Effect of the inner core on the numerical solution of the magnetohydrodynamic dynamo

Physics of the Earth and Planetary Interiors 111 Ž1999. 105–121

Effect of the inner core on the numerical solution of the magnetohydrodynamic dynamo Ataru Sakuraba ) , Masaru Kono Department of Earth and Planetary Physics, UniÕersity of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan Received 11 December 1997; accepted 31 July 1998

Abstract We report two simulation results of the magnetohydrodynamic dynamo applied to rapidly rotating spherical systems using fully nonlinear equations under Boussinesq approximation. Calculations were carried out under the same parameter conditions but for a spherical shell and a sphere. We assume that a uniform internal heat source distributed in the whole sphere drives the convection and dynamo and that the physical properties of the inner core are identical to those of the fluid outer core except for its rigidity. This treatment enables us to compare two cases under the same condition, except the existence of the inner core. Magnetic field is effectively generated by strong velocity shear and helicity of the fluid near the top Žand bottom. boundaries. A stable axial dipole field develops in the case of the spherical shell because of the steady field generation at both the outer and inner boundaries, while the magnetic field in the sphere fluctuates with time from lack of the bottom boundary before it reaches the dipole dominant state at last. This result suggests that the Earth’s magnetic field may be stabilized as the inner core grows, even though the total energy input is the same. This study provides a first step to interpret the paleointensity data from the Archaean when there was a transition due to the growth of the inner core. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Inner core; Archaean; Magnetohydrodynamic dynamo

1. Introduction Recent developments in the computational ability have enabled us to carry out numerical simulations of self-consistent, nonlinear, three-dimensional dynamos, accelerating our understanding of the mechanism of the geodynamo, the physical process in the Earth’s core by which the geomagnetic field is maintained. Numerical models of the geodynamo developed by Glatzmaier and Roberts Ž1995a,b, 1996a.

) Corresponding author. Fax: q81-3-3818-3247; E-mail: [email protected]

and Kuang and Bloxham Ž1997. produced results bearing resemblance to the present geomagnetic field and its secular variation. In particular, Glatzmaier and Roberts Ž1995b. presented the first numerical simulation of a complete field reversal, and Kuang and Bloxham Ž1997. reported that their calculated magnetic field is very similar in structure and intensity to the Earth’s field and their dynamo solution operates in an Earth-like dynamical regime. Generation of an Earth-like axial dipole field has also been demonstrated by Kageyama et al. Ž1995. and Kageyama and Sato Ž1997a,b., who simulated the dynamo process in a rapidly rotating spherical shell composed of conductive ideal gas. We can say that

0031-9201r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 1 - 9 2 0 1 Ž 9 8 . 0 0 1 5 0 - 2

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A. Sakuraba, M. Konor Physics of the Earth and Planetary Interiors 111 (1999) 105–121

our attempt to simulate the geodynamo has reached a stage where we can hope to achieve realistic behaviors by model studies. Comparison of numerical results is possible not only with the present geomagnetic field but with paleomagnetic observations. Simulation of the magnetic field reversal and its statistical behavior is certainly a most exciting problem ŽGlatzmaier and Roberts, 1995b.. However, this is still a difficult objective because of the computational cost to cover enough time interval which contains many reversals of the magnetic field. In the actual Earth in the recent past, reversals occurred every 2 = 10 5 years on average ŽCande and Kent, 1992., which is one order of magnitude longer than the magnetic diffusion time of the core. Instead, we investigate in this paper the effect of the inner core on the structure of the geomagnetic field. It is widely accepted that the Earth’s inner core was formed a few billions years ago and grew gradually to the present size Že.g., Stevenson et al., 1983; Buffett et al., 1996; Labrosse et al., 1997., which implies that the oldest Archaean paleomagnetic data may correspond to the times when there was no inner core. Paleointensity data are not very abundant for these old ages, but there is some arguments that the intensity of the Earth’s magnetic field increased drastically at about 2700 Myr ago because of the inner core growth ŽHale, 1987; but also see Kono and Tanaka, 1995.. At present, this suggestion is still speculative because of the paucity and poor quality of Archaean paleointensities, but it is very interesting to explore possibilities of the effect of the inner core formation through dynamo simulation. Once the inner core was formed, it may have affected the core dynamics in many ways. In order to clarify the effect of the existence of the inner core, we present two simulations of a self-consistent magnetohydrodynamic ŽMHD. dynamo under the same parameter conditions; one with an inner core and the other without. This approach, especially the treatment of the dynamo in a whole fluid sphere, has never been done for a fully MHD regime. Thus this report represents the first step in studying this important aspect in the truly self-consistent, three-dimensional convective dynamo. It has been argued that there are three main aspects in the role the inner core plays in the geody-

namo process. Firstly, there is a magnetically stabilizing effect due to the inner core, because its magnetic diffusion time is long enough to average out rapid advective fluctuations in the outer core ŽHollerbach and Jones, 1993a,b, 1995.. Secondly, a boundary layer may be formed in a rotating sphere because of the presence of the inner core. For example, the Stewartson layer is formed along an imaginary ‘tangential cylinder’ when the inner core rotates with an angular velocity slightly different from that of the mantle ŽStewartson, 1966; but see, e.g., Hollerbach, 1994, for the effect of magnetic field.. It has been demonstrated that a significant difference appears between the convective motions inside and outside the tangential cylinder and this may play an important role in the dynamo process ŽGlatzmaier and Roberts, 1995a.. Lastly, there is a supply of energy due to the growth of the inner core which drives the geodynamo. As it is generally believed that radiogenic isotopes are not abundant in the Earth’s core, the dynamo is driven only by secular cooling of the core in the absence of the inner core. When the core is cooled to a level in which the inner core is solidified, some additional energies localized near the inner core boundary ŽICB. become available to the dynamo process; i.e., through the generation of the latent heat ŽVerhoogen, 1961. andror the compositional buoyancy due to the expulsion of light elements from the solidifying inner core ŽBraginsky, 1964.. In the present study, we are concerned with the first two possibilities. Therefore, we take the energy input the same in the two simulations but change the size of the inner core. We believe that this is a good first step to investigate the effect of the inner core on the geodynamo process.

