Modeling of the solar activity double cycle using dynamical systems

The generation and evolution of the Sun’s magnetic field and other stars is usually related to the dynamo mechanism. This mechanism is based on the consideration of the joint influence of the α effect and differential rotation. Dynamo sources can be located at different depths of the convection zone and can have different intensities. Based on such a system, the dynamical system in the case of the stellar dynamo in a two-layer medium has been constructed with regard to meridional fluxes in order to model the double cycle that corresponds to the simultaneous presence of 22-year and quasi-biennial magnetic field oscillations. It has been indicated that the regime of mixed oscillations can originate because a dynamo wave moves oppositely to the meridional flows in the upper layer of the convection zone. This results in the deceleration of the toroidal field propagation and in the generation of slow oscillations. In deeper layers, the directions of a dynamo wave and meridional flows coincide with each other, as a result of which fast magnetic fields originate. Therefore, the total contribution of two oscillations with different frequencies corresponds to the appearance of quasi-biennial cycles against 22-year cycles. It has been indicated that the beating regime, which can be related to the secular oscillations of solar magnetic activity, originates in the system when the meridional flows are weak.     

Forecasting the parameters of sunspot cycle 24 and beyond

Solar variability is controlled by the internal dynamo which is a non-linear system. We develop a physical–statistical method for forecasting solar activity that takes into account the non-linear character of the solar dynamo. The method is based on the generally accepted mechanisms of the dynamo and on recently found systematic properties of the long-term solar variability. The amplitude modulation of the Schwabe cycle in dynamo's magnetic field components can be decomposed in an invariant transition level and three types of oscillations around it. The regularities that we observe in the behaviour of these oscillations during the last millennium enable us to forecast solar activity. We find that the system is presently undergoing a transition from the recent Grand Maximum to another regime. This transition started in 2000 and it is expected to end around the maximum of cycle 24, foreseen for 2014, with a maximum sunspot number R_max=68±17. At that time a period of lower solar activity will start. That period will be one of regular oscillations, as occurred between 1730 and 1923. The first of these oscillations may even turn out to be as strongly negative as around 1810, in which case a short Grand Minimum similar to the Dalton one might develop. This moderate-to-low-activity episode is expected to last for at least one Gleissberg cycle (60–100 years).     

分层电离层模型中的L电流体系

本文分析了电离层风发电机理论中影响电流分布的几种主要因素,认为电离层电导率模型是最主要的影响因素。在计算太阴日变化（L）电流体系时,本文放弃了过去习惯采用的无限薄球壳的电导率模型,使用了分层电导率模型。考虑电导率随高度的变化以及电导率极大值的高度随纬度的变化,得到了与观测结果较为符合的理论L电流体系。本文的结果还指出,在处理某些全球性发电机理论问题时,不能简单地假定电离层为距地面等高度的无限薄球壳,而必须同时考虑大气潮汐振荡的特性及电导率随高度的变化。由此得出结论:发展三维电导率模型对于电离层风发电机理论是必要的。     

Solar dynamo theory: a new look at the origin of small-scale magnetic fields

Fausto Cattaneo and David W Hughes delve beneath the surface of the Sun with numerical models of turbulent convection.
Although magnetic dynamo action is traditionally associated with rotation, fast dynamo theory shows that chaotic flows, even without rotation, can act as efficient small-scale dynamos. Indeed, numerical simulations suggest that granular and supergranular convection may generate locally a substantial part of the field in the quiet photosphere.     

Tuning of the mean-field geodynamo model

Parker’s two-dimensional (2D) dynamo model with an algebraic form of nonlinearity for the α-effect is considered. The model uses geostrophic distributions for the α-effect and differential rotation, which are derived from the three-dimensional (3D) convection models. The resulting configurations of the magnetic field in the liquid core are close to the solutions in Braginsky’s Z-model. The implications of the degree of geostrophy observed in the 3D dynamo models for the behavior of the mean magnetic field are explored. It is shown that the reduction in geostrophy leads to magnetic field reversals accompanied by the relative growth of the nondipole component of the field on the surface of the liquid core. The simulations with a random α-effect which causes turbulent pulsations are carried out. The approach is capable of producing realistic sequences of magnetic reversals.     

