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Towards a realistic theory of thegeodynamoS. I. Braginsky aa Institute of Geophysics and Planetary Physics, UCLA, Los Angeles,CA, 90024, USAVersion of record first published: 19 Aug 2006.

To cite this article: S. I. Braginsky (1991): Towards a realistic theory of the geodynamo, Geophysical& Astrophysical Fluid Dynamics, 60:1-4, 89-134

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TOWARDS A REALISTIC THEORY OF THE GEODYNAMO*

S. I. BRAGINSKY

Institute of Geophysics and Planetary Physics, UCLA, Los Angeles, CA 90024, USA

(Received 27 November 1990; in final form 9 January 1991)

The physics of the geodynamo is discussed. The main processes relevant for the buoyancy driven geodynamo are isolated. The successive stages of development of geodynamo theory are briefly described. The mechanism of local turbulence in the Earth’s core is explained, and an estimate is presented of the turbulent transport of density inhomogeneities in the Earth’s core. The significance of this turbulent transport to the geodynamo mechanism is stressed. The general scheme of the complete geodynamo theory of the future is outlined.

KEY WORDS Dynamo theory, geomagnetism, model-2, turbulence

1. INTRODUCTION

The theory of the hydromagnetic dynamo of the Earth (the geodynamo) has advanced a long way since Larmor (1919) made his famous hypothesis 70 years ago and since Cowling (1933), in the first theoretical dynamo paper, proved his famous negative result. Today, kinematic dynamo theory (where the fluid velocity is prescribed and only the magnetic field should be found) is highly developed. It shows with certainty that a magnetic field can be generated by the motion of a homogeneous conducting fluid. Moreover we know that many different motions can generate a field resembling that of the Earth.

It can be said the magnetic field of the Earth is produced by an “electric power station” in the Earth’s core. This power station consists of a generator, a motor and a fuel supply. Kinematic dynamo theory is the theory of the generator. Theories of the motor and of the fuel supply are also needed for the complete solution of the problem of geomagnetism.

Many different geodynamo theories? have been developed that go far beyond

*Contribution 3 of the Center for Earth and Planetary Interiors, UCLA. This article is a combination and expansion of two invited talks: one presented in Session 1.06 of the Scientific Assembly of IAGA held in Exeter, UK (July 24-August 4, 1989) and the other in Session 10 of the SEDI Symposium held in Santa Fe, New Mexico, USA (August 5-11, 1990).

?The literature on the geodynamo and related topics today contains hundreds of papers, and in the references below we mention only those that impinge directly on our lines of arguments. A few books now exist in which relevant problems are posed and numerous references are given, e.g. Krause and Radler (1980), Moffatt (1978), Roberts and Soward (1978), Soward (1983), Jacobs (1987a, 1987b). Extensive reviews of geodynamo theory by Roberts (1987b) and by Roberts and Gubbins (1987) are included in Jacobs (1987b).

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kinematic theory but none of them are complete. Some assume very simplified geometry (planar, cylindrical); others use models that are usually called “inter- mediate” in which only some of the velocity and force components are obtained from the equations while others are prescribed. There are also many papers which consider various processes relevant to the dynamics of the Earth core, such as waves and different kinds of instabilities. These theories add a lot to our understanding of the geodynamo mechanism but they leave many questions (including the most significant) unanswered. We need a fully fledged theory of geodynamo to clarify both the physical and the mathematical aspects of the whole problem.

To be more specific let us start with the question: “Why not simply compute the geodynamo?” Just this question was asked by Roberts (1988) at the outset of a review paper. And this is the question that really aims at the heart of the problem. The governing equations of the geodynamo are the well-known equations of classical fluid mechanics and electrodynamics. They include physical parameters, some of which are known with good precision while others are poorly constrained. The number of poorly known parameters of the core (e.g. the change of mass of the fluid core due to gravitational differentiation, the rate of cooling of the core, the electrical conductivities of the core and mantle, the viscosity of the core) does not look overwhelmingly large. A computer, such as the CRAY with its -100 megaflop speed, is quite a common device nowadays, and supercomputers also exist that are many times faster. So why not simply compute the geodynamo? The short answer to this question is to say that the geodynamo is a very complicated system. This answer becomes more informative (we hope) after a consideration of the physical state of the Earth’s core and the mathematical statement of the geodynamo problem given below.

In this paper only one mechanism producing the geomagnetic field is con- sidered, namely the buoyancy-driven geodynamo in which the Archimedean force drives the core into convection. The mechanism of buoyancy is clear, and plausible estimates show that it provides enough energy-this is why the hypothesis is the most popular one. We only mention here another suggestion for fueling the geodynamo exists, one that relies on external forces. It was suggested by Malkus (1963) that the core motion is stirred by the so-called “Poincare force”, F“= -a x r, resulting from the time rate of change, h, of the angular velocity vector of the Earth. Additional support for this view was given more recently by Malkus (1989). The effect of this mechanism on the creation of the geomagnetic field deserves further study, but such attempts go far beyond our present considerations.

The system of equations governing the complete three-dimensional model of the geodynamo driven by compositional buoyancy is set out in the Appendix. The region in which the geodynamo works is shown in Figure 1.

t is the time; ( r , 0,4) and (z, s, 4) are spherical and cylindrical polar coordinates in the reference

z-axis directed along the axis of rotation, z = r cos 8;

The following notation is used throughout this paper:

frame rotating with the solid Earth;

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R t l

Figure 1 The stage on which the geomagnetic dynamo acts.

s is the distance from this axis, s = r sin 6; 0 is the colatitude, 4 is the East longitude; R=0.729. 10-4s-' is the angular velocity of the Earth's rotation; a,, a,, a,, . . . are partial derivatives: a/&, a/&, 8/80,. . . ; l,, l,, 10,. . . are unit vectors in the directions of increasing r, z , 8,. . .; V is the fluid velocity; B is the magnetic field; C is the fractional excess of true fluid density, p, over the static equilibrium

density, for the latter we take po = 10.7 g/cm3. These fields will often be subdivided into two parts. One part is the average over

the longitude, 4; it is axisymmetrical about the z-axis and is marked by an overbar. The second part, which is asymmetrical, has a zero +average and is marked by a prime:

V=B+V', B = B + B , C=C+C';

t~ is electrical conductivity of the Earth's core; q=c2/4m is magnetic diffusivity of the core, c is the velocity of light; following

Braginsky (l989), we take q = 2.4. lo4 cmz/s. v is kinematic viscosity of the core; R,=6371 km, R1=3480km, R,=1222km are radii of the Earth, the fluid outer

CMB denote the core-mantle boundary, r = R,; ICB denote the inner core boundary, r = R,; ( ) is used to denote the volume average, e.g. (C)=jCdV/Vl,. Here integration

core, and the solid inner core according to Dziewonski et al. (1981);

is performed over the fluid core, and V, , = V, - V, is its volume;

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92 S. I . BRAGINSKY

L is a characteristic linear dimension of the system. In making numerical estimates we use L= lo3 km.

2. THE MAIN EQUATIONS

The set of equations governing the geodynamo must describe the convection of electrically conducting fluid driven by the buoyancy of the compositional and thermal inhomogeneities of the core, and interacting with the magnetic field generated by this convection. This set of equations can be written down in different approximations; see for example Braginsky (1964d). The simplest is the Boussinesq approximation for a liquid solution. In this “homogeneous model” the real adiabatic equilibrium distribution of density, pa(r) , is everywhere replaced by a constant, p o , except in terms in which the variation in p a is crucial. For example, the density excess is defined by C=(p-pa) /p , . The system of dynamo equations has the following form

d,V = - VP + CFK, (1)

a , ~ = v x (V x B) + ~ v ~ B , (2)

~ , c + v . ( v c + I ) = G ~ , (3)

Here d, = 3, + V - V is the motional time derivative, P is the effective pressure divided by p o , FK are the forces acting upon unit mass of fluid:

FR = 2V x R,

FB = (V x B) x B/471pO, (6b)

F’ = vV2V. (6d)

Here Fa is the Coriolis force, FB is the Lorentz (magnetic) force, Fa is the Archimedean (buoyancy) force and F‘ is the force of Newtonian viscosity, g is the acceleration due to gravity, and I is the diffusional flux of the density excess, C .

The equation of motion (1) and the induction equation (2) describe the evolution of the velocity V and the magnetic field B. They obey well-known boundary conditions: the no-slip condition of no relative motion between the fluid and solid surfaces (e.g. V=O at r = R , ) and the continuity of the magnetic field.

The derivation of (3) is based on the following assumptions: (a) core convection is associated with small deviations in all thermodynamic quantities from their values in some basic equilibrium state; (b) this basic state is well mixed by

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convection and as a result the composition and entropy are homogeneous. The basic state is therefore characterized by adiabatic gradients of density, V p , , and temperature, VT,:

Here uf is the adiabatic speed of sound, a is the coefficient of thermal expansion, and c p is the isobaric specific heat. The dimensionless combination y=uuf/cp is known as Gruneisen's parameter and is about unity in the core. The changes in p a and T, across the liquid core are relatively small (about 2079, and we neglect them using instead p o and To everywhere, except when we directly encounter the gradients V p , and VT,. The neglect of these changes is the essence of the Boussinesq approximation. Having adopted p a = p o and T,=To, we must take all thermodynamic parameters, such as a, c p etc., to be the constants.

It is worth specially noting that the second term in the expression p = p a + p o C is very small. This can be seen from gC-2RK for example, if V - 10-'cm/s, then c- lo-*.

The density p o is assumed constant in the Boussinesq approximation despite the fact that Lp, ' V p , - lo-' >> C. There is no contradiction here because Fa = gC appears directly in (1) whereas V p , would (if it were included) manifest itself only as a slight inhomogeneity in the coefficients of the equations. For example, the continuity equation V - V=O is used instead of the more precise V - (p,V) =O. Because of the spherical symmetry of the inhomogeneity, one may reasonably hope that this will not lead to the loss of significant new effects. Note that the above condition of convection in the core is in striking contrast to laboratory convection for which Lp; 'Vp,<< C.

In the simplest case, C is supposed to be caused solely by the compositional inhomogeneity: C=Cr, where C5= -El1. Here t1 =t-to is the deviation in the concentration, t, of the light constituent from its mean value, to, and Cr= - p - ' a p / d t . (ti is considered to be a constant.) It is supposed that V t o = O , so that all the inhomogeneity of t is included in 5'. In order to avoid an arbitrary constant in the definition of to and tl, we set the average of tl over the fluid core to zero: (tl) = 0. For the case of purely compositional convection, the equation governing the concentration of the admixture is

where Gr= -a t to . Taking I = -&Ir , we recover (3). The quantity Gc=Cra,t, plays the role of the effective bulk source of C which

balances its change due to fluxes of magnitude IN and ZM at the inner and outer boundaries of the core, so maintaining the average of C as zero: (C) =O. The value of poGC coincides with the slow decrease in the density of the fluid core during the evolution of the Earth over geological time. This change is responsible for the release of the gravitational energy that powers the dynamo.

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The flux 1,= -IN is a result of the liberation of the light constituent of the fluid at I = R , during inner core crystallization. The density of the fluid core is slightly (- 10%) less than the density of pure iron under the same conditions. This means that the fluid core is composed of iron and some light constituent(s). If during the crystallization of the inner core the light admixture is not accepted by the solid phase then its excess is released at the inner core boundary lowering the fluid density there. It was suggested by Braginsky (1963, 1964d) that this process is the principal cause of the negative density excess at r = R , and of the Archimedean driving force. In other words, inner core crystallization provides the “fuel supply” for the electric power station within the Earth’s core. This is “fueling from the bottom”. Another possibility is “fueling from the top”, in the form of a flux l M of heavy constituent from the mantle to the core. This was first suggested by Urey (1952), who supposed that some iron in the mantle may be sinking into the core. The quantities IN and lM are input parameters for geodynamo theory. They should be found by comparing the theory with the observations. An estimate of IN is made below.

Equations (1)-(6) describe the “homogeneous model” in which p o and other material coefficients are constants. If IN and lM are given, this model can be solved without reference to the equation governing the core temperature.

The variation of temperature in the fluid core also can be considered in the Boussinesq approximation (the “homogeneous model”). The temperature excess over the adiabat, T , = T - T,, obeys the equation

which is similar to (3). Here

where q is the complete heat flux density and q,= - p o ~ , ~ T V T , is the adiabatic heat flux, I C ~ is the molecular (electronic) thermal diffusivity of the core; the sum x Q K contains all sources of heat: radiogenic, QR; Joule, QJ; viscous, Q,; diffusional, QD (which is given by Equation (43) below) and also Q, = - V . q,. See Braginsky (1964d) for details.

