Three-dimensional domino model in geodynamo

ISSN 10693513, Izvestiya, Physics of the Solid Earth, 2013, Vol. 49, No. 4, pp. 464–473. © Pleiades Publishing, Ltd., 2013.Original Russian Text © M.Yu. Reshetnyak, 2013, published in Fizika Zemli, 2013, No. 4, pp. 18–28.

464

1. INTRODUCTION

Generation of magnetic fields is the subject of studyin dynamo theory. This theory seeks to explain the successive transformations of thermal and gravitationalenergy that is released due to the differentiation of matter into the energy of kinetic motion of conductive liquid and the subsequent conversion of the kinetic energyinto the energy of the magnetic field (Rüdiger and Hollerbach, 2004). The modern dynamo models includethe partial differential equations describing the convection and generation of the magnetic field, which shouldbe threedimensional due to some constraints. Typically, the internal magnetic fields of astrophysicalobjects are observed with an accuracy within the first10 spherical harmonics, since for the Earth, the spherical harmonics with numbers higher than 13 arescreened by the layer of conductive mantle. In the caseof other objects, the measurement accuracy is evenlower. At the same time, in order to ensure the correctproportion between the forces in the models, it is necessary that the small scales of the order of ~10–8 L (whereL ≈ 2200 km is the radius of the liquid core) are resolvable. This entails both large Reynolds numbers Re ~ 109

and a high degree of anisotropy (Hejda and Reshetnyak,2009). The anisotropy of the flows in liquid cores is dueto the fast planetary rotation, which results in the geostrophic balance of the forces, i.e., equilibrium betweenthe pressure gradient and Coriolis force (Pedlovsky,1987). The convection in the cores of planets has acyclonic character. The cyclones themselves and theanticyclones are stretched along the rotation axis, andtheir diameter is far smaller than their length. Thecyclonic convection gives rise to the largehydrody

namic helicity, which is vital for generation of largescale magnetic fields in the planetary cores. In order toreasonably approach the required regime in the liquidcore, one has to use the models with 1283 grids andlarger, which presents a severe challenge even for theparallel computing and requires months of computations. At the same time, the spatial resolution of theexisting magnetic observations, even at a few characteristic magnetic times back in the past, does not exceedthe first harmonics of spherical expansion, whichmeans that comparison of the highly accurate numerical models with the observations is ambiguous.

Notwithstanding the technical difficulties mentioned above, the modern dynamo models are still sufficiently elaborated to reproduce many features of thepresent magnetic field and paleomagnetic field including the reversals of the geomagnetic field when the magnetic dipole changes its polarity (see (Jones, 2011) formore detail). Clearly, alongside with the complexthreedimensional (3D) models that provide a detaileddescription of the spatiotemporal structure of the physical fields in the liquid core and on the surface of theplanets, it would be desirable to also have simple modelsthat are capable of reproducing the main features of thegeomagnetic field for the times long in the past. Suchmodels must rely on the modern knowledge of cyclonicconvection in a liquid core and should reasonablyreproduce the evolution of the magnetic dipole. Withthis aim in view, we consider the domino model suggested in (Nakamichi et al., 2012). The idea of thismodel is to describe the magnetic field of a planet bysuperimposition of the magnetic fields of separate spinsthat have constant amplitude but may rotate. More

ThreeDimensional Domino Model in GeodynamoM. Yu. Reshetnyak

Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, ul. Bol’shaya Gruzinskaya 10, Moscow, 123995 RussiaReceived August 18, 2012; in final form, November 29, 2013

Abstract—The Lagrangian formalism is applied to consider temporal evolution of the ensemble of interacting magnetohydrodynamical cyclones governed by Langevintype equations in a rotating medium. Thisproblem is relevant for fastrotating convective objects such as the cores of planets and a number of stars,where the Rossby numbers are far below unity and the geostrophic balance of the forces takes place. Thepaper presents the results of modeling for both the twodimensional (2D) case when the cyclones can rotaterelative to the rotation axis of the whole system in the vertical plane, and for the case of spatial rotation by twoangles. It is shown that variations in the heat flux on the outer boundary of the spherical shell modulate thefrequency of the reversals of the mean dipole magnetic field, which agrees with the threedimensional (3D)modeling of the planetary dynamo. Applications of the model for giant planets are discussed, and an explanation for some episodes in the history of the geomagnetic field in the past is suggested.

Keywords: domino model, geomagnetic field, reversals, excursions, liquid core of the Earth

DOI: 10.1134/S1069351313030130

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 49 No. 4 2013

THREEDIMENSIONAL DOMINO MODEL IN GEODYNAMO 465

details on the application of these models in other fieldsof physics can be found in (Stanley, 1971). The spinsinteract in such a way that the minimal energy of theirinteraction corresponds to the spins having identicaldirections. Rotation in the system creates a preferreddirection of the spins parallel to the rotation axis. Further, we first show how the system of these spins rotating in the vertical plane can be used to explain the existence of thermal traps for the magnetic field, when thefluctuations in the heat flux lock the magnetic dipoleclose to the poles (the regime of rare geomagnetic reversals) or near the equator (the magnetic field of Neptuneand Uranus). Then, we generalize this approach to the3D case, when the position of the spins is determined bytwo timevarying angles and the dipole may precessaround the geographical axis.

