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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. 271277.2541. INTRODUCTIONAccording 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) sinceregimes 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 APPROXIMATIONAs 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 sysGeomagnetic Reversals in a Simple Geodynamo ModelG. S. Sobko, V. N. Zadkov, D. D. Sokoloff, and V. I. TrukhinPhysical Faculty, Moscow State University, Moscow, Russiaemail: sobko@physics.msu.ruReceived January 11, 2011; in final form, July 4, 2011AbstractA 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.DOI: 10.1134/S0016793212020144GEOMAGNETISM AND AERONOMY Vol. 52 No. 2 2012GEOMAGNETIC REVERSALS IN A SIMPLE GEODYNAMO MODEL 255tem 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(1)(2)(3)(4)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 to2 21 111 1 23( 2 )2 8R b R bdaa b bdt = + ,2 21 221 2 2 1 1 2 23 ( )( ) 9 ( )2 8R R b bdab b a b b b bdt += + + + ,11 2 1( 3 ) 42Rdba a bdt= ,2223162R adbbdt= .1,a 2,a 1b 2b1athe 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 formwhere 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 magnetic field.Accepting the lowmode geodynamo model, wealso accept another assumption made when this modelwas derived: the nonlinearity resulting in dynamoaction stabilization is assumed to be simple, so that thenonlinearity itself does not result in reversals and othercomplex phenomena (Nefedov and Sokoloff, 2010).Therefore, nonlinear model solutions are either stationary or periodic at constant model parameters:oscillations (Fig. 1a) and vascillations (Fig. 1b),respectively.We should also note that the model includes a solution with specific strongly anharmonic oscillations,i.e., the socalled dynamo bursts (Fig. 1c), in additionto these modes. Such selfoscillation modes wereobtained in the dynamo experiment in (Berhanu et al.,2007) and are known for the stellar dynamo models(Moss et al., 2004); the possibility of using these modR,R1 2sin 2 sin 4B b b= + , 1 2cos cos 3A a a= ,256GEOMAGNETISM AND AERONOMY Vol. 52 No. 2 2012SOBKO et al.els in order to model the cyclic activity of certain starsis discussed (Baliunas et al., 2006; Lanza, 2010).Figure 2 shows the plane of the model parameterswith the regions corresponding to different nonlineardynamo regimes. Figure 2 indicates that we graduallypass from the regime of damping to that of stationaryconfigurations, vascillations, and, finally, dynamobursts with increasing dynamo operation intensity,i.e., with increasing dynamo number Then, theregime of damping is formed and is replaced by that ofoscillations.3. FLUCUATIONS OF AS A CAUSEOF REVERSALSFollowing the idea put forward in (Hoyng, 1993),we use fluctuations of the dynamo system parametersas a factor resulting in geomagnetic reversals. We alsoassume that the weakest link in the magnetic field selfexcitation chain (specifically, the coefficientdescribing the degree of convection mirror symmetry)fluctuates; i.e., righthand vortices predominate overlefthand ones in one hemisphere, and lefthand vor.R R (a) (b)(c)0.40.40.20.20120110100900.40.20140130120110100ta10.120.120.060.06012010090ta1 a1tFig. 1. Example of different time variations in the lowmode model nonlinear solutions: (a) oscillations, (b) vascillations, and (c)dynamo bursts. The time variations in the a1 coefficient, responsible for the systems magnetic moment, are shown. Time is shownin conditional dimensionless units. From the standpoint of paleomagnetism, the period of one oscillation in panel (b) approximately corresponds to 105 years.10008006004002000 1.00.80.60.40.2RR123451Fig. 2. Parametric space of the lowmode model in the R,R coordinates with the regions corresponding to differentregimes: damping (1), stationary solution (2), vascillations(3), dynamo bursts (4), and oscillations (5).GEOMAGNETISM AND AERONOMY Vol. 52 No. 2 2012GEOMAGNETIC REVERSALS IN A SIMPLE GEODYNAMO MODEL 257tices predominate over righthand ones in the otherhemisphere. This asymmetry of right and left originates under the action of the Coriolis force in a stratified medium. Hoyng (1993) qualitatively explainedhow helicity fluctuations result in the origination ofthe geomagnetic fields longterm evolution accompanied by numerous reversals.Pronounced fluctuations in average values (specifically, in the coefficient) are naturally present in thedynamo since the number of convective (or eddy) vortices in such problems is large, but substantiallysmaller than the Avogadro number. This explanation isbased on an analogy with the system of two weaklyrelated pendulums excited by a random force and, inour opinion, correctly reproduces many features of aphenomenon. However, this explanation ignores thefact that the geomagnetic field does not demonstratean oscillating behavior outside reversals. Ryan andSarson (2007) indicated that the threedimensionalgeodynamo model with fluctuating parameters actually shows the required behavior; however, it is difficultto reveal the reversal mechanism using this rathercomplex model.One more important difference of our model fromthe Hoyng model consists in that this researcher considered fluctuations with the characteristic timedependent on the convective vortex rotation, which isusually substantially smaller than the period ofdynamo oscillations (or vascillations). Based on theexperience in the numerical simulation of the effectorigination (Brandenburg and Sokoloff, 1993; OtmianowskaMazur et al., 2006), we assume that thesefluctuations are comparatively longterm, so that theirmemory time is comparable with the period of oscillations. This makes it possible to ignore the above unrealistic assumptions of the Hoyng model. Rayan andSarson (2007) used the cascade model of MHD turbulence (Frick and Sokoloff, 1998), the solutions ofwhich also have a long memory period, as a randomness generator.4. MODEL ANALYSIS RESULTSAt a moderate (about 1020%) standard deviationof fluctuations, we actually obtained the solutions tothe geomagnetic reversal model equations expressed in(a)(b)0.40.30.20.15900 600058505800 5950575057000.750.750.250.50059005850580057500.500.255950 6000 6050 6100ta16050 6100t432RFig. 3. Example of a reversal in the model equation solution: the time variations in the (a) a1 and (b) R coefficients. The horizontal lines separate the R ranges corresponding to different model regimes. The regimes are marked by numerals correspondingto the curves in Fig. 2.258GEOMAGNETISM AND AERONOMY Vol. 52 No. 2 2012SOBKO et al.the sign reversal of the a1 coefficient, which has thesense of a magnetic moment (Fig. 3a). Figure 3a indicates that a reversal is sudden and has a very shortperiod. On the other hand, a reversal is preceded by aseries of episodes (two episodes in this case) resembling geomagnetic excursions. During these events,the magnetic moment strongly decreases andapproaches zero; however, its sign remains unchanged.Figure 3b shows the corresponding behavior. Itis clear that a reversal originates when we pass from theregime at the boundary between the stationary regimeand regime of vascillations to the regime of dynamoR1200016000040008000200009012003060150TimeTime, MYFig. 4. Example of a polarity scale simulated by the model (left) as compared to a section of the scale constructed using paleomagnetic data (right).GEOMAGNETISM AND AERONOMY Vol. 52 No. 2 2012GEOMAGNETIC REVERSALS IN A SIMPLE GEODYNAMO MODEL 259bursts, when the magnetic moment reverses its signduring one burst as a result of a rather rare fluctuation. In this case, the time of the magnetic momentsign reversal is much shorter than the time between thebursts, i.e., 105 years (in terms of paleomagnetism).We also note that the dynamo can switch to theregime of dynamo bursts during excursion (see Fig. 3b);however, a reversal is not observed in this case. Theevent that occurs in this case (excursion or reversal) isprobably related to the depth of the systems entry intothe region of dynamo bursts in the parametric spaceand to the dynamo burst implementation during thisshort period.Figure 4 presents a polarity scale constructed onthe basis of one of the model solutions. This scale iscompared with a geomagnetic polarity scale sectionwith the same number of reversals as in the case constructed using the data presented in (Gradstein et al.,2004). It is clear that the scales are generally similarand show not only reversals, but also prolong epochswithout reversals resembling superchrons.5. CONCLUSIONSProceeding from the general equations of meanfield electrodynamics and making various simplifications, we managed to obtain a simple model reproducing the regime of geomagnetic reversals. To produce aninversion in the scope of such a model, it is not necessary to substantially change the problem hydrodynamic parameters and it is sufficient to take intoaccount effect fluctuations. If the rms deviation offluctuations accounts for 10% of the average value, areversal originates at a fluctuation of twothree standard deviations, which is well consistent with the concept that reversals are comparatively rare.The duration of reversals is much shorter than theperiod of smooth magnetic field variations outsidereversals (vascillations). In our model, this is related tothe fact that the geodynamo switches to the regime ofdynamo bursts during a reversal, and the reversal timeduring a dynamo burst is substantially shorter than theperiod between these bursts, which is approximatelyequal to the period of vascillations.The used values of fluctuations fit well in the dataon the surface magnetic field helicity in active solarregions (Sokoloff et al., 2008), according to which thedegree of mirror symmetry dependent on shows pronounced fluctuations.Epochs without reversals (superchrons) also originate in the model. The duration of these epochs is possibly shorter than the observed periods. It seems thatwe can increase the periods with very large and verysmall by introducing memory into fluctuations,not increasing the occurrence frequency of suchepochs. In this case, superchrons and epochs with veryfrequent reversals become longer. However, in thescope of this work, we do not try to compare in detailthe obtained scale with the observed one.In the corresponding range of parameters, ourmodel also reproduces the regime of a solar cycle thatis replaced by global minimums similar to the Maunder minimum (see, e.g., (Soon and Yaskell, 2008)).This conclusion is similar to the result achieved in(Moss et al., 2008); however, we use a model with afinite (and small) number of degrees of freedom,whereas Moss et al. (2008) used a simple model, butwith an infinite number of degrees of freedom.The considered model includes various complications. For example, we consider only variables corresponding to axisymmetric configurations of the magnetic field. This means that the magnetic momentshould vanish during a reversal (although the entiremagnetic field can remain nonzero). This does notnecessarily reflect the actual behavior of a magneticmoment, which can reverse its sign remaining nonzeroin the absolute value and reverse its direction. To introduce the possibility of such a behavior into the model,it is sufficient to add several nonaxisymmetric modesto the model.We emphasize that the substantial element of ourmodel is that the first stationary configuration is generated with increasing dynamo operation intensity, thegeneration is absent at higher intensities (vascillationsand dynamo bursts originate before its disappearance), and oscillations originate when the intensitycontinues increasing. In other words, the mechanismcan only operate when sufficiently severe restrictionsare imposed on the problem parameters. However, thisdoes not contradict the general impression that thegeodynamo action is generally related to a certainregion in the space of parameters outside which thedynamo acts substantially differently.We emphasize again that our model purposefullyignores many important geodynamo features. Forexample, the magnetic field in the Earth can undoubtedly have a pronounced effect on the structure of aflow in the Earths outer core. We could easily introduce similar complications in the considered model;however, it is important that the simplest modelalready shows the behavior close to the actual one.ACKNOWLEDGMENTSWe are grateful to M. Yu. Reshetnyak for useful discussions.REFERENCESBaliunas, S., Frick, P., Moss, D., Popova, E., Sokoloff, D.,and Soon, W., Anharmonicity and Standing DynamoWaves: Theory and Observation of Stellar MagneticActivity, Mon. Not. R. Astron. 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