zotero/storage/KS7NHYES/.zotero-ft-cache

This article was downloaded by: [Universite Laval] On: 09 July 2014, At: 23:08 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Geophysical & Astrophysical Fluid Dynamics
Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/ggaf20
Equations governing convection in earth's core and the geodynamo
Stanislav I. Braginsky a & Paul H. Roberts a a Institute of Geophysics and Planetary Physics, University of California , Los Angeles, California, 90024 Published online: 19 Aug 2006.
To cite this article: Stanislav I. Braginsky & Paul H. Roberts (1995) Equations governing convection in earth's core and the geodynamo, Geophysical & Astrophysical Fluid Dynamics, 79:1-4, 1-97, DOI: 10.1080/03091929508228992
To link to this article: http://dx.doi.org/10.1080/03091929508228992
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/ terms-and-conditions

Geophys. Astrophys. Fluid Dynamics, Vol. 79, pp. 1-97 Reprints available directly from the publisher Photocopying permitted by license only

Q 1995 by OPA (Overseas Publishers Association) Amsterdam B.V. Published under license by Gordon and Breach Science Publishers SA Printed in Singapore

EQUATIONS GOVERNING CONVECTION IN EARTH’S CORE AND THE GEODYNAMO
STANISLAV I. BRAGINSKY and PAUL H. ROBERTS
Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90024
(Received 27 April 1994; infinalform 15 November 1994)
Convection in Earth‘s fluid core is regarded as a small deviation from a well-mixed adiabatic state of uniform chemical composition. The core is modeled as a binary alloy of iron and some lighter constituent, whose precise chemical composition is unknown but which is here assumed to be FeAd, where Ad = Si, 0 or S. The turbulent transport of heat and light constituent is considered, and a simple ansatz is proposed in which this is modeled by anisotropic diffusion.On this basis, a closed system of equations and boundary conditions is derived that governs core convection and the geodynamo. The dual (ther-
+ mal compositional) nature of core convection is reconsidered. It is concluded that compositional convec-
tion may not dominate thermalconvection, as had previously been argued by Braginsky(Soviet Phys. Dokl., v. 149,p. 8, 1963;Geomag, and Aeron., v. 4, p. 698, 1964),but that the two mechanisms are most probably comparable in importance. The key parameters leading to this conclusion are isolated and estimated. Their uncertainties, which in some cases are large, are highlighted.The energetics and efficiency of the geodynamo are reconsidered and re-estimated. Arguments are advanced that indicate that the mass fraction of the light constituent in the solid inner core may not be smallcompared with that in the outer core,e.g. about 60%.This tends to favor silicon or sulfur over oxygen as the principal light alloying constituent.
KEY WORDS: Geomagnetism, core dynamics, compositional convection, thermal convection, geodynamo, dynamo energetics,dynamo efficiency,turbulence.

Downloaded by [Universite Laval] at 23:08 09 July 2014

1. INTRODUCTION
It is sometimes said that the equations governing the geodynamo are “well-known” and that only their solution is difficult. This statement is, however, misleading and unhelpful. It is admittedly true that the geodynamo is governed by the classical equations of fluid mechanics, electrodynamics, thermodynamics and physical chemistry, all of which were well known in the nineteenth century. But they govern physical processes that, in the context of the geodynamo, have significant roles on scales that differ by very many orders of magnitude. Moreover, the classical equations explicitly include only molecular transport processes, and these are so small in Earth’s core that diffusive mixing is accomplished not by molecular motion but by fluid motion on a variety of greatly different length scales.The net result is that the classical equations in their original form are suited for neither analytic nor numerical studies of the geodynamo. They must be transformed in such a way that physical processes operating on widely different scales are explicitly separated. To this end, the small parameters
1

2

S. I . BRAGINSKY A N D P. H. ROBERTS

Downloaded by [Universite Laval] at 23:08 09 July 2014

relevant to Earth’s core must be isolated, and their smallness used to generate a set of reduced equations that, at one and the same time, are not only geophysically sound but also simple enough to be analytically and/or numerically tractable. Our aim in this paper is to develop such a formalism for the study of the geodynamo.
Our theory recognizes at the outset that the core is an iron rich alloy. As Earth cools, iron in the fluid outer core (“FOC”)settles onto the surface of the solid inner core
(“SIC”),and the gravitational field, pressure, density . . . of Earth change. The concomi-
tant release of gravitational energy is a potent source for the geodynamo, one that we explicitlyinclude; see Appendix B. A second crucially significant source is thermal, and arises from the gradual reduction of thermal energy in a cooling Earth. We must therefore describe core convection and the geodynamo against a background of a “referencestate”, that changes slowly with time, t due to changes in composition and temperature. We find it convenient to introduce a ‘slow time’ variable, t,, that changes on the geological time scale, to and a ‘fast time’ variable, t,, that changes on the convectional time scale, 5,.
The reduced equations, which are developed in Sections 2-5, lead us to two simplified models of the geodynamo: an inhomogeneous model (Section 6) and a homogeneous model (Section 8). The former may also be described as a generalized “anelastic model” and the latter as a generalized “Boussinesq model”, generalized in each case by some additional and novel features. Many papers (e.g. Backus, 1975; Hewitt ef al., 1975; Gubbins, 1977; Gubbins et al., 1979) discuss the energy and entropy balances in the core. They derive these from the primitive equations, without approximating them by forms suitable for studies of core dynamics. Our re-discussion of these balances (Section 7) explicitly separates the effects of the slow evolution of Earth from those associated with convection.
Most discussions of the geodynamo start from systems of equations (almost invariably Boussinesq approximated) that govern core convection and magnetic field generation.The quantities arising in these theories (such as density,pressure, etc.)are in reality very small deviations from the same attributes of the background state on which the magnetoconvection takes place. The separation of the primitive fields into background and convective parts is rarely discussed, but is in fact not a trivial matter, because it demands an understanding of how small additions to a large background behave. And this inevitably introduces the complications of core turbulence. An attempt to derive the equations governing core convection driven by both compositional and thermal buoyancy was previously made by Braginsky (1964b). His procedure was much the same as that adopted here; he too separated the small convective deviations from the background, and recognized that the convective motions have a fluctuating turbulent part that significantly enhances the transport of extensive properties of the mean convective state, such as entropy and composition. His treatment was, however, too incomplete to answer satisfactorily several significant questions, which are addressed in the present paper; see particularly Section 4 and Appendix C.
The Boussinesq approximation is rather obvious in laboratory contexts, where the adiabatic gradient, V To,is associated with only minute variations across the system, i.e. T,>>LIVT,(,where T, is the departure from the reference state created by convect-

GEODYNAMO CONVECTION

3

Downloaded by [Universite Laval] at 23:08 09 July 2014

ion and L is a typical length scale of the system. The validity of the Boussinesq

approach is far less obvious for Earth's core where the variations in reference

- - state variables with depth greatly exceed those associated with convection

( p , E,LIVp,l and E,

see below). In this respect, the core resembles the

convect-ion zones of stars, to study which astrophysicists have developed mixing length

theory, in which departures from the adiabatic, such as T, = T - T,, are more signifi-

cant in the determining the convective state than T itself (see Jeffreys, 1930; Cox

& Giuli, 1968).

We are interested in this paper in making more precise the sense in which the

convection is a small deviation from the background state and in discussing in greater

detail the assumed transport processes in the core. Small parameters that arise, some of

which were referred to above, are:

E,. This measures the inhomogeneity of the basic state. It is defined in (3.8)
and -0.1; .,E This assesses the importance of centrifugal forces in determining the structure of
the background state; see (3.9). Its value -2 x
E,. This measures the relative importance of the forces that control the convection
- (e.g. the Coriolis force) to the forces determining the background state (e.g.
gravity); E, lo-'. See (3.10);
- .:E This is the ratio of convective time scale, T,, to the time scale T~ over which the background state evolves (the geological time scale); E,' lo-*. See
(3.1 1);
- cR.This is the Rossby number that measures the relative importance of inertial and Coriolis forces on the main scale of core convection; eR 10- 5 ;
$. This is the (magnetic) Ekman number, which differs from the usual Ekman
number (the ratio of viscous to Coriolis forces) in that the magnetic
q - diffusivity, q, appears in place of the kinematic viscosity, v, in its definition
(8.19); 10-9.

Even though E, >> E,, it is with core convection (which is associated with E,) that we shall be principally concerned in this paper, and not with the asymmetry of the reference state, which is measured by E, and which, as we shall argue in Section 3, is not of prime importance. To keep our model as simple as possible, we set E, = 0, i.e. we ignore the deviations in the structures of the mantle and core from spherical symmetry, even though it is known that asymmetries do exist. Taking them properly into account is a nontrivial matter that is best left for future development of the present theory.
The core is assumed in this paper to be a binary alloy, consisting mainly of iron and a single light admixture, which we need not specify. This limitation, to one light constituent, almost certainly oversimplifies a complicated core chemistry, but it suffices since it models a process that is vital to core dynamics, namely gravitational stirring by compositional convection. In the absence of detailed knowledge about the core constituents, the complications introduced by the addition of further chemical elements could not be justified even though (see Appendix D) it could easily be accomplished at the expense of introducing further unknown parameters. Our information about the physical properties of the core is far from complete, but not all of those properties

4

S. 1. BRAGINSKY AND P. H. ROBERTS

Downloaded by [Universite Laval] at 23:08 09 July 2014

are equally significant for the construction of a geodynamo model. By developing as simple a model of core dynamics as possible, we are able to assess which parameters are crucial for the construction of a geodynamo model and which are less critical. In Appendix E we have tried to estimate as many of the relevant physico-chemical parameters as we could, though we recognize that our values are rather uncertain. All significant geodynamo parameters will be determined in the future only by optimizing the fit of geodynamo models to the observational data. We wish to emphasize that our primary goals in this paper are those of developing a general theory and of establishing, in the simplest and most direct way, the main relations between the relevant physical quantities. The accurate estimation of the key parameters of the theory is of secondary importance to us; it is in any case not achievable at the present time.
The dynamics of the FOC are controlled by the SIC and the mantle. It will be sufficient for our purposes to suppose that
1. Over the geological time scale, zo, the mantle flows like a fluid to maintain hydrostatic balance so that, in the limit cn<<1 (which we adopt throughout the paper), the core-mantle boundary (“CMB”)is spherical, r = R , ;
2. The mantle is rigid on the convectional time scale, z, so that R , = R,(t,); 3. Neither the iron nor the impurity comprising the core fluid can penetrate the
mantle, so implying that the radial component, Vorl, of the fluid velocity, V,, associated with the slow evolution of Earth coincides at the CMB with the velocity, R , , of the CMB and also that the radial flux, Igl, of light admixture vanishes on the CMB. It is possible that assumptions (1)-(3) may be lifted in the future, but to do this meaningfully more geophysical information will be required. An improvement to (1) would result from the inclusion of the topography of the CMB when considering core-mantle interaction; insight into mantle rheology on time scales of order lo4yr, might lead to the abandonment of (2);information about chemical interactions on, and material exchange across, the CMB would lead to a reconsideration of (3). Modeling the SIC is the main topic of Section 5. At the present time, the solution to the geodynamo problem is hampered both by a paucity of information about the numerical values of crucial physical parameters (such as the amount of energy available) and perhaps more seriously by a lack of understanding of the principal physical ingredients of the geodynamo mechanism. One might say that the present state of geodynamo theory can be represented by the symbol-, which we use to signify uncertainty by an “order of magnitude”, i.e. by a factor of order 10. The immediate goal of geodynamo theory is to find better values for the key parameters and to improve the understanding of both the main components of the geodynamo mechanism and the structure of the physical
- fields.When this goal has been reached, one may reasonably hope to be able to replace
the symbol by the symbol z ,by which we mean that our answers would then be accurate to about 10%. This, in fact, is precisely the magnitude of E,. In the intervening period before the goal is attained, one may wonder why one should study geodynamo models at an accuracy better than E,, and this is the reason why so many studies of magnetoconvection in the core make use of the Boussinesq
approximation, in which E, = 0. It is also the reason why (Section 8) we develop

GEODYNAMO CONVECTION

5

a generalized Boussinesq model that is especially appropriate for the study of core convection. We also call this the “homogeneous model”, since the reference state is independent of position. The many complications of the inhomogeneous (or anelastic) model developed in Sections 4-6 are thereby avoided, at the expense of errorsthat are no larger than the existing uncertainties in the key parameters describing the core.
Finally, a few words about notation, Earth’s core is a complicated chemicalfluid-magnetic system involving widely disparate length and time scales. Its mathematical description, and the reduction of that description to tractable form, raise formidable notational problems. In an effort to minimize these problems, we have developed a consistent and (we hope) transparent, notation. We believe that a concise and self-explanatory system of notation is significant not only because it helps to avoid misunderstandings but also because it provides a convenient language with which to discuss and clarify the subject. Our notational scheme is set out in full in Appendix A, and it is recommended that any reader who wishes to follow our arguments in detail should consult this Appendix at the outset. We point out here only that the suffix a always signifies that the quantity concerned is evaluated in the adiabatic reference state, while (or occasionally ‘) refers to convective deviations from the reference state; the superscript t refers to the fluctuations that arise from core turbulence. The suffix 1is attached to quantities evaluated on the CMB; those evaluated on the inner core boundary (“ICB”),carry the suffix 2. In the case of ambiguity, as in the case of the density, p , which is discontinuous on the ICB, the suffix 2 refers to values on the upper (fluid)side of the ICB and the suffix N to values on the lower (solid)side.

Downloaded by [Universite Laval] at 23:08 09 July 2014

2. BASIC THEORY
2.1 Governing Equations
The core of Earth is modeled as a binary alloy consisting primarily of iron, but with a light admixture whose composition need not be specified here. This simplification of what is probably a complicated chemical mixture of many elements suffices to characterize its behavior. The governing equations for the motion of the core and the evolution of the magnetic field are, in the frame of reference rotating with the mantle, and in a notation that is set out in Appendix A,

Downloaded by [Universite Laval] at 23:08 09 July 2014

6

S. 1. BRAGINSKY AND P. H . ROBERTS

Equation (2.1) contains the effective gravitational field, g,, which includes with the true gravitational field,g = - VU, the centrifugal acceleration, -R x (R x r):

+ g, = -VU,, U , = U Un, UR= -:(n x r)2.

(2.8a, b, c)

The angular velocity S2 of the frame is attached to the mantle, which does not rotate completely uniformly. Strictly the Poincare force, - p a x r, should therefore also be added to the right-hand side of (2.1).The gravitational field, g, is mainly due to Earth itself, a fact expressed by (2.5) and the condition that U 4 0 for r -+ co.A discussion of the gravitational field and its energy is provided in Appendix B.
Equations (2.1)-(2.6) constitute 10scalar equations' for 11 scalar unknowns, namely
p, p , S, t, U and the components of V and B. They must be supplemented by boundary
conditions and by constitutive laws for the viscous and magnetic forces, F' and FB,and the fluxes of light component and entropy, I, and Is. These constitutive laws introduce a further field, the temperature 7'.It is therefore necessary to consider the thermodynamics of the fluid. This is specified by two thermodynamic variables, for example
p and S, and by the mass fraction, 5 , of the light constituent. All other thermodynamic
quantities can in principle be derived from these three variables; unfortunately, they are in practice, not well determined. We shall regard the internal energy per unit mass,
~ ' ( pS,,t),as a given function of p, S and 5. From this, p , T and the chemical potential
p are determined through the relation

+ + de' =TP d p TdS pd5, P

which implies that

g)s& g)p,<, P =P2(

(T =

($) cc =

PS

(2.10a, b,c)

It is sometimes convenient however to use p , S and 5 or p , T,and 5 in place of p, S and 5 as independent variables, in which case the enthalpy, ~ ~ (S ,p0,, or the Gibbs free
energy (also called the thermodynamic potential), ~ ' ( pT, ,t),take over the role of &'(P, s,51, where

The relation (2.9) between differentials is then replaced by

+ + + 1

1

dEH= - dp TdS pdg, dEG=- d p - SdT p d l .

P

P

(2.142.12)

'Equation (2.7) is not counted since (2.6)implies iliV.B = 0. Thus (2.7) holds for all t if it holds for any t. It therefore has the status of an initial condition.

GEODYNAMO CONVECTION

7

Further thermodynamic relations are set out in Appendix D; see also Landau and Lifshitz (1980).
We now summarize the remaining constitutive relations. For F'we have

P ~ =;v j ~ ; oi r pF' = V. E'.

(2.13)

The double overarrow is here used to signify that the symbol beneath it is a second rank tensor (here the viscous stress tensor) which can be contracted with another vector or tensor from either side. We assume that the fluid is a linear viscous(Newtonian)fluid, for which

Downloaded by [Universite Laval] at 23:08 09 July 2014

Sincethe kinematic shear viscosity,v, and the coefficient of second (bulk)viscosity,vb, are necessarily non-negative, the same is true of the rate of viscous regeneration of heat, which is

Q' =nJieji= 7r;iVjVi=2pv11(e..+~V.11VG.13.)(e..-~V.V6ij)+pv,(V(.2V.1)42).

According to (2.11)and (2.14),we have

pV.F' =V.( ?*V) - Q",

(2.15)

a fact we shall need below. The magnetic force on the fluid depends on both the magnetic field, B, and the
electric current density, J:

p F B = J x B, J = V x B/,u,.

(2.16,2.17)

Equation (2.6)is a consequence of (2.17), of Faraday's law and of Ohm's law for a dense, isotropic, moving conductor:

+ 8,B = - V x E, J = o,(E V x B).

(2.18,2.19)

Here E is the electric field, o,,= l/p,,q 2 0 is the electrical conductivity and 9 2 0 is the magnetic diffusivity. The Joule dissipation of heat is

Q J = J2/o,= (q/po)(Vx B)'. According to (2.16),(2.19)and (2.20),we have

(2.20)

~ V . =FE~. J - Q J ,

(2.21)

a fact we shall need below.
The entropy source osand the heat flux I4are related to the fluxesIs and I r of entropy
and light constituent. The form of this relationship follows from energy conservation,

Downloaded by [Universite Laval] at 23:08 09 July 2014

8

S. 1. BRAGINSKY AND P. H. ROBERTS

which takes the mathematical form afutotal + v.1total -- Q R ,

(2.22)

where

+ + + + utota=' /I(& EK ' U Q ) uB us, rtota1= p(&H + FK + UQ)V- 5P.v + IB+ 19 + 14.

(2.23a) (2.23b)

Here uB and us are magnetic and gravitational energies per unit volume, with
corresponding fluxesLB and Ig;also, E~ = V 2is the kinetic energy density relative to the
+ rotatingframe, cH= E' p / p is the enthalpy per unit mass, and Q" 2 0 is the volumetric
rate of radiogenic heating which (becauseof convective mixing) will later be assumed to be proportional to p.
Consider the time derivatives of the successiveterms of(2.23a).According to (2.9)and (2.2)-(2.4), we have

4(P&') + V*(P&'V=) (P/P)d,P + P T G + P P d L
+ = -pV.V Tos - TV*IS- ~ v . 1 ~ .

(2.24)

We write this as
+ + + + + d,(p&') V.[peHV TIS PI<] = ToS V-Vp IS.VT+ 15.Vp.

(2.25)

According to (2.1),(2.2),(2.15) and (2.21),we have
+ + 6',(pcK) V.(p&' - ?'.V) = -V.Vp - pV-VU, E - J- Q" - Q J . (2.26)

Since URis independent oft, (2.2)gives
+ d,(pUQ) V*(pUW)= p v - v u n .

(2.27)

If we multiply (2.18)scalarly by p i 'B, and apply (2.19), we obtain
+ d,uB V*IB= -E*J,

(2.28)

where

B2
uB=-,

p=-.E x B

2PO

PO

The gravitational energy balance is formulated in Appendix B as
+ drug V T = - p v . g = p ~ - v ~ ,

where

(2.29a,b) (2.30)

(2.31a, b)

GEODYNAMO CONVECTION

9

By (2.8b)and (2.25)-(2.31), we now have
+ + + + + + + dp'o'a' V.[P(€" f K U*)V - Z'.V IB I@ TIS pI'] + = TaS- Q ' - Q J + Is.VT Ic*Vp.

(2.32)

Comparing this with (2.22)and (2.23),we see that
+ I4 = TIS p15, ~8= Q' + Q~ + Q" - P - v T - F v ~ .

(2.33) (2.34)

According to (2.34),we may rewrite (2.25) as
+ + + + + + Jt(p&') V.[peHV TIS p I r ] = Q' Q J Q" V-Vp.

(2.25a)

We may recall that we assumed in Section 1that Ir*dA= 0 on the CMB and it therefore follows from (2.33)that the net flux of entropy from the core is related to 9b,the net heat flux from it, by

Downloaded by [Universite Laval] at 23:08 09 July 2014

(2.35)

(In anticipation of developments below, we have supposed here that the temperature, T,, of the CMB is predominantly the basic state temperature which is, with high precision, almost uniform over the CMB.)
Equations (2.33)and (2.34)are basic and transcend in importance any constitutive relation for Is and It. It is nevertheless essential that those relations be such that gs2 0, with equality only if the system is source-free (QR= 0), current-free ( Q J=0), in solid body rotation (Q' = 0)and in thermal equilibrium (VT = Vp = 0).The first three terms on the right-hand side of (2.34)are non-negative; only the last two terms are problematical. The answer, for the case of molecular transport of S and (, is summarized in Appendix D. We present here only the conclusions:

kh=$,

(2.36,2.36a)

+ 1

Is = -(IT T

$It),

IT = -KTVT, K T = ~ c , , K ~ ,(2.37,2.38,2.38a)

+ + + 1' = IT (p p')15, p' = pS,kS, h5.

(2.39,2.39a)

Three independent transport coefficients appear here: the diffusivity of light material, K < 2 0; the thermal conductivity, K T 2 0 (or equivalently the thermal diffusivity, K ~ ) , and the Soret coefficient, kS,. The remaining coefficients, kf,and ht = p' - &k$, are

10

S. 1. BRAGINSKY AND P. H. ROBERTS

thermodynamic properties of the fluid. It follows from (2.34)and (2.36)-(2.38) that

+ + + fJs= fJT (7r fJv+ o f f J R .

(2.40)

The individual sources of entropy comprising 0' are

(2.40a,b)

(2.40c,d, e)
which are non-negative since K~ 20, 182 0, &.2 0, Q'2 0, Qf 2 0 and Q R2 0 . It
follows that '0 2 0. In subsequent Sections, we shall frequently require integral forms of (2.2)-(2.4) and
(2.25).The FOC occupies a volume, VIZt,hat changes with time. With the help of the relation(va1id for CMB and ICB moving with velocitiesU , and U, and for an arbitrary field Q)

Downloaded by [Universite Laval] at 23:08 09 July 2014

where A , and A2 are the outer and inner boundaries of V12w, e deduce from (2.2)-(2.4) and (2.25a)that
(2.42)

I?.,, d,[y,2 pSdV =

+ + aSdV- fA, IS.dA $A, [I' pS(V - U2)]*dA,

(2.43) (2.44)

jv12 jy., 1. + + + + d, pe'dV = (Q' QJ QR V.Vp)dV - (1' pUl).dA

I,+ + [I' pU2 pP(V - U,)]*dA,
+

(2.45)

where V, is the volume occupied by the SIC. We postpone discussion of the SIC until Section 5. Equations (2.42)and (2.43)contain the statements that the total mass of each
+ constituent of the alloy in the entire core (FOC SIC) is conserved.

2.2 Continuity Conditions
Corresponding to the balance laws set out in Subsection 2.1, there are continuity demands on the ICB. These can be obtained by using a pill box argument in the usual way. The conditions are simplified by the absence of surface currents and masses. It

Downloaded by [Universite Laval] at 23:08 09 July 2014

GEODYNAMO CONVECTION

11

follows that the magnetic field must be continuous everywhere:

[B] = 0, on the ICB and CMB,

(2.46)

and that the same is true of U , and g,:

[V,] = 0, and [VUJ = 0, on the ICB and CMB; (2.47a,b)

see Appendix B. (Here [ Q ] denotes the jump in a quantity Q across the boundary concerned.) We also have

[ T ]=0, on the ICB and CMB.

(2.48)

Let n denote the unit outward normal for both the ICB and the CMB. Consider first the CMB. According to our model of the mantle (Section l),

V = U,, n.I<= 0, on the CMB.

(2.49a, b)

Energy balance requires that

[.-Iq] = 0, on the CMB.

(2.50)

In the mantle, B obeys2

8,B = - V x (qMV x B).

