Direct enstrophy cascade

On scales smaller than the forcing correlation length, the hypothesis of a constant enstrophy flux $\Pi_{\omega}(\ell) = \epsilon_{\omega}$ leads to a different scaling. The enstrophy contained in the eddies of size $\ell$ can be estimated as $Z(\ell) \sim E(\ell) / \ell^2 \sim u_{\ell}^2 \ell^2$, and its flux

$$\Pi_{\omega}(\ell) \sim {Z(\ell) \over \tau_{\ell}} \sim {u... ...^2 \over \ell^2} {u_{\ell} \over \ell} = \epsilon_{\omega}\;.$$ (1.72)

gives the following scaling for velocities:

$$u_{\ell} \sim \epsilon_{\omega}^{1/3} \ell$$ (1.73)

Therefore the velocity field in the enstrophy cascade is smooth, at variance with the velocity field in the inverse cascade. The dimensional prediction for characteristic times simply tells that there is essentially one single time scale in the direct energy cascade $\tau \sim \epsilon_{\omega}^{-1/3}$, which provides an estimate of the inverse of the Lyapunov exponent of the flow. The prediction for the energy spectrum reads

$$E(k) = C' \epsilon_{\omega}^{2/3} k^{-3}$$ (1.74)

Figure 1.5: Energy spectrum of the direct enstrophy cascade $E(k) \sim k^{-3}$. In the scaling range the enstrophy flux (shown in the inset) is almost constant.

A spectrum $k^{-3}$ means that the integral $\int_0^{\infty} k^2 E(k) dk$, which measure the mean-square shear has a logarithmic divergence in the infrared cutoff. Thus the hypothesis of locality of interactions in the cascade can be violated in the direct enstrophy cascade.

In next chapter I will discuss how the presence of a linear drag modifies this picture. We will show the presence of small-scale intermittency for the statistics of vorticity fluctuations and its dependence on the friction intensity.