2. TWODIMENSIONAL DOMINO MODEL

We consider the domino model (Nakamichi et al.,2012), which further develops the XYIsing–Heisenberg model of the interacting magnetic spins. For thedetails on classification and nomenclature, see (Stanley,1971). The idea of the domino model is to analyze theset of N coupled 2D spins Si, i = 1, …, N in the mediumthat rotates with angular velocity Ω = (0, 1). The spinsare uniformly distributed over a unit circle; in the courseof time t, angle θ that they make with the rotation axis,which is perpendicular to the plane of the circle so asSi = (sinθi, cosθi), can change. The magnetic fields generated by the cyclonic convection can be considered asthe prototypes of the spins. Each spin Si is affected by arandom force, the effective friction, and the force fromits nearest neighbors Si – 1 and Si + 1.

In order to derive the dynamic equations, we usethe Lagrangian approach. The Lagrangian of the system is equal to the sum of Lagrangians of the individualspins. We write out the Lagrangian of the ith spin in the

form i = Ki – Ui, where Ki = is the kinetic energy

of the spin and Ui = λ[(Si ⋅ Si + 1) + (Si ⋅ Si – 1)] +γ(Ω ⋅ Si)

2 is the potential energy of the spin, whichincludes the spin’s interaction with its neighbors andthe rotation effect, i.e., the tendency of a spin to alignwith the rotation axis. γ and λ are constants.

We consider the Lagrange equation for variables

(θ, ) added with the dissipation function i =

and energy source i =

(1)

12θ· i

2

θ· κ2θ· i

2

θiєχi

τ :

ddt

∂i

∂θ· i

∂i

∂θi

∂i

∂θ· i

∂i

∂θi

+ +– 0,=

where κ, є, and τ are constant parameters, and χ is arandom function. Substitution of i, i, and i into(1) gives the system of Langevintype equations:

(2)

The system of nutation equations (2) describes thetime evolution of the angles of spins’ deviation fromthe rotation axis in the form of Newton’s second lawwith periodic boundary conditions. The followingvalue corresponds to the sum axial dipole:

(3)

Nakamichi et al. (2012) solved system (2) by thefourorder Runge–Kutta method. Our numericalexperiments showed that even the use of the standardexplicit schemes of the second and the first order for thesecond and the first time derivatives in (2) and the nonlinear righthand side taken from the previous time stepprovides close results. Further, we intend to also consider a more complex numerical technique.

Time integration of (2) even at small N = 8 providesvery diverse dynamics of М, which is in many casesclose to the paleomagnetic observations of the geomagnetic dipole, including the periods of both rare and frequent reversals (Jacobs, 2005). The analysis of the timebehavior of individual spins in the domino model indicates that the spins are flipping in an avalanchelikemanner, which results in the reversal of the field, whichgave the name to the model. All the calculations are carried out for the parameters that are close to those suggested in (Nakamichi et al., 2012): γ = –1, λ = –2, κ =0.1, є = 0.65, τ = 10–2, and N = 8 with a normal random χi with zero mean and unit variance, which isupdated at each time step equal to τ. The analysis showsthat the results are not particularly sensitive to the specific shape of random noise χ, and the remainingparameters can easily be determined by fitting themodel to the observations.

The characteristic behavior of the axial dipole of themagnetic field is illustrated in Fig. 1a, which shows several reversals of the field that occur at irregular timeintervals. These reversals are accompanied by shortterm drops in amplitude М up to zero and subsequentrapid restorations, which are well known to geophysicists and referred to as geomagnetic excursions in geomagnetism. The detailed analysis and the dependenceof the solution of system (2) on the parameters can befound in (Nakamichi et al., 2012). Here, we only touchon the following points. To date, it is unclear from theobservations how the reversal of the geomagnetic fieldoccurs. Sometimes, excursions are also referred to asfailed reversals of the field. There are two probable sce

θ·· i 2γ θicos θisin λ θicos θi 1–sin θi 1+sin+( )[+–

– θisin θi 1–cos θi 1+cos+( ) ] κθ· iєχi

τ+ + 0,=

θ0 θN, θN 1+ θ1, i 1…N.= = =

M 1N θi.cos

n 1=

N

∑=

466


RESHETNYAK

narios of the reversals: either the magnetic dipole makesa rotation by 180° without a reduction in amplitude, orit first decreases in amplitude, then makes a flip, and isrestored to the previous amplitude level. In order to verify these scenarios, we consider the time evolution ofindividual spins during the reversal (Fig. 2). At themoment of the reversal М = 0, the individual spins arebroadly scattered, which results in the reduction of M.We note that the characteristic time of the reversal is farlonger than the time step in the model. The minimaltime ~Nτ/2 taken for the perturbation to propagatefrom spin Si to the other (remotest) spin is also shorterthan the time of the reversal. This agrees with the resultsof the 3D calculations, which show the spots of radialmagnetic field of a different polarity to coexist duringthe reversal. Further, we will return to this questionwhen considering the 3D domino model.

