for the tangential velocity discussed in section \[sec:wurman\]. In particular, under the assumption of axisymmetry, $v\propto r^{-\alpha}$ with $-\alpha=-\beta+1$ for $r$ between $300$ m and $9600$ m would also imply that $\zeta\propto r^{-\beta}$, which, in turn, implies . The vorticity lines’ slopes, $-\beta$, changed over the lifetime of the storms and generally were larger in magnitude for tornadic storms and smaller for nontornadic storms.

A similar approach was applied to pseudovorticity, defined as $\zeta_\text{pv}=\Delta V/L$, where $\Delta V=|(V_r)_\text{max}-(V_r)_\text{min}|$ is the difference between the maximum and minimum radial velocity of the mesocyclone and $L$ is the distance between them. The pseudovorticity can be easily obtained from a single-Doppler velocity field and thus this approach holds practical advantages over the vorticity approach. Analogous to the vorticity results, the [*pseudovorticity lines*]{}, or the lines of best fit for $\log{\zeta_{\text{pv}}}$ vs. $\log{\varepsilon}$, were found to fit the data very well (with correlation coefficients between $0.82$ for a nontornadic storm and $0.98$ for an F3 tornadic storm). This then again means that for some $\beta>0$ $$\label{eq:cai_power_law}
  \zeta_{\text{pv}}
  \propto
  \varepsilon^{-\beta}.$$ It was again observed that steeper pseudovorticity lines corresponded to stronger storms, and an empirical threshold between nontornadic and tornadic storms was determined to be the 