Mean field dynamo action in renovating shearing flows
  We study mean field dynamo action in renovating flows with finite and non
zero correlation time ($\tau$) in the presence of shear. Previous results
obtained when shear was absent are generalized to the case with shear. The
question of whether the mean magnetic field can grow in the presence of shear
and non helical turbulence, as seen in numerical simulations, is examined. We
show in a general manner that, if the motions are strictly non helical, then
such mean field dynamo action is not possible. This result is not limited to
low (fluid or magnetic) Reynolds numbers nor does it use any closure
approximation; it only assumes that the flow renovates itself after each time
interval $\tau$. Specifying to a particular form of the renovating flow with
helicity, we recover the standard dispersion relation of the $\alpha^2 \Omega$
dynamo, in the small $\tau$ or large wavelength limit. Thus mean fields grow
even in the presence of rapidly growing fluctuations, surprisingly, in a manner
predicted by the standard quasilinear closure, even though such a closure is
not strictly justified. Our work also suggests the possibility of obtaining
mean field dynamo growth in the presence of helicity fluctuations, although
having a coherent helicity will be more efficient.