(2.51)

The electrical conductivity of the mantle, l/pOqhfi,s concentrated near the CMB but even there it is much smaller than the core conductivity, i.e. q M >> q. The magnetic field in the mantle must obey (2.46) and must join continuously to a source-free potential field in the "vacuum" surrounding Earth. Similarly the U obeying (2.47) must match smoothly to a source-free potential outside Earth.
Consider next the ICB. Corresponding to (2.2), (2.1), (2.3) and (2.22) there are continuity conditions corresponding to conservation of mass, momentum, light constituent and energy. These are3

i[pn-(V- U,)] = 0,
+ [ p - n. ?'-n pn.(V - U,)n.V]I = 0,
+ [n.{15 p5(V - U,)}] = 0,
+ + [n.(Iq - ?"V - pU2 p(cH cK)(V- U2)>]= 0,

on the ICB, on the ICB, on the ICB, on the ICB.

(2.52) (2.53) (2.54) (2.55)

'The relative motions in the mantle are too small to have any inductive effect,and are omitted in (2.51). Our frame of referenceis attached to the mantle so that its velocityof rotation as a whole is zero by definition.
'Because of the simplified model we adopt for the inner core (see Section 5), we do not need to impose continuity of the tangential stress, either on the ICB or on the CMB. We have therefore excluded this from (2.53).

12

S. 1. BRAGINSKY AND P. H. ROBERTS

We shall make use of the smallness of the inertial and viscous forces in our application, as compared with the pressure gradient, to replace (2.53)and (2.55) by

[p] = 0, on the ICB, [n.(Iq - ~ E ~-(UV2 ) f ]= 0, on the ICB.

(2.56) (2.57)

By (2.33),we may write the last of these as

m [ ( E G -

wn n - u2~)a+ [P+ +P w -
+ + [[Tn.{Is pS(V - U,)}] = 0,

on the ICB. (2.58)

The ICB is a surface in phase equilibrium, at which therefore4

[ p i = 0,

- pLcJ=0. on the ICB.

(2.59,2.60)

Applying (2.48),(2.52),(2.54),(2.59)and (2.60) to (2.58),we obtain
+ [n-{Is pS(V - U2)}]= 0, on the ICB.

(2.61)

This shows that entropy is conserved at the ICB; there is no surface source of entropy corresponding to a breakdown in the continuum approximation there. (Velocities are so small that inertia is negligible, and there is no shock at the surface of discontinuity.) Conditions (2.57) and (2.61) are now seen to be equivalent, and we need use only the more convenient, which is usually (2.61).
The ICB is a no-slip surface, so that

[n x (V - U,)] = 0, on the ICB.

(2.62)

Downloaded by [Universite Laval] at 23:08 09 July 2014

3. THE REFERENCE STATE
It is extremely convenient to describe magnetoconvection in the core as a departure of core conditions from a basic reference state. The most convenient reference state is a hydrostatic, well-mixed, non-magnetic state. It is therefore governed by (2.1) with V and B set zero, by (2.5),and by statements that the state is isentropic and chemically homogeneous:
pa- ‘Vp, = - vu, = g,,
V*(PaVa=) - b a ,
vt, = 0, vs, = 0,
V2U , =4nkNp,- 2R2.
4Seefor example Loper and Roberts( 1978).Alternatively,we may recall(seeAppendix D)that p = pL - p H
+ and cG = pLtL p H t H ,where tL= [, t,, = 1 - [ and the suffices L and H refer to the light and heavy
constituents of the alloy. Thermodynamic equilibrium requires that [ p L ]= [ g H ]= 0, and these imply (2.59) and (2.60).We shall later use the fact that E‘ - pt = p,,; see footnote 5 below.

GEODYNAMO CONVECTION

13

The suffix a is used to distinguish variables in this adiabatic state. Since the gravitational field appearing in (3.1)and (3.5) is created by pa and not the full p, we have replaced the effective field, g,, and effective potential, U,, of Section 2 by g, and U , rather than by the more cumbersome g,, and U,,.
A few comments should be made about (3.1)-(3.5) as applied to the FOC. First, although the state was described as hydrostatic, it is important to incorporate the fact that it is slowlyevolving on the geological time scale:the inner core grows secularly and the concentration of light constituent in the core fluid increases as it does so. It is particularly necessary to recognize that fact in equation (3.2)of mass conservation. This
accounts for the presence of the term involving 0, on the right-hand side of (3.2)and
the term involving V, on its left-hand side. Both these terms would be absent in a truly hydrostatic state, and in the geophysical context they are minute, but necessary in order to incorporate evolution on the geological time scale. Their effect on convective and magnetohydrodynamic processes is negligible. Because V, and d,V, are so small, they can be (and have been) discarded in (3.1), resulting in the hydrostatic equation shown.
Second, since 5 , and S, are, by (3.3) and (3.4),functions o f t alone, it follows that
spatial variations in

Downloaded by [Universite Laval] at 23:08 09 July 2014

arise only because of variations in pa,the gradient of which is determined by (3.1).In this way we find that in the basic state

(3.7a,b, c)

We have here introduced the entropy coefficient of volume expansion, as, which plays a larger role in our work than the more familiar isothermal coefficient of volume expansion, a, to which it is related by

as = -p - ‘(ap/dS),,, = aTfc,;

(3.7d)

see (D18). A more familiar form of (3.7b) is

T,-‘VT, = yg,/u:, y = aug/cp,

(3.7e,f)

where y is the Gruneisen parameter. The gradients (3.7)are generally called “adiabatic gradients”.
Because of centrifugal forces, the surfaces of constant U , are not spherical, but it is clear that pa,pa and T,, are constant on surfaces of constant U,, and that they can all be labeled uniquely by that value of U,. This is also true of all other thermodynamic
parameters, such as 01, y, us, cp,.They too are, through U,,functions of position; for
notational simplicity,we have not added the suffixa to these variables in (3.7)and shall
not do so below. Since S, and 5, are constants, the density is, according to (3.3),(3.4)and
(3.6a),a function of pressure alone: pa = p(p,). When equations (3.1)and (3.5)are solved,

Downloaded by [Universite Laval] at 23:08 09 July 2014

14

S. I. BRAGINSKY AND P. H. ROBERTS

subject to suitable boundary conditions for U , and with the density a given fnction of the pressure, solutions are obtained in which the surfaces of constant U , coincide with those of constant p , and pa. This is required for self-consistency. The problem of determining such solutions is known as the problem of determining the equilibrium figure of Earth, and (3.1)and (3.5)define the part of this problem that pertains to the core. In reality the figure of Earth deviates slightly from the hydrostatic equilibrium figure. It may be noted that, if (3.1)is supplemented by the equation of heat conduction rather than (3.4),it is no longer true that p a =p(p,) and,for a general distribution ofheat sources, there is no solution in which the surfaces of constant p and p coincide. The hydrostatic problem then has no solution, as can be seen immediately by taking the curl of (3.1).This means that an imbalance of forces exists which results in some circulations if the gravitating body is fluid. In a fluid-like inner core this would result in meridional motions but, since the effectiveviscosity of the inner core is so high, these would not be significant in the leading approximation (3.1).In an elastic inner core, the imbalance of forceswould create deviations in the surfaces of constant p and p , but these would again be so small that they would easily be balanced by elastic stresses. In either case, we may safelyneglect the imbalance of forces in the SIC and use (3.1)-(3.5)there, as in the FOC. See also Section 5.
Equations (3.3)and (3.4)express the fact that we are interested only in convection that is so intense that all extensive properties of the basic state are well-mixed by the convection that is superimposed on it. Of course, this will not be true in the boundary layers at the ICB and the CMB, where the fluid moves with the adjacent solid, and vertical mixing by convection is small or absent. In the bulk of the core however, where the approximation is a good one, the thermodynamic variables in the convective state differ from their values in the basic state by so little that the differencecan be treated as a perturbation; see Section 4. In fact, the more vigorous the convection, the better the perturbation treatment works5.The choice of an adiabatic hydrostatic referencestate is usually made, though in disguised form, when modeling laboratory systems. The variation in pressure across these is so slight that the assumption of constant entropy differs little from the assumption of constant temperature, To. The difference,ATa, in the adiabatic temperature across the system is small compared with the typical temperature differences,T,, associated with the convection, in sharp contrast to Earth’s core where T,.. 10-6AT,.
We noted above that the adiabatic well-mixed reference state is close to being realized throughout the entire voiume of the fluid core apart from thin nonadiabatic boundary layers of thickness 6,, (say).If nothing special occurs near the ICB and CMB, these layers will be very thin, as the following argument shows. Suppose that the

’The system is very far from being in the steady state that might exist were convection weak (or absent)and
+ + the core close to (or precisely in) “sedimentation equilibrium”, where T, and tG- ( p U , = pH ci, are
constant. The composition and entropy are not uniform in such a state, and it cannot be regarded as
- a perturbation of a well-mixed adiabatic state. Such a quiescent ‘steady state’ is unrealistic for the core. Even
if the core were isolated, such a state would arise only after a time of order zSed= L2/d lO”yr, which is
much greater than the age of Earth. (Here K < is the molecular diffusivity.)In the well-mixed state considered
+ in this paper, (2.1l),(3.3) and (3.4) show that Vtf =‘pa- ‘Vp,,. It therefore follows from (3.1) that tf Udis
constant.

GEODYNAMO CONVECTION

15

temperature gradient deviates significantly from its adiabatic value, V,T,, over the

- length 6",. Then a temperature perturbation 6T S,,,V,T, will arise that results in

- - a fractional density perturbation of 6C u6,,,V,Ta.Equating this to C,

(a value

- - we later show is characteristic of the FOC), we obtain 6,, C,/crV,T, 1m, where we

have assumed that V,T, N 1 OK km-' = lop3"Km-' and a - lo-' OK-'. In fact,

however, special physical processes may become significant near the ICB (e.g. Loper

& Roberts, 1981, 1983)and near the CMB (Braginsky, 1993)which invalidate these

estimates of 6,, and 6C, which are found to be much too small. These special boundary

layers will not be considered in the present paper.

Let us now consider the different types of inhomogeneities that arise in the core. The

greatest inhomogeneity in the referencestate is connected with the variation across the

core of variables such as pa and T,; see (3.7).This variation can be measured by a small

parameter, E,, where

Downloaded by [Universite Laval] at 23:08 09 July 2014

- Taking the characteristic length over which the density changes to be L lo6m, the - ', - gravitational acceleration at the CMB to be g1= 10.68rnsK2, and the velocity of
longitudinal sound waves to be us 104ms- we obtain E, 0.1. The appropriate dimensionless parameter with which to assess the importance of
centrifugal forces on the structure of the reference state is

,€ = 4RZL/g,.

(3.9)

Using the values given above, we find that E, % 2 x lop3.Because of centrifugal forces,
the surfaces of constant U, are not quite spherical. They resemble more oblate
spheroids with an ellipticity of order ,E which is approximately 1/299.8 at the surface of Earth, and varies across the fluid core from e l = 1/393.0 on the CMB to e, = 1/414.9 on the ICB; see Mathews et al. (1991),a paper from which we also took the abbreviations FOC and SIC.This ellipticity is very significant for core motions that are driven by the precession and nutation of the Earth's rotation axis. It is, however, insignificant for the slow convection that drives the geodynamo. We shall therefore neglect it here and assume that all thermodynamic variables arefunctions of r alone in the reference state.
A further significant dimensionless parameter is

E, = 2RV/g,

(3.10)

where Vis a typical convective flow speed. This measures the relative importance of Coriolis forces associated with the convection and gravitational forces acting on the basic state. It is a more significant quantity in more dynamics than V2/gL,which is the ratio of inertial forces on the convection to the basic gravitational force; inertial forces play a negligible role in core convection. Traditional estimates of V are of order 3 x 10-4ms-1, based on the velocity of westward drift; integrations of model-Z
- (Braginsky, 1978) lead to values 10 times greater. If we take V = 1 0 - 3 m s - ' as
a compromise, we obtain E, lo-'. The smallness of E, is extremely significant from

Downloaded by [Universite Laval] at 23:08 09 July 2014

16

S. I. BRAGINSKY A N D P. H. ROBERTS

a dynamical point of view: E, provides an estimate of the error made in supposing that the reference state is in hydrostatic equilibrium (3.1)and in adopting (3.3) and (3.4).Its smallnessjustifies the omission of Coriolis, magnetic and buoyancy forces in modeling the reference state.
The parameter E, may be related to a further small parameter which compares the time scale z, over which the basic state evolves and the time scale 5, of core convection:

E; = 5' , f z,.

(3.11)

Depending on the physical process considered, these times span wide ranges. The time scale z, is sometimes called the "geological time scale", although this may be somewhat inappropriate since it suggests that za, is of the order of 4.5 109yr, this being the age of the Earth. In reality, the temperature of the core has diminished
- only to a small degree during its history, and zu, defined as T,/Tu, greatly exceeds
4.5 x 109yr. We expect in fact that that z, 10"yr - 10" yr. The time scale 5, is even more uncertain. At one extreme, convection associated with turbulence operates on time scales of a few years; at the other extreme, large-scale MAC
- - -- waves typically vary on periods of lo3yr, which also characteristic of convective
overturning; the time scale of the general circulation of the core is 104yr. If as a compromise we take 7, 4 x 10"yr and 5, 400yr, we obtain E; lo-*, but this value is extremely uncertain. In what follows, we shall usually not distinguish between E: and E,, writing either as E,.
Finally we list in Table 1 some properties of the core that are well determined. They are mostly taken from the PREM model of Dziewonski & Anderson (1981). For a discussion of these and other core parameters, see Appendix E.
One quantity in Table 1 deserves special comment, namely A p , the density jump at the inner core boundary. This plays a very significant role in our theory. According to

Table 1 Well-determined Parameters
R, = 6.371 x lo6m R , = 3.480 x lo6 rn R,= 1.2215 x 106m p o = 10.9 x lo3k g m - 3 p1 = 9.9035 x lo3k g m - 3 p z = 12.166 x 10' k g m - 3 p R = 1 2 . 7 6 4 ~1 0 3 k g m - 3 p(O)= 13.088 x 10'kg11-~ A p =pN -pz =0.6 x lo3kgm-' g1= 10.681ns-~ gz=4.40ms-2 p1 = 135.75GPa pz = 328.85 G P a p(0)= 363.85 G P a usl =8.065 x lO3rns-' usz= 10.356 x lo3m s - l

average radius of Earth, radius of the fluid outer core (FOC), radius of the solid inner core (SIC), mean density of the FOC, density of the F O C at the CMB, density of the F O C at the ICB, density of the SIC at the ICB, density at the geocenter. density jump at the ICB (relatively poorly known), acceleration due to gravity at the CMB, acceleration due to gravity at the ICB, pressure at the CMB, pressure at the ICB, pressure at the geocenter, longitudinal seismic velocity in the F O C at the CMB, longitudinal seismic velocity in the FOC at the ICB.

GEODYNAMO CONVECTION

17

Don Anderson (private communication), the error in the value shown can be no more
than 20%, i.e. 0.5 x lo3kgm-3 < A p < 0.7 x lo3kgm- ’. See also Jephcoat & Olson
(1987)and Shearer & Masters (1991).

4. THE NATURE O F CORE CONVECTION

4.1 The Anelastic Approximation
We stressed in Section 3 that our basic reference state depended on the presence of convection sufficiently vigorous to homogenize the entropy and chemical composition of the fluid core; see (3.3) and (3.4). In this section we study this convection explicitly.
We noted in Section 1 the existence of two distinct time scales:the slow evolutionary time scale,z,,of the reference state, and the much shorter time scale, z,,associated with convection. It is often convenient to employ a two time scale procedure in which t, denotes slow time and t, fast time. The reference state depends on t , alone; the
+ superimposed convection depends on both t, and t,, and a, = 8; a;. Wherever it
cannot lead to confusion, we replace 3; by an overdot, and omit the superfix c on 3; and d;.
We decompose all quantities into basic and convectional parts, writing

P = P . +PC,
+ p = p a p , ,

T = Ta + Tc, 5 = 4, + t c ,
+ + U = U , U , , g, = g, g,,

s = s, + s,, + V = V, V,, etc., (4.0)

where the subscript c on a quantity shows that it is associated with the convection. On substituting (4.0)into (2.1)-(2.5) and making use of (3.1)-(34, we obtain

Downloaded by [Universite Laval] at 23:08 09 July 2014

Here and in what follows, we for brevity omit the suffix a on thermodynamic
functions evaluated in the reference state, while retaining them on pa, S, and 5,.
For example, us appears below in place of us,. The superfix has been omitted from
df and a;.
In deriving (4.1)-(4.4), all terms of order E, times the corresponding terms in the reference state have been retained; those of order E,‘ have been discarded. The fact that
p J p , = O(E,)<< 1, SJS, = O(E,)<< 1, = O(E,)<< 1, allowed us to replace p in many

Downloaded by [Universite Laval] at 23:08 09 July 2014

18

S. I. BRAGINSKY AND P. H. ROBERTS

terms of (2.1)-(2.7) by pa, a simplification also made frequently below. Consider for example (4.1). The quadratic term p,g, has been discarded. In the term
+ + + pd,V =( p , p,)(dy d;)(V, V,), we recognize that dp/df and V,/V,are both O(E,)a, nd
that therefore the dominant part of pd,V is p,dfV,. The same process of linearization allows us to replace (2.16)by

p , , =~J ~x B.

(4.8)

We shall also replace (2.11)-(2.13) by the single equation

pF” = p,vV2V,

(4.9)

an approximation we discuss further below.
The constitutive relations (2.36)-(2.40) for It, Is and oSmay be simplified similarly.
For example, P K < , k ; / T and k t / p in (2.36)may be evaluated in the reference state, and
may therefore be written as pa^<, k$/Ta and k:/p, where (see above) the suffix a on KC, k$ and k i is implied but suppressed. It would however be incorrect to replace V t , V T
and V p in (2.36) by V t , =0, V T , and Vp,. Although It,\ << It,/ and I TcI<< I TaI,an
important component of the convective motions is on small length scales associated with turbulence, and it is not necessarily true that lV<,l<< lVt,l and / V T , /-K/ V T , J . Similar remarks apply to all of equations (2.36)-(2.40).
The smallness of p,, S,, t,, pc, etc. allows “thermodynamic linearization”, by
which we mean that we may, with an error only of order E,, inter-relate the deviations, p,, T,, p,, etc., created by the convection in p , T, p, etc., by applying relations
c,, such as (D5)-(D7) or (D13)-(D15), treating p,, S,, pa etc. as the infinitesimals
dp, dS, d t , d p , etc. [An example is given in (4.16) below, which follows from
(D13).] The resulting simplifications are very significant, but in Section 8 we shall reduce the complications still further by introducing what we shall call “the homogeneous model” or “the modified Boussinesq model”. In the homogeneous model, we approximate all thermodynamic coefficients, such as a, us etc. by constants, while in the present inhomogeneous model they are functions
of r. Equation (4.2) goes beyond thermodynamic linearization; it incorporates what is
generally called “the anelastic approximation”. It is justified by noting that

(4.10)
+ The replacement of p V =(pa p,)V in (2.2) by p,V in (4.2) follows as before from the
smallness of p,. The absence of the time derivative of the density in (4.2)excludes elastic waues from the solution of system (4.1)-(4.9) and explains why the approximation is termed “anelastic”. The slow motions, characterized by (4.lo), are included, while uninteresting high frequency oscillations associated with sound (seismic) waves are filtered out. If, instead of (4.2), V - V = 0, (4.9) would, for constant pv, be an exact consequence of (2.11)-(2.13); since V p , # 0, (4.9)is not precisely correct. It should be

Downloaded by [Universite Laval] at 23:08 09 July 2014

GEODYNAMO CONVECTION

19

borne in mind however that we are concerned with small viscosity flows in which F’ is significant only in thin boundary or shear layers across which p a and v vary little. The expression (4.9)therefore holds with high accuracy wherever F”is non-negligible.

4.2 A Signijicant Simplijkation

We devote this subsection to a remarkable simplification of (4.1).We introduce

p = “P
Pa

c = - ass, - art,,

(4.11,4.12) (4.13,4.14)

the last two of which are a new “effective”pressure and density; we call P “the reduced
pressure”. The quantity C plays such a central role in the theory that, in our opinion, it merits a name. We propose to call it the codensity. It determines the buoyancy force due
to the deviation of the density from the well-mixed basic state of constant Sa and tu.
Note that C is independent of pc. The first three terms on the right-hand side of(4.1)may, with the help of (4.13),(4.14)
and (3.7a),be written as

-vPe+us+ + uc> ga pcga = -v p , pega.
=
The expression for p, that follows from (D13),namely

=

1 2 p

,

-

assc -

aq,,

Pa Paus

may be written in terms of C and the effective variables (4.13)and (4.14)as

(4.15) (4.16)

(4.17a,b)

This leads to further simplifications in (4.15).We have, by (3.7a) and (4.11),

so that by (4.17b)

-VPe +Peg, = -PaVP + Pacga.

(4.

It follows from (4.15)and (4.18)that (4.1)may be written in the very simple form

+ d,V = - V P + C g a - 2&2x V + F’ FB.

(4. 9)

20

S. I. BRAGINSKY AND P.H. ROBERTS

The resemblance of (4.19)to the Boussinesq momentum equation is so striking that it is
- worth re-iterating here that (4.19)is a consequence of the assumption of an adiabatic,
well-mixed reference state. Its precision is of order E, lo-*. It should be stressed that the density inhomogeneity is taken into account in (4.19) through (3.7a).The elastic part, p,/ug, of the density perturbation has not been neglected but has been absorbed into P. Equation (4.19) shows clearly that the buoyancy force associated with deviations of order E, from a well-mixed adiabatic state is created only by the codensity through variations in entropy and composition; the buoyancy force associated with pressure variations, though it may be equally large, does not contribute because it is conservative and can be absorbed into the effective pressure to create the potential term, - VP, in (4.19).It does not contribute to the generally non-potential term, Cg,. This is the basic reason why the codensity plays such a central role in the theory and why it deserves a special name.
Through the reductions made here, the unpleasant necessity of computing U , during the process of solution is evaded; gc= -VU, has been eliminated from (4.1), though it has not been neglected. After the solution has been completed, U , can, if desired, be evaluated by solving (4.5). Though (4.19) resembles the momentum equation for Boussinesq theory, the anelastic continuity equation (4.2) is unchanged and is very different from the corresponding equation (V-V=O) of Boussinesq theory. Thus, our simplification is not tantamount to a reduction to Boussinesq theory.

Downloaded by [Universite Laval] at 23:08 09 July 2014

4.3 Core Turbulence: General Considerations
We now discuss a very significant component of the convective motions: turbulence. It is hard to doubt that the core is mixed far more effectively than molecular diffusion coefficients such as 18- 10-8v would suggest and that this is due to turbulence. Because of the notorious difficulties of turbulence theory, and because it would in any case be impractical to add such difficulties to the already formidable geophysical complexities,only a simple "engineering" approach to core turbulence has so far been contemplated. In this approach, one writes

V = ( V ) ' + V+, t, = {t,)'+t:, etc.,

(4.20)

where (V')', ((+)Ie,tc. are zero. The averages are over an ensemble of realizations of the turbulence. More practically, they are taken over the short length and/or time scales of the turbulent components. One seeks to determine the evolution of the average fields
(V)', (B)', (t)', ( S ) ' , ..., and to replace (4.1)-(4.9) and (2.36)-(2.40) by equations
governing those averages. The effects of turbulence are supposed to be local so that, as for molecular transport processes, all turbulent transport fluxes at a point are proportional to gradients at that point. For example, in the simplest ansatz, the flux I<'of mean composition and the flux 1'' of mean entropy due to turbulence are proportional to the
local gradients of (g , ) t and (S,)':

It' = - p,#'.V( t,)', IS' = -paZ'.V( S,)'.

(4.21,4.22)

Downloaded by [Universite Laval] at 23:08 09 July 2014

GEODYNAMO CONVECTION

21

The fluxes are thus not parallel to the gradients but are linearly related to them by the tensor Hf and, more importantly, the significant turbulent transport coefficients contained in H' greatly exceed the two molecular scalars.
Not surprisingly, It' and Is' are large compared with the molecular contributions 1'"
and Ism to If and IS obtained by averaging (2.36)-(2.40).6 Since the turbulence
transports 5 and S in the same way, the same tensor, E', arises in both (4.21)and (4.22).
Double diffusion processes in the core therefore differ greatly from double diffusion
processes in the laboratory of the type investigated by Cardin & Olson (1992).The
transport of (4,)' and (S,)' in the core differ not because these quantities diffuse
differently(that happens only in laboratory conditions)but because their sources are of a different nature. The light component is injected from the ICB but thermal convection is principally determined by the ability of the core to transmit heat to the mantle; the latent heat emitted during freezing at the ICB may be secondary, though it is not very
small. It is hard to avoid parametrizing turbulence in this way. Anyone who prefers
instead to employ the primitive variables, V, B, t,, S,, ..., and the corresponding
forms (2.36)-(2.40) for I(, Is and (Y' is free to do so, but he must then use values
of rcT and rcCof at most 10- 5qand lO-*q respectively, and therefore must contend with enormous Rayleigh numbers and other dimensionless parameters. The resulting flows would be turbulent and would require him to strive for impossibly high numerical resolutions. Sooner or later he would be forced to accept an engineering approxi-
- mation, probably of the type we seek to develop here. The turbulent transport
coefficients that then arise are of order t V t 1m2s-', which is many orders of
- magnitude greater than 18and is even much larger than rcT. (In making this estimate, we
have taken t 104m and have assumed the moderate value 10-4ms-1 for the rms turbulent velocity, V . )
Even within the engineering approximation, several different scenarios have
been proposed. Braginsky (1964b) and Braginsky & Meytlis (1990) supposed that motions in the core exist on essentially only two, widely disparate, scales, the
macroscale L and associated time scale z, and the microscale t and related time scale
zt. According to their theory, local turbulence consists of an ensemble of plate-like
cells having thicknesses, t,, in the s-direction much less than their other two ( z and 4) dimensions, both of which are of the order of the microscale t.They argue that, because of the smallness of t,, the turbulent microscale magnetic Reynolds
number is very small, so that microscale induction does not seriously modify Ohm's
law for the macroscale. The mean field, (B)', is therefore governed by (4.6)with the mean velocity (V)' replacing V but with the same molecular value of 4.They derive expressions for E' that are of order q, but they do not derive an approximate form for the Reynolds tensor, Z'.
The Braginsky-Meytlis picture is not the only possibility. There is a second scenario that is theoretically extremely complicated: large-scale turbulence arising from the instability of MAC waves of planetary scale. Such a turbulence

6The adiabatic gradient (3.7b)makes, however, a contribution to 1; which should not be neglected; see belw.