Kinematic dynamo induced by helical waves

We investigate numerically the kinematic dynamo induced by the superposition of two helical waves in a periodic box as a simplified model to understand the dynamo action in astronomical bodies. The effects of magnetic Reynolds number, wavenumber and wave frequency on the dynamo action are studied. It is found that this helical-wave dynamo is a slow dynamo. There exists an optimal wavenumber for the dynamo growth rate. A lower wave frequency facilitates the dynamo action and the oscillations of magnetic energy emerge at some particular wave frequencies.     

地球发电机过程的实验室模拟研究

关于地磁场起源的研究依赖于理论研究、实验室实验、数值模拟和实地观测等四个方面的工作：理论研究给出物理框架,提出物理思想;实验室实验检验理论预言,发现新现象;数值模拟可以对很大参数空间的复杂过程和对象给出多方面的过程描述;而实地观测不仅仅是上述三方面研究工作的出发点和基础,而且也是对研究结果的最终检验.本文介绍地球发电机过程实验研究的主要结果,特别是2000年后的突破性进展.目前,大多数实验研究还停留在运动学发电机水平,相对于数值模拟的巨大成就来说,实验研究需要大力发展,这就是正在发展中的第二代实验发电机.     

The flux ejection dynamo effect

Abstract

The mean-field effects of cyclonic convection become increasingly complex when the cyclonic rotation exceeds ½-π. Net helicity is not required, with negative turbulent diffusion, for instance, appearing in mirror symmetric turbulence. This paper points out a new dynamo effect arising in convective cells with strong asymmetry in the rotation of updrafts as against downdrafts. The creation of new magnetic flux arises from the ejection of reserve flux through the open boundary of the dynamo region. It is unlike the familiar α-effect in that individual components of the field may be amplified independently. Several formal examples are provided to illustrate the effect. Occurrence in nature depends upon the existence of fluid rotations of the order of π in the convective updrafts. The flux ejection dynamo may possibly contribute to the generation of field in the convective core of Earth and in the convective zone of the sun and other stars.     

The role of mean circulation in parity selection by planetary magnetic fields

Abstract

The kinematic dynamo problem is considered for certain steady velocity fields with symmetries that are plausible in a rapidly rotating convective system. By generalizing results proved for the mean field dynamo model by Proctor (1977a), it is shown that for a related “comparison problem” with modified boundary conditions, the eigenvalues are degenerate if there is no axisymmetric mean circulation, with modes of dipole and quadrupole parity excited with equal ease. The comparison problem can be shown to be closely similar to the dynamo problem when there is a region unfavourable to dynamo action surrounding the dynamo region. The near-symmetries found by Roberts (1972) for the mean field model are invoked to suggest that a close correspondence is likely even when this region is absent. It is therefore conjectured that such mean motions may be important in explaining the observed preference for solutions of dipole parity by planetary dynamos.     

Large-eddy simulations of convection-driven dynamos using a dynamic scale-similarity model

Dynamo simulations require sub-grid scale (SGS) models for the momentum and heat flux, the Lorentz force, and the magnetic induction. Previous large eddy simulations (LES) using the scale similarity model have represented many aspects of the SGS motion. However, discrepancies are observed due to interchanging the order of filtering operation and spatial differentiation. In this study, we implement a correction term for this commutation error specifically for the scale-similarity model. Furthermore, we implement a dynamic scheme to evaluate time-dependent coefficients for the SGS models. We perform dynamo simulations in a rotating plane layer with different spatial resolutions, and compare results for the time dependence of the large-scale magnetic field. Simulations are performed at two different Rayleigh numbers, using constant values for the other dimensionless numbers (Ekman, Prandtl, and magnetic Prandtl numbers). Both cases show that the dynamic LES can accurately represent the large-scale magnetic field, whereas the dynamo failed in the direct simulations without the SGS terms at the same spatial resolutions. We conclude that the dynamic versions of the SGS and commutation error correction are essential for successful dynamos on coarser grids.     

