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  • View in gallery

    Radial profiles of (a) normalized velocity functions and and (b) the product of and for selected values of . The black curve is indicated for ; the curves for are indicated by different colors: green curve for and red curve for . The gray curve with is added for comparison. Normalized radial distance from a vortex center is represented by . Black arrows are discussed in text.

  • View in gallery

    Radial profiles of normalized velocity functions and for selected values of , , and . The curve is indicated by a black curve for . At a given value of and in each panel, the curves are indicated by a green curve for , a red curve for , and a blue curve for . Normalized radial distance from a circulation center is represented by .

  • View in gallery

    Radial profile families of for selected values of , , and . Three colorful profile families in each panel are indicated by three corresponding values of . The green curves represent the modified Rankine velocity () profiles; the blue and red curves represent the non-Rankine velocity () profiles for comparison. Normalized radial distance is represented by .

  • View in gallery

    Radial profile families of pressure deficit for selected values of , , and . Three colorful profile families in each panel are indicated by three corresponding values of . The green curves represent the Rankine pressure deficit () profiles; the blue and red curves represent the non-Rankine pressure deficit () profiles for comparison. Normalized radial distance is represented by .

  • View in gallery

    Radial distributions of (a) and (c) cyclostrophic wind () and (b) and (d) corresponding pressure deficit () as functions of , and for the non-Rankine vortices A (blue curves) and B (red curves). The radial distributions of Rankine (R) tangential velocity and pressure deficit (green curves) are indicated for comparison. The profiles are normalized. Normalized radial distance is represented by .

  • View in gallery

    Radial distributions of (a) cyclostrophic wind () and (b) corresponding pressure deficit () as a function of for the non-Rankine vortices C (blue curve), D (red curve), and E (green curve) when is fixed. Radial distributions of (c) and (d) as a function of for the non-Rankine vortices F (blue curve), G (red curve), and H (green curve) as . Note that the green curve superimposes onto the blue and red curves. The profiles are normalized. Normalized radial distance is represented by .

  • View in gallery

    Radial distributions of radial velocity (), tangential velocity (), vertical velocity (), and pressure variable () derived from the Fiedler (1994) numerical simulation (green curves). Black curve represents the radial distribution of Rankine cyclostrophic pressure variable () deduced from the parametrically constructed Rankine tangential velocity (). Red-circle curve represents the radial distribution of fitted WW tangential velocity (); black-dot curve represents the radial distribution of cyclostrophic pressure variable () deduced from . The profiles are nondimensional. A horizontal dashed line separating positive values from negative values is indicated. Indicated are , , , , , RMS, CORR, and (see text). (Data courtesy of B. Fiedler of University of Oklahoma.)

  • View in gallery

    Radial distributions of (a) fitted WW tangential wind (, red curve), GBVTD-analyzed mean tangential wind (, green-dot curve) and Rankine cyclostrophic tangential wind (black curve) and (b) parametrically constructed Rankine pressure deficit (black curve) and non-Rankine pressure deficit (red curve) deduced from the fitted WW tangential wind shown in (a). In (a), a black superimposed on a green filled dot at m and a green dashed line connecting from m s−1 at m to m s−1 at m are discussed in text. Indicated are , , , , , RMS, CORR, and (see text). (Data courtesy of R. Tanamachi of NSSL.)

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A Parametric Wind–Pressure Relationship for Rankine versus Non-Rankine Cyclostrophic Vortices

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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 parametric tangential wind profile model is presented for depicting representative pressure deficit profiles corresponding to varying tangential wind profiles of a cyclostrophic, axisymmetric vortex. The model employs five key parameters per wind profile: tangential velocity maximum, radius of the maximum, and three shape parameters that control different portions of the profile. The model coupled with the cyclostrophic balance assumption offers a diagnostic tool for estimating and examining a radial profile of pressure deficit deduced from a theoretical superimposing tangential wind profile in the vortex. Analytical results show that the shape parameters for a given tangential wind maximum of a non-Rankine vortex have an important modulating influence on the behavior of realistic tangential wind and corresponding pressure deficit profiles. The first parameter designed for changing the wind profile from sharply to broadly peaked produces the corresponding central pressure fall. An increase in the second (third) parameter yields the pressure rise by lowering the inner (outer) wind profile inside (outside) the radius of the maximum. Compared to the Rankine vortex, the parametrically constructed non-Rankine vortices have a larger central pressure deficit. It is suggested that the parametric model of non-Rankine vortex tangential winds has good potential for diagnosing the pressure features arising in dust devils, waterspouts, tornadoes, tornado cyclones, and mesocyclones. Finally, presented are two examples in which the parametric model is fitted to a tangential velocity profile, one derived from an idealized numerical simulation and the other derived from high-resolution Doppler radar data collected in a real tornado.

Corresponding author address: Vincent T. Wood, NOAA/OAR/NSSL, 120 David L. Boren Blvd., Room 3921, Norman, OK 73072-7323. E-mail: vincent.wood@noaa.gov

Abstract

A parametric tangential wind profile model is presented for depicting representative pressure deficit profiles corresponding to varying tangential wind profiles of a cyclostrophic, axisymmetric vortex. The model employs five key parameters per wind profile: tangential velocity maximum, radius of the maximum, and three shape parameters that control different portions of the profile. The model coupled with the cyclostrophic balance assumption offers a diagnostic tool for estimating and examining a radial profile of pressure deficit deduced from a theoretical superimposing tangential wind profile in the vortex. Analytical results show that the shape parameters for a given tangential wind maximum of a non-Rankine vortex have an important modulating influence on the behavior of realistic tangential wind and corresponding pressure deficit profiles. The first parameter designed for changing the wind profile from sharply to broadly peaked produces the corresponding central pressure fall. An increase in the second (third) parameter yields the pressure rise by lowering the inner (outer) wind profile inside (outside) the radius of the maximum. Compared to the Rankine vortex, the parametrically constructed non-Rankine vortices have a larger central pressure deficit. It is suggested that the parametric model of non-Rankine vortex tangential winds has good potential for diagnosing the pressure features arising in dust devils, waterspouts, tornadoes, tornado cyclones, and mesocyclones. Finally, presented are two examples in which the parametric model is fitted to a tangential velocity profile, one derived from an idealized numerical simulation and the other derived from high-resolution Doppler radar data collected in a real tornado.

