Mean field dynamo action in shear flows. I: fixed kinetic helicity

We study mean-field dynamo action in a background linear shear flow by employing pulsed renewing flows with fixed kinetic helicity and nonzero correlation time ($\tau$). We use plane shearing waves in terms of time-dependent exact solutions to the Navier-Stokes equation as derived by Singh \& Sridhar (2017). This allows us to self-consistently include the anisotropic effects of shear on the stochastic flow. We determine the average response tensor governing the evolution of mean magnetic field, and study the properties of its eigenvalues which yield the growth rate ($\gamma$) and the cycle period ($P_{\rm cyc}$) of the mean magnetic field. Non-axisymmetric modes of the mean magnetic field decay as $t \to \infty$ and hence are deemed unimportant for mean-field dynamo. Both, $\gamma$ and the wavenumber corresponding to the fastest growing axisymmetric mode vary non-monotonically with shear rate $S$ when $\tau$ is comparable to the eddy turnover time $T$, in which case, we also find quenching of dynamo when shear becomes too strong. When $\tau/T\sim{\cal O}(1)$, the cycle period ($P_{\rm cyc}$) of growing dynamo wave scales with shear as $P_{\rm cyc} \propto |S|^{-1}$ at small shear, and it becomes nearly independent of shear as shear becomes too strong.This asymptotic behaviour at weak and strong shear has implications for magnetic activity cycles of stars in recent observations. Our study thus essentially generalizes the standard $\alpha \Omega$ (or $\alpha^2\Omega$) dynamo as also the $\alpha$ effect is affected by shear and the modelled random flow has a finite memory.