On the precession driven geodynamo

460

ISSN 1069-3513, Izvestiya, Physics of the Solid Earth, 2006, Vol. 42, No. 6, pp. 460–466. © Pleiades Publishing, Inc., 2006.Original Russian Text © S.L. Shalimov, 2006, published in Fizika Zemli, 2006, No. 6, pp. 14–20.

INTRODUCTION

The gravitational influence of the Sun and Moon onthe equatorial bulges of the mantle of the rotating Earthcauses the precession of the Earth at a period of about26 000 yr around the axis perpendicular to the eclipticplane. Since the influence of precession on the motionsin the Earth’s liquid core is assumed to be small com-pared to the effect of thermal and compositional con-vection, the precession driven geodynamo is usuallyignored. However, a recent laboratory simulation of theprecessing Earth (with the use of rotating water-filledspheroidal containers and with rotation and precessionparameters similar to those of the Earth) revealed[Vanyo et al., 1995; Vanyo and Dunn, 2000] (see also[Avsyuk et al., 2003]) an unexpectedly great variety offlows arising in the container (including their turbuliza-tion). Such flows are quite real in the Earth’s liquid core(because the precession itself is real); therefore, theireffects (along with hypothetical thermal and composi-tional convection) have recently received increasingattention from researchers.

The increasing interest in the precession driven geo-dynamo is accompanied by difficulties in attempting tobring the heat balance in the core into agreement withthe estimated power required for the convective geody-namo to operate in the course of the Earth’s evolution.The existing contradictions can be summarized as fol-lows: rapid cooling of the core is favorable to the oper-ation of the geodynamo, but the continuing chemicaldifferentiation causes the solid core to grow to dimen-sions that contradict the present-day observations; onthe other hand, slow cooling of the core, which isacceptable for the growth rate of the solid core, yieldsinsufficient energy for the geodynamo operation. Thecontradiction can be removed by assuming the exist-ence of radioactive heating in the core or at the core–mantle boundary (this assumption was recently ques-

tioned on the basis of experimental results [Chabot andDrake, 1999]). Yet even after this, it turns out that theage of the solid core cannot exceed 1.5 Ga. However, ageomagnetic field of approximately the same intensityhas existed for not less than 3.5 Ga [McElhinny andSenanayake, 1980]; therefore, within the convectivegeodynamo model, it is necessary to assume the exist-ence (over an interval of 2 Ga) of thermal convection inthe core with an intensity sufficient for maintaining thegeomagnetic field. Estimates show that, to create thisconvection, it is difficult to maintain the temperature atthe core–mantle boundary below the mantle’s liquidus[Gubbins et al., 2003]. Such boundary conditions in thelower mantle would have led to radical consequences inthe tectonic history of the Earth.

Thus, using the model of a purely convective geody-namo (modified by the addition of radioactive sources),it is impossible to consistently describe the Earth’s ther-mal evolution.

Traditionally, an alternative to the convective geo-dynamo has been the precession driven geodynamo[Malkus, 1968; Dolginov, 1995]. The equationsdescribing the motion of an inviscid liquid in a rotatingand precessing spheroidal shell were solved for the firsttime by Professor Sludsky (Moscow University) in1895 [Sloudsky, 1895]; later, a similar solution wasfound by the French mathematician Poincaré [1910]. Inthe coordinate system connected with a precessing con-tainer, the solution is a 2-D flow with a constant vortic-ity; it corresponds to the solid-body rotation of the liq-uid filling the container. The rotation axis of the liquiddiffers from that of the container by an angle propor-tional to the ratio of the precession velocity to the rota-tion velocity of the container (the Poincaré number)and inversely proportional to the dynamic flattening ofthe container. As the spheroid tends toward a sphere, asingularity arises in this solution due to the absence ofviscosity.

