1. Introduction
Research and development on phased-array radar (PAR) technology performed at the National Oceanic and Atmospheric Administration’s (NOAA) National Severe Storms Laboratory (NSSL) since 2003 has demonstrated great potential for weather radar applications, research, and improving warnings for severe weather (e.g., Heinselman et al. 2015; NOAA 2020). NSSL coordinated with several private sector, government, and university partners to develop the National Weather Radar Testbed (NWRT) in Norman, Oklahoma, which featured a modified single polarization, S-band SPY-1A PAR antenna. A SPY-1A system was originally tested on a U.S. Navy destroyer (Maese et al. 2001; Robinson 2002) to demonstrate the advantage of PAR technology for weather surveillance (Zrnić et al. 2007). The SPY-1A PAR antenna at NSSL was installed on a rotatable pedestal so that the array could be repositioned as storms move but was otherwise designed for stationary operations with a 90° field of view. Despite using a “passive” analog beamforming array (Palmer et al. 2022) and relatively small aperture (with an antenna beamwidth of 1.5°), the SPY-1A’s faster (∼1 min) volumetric updates relative to the current operational Weather Surveillance Radar-1988 Doppler (WSR-88D) system (5–6 min) demonstrated the potential for improved tornado and severe thunderstorm warning performance (e.g., Heinselman et al. 2015; Wilson et al. 2017) and data assimilation capabilities for high-resolution convective-allowing models (Yussouf and Stensrud 2010; Stratman et al. 2020).
In addition to being a promising replacement option for the operational WSR-88D, rapid volumetric updates provided by PAR also have potential for improving our understanding of storm evolution (Heinselman et al. 2008; Heinselman and Torres 2011; Kollias et al. 2022). Toward that purpose, this study’s primary objective is to investigate circulation and areal contraction rate analyses (developed by Davies-Jones et al. 2020; hereafter DJWR) using rapid update SPY-1A (hereafter NWRT PAR) data from a tornadic supercell thunderstorm near El Reno, Oklahoma, on 31 May 2013. This supercell produced an EF3 tornado responsible for eight fatalities and 26 injuries along a pathlength of 26 km with an unprecedented maximum path width of 4.2 km, the widest known tornado on record (NOAA/NWS 2013). Although polarimetric volume data were not available from the NWRT PAR1 in 2013, the rapid volumetric updates still delineate abrupt changes in vortex size and strength and kinematic parameters that have not been observed in previous studies.
Using a WSR-88D emulator and a case study of the 24 May 1973 Union City, Oklahoma, tornado (Lemon et al. 1978; Brown et al. 1978), DJWR advocated using the detection and measurement of circulation around and areal contraction rate of a circle as a universal method for providing advanced warnings of tornadoes. Circulation, by definition, is the line integral around a closed curve of velocity tangential to the curve, and the areal contraction rate is the line integral around the curve of velocity inward normal to the curve. Mathematically, the circulation Γ around a circle enclosing a single-vortex tornado is the same as that around a multivortex tornado provided that the circulations of the individual vortices sum to Γ. To illustrate how the circulation method works, imagine an array of wind sensors arranged in a large circle on the outskirts of a town. The data are relayed to warning meteorologists, who compute a large circulation around the circle. They realize that there is at least one cyclonic vortex somewhere in the town. They do not know if there is a broad weak vortex that could contract into a strong tornado, if a strong concentrated tornado is already present, or if there is tornado with multiple vortices, but the potential for a violent tornado is apparent and the threat could increase if there is convergent flow to further concentrate the vorticity.
Mitchell et al. (1998), Stumpf et al. (1998), Mahalik et al. (2019), and Sandmæl et al. (2023) used azimuthal gradients in Doppler velocities between the two peaks in a characteristic velocity couplet to detect and measure the strength of convergent vortices at low altitudes, but DJWR concluded that circulation and areal contraction rate may be more useful than azimuthal gradients. Closely related parameters are the rotational Doppler velocity Vrot = ΔVD/2, the azimuthal Doppler shear
By producing a simple flow model of the Union City tornado, moving it to different locations, and computing its radar signature using a virtual WSR-88D, DJWR demonstrated that circulation and areal contraction rate are better radar-based indicators of tornado threat2 than vorticity and convergence, respectively. This improvement occurs because both circulation and areal contraction rate are i) additive; ii) less scale dependent; iii) more tolerant of noisy Doppler velocity data; and iv) relatively insensitive to range, radar beamwidth, and the location of a tornado within a sampling volume compared to quantities that rely on azimuthal gradients in Doppler velocities. Furthermore, DJWR emphasized that circulation is a significant precursor parameter for anticipating tornado formation in a convergent wind field because circulation around a circle of a few kilometer radius precedes tornado formation and, outside the tornado, is large and slowly varying with distance. In contrast, the vorticity of a contracting vortex is concentrated in the core and becomes large only as the core diameter becomes small.
To illustrate the above effects, Davies-Jones and Wood (2006) used a virtual WSR-88D to obtain radar signatures of unsteady versions of the inviscid Rankine-combined and the constant-viscosity Burgers–Rott vortices. They performed experiments with different initial values of the constant eddy viscosity and uniform convergence to obtain tornadoes of different intensities, sizes, and formation and decay rates. Circulations of the vortex signatures gave good estimates of the circulations of the simulated tornadoes and mesocyclones with relatively low sensitivity to range, effective beamwidth, and stage of evolution. Thus, detection of a rotation signature with high circulation and convergence values may provide an early indication of tornadogenesis. We should note however that once a tornado has formed, large debris centrifuged outward can result in a divergent radar signature even when the air motion is convergent (e.g., Dowell et al. 2005). In a dual-polarization system, a tornadic debris signature (TDS; e.g., Ryzhkov et al. 2005; Kumjian and Ryzhkov 2008; Snyder and Ryzhkov 2015) should alert the radar meteorologist to the presence of significant debris to help resolve this paradox.
Because a Doppler radar measures only the component of 3D flow in the radar viewing direction, we assume herein that velocity normal to the radar beam is zero. This allows calculation of the so-called “Doppler circulation” (after DJWR), “Doppler mean vorticity,” “Doppler areal contraction rate,” and “Doppler mean divergence” that is detailed further in section 2. Circulation values for various vortices are documented in section 2 along with describing a bilinear interpolation method to calculate the Doppler velocity at points on the circles and a new technique to find the center of circulation. In section 3, given an assumption of axisymmetry, we list formulas for Doppler circulation and areal contraction rate, Doppler mean radial and tangential velocity components, Doppler total mean wind speed, and Doppler mean inflow angle. Section 4 briefly describes NWRT PAR data output and data processing. We use a variational method (Wood et al. 2021) that fills in missing Doppler velocity data in data voids. Section 5 presents results and interpretations of the NWRT PAR data by providing i) an overview of the El Reno, Oklahoma, tornadic supercell of 31 May 2013; ii) radius–height distributions of various circle-based Doppler kinematic quantities; iii) evolution of various vortex-based Doppler kinematic quantities over PAR scan time; and iv) verification using rapid scanning, X-band, polarimetric (RaXPol)3 observations as ground truth; and v) Doppler circulation and contraction rate analyses of a smaller, weaker Yukon, Oklahoma, tornado that happened several minutes after the El Reno tornado dissipated. In section 6, we discuss i) interpretations of the El Reno tornado in terms of known vortex dynamics, ii) comparison of measures of rotation and benefits of using circulation, and iii) limitations of measures of rotation. Conclusions of the study are presented in section 7.
