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  • View in gallery

    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.

  • View in gallery

    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.)

  • View in gallery

    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 profile is given in the appendix.

  • View in gallery

    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 profile is normalized to 1.0 for the purpose of comparing to the profiles.

  • View in gallery

    Radial profile families of for selected values of k, n, and λ. Three profile families in each panel are indicated by three different values of n. The thick black curve represents the profile that corresponds to n = k + 1 to enhance readability. The thick gray curve represents profile for comparison. Normalized radial distance is represented by ρ = r/Rx.

  • View in gallery

    Radial profile families of for selected values of k and λ. In each panel, k = 1 (solid curve), k = 3 (short-dashed curve), k = 10 (long-dashed curve), and k → ∞ (thick gray curve) are indicated. The thick gray curve represents the normalized tangential velocity profile in the stagnant core vortex model; the dotted curve represents the normalized tangential velocity profile in the Sullivan model. Normalized radial distance is represented by ρ = r/Rx. Note that the profile is normalized to 1.0 for the purpose of comparing to the profiles.

  • View in gallery

    Plots of as functions of k and n for selected values of λ. A gray shaded portion represents an area where vorticity singularities for k < 1 occur. A sloping dotted line represents k = n. A sloping dashed line represents a trough of minimum values. Several bold crosses represent the computed values of , as will be discussed in Fig. 10.

  • View in gallery

    As in Fig. 7, but for . A sloping dotted line represents k + 1 = n. A sloping dashed line represents a trough of minimum values.

  • View in gallery

    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 profile is calculated from (A5) in the appendix.

  • View in gallery

    As in Fig. 5, but for corresponding . Normalized critical points at which the corresponding and respectively occur are indicated by solid triangles. Dashed horizontal line of zero value is indicated. The thick black curve represents the vorticity profile that corresponds to n = k + 1 to enhance readability. The thick gray curve represents the profile for comparison. Normalized radial distance is represented by ρ = r/Rx.

  • View in gallery

    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 value. A horizontal dashed line representing ζ* = 0 is indicated.

  • View in gallery

    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 peaks. A horizontal dashed line representing ζ* = 0 is indicated.

  • View in gallery

    Radial profile of H(βρ2) in the Sullivan model. Normalized radial distance is represented by ρ = r/Rx.

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A New Parametric Model of Vortex Tangential-Wind Profiles: Development, Testing, and Verification

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  • 1 NOAA/OAR/National Severe Storms Laboratory, Norman, Oklahoma
  • 2 Department of Mathematics, University of Oklahoma, Norman, Oklahoma
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Abstract

A new parametric model of vortex tangential-wind profiles is presented that is primarily designed to depict realistic-looking tangential wind profiles such as those in intense atmospheric vortices arising in dust devils, waterspouts, tornadoes, mesocyclones, and tropical cyclones. The profile employs five key parameters: maximum tangential wind, radius of maximum tangential wind, and three power-law exponents that shape different portions of the velocity profile. In particular, a new parameter is included controlling the broadly or sharply peaked profile in the annular zone of tangential velocity maximum. Different combinations of varying the model parameters are considered to investigate and understand their effects on the physical behaviors of tangential wind and corresponding vertical vorticity profiles. Additionally, the parametric tangential velocity and vorticity profiles are favorably compared to those of an idealized Rankine model and also those of a theoretical stagnant core vortex model in which no tangential velocity exists within a core boundary and a potential flow occurs outside the core. Furthermore, the parametric profiles are evaluated against and compared to those of two other idealized vortex models (Burgers–Rott and Sullivan). The comparative profiles indicate very good agreements with low root-mean-square errors of a few tenths of a meter per second and high correlation coefficients of nearly one. Thus, the veracity of the parametric model is demonstrated.

