Fractal properties of geodynamo models

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  • This article was downloaded by: [Tulane University]On: 18 September 2013, At: 09:47Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

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    Fractal properties of geodynamomodelsA. Anufriev a & D. Sokoloff b ca Bulgarian Academy of Sciences, Geophysical Institute, 1113,Sofia, Bulgariab Moscow State University, Department of Physics, 119899,Moscow, Russiac Observatoire de Paris, Section astrophysique de Meudon,92195, Meudon, FrancePublished online: 19 Aug 2006.

    To cite this article: A. Anufriev & D. Sokoloff (1994) Fractal properties of geodynamo models,Geophysical & Astrophysical Fluid Dynamics, 74:1-4, 207-223, DOI: 10.1080/03091929408203639

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  • Geophys. Astrophys. Fluid Dynamics, Vol. 74, pp. 207-223 Reprints available directly from the publisher Photocopying permitted by license only

    0 1994 Gordon and Breach Science Publishers S.A. Printed in Malaysia

    FRACTAL PROPERTIES OF GEODYNAMO MODELS

    A. ANUFRIEV

    Bulgarian Academy of Sciences, Geophysical Institute, 1 1 13 SoJa, Bulgaria

    D. SOKOLOFF Moscow State University, Department of Physics, 119899 Moscow, Russia

    and Observatoire de Paris, Section astrophysique de Meudon, 921 95 Meudon, France

    (Received 7 June 1993; infinal form 7 July 1993)

    The statistical behavior of time series of reversals (TSR) of the Geomagnetic field is studied. It is demonstrated that TSR possess the self-similarity property under time scale transformations. The authors believe that such a simple statistical property should be a consequence of the unique physical mechanism leading to the reversals, and it is not necessary to find special reasons for the explanation of the long and short time intervals between them. To test this idea, TSR were analysed which result from various models of the Geomagnetic field. Simplest models like the well-known Rikitake model do not exhibit self-similarity properties despite the fact that their TSR show statistical behavior. More realistic models with significantly larger number of degrees of freedom (e.g. Hollerbach ef al., 1992) really demonstrate such self-similarity properties.

    KEY WORDS Geodynamo, magnetic field reversals, fractal, Hausdorf dimension.

    1. INTRODUCTION

    The geomagnetic field has had many alternations of polarity during its evolution. Of course, the information about late events of field reversals is much more complete than information about earlier ones. For the latest period of about 200 Ma (more precisely, up to boundary Callovian-Bathonian), however, a relatively complete magnetostrati- graphic (MS) scale exists, which gives intervals of different polarities of the geomagnetic field. Though there is some difference between various MS scales (e.g. Cox, 1982; Berggren et al., 1985) with a number of contradictions between them, the main properties of this scale can be considered as well-defined.

    The character of field reversals of the MS time scale seems to be a non-periodic chaotic process. The subjective impression is supported by statistical analysis (Cox, 1968), in which the process is treated as a Poison random process with non-constant mean period 7 between reversals.

    Let us consider in more detail the properties of a Poison random process. The probability for the magnetic field not to change its polarity during the time period t is proportional to exp (- t / t ) . The memory of the Poison process is short: after time interval t = 37 it forgets its past up to accuracy 1 - e-3( N 0.95). The total number of

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  • 208 A. ANUFRIEV A N D D. SOKOLOFF

    field reversals of the scale is about 300 in 168 Ma, so the maximal length of the interval between two reversals must not be greater than approximately 6r. Elementary analysis of the MS time scale demonstrates that such a simple estimate is completely incorrect (for instance, there exists a very long superchrone on the Cox scale in Creatceous times). That is why authors, who have studied reversals, describe this scale in terms of a Poison process with unsteady t. Other models of the scale can be constructed in which difficulties of this type do not arise.

    One of these models can be described as follows. Let us consider a one-dimensional random walk of a particle during which its displacement in both directions has equal probability 1/2. We interpret the instants of return to the origin as moments of reversals. This random walk, unlike the Poison process, has relatively long memory. If we are far away from the origin, we need a very long time to return to it, which means that in this artificial model long periods between field reversals are much more likely than for the Poison process. It is easy to estimate the probability density of the time interval A between adjacent reversals for the random walk: p(A) - A - 12. The ratio of the probability density of two given durations between field reversals for the random walk is given by p(AJp(A2) =(A2/Al)12 and does not depend on the time unit of the scale. This scale invariance implies that a long period of the scale, when considered with low resolution, is similar to a short period, when considered with high resolution.

    Mathematicians describe this scale invariance as the property of self-similarity, and objects with such properties are known as fractals. Fractal sets appear in different areas ofscience. We can also mention its poetical description in the Rhapsody On poetry by Dean Swift:

    So, naturalists observe, apea hath smallerjeas on him prey, and these have smaller yet to bite em and so proceed ad infnitum.

    Something like this seems to appear in the structure of the MS time scale. For example, Cox (1982) and Berggren et al. (1985) apply different generalizations of the scale: Cox uses all reversals from his list, but Berggren e t al. neglect the shortest intervals between reversals (Figure la, b). Nevertheless, the structures of the scales look alike: both scales contain very long, intermediate and very short intervals between reversals, which are known as superchronos, chronos and subchronos, respectively. It is important to emphasize that the relatively short (about 5 Ma) intervals on the scale with high resolution is qualitatively similar to the longer period (about 60 Ma) on the scale with low resolution. Fractal geometry studies the properties connected with this self-similarit y.

