Physical Mechanism of the Two-Dimensional Enstrophy Cascade

In two-dimensional turbulence, irreversible forward transfer of enstrophy requires anticorrelation of the turbulent vorticity transport vector and the inertial-range vorticity gradient. We investigate the basic mechanism by numerical simulation of the forced Navier-Stokes equation. In particular, we obtain the probability distributions of the local enstrophy flux and of the alignment angle between vorticity gradient and transport vector. These are surprisingly symmetric and cannot be explained by a local eddy-viscosity approximation. The vorticity transport tends to be directed along streamlines of the flow and only weakly aligned down the fluctuating vorticity gradient. All these features are well explained by a local nonlinear model. The physical origin of the cascade lies in steepening of inertial-range vorticity gradients due to compression of vorticity level sets by the large-scale strain field.

Figure 1

Enstrophy flux $Z(k)$ normalized by enstrophy dissipation $\eta$ vs $k$. The inset is the energy spectrum.

Figure 2

(a) PDF of $(Z_{\ell }-⟨Z_{\ell }⟩)/{\sigma }_Z$ with $Z_{\ell }(\mathbf{r},t)$ enstrophy flux and ${\sigma }_Z^2=⟨(Z_{\ell }-⟨Z_{\ell }⟩{)}^2⟩$, at different filter lengths $\ell$. (b) PDF’s of the true flux (solid line) and the “nonlinear model” (dashed line) at $\ell =\pi /130$. The two lines are indistinguishable.

Figure 3

(a) PDF of the angle $\theta$ between ${\mathit{\sigma }}_{\ell }$ and $\mathbf{\nabla }{\overline{\omega }}_{\ell }$ and (b) conditional mean $⟨Z_{\ell }|\theta ⟩$, for $\ell =\pi /130$.

Figure 4

Instantaneous snapshot of (a) ${\varphi }_{\ell }(\mathbf{r},t)$ and (b) $Z_{\ell }(\mathbf{r},t)$, for $\ell =\pi /130$ in a ${512}^2$ subdomain. The red regions in (a) are dominated by strain and the green by vorticity.

Figure 5

(a) PDF’s of $Z_{\ell }(\mathbf{r},t)$ and (b) PDF’s of the angle $\theta$ between ${\mathit{\sigma }}_{\ell }$ and $\mathbf{\nabla }{\overline{\omega }}_{\ell }$, $\ell =\pi /130$. The solid line indicates strain, and the dashed line indicates the vorticity region. $⟨Z_{\ell }{⟩}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}=3.7\times {10}^{-3}$ and $⟨Z_{\ell }{⟩}_{\mathrm{v}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{x}}=9.5\times {10}^{-4}$.

Figure 6

PDF’s of the angle $\chi$ between $\mathbf{\nabla }{\overline{\omega }}_{\ell }$ and ${\mathit{l}}_{\ell }^{-}$, the left eigenvector of ${\mathbf{D}}_{\ell }$ for the negative eigenvalue.