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

A New Parametric Model of Vortex Tangential-Wind Profiles:

Development, Testing, and Verification

VINCENTT. WOOD

NOAA/OAR/National Severe Storms Laboratory, Norman, Oklahoma

LUTHERW. WHITE

Department of Mathematics, University of Oklahoma, Norman, Oklahoma

(Manuscript received 22 June 2010, in final form 22 December 2010)

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.

1. Introduction

Tangential wind profiles such as those in intense at- mospheric 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 in- finitely 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).

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 in- stance, 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

Corresponding author address: Vincent Wood, NOAA/NSSL, National Weather Center, Norman, OK 73072–7323.

E-mail: [email protected]

velocities. Outside the cores, they observed Doppler velocity profiles of V } r^20.660.1, where r is the distance from the center of the tornado. Wurman et al. (2007) showed that different tornadoes have substantially dif- ferent 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 r 5 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 ad- vection 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 po- tential flow in the annular zone of tangential velocity maximum. The azimuthally averaged tangential velocity profiles derived from W-band Doppler radar measure- ments (Bluestein et al. 2007; Tanamachi et al. 2007) at low altitudes resemble the Burgers–Rott profile. Be- cause 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 in- stance, replaced Rankine’s unrealistic sharply peaked profile of tangential velocity with a more realistic, con- tinuous 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) ex- tended Sullivan’s (1959) two-celled vortex solution to the Navier–Stokes equations for a three-dimensional axisymmetric, two-celled vortex by measuring and com- paring 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 ex- ponent that controls the ‘‘peakedness’’ of the wind profile in the annular zone of tangential velocity maxi- mum. The Vatistas model, however, does not seem to have sufficient degrees of freedom fit to realistic ob- servations (e.g., one-celled and two-celled vortex con- figurations in Fig. 2) because the tangential velocity distribution of the model is constrained to solid-body rotation and potential flow. Sometimes there is a mis- match with any observed profiles of tangential velocity including those in Figs. 1 and 2 as well as other Doppler velocity profiles of V } r^20.660.1, as documented by Wurman and Gill (2000), Wurman (2002), Wurman and Alexander (2005), and Wurman et al. (2007). This pro- vides 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 pro- gram to retrieve the radial and vertical components of velocity defined in a cylindrical coordinate system.

The objective of this paper is to develop a new para- metric 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 im- prove fitting the realistic-looking profile by incorporating

FIG. 1. Radial profiles of V* in the analytical Rankine, Burgers–

Rott, and Sullivan vortex models for comparison. The computa- tions of the profiles are given in the appendix. Normalized radial distance is represented by r 5 r/Rx, where r is the radial distance from the vortex center and Rxis the radius at which the normalized tangential velocity maximum occurs.

1A one-celled vortex structure consists of a jetlike vertical ve- locity everywhere being directed upward with a maximum at the vortex axis, whereas a two-celled vortex is characterized by down- ward 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).

new parameters. The new parametric tangential-wind profile model employs five key parameters to construct a radial profile of tangential velocity. The first two physi- cal parameters include the maximum tangential velocity and the radius of the tangential velocity peak. Addi- tionally, 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 V*_qof the Vatistas model is given by

V* [_q V_q

G_‘/(2pR_x)5 r

(1 1 r^2q)^1/q, (1) where r 5 r/R_xis the dimensionless radial distance from the center of rotation, r is the radial distance from the

center, R_xis the radius at which a maximum tangential velocity occurs, V_qis the tangential velocity for a few selected values of q, G_‘is the circulation at radial in- finity, 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 gener- alization 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

V_q515 G_‘ 2pR_x

r

(1 1 r²) and (2)

V_q525 G_‘ 2pR_x

ffiffiffiffiffiffiffiffiffiffiffiffiffir 1 1 r⁴

p . (3)

Here, the q 5 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 r 5 1 in (2), the maximum tangential velocity is reduced to half of the value given by the Rankine model. The q 5 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 5 2 vortex model is physically most representative for actual vortex tangential- wind profiles in aerodynamics. In addition, they dem- onstrated that the q 5 2 tangential velocity distribution was in good agreement with their experimental data. It is interesting to note that the q 5 2 vortex is a good approximation to the Lamb–Oseen model (Lamb 1932;

FIG. 2. Based on the laboratory simulated vortices, evolutionary Vushear profiles for one-celled convective vortex (curve a), tran- sition 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.)

