On the shear-current effect: toward understanding why theories and simulations have mutually and separately conflicted

The shear-current effect (SCE) of mean-field dynamo theory refers to the combination of a shear flow and a turbulent coefficient $\beta_{21}$ with a favorable negative sign for exponential mean-field growth, rather than positive for diffusion. There have been long standing disagreements among theoretical calculations and comparisons of theory with numerical experiments as to the sign of kinetic ($\beta^u_{21}$) and magnetic ($\beta^b_{21}$) contributions. To resolve these discrepancies, we combine an analytical approach with simulations, and show that unlike $\beta^b_{21}$, the kinetic SCE $\beta^u_{21}$ has a strong dependence on the kinetic energy spectral index and can transit from positive to negative values at $\mathcal{O}(10)$ Reynolds numbers if the spectrum is not too steep. Conversely, $\beta^b_{21}$ is always negative regardless of the spectral index and Reynolds numbers. For very steep energy spectra, the positive $\beta^u_{21}$ can dominate even at energy equipartition $u_\text{rms}\simeq b_\text{rms}$, resulting in a positive total $\beta_{21}$ even though $\beta^b_{21}<0$. Our findings bridge the gap between the seemingly contradictory results from the second-order-correlation approximation (SOCA) versus the spectral-$\tau$ closure (STC), for which opposite signs for $\beta^u_{21}$ have been reported, with the same sign for $\beta^b_{21}<0$. The results also offer an explanation for the simulations that find $\beta^u_{21}>0$ and an inconclusive overall sign of $\beta_{21}$ for $\mathcal{O}(10)$ Reynolds numbers. The transient behavior of $\beta^u_{21}$ is demonstrated using the kinematic test-field method. We compute dynamo growth rates for cases with or without rotation, and discuss opportunities for further work.