3. THE EFFECTS OF THE HETEROGENEITIES IN THE HEAT FLUX

The fact that the process of the reversals depends onthe spatial pattern of the heat flux on the outer boundaryof the liquid core (Glatzmaier et al., 1999) is one of theimportant findings in the dynamo theory. The fluctuations in the heat flux on the boundary, which amount upto a few dozen percent of its average value, arise due tothe convection in the mantle and have characteristictimes of 106–107 yr, which is far in excess of the characteristic times of the processes in the liquid core(104–105 yr). In particular, it is shown in the quotedpaper that the increase in the heatflux intensity alongthe rotation axis increases the degree of axial symmetryof the whole system and impedes the process of thereversals, since, in this case, a type of thermal trap forthe magnetic field emerges. In turn, the weakening ofthe heat flux in high latitudes results in the chaoticmotion of the magnetic dipole and causes frequentreversals. The latter is associated with the violation ofthe geostrophic balance and predominance of radialbuoyancy forces. It seems fascinating to reproduce thiseffect in the simple dynamo models, which are capableof providing extensive statistics of the reversals andmore illustrative results. In some cases, these simplifiedmodels use a random parameter that mimics the smallscale fluctuations of the fields (Hoyng, 1993).

If, instead of being understood as mere magneticspin, S is treated as a magnetohydrodynamic system ofa cyclone overall, we can introduce the correction Ψ forheterogeneity of the heat flux into the potential

energy U.1 The righthand side of Eq. (2) will then con

tain the effective force Fi = whose effects on the

behavior of M(t) we consider below.

1 Had the problem been purely magnetic, this would correspondto the electromagnetic coupling between the spin and the poorlyconductive mantle of the planet, the D''.

∂Ψi

∂θi

,–

10

–1300002500015000100000 5000 20000

(а)

(b)

(c)

(d)

(e)M

t

10

–1300002500015000100000 5000 20000

M, Me

t

10

–1300002500015000100000 5000 20000

M

t

10

–1300002500015000100000 5000 20000

M

t

10

–1300002500015000100000 5000 20000

M

t

Fig. 1. The time evolution of M for (a) Сψ = 0, (b) Сψ =

0.5, and (c) Сψ = –0.5 with ψ = –cos2θ, and (d) Сψ = 10

and (e) Сψ = –9 with ψ = –cos22θ. The thick line inpanel (d) corresponds to M, and the thin line, to Mе.

10

–11–1 0

0

cosθi

M

Fig. 2. The dependence of cosθi for i = 1, …, N on M for

the reversal from Fig. 1a on the time interval t = (1–3) × 103.The dots on the curves correspond to each tenth value in thecalculations.



Let Ψ(t, θ) = Сψψ(t, θ), where Сψ is a constant andthe spatial distribution of the potential is specified in theform ψ = –cos2θ. Then, Сψ > 0 corresponds to thesteady state of the spins (cyclones) in the region of geostrophic poles θ = 0, π, so that the emergent force F =–sin2θ acting on the cyclone is directed towards thepoles. This regime is commensurate with the increase inthe heat flux in the polar regions, which causes thecyclones to stretch along the axis of rotation. The effective influence of thermal heterogeneity on the behaviorof M is illustrated in Fig. 1.

The increase of the heat flux along the rotation axis(Сψ > 0) leads to partial suppression of the magneticreversals (Fig. 1b). The number of the excursions alsodecreases: the scatter in M becomes narrower. We notethat for the shape of the potential barrier ψ selected inour analysis, the dependence for F is exactly identical tothe dependence for the term at γ in (2): the increase inthe heat flow at the poles results in the effectiveenhancement of geostrophy caused by fast diurnal rotation, which is consistent with the physics of the process.Our result agrees with the 3D calculations (see Fig. 1din (Glatzmaier et al., 1999)). The further increase in theamplitude of the barrier (Сψ ~ 2) completely halts thereversals. By increasing the amplitude of the barrier(Сψ ~ 10), one can achieve the state when M, beingalmost constant in time, has an arbitrary value |M| ≤ 1determined by the initial distribution of Si. The dispersion of M decreases. In other words, the increased heatflux results in the anchoring of spins that are not coherent. According to the observations of Shatsillo,Didenko, and Pavlov (2005), there is some evidencethat the magnetic dipole in the past was located close tothe geostrophic poles, which corresponds to the presentstate, and resided for a long time in the low latitudes. Itcannot be ruled out that such a migration could havebeen caused by the fluctuations in the thermal flux. Further, we consider a number of other distributions of theheat flux that bring about similar results.

For negative Cψ, with a relatively enhanced convection in the equatorial region and reduced geostrophy ofthe flows, we obtain the opposite effect (Fig. 1c)—theregime of frequent reversals (Fig. 1c in (Glatzmaieret al., 1999)). In this case, force F is directed away fromthe poles, the equilibrium point at the poles is unstable.The emergence of the minimum in the potential energyat the equator results in the emergence of a new attractor so that, e.g., for Cψ = –5|M| < 0.4, which correspondsto the magnetic dipole

(4)

located in the equatorial plane. Such behavior of themagnetic field is observed on Neptune and Uranus(see (Cupal, Hejda, and Reshetnyak, 2002) for moredetail).

