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 kkd is bounded from above by UkdkTheta(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-kd,k+kd) . For a given flux, say vartheta>0 , across k>kd , this bound requires Theta(k,t)> or =(varthetaUkd)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 (kd,kkappa) . 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 > or =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 vartheta are the kinetic energy spectrum and flux, respectively.