22

S. 1. BRAGINSKY A N D P. H. ROBERTS

Downloaded by [Universite Laval] at 23:08 09 July 2014

would be of the classical type, involving “cascade” from macroscale to microscale, i.e. the microscale envisaged by Braginsky & Meytlis (1990) would overlap with a macroscale, and their estimates of iz‘ would be invalid; perhaps even the forms (4.21) and (4.22) themselves would be inadmissible. Possibly a turbulent Ohm’s law (including a turbulent a-effect) would also be required, as in mean field electrodynamics. One way of investigating whether this second scenario is plausible or not would be first to solve the large-scale convection problem on the assumption that the turbulence is of Braginsky-Meytlis type, i.e. I<‘ and Is‘ from (4.21) and (4.22) would be used rather than the corresponding molecular expressions given by (2.36)-(2.40). Second, the instability of that state would be sought. If it were unstable, transition to cascading turbulence would be anticipated, i.e. the second scenario would be plausible. It seems likely, however, that the enhanced diffusion associated with R‘ would help to stabilize large-scale motions. If this were the case, the second scenario would not be plausible. Stevenson (1979) developed a heuristic theory of core turbulence, based on the assumption that all three characteristic dimensions of the cells are of the same order, L, as that of the core. His results may be relevant to the second scenario but should be treated with caution because of the possible influence of smaller scales of turbulence on the larger scales.
A third scenario has been proposed by Moffatt (1989) and Moffatt and Loper (1994). They imagine that the light material emerging from the ICB during freezing rises in discrete blobs of dimensions P between 10’m and 104m and perhaps most typically 103m. They suppose that these blobs preserve their identity as they ascend from ICB to CMB. [To the contrary, the simulations of St. Pierre (1995) suggest that the blobs will be enormously distorted after rising only a few hundred km from the ICB.] Moffatt (1989) and Moffatt and Loper (1994) argue that, as they rise, the blobs induce helicity sufficient to self-excite a magnetic field. A full statistical theory of blob motion has not yet been developed. One may imagine that at one extreme, where the blobs interact strongly with one another, such a theory would have strong points of similarity with that of Braginsky & Meytlis (1990).At the other extreme, in which the blobs interact weakly, it may be possible to develop a theory based on a rarefied “gas” of blobs. Further investigations will be required before the role of blobs in core MHD can be properly assessed, but one may again anticipate that, from a statistical mechanics of blobs, a transport theory will emerge that fits into the general framework we have developed below, albeit with a different
form for 2‘.
Which of the three scenarios is geophysically the most realistic is unknown. The Braginsky-Meytlis scenario is, at the present time, the most highly developed and (we believe)the most plausible. It does, however, rest on uncertain ground. In the absence of magnetic field, Coriolis forces impart a columnar structure to convective motions; see for example the theoretical studies of Roberts (1968), Busse (1970, 1994), and Glatzmaier & Olson (1993), and the experimental investigations of Busse and Carrigan (1974,1976) and Boubnov & Golitsyn (1986). It is plausible that Lorentz forces will stretch these structures in the direction of the magnetic field (i.e. primarily longitudinally, the +direction) and that the convective cells of core turbulence will therefore be plate-like. The uncertainties were highlighted by Braginsky (1964b),who

GEODYNAMO CONVECTION

23

concluded that additional analysis was necessary before answers could be given
to crucial questions such as: 'How long are the cells in the z-direction (i.e. parallel to
a)?'What is the mechnism that limits their length L'(<<L)in the z-direction?
The lengthening of the plate-like cells depends on the diffusivities operating on the instabilities that produce those cells. In Earth's core, the most significant compositional diffusivity is nevertheless small. It is difficult to believe that cells of a thickness,
t,, of only about 1km extend across the FOC from one hemisphere of the CMB to
the other. But what is the mechanism that "breaks-up'' these cells? At the present time this question has no satisfactory answer. Braginsky & Meytlis (1990)suggested a heuristic approach which predicts that the plate-like cells have comparable dimensions in the z- and $-directions, and about 20 times smaller in the s-direction (i.e.in the direction away from the rotation axis). A complete theory of turbulence in the presence of Coriolis, Lorentz and buoyancy forces is for the present no more than a dream.

4.4 Core Turbulence: Averaged Equations
Let us now proceed more formally to derive the average forms of (4.1)-(4.7). Most discussion centers on (4.3)and (4.4),which by (4.2)may be written as

Downloaded by [Universite Laval] at 23:08 09 July 2014

Let us focus first on (4.23).Its average is

where Irm = ( I r ) ' is the molecular flux due to the average gradients, and

It' = p a ( ty')'

(4.26)

is the turbulent flux of light component. From the average of (4.2),we see that (4.25) may be written as

+ where Ifota=l It'" I@is the total irreversible flux of admixture due to molecular + diffusion and turbulent mixing, and d, = 8: (V)'-V is the derivative following
the mean convective motion. This has exactly the same form as (4.3) but with 5,
+ replaced by (c,)', and I r replaced by I:otal,The term I<" = I F If" consists of two
parts, the first being the result of substituting Vt, ( = O ) , VT, and Vp, into
(2.36); the second arises similarly from the averaged gradients V(<,)', V( Tc)* and
- V(pc)' and is extremely small. Even the first term, Is", is minute because of the
smallness of the molecular diffusion coefficient: 10- 'q. From now on, we shall

24

S. I. BRAGINSKY AND P. H. ROBERTS

recognize that I<'is the dominating part of It5,taland shall write
+ I:otal = Itrn I" N It'.

(4.28)

This approximation was suggested by Braginsky (1964b).The final form of (4.27)is now

+ pad,( 4,)' V*15'= -pa[,.

(4.29)

The consequences of (4.4)follow similar but more complicated lines, because of the
necessityof obtaining an expression for (8)'A.s in (4.26)and (4.28),a total irreversible
+ entropy flux, ISota=l Is" Is', replaces the molecular flux, and

IS' = pa(s:v+ )',

(4.30)

+ but the molecular diffusion term Ism= 19" 1s" cannot here be omitted. The thermal
- diffusivity,tcT lO-'q, greatly exceeds the compositional diffusivity, K< N lO-*q, and
we must retain Is" in (4.30) to allow for the molecular diffusion of heat down the adiabat. On neglecting 15" and using (2.37)and (2.38),we find that the molecular flux is approximately

Downloaded by [Universite Laval] at 23:08 09 July 2014

(4.31 )

The total entropy flux is approximately
+ lSota=, T ~l -~ T I ~ ' ,

= -K~VT,,

(4.32,4.32a)

where, for brevity, IT has replaced 1:'". The total heat flux, IQotal,may now be obtained from (2.33)by making use of (4.28)and (4.32)

+ + + Ifotal= TaISotal paI:ota,= IT TaIS' paI?

(4.32b)

We may write the average of (4.4)as
+ + + pad,(S,)' V .Is' = - pasa V * [( K T/T,)VT,] (oS)',

(4.33)

but an expression for (aS)' is still lacking. Before deriving it, we raise and dismiss an apparent inconsistency that arises when we compare (4.26)and (4.30)with (4.21)and (4.22).The former expressions for I<' and Is' vanish on the walls where Vt =0, but there is no reason why the latter should; indeed, for the success of our later considerations, there is every reason why they should not! The paradox evaporates when we recognize the existence of boundary layers on the ICB and CMB. At the edges of these layers, turbulence is strong; the fluxes are nonzero and are given by (4.21) and (4.22).Within a boundary layer, the turbulent fluxes diminish to zero with the vigor of the turbulent motions as the wall is approached, but this is

GEODYNAMO CONVECTION

25

simultaneously compensated by an increase in the molecular fluxes, the gradients in
( 5 ) ' and (T)' growing to make that possible. We are not concerned here with the
detailed structures of the boundary layers, but we have to appeal to their existence in
order tojustify the application of(4.21)and (4.22)even "at the walls", by which we mean
"at the edges of the boundary layers attached to the walls".
Returning now to the evaluation of (as)', we adopt the Reynolds analogy, in which transport through the random motion of the turbulent eddies is likened to that of the
random molecular motions, although with much larger diffusion coefficients.In (2.34)
we see included, within the rate of entropy production as, contributions made by the
molecular fluxes,I5 and Is.In analogy, we use the same expression for (as)', the rate of increase of entropy created by both molecular and turbulent diffusion, i.e. we replace I5
+ and Is in (2.34)by Ifatal= I<'and I~a,a=l 1;"' I". It should be particularly noticed that,
according to the Reynolds analogy, the gradient operators scalarly multiplying these
diffusivefiuxes now act not on 5, = (5,)' + 5; and S, = ( S , )' + S: but on (5,)' and ( S , ) ' . + + Thus VTand Vp in (2.34)are nor V(Ta T,) and V(pa p,) as in the molecular case but
+ + are V( Ta ( T,)') and V(pa (p,)'). And (T, )'and (p,)' vary on the same length scale
as Taand pa,namely the macroscale L. Thus, while it was incorrect, because of the small
length scale, Ti,over which T: and p: vary, to ignore VT, and Vpcin comparison with VT, and Vpa (see above), we may make use of the smallness of I( q)'l/lTal and I(pc)'l /I paI to replace V T and Vp in (2.34)by VT, and Vpa,with an error only of order
E,. We may therefore write

Downloaded by [Universite Laval] at 23:08 09 July 2014

where by (4.32)

(4.35)

cf. (2.40a).By (3.7b,c) and (4.34),we have

+ + + + (2)=' (

( G J ) ' rJR aT a',

where

(4.36)

(4.37)

According to (4.21) and (4.22),we may rewrite (4.37)as

+ a'=P"ga-R'-( Crsvs, a'vg,)'. Ta

(4.38)

To avoid violating the positivity of entropy production, we must set R' zero at all points at which a', as calculated from (4.38), is negative. We demonstrate in

26

S. I. BRAGINSKY AND P. H. ROBERTS

Appendix C that, to a satisfactory degree of approximation, the regions where d > 0
are those that are gravitationally unstable and, as a result, are sources of turbulence. Regions in which (4.38)gives 0'< 0 are locally stable; turbulence is then absent and the
turbulent fluxes I r f and Is' are zero, as is 0'.
Most of the remaining terms in (4.1)-(4.7)are linear and easily averaged. Indeed, (4.2) and (4.5)-(4.7)are unchanged on averaging. Two further issues concerning turbulent transport arise in connection with the equation of motion (4.19).All but two terms of (4.19)are linear and easily averaged. Using (4.2),we may write the inertial term with an
+ error of order E, as pdtV = paa:V V-(p,VV). When we average we obtain

Downloaded by [Universite Laval] at 23:08 09 July 2014

where

;;Vf = -pa( VtVt)'

(4.41)

is the Reynolds stress tensor. Again using the Reynolds analogy, we may expect that

:tl = paV$,Vk( V,)',

(4.42)

where 5"' is the (fourth order) turbulent viscosity tensor, anisotropic because of the effects of Coriolis and Lorentz forces on the turbulence. It has 36 independent components, since without loss of generality V$ = v& = v$. It is also necessary that the associated entropy production, (eij)lnT = pav$Vi( Vj)'Vk(V,)', is nonnegative.
The other nonlinear term arising in (4.19)is the Lorentz force. This is similar to the inertial force and may be treated in a similar way:

p, (F')' = ( J )' x (B)' + (Jt x B')'.

(4.43)

Becausethe magnetic pressure can be absorbed into p,, (4.43)is effectivelyequivalent to

+ 1
pa( FB)'=-V.[( B)'( B)'] V.iSBf, where
PO

FBI - -(B~B+)'.
PO
(4.44,4.45)

According to the local turbulence theory of Braginsky & Meytlis (1990),the magnetic Reynolds number of the microscale motions is very small, and a linear relationship therefore subsists between Bt and Vt:

Bt = m .IJ.VJ?'

(4.46)

- According to their estimates Bt Vt; see Appendix C. It follows from (4.46)that (4.45)

can be written as

(4.47)

Downloaded by [Universite Laval] at 23:08 09 July 2014

GEODYNAMO CONVECTION

27

where

(4.47a)

has the same symmetries as v&. The similarity of (4.42) and (4.47) suggests that the last term of (4.43) should be
transferred to the viscous stress. We therefore write

paFB= (J)' x (B)', p,F" = V - K ' ,

(4.48,4.49)

where

- Since Bt Vt, the two contributions to Z' are of the same order of magnitude. We shall
assume that positivity of the entropy production, eijn; 2 0,is maintained even after the
addition of EB' to Ev';i.e. we shall suppose that pa~:jkIVVi(j)'Vk( V,)' 2 0.
+ The total viscous force is F: = F'" F'', where pF""= pavV2(Vt)' is the mean force
produced by the molecular viscosity v and the mean velocity gradients; see (4.9).
Turbulent mixing is greatly reduced near solid boundaries, and it is therefore unclear
whether v t k lis significantly greater than the molecular viscosity in the bounary layers
- - - - on the CMB and ICB. Elsewhereeven the smallest of the turbulent viscosities,vfjkl,is of
order v\ t,Vt (d',/d')q q/25 0.1m2s-', according to the estimates of Braginsky and Meytlis (1990).This is of order lo5 times greater than the molecular viscosity, if
estimated as v = 10-6m2s-'. Thus, in the main body of the core, F'" is negligible and
= F: F"'.Despite their much greater size, the effect of the turbulent viscous stresses is
scarcely more significant than that of the molecular viscous stresses. This can be seen
- - from the minute size of the turbulent Ekman number, cn' cf/25 4 x lo-", where Ef: - lo-' according to Section 1.The significant viscous stresses within the shear layer - surrounding the tangent cylinder, s = R,, are turbulent ones: 7csg = 7ces pav:VsVg. In
Sections 6-8, we shall absorb the molecular viscosity into the turbulent viscosity, make
use of

and suppose (see above) that

Q' = pavijkIVi(Vj)'Vk( V,)' 2 0.

(4.50)

- - It should be noted that, even though lBtl << I(B)'I, it is not true that lJtl IBtl/potis
much less than 1 (J)'I 1 (B)'(/poL. Thus, even though IBtI << I(B)'I, the contribution,

+ QJ= P ~ ~ ( ( J ~ )m~ad) e' ,by Jt to the total ohmic dissipation, poq( J2)r= QJ Q'is not

negligible,where QJ =poq((J)')2 is the macroscale ohmic dissipation. In fact, we show

in Appendix C that, on Braginsky-Meytlis theory, Qj= Too' accounts for all the

entropy production by the turbulence.

Downloaded by [Universite Laval] at 23:08 09 July 2014

28

S. 1. BRAGINSKY AND P. H. ROBERTS

5. THE INNER CORE

5.1 General Properties and Long Time Behavior
Earth’s core consists of two parts, the fluid outer core (FOC) and the solid inner core (SIC). We shall sometimes call the SIC “Earth’s nucleus”. The SIC occupies approximately 35% of the radius, 5% of the mass, and 4% of the volume of the core. It plays, however, a crucial part in convection by providing a source of light fluid at the ICB during the freezingof the FOC and the growth ofthe SIC. It is essential that this process is properly accounted for in modeling the geodynamo. It is the objective of this Section to consider the role of the SIC and to obtain the boundary conditions on the ICB necessary for the analysis of convection in the FOC.
As far as seismic waves and bodily tides are concerned, and indeed for all phenomena on time scales of seconds to days, the SIC responds as a solid elastic body. Over geological times however, it behaves as a fluid. This can be seen from the fact that its oblateness due to centrifugal forces associated with Earth’s rotation is close to that of a body in hydrostatic equilibrium. (Indeed, this statement is true for the entire Earth.) The geodynamo mechanism involves characteristic times ranging from about one year to -104yr and maybe more. The rheological properties of the SIC over these time intervals is poorly known. It is even uncertain whether it consists of a single phase. For instance, Fearn et al. (1981) argued that a significant fraction of the SIC consists of a matrix of iron dendritic crystals filled with liquid, i.e. that the SIC is in a mixed phase state. Fortunately, a detailed knowledge of the rheology of the SIC is not required for the goals of the present paper to be attained. The bulk of the SIC plays a somewhat passive role in our considerations.
Although the SIC is a body that has a complicated viscoelastic rheology, it behaves, for our purposes, much as a rigid solid on the short (convectional) time scale and behaves as a highly viscous fluid on the long (geological) time scale. It moves with
velocity V = V, + V, where V, is the slow velocity with which the basic state of the SIC
adjusts to changing conditions on the geological time scale and V, is a solid body rotation with angular velocity CkN(say):

V,=CEN x r.

Because of stresses exerted by the FOC across the ICB, this angular velocity may even change on the short (geodynamo) time scale.
Concerning CiN,we recall that throughout Section 4 we have relied on the smallness of ,E to neglect the oblateness of the reference state, and in particular the flattening of the ICB. Sincehowever, >> E,, the oblateness of the ICB is sufficientto strongly inhibit
the rotation of the SIC about any axis perpendicular to S2 = R1,.Rotation of the SIC
about the z-axis is however possible, i.e.f i N= QNlz.And ONis determined by the state of
- convection in the FOC and the nature of the interaction between the core fluid and the
SIC. For motions on the characteristic time scale (103-104)yr, the inertia of the SIC may be neglected, so that the SIC is in equilibrium under the action of the sum, S 2 ,of

GEODYNAMO CONVECTION

29

the z-components of all couples exerted by the core fluid on the SIC:

9,= O .

(5.2)

These couples consist of the couple created by magnetic forces,

(5.3)

the viscous couple and the topographic couple. We combine the latter two together as

Downloaded by [Universite Laval] at 23:08 09 July 2014

To compute 2;from (5.3),we must solve (4.6)and (4.7),which by (5.1) require that

+ 8,B R,d,B = - V x (qNVx B),

(5.5)

where qN is the magnetic diffusivity of the Nucleus. Solutions to (5.5) must be continuous at the ICB.
To compute 9;from (5.4),we require the coefficientof friction K,, but this is hard to estimate, especially because little is known about the topography of the ICB. If K,, is sufficiently large, (5.2) will require that 9;= 0. If K,, is sufficiently small, (5.2) will demand that 9;= 0. This may result in a significant change in the behavior of B in the core. For example, Braginsky (1964a) found, in a kinematic geodynamo model where the condition 2 ; = 0 was imposed, that the magnetic field was much changed; in particular, B, was small in the SIC. The Joule dissipation was also markedly increased. A similar effect was recently reported for the dynamo model of Hollerbach & Jones (1993).It was demonstrated in that paper that the axisymmetriczonal field was expelled from the interior of the “tangent cylinder”, i.e. the whole region s < R, that includes not only the SIC (Y < R,) but also the adjacent parts of the FOC to the North and South of it. In contrast, the influence of the SICwas found to be small for the model-Z dynamo of Braginsky (1989).It appears that the importance of the SIC and the tangent cylinder in the MHD of the core is still uncertain.
In addition to the solid body rotation (5.1), there must, as Earth evolves and the force balance slowly changes, be some slow relative motion, V,, of adjustment within the inner core. The magnitude of the radial velocity due to thermal
- expansion, V,, can be estimated as V, -$c1T2R, R2c1AT/3t, where AT is the
- change in T, during the time, t , R2/3d,, over which the inner core grows. Thus
- - - V , / d , EAT. Substituting c1 10-50K and AT N 100”K, we obtain Va/R2
In short, mass is added at the top of the SIC by freezing (and is perhaps sometimes removed by hot descending streams in the FOC) causing the ICB to advance (and
- - maybe occasionally retreat) with velocities a thousand times larger than the internal
relative motions of adjustment within the SIC. The velocity, d / L , with which the light material diffusesin the SIC is of the same order as V,; taking id lo-* m2s-’ and

30

S. I. B R A G I N S K Y AND P. H. R O B E R T S

- - - - - a characteristic length L, R,/3, we obtain d / L , R2z,rC5/R2 lOd3R2,where
z2 4 x lo9yr lo” s. Thus, the ICB moves on the z, and z,timescales and is not

precisely spherical. We denote its position by r = RN(t0, ,4)and the mass of the nucleus

by AN(tW).e write

Downloaded by [Universite Laval] at 23:08 09 July 2014

where R , and A,vary only on the evolutionary (geological) time scale; we consider the
derivatives of R,, and A,, with respect to t, to be negligibly small. According to the PREM model of Dziewonski & Anderson (1981), R, = 1221.5km =0.351 R , at the present time; Rz,(t, 0 , 4 ) and A Z c ( ta)re created by the convection and are of order E,. Differentiating (5.6),we obtain

+ where d, is not the material derivative but stands for a: a;. Although R,, is of order E,,
it varies on the z, time scale, so that d,R,, may be or order R,, and is therefore not
necessarily small. To extract R,, from R,, and to do likewise for other variables in
Section 6, it is convenient to introduce an average over the t, time scale. We shall denote the convective average of a quantity, Q(t,, t,) by Q. Clearly its time dependence is limited
to t, alone (dfQ=0).Also

afQ = 0.

(5.9)

It may be seen from (5.8) and (5.9) that

d,R,= R,.

(5.10)

The inertial forces associated with V, are completely negligible; the equation (3.1) of hydrostatic equilibrium applies also in the SIC. Deviations from hydrostatic equilibrium must, of course, exist in the SIC, but it does not matter to us whether they are equilibrated by elastic stresses or by the stresses associated with small shears in a large viscosity medium, or by some combination of these. For simplicity, we may adopt (2.1) for the SIC with the understanding that the viscosity of the SIC is so large that the convective velocities are negligible. This makes it possible for us to use the same governing equations (2.1)-(2.7) for all time scales and for the entire core.
Since the advection of material within the SIC is so insignificant, and the diffusion of the light constituent is so slight, both may be safely ignored:

v, =o, 1: =o.

(5.11, 5.12)

It might be imagined that the part of the radial flux, Z$,, that changes on the convective timescale might cause a layer of horizontally varying 5, to be deposited on the ICB.
- During the time z, lo4 yr, the thickness, 6,, of such a layer would be of order
- - 6, R,z,/z, 3 m. Even though the diffusivity xCis so small, such structures would be
- smoothed out very quickly, in a time of no more than 6;/d 30yr. The possibility of

GEODYNAMO CONVECTION

31

such layered structures can therefore be ignored. The composition inside the SIC is practically unchanging. It is, at any depth, the same as it was when the ICB passed through that level earlier in Earth's history, and when new material, with the composition appropriate to core conditions at that time, was deposited onto the ICB. During gradual freezing of the SIC, the concentration of admixture in the FOC gradually
increases, which implies that tain the SIC increases outwards. This stable stratification
of the SIC makes the possibility of (slow creeping) overturning in the SIC seem quite
implausible (Stacey, 1994). We shall suppose for simplicity that la is spherically
symmetric in the SIC; this symmetry could be at least partially brought about by horizontal motions in a boundary layer near the top of the SIC. In this context we may recall the suggestion that the SIC is anisotropic (Morelli et al., 1986),and may also be inhomogeneous. The magnitude of such deviations from spherical symmetry inferred from the observations appears, however, to be small; see also Dziewonski and Woodhouse (1987).
It follows from (5.11)and (5.12)that7

VNE Vr(R2-)=0, I$=I$(R,-)=O.