We consider ψ = –cos22θ, for which the forceF = –2sin4θ is an alternatingsign function in each

Me t( ) 1N θ t( ),sin

n 1=

N

∑=

hemisphere. In (Glatzmaier et al., 1999), this corresponds to the heterogeneity specified by the zonal

spherical secondorder harmonic where is theassociated Legendre polynomial.

For Cψ > 0, we obtain the potential barrier in themiddle latitudes, which impedes the magnetic reversals.This is confirmed by our modeling and 3D calculations(see Fig. 1e in (Glatzmaier et al., 1999)). For Cψ = 2, weobserved only one magnetic reversal. The existence ofstable equilibrium at the equator θ = π/2, where M = 0,is another remarkable feature. One might suppose theexistence of the regimes when the spins have two attractors, one in the high latitudes close to the poles andanother at the equator. Such a regime indeed takes place(Fig. 1d). Inverse transition (from the high latitudes tothe equator) was not observed. Interestingly, in thebeginning of the calculations at small M, the amplitudeof the equatorial dipole Me is large (Fig. 1d), whichmakes this scenario suitable for the interpretation of themagnetic fields of giant planets as well.

Finally, we consider the last example of Сψ < 0. Here,in addition to the attractors located at the poles (associated with the rotation of the system Ω), another twoattractors emerge in the middle latitudes (one in eachhemisphere). For Сψ = –1, we obtain the regime of frequent reversals observed in Fig. 1f in (Glatzmaier et al.,1999). Moreover, it is even possible to achieve regimes(Fig. 1d) in which the magnetic pole is located in thehigh latitudes most of the time; however, one can seethat |М| ≠ 1 is still not equal to 1 (which is indicative ofthe partial coherence of the spines). The transitions tothe unstable state М = 0, which corresponds to the magnetic dipole in the equatorial plane, are observed. Wenote that with a decreasing spatial scale of perturbationin the heat flow, the required amplitude of this perturbation increases.

4. THE THREEDIMENSIONAL MODEL

The next step in the development of the dominomodel is its generalization to the case of 3D rotation ofthe spins in the space. This transition eliminates thefollowing important drawback of the 2D model. It isclear that the nutation of the spins (the motion withvarying θ) should result in the emergence of the Coriolis force, which is perpendicular to the velocity ofmotion and direction Ω. The Coriolis force causes precession of the spins relative to direction Ω in the horizontal plane, in the manner it occurs in the case of aspinning top or, in the case of the magnetic spins, in thepresence of the external magnetic field. Formally, thisresults in the new evolution equation for the azimuthalangle ϕ. Further, we will show how this effect can betaken into account by comparing the obtained equations with the known equations for the rotating bodiesand magnetic spins in ferromagnetic objects.

In the 3D case, the direction of spin Si = (sinθicosϕi,sinθisinϕi, cosθi) is specified by two angles in local

P20, Pl

m

468


RESHETNYAK

spherical coordinates. Just as previously, we assume thatthe origins of the coordinates are uniformly distributedalong the circle located in the horizontal plane that isperpendicular to Ω. The coordinate systems are produced by parallel translation; therefore, the corresponding axes of the local Cartesian coordinate systemsare parallel.

We consider the approximation in which the terms

with and are negligible. Then, we take advantageof the fact that, in order to obtain the solution for theprecession, it is necessary to introduce a term that isproportional to cosθ into the Lagrangian and to

reject the terms that are quadratic in the velocities and (Miltat, Albuquerque, and Thiaville, 2002).Then, the Lagrangian of the ith spin is cast as

(5)

where m is an integer value. After some algebra, theterm that allows for interaction between the neighboring spins has the following form:

(6)

We confine ourselves to considering two cases. Thecase m = 1, which corresponds to the situation in theferromagnetics, when the direction of the externalmagnetic field matters, and the spins have the samedirections as Ω. The case m = 2 corresponds to the situation when it is only important that the spin is parallel to the axis Ω. This type implies that the potentialenergy associated with the rotation is symmetrical relative to the equatorial plane; this has been already usedin the 2D case (2). Since the dynamo equations aresymmetrical relative to the sign of the geomagneticfield, this case is of importance for the description ofthe geomagnetic field. As we will show below, the firsttwo terms in the righthand side of (5) at m = 1 lead tothe Landau–Lifshitz–Gilbert (LLG) equation, whichdescribes the precession of the magnetic spin in theinviscid medium in the external Zeeman magneticfield equal to Ω. The third term describes the localinteraction between the neighboring spins.

We cast the Lagrange equation for the four inde

pendent variables (θ, ϕ, ):

(7)

θ·· ϕ··

ϕ·

θ·

ϕ·

i ϕ· i θicos γ Ω ⋅ Si( )m λ Si ⋅ Si 1+( )[––=

+ Si ⋅ Si 1–( )],

i λ Si ⋅ Si 1+( ) Si ⋅ Si 1–( )+[ ]=

= λ θisin θi 1+ ϕi ϕi 1+–( )cossin([

+ θi 1– ϕi ϕi 1––( ) )cossin

+ θicos θi 1+cos θi 1–cos+( ) ].