On the Precession Driven Geodynamo

S. L. Shalimov

Schmidt Institute of Physics of the Earth, Russian Academy of Sciences,Bol’shaya Gruzinskaya ul. 10, Moscow, 123995 Russia

Received December 28, 2005

Abstract

—Formation of flow structures in the Earth’s liquid core enclosed in a precessing and rotating shell(mantle) is examined within the hydrodynamic approach. The kinematics and energetics of the motions in theEarth’s core initiated by precession allow one to regard these motions as a possible geodynamo mechanism atan early evolutionary stage of the Earth (prior to the formation of the solid core). The influence of the precessiondriven geodynamo on the stability of the geomagnetic field is discussed.

PACS numbers:

91.10.Nj + 91.25.Cw

DOI:

10.1134/S1069351306060024

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ON THE PRECESSION DRIVEN GEODYNAMO 461

A viscous laminar solution (taking into account theboundary layer in the limit of small viscosity) in a coor-dinate system connected with the precessing containerwas constructed by Busse [1968]. In the same year,Malkus [1968] advanced the idea of the Earth’s preces-sion as a possible cause of the geodynamo anddescribed the first experiments with a liquid enclosed ina rotating and precessing spheroid. In particular, in thelaminar regime, there was observed a solid-body rota-tion of the liquid with an angular momentum deviatingfrom that of the container (an analogue of the Sloud-sky–Poincaré solution), as well as a westward drift ofthe liquid rotation axis (a lag behind the containermotion) and jet streams of opposite directions at a(dimensionless) distance of 0.866 from the rotation axisof the liquid. The latter two features of the flow werealso characteristic of the Busse solution. With increas-ing precession velocity in the experiments of Malkus,the flow became unstable and subsequently turbulized.Assuming that it is the turbulent regime that character-izes motions in the core, Malkus estimated (up to anorder of magnitude) the power of the precession drivengeodynamo (

2.3

×

10

10

W). This estimate is one to twoorders of magnitude smaller than the current estimates(which have increased by about one order of magnitudeover the last ten years) of the efficiency of the convec-tive geodynamo.

In critical comments [Rochester et al., 1975; Loper,1975] on the work of Malkus [1968], it was demon-strated [Rochester et al., 1975], in particular, that a flowwith the structure of the Sloudsky–Poincaré solution is,for symmetry considerations, unfavorable to the gener-ation of the geomagnetic field. Moreover, it was con-cluded that laminar flows are insufficiently efficientaltogether for transforming the mantle rotationalenergy into the geodynamo. Thus, the power of the lam-inar regime was estimated at

10

8

W [Rochester et al.,1975] and

3

×

10

7

W [Loper, 1975]. The turbulentregime of flows in the core remained beyond criticism.However, based on a rigorous analysis, Kerswell [1996]obtained an estimate from above for the power of bothlaminar and turbulent regimes of dissipation initiatedby precession. For the turbulent regime, the power doesnot depend on the Poincaré and Ekman numbers and isclose to

10

21

W, which is 14 orders of magnitude greaterthan the estimate of Loper [1975]. Therefore, the state-ment that precession driven flows are unsustainable interms of energy, which has prevailed for 30 years,should now be transformed into the question: Can pre-cession initiate turbulence in the core (it is sufficient toturbulize even a small region of the core) and/or flowswith a structure favorable to the generation of the geo-magnetic field? The goal of this paper is to justify a pos-itive answer to the second part of the question.

LABORATORY FLOW STRUCTURE

The Earth is approximately a flattened spheroid withthe axis of symmetry and rotation oriented nearly alongthe geographic N–S direction. Its ellipticity is about1/298 at the Earth’s surface and about 1/400 at thecore–mantle boundary. The equatorial bulge of themantle is inclined at an angle of

23.5°

to the ecliptic.The mantle response to the gravitational moment offorces of the Moon and Sun generates precession aboutthe axis perpendicular to the ecliptic at a period ofabout 26 000 years. The ratio of the Earth’s rotationperiod (1 day) to the precession period (26 000 years),i.e., the Poincaré number, is equal to

10

–7

. The ratio ofthe viscous force to the Coriolis force in the Navier–Stokes equation, i.e., the Ekman number, for the Earth’score has not been accurately determined (because of theuncertainty in the viscosity value), and, therefore,researchers use values ranging from

