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ISSN 0016�7932, Geomagnetism and Aeronomy, 2012, Vol. 52, No. 2, pp. 254–260. © 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. 271–277.

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1. INTRODUCTION

According to the present�day concepts, geomag�netic reversals (polarity reversals), which repeatedly tookplace during the Earth’s geological history, are one of themost dramatic events studied in paleomagnetism (Chris�tensen 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 Earth’s 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 direc�tion of the present�day geomagnetic field or were anti�parallel to this direction. Such NRM direction alter�nations are global, which is related to the assumptionthat NRM reversals are caused by geomagnetic rever�sals. 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 magnetizationself�reversal 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 thegeodynamo�specific 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 sim�ple 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)); how�ever, 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. LOW�MODE APPROXIMATION

As a basis for the required model, we use a low�mode approximation for the dynamo in a thin spheri�cal 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 mag�netic 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 sev�eral time�dependent 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, Russiae�mail: [email protected]

Received January 11, 2011; in final form, July 4, 2011

Abstract—A simple finite�dimensional geodynamo model, obtained from the equations of the mean fieldelectrodynamics and reproducing the phenomenon of geomagnetic reversals, is proposed. It has been indi�cated that the reversal scale obtained in the scope of this model is rather close to the observed scale in its prop�erties. The reversal mechanism is related to the α�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 α�effect fluctuations into account. If the rms deviation of fluctuationsaccounts for 10% of the average α value, a fluctuation of two–three 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 dif�ferent behaviors in different regions of the parametric space in linear mode: monotonically increasing fieldsand fields increasing with oscillations.

DOI: 10.1134/S0016793212020144

GEOMAGNETISM AND AERONOMY Vol. 52 No. 2 2012

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 con�tains (if the set of parameters is appropriate) self�exci�tation of an initially weak magnetic field. In addition,in nonlinear mode, the model should give stationarysolutions or solutions with the so�called 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 self�oscillations 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 theEarth’s and Sun’s 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 corre�spond 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 self�excitation process, whereas the nonlinear termsdescribe stabilization due to the nonlinear suppressionof helicity. Magnetic field self�excitation 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 so�called α�effect relatedto the convection mirror symmetry breakdown due to

2 21 111 1 2

3( 2 )

2 8

R b R bdaa b b

dtα α

= − − + ,

2 21 221 2 2 1 1 2 2

3 ( )( ) 9 ( )

2 8

R R b bdab b a b b b b

dtα α

+= + − − + + ,

11 2 1( 3 ) 4

2

Rdba a b

dtω

= − − ,

222

316

2

R adbb

dtω

= − .

1,a 2,a 1b 2b

1a

the Coriolis force action in a stratified medium (see,e.g., (Parker, 1979)).

The and quantities, nondimensionalized bythe eddy diffusion coefficient and the problem’s geo�metric parameters, are included in the set of equations(1)–(4) as governing parameters. These quantitiescharacterize the amplitudes of the α�effect and differ�ential rotation, respectively. After nondimensionaliza�tion, time is measured in conditional dimensionlessunits.

The period of geomagnetic field vascillation is usu�ally 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 cer�tainly included in more detailed solar dynamo models.

The terms describing how a toroidal field is trans�formed into a poloidal one with the help of the α�effect are also omitted in the model equations sincethe effect of differential rotation on this transforma�tion is much more intense (the so�called dynamo)(Crause and Rädler, 1980). In our approximation, atoroidal field is always much stronger than a poloidalone; therefore, the nonlinear terms that include poloi�dal 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 θ 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 low�mode 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 sta�tionary or periodic at constant model parameters:oscillations (Fig. 1a) and vascillations (Fig. 1b),respectively.

We should also note that the model includes a solu�tion with specific strongly anharmonic oscillations,i.e., the so�called dynamo bursts (Fig. 1c), in additionto these modes. Such self�oscillation 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 mod�

Rα

,Rω

αω

ξ

1 2sin 2 sin 4B b b= − θ + θ, 1 2cos cos 3A a a= θ − θ,

256


SOBKO 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 α AS A CAUSEOF REVERSALS

Following 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 self�excitation chain (specifically, the α coefficientdescribing the degree of convection mirror symmetry)fluctuates; i.e., right�hand vortices predominate overleft�hand ones in one hemisphere, and left�hand vor�

.R Rα ω

(a) (b)

(c)

0.4

–0.4

0.2

–0.2

012011010090

0.4

0.2

0140130120110100

t

a1

0.12

–0.12

0.06

–0.06

012010090

t

a1 a1

t

Fig. 1. Example of different time variations in the low�mode model nonlinear solutions: (a) oscillations, (b) vascillations, and (c)dynamo bursts. The time variations in the a1 coefficient, responsible for the system’s magnetic moment, are shown. Time is shownin conditional dimensionless units. From the standpoint of paleomagnetism, the period of one oscillation in panel (b) approxi�mately corresponds to 105 years.

