Parker's Dynamo Research Articles

Parameter Regimes of Hemispherical Dynamo Waves in a Spherical Shell From 3D MHD Simulations

AbstractThe MHD dynamo process, which generates a magnetic field through turbulent motion inside an electrically conducting fluid layer, is of fundamental importance to earth and planetary science. Numerical simulations have been employed to investigate the MHD dynamo process for several decades and can reproduce various observed magnetic field properties, including dipolar and multipolar magnetic fields with different field strengths. In this work, we focus on a particular type of dynamo solution where the magnetic fields are concentrated in either the northern or the southern hemisphere and show wave‐like regular reversals and propagations. We surveyed an extensive parameter range and found that these dynamo waves are the predominant solution when the thermal drive is sufficient to maintain a magnetic field but insufficient to drive a chaotic multipolar dynamo. These waves seem to require moderate convection forcing, characterized by a local Rossby number between 0.04 and 0.1. Further investigation shows that the waves likely are αΩ Parker dynamo waves, based on the consistency of the simulated wave frequency with the mean‐field dynamo theory. Our results have implications for the observed dichotomy of planetary magnetic fields as well as the possibility of the absence of paleomagnetic records while the dynamos were still operating.

ON THE EVOLUTION OF ANGULAR MOMENTUM, MAGNETIC ACTIVITY AND MASS LOSS RATE OF LATE TYPE MAIN SEQUENCE STARS

With the best data, I find that nearly all 0.5 to 1.2 M main sequence stars converge to a single rotational mass-dependent sequence after 750 Myr; when M > 0.8 M_, most of them converge in ≈ 120 Myr. If stars rotate as rigid bodies, most have angular momenta within clear bounds. The lower bound defines a terminal main sequence rotational isochrone, the upper one coincides with slow rotators from the Pleiades; stars from Praesepe delineate a third one. Mass dependent exponential relationships between angular momentum and age are determined. Age estimates based on the angular momentum are acceptable for stars older than 750 Myr and with M > 0.6− 0.7 M_⨀. The Rossby number indicates that the Parker dynamo may cease early on in stars with M/M_≥ 1.1. An empirical formula and a model for the torque, and a relation between rotational period and magnetic field, lead to a formula for the evolution of the mass loss rate; the present solar rate is near a minimum and was about five times larger when life on Earth started.

Growth of the inner core in the mean-field dynamo model

Application of Parker's dynamo model to the geodynamo with the growing inner core is considered. It is shown that decrease of the inner core size, where intensive magnetic field generation takes place, leads to the multi-polar magnetic field in the past. This effect reflects the decrease of the region of the effective magnetic field generation. The process is accompanied by increase of the reversals number and decrease of intensity of the geomagnetic field. The constraints on the mechanisms of convection in the liquid core are discussed.

Phase space of a two-dimensional model of a Parker dynamo

The dependences of the magnetic-field strength, variations of the magnetic field, and the multipole level on the amplitudes of the α and ω effects are considered using a two-dimensional model for a Parker dynamo in a spherical layer. Calculations have been carried out for both traditional spatial distributions of α and ω and geostrophic regimes obtained from three-dimensional modeling of thermal convection. Two-dimensional distributions of the dynamo-wave velocities in the zone where magnetic field is generated are presented. Comparisons with the solar and planetary dynamos are considered.

Inverse problem in Parker's dynamo

The inverse solution of the 1D Parker dynamo equations is considered. The method is based on minimization of the cost-function, which characterize deviation of the model solution properties from the desired ones. The output is the latitude distribution of the magnetic field generation sources: the αand ω-effects. Minimization is made using the Monte-Carlo method. The details of the method, as well as some applications, which can be interesting for the broad dynamo community, are considered: conditions when the invisible for the observer at the surface of the planet toroidal part of the magnetic field is much larger than the poloidal counterpart. It is shown that at some particular distributions of α and ω the well-known thesis that sign of the dynamo-number defines equatorial symmetry of the magnetic field to the equator plane, is violated. This is the e-book version of the article, published in Russian Journal of Earth Sciences (doi:10.2205/2015ES000558). It is generated from the original source file using LaTeX’s epub.cls class. It is also demonstrated in what circumstances magnetic field in the both hemispheres have different properties, and simple physical explanation of this phenomenon is proposed.

