When will the spectral transfers of the two-dimensional passive scalar present kinetic energy-enstrophy duality?

Abstract

We offer four arguments to collectively support the conjecture of the possible non-universal genuine transfer directions of two-dimensional (2D) passive scalar energy,especially the possibility of genuine inverse transfer to large scales which may co-exist with the small-scale forward cascade: 1) The inverse transfer of the magnetic potential energy, in two-dimensional magnetohydrodynamic turbulence, already suggests possibly similar behavior of the corresponding passive scalar, at least when advected by the same/similar velocity and pumped by the similarly correlated (to the active/passive scalar) forcing, if indeed the effect of correlations among Lagrangian paths and the scalar pumping is the key actor. 2) The cross-correlation \(\mathcal{C}_{\theta\zeta}\) of the passive scalar \(\theta\) to the two-dimensional (2D) vorticity \(\zeta\) of the advecting velocity field is a quadratic invariant. This passive cross correlation results in a large-scale concentration of passive-scalar energy in the absolute equilibrium state, with also equipartition at large wavenumbers, thus indicating the \textit{energy-enstrophy duality} of the passive scalar variance. 3) Recent numerical data of \textit{rotating flows} with moderate Rossby number showing inverse transfers of the vertically-averaged vertical velocity, could be related in a subtle way, according to resonant-wave theory and under the speculation of spontaneous mirror symmetry breaking (chirality) between the weakly coupled fast (wave) and slow (vortex) modes. The theory applying the absolute equilibrium does not directly and fully explain these numerical observations, but can play a crucial role. 4) The preliminary calculations of a modification of the Kraichnan passive scalar model indicates that some details of the passive scalar pumping, taken here passively correlated to/dependent on the vorticity, indeed matter.