We study large-scale dynamo action due to turbulence in the presence of a
linear shear flow, in the low conductivity limit. Our treatment is
nonperturbative in the shear strength and makes systematic use of both the
shearing coordinate transformation and the Galilean invariance of the linear
shear flow. The velocity fluctuations are assumed to have low magnetic Reynolds
number (Rm) but could have arbitrary fluid Reynolds number. The magnetic
fluctuations are determined to lowest order in Rm by explicit calculation of
the resistive Green's function for the linear shear flow. The mean
electromotive force is calculated and an integro-differential equation is
derived for the time evolution of the mean magnetic field. In this equation,
velocity fluctuations contribute to two different kinds of terms, the C and D
terms, in which first and second spatial derivatives of the mean magnetic
field, respectively, appear inside the spacetime integrals. The contribution of
the D terms is such that the time evolution of the cross-shear components of
the mean field do not depend on any other components excepting themselves.
Therefore, to lowest order in Rm but to all orders in the shear strength, the D
terms cannot give rise to a shear-current assisted dynamo effect. Casting the
integro-differential equation in Fourier space, we show that the normal modes
of the theory are a set of shearing waves, labelled by their sheared
wavevectors. The integral kernels are expressed in terms of the velocity
spectrum tensor, which is the fundamental quantity that needs to be specified
to complete the integro-differential equation description of the time evolution
of the mean magnetic field.Comment: Near-final version; Accepted for publication in the Journal of Fluid
Mechanics; References added; 22 pages, 2 figure
