1. Introduction
Tangential wind profiles such as those in intense atmospheric vortices arising in dust devils, waterspouts, tornadoes, mesocyclones, and tropical cyclones often are approximated by continuous functions that are zero at the vortex center, increase to a maximum at some radius, and then decrease asymptotically to zero infinitely far from the center. It is generally believed that three types of tangential wind distributions resembling the observed profiles of tangential winds in different vortices are those predicted by the idealized, inviscid Rankine (Rankine 1882), the viscous Burgers–Rott (BR) (Burgers 1948; Rott 1958), and the Sullivan (Sullivan 1959) analytical vortex models (Fig. 1).
Radial profiles of V* in the analytical Rankine, Burgers–Rott, and Sullivan vortex models for comparison. The computations of the profiles are given in the appendix. Normalized radial distance is represented by ρ = r/Rx, where r is the radial distance from the vortex center and Rx is the radius at which the normalized tangential velocity maximum occurs.
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
Attempts to measure observed tangential wind fields and other characteristics from available observational data have been made using the above analytical vortex models. In his analysis of the wind distributions in the Dallas tornado of 2 April 1957, Hoecker (1960), for instance, constructed to fit a few pieces of the Rankine’s outer wind profiles to the radial profiles of tangential wind measurements at a few heights above the ground for his purpose of estimating a decay exponent in the Rankine model. The tangential wind measurements of waterspouts (Leverson et al. 1977; Schwiesow 1981), dust devils (Sinclair 1973; Bluestein et al. 2004; Toigo et al. 2003; Tratt et al. 2003; Kanak 2005; Cantor et al. 2006, among others), and laboratory-simulated vortices (Church et al. 1979), to a first approximation, were modeled as the Rankine vortex in spite of the tangential velocity cusp at its core radius in the model. In their proximity radar observations of tornadoes by mobile, high-resolution Doppler radars, Wurman and Gill (2000), Wurman (2002), and Wurman and Alexander (2005) used the tangential wind profiles of the Rankine model to closely match the inner cores of solid-body rotation—in some cases, to the inner radial profiles of Doppler velocities. Outside the cores, they observed Doppler velocity profiles of V ∝ r−0.6±0.1, where r is the distance from the center of the tornado. Wurman et al. (2007) showed that different tornadoes have substantially different tangential wind field structures, and both the horizontal and vertical distributions of winds differed markedly from one tornado to another.
In view of the above, although the inner and outer tangential wind profiles of the Rankine model fit fairly well the available observational data, the profile’s cusp at ρ = 1 remains unchanged and is ignored. The result is that the discontinuous tangential velocity peak is overestimated and is not matched with the observed, continuous tangential wind maximum. The problem with the cusp in the Rankine model has been improved by solving for the radial balance between inward advection and outward diffusion of angular momentum in the presence of sink flow with a constant viscosity to obtain the viscous steady-state Burgers–Rott (Burgers 1948; Rott 1958) vortex model. The model yields a smooth transition between solid-body rotation and potential flow in the annular zone of tangential velocity maximum. The azimuthally averaged tangential velocity profiles derived from W-band Doppler radar measurements (Bluestein et al. 2007; Tanamachi et al. 2007) at low altitudes resemble the Burgers–Rott profile. Because of constraints in the Burgers–Rott profile, the profile sometimes does not match some other Doppler-derived tangential wind profiles very well.
Although the Rankine model is viable for different applications because of its mathematical simplicity, the model does not seem to be a good starting point for the examination of any atmospheric vortex for which there is incomplete observational data. Snow (1984), for instance, replaced Rankine’s unrealistic sharply peaked profile of tangential velocity with a more realistic, continuous tangential velocity profile for his investigation of the formation of particle sheaths in one- and two-celled model vortex structures.1 Leslie and Snow (1980) extended Sullivan’s (1959) two-celled vortex solution to the Navier–Stokes equations for a three-dimensional axisymmetric, two-celled vortex by measuring and comparing the radial pressure profiles to those produced by various swirl ratios in laboratory tornado simulators. The lack of versatility in the Rankine model highlights the need for a more realistic alternative.
