Disk Dynamo Research Articles

Structural instability of the Rikitake Disk Dynamo

The Rikitake‐type disk dynamo, consisting of two identical single Faraday‐disk dynamos of the Bullard type coupled together, has been analyzed intensively for more than thirty years by: (a) mathematicians interested in its ability to behave chaotically, and (b) geophysicists needing low‐dimensional electromechanical analogues of the geodynamo in their efforts to elucidate the complex time series of geomagnetic polarity reversals over geological time. Implicit in the theory is what now appears to be an unwarranted assumption, namely that mechanical friction is unimportant both qualitatively and quantitatively as a dissipative agency in comparison with ohmic effects, and can therefore safely be neglected. The assumption is inconsistent with the findings of the present study, which shows that mechanical friction can render the Rikitake dynamo “structurally unstable” and consequently incapable of producing chaotic oscillations. Research on low‐dimensional analogues of the geodynamo should concentrate attention on structurally stable systems, with properties less dependent on the starting conditions.

Dynamo theory and paleomagnetism

For more than 200 years the origin of Earth's magnetic field was attributed to permanent magnetization. Even today no single argument (e.g., that Earth's deep interior is too hot to sustain permanent magnetization) conclusively rules out the permanent magnetization hypothesis. Nevertheless, when all the evidence is considered, this hypothesis can be safely discarded and replaced with an electric current (dynamo) hypothesis. Surprisingly, this can be done even though there is no adequate dynamo model for Earth. The development of geodynamo models began with the disk dynamo of Larmor in 1919 and expanded to include many classes of models, such as αω, α2, α2ω, Taylor state, and Model Z dynamos. Because of mathematical difficulties associated with solving the many coupled partial differential equations of dynamo theory, numerous simplifying assumptions are made. The majority of numerical dynamo models assume a three‐dimensional velocity field in an inviscid fluid and use mean field theory to solve for axisymmetric magnetic fields. There is also an increasing number of intermediate and strong field models emerging, in which feedback from the magnetic field to the velocity field is permitted. Nevertheless, these models still require several simplifying assumptions and there are many additional problems. For instance, many core parameters are difficult to estimate; there is debate on whether the top of the core is stably stratified and on the effects such stratification might have; what effects the presence of an inner core have; and whether the coupling across the coremantle boundary significantly affects the geodynamo. Perhaps it is not surprising that dynamo theoreticians, faced with large difficulties in mathematics and many uncertainties in physics, essentially choose to ignore input from fields such as paleomagnetism. However, it is precisely because of such difficulties that paleomagnetism can provide valuable constraints to narrow the range of viable dynamo models. For example, paleomagnetism ultimately should provide constraints on the velocity and magnetic field symmetries of dynamos; determine whether the geodynamo is in the weak, intermediate, or strong field regime; determine if there is a fundamental difference in dynamo processes during superchrons when reversals of the magnetic field essentially cease; and provide valuable information on the growth of the inner core and its possible stabilizing effects on geodynamo processes.

Chaos in geophysical fluids I. General introduction

Irregular buoyancy-driven flows occur in the atmospheres and fluid interiors of the Earth and other planets, and of the Sun and other stars, where they influence and often control the transfer of heat. Their presence is manifest in or implied by a wide variety of observed phenomena, including external magnetic fields generated by self-exciting magnetohydrodynamic (MHD ) dynamo action. Based on the laws of classical mechanics, thermodynamics and, in the case of electrically conducting fluids, electrodynamics, the governing mathematical equations are well known, but they are generally intractable owing to their essential nonlinearity. Computers play a key role in modern theoretical research in geophysical and astrophysical fluid dynamics, where ideas based on chaos theory are being applied in the analysis of models and the assessment of predictability. The aim of this paper is to provide a largely qualitative survey for non-specialists. The survey comprises two parts, namely a general introduction (Part I) followed by a discussion of two representative areas of research, both concerned with phenomena attributable to symmetry-breaking bifurcations caused by gyroscopic (Coriolis) forces (Part II), namely (a) large-scale waves and eddies in the atmospheres of the Earth, Jupiter and other planets (where, exceptionally, laboratory experiments have been influential), and (b) MHD dynamos. Various combinations of Faraday disc dynamos have been studied numerically as low-dimensional nonlinear electromechanical analogues of MHD dynamos, particularly in efforts to elucidate the complex time series of geomagnetic polarity reversals over geological time. The ability of the intensively studied Rikitake coupled disc dynamo system to behave chaotically appears to be a consequence of the neglect of mechanical friction, the inclusion of which renders the system structurally unstable.

