Investigations of Cai's Power Law for Strong Tornados

Case studies have indicated that there may be values of exponents that are strongly correlated with thresholds of tornado activity. For tornadic thunderstorms one can estimate the relationship between vorticity and the length scale using Doppler radar data. In a recent study, Cai [1] defines the psuedovorticity by ζ = DV/L , where DV = |(V_r)_max - (V_r)_min| is the velocity difference between the maximum and minimum radial velocity of the mesocyclone 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 sets of radar data corresponding to different length scales ( ε is the finest resolvable scale of the filtered radar data). From the filtering of the data he obtained data points (ln(ε), ln(ζ )). He plots the points and finds best linear fit to them. 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 tornado genesis 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. It has been observed that the exponent can be thought of as measuring 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 radius of the vortex. Using Mathematica we revisited Serrin's model [3] and attempted to find solutions to the Navier Stokes equations in spherical coordinates, where b is not necessarily -1. We also considered solutions to the Euler equations as well. A possible explanation for the fractal dimension of the vortex is the fractalization of vortex lines due to intense stretching by the updraft. Recent studies of radar data [4] and numerical simulations [5] have produced arching vortex lines in the rear flank of supercell storms. These appear to be correlated with tornado genesis. While fractalization of vortex lines has not been observed, turbulence theory suggests as the vortex lines are stretched they kink, becoming fractalized [2]. As more and more vortex lines are produced in this region, viscous interactions between neighboring vortex lines would lead to mergers and an increase in the dimension of the vortex [2]. This should also lead to a strengthening of the vortex and increase in the vorticity as well. We give a heuristic argument to support Cai's power law for strong tornados, ζ∝r ^-1.6 .

[1] H. Cai, Monthly Weather Review 133 (2005), 2535–2551.

[2] A. J. Chorin, Vorticity and Turbulence, Springer, New York (1994)

[3] J. Serrin, Phil. Trans. Roy. Soc. London, Series A, Math & Phys. Sci. 271(1972), 325–360.

[4]Markowski, P. M., J. M. Straka, E. N. Rasmussen, R. P. Davies-Jones, Y. Richardson, and J. Trapp, 2008: Vortex lines within low-level mesocyclones obtained from pseudo-dual-Doppler radar observations. Monthly Weather Review, 136, 3513–3535.

[5]Straka, J. M., E. N. Rasmussen, R. P. Davies-Jones, and P. M. Markowski, 2007: An observational and idealized numerical examination of low-level counter-rotating vortices toward the rear flank of supercells. Electronic Journal of Severe Storms Meteorology, 2(8), 1–22.