FIG. 3. Radial profiles of V* for the Scully (q 5 1, short-dashed curve), Vatistas (q 5 2, long-dashed curve), and Rankine (q / ‘, solid curve) models. Normalized radial distance is represented by r 5r/Rx. Dotted curve represents V* of the Burgers–Rott or Lamb–

Oseen model for comparison. Note that the calculation of the V*_BR profile is given in the appendix.

Oseen 1911) model and hence the Burgers–Rott (Burgers 1948; Rott 1958) model (Fig. 3).

As q / ‘ in (1), the q / ‘ vortex approaches the Rankine vortex (Fig. 3) and is given as

V_q!‘5 G_‘

2pR_xr, r #1, (4)

G_‘

2pR_xr^1, r .1. (5) 8>

>>

<

>>

>:

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, exclu- sive 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 r^kso that the new power-law exponent k controls the shape of the velocity profile near r 5 0.

When k . 1 (k , 1), the radial profile has positive (negative) curvature, meaning that the curve turns to the left (right) with small r.

When k 5 1, the V-shaped profile of tangential ve- locity near r 5 0 has zero curvature with increasing r until the profile reaches at r 5 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., V_q } r²¹), we seek to modify the profile by in- troducing a new function (1 1 r^n/l)^l, where a new sec- ond power-law exponent n dictates the tangential velocity decay beyond r 5 1. The larger the n value is, the faster the outer profile decays with increasing r.

Additionally, a new third power-law exponent l 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, l) will be described in detail in the subsequent sections.

To tailor the Vatistas model for application to re- alistic-looking tangential wind profiles, we propose that (a) r in the numerator of (1) is replaced by r^kand (b) the denominator is replaced by (1 1 r^n/l)^lto represent one- and two-celled cyclonic (or anticyclonic) vortices de- fined in cylindrical coordinates. As a consequence, we define a new tangential-velocity function x where

x(r) [ V

V_x5 r^k

(1 1 r^n/l)^l, 0 , k , n. (6)

Further, the relation of k to n is constrained by the condition that the function x(r) goes to zero as r / ‘.

Hence, we impose the condition 0 , k , n. To scale the function such that its maximum is at r 5 r_x 51, we observe that the normalized radius at which the maxi- mum x occurs is given by

r_x5 k n k

 l/n

. (7)

To obtain the maximum at r 5 r_x51, we modify (7) and define the function

x₁(r) 5 x r k n k

 l/n

" #

5

r k

n k

 l/n

" #k

1 1 r k n k

 l/n

" #n/l

8<

:

9=

;

l.

(8) Hence, (8) becomes

x₁(r) 5(n k)^(1k/n)lk^(kl/n)r^k

[n k 1 kr^(n/l)]^l . (9) By letting r 5 1 in (9), we note that

x₁(1) 5(n k)^(1k/n)lk^(kl/n)

n^l . (10)

By dividing x₁(r) by x₁(1), we obtain a function c(r; k, n, l) 5 x₁(r)/x₁(1), with the maximum value normalized to 1 at r 5 1. As a consequence, the new parametric tangential velocity distribution²is given by

V_WW* [V_WW(r; m)

V_x 5 c(r; k, n, l)

5 r^k

[1 1 kn^1(r^n/l 1)]^l, 0 , k , n, l . 0, (11) where m 5 [Vx, R_x, k, n, l]^Trepresents a model vector of the five parameters, and the subscript WW represents

2When l 5 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.

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): c(0; k, n, l) 5 0, c(1; k, n, l) 5 1 (where c attains its maximum value), c(‘; k, n, l) 5 0, c(r; k, n, l) . 0 for 0 , r , ‘, ›c/›r . 0 for r , 1, ›c/›r 5 0 at r 5 1, and ›c/›r , 0 for r . 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 G_‘and Rxmay be rewritten as

V_WW* [V_WW(r; m)

G_‘/(2pR_x) 5 r^k

[1 1 kn^1(r^n/l 1)]^l, (12) which is comparable to (1).