    Different quantitative characteristics for the fractal set and especially for the MS time scale can be constructed. One such parameter, namely the Hausdorf dimension of the scale, is about 0.9 (see Section 2). The random walk model also corresponds to a fractal scale though its dimension is not 0.9, but 1/2 (see Section 3). The scale constructed with the Poison process is a nonfractal one: its dimension is exactly 1.

    Here we want to discuss the fractal properties of various models of the MS time scale starting from very simple (such as the Poison process or the random walk model), which are not concerned with the physical processes leading to reversals, and finishing with rather sophisticated models taking into account the magnetohydrodynamics of the Earth core.

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  • FRACTAL PROPERTIES OF THE GEODYNAMO 209

    '-0 Ma

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    Figure 1 Magnetostratigraphic time scale: (a)-the Cox scale, (6)-the Berggren et a!. scale.

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  • 210

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    A. ANUFRIEV AND D. SOKOLOFF

    40

    60 i b

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    Figure 1 (Continued.)

    It will be demonstrated here that the simple models of geomagnetic dynamo with field reversals, like the well-known Rikitake model, have no fractal behavior. More developed models of geodynamo (e.g. Hollerbach et al., 1992) really have fractal properties. However, although the am-dynamo model of Hollerbach et al. has the Hausdorf dimension that compares for some values of dimensionless parameters with the Hausdorf dimension of Cox scale, the situation is not entirely satisfactory. The solutions of this dynamo model oscillate far too quickly to represent geodynamo realistically (see Section 6).

    The property of self-similarity for the geomagnetic scale is valid of course only approximately. Neither very short nor very long time intervals between reversals can be revealed. The first are impossible because of the inertia of the physical processes taking part in the Earth liquid core. The second also cannot be found because of the limitation of the observational data.

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  • FRACTAL PROPERTIES OF THE GEODYNAMO 21 1

    2. FRACTAL PROPERTIES OF THE MAGNETOSTRATIGRAPHIC SCALE

    Let us introduce a resolution time A. We shall consider two field reversals as unresolvable, if the time interval between them is shorter than A. The total MS time scale is divided into intervals of length A, the total number of such intervals being N = T/A. Let us call N(A) the number of intervals in which at least one reversal occurs. Consider now the qualitative behavior of the function N(A). If A equals T , there is a single interval on the scale, so N(A) = 1. In the opposite case of very small A (equal to or smaller than the time resolution for the list of reversals used by Cox for construction of the scale), N(A) coincides with the total number N of reversals. There is an intermediate region between these asymptotic regimes which might possess fractal properties. If the case of fractal scale takes place, then in the intermediate region N(A) has power-law behavior:

    N(A) N MA-d

    with non-integer d. Equation (1) represents a description of the self-similarity of the MS time scale in terms of the function N(A). The value d is known as the Hausdorf dimension of the scale and M is its Hausdorf measure. This notation is connected with the name of the German mathematician F. Hausdorf who introduced this idea into geometry in the beginning of our century. It was introduced into physics by the famous book of B. Mandelbrot, 1982.

    To clarify the concept of the Hausdorf dimension, we consider a scale with approximately equidistant reversal points. If the time resolution is smaller, i.e. the time interval A is longer than the average temporal distance between the reversals, then, roughly speaking, there is at least one reversal on each time interval A. Since the total number of such intervals on the scale with given time T equals T/A, then N(A) N T/A - A-'. Thus N(A) has the power-law behavior (1) with index d = 1. This behavior corresponds to a straight-line profile when plotted on a log-log scale. According to Hausdorf, the dimension d = 1 means that this scale, when resolved on the interval A, can be considered as a one-dimensional manifold (a line). If, on the other hand, the time resolution is very high, then, evidently, the number of intervals containing reversals coincides with the total number of reversals N. Then the function N(A) is approximately constant

    N(A) = NAO.

    It means that the index d = O and therefore we must consider our scale as a set of isolated points (with zero dimension) rather than a line. Both of these objects are non-fractal because their dimension is an integer number. Fractal is a set with non-integer fractal Hausdorf dimension d. The Hausdorf measure M is a value such as curve length or surface area. For a fractal set with dimension d the unit of Hausdorf measure is (Ma)d.

    The Cox scale is only approximately self-similar. It implies that the value A in (1) can vary over a wide but limited region: Bmin < A < d,,, (in particular, d,,, < T and Bmin 2 To, where To is a typical period of the magnetic field oscillations - lo4 yrs). For

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  • 212 A. ANUFRIEV AND D. SOKOLOFF

    this reason the asymptotic formulae ofsuch type are known as intermediate (Barenblatt and Zeldovich, 1971).

    We have studied fractal properties for the Cox scale with N = 296 and T = 168 Ma. The results are summarized in Figure 2. The Hausdorf dimension of the Cox scale with accuracy about 0.02 is given by

    d = 0.89.

    The Hausdorf measure is approximately equal to

    M = 96.6(Ma)d.

    Let us stress that our investigations of fractal properties of Cox scale and MS scales obtained in MHD simulations (see below) are restricted by available information. Only one realization of each of MS scales which are really random ones are available. The total number of observed and simulated reversals is relatively small for fractal tests and we observe some fractal properties due to good luck. It is impossible in such a situation to apply a number of natural tests to check the reliability of fractal behavior of a given MS scale. The uncertaincies in Hausdorf dimension correspond only to uncertaincies in the determination of the straight-line region on Figure 2. It is why we use the most primitive method of obtaining Hausdorf dimension-the technique of the naked eye. The situation can be improved at least for the simulated scales, provided more rich numerical information can be obtained. Until then our results must be considered qum grano salus.

    Figure 2 The Cox scale S, has fr...