Mean field dynamo action in shearing flows. II: fluctuating kinetic helicity with zero mean

Here we explore the role of temporal fluctuations in kinetic helicity on the generation of large-scale magnetic fields in presence of a background linear shear flow. Key techniques involved here are same as in our earlier work \citep[][hereafter paper~I]{JS20}, where we have used the renovating flow based model with shearing waves. Both, the velocity and the helicity fields, are treated as stochastic variables with finite correlation times, $\tau$ and $\tau_h$, respectively. Growing solutions are obtained when $\tau_h > \tau$, even when this time-scale separation, characterised by $m=\tau_h/\tau$, remains below the threshold for causing the turbulent diffusion to turn negative. In regimes when turbulent diffusion remains positive, and $\tau$ is on the order of eddy turnover time $T$, the axisymmetric modes display non-monotonic behaviour with shear rate $S$: both, the growth rate $\gamma$ and the wavenumber $k_\ast$ corresponding to the fastest growing mode, first increase, reach a maximum and then decrease with $|S|$, with $k_\ast$ being always smaller than eddy-wavenumber, thus boosting growth of magnetic fields at large length scales. The cycle period $P_{\rm cyc}$ of growing dynamo wave is inversely proportional to $|S|$ at small shear, exactly as in the fixed kinetic helicity case of paper~I. This dependence becomes shallower at larger shear. Interestingly enough, various curves corresponding to different choices of $m$ collapse on top of each other in a plot of $m P_{\rm cyc}$ with $|S|$.