Now V*_WW is compared to V*_qby evaluating (11) for k 5 1 and n 5 2. It is given by

V_WW* 5 2^lr

[1 1 r^(2/l)]^l 52^lV_q*. (13) The difference between (1) and (13) is the inclusion of 2^lin (13) or the exclusion of 2^lfrom (1). Note that the denominator in (13) plays the same role as that in (1). If we compute and plot 2^2lV*_WWin (13), the V*_WWprofiles exactly coincide with the V*_qprofiles (Fig. 3).

When l 5 1 in (13), (13) is reduced to a simple model tangential wind profile, 2r/(1 1 r²) (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 2r/(1 1 r²) 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.

Further insight into the relevant tangential velocity kinematics is obtained by comparing the V*_WWprofile to the V*_RVprofile in the idealized Rankine model (Fig. 4).

Taking the limit of (13) as l / 0, V*_WW/rfor r # 1, which means that V*_WW approaches the inner core of solid-body rotation (Fig. 4), thereby agreeing with (A1) (see the appendix). Furthermore, as l / 0, V*_WW/r²¹ for r . 1, indicative of the fact that V*_WWdecreases and

tends asymptotically to the value given by potential flow in which V*_WW} r²¹and concurs with (A1). Hence, the V*_WWprofile exactly coincides with the Rankine profile when l / 0. The Rankine model may be viewed as a limiting case for the Wood–White model as l / 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 l on the physical behaviors of the radial profile families of tangential ve- locity. To facilitate comparison with the profiles, nor- malized composites are constructed that beneficially preserve the underlying tangential wind structure. Each individual profile is expressed in the convenient di- mensionless form utilizing the typical scales Vxand Rx. The radial profile families of normalized tangential ve- locity as functions of k, n, and l 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 prog- resses from the top to the bottom panels, k and n remain unchanged with decreasing l.

Varying the degree of the power-law exponent k in r^k of (11) primarily governs the shape of the tangential ve- locity profile near r 5 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 5 1 and k . 1, respectively. The reason for not showing k , 1 in this figure will be dis- cussed subsequently.

FIG. 4. Radial profiles of V* for the Wood–White (l 5 1, short- dashed curve; l 5 0.5, long-dashed curve) and for the Rankine (l / 0, solid curve) models. Normalized radial distance is repre- sented by r 5 r/Rx. The dotted curve represents the normalized tangential velocity profile of the Burgers–Rott or Lamb–Oseen model for comparison. Note that the V*_BRprofile is normalized to 1.0 for the purpose of comparing to the V*_WWprofiles.

As already mentioned previously, the decay exponent n in (11) mainly governs the outer profile that decays with increasing r. The lower (higher) the n value, the more slowly (rapidly) the decaying tangential velocity decreases with r (Fig. 5). It is important to recall in section 2 that the velocity profile exists only for k , n.

When k 5 n, the profile is perfectly flat and the vortex cannot exist.

The last power-law exponent, represented by l 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 tan- gential wind can be made either broader or narrower. As lchanges 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 l around the radius at which the maximum occurs. As l / 0, three radial profiles for very differ- ent n values merge together in each panel to form one superimposed radial profile at r , 1.

When l / 0 in and beyond Figs. 5g–i, (11) exactly coincides with the modified Rankine tangential velocity and is given by

V_WW* 5 r^k, 1, r^(kn),

r ,1, r 51, r .1, 8<

: 0 , k , n. (14)

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

FIG. 5. Radial profile families of V*_WWfor selected values of k, n, and l. 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 5 k 1 1 to enhance readability. The thick gray curve represents V*_RVprofile for comparison. Normalized radial distance is represented by r 5 r/Rx.

easily are determined by setting (14) equal to (A1)—

that is, V*_WW 5 V*_RV—and then by taking the natural logarithm of the result. Consequently, the power-law exponents are given as

k, k n,



5 g, r #1, r .1,

g 51, g 51.



(15)

The radial variations of V*_WW5V*_RVcan be modified for any g values. When g 5 k 5 1, for example, the tan- gential velocity increases linearly from a circulation center to a core radius (r 5 1) where the velocity attains its maximum. When g 5 21 and k 5 1 and hence n 5 k 2 g 5 2, the velocity is inversely proportional to dis- tance beyond the core radius (potential flow).