1000

800

600

400

200

0 1.00.80.60.40.2Rα

Rω

12

34

5

1

Fig. 2. Parametric space of the low�mode model in the Rα

,Rω

coordinates with the regions corresponding to differentregimes: damping (1), stationary solution (2), vascillations(3), dynamo bursts (4), and oscillations (5).



tices predominate over right�hand ones in the otherhemisphere. This asymmetry of right and left origi�nates under the action of the Coriolis force in a strati�fied medium. Hoyng (1993) qualitatively explainedhow helicity fluctuations result in the origination ofthe geomagnetic field’s long�term evolution accompa�nied by numerous reversals.

Pronounced fluctuations in average values (specif�ically, in the α coefficient) are naturally present in thedynamo since the number of convective (or eddy) vor�tices 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 three�dimensionalgeodynamo model with fluctuating parameters actu�ally 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 con�sidered 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 α�effectorigination (Brandenburg and Sokoloff, 1993; Otmi�anowska�Mazur et al., 2006), we assume that thesefluctuations are comparatively long�term, so that theirmemory time is comparable with the period of oscilla�tions. This makes it possible to ignore the above unre�alistic assumptions of the Hoyng model. Rayan andSarson (2007) used the cascade model of MHD turbu�lence (Frick and Sokoloff, 1998), the solutions ofwhich also have a long memory period, as a random�ness generator.

4. MODEL ANALYSIS RESULTS

At a moderate (about 10–20%) standard deviationof α fluctuations, we actually obtained the solutions tothe geomagnetic reversal model equations expressed in

(a)

(b)

0.4

0.3

0.2

0.15900 600058505800 595057505700

0.75

–0.75

0.25

–0.50

05900585058005750

0.50

–0.25

5950 6000 6050 6100

t

a1

6050 6100t

4

3

2

Rα

Fig. 3. Example of a reversal in the model equation solution: the time variations in the (a) a1 and (b) Rα

coefficients. The hori�zontal lines separate the R

α ranges corresponding to different model regimes. The regimes are marked by numerals corresponding

to the curves in Fig. 2.

258


SOBKO et al.

the sign reversal of the a1 coefficient, which has thesense of a magnetic moment (Fig. 3a). Figure 3a indi�cates 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) resem�bling 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 dynamo

Rα

12000

16000

0

4000

8000

20000

90

120

0

30

60

150

Tim

e

Tim

e, M

Y

Fig. 4. Example of a polarity scale simulated by the model (left) as compared to a section of the scale constructed using paleo�magnetic data (right).



bursts, when the magnetic moment reverses its signduring one burst as a result of a rather rare α fluctua�tion. 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 system’s 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 con�structed 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. CONCLUSIONS

Proceeding from the general equations of meanfield electrodynamics and making various simplifica�tions, we managed to obtain a simple model reproduc�ing the regime of geomagnetic reversals. To produce aninversion in the scope of such a model, it is not neces�sary to substantially change the problem hydrody�namic parameters and it is sufficient to take intoaccount α�effect fluctuations. If the rms deviation offluctuations accounts for 10% of the average α value, areversal originates at a fluctuation of two–three stan�dard deviations, which is well consistent with the con�cept 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 α 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 α shows pro�nounced fluctuations.

Epochs without reversals (superchrons) also origi�nate in the model. The duration of these epochs is pos�sibly shorter than the observed periods. It seems thatwe can increase the periods with very large and verysmall α by introducing memory into α 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 Maun�der 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 complica�tions. For example, we consider only variables corre�sponding to axisymmetric configurations of the mag�netic 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 intro�duce 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 gen�erated with increasing dynamo operation intensity, thegeneration is absent at higher intensities (vascillationsand dynamo bursts originate before its disappear�ance), 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 undoubt�edly have a pronounced effect on the structure of aflow in the Earth’s outer core. We could easily intro�duce similar complications in the considered model;however, it is important that the simplest modelalready shows the behavior close to the actual one.

ACKNOWLEDGMENTS

We are grateful to M. Yu. Reshetnyak for useful dis�cussions.

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