In the aerodynamics community, Vatistas et al. (1991, 2006) sought a better approximation than the Rankine model by proposing an alternative formulation that gives rise to a “q family” of smooth tangential velocity distributions having solid-body rotation and potential flow modes as asymptotes. The Vatistas profile employs three key parameters: maximum tangential wind, radius of maximum tangential wind, and q, a power-law exponent that controls the “peakedness” of the wind profile in the annular zone of tangential velocity maximum. The Vatistas model, however, does not seem to have sufficient degrees of freedom fit to realistic observations (e.g., one-celled and two-celled vortex configurations in Fig. 2) because the tangential velocity distribution of the model is constrained to solid-body rotation and potential flow. Sometimes there is a mismatch with any observed profiles of tangential velocity including those in Figs. 1 and 2 as well as other Doppler velocity profiles of V ∝ r−0.6±0.1, as documented by Wurman and Gill (2000), Wurman (2002), Wurman and Alexander (2005), and Wurman et al. (2007). This provides a motivation to modify the Vatistas et al. (1991) model and to determine if the modified model can better fit realistic vortices. Our estimated tangential velocity field obtained in this model is a part of our future program to retrieve the radial and vertical components of velocity defined in a cylindrical coordinate system.
Based on the laboratory simulated vortices, evolutionary Vθ shear profiles for one-celled convective vortex (curve a), transition vortex (curve b), two-celled turbulent vortex (curve c), and transition into multiple vortex structures (curve d) as a function of r. The gray hatched bar represents the region of critical shear for curve d. (Courtesy of E. M. Agee of Purdue University.)
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
The objective of this paper is to develop a new parametric tangential-wind profile model by adopting and modifying the Vatistas et al. (1991) model. Section 2 describes how the Vatistas model may be tailored to improve fitting the realistic-looking profile by incorporating new parameters. The new parametric tangential-wind profile model employs five key parameters to construct a radial profile of tangential velocity. The first two physical parameters include the maximum tangential velocity and the radius of the tangential velocity peak. Additionally, the last three parameters include new power-law exponents that aid in controlling the shape of different portions of the velocity profile. Section 3 investigates the roles of varying model parameters in modifying the radial profiles of tangential velocity and corresponding vertical vorticity. In section 4, the analytical vortex models of Burgers (1948) and Rott (1958) and of Sullivan (1959) are used to test and verify the new parametric model. A summary and discussion of future work are given in section 5.
2. Parametric tangential-velocity profile formulation
Radial profiles of V* for the Scully (q = 1, short-dashed curve), Vatistas (q = 2, long-dashed curve), and Rankine (q → ∞, solid curve) models. Normalized radial distance is represented by ρ = r/Rx. Dotted curve represents V* of the Burgers–Rott or Lamb–Oseen model for comparison. Note that the calculation of the 
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
Since the Vatistas et al. (1991) profile inside the radius of maximum tangential velocity is constrained to solid-body rotation, we seek to relax the profile by introducing a new parameter ρk so that the new power-law exponent k controls the shape of the velocity profile near ρ = 0. When k > 1 (k < 1), the radial profile has positive (negative) curvature, meaning that the curve turns to the left (right) with small ρ.
When k = 1, the V-shaped profile of tangential velocity near ρ = 0 has zero curvature with increasing ρ until the profile reaches at ρ = 1. This means that the tangential velocity increases linearly, indicative of solid-body rotation of fluid in a vortex core with constant angular velocity. Varying radial profiles corresponding to various k values will be shown in the subsequent figures.
