On passive scalars advected by two-, three- and $d$-dimensional incompressible flows

Jian-Zhou Zhu

Published 2018-04-16Version 1

When a passive scalar in the incompressible two-dimensional (2D) flow is so smart as to track the vorticity field to have a rugged correlation, it may form large scale patterns as the vortex with inverse transfer of the scalar energy, together with the 2D kinetic energy. For the $1$-form velocity $\verb"U"$ solving the 4D ideal Euler equation, L. Tartar's relevant invariant $\mathscr{N}=\int_{\mathcal{D}} d\verb"U"\wedge d\verb"U"$ vanishes in the case of no boundary contribution from the integration by parts (typically for $\mathcal{D} = \mathbb{T}^4$), becoming a null correlation constraint for the corresponding 3D passive scalar $\theta$ from the cylinder condition, so the absolute-equilibrium energy of $\theta$ is expected to be equipartitioned and transferred forwardly to smaller scales in turbulence. General situation of dimension $d$, odd or even, is also remarked.