ISSN 00167932, Geomagnetism and Aeronomy, 2012, Vol. 52, No. 2, pp. 254260. Pleiades Publishing, Ltd., 2012.Original Russian Text G.S. Sobko, V.N. Zadkov, D.D. Sokoloff, V.I. Trukhin, 2012, published in Geomagnetizm i Aeronomiya, 2012, Vol. 52, No. 2, pp. 271277.
According to the presentday concepts, geomagnetic reversals (polarity reversals), which repeatedly tookplace during the Earths geological history, are one of themost dramatic events studied in paleomagnetism (Christensen et al., 2010; Hulot et al., 2010). Several days beforereversals, it is possible to reproduce the geodynamo in thescope of a direct numerical simulation (Olson et al.,2010), and similar phenomena are encountered indynamo experiments (Berhanu et al., 2007).
At the same time, the nature of reversals (if theyexisted in the Earths history) remains unclear in manyrespects. The fact is that geomagnetic reversals havenot been directly observed by researchers. They wereonly found when reversals of the natural remanentmagnetization (NRM) of igneous and sedimentaryancient rocks were registered during paleomagneticstudies. In geological sections, time variations in therock NRM direction either corresponded to the direction of the presentday geomagnetic field or were antiparallel to this direction. Such NRM direction alternations are global, which is related to the assumptionthat NRM reversals are caused by geomagnetic reversals. However, when natural ferrimagnetic mineralswere studied, it was detected that these minerals canacquire thermal magnetization directed both along themagnetizing field and against this field (Trukhin et al.,2006). This phenomenon was called magnetizationselfreversal and is an alternative to geomagneticreversals.
In addition to the interpretation of paleomagneticdata (Hulot et al., 2010; Trukhin et al., 2006), there areunclarified questions related to the detection of thegeodynamospecific features resulting in reversals (ifgeomagnetic reversals nevertheless existed) since
regimes with magnetic field time reversals areunknown for other natural dynamos (Christensenet al., 2010; Hulot et al., 2010). It is difficult to detectthese specific features using only methods of directnumerical simulation because these methods areaimed at reproducing a phenomenon in all detailsrather than at elucidating its individual features.
Therefore, it seems reasonable to complete a directnumerical simulation with the construction of a simple phenomenon model that makes it possible tounderstand the phenomenon qualitative features.Similar models are well known in the literature (see,e.g., (Wicht et al., 2010; Roberts and Glatzmaier,2000; Dormy and Soward; Ershov et al., 1989)); however, these models are illustrative since they onlyreproduce the desirable behavior of the magnetic field,not pretending to the possibility of deriving thesemodels from complete geodynamo models in thescope of any explicitly described approximations. Ouraim is to obtain a similar model from the equations ofmean field electrodynamics and to study the modeldynamics.
2. LOWMODE APPROXIMATION
As a basis for the required model, we use a lowmode approximation for the dynamo in a thin spherical shell proposed in (Nefedov and Sokoloff, 2010).The essence of this approximation is that the meanfield dynamo equations (after various simplifications)are mapped onto the minimum possible system of thefirst several eigenfunctions for the problem of magnetic field damping in the absence of generationsources. This minimum set of functions is selected sothat the solution, which is now the set of the first several timedependent Fourier coefficients for the sys
Geomagnetic Reversals in a Simple Geodynamo ModelG. S. Sobko, V. N. Zadkov, D. D. Sokoloff, and V. I. Trukhin
Physical Faculty, Moscow State University, Moscow, Russiaemail: firstname.lastname@example.org
Received January 11, 2011; in final form, July 4, 2011
AbstractA simple finitedimensional geodynamo model, obtained from the equations of the mean fieldelectrodynamics and reproducing the phenomenon of geomagnetic reversals, is proposed. It has been indicated that the reversal scale obtained in the scope of this model is rather close to the observed scale in its properties. The reversal mechanism is related to the effect fluctuations. It is not necessary to substantiallychange the hydrodynamic parameters of the problem so that a reversal originates in the scope of such a model,but it is only sufficient to take the effect fluctuations into account. If the rms deviation of fluctuationsaccounts for 10% of the average value, a fluctuation of twothree standard deviations is sufficient for theorigination of a reversal, which quite agrees with the concept that reversals are rather rare phenomena.Another factor resulting in the regime with reversals is that the model can generate magnetic fields with different behaviors in different regions of the parametric space in linear mode: monotonically increasing fieldsand fields increasing with oscillations.