2. Kinematic quantities of the Doppler velocity field
DJWR calculated circulation around and contraction rate of several specified circles quickly and easily from Doppler radar data. The measurable radar coordinates (r, α, β) are slant range r, azimuthal angle β measured clockwise from due north, and the elevation angle α of a ray. During a volume scan of the Doppler radar, the Doppler velocity data VD are recorded on a three-dimensional grid. The grid points in a surface of constant elevation angle are located at r = ri, β = βk, where subscripts i and k are, respectively, used to label the discrete ranges and azimuths, and ri = iΔr. The ik grid cell is defined as the area in the surface enclosed by successive slant-range circles ri and ri+1 and successive radials βk and βk+1. The data are collected sequentially along radials at constant range-gate spacing Δr = ri+1 − ri, then at successive azimuth angles, and finally at different elevation angles specified by the particular volume coverage pattern (VCP) in use. Note that the azimuthal spacing Δβ = βk+1 − βk in the azimuthal direction varies for the phased-array radar. The subscript i is the index in the range direction and k is the index in the azimuthal direction.
a. Doppler circulation and mean vorticity
The Doppler circulation of a tornadic vortex signature (TVS; Brown et al. 1978; Brown 1998) at the radius rmax of maximum tangential wind υmax is πrmaxυmax. Figure 1 contains graphs of Doppler circulation as functions of υmax for various rmax. Wide strong vortices have much larger circulations than do narrow weak ones.
Doppler circulation ΓD(rmax) at the radius rmax of maximum tangential wind υmax as a function of υmax for various rmax. For a Rankine combined vortex, ΓD(r) at r = ρ is ΓD(rmax) for ρ ≥ rmax and
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
Some actual (not Doppler) circulation values for mesocyclones, tornadoes, and waterspouts have been documented extensively—values and references from which they are taken are given in Table 1. Listed tornado cases are those for which we could compute the circulation. There were a few cases in the 1950s–1970s, most of which were lower-circulation tornadoes. In addition, more recent papers presenting mobile radar observations of tornadoes usually report ΔVD instead of circulation. There is also uncertainty as to how representative the listed circulation values relate to the F- or EF-scale rating assigned to the tornadoes given how the intensity and size of tornadoes vary during their life cycle versus how the maximum damage rating seen during a tornado’s life is determined, which may not have occurred when the estimated circulation was provided.
The actual circulation values are given for various mesocyclones, tornadoes, and waterspouts.
The 1973 Union City, Oklahoma, violent tornado had a photogrammetric and radar-derived circulation of 1 × 105 m2 s−1 (DJWR). This tornado had a TVS. Since circulation is constant outside the core of a Rankine combined vortex (i.e., for r ≥ rmax), the circulation can be estimated from Γ = πVrotD where Vrot is the rotational velocity and D is the distance between velocity peaks. For Union City, these yield Γ = π (28.5 m s−1) (890 m) = 8 × 104 m2 s−1, which is slightly less than the circulation of 1 × 105 m2 s−1 estimated by the DJWR method. Also, in the DJWR study, analysis of data from the KOUN (Norman, Oklahoma) WSR-88D revealed that the circulation of the El Reno tornado, respectively, was 3 × 105 and 3.4 × 105 m2 s−1 at 2303 and 2311 UTC.
b. Doppler areal contraction rate and mean divergence
Figure 2 shows how the Doppler contraction rate varies with circle radius ρ, in idealized flows with uniform convergence. The different graphs are for convergences of 5, 10, and 15 × 10−3 s−1. Since
Doppler areal contraction rate
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
Circle radius, estimated contraction rate of the circle, and associated estimated mean convergence for three tornadoes (the estimated values are twice the corresponding Doppler ones).
c. Interpolation of Doppler velocity to circle points
We used bilinear interpolation to calculate the Doppler velocity at points on the circles. Figure 3a depicts the interpolation points used around Cα. We used the formulas in DJWR for calculating these points. As in DJWR, we interpolated Doppler velocity data bilinearly to these points prior to computing Doppler circulation around Cα and Doppler areal contraction rate of Cα. The circle points are spaced equally in slant range because this is optimal for measuring (Γα)D. The points on the circle are farther apart where the tangential velocity of a vortex is perpendicular to the radar viewing direction, and they are closer together where the tangential velocity is well observed. Although this arrangement is not optimal for areal contraction rate, the results for Doppler areal contraction rate [denoted by
(a) Red dotted curves are concentric circles of radii ρ1, ρ2, and ρ3 centered on black × at radar coordinates (r0, α, β0). The total number of computation points (red dots) at each circle perimeter is 120 after DJWR. Each gray rectangle depicts a grid cell. (b) An enlargement of a blue grid cell with corner points (ri, βk), (ri, βk+1), (ri+1, βk+1), and (ri+1, βk) is shown. Black arrows away from and toward the radar at the right-side and left-side corner points, respectively, exemplify different positive and negative Doppler velocities. The circulation value at the blue grid cell center is computed from the observed Doppler velocities at the corner points and [(8)].
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
We did not use circles with radii less than 1 km because there were too few data points in their vicinity to have confidence in the computed quantities (Carbone et al. 1985; Tanamachi et al. 2007). Circles with radii greater than 4 km (and sometimes smaller circles than this) are problematic because parts of their perimeters are often outside the echo and thus there may be insufficient nearby data points to accurately interpolate.
d. Center-finding technique for Doppler vortex signature center
For a large tornado, the circulation center (CC) is precisely defined only if the flow is axisymmetric. Due to asymmetries, pinpointing an exact axis of rotation is impossible even using a hypothetical radar with infinitely fine resolution. Owing to different viewing angles, ranges, and resolution, the CC obtained from one radar will differ from that found from another. We now describe a method to determine a circulation center objectively (see appendix A).
DJWR generated plan position indicator displays of mesocyclones with the cells outlined and the circulation values of each cell displayed. For example, the fields of Doppler velocities VD and corresponding Doppler cell circulations d(Γα)D for the El Reno tornadic storm are shown in Fig. 4. The plots here are similar to Fig. 3 of DJWR. We can easily calculate the observed circulation around a group of cells because circulations (and line integrals in general) are additive (e.g., Petterssen 1956, p. 127). The circulation around the boundary of any area that is partitioned into subareas is equal to the sum of the circulations around the perimeter of each subarea because the line integrals along interior sides (ones that are shared by two of the subareas) cancel (Fig. 3b).
NWRT PAR scans at (a) Doppler velocity field (VD, m s−1) and (b) Doppler cell circulation field (dΓD, 103 m2 s−1) at an elevation angle of 0.5° at 2324:07 UTC for the El Reno tornadic storm on 31 May 2013. The CC (r0, α, β0) calculated at the range of 48.1 km and the azimuth of 303.5° is indicated by a black star. ND stands for no data. In (a), two yellow circles of specified radii (from 1 to 4 km) are concentric with the CC (black star). A distance scale is included.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
Fortunately, because of the aforementioned properties of circulation, Doppler circulation around a circle is insensitive to the precise location of the CC when there are several grid cells within the circle. For circles nearly the size of the grid cells, we can use the procedure in appendix A to locate the CC more accurately within the grid cell with the maximum circulation. This method utilizes the circulations of the one with the maximum and of its four nearest neighbors. Interpolation of grid cell circulation data to refine the circulation center position is generally unnecessary based on testing the method on the Union City TVS (Fig. 4 in DJWR).