The expression for the density excess that includes both compositional and thermal effects is

The complete expression for C would also include a pressure term, C P = p / K T , where K T is the isothermal incompressibility and p is the thermodynamic pressure, which slightly differs from the effective pressure pe=poP. The term C p should however always be neglected in the Boussinesq approximation. One can estimate that p-pogCL and K T -p0uS. Hence, according to (7a), Cp/C-gL/u,2 - LVp,/p,;

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GEODYNAMO THEORY 95

but this is just what has been neglected in this approximation. In short, the Boussinesq approximation supposes that K T -P 00.

In conditions of compositional convection, the heat production term, Q = pOcpGT, in (8) can be roughly estimated to be the rate of release of gravitational energy: Q-poGCgL. Because of the similarity between (3) and (8), this gives CT/Cs - uGT/GC - CrgLJc, - LVTJT,, which is negligible. In fact, the negative term q. makes CT even smaller. Only when the sources of compositional density excess are extremely small can the thermal term CT in (11) be significant. It was shown by Braginsky (1964d) and by Loper and Roberts (1983) that, in the conditions prevailing in the Earth core, the compositional term in (1 1) dominates. (See also Section 6.) The thermal term should be omitted in the Boussinesq approximation, along with the pressure dependence of C . The approximation has an accuracy of about 1&20 %.

Even in the approximation in which the term C T = -aT, in (11) is neglected, the evolution of the temperature can have some interesting consequences for the geodynamo. As noted by Braginsky (1964d), thermal effects play a regulating role in the geodynamo, because it is just the cooling of the core that determines the rate of inner core crystallization. For example, if the magnetic field increases too much, Joule heat production will also increase. This will result in a reduction in the cooling rate of the core, because the removal of heat from the core by the mantle takes a longer (geological) time; it cannot adjust quickly to the increased Joule heating. The reduction in the cooling rate retards the crystallization of the inner core, and IN therefore diminishes. Hence C and the Archimedean driving force decrease as well, and the entire dynamo process weakens. This leads to a decrease in B and in QJ. This mechanism could regulate the general level of dynamo activity on the long time scale; it may be responsible for the observed very slow variation in the geomagnetic dipole moment.

3. GENERATOR WITH SELF-EXCITATION IN THE EARTHS CORE

Cowling’s theorem forbids an axially symmetric dynamo. The “working” dynamo must therefore be three-dimensional, and hence rather complicated. It was, however, shown by Braginsky (1964a, b, c) that the self-excitation of a magnetic field having a small deviation from axisymmetry is possible if the magnetic diffusivity pl is small enough, i.e. if the magnetic Reynolds number, W= LV/pl, is large enough. Indeed, adopting a common estimate of the core conductivity 0-3. lo5 S/m, and taking V-3*1O-’cm/s and L - 108cm for velocity and length, we obtain W - 10’. In such a “nearly axisymmetric dynamo”, the axisymmetric parts of the velocity and the magnetic field, P and B, obtained by averaging V and B over the longitude 4, are much greater than the asymmetrical components V and B’. The averaged components may be written in the form satisfying continuity equations:

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B = 1,B, + B,’ Bp = vlj x 1,s- 1. (1 2b)

Here t,b = sA, where A is the magnetic vector-potential.

(1964a, b, c): The following order of magnitude relations were assumed by Braginsky

and by expansion of Eq. (2) in the small parameter 9-l” the “equations of generation” were obtained in two spatial variables, e.g. z, s or r, 8. Let us choose appropriate units: for length LI = R , ; for time t , = R f / v ; for velocities V,, = R l / t , = q /R , , V,, = V,,B (so that 9 = R1VbI/q); and for magnetic fields B,,=9Bp,, where B,, is yet to be defined. The non-dimensional equations of generation then have the form

a,A, + - ‘0, . v ( d , ) - d%i, = rB,, (13)

Here A ( 1 ) = V z - s - 2 and c = V,/s. The function T(r,O) was expressed explicitly as the 4 averages of some combinations of the components of the asymmetrical velocity, V‘. With the scaling (12c), the dimensionless coefficient of generation, r, is of order of unity. However r depends on the gradients of V‘. If V‘ varies rapidly in a thin layer of thickness - 6 , then r-8-l; the integral of r over the layer is then finite ( - 1). Such “concentrated generation” was analyzed by Braginsky (1964a). If V’ has the form of travelling wave, concentrated generation develops in the layer of thickness c ~ - W - “ ~ on the surface, where the wave velocity coincides with the velocity of fluid. If V’=O, (13) has no source term, and therefore A,+O, B,+O as t+w. Then the source of B, in (14) vanishes and hence B,+O. This is Cowling’s theorem.

The nearly axisymmetric dynamo mechanism with B>>1 consists of the following two steps. The stretching of the field lines of “poloidal” field Bp by the fast non-uniform rotation, c = V,/s, produces a large “toroidal” field B, - 9B,. The action of the source term, rE,, often called the a-effect, then creates the poloidal field B, from B,, thus completing the generation loop. It should be noted that the large toroidal field, B, - BEp, is invisible; only the poloidal field, B,, reaches the Earth’s surface r = R,. Such a dynamo is often referred to as an “am-dynamo” or as a “strong field dynamo”.

Generation by a motion in which V’ has a short characteristic length scale, 1<<L, was considered by Parker (1955) and by Steenbeck, Krause and Radler (1966); see also Moffat (1970, 1978), Krause and Radler (1980). Two-dimensional equations similar to (13) and (14) were obtained for the mean magnetic field in this case also. The name a-effect was introduced by Steenbeck, Krause and Radler (1966) for the generation of a mean electric current parallel to the mean magnetic field by the term Vx(aB), obtained after averaging (2) over the small length scale. The

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GEODYNAMO THEORY 91

quantity a may be a tensor in the case of anisotropy. In the am-dynamo we have aB=lJB,, but the concept of an a-effect is applicable also for the case B,-B,, in which a significant a-effect term appears also in Eq. (14) for Do. The dynamo with 8,-Bp where both Bp and B, are generated by a-effect while rotation 5 is insignificant, is referred to as an “a2-dynamo” or a “weak field dynamo”. The self-explanatory label “a2m-dynamo” is also used when both shear and a-effect creates toroidal field from poloidal. A more general derivation of the generation equations, one that links the different approaches together, was provided by Soward (1972, 1990).

By numerical solution of the two-dimensional equations (13) and (14) with some plausible expressions for r, [ and V,, Braginsky (1964~) obtained the first “working” kinematic models of the geodynamo that gave fields similar to that of the Earth. Many different models of both am and a2-types were obtained numerically with the generation equations by Roberts (1972). Numerical solutions of the three-dimensional equation (2) were performed by Kumar and Roberts (1975). They obtained a family of three-dimensional kinematic dynamo solutions of differing Reynolds number, and they demonstrated that, as 9 and V, became large enough, these tended to the corresponding solution obtained from the two- dimensional equations (13) and (14).

Kinematic theory is now broadly developed (e.g. see Moffatt, 1978; Krause and Radler, 1980), but this is only the first step in geodynamo theory. Kinematic models are based on Eq. (2), which is linear and homogeneous in B, and which leaves its amplitude undetermined. The amplitude of Bp in such models of the geodynamo is obtained from the observed geomagnetic dipole moment. The amplitude of the greater component B,, which is hidden in the core, can be determined from the model, thus making the calculation of the Joule heat dissipated by the geodynamo possible. This dissipation has a magnitude of the order of QY - V,y,(v/R:) (B:)/47~ where V , is the core volume, y J is a constant depending on the model, and ( B : ) is the average of B: over the volume of the core. Numerically QY- lO”erg/s (with yJ- lo2); this determines the scale of the energetics of a geodynamo of strong field type.

Note that well-developed models of weak field geodynamos exist that go far beyond kinematic theory; these are not considered here. See for example Busse (1975), Cuong and Busse (1981), Zhang and Busse (1989). Only strong field geodynamos are studied in this paper.

4. MECHANICS OF AXISYMMETRIC MOTION

The natural way to develop the two-dimensional dynamo further is by adding the axisymmetric component of equation of motion (1) to the system (13) and (14). The axisymmetric velocity 0 is then to be found from that equation instead of being prescribed.

To concentrate on the mechanical part of the problem, we may simply consider the axisymmetric V and B but with a prescribed a-effect term rB, added into (13). Such an “intermediate” model is the next step after kinematic theory, but it is

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much less than a complete one. To simplify the problem even more, the axisymmetric Archimedean force gC in (1) may also be prescribed, thus avoiding the need to analyze the complicated processes of the production and transfer of the density excess C. This intermediate model of am-dynamo, driven by (pre- scribed) buoyancy, was considered by Braginsky (1972, 1975, 1976, 1978, 1988, 1989, 1990), Braginsky and Roberts (1987), and Roberts (1988, 1989a).

The inertia term d,V in (1) is much smaller than the Coriolis term, Fn = 2V x a, and may be neglected. Note that we lose in this way an interesting class of solutions, namely the torsional oscillations; see Braginsky (1970, 1980, 1989). These could be re-instated by retaining the term C?,V, in (1). After neglecting d,V, we reduce (1) to an equation of equilibrium from which the fluid velocity at any moment can be expressed in terms of the forces at the same instant. The viscous force inside the main body of the core, F’-vVjL2, is also negligibly small for ~ < < Q L ~ - l O ’ ~ c m ~ / s , which is true for any reasonable estimate of v. The 4- projections of (1) and of its curl are

2!2 Vs = q, (154

2!2C?zV,= -(VxFB),+(gxVC),, (1 5b)

and give us Vs (hence x, 0,) and [= V&. The problem is now a nonlinear one. A natural unit of magnetic field strength is

B , =(47~p,2!2q)”~. (16)

Taking 7 = 2.4.104 cmjs and po = 10.7 gjcm, we obtain B, = 21.7 G. The correspond- ing Alfvtn velocity, V, =(2!2q)’”, is V, = 1.87 cm/s.

To make the equations nondimensional we use the same units as for the nearly axisymmetric kinematic models (Braginsky, 1964c), namely length L, = R, ; time t , = R:/q; velocities V,, = q /R1 and V,, =BV,, so that cl =B/t,; and magnetic fields B,, = B , K ’ 1 2 and B,, = B,B112. We assume B,/B,-B>> 1, and this simplifies the expression for the magnetic force. The effective magnetic Reynolds number, 92, is chosen to be of the same order of magnitude as the large ratio of the fields, B-B,/B,. The units defined above make nondimensional values “of order of unity”. Here the quotation marks indicate that order 1 is not really achieved, but that scaled values too far from unity are not expected. These values are also functions of other model parameters. Especially significant is their dependence on the small parameter cJ (see below).

The nondimensional expressions for velocities are

vs=s-lv * (sB,B,), (174

The term 5, in (17b) is assumed to be given. It can be directly expressed by integration of 8,C over z:

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where the subscript 1 corresponds to r = R , and ~ = z , = ( R ~ - s ~ ) ” ~ ; we set [,=O at r = R, . It is assumed that g= - lrg lr /Rl , where g, is constant, and C,= e / g l R l is used as the unit of C.

The analog of 5, in dynamical meteorology is called “the thermal wind”, and by analogy iB = B:/s2 is often called “the magnetic wind”. The term iC(s) is called the “geostrophic shear”; so far it is an arbitrary function of s which cannot be determined from (15b). It is simply related to the velocity shear on the core-mantle

To find C 1 (and rG), let us consider the interaction of the core fluid with the mantle. The tangential stresses subjected by the wall onto the fluid may be expressed as the sum of coupling forces of different origin. The three main components considered are viscous, magnetic and topographic couplings, and there is now a vast literature on this subject; see e.g. the review of Roberts (1989b). The magnitudes of all three kinds of coupling are rather poorly determined, but all of them are weak in comparison with magnetic interactions within the core, where the fluid has a large electrical conductivity. Simple expressions for the couplings and estimates of their magnitudes may be made in the following way.

It is well known that so-called “Ekman layers” develop on the boundaries of a rotating vessel where the fluid velocity adjusts itself to the no-slip condition. These layers have a thickness of the order of ~ , = ( V / Q , , ) ’ ~ ~ where v is the fluid viscosity and n is the normal to the boundary. In our case the corresponding viscous stress in the +-direction is

boundary, 5 1 = &l, 4, by 5 l b ) = CBl(s) + CG(S).

where 8,=(v/Qcos 6)1/2. Magnetic stresses, B,B,, act across the boundary because the component B, of

the magnetic field diffuses from the core into the mantle due to the (small) mantle conductivity, and this gives rise to magnetic coupling. The electrical conductivity of the mantle is mainly concentrated in a thin layer near the CMB, and the magnetic coupling therefore has a local character like that of the viscous coupling. Hence the coupling may be approximated by the expression -fV+ where the effective coefficient of friction, f , may be a function of position and magnetic field.