Corresponding author address: Vincent Wood, NOAA/NSSL, National Weather Center, Norman, OK 73072–7323. E-mail: vincent.wood@noaa.gov

Abstract

A new parametric model of vortex tangential-wind profiles is presented that is primarily designed to depict realistic-looking tangential wind profiles such as those in intense atmospheric vortices arising in dust devils, waterspouts, tornadoes, mesocyclones, and tropical cyclones. The profile employs five key parameters: maximum tangential wind, radius of maximum tangential wind, and three power-law exponents that shape different portions of the velocity profile. In particular, a new parameter is included controlling the broadly or sharply peaked profile in the annular zone of tangential velocity maximum. Different combinations of varying the model parameters are considered to investigate and understand their effects on the physical behaviors of tangential wind and corresponding vertical vorticity profiles. Additionally, the parametric tangential velocity and vorticity profiles are favorably compared to those of an idealized Rankine model and also those of a theoretical stagnant core vortex model in which no tangential velocity exists within a core boundary and a potential flow occurs outside the core. Furthermore, the parametric profiles are evaluated against and compared to those of two other idealized vortex models (Burgers–Rott and Sullivan). The comparative profiles indicate very good agreements with low root-mean-square errors of a few tenths of a meter per second and high correlation coefficients of nearly one. Thus, the veracity of the parametric model is demonstrated.

Corresponding author address: Vincent Wood, NOAA/NSSL, National Weather Center, Norman, OK 73072–7323. E-mail: vincent.wood@noaa.gov

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).

Fig. 1.
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 Vr−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 Vr−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.

Fig. 2.
Fig. 2.

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

Vatistas et al. (1991) proposed a q family of algebraic tangential velocity profiles for vortices with continuous distributions of the flow quantities. The normalized tangential-velocity function of the Vatistas model is given by
e1
where ρ = r/Rx is the dimensionless radial distance from the center of rotation, r is the radial distance from the center, Rx is the radius at which a maximum tangential velocity occurs, Vq is the tangential velocity for a few selected values of q, Γ is the circulation at radial infinity, and q is a power-law exponent that governs the shape of the velocity profile in the annular region of tangential velocity peak. The Vatistas model is a generalization of a few well-known vortex tangential-velocity profiles in the aerodynamics community. For example, a series of tangential velocity profiles for two specific vortex models in (1) is given by the relations
e2
e3
Here, the q = 1 vortex in (2) is commonly known as the Scully model, from a study of helicopter aerodynamics (Scully 1975), and sometimes is called the Kaufmann model (Kaufmann 1962). A normalized radial profile of tangential velocity in the Scully model is shown by the short-dashed curve in Fig. 3. When ρ = 1 in (2), the maximum tangential velocity is reduced to half of the value given by the Rankine model. The q = 2 vortex in (3) belongs to the Vatistas model, (shown by the long-dashed curve in the figure). Scully (1975) and Vatistas et al. (1991) have found that the q = 2 vortex model is physically most representative for actual vortex tangential-wind profiles in aerodynamics. In addition, they demonstrated that the q = 2 tangential velocity distribution was in good agreement with their experimental data. It is interesting to note that the q = 2 vortex is a good approximation to the Lamb–Oseen model (Lamb 1932; Oseen 1911) model and hence the Burgers–Rott (Burgers 1948; Rott 1958) model (Fig. 3).
Fig. 3.
Fig. 3.

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 profile is given in the appendix.

Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1

As q → ∞ in (1), the q → ∞ vortex approaches the Rankine vortex (Fig. 3) and is given as
eq1
It is clear that the gradient of the Rankine velocity profile is discontinuous at the boundary between solid-body rotation and potential flow, while varying q, exclusive of infinity, assures a smooth, continuous transition at this boundary in the Vatistas model.

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., Vqρ−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.

