Parker’s model in geodynamo

ISSN 1069�3513, Izvestiya, Physics of the Solid Earth, 2012, Vol. 48, No. 4, pp. 326–334. © Pleiades Publishing, Ltd., 2012.Original Russian Text © M.Yu. Reshetnyak, 2012, published in Fizika Zemli, 2012, No. 4, pp. 44–52.

326

1. INTRODUCTION

Over the past few decades, geodynamo models havemade great progress: having started from simple one�dimensional (1D) considerations (see, e.g., Rikitake,1966), they developed into models describing thedynamics of the mean field (Krause and Rädler, 1980;Braginskii, 1978) and then matured into three�dimen�sional (3D) dynamo models, which were validated bynumerous comparisons with the observations (Schu�bert, 2011). The modern dynamo models present a seri�ous test for the physical models of the Earth and includethe convection equation, the equations of state andentropy, and the induction equation that describes theconversion of kinetic energy to magnetic energy. Thereis no doubt that 3D calculations on parallel computerswere and will be the main source of information on theconvection and generation of a magnetic field inside theliquid core of the Earth. As of now, the computationsprovide much finer details than the scale of the spa�tiotemporal resolution of the paleo� and archaeomag�netic data; moreover, the degree of detail of the model�ing has consistently increased. In turn, the appearanceof new information that is directly inaccessible by othermethods (for example, the data concerning the charac�ter of convection in the liquid core of the Earth) stimu�lates the development of simple dynamo models reflect�ing the key features of the geomagnetic field. Being wellsuited for describing complex processes without goingdeep into detail, these models are in demand by geo�physicists engaged in observations; besides, providing anew vision on the known phenomena, these models arealso attractive for specialists in 3D dynamo modeling.In our opinion, the iterative process of alteration of thesimple and complex models will provide a better under�standing of the subject.

The influence of the magnetic field on flows is one ofthe fascinating issues in dynamo theory. Apparently, themagnetic energy contained in the liquid core, which

exceeds the energy of the convective motion, maystrongly affect the dynamics of the fluid (see (Reshet�nyak and Sokolov, 2003) for details). However, the 3Dcalculations contradict this fact: although becomingless regular, the geostrophic convection in the core var�

ies weakly (Jones, 2000).1 In other words, some features

observed in the magnetic field at its generation thresh�old remain the same even when the model fieldapproaches a non�linear regime. This gives us hope thateven simple kinematic dynamo models will be suitablefor predicting the key properties of the magnetic field.Significant progress in this field was achieved in thelow�mode stellar dynamo modeling (Sokolov andNefedov, 2007; Nefedov and Sokolov, 2010) based onthe expansion of the sought solution in the first few freedamping modes. Using this approach, one may calcu�late various modes of generation of the solar magneticfield and construct realistic butterfly diagrams. Anothertrend in using the low�mode models is their applicationin the geodynamo (Sobko et al., 2012). In that work, theauthors used the Galerkin decomposition in the firsttwo damping modes in Parker’s equation to show thatintroducing stochasticity in α�effect (Hoyng, 1993)yields the scale of geomagnetic reversals close to thatobserved for the last 160 Ma. Below, we will show whichfeatures in the behavior of the geomagnetic field stemfrom the structure of the spectrum of the geomagneticfield in the linear kinematic approximation. Thisapproach is insensitive to the chosen basis functions ofthe expansion: the form of the solution depends on thesolution of the eigenvalue problem and on the selectedform of nonlinearity, which is absent in this problem.We also consider the obtained results in view of themodern 3D geodynamo models, which provide an ideaof the hydrodynamics of the Earth’s core.

1 The configurations of the magnetic field provided by these simu�lations are force�free.

Parker’s Model in GeodynamoM. Yu. Reshetnyak

Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, ul. Bol’shaya Gruzinskaya 10, Moscow, 123995 RussiaReceived January 19, 2011

Abstract—The eigenvalue problem for Parker’s dynamo model is considered. We study how the intensity ofconvection in the liquid core of the Earth affects the generation of the geomagnetic field with different direc�tions of latitudinal field propagation. The scenarios of transition of the geomagnetic field from frequent torare reversals are suggested.

Keywords: low�mode dynamo models, reversals, excursions.