The approximate expression for magnetic coupling can be written as

where oM is the electrical conductivity of the mantle, effectively concentrated in the layer of thickness LM, and z,=(c2/4710,)L~ is the characteristic time of screening of electromagnetic signals by this conducting layer. The magnitudes of g M and LM are poorly known. Let us assume that T~ = 7 y, which still does not screen decade variations too much, and take L,-2.102km. With these parameters the ratio of viscous friction to the magnetic friction is fv/fM=(v/vM)1/2, where v,-0.7. B:; here v y is in m2/s and B, is in G. If B,- 3.5 G then vM - 1 m2/s.

The unevenness of the core-mantle boundary is also supposed to produce a

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coupling analogous to the frictional force exerted by a wind blowing over the Earth’s topography, a phenomenon known in meteorology. This mechanism is more complicated in the Earth’s core (see, for example, Anufriev and Braginsky, 1977) because of the influence of magnetic forces, the finiteness of the region of flow, and possibly the effect of the density stratification.

Let us consider now the equilibrium of the z-projection of the angular momentum of the forces acting on the thin cylindrical shell within the core that has generators parallel to the z-axis, and which occupies the space between distances s and s+ds from that axis. The “top” ___ (z=zl) and “bottom” (z= -zl) boundaries of the ~ shell experience stresses B,B, where they meet the mantle. Magnetic stresses B,B, act across the side surfaces. Their total momentum on one side of the shell is proportional to the so-called “Taylor integral”:

~ __ Here B,B,=B,B,+B,B: should appear, but the second term is not included in (19) for our simplified model. The angular z-momentum of the Coriolis force is zero because it is proportional to the integral of Vs over dz, which is equal to a net mass flux through the surface of the shell. The Archimedean force has no 4- component at all. If therefore there were no direct core-mantle interaction (“friction”) on the CMB then the equation of equilibrium for the shell would be

This is the so-called “Taylor condition”. In reality, however, core and mantle are coupled viscously, magnetically and topographically at z = f z,; the shell equili- brium condition is therefore

Here f(s) is a function of order unity depending on position and on the (nondimensional) magnetic field. If magnetic friction acts, then (19b) contains an additional term proportional to B,B, on the boundary, see e.g. Braginsky (1975). The left hand side of (19b) contains a small parameter Ef which reflects the relative weakness of core-mantle coupling. Its order of magnitude is This is another undetermined parameter of geodynamo theory which should be chosen by fitting the theory to the observations. If it could be found, it would provide valuable information about the properties of the CMB.

5. MODEL-Z SOLUTION

The presence of the small parameter cf in the dynamo model invites the question,

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“Is this parameter significant?”; in other words, can we take E f =O in the leading approximation?

Taylor (1963) considered __ the case E ~ = O [and in (19) he also included both contributions to B;B:]. He suggested that the function cl(s) will automatically adjust itself in such a way that the equilibrium condition (19a) is fulfilled: $(s)=O. If this were true, would be of order unity because there are no small parameters. We may, however, also anticipate a priori, another possibility: F N 67,

where 0 < n < 1. Then il - Y/E N ~ 7 - l is very large. Which of these possibilities is adopted by the geodynamo? The Taylor integral (19) can be made small by diminishing B, and/or by cancellation of the contributions from different parts of the region of integration in (19). The question is: what is the extent of this cancellation and, if T-E;, what is the value of n? Up to the present time, there has been no general answer to this question. Braginsky (1975) suggested a form of the solution that corresponds to the second alternative, >> 1. In this solution the field lines of Bp are predominantly parallel to the z-axis in order to diminish B, and F. A sharp “break” in the lines of force of Bp is necessary near the core-mantle boundary where the internal field has to match to the field in the poorly conducting mantle. Braginsky named this hypothetical dynamo “model-Z”.

The direct, but insufficiently general, way to decide between the two possible scenarios is by integrating the system of Eqs. (13), (14), (17a,b), (19b) forward in time, starting from some arbitrary initial state. By making E~ small but finite, the system can approach either of the two possible states. This program was realized by Braginsky ( 1978), who integrated the equations numerically by time stepping the functions $(t , r , 8) and B,(t, r, 8) starting from an arbitrary smooth initial state, $(O, r, 8) and B,(O, r, 8). The solution was defined on a numerical grid ( r i , ej) having a total of 750 grid points. The advection term was calculated explicitly while the diffusion term was integrated implicitly using the “method of splitting” for the variables r and 8.

For simplicity, only viscous friction was assumed in this work. This implies the boundary condition: B, = 0 at r = 1 and cG = il. Computational expediency dictated the choice E f =~,=0.01. This is an order of magnitude greater than the realistic c f - but nevertheless a rather thin boundary layer develops in the solution. For a better resolution of the layer, the radial grid points of the numerical mesh were chosen to crowd together as the core surface was approached.

The existence of a nontrivial solution for $ and 8, depends significantly not only on the amplitude of the functions r and 5, but also on their form. After numero.us unsuccessful attempts, in which the initial field decayed even though large amplitudes were chosen for r and c,, favorable forms for these functions were hit upon. The driving wind, c,, was chosen proportional to -s2(1 - r2 ) . This corresponds to a,C<O, so that the fluid is lighter on the equatorial plane than on the polar axis.

Surprisingly, the existence of a solution turned out to be highly sensitive to the form chosen for r. The successful cr-effect coefficient, r, was zero everywhere except in the equatorial region 0.8<s< 1.0. All functions, r, that are smooth in the main bulk of the core (including those successfully used previously in kinematic models) encountered a “displacement effect” for the field B,. In the regions in

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I02 S. I . BRAGINSKY

which the sign of rB, corresponds to an intensification of Bp, the field B, weakens rapidly and may even change sign. The field Bp then reverses its initial amplifica- tion, and with its demise the dynamo decays away. This mechanism of displace- ment of 8, from a region of generation is poorly understood. It is a direct consequence of the expressions (17) for the velocities, and was never encountered in integrations of kinematic models. The function, r, used by Braginsky (1978) was empirically selected to defeat this mechanism.

The process of step by step integration in time led to a stationary state of model-Z type, in which B,<<B, over most of the volume of the core, and which had a current layer over most of the core surface. The geostrophic velocity turned out to be very large: Itl 1000 at s=O and a second maximum at s%0.82 of about 260 nondimensional units. Its viscous dissipation was of the same order as the total Joule heating: ( Q , ) - ( Q J ) . This stands in striking contrast to the Taylor state, for which ( Q r ) < < ( Q J ) . The ratio ( Q f ) / ( Q J ) can be used as a discriminator between model-Z and Taylor-type solutions. A ratio of order unity corresponds to model-Z, and a small value indicates a Taylor state.

Many numerical solutions of a model-Z dynamo were obtained by Braginsky and Roberts (1987), in which E , and other parameters of the model were varied. Different starting conditions were tried, and they always led to the same stationary solution of model-Z type. A “rough” numerical grid with a total of 1089 grid points, and a “fine” grid with 4225 grid points were employed. The results for these two grids were very close to one another, so confirming the reliability of the numerical solution.

A few variants of the model-Z solution were obtained by Braginsky (1988) in which magnetic friction between core and mantle was introduced. The solutions were of broadly the same form as they had been when the coupling was viscous. However, to prevent the decay of the dynamo, it turns out to be necessary to add to the model (in a rather artificial ~ way) an additional magnetic coupling term which is a substitute for the E,BL stresses on the boundary. This term is mostly significant in the equatorial region, where the magnetic coupling proportional to B,B, is small because B,=O and B,=O when % = $ E When the mantle is conducting, the toroidal field, B , is not zero on the core-mantle boundary, and heat of magnetic friction (which is Joule heat in this case) is deposited in the lower mantle.

Braginsky (1989) found that the main characteristics of model-Z were unaltered when a solid inner core was included. A nonstationary model-Z-type solution was also reported in this paper, which was obtained by a somewhat artificial trick. To turn a stationary solution into an oscillatory one, an “on-off switch” was added to the generation mechanism. The coefficient r in (13) was set to zero when the dipole moment exceeded some upper bound, and was restored when the dipole moment fell below some lower bound. These bounds were chosen to agree with the observational (archeomagnetic) data. By comparing the observed period, To = 7.7 * lo3 y, of the “dynamo’s fundamental oscillation” with the nondimensional period Tnd=4.8. lop2 defined by the model, it was possible to calculate the magnetic diffusivity of the core, by applying the relation To= TndRf/q. The result was q = 2.4 m2/s, which corresponds to a core conductivity of c = 3.3. lo5 S/m. The

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reasonable magnitude for the conductivity is a consequence of the very short nondimensional period, Tnd, which is a characteristic property of an oscillating model-Z.

Adopting the estimate q = 2.4m2/s, and taking po = 10.7 g/cm3, we find from (16) that B , = 21.7 G. From the nondimensional Gauss coefficient, (g(&d, supplied by the model and the value gy=0.3G obtained from the observations, we can determine the unit. Bpi =g!/(gy)nd and B,, = Bi /Bp , . We can then find W - B+l /Bpl . The time and velocity units are tr=R:/q= 1.6. 105y, vp/p,=Rl/t~=0.7’ 10-4cm/s, and V,! = W V,, .

Braginsky (1978) estimated for his stationary model-Z that the effective magnetic Reynolds number is W= 100, the averaged and maximum magnetic fields are (B$)1/2 = 1.10’ G and B,,,,= 3.3. lo2 G, the maximum velocity is sll x 1 cm/s, and the total dissipation rate in the dynamo is Qto‘=2.9. lO”erg/s. From his oscillating model-Z (Braginsky, 1989), a smaller value of W = 30.5 was obtained and q=2.4m2/s. For a phase of the dipole oscillation somewhere between maximum and minimum, he estimated that (8:)’12 = 1. lo2 G, B,,,, = 2.3. lo2 G, ~~~, , , zO.35cm/s and Qto‘=0.72. lO”erg/s.

It is worth noting that, although the parameter 92 has a rather moderate value in this case, the dynamo model definitely is of model-Z type: il(s) is large with i3,~1>0 over most of the s-interval; the current layer at the CMB is pronounced; and (the main feature) Qf - QJ.

It was shown by Braginsky (1975) that, if one solution of (13), (14), (17)319) is known, one can obtain from it an entire family of solutions by the following scaling:

0, is unchanged. The parameter W which scales the values in the model is a specific example of such a A.

tends to decrease when B, ( ~ 9 ” ~ ) increases in the strong field dynamo. Intensification of the fueling of the geodynamo therefore increases C and B$ but tends to decrease the observed geomagnetic field, Bp.

It is interesting to note that the instantaneous switching of the cr-effect in the oscillating model-Z dynamo creates a large perturbation, the decay of which is not monotonic, but is accompanied by oscillations having periods an order of magnitude less than Tnd. This shows that the axially symmetric field structure of a model-Z dynamo is stable and is associated with a large effective elasticity. In the real dynamo, such oscillations can interact with the MAC waves, which have periods of the same order of magnitude.

Roberts (1989a) derived numerous intermediate dynamo models (with viscous coupling) for widely different choices of defining parameters. He traced a gradual change from Taylor-type solutions (characterized by small i1 and by (Q,)<< (QJ)) for cases in which the cr-effect is weak, to model-Z-type solutions in which r is large. This demonstrates that different types of solution can arise when the parameters defining the model are chosen differently. At the present time, no

It is worth observing that B p

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I04 S. I . BRAGINSKY

uniqueness theorem is known that precludes one or other type of solution under the same conditions.

All of the models described above used the same forms for the functions r and c,, though their amplitudes were varied.

An intermediate model of a quite different type was studied by Malkus and Proctor (1975) and Proctor (1977). This model is of a2-type, that is the a-effect is retained not only in (13), but also in (14). Generation by V' x B induction 's now responsible for the both parts of the generation loop, and creates not only B, %om B, but also B, from B,. The a-effect coeficient was taken to be time indepent ent. The inertial term and the B" stresses were neglected. In contrast, however, with the buoyancy driven am-dynamo considered above, the a2-model of Malkus and Proctor (1975) neglects also the Archimedean force, so that it is driven by the a- effect alone. It is therefore a weak field dynamo with B, -BP-B* . Both cases of large and small viscosity were considered. In the latter, a velocity, il(s), of order unity develops that fulfils Taylor's condition F = 0 or, more accurately, S - c f . The nonlinear self adjustment of the axisymmetric velocities resulted in a stationary dynamo in which B grows exponentially when it is small but then levels off. Recently, Hollerbach and Ierley ( 1991) calculated numerically similar inter- mediate az-models for a variety of choices of the a-effect coefficient. All models yielded Taylor-type solutions that were either steady or oscillatory.