Corresponding author address: Vincent T. Wood, NOAA/OAR/NSSL, 120 David L. Boren Blvd., Room 3921, Norman, OK 73072-7323. E-mail: vincent.wood@noaa.gov

1. Introduction

Attempts to interpret observed tangential wind and pressure distributions arising in dust devils, waterspouts, tornadoes, and tornado cyclones from available observational data (e.g., mobile Doppler radar) have been made using an idealized, inviscid Rankine (Rankine 1882) combined vortex1 model. The simple Rankine vortex (RV) model coupled with the cyclostrophic balance assumption has been widely used by numerous investigators to provide a diagnostic tool for analyzing and interpreting the observed tangential wind and deduced pressure structures in dust devils (Sinclair 1973; Cantor et al. 2006), waterspouts (Leverson et al. 1977), tornadoes (Hoecker 1961; Wakimoto and Wilson 1989; Winn et al. 1999; Lee et al. 2004; Wurman and Samaras 2004; Lee and Wurman 2005; Tanamachi et al. 2013) and misocyclones (Inoue et al. 2011). The model's inner and outer tangential wind profiles can be modified to fit fairly well the available radar-derived observational data; however, the unrealistic sharply peaked wind profile at the core radius remains unchanged and is usually ignored. The result is that the tangential velocity peak is not well matched with the observed continuous tangential wind maximum. This problem gives rise to a vital question as to how varying the radial profiles of the RV tangential wind affects the deduced pressure deficits within the vortex core region when the sharply peaked tangential wind peak does not match the broadly peaked observational wind profile. The RV model with cyclostrophic flow, for instance, explained about 75% of the in situ measurements of pressure deficit in the vicinity of dust devils (Sinclair 1973) and waterspouts (Leverson et al. 1977). This discrepancy could be attributed to the neglect of radial and vertical components of flow in the model. For a given tangential velocity maximum, Fiedler (1994), however, showed that his non-Rankine vortex2 (non-RV) has a central pressure deficit twice that of the RV, suggesting that varying tangential wind profiles in the non-RV model have an important modulating influence on the behavior of deduced pressure profiles. The profiles underscore the limitation of the RV model for many applications because of the mathematical simplicity in the model. The problem with the sharply peaked tangential velocity profile peak 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. Some unrealistic aspects of the Burgers–Rott vortex, however, were that the radial and vertical components of velocity increased linearly to infinity, and the strength of the circulation was decoupled from these velocity components (Davies-Jones 1986). Additionally, the outward radial (vertical) integral of pressure from the vortex's stagnation point at () approached infinity unrealistically as increased unbounded. This implies that the flow was driven by infinitely high (low) pressure at radial (vertical) infinity, according to Davies-Jones (1986). Consequently, the inadequacy of the Rankine and Burgers–Rott vortex models provides motivation to apply the Wood and White (2011) parametric tangential wind profile model to a wind–pressure relationship and to determine whether the model can provide good potential for diagnosing the pressure features arising in dust devils, waterspouts, tornadoes, tornado cyclones, and mesocyclones. By tailoring the Wood and White model for tropical cyclone applications, Wood et al. (2013) developed parametric tropical cyclone wind profiles for depicting representative surface pressure profiles that corresponded to the multiple-maxima tangential wind profiles associated with three complete rings of enhanced radar reflectivity (i.e., inner eyewall, first and second outer eyewalls). The profiles favorably resembled the observed profiles of flight-level tangential wind in several tropical cyclones.

The objective of this paper is to develop a diagnostic tool by applying the Wood–White (WW) model to the cyclostrophic wind–pressure relationship of an axisymmetric vortex. The model employs five key parameters per wind profile: tangential velocity maximum, radius of the maximum, and three shape parameters that control different portions of the profile. Understanding properties of the parameter dependence aids in developing intuition that is critical in defining admissible parameter sets in different applications. Analytical tangential wind profiles in the RV and non-RV (WW) models are described in section 2. In section 3, a description of the wind–pressure relationship for the Rankine and non-Rankine vortices is presented. Section 4 elucidates the roles of the shape parameters in the behaviors of the radial distributions of tangential winds and pressure deficits of the RV versus non-RV for comparison. A special case of a stagnant core vortex model of tangential wind and pressure deficit distributions is described in section 5. Section 6 presents two illustrative examples in which the parametric model is fitted to a tangential velocity profile, one derived from an idealized numerical simulation and the other derived from high-resolution Doppler radar data collected in a real tornado. The cyclostrophic pressure deficit profiles are then calculated from the fitted profiles of tangential velocity. A summary and discussion are presented in section 7. Section 8 describes future work.

2. Analytical tangential wind profiles

a. RV tangential wind profile

A classic model of inviscid vortex flow is the idealized, steady-state Rankine vortex that is frequently used as a first approximation to an atmospheric vortex. It consists of tangential velocity () that increases linearly from zero at the center of the vortex to a maximum value at the core radius (solidly rotating core region) and then decreases, with velocity being inversely proportional to distance from the center. The normalized tangential velocity distribution () is expressed as
e1
where is the peak tangential velocity that occurs at the core radius , is the radius from the vortex center, is the dimensionless radius, and is an exponent that governs the inner (subscript i) velocity profile within the core region () and is another exponent that governs the outer (subscript o) velocity profile beyond the core region (). Other than and , the exponents can be modified to describe different shapes of the inner and outer velocity profiles; however, the pointed curve that peaks at remains unchanged. 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 (i.e., )—in some cases, to the inner radial profiles of Doppler velocities. Outside the cores, they observed Doppler velocity profiles of , where . The Rankine vortices with varying and exponents will be used to compare against the non-Rankine vortices that are described in the subsequent subsections.

b. Wood–White parametric tangential wind profile

The simple non-Rankine vortex may be modeled using the parametric tangential wind () profile of Wood and White (2011). The profile for inviscid, axisymmetric flow is expressed by
e2
The WW profile employs a model vector of five key parameters: , where is the maximum tangential velocity and is the radius of the maximum. The three shape parameters () are related to different shapes of the velocity profile. Note that and are the same parameters k and n as initially developed by Wood and White (2011). Tangential wind in (2) assumes a circular wind flow pattern at a given height level and does not adequately depict the actual surface boundary layer winds.
Interested users and analysts may have some difficulty in choosing the adjustable shape parameters () in (2) in order to obtain desired results. We want to assist users and analysts interested in applying (2) for their different applications, such as calculating axisymmetric vertical vorticity or radial pressure profiles deduced from the model azimuthal-mean tangential wind data fitted to observed data, or vortices initialized in numerical or idealized model or theoretical vortex studies. To facilitate the physical interpretation of each parameter in (2), Wood and White (2011) showed that the parametric model in (2) coincides with the Rankine model in (1) by taking the limit of (2) as . This is given by
e3
The values of and are easily determined by setting and in (1) equal to (3) and then taking the natural logarithm of the result, which yields for and for . Substituting and into (2) results in a simplified Wood and White parametric model, given by
e4
In this formulation, the definitions of and for the non-RV model are the same as those for the RV model. In other words, the tangential velocity profile is defined by (i) the growth parameter , which predominantly dictates the inner profile near the vortex center; and (ii) the decay parameter , which primarily governs the outer profile beyond the radius of the tangential wind maximum. The main reason for using the formulation (4) instead of (2) is that the definitions of the Rankine parameters and are more intuitive than and . This is because (i) the means to choose and appear to be much easier to understand than those of and ; (ii) and are independent of each other; and (iii) the Rankine parameters are free, whereas the relation of to is constrained by the condition that . The reason for defining in (4) is to avoid a vorticity singularity as (Wood and White 2011).

A simple technique is described below whereby investigators can learn how to choose which of the adjustable parameters in (4) to start without having to work through a large number of plots to get their desired results. The first step the investigator can do is to conveniently partition into two normalized velocity functions: , where and . The function may be thought of as the “growth” function because the parameter controls a linearity or nonlinearity of the inner velocity profile (Wood et al. 2013). The function represents the so-called decay function because the parameter decays the outer velocity profile with increasing . The parameter in may be thought of as the “size” parameter because it controls the radial width of the overall velocity profile straddling the wind maximum. When is infinitesimally small (becomes large), the profile is sharply (broadly) peaked at , representing an idealized Rankine (non-Rankine) vortex.

In the next step, the investigator can construct a nomogram consisting of a family of and curves, following a simple example shown in Fig. 1. The profiles of and are plotted in the left panel of the figure; the distribution of is depicted in the right panel.

Fig. 1.
Fig. 1.

Radial profiles of (a) normalized velocity functions and and (b) the product of and for selected values of . The black curve is indicated for ; the curves for are indicated by different colors: green curve for and red curve for . The gray curve with is added for comparison. Normalized radial distance from a vortex center is represented by . Black arrows are discussed in text.

Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-13-00041.1

In the third step, the investigator then examines for and for when . One now focuses on because it is easy with which to work. Recognizing that the definition of for the Rankine vortex is the same definition for the non-Rankine vortex, linearity of the profile indicates solid-body rotation (Fig. 1a) when . Between the vortex center and , the curve is flat because as . As a result, tends to dominate, while is relatively unimportant in contributing to (Fig. 1a). As one approaches and beyond, tends to dominate with increasing , while continues to increase linearly, because in (4) becomes large at and beyond . As a consequence, drops off more rapidly than increases linearly, thus producing a narrow, sharply-peaked profile (Fig. 1b). The profile is identical to the profile for . Varying does not alter the peak at .

Conversely, when is varied from approximately zero to 1.0, the flat profile of inside changes to a Gaussian-like or bell-shaped profile, as indicated by the vertical arrows in Fig. 1a. At the same time, the profile transitions from a sharply to broadly peaked profile in the annular zone of the maximum velocity by increasing the radial width of the profile at a given tangential wind level (indicated by horizontal arrows in Fig. 1b). Hence, the size of the non-RV velocity profile is wider than that associated with the RV velocity profile.

Now that we understand how constant values of and and two various values of in and control the shape profiles of (Fig. 1), we further investigate the role of varying and at three given values of in influencing the and curves in each panel of Fig. 2 and the radial distributions of corresponding in each panel of Fig. 3. There are specific considerations for the use of the selected parameter values in the figures. In the left columns of Figs. 2 and 3, the value of is elected because the inner velocity profile is characterized by a core of solid-body rotation. In the middle and right columns of Figs. 2 and 3, the profile is not characterized by this core because the curvature of the profile near the vortex center changes its direction from zero to positive as is increased. The values of are chosen because we seek to examine the effect on the behavior of the radial pressure deficit profile that corresponds to the inner tangential velocity profile that bears a resemblance to the Sullivan (1959) tangential velocity profile. 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. 2.
Fig. 2.

Radial profiles of normalized velocity functions and for selected values of , , and . The curve is indicated by a black curve for . At a given value of and in each panel, the curves are indicated by a green curve for , a red curve for , and a blue curve for . Normalized radial distance from a circulation center is represented by .

Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-13-00041.1

Fig. 3.
Fig. 3.

Radial profile families of for selected values of , , and . Three colorful profile families in each panel are indicated by three corresponding values of . The green curves represent the modified Rankine velocity () profiles; the blue and red curves represent the non-Rankine velocity () profiles for comparison. Normalized radial distance is represented by .

Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-13-00041.1

The value of shown in Figs. 2a–c and 3a–c is chosen because we seek to compare the outer profiles of to the observed Doppler velocity profiles outside the cores (i.e., , where ) in their proximity radar observations of tornadoes by mobile, high-resolution Doppler radars (Wurman and Gill 2000; Wurman 2002; Wurman and Alexander 2005). The value of is chosen because the outer velocity profile is characterized by potential flow, as shown in the middle panels of Figs. 2 and 3. As depicted in the bottom panels, we choose because we seek to determine what the effect on the pressure deficit distributions would be if there were a rapid decay of the outer velocity profiles.

In each panel of Figs. 2 and 3, three varying values of are presented to compare the impact of changing one value on three different radial profiles, while each value of and is held constant. Excluding , the and profiles shown in Fig. 2d are identical to those in Fig. 1a; the profiles depicted in Fig. 3d are equal to those in Fig. 1b.