θ· , ϕ·

ddt∂

∂θ· i

∂∂θi

∂

∂θ· i

+–∂i

∂θ+ 0,=

ddt∂

∂ϕ· i

∂∂ϕi

∂∂ϕ· i

∂i

∂ϕi

+ +– 0,=

where

(8)

and ψ is a random function.Substitution of (5) and (8) into (7) for m = 2 gives the

system of equations:

(9)

where i = λ[(Si ⋅ Si + 1) + (Si ⋅ Si – 1)]. The exact formof the terms with the derivatives i is specified by thefollowing relationships:

(10)

As previously in (2), we use periodic boundary conditions

(11)

Since the Lagrangian in form (5) does not take intoaccount the quadratic terms that are proportional to

and then, the evolution equation (9) is a forcebalance equation. From the standpoint of dynamics,this is expressed by the sharper response to the perturbations introduced by a random force, which managesto relax in (2).

Prior to passing to the analysis of system (9), weconsider the case with m = 1, for which the term2γcosθ in (9) is changed by γ. Setting λ = 0, κ = 0, and

є = 0 gives = 0, = –γ, i.e., the precession aroundthe vertical axis, which corresponds to the solution ofthe wellknown LLG equation without dissipation forferromagnetics:

(12)

which has the motion integral = 0 and, with

timeindependent Ω, another integral = 0,

whence it follows that = 0. The ϕcomponent ofEq. (12) gives the evolution equation for the azimuthalangle = –γ, which we have already obtained previ

iκ2 θ· i

2θ2

iϕ· i2

sin+( ), iє

τ θiχi ϕiψi+( ),= =

θ· i κ θiϕ· isiniϕ'

θisin

єψi

θi τsin––– 0,=

ϕ· i κ θ· i

θisin 2γ θicos

iθ'

θisin

єχi

θi τsin+ +–+ 0,=

iϕ'

θisin– λ θi 1+sin ϕi ϕi 1+–( )sin[=

+ θi 1–sin ϕi ϕi 1––( )sin ],

iθ'

θisin λ θicot θi 1+ ϕi ϕi 1+–( )cossin([=

+ θi 1–sin ϕi ϕi 1––( )cos ) θi 1+cos θi 1–cos+( ) ].–

θ0 θN, θN 1+ θ1, ϕ0 ϕN,= = =

ϕN 1+ ϕ1, i 1…N.= =

θ·2

ϕ· 2

θ· ϕ·

S· i γSi– Ω,×=

∂∂dSi

2

∂∂d Si Ω⋅( )

θ·

ϕ·



ously. As mentioned earlier, the existence of the precession distinguishes the 3D case from the 2D case,where the rotation only caused the attraction of thespins to the poles (2). From a formal standpoint, the

difference lies in the fact that the derivative has

passed from the evolution equation for θ to the equation for ϕ.

For the case with m = 2, we have the precession

equations in the form = 0, = –2γcosθ, which predict the reversal of the angular velocity of the spins atthe transition through the equator. At the equatoritself, = 0. The obtained asymmetry should betreated in the following way. Clearly, if a system has apreferred direction associated with the rotation, thissystem is not symmetrical relative to the reversal of thecoordinate z → –z, and violation of mirror symmetryin the case with m = 1 results in the preferred polarity,i.e., the preferred value of angle θ. In the case withm = 2, we have an equally probable state for θ, but theasymmetry is reflected in the change of the sign of at the transition through the equator.

It is instructive to draw the analogy with the termsof the Navier–Stokes equation with rotation. Byneglecting the terms with acceleration and spatialderivatives and setting the radial velocity Vr = 0, wecome to the following relations for the velocity components and :

(13)

where H is the amplitude of the Coriolis force, and thecomponents of the tangential velocity (Vθ, Vϕ) are

expressed through the angles as ( sinθ). Then,

= –H, = –H, (14)

which corresponds to the case with m = 1 considered

above.2 As we can see, the condition of equally proba

ble normal and inverse polarity, which was taken intoaccount in the case with m = 2, substantially changesthe situation.

Now, we return to system (9) with m = 2. Since wedisregarded the terms that are quadratic with respect tothe velocities, this approximation is valid for the slowsolution, when the spins are precessing around the rotation axis and do not experience sudden motions; therefore, this approximation is of little value for analyzingthe reversals. For this purpose, further, we will considerfull equations with the retained quadratic terms. Meanwhile, let us examine how the action of a random forceaffects the predicted asymmetry of the precession in different hemispheres.

It is worth noting that we are now dealing with twoconjugate equations, which require considering the

2 The value of Vθ remains undetermined.

∂∂θ

θ· ϕ·

ϕ·

ϕ·

Vθ

– Vϕ

–

Vϕ

2 θcot– HVϕ θ, V0Vϕ θcotcos HV0 θ,cos–= =

θ· , ϕ·

ϕ· ϕ·

energy balance between the equations and applyingmethods that improve the stability of the solution, sincethese equations contain the terms with sinθ in thedenominator, which vanishes on the axis. For this purpose, we use the equations in the residual form andapply the Newton–Rapson technique for solving thesystem of linear equations (see Appendix).