2. The model and the numerical method Let us consider an electrically conductive fluid spherical shell with outer radius R o and inner radius R i . The fluid is covered by an insulating solid mantle and includes a solid inner core of which the physical properties are assumed to be identical to those of the fluid outer core except that it is solid. The case of no inner core is also included as the special case of R i s 0. The mantle rotates at a constant angular

A. Sakuraba, M. Konor Physics of the Earth and Planetary Interiors 111 (1999) 105–121

velocity V s V e z , where e z is the unit vector parallel to the rotation axis. The inner core is not required to co-rotate with the mantle; instead, its rotation rate is determined by viscous and magnetic torques acting on the ICB. We assume that a uniform heat source distributed in the whole sphere drives the dynamo and the temperature at the bottom of the mantle is fixed to a constant Tm . This assumption is adequate for the planetary convection and dynamo even if radiogenic heat source is completely absent in the core, because putting a uniform heat source in a medium with a constant mean temperature is equivalent to the situation in which the medium cools at a constant rate and has no internal heat source. Such an idea was used by Krishnamurti Ž1968. in a study of Benard ´ convection, and the formulation of complete dynamo equation ŽBraginsky and Roberts, 1995. also includes this term after the two time scales of convection and core growth are separated. We adopt the simplest Boussinesq approximation ŽChandrasekhar, 1961.; that is, the fluid is incompressible, the physical properties such as the density — r 0 , kinematic viscosity— y , coefficient of thermal expansion— a , and thermal and magnetic diffusivities k and h are constants everywhere. The viscous and Joule dissipations are neglected and the density fluctuation D r ' r y r 0 due to thermal expansion is considered only in the buoyancy term. Variables in a hydrostatic reference state will be designated by overbars. The reference state velocity —u, temperature—T, pressure—p, density change —D r , and magnetic field—B satisfy u s 0, B s 0, y=

ž

< V=r<

p

r0

q 2

2

/ ž

q 1q

Dr r0

The time-dependent perturbations from the reference state are denoted by unbarred variables in the following formulas. We use the following quantities for the unit of variables to make them non-dimensional: length—R o , time—R o2rh , velocity— hrR o , pressure— r 0 yhrR o2 , temperature— b R o hrk and magnetic field— Ž2 Vr 0 m 0 h .1r2 ; where m 0 is the magnetic permeability and b s R o erŽ3 k . is the temperature gradient at the CMB in the reference states; i.e., = T s yb Ž rrR o .. By this definition, velocity and magnetic field measure, respectively, the local magnetic Reynolds number and the local Elsasser number Žthe ratio of Coriolis to Lorentz forces.. In the fluid outer core, we solve the following non-dimensional perturbation equations: Pmy1

g

2

s 0, k= T q e s 0, Dr s yar 0 Ž T y Tm . , in a frame of reference co-rotating with the mantle. Here e represents the homogeneously distributed internal heat source, r is the position vector with respect to the center of the sphere, g s yg o Ž rrR o . is the acceleration due to the self-gravitation of the core material, and g o is its value at the core-mantle boundary ŽCMB.. Note that the heat source e can be interpreted as a constant secular cooling rate of the mantle as discussed above.

Eu Et

q Ž u P = . u s y= p q Ey1 u = e z q R aTr q Ey1 Ž = = B . = B q = 2 u,

Pky1

EB Et

ET Et

quP= T suPrq= 2T ,

s = = Ž u = B . q = 2 B,

Ž 1. Ž 2. Ž 3.

= P u s 0,

Ž 4.

= P B s 0.

Ž 5.

In the solid inner core, we solve Pky1

/

107

EB Et Pmy1

ET

q Ž V I =r . P= T s= 2T ,

Ž 6.

s = = Ž V I = r . = B q = 2 B,

Ž 7.

Et

EV I Et

15 s

8pg 2

HH y Ž X sin f q Ycos u cos f . e

x

q Ž Xcos f y Ycos u sin f . e y q Y sin u e z =sin u d u d f ,

Ž 8.

where V I is the angular velocity of the inner core relative to the mantle, 8pg 5r15 is the moment of inertia of the homogeneous inner core, Ž r, u , f . is the spherical polar coordinates, e i is the unit vector

A. Sakuraba, M. Konor Physics of the Earth and Planetary Interiors 111 (1999) 105–121

108

parallel to the i-axis, g ' R irR o is the nondimensional radius of the inner core, and

E Xs r

Ys r

uu

E ž / E / E ž r

r

uf

r

r

q Ey1 Br Bu

, rs g

q Ey1 Br Bf

. rs g

The nondimensional parameters, Rayleigh number— R a , Ekman number—E, magnetic Prandtl number— Pm and the ratio of diffusivities—Pk , are defined, respectively, by Ra s

ab g o R 4o ky

y , Es

2 V R o2

y , Pm s

h

,

k Pk s

h

.