Overall, three model parameters (k, n, l) do not change magnitude at V*_WW51 and r 5 1. This effect is shown in Figs. 4 and 5.

It is worthwhile to compare the WW tangential ve- locity 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 5 k 1 1, given as

V_WW* 5 (k 1 1)^lr^k

[1 1 kr^(k11)/l]^l. (16) Figure 6 plots the radial profile families of V*_WWfor se- lected k and l values. When k 5 3.0 and l 5 0.5, the V*_WWprofile closely matches the Sullivan (Sullivan 1959) profile (Fig. 6b). A good example of the Sullivan profile is the W-band Doppler-derived azimuthal velocity pro- file of a Texas dust devil (Bluestein et al. 2004).

Taking the limit of (16) as k / ‘, an area of zero V*_WW values increases from r 5 0 to r , 1; beyond r $ 1, the nonzero V*_WWvalues approaches a potential flow. At the same time, the cusp in the annulus of maximum V*_WW becomes increasingly sharp as l / 0 (Fig. 6). Conse- quently, (16) is simplified to

V_WW* 5 0, r ,1 r^1, r $1



. (17)

Equation (17) is equivalent to those of Fiedler and Rotunno (1986), Snow and Lund (1989), and Fiedler

FIG. 6. Radial profile families of V*_WWfor selected values of k and l. In each panel, k 5 1 (solid curve), k 5 3 (short-dashed curve), k 5 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 ve- locity profile in the Sullivan model. Normalized radial distance is represented by r 5 r/Rx. Note that the V*_Sprofile is normalized to 1.0 for the purpose of comparing to the V*_WWprofiles.

(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 re- sults 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 (r 5 1), a zero tangential velocity within the core (r # 1), and omission of diffusion are suspect for intense atmo- spheric vortices. In real fluids, some radial profiles of V*_WWfor some finite high k and low l values are prob- ably realistic (e.g., Fig. 6).

b. Vertical vorticity

In the last subsections, we computed and plotted ra- dial profiles of V*_WWin order to explain how each model parameter (k, n, l) plays a key role in shaping the pro- files. 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 examina- tion 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 z is the vertical component of the curl of the velocity vector field and is a microscopic measure of rotation of a fluid flow. Here z is expressed for axi- symmetric flow as

z 5›V

›r 1V

r, (18)

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 v 5 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

z_WW5kv 1 kn^11 r^n/l(kn^1 1) 1 1 kn^1(r^n/l 1)

" #

1 v,

1 # k , n, l .0, where (19)

v 5 V_xR^1_x r^k1

[1 1 kn^1(r^n/l 1)]^l. (20) For a cyclonic vortex (Vx.0), the shear vorticity con- tributing to z is positive, zero, and negative for r , 1, r 51, and r . 1, respectively. Additionally, the curva- ture vorticity contributing to z is always positive for 0 # r , ‘, with its value being equal to V_x/R_xat r 5 1.

On the other hand, for an anticyclonic vortex (V_x,0),

›V/›r contributing to z is negative, zero, and positive for r ,1, r 5 1, and r . 1, respectively. Furthermore, v contributing to z is always negative for 0 # r , ‘, with its value being equal to V_x/R_xat r 5 1.

The exponent k 2 1 in r^k21in (20) reveals that if 0 , k , 1, the z_WWprofile produces a singularity as r / 0.

The profile has the undesirable property that, for 0 , k , 1, zWW 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 r / 1 (e.g., Fiedler and Rotunno 1986). The vorticity profile, how- ever, produces an unrealistic discontinuity in the in- finitesimal radial thickness of tangential velocity peak (not shown). To evade the undesirable vorticity dis- continuity at r / 1, we suggest that k be limited to some large values, perhaps from 10 to 50, and l 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 r and setting ›zWW/›r to zero. As a consequence, the result is given by

k²(1 1 l^1)a² k(2k 1 nl^1)a 1 (k² 1) 5 0, (21)

where a 5 r^n/l/[1 1 kn²¹(r^n/l21)]. Since (21) is qua- dratic in a, it can easily be solved using the quadratic formula, and critical parameters a₆are expressed as

a₆5k(2k 1 nl^1) 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [k(2k 1 nl^1)]² 4k²(1 1 l^1)(k² 1) q

2k²(1 1 l^1) .0, l .0. (22)

Taking the limit of (22) as k / 1, a₂ / 0 and also a₁/‘as k / 0. There are two distinct real roots for

a₆that yield r. Two (normalized) explicit solutions of r_z

max and r_z

minare thus given by

r_z

max[ R_z

max

R_x 5 (n k)a_ n ka_

 l/n

,1, n, n ka_.0, n . k $ 1, and (23)

r_z

min[ R_z

min

R_x 5 (n k)a₁ n ka₁

 ^l/n

.1, n, n ka₁.0, n . k 1 1 . 0, (24)

Here, r_z

max(r_z

min) 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, R_z

maxand R_z

minrespectively represent the dimensional critical radial distances of z_maxand z_min. Having available r_z

max

and r_z

min in (23) and (24), R_z and R_z max

min

easily are calculated if R_xis known; z_maxand z_min readily are computed in (19). The advantage of using (23) and (24) without a required knowledge of z_max and z_minis evident.

Figure 7 presents the computation of r_z

max that varies with k and n for selected values of l. A shaded portion is

bounded by k 5 0 and k , 1; this represents the area where vorticity singularities occur and r_z

maxthus cannot be computed. A sloping dashed line represents a ridge at which the maximum (r_z

max

) values occur at halfway be- tween the k 5 1 and k 5 n lines. Approaching from the ridge toward the k 5 1 line, r_z

max! 0 as k / 1. Addi- tionally, r_z

max ! 0 as k / n, when approaching from the ridge toward the k 5 n line (sloping dotted line in Fig. 7).

As one progresses from Fig. 7a to Fig. 7d (i.e., l / 0), r_z

max ! 1 for all n . k $ 1 values. Several boldface crosses represent the computed values of r_z

max(as will be subsequently shown in Fig. 10).

FIG. 7. Plots of r_z

maxas functions of k and n for selected values of l. A gray shaded portion represents an area where vorticity singularities for k , 1 occur. A sloping dotted line represents k 5 n. A sloping dashed line represents a trough of minimum r_z

maxvalues. Several bold crosses represent the computed values of r_z

max, as will be discussed in Fig. 10.

In a manner analogous to the discussion in Fig. 7, r_z in (24) varies with k and n for selected values of l and ismin

presented in Fig. 8. A trough at which the minimum (r_z

min) values occur at halfway between the k 5 0 and k 1 1 5 n lines is represented by a sloping dashed line. As can be clearly seen in the figure, r_z

min ! 0 as k / 0 as one approaches from the trough toward the k 5 0 line, although the k , 1 values are not permitted. Addition- ally, r_z

min! 0 as k 1 1 / n as one approaches from the trough toward the k 1 1 5 n (dotted) line. As one progresses from Fig. 8a to Fig. 8d, r_z

min! 1 as l / 0 for all n $ k 1 1 . 0 values. A few boldface crosses rep- resent the calculated values of r_z

min

(as will be later shown in Fig. 10).

In a way similar to the previous comparison of the V*_WWprofiles to the V*_qprofiles (Figs. 3 and 4), the nor- malized Wood–White vertical vorticity z*_WW is com- pared to the normalized Vatistas vertical vorticity z*_q by evaluating (19) and (20) for k 5 1 and n 5 2. The resulting equation is expressed in a dimensionless form as

z_q* 5 2

(1 1 r^2q)^(11q)/q52^lz_WW* , (25) where z*_WW[ z_WW/(V_x/R_x) 5 [2/(1 1 r^2/l)]^l11. Figure 9 compares the normalized vorticity profile families be- tween z*_WWand z*_q. The profiles decrease monotonically and asymptotically to zero with increasing r. If we were to compute 2^2lz*_WWin (25), the z*_WWprofiles would have exactly concurred with those of z*q(Fig. 9b).

We consider a special case in which the normalized vertical vorticity profiles in the Wood–White and Vati- stas models in (25) are compared to the normalized Rankine vorticity z*_RVprofiles. When r 5 0 in (25) and (A2), (25) is simplified to z*_q52 and also to z*_WW52^l11 (Fig. 9), which are compared to z*_RV 52. As l / 0, z*_WW / 2, which exactly coincides with z*_RVat r 5 0 (Fig. 9a). When r . 0, taking the limit of (25) as l / 0 and q / ‘, the bell-shaped vorticity profiles of z*_WWand z*_qevolve to a top-hat distribution of z*_RV(Fig. 9). As is well known, the Rankine vorticity suffers from the fact that the vorticity profile has an unrealistic discontinuous

FIG. 8. As in Fig. 7, but for r_z

min. A sloping dotted line represents k 1 1 5 n. A sloping dashed line represents a trough of minimum r_z

minvalues.

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 poten- tial flow is not physically realistic and cannot be a natural entity for viscous vortex flow. Owing to the discontinu- ity, (A1) is not a global solution and only satisfies the Navier–Stokes equations of motion if a rigid boundary is present at r 5 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 z*_WWpass through a point at which z*_WW51.0 at r 5 1, where the shear vorticity in the Wood–White and Vat- istas models always is zero. Not all of the vorticity pro- files of z*_q, however, pass through that point. Note that z*_q5150.5 at r 5 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 / ‘, z*_q/‘increases and tends asymptotically to z*_RV51 at r 5 1 (Fig. 9b).

The hurricane aircraft flight-level data analysis of Mallen et al. (2005) revealed that their computed rela- tive vorticity values, when normalized, have been shown to be 1.0 at r 5 1. This is because v in (20) becomes dominant and equal to 1.0 at r 5 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 V*_WWfor selected values of k, n, and l in Fig. 5. We now investigate how z*_WWprofiles behave in response to the normalized tangential velocity profiles in the WW model. The vorticity profiles, calcu- lated 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.

When k 5 1.0 and n 5 1.1, the maximum values of z*_WWat r 5 0 are 22.0 for l 5 1.0 and 6.6 for l 5 0.5 (not shown in Figs. 10a and 10d). As one progresses from k 5 1 to k . 1, vorticity concentrations are pro- gressively displaced away from r 5 0 toward the stron- gest gradient of the V*_WWprofiles. The annular pattern of z*_WW 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 pe- ripheral circulation may be small compared to that of the vortex core. The slow underlying monotonic de- crease of z*_WWensures appreciable vorticity out to large r until z*_WWasymptotes to zero at radial infinity. Approxi- mate 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

FIG. 9. Radial profiles of z* for (a) the Wood–White (l 5 1.0, short-dashed curve; l 5 0.5, long-dashed curve; l 5 0.2, medium- dashed curve; l / 0.0, solid curve) and (b) the Scully (q 5 1.0, short-dashed curve), Vatistas (q 5 2.0, long-dashed curve; q 5 5.0, medium-dashed curve), and Rankine (q / ‘, solid curve) models.

Dotted curve represents the z* profile of the Burgers–Rott or Lamb–Oseen model for comparison. Normalized radial distance is represented by r 5 r/Rx. Note that the z*BRprofile is calculated from (A5) in the appendix.

radius of maximum wind, and hence by a corresponding cyclonic vorticity skirt (Mallen et al. 2005). The realistic- looking z*_WWprofiles beyond r 5 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 ra- dius of tangential velocity peak, but its magnitude is small and approaches 21.0 (0.0) asymptotically as r / 1 (r / ‘) when taking the limit of (19) and (20) as l / 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 r more rapidly than r²¹, which is consistent with the findings of Lee and Wurman (2005).

With decreasing l, the critical points where z*_maxand z*_min occur are displaced toward each other at r 5 1 (solid triangles in Fig. 10). That is, taking the limit of (23)

and (24) as l / 0, r_z

max, r_z

min ! 1, as can be clearly seen in Figs. 7 and 8. At the same time, the more rounded peaks in the vorticity profiles become increasingly lo- calized around r 5 1 as the profiles correspond to the tangential velocity profiles (Fig. 5).

