Disk Dynamo Model Research Articles

Insights into pre-reversal paleosecular variation from stochastic models

To provide insights on the paleosecular variation of the geomagnetic field and the mechanism of reversals, long time series of the dipolar magnetic moment are generated by two different stochastic models, known as the “domino” model and the inhomogeneous Lebovitz disk dynamo model, with initial values taken from the from paleomagnetic data. The former model considers mutual interactions of N macrospins embedded in a uniformly rotating medium, where random forcing and dissipation act on each macrospin. With an appropriate set of the model’s parameters values, the series generated by this model have similar statistical behaviour to the time series of the SHA.DIF.14K model. The latter model is an extension of the classical two-disk Rikitake model, considering N dynamo elements with appropriate interactions between them. We varied the parameters set of both models aiming at generating suitable time series with behaviour similar to the long time series of recent secular variation (SV). Such series are then extended to the near future, obtaining reversals in both cases of models. The analysis of the time series generated by simulating the models show that the reversals appears after a persistent period of low intensity geomagnetic field, as it is occurring in the present times.

MAGNETIC FIELD REVERSALS OF THE EARTH: A TWO–DISK RIKITAKE DYNAMO MODEL

The Sun and the Earth possess dipolar magnetic fields that exhibit polarity reversals. Recent works, based on numerical simulations and laboratory experiments, found similar dynamical behaviours. We present results of a statistical analysis of a numerical simulation based on a generalized two–disk dynamo model. From a first investigation, we found that the dynamics of the system is controlled by the variations of the ratio of the torques and we observed different dynamical regimes characterized either by bursts or reversals, which can be periodic or random, of the magnetic field.

A multi‐scale disk dynamo model

AbstractWe suggest to combine the mean‐field description of the galactic dynamo with a shell description of small‐scale turbulence for high values of hydrodynamic and magnetic Reynolds numbers. The model allows us to keep the balance of energy and realizes the kinematic and non‐linear regimes of the magnetic field generation. The suggested approach gives a possibility to study the role of magnetic helicity in the generation process. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Interpretation of Geomagnetic Field Reversal on the Basis of Fluctuation Growth by Using Disk Dynamo Model.

Based on the viewpoint of fluctuation growth, we analyze and discuss the geomagnetic field reversal, focusing on a three-disk dynamo model, mainly on the effect of mutual inductance on dynamic behaviors. If mutual inductance is large, which means strong interaction between neighboring disks and their associated circuits, then a very slight fluctuation of electric current is found to be greatly enhanced after a short incubation time. On the contrary, after a long incubation time, fluctuation growth is very slow when mutual inductance is small. Therefore the degree of interaction is related to the length of incubation time: the stronger the interaction, the shorter incubation time. The analogy between the disk dynamos and the geomagnetic field reversal patterns implies slight fluctuations are greatly enhanced during the period for the past 25 Ma, whereas fluctuation growth is very slow during the period of the Cretaceous Superchron, implying that mutual inductance is dependent on time. Hence, disk dynamo calculations including time dependence of mutual inductance are found to explain geomagnetic field reversal pattern qualitatively. From the mathematical viewpoint, numerical solutions for time dependent mutual inductance is found to be obtained by frequency modulation (FM) over those for time independent mutual inductance.

Poissonian asymptotics of a randomly perturbed dynamical system: Flip-flop of the stochastic disk dynamo

A dynamical system with two stable equilibrium points will show a flip-flop motion between the neighborhoods of the two points when it is perturbed by small random noises. A typical example is the stochastic disk dynamo model where the two equilibrium points correspond to the two polarities of the earth's magnetic field. We prove what has been suggested by a computer simulation, that the counting process of the flips or the reversals of the earth's field converges to a standard Poisson process if the time is suitably scaled.

The disk-dynamo model for natural dynamos

We analyze the kinematic and the dynamic self-excited disk dynamo models for natural dynamos. In the kinematic model (given disk angular velocity), the magnetic field can either increase or decrease exponentially with time, or it can be constant if the angular velocity has a specific value that is a function of the geometry and of the resistance of the circuit. With the dynamic disk dynamo (given mechanical power input), there are both transient and steady-state solutions, and the steady-state magnetic flux density is proportional to the square root of the power input. The polarity of the induced magnetic field in either dynamo is the same as the seed field, and erratic polarity reversals can be expected if the power input is erratic.

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).

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.

A perturbed disk dynamo model and polarity reversals of the earth's magnetic field.

A perturbed disk dynamo model and polarity reversals of the earth's magnetic field.

Ergodicity of randomly perturbed Lorenz model

We show, by using the Liapunov method, that the Lorenz model perturbed by Gaussian white noise is ergodic for any Rayleigh number. Our theory confirms two properties which have been found by numerical calculation. We also discuss the ergodicity of some other randomly perturbed dissipative systems, a one-dimensional laser, and a homopolar disk dynamo model of the geomagnetism.

Precession in a Disk Dynamo Model of the Earth's Dipole Field

THE Earth's dipole field has undergone more polarity reversals than was hitherto supposed, according to recent studies of geomagnetic data1,2. It has been estimated2 that the Laschamp polarity interval lasted only about 20,000 yr, and Cox3 has predicted that several intervals shorter than 50,000 yr remain to be discovered. I suggest here that such polarity intervals may be connected with the effects of precession acting on the Earth's dynamo process.