10

–15

to

10

–6

.In the first experiments with rotating and precessing

spheroids filled with water [Malkus, 1968] intended tosimulate flows in the liquid core, the ellipticity of con-tainers was 1/10 and 1/25. In the recent experiments ofVanyo [Vanyo et al., 1995; Vanyo and Dunn, 2000], theellipticity reached 1/100 and 1/400. The smallestPoincaré number attained in Malkus’s and Vanyo’sexperiments was, respectively, 1/80 and

10

–6

, and theEkman number,

3.6

×

10

–6

and

2

×

10

–7

. These differ-ences in the parameters of laboratory design substan-tially influenced the researchers’ conclusions on thestructure of the flows initiated in the containers. Forexample, it was found that, as the values of the elliptic-ity and precession velocity of the container approachthe Earth’s parameters, the flows become “smoother”and are never turbulized. It is natural to expect that thesame states are realized in the precessing Earth’s core.Therefore, the conclusions of Malkus [1968] on thegeomagnetic dynamo, based on the assumption of thecore turbulent state initiated by precession, can becalled into question.

Now, we discuss the experimental results of Vanyo’sgroup [Vanyo et al., 1995; Vanyo and Dunn, 2000].Their main conclusions on the steady motion of the liq-uid are as follows: (i) the liquid in a precessing androtating container with an ellipticity of 1/400 movesalmost as a solid body; (ii) the rotation axis of the pre-cessing liquid lags (by a very small angle) behind thatof the precessing container; (iii) the liquid drifting as awhole also lags behind the container; (iv) this motion ofthe liquid results in boundary layers of complex struc-ture; (v) the motion of the liquid as a whole inside thecontainer is stratified into flows in the form of cylinderscoaxial with the rotation axis of the liquid; (vi) the cyl-inders rotate relative to each other, as well as relative tothe container; (vii) laminar flows of opposite directions(to the south and to the north) coexist within the cylin-ders; and (viii) vortices and columns (with variable vor-ticity and parallel to the rotation axis of the liquid) thatresemble the Busse columns [Busse, 1970] used in the

462

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SHALIMOV

geodynamo theory form between the differentiallyrotating cylinders.

The aforementioned features of flows initiated in aprecessing and rotating container are schematicallyillustrated in Fig. 1 (modified after [Vanio et al., 1995]).

VORTEX SOLUTIONS

Below, we address an incompressible liquid of adensity

ρ

and a kinematic viscosity

ν

in a sphericalshell of radius

R

rotating about the axis of symmetry atan angular velocity

ω

d

; the rotation axis (inclined at anangle

α

to the vertical axis) precesses at an angularvelocity

ω

p

. The Navier–Stokes equations in a coordi-nate system connected with the rotating shell (mantle)and the condition of incompressibility have the form(e.g., see [Pedlosky, 1981])

(1)

The concrete form of the reduced pressure

p

is notwritten out because it is not required here.

An analytical solution for the flows in a rotating andprecessing spheroid that have the form of nested coax-ial cylinders (Fig. 1) apparently has not yet been

∂v∂t------ v∇( )v 2 wd wp+( ) v×++

= 1ρ---∇p– ν∆v+ wp wd×( ) r,×–

∇ v⋅ 0.=

obtained. However, we will take a cue from experimentand examine the flow structure against the backgroundof existing differentially rotating cylinders; for simplic-ity, the deviation of the liquid axis from the containeraxis is neglected below.

We apply the curl operation to both sides of theNavier–Stokes equation (1) and project the result ontothe rotation axis in the approximation of the

β

-plane(i.e., processes are examined not on a spherical surfacebut rather on a plane tangent to this surface). We obtain

(2)

where

ω

=

∆ψ

is the vorticity of the rotating liquid;

ψ

is the 2-D function of the current related to the liquidvelocity by the formulas

v

x

= –

∂ψ

/

∂

y

,

v

y

=

∂ψ

/

∂

x

; theparameter

β

= 2

∂

(

w

d

+

w

p

)

z

/

∂

y

accounts for the latitudedependence of the rotation velocity;

[

a

,

b

] =(

∂

a

/

∂

x

)(

∂

b

/

∂

y

) – (

∂

a

/

∂

y

)(

∂

b

/

∂

x

)

(the Poisson brackets);and the

x

and

y

coordinate axes are directed to the eastand north, respectively.