Since a cyclostrophic vortex is only approximately in cyclostrophic balance, we need to examine only the cylindrical form of the cyclostrophic wind equation in azimuthal mean and integral form to better understand what factors determine the central pressure deficit in the cyclostrophic vortex having varying tangential wind profiles. This will be discussed in the following section.

3. Wind–pressure relationship for a cyclostrophic vortex

In an axisymmetric vortex, the assumption of balance between dynamic pressure drop and wind speed, termed cyclostrophic balance, is given by
e5
where is the cyclostrophic (tangential) velocity, is the radial pressure perturbation with respect to the motionless base state at radial infinity, and is the constant specific volume of air. The pressure deficit is obtained by integrating (5) radially inward from an environmental pressure at which the tangential wind decreases asymptotically to zero infinitely far from the vortex center, and is given by
e6
Hence,
e7
where is a dummy variable for the integration. Integration of (7) is done numerically in all but simple cases; (7) involves the inward integral, which is calculated using the trapezoidal rule (e.g., Press et al. 1992, 125–126). Using (7), the cyclostrophic wind balance for the vortex is employed to derive a pressure profile from a varying tangential wind profile in the WW model, as will be presented in the subsequent section.
To facilitate comparison with the profiles, normalized composites were constructed that preserved the underlying tangential wind and pressure structures. Each individual profile was expressed in the convenient dimensionless form utilizing the typical scales and . A profile of for the non-RV is obtained by incorporating in (4) into (7), dividing the result by , and integrating the result. Thus, it is expressed by
e8
To determine whether the integral in (8) is convergent or divergent, we begin with outward radial integration (see the appendix) and observe that
e9
It has been verified that the integral in (9) is convergent to zero as , regardless of the choice of the free parameters. This is because approaches zero very rapidly as .
A normalized pressure deficit () for the RV is derived by incorporating in (1) into (8) and integrating the result inward radially in a piecewise manner. Thus, is obtained as, via (4),
eq1
Note in (10a) that must change continuously at between the inner () and outer () tangential velocity profiles. At the RV center (), the normalized central pressure deficit in (10a) is given by .

Figure 4 presents the radial profile families of normalized RV and non-RV pressure deficits as functions of , , and that correspond to the radial profile families of normalized RV and non-RV tangential velocities in Fig. 3. The non-RV and RV profiles of pressure deficits were calculated from (8) and (10), respectively. The roles of the shape parameters (, , and ) in the behaviors of the radial distributions of normalized tangential velocities and pressure deficits of the RV versus non-RV will be described in the subsequent sections.

Fig. 4.
Fig. 4.

Radial profile families of pressure deficit for selected values of , , and . Three colorful profile families in each panel are indicated by three corresponding values of . The green curves represent the Rankine pressure deficit () profiles; the blue and red curves represent the non-Rankine pressure deficit () profiles for comparison. Normalized radial distance is represented by .

Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-13-00041.1

4. Radial profiles of tangential velocity and pressure deficit in the Rankine and non-Rankine vortices

This section uses a simple vortex simulator to provide what each input parameter (, , , , and ) may be able to deduce about a non-Rankine cyclostrophic vortex structure in terms of intensity. Intensity is measured by the central pressure minimum or maximum tangential velocity in the vortex core. We use the central pressure minimum as a proxy for intensity because it is our interest to investigate how a change in the shape and distribution of tangential wind directly affects intensity when comparing against the radial distributions of the RV tangential velocity and pressure deficit. We perform comparative cases by varying one parameter while keeping other parameters unchanged in the radial distributions of the non-Rankine tangential velocity and pressure deficit. Finally, we present a detailed discussion on the manner in which the radial profiles of normalized tangential velocity and pressure deficit are varied and compared to those of the RV. Our approach was similar to that of Knaff et al. (2011) and Wood et al. (2013), who investigated the effects of fine-tuning the different parameters on their different models of tropical cyclone intensity, strength, and size.

We assumed that a non-Rankine cyclostrophic vortex was steady state, axisymmetric, columnar, and with azimuthal (tangential) velocity independent of height. The cyclostophic wind balance was not a good approximation in a surface boundary layer, owing to the relative importance of turbulent friction. Additionally, the balance was applied to the core flow in the portion of the vortex above the boundary layer in which the balance was destroyed.

a. Influence of on the wind–pressure profiles

To begin with a simple Rankine vortex and two non-Rankine vortices A and B (Table 1), we find that the radial profiles of tangential velocity and pressure deficit are calculated from (1), (4), (8), and (10) and are given by

RV:
eq2
non-RV A:
eq3
and non-RV B:
eq4
Equation (11b) concurs with the findings of Winn et al. (1999), Alekseenko et al. (2007), Markowski and Richardson (2010), and others. Also, (13a) has been widely used by several other investigators for their different applications (e.g., Jelesnianski 1966; Ooyama 1969; Houston and Powell 1994; Dowell et al. 2005; among others).
Table 1.

The selected , , and values that produced the radial profiles of and are given in the eight experiments defined by non-RV identifications (IDs). The central pressure deficit values are indicated. The calculated values of are indicated.

Table 1.

The resultant profiles calculated from (11) – (13) are depicted in Figs. 5a and 5b. The green curve in Fig. 5a represents the normalized tangential velocity () of a classic Rankine vortex that has long been of interest to scientists, who have constantly strived to duplicate naturally occurring atmospheric vortices. The motion in the vortex creates a dynamic pressure that is lowest in the core region. The radial distribution of normalized pressure deficit () is characterized by a drastic decrease in the RV core (Fig. 5b).

Fig. 5.
Fig. 5.

Radial distributions of (a) and (c) cyclostrophic wind () and (b) and (d) corresponding pressure deficit () as functions of , and for the non-Rankine vortices A (blue curves) and B (red curves). The radial distributions of Rankine (R) tangential velocity and pressure deficit (green curves) are indicated for comparison. The profiles are normalized. Normalized radial distance is represented by .

Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-13-00041.1

In contrast to the and profiles of the RV, the non-RV model admits a great variety of the normalized tangential velocity () and pressure deficit () distributions depending on the values of the size parameter . When , , and remain unchanged, the broadly peaked distribution of non-RV A approaches solid-body rotation for very small (Fig. 5a) and converges to a potential vortex for large . In the extreme case (), non-RV A evolves through non-RV B to the RV by transitioning the broadly peaked profile to a sharply peaked profile. The evolving distribution produced by an increase (a decrease) of is lowered (raised) as it departs from (approaches) that of the RV (Fig. 5b), thereby producing an increase (a decrease) in vortex intensity.

Fiedler (1994) showed that for a given maximum tangential velocity, his non-Rankine vortex would have twice the central pressure deficit of the Rankine vortex. It might be useful to introduce a new parameter as the ratio of the non-RV's central pressure deficit at to that of the RV at , and is given by
e14
where the subscript refers to the tangential velocity peak. In Fielder's instance, and , thus yielding for Fiedler's non-RV B. This means that for a given tangential velocity maximum (i.e., ), the non-RV B has twice [i.e., ] the central pressure deficit of the RV (Fig. 5b).
Conversely, for a given central pressure deficit (Fig. 5d), Fiedler's (1994) non-RV B would have a maximum tangential wind speed 0.71 times that of the RV (Fig. 5c). This numerical value of 0.71 is the ratio of to and is, via (14), calculated from
e15
where for this case of the non-RV B. As , , meaning that the non-RV coincides with the RV.

b. Influence of on the wind–pressure profiles

In the last subsection, the role of in the behavior of the radial distributions of and corresponding has been discussed. We now investigate how various values can have an impact on the behavior of the pressure deficit profiles. We choose Figs. 3d–f and 4d–f to discuss the investigation. At given values of and , a transition from the right half of a V-shaped profile to the right half of a U-shaped profile of tangential velocity near the vortex center produces a change in the corresponding pressure deficit profiles as progresses from 1.0 through 3.0–5.0. Between the vortex center and , the RV and non-RV pressure structures have a flat minimum for (Figs. 4e and 4f). The radial width of the flat pressure profile associated with the RV is greater than that associated with the non-RV. Additionally, the RV has a lesser, flat central pressure minimum inside when comparing to those of the non-RVs. At given values of and , the non-RV tangential velocity and pressure deficit distributions coincide with those of the RV as . An analogous description can be applied to and , as shown in the top and bottom rows of Figs. 3 and 4, respectively. Based on the simulation results, an increase (a decrease) in tends to produce a decrease (an increase) in vortex intensity and a rise (fall) in the central pressure minimum.

c. Influence of on the wind–pressure profiles

Now that we comprehend how the different and values control the shape profiles of and , we further explore the role of the decay parameter () in influencing the profiles. We want to focus on the left panels of Figs. 3 and 4. Using the selected values of , , and in these panels, we applied them to (1), (4), (8), and (10) to simulate the radial profiles of and corresponding in the Rankine and non-Rankine vortices. The outer profiles of and the radial distributions of are changed by different parameter values of , as a comparison between the left panels of Figs. 3 and 4 illustrates. The values are relatively insensitive to the inner profiles because is dominant at . Decreasing produces a decay of the outer profile and causes the profile to rise, and hence decreases vortex intensity. A similar explanation can be applied to the middle and right panels of Figs. 3 and 4.

d. Influence of and on the wind–pressure profiles

As expected, an increase in vortex tangential velocity peak is strongly correlated with vortex intensity and pressure fall when other free parameters (, , , and ) are held constant. The central pressure minimum appears to be relatively insensitive to variations in . This finding was consistent with Holland (1980), Weatherford and Gray (1988), Lee and Wurman (2005), Knaff et al. (2011), and Wood et al. (2013), who showed that the size of any atmospheric vortex is irrelevant in computing the central pressure deficit.

5. Stagnant core vortex model

In this section, we present a special case for a theoretical stagnant core vortex (SCV) model used by Fiedler and Rotunno (1986), Snow and Lund (1989), and Fiedler (1994). We seek to compare the SCV velocity and pressure deficit profiles with those of the RV. The SCV model qualitatively describes a two-celled columnar vortex in which no tangential velocity exists within the core and a potential vortex occurs outside the core. In an SCV, the inner and outer velocity profiles are modified, but the profile's sharply peaked at remains unchanged. The SCV tangential velocity and associated pressure deficit distributions can easily be modeled using (1) and (10). At a given and and as , the inner velocity profiles evolve from the right half of the V-shaped profile (Fig. 6a) through the right half of the U-shaped profile to the backward L-shaped profile. This transition results in the SCV E. Simultaneously, the inner pressure deficit profiles change from the right half of the bowl-shaped profile to the flat profile inside , and the central pressure minimum rises (Fig. 6b). Additionally, the outer pressure deficit profiles remain unaffected beyond . As a consequence, (1) and (10), respectively, are simplified to

SCV:
eq5
With the selected values of and (Table 1), the radial profiles of and corresponding to the SCVs F, G, and H are depicted in Figs. 6c and 6d, respectively. Equation (16) shows that the inner core is stagnant and the pressure deficit profiles inside the radius of the maximum tangential velocity are flat, owing to the absence of tangential motion inside . Note that the and profiles of the SCV G are identical to those of Fiedler (1994, his Fig. 1; (1)].
Fig. 6.
Fig. 6.

Radial distributions of (a) cyclostrophic wind () and (b) corresponding pressure deficit () as a function of for the non-Rankine vortices C (blue curve), D (red curve), and E (green curve) when is fixed. Radial distributions of (c) and (d) as a function of for the non-Rankine vortices F (blue curve), G (red curve), and H (green curve) as . Note that the green curve superimposes onto the blue and red curves. The profiles are normalized. Normalized radial distance is represented by .

Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-13-00041.1

As decreases (from non-RV F to H in Table 1), the outer velocity profiles decay increasingly with (Fig. 6c) for an infinitely large value of . Concurrently, these decaying profiles cause the pressure deficit profiles, particularly the flat ones inside , to rise significantly. For a given tangential velocity maximum in Table 1, the SCV would have times the central pressure deficit of the RV (Fiedler 1994).

