I will illustrate two uses of simplified models for turbulent
diffusion to address practical issues in modeling the effects of
subgrid scale turbulence on the transport of passive scalar
fields (such as heat or concentration density of pollutants).
First, I will describe some joint work with Andy Majda and Eric
Vanden-Eijnden in which we analyze some standard closure
approximations used in turbulence which are based on renormalized
perturbation expansions (such as the direct interaction
approximation (DIA)). We examine their predictions in a model
flow consisting of a random shear flow with fluctuating cross
sweep. Though this model velocity field has nontrivial features,
such as unequal Eulerian and Lagrangian velocity correlations,
and induces intermittent behavior in the passive scalar field, it
is still simple enough to permit the exact computation of many
passive scalar statistics against which we can compare the
predictions of the closure approximations. Our study indicates
some of the strengths and weaknesses of the closure
approximations and points out the physical phenomena that these
approximations are able or not able to describe properly. We
find in particular that the DIA performs much less succesfully in
this model than some simpler closure approximations (Modified
Direct Interaction Approximation (MDIA) and Lagrangian
Renormalized Approximation (LRA)).
In the second part of the talk, I will discuss some ongoing work
with Grigorios Pavliotis on the interaction between large-scale
mean flows and small-scale turbulent fluctuations. We consider a
simple model in which the turbulent fluctuations are approximated
as a single-scale periodic velocity field, and the mean flow is
allowed to vary in space and time in a general smooth way. While
this model incorporates only some aspects of turbulence, it
allows us to derive and study in a precise fashion the effective
large-scale equations for the transport of a passive scalar field
in which the small-scale velocity fluctuations manifest
themselves both through an enhanced effective diffusivity and a
"skew-flux" correction to the drift. These effective transport
coefficients can exhibit dramatic variability in space and time
due to the nonlinear interaction between the large-scale mean
flow and the small-scale fluctuations. I will present numerical
computations for some simple examples and provide physical
explanations for some of the interesting features which appear.
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