We assume impermeable, no-slip boundary conditions at both spherical boundaries. At the CMB, the perturbation of the temperature is fixed to zero and the magnetic field continues to the potential field outside the core. On the other hand, temperature, heat flow, magnetic field and electric current are forced to be continuous at the ICB because we assumed the same physical parameters in both the outer and inner cores. We impose the condition that all the variables should be regular at the center of the sphere. Eqs. Ž1. – Ž7. are solved in the spherical polar coordinates using the spectral transform method similar to the one used by Glatzmaier Ž1984.. The vector variables are decomposed to toroidal and poloidal scalars by virtue of the conditions Ž4. and Ž5., and all scalar variables are expanded into spherical harmonics Yl m Ž u , f . and Chebyshev polynomials TnŽ j . to resolve their horizontal and radial structures, respectively. The variable j is defined by j s Ž2 r y 1 y g .rŽ1 y g . to resolve the outer core and by j s Ž2 r y g .rg to resolve the inner core. We use Chebyshev-tau method ŽCanuto et al., 1987. and Crank– Nicolson time-integration scheme for the diffusion terms, and Chebyshev–Legendre–Fourier transform method and second-order Adams–Bashforth scheme for the other terms. Time integration was carried out for two cases with the same nondimensional parameters Ž E s 3.16

= 10y5 , R a s 1 = 10 7, Pm s Pk s 20. except that one has an inner core with nondimensional radius g s 0.4 and the other does not Žg s 0.. The degrees and orders of spherical harmonics Yl m and Chebyshev polynomials Tn included in the analysis satisfy 0 F m F l F L and 0 F n F N, where L and N are fixed integers. In the case of the spherical shell with a finite inner core, we set Ž L, N . s Ž43,32. for the fluid layer and Ž L, N . s Ž43,10. for the solid inner core. In the other case, we used Ž L, N . s Ž43,43. for the whole sphere. A fully dealiasing technique known as the 3r2-rule ŽCanuto et al., 1987. was used for the spectral transformation. Calculations were performed with these truncation levels using the prescribed hyper-diffusivity y , k , h 4 s y 0 , k 0 , h 0 4w1 q Ž lr20. 3 x which artificially dumps higher-degree modes ŽGlatzmaier and Roberts, 1995a.. Time step in integration was taken to be f 2 = 10y7 in nondimensional unit, which amply satisfies the usual Courant conditions.

3. Results for the spherical shell All the simulation was started from the stationary reference state. A random small amplitude perturbation in temperature was given at grid points and the evolution of velocity and temperature was followed. Convection rapidly grew from this perturbation and reached a quasi-stable state typically in about 0.005 time units. After the convection has fully grown up, random magnetic perturbation of small amplitude was applied and integration in time was continued for velocity, temperature and magnetic field. In both the spherical shell and the sphere, the magnetic energy ŽME. grew exponentially while it was smaller than the kinetic energy ŽKE., and the ME eventually reached a constant level when it became about 30 times larger than the KE. There are many features in common in the two cases. We will first examine the case of spherical shell in detail. 3.1. Basic conÕectiÕe state As it is well known from theoretical ŽRoberts, 1968; Busse, 1970. and numerical studies ŽZhang

A. Sakuraba, M. Konor Physics of the Earth and Planetary Interiors 111 (1999) 105–121

and Busse, 1987; Sun et al., 1993., convection in rapidly rotating spheres and spherical shells is dominated by the effect of Coriolis force, which results in the formation of a number of convective cells aligned parallel to the rotational axis. These cells are usually called convective rolls because of their elongated shapes. Following Kageyama et al. Ž1993., these convective rolls can be classified as either cyclonic or anticyclonic rolls depending on the sense of the fluid circulation. When viewed from the north, the cyclonic Žanticyclonic. roll is a cell in which the fluid circulates in the counterclockwise Žclockwise. direction and has the axial velocity component directed to Žaway from. the equatorial plane. In other words, the axial component of the vorticity w= = u x z is positive Žnorthward. in cyclonic rolls. In our simulation, we typically observed six or seven pairs of such rolls; an example of the vorticity distribution in the equatorial plane is shown in Fig. 1Ža.. The axial flow in the convective rolls is essentially caused by the Ekman pumping in the viscous boundary layer. As the pressure in the cyclonic convective rolls is lower than the surrounding region, the Ekman pumping generates a converging flow in the boundary layer. The equatorward flow in cyclonic rolls is induced to conserve the equation of continuity. The situation in the anticyclonic rolls is exactly opposite to the above description ŽKageyama et al., 1993.. Because of this correlation between the axial components of the velocity and the vorticity, the convective motion in the northern Žsouthern. hemisphere has negative Žpositive. helicity H s u P Ž= = u. ŽFig. 1b.. In the absence of the magnetic field, cyclonic convective rolls tend to become larger than anticyclonic ones because of this Ekman pumping ŽKageyama et al., 1993.. This tendency can be seen in Fig. 1Ža.. In consequence, it follows that the mean azimuthal velocity Ži.e., differential rotation. arises, because the predominance of cyclonic rolls produces a net westward zonal flow deep inside the core and eastward flow in the shallower parts of the core. Fig. 1Žc. shows a meridional section of the differential rotation in a basic convective state, illustrating that the main part of the core containing convective rolls rotates slower than the mantle, while the outer part near the equatorial Ži.e., low-latitude. region shows superrotation. The contour lines in this figure are

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Fig. 1. Structures of velocity field in the basic convective state. Ža. Z-component of vorticity w= = u x z in the equatorial section viewed from the north. Solid Žbroken. lines represent positive Žnegative. values, i.e., counterclockwise Žclockwise. or cyclonic Žanticyclonic. convective rolls. Contour interval is 5000 in non-dimensional unit. Žb. Kinetic helicity uPŽ= = u. and Žc. zonal velocity uf in the meridional section averaged in longitude. Solid contours represent positive values; positive helicity in Žb. and eastward mean zonal flow in Žc.. Contour intervals are 10 6 in Žb. and 50 in Žc. in non-dimensional units.

nearly parallel to the z-axis except in the thin Ekman layers. The rapid change of zonal flow in the boundary layers causes a strong shear of fluid which plays an important roll in dynamo action.