6. Case studies

We present two illustrative examples in which the parametric model is fitted to a tangential velocity profile. One is derived from an idealized numerical simulation of a tornado-like vortex, and the other is derived from high-resolution Doppler radar data collected in a real tornado. We seek to examine the parametric profile's realism by assessing how well the suggested WW model, coupled with the cyclostrophic balance assumption, is able to produce the deduced profiles of pressure deficit from (i) the numerical tangential wind output and (ii) Doppler radar–derived tangential wind output. The Levenberg–Marquardt (LM) (Levenberg 1944; Marquardt 1963) optimization method is employed, which is a standard technique used to solve unconstrained nonlinear least squares problems for curve-fitting applications. The algorithm for implementing the LM method was described in Press et al. (1992).

a. Numerical modeling of tornado-like vortex

We used output from the Fiedler (1994) numerical simulation of tornado-like vortex as “truth.” Radial distributions of nondimensional, gridded radial velocity (), tangential velocity (), vertical velocity () and pressure variable () data are presented in Fig. 7. Using numerical data provided by Fiedler (1994), Fig. 7 shows the same data as in Fig. 12a of Rotunno (2013).

Fig. 7.
Fig. 7.

Radial distributions of radial velocity (), tangential velocity (), vertical velocity (), and pressure variable () derived from the Fiedler (1994) numerical simulation (green curves). Black curve represents the radial distribution of Rankine cyclostrophic pressure variable () deduced from the parametrically constructed Rankine tangential velocity (). Red-circle curve represents the radial distribution of fitted WW tangential velocity (); black-dot curve represents the radial distribution of cyclostrophic pressure variable () deduced from . The profiles are nondimensional. A horizontal dashed line separating positive values from negative values is indicated. Indicated are , , , , , RMS, CORR, and (see text). (Data courtesy of B. Fiedler of University of Oklahoma.)

Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-13-00041.1

The numerical wind–pressure balance is nearly cyclostrophic, as shown in Fig. 7. To explain why this balance occurs, we adopted the approach of Lee and Wurman (2005) in which the inviscid radial pressure () equation in the cylindrical coordinates was partitioned into advection () and cyclostrophic () pressure gradients. The gradients are given by
e17
where is balanced with the advection terms (), and is balanced with the centrifugal term (). Since the radial velocity () is very small at the height of maximum tangential velocity (Fig. 7), the radial gradient of is negligibly small. As a result, is very small compared to . We can see that to a good approximation, the wind–pressure balance is virtually cyclostrophic (i.e., ~ ).

The LM algorithm required an initial guess for a model vector of five key parameters to be estimated for initializing a starting vector. Some judgment is needed to use the initial guesses. The guesses must be near enough to a solution to give convergence. An investigator may wonder what the initial values of , , and should be used. Figure 3 may be used as a guide for estimating the guesses. First, one searched the maximum value of a numerical (normalized) tangential velocity () by scanning the numerical profile completely. When was found after a complete scan, at which occurred was obtained. Second, it appears that the numerical profile of tangential velocity may follow a curve between the blue and red curves shown in Fig. 3g. If the inner profile near a vortex center (Fig. 7) indicates that it is linear, then one should choose to set to be 1.0. If the outer profile beyond the radius of maximum tangential velocity appears to decay quickly, then one may reasonably set to be . Last, since the profile apparently shows to be very broad in the annulus of the numerical tangential velocity maximum, then it is reasonable to set .

The fitted model parameters generated by the LM algorithm were , , , , and . Having the fitted parameters, we used (8) to integrate the pressure inward radially and to parametrically construct the radial distributions of the Rankine tangential velocity and pressure deficit (Fig. 7). Comparing the fitted profiles of and to the same output from the numerical simulation indicates good agreement with a low root-mean-square (RMS) error (0.008) and a high correlation (CORR) coefficient value (0.999).

For a given , the numerical non-RV vortex has a central pressure minimum of about one and a half () times that of the parametrically constructed Rankine vortex. Conversely, for a given pressure deficit, the numerical vortex would have a maximum tangential wind speed of approximately (~0.8) times that of the Rankine vortex (not shown but similar to Figs. 5c and 5d).

It is suggested that the WW parametric model retains much of the high-order accuracy inherent in the numerical model while avoiding the extensive computational requirements of this model. Comparisons between the parametric and numerical model results imply that the parametric model performs well.

b. Doppler observations of the Stockton, Kansas, tornado of 15 May 1999

Tanamachi et al. (2007) described the collection of high-resolution, W-band Doppler radar data in a tornado that occurred near Stockton, Kansas, on 15 May 1999. A Ground-Based Velocity Track Display (GBVTD; Lee et al. 1999) analysis of the data of the Stockton tornado produced the axisymmetric component of the azimuthal wind profile of the tornado. The radial distribution of the GBVTD-analyzed mean tangential wind at 2003:01 central daylight time (CDT; or 0103:01 UTC) 16 May 1999, when the tornado attained its peak intensity (Tanamachi et al. 2007), is presented (green dotted-line curve in Fig. 8a). The wind profile resembled that of a Burgers–Rott vortex. The maximum GBVTD-analyzed azimuthally averaged tangential wind, m s−1, occurred at the radius of about 75 m from the vortex center. The Stockton tornado received a rating of F2 on the Fujita scale (Fujita 1981) as a result of damage to farmstead outbuildings and fences (NCDC 1999). Herein, we use this azimuthally averaged tangential wind speed () profile to assess the WW tangential velocity () profile's realism by fitting the profile to the profile and then computing the pressure deficit distribution deduced from the cyclostrophic wind balance (8), using the fitted profile. At m, one zero data point (i.e., , as indicated by a green filled dot in Fig. 8a) connecting to m s−1 at m (green dashed line) was regarded with reduced confidence because relatively few Doppler radar data points (6) were used in the GBVTD analysis at this small radius (Carbone et al. 1985). A black X superimposed on the green filled dot at m indicates that the zero data was not used during the fitting procedure.

Fig. 8.
Fig. 8.