Nature of the sunrise effect in daily electric field variations at Kamchatka: 2. Electric field frequency variations

The power spectra of time variations in the electric field strength in the near-Earth’s atmosphere and in the geomagnetic field horizontal component, which were simultaneously observed at the Paratunka observatory (φ = 52°58.3′ N; λ = 158°14.9′ E) in September 1999, have been studied. The periods of the day (including sunrise, sunset, and night) have been considered. It has been indicated that oscillations with periods T ～ 2.0–2.5 h are present in the power spectra of these parameters during the day. The intensity of these oscillations increases noticeably and the oscillations in the band of periods T < 1 h increase simultaneously in the field strength power spectra at sunrise. The variations in the argument of the cross-spectrum of these parameters indicated that oscillations in the 2.0–2.5 h period band are caused by sources that are located above the ionospheric dynamo region; at the same time, oscillations in the 0.5–1 h period band are caused by sources in the lower atmosphere. A possible mechanism by which these oscillations are generated, related to the vortex motion of convective cells that originate at sunrise in the boundary atmospheric layer, is proposed.     

Origin of the magnetic fields in the giant planets

Recent observations and progress in the understanding of various requirements for the generation of magnetic fields permit much more definite conclusions to be drawn about the fields of the giant planets than was possible until quite recently. The Jovian magnetic field of about 4 gauss could be either of primordial origin or generated by a thermally driven dynamo. The expected Saturnian field of about 1 gauss can be similarly accounted for either by a thermally or by a precessionally driven dynamo. The presence of a field on Uranus of perhaps 0.1 gauss presents a problem because although it could be accounted for by a thermally driven dynamo operating in a highly conductive shell of hydrogen, the so far unobserved thermal flux and convection may be too low. If such a dynamo were to operate then one would expect the field to show seasonal variations. A precessional dynamo driven by Miranda seems to be marginally possible. On Neptune a conductive shell similar to that on Uranus appears to be much thinner, which perhaps explains the absence of an active dynamo driven either thermally or precessionally by Triton. It is, however, very likely that Neptune does have a magnetic field but that it is too weak to lead to observable electromagnetic radiations.     

Properties of a nonlinear solar dynamo model

Abstract

A simple nonlinear model is developed for the solar dynamo, in which the real convective spherical shell is approximated by a thin flat slab, and only the back-reaction of the field B on the helicity is taken into account by choosing the simple law α = α(1-ζB ²), where α and ζ are constants, to represent the decrease in generation coefficient ζ with increasing field strength. Analytic expressions are obtained for the amplitude of the field oscillation and its period, T, as functions of the deviation d - d_CT of a dynamo number d from its critical value d_cr for regeneration. A symmetry is found for the case of oscillations of small constant amplitude: B(t+½T)= -B(t). A Landau equation is obtained that describes the transition to such oscillations.     

A unified approach to a class of slow dynamos

Abstract

Dynamo action in a highly conducting fluid with small magnetic diffusivity η is particularly sensitive to the topology of the flow. The sites of rapid magnetic field regeneration, when they occur, appear to be located at the stagnation points or in regions where the particle paths are chaotic. Elsewhere only slow dynamo action is to be expected. Two such examples are the nearly axially symmetric dynamo of Braginsky and the generalisation to smooth velocity fields of the Ponomarenko dynamo. Here a method of solution is developed, which applies to both these examples and is applicable to other situations, where magnetic field lines are close to either closed or spatially periodic contours. Particular attention is given to field generation in the neighbourhood of resonant surfaces where growth rates may be intermediate between the slow diffusive and fast convective time scales. The method is applied to the case of the two-dimensional ABC-flows, where it is shown that such intermediate dynamo action can occur on resonant surfaces.     

The initial temporal evolution of a feedback dynamo for Mercury

Various possibilities are currently under discussion to explain the observed weakness of the intrinsic magnetic field of planet Mercury. One of the possible dynamo scenarios is a dynamo with feedback from the magnetosphere. Due to its weak magnetic field, Mercury exhibits a small magnetosphere whose subsolar magnetopause distance is only about 1.7 Hermean radii. We consider the magnetic field due to magnetopause currents in the dynamo region. Since the external field of magnetospheric origin is antiparallel to the dipole component of the dynamo field, a negative feedback results. For an αΩ-dynamo, two stationary solutions of such a feedback dynamo emerge: one with a weak and the other with a strong magnetic field. The question, however, is how these solutions can be realized. To address this problem, we discuss various scenarios for a simple dynamo model and the conditions under which a steady weak magnetic field can be reached. We find that the feedback mechanism quenches the overall field to a low value of about 100–150 nT if the dynamo is not driven too strongly.     