3. Other kinematic quantities associated with Doppler circulation and areal contraction rate
The
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
As a simple illustration, Fig. 6 shows the absolute value of inflow velocity as a function of ρ in flows for three different values of uniform convergence magnitude. For a given convergence, increasing ρ is associated with increasing inflow velocity; for a given ρ, increasing convergence is associated with increasing inflow velocity. From (10), the inflow velocity is the actual areal contraction rate divided by 2πρ. To use three specific examples, in the Union City tornado, the inflow velocity is 8 m s−1 at 3 km (DJWR); for the Dodge City tornado, it is 5 m s−1 at 1.75 km (Wood et al. 2021), and for the El Reno tornado at 2311 UTC, it is a large 24 m s−1 at 5 km (Fig. 21 in DJWR). From Figs. 1, 2, 6, and 10–19 below, the El Reno case is clearly an outlier regarding circulation magnitude and vortex size.
Inflow velocity as a function of the circle radius ρ for theoretical flows with constant convergence C. The graphs also apply to divergent flow if outflow replaces inflow and −C replaces C. The brackets represent the absolute value signs.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
There may be significant differential motion between the radar targets and airflow. We estimate the outward terminal velocity VC of a target with fall velocity VF due to centrifuging as follows. The ratio of centrifugal to gravitational acceleration g at a radial distance ς from the axis of rotation is
4. Data and data processing
On 31 May 2013, the NWRT PAR sampled the supercell that produced an EF3 tornado near El Reno, Oklahoma (e.g., Wurman et al. 2014; Bluestein et al. 2015; Kuster et al. 2015). The application of the Doppler circulation and Doppler areal contraction rate calculations to the rapid scan NWRT PAR data yielded a set of analyses for all 168 sector scans. The scans contained surfaces of four selected elevation angles (0.5°, 0.9°, 1.3°, and 1.8°) between 2258:03 and 2346:46 UTC. The vertical offsets of the Doppler velocity signature centers between two adjacent elevation angles vary from 0.4 to 0.5 km at the farthest range from the NWRT PAR (about 63 km at 2258 UTC) to 0.3–0.4 km at the nearest range (about 45 km at 2347 UTC) (not shown). The minimum heights at which the actual Doppler velocity and reflectivity data were collected are 0.5 km ARL at the farthest range and about 0.35 km ARL at the nearest range (assuming standard atmospheric propagation). The volume update time was 1.18 min for a 90° sector. National Center for Atmospheric Research (NCAR) Solo 3 software (Oye et al. 1995) was used for manually dealiasing the Doppler velocity field and removing poor-quality data. We extracted data from a 20 km (range) × 20 km (azimuth) sector for all 168 sector scans using the Warning Decision Support System–Integrated Information (WDSS-II; Lakshmanan et al. 2007) to perform Doppler circulation and contraction rate analyses. The sector was approximately centered on the vortex signature center at each elevation angle. Due to poor data quality, primarily resulting from the NWRT PAR antenna’s higher sidelobes relative to the WSR-88D, some data were removed in the Doppler velocity fields prior to 2258 UTC. Because some of the PAR data were missing during the first half of the tornado’s life cycle, we refer to the DJWR analyses of data that were collected at 2303 and 2311 UTC by the KOUN WSR-88D located very close to the NWRT PAR. At these times, the KOUN radar was about 60 km from the tornado. The PAR perspective does not tell us when the tornado formed, so we refer to the Bluestein et al. (2019) field observations to establish the time of tornado formation since the PAR data are incomplete at that time.
Some preprocessing of the NWRT PAR data was required to facilitate the calculation of the Doppler circulation and areal contraction rate analyses. Doppler velocity data may be missing at “grid points” within an important region of a storm (e.g., near a hook echo) because of low return power, low signal-to-noise ratio, sidelobe contamination, ground clutter associated with normal and anomalous propagation, and missing radials associated with partial or total beam blockage. We use a variational technique (Wood et al. 2021) to fill in data voids with values that minimize the average gradient of Doppler velocity to avoid the creation of spurious signatures (Fig. 7). We choose a sector (defined by beginning and ending ranges and azimuths as shown by yellow rectangles in Fig. 7) and flagged the points with a missing data parameter (e.g., 999 in Fig. 7b). At grid points with data, the Doppler velocity is the observed value. At interior grid points where data are missing, the Doppler velocity is related to that at neighboring points according to an elliptic Euler–Lagrange equation. The associated second-order-centered finite-difference equation is a five-point formula that relates the value at the missing data point to the values at adjacent points along the radial and range circle through the point. At boundary points with missing data, closure is achieved by applying the natural boundary condition, namely, that the normal gradient of Doppler velocity is zero. This condition relates the Doppler velocity at the exterior point in the five-point scheme to the Doppler velocity at the interior point. At corner points with missing data, the five-point scheme has two exterior points, so closure requires application of the boundary condition at each of the intersecting boundaries. These relationships yield explicit formulae for the values at all isolated missing data points (i.e., those whose four adjacent neighbors in the five-point scheme all have good data). Data are filled in at these points to speed up the computations, resulting in an “updated” system of finite-difference equations. Finally, all the larger data-void regions are filled simultaneously by solving this entire system using a line successive relaxation method (Lapidus and Pinder 1982, p. 418). This reduces the problem to a tridiagonal system of simultaneous linear algebraic equations (some of which are in one variable only). This system is solved by a standard technique (Press et al. 1996, p. 40). The method always converges to a result regardless of how much data are missing and takes little computational time.
(a) NWRT PAR scans of ground-relative, mean Doppler velocity (VD, m s−1) of the El Reno tornado as collected at the 1.8°-elevation angle at 2258:03 UTC 31 May 2013. Range (m), azimuths (°), and height (m) from the NWRT PAR are labeled. ND (black pixel) stands for not determined. The yellow rectangle surrounds missing data referenced in the discussion of the variational gap-filling method. (b) The field of actual, mean Doppler velocities (m s−1) at 1.8° elevation angle enclosed by the yellow rectangle in (a). Missing data are labeled 999. (c) As in (b), but after the variational method has filled the missing data.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
Wood et al. (2021) studied the effects of randomly voiding up to 20% of the data collected by KOUN in the El Reno storm and then filling data back in using the algorithm. They concluded that the circulation and contraction rate were minimally affected by this experiment.
5. Results and interpretations of the NWRT PAR data
a. Overview of the El Reno, Oklahoma, tornadic supercell of 31 May 2013
Tornadogenesis, tornado evolution, photogrammetric and polarimetric analyses, aerial damage surveys, mobile research and fixed-site radars, lightning detection networks, and societal impacts of the El Reno tornadic supercell have been extensively documented by Snyder and Bluestein (2014), Wurman et al. (2014), Wakimoto et al. (2015, 2016), Bluestein et al. (2015, 2016, 2018, 2019), and Hatzis and Klockow-McClain (2022), respectively. Kuster et al. (2015) showed that, to a warning forecaster, the ∼1-min NWRT PAR data provided advantages compared to ∼5-min KTLX (Oklahoma City, Oklahoma) WSR-88D data. Specifically, the warning forecaster indicated that the NWRT PAR data better depicted i) occurrences of low- and midlevel mesocyclones and associated changes in the extent and magnitude of storm inflow, ii) rapid intensification of the low-level mesocyclone (LLM) and embedded TVS preceding the EF0-rated Calumet and EF3-rated El Reno tornadoes, and iii) abrupt, short-term changes in the motion of the tornadic circulation leading to more precision in determining its location.