On the resonance of galactic dynamos with density waves

The coupling between dynamo and density waves in galaxies is reinvestigated analytically and numerically. Applying the method of multiple time-scales to the marginal state of the disc dynamo, we derive an expression for the additional growth rate of the magnetic field as a result of coupling. It is shown that the first-order resonance in the density wave perturbation σ/σ 0 , and thus the occurrence of linear swing excitation, is possible. Moreover, the final linear growth rate of the magnetic field is always positive, and does not depend on the initial phase difference between the density wave and the magnetic field wave. Both the numerical and the analytical calculations show that ω=2ω 0 (ω=density wave frequency; ω 0 =dynamo frequency) is still the best condition for resonance, due to the above linear effect of swing excitation

Effect of advected fields on accretion disk dynamos

We present calculations of favored dynamo modes when advection of ambient magnetic fields onto accretion disks is important. These models are relevant for compact binary systems and young stellar objects and can be extended to active galactic nuclei (AGNs). The dynamo equation, including the standard alpha-effect, is modified to take into account advected magnetic fields. Vacuum boundary conditions are assumed outside the disk and the dynamo number switches sign across the equatorial plane. For the local steady state problem, critical dynamo numbers for various modes are obtained analytically. Our motivation is to investigate whether the dominant dynamo generation of quadrupolar magnetic fields and accretion of dipolar magnetic fields is likely to lead to particle acceleration in the form of jets. The results shown here are for a particlular choice of boundary conditions and geometry of the advected field. Besides examining other choices, we shall calculate growth rates for different modes, and the influence of the initial seed field configuration on the evolution of the magnetic fields in subsequent work.

Effect of Advected Fields on Accretion Disk Dynamos

AbstractWe present calculations of favored dynamo modes when advection of ambient magnetic fields onto accretion disks is important. These models are relevant for compact binary systems and young stellar objects and can be extended to active galactic nuclei (AGNs). The dynamo equation, including the standard α-effect, is modified to take into account advected magnetic fields. Vacuum boundary conditions are assumed outside the disk and the dynamo number switches sign across the equatorial plane. For the local steady state problem, critical dynamo numbers for various modes are obtained analytically. Our motivation is to investigate whether the dominant dynamo generation of quadrupolar magnetic fields and accretion of dipolar magnetic fields is likely to lead to particle acceleration in the form of jets. The results shown here are for a particular choice of boundary conditions and geometry of the advected field. Besides examining other choices, we shall calculate growth rates for different modes, and the influence of the initial seed field configuration on the evolution of the magnetic fields in subsequent work.Subject headings: acceleration of particles — accretion, accretion disks — binaries: close — galaxies: nuclei — MHD — stars: pre-main-sequence

The Internal Wave Driven Dynamo in Accretion Disks

We present the results of recent work on a model of angular momentum transport in thin, ionized, accretion disks. In particular, we consider three physical effects, each of which can be represented in terms of a local MHD mode in such a disk. First, we discuss the generation and propagation of internal modes within accretion disks, pointing out certain features which make them particularly promising as the driving force behind a strong, fast dynamo in accretion disks. Second, we point ou that the magnetic shearing instability (MSI) first discussed by Velikhov, and more recently by Balbus and Hawley in the specific context of these disks, provides a natural saturation mechanism for any disk dynamo, leading to an approximate equality between the dimensionless viscosity and the ratio of the dynamo growth rate to the local shear. Third, we argue that magnetic buoyancy is largely suppressed by the turbulence generated by the shearing instability. This prevents it from removing magnetic flux from the disk any faster than random turbulent diffusion. We find that the dimensionless viscosity α scales as (H/r)4/3, where H is the disk height and r is its radius.