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GEOMAGNETIC REVERSALS IN A SIMPLE GEODYNAMO MODEL 255
tem of basis functions, would generally reproduce thebehavior of the field of the studied object in the casewhen generation sources are taken into account andcould not reproduce this behavior if the set wassmaller.
In this case, we require that such a solution contains (if the set of parameters is appropriate) selfexcitation of an initially weak magnetic field. In addition,in nonlinear mode, the model should give stationarysolutions or solutions with the socalled vascillations(periodic oscillations of parameters when their signremains constant). These solutions correspond to thegeomagnetic field behavior between reversals. Finally,in nonlinear mode, the model should have (certainly,in a different range of its parameters) solutions in theform of selfoscillations about zero average value,which correspond to the solar magnetic field behaviorduring a solar cycle. We certainly require that thismodel gives solutions with a nonzero magneticmoment of the system since precisely this moment isfirst of all observed in geo and paleomagnetic studies(Christensen et al., 2010; Hulot et al., 2010). Thus, itis necessary that the similarity in the geometry of theEarths and Suns shells affected by convection, as wellas the difference in the behavior of magnetic fields ofthese bodies, were reflected in the model.
The model proposed in (Nefedov and Sokoloff,2010) has all these properties, and the set of equationsdescribing this model has the form
Four Fourier coefficients ( , and ) are themodel parameters. The first two coefficients correspond to the first two modes of a poloidal field, and the
coefficient is proportional to the magnetic moment.The second two coefficients correspond to the first twomodes of a toroidal field. Nefedov and Sokoloff (2010)indicated that a smaller set of variables is insufficientfor us to construct the model of interest in contrast tothe prevailing opinion.
The linear terms of this model describe the selfexcitation process, whereas the nonlinear termsdescribe stabilization due to the nonlinear suppressionof helicity. Magnetic field selfexcitation is normallyrelated to the processes of poloidal magnetic fieldtransformation into a toroidal field due to differentialrotation and toroidal magnetic field transformation ina poloidal field owing to the socalled effect relatedto the convection mirror symmetry breakdown due to
2 21 111 1 2
3( 2 )
R b R bdaa b b
= + ,
2 21 221 2 2 1 1 2 2
3 ( )( ) 9 ( )
R R b bdab b a b b b b
+= + + + ,
11 2 1( 3 ) 4
Rdba a b
1,a 2,a 1b 2b
the Coriolis force action in a stratified medium (see,e.g., (Parker, 1979)).
The and quantities, nondimensionalized bythe eddy diffusion coefficient and the problems geometric parameters, are included in the set of equations(1)(4) as governing parameters. These quantitiescharacterize the amplitudes of the effect and differential rotation, respectively. After nondimensionalization, time is measured in conditional dimensionlessunits.
The period of geomagnetic field vascillation is usually taken equal to 105 years so that the results could becorrelated with observational data (Christensen et al.,2010; Hulot et al., 2010), whereas the solar activity(oscillation) period is 22 years. Together with theseparameters, the parameters characterizing the spatialdistribution of generation sources and other importantdetails omitted in this simplest approximation are certainly included in more detailed solar dynamo models.
The terms describing how a toroidal field is transformed into a poloidal one with the help of the effect are also omitted in the model equations sincethe effect of differential rotation on this transformation is much more intense (the socalled dynamo)(Crause and Rdler, 1980). In our approximation, atoroidal field is always much stronger than a poloidalone; therefore, the nonlinear terms that include poloidal modes are eliminated from the model.
For definiteness, we measure the magnetic field interms of the field at which the effect of the magneticfield on a flow becomes substantial; i.e., we assumethat the coefficient from (Nefedov and Sokoloff,2010) is equal to unity.
The latitudinal distribution of the magnetic fielddescribed by our model has the form
where is the latitude measured from the equator, B isthe toroidal component of the magnetic field, and A isthe toroidal component of the magnetic potentialresponsible for the poloidal magneti