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Edited by: Bernard A. Housen, Western Washington University, USA

Reviewed by: Oscar Pueyo Anchuela, Universidad de Zaragoza, Spain; Peter Aaron Selkin, University of Washington, USA

*Correspondence: Klaudio Peqini, Department of Physics, Faculty of Natural Sciences, University of Tirana, Bulevardi Zogu I, Nr. 25, 1005 Tirana, Albania

This article was submitted to Geomagnetism and Paleomagnetism, a section of the journal Frontiers in Earth Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

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 paleomagnetic data. The former model considers mutual interactions of

The Earth's magnetic field is one of the most representative characteristics of our planet. There is a widespread consent that the main geomagnetic field is of internal origin and is created and maintained by a dynamo mechanism in the molten outer core of the Earth (Moffatt,

The field observed at the Earth's surface is mainly dipolar, at least for the last five Myr (Heimpel and Evans, ^{−3} s to 10^{8} years. The time variations with time scales shorter than several months are considered to have an external origin, while the time variation with a longer time scale are mainly of internal origin. The time variation with time scale from months to thousands of years is termed secular variation (SV) contains a small contribution from external origin, especially those variations of time scales of months to 10 years. Variations with longer time scale, namely geomagnetic field reversals or excursions, are properly due to internal sources (Merrill and Mcfadden,

A reversal is a complete flip of the polarity of the magnetic dipolar moment of the geomagnetic field (Krijgsman and Kent,

Despite of the increasing data which enhance our understanding on SV, especially abrupt changes such as jerks (Duka et al.,

Measurements of the Earth magnetic field show that in the last centuries the dipolar moment has decreased and such decreasing is lasting for longer periods as given by different SV models like CALS7K.2, CALSK.10b, or SHA.DIF.14K (Korte and Constable,

Much of the magnetic field variations are thought to have a stochastic nature and almost all kinds of different models: phenomenological, experimental, or numerical (Petrelis et al.,

Numerical simulations are a very helpful tool to study the statistical behavior of the geomagnetic field, to provide some insights in its dynamics and probably to predict near future changes. The numerical models span from models based on magneto-hydrodynamics (MHD) partial differential equations (Rüdiger and Hollerbach,

The complexity of the MHD equations requires powerful computational resources that often are inaccessible, whereas “toy” models can be simulated easily with a PC. Despite their simplicity, “toy” models seem to reproduce quite accurately many features of the geomagnetic field temporal variations. One of these models, the “domino” model, appears with several versions, but two of them are of particular interest: the Short-range Coupled Spins (SCS) model (Mori et al.,

In the dynamical equations of the SCS and LCS models, there is an important random term (see Section Statistical Models below), which makes the name stochastic model quite accurate. As a consequence, the magnetisation time series generated by these models show random polarity flips (Duka et al.,

Similarly to Duka et al. (

The structure of the paper is as follows: in Section Statistical Models we shortly describe both stochastic models together with the respective dynamical equations; in Section Some Results and Statistics we will present some preliminary results, comparing the statistical behavior of dipolar magnetic field generated by both models; in Section SV-like Time Series we present the results of the simulations of the long time series of SV of the dipolar geomagnetic field. In the last section we give our conclusions. We complete the article with an Supplementary Material where we include the MATLAB lists of programs used to generate the synthetic time series of the different models, whose main characteristics and results are explained in the article.

The fluid flow in the Earth's outer core is organized in well-defined columns known as convective columns outside the Taylor cylinder (Kageyama and Sato, _{i} (_{i} that it makes with the rotational axis, i.e., _{i} = (sinθ_{i}; cosθ_{i}). The kinetic energy

The potential energy _{i+1} = _{i} when

The Lagrangian

A Langevin-type equation is set up as follows:
_{i} is a Gaussian-distributed random number with zero mean and unit variance which is updated after each correlation time τ. The substitution of Equations (3) into (4) yields the dynamical equations:
_{0} = θ_{N} and θ_{N+1} = θ_{1}. (In both models (SCS and LCS), periodic boundary conditions are applied).

The LCS model differs from SCS model in the spin-spin interaction term and all the other terms in Equations (1) and (2) remain identical. The modified potential energy is:

The Equations (5) and (7) are integrated using a 4th-order Runge-Kutta subroutine by applying an internal function of MATLAB (ode45). The initial values of θ_{i} and time derivatives

The output of each simulation is the cumulative axial orientation of all macro-spins or axial synchronization, simply named magnetization:

It corresponds to the normalized axial dipolar moment (ADM) of the geomagnetic field or the first Gauss coefficient

The first example of a rigid rotating system where the dynamo process effect takes place is the homopolar disk dynamo (Moffatt,

Constructing more complicated systems with disk dynamos, one can get interesting results. The first example is the Rikitake two-disk dynamo, widely studied especially from the nonlinear dynamics perspective (Shimizu and Honkura,

Complex magnetic field time series can be obtained by constructing more complex systems with disk dynamos. Shimizu and Honkura (