The nonlinearity of this intermediate model is due to the action of the Lorentz force (which is quadratic in B) on the large scale axisymmetric fluid velocity, 9. Another source of nonlinearity is the action of that force on the small-scale and large-scale asymmetric flows and hence on the a-effect. These very complicated processes are far from being understood at the present time. One possible way to deal with them is by simple parametrizations of the a-effect coefficient, I-. For example, Olson and Hagee (1990) constructed a numerical kinematic model (with prescribed 8=1@3 and r) in which rW(1 +B2/B*2)-1. They applied this model to the study of geomagnetic reversals.

A dilemma exists at the present time. It is unclear whether Taylor states or model-Z states typify small E~ solutions more faithfully, and which corresponds better to the actual state of the Earth's core. It should be stressed that the velocity V,, as determined by (17a), is just as significant for the model-Z solution as (19b), that gives an expression for C l which has the small parameter in the denominator. Models that neglect V, are oversimplified and cannot resolve the dilemma.

One may suppose that the model-Z state is more typical for a buoyancy-driven am-dynamo that is working far beyond its excitation threshold.

It is obvious that intermediate models of every type are, by their very nature, incomplete. They invite a more complete approach to the problem of the geod ynamo.

6. CREATION OF BUOYANCY

Let us introduce some convenient labels. Intermediate models are described by

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two independent functions: A, (or II/ = d,) and B6; all other quantities, such as Bp and V, are expressible in terms of these. We will attach the label “AB” to these models. By adding the axisymmetric part of (3), we obtained more general “ABC”- models in which the density excess and the buoyancy force are not prescribed but are to be computed.

An additional equation is obtained from (3):

__ The term V ’ C is omitted here, in just the same way as like terms, quadratic in primed variables, were omitted in the AB-problem.

To use (21) we needed expressions for GC and 1. Applying the condition (C) = O to (21) and assuming that V,=O on the boundaries, we obtain a relation between the homogeneous volume source and the downflowing fluxes at the boundaries:

Here Sk=4xRE, vk=(47-r/3)R;, and Vl2=VI-V2, where the subscripts k = l and 2 refer to the CMB and the ICB, respectively. If IM =0, then

INri=GCR1(l -r;)/3zGCR,/3, (224

where r,=R,/Rlz0.35. Equation (22) has a clear meaning: the removal or addition of mass at the ICB and CMB changes the mass of the fluid core in the volume V , , = V , - V,. The new material is homogeneously distributed over the core volume.

The sources of IM are rather obscure because of the very complicated physical state of the CMB, but the creation of I N through the crystallization of the inner core is clearer. Any estimate of I N depends crucially on the density excess, Ap,, of the solid core over the density of the fluid core. It is also proportional to the rate of inner core growth:

Here we will assume that at V2 N V2/t2, where t, = 4.10’ y. To make the ABC-equations nondimensional we supplement the units for A and

B previously used by the proper unit for the density excess: C,=(2Rq/g,Rl)W. Correspondingly, C,/t, is the unit of GC and C,Rl/t, is the unit of flux 1. Substituting q=2.4m2/s and g, = 10m/s2 we have C,= 1.2.10-”W. If I M = O then (22a) and (23) provide

GC = 3lNr:(l - r : ) - ’ z 1.8. 1 0 5 ( A p 0 / p 0 ) K (24)

as the nondimensional value of the effective volume source of density excess. The commonly used Apo/po x 1/20 implies the very large value

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106 S. I . BRAGINSKY

The hydromagnetic dynamo of the Earth operates far beyond the threshold of self-excitation because its source of buoyancy is very strong.

It is now clear that the large value of W that led to the nearly axisymmetric dynamo is the direct consequence of the large flux, I”, feeding the geodynamo. This large W should be adopted in order to make the nondimensional variables of the ABC-problem “of order of unity”. This statement should be taken “with a grain of salt”, because large numerical coefficients (even as large as lo2) arise in the problem, mainly because the relevant characteristic length is a few times smaller than R,. In practice, W- lo2 or an even smaller value should be chosen.

Extremely complicated questions arise just about the correct expression for the flux I. It is obvious that this diffusional transport cannot be performed by the molecular diffusivity of the admixtures, I C ~ , which is extremely small. Braginsky (1964d) estimated that 18- lO-’’q. Loper and Roberts (1981) gave a value that is 30 times greater but which nevertheless is very small. Such estimates, though rather uncertain, establish that molecular transport is too slow. The thermometric diffusivity can be estimated by using Wiedemann-Franz’s law and also turns out to be very small, 1 ~ ~ - 2 . 1 0 - ~ 7 .

On the other hand a simple estimate shows that, even with a small mixing length I and a moderate turbulent velocity v, the turbulent diffusivity IC‘ - lu is rather large. For example, if 1- lOkm and v - lO-‘cm/s, we have K ‘ - I U - 1 m2/s - 7.

Although a detailed quantitative theory of the turbulent transport is lacking at this time, it is obvious that the strong magnetic field and the fast rotation should make the process anisotropic. A simple diffusion-like expression of the form

should therefore apply. It is convenient to write ~ i ~ = q D ~ ~ where D , is a nondimensional tensor diffusivity. Then Ti= -DijdjC should be used in the nondimensional equation (21).

The “diffusional” parametrization of the transport of density excess is not the only one conceivable. Quite different possibilities are also under discussion. Frank (1982) suggested that very small light droplets (- lo-’ cm in diameter) are continually ascending from the ICB and are preserved from disintegration by surface tension. Moffatt (1988) proposed very large ( - 10 km in diameter) light blobs that rise from ICB, stirring the core as they do so. The way in which the light fluid is released on the ICB depends on the detailed process of crystallization. These were considered by Loper and Roberts (1981, 1983), who argued for the presence of a “mushy layer” on the ICB and who also hypothesized that the entire inner core might be a mixture of solid and liquid. There is still much room for discussion before the realistic scenario for transfer processes in the core is known.

The existence of small-scale turbulent motions in the core seems most plausible. These would mix the core efficiently, so removing the tremendously large C and anomalous Archimedean force gC that would otherwise be present. It should be

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noted here that, as compared with the “normal” magnitude C- a value such as C - lop6 (say) is “tremendously large”. The assumption of efficient mixing is implicit in the fundamental approximations made in deriving (3). We shall make this assumption.

The estimate (24) shows that compositional convection can drive the geo- dynamo. The prevailing hypothesis of thermal convection would be valid only if Ap, were very small. It is nevertheless interesting to discuss briefly also the case of combined thermal and compositional convection.

It is easy to see that an equation of the form (3), together with (1 1) and (25), can be used also for the more complicated case of this combined convection: both the admixture and the heat are transported by the same turbulent motion. Their fluxes are proportional to the gradients of the scalars t1 and Tl with the same turbulent diffusivity. Hence (25) follows from 15= -K‘ - V t l and I T = -K‘ - VT,, together with expression (11). The left-hand side of (3) follows immediately from (3a) and (8) together with (11). The right-hand side, GC, has now, however, a more complicated form: GC = GC‘ + GCT. Here GCT = - aGT is determined from (lo), and GC‘ = - CGr can be estimated from (24).

It is useful to transform expression (10) by excluding the time derivative a,T,. Applying the condition of zero average, ( Tl) = 0, to (8) we obtain

where as before the subscripts 1 and 2 refer to the CMB and the ICB. Substitution of (26) into (10) gives

Terms describing the homogeneous production of heat do not appear in (27). For example, the radiogenic term QR is absent; this may be compared with the analogous statement by Loper (1985) concerning mantle convection. The adiabatic gradient VT, is proportional to g - r in the homogeneous model approximation. Thus Q,= -V q, is also constant and absent from (27). The term I r 2 S , is approximately equal to the rate of latent heat release at the ICB divided by the specific heat; this supplies the thermal buoyancy.

What is really significant for the combined thermal and compositional convec- tion is the removal of heat from the core by the flux 4,. Only the excess q1 -qal = p o c p I r l over the adiabatic flux qal influences the thermal convection. If q l < q a l , compositional convection has to pump heat down the adiabat as was explained by Loper (1978), and thermal effects hinder convection instead of driving it. Estimates of core cooling are not very precise because they involve complicated questions of mantle convection and of the Earth’s thermal history. According to the estimates of Loper (1978, 1988), q l < q a l . Nevertheless, to obtain a numerical estimate, let us for example imagine that poc,llTl I - I q1 -q,, I “0.3q,1. Then, using (23) and (27), we can estimate the ratio GCT/GCS to be

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Substituting here ~ = 7 . 1 0 - ~ K - ' , ~ '=0 .05cm~/s and IVT,,I=(yg,/uf)T,, = 0.8K/km (see e.g. Loper and Roberts, 1983), we then obtain

We may conclude that the compositional buoyancy is the most plausible driving force of the geodynamo and that the simplest model, that of Section 2, is the best available. Thermal effects are not extremely small, but they are rather uncertain and are significantly smaller than compositional effects.

The solution of the ABC-problem should help us to understand how the process by which buoyancy is created influences magnetic field generation. Some infor- mation about the physics of the Earth's interior could also (hopefully) be obtained. Imagine, for example, that solutions for the fueling from the top (IM # 0, IN = 0) led to oscillations having a period Tnd much longer than the Tnd- 1/20 obtained by Braginsky (1989); see Section 5. Then this kind of fueling would have to be abandoned because it would demand too large a value of 9.

The set of ABC-equations consists of those governing the AB-problem together with (21) and (22); here the parametrization (25) should be specified. The AB- problem supplies the velocity Vp for (21), while the driving wind for AB is calculated from C by using (17c).

It may be shown (see Braginsky, 1975) that the growth of magnetic energy is determined by

where the rate of working of the Archimedean force and the total rate of heat production can be expressed in the following alternative (non-dimensional) forms:

Equation (29c) applies only when core-mantle friction is viscous and the inner core is absent; for the case of magnetic friction, see Braginsky (1975).

The author is now investigating the ABC-problem numerically by a method similar to the one he previously used for the AB-problem, and with the same expression for the a-effect coefficient r. The treatment of the terms V, . V C and DV'C is the same as that described by Braginsky (1978) for similar terms arising in the AB-problem. Unfortunately, satisfactory results have not so far been obtained. The following summarizes what he has so far attempted.

The mathematically simplest (but not the physically realistic) choice, Dij = Dd,, of isotropic diffusivity was made, where values of D ranging from 0.1 to 1.0 were tried.

First only the evolution of C was explored, for different magnitudes of D and IN

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GEODYNAMO THEORY 109

(or ZM) and with the velocity O,, obtained from one of the AB solution of model-Z type, held constant. The computation always proceeded quite stable with the time step h,= and led to a stationary solution. If 9-30 (see Braginsky, 1989), then r i Z N - 10’ was obtained from (24a). With IN (or ZM) fixed, it was easy to fit D in such a way that, in the stationary state, C<O near the equator and C>O near the z-axis (d,C<O); moreover the values of ((,) and (5;) were then approxim- ately the same as those used in the model-Z calculations. The detailed form of la was however quite different from the expression, -s’(l-r’), assumed in the model-Z calculations. In particular, the values of la obtained for small s were too large.

Time-stepping of the whole ABC-system gave rise to additional difficulties, connected with the numerical stability of the integrations, when GC was rather large. In all solutions studied, the magnetic field decayed during a time interval of order of unity and the stationary solution with C=C(r ) , but with V=O and B = O , eventually became established. In the transition period before this happened, (Qr) - ( Q J ) , so that the transient solution might be described as being of model-Z type (even though the ratio (Bf)/(BS) was not small). A question may be asked here: “Can an ctw-dynamo be self-sustained in the framework of the ABC model when the diffusion coefficient, D, is constant?”

7. LOCAL TURBULENCE

It is quite obvious that the transport of heat and compositional inhomogeneities in the core is accomplished by mixing due to turbulent fluid motions. But the real picture of these motions, their form and amplitude is far from obvious. Unfortun- ately, in contrast to the Earth’s atmosphere and oceans, direct observations of the turbulent velocities in the Earth’s core are impossible. The experimental study of turbulent transfer in a conducting fluid in the presence of a strong magnetic force and a fast rotation is probably possible, but is by no means an easy task. At this time, the picture of core turbulence can only be painted with a theoretical brush.