Since the Vatistas profile at radial distances of greater than a few core radii is restricted to the potential flow (i.e., Vq ∝ ρ−1), we seek to modify the profile by introducing a new function (1 + ρn/λ)λ, where a new second power-law exponent n dictates the tangential velocity decay beyond ρ = 1. The larger the n value is, the faster the outer profile decays with increasing ρ. Additionally, a new third power-law exponent λ controls the profile shape of the highest tangential velocity that can be made either broadly or sharply peaked. Three new power-law exponents (k, n, λ) will be described in detail in the subsequent sections.
When λ = 1 in (13), (13) is reduced to a simple model tangential wind profile, 2ρ/(1 + ρ2) (Fig. 4). This wind profile was commonly used by several investigators for different applications. For instance, Jelesnianski (1966) used this profile to develop a tropical cyclone model for application with a numerical storm surge model. Ooyama (1969) used this profile to initialize his two-dimensional, axisymmetric numerical simulation of the life cycle of tropical cyclones. Houston and Powell (1994) simulated this profile to analyze and compare to surface wind fields in tropical storms and hurricanes. Dowell et al. (2005) used Fiedler’s (1989, 1994) tangential wind profile identical to 2ρ/(1 + ρ2) to initialize their high-resolution, axisymmetric numerical models and study how one- and two-dimensional distributions of particle motions and concentrations responded to evolving tornado-like vortex flows.
Radial profiles of V* for the Wood–White (λ = 1, short-dashed curve; λ = 0.5, long-dashed curve) and for the Rankine (λ → 0, solid curve) models. Normalized radial distance is represented by ρ = r/Rx. The dotted curve represents the normalized tangential velocity profile of the Burgers–Rott or Lamb–Oseen model for comparison. Note that the 
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
Further insight into the relevant tangential velocity kinematics is obtained by comparing the 
3. The physical behaviors of the parametric profiles
a. Tangential velocity
The plots in Fig. 5 were prepared to elucidate the roles of the model parameters of k, n, and λ on the physical behaviors of the radial profile families of tangential velocity. To facilitate comparison with the profiles, normalized composites are constructed that beneficially preserve the underlying tangential wind structure. Each individual profile is expressed in the convenient dimensionless form utilizing the typical scales Vx and Rx. The radial profile families of normalized tangential velocity as functions of k, n, and λ are represented in Fig. 5. In each panel of the figure, three varying values of n are presented for each selected value of k. As one progresses from the top to the bottom panels, k and n remain unchanged with decreasing λ.
Radial profile families of 
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
Varying the degree of the power-law exponent k in ρk of (11) primarily governs the shape of the tangential velocity profile near ρ = 0. As one progresses from the left to the right panels of Fig. 5, the curvature of the tangential velocity profile progressively changes its direction from zero to positive as k = 1 and k > 1, respectively. The reason for not showing k < 1 in this figure will be discussed subsequently.
As already mentioned previously, the decay exponent n in (11) mainly governs the outer profile that decays with increasing ρ. The lower (higher) the n value, the more slowly (rapidly) the decaying tangential velocity decreases with ρ (Fig. 5). It is important to recall in section 2 that the velocity profile exists only for k < n. When k = n, the profile is perfectly flat and the vortex cannot exist.
The last power-law exponent, represented by λ in (11), basically controls the peakedness of the tangential wind profile in the annular zone of tangential velocity maximum, meaning that the region of the highest tangential wind can be made either broader or narrower. As λ changes from 1.0 to approximately 0, a broadly peaked profile of tangential velocity transitions to a sharply peaked profile. As one progresses from the top to the bottom panels of Fig. 5, a more rounded maximum tangential wind becomes increasingly localized with decreasing λ around the radius at which the maximum occurs. As λ → 0, three radial profiles for very different n values merge together in each panel to form one superimposed radial profile at ρ < 1.
Overall, three model parameters (k, n, λ) do not change magnitude at 
Radial profile families of 
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
The restrictive assumptions that are implied by the stagnant core vortex model should be kept in mind when relation (17) is used. In particular, the assumptions of unrealistic discontinuity at the core boundary (ρ = 1), a zero tangential velocity within the core (ρ ≤ 1), and omission of diffusion are suspect for intense atmospheric vortices. In real fluids, some radial profiles of
b. Vertical vorticity
In the last subsections, we computed and plotted radial profiles of
The exponent k − 1 in ρk−1 in (20) reveals that if 0 < k < 1, the ζWW profile produces a singularity as ρ → 0. The profile has the undesirable property that, for 0 < k < 1, ζWW is infinite along the vortex axis. To avoid the unwanted singularity, we propose that k ≥ 1. When applying (19) to the stagnant core vortex model (16) and taking the limit of (19) as k → ∞, all the vorticity is confined to a cylindrical vortex sheet at ρ → 1 (e.g., Fiedler and Rotunno 1986). The vorticity profile, however, produces an unrealistic discontinuity in the infinitesimal radial thickness of tangential velocity peak (not shown). To evade the undesirable vorticity discontinuity at ρ → 1, we suggest that k be limited to some large values, perhaps from 10 to 50, and λ be limited to no less than 0.1.
Figure 7 presents the computation of 
Plots of 
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
In a manner analogous to the discussion in Fig. 7, 
As in Fig. 7, but for 
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
Radial profiles of ζ* for (a) the Wood–White (λ = 1.0, short-dashed curve; λ = 0.5, long-dashed curve; λ = 0.2, medium-dashed curve; λ → 0.0, solid curve) and (b) the Scully (q = 1.0, short-dashed curve), Vatistas (q = 2.0, long-dashed curve; q = 5.0, medium-dashed curve), and Rankine (q → ∞, solid curve) models. Dotted curve represents the ζ* profile of the Burgers–Rott or Lamb–Oseen model for comparison. Normalized radial distance is represented by ρ = r/Rx. Note that the 
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
We consider a special case in which the normalized vertical vorticity profiles in the Wood–White and Vatistas models in (25) are compared to the normalized Rankine vorticity 
The Rankine vortex model in (A1) and (A2) of the appendix reveals that the model is the combinations of solutions for a solid-body rotation and a potential flow. The Rankine equation is not an exact solution to the Navier–Stokes equations because it only satisfies them in a piecewise sense (Schwiesow 1981; Lugt 1983, 1996; Davies-Jones 1986; Kanak 2005; Cantor et al. 2006). The discontinuity of the first and second tangential velocity derivatives in the second Navier–Stokes equation at a point of transition from solid-body rotation to potential flow is not physically realistic and cannot be a natural entity for viscous vortex flow. Owing to the discontinuity, (A1) is not a global solution and only satisfies the Navier–Stokes equations of motion if a rigid boundary is present at ρ = 1. Therefore, the model is unrealistic for atmospheric vortices, but is still useful, however, because it does closely approximate many atmospheric vortical flows in a piecewise sense and thus can be a useful first step for analyses.
Figure 9 shows that all normalized vorticity profiles of 
The hurricane aircraft flight-level data analysis of Mallen et al. (2005) revealed that their computed relative vorticity values, when normalized, have been shown to be 1.0 at ρ = 1. This is because ω in (20) becomes dominant and equal to 1.0 at ρ = 1, where ∂V/∂r always is zero. Their calculated profiles of hurricane vorticity favorably agree with Fig. 9a.
As mentioned earlier, we computed and plotted the radial profile families of
As in Fig. 5, but for corresponding 
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
When k = 1.0 and n = 1.1, the maximum values of
Relatively slow tangential velocity decay beyond the radius of maximum tangential velocity results in a skirt of nonzero vorticity (Fig. 10) in the sense that the peripheral circulation may be small compared to that of the vortex core. The slow underlying monotonic decrease of
The vorticity skirt becomes negative beyond the radius of tangential velocity peak, but its magnitude is small and approaches −1.0 (0.0) asymptotically as ρ → 1 (ρ → ∞) when taking the limit of (19) and (20) as λ → 0 (Fig. 10). The negative vorticity is the result of the shear vorticity dominating the curvature vorticity. This is because the tangential velocity decreases with ρ more rapidly than ρ−1, which is consistent with the findings of Lee and Wurman (2005).