Probing latitudinal variations of the solar magnetic field in cycles 21–23 by Parker's Two-Layer Dynamo Model with meridional circulation

Abstract. Principle component analysis (PCA) of the solar background magnetic field (SBMF) measured from Wilcox Solar Observatory (WSO) magnetograms revealed the following principal components (PCs) in latitudes: two main symmetric components, which are the same for all cycles 21–23, and three pairs of asymmetric components, which are unique for each cycle. These SBMF variations are assumed to be those of poloidal magnetic field travelling slightly off-phase from pole to pole while crossing the equator. They are assumed to be caused by a joint action of dipole and quadruple magnetic sources in the Sun. In the current paper, we make the first attempt to interpret these latitudinal variations in the surface magnetic field with Parker's two-layer dynamo model. The latitudinal distributions of such waves are simulated for cycles 21–23 by the modified Parker's dynamo model taking into account both α and ω effects operating simultaneously in the two (upper and lower) layers of the solar convective zone (SCZ) and having opposite directions of meridional circulation. The simulations are carried out for both dipole and quadruple magnetic sources with the dynamo parameters specifically selected to provide the curves fitting closely the PCs derived from SBMF variations in cycles 21–23. The simulations are optimised for matching the positions of maximums in latitude, the number of equator crossings and the phase difference between the two dynamo waves operating in the two layers. The dominant pair of PCs present in each cycle is found to be fully asymmetric with respect to the magnetic poles and produced by a magnetic dipole. This pair is found to account for the two main dynamo waves operating between the two magnetic poles. There are also three further pairs of waves unique to each cycle and associated with multiple magnetic sources in the Sun. For the odd cycle 21 the simulated poloidal field fits the observed PCs, only if they are produced by magnetic sources with a quadruple symmetry in both layers, while for the even cycle 22 the fit to the observed PCs is achieved only in the case of quadruple magnetic sources in the upper layer and dipole sources in the inner layer. For the other odd cycle 23 the fit to observation is obtained for the quadruple magnetic sources in the inner layer and the dipole sources in the upper layer. The magnitudes of dynamo numbers D defining the conditions (depth and latitude) of a magnetic flux formation and the numbers N of zeros (equator crossings by the waves) are found to increase and the meridional circulation speed to decrease with a cycle number increase (D = −700, N = 3 for cycle 21 and D = −104, N = 9 for cycle 23). The phase delays between the waves in each unique pairs are also found to increase with the cycle number from ~9° in cycle 21 to ~13° in cycle 23.

A dynamical system for the parker dynamo in the case of quadrupole symmetry of the magnetic field

Adynamical systemin the case of quadrupolar symmetry of the toroidal field is constructed. The system obtained reproduces the regimes of oscillations, vacillations, dynamo bursts, and a configuration with monotonically growing magnetic fields reaching stationary regimes. The ranges of dynamo numbers corresponding to these regimes are found. Butterfly diagrams for the steady-oscillation regime are constructed for both the poloidal and toroidal components of the magnetic field.

A low-mode dynamo with meridional circulation and dipolar symmetry of the magnetic field

A dynamic system for the Parker dynamo including meridional circulation that is applicable to astrophysical objects is constructed. The meridional circulation is able to control the regimes for the generation of magnetic fields. If the meridional flows are weak, regimes with steady oscillations, dynamo bursts, fluctuations, and chaotic components are possible. When the meridional circulation is strengthened, the range of dynamo numbers required for fluctuations and dynamo bursts is reduced, and gradually vanishes; at the same time, the range required for oscillations is increased and raised to higher dynamo numbers. The latitude-time distributions of the toroidal and poloidal magnetic fields for steady oscillations are presented.

