On the Possibility of a Dual Cascade in Three-Dimensional Incompressible Turbulent Flows

Models for dual cascades in power-spectra for fully three-dimensional (3D) incompressible turbulence are reviewed and summarized. Special attention is given to analyses where the basic equations for 3D incompressible flows are expanded in terms of the eigenfunctions for the curl-operator. The possibilities for forward and inverse cascades in 3D fluid turbulence are illustrated and quantified. Conditions for dual- and forward-energy cascades in wavenumber space are presented. The forward or unidirectional cascade is found to dominate, a result consistent with the basic physical arguments formulated by vortex-line stretching. The analysis gives additional details to quantify the cascade conditions including dual cascades. Selected initial or boundary value conditions can give transient space or time intervals, where a dual cascade is dominating.