Foundations

We consider a class of strongly mixing sequences with infinite second moment. This class contains important GARCH processes that are applied in econometrics. We show the relative stability for such processes and construct a counterexample. We apply these results and obtain a new CLT without the requirement of exponential decay of mixing coefficients, and provide a counterexample to this as well.

This study investigates the origin and amplification of magnetic fields in planets and galaxies, emphasizing the foundational role of a seed magnetic field (SMF) in enabling dynamo processes. We propose a universal mechanism whereby an SMF arises naturally in systems where an orbiting body rotates non-synchronously with respect to its central mass. Based on this premise, we derive a general equation for the SMF applicable to both planetary and galactic scales. Incorporating parameters such as orbital distance, rotational velocity, and core radius, we then introduce a dimensionless factor to characterize the amplification of this seed field via dynamo processes. By comparing model predictions with magnetic field data from the solar system and the Milky Way, we find that the observed magnetic fields can be interpreted as the product of a universal gravitationally induced SMF and a body-specific amplification factor. Our results offer a novel perspective on the generation of magnetic fields in a wide range of astrophysical contexts and suggest new directions for theoretical investigation, including the environments surrounding black holes.

Using the results of numerical simulations and solar observations, this study shows that the transition from deterministic chaos to hard turbulence in the magnetic field generated by the emerging small-scale, near-surface (within the Sun’s outer 5–10% convection zone) solar MHD dynamos occurs through a randomization process. This randomization process has been described using the concept of distributed chaos, and the main parameter of distributed chaos $\beta$ has been employed to quantify the degree of randomization (the wavenumber spectrum characterising distributed chaos has a stretched exponential form ). The dissipative (Loitsianskii and Birkhoff–Saffman integrals) and ideal (magnetic helicity) magnetohydrodynamic invariants govern the randomization process and determine the degree of randomization at various stages of the emerging MHD dynamos, directly or through Kolmogorov–Iroshnikov phenomenology (the magnetoinertial range of scales as a precursor of hard turbulence). Despite the considerable differences in the scales and physical parameters, the results of numerical simulations are in quantitative agreement with solar observations (magnetograms) within this framework. The Hall magnetohydrodynamic dynamo is also briefly discussed in this context.