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A. Sakuraba, M. Konor Physics of the Earth and Planetary Interiors 111 (1999) 105–121

3.2. Growth of the magnetic field As stated earlier, the magnetic field grows nearly exponentially while the magnetic energy ŽME. is less than the kinetic energy ŽKE.. The growth of the magnetic field can clearly be seen by the time development of the total ME and KE integrated over the fluid spherical shell ŽFig. 2a.. In this weak field regime Ži.e., ME - KE., the inner core slowly rotates westward because of the viscous torque generated by the westward mean zonal flow near the inner core ŽFig. 2b.. Fig. 3 shows the azimuthally averaged magnetic field at time t s 0.0375, which represents a typical structure in the weak field regime. The velocity field is very similar to the case of no magnetic field ŽFig. 1.. In the poloidal component ŽFig. 3a., the terms with dipole symmetry Že.g., axial dipole. seem to be preferred by the system, though the simulation started from a random seed magnetic field. The toroidal field is also antisymmetric with respect to the equatorial plane as shown in Fig. 3Žb.. As the back reaction of the magnetic field to the velocity field is small in the weak field regime, we can conclude that this preference of a symmetry is the inherent property of the velocity field.

Fig. 3. Longitudinally averaged Ža. poloidal and Žb. toroidal magnetic fields at t s 0.0375 Žweak field regime. projected on a meridional plane. Poloidal field is represented by lines of force, broken lines being counterclockwise. Maximum field strength is 0.176 in non-dimensional unit. Toroidal field is represented by contours of equal intensity, solid lines being eastward. Contour interval is 0.1.

The mechanism of the generation of the magnetic field may be summarized as follows. Due to the existence of the westward mean zonal flow in the

Fig. 2. Ža. Time development of the non-dimensional kinetic and magnetic energies ŽKE and ME. integrated over the fluid spherical shell. Žb. Time development of the non-dimensional angular velocity of the inner core around the z-axis relative to the mantle. Note the abrupt change to positive Žeastward. velocity at t f 0.06, the time when the ME reached the same level as the KE.

A. Sakuraba, M. Konor Physics of the Earth and Planetary Interiors 111 (1999) 105–121

bulk of the fluid layer ŽFig. 1c., a new toroidal field is induced in the core by the v-effect. The sign of the toroidal field is as expected from the stretching of a poloidal field Žessentially from south to north in this simulation. by the differential rotation ŽElsasser, 1946.. It is notable that two distinct foci of toroidal magnetic field are seen in each hemisphere, one below the CMB and the other above the ICB ŽFig. 3b.. Particularly, the one below the CMB is quite strong and localized. This fact suggests that the existence of the boundary may be very important in the dynamo process. As seen in Fig. 1Žc., there exists a strong shear of zonal flow in the Ekman boundary layers, supporting the localization of toroidal field near the boundaries because of its strong v-effect. The toroidal field above the ICB is rather weak, but its existence is meaningful. Vigorous convective motion occurs outside the axial cylinder attached to the inner core Žtangential cylinder., while the fluid motion inside this cylinder is very weak. A shear layer thus appears along the cylindrical surface, similar to the formation of the Stewartson layer ŽStewartson, 1966.. The existence of the concentrated magnetic field above the ICB can be explained by the v-effect in this shear layer. We show a contrasting case of the sphere in a later section. The v-effect itself is not sufficient for the generation of dynamo. To complete the dynamo process, generation of poloidal field from toroidal field Ž a-effect. is required. The importance of motion with helicity in the magnetic field generation was first pointed out by Parker Ž1955.. A close relation between the helicity and the a-effect has also been noted in the mean field dynamos ŽKrause, 1967; Moffatt, 1978.. Even in dynamos without prescribed a-effects, fluid motions which have larger averaged hemispherical helicities were found to reach the critical state for the dynamo generation at smaller velocity ŽNakajima and Kono, 1993.. This result was obtained from a study of kinematic dynamos, but it still suggests the importance of the helicity in the generation of magnetic field. The distribution of helicity in the weak field regime is essentially the same as the one without the magnetic field; negative in the northern hemisphere and positive in the southern hemisphere. A remarkable concentration of the contour lines is seen just

