Local transfer and spectra of a diffusive field advected by large-scale incompressible flows

This study revisits the problem of advective transfer and spectra of a diffusive scalar field in large-scale incompressible flows in the presence of a (large-scale) source. By "large scale" it is meant that the spectral support of the flows is confined to the wave- number region k < k(d), where kd is relatively small compared with the diffusion wave number k(kappa). Such flows mediate couplings between neighboring wave numbers within k(d) of each other only. It is found that the spectral rate of transport (flux) of scalar variance across a high wave number k > k(d) is bounded from above by Uk(d)k Theta(k,t), where U denotes the maximum fluid velocity and Theta(k,t) is the spectrum of the scalar variance, defined as its average over the shell (k- k(d), k+ k(d)). For a given flux, say theta > 0, across k > k(d), this bound requires Theta(k, t)>=(theta/Uk(d))k(-1). This is consistent with recent numerical studies and with Batchelor's theory that predicts a k(-1) spectrum (with a slightly different proportionality constant) for the viscous-convective range, which could be identified with (k(d),k(kappa)). Thus, Batchelor's formula for the variance spectrum is recovered by the present method in the form of a critical lower bound. The present result applies to a broad range of large- scale advection problems in space dimensions >= 2, including some filter models of turbulence, for which the turbulent velocity field is advected by a smoothed version of itself. For this case, Theta(k, t) and theta are the kinetic energy spectrum and flux, respectively.