To tailor the Vatistas model for application to realistic-looking tangential wind profiles, we propose that (a) ρ in the numerator of (1) is replaced by ρk and (b) the denominator is replaced by (1 + ρn/λ)λ to represent one- and two-celled cyclonic (or anticyclonic) vortices defined in cylindrical coordinates. As a consequence, we define a new tangential-velocity function χ where
e6
Further, the relation of k to n is constrained by the condition that the function χ(ρ) goes to zero as ρ → ∞. Hence, we impose the condition 0 < k < n. To scale the function such that its maximum is at ρ = ρx = 1, we observe that the normalized radius at which the maximum χ occurs is given by
e7
To obtain the maximum at ρ = ρx = 1, we modify (7) and define the function
e8
Hence, (8) becomes
e9
By letting ρ = 1 in (9), we note that
e10
By dividing χ1(ρ) by χ1(1), we obtain a function ψ(ρ; k, n, λ) = χ1(ρ)/χ1(1), with the maximum value normalized to 1 at ρ = 1. As a consequence, the new parametric tangential velocity distribution2 is given by
e11
where m = [Vx, Rx, k, n, λ]T represents a model vector of the five parameters, and the subscript WW represents the Wood–White model for the purpose of discussing and comparing with other theoretical vortex models in this study. The following properties are immediate from (11): ψ(0; k, n, λ) = 0, ψ(1; k, n, λ) = 1 (where ψ attains its maximum value), ψ(∞; k, n, λ) = 0, ψ(ρ; k, n, λ) > 0 for 0 < ρ < ∞, ∂ψ/∂ρ > 0 for ρ < 1, ∂ψ/∂ρ = 0 at ρ = 1, and ∂ψ/∂ρ < 0 for ρ > 1. Additionally, the restriction that 0 < k < n in (11) ensures that a perfectly flat tangential velocity profile in which a vortex cannot exist is not permitted.
The alternative form of (11) in terms of Γ and Rx may be rewritten as
e12
which is comparable to (1).
Now is compared to by evaluating (11) for k = 1 and n = 2. It is given by
e13
The difference between (1) and (13) is the inclusion of 2λ in (13) or the exclusion of 2λ from (1). Note that the denominator in (13) plays the same role as that in (1). If we compute and plot in (13), the profiles exactly coincide with the profiles (Fig. 3).

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.

Fig. 4.
Fig. 4.

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 profile is normalized to 1.0 for the purpose of comparing to the profiles.

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 profile to the profile in the idealized Rankine model (Fig. 4). Taking the limit of (13) as λ → 0, for ρ ≤ 1, which means that approaches the inner core of solid-body rotation (Fig. 4), thereby agreeing with (A1) (see the appendix). Furthermore, as λ → 0, for ρ > 1, indicative of the fact that decreases and tends asymptotically to the value given by potential flow in which and concurs with (A1). Hence, the profile exactly coincides with the Rankine profile when λ → 0. The Rankine model may be viewed as a limiting case for the Wood–White model as λ → 0 and also for the Vatistas model as q → ∞.

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.

Fig. 5.
Fig. 5.

Radial profile families of for selected values of k, n, and λ. Three profile families in each panel are indicated by three different values of n. The thick black curve represents the profile that corresponds to n = k + 1 to enhance readability. The thick gray curve represents profile for comparison. Normalized radial distance is represented by ρ = r/Rx.

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.

When λ → 0 in and beyond Figs. 5g–i, (11) exactly coincides with the modified Rankine tangential velocity and is given by
e14
Equation (14) is applicable only to the sharply peaked profiles of tangential wind. For those who are familiar with the Rankine model and interested in comparing the WW model with their data, estimated values of k and n easily are determined by setting (14) equal to (A1)—that is, —and then by taking the natural logarithm of the result. Consequently, the power-law exponents are given as
e15
The radial variations of can be modified for any γ values. When γ = k = 1, for example, the tangential velocity increases linearly from a circulation center to a core radius (ρ = 1) where the velocity attains its maximum. When γ = −1 and k = 1 and hence n = kγ = 2, the velocity is inversely proportional to distance beyond the core radius (potential flow).

Overall, three model parameters (k, n, λ) do not change magnitude at and ρ = 1. This effect is shown in Figs. 4 and 5.

It is worthwhile to compare the WW tangential velocity profiles with the theoretical tangential velocity distributions of a two-celled vortex model (e.g., Sullivan 1959) and also of a theoretical stagnant core vortex model (e.g., Fiedler and Rotunno 1986; Snow and Lund 1989). Such comparisons are obtained by evaluating (11) for n = k + 1, given as
e16
Figure 6 plots the radial profile families of for selected k and λ values. When k = 3.0 and λ = 0.5, the profile closely matches the Sullivan (Sullivan 1959) profile (Fig. 6b). A good example of the Sullivan profile is the W-band Doppler-derived azimuthal velocity profile of a Texas dust devil (Bluestein et al. 2004).
Fig. 6.
Fig. 6.

Radial profile families of for selected values of k and λ. In each panel, k = 1 (solid curve), k = 3 (short-dashed curve), k = 10 (long-dashed curve), and k → ∞ (thick gray curve) are indicated. The thick gray curve represents the normalized tangential velocity profile in the stagnant core vortex model; the dotted curve represents the normalized tangential velocity profile in the Sullivan model. Normalized radial distance is represented by ρ = r/Rx. Note that the profile is normalized to 1.0 for the purpose of comparing to the profiles.

Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1

Taking the limit of (16) as k → ∞, an area of zero values increases from ρ = 0 to ρ < 1; beyond ρ ≥ 1, the nonzero values approaches a potential flow. At the same time, the cusp in the annulus of maximum becomes increasingly sharp as λ → 0 (Fig. 6). Consequently, (16) is simplified to
e17
Equation (17) is equivalent to those of Fiedler and Rotunno (1986), Snow and Lund (1989), and Fiedler (1994), who described the stagnant core vortex model in which no tangential velocity exists within the core and also a potential vortex occurs outside the core. Fiedler and Rotunno (1986) used the stagnant core vortex in a simple analysis of vortex breakdown, and it gave results similar to the more realistic profile (B. Fiedler 2010, personal communication).

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 for some finite high k and low λ values are probably realistic (e.g., Fig. 6).

b. Vertical vorticity

In the last subsections, we computed and plotted radial profiles of in order to explain how each model parameter (k, n, λ) plays a key role in shaping the profiles. The choices of free parameters thus far have not been determined. It is reasonable to determine the ranges of free parameters by evaluating and critically examining a parametric vertical vorticity’s realism. Such examination allows us to restrict the parameters to values for which a vorticity singularity and discontinuity are not permitted, as will be presented in this study.

Vertical vorticity ζ is the vertical component of the curl of the velocity vector field and is a microscopic measure of rotation of a fluid flow. Here ζ is expressed for axisymmetric flow as
e18
where ∂V/∂r is the shear vorticity that represents the angular velocity of a fluid parcel produced by distortion due to horizontal velocity differences at its boundaries, and ω = V/r is the curvature vorticity that represents the angular velocity of rotation about a vertical axis through the instantaneous center of curvature. Substitution of (11) into (18) yields
e19
e20
For a cyclonic vortex (Vx > 0), the shear vorticity contributing to ζ is positive, zero, and negative for ρ < 1, ρ = 1, and ρ > 1, respectively. Additionally, the curvature vorticity contributing to ζ is always positive for 0 ≤ ρ < ∞, with its value being equal to Vx/Rx at ρ = 1. On the other hand, for an anticyclonic vortex (Vx < 0), ∂V/∂r contributing to ζ is negative, zero, and positive for ρ < 1, ρ = 1, and ρ > 1, respectively. Furthermore, ω contributing to ζ is always negative for 0 ≤ ρ < ∞, with its value being equal to Vx/Rx at ρ = 1.

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.

A critical point of positive (negative) vorticity peak is defined as the location of maximum (minimum) vorticity value and is obtained by differentiating (19) with respect to ρ and setting ∂ζWW/∂ρ to zero. As a consequence, the result is given by
e21
where α = ρn/λ/[1 + kn−1(ρn/λ − 1)]. Since (21) is quadratic in α, it can easily be solved using the quadratic formula, and critical parameters α± are expressed as
e22
Taking the limit of (22) as k → 1, α → 0 and also α+ → ∞ as k → 0. There are two distinct real roots for α± that yield ρ. Two (normalized) explicit solutions of and are thus given by
e23
e24
Here, represents a normalized critical point of positive (negative) vorticity peak that is defined as the location of maximum (minimum) vorticity value. At the same time, and respectively represent the dimensional critical radial distances of ζmax and ζmin. Having available and in (23) and (24), and easily are calculated if Rx is known; ζmax and ζmin readily are computed in (19). The advantage of using (23) and (24) without a required knowledge of ζmax and ζmin is evident.

Figure 7 presents the computation of that varies with k and n for selected values of λ. A shaded portion is bounded by k = 0 and k < 1; this represents the area where vorticity singularities occur and thus cannot be computed. A sloping dashed line represents a ridge at which the maximum values occur at halfway between the k = 1 and k = n lines. Approaching from the ridge toward the k = 1 line, as k → 1. Additionally, as kn, when approaching from the ridge toward the k = n line (sloping dotted line in Fig. 7). As one progresses from Fig. 7a to Fig. 7d (i.e., λ → 0), for all n > k ≥ 1 values. Several boldface crosses represent the computed values of (as will be subsequently shown in Fig. 10).

Fig. 7.
Fig. 7.

Plots of as functions of k and n for selected values of λ. A gray shaded portion represents an area where vorticity singularities for k < 1 occur. A sloping dotted line represents k = n. A sloping dashed line represents a trough of minimum values. Several bold crosses represent the computed values of , as will be discussed in Fig. 10.

Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1

In a manner analogous to the discussion in Fig. 7, in (24) varies with k and n for selected values of λ and is presented in Fig. 8. A trough at which the minimum values occur at halfway between the k = 0 and k + 1 = n lines is represented by a sloping dashed line. As can be clearly seen in the figure, as k → 0 as one approaches from the trough toward the k = 0 line, although the k < 1 values are not permitted. Additionally, as k + 1 → n as one approaches from the trough toward the k + 1 = n (dotted) line. As one progresses from Fig. 8a to Fig. 8d, as λ → 0 for all nk + 1 > 0 values. A few boldface crosses represent the calculated values of (as will be later shown in Fig. 10).

Fig. 8.
Fig. 8.

As in Fig. 7, but for . A sloping dotted line represents k + 1 = n. A sloping dashed line represents a trough of minimum values.

Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1

In a way similar to the previous comparison of the profiles to the profiles (Figs. 3 and 4), the normalized Wood–White vertical vorticity is compared to the normalized Vatistas vertical vorticity by evaluating (19) and (20) for k = 1 and n = 2. The resulting equation is expressed in a dimensionless form as
e25
where . Figure 9 compares the normalized vorticity profile families between and . The profiles decrease monotonically and asymptotically to zero with increasing ρ. If we were to compute in (25), the profiles would have exactly concurred with those of (Fig. 9b).
Fig. 9.
Fig. 9.

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 profile is calculated from (A5) in the appendix.

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 profiles. When ρ = 0 in (25) and (A2), (25) is simplified to and also to (Fig. 9), which are compared to . As λ → 0, , which exactly coincides with at ρ = 0 (Fig. 9a). When ρ > 0, taking the limit of (25) as λ → 0 and q → ∞, the bell-shaped vorticity profiles of and evolve to a top-hat distribution of (Fig. 9). As is well known, the Rankine vorticity suffers from the fact that the vorticity profile has an unrealistic discontinuous jump from constant to zero in the infinitesimal radial thickness of tangential velocity maximum.

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 pass through a point at which at ρ = 1, where the shear vorticity in the Wood–White and Vatistas models always is zero. Not all of the vorticity profiles of , however, pass through that point. Note that at ρ = 1 is lower than the Rankine vorticity value of 1.0 because the peak tangential velocity in the Scully model reduces to half of the tangential velocity value given by the Rankine model [e.g., see (2) and Fig. 3]. As q → ∞, increases and tends asymptotically to at ρ = 1 (Fig. 9b).