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below the CMB ŽFig. 1b.. This concentration is again due to the effect of the Ekman boundary layer. As the velocity changes very rapidly in this thin layer, the vorticity becomes quite large since it involves the space derivative of the velocity, with the result that the helicity also attains a large value near the boundary. We suggest that the no-slip boundary Žthe CMB. plays an essential role in the field generation, first by the concentration of the toroidal field, and then by the concentration of helicity and fluid shear which can be regarded as signifying the transformation from toroidal to poloidal field and vice versa. 3.3. Dynamo in the fully grown stage Fig. 4 shows a snapshot of the axisymmetric structure of the magnetic and velocity fields at t s 0.2525. When the ME becomes comparable to the KE, the velocity field is no longer free to maintain the original structure. A most prominent change in the velocity field can be seen in the mean azimuthal velocity ŽFig. 4b.. The sense of the differential rotation is essentially reversed: the low- and middle-latitude region near the CMB now lags behind the mantle, while most of the inner part of the outer core rotates faster than the mantle. The transition to this new structure reflects the growing importance of Lorentz force. The poloidal magnetic field is swept by the converging Ekman flow in the cyclonic convective rolls at the CMB as seen in Fig. 5Ža.. At the equatorial plane, on the other hand, the anticyclonic rolls have the converging horizontal flow Žbecause the velocity component parallel to the rotational axis is away from the equatorial plane in both hemispheres.. Because of this polarity change, the magnetic flux tends to be concentrated inside the anticyclonic rolls near the equatorial plane. Fig. 5Žb. and Žc. show, respectively, the equatorial sections of axial vorticity w= = u x z and axial magnetic field Bz at the same time as Fig. 4. The significant concentration of Bz around f s 1808 ŽFig. 5c., for example, occurs exactly in the large anticyclonic roll Žw= = u x z - 0. in this place as seen in Fig. 5Žb.. Other foci of magnetic flux can be explained in a similar way. These situations are nearly the same as described by Kageyama and Sato Ž1997a..

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A. Sakuraba, M. Konor Physics of the Earth and Planetary Interiors 111 (1999) 105–121

force, which points inwards in the case of the anticyclonic rolls ŽFig. 6b.. It appears from Fig. 6 that the

Fig. 4. An example of the longitudinally averaged magnetic and velocity fields in the meridional section in the strong field regime Ž t s 0.2525.. Ža. Helicity and Žb. zonal velocity are shown in the same manner as Fig. 1Žb,c.. Žc. Poloidal and Žd. toroidal magnetic fields are to be compared with Fig. 3Ža,b., but are much stronger now. Maximum intensity of axial poloidal field is 3.60 and contour interval of toroidal field is 0.5 in this figure.

However, we observed additional effects of the flux concentration in our simulation. Near the equatorial plane, the magnetic flux concentrated in the anticyclonic convective rolls is essentially parallel to and the electric current associated with it Ž= = B . is perpendicular to the z-axis, so that the Lorentz force points away from the center of the anticyclones. Fig. 6Žc. clearly shows that the strong Lorentz force exists in an outward direction at the mid-depth of f s 1808 where the concentration of B z is observed. This strong outward force is opposed by the Coriolis

Fig. 5. Another views of magnetic and velocity fields at the same time as Fig. 4. Ža. The radial magnetic field Br at the CMB in equal-area projection, and the axial components of Žb. vorticity and Žc. magnetic field in the equatorial plane. Contour intervals are Ža. 1.5, Žb. 6000 and Žc. 1.5. In Ža. the central meridian is f s1808 Žthe longitudes are arbitrary, but are common in Fig. 5Fig. 6.. Solid Žbroken. contours indicate positive Žnegative. Br , w= = u x z ) 0 and B z , respectively.

A. Sakuraba, M. Konor Physics of the Earth and Planetary Interiors 111 (1999) 105–121

Fig. 6. Distribution of Ža. velocity, Žb. Coriolis force and Žc. Lorentz force in the equatorial plane at t s 0.2525, with their magnitudes represented by the lengths of arrows. Only half hemisphere Ž1208F f F 3008. is shown. The scale of arrows is arbitrary, but is common between Žb. and Žc.. Maximum values in three figures are Ža. 1560, Žb. 1.07=10 9 and Žc. 0.988=10 9.

shape and strength of the convective cells are determined so that a balance between these two forces is established. As the magnetic field grows, the anticyclonic circulation becomes stronger so as the Coriolis force can be strong enough to oppose the Lorentz force ŽFig. 6a.. Consequently, the anticyclonic rolls grow much larger than the cyclonic ones, and their effect dominates in the azimuthally averaged zonal flow. Since the flow around the anticyclones is clockwise when viewed from the north, the mean zonal flow is eastward near the rotation axis and westward in the outer parts of the sphere ŽFig. 4b..

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In the case of Kageyama and Sato Ž1997a., such effect was not observed because the Lorentz force was not strong enough in their model. One of the consequences of the change of the mean zonal flow is the development of a toroidal field in the polar regions ŽFig. 4d.. Since the sense of the v-effect has changed in this region, the toroidal field has the opposite polarity to the one near the boundary Žthe CMB and ICB.. Although the mean azimuthal flow is reversed and a new toroidal field develops around the polar regions, it seems that an essentially same mechanism is operating in the magnetic field generation as we have seen for the weak field regime ŽSection 3.2.. For example, Fig. 4Ža. shows that the helicity is antisymmetric about the equator even in the strong field regime, and it appears that the axial dipole field dominates in the poloidal field as well ŽFig. 4c.. Furthermore, there still remains the same type of toroidal field concentrated below the CMB at the equatorial region and above the ICB ŽFig. 4d.. Taking these facts into consideration, we can conclude that the main feature of the dynamo process is not much different from the one operating in the weak field regime, except that stronger perturbations sometimes make a clearcut interpretation difficult. The role of the additional toroidal field around the polar regions is not clear at present. However, since the dynamo generation mechanism in the strong field regime appears to be essentially the same as in the weak field regime, we consider that they play only a secondary role in the dynamo process. Because the mean zonal velocity in the deeper parts of the core is eastward in the presence of the strong magnetic field, the inner core threaded by the magnetic flux passing through these parts will receive strong torque if it has a velocity much different from the mean fluid velocity near the ICB. As a consequence, the inner core rotates faster than the mantle in this regime. It is very interesting that the change from westward to eastward rotation takes place very abruptly at t f 0.06, which is the time when the ME reached the level of the KE ŽFig. 2b.. For t ) 0.06, the rotation rate of the inner core fluctuates with time, but its average is about 0.68 per one rotation period of the mantle. If we dimensionalize the time with respect to the Earth’s rotation rate, the rate of the inner core rotation of our model