With decreasing λ, the critical points where 
4. Testing and verification
Prior to eventually applying the Wood–White parametric model with real data, it is reasonable to examine the parametric profile’s realism by fitting the parametric profiles to the idealized tangential velocity and corresponding vertical vorticity profiles. Experimental tests were performed using two tangential velocity solutions of the analytical Burgers–Rott (Burgers 1948; Rott 1958) and Sullivan (Sullivan 1959) vortex models. We used the Levenberg–Marquardt (LM) (Levenberg 1944; Marquardt 1963) optimization method, a standard technique used to solve unconstrained nonlinear least squares problems for curve-fitting applications. The algorithm for implementing the LM method was described in Press et al. (1992).
a. The Burgers–Rott model
Figure 11 assesses and compares radial variations of normalized tangential velocity and corresponding vertical vorticity between the BR and WW models. Equations (A4) and (A5) were used to simulate the normalized tangential velocity and vorticity data in the form of radial profiles. The LM fits to the tangential velocity data presented a rigorous test of the WW tangential wind profile’s realism. In constrained fits, the maximum value of
Radial profiles of (a) the Burgers–Rott tangential velocity (solid curve) and the fitted tangential velocity (dotted curve) and (b) the corresponding Burgers–Rott vertical vorticity (solid curve) and the corresponding fitted vertical vorticity (dotted curve). All profiles are normalized. Normalized radial distance is represented by ρ = r/Rx. The fitted model parameters (k, n, λ), RMSE, and CC are indicated. In (b), a downward-pointing arrow indicates the location of the minimum
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
The maximum value of 
b. The Sullivan model
The comparative radial variations of normalized tangential velocity and corresponding vorticity between the Sullivan and Wood–White models are presented in Fig. 12. At 
Radial profiles of (a) the Sullivan tangential velocity (solid curve) and the fitted tangential velocity (dotted curve) and (b) the corresponding Sullivan vertical vorticity (solid curve) and the corresponding fitted vertical vorticity (dotted curve). All profiles are normalized. Normalized radial distance is represented by ρ = r/Rx. The fitted model parameters (k, n, λ), RMSE, and CC are indicated. In (b), two downward-pointing arrows indicate the locations of the positive and negative
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
Overall, a good agreement between the parametric and theoretical tangential velocities and vorticities indicates that the WW model has shown considerable success in reproducing the tangential velocity and vorticity profiles that favorably concur with those of the Burgers–Rott and Sullivan models. As revealed in Figs. 11 and 12, low root-mean-square values and high correlation coefficient values of nearly one indicate the limitations but still the effectiveness and versatility of the Wood–White parametric model.
5. Concluding discussion and future work
The Wood–White parametric model is designed as a modification of the Vatistas et al. (1991) model to depict a realistic-looking tangential wind profile that can better fit a realistic vortex. The Wood–White profile employs five key parameters: maximum tangential wind, radius of maximum tangential wind, and three new power-law exponents that shape different portions of the velocity profile, including the sharply or broadly peaked region of the highest tangential wind. Experimental studies have been carried out by varying the model parameters to explore and understand the behaviors of tangential wind and corresponding vertical vorticity profiles. The effectiveness and versatility of the parametric model were successfully tested and validated against the exact tangential velocity solutions of the Burgers–Rott (Burgers 1948; Rott 1958) and Sullivan (Sullivan 1959) vortex models. Detailed comparisons between the parametric and theoretical tangential wind and corresponding vorticity profiles suggest that the WW model performs well with low values of root-mean-square errors and high values of correlation coefficients. Hence, the capability of the WW parametric model to reproduce the different profiles of tangential wind and vertical vorticity that accurately coincide with those in the Rankine, Burgers–Rott, Sullivan, and stagnant core vortex models (Fig. 1) has been examined.