Basic Properties of Swing-Excitation Mechanism

We describe the basic process of coupling between dynamo and density waves in galaxies. The growth rate of magnetic field as a result of coupling is derived, applying the method of multiple time scales to the marginal state of disk dynamo. It is shown that the 1st-order resonance in a perturbation of density and thus the linear swing excitation, is possible. Moreover, the growth rate of magnetic field is always positive and does not depend on the initial phase difference between the magnetic and density waves. Both the numerical and analytical calculations show that ω = 2ω0 (ω: density-wave freq., ω0: dynamo freq.) is still the best condition for resonance due to the linear effect of swing excitation.

Galactic Dynamics and Magnetic Field Amplification

Stimulated by recent high frequency radio polarization measurements of M83 and M51, we consider the influence of non-axisymmetric features (bars, spiral arms, etc…) on galactic magnetic fields. The time scale for the field amplification due to the non-axisymmetric velocity field is related to the time scale of angular momentum transport in the disk by the non-axisymmetric features. Due to its dissipational character (cooling and angular momentum transport) the gas plays a major role for the excitation of non-axisymmetric instabilities. Since it is the gaseous component of the interstellar gas in which magnetic field amplification takes place we consider the interplay of gasdynamical processes triggered by gravitational instabilities and magnetic fields. A comparison with the time scale for dynamo action in a disk from numerical models for disk dynamos gives the result that field amplification by non-axisymmetric features is faster in galaxies like M83 (strong bar) and M51 (compagnion and very distinct spiral structure), than amplification by an axisymmetric dynamo. Furthermore, we propose that axisymmetric gravitational instabilities may provide the turbulent magnetic diffusivity ηT. Based on standard galaxy models we obtain a radially dependent diffusivity whose numerical value rises from 1025cm2s−1 to 1027cm2s−1, declining for large radii.

A steady state of the disc dynamo

Abstract We discuss the steady states of the αω-dynamo in a thin disc which arise due to α-quenching. Two asymptotic regimes are considered, one for the dynamo numberD near the generation thresholdD 0, and the other for |D| ≫ 1. Asymptotic solutions for |D—D 0| ≪ |D 0| have a rather universal character provided only that the bifurcation is supercritical. For |D| ≫ 1 the asymptotic solution crucially depends on whether or not the mean helicity α, as a function ofB, has a positive root (hereB is the mean magnetic field). When such a root exists, the field value in the major portion of the disc is O(l), while near the disc surface thin boundary layers appear where the field rapidly decreases to zero (if the disc is surrounded by vacuum). Otherwise, when α = O(|B|−s) for |B| → ∞, we demonstrate that |B| = O(|D|1/s ) and the solution is free of boundary layers. The results obtained here admit direct comparison with observations of magnetic fields in spiral galaxies, so that an appropriate model of nonlinear galactic dynamos hopefully could be specified.

Dynamo magnetic field modes in thin astrophysical disks - an adiabatic computational approximation

An adiabatic approximation is applied to the calculation of turbulent MHD dynamo magnetic fields in thin disks. The adiabatic method is employed to investigate conditions under which magnetic fields generated by disk dynamos permeate the entire disk or are localized to restricted regions of a disk. Two specific cases of Keplerian disks are considered. In the first, magnetic field diffusion is assumed to be dominated by turbulent mixing leading to a dynamo number independent of distance from the center of the disk. In the second, the dynamo number is allowed to vary with distance from the disk's center. Localization of dynamo magnetic field structures is found to be a general feature of disk dynamos, except in the special case of stationary modes in dynamos with constant dynamo number. The implications for the dynamical behavior of dynamo magnetized accretion disks are discussed and the results of these exploratory calculations are examined in the context of the protosolar nebula and accretion disks around compact objects.

