  We study large-scale dynamo action due to turbulence in the presence of a
linear shear flow. Our treatment is quasilinear and equivalent to the standard
`first order smoothing approximation'. However it is non perturbative in the
shear strength. We first derive an integro-differential equation for the
evolution of the mean magnetic field, by systematic use of the shearing
coordinate transformation and the Galilean invariance of the linear shear flow.
We show that, for non helical turbulence, the time evolution of the cross-shear
components of the mean field do not depend on any other components excepting
themselves; this is valid for any Galilean-invariant velocity field,
independent of its dynamics. Hence, to all orders in the shear parameter, there
is no shear-current type effect for non helical turbulence in a linear shear
flow, in quasilinear theory in the limit of zero resistivity. We then develop a
systematic approximation of the integro-differential equation for the case when
the mean magnetic field varies slowly compared to the turbulence correlation
time. For non-helical turbulence, the resulting partial differential equations
can again be solved by making a shearing coordinate transformation in Fourier
space. The resulting solutions are in the form of shearing waves, labeled by
the wavenumber in the sheared coordinates. These shearing waves can grow at
early and intermediate times but are expected to decay in the long time limit.
