Nonlinear Superposition of Direct and Inverse Cascades in Two-Dimensional Turbulence Forced at Large and Small Scales

We inquire about the properties of 2D Navier-Stokes turbulence simultaneously forced at small and large scales. The background motivation comes by observational results on atmospheric turbulence. We show that the velocity field is amenable to the sum of two auxiliary velocity fields forced at large and small scale and exhibiting a direct enstrophy and an inverse-energy cascade, respectively. Remarkably, the two auxiliary fields reconcile universal properties of fluxes with positive statistical correlation in the inertial range.

Figure 1

Average energy spectrum $E_v(k)$ (symbols). $E_{1,2}^v(k)$ labels the energy spectra associated with the auxiliary velocities ${\mathit{v}}_{1,2}$ and suggests a direct (inverse) cascade for ${\mathit{v}}_1$ (${\mathit{v}}_2$), see text. Arrows mark the forcing bands at $k_L\approx 7$ and $k_{\ell }\approx 240$. Dotted lines show the slopes $-3$ and $-5/3$. Inset: energy ${\Pi }^v(k)$ and enstrophy ${\Pi }^{\omega }(k)$ fluxes normalized by their average values ($ϵ$ and $\eta$, resp.). DNS of Eqs. (1, 2, 3) were done with a standard $2/3$-dealiased, pseudospectal method with ${1024}^2$ collocation points.

Figure 2

Dynamical fluxes of (a) energy ${\Pi }_{0;jj}^v(k)$ and (b) enstrophy ${\Pi }_{0;jj}^{\omega }(k)$ (defined in 29). Light grey symbols in (b) refer to passive scalar flux ${\Pi }^{\theta }$, see text [cf. Eq. (10)]. The solid curves show energy and enstrophy fluxes obtained by integrating (2) with $f_L=0$ (a) or $f_{\ell }=0$ (b), their superposition to ${\Pi }_{0;00}^v$ and ${\Pi }_{0;00}^{\omega }$, respectively, further demonstrates the cascades overlap when both forcings are present. Notice that ${\Pi }_{0;22}^v\approx {\Pi }_{0;00}^v<0$ and ${\Pi }_{0;11}^v\approx 0$ making evident that ${\mathit{v}}_2$ are the degrees of freedom associated with the inverse cascade. Similar considerations apply to vorticity, though ${\Pi }_{0;11}^{\omega }>{\Pi }_{0;00}^{\omega }$, which will be discussed later in relation to Fig. 4.

Figure 3

Third structure functions: (a) for velocity $S_{000}^L(r)$ compared with the prediction (8) (solid red line) and $S_{012}^L(r)$; (b) for vorticity $-Z_{0;11}(r)\equiv -\mathbf{r}·{\mathcal{Z}}_{0;11}(\mathbf{r})/r$ and $Z_{0;12}(r)$, the black lines show linear behavior in $r$. The negative linear behavior of $Z_{0;11}(r)$ links to the direct cascade of $\prec {\omega }_1^2\succ$; (c) ${\mathcal{C}}_{12}^{\omega }(r)=\prec {\omega }_1(0){\omega }_2(\mathbf{r})\succ$ vs $r$. The behaviors of $Z_{0;12}(r)$ (which should vanish in the ideal limit) and of ${\mathcal{C}}_{12}^{\omega }(r)$ are evidence of the sensitivity of directly cascading degrees of freedom to the hypofriction, see text [cf. Eq. (9)].

Figure 4

Fluxes of energy and enstrophy defined in 29: (Left) energy fluxes ${\Pi }_{i;jj}^v$ of ${\mathit{v}}_j$ ($j=0$, 1, 2 from top to bottom) due to the transport by ${\mathit{v}}_i$ ($i=0$ black semifilled circles, $i=1$ red empty circles, $i=2$ blue filled circles); (Right) Enstrophy fluxes ${\Pi }_{i;jj}^{\omega }$: panels, symbols, and colors follow the same convention of Left panel.