Basically the IL model is a modification of the Lebovitz model. In this last model, ^{th} disk surrounds the axle of the i^{th} +1 disk (Figure _{i} and angular velocities Ω_{i}, where _{i} denotes the non-dimensional current intensity of _{i} denotes the non-dimensional angular velocity of the same disk. The only model parameter μ is a non-dimensional quantity that results from the non-dimensioning procedure and characterizes the contribution of the current that flows in the

If we replace one of the disk dynamos with another one with different physical parameter values, the homogeneity is broken. Let us choose the

Here the parameters

The output of the system is the total magnetisation or the normalized sum of the axial magnetic fields generated by all the disks. The magnetic dipole moment generated by a current

The SCS and LCS models have the same six independent parameters. It would be a great challenge to exploit the whole parameter space of the model. Duka et al. (

In order to compare results of the two models, the same set of parameter values are used, precisely: γ = −1, λ = −2, κ = 0.1, ε = 0.4, _{i} is updated after each time step Δ_{i} uniformly distributed in the interval (0, 2π) and

The full run comprises 30,000,000 time steps and we print the output every 100 time steps, i.e., the full time series has 300,000 time steps. In Figures

The typical feature in both series is the apparently random variance of magnetisation and random change of polarity. It is Power Spectral Density (PSD) of the time series that supplies very valuable information about the statistical behavior of the system in different frequency ranges. The PSDs calculated for long time series (300,000 values) of both SCS and LCS models are shown in Figures

In Figures

In order to determine which of the models is statistically closer to the paleomagnetic series of reversals (Cande and Kent,

It can be seen by Equation (10) that the IL model has seven independent parameters, where six of them characterize the physical quantities of the disk dynamos (we use the values μ = 1.0,

We studied some dependencies of the IL model from the parameter values, but our study does not completely explore all options for the parameter space variations. Especially, we studied the dependency from the number of disks

Simulations with different values of the free parameters of the IL model show some interesting results. The increase of μ results in an increase of

The PSDs of all series of magnetisation are qualitatively similar: all of them have a three slope pattern. Quantitatively there are differences especially in the magnitude of the second slope. This part of the power spectra is important because it comprises variations that occur in the time scale from thousands to hundreds of thousands of years, i.e., the time scale of SV. The system with the statistical behavior closest to SHA.DIF.14K model (Pavon-Carasco et al.,

The reversal statistics (Figure

We generated by the LCS and IL models the time series of magnetisation that are statistically closest to the long time series of SV of the observed Axial Dipolar Moment (ADM) according to paleomagnetic models like as SHA.DIF.14K, at least in the range of parameter values of both models that we have investigated.

In order to compare the results of different models regarding the SV, the time series of axial magnetisation generated by LCS and IL models with proper parameter values and with initial values according to the series of SHA.DIF.14K model that starts at 14,000 BP. The full run comprises 100,000 years, from 14,000 BP to 86,000 terrestrial years in the future (after present, AP). This period of time corresponds to the time-scale of SV. The appropriate parameter values of the LCS model which produce a statistically similar time series with the SHA.DIF.14K time series, found empirically are (Duka et al., _{i}, uniformly randomly generated, such that their sum is equal to the initial value of the magnetic dipolar moment (ADM) of the SHA.DIF.14K time series. The initial angular velocities

A similar approach is applied for the IL model. The set of parameter values we used is μ = 0.8,

In both cases we obtained 30 time series respectively. Then we calculated the averaged time series. The SHA.DIF.14K time series and the averaged time series generated by LCS and IL models are all shown in Figure

The time series generated by the LCS model approximately reproduces the patterns of SV not only from the statistics point of view, but also by the time dependence of the ADM magnitude. Despite of the quantitative differences, the series have similar statistical behavior (Figures

The long SV time series is practically a future extension of the actual 14 kyr period initially based on the SHA.DIF.14K model. We performed 30 runs to obtain the averaged long SV time series. The series generated by both models and the respective power spectra are shown in Figures

We studied the time series generated by two stochastic models, the LCS model (a version of the “domino” model) and IL model (an extended version of the Rikitake two disk dynamo model). These models consider two distinct ways of dipolar magnetic field generation through collective interaction of dynamo elements: a global interaction of macrospins (LCS model) and a one-sense interaction among neighboring disk dynamos (IL model). In the former case there are implemented secondary interactions like friction and random forces, whilst in the latter case the magnetic field generation is based on interactions among disk dynamos including electric resistance that is analogous to the dissipation in the LCS model. However based on the IL model, more complicated systems including secondary interactions can be explored. The statistical analysis of the magnetisation series generated by both models suggests that the LCS model is more appropriate than IL model to simulate the low frequency processes of the geomagnetic field, i.e., reversals. On the other hand, the power spectra study shows no significant difference between the IL model and LCS model for higher frequency variations. Despite of this, the time series generated by the models showed that the IL model, at least for the considered range of parameters and periods of time of several millenia, is not reliable because of the large discrepancies between the time series of the IL model and paleomagnetic models like SHA.DIF.14K model (Pavon-Carasco et al.,

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Some financial support was provided by the Istituto Nazionale di Geofisica e Vulcanologia (INGV) through the funded project LAIC-U.

The Supplementary Material for this article can be found online at: