Multi-branch Shell Models of Two-Dimensional Turbulence exhibit Dual Energy-Enstrophy Cascades

Abstract

Classical shell models of turbulence do not display dual cascade - inverse of energy and direct of enstrophy - because they fail to reproduce the right thermal spectra. We propose here a multi-branch shell model, including a geometry hierarchically organized across scales, in order to overcome this limitation. For this model, we demonstrate numerically both the agreement of the thermal spectra with those of two-dimensional fluid equations and the emergence of a statistically stationary dual cascade. This construction also allows us to study local transfers and to investigate both self-similarity and non-Gaussianity.

Figures (4)

• Figure 1: Schematic representation of the dyadic shell model topology. Each level $\ell$ corresponds to a shell with characteristic wavenumber $k_\ell=\lambda^\ell$, while the index $n$ labels spatial substructures of the same characteristic length scale.
• Figure 2: Thermal spectra obtained from the inviscid and unforced dynamics for $D=2$. At equilibrium, the scaling (\ref{['eqScaling']}) is recovered. Left: initial condition with energy concentrated at small scales. Right: initial condition with energy concentrated at large scales.
• Figure 3: Top: mean energy spectrum. The forcing band is shaded in grey. At scales larger than the forcing scale, a $k_\ell^{-2/3}$ scaling consistent with an inverse energy cascade is observed, while at smaller scales the spectrum approaches a $k_\ell^{-2}$ scaling consistent with a direct enstrophy cascade. Bottom: mean fluxes of the quadratic invariants, energy (red) and enstrophy (blue).
• Figure 4: Probability density functions $f$ of the local energy flux ${\Pi^e}_{\ell,n}$ for $\ell$ within the inverse inertial range, rescaled by their maxima $f^*$. The Gaussian reference is shown in black. Inset: scaling exponents computed from the local energy flux structure functions; the dashed line indicates the Kolmogorov dimensional prediction.