(5.11a, 5.12a)

We may now appeal to (2.52)and (2.54).As in (5.Q we write
+ + + + Vr(RN ) = V 2 V,,, I,5'(RN ) = I , 12,.
Since U 2 ,= d , R N ,we have

(5.13a,b)

Downloaded by [Universite Laval] at 23:08 09 July 2014

where Ap = pN - p, is the discontinuity in pa at the ICB, and
is the mass fraction of light constituent that is rejected from the solid and is added to the FOC when core fluid freezes onto the ICB. We call rFs the rejectionfactor; the sufficesFS stand for Fluid and Solid. It is determined by the form of the phase diagram of the alloy; see Appendix E.
The velocities (5.14a,b) are of order lo6 times smaller than the characteristic poloidal convectional velocity, V,, in the FOC, and when we apply (5.14a,b) as boundary conditions on the ICB, we make negligible error if we replace both of
'By our notational convention, the subscript N distinguishes values of variables at the top of the SIC in the basic state (adiabatic reference state), while ,, and not the more cumbersome ,, denotes the values of the same variables at the bottom of the FOC. An exception is made in the case of R , and R,; see (5.8) above. Since neither the ICB nor the CMB are precisely spherical, conditions (5.1la) and (5.12a)are slightly inaccurate, but similar simplifications are frequently made in this paper. The concomitant errors are negligibleto the order to which we are working, as are the errors we make when,as we shall, we set = I,, the unit vector in the radial direction.

32

S. I. BRAGINSKY A N D P. H. ROBERTS

them simply by

V,=O, on r = R,.

(5.16)

In contrast, the typical magnitude of It in the FOC is of the same order as (5.14d) and it would be incorrrect to replace (5.14d) by I:, = 0.

Downloaded by [Universite Laval] at 23:08 09 July 2014

5.2 Hear conduction in the SIC
Any definition of a basic state for the SIC is to some extent arbitrary. Unlike the FOC, nothing changes in the SIC on a fast time scale (apart from changes imposed on it by the FOC) and there is therefore no unique way of extracting a reference state for the SIC. As there is no vigorous mixing in the SIC, its temperature is determined by heat conduction, and we have no strict foundation for assuming (3.4). We may nevertheless use (3.4) to define a reference state and, because the temperature varies little within the SIC, this should differ only slightly from the actual temperature of the SIC. We believe that assumption (3.4) is adequate and more practical than alternatives. Moreover, the negative slope of the adiabat (V,Ta< 0) necessarily agrees a little better with the negative slope of the actual temperature distribution than a constant reference temperature would. It is certainly true however that S deviates from uniformity in the SIC far more strongly than it does in the FOC. It is possible, as we shall now describe, to take into account the heat sources and heat flux in
the SIC and to derive corrections, T, = T - T, and S, z S - S,, to the reference
temperature and entropy. (In this Subsection, the subscript c will be used to denote the deviation from adiabaticity created by conduction. The small amendments p c and g, to the density and gravitational field will be ignored.) If the SIC is a mixed phase region, then some small scale convection is also possible within the solid matrix. These might convect heat, and so markedly increase the effective thermal conductivity of the SIC, at the same time reducing the temperature gradient within it. Large scale circulations within the SIC are, however, strongly impeded by having to take place through a porous matrix. In any case, the diffusivity of heat greatly exceeds that of composition, and we cannot ignore heat conduction in the SIC.
We have argued above that the density and composition of the SIC can only change on the geological time scale. We therefore have

5, =0,

(5.17)

and only two thermodynamic parameters are therefore required to describe the thermal state of the SIC. By (D5)and (D13)we have

(5.18)

No significant relative movement can take place on the convective time scale of lo4yr or less, i.e. (5.1 1)holds. Equation (4.4)therefore gives

paTad,S,= - V . I T -paTaSa+ Q" + QJ,

(5.19)

GEODYNAMO CONVECTION

33

where only thermal conduction transports heat:
+ I ~ = - KTV(T, T,).

(5.20)

Equation (5.19)can be transformed into an alternative, and more convenient, form by using (D6):

(5.21)

To determine the pressure variation, p,, we should specify the SIC model more precisely. Fortunately, this complicated task can be side-stepped because the last term
- - in (5.21)is much smaller than the others and can be neglected. In order of magnitude,
pc/R2 gp, and pJpa uT,, so that the ratio of the two terms on the right-hand side of
(5.21) is

Downloaded by [Universite Laval] at 23:08 09 July 2014

(5.22)

We may therefore write S , = (c,/T,)T,,

and similarly

s, =(cp/T,)f,.

(5.21a,b).

Neglecting also the variation in cp/T, across the SIC, we obtain from (5.19) and (5.21a, b)

+ + p,c,d,T, = - V . I T - p , c P f , Q" QJ.

(5.23)

- The basic (adiabatic) temperature contrast is of order AT, 100"K, which is an

order of magnitude smaller than that across the FOC. Because there is no turbulent

transport of heat in the SIC, the deviation. T,, of the basic temperature from that of the

basic state is much greater in the SIC than in the FOC. Let us for example estimate the

- contribution made to T, by Joule heating, using p,cpT, QJ7KNw, here t K N= L ; / K ~

- is the thermal timeconstant ofthe nucleus. We take K ;

m2s-.' to be the thermal

diffusivity of the nucleus and L, = R,/n as its characteristic length scale, so obtaining

- - z K N 5 x 108yr, a very long time. The estimate QJ q N B 2 / p o L i leads to

- - T, ( ~ N / ~ ~ ) B 2 / p owphioch~,pfor B = 50G, gives T, 1°K. Comparing this with

- a typical value, T,

OK for the FOC, we see that the FOC provides an almost

isothermal environment for the SIC.
Other contributions to T, in the nucleus are Q" and ?,; see (5.23). These are an

order of magnitude greater than QJ,and are spherically symmetric and stationary

(on the t , timescale). The T, due to QJ could depend on t , but only very weakly.

- - If Joule heating has a component, Q i , varying with frequency 0 , then a time
varying temperature component is generated of order T,, Q:/p,c,o

- ( Q i / Q ' ) T c / ~ t KINf .Q i - Q J , then the ratio of the varying component, T,,, of T,
to the stationary one is very small: Tc,/Tc -(cotKN)-' 3 x low5.This means

34

S. I. BRAGINSKY AND P. H. ROBERTS

that T, in the nucleus can be considered to be stationary on the t, time scale, and (5.23) may therefore be replaced by
+ + + pacpTa= -V*I* Q” QJ, where IT= - p,c#V(T, T,). (5.24,5.25)

Solutions to (5.24)must satisfy a boundary condition on r = R,. Since it is convenient
- to match the basic adiabatic temperatures on the ICB, continuity of T implies
continuity of T,. But T,is very small in the FOC (T, l o p 3OK), so that the temperature differences over the ICE are about 4 orders of magnitude smaller than elsewhere in the SIC. In effect, the FOC provides a uniform temperature “heat bath” in which the SIC lies.And (5.24)must therefore be solved subject to the spherically-symmetric boundary condition

T, = 0.

(5.26)

Such a solution provides the thermal flux IT in the SIC. Because Q’ is relatively
small, ITis nearly spherically symmetric and depends only on t,, this despite the fact that the state on the fluid side of the ICB is neither spherically symmetric nor independent oft,. We have

(5.27)

which is nearly independent of 9, Cp and t,.

Downloaded by [Universite Laval] at 23:08 09 July 2014

6. THE CONVECTIVE STATE: INHOMOGENEOUS MODEL (ANELASTIC THEORY)
In this section we use the theory developed in Section 4 to formulate a model of core convection.
We start by summarizing the basic equations derived in Section 4. The angle brackets ( and)‘ of Section 4 will be omitted wherever feasible, as will the
superscript c from d f and a:, but it should be understood that we are now dealing
with turbulently averaged quantities. We write (4.19), (4.2), (4.29), (4.33) and (4.5)-(4.7) as

GEODYNAMO CONVECTION

where

c= -MSSc -M y c ,

is the codensity. By (4.48)and (4.49),we have

35
(6.la)

Downloaded by [Universite Laval] at 23:08 09 July 2014

where Vijkl is the total viscosity, given by (4.50a).The effective sources appearing in (6.3) and (6.4) are seen from (4.29),(4.33)and (4.37)to be

~2 = - Pat,,

+ + 0;= - pas, a: o;,

(6.10,6.11)

where

= + oT,

= T ~ -l ~ D ,

(6.12,6.13)

and

o R =Ta-'QR, aT = - T,-'V.IT, I T = -KTVTa,
Q ~ Q=" + Q~ + Q',
+ + Q' = - (Is'.V Ta Irt.Vpa=) - ga*(cPIs' ~81").

(6.12a, b,c) (6.13a)
(6.14a, b)

For brevity, we here and below replace the ITmof Section 4 by IT.Corresponding to (6.8) and (6.9),

Q" = PaVijkl(Vivj)(VkT/I) 2 0,
Also, by (4.21)and (4.22),we have

QJ = pof'/J22 0.

(6.15a, b)

15' = - paPt*v{c, IS' = - p,P'*VS,.

(6.16a, b)

' It may be noted that oT is not the rate of entropy production by conduction down the
adiabatic gradient, which is oT = - Ta-'IT-VTa= KT(T; VTa)22 0; see (4.35).It is, in fact, a combination of that term and the divergence of the entropy flux down the adiabatic gradient, i.e. V V I :=~ V.(T,-'IT); see (4.32).It therefore need not be positive and, in the geophysical context is, in fact, negative. In (6.12) we recognize two
well-known effects acting on the reference state: radioactive heating, QR,which tends to
promote convection and -V*ITwhich, by diminishing the effectivenessof QR,tends to suppress convection. In (6.13) we see sources that arise from convection alone; they cannot therefore be a primary cause of convection.
An equation governing the evolution of the codensity, C, can be obtained by multiplying (6.3)and (6.4) by -a5and - respectively, and by adding corresponding sides:

+ + p,d,C v-IC= 02 ,:a

(6.17)

36

S. I . BRAGINSKY AND P. H. ROBERTS

where

I= = - &5' - $IS'

0;= - QTOTe - sGse?

(6.18a, b)

+ + 0: = - (5,paV I").Va' - (ScpaV IS').VaS.

(6.18~)

The source OF arises because of inhomogeneities in uCand/or as.
Equations (6.1)-(6.16), together with boundary conditions on the ICB and CMB, define the inhomogeneous model. On solid boundaries, V must obey the no-slip conditions:

V(R,) = 0, V(R,) = V,, V, =fiN x r,

(6.19a,b,c)

where SL, is the angular velocity of solid-body rotation of the Nucleus (SIC). The magnetic and gravitational fields are continuous; see (2.46)and (2.47).
Conditions (2.49b),(2.33)and (2.50)give

Downloaded by [Universite Laval] at 23:08 09 July 2014

(6.20a,b)

where 1: = ZT(R,) is the heat flux down the adiabat at the CMB, and I& = Z:(R,) is the

heat flux from the core to the mantle. These are determined by conditions in the mantle,

namely by the temperature distribution and the state of convection there. They change

on the slow geological time scale, and are regarded here as being prescribed quantities.

In a broader statement of the problem, the core and mantle should be considered

together in determining the thermal history of Earth. The flux of heat from the core, I&,

is primarily determined by convection in the mantle, and in its turn that determines the

intensity of all dynamical processes in the core. These two subsystems, the core and the

mantle, are separated by the D" layer, where a rather large decrease in temperature

(about 1000OK) occurs. This decrease is possible because the thermal conductivity of

the mantle is about ten times smaller than that of the core. Each subsystem adjusts to

the other, and each evolvesin its own (but mutually coupled) way. Even the characteris-

- tic time, Z" L ~ / Kof ~the, D"-layer is much longer than the magnetic diffusion time,

- zP,of the core. If we take L, 105m as the characteristic scale of the layer and

- - - tcM

m2s- as the thermal diffusivity of the mantle, we obtain Z" 3 x lo8yr

3 x 1042,.Nevertheless, z'' is much less than the diffusivetime scale L , / K ~of the mantle

as a whole.

The conditions on the composition and temperature (entropy) at r = R, are more

complicated than (6.20). Let us consider first composition. To solve (6.3)we need an
expression for &; see (6.10).This can be obtained by conservation of light component in

the FOC. Integrating (6.3) over the FOC, we obtain

1.* i6,, I?.,? tcpad~+
atL,

tcpaV.dA +

1 ' t . d ~= -

tapadv-

(6.21)

We average this integral balance over the convective time scale. The first term on the left-hand side disappears identically by (5.9), and the second term vanishes because V, is zero on the CMB and ICB by (6.19). Because of (6.20a), the third term is

GEODY NA M O CONVECTION

31

integrated only over A,, and to perform this integration we apply the boundary conditions (5.14c,d) on

+ Ip(R2+) = 1: lzc,

(6.22)

namely

' 5 = PNt2NR2, '$c= PN52NatR2c'

(6.22a, b)

Noting that the convectional average of is zero and that I , is independent of 0 and 4,
we see from (6.21)that

*

A2

5 4 =-

I:,

or

. 4
t4=-52NPNR2,

-A12

A12

(6.23a, b)

where A 1 2= jy ,,padVis the mass of the FOC and A , = 4nR; is the area of the ICB. It
follows from (6.10)and (6.22)that

where

CT: = 05,

(6.24)

(6.24a,b)

Downloaded by [Universite Laval] at 23:08 09 July 2014

(6.25)
is a non-dimensional function that describes the form of the density distribution in the FOC. It was here convenient to introduce a mean density, po; later we shall need a mean temperature, To,also. These are defined by

Po = (Pa>',

To = (To>",

where the volumetric average (Q)' of a quantity Q is given by

(6.25a,b)

(Q)'=-J 1

QdV.

"y-12 Y - , 2

(6.26)

Condition (6.23)can be obtained more easily if we recall that the total mass of light constituent,

and the total mass,

(6.27a) (6.27b)

Downloaded by [Universite Laval] at 23:08 09 July 2014

38

S. I. BRAGINSKY AND P. H. ROBERTS

+ of the core are constant. Operating with 8: therefore gives dpA5f [ a A 1 2 t a k 1+2 in, = o or, since <kI=2- k,,

and, substituting

we obtain (6.2313).It may be noted that, although Z?(R, +) varies on the t, time scale, of does not, because o: = - pata; see (6.10).The slowly varying surface flux corre-
02 sponding to of is If;see (5.14~)I.n essence, the volumetric source of 5, arises from c, a surface source of on the ICB, and this is the reason we have introduced the suffix
2 on ofin (6.24)and (6.24a).The light fluid, released when the heavy constituent of core fluid freezes onto the ICB, rises and distributes itself homogeneously throughout the volume Y,,, so producing an effective sink within it represented by of = CJ$< 0. The thermal parallel is a fluid core, heated from below, and cooled homogeneously within, both factors being favorable for convective instability. We could also easily allow for sources at the top of the layer. For example, a flux of iron into the core from the mantle would introduce a contribution, a!, to ,:rc but we do not consider such effects in this paper.
Consider next entropy. Condition (2.61) of entropy conservation may be written as

where (2.52) has been used and VN has been neglected. Using (4.32) and
U, = R , + d,RZowe separate

+ Z 3 R 2+) = I ; I;,,

(6.29)

into

In (6.29a)and (6.29b),hNis the heat released at the ICB due to the freezing of the ICB. In view of the presence of the light element, this differs from the latent heat of melting, h,, as it is usually defined:
see Appendix D. We could also use the continuity condition (2.57)for the total heat flux (4.32b)instead of (2.61).The results of doing so are identical because of (6.30a,b) and the equation (2.54)expressing conservation of admixture.
The arguments leading to (6.24) and (6.24a) show that the effective source, a:,
of 5, arises from the need to conserve light material, which requires a nonzero

GEODYNAMO CONVECTION

39

flux, 15, at the ICB. A similar, but more complex, connection can, and must now,

s, be established between the volumetric and surficial sources of S,. The derivation of
(6.23b)and (6.24)is a good preparation for the corresponding derivation of and 0:. Integrate (6.4) over V12and average over t,. As before, the first term on the

left-hand side vanishes by (5.9).We therefore obtain

z)“, + + 1
p o i a = - -(ZSA, - ZSA,) (0,”

(6.31)

v12

where

(6.31a)

Downloaded by [Universite Laval] at 23:08 09 July 2014

(6.3lb)

cf. (6.29a).The convective average of the flux (6.29b)is zero, and it therefore does not appear in (6.31).The expressions (6.29b)and (6.31a,b) provide boundary conditions for
s,. equation (6.4). We now substitute expression (6.31)into (6.1l), thereby eliminating Some insight is gained by dividing the resulting “effective”entropy source, 0;into three parts:

+ + 0;= 0; 0; 0SZ,

(6.32)

where by (6.1l), (6.29b)and (6.31a,b)

(6.32a)

(6.32b)

+ +Z)”. p ( @S12 -- s 0;- 0,”

(6.32~)

The second forms in (6.32a,b) were obtained from (6.31a,b). The three volumetric sources appearing in (6.32)arise respectively from the CMB,
the ICB and the bulk of the FOC. It may be particularly noted that, according to (6.32a),only the excess of I$ over the adiabatic heat flux, IT, enters 0;.If the flux of heat
down the adiabat is too great, that Z$ < IT. Then 0;<0, and the situation resembles
that arising in thermal convection when a layer is heated from above and cooled from
within. Compositional buoyancy arising from the light fluid source 05 will, if large
enough, drive convection even if 0; is negative, although the magnitude of C will be reduced (Loper, 1978).The heat flux from the ICB is associated with a negative 0; and
a positive I s ; see (6.31b).It promotes convection. The terms 07 and 0; do not vary on
the t, time scale. Consider next os, given by (6.32~)T. he two contributions, g R and oT_,to 0: given by
(6.12a,b) act in opposite directions, oR to assist convection and a? to oppose it, but a net effect can arise only through the radial inhomogeneity of their sum. To see this,

Downloaded by [Universite Laval] at 23:08 09 July 2014

40

S. 1. BRAGINSKY AND P. H. ROBERTS

note that, if either a,”or a,”were proportional to pa, it would make no contribution to aT2.For this reason, we expect that oRwill play a small part: convective mixing makes QR/pa uniform and, if the core were uniform in temperature, aRwould be proportional to pa. Similarly, bearing in mind that 9, is approximately proportional to r, we see that,
in a uniform core ( p = l), T,aT would, like QR, be constant by (3.7b) and (6.12b,c).
Again, a: would not contribute to a;,. Only through the inhomogeneity of T , can aR
and a? contribute terms to as2,but even then these terms are only of order E,. The term R , appears in (5.14c), (6.23b), (6.31b) and (6.32b); it should now be
evaluated. The ICB is a surface in phase equilibrium, and must therefore be at the melting temperature, corresponding to the liquidus ‘curve’ (really a surface in
+ + p<T - space), Tm(p0, ,for the pressure p =pa p , and composition 5 = ta 5, on the
fluid side of the ICB:

The corresponding composition for the SIC is given by the solidus ‘curve’.Substituting
+ + + R, = R , R2, into (6.33) and writing T = T, ?;., p = p , p , and 5 = ta+ &, we
obtain in the zeroth approximation

We consider (6.34)to be the definition of R,. Differentiating (6.34)with respect to f a , we find that
(6.35)
where V,T, = (V,Ta), and V,p, = (VrpJ2.Differentiating Ta= T(p,,S,, t a )with respect
to t,, we obtain

Using this to eliminate T2from (6.35)and recognizing that V,p, = - p 2 g 2 ,we see that

R2IR2 = r 2 p P 2 - r2sSn - r 2 5 t u 2 where, by (D14),(D18) and (3.7e).

(6.37)

(6.37d)
Here y2 is the Griineisen parameter (3.7f) evaluated at r = R,; the dimensionless parameter Am, is proportional to the differencebetween the melting point gradient and

GEODYNAMO CONVECTION

41

the adiabatic gradient at r = R,. We note that d T a / a p , = cls/p2, according to (D18). Working to the next approximation, we obtain from (6.33) a form for R,, similar to (6.37):

4R2cIR2 = r2pPc - r2ssc - r 2 &

(6.38)

The hydrostatic pressure, p,, changes because of mass redistribution. That due to

processes outside the core was considered by Gubbins (1983),who found it too small to

affect significantly either convection in the bulk of the FOC or freezing of the SIC. As

mentioned in Section 1, we exclude from consideration all such processes, except the

one responsible for the heat flux 1; is emanating from the core. The change in p , due to

the redistribution of mass within the core is small, being proportional to the density

drop, Ap, at the ICB; in fact, pZ N g2R2Ap8The term r Z p p 2in (6.37)is therefore of order

Ap/po-6 x we may neglect

timessmallerthan r Z p p 2A. ccording to

theleft-handsideof(6.37).SinceAp/po-6 x
(6.22), t ois proportional to R2,and (6.37) can

be

written in the simple form

(6.39)

Here A2 is a new dimensionless parameter defined by

Downloaded by [Universite Laval] at 23:08 09 July 2014

where by (6.37~w) e have

(6.41)

If pc in (6.38)is similarly negligible, we also have

d,Sc(R,) =

-

c

A

4R2,
2R,'

(6.42)

An argument that leads to the estimate A, z 0.05 is presented in Appendix E. It should

be stressed that this value is very uncertain.

Equation (6.42)shows that too strong a heating of the core results in a negative d,R2,

and hence in reductions in the fluxes 15' and I:, according to (5.14~a)nd (6.29a) or

(6.31b);similarly, too weak a heating leads to growth of these fluxes. This favors the

(z)" establishment of some 'average' level in the intensity of convection. Especially the term

~7: -

in (6.40~p) roduces such a stabilizing effect: if convection is too intense,

dissipation rises and enhances S,, resulting in a diminishment in the sources of

convection. This stabilizing effect was noted by Braginsky (1964b).

8See Appendix B. It should perhaps be noted that p 2 is the Eulerian derivative of pa at r = R , and not the rate of change of p 2 following the motion of the ICB, which is of order g z R 2 p 2 ,i.e. is much larger.

Downloaded by [Universite Laval] at 23:08 09 July 2014

42

S. I. BRAGINSKY AND P.H. ROBERTS

Through (6.39)we can write cs in a third form which is particularly convenient if Q"
is known. While it is true that cooling of the core by the heat Zk lost to the mantle is the primary cause of both thermal and gravitational stirring of the core, the value of ZG is
not directly observable and is poorly known. It can be obtained only indirectly through investigations of the thermal evolution of the coupled core-mantle system. The age, t,, of the inner core is comparatively better known: on the one hand, t , cannot be very small because the birth and growth of the inner core would then be evident from paleomagnetic data; on the other hand, the creation of the inner core could easily be missed if it occurred in the remote geological past, for which paleomagnetic information is comparatively scant. We shall estimate t , to be 4 x lo9yr, but we recognize that this may be too large by a factor of order 2. If we assume that the age of the core is t , and
-that it has been growing at an approximately uniform rate ever since, we obtain
.k2 A?,/t, and an estimate of R , which, though rough, is probably more reliable
than any estimate of I&. Moreover, it may in principle be made more precise by detailed investigation of the evolution of Earth. It seems therefore reasonable to use R , as the main parameter determining the amplitude of the power source fueling the geodynamo, in the case of dominating compositional convection, when the thermal source can be neglected. In this way, if we still wish to derive S,, we obtain from (6.11)and (6.39)a new expression for cs,namely

-+ G: = pacpA2RJ, R , -I- ez c,".

(6.43)

To use this, we need ,:c which requires knowledge of the magnitude of Q". The expression for the boundary condition on l ~ ' = I s ' ( R , )in terms of R, is
a complicated matter. This is because Z4,can be linked to S, only through the entropy balance (6.11)and (6.31). This gives

(6.44)

The flux Is is given by (6.29a)in terms of R2 and, provided we know Q", we can evaluate
the final term in (6.44) by averaging (6.43).
In summary, the intensity of convection is determined by (in addition to the
physical properties of the core) just two parameters: R , and ZL -IT, the former
of which can be roughly estimated. The radioactive heating, Q", can be expressed in terms of the other two parameters, and (supposing that R , 2 0 ) Q R Y - ,5 All$. According to (6.32c), Q" is relatively ineffective as a source of entropy but it does influence the size of IG - IT.

7. DYNAMO ENERGETICS AND EFFICIENCY
7.1 Energetics of a Heat Engine There are significant points of difference between the geodynamo system and a common heat engine. To understand the former better, let us start from a heat engine operating steadily in the well-known Carnot cycle. In this classical device, the heat

GEODYNAMO CONVECTION

43

input 2 + ,is provided at a higher temperature, T , ,than the temperature, T _,at which heat (2-,say) is extracted. Averaged over the working cycle, the rate at which the machine does useful work is

d = 2 +- 2 - .