DOI: 10.1134/S1069351312040076

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 48 No. 4 2012

PARKER’S MODEL IN GEODYNAMO 327

2. THE DYNAMO EQUATIONS

We consider Parker’s model, which is widely appliedfor describing the processes of the magnetic field gener�ation in different astrophysical objects (Parker, 1955;Braginskii, 1964; Ruzmaikin, Sokolov, and Shukurov,1988). Hereinafter, we will use the following equationsin a thin spherical shell:

(1)

where � and � are the azimuthal components of thevector potential of the magnetic field (B = curlA) andof the magnetic field B, respectively; α and Ω are theα�effect and the differential rotation, which dependon the angular coordinate ϑ; and � is the dynamonumber, which is proportional to the product of theamplitudes of the α� and ω�effects. The prime symboldenotes the derivative with respect to ϑ. At the poles,ϑ = 0, π, the boundary conditions � = 0 (which followfrom the uniqueness of the axisymmetric azimuthalmagnetic field at the pole) and � = 0 are satisfied. Thelogic behind this condition is the following. The Brcomponent of the magnetic field in the spherical coor�dinates (r, ϑ, ϕ) is cast in terms of potential as

=

Since the second term is finite, it follows that � = 0.Historically, the second term in the approximation of Bris often omitted; further, we will also do so.

The solution of Eq. (1) has the following form:

(2)

which corresponds to the following eigenvalue prob�lem:

(3)

where γ is a complex rate of growth; the amplitudes are complex, too. Let θ = 90 – ϑ be the latitude.

From the general considerations, it follows that thepseudoscalar quantity α(θ) is antisymmetric withrespect to θ about the equator: α(–θ) = –α(θ). Then,the solution can be divided into two classes: the dipolesolution (D) such that Br(–θ) = –Br(θ), Β(–θ) =–B(θ), and the quadrupole solution (Q) such thatBr(–θ) = Βr(θ), B(–θ) = Β(θ). It is remarkable that theabove division in terms of the type of symmetry does notcoincide with the common notions of a dipole and qua�drupole, as these are traditionally understood in geo�magnetism. The notion of a dipole D as introducedabove is applicable to all odd harmonics l (includingthose which are not axisymmetric with 0 < m ≤ l) in theexpansion in terms of the associated Legendre polyno�

mials likewise, the notion of a quadrupole Q relatesto all harmonics with even l, again with 0 < m ≤ l.

∂= α +

∂

∂= Ω + ,

∂

''

' ''

t

t

� � �

� � � �

= ϑθ

'1 ( sin )sin

rBr

� ' cot .r+ ϑ

� �

γ= ϑ , ϑ ,,( ) ( ( ) ( ))te A B� �

'';

' ''

A B A

B A B

γ = α +

γ = Ω + ,�

( )A B,

,mlP

Below, we will consider how the first three eigenval�ues and the solutions depend on thedynamo number � for the most common distributionsof α and Ω: α = cosϑ and Ω = sinϑ; we will also checkthe solutions for parity. Of course, the solution of thelinear problem only gives a hint of what may occur inthe nonlinear case; however, in many cases, the non�linear solution is governed by the dominant mode ofthe linear solution (Brandenburg et al., 1989).

3. THE EIGENSOLUTIONS

3.1. � > 0

The behavior of the complex eigenvalue γ(�) isshown in Fig. 1. Let us consider its real part, �γ. Thepositive values of �γ correspond to the growing solu�tions. Figure 1a presents the first three solutions withthe maximal growth rates sorted for each � in such away that the circles always correspond to the fastestgrowing mode; the circles are followed by squares andtriangles.

For positive �, the mode that emerges first is sta�tionary (the imaginary part of the growth rate = 0,Fig. 1b) and quadrupole (see Fig. 2, which presents the

butterfly diagrams =2.

As � increases, the growth rate �γ of the secondmode, which is oscillating and dipole, eventually sur�passes the growth rate of the quadrupole mode. Thesecond (dipole) mode becomes the leading mode, andwithin some interval of �, both modes coexist simul�taneously (Fig. 3). The symmetric branches of theimaginary part (�) = – (–�) in Fig. 1b, whichcorrespond to the frequencies of oscillations, reflectcomplex conjugate solutions. The oscillatory mode ischaracterized by the wave propagation from the equa�tor towards the poles. With the further increase in �,the stationary quadrupole solution stops growing whilethe new oscillatory quadrupole mode appears (Fig. 4;the solutions are ordered with increasing �γ). In thiscase, again, the waves propagate poleward.

