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 $k_d$ is relatively small compared with the diffusion wave number $k_{\kappa }$. Such flows mediate couplings between neighbouring 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 $U{k}_{d}k\mathrm{\Theta }(k,t)$, where $U$ denotes the maximum fluid velocity and $\mathrm{\Theta }(k,t)$ 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 $\vartheta >0$, across $k>k_d$, this bound requires $$\mathrm{\Theta }(k,t)\ge \frac{\vartheta }{U{k}_d}{k}^{-1}\mathrm{.}$$ 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 $\ge 2$, including some filter models of turbulence, for which the turbulent velocity field is advected by a smoothed version of itself. For this case, $\mathrm{\Theta }(k,t)$ and $\vartheta$ are the kinetic energy spectrum and flux, respectively.