Nonrobustness of the two-dimensional turbulent inverse cascade

The inverse energy cascade in two-dimensional Navier-Stokes turbulence is examined in the quasisteady regime, with small-scale, band-limited forcing at scale $k_f^{-1}$, with particular attention to the influence of forcing Reynolds number Re on the energy distribution at large scales. The strength of the inverse energy cascade, or fraction of energy input that is transferred to larger scales, increases monotonically toward unity with increasing $\mathrm{Re}\propto k_{\max }^2∕k_f^2$, where $k_{\max }$ is the maximum resolved wave number. Moreover, as Re increases beyond a critical value, for which a direct enstrophy cascade to small scales is first realized, the energy spectrum in the energy-cascading range steepens from a $k^{-5∕3}$ to $k^{-2}$ dependence. The steepening is interpreted as the result of a greater tendency for coherent vortex formation in cases when forcing scales are adequately resolved. In spectral space, it is associated with nonlocality of the inverse energy transfer.

Figure 1

Snapshot of the vorticity field $\zeta (x,y)$ at quasiequilibrium for $k_{\max }∕k_f=\left(\mathrm{a}\right)$ 8 and (b) 64 (where $k_{\max }=N∕3$). The spatial domain shown is $\mid x,y\mid ⩽21\pi ∕k_f$, so the forcing scale appears the same in each panel.

Figure 2

Energy and enstrophy flux for $k_{\max }∕k_f=4,8,16$ ($k_f=85$, $N=3k_{\max }=1024,2048,4096$). Inset: $k_{\max }∕k_f=4$ on rescaled axes.

Figure 3

Energy spectrum for the cases shown in Fig. 2. Inset: $k^{2}E\left(k\right)$ for the case $k_{\max }∕k_f=16$ at three separate times.

Figure 4

Energy spectrum for $k_{\max }∕k_f=16$ but with $k_f=170$ and $N=8192$. Inset: $k^{2}E\left(k\right)$ (solid) and $k^{5∕3}E\left(k\right)$ (dashed).

Figure 5

Inverse cascade strength, $r=(\epsilon -2\nu \mathcal{Z})∕\epsilon$ against Re for $4⩽k_{\max }∕k_f⩽64$ $(512⩽N⩽4096)$. Numbers indicate the slopes of inverse and direct cascades. Open circles, $k_f=21$; filled circles, $k_f=85$; diamond represents a hyperdiffusion case with $k_f=21$ and $N=4096$, for which $r$ is defined as the average ${\Pi }_{\epsilon }$ in the inverse cascading regime normalized by $\epsilon$.

Figure 6

Vorticity probability distribution function, normalized by standard deviation, for cases $k_{\max }∕k_f=4,8,16,32,64$: $k_f=85$ (solid), 21, (dotted), and 170 (bold).