Steepening of the vorticity spectrum

The steepening effect on the vorticity spectrum has been investigated by performing a set of simulations with increasing intensity of the friction $\alpha = 0.15, 0.23, 0.30$.

According to the picture drawn in Section 2.2, at every every step of the cascade process, the friction removes part of the vorticity, thus at higher values of friction the fraction of vorticity which is transported to small scales is reduced, and the resulting filamental structures are less intense, as it can be seen in Figure 2.3. Consequently one observes steeper vorticity spectra at increasing values of friction (see Fig. 2.4), in agreement with Eq. (2.34).

Figure 2.4: The vorticity spectrum $Z(k) \sim k^{-1-\xi }$ steepens by increasing the friction coefficient $\alpha$. Here $\alpha = 0.15$ (solid line), $\alpha = 0.23$ (dashed line), $\alpha = 0.30$ (dash-dotted line). In the inset, the exponent $\xi$ as a function of $\alpha$.