We are interested in the vortex solutions of Eq. (2).The presence of an Ekman boundary layer at the liquid–shell boundary is analogous to the effect of weak strat-ification in shear flows of an incompressible liquid[Howard, 1963]. If the characteristic scales of vortexstructures are considerably larger than the thickness ofthe Ekman layer, the flow can be treated as two-dimen-sional. Neglecting viscous dissipation on the right sideof Eq. (2), we seek its solutions in the form

ψ

=

ψ

(

ξ

,

y

)

,

ξ

=

x

–

ut

, where

u is the velocity of motion along theparallel. Then, Eq. (2) takes the form

(3)

which yields

Here, f = f(η) is an arbitrary function. Assuming that thevelocity of motion coincides in modulus with the veloc-ity of Rossby waves, u = uR = β/k2 (on scales smaller

than the barotropic Rossby radius), where k2 = + is the wave number, and taking the arbitrary function inthe form

we can write Eq. (3) as

(4)

The solution of this equation has the form (e.g., see[Bitsadze, 1982])

(5)

and the corresponding flow structure is presented inFig. 2. It has the form of a vortex street (the so-called

∂ω∂t------- ψ ω,[ ] β∂ψ

∂x-------+ + ν∆ω,=

ω βy ψ uy+,+[ ] 0,=

∆ψ βy– f ψ uy+( ).+=

kx2 ky

2

f η( ) βy 1 ε2–( ) 2η–( ),exp+=

∆ ψ uRy+( ) 1 ε2–( ) 2 ψ uRy+( )–[ ].exp=

ψ uRy+ cosh y( ) ε ξ( )cos–( ),log=

4

2

1

3

N ωd

ωp

23.5°

Fig. 1. Schematic diagram of flows (not to scale) observedin a precessing and rotating container: (1) rotation axis ofthe container; (2) precession axis; (3) rotation axis of theliquid; (4) typical flow structures.

IZVESTIYA, PHYSICS OF THE SOLID EARTH Vol. 42 No. 6 2006


cat’s-eye structure). At ε = 0, solution (5) takes the formof a zone flow:

For our purposes, it is important that, in the flows ofa conducting liquid with a 2-D cat’s-eye structure, akinematic dynamo was found by methods of numericalsimulation [Courvoisier et al., 2005]. For the field to begenerated, the existence of a vertical velocity havingthe dependence vz = vz(ψ) was admitted. In this case,the generation of the field is concentrated on separa-trices and at hyperbolic points.

Taking now the arbitrary function in the form

we obtain, instead of (4), the equation

and its solution is [Milne-Thomson, 1960]

(6)

The flow structure corresponding to this solution isshown in Fig. 3. In a coordinate system moving at theRossby velocity, it takes the form of a chain of vorticesof variable vorticity: cyclonic and anticyclonic. It iswell known that the mechanism of Ekman pumpingoperating in the boundary layer (at the liquid–shellboundary) ejects the liquid in cyclonic regions andsucks it in anticyclonic regions [Pedlosky, 1981]. Dueto this mechanism, the motion of the liquid takes theform shown in Fig. 4a and, according to observations[Vanyo et al., 1995], resembles the Busse convectivecolumns.

ψ uRy– cosh y( )( ).log+=

f η( ) βy + a2 1–2

---------------------------sinh 2η( ),=

∆ ψ uRy+( ) a2 1–2

--------------sinh 2 ψ uRy+( )( ),=

ψ uRy+ 2arctanha xcos

cosh ay( )----------------------⎝ ⎠

⎛ ⎞ .–=

For the flow structure shown in Fig. 4a, the poloidalcomponent of the magnetic field can be generated fromits toroidal component and vice versa, as shown inFig. 4b. It is this mechanism of the α2 dynamo that wasused to interpret the generation of the dipole field innumerical calculations [Kageyama and Sato, 1997].