The phase diagram of the dependence of the pre

cession velocity on the polarity of dipole М for asequence of the reversals is shown in Fig. 3b. Evidently, the introduction of a random force eliminatesthe asymmetry that takes place in the absence of dissipation and external forces. As we can see, the azimuthal velocities can be very large, resulting in themodel moving beyond the scope of the assumptionsmade when deriving the equations. However, in ouropinion, the precession approximation consideredabove is quite instructive in terms of the analogy to theLLG equation for ferromagnetics, where, at a temper

ature below the Curie point, is small. A similarapproximation is also used in astronomy, where thenutation effects may also be quite weak.

We consider the case when and are not equalto zero. We draw the analogy with the spinning top

Vϕ

θ·

θ·· ϕ··

1

0

–11045 × 1030

M

t

(а)

(b)

10

0

–10100

Vϕ

M–0.5 0.5

5

–5

Fig. 3. (a) The dependence of the precession velocity Vϕ onthe polarity of the dipole M; (b) the time evolution of thedipole M.

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RESHETNYAK

with the unit moment of inertia (Landau and Lifshitz,1988) by casting the Lagrangian in the form

(15)

where is the angular velocity of the spinning top.The analogy with the spin model lies in the fact that

that is a given constant value, which is specific to theobject. For convenience, we assume it to be unity:

The potential Ψ will be of use whendescribing the external impacts on the system.

i12 θ· i

2θ2

iϕ· i2

sin ϕ· i θicos ζ· i+( )2

+ +( )=

– γ Ω Si⋅( )2 i Ψi,+–

ζ· i

ζ·

ζ· i2

Si2≡ 1.=

Then, after simplifying the formulas, we have

(16)

where the constant is discarded since the Lagrangeequation only contains the derivatives of . Discarding the terms in (16) that are quadratic with respect tothe velocities, we arrive at the simplified form of theLagrangian (5), which was used previously.

Substituting (15) into (7), we obtain the dynamicequations in the following form:

(17)

A simple check shows that discarding the secondorder time derivatives and the terms with Ψ gives system (9). The numerical method for the solution of system (17) is described in the Appendix.

In order to present a model example, we explorewhether it is possible to predict the reversals of thegeomagnetic field by analyzing the changes in themagnetic variations before the reversal (Jacobs, 2005),which is important for geomagnetism. We assume theprecession amplitude of the magnetic dipole Vϕ to be ameasure of the variations. The model series of thereversals of geomagnetic field is presented in Fig. 4a.We focus on the set of the five reversals shown inFig. 4b. A characteristic feature of the domino modelis the presence of a transient regime at small |М|. Whilein this state, the system does not know in which direction to move, which manifests itself in the presenceof numerous magnetic excursions (Fig. 4a). In orderto answer our question, we consider the behavior of

and averaged over five realizations (Figs. 4cand 4d). It can be seen that the change in the polaritydoes not cause any significant variations in the ampli

tude of which renders this criterion unsuitablefor predicting the reversals.

The domino model, in which the interactionbetween the spins is implied to be longrange, does notprovide a correlation between the position of a spin andthe 3D magnetic field generated by it, which is a seriousdisadvantage of this model. It is because of this featurethat until recently we used a simplified integral characteristic of the field М. The next step is to introduce thefield of a spin, through which the interspin interactionis implemented. Prior to elaborating the model, wesummarize our requirements for it: (i) the potentialenergy of the spins should have a minimum when thecyclones are aligned with the rotation axis, and the spinsshould be codirectional; (ii) the model should be suitable for calculating the 3D magnetic field at each pointof the environment.

By identifying the spins with the magnetic dipoles,Nakamichi et al. (2012) managed to derive the 3D dis

i12θ· i

2 12ϕ· i

2 ϕ· i θicos γ θ2icos i Ψi,+––+ +=

θ·· i ϕ· i θisin γ 2θisin iθ' κθ· iєχi

τ Ψiθ'–+ + +–+ 0,=

ϕ·· θiθ·

isin iϕ' κ θ2iϕ· i

єψi

τ Ψiϕ'–+sin+ +– 0.=

M Vϕ

Vϕ ,

0.3

0.1

010–1

(а)

(b)

(c)

(d)

M

1

0

–15030200 10 40

M

t

0.2

M

|Vϕ|

1

0

–15030200 10 40

t

1

0

–12 × 1040 104

t

M

Fig. 4. (a) The time evolution of M for γ = 1, λ = –4.8, є =0.8, and κ = 0.2; (b) the reversals of the field (in the orderof thickening of the line) for five time intervals: (9640,9690); (12000, 12050); (12260, 12310); (17840, 17890);and (18670, 18720). For the third and fifth intervals, thevalue of M is taken with the opposite sign; (c) the mean

value for the case shown in panel (b); (d) the mean

value of the magnitude of the precession velocity as a

function of

M

Vϕ

M.



tribution of the magnetic field for the system of spins.These authors replaced the spines by the magneticdipoles with the field

(18)

where d is the magnetic moment of a dipole at thecoordinate origin and r is the radius vector of theobservation point. However, as follows from (18), thestable state for the two dipoles that are located at distance R from each other and have interaction energy

(19)

is the quadrupole configuration with oppositelydirected magnetic moments d1 and d2. At the sametime, in order to account for the existence of the average fields in ferromagnetics, when the magneticmoments of the domains are codirectional, it is vitalthat the quantum effects (the energy of exchangeinteraction) are taken into account. In other words,the expression used in Nakamichi et al. (2012) for calculating the potential energy of spin interaction wasadopted from quantum theory, whereas the spatial distribution of the 3D magnetic field was calculated forthe ensemble of magnetic dipoles in classical electrodynamics. Since the results in the domino modelappear to be quite attractive, we will further also keepto this scenario without introducing any additionalcomplications to the model. We note that, for obtaining the formula for U that is used in (5), one shouldchange the sign in (19) and reject the first term, whichis small in the time intervals between the reversals.