A. Sakuraba, M. Konor Physics of the Earth and Planetary Interiors 111 (1999) 105–121

114

corresponds to 0.68rday. However, because our model is not exactly a scale model of the Earth, there is different ways of making correspondences. If we use the magnetic diffusion time for adjustment, the result is only 0.28ryear Žfor h s 3 m2rs and R o s 3.48 = 10 6 m.. Glatzmaier and Roberts Ž1995a. also found the superrotation of the inner core with a rate of about 1–38ryear. As their model is more ‘Earth-like’, this value may be used in comparison with observation. It turned out that their value is very similar to the rate of rotation found by recent studies of the seismic anisotropy of the inner core ŽSong and Richards, 1996; Su et al., 1996.. Glatzmaier and Roberts Ž1996b. investigated the cause of this rotation, and concluded that the magnetic torque is responsible for this effect.

4. Results for the sphere Simulation of the dynamo process in the fluid sphere was carried out in the same way as described for the spherical shell in the last section. The results are also quite similar in many ways. Since we have already examined the dynamo action in the spherical shell in detail, we will focus on the results bearing significant difference from the previous result. 4.1. Transient stage Thermal convection in the absence of the magnetic field has features similar to the case of the spherical shell, except for the lack of the flow structures associated with the inner boundary layer. Fig. 7

shows the time development of the total ME and KE integrated over the entire sphere. After the exponential growth of the ME, the energy level fluctuates violently for a while in contrast to the field growth in the spherical shell ŽFig. 2.. From around t s 0.15, the ME starts to grow again and reaches the same level as that in the spherical shell, about 30 times greater than the KE. Here we give a brief description of the growing and transient stage Ž t - 0.15. to make clear the basic mechanism of the field generation in the sphere. The issue of the fully developed dynamo will be presented in the next subsection. When a seed magnetic field is given in a random fashion, preference of the dipole-type field is not apparent for quite a long time. In fact, the growing field contains both the dipole-type and quadrapoletype fields, and their relative magnitudes change with time. This situation continues even when the ME becomes comparable with the KE. The fluctuation in the intermediate stage Ž0.07 - t - 0.15. can be interpreted as competition between the different types of the magnetic field. Fig. 8 shows a snapshot of the axisymmetric part of the magnetic and velocity fields at t s 0.0525. Because of the existence of two types of fields, no clear symmetry is observed with respect to the equatorial plane for both poloidal and toroidal magnetic fields ŽFig. 8c and d.. It is remarkable that the toroidal field is localized only near the CMB ŽFig. 8d., in contrast to the case of the spherical shell. We have seen that the strong fluid shear associated with the boundary layers produces the foci of the toroidal field through the v-effect. In this respect, it is natural that the toroidal field exists only near the outer boundary Žthe CMB. and is very weak in the central

Fig. 7. Time development of the non-dimensional kinetic and magnetic energies ŽKE and ME. integrated over the fluid sphere.

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sphere has no source region in the central part to enhance the dynamo action. 4.2. Fully grown stage The competition between the two different types of magnetic field comes to an end at last and the dipole-type field dominates from about t s 0.15. After that, the energy level of the magnetic field becomes stable with time. Fig. 9 shows a snapshot of the axisymmetric structures of the magnetic and velocity fields at time t s 0.25. The poloidal magnetic field is of a dipole type and is stable to the end of the simulation ŽFig. 9c.. Its structure at the CMB

Fig. 8. Meridional sections of magnetic and velocity fields in the sphere Ž t s 0.0525.. This figure shows the weak field regime, and can be compared with Fig. 1Žb,c. and Fig. 3Ža,b. Žthe contour intervals are the same..

part of the core in the case of the sphere because of the lack of the bottom boundary. Comparing the flow structures to those observed in the spherical shell ŽFig. 1., we see that the helicity around the central part of the sphere is very small ŽFig. 8a., and the contour lines of the mean zonal flow are almost straight without any distortion by the inner boundary ŽFig. 8b.. These tendencies confirm the idea that the magnetic field is effectively generated in the boundary layer and no effective field generation occurs around the center of the sphere. To understand the characteristics of the dynamo process in a fluid sphere, we must direct attention to the fact that the

Fig. 9. As Fig. 8 but in the strong field regime Ž t s 0.25.. The contour intervals are the same as in Fig. 4.