An attempt to develop a heuristic theory of turbulence in the Earth‘s core was initiated by Braginsky (1964d) and developed by Braginsky and Meytlis (1990). The simplest model was used, based on the fundamental assumption that the turbulence is local in character, its spatial spectrum being dominated by small scales, of order l c L . Indeed, the simplest estimate of the turbulent diffusivity by the “mixing-length formula” K‘ - lo shows that K‘ is of order of r] when 1 is rather small. Let the turbulent velocity be u- lO-’cm/s (an estimate often made of the core’s velocity); then lu - for 1 - 10 km c<L.

The reason for turbulence is always some kind of instability and, to excite local turbulence in the Earth’s core, the basic state of the core should be prone to a local instability. As was shown by Braginsky (1964d), such a possibility exists: the strong field dynamo driven by buoyancy should be locally unstable. The mech- anism is very simple: the upwelling of light fluid parcels and the descent of heavy ones under the action of the Archimedean force in the top heavy density

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distribution. Coriolis and magnetic forces retard these motions and reduce the growth rate of the instability but they cannot stabilize it. This buoyancy instability of the core is highly supercritical and results in fully developed small-scale turbulence.

The model consists of a Boussinesq fluid with a constant imposed gradient of excess density, VC=l,d,C, which is parallel to a uniform gravity, l,g,, and to the angular velocity vector, l,Q, a constant horizontal magnetic field, 1 ,B, simulates the main toroidal field, 1,B,, of the geodynamo.

Let us denote the perturbation of this basic state V, B, C , .. . in velocity, magnetic field, density excess, . . . by corresponding small letters: v, b, c, . . .. In the linear approximation, the perturbations have the form of periodic cells defined by the wave vector k = ( k , , k,, k J , where k,L>> 1, k,L>> 1, k,L>> 1. The dissipative terms due to (molecular) viscosity, magnetic diffusivity and admixture diffusivity are proportional to the following dissipative frequencies:

where k2 = k: + k,” + k; = k: + k; and k: = k: + k; .

y = y, - yK, where y K is very small and Linear stability analysis shows the growth rate of small perturbations is

0, = ( - g , d,C)”2, (32)

R, = 2Rk,/k, (33)

Here w, is the buoyancy frequency (corresponding to the Brunt-Vaisala frequency that would arise for stable stratification), and ys is the coefficient of magnetic friction.

For small scale motion, y,, is very large, and the induction term, d,b, is therefore much less than the magnetic diffusion term, y,,b, and can be omitted. In this case, the magnetic field can be expressed linearly in terms of the velocity at the same moment, and the magnetic force becomes a frictional drag

This magnetic friction is very significant for the local turbulence. When making numerical estimates here and below, we take B = 102G, which gives y B - 4.10-3 s-’ . It should be noted that y B is the largest frequency characteristic of the strong field dynamo; it is even greater than the Earth’s rotation frequency:

The elongation of a cell in the y-direction diminishes the magnetic friction; see (35). Its elongation in the z-direction reduces the influence of the Coriolis force

yB/2R - B2/B: - W >> 1.

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(33). A cell grows most rapidly if both k,/k and k , /k are small; then k x k , z k k , . According to (31) the maximum y a arises when

The plate-like cells obeying (36) are the fastest growing, and we suppose that they dominate in the spectrum of the turbulence. Their growth rate is given by

yo = 0,2/2y* = W,2/2R*. (37)

Condition (36) may be written (with y y neglected) as

where

For the strong-field dynamo ~*-9-’ , so that ~ * - 4 . 1 0 - ~ . Braginsky and Meytlis (1990) supposed that just these dominating plate-like

cells determine the properties of local turbulence. They estimated the parameters of the dominating cells, postulating that the amplitude of each cell grows until a nonlinear process comes into play which first limits and then destroys the cell. Therefore the whole space is filled by turbulent perturbations in which packets of plate-like cells are born, grow exponentially, decay, and finally disappear in a never ending sequence. Braginsky and Meytlis made the fundamental assumption that more than one mechanism of cell destruction act on a dominating cell at the same speed, and obliterate it over the same interval of time, namely a time of order of ya-’.

Local turbulence is highly anisotropic because of the influence of the Coriolis force and magnetic friction. That is why, even for the crudest picture of one-mode turbulence, one must estimate four parameters: the amplitude of the cell and its three characteristic dimensions, k,, k , and k,. Three conditions must be added to (36) to determine these parameters. Two mechanisms of cell destruction are simply its geometrical distortion by the turbulent velocities, uy or u,. The third condition arises from a quite nonlinear type of motion produced by nonlinear xy stresses of magnetic and kinetic origin. These stresses averaged over y and z produce a geostrophic velocity, I?,, associated with a packet of turbulent cells.

The results of applying these four conditions are different in situations of small and large viscosity, cases separated by a viscosity of order v 1 - 20cm2/s. Qualita- tively these two cases are similar. Only the results for the (simpler) low viscosity case are presented here. (See Braginsky and Meytlis (1990) for further details.)

The dimensions of the dominating cells are

k , - 252(4~p,)”~/B, (394

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The characteristic time scale is given by the inverse of the optimum growth rate

The amplitude of the cell is given by

It is convenient to exclude the quantity a,C by using the relation I , = - K : ~ , C , and to express the turbulent diffusivity in the z-direction, K: - ya/k,2, through the flux I,. This gives

ti:/q - (Z,g/8R2~)"ZB4/B4 *' (42)

In order of magnitude we have K: = K: and ti:<< ti:.

Numerical estimates with B - 100 G give 1, = n/kz - 50 km. I , = n/k , - 2 km and, with estimates from Section 6, namely 1- LGC and GC- 1.7. 10-20s-1, we obtain K: - r] . This value depends strongly on, and increases rapidly with, the magnetic field, B. The strong dependence of K'(B) on B makes turbulent transport an influential factor in determining the amplitude of the field produced by the geodynamo. An increase of B enhances the turbulent transport and diminishes both VC and the driving Archimedean force, thus producing a tendency to equilibrate the dynamo field.

It should be stressed that the influence of local turbulence on the magnetic diffusivity and on the cr-effect is negligible because both of these are proportional to the square of the magnetic Reynolds number, W,=v, /k ,q (e.g. see Moffatt, 1978), which is extremely small for the plate-like cells: B t - k , / k , - ~ , . This is in strong contrast to the crucial role played by local turbulence in the transport of the density inhomogeneities and which implied that K' - r] .

Local turbulence dissipates energy at the same rate as does the main geo- dynamo mechanism. It was shown by Braginsky (1964d) that the heat released by turbulent diffusion is

This should be compared with the rate of working of the Archimedian force driving the geodynamo:

(44) dtot= JpoCg * V,dV.

If I - d C I L and Vp-q/L, which is normal for the dynamo mechanism, and if K' - r ] , then Q , - d. The energy dissipated by local turbulence is, however, not extracted from the large-scale magnetic field (as is done by magnetic diffusion). This energy is drawn directly from the gravitational energy released by the rate of working of the Archimedean force on the velocity of small-scale turbulent motion;

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it is dissipated by magnetic friction. In fact, Q D is the Joule heating of electric currents having the same length scale as the turbulence. The density of these currents is about k,b, which is large because k,L>> 1. Thus, k,b - B/L, even though b c B .

A few simple estimates can be made of the turbulent viscosity, v', in the Earth's core. The Ekman number, E=v/RL2, with L-103 km can be written as E - lO-'(v/q). Even for v - v f - q , the value of E is very small and viscosity therefore plays a negligible role in the bulk of the core. The mixing length in the Ekman layer cannot be greater than its thickness, so that ~ , - (V/R)"~ . Let v-vf-u6:, then 6: - (vt/i2)'/' - (U~:/Q)'/~; therefore 6: - u/Q. For u - V , - 2 cm/s which is equivalent to the Alfvh velocity for the field B,, this would give 6:-2.1O2rn and vf-q. Such a layer could be significant for core-mantle friction. However no mechanism is known of creating in the Ekman layer either such a large turbulent velocity or a varying field of strength B,.

8. MAC-WAVES

The asymmetric fields V', B and C' are perhaps the least understood ingredients of the geodynamo. The top-heavy inhomogeneity, C, in conjunction with the large magnetic field, B,, and the fast rotation, give rise to a large-scale instability which manifests itself through waves travelling in the +-direction. These waves have been called MAC-waves because magnetic, Archimedean and Coriolis forces (together with the pressure gradient) are mutually balanced in them. The instability mechanism was demonstrated with a crude Cartesian model by Braginsky (1964d), and in greater detail by Braginsky (1967, 1980). The waves have also been considered by Hide (1966), Malkus (1967) and other authors, see for example the reviews by Hide and Stewartson (1972), Roberts and Soward (1972), Soward (1979) and Roberts (1987).

MAC-waves provide the velocity V' needed to evade Cowling's theorem and generate Bp from 8,. The field, B', of these waves manifests itself in the (archeomagnetically observed) geomagnetic secular variation with time scales of order 103y, which is significantly shorter than the fundamental period of dynamo oscillation, To z 8. lo3 y. A crude estimate of MAC-waves periods can be obtained by equating in order of magnitude the magnetic and Coriolis forces: B2/4nL- p02QwL, whence

For B - 2 . 1 0 2 G and L-103km we have 0-2.10-'~s-' and T-2n/o-103y. It is not out of place to note here that there is only one direct observational fact

that supports the hypothesis that a strong-field dynamo is at work in the Earth's core, namely the agreement between the order of magnitude estimate (45) and the archeomagnetically observed periods of the geomagnetic SV. Another (indirect) indication of the strong field character of the geodynamo is the small value of the observed dipole field: B p / B , - 9%- ' I 2 << 1 ( - 0.2), hence W >> 1 ( - 30).

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I14 S . I . BRAGINSKY

The amplitudes of the MAC-waves are limited by dissipation and by the nonlinear interaction between them. They are also influenced by the inhomo- geneity of the background, 0, B, and e. They, in their turn, react on and modify that background. It was shown by Braginsky (1980), see also Fearn and Proctor (1983), that the inhomogeneity of rotation V4 suppresses MAC-waves. This explains why, under the conditions of model-Z, MAC-waves are more prominent in the equatorial belt and are relatively suppressed in regions where V l l is rather large. The back reaction of waves on the axisymmetric background was demon- strated by Roberts and Soward (1972). Nonlinear processes working on a complicated background and in the spherical geometry are, however, very hard to analyze. At the present time they are poorly understood (if at all).

Fearn and Proctor (1987) attempted to develop a self-consistent spherical dynamo model powered by thermal buoyancy. The computation of V’ and B was an essential part of their iterative scheme of solution of the dynamo equations. Unfortunately no converged results were obtained. It is therefore difficult to form a definite opinion about the properties of the asymmetric velocities and fields from their paper.

A crucial question about the MAC-waves of the geodynamo is, “How chaotic is the pattern of mutually interacting MAC-waves?” Two extreme scenarios may be imagined. In the first scenario, the nonlinear MAC-waves fall into a small number of rather stable clusters. During the fundamental dynamo oscillation, these clusters gradually evolve in concert with the slow evolution of the axisymmetric back- ground. In the second scenario, a chaotic set of MAC-waves develops that defines a type of large-scale wave-turbulence in the core, one that overlaps with the small- scale turbulence considered in the last section. In the first scenario, the geodynamo is predominantly laminar with the local turbulence superimposed on that laminar flow. This small-scale turbulent motion replaces molecular diffusion, but the dynamo generation mechanism is maintained by the smooth regular motions. The dynamo is then most probably of No-type. In the second scenario, the geodynamo is predominantly chaotic, and possesses a turbulence of rather large linear scale, which performs both the tasks of diffusing the density inhomogeneities and of generating the magnetic field. This dynamo is of do-type. A significant fraction of the gravitational energy is released through the asymmetric V’-motion and drives the dynamo by the cr-effect, as in the model of Malkus and Proctor (1975).

Which scenario best describes the geodynamo? A definite answer cannot be given at this time.

It should be stressed that local turbulence is very significant in both scenarios. Particularly on the large scale instability, the local turbulent diffusivity, IC‘, works effectively alone; the very small molecular diffusivities K~ and K < do little. The diffusivity K‘ is rather large, and this favors a stable pattern of MAC-waves.

It was shown by Acheson (1983), see also Roberts and Loper (1979), that a local criterion exists for MAC-wave instability of the form

In model-Z, the left-hand side of (46) is very large and positive over most of the

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volume of the core, so that the state is there locally stable. This is a further argument in favor of the first scenario: a stable pattern of MAC-waves.