By using expression (18) for the ensemble ofN dipoles distributed along the unit circle, we calculate the distributions of the three magnetic components. Before proceeding to the analysis of the maps ofthe field, let us examine how the spectral characteristics of the field vary with time and consider a sequenceof five magnetic reversals (Fig. 5a), which are derivedfrom the calculations presented in Fig. 4a. Thesereversals are represented in the form of the evolution

of the axial dipole Gauss coefficient in the sphericalexpansion at a distance of three radii of the circle onwhich the spins are distributed. Overall, this distribution is characterized by equal times of falling in areversal and resuming after the reversal, the closeamplitudes of the fields of different polarity, and thebuildup of the field immediately before and after thereversal. One should bear in mind that, as we are dealing with spins that do not change their amplitudes withtime, the magnitude of the integral flux of the magnetic field through the arbitrary surface encompassingour system is also constant. The same applies to theintegral over the spectrum of the magnetic field, e.g.,during the reversal of the field. However, according tothe calculations (Fig. 5b), the intensities of the entire

B 3r d r⋅( ) r2d–

r5,=

U3 d1 R⋅( ) d2 R⋅( ) d1 d2⋅( )R2–

R5,=

g10

dipole and quadrupole are very close, and theenergy is redistributed on smaller scales. Interestingly,magnetic energy redistribution during the reversals isalso observed in the 3D calculations. This is due to thefact that at large Reynolds numbers, ohmic dissipationis low and the magnetic energy in the magnetostrophicsystems in the rotating coordinates is far larger thanthe kinetic energy. This means that the increasedkinetic energy of the flows does not account for thedrop in the magnetic energy of a dipole during thereversal.

Figure 6 displays the distributions of the Br component of the magnetic field during one of the reversals.We see that the ability of the spins to rotate alongangle ϕ creates a preferred corridor for the changes inthe polarity (Fig. 6b). This phenomenon has long beendebated among the paleomagnetologists (Jacobs,2005), and the results are frequently specific to themethod applied to the processing of the records of theancient magnetic field.

5. DISCUSSION

Evidently, against the modern 3D models, thedynamo model considered in this study is very simple.However, many of its components can still withstandsevere criticism. In order to illustrate this, we draw ananalogy with the wellknown Parker’s model (Parker,1955), which needs no advertising and has served as aprototype for designing the mean field dynamo models(Krause and Rädler, 1980), e.g., the Zmodel of geodynamo developed by Braginsky. In both the dynamo andParker’s model, the notion of cyclonic convection isvital. In the mean field models, rotation is the mecha

(а)

(b), 0.3

0300020001500

t2500

0.2

–0.2300020001500

t2500

0

g10

Fig. 5. The evolution of (a) ; (b) =

(the solid line), = (the

circles).

g10

2 g10

2

g11

2

h11

2

+ +

3 g20

2

g21

2

h21

2

g22

2

h22

2

+ + + +

472


RESHETNYAK

nism that leads to the violation of mirror symmetry,generates helicity and the αeffect, and creates the prerequisites for the emergence of the largescale magneticfields. However, at the level of applications, the apparatus of the theory loses its advantages of simple and practical use because several constraints on the uniformityand isotropy of the fields are violated. As a result, anumber of the key parameters used in the theory requireadaptation to the strongly anisotropic conditions in thegeostrophic regimes.

At the same time, the starting point in the 3Ddynamo modeling in planetary cores is the generationof cyclonic convection. The magnetic field in the firstapproximation does not alter the structure of theseflows, and, evidently, if a simplified model uses suchinformation, the suggested scenarios of the physicalprocesses become even more clearly pronounced.Interestingly, the scale redistribution, which is necessary in the mean field theory, is automatically imple

mented for the spatial scales of the domino modelbecause the energycarrying scale of turbulence isdetermined by the number of cyclones N. We notethat the introduction of nonlocal spin interactionprovides an extended spectrum of variations of thefield (Nakamichi et al., 2012). We also expect ourreaders to leave some leeway in the definition of acyclone: actually, during the reversal, only the direction of the magnetic field of an individual cyclonechanges, while the hydrodynamics of the cycloneremain unchanged. Since the Lorentz force is quadratic in terms of the magnetic field, this does notcontradict the laws of electrodynamics.