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resembles the field structure in the spherical shell; that is, several patches of a concentrated radial magnetic flux are observed on the CMB due to the converging flow at both ends of the cyclonic rolls ŽFig. 10a.. The toroidal part, on the other hand,

Fig. 10. Another views of magnetic and velocity fields at the same time as Fig. 9. Compare with Fig. 5, which are shown with the same contour intervals.

shows a different feature to that observed in the spherical shell. In Fig. 9Žd., two foci of the toroidal field exist beneath the lower latitude region of the CMB similar to that observed in the spherical shell, while the field near the rotation axis is complex with less symmetry. The close similarity of the poloidal magnetic field at the CMB for both sphere and spherical shell implies that one cannot tell the existence of the inner core from the observation of the potential field outside the core despite a big difference in the structure of the toroidal field inside the core. Fig. 9Ža. and Žb. show the longitudinally averaged helicity and zonal velocity, respectively. The distribution of helicity has a similarity to that observed in the weak field regime; it is antisymmetric with respect to the equatorial plane with negative Žpositive. helicity in the northern Žsouthern. hemisphere. Concentration of helicity to the outer boundary layer still continues to exist. The direction of the mean zonal flow, on the other hand, has changed after the full development of the magnetic field, eastward near the rotation axis and westward in the outer region. The mechanism to maintain this zonal flow can again be found in the predominance of the anticyclonic convective rolls due to the strong Lorentz force. Comparison of the equatorial sections of axial vorticity and axial magnetic field ŽFig. 10b and c. shows that a number of concentrated magnetic flux reside in the enlarged anticyclonic rolls. The generation of the strong axial dipole field and the localized toroidal field beneath the CMB is, therefore, due to the same mechanism as that described in the case of the spherical shell; shear and distortion of fluid in the boundary layer significantly contributes to the dynamo process. The toroidal field near the rotation axis is very complicated and difficult to interpret. The eastward flow near the axis is similar to the case of the spherical shell, and has a potential to generate a new toroidal field with the opposite polarity. However, there is no generating source around the center of the sphere as the inner boundary is lacking in this case. Because the combination of the axial dipole field and the toroidal field beneath the CMB is quite stable in time, the main part of the dynamo process in the sphere does not require the existence of the additional toroidal field near the z-axis. As we have

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mentioned in the case of the spherical shell, the role of the additional field is only of a secondary importance.

5. Discussion We have developed a dynamo model which can include the inner core of any size. In order to discuss only the geometrical effect of the inner core, we assumed a uniform heat source in the core and the same physical properties both in the outer and inner cores. Numerical simulations are performed for two cases, one with an inner core and another without. The nondimensional parameters used in the simulations are R a s 1.0 = 10 7, E s 3.16 = 10y5 and Pm s Pk s 20. This Ekman number indicates a thermal convection under a strong influence of rotation. The linear stability analysis ŽArdes et al., 1997. shows that the asymptotic relation in a rapidly rotating system is well established for this Ekman number. Though the Ekman number in the Earth’s core is considered to be at least four orders smaller than the value used here even for the eddy viscosity, we believe that a general tendency expected in a rapidly rotating system is preserved in the present calculations. The magnetic Prandtl number in our simulation may be significantly larger than the actual value applicable to the core. Large Pm and Pk were used to drive the dynamo under a moderate Rayleigh number. In fact, this Rayleigh number is about twice as large as the critical Rayleigh number for the onset of convection Žf 5 = 10 6 . obtained by the same code as used in dynamo simulation. The parameter regime represented by our simulation is significantly different from the one employed by Glatzmaier and Roberts Ž1995a, 1996a. or by Kuang and Bloxham Ž1997.. Especially, the diffusivity ratios Ž Pm , Pk . of our values are quite different from theirs. The Rayleigh number is also different. The parameter regime we employed is more similar to the one studied by Zhang and Busse Ž1988. or Jones et al. Ž1995.. Numerical results for the two simulations bear some resemblance to one another. Especially the magnetic field intensity and structure on the CMB has similar appearance after the magnetic field reached the complete saturation level. In both cases,

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zonal magnetic field antisymmetric with respect to the equatorial plane develops around the low latitudes beneath the CMB. The direction of the toroidal field is determined by the v-effect of the westward differential rotation in the bulk of the fluid layer. The combination of the zonal magnetic field and the main dipole field is a key constituent of the whole dynamo process. An interesting point is that the toroidal magnetic field is localized beneath the CMB. We consider that strong shear and distortion in the boundary layer play an important role in the generation of the toroidal field. Difference between the two numerical results is also apparent, especially in the structure of toroidal magnetic field near the rotation axis. In the spherical shell, there exist two toroidal fields in each hemisphere, one of which develops above the ICB with the same polarity and another develops around polar region with the opposite polarity to the field beneath the CMB. The opposite polarity is produced by the eastward differential rotation near the rotation axis. The eastward flow is generated by the dominating effect of the Lorentz force, in other words, because of the predominance of anticyclonic convective rolls due to the concentration of magnetic flux in them. The toroidal field above the inner core is generated by the same mechanism as that of the field beneath the CMB. In the sphere, on the other hand, the toroidal field near the rotation axis has very complex structure. This can be interpreted by the lack of the source of magnetic field at the bottom boundary. In order to show the difference of the field structure in the core, we analyzed the temporal development of the longitudinally averaged zonal field Bf on a cylindrical surface s s r sin u s const., where the cylindrical polar coordinates Ž s, f , z . is temporarily used. Figs. 11 and 12 show the temporal development of Bf in the spherical shell and the sphere, respectively. In both figures, the zonal field on s s 0.8 is very stable in time, indicating that the field beneath the CMB stably continues to exist. This is the result of the continuous field generation in the outer boundary layer. The field structure on s s 0.3 in the spherical shell ŽFig. 11b. is also nearly timeindependent; the toroidal fields above the inner core and around the polar region are stable in time. In the sphere, on the other hand, Bf on s s 0.3 is very irregular ŽFig. 12b.. This fact clearly shows the

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Fig. 11. Time development of the axisymmetric zonal magnetic field Bf on cylindrical surfaces Ža. s s 0.8 Žy0.6 F z F 0.6. and Žb. s s 0.3 Žy0.954 F z F 0.954., where Ž s, f , z . is the cylindrical polar coordinates. Solid contours denote Bf ) 0 Žeastward.. Contour intervals are 0.75 in both figures. The top and bottom boundaries correspond to the core–mantle boundary ŽCMB. in both figures. The inner core boundary ŽICB. is represented by two horizontal broken lines in Žb..

difference of the field structures in the spherical shell and sphere. In the spherical shell, the inner boundary

layer always generates a new toroidal field and stabilizes the field structure. However, the sphere

Fig. 12. As Fig. 11 but in the case of the sphere.