Although only two test cases were presented in this study, it is our desire to continue testing and validating the Wood–White parametric model against numerical tangential-wind measurements derived from high-resolution, two-dimensional, axisymmetric, numerical vortex models that represent evolving simulated vortices. This will allow us to examine and assess how well the Wood–White parametric model is capable of reproducing and comparing to the complex tangential wind fields defined in a cylindrical coordinate system. If the model parameters in the Wood–White model are not accurately determined or do not have sufficient degrees of freedom to fit to realistic observations, then the model may be enhanced by incorporating some new parameters and/or modifications or by developing a different mathematical model from scratch to improve the realistic-looking WW parametric profiles that very closely coincide with the observed profiles. This approach carrying out this analysis has been developed in this work and can be applied in other contexts. In fact, this is an important part of our future program to retrieve the radial and vertical components of velocity as well as pressure defined in the cylindrical coordinate system.
Our near-future work will include application of the WW parametric model to aircraft flight-level tangential wind and pressure profiles in tropical cyclones. Detailed inspection of the profiles allows us to examine the parametric profile’s realism to determine if the model is capable of replicating the observed profiles of gradient wind and pressure in different stages of tropical cyclones ranging from tropical storms having nearly flat tangential-wind profiles to intense hurricanes exhibiting single- and dual-maximum eyewall tangential-wind profiles.
The authors thank Qin Xu and John Lewis of NSSL and Alan Shapiro, Brian Fiedler, Katharine Kanak, and John Snow of the University of Oklahoma for reading and making many useful suggestions in the earlier version of the paper. Particular thanks are extended to Kim Elmore of NSSL for providing editorial assistance in this version. The authors are grateful to Lynn Greenleaf of the University of Oklahoma for providing assistance in the computation of the Sullivan tangential velocity profile. The authors appreciate the efforts of the anonymous reviewers for reviewing and providing helpful comments, insights, and suggestions that led to an improved manuscript. The lead author would like to thank Ernest Agee of Purdue University for providing his figure (which appears herein as Fig. 2 for illustrative purposes). The author also is indebted to George Vatistas of Concordia University (Montreal, Quebec, Canada) for answering the author’s questions about the Vatistas and Sullivan models.
APPENDIX
Idealized Tangential Wind and Vorticity Profiles
The idealized vortex profiles of the Rankine (Rankine 1882), Burgers–Rott (Burgers 1948; Rott 1958), and Sullivan (Sullivan 1959) vortex models discussed in this study that represent the radial structure of an axisymmetric vortex are presented here in a convenient dimensionless form. These three profiles of tangential velocity and vertical vorticity will be used to compare with those of the Wood–White model.
a. Rankine model
b. Burgers–Rott model
c. Sullivan model
Radial profile of H(βρ2) in the Sullivan model. Normalized radial distance is represented by ρ = r/Rx.
Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1
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A one-celled vortex structure consists of a jetlike vertical velocity everywhere being directed upward with a maximum at the vortex axis, whereas a two-celled vortex is characterized by downward motion along the axis in a circular region surrounded by an annular region of upward motion at outer radius, as described by Snow (1982, 1984), Davies-Jones (1986), Pauley and Snow (1988), Church and Snow (1993), and Davies-Jones et al. (2001).
When λ = 1 in (6) and (11), we formulated our old model several years ago before the lead author found out about the Vatistas et al. (1991) model online. Because our old model failed to produce a sharply peaked tangential velocity profile, we decided to adopt the Vatistas model, reformulated [starting with (6)] that led to (11); we further tested and verified our new model [(11)] by comparing satisfactorily radial profiles of parametric tangential velocity with those of the idealized Rankine, Burgers–Rott, and Sullivan vortex models, as will be shown in this study.
Sometimes called the Rankine-combined vortex by some investigators because the vortex’s inner and outer tangential velocity profiles are combined. The word “combined” is dropped for the sake of convenience.