Reduced description of weakly non-linear -dynamos

A new local approach for non-linear (MHD) αω-dynamos based on the assumptions of weak non-linearity and axially symmetric magnetic field is presented. The non-linear behaviour of magnetic fields is found to be described by a generalized time-dependent Ginzburg-Landau equation. From this equation, the steady amplitudes of magnetic fields and their modulational stability are determined. Numerical examples are given for two cases simultaning a galactic dynamo and an accretion disc dynamo. In both cases, steady amplitudes exist because of non-linear feedback originating from the compressibility of conducting gaz, and they have turned out to be modulationally stable.

Generation of dynamo magnetic fields in thin Keplerian disks

view Abstract Citations (40) References (32) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Generation of Dynamo Magnetic Fields in Thin Keplerian Disks Stepinski, T. F. ; Levy, E. H. Abstract The combined action of nonuniform rotation and helical convection in protoplanetary disks, in the Galaxy, or in accretion disks surrounding black holes and other compact objects, enables an alpha-omega dynamo to generate a large-scale magnetic field. In this paper, the properties of such magnetic fields are investigated using a two-dimensional, partially numerical method. The structures of the lowest-order steady state and oscillatory modes are calculated for two kinds of external boundary conditions. A quadruple, steady state, highly localized mode is the most easily excited for low values of the dynamo number. The results indicate that, except under special conditions, disk dynamo modes tend to consist of relatively localized rings structures. For large values of the dynamo number, the magnetic field consists of a number of quasi-independent, spatially localized modes generated in various concentric rings filling the disk inward of a dynamo generation 'front'. Publication: The Astrophysical Journal Pub Date: October 1990 DOI: 10.1086/169268 Bibcode: 1990ApJ...362..318S Keywords: Accretion Disks; Black Holes (Astronomy); Dynamo Theory; Interstellar Magnetic Fields; Kepler Laws; Boundary Conditions; Milky Way Galaxy; Rotating Disks; Steady State; Astrophysics; ACCRETION; HYDROMAGNETICS full text sources ADS |

Poloidal Magnetic Fields in Galactic Central Regions

We discuss the origin of the observed strong poloidal fields Bz in the central regions of galaxies which have gaseous rings. In the context of galactic disk dynamo models only weak poloidal fields but strong toroidal fields result (Bϕ > Bz). Therefore we tie the strength of the poloidal fields to the central activity and apply known and tested ideas rigorously. A battery process on galactic scales is discussed which ensures the existence of a large scale magnetic field in the inner galactic region. The frozen-in field may be amplified by compression and turbulent stretching; the resulting field is poloidal. The central activity provides a nonaxisymmetric flow field which, just as the α – ω dynamo produces mostly Bϕ > Bz, can produce Bz ≥ Bϕ. This model explains the structure and strength of the magnetic field in star-burst galaxies like M82 (Lesch et al., 1989).

How well developed is the dynamo theory of flat objects?

Abstract The existing models of disk dynamos are reviewed and an attempt is made to assess the reliability of their predictions. We analyze key questions such as whether the infinitely extended layer is a proper approximation of the finite disk, or whether it overlooks the possibility of ‘‘forgotten modes'', i.e. whether there is a guarantee that, when the model is used in applications, the mode it predicts as the most easily excited is truly such. What we find is that all existing models of disk dynamos have shortcomings in these respects, so that the analysis of improved models is worth the effort.

The Shunted Homopolar Dynamo—An Analytic Approach to a Poincaré Map

We study the set of ordinary differential equations governing the homopolar disk dynamo. It is found that this set, which is a modification of the Lorenz system, has strange attractors of Lorenz type when R (which corresponds to the Rayleigh number of the Lorenz system) tends to infinity. A central aspect of this study is that the Poincare map for this limit can be obtained through Melnikov's perturbation method, in contrast to the usual dependence on numerical computation.

