_{pv}= ΔV/L where ΔV = |(V

_{r})max -(V

_{r})min| is the difference between the maximum L and minimum radial velocity of the mesocyclone (rotating updraft) and L is the distance between them. Cai filtered the data from the highest resolution to the smallest resolution determined by the diameter of the mesocyclone. He filtered the radar data to obtain data sets corresponding to different length scales ( ε is the finest resolvable scale of the filtered radar data). By filtering, he obtained data points (ln(ε),ln(ζ

_{pv})) He plotted the points and obtained the best linear fit. Cai′s study comparing mobile Doppler radar data from tornadic and non-tornadic storms indicates that the steeper slopes (smaller negative values) are indicative of tornadic storms. As those mesocyclones that produced tornados become stronger approaching tornadogenesis the slope of the line decreased. Cai found the threshold for strong tornados was slope m = -1.6. For tornadic mesocyclones, this suggests a power law of the form, ζ ∝ r

^{b}, where r is the radius of the vortex. Cai observed that the exponent could be thought of as a fractal dimension associated with the vortex. For high-resolution mobile Doppler radar data, there has been some attempt to interpret this as a giving a power law for the drop-off of the velocity as a function of the radius of the vortex. In several papers devoted to analyzing mobile radar data associated to strong or violent tornados Josh Wurman computed v ∝ r

^{b}. Wurman(2000) has calculated the exponent b in v ∝ r

^{b}from high-resolution mobile Doppler radar data obtained from a tornado, rated F2-F4, and found b varying from -0.5 to -0.7. From data he obtained in the intercept of the Spencer, South Dakota tornado (May 31, 1998), Wurman (2005) calculated the value of b=-0.67. The tornado was rated EF4. These values were calculated from the data taken at one instant during the tornados existence. There may have been considerable variation during the tornados lifespan. It is interesting to note that the (threshold) power law for vorticity in strong tornados given by Cai differs from the power law Wurman calculated for the velocity of the two strong/violent tornados by 1. This is consistent with the vorticity being the derivative of the velocity. Hence this suggests that the results are consistent. Power laws suggest self-similarity. We agree that to truly understand strong atmospheric vortices their self-similarity is a fact that needs to be exploited.

Preliminary results from numerical simulations using ARPS have produced results similar to those of Cai and Wurman. We did several nested grid runs consisting of only two grids, the coarse grid of 100km by 100km on the side, 15km in the vertical, and Δx = Δy = 1000m, Δz = 300m, and a second grid with 80km by 80km on the side, 15km in the vertical, and Δx = Δy = 333m, Δz = 300m. Using the May 20th 1977, Del City sounding, we did runs of four hours, and compared the results of the runs with the results of Adlerman's thesis. Let ζ _{ ε }, be the low-level vertical vorticity maximum corresponding to the grid scale ε at a given time in the run. We considered the points (ln(ε),ln(ζ _{ ε })) corresponding to the different scales. We computed the slope of the vorticity lines as follows: m = - ln(ζ _{333}/ζ _{1000})/ ln(3). We formed a time series for the 4-hour runs with Δt = 60sec. The slopes with the most negative values corresponded roughly with the tornado cyclones in Adlerman's thesis. The values of the slopes varied from -1.2 to roughly -1.45 during the short tornado phases. The tornados in Adlerman's thesis had wind speeds in the EF2 EF3 range. Spikes in the vorticity were noted at times delayed from the times in the simulation in chapter 4 of Adlerman's thesis, by about 10 minutes. We think this delay is due to the coarser grid used in our simulations. Also, the strongest tornado in the runs of Adlerman occurred during a period where the fractal dimension of the vorticity was roughly -1.2. This suggests that one might use a Cai/Wurman type scaling criterion from the vorticity (velocity) data generated by nested grids in a mesoscale numerical weather forecast to predict tornados.