Non-helical turbulence within a linear shear flow has demonstrated efficient amplification of large-scale magnetic fields in numerical simulations, but its precise mechanism remains elusive. The incoherent alpha mechanism proposes that a zero-mean fluctuating transport coefficient alpha (linked to kinetic helicity) in the shear flow is a candidate driver. Previous renovating-flow models have proposed that the correlation time of helicity fluctuations must be sufficiently extended to overcome turbulent magnetic diffusivity, yet only empirical validation of this concept has been obtained. In this study, we conduct direct numerical simulations of weakly compressible non-helical hydrodynamic turbulence. We scrutinize the correlation times of velocity and kinetic helicity fluctuations in distinct flow configurations, including rotation, shearing and Keplerian flows, as well as the shearing burgulence counterpart. Our findings indicate that rotation contributes to a prolonged correlation time of helicity compared with velocity, particularly notable in auto-correlations of both volume-averaged quantities and individual Fourier modes due to the formation of large-scale vortices. In contrast, moderate shear strength does not exhibit significant scale separation, with shear flows elongating vortices in the shear direction. Shearing burgulence, characterized by shorter helicity correlation times, appears less conducive to hosting the incoherent alpha effect. Notably, at modest shear rates, only Keplerian flows exhibit sufficiently coherent helicity fluctuations, in contrast to shearing flows. However, the relative strength of helicity fluctuations compared with turbulent diffusivity is significantly lower, raising doubts about the viability of the incoherent alpha effect as a potential dynamo driver in the subsonic flows examined in this study.

The magnetic fields in galaxy clusters and probably also in the interstellar medium are believed to be generated by a small-scale dynamo. Theoretically, during its kinematic stage, it is characterized by a Kazantsev spectrum, which peaks at the resistive scale. It is only slightly shallower than the Saffman spectrum that is expected for random and causally connected magnetic fields. Causally disconnected fields have the even steeper Batchelor spectrum. Here, we show that all three spectra are present in the small-scale dynamo. During the kinematic stage, the Batchelor spectrum occurs on scales larger than the energy-carrying scale of the turbulence, and the Kazantsev spectrum on smaller scales within the inertial range of the turbulence - even for a magnetic Prandtl number of unity. In the saturated state, the dynamo develops a Saffman spectrum on large scales, suggestive of the build-up of long-range correlations. At large magnetic Prandtl numbers, elongated structures are seen in synthetic synchrotron emission maps showing the parity-even E polarization. We also observe a significant excess in the E polarization over the parity-odd B polarization at subresistive scales, and a deficiency at larger scales. This finding is at odds with the observed excess in the Galactic microwave foreground emission, which is believed to be associated with larger scales. The E and B polarizations may be highly non-Gaussian and skewed in the kinematic regime of the dynamo. For dust emission, however, the polarized emission is always nearly Gaussian, and the excess in the E polarization is much weaker.

The Saffman helicity invariant of Hosking & Schekochihin (Phys. Rev. X, vol. 11, issue 4, 2021, 041005), which we here call the Hosking integral, has emerged as an important quantity that may govern the decay properties of magnetically dominated non-helical turbulence. Using a range of different computational methods, we confirm that this quantity is indeed gauge invariant and nearly perfectly conserved in the limit of large Lundquist numbers. For direct numerical simulations with ordinary viscosity and magnetic diffusivity operators, we find that the solution develops in a nearly self-similar fashion. In a diagram quantifying the instantaneous decay coefficients of magnetic energy and integral scale, we find that the solution evolves along a line that is indeed suggestive of the governing role of the Hosking integral. The solution settles near a line in this diagram that is expected for a self-similar evolution of the magnetic energy spectrum. The solution will settle in a slightly different position when the magnetic diffusivity decreases with time, which would be compatible with the decay being governed by the reconnection time scale rather than the Alfven time.

Despite spatial and temporal fluctuations in turbulent astrophysical systems, mean-field theories can be used to describe their secular evolution. However, observations taken over time scales much shorter than dynamical time scales capture a system in a single state of its turbulence ensemble. Comparing with mean-field theory can falsify the latter only if the theory is additionally supplied with a quantified precision. The central limit theorem provides appropriate estimates to the precision only when fluctuations contribute linearly to an observable and with constant coherent scales. Here, we introduce an error propagation formula that relaxes both limitations, allowing for non-linear functional forms of observables and inhomogeneous coherent scales and amplitudes of fluctuations. The method is exemplified in the context of accretion disc theories, where inhomogeneous fluctuations in the surface temperature are propagated to the disc emission spectrum - the latter being a non-linear and non-local function of the former. The derived precision depends non-monotonically on emission frequency. Using the same method, we investigate how binned spectral fluctuations in telescope data change with the spectral resolving power. We discuss the broader implications for falsifiability of a mean-field theory.

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 favourable 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 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(rms) similar or equal to b(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 versus the spectral-tau closure, 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 O(10) Reynolds numbers. The transient behaviour 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.