Velocity, reflectivity, and spectrum width data that were likely associated with the NWRT PAR’s sidelobes mentioned previously in section 4 were manually removed for our analysis times (2258:03–2346:46 UTC). Sidelobe echoes in areas with low signal-to-noise ratio (SNR) were removed, whereas much of the data within the storm in areas with higher reflectivity were preserved. Removing some data, however, did seem to be detrimental to our Doppler circulation and areal contraction rate analyses. Because some of the PAR data were missing during the first half of the tornado’s life cycle, we followed the DJWR analyses of data that were collected at 2303 and 2311 UTC by the KOUN WSR-88D located very close to the NWRT PAR and approximately 60 km to the tornado. Unfortunately, the PAR perspective does not tell us when the tornado formed due to this missing data, so we refer to the Bluestein et al. (2019) RaXPol field observations (discussed in greater detail in section 5d) to establish the time of tornado formation.
At 2300 UTC, RaXPol data show that a large, broad LLM is present, but the actual circulation that ends up making the tornado forms around 2302 UTC as a new curl in reflectivity and Doppler velocity couplet on the easternmost portion of the broad LLM (Bluestein et al. 2019). The El Reno tornado formed sometime between 2302 and 2304 UTC. Bluestein et al. (2019) and Seimon et al. (2016) report a condensation funnel in contact with the ground around 2302 UTC away from the CC, but a deeper vortex was not observed on radar until 2304 UTC. Thus, it appears that the tornado may have formed from the ground up (Bluestein et al. 2019) and joined with the broader vortex aloft as in the Davies-Jones (2008) simulation. Based on their RaXPol analyses, Wakimoto et al. (2016) reported that at 2304:41 UTC, there was a pronounced TDS in the El Reno storm.
The El Reno vortex was on the scale of a LLM and had the intensity of a strong tornado. We decided to call the overall vortex at and after 2304 UTC a tornado with resolvable multiple vortices at times. Circulation analysis (DJWR) identifies only one circulation about the CC, so we believe that the contraction of the LLM was associated or coincident with the development of the tornado.
Figure 8 presents photographs of the El Reno tornado at 2300, 2304, 2307, 2309, 2311, and 2325 UTC 31 May 2013 (Bluestein et al. 2015). At ∼2300 UTC, a wall cloud documented by Wakimoto et al. (2016) was observed prior to tornadogenesis (Fig. 8a). By 2304 UTC, the tornado had formed and transitioned from a primarily single-vortex structure (not shown) to a multiple-vortex structure (Fig. 8b; Wakimoto et al. 2016). A video4 taken at the site of a RaXPol deployment shows that the elevated condensation funnel of the subvortex on the right in Fig. 8b extended to the ground slightly prior to 2304 UTC. By ∼2309 UTC, the tornado under the wall cloud had transitioned back to a single-vortex structure (Fig. 8d) in appearance. By ∼2311 UTC, a clear slot in close proximity to the tornado (Fig. 8e) visually manifested subsiding air within the rear-flank downdraft (RFD) (Lemon and Doswell 1979; Davies-Jones 2008). The tornado grew considerably through ∼2325 UTC (Fig. 8f). Thereafter, additional photographs are unavailable because the tornado was obscured from view by curtains of rain. Generally, the larger the visual tornado, the greater the circulation [as indicated by the Ward (1972) tornado simulator and some photogrammetric studies, e.g., Golden and Purcell 1978]. Thus, the photographs suggest that the circulation is increasing with time (Fig. 8). Mobile radars and photography detected multiple vortices from 2318 to 2328 UTC (Bluestein et al. 2015, 2018; Snyder and Bluestein 2014; Wakimoto et al. 2016; Wurman et al. 2014), though the tornado may have had multiple vortices during other times as well.
Photographs of the El Reno, Oklahoma, tornado of 31 May 2013 at (a) 2300, (b) 2304, (c) 2307, (d) 2309, (e) 2311, and (f) 2325 UTC, west of El Reno. In (a), there is a wall cloud but no tornado. In (b) and (c), multiple vortices are clearly seen. In (e), a clear slot near the tornado is evident (photography courtesy of J. Snyder).
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
The tornado damage path is mapped in Fig. 9. We have superimposed both the path of the vortex signature observed by RaXPol and the track of the grid cell with the greatest Doppler cell circulation (dΓD) in the 0.5° elevation angle surface. The latter is generally in good agreement with the path of the vortex signature and the ground survey.
Calculated Doppler cell CCs (r0, α, β0) are indicated by black dots and are connected by white line (approximately 1 min apart). Times (UTC) are listed at approximately 5-min intervals. The distance (km) scale is shown at the lower-right side. The swath/extent and estimated centerline of the large tornado, as assessed using a combination of ground survey (G. Garfield 2013, personal communication) and RaXPol data, are shown as orange shading and a dark orange curve, respectively. The dark orange curve is the tornado centerline informed by damage assessment and RaXPol data. The yellow curve to the east of the main cyclonic El Reno tornado is the approximate path of the anticyclonic tornado during 2329–2341 UTC. More information concerning tornado paths and mobile radar deployments is given in Bluestein et al. (2015). Damage tracks and enhanced Fujita scale ratings are from the NWS DAT (2013) (courtesy of the Norman NWS).
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
Changes in circulation occur very quickly. At a given time, circulation immediately outside a tornado tends to be constant with height as shown in DJWR. Angular momentum is advected upward rapidly in a tornado, so it is not surprising that changes in circulation near the ground are transmitted upward faster than typical radar data acquisition times. Coincident with rapid changes in circulation, the rapid development of tornadoes can be associated with rapid changes in vertical vorticity. Using Eqs. (10.6) and (10.7) of Davies-Jones (1986) with reasonable convergence and vertical vorticity values, we can calculate that vertical vorticity in a tornadic updraft can triple in about 3 min (see appendix B).
In a 3-min time span, 2–3 full volumetric updates could be obtained from the NWRT PAR. However, 3 min is shorter than the time needed for a WSR-88D to complete a volume scan. The use of Supplemental Adaptive Intravolume Low-Level Scans (SAILS; Crum et al. 2013) or the Mid-Volume Rescan of Low-Level Elevations (MRLE; NOAA/NWS/ROC 2022) can help provide faster WSR-88D update times of around 1.5–2.5 min at the 0.5° elevation angle (SAILS) or up to the lowest four elevation angles (MRLE). However, these scanning techniques would not improve the update rates of all other mid- and upper-level elevation angles (e.g., Kingfield and French 2022) like can be achieved with PAR. Rapid scan mobile radars have observed rapid changes in vortex intensity over a 3-min time span (e.g., French et al. 2014; Houser et al. 2015; Bluestein et al. 2019; Houser et al. 2022). Houser et al. (2015) noted rapid evolution of tornado structure over even shorter time scales (10 s–1 min), probably due to locally high convergence of around 0.1 s−1 very close to the tornadic vortex. The use of more rapid and adaptive5 scanning technologies available with PAR may reveal that the nature of tornadogenesis is different than that implied when slower scanning radars are used for data analysis (e.g., see Fig. 20 in French et al. 2013).
Although the NWRT PAR could observe rapid changes in circulation at 1–2-min intervals, the radar’s data acquisition is not fast enough to differentiate between bottom-up, top-down, or simultaneous formation. The RaXPol data (Bluestein et al. 2019) showed that the main circulation associated with the El Reno tornado formed near the ground initially, ∼90 s prior to the development of the vertically coherent vortex. This vortex built upward through a vertical column of at least 3.5 km in less than 20 s, essentially the volumetric update time of the RaXPol data (∼29 s during the time preceding tornadogenesis through the first few minutes of the tornado’s life) and considerably faster than the PAR’s volumetric update time of ∼71 s. Advanced PARs, with the ability to more adaptively and quickly collect azimuthal sectors of high-resolution, nearly instantaneous RHIs, may provide very valuable insight into the vertical evolution of tornado development (and tornado dissipation, for that matter) that eludes even rapid scan dish-based radars like RaXPol.
b. Radius–height distributions of various circle-based Doppler kinematic quantities
We now present the vertical (radius–height) distributions of eight circle-based parameters at selected times (Fig. 10). The heights ARL are computed from the four elevation angles (0.5°, 0.8°, 1.3°, and 1.8°) at a given Doppler velocity signature center. Circle radii ρ vary from 1 to 4 km in increments of 0.2 km. Fields for each parameter are calculated on the resulting 4 × 16 grid and contoured. Green dots denote missing analyses due to insufficient reflectivity data. The evolutions of Doppler velocity and reflectivity fields are given in Figs. S1 and S2 in the online supplemental material.
