Parker’s dynamo near the excitation threshold

A method of constructing asymptotic solutions for nonlinear mean-field dynamo equations near the excitation threshold is developed and applied to equations describing the solar dynamo in a Parker model. The form of solution obtained corresponds to the eigensolution for a kinematic dynamo, for the intensity of the generation sources at which self-excitation of the magnetic field begins (the so-called marginally stable eigenfunction). The wave amplitude is calculated.     

Solar Grand Minima and Random Fluctuations in?Dynamo Parameters

We consider to what extent the long-term dynamics of cyclic solar activity in the form of Grand Minima can be associated with random fluctuations of the parameters governing the solar dynamo. We consider fluctuations of the alpha coefficient in the conventional Parker migratory dynamo, and also in slightly more sophisticated dynamo models, and demonstrate that they can mimic the gross features of the phenomenon of the occurrence of Grand Minima over suitable parameter ranges. The temporal distribution of these Grand Minima appears chaotic, with a more or less exponential waiting time distribution, typical of Poisson processes. In contrast, however, the available reconstruction of Grand Minima statistics based on cosmogenic isotope data demonstrates substantial deviations from this exponential law. We were unable to reproduce the non-Poissonic tail of the waiting time distribution either in the framework of a simple alpha-quenched Parker model or in its straightforward generalization, nor in simple models with feedback on the differential rotation. We suggest that the disagreement may only be apparent and is plausibly related to the limited observational data, and that the observations and results of numerical modeling can be consistent and represent physically similar dynamo regimes.     

Superflares on Giant Stars

The Kepler mission has identified huge flares on various stars, including some solar-type stars. These events are substantially more energetic than solar flares, and are referred to as superflares. Even a low probability of such a superflare occurring on the Sun would be a menace to modern society. A flare comparable in energy to that of superflares was observed on September 24 and 25, 1989 on the binary HK Lac. Unlike the Kepler stars, observations of differential rotation are available for HK Lac. This differential rotation appears to be anti-solar. In the case of anti-solar differential rotation, dynamo models can producemagnetic-activity waves with dipolare symmetry, as well as quasi-stationary magnetic configurations with quadrupolar symmetry. The magnetic energy of such stationary configurations is usually about two orders of magnitude higher than the energy associated with activity waves. We believe that this mechanism could provide sufficient energy to produce superflares on late-type stars. Some simple models in support of this idea are presented.     

Resonances in Spherical Dynamos and Super-Flares

Stars similar to the Sun demonstrate super-flares, which are considerably more powerful than solar flares. It is believed that the magneticfield energies of these stars are much higher than that of the Sun. The present study attempts to explain such an anomalously high magnetic energy by resonance phenomena related to the stellar dynamo, which involve significant changes in the behavior of the solutions subject to certain external effects and satisfy certain parametric relationships. These resonance phenomena are studied using low-mode models for a dynamo occurring in two or one spherical shells. It is shown that resonance effects arising in these models can result in increases of the magnetic energies by one and a half orders of magnitude compared with nonresonance cases. It is also shown that resonance dynamo conditions can differ considerably from the simple resonance conditions used for oscillating systems. This can probably be explained by the fact that the excitation and propagation of magnetic waves in dynamo problems are closely connected with each other, so that the resonance equations remain nonlinear even when they are maximally simplified.     

Solar cycle according to mean magnetic field data

To investigate the shape of the solar cycle, we have performed a wavelet analysis of the large–scale magnetic field data for 1960–2000 for several latitudinal belts and have isolated the following quasi-periodic components: ∼22, 7 and 2 yr. The main 22-yr oscillation dominates all latitudinal belts except the latitudes of ±30° from the equator. The butterfly diagram for the nominal 22-yr oscillation shows a standing dipole wave in the low-latitude domain  (∣θ∣≤ 30°)  and another wave in the sub-polar domain  (∣θ∣≥ 35°)  , which migrates slowly polewards. The phase shift between these waves is about π. The nominal 7-yr oscillation yields a butterfly diagram with two domains. In the low-latitude domain  (∣θ∣≤ 35°)  , the dipole wave propagates equatorwards and in the sub-polar region, polewards. The nominal 2-yr oscillation is much more chaotic than the other two modes; however the waves propagate polewards whenever they can be isolated.
We conclude that the shape of the solar cycle inferred from the large-scale magnetic field data differs significantly from that inferred from sunspot data. Obviously, the dynamo models for a solar cycle must be generalized to include large-scale magnetic field data. We believe that sunspot data give adequate information concerning the magnetic field configuration deep inside the convection zone (say, in overshoot later), while the large-scale magnetic field is strongly affected by meridional circulation in its upper layer. This interpretation suggests that the poloidal magnetic field is affected by the polewards meridional circulation, whose velocity is comparable with that of the dynamo wave in the overshoot layer. The 7- and 2-yr oscillations could be explained as a contribution of two sub-critical dynamo modes with the corresponding frequencies.     

Excitation of non-axially symmetric modes of the Sun's mean magnetic field

The kinematic dynamo equations for the mean magnetic field are solved with an asymptotic method of the WKB type. The excitation conditions and main characteristics of the non-axially symmetric modes for a given distribution of the sources are obtained. Utilization of the helioseismologic data on the Sun's internal rotation permits an explanation, within the framework of dynamo theory, of the excitation of the main non-axially symmetric modes revealed in the Sun's magnetic field sector structure.     

The saturation of galactic dynamos

We summarize the current state of the long term discussion about the saturation mechanisms associated with rapid growth of small‐scale magnetic field, that operate in large‐scale galactic dynamos, and related problems with magnetic helicity conservation. Our general conclusion is that, taking into account magnetic helicity fluxes, large‐scale magnetic field can be amplified up to about the equipartition level. In contrast, models without helicity fluxes give an initial temporal magnetic field growth, but then decay. In our opinion, it is more appropriate to refer to the situation as a “potentially catastrophic scenario” rather than as “catastrophic α‐quenching” (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)