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SUMMARY:Local transfer and spectra of a diffusive field advected by large-
 scale incompressible flows - Tran\, CV (St Andrews)
DTSTART:20081208T160000Z
DTEND:20081208T163000Z
UID:TALK15602@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:This study revisits the problem of advective transfer and spec
 tra of a\ndiffusive scalar field in large-scale incompressible flows in th
 e presence\nof a (large-scale) source. \nBy ``large scale'' it is meant th
 at 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 wav
 e number $k_kappa$. Such flows mediate couplings between neighbouring wave
  numbers within $k_d$ of each other only. It is found that the spectral ra
 te of transport (flux) of scalar variance across a high wave number $k>k_d
 $ is bounded from above by $Uk_dkTheta(k\,t)$\, where $U$ denotes the maxi
 mum fluid velocity and $Theta(k\,t)$ is the spectrum of the scalar varianc
 e\, defined as its average over the shell $(k-k_d\,k+k_d)$. For a given fl
 ux\, say $artheta>0$\, across $k>k_d$\, this bound requires $$Theta(k\,t)
 ge rac{artheta}{Uk_d}k^{-1}.$$ \nThis is consistent with recent numerica
 l studies and with Batchelor's theory that predicts\na $k^{-1}$ spectrum (
 with a slightly different proportionality constant)\nfor the viscous-conve
 ctive range\, which could be identified with\n$(k_d\,k_kappa)$. Thus\, Bat
 chelor's formula for the\nvariance spectrum is recovered by the present me
 thod in the form of a\ncritical lower bound. The present result applies to
  a broad range of\nlarge-scale advection problems in space dimensions $ge2
 $\, including\nsome filter models of turbulence\, for which the turbulent 
 velocity field\nis advected by a smoothed version of itself. For this case
 \, $Theta(k\,t)$\nand $artheta$ are the kinetic energy spectrum and flux\
 , respectively.\n\n\n
LOCATION:Seminar Room 1\, Newton Institute
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