3.2. � < 0

For � < 0, the stationary dipole (Fig. 5) is the solu�tion that appears first. After it, the oscillating quadru�pole emerges, for which the wave solution propagatesequatorward from the poles (Fig. 6), as it is the case withthe Sun. The growth rate of the quadrupole becomeshigher than that of the dipole (Figs. 1a and 7). At largenegative �, there are three modes that coexist simulta�neously (Fig. 8): the oscillating dipole (Figs. 8a and 8b)and quadrupole (Figs. 8c and 8d) modes with close γand the stationary dipole mode which is far weaker(Figs. 8e and 8f). The latter mode could introduce theasymmetry of the hemispheres at large times.

2 For simplicity, the symbol ^ is further omitted.

1 3…γ 1 3( )r …B B,

γ�

�

�( )( )rB B t, ϑ,

γ,( ( ))i t

re B B��

γ� γ�

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RESHETNYAK

4. DISCUSSION

The geomagnetic field has quite a complex struc�ture (Langel, 1987; Petrova, 1992). In the frameworkof the dynamics of the mean fields, which includesParker’s model used above, it is possible to describe thelarge�scale component of the magnetic field, in partic�ular, the flips in polarity of the geomagnetic field. Thispossibility was first noted by Braginskii (1964). In thatwork, the dipole oscillating solution was interpreted asa regime of frequent reversals. In order to allow for rarereversals, Braginskii introduced the meridional circu�lation (the terms that have the form and ,where is the flow velocity in the direction ϑ) intothe right�hand side of (1). Formally, this procedurecorresponds to the renormalization of . By select�

V Aϑ− ' V Bϑ− 'Vϑ

γ�

ing suitable Vϑ, one can set = 0 and pass to theregime without reversals. It has long been unclear, onwhat the selection of Vϑ depends. Popov (2008)showed that, in terms of the Wentzel�Kramer�Bril�louin (WKB) theory, there is a solution for a broadrange of Vϑ that provides a regime without reversals. Inthe phase space, it corresponds to the attractor: start�ing from a certain value of Vϑ and during its furtherincrease, there are no reversals. Thus, we hope toaccount for the frequency of the reversals by using onemode only. The variations in the value of the meridi�onal circulation are extensively invoked to explain theMaunder minimum (Karak, 2010). In this relation, itis remarkable that from the nonlinear analysis of theαω�dynamo equations, it follows that in the case of

γ�

10

5

0

–5

–1010005000–500–1000

4

0

–4

–83000–300

(а)

�

ℜγ

16

0

–163000–300

(b)

ℑγ

30

15

0

–15

–3010005000–500–1000

�

ℜγ

�

�

ℑγ

Fig. 1. The dependences of (a) and (b) on �. In ascending order of : the circles, the squares, and the triangles.The enlarged images are shown in the insets.

γ� γ� γ�

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

0.3 0.6

ϑ

t

(а) (b)

Fig. 2. The stationary quadrupole solution for � = 102 (corresponds to the circles in Fig. 1).



0–90

–45

0

45

90

1 2

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

1 2

ϑ

t

1.0

0.5

0

–0.5

–1.0

(а) (b)

0–90

–45

0

45

90

1 2

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

1 2

ϑ

t

1.0

0.5

0

–0.5

–1.0

(c) (d)

Fig. 3. (a) and (b): the oscillatory dipole solution; (c) and (d) the stationary quadrupole solution for �= 2 × 102. The dipole andthe quadrupole correspond to the circles and the squares in Fig. 1, respectively.

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

(а) (b)

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

(c) (d)

Fig. 4. The oscillatory (a), (b) dipole and (c), (d) quadrupole solutions for � = 103. The dipole and the quadrupole correspondto the circles and the squares in Fig. 1, respectively.

330


RESHETNYAK

deviation from a strictly oscillatory process accompa�nied by a phase shift φ, the expression isvalid (Hoyng, 1993).

