![Time series of (a) maximum surface vorticity (s$^{-1}$), (b) $V_T$ (m s$^{-1}$), and (c) vorticity line slope magnitude.[]{data-label="fig:timeseries-ok-nssl"}](timeseries-ok-nssl.pdf){width="60.00000%"}

In agreement with [@cai05], a power law for vorticity appears to hold for scales exceeding that of the low-level mesocyclone core, but breaks down at smaller scales. In [@cai05], this was attributed to smaller scales being more poorly resolved in the radar dataset. A similar effect occurs in our scenario: the effective model resolution of [@frehlich2008] artificially decreases the energy contained at scales approaching the grid spacing. In the absence of positive evidence that the vorticity power law indeed extends to tornadic and smaller scales, we can only offer this as a speculative explanation for the flattening of the vorticity lines at sub-mesocyclone scales.

Power laws in energy spectrum {#sec:energy_spectrum}
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We now give an explanation of an increase in vorticity of a developing tornado that is alternative to the power laws of Cai and Wurman. We argue that tornadoes have approximate fractal cross sections and negative temperature (defined as the reciprocal of the rate of change of entropy with respect to energy). We give a power law that relates the increase of the approximate fractal dimension of the cross section of 