When the archeomagnetic data for the scalar variation of the geomagnetic field is spectrally analyzed, the resulting spectrum shows peaks. This is yet another argument in favor of the regularity of the geodynamo. Unfortunately, the archeomagnetic data are sparse and approximate, so reducing the reliability of this observational argument.

The difficult question of the stability of the MAC-wave pattern is obviously one of the most significant in geodynamic theory.

9. OBSERVATIONAL CONSTRAINTS

The geodynamo is an auto-oscillating magneto-fluid system with a few interacting components that exhibit both regular and stochastic (turbulent) behavior. There is no hope of building a realistic theory of such a complicated system purely by mathematical reasoning. Geodynamo theory will inevitably include some unknown parameters which can only be determined by fitting the theory to the observations. The number of significant constraints extracted from the observations should exceed the number of fitted parameters, in order that the completed theory may be checked. Verification of the finished theory is, however, not the only use to which the observational constraints can be put. Additional information from obser- vations and experiments should be taken into account during the construction of the theory so that right approximations can be selected, those that correspond to the real geodynamo.

The main body of information about the geodynamo comes from observations of the geomagnetic field and its secular variation (SV). The data about the modern field are precise, but the SV data are less accurate, and tend to be scarce and approximate for remote epochs. Geomagnetic data has been analyzed in so many works that their review is far beyond the scope of this paper. Only a few relevant questions of a general nature are raised below.

The observed SV is a complicated mix that results from several different mechanisms. They can be sorted by their characteristic periods, T, in the following way: short-period (decade) variations (SSV) with T- Tk< lo2 y, medium-period variations (MSV) with T - T,- lo3 y, and the fundamental oscillation (FO) with period To ~ 0 . 8 . lo4 y. The shorter the characteristic periods of the SV, the smaller are their amplitudes. Roughly speaking, the amplitudes, Sgr and ah;, of variations in the (first) Gauss coefficients for the SV having periods of order of lo4, lo3 and < lo2 y are of order 100, 10 and 1 mG. The amplitudes of the time derivatives, B-SB/T, are therefore of the same order of magnitude for all three components of the SV. That is why the mix of mechanisms creating the SV is so difficult to disentangle.

Since the papers of Roberts and Scott (1965) and Backus (1968), observations of the geomagnetic field on the short (decade) time scale have often been interpreted through the frozen field approximation. The validity of the supposition that

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1 I6 S. 1. BRAGINSKY

magnetic field lines are frozen into the moving fluid does not, however, depend on the length of the time interval considered. Rather, it depends on the SV- mechanism; more specifically, it depends on the characteristic time of the mechanism involved. The magnetic field cannot be considered to be frozen during the FO, but it may be assumed frozen during SSV processes. The thickness, 6=(2q/w)’’2, of a skin-layer is about 40km for SSV frequencies, w-w1 = 2n,/Tl ~ 3 . 1 0 - ~ s - ’ , and it is about 150km for the MSV. The latter is smaller than the typical characteristic length L- lo3 km, but it is of just the same order as the thickness of the current layer in model-Z. The FO involves a significant reordering of the geomagnetic field that requires magnetic diffusion to act. The frozen-field approximation is therefore not applicable to the FO even over a short time interval. Though applicable to the SSV, its validity for the MSV is a more complicated and still unresolved question. It is clear however that, for the purposes of interpretation, the three types of SV should be separated.

Geodynamo theory benefits little from the myriads of observational data gathered during the past century and from the many thousands available for earlier epochs, until that data has been properly digested and reduced. To check the theory we need a few dozens of physically sensible parameters, e.g. those describing the form of magnetic waves, and oscillations in the core, and their frequencies, amplitudes and phases. To obtain a set of such “observed parameters”, the data should be analyzed and interpreted. It would be wonderful if, as a preliminary, one could first convert the observational data into an “observed function”, B(t, 8,4), describing the magnetic field over the Earth’s surface (during a long enough time interval), and then to proceed to the desired interpretation from this function. Unfortunately, the data available for past epochs are too scarce to make this possible. One should introduce some theoretical constraints directly into the process of obtaining B(t,8,$) from the data, e.g. the data for the past few millenia. Even the data for the past few centuries (except the most recent one) are rather crude. The simplest constraint (one that could be described as “mathemati- cal”) is the assumption of smoothness, which permits us to interpolate across gaps in the record. If the data are poor (which they certainly are!), stricter “physical” constraints are needed, based on the theoretical analysis of the dynamics of the relevant physical processes generating the SV. The same is true for the more precise data of recent times if we intend to isolate SSV processes having small amplitudes.

Since the physical mechanisms creating the SSV, MSV and FO are quite different, each should be isolated from the others, and analyzed separately in terms of its own theory. After this division, the long period SV appears in the time evolution of the SSV as a smooth trend, having an amplitude of the same order as the SSV itself. It is therefore necessary to make the separation accurately, in order to exclude the trend when applying the frozen field approximation to the SSV.

Additional constraints can be derived from complementary parts of geophysics. There are, for example, good reasons for believing that the SSV is tightly connected to variations in the Earth’s angular velocity of rotation, that is to the length of day (1.o.d.). These are known more precisely, and over a longer time interval, than the SSV. It is therefore useful to determine the characteristic

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frequencies of both these variations from the 1.o.d. data alone. Braginsky (1984) did just this, and extracted periods of TI = 65 y nd T2 = TI for the SSV.

What dynamical mechanisms do these periods of the SSV portray? Up to now, only one mechanism is known, namely the magnetohydrodynamic torsional oscillation (TO) of the Earth's core; see Braginsky (1970, 1974, 1984). The possibility of a TO rests on a phenomena well known since the work of Alfven: the elasticity of frozen-in magnetic field lines and the inertia of core fluid in torsional motion. Their interplay results in oscillation modes with periods of the order of a decade. The TO produces the SSV directly, by dragging back and forth the magnetic field lines near the CMB. The magnetic elasticity can be directly expressed through the field strength. In this case, only the magnitude of Bs is relevant because the oscillating angular velocity, r, in a TO is independent of the z coordinate: f= r(t, s) thus excluding the action of the much larger Coriolis force. Equilibrating the inertia term, pow2r, and the magnetic elasticity, cBi/4nL2 (where

is here used instead of the fluid azimuthal displacement), we obtain an estimate of the TO-frequency: o-Bs(4npoLZ)-'12. If B,-3G then ~ - 3 . 1 0 - ~ s - ' and 2 n / ~ N 70 y.

Unfortunately, the source of TO excitation is still unknown. Moreover, other causes for the SSV are anticipated, and the form of the slower (non-oscillatory) motion near the CMB also influences the SV significantly. It is necessary to glean more information about these additional factors, and to take them properly into account prior to fitting the TO-produced SSV to the observational data. Braginsky (1984) attempted, with the help of a simple model of the TO, to fit the geomagnetic SV for the 20th century (given by the known Gauss coefficients gr and hr at a sequence of different times) by TO-produced SSV. The amplitude of the TO and its frequencies were determined from the 1.o.d. data. Only the order of magnitude of the effect turned out correctly; the form of the calculated SSV differed significantly from the one observed. The comparative failure of this impatient attempt shows that other mechanisms besides the TO are also signifi- cant. One such mechanism, considered by Braginsky and Fishman (1989), may be the hydromagnetic instability of a current layer adjacent to the CMB.

Let us now turn to the longer period SV. It is natural to suppose that MAC- waves provide both the generating velocities, V', for the geodynamo mechanism and the MSV observed by archaeo- and paleo-magnetic methods. Their periods can be roughly estimated by (45). A satisfactory theory of MAC-waves should emerge as an integral part of the complete dynamo model. No such accurate theory is yet available. Some helpful information about the properties and structure of MAC-waves can, however, be obtained even from intermediate models of the geodynamo. They tell us that MAC-waves travel both westward and eastward, but they do not provide clear information about the forms of the asymmetric fields, B&(O, d), they create.

With this small amount of theoretical information, Braginsky (1974) attempted to fit the SV for the last 2000y by a theoretical model that included both MAC- waves and the fundamental oscillation of the geomagnetic dipole. He used 11 sets of Gauss coefficients to represent the SV during the last 500y, together with worldwide archeomagnetic data for 500-2000 BP. The geomagnetic potential was

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approximated by a truncated Gauss series that retained the n, m 5 2 terms, and which therefore included the main axial dipole (m = 1, n = 0), the equatorial dipole (n= 1, m = 1) and the quadrupoles ( n = 2). It therefore also retained not only the inclination of the geomagnetic axis but also the largest scale features of the non- dipole field. The longitudinal (4) dependence of the field was taken to be the sum of terms proportional to cos (m4 - ot - a), corresponding to travelling waves. (Here 6 is a constant phase.) The Gauss coefficients in the model were expressed in the form

Here t = O is the year 1900 and F ( t ) = t ( t + To) (2t + To)/T& where To = 8.103 y. The parameters to be fitted are orfl, Srn, f Z , g': and fy. Each g," and h," was represented by one wave travelling to the East, one wave traveiling to the West, and one standing wave (w=O). The existence of the standing SV was first demonstrated by Yukutake and Tachinaka (1969). It turns out that the fit to the data loses negligible accuracy if the following resonance conditions between the waves are adopted:

This greatly reduces the number of parameters to be fitted. All frequencies are represented as a simple combination of w1 and w 2 where wI=23"/century (27c/w, = 1560y) and w2 =$w,.

Unfortunately the lack of all but scarce and dispersed archeomagnetic data for an all too short time interval (2.103y<<T0) combined with this all too simple theoretical model did not lead to convincing results.

Both of these examples of data fitting (for the short and long periods SV), though they were not totally successful, serve here to illustrate what we mean by combined theoretical/observational investigations of the SV. They are reconnais- sance works that make clear what is meant by the statement that theoretical constraints from dynamical models of the SV must be introduced directly into the analysis of observational data. Obviously both the theoretical models and the data sets will have to be greatly improved before convincing results will be obtained. Those results will provide not only the parameters specifying the dynamical SV mechanisms but also those describing the function d,B(t, 0,b).

A highly significant and exciting problem is the determination of the parameters describing the FO. At the present time, even the existence of the FO itself is in doubt.

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Bucha (1969) and Burlatskaya (1970) adduced evidence that the geomagnetic dipole oscillates with a period of approximately 7-9.103 years above its average value, An”, and with an amplitude of about 0.4-0.5Aa,. These authors based this conclusion on archeomagnetic data predominantly from the northern hemisphere. They demonstrated that, when smoothed over 500 y intervals, A ( t ) , exhibits pronounced maxima at about 2.103 BP and 9.103 BP. McElhinny and Senanayake (1982) obtained A ( t ) from an enlarged archeomagnetic data base with the significant inclusion of data from the southern hemisphere. The southern hemi- sphere data alone gave a comparatively low value of the dipole moment in remote epochs. As a result, when A was averaged over the entire terrestrial globe, the maximum near the 9.103 BP was no longer in evidence. Moreover, McElhinny and Senanayake (1982) observed that the archaeomagnetic data for the period from about 15 to 50.103 BP show an even smaller dipole moment. They concluded that “the time scale of changes in the Earth’s dipole moment must be very much longer than has previously been supposed and must be at least lo5 years”.

The results of McElhinny and Senanayake can, however, be reinterpreted in the following way. Let us assume that &(t )=Ao( t )+Al ( t ) , where A,(t) is from the slow evolution of the geodynamo while Al(t) is caused by the FO. The term A o ( t ) can be selected (see below) in such a way that A l ( t ) exhibits an oscillatory behavior with slowly evolving period and amplitude. This reinstates the concept of the FO.

Braginsky (1989) represented A,( t ) by a linear trend: A,, = 8.8 +0.2t, where the units of time and of dipole moment are lo3 BP and 1022Am-’. After removal of this trend, the dipole moments of McElhinny and Senanayake, which were smoothed over lo3 y intervals, show an oscillation Al(t) of comparatively small amplitude, namely about 0.2 of the period-averaged value, Aav. The period of oscillation is To=7.7. 10’~ . It was just this result that Braginsky (1989) compared with his oscillatory model-Z in order to estimate the magnetic diffusivity of the core. This led to r] =2.4m2/s, and a core conductivity of a=3.3 . lo5 S/m. These reasonable values qualitatively confirm the internal consistency of the whole procedure.

This example once more demonstrates that, to obtain a convincing interpre- tation of data, one should combine together the observed geomagnetic data with theoretical models. Both these components have to be signficantly improved before a realistic geodynamo can be constructed.