(7.1)

In the absence of any losses due to imperfections in the engine, the entropy input and output are equal in the steady state:

The efficiency of a perfect engine is therefore

Downloaded by [Universite Laval] at 23:08 09 July 2014

where

T

qc=

1

I-
--

T+

(7.3b)

is the “Carnot Efficiency”. Applying similar ideas to Earth’s core, considered as a system that is stationary on
average, Backus (1975)and Hewitt et al. (1975)noted significant differences. First, it is no longer clear what should be classed as the “useful work” done by the engine. They definedit, as we shall, to be the rate, 9Jo,f production of large-scale magnetic energy by large-scale fluid motions. This energy is ohmically degraded into heat. The engine must make good not only this energy loss but also the energy, uselessly dissipated at the rate 2F(say),by internal friction. It follows that

d =gD,

(7.4a)

where

p = 9J+ 2 F

(7.4b)

is the total dissipation. Second, both the Joule and frictional heat reappear within the fluid; they must be regarded as part of the energy source driving the engine. The energy balance is therefore

2!++2?D=2- +d, or, by (7.4a),
2+= 2 - , The entropy balance is expressed by

(7.5) (7.5a)

44

S. I. BRAGINSKY AND P. H . ROBERTS

where T, is the effective temperature at which 2 , is produced. This is related to the effectivetemperatures, TJand TF, at which 2?Jand 2?F are dissipated by

-2_, -- 2?J +-2-?F.
TD TJ TF
According to (7.5a),we may rewrite (7.6)as

(7.6a)

(7.6b)

Downloaded by [Universite Laval] at 23:08 09 July 2014

It may be seen from (7.4a) and (7.6) that heat is needed not to maintain the energy balance [note particularly that (7.5a)does not contain 2?J or SFat all] but to preserve the entropy balance.
Regarding the magnetic energy production t o be the only work that the engine does usefully, we may write the dynamo efficiency in the form

(7.7a)

where the “frictional factor”, qF gives the fraction of the energy dissipation that is “useful dissipation”:

(7.7b)

while the factor

(7.7c)

represents an “ideal efficiency” which cannot be exceeded, even if there is no source of internal friction. Since T- 5 To I T,, it follows from (7.7~t)hat

ilC 5 ‘I1 5 q S ,

where

qe =- T+- 1; T-

(7.8a,b)

see Backus (1975). This example demonstrates clearly that the necessity of preserving the entropy
balance limits the efficiency of the device in producing magnetic power. It suggests that, in analogy with the oft encountered phrase ‘the available energy’, it is useful to call a quantity such as “the available dissipation”, of which only the fraction qF goes into useful work, S J ,while the remainder, 2IF, represents work wasted by the engine. The crucial importance of “the entropy balance in relating magnetic field to the energy sources” was stressed by Gubbins et al. (1979).

GEODYNAMO CONVECTION

45

7.2 Energetics of the Inhomogeneous Model

Let us now turn to the core. This differsfrom the examplejust considered in an essential way: it is an evolving system that receives its energy from that evolution, namely through its cooling and the resulting gravitational settling. We shall completely confine this non-stationarity to the reference state, and shall consider the superimposed convection as cyclic,i.e. one that, when averaged over a time z, equal to the period of the convection cycle, varies only on the z, time scale. To provide a suitable definition of geodynamo efficiency,we must consider the balances of energy and entropy. The total kinetic and magnetic energies associated with the macroscale are

&B=-

[ B'dV.

2Po Y ,

(7.9,7.10)

The viscous and Joule dissipations areg

p;'q(V x B)'dV, (7.11,7.12)
Y1
and the rate of working of the fluid on the large scale magnetic field is

(7.13)

By multiplying (6.1)scalarly by p,V, applying (6.2)and integrating over Y12w, e obtain the macroscale kinetic energy balance

Downloaded by [Universite Laval] at 23:08 09 July 2014

which a, replaces 8; and dcis the rate of working of the buoyancy force (the
Archimedean power):

(7.15)

By multiplying (6.6) scalarly by &'B an integrating over all space, we obtain the macroscale magnetic energy balance:

a,&B = d~- 2 ~ .

(7.16)

'Strictly, (7.12) is correct only if the mantle is an insulator. In the more realistic case of an electrically conducting mantle, an additional term, 9JMa,rises. This is the Joule dissipation associated with electric currents that either leak from core to mantle or that are induced in the mantle by time varying fields in the core. For brevity,we shall not writedown an explicitexpression for 2JMb,ut shall consider that it is implicitly
+ included in the viscousterm, 2', a procedure that is possible since 9'and 9'M never appear separately in the
energy balance, but only as a sum 1' 2'M.

46

S. I. BRAGINSKY AND P.H. ROBERTS

Averaging this over the convection cycle, we obtain
2 B = 2J
It follows from (7.14)and (7.16)that
a,(&" + € B ) = d C - L?J - L?v,

and after averaging over the convective cycle we obtain

(7.16a) (7.17)

This is the analogue of (7.4a,b), in which the quantities shown are in fact also averages over the working cycle. A result similar to (7.17a)holds for the microscale. Recalling results from Appendix C , we see that

- -.
9'zi 2J,

(7.18)

where

Downloaded by [Universite Laval] at 23:08 09 July 2014

are respectively the rate of working of the buoyancy force on the microscale motions and the Joule dissipation of the microscale currents.
An energy balance for the core similar to (7.5)and (7.5a)can be obtained by applying (6.3),(6.4)and (7.17a). We multiply (6.3) and (6.4) by pa and Ta respectively, sum the two resulting expressions, and simplify using

+ paPaV.Vtc+ TaPaV.'Sc = V . C p a ( ~ a S c TaSc)VI + PaCV*ga,

which follows from (6.la),(6.2)and (3.7b,c);the last term here is the volumetric rate of working of the buoyancy force. In this way we obtain, using (6.10)-(6.14),

+ + + + + + P a b a a i S c 'adisc) V.C(Pa5c TaSJpaV par'' Tars' IT]
+ + + + = - pa(pa,8, TaSa) Q" Q" QJ-paCV.ga.

(7.19)

Integrate this over Y,,, take the convective average, and use (6.19a,b,c), (6.20a,b), (6.22a),(6.31a,b) and (7.17a)to obtain

+ + + + (pata TaSa)padV= - I L A , (1: If;)Az- 2; 2T2,
I13
where [see (5.14~a)nd (6.30a)l Zi is the flux of latent heat from the SIC,

(7.20) (7.21)

Downloaded by [Universite Laval] at 23:08 09 July 2014

GEODYNAMO CONVECTION

41

(7,20a,b,c)

We obtain an expression for the heat balance for the SIC from (5.19):

(7.22)

Adding (7.20)and (7.22), we obtain the heat balance for the entire core:
n
J + (pata T,S,)p,dV = 2R+ 2L- j L , Yl
where 2L= I$Az is due to the latent heat of SIC crystallization and

(7.23)

(7.23a,b, c)

Equation (7.23)can be obtained directly by integrating (7.19) over the entire core; the
term 2Lthen arises as an internal heat source. From now onwards in this Section and
throughout Section 8, we shall consider exclusively convectively averaged balances,
and without risk of ambiguity we may (and shall) omit the overbar on 2JH, D , etc.
Like the earlier simple example (7.5a), the balance law (7.23) does not involve the Joule dissipation, but there is a crucial difference between (7.5a) and (7.23):the heat
engines most commonly considered are on average in a steady state but, because
Earth’s core is evolving,(7.23)-unlike (7.5a)-involves time derivatives, namely S, and
(,, The geodynamo is fed not merely by the heat source QRbut also by the changing
state of the core. The left-hand side of (7.23)is in fact a potent source of energy for the
geodynamo. The heat balance (7.23) can be transformed into a more familiar statement
expressing internal energy balance. The time derivative of the internal energy of the
basic state is

Substituting the expression for ifi implied by (2.9), and introducing the enthalpy
+ = p/p, we may write this as

48

S. I. BRAGINSKY AND P. H. ROBERTS

Downloaded by [Universite Laval] at 23:08 09 July 2014

where we have applied the divergence theorem; V denotes the radial component of V.
Conservation of mass at the ICB requires that p,(R, - V,) = p2(d2- V 2 )and, since V, = d , , we may combine (7.24b)and (7.24~t)o give

jvl + + + &: = (patO T,S,)p,dV d: d;-SL.

(7.25)

We have here ignored V , in comparison with R 2 (see Section 5 ) and have recognized that h,p,A,R, = 2'. We have also defined

{ lVl dga= V,.Vp,dV = p,V;g,dV, YI

222; = - fAl p,V;dA.

(7.25a.b)

The two expressions for dz are equal in virtue of the equation (3.1) of hydrostatic equilibrium. Equations (3.1) and (7.25a) show that d: is the work done by the gravitational field through the geologically averaged motion of the core.
Using (7.25)we may now write (7.23)in the form

+ + &: = ..I: d7 SR- L2&.

(7.26)

This may be compared term by term with Eqn. (5) of Gubbins et al.(1979).A significant difference is that we have separated the slow evolutionary effects of the evolving background from short time scale processes, whereas only one time scale is explicitly included in their analysis. The changes occurring on the convective time scale were filtered out by them from those occurring on the geological time scale in their subsequent discussion. It may be noted that the first two terms in (7.26) may be combined into one term describing the rate of working of the pressure on the fluid:

(7.2%)

To derive the entropy balance analogous to (7.6),we integrate (6.4)over V1and take the convective average of the result, so obtaining

+ + SR
SapadV =-

-2

-+ ---,

TR

TD T2

(7.27)

where

GEODYNAMO CONVECTION

49
(7.29a, b) (7.30a, b)

Downloaded by [Universite Laval] at 23:08 09 July 2014

In the derivation of(7.27)we have substituted a? = aT- V*(Ta-'ITa)nd have used the boundary conditions (6.3la,b) along with (7.23b,c) and the expression

aT= IT.VTa-' = KT(VTa/Ta)2,

(7.31a)

which followsfrom (6.12b,c).BecauseVT,is not necessarily negligibly small in the SIC, the slightly different expression

(iT =K ~ ( v T ) ~ / T , I ,

(7.31b)

where T = Ta+ T,, should be used in preference to (7.31a)in the SIC. In deriving (7.28) we have used h, = h, - p 2 5 2 Nwhere < 2 N = ta- tN;see (6.30a).
The heat balance (7.23) can be given a different form in which the heat flux to the
mantle, 24,,is expressed as a sum of sources arising from gravitational differentiation,
dc,the decreasing entropy of the core, P,the radiogenic heating, bR,and the growth of
the nucleus, 2,:

24, = dt + P +2 R + p.

(7.32)

Here

(7.33)

(7.34)

To derive (7.32)from (7.23),we have used (6.28a)and (7.28). By multiplying the entropy balance equation (7.27)by T Iand subtracting the result
from (7.32),we may eliminate the unknown 24,and obtain an expression for bD.This quantity, representing the 'available dissipation' is proportional to the sum of the compositional and thermal terms:

( T , / T ~=)dc~ +~ LP,

(7.35)

Downloaded by [Universite Laval] at 23:08 09 July 2014

50
where

S. I. BRAGINSKY A N D P. H. ROBERTS

(7.36)

J * % 9' - - (T,- Tl)S,p,dV. 1-

(7.37)

All terms except 2?Ron the right-hand side of (7.32) and (7.36) may be estimated by
- writing k 2= M 2 / t 2and by assuming a reasonable value for t,. Comparison of (7.34)
and (7.37) shows that 2?;/2' ATa/T,. The effect of core cooling is therefore more
significant in the heat balance (7.32) than in the dissipation (7.35).
Expression (7.36)may be rewritten as

(7.38)

where the following temperature differences have been introduced:

ATol = To- T I ,

ATz1= T2- T I ,

(7.39a' b)

(7.39c,d)

where ZTis given by (7.31)and 2: = AIZT.
By (6.31a), the total rate, 21D, of energy dissipation associated with core convection is the sum of contributions 2" from viscosity, 3Jfrom magnetic field
generation, and 9 from turbulence, the last of which is mainly due to magnetic
friction. It is also the total rate of working of the buoyancy forces, averaged over
the convection cycle. This is the sum oi the averaged contributions, dcand P,
from macroscale and microscale, respectively:

It does not contain the term TCT,corresponding to the entropy sources (7.31)appearing in (7.27),a term independent of the convection; only the Q J part of QDis used to power the geodynamo while 2"and 2' represent energy wasted in useless 'friction'.
We may define dynamo efficiency to be
(7.41)
which relates the total 'effective' energy supply (7.32) to the useful work d B( = 2'), as given by (7.16a). It should be stressed that qDis defined using the most significant quantity, QL, the total heat flux from core to mantle. It may be compared with 3-in the example of Subsection 7.1, which is there (but not here) equal to 2 + .We may

GEODYNAMO CONVECTION

51

rewrite (7.41) as

Downloaded by [Universite Laval] at 23:08 09 July 2014

where the frictional factor is

(7.43)

This represents the attenuation in efficiency arising from friction, both macroscopic (because of viscosity) and microscopic (because of turbulence). By (7.32) and (7.39, the ‘ideal geophysical efficiency’, analogous to the ideal efficiency of Subsection 7.1, is

2?D TD d t + i 2 H ‘Ic=2-=&-. TI d 5 + i 2 s + 2 R + 2 N ‘

(7.44)

This expresses the way that the efficiency depends on which of the energy sources dominates. It is difficult to evaluate gc accurately at present. If compositional
- convection dominates thermal convection then g c will be close to unity (Braginsky,
1964b) but in the reverse case g, AT,/T,. More generally ATJT, 5 g c 5 1. Even if the molecular diffusivity is negligible and compositional convection dominates, the total efficiency, qD, cannot be close to unity, because the turbulent
-4 losses, 2,are not small. Perhaps qF is a reasonable guestimate.
- The relative importance of the compositional and thermal contributions to d D
is assessed in Section 8, and it is concluded that d5 g H .We infer that, in all probability, both driving mechanisms are significant in Earth’s core but that this cannot be established with certainty until the values of key parameters are known with greater precision.
In all our calculations of the sources of convection, such as (6.22) and (6.31b), we considered R, to be a prescribed quantity. Speaking more physically, it would be natural to take the heat flux, 2&,as the prescribed quantity, determined by the way that the mantle extracts heat from the core. This flux is, however, poorly known at present. It is related to R, by the condition (7.32) of heat balance, in which the terms d 5 ,2’ and 2?Nare proportional to R,, according to (7.28), (7.33),(7.34)and (6.39).We shall write their sum as

+ d 5+2‘ .L2N= 2?2.(3R,t,0/R,),

(7.45)

where 2. is a convenient constant with which to measure power and, in anticipation of Section 8, we have arbitrarily introduced a nominal magnitude, t,,, for the age of the SIC. After this has been selected (e.g. t,, = 4 x lo9yr), 2* becomes unique; it is a very convenient parameter with which to assess the importance of the terms on the left of (7.45). We may rewrite (7.32)as

R. = >dR= 2 M i q - 2 R.-R ,

2 - dt,

2. 3t,,’

(7.46)

52

S. I . BRAGINSKY AND P. H. ROBERTS

a differential equation determining R,(t,) from 2; - d R . We do not attempt to solve this; we simply assume that $”2 is constant and therefore replace R, by R2/3t,, so that

t& = (2; - P ) / d * .

(7.47)

The considerable uncertainty in t , is matched by a like uncertainty in L2& - L2R. The
quantities d,, 9; and L2R are related by (7.46), so that the thermal input into the
core is determined by only two independent parameters. Numerical values in this relation are considered in Section 8. While L2& - 3!Rdecides the thermal balance and the rate of cooling of the core, as (7.47)illustrates, it is 2; - 9;that determines how strongly the thermal sources drive core convection.
In conclusion, we reiterate that all forms of core convection are essentially due to thermal effects, both
(I) directly, through the thermal codensity, assc,and (2) indirectly, through the general cooling of the core and the concomitant
growth of the nucleus by freezing, thus producing the computational codensity a5tC.

Downloaded by [Universite Laval] at 23:08 09 July 2014

8. THE CONVECTIVE STATE: HOMOGENEOUS MODEL (MODIFIED BOUSSINESQTHEORY)
8.1 Basis of the Homogeneous Model
In Section6 we constructed a rather general model of core convection and the geodynamo. Because the values of key parameters in Earth’s core are so uncertain, this “inhomogeneous model” is perhaps too sophisticated for use in numerical geodynamo calculations. In this section, a simpler, and perhaps even simplistic, model is developed that is hopefully of some practical utility. It may also be the simplest possible model that retains all the main features of the geodynamo
- - ’, - mechanism. Three small parameters were introduced in Section 3: E, 10- E, 2 x and cc lop8. The smallest of these determines how far convection causes the configuration of the core to differ from the basic reference state; the smallness of E, was exploited in Sections4 and 5. The parameters ,E and E, measure inhomogeneities of the reference state. The asphericity of that state created by centrifugal forces is of order E,. Asphericity has a very small effect on the convective motions on the time scale of hundreds of years and longer, and we will continue to neglect it. The parameter E, measures the radial gradients in quantities
- such as p , and p, arising from the gravitational compression of the core. It is not
very small (6, O.l), but in this section we exploit its supposed smallness in order to simplify the model introduced in Section 6. In other words, we develop a “homogeneous model” of core convection. More precisely, since small variations in density are essential in order to retain the buoyancy forces driving the geodynamo, we construct a Boussinesq model of core convection.

Downloaded by [Universite Laval] at 23:08 09 July 2014

GEODYNAMO CONVECTION

53

In non-dimensional units, the governing system of equations of our model is

<(d,V - F")= - VP- Cr -1, x V +FB,

(8.1)

F"= (v/q0)V2V, FB= J x B = B-VB-$VB2,
c=-x-Y,

(8.la, b) (8.1~)

v - v= 0,
+ d,X V.1' = ox, where I' = - b-VX,
+ d,Y V.1' = o', where I' = - b . V Y ,

(8.2) (8.3,8.3a) (8.4,8.4a)

v2uc= 3pc,
d,B = B-VV + V2B,

V.B = 0.

(8.7)

As for any Boussinesq model, the essence of (8.1)-(8.7) is that all basic variables for which it is meaningful to do so are assumed to be constant. We have replaced pa by
po = Al2/^Y(li.2e.we have taken p = 1)and have replaced T, by Toeverywhere except
in places where the inhomogeneity of T, enters the theory directly, as in the expression for ITand in expression (8.27a)below for 0,'. The coefficients,a5and as,are defined on the reference state and are no longer functions of r; they are constants. We used Ap as a surrogate for a<in the following way. The density discontinuity is the sum of Asp,the change in the density of pure iron on solidification, and Acpthe change in density due to the discontinuity in composition at the ICB; see (D48). We suppose (see Appendix E) that Asp is negligible. Then (D48c) gives

a5= APIPN52N.

(8.8)

A representative average value, a: and as was chosen by integrating relation (3.7b) approximately using (8.42) below. This led to a value of as intermediate between as and a;:

(8.9)

We accept the assumed constancy of the coefficients as providing a useful but somewhat crude approximation. It now follows that we can write
aed,(, = - d,Cr, where C5= - d~&, (8.10,8.10a)

a$,SC = - d,CS, where Cs= - a:Sc.

(8.11,8.11a)

Neither of these simplifications is precisely correct, the second being more in error than the first since as varies by about 30% across the core whereas a5 changes perhaps by only a few percent. The variables X and Y are defined by

54

S. I . BRAGINSKY A N D P. H. ROBERTS

To make the system (8.1)-(8.7)non-dimensional, we have introduced a characteristic magnetic field and corresponding Alfven velocity by

V , = (2sZq0)1'2, B, = (popo)' 'V*.

(8.14a, b)

We have then taken L, = R , as unit of length, f, = R f / v o as unit of time and V , = L J t , = qO/R,as unit of velocity, B, = B , as unit of field, J, = B,/poR1 as unit of electric current density, and poC, as unit of density perturbation (p,), where C, = 222V,/~/,= V,2/g,Rl is the unit of codensity; g, = 10.68msf2. Taking qo = 2 m 2 s f 3and po = 10.9 x 103kgm-3,we find that V , = 1.71 x lO-'ms-', B , = 2 0 G , tI=6.055x lo'* s=1.92x 105yr, V1=5.74x 10-7ms-' and C,=O.785x lo-". Note that V , >> V,. Energy densities per unit mass are measured in units of V i ;this is also the unit of P and U,. Power density is measured in units of Q, where

(8.14~)

This unit also appears in a combination that is often used in the nondimensionalization:

(8.14d)

ITis measured in units of Q,R,. The fluxes are made dimensionless in a similar way:

(8.15,8.16)

Downloaded by [Universite Laval] at 23:08 09 July 2014

(8.17,8.18)

Comparing (8.1)-(8.7) with (6.1)-(6.7),we see that the dimensionless fluxes and sources
< labeled with superscripts X and Y correspond simply and directly with the correspond-
ing variables labeled with superscripts and S. This correspondence between Sections 6 and 8 will arise many times below.
The "Ekman number" used in (8.1) is based on magnetic diffusivity rather than viscosity:

Ef: = qo/2QRf.

(8.19)

It is very small (Ef: %

and the more usual (viscous) Ekman number, v0/222R:, is

even smaller. The Ekman layers at r = R and r = R, can be described using the simple

isotropic expression (4.9)for the viscous force, with appropriate molecular viscosities,

v, and v,. A turbulent viscosity should be used to describe internal shear layers inside

the main volume of the core (seeAppendix C).In the bulk of the core, F"is insignificant.

The dimensionless turbulent diffusivity tensor,

li = 2'/qo,

(8.14e)

GEODYNAMO CONVECTION

55

Downloaded by [Universite Laval] at 23:08 09 July 2014

is poorly known at present. It is hoped that future studies of the theory of local turbulence will eventually rectify this. At the present time it is necessary to make assumptions that, though unsubstantiated, are plausible. Numerical dynamo
integrations with different choices of 6 will hopefully, when compared with the
observed geophysical data, provide information about both fi and local turbulence
in the core. We shall find that consequences of the theory rely particularly heavily on five poorly
known parameters: the rate of evolution of the core, expressed through the time scale t , = V , / V ,, the heat released at the ICB during freezing of the formation of the SIC, h,, the heat flux, d&,from the core into the mantle and the parameters cd(,, and A,; see (8.8)and (6.40).To expose the effects of the uncertainties more clearly, we introduce four “nominal values”

t,, = 4 x 109yr, h,, = lo6J kg-I, A p , = 0.6 x lo3kgm-3, A,, = 0.05. (8.20a,b, c, d)

The value of A p , given in (8.20~)is the seismically inferred value of Dziewonski &

Anderson (1981). If Asp and A,p were comparable, we would have to assume smaller

Ap. For h, in (8.20b)we have taken the value of h, given in Appendix E; it seems that

this estimate may be uncertain by a factor of 2 either way. Our value (8.20a)of t z ois

comparable with the age of Earth. We cannot, however, rule out the possibility that the

SIC is significantly younger than this, and the ratio t,/rzo through which we express our

uncertainty might well be 0.5 rather than our preferred value of 1. The parameter A, is

the worst determined of all. The value shown in (8.20d) is defended in Appendix E.

Taken with the estimate (8.20a)of t,,, it implies a geophysically acceptable estimate of

the core cooling rate; see (8.41a) below.

Whenever t,, h,, Ap and A2 arise in the theory, we give them the values I,,,

h,,, Ap, and A2, shown in (8.20a,b,c,d), but we also include ratios t,/t,,, h,/h,,,

AplAp, and A,/A2,, so that the cause of uncertainty can be readily identified. We refer

to a unit value for any ratio as its “preferred value” (by which we mean, of course,

only “preferred by us here”).

The heat flux from the core, 2&, is replaced in this section by the Nusselt

number, Nu=A?&/d;, where 2; is the heat flux along the adiabat at the CMB,

which can be estimated easily using numerical values listed in Appendix E where
- - it was found that K T 40 W m - ‘ OK-’ and VrTa - 0.89 “Kkm-I, so that IT =
- - KTVrTa 0.0356 W rn-‘ and ST = 4nR;IT = 5.42 x 10l2W. Nevertheless, N u ,

though very significant for the present theory, is largely unknown. Its uncertainty

is not related to the uncertainty of t,, which depends on 4&-dR. Even if the

estimate t , = 4 x lo9 yr is accepted, the uncertainty in the radioactive power supply,

2R,translates this into a corresponding uncertainty in Nu. It should be emphasized

that in the core, unlike the laboratory, a small value of N u - 1 does not mean that

convection is weak. This is because conduction of heat is driven by a temperature

contrast across the core of order T2- TI = 1300”K, while convection of heat,

- through vigorous turbulent motions, is driven by temperature differences of order

only T,

OK. In the laboratory but not in the core, \ N u - 1I << 1 means that

convection is weak.

Downloaded by [Universite Laval] at 23:08 09 July 2014

56

S. 1. BRAGINSKY AND P. H . ROBERTS

Equations (8.1)-(8.7),supplemented by appropriate boundary conditions, define the

"homogeneous model" of the geodynamo. It may be noted that the factor 3 in

(8.5), which arises from taking g1 = 4nk,poR,/3 and g = -glr/Rl for this model,

is slightly inaccurate-it should be multiplied by 1 - (R2/R1)3(~z%'Z/pOY"-Z 1).

This factor differs from unity by only approximately

which we ignore. It

may be recalled from Section 4 that we need the functions p, and U , only if we

wish to evaluate the pressure perturbation p , = p , ( P - U,). It should be noted

that the magnetic Reynolds number does not appear in (8.6); it has been absorbed

into the magnitude of V, which may be large in amplitude when the unit of

velocity is chosen as we have done. We continue to ignore any changes in the

mass distribution of the mantle, and assume that R,, =0. We retain, however, the

crucial change in radius of the ICB: R, # 0.