Now, confining ourselves to the case Vϑ = 0, weconsider another aspect: how would the changes in |�|affect the behavior of the geomagnetic field? In termsof our model, it is not only the evident increase in theconvection intensity, e.g., due to the approximate 20%variation in the regime of the heat transfer in the D''layer (Olson et al., 2010). We also mean the changescaused by random variation in the averaged parame�ters α and Ω (Hoyng, 1993) which are associated with

ln 0B + φ ≈

a finite number of the convective cells in the liquidcore of the Earth and with relatively low magnetic

Reynolds numbers ( ) characterizingthe number of the vortices . For example,Gissinger (2010) derives the chaotic flips of the polar�ity in the dynamo experiment from the assumptionthat the system is close to the transition of the dipolemode to the quadrupole mode.

For the further analysis, we have to select the signof the dynamo number �, which defines the propaga�tion direction of the dynamo waves. In the case of the

2 310 10−∼Rm3 4n /

∼ Rm

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

(а) (b)

Fig. 5. (a), (b) the stationary dipole solution for � = –50. The dipole corresponds to the circles in Fig. 1.

0–90

–45

0

45

90

1 2

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

1 2

ϑ

t

1.0

0.5

0

–0.5

–1.0

(а) (b)

0–90

–45

0

45

90

1 2

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

1 2

ϑ

t

1.0

0.5

0

–0.5

–1.0

(c) (d)

Fig. 6. (a), (b) the stationary dipole and (c), (d) the oscillatory quadrupole solutions for � = –150. The dipole and the quadrupolecorrespond to the circles and the squares in Fig. 1, respectively.



solar dynamo, the selection is unique: the solar spots

move equatorward3 and � < 0. The situation for the

Earth has not been studied as thoroughly: in fact, thespatiotemporal resolution of the geomagnetic field israther low on times commensurable with the intervalsbetween the reversals. Of course, attempts were madeto determine the latitudinal propagation of the geo�magnetic field structures: Reshetnyak (1995) analyzedthe motion of the foci of the secular variation (the timederivative of the normal component of the magnetic

field, ) for the past few centuries and demonstratedthe polarward motion. This result is consistent withthe ideas suggested in (Olson and Hagee, 1987); fordetails, see (Reshetnyak and Sokolov, 2003). However,these inferences should be treated with caution: theobserved foci of the secular variation may well be ofanother nature, being due to the heterogeneities at theboundary between the core and the mantle.

Due to the personal preferences of the author, itseems to be more instructive to consider the results of3D modeling of thermal convection in a sphericalshell. For this, we have to know the distribution ofkinetic helicity which is related to the α

3 Probably, except for the high latitudes where the dynamo wavespropagate in the opposite direction. See (Kuzanyan, 2004) fordetails.

Z·

curl ,χ = V V

effect as α = –�χ4, the distribution of the azimuthal

component of the velocity, Vϕ, which determines the

sign of the differential rotation: We take

these distributions for the typical regimes of thermalconvection in the modern 3D geodynamo models(Fig. 9) from (Reshetnyak, 2010). The product � =αΩ provides the measure for the intensity of the con�vection, which is specified by the value of the dynamonumber � in Eq. (1). This regime is close to thethreshold of generation of thermal convection in thespherical shell. It is remarkable that α is localized closeto the Taylor cylinder (the cylinder cut by the innercore). This somewhat differs from the simplified formselected above, α = cosϑ, which has maxima at thepoles. The product � has its maxima in the samedomain. As mentioned above, we are apt to assumethat � > 0 for the Earth (Fig. 9c). Unfortunately, theBoussinesq approximation, which is used in the bulkof the models, is unsuitable for providing large�scaledistributions of helicity χ at large Rayleigh numbersRa, which would be typical for the convection in theEarth’s core. As Ra increases, the scale of χ becomesfiner, and χ localizes close to the rigid boundaries evenif the nonviscous boundary conditions are used.Therefore, the product � becomes small. The analysis

4 Hereinafter, we assume � = 1.

,V

rϕ∂

Ω∂

∼

0–90

–45

0

45

90

1 2

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

1 2

ϑ

t

1.0

0.5

0

–0.5

–1.0

(а) (b)

0–90

–45

0

45

90

1 2

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

1 2

ϑ

t

1.0

0.5

0

–0.5

–1.0

(c) (d)

Fig. 7. (a), (b) the oscillatory quadrupole and (c), (d) the stationary dipole solutions for � = –225. The quadrupole and the dipolecorrespond to the circles and the squares in Fig. 1, respectively.

332


RESHETNYAK

of this case is beyond the scope of the present paper,and we will analyze it later. It is quite possible that, byallowing for the compressibility of the liquid core (thedensity of the liquid core is known to radially increaseby 10% with the increasing depth), we will obtainlarger values of �.

Turning back to the results of modeling and assum�ing � > 0 (scenario I), we find ourselves in quite a diffi�cult situation: the only way to obtain the solution interms of oscillations (the oscillatory regime withoutchanges in polarity) is to assume the superposition ofthe oscillatory dipole and the stationary quadrupole(Fig. 3). This regime might correspond to a regimewithout reversals. The regime of rare reversals corre�sponds to a certain finite interval of the numbers �,after which the reversals again start to appear at larger�, because �γQ � �γD (see the inset in Fig. 3a).

As � further increases, the stationary quadrupole solu�tion is changed by the oscillatory solution and, corre�spondingly, by the regime with frequent reversals(Fig. 4). The difficulty stems from the fact that thesuperposition of the dipole and quadrupole solutions isweakly consistent with the observations at large times,which imply the predominance of the dipole mode and,correspondingly, superposition of the stationary andoscillatory dipole modes.

Summarizing the above, we obtain that for � > 0,the transition from frequent to rare reversals corre�sponds to the better�developed convection. Then,both modes (the dipole and the quadrupole) start tooscillate (Fig. 4), and the intensity of the dipole mag�netic field increases due to the increase in �γD

(Fig. 1). This scenario is close to that discussed in(Reshetnyak and Pavlov, 2000), which is based on a

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

(а) (b)

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

(c) (d)

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

0–90

–45

0

45

90

0.3 0.6

ϑ

t

1.0

0.5

0

–0.5

–1.0

(e) (f)

Fig. 8. (a), (b) the oscillatory quadrupole solution, (c), (d) the oscillatory dipole solution, and (e), (f) the stationary dipole solu�tion for � = –103. The solutions correspond to the circles, the squares, and the triangles in Fig. 1, respectively.



series of works on 2D αω modeling resembling theknown �model suggested by Braginskii (see the ref�erences in the paper cited above).

It is interesting that the solution for � < 0 (scenarioII), which is often considered as being associated withthe solar dynamo, is more suitable for interpreting thegeomagnetic data: as � increases, the dipole field,which is stationary at small � (Fig. 5), starts to beaccompanied by some ancillary variations (the oscilla�tory quadrupole solution which is interpreted, e.g., asthe increased frequency of reversals) (Figs. 6 and 7), andthen it passes to the mode with frequent reversals(Fig. 8). However, in this case, again, we should expectthe solution to be somewhat asymmetric due to thepresence of the oscillatory quadrupole and the station�ary dipole. Obviously, the third (dipole) mode, which isconfined to the equatorial zone, has a maximal rate ofdecay when propagating towards the Earth’s surface. Inother words, according to this scenario, the increase in� causes the transition from rare to frequent reversals,and this scenario was selected as the basis for (Sobkoet al., 2012). Generally, the two scenarios do not con�tradict each other if we assume that the transition fromthe regime of rare reversals to the regime of frequentreversals in scenario I corresponds to the second transi�tion associated with the increase in �. In this case, theincreased intensity of the geomagnetic field at the tran�sition to frequent reversals can also be explained (Resh�etnyak and Pavlov, 2000). Probably, the emerging con�tradictions associated with the asymmetry of the hemi�spheres could be settled by changing the spatialconfiguration of α and Ω and by taking into account thenonlinearities. We intend to do this in a future work.

ACKNOWLEDGMENTS

The author is grateful to D.D. Sokolov for his dis�cussions and the interest shown in the work.

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Z

(а)

(b)

(c)

Fig. 9. The distributions of the averaged (over ϕ (a) kinetichelicity χ, (b) differential rotation Ω, and (c) their productin the 3D model of thermal convection (Reshetnyak,2010) for the Rayleigh number Ra = 102. The maximal val�ues are shown in white, and the minimal values, in black.The dashed contours denote negative values.

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Parker’s model in geodynamo