4. Testing and verification

Prior to eventually applying the Wood–White para- metric 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 corre- sponding 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

FIG. 10. As in Fig. 5, but for corresponding z*_WW. Normalized critical points (r_z

* and rmin zmax*) at which the corresponding z*_minand z*_maxrespectively 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 5 k 1 1 to enhance readability. The thick gray curve represents the z*_RVprofile for comparison. Normalized radial distance is represented by r 5 r/Rx.

1963) optimization method, a standard technique used to solve unconstrained nonlinear least squares problems for curve-fitting applications. The algorithm for implement- ing the LM method was described in Press et al. (1992).

We calculated the root-mean-square error (RMSE) and correlation coefficient (CC), given by

RMSE 5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

N_obs

å

and (26)

CC 5

å

å

å

q , (27)

where the sums extend over the N_obsanalysis grid points for which both observed (obs) and fitted (fit) tangential velocity variables are available, and the overbar repre- sents 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 ver- tical vorticity between the BR and WW models. Equa- tions (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 pre- sented a rigorous test of the WW tangential wind profile’s realism. In constrained fits, the maximum value of V*_WW 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 l were finalized. Then, the fitted model parameters (k, n, l) were used to compute and plot V*_WW, z*_WW, r_z

max, and r_z

min(Fig. 11).

The maximum value of z*_WWat r_z

max 50.0 is found to be 2.32, comparing to the z*BRvalue of 2.51. The mini- mum value of z*_WWat r_z

min 53.13 has been shown to be 20.015. As can been evidently seen in Fig. 11, the radial profiles of fitted V*_WWand z*_WWlie 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 tan- gential velocity and corresponding vorticity between the Sullivan and Wood–White models are presented in Fig. 12. At r_z

max 50.70, the calculated peak values of z*_WW is shown to be 2.55, favorably comparing to the maximum value of z*_S52.62 at r 5 0.67. Furthermore, the value of the z*_WWminimum is 20.007 at r_z

min 51.97.

It is of interest to note that at r 5 0, the maximum value of z*_Sis 0.33, comparing to z*_WW5 0.0. The reason why z*_WWdoes not closely coincide with z*_S 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

FIG. 11. Radial profiles of (a) the Burgers–Rott tangential ve- locity (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 r 5 r/Rx. The fitted model parameters (k, n, l), RMSE, and CC are indicated. In (b), a downward-pointing arrow indicates the location of the minimum z*_WWvalue. A horizontal dashed line representing z* 5 0 is indicated.

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).

Overall, a good agreement between the parametric and theoretical tangential velocities and vorticities indi- cates that the WW model has shown considerable suc- cess 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 co- efficient 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 ve- locity profile, including the sharply or broadly peaked region of the highest tangential wind. Experimental studies have been carried out by varying the model pa- rameters to explore and understand the behaviors of tangential wind and corresponding vertical vorticity pro- files. 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 para- metric 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 ac- curately 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 numeri- cal tangential-wind measurements derived from high- resolution, two-dimensional, axisymmetric, numerical vortex models that represent evolving simulated vorti- ces. This will allow us to examine and assess how well the Wood–White parametric model is capable of repro- ducing 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 param- eters and/or modifications or by developing a differ- ent mathematical model from scratch to improve the realistic-looking WW parametric profiles that very closely coincide with the observed profiles. This approach car- rying 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 ra- dial and vertical components of velocity as well as pres- sure 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 para- metric 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

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 pro- files are normalized. Normalized radial distance is represented by r 5r/Rx. The fitted model parameters (k, n, l), RMSE, and CC are indicated. In (b), two downward-pointing arrows indicate the loca- tions of the positive and negative z*_WWpeaks. A horizontal dashed line representing z* 5 0 is indicated.

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 sug- gestions in the earlier version of the paper. Particular thanks are extended to Kim Elmore of NSSL for pro- viding 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 ap- pears 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 axisym- metric vortex are presented here in a convenient dimen- sionless 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 tangen- tial wind V_RVis inversely proportional to radial distance r from a center of rotation. The normalized tangential velocity V*_RVof the Rankine model is plotted in Fig. 1, using the formula

V*_RV[V_RV

V_x 5 r^g, (A1)

where V_xis the tangential velocity peak that occurs at its core radius R_x, r 5 r/R_x the dimensionless radial dis- tance from the center, and g the power-law exponent that governs the velocity profile with g 5 1, 0, 21 for r , 1, 51, .1, respectively.

The normalized Rankine vorticity z*_RVis obtained by substituting (A1) into (18) and is given by

z*_RV5(g 1 1)r^g15

2, r ,1 1, r 51 0, r .1 , 8<

: (A2)

where z*_RV[ z_RVR_x/V_x5 z_RV2pR_x²/G_‘.

b. Burgers–Rott model

A tangential velocity VBRdistribution of the Burgers–

Rott (Burgers 1948; Rott 1958) vortex solution is given by

V_BR5 G_‘

2pr 1 exp ar² 2n_e

 

 

, (A3)

where G_‘ is the circulation at radial infinity, 2a is the horizontal convergence, n_eis a constant eddy viscosity, and r is the radial distance from the vortex center. The normalized tangential velocity V*_BR in (A3) is simpli- fied to

V*_BR[V_BR V_x 51

r[ 1 exp(Kr²)], (A4) where V_x 50.715G_‘/(2pR_x) is the tangential velocity maximum, K 5 1.2564, r 5 r/R_x, and R_x 5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Kn_e/a p is the radius at which V_xoccurs. 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 Kr² without a required knowledge of a and n_ein (A3) is evident.

The normalized vertical vorticity z*_BR profile in the Burgers–Rott model is obtained by applying (A4) to (18) and is given by

z_BR* [ z_BR

V_x/R_x52K exp(Kr²). (A5) Equation (A5) predicts that the bell-shaped profile at- tains its maximum at r 5 0 and approaches zero asymp- totically as r / ‘.

3Sometimes called the Rankine-combined vortex by some in- vestigators because the vortex’s inner and outer tangential velocity profiles are combined. The word ‘‘combined’’ is dropped for the sake of convenience.

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

V_S(r) 5 G_‘ 2pr

H(j) H(‘)

 

, (A6)

where j [ ar²/(2n_e) and H(j) is a function given by

H(j) 5 ðj

0

exp t 1 3 ðt

0

1 exp(s) s

 

ds

 

dt . (A7) To normalize (A6) for the sake of simplicity, we sub- stitute j [ aR_x²r²/(2n_e) 5 br²with the aid of r 5 r/R_x in (A6) and divide the result by G_‘/(2pR_x). Thus,

V_S* 51 r

H(br²)

r!‘limH(br²) 2

4

3

5. (A8)

Since (A8) cannot be obtained explicitly, an ordinary differential equation in (A8) is solved implicitly using the Runge–Kutta–Nystro¨m method (Kreyszig 1988, p. 1078–1081), which is a generalization of the Runge–

Kutta method. Note that lim

r!‘H(br²) 5 37.9043, 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(br²), which favorably agrees with Fig. 2 of Leslie and Snow (1980).

Following the Vatistas (1998) approach, a maximum tangential velocity V*_Sat r 5 1 in the Sullivan model is obtained by differentiating (A8) with respect to r, given by

›V_S*

›r

 

r51

52br²›H(br²)

›r  H(br²) 5 0, (A9) which therefore becomes

2b›H(b)

›r  H(b) 5 0. (A10)

The value of b is the root of (A10); (A10) is equivalent to

2b exp b 1 3 ðb

0

1 exp(s) s

 

ds

 

 ðb

0

exp t 1 3 ðt

0

1 exp(s) s

 

ds

 

dt 5 0.

(A11)

In (A11), the value of b has been found to be 6.238 numerically, and ensures that the V*_Smaximum occurs at r 51 (Vatistas 1998).

The normalized vertical vorticity z*_S in the Sullivan model is obtained by applying (A8) to (18). It is given by

z_S* [ z_S G_‘/(2pR²_x)

5 2b

r!‘limH(br²)exp br²13 ðbr²

0

1 exp (s) s

 

ds

( )

.

(A12) Equation (A12) is implicitly solved using the Runge–

Kutta–Nystro¨m method.

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FIG. A1. Radial profile of H(br²) in the Sullivan model. Normalized radial distance is represented by r 5 r/Rx.