ENERGY ESTIMATESUsing a rigorous approach, Kerswell [1996]

obtained an upper bound for the rate of viscous dissipa-tion in a liquid enclosed in a rotating and precessingspheroid.

At a sufficiently low precession velocity (such thatΩ < Ek1/2, where Ω = ωp/ωd; Ek = ν/ωdR2 is the Ekmannumber), the restriction on the rate of viscous dissipa-tion D has the form

(7)

where c is the flattening of the shell.Under the condition Ω ≥ O(Ek1/2) (the condition of

the loss of stability of the Sloudsky–Poincaré solution[Kerswell, 1993]), the dissipation ceases to depend onthe precession velocity and the viscosity and the upperbound for the rate of viscous dissipation takes the form

(8)

For the Earth’s core, we have ρ = 104 kg/m3, ωd = 7 ×10–5 1/s, R = 3.5 × 106 m, c = 1–1/400, Ω ≈ 10–7, and Ek =10–15–10–6 (it is seen from this how important it is toevaluate the viscosity of the liquid core). Comparisonwith experimental data showed that, although theexperimental dependence of the dissipation rate on theprecession velocity and the above relations are qualita-tively similar, it is necessary to decrease the latter by102 and 103 times, respectively [Kerswell, 1996]. As a

D116πc5

105 4c4 3c2 3+ +( )----------------------------------------------Ω2

Ek------ρωd

3 R5,≤

D2 0.43ρωd3 R5.≤

–3–8

–2

–1

0

1

2

3

–4 0 4 8

y

ξ –4

–4

–2

0

2

4

–2 0 2 4

y

ξ

Fig. 2. Cat’s-eye flow structure (ε = 0.87). Fig. 3. Vortex-chain flow structure (a = 0.9).

464


SHALIMOV

result, we obtain D1 ≤ 3 × 1012 W at a low precessionvelocity and D2 ≤ 1021 W at a high precession velocity(the turbulent regime). Even in this “reduced” variant,the rate of turbulent dissipation of precessional motionis 14 orders of magnitude higher than the previous(laminar) estimate [Loper, 1975], which has prevailedfor 30 years and has been responsible for the negativeattitude of geophysicists toward the energy possibilitiesof the precession-driven geodynamo. Of course, thereis no experimental evidence for such a high dissipationrate, and the estimate on the order of 1012 W derivedfrom the observed variation in the length of day[Mound and Buffett, 2003] apparently corresponds bet-ter to the upper bound of the power of the motions ini-tiated by precession in the core. However, estimate (8)clearly shows the potentialities of precession.

The power of precessional motion spent on ohmicheating can be estimated by using the standard scalingprocedure. The balance of the radial components of theCoriolis and Lorenz forces (the so-called strong-fieldapproximation [Braginsky and Roberts, 1995]) yields

(9)

where VT and BT are the toroidal components of the flowvelocity and the magnetic field and L is the characteris-tic scale.

On the other hand, in the steady-state regime, theequation of magnetic induction implies that the diffu-sion and induction terms have the same orders of mag-nitude, so that

(10)

where η is the coefficient of magnetic diffusion and BP

is the poloidal component of the magnetic field.

2ωdVT

BT2

4πρL-------------,≈

ηBT

L2----------

VT BP

L-------------,≈

According to (9) and (10), the momentum fluxthrough a unit area per unit time at the boundary of theshell is

(11)

As a result, the work per unit time done by the shellabove the conducting liquid is estimated as

(12)

where Rm = VTL/η is the magnetic Reynolds number.