The non-dimensional expressions for the sources oxand'0 can be obtained from the

the 0: and 0;derived in Section 6. In the homogeneous model,'TC = 0: is a constant

given by (6.24b),(8.8)and (8.16)as:

where

(8.21) (8.21a)

Here V2/V1=2 r:/(1 - r i ) =0.0452, r2 = R,/R, = 0.351. We have also taken A,R2 =
9,z Y2/tw, ith t,, = 4 x lo9yr. We have (seeabove)cast (8.21)into a form where the
dependence of 0; on poorly known parameters like Ap and t2 is explicitly shown, while its dependence on better known parameters is implicitly contained in .fob. Other expressions, such as CT'; and a;, are treated similarly below.
By (6.24a)and (8.15),the volume source of light fluid, CT;, is associated with a flux at the ICB of

A transparent relation similar to (6.24a)follows from (8.21)and (8.22):

(8.23a,b)

where A , = 3Y,/r2 (the dimensionless form of A , = 3V2/R2).The large constants appearing in (8.21)and (8.22)betoken a plentiful source of light fluid that drives core convection powerfully. They also strongly suggest that the convection is rather far beyond threshold, a fact that was noted by Braginsky (1991).
We may use (6.32) and (6.32a,b, c) to obtain the non-dimensional entropy source as

+ + '0 = 0'; 0; 0;,,

(8.24)

GEODYNAMO CONVECTION

57

the suffices being in 1-1 correspondence with those appearing in (6.32).According to (6.32a), the first term on the right-hand side of (8.24) depends on the difference between 14, and I T , the former of which is unknown, while (see above) I T =0.0356 W m - 2 . According to (6.32b), the second term on the right-hand side of (8.24) involves the difference between 1; and 1; which we neglect, and a term proportional
- to R 2 that can be conveniently expressed in terms of 0; and hence evaluated with the
help of the estimate h, zh, lo6J kg- [see (8.20b)l. In this way, using also (8.18), we obtain

(8.25,8.26)

(8.25a)

Downloaded by [Universite Laval] at 23:08 09 July 2014

In making these estimates, the values given in Appendix E were adopted.
The parameters ct, 01; and CT; are the dimensionless numbers that characterize
the nature of the convection; they play roles similar to that of the Rayleigh
number” in classical thermal convection theory . Note that 0; is not small in
comparison with 0;; on the contrary, a ~ , - 3 a t 0 . The magnitude of the ‘source’ CT; depends on the factor N u - 1, which is poorly known. We are not even certain
of its sign; .yo might be negative, i.e. a ‘sink’! On the one hand, 0; and 1; are
proportional to A p / t , and may be significantly changed if these poorly known values are re-estimated; on the other hand, 0: will be markedly altered if the poorly known ratio 14,lIT = N u is changed.
The term 01, originates from rsf, in (6.32c), that is composed of two very different parts 0: and 0;. The former is proportional to the inhomogeneous part of the basic quantity (QR- V . IT)/T,, while the latter depends on convective quantities. Using also (8.18), we have, in non-dimensional terms,
l o The Rayleigh number as usually defined can be written in the form Ra = o ~ T , T , , where T,, = L2/v and T. = L ’ / K ~are the viscous and thermal diffusion time scales and wf = @T/L is the square of the buoyancy frequency. Since magnetic diffusion is more significant to us than viscous or thermal diffusion, we may replace T~ and T~ and T,, = L2/qand Ra and Ra, = w f t i . We have taken L = R , above. Our non-
dimensional parameters u:, u:.. .. are proportional to the contributions they make to o:,but they are less than the corresponding Rayleigh numbers by the factor 2Rr,- lo9. Let us, for example, take
- - - - C &u$T,/~,,;see (6.3). Then w i g C / R , a:a~gr,/p,R, and hence Ra, cfCu~gT~/p,RA,.ccording
- - to (8.16). we have us = o x p O C , / z c ~a,nd C, = 2QV1/g,. it follows that Ra, u X g C , r i / R , 2Quxr,.
Our non-dimensional numbers C T ~ ,cry,. . . are therefore similar to the so-called ‘modified Rayleigh number’, Ramod= w,25,/2Q.After reducing them by about two orders of magnitude (numerical factors of order lo2 arise if we take L - R , / n - 106m instead of L= R , ) , they provide measures of how far core convection is operating beyond critical.

58
where'

S. 1. BRAGINSKY AND P. H. ROBERTS

Downloaded by [Universite Laval] at 23:08 09 July 2014

(8.27b)

Here p = p a / p o and 1 / 7 = T,/T,; (3.7b) was used to obtain (8.27b). The radioactive
source, QR,is proportional to pa. If we adopt the Boussinesq approximation, all thermodynamic quantities are nearly
uniform (E, << 1) so that I 1 - 71<< 1, and the terms in round brackets in (8.27a)are small
compared with unity, so that 0,' may be neglected in comparison with 0:. This greatly simplifies the theory. Some contributions to the dissipation QDare relatively concentrated, and their averages may then to a good approximation be omitted in (8.27b).In the resulting simplified model, the sources cr: and 0; of thermal codensity, Y , are constants. The source, o;, of compositional codensity X , always promotes convection,
as does 05. The term 0: may assist convection or oppose it, depending on whether
N u > 1 or N u < 1. The role of is stabilizing, as was discussed in Section 6.
Boundary conditions for the system (8.1)-(8.7)may be derived from the corresponding conditions obtained in Section 6. Those applying to V, B and C; follow from (2.46), (2.47)and (6.19a,b, c); those required of the fluxes at r = 1follow from (6.20a,b) and are

I,X(l)= 0, I,Y(l)= a ~ , ( ~ . ' , 2 / A l R , ) (-N1u).

(8.28,8.29)

More complicated conditions arise at r = r2. Corresponding to (6.22),the composi-

+ tionalflux on the ICB can be written as 1; = 1;

The boundary condition(8.23a, b)

on 1; may be written as

here 0': < 0; see (8.21).Similarly, by (6.32b),the averaged flux of entropy corresponding to the source 0: is

Expressions for 0; and 0; are given by (8.21)and (8.26).
- - Assuming that ic' yl, we estimate the diffusional operator to be V . 5 - V
- (ic'/yl)Vz 10-30 (a few multiples of n2), and recalling that the non-dimensional
sources of X and Y are of order lo4, we may expect from (8.3) and (8.4) that
" In (8.27a) we have restored 6, tand y, despite having stated earlier that we would set these to unity in this
section. This is because differences in P/?from its average enter this formula, and not itself.

GEODYNAMO CONVECTION

59

- - - - X Y- 300-1000. Taking Cr = 0.785 x lo-", we estimate the codensity from (8.12)

and (8.13) as C 3 x lo-' - lop8.The same value is obtained by comparing the

buoyancy and Coriolis forces: C 2RV/g, N 3 x

- lo-* for I/ 3 x

- - ms-'. This provides some qualitative support for the heuristic theory of
Braginsky & Meytlis (1990),on which the assumption, K' q, is based.

Let us now consider the oscillating fluxes of codensity. By (6.22b)and (6.29b),these

may be written as

(8.32,8.33)

Downloaded by [Universite Laval] at 23:08 09 July 2014

Using (6.42)and (8.13),we see that

(8.34)

where Y2 = Y ( r 2 , t )and

- y 2 , = 3t2 = c - 1.9 x 10. - 4-t.-2. AI 20

tr cp4A2

t 2 0 A2

(8.34a)

Here we have taken Azo =0.05 and cpas= a2T, = 5.3 x

see Appendix E. Let us

write a,Y2 = ( c ~ ~ t ~ w) ?he~re, Y2 is the amplitude of the Y, oscillation and

coo = 2n/(8 x lo3yr) is the fundamental frequency of the geomagnetic field; then

- cootrs J . 5 x 10' and r2,dtY2

y2.According to (8.34),a,R2,/R2 is of order unity

- when Y, 30, and such a value is quite probable since, according to our estimate,

Y-3 x lo2.

8.2 Energetics of the Homogeneous Model
The principal integral relations expressing the energetics of the geodynamo were obtained in Section 7 but the calculation of specific numerical coefficients was postponed until the present Section. To avoid unnecessary compiications, we derive here the energy balance for the homogeneous model. We also calculate the coefficient A, which determines the rate, R 2 ,at which the inner core grows, and which is used to
+ + estimate the sum, d t 22' 2?No,f three significant terms of the energy balance; see
(7.45). To estimate d r ,we note that, for the homogeneous model,

pa - p2 = uqu, - U,), T, - T , = - aqua- U1),

(8.35,8.36)

can be obtained by integrating (3.7b,c). Also, expressing k 2in terms of R, and using
(6.28a)and (8.8),we obtain from (7.33)

d r= Ap((U,)' - U 2 ) V 2 / t 2= 0.250ApglR, V2/t2.

(8.37)

We here used the fact that, apart from an irrelevant additive constant, U , = g1r2/2R, so that we may replace ( U , ) ' - U , by 0.2509, R,. Substituting our

60

S. I. BRAGINSKY AND P. H. ROBERTS

preferred values (8.20),we find that

d' = 0.34--A10P12t 2 0 W. APO t2

(8.37a)

It may be seen that, although clr, laand C N are poorly known, we are fortunately able,
with the help of the simplified model of Appendix B, to express xi'< directly in terms of the much better determined quantity Ap.
Using the relation (6.39) for S,, replacing Ta by To, and ignoring the difference
between 3, in the SIC and in the FOC, we may derive

2 2 ' ~ A2c,ToAl/3t2

(8.38)

from (7.34).In similar fashion, we find from (7.28)that

22' = k 2 h N= h N p ~ v 2 / t 2 .

(8.39)

Substituting our preferred values (8.20)into (8.38) and (8.39),we find

p=1 . 0 ~ . f 2 0 1 0W~, ~9N= 0 . 7 7 h , - k 1 0 1 2 W. (8.38a,8.39a)

A20 t2

hNO t2

Collecting together (8.37a),(8.38a)and (8.39a),and substituting into (7.49, we obtain

Downloaded by [Universite Laval] at 23:08 09 July 2014

(8.40)

The origin of each term in (8.40)should be obvious from the corresponding poorly known ratio attached to it. With the preferred unit values of the ratios, we have

9*= 2.1 x 10l2w.

(8.40a)

It may be clearly seen that, according to the 22. given in (8.40),thermal effectsdominate in determining the rate of growth of the inner core. We will find below, however, that the compositional part of 2?Dis, in order of magnitude, as potent as the thermal part in powering the geodynamo.
The rate of cooling of the core can easily be estimated from (6.39) and the
Ea. approximation Taz (Ta/cP)gaw, hich neglects a term proportional to This gives

T, z - TaA2/3t2,

(8.41)

and for the preferred values of the parameters this gives

F a = = - - -8-0-°K - 20°K

t2

109 yr'

(8.41a)

Downloaded by [Universite Laval] at 23:08 09 July 2014

GEODYNAMO CONVECTION

61

- It is interesting to see that the characteristic time of cooling of the core is not t , but is of
order T a / z 3t,/A, N 60 t,. Now we turn to the energy balance. Equations (7.35) and (7.32),which express the
buoyant power balance and the heat balance, can be rewritten in simplified forms appropriateto the homogeneous model. At the expense of a slight loss of precision, the expression (7.44)for the geophysically perfect efficiencycan be cast into a simple and transparent form. To do this, we roughly estimate the different averaged temperatures in (7.38),by assuming that the temperature, T,, follows a simple parabolic law which is a consequence of the approximations us = u; and g, = - glr:

+ T, = T , Td(l- r2), Td= ia;glRl,

(8.42,8.42a)

where Td= (T, - T l ) / ( l- T I ) = 1483°K and r2 = R 2 / R l= 0.351.
Using (8.42), we find that To=4558°K and

A T , = ( T , - T,)"=ATol =558"K.

(8.43)

This value of ATadiffers by less than 1% from that implied by Table E2. The ratio ATol/Toz 0.122 nearly coincides with iAT21/Tz2 0.123. Supposing, in the spirit of the Boussinesq approximation, that AT2l= T2- TI<< Tl is infinitesimally small (although in reality AT,,/T, =0.325), the average of every AT appearing in (7.38), no matter how weighted (provided that the weighting factor is close to unity) is unique and equal to AT,. Calculations made using (8.42)and (8.42a)give approximately the same results. Equation (7.38)then takes the form (with all terms having a precision of 5% or better)

= ( A T J T ~ +) (9~s-~2; +

(8.44)

and, by substituting 2'+ 9Rfrom (7.32),we obtain
2H= (AT,/To)(2&- 2; + 2N- at").
+ Substituting To= T, ATa here, we find from (7.35)that

(8.44a)

" [ 1 '- 9 --TO d5+-A(T27l'L' -9;+9N)

.

(8.45)

This provides the simple expression for the geophysically perfect efficiency referred to above:

(8.46)

where [see (7.32)]

(8.47)

Downloaded by [Universite Laval] at 23:08 09 July 2014

62

S. I. BRAGINSKY A N D P. H. ROBERTS

This may also be written as

+ 9& = L?R 2.t2,/t2.

(8.47a)

The relative importance of compositional and thermal driving is clearly exhibited in (8.46);it depends crucially on the difference 24, - 9;= =%?:(Nu- l), and on the ratio of d 5and 2; - L?;, the latter being reduced by the factor AT,/T, z 0.14. One of these magnitudes can be easily estimated: ST= 5.42 x lo’, W; see Appendix E. It follows that

(AT,/T,)(94,- 2;)= 0.76 x lo’, ( N u - 1)W.

(8.46a)

’

Similarly, taking h, = h,, and t , = t,,, we obtain from (8.39a)

(AT,/T1)2’ = 0.11 x lo’, W.

(8.46b)

The term 9, in (8.46) is small compared with 2;. Let us temporarily ignore it. Comparing (8.46a) and (8.37a), we observe that, despite the small factor ATJT,, the thermal terms (8.46a)in (8.45)is about twice the compositional term (8.37a).The role of
the factor N u is crucial. If [ N u- 112i,the thermal driving dominates compositional
driving and either assists ( N u > 1) or impedes (Nu < 1) dynamo action. It is appropriate here to make the following point. Our Boussinesq approximation
does not correspond to the limit E, +O, with heat sources and sinks, like QT,Q& and so
on, held fixed-this is the appropriate for the usual ‘laboratory case’. Rather, we consider situations in which E,QT,E,Q& and so on are held constant as E, +O-this is a Boussinesq approximation tailored to the case of Earth’s core.
We havejust obtained the Boussinesq form for !lDby approximating the corresponding expression from Section 6. It is possible also to extract it directly from the equations governing the homogeneous model in the following way.
We multiply (8.3) by U , - U,(r) and (8.4) by U , - U,(r), average the results over a cycle of the convection, average them over V12,and add them together. The non-dimensional gravitational field in the core is approximated by g, = - r, so that the
gravitational potential is, apart from an irrelevant constant, U , = i r 2 .Hence

U , - U , =+(r; - r2), U , - U , =+(I -r’).

(8.48a, b)

Recalling (6.26)and integrating by parts, we obtain on the left-hand side

(8.49)

where we have used (7.15),(7.18a),(7.35),(8.31),and

SZ!;~= -Iy,,(X+Y)V*rdV, S2fnd=- ( I x + I y ) * r d V . (8.49a,b)

GEODYNAMO CONVECTION

63

Here the subscripts nd stands for "non-dimensional". To obtain dimensional values, it is necessary to multiply by QrY1,=0.884 x 108W;see (8.14~).
+ On the right-hand sides of (8.3)and (8.4)we have constant terms,:a G'; a; and the + variable term r ~ ' ;=~ 0: a,'. The latter consists of two parts, both of which are of order
ATIT,, and we neglect them here. This is a somewhat crude simplification, but significant magnitudes are not dangerously distorted, and the calculations are greatly simplified; the result is

2fd= 2; + 2; + 2;.

(8.50)

Here

2x--12 ( r 2 - r22 }v lax21 -- 0.250g-1--R*214J 9," -0.38 x lo4-A--P-.--t2, 0

QiVY-12 t 2

APO t2

(8.50a)

( 8 . 5Ob)

-.-ATa ST

2';= + ( I -r">"G; =

( N u - 1) % 0.86 x 104(Nu- 1). (8.50~)

Ti QI'Y12

To obtain these results, the following relations were used [see (8.42)and (8.42a)l:

(8.51a)

Downloaded by [Universite Laval] at 23:08 09 July 2014

:(l - r2 )v LsxoglRl =AT,,, = ATa,

(8.51~)

together with the approximate but very accurate relation AT,,/T2 = ATa/Tl. If we multiply (8.50)by Q1Y12it,coincides term by term with (8.45), where TD= Tois
assumed. Expression (8.50a),when multiplied by QrV12coincides exactly with (8.37). After multiplying by QIV12the expressions (8.50b)and (8.50~c)oincide with 9,and 2?T(Nu- 1) respectively, when multiplied by ATa/T,.
The term 2; is associated with the latent heat released on the ICB through the crystallization of the SIC; the existence of this source of thermal forcing was first pointed out by Verhoogen (1961). The term 2: is associated with the gravitational energy releasedue to the flux of light admixturefrom the ICB during the same crystallization process; the existence of this source of compositional forcing was first pointed out by Braginsky (1963).The term 2;is associated with the cooling of the core through heat conduction to the mantle from the superadiabatic temperature gradient alone.
Equation (8.50)clearly demonstrates the relative significance of the compositional and thermal sources of convection in Earth's core. Values of the key parameters are given in (8.50a)-(8.50~)b, ut these are poorly known. If Ap = Apo,t, = f, and h, = h,,, the contributions made by compositional and thermal convection to 2' are approxi-
mately equal if Nu = 1.3.

64

S. I. BRAGINSKY AND P. H. ROBERTS

Downloaded by [Universite Laval] at 23:08 09 July 2014

The homogeneous (Boussinesq) model governed by (8.1)-(8.7) can be simplified even further by excluding (8.4). Though the source of compositional codensity is a consequence of thermal processes that results in the freezing of the inner core, the
source terms of the model are expressed explicitly through d , R , =.R, + btRz,, the
growth rate of the SIC. It is possible therefore, by prescribing R , and ignoring 6,R Z c ,to separate completely the compositional effect, X , from the thermal effect, Y. Then, setting Y = O in (8.1~a) nd omitting (8.4), we obtain the simiplest possible self-sustaineddynamical model of core convection and the geodynamo, which we may call “the compositional geodynamo”. This model, which has only one source of buoyancy, namely the compositional codensity, C = - X , was considered by Braginsky (1991).
Geodynamo theory is very complicated mathematically, and the compositional model, though possibly over-simplified, recommends itself through its comparative simplicity. It is physically sound, but is it at least qualitatively correct? The answer depends on the numerical values of the parameters relevant to Earth’s core, and on the sensitivity of the features of the geodynamo to details of the convective sources. It is difficult at the present time to be dogmatic. There is no doubt, however, that, if t , is smaller than our nominal value, the model is qualitatively correct. For example, if t , = 2 x lo9yr or less, and A p > Ape, compositional driving makes a greater contribution to A?Dthan thermal driving; the simplest compositional model would then become qualitatively realistic. If the values we have taken are considered more plausible, however, the compositional and thermal codensities are comparable, and their interplay can generate interesting effectsthat are absent from the compositional model.
This interplay requires special study, and it depends of course on unknown details of the geodynamo process and of the mechanism of heat transfer through the D” layer and mantle. The conducting and convecting D” layer is some kind of complicated thermal valve. Here, in the low viscosity region of the mantle on the side of the D” layer adjacent to the core, mantle plumes originate, according to Stacey and Loper (1983) and Loper and Stacey (1983), who stressed the crucial role of plumes in cooling the core. A stably stratified layer may also exist at the top of the core that plays a significant role in the exchange of heat between mantle and core (Braginsky, 1993). A complete understanding of the thermal coupling of core and mantle is still lacking. The following speculations may, however, be of some interest. The coefficient appearing in (8.25) is a few times greater than the coefficients appearing in (8.21) and (8.26). The rather large prominence of the thermal terms is the basic reason why both the intensity of convection and the field generation mechanism depend sensitively on N u - 1. This dependence is particularly strong when N u < 1. The compositional source of convec-
T - - tion is then partially spent in overcoming the negative (stabilizing) influence
of the thermal sources, as was pointed out by Loper (1978). When N u - 1 -+,
as happens when 2? $9& the effective ‘heating from the top’ is so great that it
may even stifle core convection and magnetic field generation completely. This indicates that core convection depends sensitively on heat transport through the mantle. One may therefore suspect that the factor N u - 1 establishes itself at rather a small value, though the mechanism through which this adjustment is

GEODYNAMO CONVECTION

65

effected is unknown. A sensitivity of core processes to N u - 1 might explain the
- observed variation in both the geomagnetic field intensity and the frequency of
reversals over the geologically long period, zG 2 x 10’ yr.

A little support for these speculations is provided by the fact that the thermal

- time constant, t” L“’/K, of the D layer is of the same order as zG. Here L”

- is a characteristic dimension of the layer and K ,

m2 s-’ is the thermal

- diffusivity of the mantle. It follows that t” zGfor L” z 80km, which is comparable

with the thickness of the D” layer. To establish the plausibility or implausibility of the

ideas advanced here, it would be necessary to treat the core and mantle as a coupled

system-the mechanism does not operate when, as in this paper, 2 ; is specified and

the core alone is considered.

If the value of N u were known, we would be able to estimate the radioactive heat

production in the core by using (7.47),which can be written as

(8.52)

With N u = 1 and our previous estimates of A?* =2.1 x 10” W, we obtain 2?’=3.3 x 1012 W for t,=t,,, 2 R =1.2 x l o i 2 W for t , = $ r 2 , , and ~ 2 ~ f=or0 t , = 0.4t2,. It t , = 2 x lo9 yr instead of 4 x lo9yr, then T R= 0 is attained if N u z 0.8.
Then [ N u- 1I z 0.2, which is rather small. Lacking precise values for the crucial
parameters, we may only suggest that 2’ is not significantly greater than about
3 x lO”W, but may also be much smaller (including zero).