Structure of magnetic fields in spiral galaxies: global properties of the turbulent dynamo

General considerations are given to the global modes of magnetic fields generated by the galactic dynamo. The standard disc dynamo equations are extended to include the generation of azimuthal field components via the α-effect (i.e. the α2-dynamo), which is important in central regions of some galaxies (e.g. M51) where the shear of galactic rotation is weak. We focus our attention on the non-axisymmetric modes (m = 1 modes) to compare them with the observed bisymmetric magnetic fields. Although growing m = 1 modes do exist, they tend to grow more slowly than the axisymmetric modes with m = 0. Their eigenfunctions are confined within narrow regions in the radial direction of a disc. Thus, using the standard galactic dynamo, it is still difficult to explain the dominance of the bisymmetric fields and to reproduce their coherent structure extending over a disc by a single m = 1 mode. The growth rate of m = 1 modes depends on the thickness of the disc; it is higher in a thin disc than in a thick disc. As a general property of the field configuration generated by the galactic dynamo, the field lines show a more tightly wound appearance in the outer region of the disc, in spite of the decrease of shear strength in the galactic rotation.

Fluctuation of the nondipole magnetic field and its implication for the process of geomagnetic polarity reversal in the Cox model

In the Cox model on the mechanism of polarity reversal of the Earth's magnetic field, the nondipole magnetic field is assumed to behave differently from the dipole magnetic field; the former fluctuates at random with a time scale of the order of 1000 years, whereas the latter varies systematically with a longer time scale. Our analyses of Gauss coefficient data for various epochs show that the characteristic time of nondipole fluctuation is about 500 years, while paleomagnetic studies for lake sediments indicate 1000–2000 years. These estimates are comparable to the time scale for nondipole field fluctuation presumed in the Cox model. This result supports one aspect of the Cox model. It is also important to examine whether the Cox model actually presents a physically plausible mechanism of polarity reversal, in particular from the viewpoint of nonlinear dynamo action. In this respect, we first show that the Cox model is represented by a disk dynamo which is subject to magnetic perturbations exerted externally. Then we solve disk dynamo equations numerically and examine time evolution of the magnetic field of the disk dynamo with special attention to polarity reversals. We finally conclude that the mechanism of polarity reversal presented in the Cox model is basically operative. It is surprising, however, that excursionlike variations also appear as often as polarity reversals in this specific dynamo model. Detailed examination shows that the probability of magnetic field growth from the transitional state either in the previous direction (excursion) or in the opposite direction (reversal) is 1/2. We also conclude that the Cox model is capable of accounting for the exponential distribution of polarity intervals which has been found from paleomagnetic data.

Rikitake two‐disk dynamo system: Statistical properties and growth of instability

The apparent randomness in the observed behaviors of the geomagnetic field (polarity reversals, intensity fluctuations, etc.) can be simulated either by simplified physical models (e.g., disk dynamo models) or by stochastic models (e.g., the Cox model). The Rikitake two‐disk dynamo is particularly interesting as a simulator, since it displays aperiodic changes of current direction resembling the geomagnetic reversals. The behaviors of the Rikitake system are investigated in some detail, with various values for parameters (μ, k). The current intensity fluctuates around one of the two stationary states, and the amplitude of oscillation grows monotonically with time, following a growth curve common to models with the same k. When the amplitude becomes so big that the peak of oscillation exceeds some threshold value, the system flips to the opposite polarity and the oscillation resumes around the new stationary state. The growth of the instability and the eventual flip of the polarity occur as deterministic process as dictated by the differential equation governing the system, and yet the system behavior appears almost random for a wide range of parameters (μ, k). The randomness enters as the growth rate of the amplitude is exceptionally large just before the polarity reversal, and the system behavior in the next polarity interval depends critically on the largest amplitude gained in the previous interval. The two statistical distributions, those of polarity intervals and field intensity, can be classified into three types which are distinctly different from each other. The transition between these types occur almost in parallel in the two distributions in the (μ, k) space.

Stochastic disk dynamo as a model of reversals of the earth's magnetic field

A stochastic model is given of a system composed ofN similar disk dynamos interacting with one another. The time evolution of the system is governed by a master equation of the class introduced by van Kampen as relevant to stochastic macrosystems. In the model, reversals of the earth's magnetic field are regarded as large deviations caused by a small random force ofO(N −1/2) from one of the field polarities to the other. Reversal processes are studied by simulation, which shows that the model explains well the activities of the palaeomagnetic field inclusive of statistical laws of the reversal sequence and the intensity distribution. Comparisons are made between the model and dynamical disk dynamo models.