According to theory [Stewartson and Roberts,1963], the maximum velocity V of perturbed motion ofa liquid in a spheroidal rotating and precessing shell(i.e., the motion driven by precession and superimposedon the Sloudsky–Poincaré solid-body rotation) is esti-mated as

(13)

where α is the precession amplitude and e is thedynamic flattening of the shell. Using the above param-eters of the Earth’s core, assuming V ≈ VT, and substi-tuting η ≥ 1 m2/s into (12), we obtain the estimate D3 ≈1012 W. This value is well consistent with the currentestimates of the power of the geodynamo determinedfrom ohmic losses, although these estimates haveincreased by an order of magnitude over the lastdecade, from (0.1–0.3) × 1012 to (1–2) × 1012 W (takinginto account the contribution of the small-scale field)[Buffett, 2002; Labrosse, 2003]. Note that, if the large-scale field is produced by an inverse cascade, the prob-

BT BP

4π------------- 2ωdρη.≈

D3

BT BP

4π-------------⎝ ⎠

⎛ ⎞ VT4πR2≈ 8πρωdηVT R2=

= 8πρωdVT

2 R3

Rm------------------------------ L

R---,

VωpR αsin

e----------------------,=

1

2

1

2

(a) (b)

Fig. 4. (a) Development of columns from a vortex chain due to the mechanism of Ekman pumping. (b) Generation of the poloidalcomponent of the magnetic field from its toroidal component due to the flows shown in Fig. 4a. (1) Ekman layer; (2) equator.



lem of separating the contributions of large and smallscales to the energy becomes complicated.

DISCUSSION AND CONCLUSIONS

The geomagnetic field has existed for no less than3.5 Ga and has had approximately the same intensity[McElhinny and Senanayake, 1980]. Its generation hasbeen attributed to convection (thermal and composi-tional). However, the thermal convection alone (with-out compositional) is not capable of supplying the nec-essary energy to the geodynamo [Gubbins et al., 2003].On the other hand, compositional convection is associ-ated with the existence of the solid core, whose esti-mated time of formation suggests its relatively youngage (≤1.5 Ga [Labrosse, 2003; Nimmo et al., 2004]).Under these conditions, it is natural to ask what mech-anism could have maintained the geomagnetic fieldbefore the solid core began to form (i.e., for a period ofno less than 2 Ga). This mechanism can be due to pre-cession because, as is evident from the above argu-ments, the motions in the liquid core driven by preces-sion are quite comparable in their kinematic and energypossibilities to the convective motions.

If precession plays a significant role in the genera-tion of the magnetic field, paleomagnetic measure-ments should contain information about the influenceof precession on the geomagnetic field. In particular,the dependence of the power of ohmic losses on the pre-cession amplitude (see formulas (12) and (13)) shouldaffect the intensity of the geomagnetic field and,accordingly, its stability. For example, it is known fromthe convective geodynamo theory [Jones, 2000; Resh-etnyak, 2005] that a weakening of the geomagneticfield reduces the flow scales in the Earth’s liquid core(in the plane perpendicular to the rotation axis). Thisleads to a more rapid rate of their ohmic dissipation,and the further generation of the field stops. The recov-ery of the previous field intensity is apparently due tononlinear effects (e.g., see [Shalimov, 2004]). With thecombined action of the convective and precessiondriven geodynamos, this scenario of the magnetic fieldinstability is quite possible. In other words, excursionsand reversals of the geomagnetic field must correlatewith low precession amplitudes. Note that, according toobservations [Lund et al., 1998], the normal state of thegeomagnetic field (with the predominant dipole com-ponent) is interrupted at intervals of a few tens of thou-sands of years, but the direct comparison between thenormal state interruption times and precession ampli-tude minimums requires a special study.

In conclusion, we should make a comment on themechanism by which the diurnal fluctuations of theforce (wp × wd) × r in Eq. (1) can be transformed intothe slow movements required for the generation of thegeomagnetic field. One of the mechanisms can be dueto the excitation of pairs of inertial waves: if the differ-ence between their frequencies coincides with the fre-quency of the driving force, a secular increase (at times

of the order of ohmic dissipation) in the amplitude ofthe waves is possible [Kerswell, 1993].

ACKNOWLEDGMENTS

This work was supported by the Russian Foundationfor Basic Research, project no. 04-05-64862.

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On the precession driven geodynamo