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

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 $k<k_d$, 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 $U{k}_{d}k\Theta (k,t)$, 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 $\vartheta >0$, across $k>k_d$, this bound requires $$\Theta (k,t)\ge \frac{\vartheta }{U{k}_d}{k}^{-1}.$$ 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 $\ge 2$, including some filter models of turbulence, for which the turbulent velocity field is advected by a smoothed version of itself. For this case, $\Theta (k,t)$ and $\vartheta$ are the kinetic energy spectrum and flux, respectively.

arXiv comment6 journal pages, 1 "cartoon" figure, to appear in PRE

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Cited by in corpus (1)

• Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence