Physical Mechanism of the Two-Dimensional Inverse Energy Cascade

We study the physical mechanisms of the two-dimensional inverse energy cascade using theory, numerics, and experiment. Kraichnan’s prediction of a $-5/3$ spectrum with constant, negative energy flux is verified in our simulations of 2D Navier-Stokes equations. We observe a similar but shorter range of inverse cascade in laboratory experiments. Our theory predicts, and the data confirm, that inverse cascade results mainly from turbulent stress proportional to small-scale strain rotated by 45°. This “skew-Newtonian” stress is explained by the elongation and thinning of small-scale vortices by large-scale strain which weakens their velocity and transfers their energy upscale.

Figure 1

Energy spectra $E(k)$ vs $k$ at steady state: (a) DNS and (b) experiment (cgs units) with dotted line of slope $-5/3$. Insets: spectral energy flux $\Pi (k)/ϵ$ vs $k$ for (a) DNS and (b) experiment. $ϵ$ is the net energy dissipation at all scales.

Figure 2

Instantaneous snapshot of filtered energy flux. Exact flux $\Pi$ from (a) DNS, $l=\pi /15$ and (b) experiment, $l=3\mathrm{cm}$ and the theoretical MSG flux ${\Pi }_{\mathrm{MSG}}$ (defined below) at the same length-scales $l$ for (c) DNS and (d) experiment. Negative flux is indicated by blue-green (DNS) and blue (experiment), while positive flux is indicated by red-orange (DNS) and red (experiment).

Figure 3

Vortex-thinning mechanism: (a) small-scale circular vortex (blue) in a large- scale strain-field ${\mathbf{S}}^{(0)}$ (red); (b) small-scale vortex drawn out into a narrow shear layer with weakened velocity $u^{(n)}<U^{(n)}$. The axes of the small-scale stress tensor ${\mathbf{\tau }}^{(n)}$ (black) are aligned with those of the large-scale strain ${\mathbf{S}}^{(0)}$ whereas the small-scale strain ${\mathbf{S}}^{(n)}$ axes are rotated $\pm 45°$ with respect to ${\mathbf{S}}^{(0)}$.

Figure 4

Normalized PDFs of true energy flux $\Pi$ and the model flux ${\Pi }_{\mathrm{MSG}}$, using the same filter scales $l$ as in Fig. 2, for (a) DNS and (b) experiment.

Figure 5

Flux fraction $⟨{\Pi }^{(n)}⟩/⟨\Pi ⟩$ from each scale $l_n$. Shown are the exact and MSG results from DNS, and, for comparison, the TFM predictions.