10. EVENTS ON THE CORE BOUNDARIES

The broad range of complicated phenomena occurring near the CMB deserves special attention. Complications arise on each side of the boundary, in both the solid phase and the liquid one. On the solid side of the CMB, there is observational evidence that the D”-layer, adjacent to the core, is highly inhomo- geneous; see for example Stacey (1990). The processes at work in this layer are responsible for removing heat from the core, and they therefore determine the rate of growth of the inner core and the production of the density deficiency, C, at the

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ICB. On the liquid side of the CMB, the large fluid velocity, V4, creates the friction which is so significant in determining the mechanical balance of the geodynamo. The source of the geomagnetic SSV lies here, in the electrodynamic skin-layer on the CMB.

A density stratification near the CMB can significantly influence the fluid motion. The implications of a strong density stratification on the SV was considered in many papers; see for example the pioneering work by Whaler (1980). The nature of such a stably stratified layer on the CMB, its defining parameters, and even its very existence are, however, as yet uncertain.

One possibility, which is directly linked to the main cause of convection, was suggested by Braginsky ( 1 984). Once one believes that, as heavy solid precipates onto the ICB, the light fluid residue rises, he finds it hard to resist the suggestion that some fraction of that light fluid accumulates at the top of the core. This accumulation takes place thanks to the Archimedean force, which presses the light fluid against the solid mantle. A stably stratified layer on the CMB results. Let us call it “the H-layer” and denote its thickness by H, its (negative) density excess being of order C,.

The density excess, -Co, driving the geodynamo is very small: Co-251V/g- for V - lo - ’ cm/s. Even if C, is a few orders of magnitude greater than C,,

it is still small and cannot be detected seismically. It is, however, large enough to change drastically the dynamics of the fluid near the CMB.

The H-layer may be imagined as an inverted ocean of light fluid. The hard CMB is its “bottom”, and its “top” merges into the main body of the core. The arrival of light component from the main core volume into the H-layer can be pictured as a constantly falling “rain”. The removal of light fluid, as the turbulent motion mixes it with the main fluid core “above”, acts like “evaporation”. The magnitudes of H and C , are determined by a balance between “rain” and “evaporation”, brought about by a so far unknown mechanism. To estimate H and C,, we need to make some assumptions. Braginsky (1984) supposed that H is of the order of the turbulent mixing length, which he estimated to be H - 20 km. He also supposed that C , is approximately given by 0 ~ ~ 5 1 , where W H N ( g C H / H ) 1 ‘ 2 is the Brunt-Vaisala frequency in the H-layer. This condition is reminiscent of corresponding equalities in the oceans and atmosphere, which are known to be brought about by the joint action of stable stratification and Coriolis force through manifold processes (such as the baroclinic instability, Rossby waves, synoptic eddies, etc.). One may also postulate similar manifold processes in the H- layer, and adopt oH -51. These processes might establish the balance between “rain” and “evaporation”. It now follows that CH-QZH/g- so that the density deficit in the H-layer is only about - g/cm3. The corresponding stratification in the H-layer is, however, much greater than in the main body of the core: CH- 103C0 and C, /H- 105C0/L. Braginsky (1984, 1987) studied simple models of the H-layer and demonstrated that both short-scale and global baroclinic waves can propagate there. The structure of the waves is complicated by the presence of the magnetic field, which also contributes to their attenuation. These waves have periods of the order of a decade, and may therefore participate in the SSV.

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The postulated value of H is of the same order of magnitude as the skin-depth for the decade SV. The H-layer therefore does not screen them, and the varying magnetic fields created within the layer can reach the core surface.

The parameters of the H-layer and its very existence invite further investigation. The H-layer will, if it exists, significantly influence the SV and other events on

the CMB. It should greatly reduce the effects of inhomogeneities on the CMB. For example, the topography of CMB would be “covered” by a stably stratified liquid in which the gradient, d,C - 10’ COIL, is large. As far as its interaction with nearby fluid flows is concerned, the topography is effectively smoothed out, and topo- graphic coupling between core and mantle is significantly reduced. (This idea was born during a discussion between the author and Adam Dziewonski in the autumn of 1988.)

Thermal processes near the CMB are also significantly influenced by the H - layer. Even if conditions in the mantle, on the solid side of the CMB, were uniform, various inhomogeneities could still exist in the fluid on the other side of the boundary.

The heat generated viscously by the geostrophic flow is released in the Ekman layer on the CMB, and this heating depends on latitude. Even more heat is deposited there through the core motions generated by the PoincarC force corresponding to the 26000 year precession of the Earth. This motion is one of solid body rotation about a rotating axis. In the coordinate frame fixed to the mantle, this flow oscillates with the frequency Q. This simple oscillatory motion cannot act as a dynamo. Its amplitude is, however, extremely high (-OScm/s), and can therefore heat the Ekman layer rather strongly. Precessional heating was considered by Stewartson and Roberts (1963). It depends on latitude and, averaged over a period of one day, it is greater in polar regions than near the equator.

The amount of heat released in the Ekman layer depends on the viscosity of the core, which unfortunately is very uncertain because of the difficulty in estimating the dependence of viscosity on pressure. Buffett et al. (1990) estimated core-mantle friction from the Earth’s forced nutation which also is an oscillatory movement with the frequency R. They interpreted the estimated friction as being due either to magnetic friction (although in this case the existence of a highly conducting metallized layer at the base of the mantle should be allowed for) or to viscous friction with a viscosity, v, of order lo3 cm’/s. Accepting the second interpretation or allowing an equivalent friction due to the effects of topography, we estimate the precessional heating of the fluid at the CMB to be -3.101’erg/s, which is only one order of magnitude less than the heat released by the main geodynamo mechanism.

The variations of temperature in the core are very small and, when considering thermal convection in the mantle, one may therefore adopt the boundary condition, T= constant on r = R,. This condition is, however, physically incorrect when applied to the problem of thermal convection in the core. Instead, it is necessary to prescribe the heat flux on the CMB that is consistent with the solution of the mantle convection problem. The simplest condition that is physically meaningful is the prescription of a constant heat flux on r = R , . The most realistic condition is as yet unclear both because of the presence of

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inhomogeneities in the mantle and because it is still not known whether the heat flux is redistributed by thin thermal layers at the CMB.

Fortunately, core convection is predominantly compositional for which the most realistic boundary conditions are also the simplest. One may assume that I , = - I N at the ICB, where I N is prescribed, and make the simplest assumption, I,=O, on the CMB.

When averaged over very long (geological) times, the geomagnetic field is approximately that of an axial dipole. Integrated over geological time, the dipole spends approximately as much time in one polarity state as in the other. Examining the long-term ( - 10’ y) behavior of the geomagnetic field, Merrill et al. (1979) found evidence for differences in the North-South asymmetries of the direct and reversed polarities. They described these asymmetries as “small but signifi- cant”. From their smallness, we may say that the geodynamo is approximately North-South symmetrical. This encourages one to believe that the simple bound- ary conditions for compositional convection are good first approximations to reality, and that the corrections produced by thermal convection are of secondary significance.

One can seek the reason why the polarity in the system changes despite its symmetry about the equatorial plane. Though the reason is not yet definitely known, the following comments may be made. The behavior of the geodynamo is sensitive to changes in core-mantle coupling. Magnetic friction is proportional to the square of the radial magnetic field on the CMB. This field can occasionally vary due to some instabilities near the CMB, or due to some fluctuations in the rate of growth of MAC-waves, or due to other chance perturbations in the dynamo. The concomitant changes in core-mantle coupling may lead sometimes to a complete restructuring of the geodynamo, that is to a change of polarity. The complicated dependence of the generation coefficient, r, on the form of the asymmetric velocities, V‘, increases the chances of pole inversion.

11. CONCLUSIONS

We are now in a position to provide a more detailed answer to the question posed in Section 1: “Why not simply compute the geodynamo?” It can indeed be computed but the task is not easy. Moreover the problem is not a purely mathematical one. Additional information from geomagnetic observations and from cognate branches of geophysics must be injected, and used in conjunction with computations of the model. The computations themselves are difficult not only because the geodynamo model is governed by a system of nonlinear partial equations in three dimensions, not only because interesting solutions that truly reflect reality are essentially non-stationary, but also because the system contains two small parameters: Ef and ~*=9-’.

The smallness of the first parameter, ef, reflects the weakness of core-mantle coupling. The parameter appears in the denominator of the expression determining il(s), which is therefore sensitive to small changes in the magnetic field within the

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core. The numerator of the expression should be calculated with sufficient precision, bearing in mind the partial cancellations that occur as T(s) is computed. Another consequence of the smallness of E~ is the singular character of the solution, evinced by the presence of the thin “current layer” near the CMB.

The second small parameter, E * , implies the approximate alignment of the velocity and magnetic field in the main body of the core. It is, however, precisely the small deviation in alignment (as measured by V x B) that is responsible for the electromagnetic induction from which the geodynamo originates. The small non- alignment should therefore also be calculated with sufficient precision when the dynamo equations are solved.

Both these demands for precision make the computations difficult. Moreover, the resulting sensitivity of the equations to small perturbations favors the development of numerical instabilities. Special care is necessary in order to secure the numerical stability necessary for a robust computational algorithm. Further difficulties may arise through the possible occurrence of real instabilities in some regions of the core during some time intervals, where by “real” we mean created physically rather than numerically. These instabilities may either be legitimately suppressed, or be filtered out (as for example in the case of the short-period torsional oscillations), or be accurately included in the general computational process. One physical instability which is essential for dynamo generation is that of the MAC-waves. These waves develop on time scales that are one order of magnitude less than the main dynamo period. Hopefully, they establish an organized wave pattern on an equilibrium level that is determined by variable nonlinear processes. The parameter E, is not really very small and often appears in fractional powers such as ~;’~-0.2(say) or E:”. This is inconvenient both for the application of asymptotic methods and for numerical calculations. In practice, one has to compute solutions of the geodynamo equations numerically in a way that makes allowance for several different length and time scales present in the solution.

One does not necessarily make the task easier by supposing that E~ is irrelevant and may be eliminated by using Taylor condition. Even if a Taylor type solution is found, it is still necessary to confirm its eligibility. The confirmation includes a difficult proof of its stability. This task is automatically fulfilled if we time-step dynamo equations that use the generalized momentum balance, i.e. which include both the Taylor term and core-mantle friction.

Additional complications arise from the presence of processes that act over very small length scales. Their representation requires a finer resolution than the global dynamo model can provide. The complicated events on the CMB and the small scale turbulence are two examples of such processes. These should be investigated separately and then included in the model by appropriate parametrization.

The diffusional transport of heat and admixtures in the core raises rather difficult issues. Even when the flux of density excess is parametrized in the simplest way, namely by I = -K‘ - VC, we need a reliable theory of turbulence to estimate the turbulent diffusivity tensor, K‘, which is inhomogeneous, anisotropic and strongly dependent on the magnetic field. The turbulence mechanism involves a number of different physical effects and corresponding parameters (B, VC, etc.) that

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differ in different parts of the core. As a result, even the mechanism creating the turbulence may not be the same everywhere in the core. To make sensible simplifications and a correct parametrization of K', we require a better under- standing of turbulence which is strongly influenced by Coriolis and magnetic forces. This is especially significant because different expressions for the diffusivity tensor can lead to widely different solutions. The strong dependence of K' on B implies that the mechanism of turbulent transfer plays a part in equilibrating the amplitude of the dynamo field.

It would be a very strange coincidence if the Earth's cooling rate determined a level of buoyant forcing in the core that lay near the threshold for the self- excitation of the magnetic field. In fact, the geodynamo is working far beyond that threshold. The coupling between the magnetic field and the cooling of the core (see Section 2) arises only after the magnetic field becomes very strong. Moreover this back reaction is weaker than the main processes that regulate the Earth's cooling. Perturbation methods for deriving solutions near the linear excitation threshold are therefore inadequate. The geodynamo equilibrates through fully developed, non-linear processes involving large scale adjustments of the velocities, non-linear interactions between MAC-waves, and the development of small-scale turbulence. Can the motions and fields in this equilibrium follow a rather deterministic path, or must the evolution be chaotic? This is a central question of geodynamo theory.

It is obvious that such a complicated theoretical picture must be confirmed by a detailed comparison with the observed facts. As explained in Section 9, two major fitting operations seem to be called for, one for the MAC waves and the fundamental oscillation, the other for the short period oscillations and related phenomena. During these fitting operations, appropriate physical models should play an integral part in the data analysis.

