capturing the overall qualitative radial structure as well as the quantitative structure at both small and large radii.

### b. Outer model

The outer wind model of Eq. (2) is physically valid for the region beyond the storm inner core where convection is absent. Approaching very large radii, however, it is increasingly likely that QuikSCAT data coverage becomes highly asymmetric (the maximum possible cross-swath radius of perfectly symmetric data is 900 km) and the assumption of constant background flow loses its validity. Furthermore, the large variability in storm size implies that fixed radial bounds cannot be used to define the outer region across storms. With these issues in mind, we take a simple approach based instead on *V*, which has a monotonic, one-to-one relationship with *r* beyond

A model solution requires a single point (integration constant) and a value of ^{−1} that remains nearly constant across intensity bins (

*θ*is potential temperature, and

*z*is altitude. For an atmosphere corresponding to a parcel lifted along a reversible moist adiabat from the sea surface of temperature of 302 K, corresponding to a typical tropical warm pool temperature, with 80% boundary layer relative humidity and a characteristic atmospheric radiative cooling rate of 1 K day

^{−1}, Eq. (12) gives a value of

*W*

_{cool}= 1.9 mm s

^{−1}, nearly uniform with altitude below

*z*= 10 km. This prediction is very close to the median value estimated from our model fit. This value is also very close to that found in the simulation of Davis (2015). We note that this value is significantly smaller than that used in Chavas and Emanuel (2010), indicative of the long tail in the wind profile at radii approaching

Using the optimized values of ^{−1} at all radii of wind speeds less than ^{−1} at

In combination, Figs. 5 and 6 indicate that the simple outer wind model given by Eq. (2) appears to successfully capture the fundamental physics of the radial structure of the broad outer descending region in tropical cyclones in nature. The intrabin variance of

### c. Inner model

The inner wind model of Eq. (6) represents a solution for the complete radial wind profile, but its underlying physics are valid only in the ascending inner core. Given a value for

First, though, we seek the optimal estimate of

The result is displayed in Fig. 7, which shows the variation of ^{−1}. At lower intensities, with ^{−1}, the values of

The upward trend of

Finally, the mismatch may indicate a flaw in the underlying physics of the model, given that the optimized values of

Ultimately, for our purposes, we choose to treat

Returning to the radial wind structure, Fig. 8 displays the statistics of model error as a function of normalized radius, binned by intensity, where ^{3} As such, the inner model performs well for ^{−1}. Meanwhile, at larger radii, the model increasingly underestimates the observed wind field, with (negative) errors of approximately 5 m s^{−1} at

### d. Complete model

We now test our complete model merging Eqs. (6) and (2), where the same inputs ^{−1} found in Fig. 5. Figure 9 displays the error profile binned by best-track intensity. The model now does an admirable job at all normalized radii and across all intensity bins, providing a substantial reduction in model error beyond ^{−1} in the vicinity of

The relative constancy of model error with radius at large normalized radii in Fig. 9 mirrors the good fit of the outer model to the QuikSCAT observations shown in Fig. 6. However, there remains an overall negative bias in the much of the outer region of the storm across all intensity bins. Despite the good performance of the inner and outer models within their respective regions of validity (Figs. 6 and 8), the existence of a negative bias in the merged model indicates that this model is not quite sufficient to capture the structure of the storm at all radii simultaneously. Indeed, merging the two models eliminates the constraint that the outer model be fixed to an observed wind radius in the outer region (*V*, which is equivalently a negative bias in *r*, is evidence for the existence of a transition region of nonnegligible width between the strongly convecting inner region and nonconvecting outer region not captured with this model. Notably, this bias decreases with intensity in the far outer region, again consistent with expectations that the model assumptions are most valid at high intensity when the storm is more axisymmetric and convection is more sharply confined to the vicinity of

For the purpose of demonstration, Fig. 10 displays the observed median radial structure and model fit within each intensity bin. The observed profiles are calculated by first taking the median profile of *V* using median parameter values of *f* within each intensity bin. Meanwhile, the corresponding model profiles are taken as the solution using the same median parameter values for each bin. The salient features of the error profiles of Fig. 9 are evident, including both the underestimation at intermediate radii and the improved model performance over the inner model at large radii, especially at high intensities.

To further probe the radial extent of convection, Fig. 11 displays the characteristic radial structure of QuikSCAT-estimated precipitation rate *P* and corresponding merge point radius

Overall, the complete model appears capable of credibly capturing much of the characteristic radial structure of a tropical cyclone, particularly at higher intensity consistent with its underlying assumptions. The model also performs surprisingly well at low intensity despite the broader and more disorganized field of convection characteristically associated with weaker systems, seemingly at odds with the assumptions and approximations of our model that are expected to break down in such conditions. As a result, we view this latter result as largely fortuitous, though nonetheless beneficial for practical applications.

## 6. Summary and conclusions

We have mathematically merged existing theoretical solutions in the inner ascending region characterized by persistent strong convection (ER11) and the outer descending region where convection is absent (E04) to create a model for the complete radial wind structure of the tropical cyclone wind field at the top of the boundary layer. The outer solution is found to reproduce the observed outer wind field of the storm within the broad annulus bounded by wind radii ^{−1} at most radii except for slightly larger underestimation near

Overall, despite its simplicity, the model appears to credibly capture the characteristic radial structure of the tropical cyclone wind field in nature and in a manner consistent with existing, distinct theories for the ascending and descending thermodynamic regimes. Though the direct merger of the inner ascending and outer descending solutions is a product of mathematical convenience, this work indicates that the physical world is not far off for reasonably intense storms, suggesting that the model may represent the attractor state toward which storm structure evolves in conjunction with the analytical solution for the attractor of storm intensity derived in Tang and Emanuel (2010).

Though these apparent successes are encouraging, the model and its evaluation are also subject to limitation. First and foremost, the analysis neglects the role of the boundary layer as an intermediary between the gradient level and the surface. Our results suggest that its effect on the characteristic radial wind structure outside of the inner core may be relatively small on average, though we cannot rule out its significance. In the inner core, though, the mismatch between the predicted value of

Nonetheless, for practical purposes the model offers a simple theoretical basis of comparison for future observational and modeling studies of tropical cyclone size and structure. Furthermore, this work offers the potential to place individual wind radii within a holistic, physics-based structural framework. Finally, comparison of the performance of this wind profile model against existing parametric models (e.g., Holland 1980; Willoughby et al. 2006) is a logical extension of this work for risk analysis applications. Part II explores the modes of variability inherent to the model in the case of input

## Acknowledgments

This material is based upon work supported by the National Science Foundation under Awards AGS-1331362 and AGS-1032244. Very special thanks to Bryan Stiles, Svetla Hristova-Veleva, and their team at the NASA Jet Propulsion Laboratory for their help working with their excellent QuikSCAT tropical cyclone database. Thanks to Mark Donelan for sharing the drag coefficient data from his 2004 paper. We thank Mark Powell for his work developing and maintaining the HWind database. Finally, we thank three anonymous reviewers for helping to improve this work.

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^{1}

A methodological note: the values of maximum wind and radius of maximum wind are equal to the parameters

^{2}

Typically wind speeds are simply reduced from gradient level by a factor of

^{3}

The eye is often modeled using a linear or slightly superlinear profile of