Circle radius–height distributions of (a) Doppler circulation (α)D (103 m2 s−1), (b) Doppler mean vorticity
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
At 2303 UTC, KOUN radar data revealed favorable conditions for tornadogenesis. There were large contraction rate of and significant circulation around circles of 4–5-km radius (DJWR, 1130–1131 and their Figs. 18 and 19). The mean convergence within a circle of 5-km radius was 0.006 s−1 at 2303 UTC and 0.01 s−1 at 2311 UTC. The mean inflow velocity at 5 km from the tornado was very strong (24 m s−1) at 2311 UTC. DJWR (p. 1130) also showed that the mesocyclone was a two-celled vortex with a central divergent downdraft (Trapp 2000) just prior to tornado formation and speculated that the tornado probably formed away from the mesocyclone center. The El Reno tornado was so large that on PAR it had a tornado signature6 at 50-km range, not a TVS. Figures 10, 18, and 19 in the present paper did not reveal this because of missing data early in the PAR data collection that prevented calculation of circulation and contraction rate for the larger circles and because the circle radius was cut off at 4 km. Since the smaller circles were in divergent flow within the tornado itself, they were expanding rather than contracting and possessed relatively weak circulation.
By 2319 UTC (Fig. 11), the Doppler circulation (Γα)D, the Doppler mean vorticity
As in Fig. 10, but for 2319:22–2319:40 UTC. In (e), the approximate radius at which the
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
By 2322 UTC (Fig. 12), the apparent radius of maximum tangential velocity expanded to its maximum size, 2.2 km, while the tangential velocity peak (Fig. 12e) was virtually unchanged from 2319 UTC. The actual circulation had grown to its maximum value, 9 × 105 m2 s−1 (Fig. 12a). This is very large compared to the circulation of a violent tornado (1 × 105 m2 s−1) (DJWR) or a typical mesocyclone (3 × 105 m2 s−1 based on a maximum tangential wind of 23 m s−1 at 2 km from the axis). Within the tornado, the vortex flow was divergent (Figs. 12c,d) with mean outward velocities up to 18 m s−1 (Fig. 12f). This radial outflow was too large to be explained by debris centrifuging and was associated with the tornado expanding. The radius of the downdraft appears larger in Fig. 12d than it is because Fig. 12d plots the areal average of divergence for each circle, not the divergence as a function of radial distance from the circulation center. Currently, it seems likely that the tornado has a central downdraft at least at the observation heights. Mean inflow was detected for low-level circles with radii greater than 3 km.
As in Fig. 10, but for 2321:45–2322:02 UTC.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
At 2324 UTC, the visual tornado was very large (Fig. 8f). The actual circulation was still more than 6 × 105 m2 s−1 (Fig. 13a). All the circles with radii of 4 km or less had positive expansion rates (Fig. 13c). The largest value, 1.56 × 105 m2 s−1, of Doppler areal expansion rate
As in Fig. 10, but for 2324:07–2324:24 UTC.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
The circulation weakened at 2329 UTC (Fig. 14a), the radial flow was convergent once more (Fig. 14f), and the radius of maximum tangential wind (Fig. 14e) decreased as a result. The maximum mean tangential velocity remained above 50 m s−1 as it had been since 2319 UTC (Figs. 11e–15e). The anticyclonic tornado (AT), whose track is shown in Fig. 9, formed at this time and persisted until 2341 UTC. The NWRT PAR also shows that it had a strong vortex signature (Fig. 16 and the animations between 2327:40 and 2334:46 UTC in the supplemental material). Initially, the AT was located about 3 km from the main cyclonic tornado, and it gradually drifted further away (Fig. 16). On rare occasions, anticyclonic tornadoes form at the trailing end of a rear flank gust front in supercells dominated by mesocyclones as a result of a surge in the RFD outflow (Davies-Jones et al. 2001; Bluestein et al. 2015, 2016). The anticyclonic circulation was probably too weak to significantly reduce the circulation of circles around the cyclonic tornado.
As in Fig. 10, but for 2328:51–2329:09 UTC.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
As in Fig. 10, but for 2332:24–2332:42 UTC.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
NWRT PAR scans at Doppler velocity fields (VD; m s−1) and reflectivity fields (Z; dBZ) at elevation angles of (a),(b) 1.8°, (c),(d) 1.3°, (e),(f) 0.9°, and (g),(h) 0.5° at 2328:51–2329:09 UTC for the El Reno tornadic storm on 31 May 2013. Black dotted curves are concentric circles of radii ρ = 1, 2, 3, and 4 km centered on the central points (r0, α, β0) of Doppler velocity signature. On top of each sector, the character string “number/number/number” represents the range(km)/azimuth(°)/height(km) to the signature center. Black dotted curves indicate measured Doppler circulations and Doppler areal contraction rates. Doppler velocity and reflectivity scales, respectively, are indicated at left and right. ND stands for no data. The AT is indicated by a yellow circle; see the yellow tornado track in Fig. 9.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
At 2332 UTC, the circulation and tangential winds declined markedly (Figs. 15a,e), and the mean radial flow became outward once again (Figs. 15d,f,h), probably as a result of an RFD overtaking the tornado. As the tornado neared the end of its life cycle (2337 UTC), it narrowed, and the peak tangential winds around the tornado (Fig. 17e) were slightly stronger than at 2332 UTC. At this stage, the tornado had become surrounded by weak rainy outflow (see animations in the supplemental material).
As in Fig. 10, but for 2337:08–2337:26 UTC.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
From 2322 UTC onward, circulation as a function of circle radius ρ decreased at large ρ (Figs. 12a–15a). This is probably due to air with anticyclonic vorticity surrounding the tornado (Hoecker 1960) as reproduced in the Davies-Jones (2008) model. Since circulation is additive and is the areal integral of vorticity, the circulation around a larger circle can only be smaller than that around a smaller circle if the vorticity in the annular region between the circles is anticyclonic.
Simultaneously or shortly after the tornado’s maximum Doppler circulation at 2322 UTC (Fig. 12a), the tornado’s largest path width (Fig. 9) and maximum tangential wind (Fig. 12e) occurred, the tornado’s radius of maximum tangential wind was largest (e.g., Fig. 12e), and there was strong divergence in the vortex (e.g., Figs. 12c,d,f,h). All this implies a strong central downdraft that was driven by very low pressure at low heights above ground.
In Figs. 13a–15a, there is some decline of the circulation with height above 1 km. This indicates vortex widening with height. There are many reasons for vortex spreading. These include frictionally caused inflow next to the ground as in Lewellen et al. (2000, see their Fig. 1), vortex breakdown above the ground, an axial downdraft associated with a downward pressure-gradient force, and a tornado forming ground up (Davies-Jones 2008; Bluestein et al. 2019) to join up with a broader vortex aloft. The friction layer is probably below the radar horizon, so the friction effect does not show up in the circulation analyses.