The dynamo basis of solar cycle precursor schemes

We investigate the dynamo underpinning of solar cycle precursor schemes based on direct or indirect measures of the solar surface magnetic field. We do so for various types of mean-field-like kinematic axisymmetric dynamo models, where amplitude fluctuations are driven by zero-mean stochastic forcing of the dynamo number controlling the strength of the poloidal source term. In all stochastically forced models considered, the surface poloidal magnetic field is found to have precursor value only if it feeds back into the dynamo loop, which suggests that accurate determination of the magnetic flux budget of the solar polar fields may hold the key to dynamo model-based cycle forecasting.     

Nonlinear dynamics of a stellar dynamo in a spherical shell

Abstract

A simple mean-field model of a nonlinear stellar dynamo is considered, in which dynamo action is supposed to occur in a spherical shell, and where the only nonlinearity retained is the influence of the Lorentz forces on the zonal flow field. The equations are simplified by truncating in the radial direction, while full latitudinal dependence is retained. The resulting nonlinear p.d.e.'s in latitude and time are solved numerically, and it is found that while regular dynamo wave type solutions are stable when the dynamo number D is sufficiently close to its critical value, there is a wide variety of stable solutions at larger values of D. Furthermore, two different types of dynamo can coexist at the same parameter values. Implications for fields in late-type stars are discussed.     

中性风与等离子体行星波的同步振荡

本文是“赤道异常的行星波振荡”的续篇。提供新的事实以说明:由于中层大气的行星波振荡会调制到潮汐风上面,潮汐风上升到E层发电机区域,通过发电机效应产生电场及S_q电流体系的行星波振荡。电场沿磁力线传到F_2层,通过“喷泉效应”引起电离层赤道异常的行星波振荡。所有这些振荡来源相同,所以是同步的。     

The weak taylor state in an αω-dynamo

Abstract

A spherical αω-dynamo is studied for small values of the viscous coupling parameter ε ～ v^1/2, paying attention particularly to large dynamo numbers. The present study is a follow-up of the work by Hollerbach et al. (1992) with their choice of α-effect and Archimedean wind including also the constraint of magnetic field symmetry (or antisymmetry) due to equatorial plane. The magnetic field scaled by ε^1/2 is independent of ε in the solutions for dynamo numbers smaller than a certain value of D _b (the Ekman state) which are represented by dynamo waves running from pole to equator or vice-versa. However, for dynamo numbers larger than D _b the solution bifurcates and subsequently becomes dependent on ε. The bifurcation is a consequence of a crucial role of the meridional convection in the mechanism of magnetic field generation. Calculations suggest that the bifurcation appears near dynamo number about 33500 and the solutions for larger dynamo numbers and ε = 0 become unstable and fail, while the solutions for small but non-zero ε are characterized by cylindrical layers of local maximum of magnetic field and sharp changes of geostrophic velocity. Our theoretical analysis allows us to conclude that our solution does not take the form of the usual Taylor state, where the Taylor constraint should be satisfied due to the special structure of magnetic field. We rather obtained the solution in the form of a “weak” Taylor state, where the Taylor constraint is satisfied partly due to the amplitude of the magnetic field and partly due to its structure. Calculations suggest that the roles of amplitude and structure are roughly fifty-fifty in our “weak” Taylor state solution and thus they can be called a Semi-Taylor state. Simple estimates show that also Ekman state solutions can be applicable in the geodynamo context.     

An extension of the Toroidal Theorem

The well-known “toroidal theorem” of Elsasser and Bullard and Gellman rules out dynamo action in a conducting sphere when the velocity field has no poloidal part. It is here shown that for a fixed toroidal velocity field any poloidal velocity must attain a finite size if dynamo action is to be possible. The resulting “anti-dynamo” theorem generalises the earlier result of Childress by giving a bound on the product of the suprema of the toroidal and poloidal velocities.