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

Tran, CV *(St Andrews)*

Monday 08 December 2008, 16:00-16:30

Seminar Room 1, Newton Institute

#### 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 $Uk_dk\Theta(k,t)$, 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 $$\Theta(k,t)\ge \frac{\vartheta}{Uk_d}k^{-1}.$$
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 $\ge2$, 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 $\vartheta$ are the kinetic energy spectrum and flux, respectively.

#### Presentation

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