Connections beween photospheric current helicity, flares and solar subsurface kinetic helicity

The α-effect is central to magnetic field generation in αΩ dynamo theory (Parker, 1955), however, it lacked observational and experimental supports over a long term (Sokoloff, 2007). Mirror asymmetries of magnetic field (magnetic helicity) were not observed until the recent 20 years (Seehafer, 1990; Pevtsov et al., 1995, Bao & Zhang, 1998). Later, the advancement in local helioseismology has made possible the measurements of solar subsurface velocity field and the vorticity (or kinetic helicity) as well (Zhao & Kosovichev, 2003). Apparently, frozen flow with the magnetic field in the fluid of high magnetic Reynold number is a necessary condition for the α-effect in the framework of Parker's dynamo model.

Effect of stochasticity in mean field dynamo models

We present a comprehensive investigation into the effect of choosing the stochastic control parameters in a simplified—Parker dynamo model. Through considering the manifold of marginal stability, i.e., the region of parameter space where the mean growth rate is zero, we show that stochastic fluctuations are not prohibitive to dynamo. Furthermore, by directly comparing results obtained by periodic and Gaussian coloured noise alpha with identical characteristic time-scales and fluctuating amplitudes, we find that the transition to dynamo is significantly eased for stochastically fluctuating alpha. The effect of stochasticity in magnetic diffusion is also investigated, highlighting the importance of resonance between poloidal and toroidal magnetic fields on the growth rate. Furthermore, we show that probability density functions of the growth-rate, magnetic field, and magnetic energy can provide a wealth of useful information regarding the dynamo behaviour/intermittency. Finally, the statistical properties of the dynamo such as temporal correlation and fluctuating amplitude are found to be dependent on the distribution of the fluctuations in stocastic parameters.

The properties of a dynamo wave in the case of intense meridional circulation

The solution of a Parker dynamo was constructed for the case of intensive meridional circulation using a method similar to WKB. We show how to build a solution in the transition from the traveling-wave regime to the regime of stationary magnetic field configuration. A solution of the Hamilton-Jacobi equations was discovered for the problem of a dynamo containing a triple point in the complex plane wave vector.

Time-lag effects in Parker’s dynamo

The effects of a time lag in the magnetic quenching of the α effect is considered for an oscillating magnetic field in a Parker dynamo. The hypothesis of a parametric resonance in the system is justified, a modification of the solution is found, and the appearance of processes with periods much longer than the fundamental oscillation period is demonstrated.

Investigation of the helicity of magnetic fields on the Sun within the Parker dynamo model

Construction of the butterfly diagrams for the magnetic helicity in several approximations of a Parker dynamo has been carried out. The diagrams are constructed both for the cases of efficient generation of the magnetic field (large dynamo numbers) and for the weak generation (a small dynamo number). The corresponding asymptotic solution to the solar dynamo is used in the first case. The butterfly diagrams for different values of the meridional circulation were studied due to this solution. The butterfly diagrams are constructed and based on the few-mode approximation, which is valid for moderate dynamo numbers. The issue of which butterfly diagram features are common in all these approximations and can be compared with observational data is discussed.

Stellar Dynamo Waves: Asymptotic Configurations

We investigate the parameter space of a Parker dynamo with a simple alpha quenching nonlinearity, taking as governing parameters the dynamo number D (D<0) and the ratio of diffusion times in the radial and latitudinal directions in the convective zone. The latter parameter, μ, is connected with the aspect ratio (dimensionless thickness) of the convective zone. We isolate two asymptotic configuration of the dynamo waves excited by the Parker dynamo in the limiting case of strong generation. Apart from the standard case with the solar type dynamo wave travelling from mid-latitudes to the equator, we describe a form of dynamo activity which is basically an anharmonic standing wave. The first situation occurs when μ increases with |D|. With μ fixed and |D| increasing, the second asymptotic configuration occurs. We discuss possibilities of identifying these asymptotic configurations with various types of stellar activity as traced by stellar CaII data.

Magnetic helicity and solar activity cycle: observations and dynamo theory

We study a simple model for the solar dynamo in the framework of the Parker dynamo, with a nonlinear dynamo saturation mechanism based on magnetic helicity conservation arguments. We find a parameter range in which the model demonstrates a cyclic behaviour with properties similar to that of Parker dynamo with the simplest form of algebraic -quenching. We compare the nonlinear current helicity evolution in this model with data for the current helicity evolution obtained during 10 years of observations at the Huairou Solar Station of China. We conclude that, in spite of the very preliminary state of the observations and the crude nature of the model, the idea of using observational data to constrain our ideas concerning magnetic field generation in the framework of the solar dynamo appears promising.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

Magnetic helicity evolution during the solar activity cycle: Observations and dynamo theory

We study a simple model for the solar dynamo in the framework of the Parker migratory dynamo, with a nonlinear dynamo saturation mechanism based on magnetic helicity conservation arguments. We find a parameter range in which the model demonstrates a cyclic behaviour with properties similar to that of Parker dynamo with the simplest form of algebraic alpha-quenching. We compare the nonlinear current helicity evolution in this model with data for the current helicity evolution obtained during 10 years of observations at the Huairou Solar Station of China. On one hand, our simulated data demonstrate behaviour comparable with the observed phenomenology, provided that a suitable set of governing dynamo parameters is chosen. On the other hand, the observational data are shown to be rich enough to reject some other sets of governing parameters. We conclude that, in spite of the very preliminary state of the observations and the crude nature of the model, the idea of using observational data to constrain our ideas concerning magnetic field generation in the framework of the solar dynamo appears promising.

A Model of the Double Magnetic Cycle of the Sun

It has been argued that the solar magnetic cycle consists of two main periodic components: a low-frequency component (Hale's 22 yr cycle) and a high-frequency component (quasi-biennial cycle). The existence of the double magnetic cycle on the Sun is confirmed using Stanford, Mount Wilson, and Kitt Peak magnetograph data from 1976 to 1996 (solar cycles 21 and 22). In the frame of the Parker's dynamo theory, a model of the double magnetic cycle is presented. This model is based on the idea of two dynamo sources separated in space. The first source of the dynamo action is located near the bottom of the convection zone, and the second operates near the top. The model is formulated in terms of two coupled systems of nonlinear differential equations. It is demonstrated that in the case of a weak interaction between the two dynamo sources, the basic features of the double magnetic cycle, such as the existence of two components and the observed temporal variations of the high-frequency component, can be reproduced.

Simulation of geomagnetic polarity reversals by a model of interacting dipole sources

We describe a simple model, derived from Parker's dynamo theory. The field source consists of N interacting point dipoles, characterized by a finite lifetime. A Monte Carlo process gives the temporal evolution of the system. When the number of dipoles is of the order of the elementary cells present in Parker's theory, for a wide range of coupling parameters between the dipoles the thermodynamic fluctuations may generate excursions and polarity reversals, whose morphology and temporal distribution may be compared with those observed for the geomagnetic field.

Periodic and aperiodic dynamo waves

Abstract In order to show that aperiodic magnetic cycles, with Maunder minima, can occur naturally in nonlinear hydromagnetic dynamos, we have investigated a simple nonlinear model of an oscillatory stellar dynamo. The parametrized mean field equations in plane geometry have a Hopf bifurcation when the dynamo number D=1, leading to Parker's dynamo waves. Including the nonlinear interaction between the magnetic field and the velocity shear results in a system of seven coupled nonlinear differential equations. For D>1 there is an exact nonlinear solution, corresponding to periodic dynamo waves. In the regime described by a fifth order system of equations this solution remains stable for all D and the velocity shear is progressively reduced by the Lorentz force. In a regime described by a sixth order system, the solution becomes unstable and successive transitions lead to chaotic behaviour. Oscillations are aperiodic and modulated to give episodes of reduced activity.

Periodic and Aperiodic Behaviour in Stellar Dynamos

We have constructed a simple parametrized mean field dynamo model that includes the dynamical interaction between the magnetic field and differential rotation. This system of seven coupled nonlinear ordinary differential equations has finite amplitude oscillatory solutions (corresponding to Parker's dynamo waves) when the dynamo number D&gt;1. We have studied two regimes. In the first, the velocity shear is reduced by the Lorentz force and there are stable periodic solutions for all D&gt;1. In the second there is a transition from strictly periodic oscillations to aperiodic (chaotic) behaviour as D is increased. This simple example shows that nonlinear hydromagnetic dynamos can produce aperiodic cycles, with Maunder minima, as observed in the sun and other late-type stars.