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 for selected values of k, n, and λ in Fig. 5. We now investigate how profiles behave in response to the normalized tangential velocity profiles in the WW model. The vorticity profiles, calculated directly from (19) and (20), are plotted in Fig. 10. In each panel of the figure, three different values of n are presented at each selected value of k to examine how the corresponding vorticity profiles behave. Furthermore, the critical points of positive and negative peaks of the normalized vorticity, if present, are indicated by solid triangles in Fig. 10 and are directly computed from (23) and (24) as well as Figs. 7 and 8.

Fig. 10.
Fig. 10.

As in Fig. 5, but for corresponding . Normalized critical points at which the corresponding and respectively occur are indicated by solid triangles. Dashed horizontal line of zero value is indicated. The thick black curve represents the vorticity profile that corresponds to n = k + 1 to enhance readability. The thick gray curve represents the profile for comparison. Normalized radial distance is represented by ρ = r/Rx.

Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1

When k = 1.0 and n = 1.1, the maximum values of at ρ = 0 are 22.0 for λ = 1.0 and 6.6 for λ = 0.5 (not shown in Figs. 10a and 10d). As one progresses from k = 1 to k > 1, vorticity concentrations are progressively displaced away from ρ = 0 toward the strongest gradient of the profiles. The annular pattern of is characteristic of a two-celled structure, as is consistent with the numerical model findings of Rotunno (1977, 1984) and the observed findings of Bluestein et al. (2004), Lee and Wurman (2005), and Mallen et al. (2005).

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 ensures appreciable vorticity out to large ρ until asymptotes to zero at radial infinity. Approximate potential flow is found just outside the annulus of maximum tangential velocity (thick solid curves in Figs. 10a–f). Tropical cyclones are often characterized by a relatively slow decrease of tangential wind outside the radius of maximum wind, and hence by a corresponding cyclonic vorticity skirt (Mallen et al. 2005). The realistic-looking profiles beyond ρ = 1 look more flexible than those of the Smith et al. (1990) model (e.g., Fig. 1 of Mallen et al. 2005).

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 and occur are displaced toward each other at ρ = 1 (solid triangles in Fig. 10). That is, taking the limit of (23) and (24) as λ → 0, , , as can be clearly seen in Figs. 7 and 8. At the same time, the more rounded peaks in the vorticity profiles become increasingly localized around ρ = 1 as the profiles correspond to the tangential velocity profiles (Fig. 5).

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).

We calculated the root-mean-square error (RMSE) and correlation coefficient (CC), given by
e26
e27
where the sums extend over the Nobs analysis grid points for which both observed (obs) and fitted (fit) tangential velocity variables are available, and the overbar represents the sample means of each variable. The two indices in (26) and (27) will be applied to evaluate the accuracy of the fitted and analytical tangential wind and vorticity profiles.

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 initially was set to the observed value (i.e., Fig. 11a) and k initially was set to 1.0. When convergence in the LM algorithm was achieved, fitting was completed so that the updated n and λ were finalized. Then, the fitted model parameters (k, n, λ) were used to compute and plot , , , and (Fig. 11).

Fig. 11.
Fig. 11.

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 value. A horizontal dashed line representing ζ* = 0 is indicated.

Citation: Journal of the Atmospheric Sciences 68, 5; 10.1175/2011JAS3588.1

The maximum value of at is found to be 2.32, comparing to the value of 2.51. The minimum value of at has been shown to be −0.015. As can been evidently seen in Fig. 11, the radial profiles of fitted and lie very close to those in the Burgers–Rott model with low RMSE values and high CC values of nearly 1. Careful inspection of the dotted curve passing through the solid curve (Fig. 11a) reveals that the solid curve representing the BR profile runs not only through the center but also at edge of the dotted curve representing the WW profile. As expected, the WW profile does not exactly coincide with the BR profile because the BR vortex is an exact solution of the Navier–Stokes equations of motion. The main weakness of the WW parametric model is that it is not the solution of such equations. The model, however, can calculate an approximation of the BR solution only if the driving model parameters are accurately determined.

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 , the calculated peak values of is shown to be 2.55, favorably comparing to the maximum value of at ρ = 0.67. Furthermore, the value of the minimum is −0.007 at . It is of interest to note that at ρ = 0, the maximum value of is 0.33, comparing to . The reason why does not closely coincide with is that the Sullivan vortex, just like the exact solution of the BR vortex, is also an exact solution of the Navier–Stokes equations of motion and the WW model is not. Comparisons of the fitted profiles with the idealized profiles indicate good agreement with low values of RMSE values and high CC values (Fig. 12).

Fig. 12.
Fig. 12.

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 peaks. A horizontal dashed line representing ζ* = 0 is indicated.

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.

Acknowledgments

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

A classic model of inviscid vortex flow is the idealized, steady-state Rankine (Rankine 1882) vortex model. The Rankine vortex3 (RV) consists of a core in solid-body rotation surrounded by potential flow where the tangential wind VRV is inversely proportional to radial distance r from a center of rotation. The normalized tangential velocity of the Rankine model is plotted in Fig. 1, using the formula
ea1
where Vx is the tangential velocity peak that occurs at its core radius Rx, ρ = r/Rx the dimensionless radial distance from the center, and γ the power-law exponent that governs the velocity profile with γ = 1, 0, −1 for ρ < 1, =1, >1, respectively.
The normalized Rankine vorticity is obtained by substituting (A1) into (18) and is given by
ea2
where .

b. Burgers–Rott model

A tangential velocity VBR distribution of the Burgers–Rott (Burgers 1948; Rott 1958) vortex solution is given by
ea3
where Γ is the circulation at radial infinity, 2a is the horizontal convergence, νe is a constant eddy viscosity, and r is the radial distance from the vortex center. The normalized tangential velocity in (A3) is simplified to
ea4
where Vx = 0.715Γ/(2πRx) is the tangential velocity maximum, K = 1.2564, ρ = r/Rx, and is the radius at which Vx occurs. The numerical value of K has been used by Vatistas (1998), Davies-Jones and Wood (2006), and Alekseenko et al. (2007, 149–152). In (A4), the advantage of using without a required knowledge of a and νe in (A3) is evident.
The normalized vertical vorticity profile in the Burgers–Rott model is obtained by applying (A4) to (18) and is given by
ea5
Equation (A5) predicts that the bell-shaped profile attains its maximum at ρ = 0 and approaches zero asymptotically as ρ → ∞.

c. Sullivan model

Sullivan (1959) obtained an exact solution to the Navier–Stokes equation in the form of a steady two-celled vortex. The tangential velocity VS distribution in the Sullivan model is expressed as
ea6
where ξar2/(2νe) and H(ξ) is a function given by
ea7
To normalize (A6) for the sake of simplicity, we substitute with the aid of ρ = r/Rx in (A6) and divide the result by Γ/(2πRx). Thus,
ea8
Since (A8) cannot be obtained explicitly, an ordinary differential equation in (A8) is solved implicitly using the Runge–Kutta–Nyström method (Kreyszig 1988, p. 1078–1081), which is a generalization of the Runge–Kutta method. Note that , which is identical to the numerical value used by Leslie and Snow (1980), Lugt (1996, p. 81), Vatistas (1998), Vatistas and Aboelkassem (2006), and Wu et al. (2006, p. 267). Figure (A1) plots the radial profile of H(βρ2), which favorably agrees with Fig. 2 of Leslie and Snow (1980).
Fig. A1.
Fig. A1.

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

Following the Vatistas (1998) approach, a maximum tangential velocity at ρ = 1 in the Sullivan model is obtained by differentiating (A8) with respect to ρ, given by
ea9
which therefore becomes
ea10
The value of β is the root of (A10); (A10) is equivalent to
ea11
In (A11), the value of β has been found to be 6.238 numerically, and ensures that the maximum occurs at ρ = 1 (Vatistas 1998).
The normalized vertical vorticity in the Sullivan model is obtained by applying (A8) to (18). It is given by
ea12
Equation (A12) is implicitly solved using the Runge–Kutta–Nyström method.

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1

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).

2

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.

3

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.

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