Radial distributions of (a) fitted WW tangential wind (, red curve), GBVTD-analyzed mean tangential wind (, green-dot curve) and Rankine cyclostrophic tangential wind (black curve) and (b) parametrically constructed Rankine pressure deficit (black curve) and non-Rankine pressure deficit (red curve) deduced from the fitted WW tangential wind shown in (a). In (a), a black superimposed on a green filled dot at m and a green dashed line connecting from m s−1 at m to m s−1 at m are discussed in text. Indicated are , , , , , RMS, CORR, and (see text). (Data courtesy of R. Tanamachi of NSSL.)

Citation: Journal of Atmospheric and Oceanic Technology 30, 12; 10.1175/JTECH-D-13-00041.1

After fitting our parametric model to the radial distribution of , the fitted model parameters generated by the LM algorithm are found to be = 46 m s−1, = 71 m, = 1.12, = −0.97, and = 0.66. Using these parameters, the fitted profile of the non-RV vortex was calculated, via (4), and is plotted in Fig. 8a. The non-RV profile was compared to the RV distribution, when the fitted parameters of , , , and were computed from (1) to parametrically construct the RV profile. Comparison of the fitted non-RV profile with the GBVTD-analyzed profile indicates good agreement with a low root-mean-square error (0.74) and a high correlation coefficient value (0.998). Next, the radial non-RV pressure deficit distribution (Fig. 8b) associated with the fitted tangential velocity data (Fig. 8a) was calculated via cyclostrophic balance in (8). The derived central pressure deficit of the non-RV was −40 hPa, which was much lower than the −23.7 hPa associated with the parametrically constructed RV (Fig. 8b). In situ measurements of pressure at the analysis height in the Stockton tornado were unavailable that would have needed to verify the deduced pressure deficit profile of the non-RV. Such in situ measurements (e.g., Karstens et al. 2010) are needed to substantiate the suggested model.

For a given , the GBVTD-analyzed non-RV vortex has a central pressure minimum times that of the parametrically constructed Rankine vortex. The value of is similar to that shown in Fig. 7.

We have demonstrated the application of our model to high-resolution Doppler velocity data collected by mobile Doppler radars within close range of an intercepted tornado. Such an application could conceivably be expanded to include a comparative range of observed pressure deficits for a wide variety of tornado sizes and intensities.

7. Summary and discussion

The Wood–White parametric tangential wind profile model coupled with the cyclostrophic balance assumption has been developed and offers a diagnostic tool for estimating and examining a radial profile of pressure deficit deduced from a theoretical tangential wind profile in an axisymmetric, cyclostrophic vortex. The sensitivity of the wind–pressure relationship between parametrically constructed Rankine and non-Rankine cyclostrophic vortices is examined comparatively. The experiments and case studies demonstrate that the parametric WW model may be capable of explaining significant fluctuations in tangential wind speeds and pressure deficits, owing to substantial flexibility of the model shape parameters (). The main conclusions of this study are as follows:

  1. The shape parameters () are shown to play a vital role in modulating various portions of the radial profile of the azimuthal (tangential) velocity. In our parametric formulation, the profile is defined by (i) the growth parameter , which predominantly dictates the inner profile near the vortex center; (ii) the decay parameter , which primarily governs the outer profile beyond the radius of the tangential wind maximum; and (iii) the size parameter , which mainly determines the radial width of the profile spanning the maximum. Although the first two free parameters ( and ) for the non-RV are the same as for the RV, the definitions of , , and for the non-RV are more intuitive than those for the RV because the choice of the three free parameters appears to be much easier to understand than those used for the RV. The model may relate the choice to the properties of real-world wind profiles well.
  2. The pressure deficit in the vortex core is shown to be sensitive to the shapes of the radial profiles of the tangential velocity. For a given tangential velocity maximum, a decrease (increase) in narrows (broadens) the tangential wind profile straddling the maximum by decreasing (increasing) the radial width of the tangential velocity at a given velocity level. It also raises (lowers) the corresponding pressure deficit profile and hence fills (deepens) the central pressure minimum. Increasing (decreasing) the parameter changes the inner radial profile of the tangential velocity in such a way as to not only raise (lower) the pressure deficit profile but also increase (decrease) the central width of the profile inside the radius of the tangential velocity peak. As tends to infinity, the central pressure deficit profile approaches flatness, owing to the absence of tangential motion inside the core radius, as seen in the illustrative example of the SCV model. Increasing (decreasing) the parameter raises (lowers) the radial wind profile beyond the radius of the maximum tangential velocity and hence lowers (raises) the corresponding pressure deficit profile. For given values of , , , and , the central pressure deficit at remains unchanged, regardless of the vortex core size . Contrary to , an increase in vortex tangential velocity maximum , as expected, is strongly correlated with the vortex intensity and pressure fall when the free parameters remain unchanged.
  3. Regardless of the choice of the free parameters, all integrals of the parametrically constructed pressure deficit distributions of the non-RVs are shown to be convergent as tends to infinity.
  4. The parametrically constructed non-RVs are shown to have a larger central pressure deficit compared to the RV. For given values of in the non-RV, the central pressure deficit would have (1) times the central pressure deficit of the RV when is large. Conversely, for a given central pressure deficit and a large value of , the non-RV would have ( 1) times the tangential wind speed maximum of the RV. As , , and , the central pressure deficits and maximum tangential wind speeds of the non-RV coincide with those of the RV, respectively.

We have shown that the sensitivity of pressure deficit in the vortex core is related to the choice of the free parameters that control the shapes of the radial profiles of the tangential velocity. However, it is worthwhile to examine the finer properties of the dependencies. Modeling and data assimilation are iterative processes. Understanding the finer properties of the parameter-to-state dependency aids in developing intuition that is in turn valuable in the modeling process. Understanding the parameters-to-state behavior facilitates us to specify more precisely admissible sets of free parameters for use in data assimilation procedures. In the context of estimation, this represents a priori knowledge that is important in efficient estimation. From a probabilistic perspective, this analysis is an important step in specifying the a priori information in a Bayesian approach. This information may enable the use of the shape parameters to structure appropriate wind and pressure profiles of rotating systems, particularly in numerical or theoretical vortex models.

