Vorticity vs. passive scalar statistics

To directly check whether small-scale vorticity can be considered as passively advected by velocity, we integrated at the same time Navier-Stokes equation for the vorticity field (Eq. (2.6)) and the equation of transport of passive scalar with a finite lifetime (Eq. (2.7)) advected by the velocity fields resulting from parallel integration of Navier-Stokes equation.

The parameters appearing in Eqs. (2.6) and (2.7) are the same ($\nu = \kappa$ and $\alpha = 1 / \tau = 0.15$), yet the forcings $f_{\omega}$ and $f_{\theta}$ are independent processes with the same statistics. As already discussed the independence of the two forcings is crucial: were the two fields forced with the identical random process, the difference field $\theta - \omega$ would be free-decaying, and after a short time the passive field would became identical to the active one.

According to the picture drawn in previous section we expect to observe the same statistics for the fluctuations of vorticity $\delta_r \omega$ and passive scalar $\delta_r \theta=\theta({\mbox{\boldmath$x$}}+{\mbox{\boldmath$r$}},t)-\theta({\mbox{\boldmath$x$}},t)$ on scales $r$ small enough to ensure that the passive condition (2.37) for vorticity is satisfied.

Figure 2.6: Power spectra of passive scalar ($\times$) and vorticity ($+$). Here $\alpha = 0.15$. In the inset we show the ratio $Z(k)/E_{\theta }(k)$, which approaches a constant for large $k$

In Figure 2.6 we show the power spectra of vorticity, $Z(k)$, and of passive scalar $E_{\theta}(k)$. The estimate of the range of wave-numbers at which the statistics of vorticity and passive scalar are expected to be coincident, is $k \gg k^* \simeq k_f \exp (\lambda / \alpha)$. With the actual values $k_f = 8$, $\alpha = 0.15$ and $\lambda = 0.16$ (see Figure 2.9) we have $k^* \simeq 23$. The two curves in Figure 2.6 are indeed parallel at large $k$ ($k \gg k^*$), in agreement with the expectation $\zeta^{\omega}_2=\zeta^{\theta}_2$. At smaller wave-numbers we observe a big bump in $Z(k)$ around $k=k_f$ which has no correspondent in $E_{\theta}(k)$. This deviation is most likely associated to the presence of an inverse energy flux in the Navier-Stokes equation, a phenomenon that has no equivalent in the passive scalar case. Due to this effect, the scaling quality of $S^{\omega}_p(r)$ is poorer than the $S^{\theta}_p(r)$ one, and a direct comparison of scaling exponents in physical space is even more difficult.

Figure 2.7: Probability density functions of vorticity differences (solid line) and of passive scalar ones (dashed line), normalized by their respective standard deviation, at different scales $r$ within the scaling range.

However, we observe in Figure 2.7 that the probability density functions of vorticity and passive scalar increments, once rescaled by their root-mean-square fluctuation, collapse remarkably well onto each other. That is sufficient to state, along with the result $\zeta^{\omega}_2=\zeta^{\theta}_2$ obtained from Fig. 2.6, the equality of scaling exponents of vorticity and passive scalar at any order: $\zeta^{\omega}_p=\zeta^{\theta}_p$.

The excellent collapse of the probability distribution of fluctuations confirms that the underlying mechanism which generates intermittency, that is the competition between exponential separation of Lagrangian trajectories and exponential decay of fluctuations due to the linear damping, is the same both for vorticity and passive scalar.