APPENDIX

Reduce (17) to the system of the firstorder differential equations

(A1)

with respect to the vector

(A2)

where

(A3)

Using the implicit Euler scheme on the nth step oftime, we introduce for (A3) the vectors in the residualform:

(A4)

Consider interactive process according to the Newton–Rapson method for the pth iteration:

(A5)

Vi θ· i– 0, Wi ϕ· i– 0,= =

V· κV W θisin A+ + + 0,=

W· θiVsin κ θ2iWsin B+ +– 0,=

yn

Vn

Wn

θn

ϕn⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

,=

A γ 2θisin– iθ'єχi

τ Ψiθ' ,–+ +=

B iϕ'єψi

τ Ψiϕ' .–+=

en

Vndt θn θn 1–+–

Wndt ϕn ϕn 1–+–

Vn Vn 1– κVndt Wn θinsin dt Andt+ + +–

Wn Wn 1– θinsin Vndt( ) ––

+ κ θ2 nisin Wndt Bndt+⎝ ⎠

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

.=

ypn yp 1–

n ∂ep 1–n

∂yp 1–n

⎝ ⎠⎜ ⎟⎛ ⎞

1–

ep 1–n

,–=

(а)

(b)

(c)

Fig. 6. The distribution of the Br magnetic component inthe Mollweide projection at the time moments (a) t =2500, (b) t = 2600, and (c) t = 2700. The white and blackcolors correspond to the positive and negative values of thefield.



where the Jacobi matrix has the following form:

(A6)

where

For each moment of time n and for each spin i,interactive process (A3) with the updated values forthe other spins was implemented. Once the convergence for all spins is achieved, the process passes to thenext time step n + 1. At each time step, the conditionsθi ∈ (0, 2π), ϕi ∈ (0, 2π) are checked and, if necessary,the corresponding correction for the periodicity isintroduced. This algorithm is quite stable with respectto singularities on the axis. It can easily be tailored forthe secondorder accuracy by using the Crank–Nicolson scheme instead of the Euler scheme. We note thatthe presence of singularity on the axis, which arisesdue to using the spherical coordinates, does not bringabout instabilities, as it often occurs in the problemsdealing with spatial derivatives. In our problem, as thepole is approached, the increments of the velocityincrease, and the spin is displaced from the polarregion. This situation crucially differs from the partialderivative problem where one has to exclude theparaxial region or to apply the nongrid methods.

REFERENCES

Cupal, I., Hejda, P., and Reshetnyak, M., Dynamo Modelwith Thermal Convection and with the FreeRotating InnerCore, Planet. Space Sci., 2002, vol. 50, pp. 1117–1122.Glatzmaier, G.A., Coe, R.S., Hongre, L., and Roberts, P.H., The Role of the Earth’s Mantle in Controllingthe Frequency of Geomagnetic Reversals, Nature, 1999,vol. 401, pp. 885–890.

Hejda, P. and Reshetnyak, M., Effects of Anisotropy in theGeostrophic Turbulence, Phys. Earth Planet. Inter., 2009,vol. 177, pp. 152–160.Hoyng, P., Helicity Fluctuations in Mean Field Theory: AnExplanation for the Variability of the Solar Cycle?, Astron.Astrophys., 1993, vol. 272, pp. 321–339.Jacobs, J.A., Reversals of the Earth’s Magnetic Field, Cambridge: Cambridge Univ. Press, 2005.Jones, C.A., Planetary Magnetic Fields and Fluid Dynamos,Annu. Rev. Fluid Mech., 2011, vol. 43, pp. 583–614.Krause, F. and Rädler, K.H., Mean Field Magnetohydrodynamics and Dynamo Theory, Berlin: AkademieVerlag, 1980.Landau, L.D. and Lifshitz, E.M., Mechanics, Oxford: ButterworthHeinemann, 1976, 3rd ed.Miltat, J., Albuquerque, G., and Thiaville, A., An Introduction to Micromagnetics in the Dynamic Regime, Topics Appl.Phys., 2002, vol. 83, pp. 1–34.Nakamichi, A., Mouri, H., Schmitt, D., FerrizMas, A.,Wicht, J., and Morikawa, M., Coupled Spin Models forMagnetic Variation of Planets and Stars, Mon. Not. R. Astron.Soc., 2012, vol. 423, no. 4, pp. 2977–2990.Parker, E.N., Hydromagnetic Dynamo Models, Astrophys. J., 1955, vol. 122, pp. 293–314.Pedlosky, J., Geophysical Fluid Dynamics, New York:Springer, 1987.Rüdiger, G. and Hollerbach, R., The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory, Weinheim: Wiley,2004.Shatsillo, A.V., Didenko, A.N., and Pavlov, V.E., Two Competing Paleomagnetic Directions in the Late Vendian: NewData for the SW Region of the Siberian Platform, Rus. J.Earth Sci., 2005, vol. 7, no. 4, pp. 3–24.Stanley, H.E., Introduction to Phase Transitions and CriticalPhenomena, Oxford: Clarendon Press, 1971.

J∂ep

n

∂ypn

dt 0 1– 0

0 dt 0 1–

1 κdt+ θindtsin Wn θi

ndt ∂An

∂θdt+cos ∂An

∂ϕndt

θindtsin– 1 κ θ2 n

i dtsin+ J43∂Bn

∂ϕndt

,= =

J43 θinVndtcos κ 2θi

nWndtsin ∂Bn

∂θndt.+ +–=

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Three-dimensional domino model in geodynamo