What types of wind and pressure measurements would be needed to verify the WW model in real atmospheric vortices like tornadoes? Both in situ and radar observations have inherent advantages and limitations. Karstens et al. (2010) summarized historical near-ground tornado measurements as shown in their Table 1. The measurements should be used with some caution because the shapes of the radial profiles of tangential winds and the approximate radial distances at which the measured wind speed maxima occurred were either approximated or unknown. The most comparable tornadoes to the Stockton tornado of 15 May 1999 (Fig. 8) were probably the ones on 15 May 2003 (Stratford, Texas); 22 April 2007 (Tulia, Texas); 23 May 2008 (Quinter, Kansas); and 30 May 2008 (Tipton and Beloit, Kansas). In the three May 2008 tornadoes (last three lines of Table 1 of Karstens et al.), the measured azimuthal wind maxima were 40–46 m s−1, while the recorded pressure deficits were 13–15 hPa. If the suggested WW model were to apply to these tornado data, the model would overestimate the pressure deficit by a factor of 2–3 (R. Tanamachi 2013, personal communication) when compared to Fig. 8. It is likely that the cyclostrophic imbalance in the surface layer will have an effect on the resulting shapes of in situ pressure measurements (C. Karstens 2013, personal communication). Interestingly, Lewellen et al. (1997) noted that if a successful high-frequency recording of a high-resolution simulated tornado were obtained while it passed directly over the surface instrument, it would underestimate the minimum instantaneous pressures that occurred at an altitude of ~30 m by an average of 80%. In contrast, a pressure deficit of 194 hPa and peak winds of 50 m s−1 were measured in an EF2 tornado in Tulia, Texas (Blair et al. 2008). However, as indicated in Karstens et al. (2010), this pressure deficit measurement may be erroneous. In this instance, the WW model would underestimate the central pressure deficit by a factor of 3–4. Measured by Wurman and Samaras (2004), a maximum wind peak of 53 m s−1 and a pressure deficit of 41 hPa in the Stratford, Texas, tornado of 15 May 2003 [see Table 1 of Karstens et al. (2010)] were closely compared to the Doppler-derived tangential wind peak and deduced pressure deficit estimates found in the Stockton tornado (Fig. 8). It is suggested that the WW model did a good job of calculating the pressure deficit from the fitted non-RV tangential wind profile.

In view of the above, there are a number of possible explanations for the apparent discrepancies, however. First, in the Stockton case, we did not take the effects of surface friction into account, and the Stockton analyses were conducted well above the boundary layer (at 150–250 m AGL). Second, although Karstens et al. (2010) document some intercepts as “near direct” transects of a tornado by the instruments, it is not whether the core of the tornado (i.e., the region inside the radius of maximum wind) passed precisely over the instruments. Displacement of the instruments only a few meters to the right or left of the vortex center would have resulted in a substantial reduction in the observed winds and pressure deficit, as evidenced by a strong gradient of damage to intercept vehicles (R. Tanamachi 2013, personal communication). Third, in one of the above-mentioned instances (the Tipton, Kansas, tornado of 30 May 2008), the wind measurements were collected more than 200 m away from the pressure measurements. Fourth, the sampling frequency of the mobile mesonet instruments was 0.5 or 1 Hz (Karstens et al. 2010), while the tornado took only ~10–30 s to pass over the instruments. It is possible that the true minimum pressure and maximum winds in the vortex were not captured, even with 5–30 samples collected in the vortex core, as suggested by the 194-hPa pressure deficit measured in the Tulia, Texas, tornado (Blair et al. 2008). Increasing the frequency of sampling perhaps to 10 or more samples per second [e.g., Hardened In Situ Tornado Pressure Recorder (HITPR) probes described by Karstens et al. 2010] during intercept would make it more likely that these extrema are captured.

8. Future work

Our parametric wind–pressure relationship in the WW parametric model is part of our ongoing research to provide a diagnostic tool for analyzing and interpreting the observed tangential wind and deduced pressure deficit structures in tornadoes, tornado cyclones, and mesocyclones. Our potential future application includes high-resolution Doppler radar data collected by mobile radars within close range of an intercepted tornado. Such an application allows us to conceivably expand our study to include a comparative range of observed pressure deficits for a wide variety of vortex sizes and intensities. It is important to acknowledge that the pressure profiles presented in this study are likely more representative of pressure profiles a few tens of meters above the surface, away from the surface layer and away from the effects of friction, thus allowing cyclostrophic balance to have more validity. Additionally, a comprehensive comparison of pressure deficit structures with a detailed damage survey of a tornado track is highly recommended. In future research, efforts should be made to measure shapes of the radial profiles of tangential winds and pressure deficits, and the estimated radial distances at which the wind speed maximum occurs, in order to advance our understandings of tornado dynamics and improve comparisons made to laboratory and numerical simulations.

Acknowledgments

The authors thank Lou Wicker, Pam Heinselman, and Qin Xu of NSSL, and Brian Fiedler, Alan Shapiro, and John Snow of the University of Oklahoma for reading and making useful suggestions to the earlier version of the paper. The authors also thank Brian Fiedler for providing us his numerical data (given in Fig. 7) for our testing and verification of the simplified WW parametric model. Furthermore, the authors are grateful to Robin Tanamachi of NSSL for providing her GBVTD-analyzed mean tangential velocity data for our calculation of the fitted radial profile of tangential velocity and the radial profile of deduced pressure deficit given in Fig. 8. The authors appreciate the constructive comments provided by Robin Tanamachi and Christopher Karstens of NSSL and the three anonymous reviewers, thus leading to an improved manuscript.

APPENDIX

Convergent Solution of Integral in (9)

The purpose of this appendix is to show that the integral in (9) is convergent to zero as . Let us begin with outward radial integration of (8), given by
ea1
Since , we show in (A1) that
ea2
ea3
Hence,
ea4
which is thus obtained as (9). The right-hand side of (A4) is always convergent to zero as , no matter what the choice of the free parameters is.

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1

The Rankine combined vortex is characterized by a core of solid-body rotation [wherein tangential velocity () ~ radius ()], surrounded by an outer region of potential flow (wherein ~ ). For convenience, the word “combined” may be dropped.

2

The “non-Rankine vortex” may be defined as a viscous vortex that exhibits a smooth transition between solid-body rotation and potential flow that encompasses the annular zone of the velocity maximum, resembling the Burgers–Rott (Burgers 1948; Rott 1958) tangential velocity profile.

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