Downloaded by [Universite Laval] at 23:08 09 July 2014

CONCLUSIONS
A proper foundation has been laid in this paper for studying core convection and the geodynamo; a complete set of workable equations has been consistently derived from first principles. We have formulated the MHD theory for the motion of a binary alloy of iron and some light admixture, in which the momentum balance is simplified by an
+ anelastic approximation. The dual (thermal compositional) character of core con-
vection has been properly recognized. Although compositional and thermal driving depend significantly on the thermal interaction of core and mantle and on the thermal history of Earth, neither of which are yet known with any precision, we can (and have) introduced the dimensionless parameters that appropriately measure the relative importance of the key physical mechanisms. We have argued that the geomagnetic field is a bye-product of large-scale magnetoconvection in the core, but the important role of small-scale motions has also been recognized through the introduction of a local turbulence model. We have seen that the existence of turbulent mixing is essential for the existence of the basic state of uniform entropy and composition.
Amongst the novelties and achievements of the paper, we wish to draw particular attention to the following:
1. A significant simplification of the anelastic equations has been established in Subsection 4.2, where the momentum equation was transformed without approximation into Boussinesq form. We there introduced a new quantity,

66

S. I. BRAGINSKY AND P. H. ROBERTS

Downloaded by [Universite Laval] at 23:08 09 July 2014

which we have christened ‘the codensity’, and which determines the nonconservative buoyancy force resulting from small perturbations of the wellmixed adiabatic state associated with the convection. This allows the irrelevant complications, created both by the pressure variations, and by the changes in gravity associated with the convective motions, to be filtered out, leaving behind only the crucial compositional and thermal buoyancy forces. This simplifies the re-assessment of the relative importance of these forces; 2. Emphasis has been placed on the probable dominance of core turbulence in the transport of mean large-scale fields such as entropy and chemical composition. This emphasis is not new, but goes beyond the early ideas adumbrated by Braginsky (1964b). The formalism developed here does not require one particular description of core turbulence rather than another; it does however suppose that the turbulence can be adequately described by a local theory, i.e. that, to a first approximation, the fluxes of mean fields (such as entropy and composition) can, at every point, be expressed as a linear combination of the gradients in those fields at that point. The coefficients in these relationships are the turbulent transport coefficients, which are expected to be very much larger than the corresponding molecular coefficients. The theory that is developed here has some points of similarity with the mixing-length theory used by astrophysicists in studying the convection zones of stars. See, for example, Ch. 14 of Cox and Giuli (1968); 3. We have stressed that, because the microscale magnetic Reynolds number is. very probably small, turbulence in the core is likely to be quite different from ‘classical’turbulence, in which inertial forces are all important. We have argued, however, that Coriolis and Lorentz forces are so potent that the turbulent cells in the core have a plate-like strucutre, so that the associated turbulent transport of macroscopic quantities by the turbulence is strongly anisotropic. We have made use of the turbulence model of Braginsky and Meytlis (1990) to estimate that transport. We have also emphasized that in the core, unlike the laboratory, the (tensor) turbulent diffusivities of entropy and composition are identical to one another;
4. We have provided an expression for the entropy production rate, d,due to
the turbulence. We have shown that this is simply - g * I c , where I c = - a V - a S I S is the flux of codensity. Translated into simple terms, the source oi turbulent energy is not inertial cascade from the macroscale, but is the gravitational instability associated with mean gradients of composition and entropy. According to all estimates, the largest of the molecular diffusivities in the core is the magnetic diffusivity. Assuming that this is the principal diffusivity that affects core turbulence, it is shown that the entropy production by the turbulence arises entirely from the Joule dissipation of the microscale electric currents, Q’, i.e. Q’ = Q’. The rate of mean field entropy production due to turbulent processes must be positive definite (or, more precisely, it must be non-negative); in local turbulence theory it must be non-negative, at every point in space. Consequently, as is shown here, the notion, that a simple enhancement of the transport coefficients is all that is required to incorporate the effects of turbulent diffusion, is incorrect. Such

GEODYNAMO CONVECTION

61

Downloaded by [Universite Laval] at 23:08 09 July 2014

an idea would lead to locally increasing entropy everywhere, including regions that are locally stable. At a point of local stability, turbulent diffusion will be absent, according to any local theory of turbulence, and only molecular transport can take place. This creates a positive but very small entropy production. It should be stressed that, unlike the corresponding astrophysical application mentioned above, the boundaries separating unstable regions of the core from stable regions is not known, and may not even be spherical. 5. The relative importance of thermal buoyancy (from the cooling of the core and the release of latent heat at the inner core boundary) and compositional buoyancy (from the release of the light constituent of the alloy at the inner core boundary) in driving core convection has been estimated, using modern geophysical data. Braginsky (1963, 1964b) argued that compositional convection dominates thermal convection in Earth’s core. His arguments were subsequently examined by a number of authors (e.g. Gubbins, 1977; Loper 1978; Gubbins et al., 1979; Loper & Roberts, 1983) who confirmed that compositional driving was an effective mechanism for stirring the core. In Section 8 we have found that, most plausibly, the contributions made by compositional and thermal sources to the codensity are comparable and that some interplay between these two mechanisms must be expected. This opinion depends on the spatial distribution and magnitude of the two sources and on the sizes of various parameters that are poorly known today, and it is therefore impossible to be dogmatic about this matter. The flux of heat from core to mantle is crucially important for both convection mechanisms, and the connection between convection processes in mantle and core significantly influences the geodynamo. Nevertheless, we argue that compositional buoyancy is especially significant since it admits the possibility that heat is pumped downwards, against the adiabat (see point 7); 6. In Sections 7 and 8, we have given new expressions for the efficiency, qD, of the geodynamo, considered as a heat engine. These differ from earlier derivations in that the effect on the efficiency of the slow evolution of the core is explicitly separated from contributions made on the convective time scale. We expressed qD as a product, qD= qFqG,where qFis the frictional factor (7.43) and is the ratio of the Joule dissipation of the geodynamo to the total dissipation of the core, arising from Joule dissipation and all forms of friction, and qGis the geophysically ideal efficiency (7.44),which replaces the Carnot efficiency of a traditional heat engine, and which represents the maximum attainable efficiency of the geodynamo. We have also presented new arguments that relate the magnitude of the geomagnetic field to the available power. (Unfortunately, it is again hard to apply these arguments with confidence to the core because of the-uncertainty with which some key parameter are known). 7. Many parameters important for the description of core convection are poorly known. Even the heat flux from the core, which is central to the character and vigor of core convection, is so badly determined that it is even uncertain whether it exceeds, or is less than, the heat conducted down the adiabat. The rate at which the inner core grows through freezing, i.e. R,, is also not reliably known but (we argue) it can be more accurately estimated than the heat flux. We have therefore,

Downloaded by [Universite Laval] at 23:08 09 July 2014

68

S. I. BRAGINSKY AND P. H. ROBERTS

where possible, cast the theory into a form in which badly determined parameters

are removed in favor of quantities dependent on R,. Where this was not possible,

we expressed a poorly determined quantity as the product of a nominal value for

that quantity (which we used to evaluate our formulae) and the ratio of the

quantity to that nominal value. For example, where the age, t , of the solid inner

core appears, we took t,, = 4 x lo9 yr as the nominal value, used it to evaluate

expressions in which t , appears, but retained the ratio t,/t,, in those expressions.

In this way not only were the consequences of the uncertainties oft, made evident,

but also anyone who prefers to take some value of t , other than 4 x 109yrcan

easily see the implications of his choice. This illustrates what we have striven to do

in .this paper: we have tried to describe clearly what is a very complex physical

c N , situation;
8. The mass fraction,

of light constituent, Ad, in the inner core has been

cN)/c,, estimated using a simple model of a binary alloy. We have defined a

rejection coefficient, rFs= (5, -

and have derived a simple expression

for that coefficient that relates it to the well determined density jump, Ap, at

the inner core boundary. Adopting the PREM value of 0.6 x lo3kg m - 3 for

Ap, we find that rFSz 0.4. Thus, most of the light material is retained by

core fluid when it solidifies. Once the phase diagrams of the relevant alloys

under high pressure become better known, it should be possible to use this

value of rFS as a means of determining which alloying element is most

abundant in the core. For the present, we prefer models that take Ad = S or

Si rather than Ad = 0.

Let us suppose that the density discontinuity, Ap, at the inner core boundary is 0.6 x lo3kg m - 3 and that the age of the inner core is 4 x lo9 yr. Let us further suppose that the heat flux, 2&2&f,rom the core is that conducted down the adibat, 2:. (In reality, it is not known which is the larger.) Then according to (8.50)(8.50c),

2~=0.5 x lo4 2, z 4.4 x ioll w

+ Here 2, = Q, V12z 0.88 x 108W,where QI is the basic unit of dissipation per unit
volume in the core, defined in (8.14~)T. he dissipation rate, 2 D= d c 2',given in (9.1) includes both Joule and viscous losses (including friction between core and mantle) from both the large-scale ( d ca)nd turbulent ( 2 ' )fields and motions. The
+ macroscale parts 2" S J ,are provided by the Archimedean power, d cd,riving the
dynamo, and can be estimated from integrations of kinematic or intermediate dynamo models:

where ya is a dimensionless constant that is model dependent, Bayis the rms toroidal field, averaged over the volume of the core, t , = R : / q z 1.92 x lo5 yr is the electromagnetic time constant of the core, and r] is its magnetic diffusivity,which we assume

GEODYNAMO CONVECTION

69

is q z 2 m 2s-’. With these values (9.2)may be written as

Downloaded by [Universite Laval] at 23:08 09 July 2014

It is here supposed that B,, is measured in Gauss so that the constant multiplying B:, in (9.2)has units of WG-’. Suppose that B,, = 100G and d cx 0.721D z 3 x lo1’W, so that 0.3 remains to supply the turbulent dissipation, 2?f. Then we find from (9.2a) that y A = 130, which happens to be close to the value given by the Kumar-Roberts dynamo model; see the final column of Table 6 of Kumar and Roberts (1975).Model-Z dynamos require y, of order twice as large; ya tends to be rather smaller than 130 for Taylor-type models. It is much smaller for the free decay of either the poloidal dipole ( y A = n2)or the toroidal quadrupole ( y A z 20), both of which are often used in similar
- calculations. Despite the uncertainty in the way that ?,varies from one model to
another, we may say with some confidence that a geodynamo in which B,, 100G can be maintained in Earth‘s core. This is a typical magnitude for the toroidal field in the so-called “strong field dynamo”; although the field is not extremely strong, it is much greater than the poloidal field, which is the only magnetic field seen at Earth’s surface.
Theories of the geodynamo should rest on equations that are both geophysically realistic and sufficiently tractable for theoretical progress to be possible. The conflict between these two desiderata has led us to develop models at different degrees of complexity. These are roughly of three types: in the order of increasing simplicity but decreasing realism, they are
(I) The inhomogeneous model (Section 6); (11) The homogeneous model (Section 8); (111) The compositional model (Section 8).
Model I seems to provide a rather satisfactory basis for the study of core magnetoconvection, a framework on which further improvements can be constructed. Model I1 is much simpler than model I but it employs the rough Boussinesq approximation (E, +0).It isjust this model that has been used here to provide numerical estimates with a minimum of complications. It is worth remarking that it is possible to define models that are intermediate between models I and I1and that these models are almost as easy to employ in massive numerical computations as model 11. For example, one could
incorporate the inhomogeneities of pa and T,, as given for example by the PREM
model, but continue to suppose that at and as are constants. Then instead of (8.2),i.e. V.V = 0, one would use the anelastic approximation (6.2),i.e. V. ( p o V )=0. The effect of the small spherically-symmetric deviation from incompressibility on field generation could then be investigated, and the consequences of the limit set by the Carnot factor, AVT, could also be studied directly. This could be done with little added computational effort.
Model I11 is obtained by omitting thermal forcing that, though poorly known, may be as large as compositional forcing; it therefore also rules out all effects that arise through the interplay between these two mechanisms. It is however much simpler than models I and I1 because it uses only one equation governing the codensity together with a simple boundary condition. It evades most of the uncertain-

70

S. I. BRAGINSKY AND P. H. ROBERTS

Downloaded by [Universite Laval] at 23:08 09 July 2014

ties in the physical chemistry of the core. Moreover, the number of parameters that must be assigned is at a minimum:d,R,, K‘ and the strength of core-mantle friction. (The interaction between solid inner core and fluid outer core across the inner core boundary can be approximately incorporated without the addition of another parameter-see Braginsky, 1989.)This simplest model could, as suggested by Braginsky (1991), be the best weapon to wield at this time in the formidable battle of finding a self-consistent geodynamo.
It is possible, within the framework of model 111, to investigate in a self-consistent way all the main elements of the geodynamo mechanism, namely:
(A) magnetic field generation resulting in the mutual excitation of the mean (0-
averaged) poloidal and toroidal field components; (B) generation of MAC waves, that account for the existence of the asymmetric fields
and velocities which allow the dynamo to evade the restrictions of Cowling’s theorem; (C) the advection and turbulent diffusion of the mean codensity; (D) the local turbulence mechanism which creates the diffusional transport of mean quantities.
All these four processes are nonlinear and interact with one another, thus turning the geodynamo into an auto-oscillating system. Such a system holds promise of exciting developments in the future. Perhaps a little hopefully, one may imagine that geodynamo theory will throw light on the composition of the core, and in particular on which alloying element is its principal light constituent. This would be achieved by solving the geodynamo equations for many choices of the key, but poorly determined, parameters we have isolated above, and by deciding which model fits best the geomagnetic observations and all other relevant geophysical data. These ‘best values’ for the poorly determined parameters would provide information, unavailable from any other source at present, about the composition of the core.

Acknowledgements
We thank NASA for grant NAGWl2546, during the tenure of which this research was initiated and completed. One of us(S1B)is also very grateful to the Alexander von Humboldt Foundation for the award of a Fellowship that enabled him to benefit from the hospitality of Professor Freidrich Busse at Bayreuth University, and to bring this paper t o fruition. We wish also to acknowledge helpful discussions with Professors Orson Anderson, Bill McDonough and Frank Stacey. We are grateful to David Stevenson and the other (anonymous)referee for their critical comments on the first submission of this paper. David Stevenson’s remarks led to improvements in the presentations of Section 9. The main contents of this paper were presented at the SEDI meeting at Whistler in August 1994.

References
Anderson, O.L., “Physics of iron: showdown in Colorado Springs,” E O S . trans. AGU, 237-238 (1994). Anderson, O.L., “Mineral physics of iron and the core.” submitted to Reo. Geophys, (1995). Anderson, O.L. and Young, D.A., “Crystallization of the Earth’s inner core,” in:Structure and Dynamics of
Earth’s Deep Interior, (Eds. D.E. Smylie and R. Hide). AGU Geophysical Monograph 46, pp. 83-89 (1988). Backus, G.E., “Gross thermodynamics of heat engines in the deep interior of the Earth, Proc. Nnf. Acnd. Sci. W s h .72, 1555-1558(1975).

GEODYNAMO CONVECTION

71

Downloaded by [Universite Laval] at 23:08 09 July 2014

Boehler, R., “Temperatures in the Earths core from melting-point measurements of iron at high static pressures,” Nature 363,534-536 (1993).
Boubnov, B.M. and Golitsyn, G.S., “Experimental study of convective structures in rotating fluids,” J . Fluid Mech. 167, 503-531 (1986).
Boness, D.M. and Brown J.M., “The electronic band structure of Iron, Sulfurand Oxygen at high pressures and the Earth’s core,” J . Geophys. Res. 95,21721-21730 (1990).
Braginsky, S.I.,“Structure of the F layer and reasons for convection in the Earth’s core,” Soviet Phys. Dokl. 149,8-10 (1963).
Braginsky, S.I., “Kinematic models of the Earth’s hydromagnetic dynamo,” Geomag. Aeron. 4, 572-583 (1964a).
Braginsky, S.I.,“Magnetohydrodynamics of Earth’s core,” Geomag. Aeron. 4,698-712 (1964b). Braginsky, S.I., “Nearly axially symmetric model of the hydromagnetic dynamo of the Earth,” Geomag.
Aeron 18,225-231 (1978). Braginsky, S.I.,“The Z model of the geodynamo with an inner core and the oscillations of the geomagnetic
dipole,” Geomag. Aeron. 29, 121-126 (1989). Braginsky,S.I.,“Towards a realistictheory of the geodynamo”, Geophys.Astrophys. Fluid Dynam.60,89-134
(1991). Braginsky, S.I., “MAC-oscillations of the hidden ocean of the core,” J . Geomag, Geoelectr. 45, 1517-1538
(1993). Braginsky, S.I. and Meytlis, V.P., “Local turbulence in the Earths core,” Geophys, Astrophys. Fluid Dynam.
55,71-87 (1990). Brown, J.M. and McQueen, R.G., “Phase transitions, Gruneisen parameters and elasticity for shocked iron
between 77 GPa and 400 GPa,” J . Geophys. Res. 91,7485-7494 (1986). Busse, F.H., “Thermal instabilities in rapidly rotating systems,”J . Fluid Mech. 44,441-460 (1970). Busse, F.H., “Convection driven zonal flows and vortices in the major planets,” CHAOS 4, 123-134
( 1994). Busse, F.H. and Carrigan, C.R., “Convection induced by centrifugal buoyancy,” J . Fluid Mech. 62,579-592
(1974). Busse, F.H. and Carrigan, C.R., “Laboratory simulation of thermal convection in rotating planets and stars,”
Science 191, 81-93 (1976). Cardin, P. and Olson, P., “An experimental approach to thermochemical convection in Earth’s core,”
Geophpys. R e s . L e t t . 19, 1995-1998 (1992). Cox, J.P. and Giuli, R.T., Principles of Stellar Strucfure,1.New York:Gordon and Breach (1968). Dziewonski, A.M. and Anderson, D.L., “Preliminary reference Earth model,” Phys. Earth Planet. Inter. 25,
297-356 (1981). Dziewonski, A.M. and Woodhouse, J.H., “Global images of the Earth’s interior,” Science 236, 37-48
(1987). Fearn, D.R., Loper, D.E. and Roberts, P.H., “Structure of the Earth’s inner core,” Nature 292, 232-233
(1981). Frenkel, Ya.1.. Introduction to the 7heory ofhlerals, Moscow: Gostechisdat (in Russian) (1958). Gans, R.F., “Viscosity of the Earth;s core,” J . Geophys. Res. 77, 360-366 (1972). Glatzmaier, G.A. and Olson, P., “Highly supercritical thermal convection in a rotating spherical
shel1:centrifugalvs. radial gravity,” Geophys. Astrophys. Fluid Dynam. 70, 113-136 (1993). deGroot, S.R. and Mazur, P., Non-equilibrium Thermodynamics. Amsterdam:North Holland (1962). Gubbins, D., “Energetics of the Earth’s core,” J . Geophys. 43,453-464 (1977). Gubbins, D., “The influenceof extrinsic pressure changes on the Earth’s dynamo,” Phys. Earth Planet. Inter.
33,255-259 (1983). Gubbins, D., Masters, T.G. and Jacobs, J.A., “Thermalevolution of the Earth‘s core,” Geophys. J . R. astr. Soc.
59, 57-99 (1979). Hewitt, J.M., McKenzie,D.P. and Weiss, N.O., “Dissipative heating in convectiveflows,” J . Fluid Mech., 68,
721-738 (1975). Hollerbach, R., “Magnetohydrodynamic Ekman and Stewartson layers in a rotating spherical shell,”
Proc. R . Soc. Lond., A#, 333-346 (1994). Hollerbach, R. and Jones, C.A., “Influence of the Earth’s outer core on geomagnetic fluctuations and
reversals,” Nature 365,541-543 (1993).

72

S. I. BRAGINSKY AND P. H. ROBERTS

Downloaded by [Universite Laval] at 23:08 09 July 2014

Jeffreys, H., “The instability of a comprehensible fluid heated below,” Proc. Camb. Phil. Soc., 26, 170-172 (1930).Reprinted in Saltzman (1962).
Jephcoat, A. and Olson, P.. “Is the inner core of the Earth pure iron?” Nature 325, 332-335 (1987). Keeler, R.N. and Mitchell, A.C.. “Electrical conductivity, demagnetization, and the high-pressure phase
transition in shock-compressed iron,” Solid State Comm. 7, 271 -274 (1969). Kleeorin, N., Rogachevskii, I. and Ruzmaikin, A,, “Magnetized flow between two rotating spheres,”
submitted (1995). Kubaschewski, 0.and Alcock, C.B.,Metallurgical Thermochemistry, Srhedition, Oxford: Pergamon (1979). Kumar, S. and Roberts, P.H.“,A three-dimensional kinematic dynamo,” Proc. R. Sue. Lond. A344,235-258
(1975). Landau, L.D. and Lifshitz. E.M.. Statistical Physics, Part 1 . Y dEdition. Oxford: Pergamon (1980). Landau, L.D. and Lifshitz, E.M., Fluid Mechanics. Znd Edition. Oxford: Pergamon (1987). Loper, D.E., “The gravitationally powered dynamo,” Geophys. J . R. astr. SOL..54,389-404 (1978). Loper, D.E. and Roberts, P.H., “On the motion of an iron-alloy core containing a slurry. I. General Theory,”
Geophys. and Astrophys. Fluid Dynam. 9. 289-321 (1978). Loper, D.E. and Roberts, P.H.. “A study of conditions a t the inner core boundary of the Earth,” Phys. Earth
Planet. Inter. 24, 302-307 (1981). Loper, D.E. and Roberts. P.H., “Compositional convection and the gravitationally powered dynamo,” in:
Stellarand P1anetar.v Dynamos,(Ed. A.M. Soward). pp. 297-327. New York:Gordon and Breach (1983). Loper, D.E. and Stacey. F.D., “The dynamical and thermal structure of deep mantle plumes,” Phys. Earth
Planet. Infer.. 33,304-317 (1983). Matassov. G.,The Electrical Conductivity oflron-SiliconAlloys at High Pressures and the Earth’s Core, Ph.D.
thesis, Lawrence Livermore Laboratory. Rpt. UCRL-52322 ( 1 977). Mathews, P.M., Buffet, B.A., Herring, T.A. and Shapiro, I.I., “Forced nutation of the Earth: influence of inner
core dynamics,” J . Geophys. Res. 96B,8243-8257 (1991). Moffatt, H.K., “Liquid metal M H D and the geodynamo,” pp. 403-412 in Liquid Metal Magneto-
hydrodynamics. (Eds. J. Lielpeteris and R. Moreau). Dordrecht: Kluver (1989). Moffatt, H.K. and Loper, D.E.,“Hydromagnetics of the Earth’score, I. The rise of a buoyant, blob,” Geophys.
J . Intern. 117,394-402 (1994). Morelli, A,, Dziewonski, A.M. and Woodhouse, J.H., “Anisotropy of the inner core inferred from PKIKP
travel times,” Geophys. Res. Lett., 13, 1545-1548 (1986). Poirier, J.P.,“Transport properties of liquid metals and viscosity of the Earths core,” Geophys. J . Intern. 92,
99-105 (1988). Poirier, J.-P. and Shankland, T.J., “Dislocation melting of iron and the temperature of the inner core,
revisited,” Geophys. J . Intern. 115, 147-151 (1993). Pollack, H.N., Hurter, S.J. and Johnson, J.R. “Heat flow from the Earth’s interior: analysis of the global data
set,” Rev. Geophys. 31, 267-280 (1993). Roberts, P.H., “On the thermal instability of a rotating-fluid sphere containing heat sources,” Phil. Trans. R.
Soc. Lond. A263,93-117 (1968). Ruzmaikin, A. “On the role of rotation of the internal core relative to the mantle,” in:Solar and Planetary
Dynamos. (Eds. M.R.E. Proctor, P.C. Matthews and A.M. Rucklidge), pp. 265-270. Cambridge UK: University Press (1993). Saltzman, B., Selected Papers on the Theory of Thermal Convection with Special Application t o the Earth’s planetary Atmosphere New York: Dover (1962). Shearer,P. and Masters, G.,“The density and shear velocity contrast at the inner core boundary,” Geophys. J . Intern. 102,491-496 (1991). St. Pierre, M.G., “On the local nature of the fluid flow in Earth’s outer core”. Submitted (1995). Stacey, F.D., Physics of the Earth, Brisbane: Brookfield Press, 3‘d Edition (1992). Stacey, F.D., “Theory of thermal and elastic properties of the lower mantle and core,” submitted to Geophys. Suru. (1994). Stacey, F.D. and Loper. D.E., “ The thermal boundary layer interpretation of D” and its role as a plume source,” Phys. Earth Planet. Inter. 33.45-55 (1983). Stevenson, D.J., “Turbulent thermal convection in the presence of rotation and a magnetic field: a heuristic theory,” Geophys. Astrophys. Fluid Dynam. 12, 139-169 (1979). Verhoogen, J., “Heat balance in the Earths core:, Geophys. J . 4,276-281 (1961).