Analyses of the eight circle-based parameters have provided valuable insights into the rapid evolution of the tornado.
c. Evolution of various vortex-based Doppler kinematic quantities over PAR scan time
We presented radius–height plots to reveal the structure of the tornado during the time period from 2258 to 2347 UTC. We now create time–circle radius plots to highlight rapid changes between PAR scan times. Figures 18 and 19 depict the evolution of the various vortex-based Doppler kinematic parameters as functions of i) circle radius ρ and ii) PAR scan time at 0.5° and 1.8° elevation angles, respectively. The evolutions of these parameters in the 0.9° and 1.3° elevation-angle surfaces (not shown) were similar to those in the 0.5° and 1.8° surfaces. The horizontal cross sections contain large regions of missing data (green dots) owing to the sparsity of reflectivity near the circumferences of the mid-to-large circles between ∼2301 and ∼2320 UTC.
Horizontal cross sections of (a) Doppler circulation (Γα)D (103 m2 s−1), (b) Doppler mean vorticity
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
As in Fig. 18, but for the elevation angle of 1.8°.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
For the smaller circles at least, the circulation (Figs. 18a and 19a) increased with ρ as expected in regions of cyclonic vorticity. After the tornado reached peak intensity, anticyclonic air infiltrated the larger circles, and at a fixed time, the maximum circulation was located at an intermediate value of ρ.
The Doppler mean vorticity (Figs. 18b and 19b) is Doppler circulation divided by circle area (per Stokes’ theorem) and decreased with increasing ρ. The mean vorticity within the ρ = 1 km circle exhibited several maxima of roughly 5 × 10−2 s−1. The mean tangential velocity is actual circulation divided by 2πρ, and its peaks occurred for the smaller circles (ρ < 1.5 km). Maximum values of over 50 m s−1 occurred during the tornado’s mature stage.
Areal contraction rate, mean divergence, and mean inflow velocity are shown in panels (c), (d), and (f), respectively, of Figs. 18 and 19. The mean inflow velocity and mean convergence are the actual areal contraction rate divided by 2πρ and by πρ2, respectively. The fields exhibit alternate episodes of inflow and outflow. The outflow has three possible causes. The first reason is the vortex spreading with height. Inflow may have been confined to a shallow layer below the 0.5° α surface with divergence above the vortex boundary layer. The second reason could be the development of an axial downdraft driven by a downward-directed vertical perturbation pressure-gradient force. Tornadoes broaden with height in theory (Long 1958; Morton 1966; Fiedler and Rotunno 1986), in visual observations (e.g., Burggraf and Foster 1977), and in radar observations (e.g., Bluestein et al. 2007). Often there is a vortex breakdown, where the tornado transitions from a narrow, high-speed vortex jet below to a broader weaker vortex aloft (Burggraf and Foster 1977). The latter may have a central downdraft. The vortex breakdown may lower to the surface, and in this case, the central downdraft reaches the ground.
The third possibility for the appearance of radial outflow is debris centrifuging, which can lead to a false divergence signature for air motion that is in fact convergent because the radar measures the reflectivity-weighted mean radial velocity of everything in the radar resolution volume, not just the actual local mean air velocity (e.g., Snyder and Bluestein 2014). Based on the estimate of terminal outflow speed for large drops in section 3, the larger outflow velocities in Figs. 18f and 19f were too strong to be explained by centrifuging of large drops. We cannot rule out the possibility that part of the observed outflow was due to the centrifuging of large hail or large debris.
The outflow velocities were small compared to the tangential velocities, so the contours of mean wind speed (Figs. 18g and 19g) resembled those of mean tangential velocity (Figs. 18e and 19e), and the Doppler mean inflow angles (Figs. 18h and 19h) did not depart far from 90°.
Between 2258:03 and 2300:26 UTC, the analyses successfully detected large circulation and convergence. This is consistent with stretching and amplification of the mesocyclone prior to tornadogenesis (Figs. 18a–d). The importance of the rapid updates (∼1 min per sector volume) is highlighted by how readily the rapid evolution of the Doppler areal contraction rate at mid-to-large circle radii may be observed by the NWRT PAR.
In contrast, during the tornado’s mature stage, from 2321 to 2333 UTC, the circulation was expanding and weakening, followed by contracting and intensifying, and finally expanding and weakening again (Figs. 18c,d,f). This oscillation of the vortex flow over a period of 12 min is reminiscent of similar behavior in the 1998 Spencer, South Dakota, tornado (Alexander and Wurman 2005) and could be due to changes in the swirl ratio (e.g., Davies-Jones et al. 2001). Increases and decreases in the swirl ratio can cause the height of the vortex breakdown to move down and up, potentially changing the sign of the central vertical velocity and the nature of the quasi-horizontal divergence or convergence.
We have shown that the NWRT PAR’s rapid updates (e.g., Figs. 18 and 19) delineated abrupt changes in the tornado’s characteristics that may provide valuable updates of the circulations and locations of tornadoes. Abrupt changes in tornado’s characteristics can and do occur [e.g., see vortex breakdown in Burggraf and Foster (1977) and many examples of tornado decay (e.g., Golden and Purcell 1978)], and volumetric updates faster than that available from existing operational weather radars are needed to capture these rapid changes. Future work may evaluate polarimetric PAR data currently being collected by the ATD, once a similar (significant or violent) tornadic supercell case is observed.
d. Verification using RaXPol observations as ground truth
RaXPol collected data from east and northeast of the tornado between 2300 and 2314 UTC before moving for safety to avoid hail, heavy rain, and a potentially close encounter with the growing tornado (e.g., Bluestein et al. 2019). During the deployment, the distance of the tornado from the radar decreased from ∼9.9 to ∼4.6 km. Visually, the tornado was multivortex early and single vortex later in this time interval (Fig. 8), although heavy rain and the condensation funnel itself may have prevented eyewitnesses from seeing multiple vortices if they were present. From these data, Thiem (2016) and Bluestein et al. (2019) tabulated ΔVD (from which we can calculate the rotational velocity, Vrot = ΔVD/2) and the distance between velocity peaks, D, as functions of time. These fine-scale data might be regarded as the “ground truth” for the broader-scale PAR circulation analysis. The PAR was about 50-km east-southeast of the tornado. At 50-km range and 0.5° elevation angle, the axis of the PAR beam is at 600 m above radar level. The beamwidth of the PAR at normal incidence is 1.5° or 1.3 km at 50-km range, so very narrow and low-altitude vortex signatures in the RaXPol data will be invisible to the PAR.
The RaXPol rotational velocity increased quite steadily at all elevation angles during this interval (Fig. 20; Figs. 5a and 7a in Bluestein et al. 2019). It was slightly larger at 0° elevation angle but otherwise quite constant with height. Since we could not compute the circulation around fixed circles from the RaXPol tabulated results, we calculated instead the “Doppler circulation of the vortex signature,” defined as πVrotD/2. This is shown as a function of time in Fig. 21, along with the rotational velocity at 2° elevation and D. We note that the comparison of the PAR-estimated circulation to the Doppler circulation estimate from RaXPol is admittedly suboptimal because the latter, based on only two quantities (D and Vrot) at each time, is not the same as the circulation around fixed circles of specified radius (which is based on the Doppler velocities on each circle), particularly because of the presence of multiple vortices during the mutual observation period. The proxy circulation is based on the concept of a single vortex, not a well-resolved multivortex tornado, whereas the observed signature on which the RaXPol D and Vrot data are calculated may be affected by a single transient subvortex. This misfit of model to data may explain the large oscillations in the RaXPol D values in Fig. 21. In contrast, the PAR-estimated circulation around a fixed circle estimates the circulation of the overall tornado, provided that the circle encompasses all the subvortices. The Doppler circulation exhibited considerable oscillations superposed on an overall decrease. Since the rotational velocity was increasing steadily, this decrease in Doppler circulation was associated with a major decrease in D (Fig. 21; Figs. 5b and 7b in Bluestein et al. 2019), which would occur if the flow were strongly convergent. If angular momentum were conserved, large oscillations in D would be accompanied by large opposing oscillations in Vrot.
Time series (hhmm:ss, UTC) of RaXPol rotational velocity (m s−1) for different elevation angles (°).
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
Time series (UTC) of RaXPol πVrotD/2 (i.e., the Doppler circulation of the signature, m2 s−1, red curve), Vrot (i.e., rotational velocity; m s−1; blue curve), and one-half of D (i.e., the half-distance between velocity peaks, m, green curve). Data are at 2° elevation angle.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
Figure 22 shows a time series of the RaXPol rotational velocity at 2° elevation and the PAR rotational velocity at 0.5° elevation. At times, the two rotational velocities agree closely, although those from RaXPol were consistently higher than those from the PAR, likely owing to differences in the resolution of data from each radar. When the beamwidth divided by the vortex core diameter D is greater than ten, the ratio of rotational velocity to the vortex’s maximum tangential velocity is less than 0.1 (see Fig. 7 in Brown et al. 1978). Since the ratio of PAR to RaXPol rotational velocity is much greater than 0.1 (Fig. 22) when D is less than 130 m (one-tenth the PAR beamwidth; Fig. 21), we must conclude that at these times the PAR was measuring a larger-scale vortex than the narrow vortex observed by RaXPol. In corroboration, Wurman et al. (2014) report quasi-concentric wind field maxima at 2314 UTC, one with a diameter, from peak inbound to peak outbound Doppler velocity, D, of 1–2 km and the other much smaller. The small vortex was observed by Doppler on Wheels radars but was unresolved by and perhaps below the radar horizon of the WSR-88D radars 60 km away. Regardless of the differences in the magnitudes of rotational velocity from RaXPol and the PAR, the behavior of the PAR-determined circulation at ρ = 1 km (gray dots and maroon line in Fig. 22) nearly paralleled the behavior of the RaXPol rotational velocity between 2300 and 2314 UTC, potentially indicating an ability of the circulation technique to directly represent what is happening on the tornadic vortex scale.
Time series (hhmm:ss, UTC) of RaXPol rotational velocity (blue dots) at 2° elevation angle, PAR rotational velocity (orange dots), and PAR circulation (maroon line with gray dots) at 0.5° elevation angle. PAR circulation is for the circle with ρ = 1 km.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
e. Doppler circulation and contraction rate analyses of the Yukon, Oklahoma, tornado
Whereas the main “El Reno” tornado was unprecedented in size, we present here a case where the Doppler-circulation method is applied to a smaller, weaker tornado. Figure 23 shows the Doppler circulation, areal contraction rate, and other fields for the EF1 Yukon, Oklahoma, tornado at 2351 UTC. The Yukon tornado formed after the El Reno tornado dissipated at 2347 UTC; it existed from 2351 to 0009 UTC and traveled ∼9 km. The damage rating and track width during its early stages were EF0 and 550 m (NWS DAT 2013). The fields were weak compared to the corresponding fields during the mature stage of the El Reno tornado (Figs. 11–14) but similar to the El Reno cases’ magnitude immediately before tornadogenesis (Fig. 10). The analysis indicates a broad vortex with central divergent flow surrounded by convergent flow. This structure is reminiscent of the El Reno tornado. The main difference between the Yukon and El Reno tornadoes near their times of formation was the much stronger central divergent flow in the El Reno tornado. Magnitudes were about 65%–75% lower for all parameters except the Doppler mean inflow angle. This suggests much lower central pressure deficit near the ground in the El Reno case.
As in Fig. 10, but for the smaller, weaker Yukon tornado that formed later at 2351 UTC.
Citation: Monthly Weather Review 153, 3; 10.1175/MWR-D-23-0110.1
6. Discussion
a. Interpretations of the El Reno tornado in terms of known vortex dynamics
We can explain the properties of the extremely large El Reno tornado using the Ward (1972) laboratory model. This simulator produces a wide range of tornado-like vortices. Davies-Jones (1973) showed that the core radius of the miniature vortices was a function of the swirl ratio, which is the circulation times updraft radius divided by the updraft’s volume flow rate. The swirl ratio increases with circulation with other conditions remaining the same. At high swirl ratio, the simulator produces either a very wide vortex or multiple vortices. At a high value of circulation, the overall vortex is wide owing to the large central pressure deficit. The vertical perturbation pressure-gradient force along the axis drives a central downdraft, which prevents contraction to a small radius. Outflowing air that has descended in the vortex is separated from the inflowing air by a vortex sheet. Although a concentrated vortex does not form on the axis, smaller-scale vortices can still form along this vortex sheet. The sighting of multiple vortices and the large visual size of the tornado at El Reno are indications of high swirl ratio.
As in the Ward simulator, the divergence within the tornado increased as the tornado’s size increased (Figs. 8, 11d–13d). The tornado structure changed rapidly between 2321 and 2333 UTC. This could be due either to vortex breakdown (Burggraf and Foster 1977) or changes in the surrounding flow [see tornado review articles by Davies-Jones et al. (2001), Davies-Jones (2015), and Rotunno (2013)].
b. Comparison of measures of rotation and benefits of using circulation
Circulation and contraction rate have several advantages over widely used TVS parameters such as ΔV, rotational velocity Vrot, and azimuthal shear S (
Because circulation tends to be fairly constant for circles circumscribing the whole of a vortex core, it is far less dependent on range than the other warning parameters (cf. Figs. 13 and 11 in DJWR) and thus is a better standard measure of vortex strength. It also may have some predictive values because of the tendency for angular momentum to be conserved during vortex contraction. Because circulation and contraction rate on a larger scale tend to be prerequisites for tornadogenesis, they may be able to be used for advance warning as demonstrated analytically in Davies-Jones and Wood (2006) and in DJWR and from forecasts by Homeyer et al. (2020) and Murphy and Homeyer (2023). In contrast, TVS parameters will not provide as much lead time because they do not measure what is happening on the mesocyclone scale, and TDSs provide no lead time at all.
An advantage of the TVS parameters is that they are independent of the CC location. However, as shown in section 2d, circulation and contraction rate are quite insensitive to the exact location of the CC within the grid cell of maximum cell circulation. On the other hand, circulation is more forgiving than the TVS parameters regarding missing data. TVS parameters will be undefined if there is missing data at one or both sides of the TVS whereas circulation on a larger scale will still be detectable.
