Abstract

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 $k k_d$ is bounded from above by $Uk_dkTheta(k,t)$, 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 $artheta>0$, across $k>k_d$, this bound requires $$Theta(k,t)ge rac{artheta}{Uk_d}k^{.$$ 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 $ge2$, 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 $artheta$ are the kinetic energy spectrum and flux, respectively.