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lacks any continuous generation mechanism around its central part because it has no inner boundary layer, resulting in the turbulent magnetic field near the rotation axis. We come to the conclusion from the simulation results that strong shear and distortion in and near the boundary layers play a significant role in the dynamo process. Hollerbach and Jones Ž1993a,b. demonstrated that the inner core stabilizes the fluctuations of magnetic field outside it because of the magnetic diffusion in the conductive inner core. In their simulation of the mean field dynamo, it was observed that the axisymmetric poloidal field penetrating the inner core prevented the field reversal triggered by the fluctuation of toroidal field outside the tangential cylinder. In the present simulations, however, poloidal field is mostly confined to anticyclonic convective rolls outside the tangential cylinder, and the toroidal field beneath the equatorial part of the CMB is very stable in time. Consequently, the stabilizing role of the inner core due to its diffusivity has less significance in our numerical results. Instead, the inner core has another stabilizing effect due to a continuous magnetic field generation in the boundary layer above its surface. Kuang and Bloxham Ž1997. presented a geodynamo model using either stress-free or no-slip boundary conditions at the spherical boundaries. The toroidal magnetic field generated in the stress-free model was distributed over the entire region of the core, while the toroidal field in the no-slip model was confined to the regions beneath the equatorial part of the CMB and inside the tangential cylinder. This field structure of their no-slip model is similar to the features shown by the Glatzmaier–Roberts model ŽGlatzmaier and Roberts, 1996a.. Kuang and Bloxham concluded that the stress-free model was more appropriate to apply to the actual geodynamo, firstly because the force balance realized in their simulation was appropriate for the Earth’s core where viscous effect was negligible and secondly because it was unlikely that the thin viscous boundary layers were important in generating a planetary scale magnetic field. However, we cannot neglect strong shear and distortion arising in the boundary layers however thin they are. Of course, our simulations assumed a significantly large Ekman number and does not exactly correspond to the situation in the real geody-

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namo. Importance of the boundary layers in generating the geomagnetic field should be investigated more in detail in future works.

6. Conclusions Our dynamo is driven only by secular cooling of the core and mantle and does not depend on the existence of the solid inner core. This treatment enables us to compare the dynamo processes in the spherical shell and the sphere under the same conditions. In both cases, the generated magnetic field was an Earth-type dipole field and had common features outside the core. We investigated the dynamo processes in both weak field and strong field regimes. In the weak field regime, a remarkable point is that the toroidal field is localized near the boundaries where strong fluid shear and helicity exist. We conclude that the thin boundary layers are important in the magnetic field generation. In the strong field regime, a quite similar mechanism seems to operate in the magnetic field generation. The most dramatic change in the force balance in the core is the predominance of the anticyclonic rolls caused by the increase of the Lorentz force. We found that the magnetic flux is localized in the anticyclonic rolls near the equatorial plane. The clockwise circulation is intensified so that the Lorentz force is balanced by the Coriolis force. The magnetic field in the sphere is less stable because it has no persistent source of field generation in its central region, while the field in the spherical shell is more stable. Other than this distinction, there seems to be very little difference between the fields outside the core with or without the inner core. Our results do not support the conjecture that a drastic change may occur in the field intensity when the inner core was formed ŽHale, 1987.. However, the present conclusion is just from our simulations and may not be applicable to general situation. This limitation is caused by a few factors. First, we assumed the same energy inputs for the spherical shell and sphere. The result might have been quite different if we included an additional energy source on the inner core surface by modeling the effect of latent heat or compositional convection. Second, the diffusivity ratios Ž Pm , Pk . used in our simulation are quite different from the ones applicable to the Earth.

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Third, the simulation covers about three times the decay time of the dipole, but probably is still insufficient for the perfect description of the dynamo solutions. For example, we cannot exclude the possibilities that the field reversal occurs or a magnetic field of different type dominates at some time. However, the tendencies observed in the simulations Žespecially in the growing stage of the magnetic field. seem to be quite general. These include the fact that the boundary layers play an important role in magnetic field generation, and the stability of the magnetic field in the deeper parts of the core depends on the existence of the inner core. These are the inherent features of the present model.

Acknowledgements We are grateful to Chris Jones ŽUniversity of Exeter. and an anonymous reviewer for constructive criticisms which resulted in a substantial improvement of this paper. We thank Akira Kageyama and Keisuke Araki ŽNational Institute for Fusion Science. for preprints and for useful discussion. We also appreciate the help provided by Mizuho Ishida and Takao Eguchi ŽNational Research Institute for Earth Science and Disaster Prevention, NIED. and Takashi Tanaka ŽCommunications Research Laboratory. in the course of numerical simulation. The computing resources for the simulations were made available to us by NIED under a special project of ‘Computational Science’. AS was supported by the Ocean Hemisphere Project for presenting this paper at IAGA Assembly in Uppsala.

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