Whereas a TVS may indicate strong rotation, the deployment of polarimetric radars has brought forth another very valuable tool for tornado detection. Unlike circulation, a TDS, by definition, is essentially a sure sign that a tornado has formed. However, the absence of a TDS does not imply the absence of a tornado. Furthermore, a TDS does not offer advance warning, and one may be absent due to a lack of debris sources or debris that is not lofted to the height of the radar beam. The tornado may also be too short lived or too weak to loft debris to the radar volume(s) at the time the radar is scanning the tornado. Because it depends on damaging winds, availability of debris, and time for the debris after generation to reach altitude, it has a negative lead time. For example, it would take a minute for debris traveling upward at 30 m s−1 to reach 2 km AGL and additional time to disseminate a warning to the public. In a few minutes, a fast-moving tornado that forms in a city could travel several kilometers.
c. Limitations of measures of rotation
There are, of course, good reasons to use all available signatures in a tornado probability algorithm. Circulation/contraction rate is not intended to be the only pieces of information that a forecaster or algorithm uses in a decision whether a tornado warning is warranted. Because tornadoes come in a wide range of sizes and strengths and form in updrafts of various sizes and intensities, there are no magic thresholds for circulation/contraction rate (as also for rotational velocity or azimuthal shear) that discriminate between tornadic and nontornadic storms. A circulation threshold is not possible because tornadoes, landspouts, and waterspouts have widely differing circulations (section 2c). Large circulations and large contraction rates are warning signs that the processes that initiate mesocyclone-associated tornadoes are underway. The contraction rate governs the speed at which a tornado will form if the existing convergence is maintained. In essence, the larger the circulation, the greater the chance of a large, intense tornado happening.
All tornado detection methods, regardless of whether they are based on TVS parameters, TDS, or circulation, will fail to detect some tornadoes. The smaller and weaker the tornado and the further it is from the radar, the more likely it is to be undetected because of limited resolution and/or poor (or nonexistent) low-level sampling. Doppler circulation was run on several cases of analytical tornadoes at different distances from the radar in DJWR, and it was concluded that it was far less range sensitive than rotational velocity (cf. DJWR Figs. 11 and 13).
Recently, Homeyer et al. (2020) and Murphy and Homeyer (2023) have found that the main difference between nontornadic and tornadic storms was that, in tornadic storms (regardless of convective mode), the low-level and midlevel mesocyclones were vertically aligned. This alignment, which is associated with the vertical stretching of vorticity in convergent flow with enhanced circulation, was present both 20 min prior to and at the time of tornado formation. Therefore, the circulation method may be improved by including a measure of the verticality of the mesocyclonic rotation. However, this is not to say that all tornadoes are vertically aligned—indeed, many tornadoes may have significant tilt away from the vertical (particularly, it seems, leading up to and during tornado dissipation), as some close-range observations from mobile radars have shown (e.g., Wurman and Gill 2000; Tanamachi et al. 2012; French et al. 2014, 2015; Griffin et al. 2019; Snyder et al. 2020).
7. Conclusions
Many previous observational studies of mesocyclones and tornadoes that were made with WSR-88D radars emphasized the kinematic properties of storms such as differential and rotational Doppler velocities and azimuthal Doppler shear (e.g., Mitchell et al. 1998; Stumpf et al. 1998; Mahalik et al. 2019; Sandmæl et al. 2023). These kinematic properties diminish with range and vary with azimuth angle (relative to the broadside direction) for a PAR. This study uses PAR data to assess the dynamical properties of tornadoes and LLMs. We have obtained strong evidence that Doppler-derived kinematic quantities like circulation, areal contraction rate, mean vorticity, and mean divergence may help us move toward more physically meaningful radar-based characterization of rapidly evolving mesocyclones and tornadoes. We analyzed rapid scan PAR data collected from the significant El Reno, Oklahoma, supercell of 31 May 2013 using a modified single-polarization S-band SPY-1A PAR antenna located at the NWRT in Norman, Oklahoma. Analyses of these quantities could provide a means of supporting or refuting theories about the 3D structures of evolving mesocyclones and their associated tornadoes.
The conclusions from these analyses are as follows:
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Rapid evolution in the kinematic structures of the mesocyclone and associated tornado highlights the importance of the rapid updates provided by the NWRT PAR.
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Large circulation values and convergent flow in the LLM just prior to the tornado indicate that a tornado may be imminent.
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At its maximum size, the tornado had its largest estimated circulation, 9 × 105 m2 s−1, which is much larger than most tornadoes.
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Changes in circulation can be rapid and correlate well with changes in damage path width and the visual size of the tornado.
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Changes in tornado circulation occur almost simultaneously at all heights.
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Anticyclonic vorticity surrounding the tornado decreases circulation at large circle radii.
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The tornado expanded–contracted–expanded with associated outflow–inflow–outflow and divergence–convergence–divergence signatures in 12 min.
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These oscillations may be explained by changes in the height of a vortex breakdown (Burggraf and Foster 1977).
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Expansion of the vortex was associated with divergence within it.
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The largest estimated areal expansion rate, 3 × 105 m2 s−1, occurred just after the tornado was at its maximum size and intensity.
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The path of the tornado through open country and the size of areal expansion rates make it unlikely that observed large outflow velocities can be explained by centrifuging of large debris.
We believe that these scientific conclusions gleaned just from a single distant, fixed radar justify at least the continued exploration of the use of the circulation method to identify low-level mesocyclones that could produce intense tornadoes, which is of great importance to the operational community and to the general public.
The circulation method has been tested on just a few actual cases (DJWR; Wood et al. 2021; this paper), some tornadic vortex signatures obtained by moving a successful simulation of the Union City TVS to many different ranges (DJWR), and some analytical vortices that evolve from a LLM to a tornado (Davies-Jones and Wood 2006). More cases are needed to generate statistics of circulation versus tornado occurrence and intensity, particularly with regard to specific values of circulation and areal contraction rate that may be useful for discriminating tornadic LLMs from nontornadic LLMs. The data-filling procedure, the technique for finding the center of circulation, and the circulation analysis can be performed quickly and should not be an impediment to eventual operational use.
Acknowledgments.
The authors are grateful for the phased-array radar data collection efforts of NSSL/CIWRO staff. We thank Drs. Corey Potvin, Erik Rasmussen, Anthony Reinhart, and Jidong Gao of NSSL and Terry Schuur of NSSL/CIWRO for reviewing and making helpful discussions on the early version of the manuscript. The authors thank Robert Toomey and Jeff Brogden of NSSL/CIWRO for WDSS-II help and support. The lead author is deeply indebted to Charles Kuster of NSSL for explaining WDSS-II processes and helping the lead author improve his skill and knowledge in WDSS-II. We thank Kyle Thiem of the National Weather Service, Peachtree City, Alabama, and Howard Bluestein of the School of Meteorology at the University of Oklahoma, Norman, Oklahoma, for helping to provide the RaXPol data. Furthermore, we thank the three anonymous reviewers for their comments, queries, and suggestions.
Data availability statement.
NWRT PAR data used in this study are available at https://data.nssl.noaa.gov/thredds/catalog/RRDD/MPAR/2013/catalog.html and by request. Computer algorithms used in this case study are available online upon request to NSSL at https://www.nssl.noaa.gov/contact.php. Build 3 August 2020 of the Warning Decision Support–Integrated Information (WDSS-II; https://www.wdssii.org/) software was used for some data analysis.
Footnotes
The SPY-1A was decommissioned in 2016 and replaced by the Advanced Technology Demonstrator (ATD) in 2018, which is an S-band, dual polarization PAR that uses an “active” hybrid analog/digital subarray beamformer (Palmer et al. 2022).
Tornado threat means that there is a high probability of a tornado happening in the future or already present.
RaXPol (Pazmany et al. 2013; Bluestein et al. 2015) is a mobile rapid scan X-band (3-cm wavelength) polarimetric radar system that was developed for severe weather research.
An adaptive scanning technology is one that automatically tailors the scan strategy to the phenomena being observed.
This signature is “a rare vortex signature of extreme Doppler velocity values (of opposite sign) separated by at least several beamwidths in the azimuthal direction, which arises when the tornado is within a few kilometers of the radar and the tornado is larger than the radar beam.” (Brown 1998).
APPENDIX A
Locating the Circulation Center
APPENDIX B
Estimating the Time of Rapid Tornado Formation
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