The workings of the strong field cro-dynamo based on the associated poloidal field generation by the MAC-waves are shown schematically in Figure 2. The mathematical expression of this scheme is implemented by the system of equations given in the Appendix.

The theory of the axisymmetric field (the AB-problem) has been intensely studied, as has the theory of the a-effect produced by prescribed asymmetric velocities, V'. They have been considered both in the kinematic approximation and as part of the intermediate model approach. The ABC problem turns out to be the more difficult, and has not been investigated at the present time. Up to now, the linear theory of MAC-waves has been developed only slightly, and the properties of finite amplitude MAC-waves are completely unknown. The central role of these ingredients of geodynamo theory is widely recognized. The significance of the lower right corner of Figure 2, namely the theory of local turbulence and turbulent transport, should also be stressed. This theory is in its infancy at present, but deserves the same attention as all other components of the geodynamo.

It should be stated in conclusion that this paper is not intended to be a broad review, but a personal one. It paints the picture of geodynamo which the author considers to be the most plausible. Many gaps in understanding remain. They are bridged today by intuition and imagination alone. Whether these guesses are correct or not is for the future to tell. When will that happen? There is little doubt

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MAIN COMPONENTS OF THE QEODYNAMO

AXISYMMETRIC NON - AXISYMMETRIC

c - - - - - - - - - - - 7 7

I25

smal-scale v turbulence

density -< diffusional C excess - - transport

Figure 2 The general scheme of the geodynamo’s operations: The mutual interaction of its axi- symmetric components ( A B + C problem), the MAC-waves and the small-scale turbulence.

that satisfactory numerical codes for computing three-dimensional dynamo models will be developed during the next few years. Unfortunately, the accumulation of observational data is a slower process. One may only hope that the addition of unattainably numerous data will not be required before convincing results emerge from a reasoned combination of theory and observation.

Acknowledgements I am grateful to Professor Paul Roberts for improving the presentation of this paper and to the Institute of Geophysics and Planetary Physics for its support.

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standing parts,” Bull. Earthquake Res. Inst. (Tokyo Univ.) 47, 65-97 (1969).

shells,” Geophys. Astrophys. Fluid Dynam. 49, 97-1 16 (1989).

APPENDIX: A THREE-DIMENSIONAL MODEL OF A GEODYNAMO DRIVEN BY COMPOSITIONAL BUOYANCY

The general scheme of operation of the geodynamo model is shown in Figure 2. It consists of interacting axisymmetric and asymmetric parts, with small-scale turbulence providing diffusional transport for both. Each part includes kinematic equations (electrodynamics), equations of fluid mechanics (magnetogeostrophic equilibrium) and buoyancy creation (transport of the density excess). To make the equations dimensionless, we select the simplest possible set of units:

where the acceleration due to gravity is taken to be g= - l,(r/Rl)gl. Unlike the units adopted below (12c), the same scaling is adopted here for all components of

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GEODYNAMO THEORY I29

V and B; the parameter 92 is not involved. The inequalities B,>>B, and V@>>T/,, though they are true in the Earth's core, are not utilized. The a-effect is not invoked. Instead, the induction term, V x B , is included in the kinematic equations. Equations (1)-(5) are separated into symmetric parts (indicated by overbars) and asymmetric parts (distinguished by dashes). They are presented below.

THE AXISYMMETRIC PART

Kinematics

The magnetic field B in the fluid core is governed by equations

The same functions in the solid inner core are described by

a & - ( q ~ / q ) A " ' B ~ = o , (A.2a)

where A(') = V2 - s - '. The diffusivity of the solid core, qN, is unknown. Probably the assumption q N = q is not far from truth.

The effective electromotive force is

Z - V ' X B . (A.3)

The velocity and field components are expressed in the forms (12a,b) to fulfil the continuity equations (4) and (5):

P=l,V,+O,, v ,=vxx 1,s-l, (A.4a, b)

B=l,B,+B,, B , = V * x l g S - l . (ASa, b) Here [= V,/s, and +=sA,.

The boundary conditions for the magnetic field, B, in the fluid core are continuity to the external field, B , in the mantle and to the field, B', in the inner core. The mantle conductivity ciM is much smaller than r ~ , therefore one can assume that (approximately)

The toroidal field is zero in an insulating mantle and, when a small mantle conductivity is to be allowed for, we adopt the simplified condition given by Braginsky (1975, 1988):

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130 S. I. BRAGINSKY

The subscript 1 signifies values on the CMB, i.e. r = 1; E~ is a small parameter proportional to the integral mantle conductivity

eM = (oR 1) - J oM(r) dr. (A.2c)

Expression (A.2b) is a consequence of the continuity of the &component of the electric field, E,=a-'J,-(V x B),. It is supposed that the conductivity of the mantle is concentrated in a thin l a y c a t the CMB. The field B , in this layer increases from zero to B,l. The term BiV, is omitted from (A.2b).

Let us note that only fields and velocities that are mirror-antisymmetric were contemplated in the intermediate models described in Section 5. Antisymmetry with respect to the equatorial plane, z = 0 is implied; e.g. A,( - z) = A,(z), B,( - z) = - B,(z), x( -z) = - ~ ( z ) . In the real Earth, (smaller) symmetric components are also present.

Mechanics

v, = F @ , (A.6)

where

F +- -s- 'v . (sB,B,)=s- 'V. (sB,B,+sB&Bb), (Aha)

(A.7b)

(A.7c) z

Here z = (1 - s ~ ) ~ ' * , 3 = V x B, J' = V x B . The integrands (V x F)+ and 8,C in (A.7b,c) are supposed to be functions of s and z. If only the field B, were taken into account in F then i B = ( B : - B i l ) / s 2 . It is supposed that iBI =0 in (A.7b); the geostrophic term in (A.7) is therefore il(s) =i(s, zl).

The flux function x is obtained from 8,x= -sVs by integration over dz. There are significant differences in the expressions for x and c l for the region r 2 < s 5 1 and the region Osssr,. Only the formulas for the (simpler) former case are written down below.

The expression for x(s,z) has the form

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GEODYNAMO THEORY 131

(A.8)

The superscripts s and a distinguish parts that are respectively symmetric and antisymmetric with respect to z=O; e.g.

F" Q -1 - 2 [F,( s, z ) + F,( s, - z ) ] , 4 = + [F,(s, 2) - F,( s, - z )] .

From conservation of mass, the quantity

(A.8a)

provides the flux integrated over the Ekman layer: f I/e dr = f Vs dz = xl/s. Ekman layer theory connects x, and by the relation

~1 = - E v f a s 2 i l , (A.8b)

where E ~ = ( V / Q R ~ ) ~ ~ ~ and f a = i z l l i z .

The angular momentum balance for the shells s and s+ds obtained above for an axisymmetric field can be generalized in a straightforward way to the case when the field B is also present:

~

s - ' J ( s ~ F ) / J s + z ; ~ ( B , B , ) ; -E,fas2[l =0, (A.9) __ ~

where (B,B,): =(B,Br+ B,B:); and

(A.9a)

Equation (A.9) can also be obtained from (A.6a) and (A.8a, b). Using (A.2b) and the similarly simplified expression

one may rewrite (A.9) in the form

~ 1 . fs(1 = s - ~ J ( s2F) /ds -Fd , (A . lO)

where E, f = E , f, + E~ f,, and

f*=Z;l(B,Z+B:Z);, (A. 1 Oa)

The magnetic field in (A.lOa, b) is averaged over longitude and is symmetrized over z= k z , .

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132 S. I . BRAGINSKY

The procedure for determining x and in the presence of the inner core is not quite clear because of uncertainties about the nature of the interaction on the ICB. Some simplified expressions have been adopted for the intermediate models. See, for example, Braginsky (1978, 1988, 1989). A better understanding is still required for the complete three-dimensional model.

Buoyancy Creation

The transport of the axisymmetric density excess is governed by ____ a,c + vp . vc + v . ( v ’ c +I) = GC. (A.11)

Here GC is the constant determined by the prescribed fluxes of the density excess on the inner core I N and on the mantle I M . The flux I can be parametrized as

The form of matrix, D,,, of turbulent diffusivities is open to discussion but, rather approximately, it may be supposed D z z - D,,- 1, Dss<< 1. In reality, D i j contains a few other parameters that should be fitted to the geodynamo model.

THE ASYMMETRIC PART

Kinematics

The asymmetric parts of (2) and (5) are

a , B - V 2 B = V x (G’+ P x B’+ V’ x B), (A’. 1)

v * B = O , (A‘. 2)

where the quadratic induction term, G , of zero average, is

G = V’ x B - 8. (AI.3)

Only two of three components of the field B’= I,$$ + Bb are independent because of condition (A’.2). The formulae become simpler if the meridional components B,, are, following Braginsky (1964a, b), taken to be independent functions. Then (A’.l) takes the form

8, B,, + [ 8 , Bp - ( V2 B)p = s - ’ B, d ,V; + (V x G’),,

+(Bp - V)Vb-(Vb * V)B,,+(B,, * V)V,-(Pp * V)B,,. (A.4)

Here a,, denotes the differentiation with respect to 4 with the unit vectors 1, and 1, kept constant, e.g.

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GEODYNAMO THEORY I33

a l ,B = 1, a,B, + 1, a , ~ : + 1, a, B,.

The component Bb can be obtained from (A'.2) which we rewrite as

a,B,= -sv - Bp. (A'S)

For the field in the inner core we have

a,B" - ( yIN/yI) VZB" = 0. (A'.la)

For the insulating mantle

VZBe=O. (A'. 1 b)

Its solution is determined by the value of the normal component, Bbl, at r = 1 and the condition that Be+O as r+m.

In writing (A'.4), the continuity equation (4) was used in the forms

v - V'=O, (A'.6a)

a,v,= + v . v;. (A'.6b)

On the CMB it can be assumed that V:=O, while V, and V, have finite jumps across the Ekman layer. The field B should match to the external field, Be, which differs little from the field that would be present if the mantle were insulating. This is the boundary condition for the field B'. For an ideally conducting core (to which therefore the magnetic field lines are frozen), the condition V:=O would imply B: = 0 so that B e = 0 but this is not the case for a core of finite conductivity. It was shown by Braginsky (1964a, b) that, in the real fluid core, for which the conductivity is high ( B D ~ ) , the asymmetric field in the core is large: B/B,- B1i2>>1. However only its small fraction (-W-') penetrates the CMB and can be observed: B'e/Bp-W)-1i2 . The small inclination of the dipole axis (about 0.2 radians) provides an estimate of W- ' I 2 .

Mechanics

By taking the curl of the equation of mechanical balance (2) to eliminate the term VP', we obtain its meridional projection in the form

a,v; = z;, (AI.7)

zp = le(r/s) a,@ - (v x F),, (A'.8)

(AI.9)

Here a, = 1, - V so that (A'.7) has a simple form in cylindrical coordinates:

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134 S. I . BRAGINSKY

a, v;. = z:, a, v, = 2:. (A‘.7s, z)

The magnetic force density, F’, can be expressed (e.g. by using the current density J = V x B ) in the form F’=(JxB)’ or

F’= J x B’+ J’ x B + J’ x B‘- J’ x B . (A’. 1 Oa)

By absorbing the magnetic pressure iVB2 into the term VP, it may be also expressed in the form F’=(B . VB)’ or

F‘=B - VB’+B’ * VB+B * VB‘-B * VB’. (A’. 1 Ob)

The velocity, Vl,, may be uniquely obtained from (A’.7) by the direct integration of the Z; over dz, as was shown by Taylor (1963). The boundary condition of the vanishing of the normal (radial) component of Vl, on the solid surface should be used in this operation. Therefore V‘ can be expressed in terms of B and C‘ at the same moment in time.

Buoyancy Creation

The evolution of the density excess, C‘, is governed by

Here I’ can be parametrized in the form

The only source of C in (A.11) is the one on its right-hand side, because the sources, I , , on the boundaries are supposed to be constant.

The equations for the primed quantities have solutions in the form of complicated non-linear waves-the MAC-waves. Equations (A’.4) and (A’.l l), which contain the operator d,+(d,, permit us to trace the evolution of MAC- waves from some arbitrary initial state and determine the pattern that becomes established.

The approach adopted here is one in which the natural evolution of the dynamo is followed. This evolutionary approach does not impose any artificial conditions on the solution, such as the demand of stationarity or of harmonic oscillation. Relatively stable solution(s) is (are) automatically selected by the system if the governing parameters are properly chosen. The latter should be confirmed by a comparison of the behaviour of the model with observations of the geomagnetic field. Events such as polar reversals can also be naturally described by the evolutionary model.

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