in the fractal dimension of the cross-sectional area may be associated with tornadogenesis or strengthening of an existing tornado. Note that this is analogous to Cai’s power law, in which a decrease in the (negative) exponent in the power law leads to tornadogenesis or a stronger tornado.

To gain further insight into the processes that might contribute to tornadogenesis, we consider the effects of helicity in the simplified case of homogeneous isotropic turbulence [@lilly86a; @lilly86b; @andre77]. Writing the Navier–Stokes equation in energy form and taking the Fourier transform, one obtains $$\partial_t\langle E\rangle(k)
  +
  2k^2R^{-1}\langle E\rangle(k)
  =
  Q(k),$$ where $Q(k)$ comes from the nonlinear term in the Navier–Stokes equation [@chorin]. This term represents the transfer of energy between wave numbers and has been studied for the case of homogeneous turbulence [@waleffe92]. Certain terms have been singled out and studied in relation to inverse energy cascades. These interactions involve three wave numbers. It was found that the net effect of the so-called nonlocal interactions is to transfer energy from intermediate scales to larger scales. These interactions occur between modes with helicity of the same sign. Flows without helicity and dissipation of energy as shown in Figure \[fig:lilly-13\], and flows with helicity and 