GEODYNAMO CONVECTION

73

APPENDIX A: NOTATION

Downloaded by [Universite Laval] at 23:08 09 July 2014

Four abbreviations occur frequently in the text:
CMB = core mantle boundary,
ICB = inner core boundary,
FOC = fluid outer core,
SIC = solid inner core.
We use the word ‘Nucleus’as an alternative to SIC, the suffix N then being attached to quantities evaluated in the SIC or on its surface. In an effort to make our notation self explanatory, we have adopted a few simple rules: To track the numerous physical variables that arise in our work, we have distinguished each by a combination of a letter and a suffix that have unique meanings. For example, p is used everywhere for density while a always refers to the adiabatic reference state. The significance of pa is therefore immediately obvious. The following should also be noted:
1. All small parameters are denoted by E, with an appropriate subscript; 2. Energies per unit mass (with the exception of the gravitational potentials Cr) are
denoted by E, with an appropriate superscript; 3. All energies per unit volume by u, with an appropriate superscript; 4. All rates of dissipation of volumetric energies are denoted by Q, with ah
appropriate superscript; all rates of volumetric entropy production are denoted by 0, with an appropriate superscript; 5. When an extensive quantity is integrated over a volume, such as the volume of the core, it is denoted by a calligraphic letter. For example, the integral of a mass density p is denoted by JN, the integral of a volumetric energy density u is denoted by 6,and the integral of the rate of dissipation of such an energy density, Q, is denoted by 9;in Section 7, s4 is the rate at which “useful work” is done and Y denotes couple; 6. Material fluxes are denoted by I, with an appropriate superscript; electric current density is denoted by J; 7. Some subscripts and superscripts permanently have specificmeanings; these are
, often omitted when vectors or tensors are written in component form. In particular:
(a) The suffices and refer respectively to the CMB and ICB, the volumes they contain being denoted by V 1(the entire core) and V z(the SIC); between them lies V12(the FOC). Where a quantity carries the suffix 1, it is to be evaluated on the CMB; 2 means it is evaluated on the fluid side of the ICB-if the quantity is discontinuous there, means that it is evaluated on the solid of the ICB;
(b) The subscript a refers to the basic adiabatic state. Note however that
properties of the fluid such as a, c,, q,... do not usually carry suffices even
when they are evaluated for the basic state. The subscript a is also omitted when it occurs in conjunction kith the suffices and that refer to CMB and ICB. We therefore write R , and R, in place of R,, and R2,,, TI and T, instead of T,, and T2,,1; instead of I;,, and so forth;

74

S. 1. BRAGINSKY A N D P. H. ROBERTS

Downloaded by [Universite Laval] at 23:08 09 July 2014

(c) For the FOC, the increment in a variable, through the action of convection, over its value in the basic state carries the subscript c. The fields V and B arise only in the convective state and do not carry the subscript c. For the SIC, the suffixc on a variable signifiesthat it is the deviation from its value in the basic state which is mainly due to the thermal conduction;
(d) The superscript t is used to distinguish transport coefficientsassociated with turbulent processes from the corresponding molecular coefficients. The turbulent contributions to other fields are denoted by daggers, e.g. Vt, Bt,
Ct,. .. are the fluctuating parts of V, B, C,. ... [For brevity, Vt, Bt,and Ct are
replaced by v, b J ( p o p o ) and c in Appendix C; (e) Other superscripts and subscripts that appear are
0 which stands for average over the FOC B which stands for magnetic field e which stands for effective J which stands for Joule K which stands for kinetic M which stands for mantle N which stands for nucleus (SIC) nd which stands for non-dimensional P which stands for pressure S which stands for entropy T which stands for temperature V which stands for volume or for velocity, depending on context V which stands for viscosity
5 which stands for composition
8. Time derivatives of basic quantities are denoted by an overdot, e.g. we write pa in
place of dp,ldt; 9. There are types of average: ( Q ) ' , Q, (Q)'. These are introduced in Subsections
4.2 and 5.1 and in (6.26) and are respectively averages over the turbulent ensemble, over large-scale convection and over Volume. In (6.25a, b) we introduce p o and Toin place of ( p a ) " and ( T,)"; 10. Double square brackets are used to denote the discontinuity of any field at
a surface,the location of which is specified.For example, "[t,] at r = R,"denotes 5 , - tN,where t2= t a ( R 2+) and tN= t a ( R 2-) are the concentrations of
admixture at r = R, in the FOC and SIC, respectively.We denote this particular jump also by (2N.
Table A1 Key to Notation

Quantity Name

Units

Definition

1,

Unit vector in direction of increasing coordinate q None

a1

Eulerian time derivative

dl

Lagrangian time derivative

s-

aiat

S-l

a, +v.v

d

Various energies

J

a

Thermal coefficient of volume expansion

"K- I

(D8)

as

Entropy coefficient of volume expansion

k g J - 1 " K = s Z m - 2"K (3.7d)

Downloaded by [Universite Laval] at 23:08 09 July 2014

GEODYNAMO CONVECTION

Table AI (Continued)

Quantity Name

Units

Isothermal compositional expansion coefficient Adiabatic compositional expansion coefficient Gruneisen parameter Kronecker delta Various small parameters Various energies per unit mass Magnetic diffusivity ( 20) Various efficiencies( X = B, C,D,E , F , G, I) Colatitude Molecular thermal diffusivity ( 20) Molecular compositional diffusivity (20) Turbulent diffusivity tensor Permeability of free space (437 10- ') Chemical potential Isothermal compositional derivative of chemical potential Adiabatic compositional derivative of chemical potential Kinematic shear viscosity ( 2 0) Kinematic second (bulk) viscosity ( 20) Viscosity tensor Mass fraction of light component of alloy Stress tensor Density Electrical conductivity (20) Entropy production per unit volume Various time scales East longitude Angular speed of reference frame Angular velocity of reference frame Area of CMB, area of ICB
+ Area A , A , of boundaries of FOC
Rate of working Magnetic field Specificheat at constant pressure ( 20) Specificheat at constant volume Codensity (Fractional density change at constant pressure) Rate of strain tensor Dimensionless turbulent diffusivity tensor Electric field Body force per unit mass Gravitational field (-VU) Effectivegravitational field ( - VV,) Heat of reaction Latent heat Generalized latent heat Various fluxes Electric current density Newtonian constant of gravitation (6.673 lo-")
Pressure coefficient

None None None None None m,s-,=Jkg-l m's-' None None m2s-' m's-' m's-' H m-' m2s1'
m2s1'
m's-' mZs-l mzs-' m's-' None Nm-' kgrK3 Sm-' Wm - 3 OK-1
S
None
S-'
S-'
m2 m2 W T
J kg- ' "K
J kg-' OK-'
None sNone Vm-' ms-' ms-, ms-2 Jkg-' Jkg-' J kg-' Various Am-' kg-' m 3s - ~
None

75
Definition (D9) (~17) (3.7f) (3.8)-(3.11) Section 7 (2.38a) (D31) (D4) (Dl01 (D16) (2.13a) (2.13a) (2.13)
(4.12) (2.13b) (8.14e)
(D1.2) (6.30b) (6.30a,c)

76

S. I. BRAGINSKY AND P. H. ROBERTS

Table A1 (Continued)

Quantity

Name
Soret coefficient
Thermal conductivity ~ c , (K20~)
Incompressibility puf Couple Mass of entire core, mass of SIC
Mass of FOC Pressure Reduced pressure Energy dissipation per unit volume Energy dissipation Radius vector from geocenter Distance from geocenter Rejection coefficient Distance from the polar axis Time Time on geological scale Time on convectional scale Specific entropy Temperature (Adiabatic) velocity of sound Isothermal velocity of sound Potential of gravitational field Velocities of CMB, ICB Effectivegravitational potential Fluid velocity Volume of entire core, volume of SIC Volume of FOC Distance northwards from equatorial plane

Units

Definition
PJP,
Irl
(5.15)
(2.8b)

Downloaded by [Universite Laval] at 23:08 09 July 2014

APPENDIX B: GRAVITATIONAL ENERGY

The theory of Newtonian gravitation bears a close relationship with that of electrostatics; there are also significant differences, of which the opposite sign (attraction of masses rather than repulsion of like charges) is not alone. The basic field equations are

v X g=o, v.g= - 4 x k ~ p .

(BL21

As in electrostatic theory, ( B l )and (B2)are the pointwise forms of more general integral statements

v, $cg .dC = 0, fA g'd* = -4nkN pd

(B374)

where A is any closed surface containing a volume V ,and C is any closed curve. When applied at a surface where p changes discontinuously, they imply that

[n x g] = 0 , [n-g] = - 4 n k N p A , on A ,

(B5,6)

GEODYNAMO CONVECTION

77

where pa is the concentrated surface mass density and n is the unit outward normal to A. In most models of gravitational phenomena, pa is zero, and then (B5)and (B6)give

[g] = 0 on A provided pa = 0.

(B6a)

When mass is contained only in a bounded volume, V ,surrounded by vacuum, g”w,e
have

g = O W 2 ) , for r + w .

(B7)

Equations (Bl)and (B2) allow one to write the gravitational force per unit volume as

Downloaded by [Universite Laval] at 23:08 09 July 2014

is the gravitational stress tensor. With the help of (BS), the gravitational force and couple on a body .Ir are readily expressed as integrals over its surface A . It is easy to show from (B5), (B6),(B7),(B8) and (B8a) that the self-force and self-couple on V are zero. According to (BI) and (B2), we have

g = - VU, V2U = 471kNp,

(B9,lO)

and (B.5)and (B6) are satisfied if we apply
[ u ]= 0, [n-VU] = 4 n k N p , , on A,

(B11,12)

while (B6a) becomes

[ U ] = O , [n.VU]=O, on A provided pa=O.

(Blla,12a)

Condition (B7) reduces to

U = o(r-’), for r-co.

(B13)

The gravitational energy of a mass distribution is defined to be the energy required to assemble it from masses brought “from infinity”. It is (for pa =0)

gg=+SpUdI/.

(B14)

This is in fact negative since energy is extracted during the process of assembly. By using

(BlO),(B1la) and (B12a), we find that (B14) can be written as

8g=--[1

UV2UdV=-

$ UdA.VU--

1 (VU)’dV,

8% Y

8 n k N Am

871kN Y ,

(B15)

78

S. 1. BRAGINSKY AND P. H. ROBERTS

where the surface integral is taken at infinity and vanishes by (B13);f m is all space. We then obtain an alternative to (B14),namely

This provides a definition of the energy density for gravitation that is, apart from sign, exactly analogous to the energy density that arises in electrostatics. It is interesting to note that even a second alternative expression for d gexists. By (B7),(B8),(B8a)and an application of the divergence theorem, we have

Downloaded by [Universite Laval] at 23:08 09 July 2014

where the volume integrals are taken over all space so that the surface integral is at infinity. By (B16,B16a) we now see that

We may use (B16a) to derive a pointwise expression of gravitational energy conservation. By (B9),(B10) and (2.2),we have

drug=

--4n1kNy.'a'g'.

=

-

1 v,uv,(a,u
~
4nkN

) =

-- 1 vi(ua,viu 4nkN

)+- 1 u 4nkN

a

p

u

which may be written as where

+ v pv a , u g

.19 = - -g,

4nkN

(B20a)

Despite the possible discontinuity in p U n - V at a surface A of discontinuity in p, we have

[n*Ig]=OO, on A.

(B21)

To see this, we take the motional derivative of (B11) and (B6) with respect to the velocity, n . V,, of A along its normal and obtain

[a,U]=-n.V,[[n.VU], on A,

(B22)

[n.a,g] = -n*V,[n-Vg] -4nkNPA, on A.

(B23)

On applying (B12) and (B2) and using Pa = - i[pn.(V - V,)], we reduce these to

[[la,V ]= - 4 n k N p A n - V A , [rn-d,gTI=4nkNlpn.V1, on A. (B22a,23a)

Downloaded by [Universite Laval] at 23:08 09 July 2014

GEODYNAMO CONVECTION

79

Taking the scalar product of (B20a)with n and again applying (Bll),we obtain (B21)
from (B23a).(In fact,Ig= Ig 1, vanishes identically for a spherically symmetric system.)
It is also worth noticing that, according to (B7)and (B13),

I g = O ( F ~ ) , for r + 00.

(B24)

By a transformation similar to (B19),it is possible to show that (B20)also holds when (B16a)and (B20a)are replaced by

+ u, ug = ; p

19 =p uv

1 -8(ngkaNt

u - ua,g).

(B25,26)

- Though more symmetrical, these are perhaps not quite convenient as the other forms,
since n Ig would in general be discontinuous on A .
Many of these results have been derived on the assumption that the gravitational field is self-generated;see (B7)and (B13).If an externally generated field,

g e x t = - VU'"',

(I3271

is present in addition, it is necessary to add ugextto the right-hand sides of (B16a)and (B25) and to add Igext to the right-hand sides of (B20a) and (B26),where

p x t = Uext,

~ g e x =t u e x t v v .

(B28,29)

Equation (7.33) provided an estimate of the power, at<r,eleased by gravitational
settling. This estimate presumed, consistently with the basis of the Boussinesq model of Section 8, that the density of the FOC is almost constant. We conclude this Section by
- attempting, again through the use of (7.33),to derive a more accurate estimate of at<I.t
is possible to represent the density of the FOC with an inaccuracy of at most 0.5% by
a simple parabolic law which, replacing r by the non-dimensional r/Rl, is

where p1 and p2 are given by PREM and are listed in Table 1, so that pdE 2580.81kgm-3. The resulting mass, M I 2= 1.8367 x loz4kg, of the FOC agrees well with the value 1.8411 x loz4kg given by PREM. We adjust the mass of the SIC to give the g2 obtained from PREM and listed in Table 1. The value, g1 = 10.66msP2 implied by (B30) then agrees well with the PREM value listed in Table 1. Since aU/& = g = k, A(r)/r2, where M(r) is the mass contained in the sphere of radius r centered on 0,we find that

From (7.33) we have

Downloaded by [Universite Laval] at 23:08 09 July 2014

80

S. I. BRAGINSKY A N D P. H. ROBERTS

We have here appealed to (8.35), which holds for nonuniform p, provided that we
assume, as we shall, that is constant in the FOC; we have also used (6.23b) and
(D48c).The final integral in (B32) is easily evaluated with the help of (B30) and (B31).
Writing also ?= Y"/t2, we find that

d'=0.38-- AP t20 10l2W. APO t 2

(B32a)

We have used the full Apo =0.6 x lo3k g m - 3 in this evaluation instead of A'p ~ 0 . x5 103kgm-3. The coefficient in (B32a) is about 10% greater than that appearing in (8.37a), where the approximations p = po and g = gl r were used.
It is clear from the derivation of (7.32)that, in general, dt# bg.Changes in p (and therefore in € g ) arise from variations in p , S and (. All three are properly accounted for in Sections 6-8, but d' involves only the (-created p-variations. It is also apparent that, if at is not constant, even variations in ( will cause the volume of the core and therefore R, to change, with a concomitant modification to the distribution of p in the mantle that will make a nonzero contribution to &g. When at is constant however, the mantle and SIC do not contribute to &g, and (B32)has a simple interpretation. In time lit, a fraction a t ( 2 N6t of the mass in a unit volume situated at a distance of r from the geocenter is effectively carried to the ICB, releasing gravitational energy of paat(2N6t [U,(r)- U,]. Integrating this over the FOC, we obtain (B32)as the total gravitational
energy release. In this case therefore = k g .
We may use the constant-a' model to estimate dtp2,a quantity arising in Section
6 (see footnote 8). According to this model R, and p , are independent o f t and 8,pa is
independent of r in the FOC. We have

3r2 i, a 1 p a = - q AP,

a19-

4nkNR, 3r (1 - r 3 )8, Pa.

(B33,34)

Differentiating the equation, drp = - pg, of hydrostatic equilibrium with respect to t and integrating over r, we find that

where we have used (B9) and a, p1 =0. Combining (B33) and (B35),we obtain
where, for the model defined by (B30),we have by (B31)

i.e. f ~ 0 . 6 9 .

(B36a)

Downloaded by [Universite Laval] at 23:08 09 July 2014

GEODYNAMO CONVECTION

81

APPENDIX C: LOCAL TURBULENCE

Turbulence plays a crucial role in the MHD of Earth’s core, but no theory has yet been developed to describe it fully. The discussions that have so far been published might be better described as intelligent speculations than as deductive theories. As we have mentioned in Section 4, proposals have fallen into two main categories: developments of the classical ideas of large-scale turbulence in which energy is injected at a macroscale and cascades to dissipation at a microscale, and new ideas concerning local turbulence in which energy enters at the microscale level, at which it is also dissipated (see Braginsky 1964b; Braginsky & Meytlis, 1990). Both proposals fully recognize the importance of Coriolis and Lorentz forces, and both therefore visualize turbulence that is far from isotropic. Despite its small length scale, turbulence of the second kind enormously enhances the diffusion of heat and composition. We speculated in Section 4 that this diffusion is so large that it would quench instabilities on the macroscale that might otherwise have been expected to provide the source of classical turbulence of the first kind. We therefore concentrate in this appendix on turbulence of BraginskyMeytlis type. We summarize their proposal, which they call “local turbulence”, and derive a new relationship that has considerable bearing on the arguments of Sections 4 and 6.
Local turbulence is caused by a simple local instability: that of a heavy fluid overlying a lighter one. Because of the strong influence of Coriolis and Lorentz forces, this instability has a fundamentally different character from the usual buoyancy-driven instability. We will exhibit the instability through the simplest possible example; namely the growth of a disturbance in a plane layer in which 2! and g are constant and parallel: R = R 1 , , g = - g l z , and in which the main flow and magnetic field are
constant and horizontal: V = Vl,, B = Bl,. The cylindrical coordinates (s. 4,z ) for the
core therefore correspond to (x,y, z), in that order. We suppose that this equilibrium
state is slightly perturbed, so that V, B, C and P become V + v, B + b, C + c and P + p ,
where v, b, c and p are infinitesimal quantities whose squares and products can be neglected, i.e.we appeal to linear stability theory and we seek to find the growth rate, ya, of the resulting motion. In the notation of Section 4, v, b, c and p correspond to Vt,
Bt(pop o ) - l”, Ct and Pt. Since we have no small letter counterparts for S and t (sbeing used-see above-for cylindrical radius), we continue to use St and t’ for the
departures of S and 5 from their equilibrium values. The effect of V is merely to Doppler
shift Im y,, and we shall for simplicity transform to the frame in which V = 0. The linearized equations (6.1)-(6.4), (6.6)and (6.7)give

+ + O = - V d - 2S‘l x v cg B ’ V b ,

(C1)

0 = B . Vv + VV’b,

(C2)

V*v=O, V*b=O,

(C4,5)

+ where d = p B * b. We have assumed here that the microscale magnetic Reynolds
number is negligibly small so that d, b, which would otherwise have replace 0 on the

82

S. I. BRAGINSKY A N D P. H. ROBERTS

left-hand side of (C2),has been discarded. We have divided B and b by J(po po),so that these fields now have dimensions of velocity; 1cms-' is equivalent to 11.7 G. The
- Lorentz force has been written as the divergence of the magnetic stress tensor, and the
term V(B b) has been absorbed into the pressure gradient. We have neglected the
molecular fluxes 1' and Is appearing in (6.3) and (6.4), and t and S therefore obey
equations of identical form. This marks it possible to combine them together into a single equation governing the perturbed codensity,

r,' instead of two equations governing and S: separately; see (6.17).The linearized form
of this single equation is (C3).
Solutions to (Cl)-(C5) can be sought as a superposition of elementary perturbations
that, near the point x = xo, have the form

where k is a constant wave vector; the suffix k will usually be omitted in what follows and V will often be replace by ik. Substituting (C7)into (C2)we obtain a linear relation between b and v:
1 b =- ( B - V ) V.
rl k2
The Lorentz force can then be expressed as an anisotropic frictional force:

Downloaded by [Universite Laval] at 23:08 09 July 2014

where

- In conditions prevailing in Earth's core, the 'coefficient of magnetic friction', y B , is very

large. For example, if B N 10' G =8.5 cm s - l , then y B 4 x

s-l, which is much

greater than even the frequency R of the Earth's rotation, which itself is so large that

Coriolis forces dominate large scale motions in the core.

Operating on (Cl)by V x in order to eliminate the pressure gradierlt, we may express

v in terms of c:

where
+ k', =k; k : ,

+ V, = 1,V, 1,V, =V - 1,V,,

R, = 2Rkz/k. (ClOa, b, c)

GEODYNAMO CONVECTION

83

Applying (C3) we now see that the perturbation grows exponentially as exp(y,t), where

(C11,12)

Downloaded by [Universite Laval] at 23:08 09 July 2014

The assumption VC = 1,V, C is used here. The Archimedean frequency, w,, is the Brunt
frequency, but for unstable rather than stable situations. The growth rate y, is real and depends on the squares of k,, k , and k,. Hence, from the
elementary solutions (C7),growing cells can be constructed with sinusoidal coordinate
- - dependence,i.e. standing rather than progressing cells.The magnitude of w, in the core
can be estimated as gC,/L, where C, lo-*. This gives mu- 3 x lO-’s-’ (or 27c/w, 1yr) which is much smaller than both 22 and y,. If all components of k were of the same order of magnitude, y, would be of order w:/yB or (in the case of small k , ) of order w,’/22, i.e. the characteristic time over which such cells would grow would be of the order of a thousand years. In other words, the magnetic frictional force and the Coriolis force both strongly suppress the growth of instabilities that are more or less isotropic, It is well known however that Coriolis forces have a much weaker influence on perturbations that are elongated in the direction of f2,and it is immediately seen
a,,., from (ClOc)that, if k , << k, then the that enters (Cll)is much smaller than 222. In an
analogous way, perturbations that are elongated in the direction of B (i.e. those for which k, << k ) experience less magnetic friction; for these y, <<ye. It follows that plate-like cells, that are elongated in the directions both of and of B, are less suppressed by Coriolis and magnetic forces and will grow fastest. This conclusion was reached by Braginsky (1964b), who argued that small-scale turbulence in the core consists of a collection of such plate-like cells. This simple idea, of elongation of turbulent cells in directions parallel to both f2 and B, provides us with a plausible qualitative picture of anisotropic local turbulence.
A quantitative theory of turbulence based on these ideas is still lacking, but
- - a ‘heuristic theory’ was developed by Braginsky & Meytlis (1990).They argued that,
for the dominating cells, k, k , << kx so k , k , N k. They concluded that
the relative dimensions of the cells in the three coordinate directions is given by y,-R,, which according to (C11) maximizes the growth rate y,, and which implies that

(C 13a, b)

where

-E, -2,Q YE

- that is

B2 E, 2 B2’

(C14a, b)

Here B , = ,/(222r]), which corresponds to approximately 20G, is the natural scale for
- measuring dynamo created field strength. Considered as a function of position in the
core, E, varies strongly; E, 1/25 might be taken as representative of the FOC as

84

S. I. BRAGINSKY AND P. H. ROBERTS

a whole. For the modes (C13a,b), (C11) may now be rewritten as

(C1la)
- Equation (C8)may be written, in order of magnitude, as b ( k , / k : ) ( B / q ) vand, taking
into account (C13a) and k , = k,, we obtain

Downloaded by [Universite Laval] at 23:08 09 July 2014

On the basis of qualitative arguments, Braginsky & Meytlis (1990) concluded that a typical turbulent velocity, v,, is of order y a / k , .
It will be seen that nothing in the foregoing discussion determines the absolute dimensions of the cells. This would not have been the case had we included thermal and/or viscousdiffusion,but the molecular transport coefficientsfor these processes are so small that they are surely irrelevant even on the dimensions of the turbulent cells. It seemsmuch more likely that the dimensions of the cells are determined by the nonlinear processes that have been omitted from this linear stability analysis. The inclusion of these processes adds considerable complexity, and so far their effects have not been quantitatively analyzed. In applying their qualitative analysis to core turbulence, Braginsky & Meytlis (1990)visualized a statistical ensemble of cells of all shapes and
- sizes but predominantly those having dimensions t, n / k , = t,, and
- - - - - - - t( t y n / k y /, n/k,),with characteristic velocities vI v, and v vy vz;corres-
- - pondingly b by b,, On the basis of heuristic arguments, Braginsky & Meytlis (1990)
concluded that, if the viscosity is as small as that typically quoted for the core (see Appendix E), approximate equipartition will establish itself in local core turbulence:
- - - - Applying this to (CSa), we obtain k , 2R/B or t, 2 km; (C13a,b) give
t t , / E , = 50 km.Using v yJk, (see above) and (C1la), we can estimate v. Based on
the arguments adumbrated above, Braginsky & Meytlis (1990)estimated that

Clearly icix is much smaller than ci,; but it nevertheless greatly exceeds the molecular
- - diffusivitiesK~ 10- m2s - and lo-' mz s - '. The magnetic Reynolds number, - - - 9?/ v t J q E,, of this turbulent motion is approximately 9?( t,/t << 1.Thisjustifies - - - - the neglect of d, b in (C2) because d, b/qV2b y,/qkt k,v/qkt B(t,/t 9;<< 1.
It may be recalled that expression (C11) rested on the assumption that VC = l , V , C ,
+ - so that terms perpendicular to 1, disappear when we calculate V - V C .For plate-like
perturbations and for VC = l,V,C l , V , C this term is of order v,/v, k,/k,, i.e. small,and the result (C11)is still approximately correct; only the component V ,C of VC is influential in exciting the instability. The model on which the present discussion is
based is therefore not quite as narrow as might at first sight appear.

GEODYNAMO CONVECTION

85

Assuming that turbulence in the core is of this type, it is possible to relate the entropy
- - source, 8,directly to the ohmic dissipation, Q', of the microscale currents. It should be
stressed here that, although Ibl<<IBI, it is not true that (V x bl Ibl/L, is small compared with IV x BI /B//L.Thus Qjis not a negligiblysmall part of the total ohmic
+ heating, QJ Qj. To establish the stated relationship, we first note that the neglect of
8,b in (C2)is equivalent to assuming that the microscale electric field,e,is dominated by its potential part, which we write here as -V [J(po po)cp], i.e. (C2)is a consequence of
Ohms law in the form

Downloaded by [Universite Laval] at 23:08 09 July 2014

where in present units j = V x b. It follows from this expressions that qj2 = B.(j x v) -V-(cpj).
The equation of motion (Cl)and (C4) give
+ cv * g = B.(j x v) V * (pv).
The ohmic dissipation due to the microscale currents j is

(C17) (C 18)

By (C17) and (C18),we have

The final term vanishes when we interpret the average over the turbulent ensemble as a local average over space. We therefore have

where

Ic = - aspo(Siv)' - a'po ((,'v)' = - asI" - at I@.

(C20a)

We have here used (C6),(4.26)and (4.30);see also (6.18a).It followsfrom (C20)and (4.37) that

a'=Qj/T,, or a'=g-IC/T,.

(C21,21a)

It should be no surprise that d is given by the Joule dissipation of the electric
currents associated with the turbulence. The right-hand side of (C20) is the rate of working of the gravitational force on the rising mass flux, 1'. All this energy is dissipated locally into heat, in this case through Joule heating. If other dissipation mechanisms acted, e.g. the viscous regeneration of heat, Q', by the microscale motions, they would also have t o be included, along with Q', in (C21). According t o

86
(C19) and (C15),

S. 1. BRAGINSKY AND P.H. ROBERTS

so that Q" << Qj for the linear plate-like cells under consideration. i f however strong nonlinearities developed in the small-scale turbulence and as a result local viscous dissipation became of the same order as the rate of working, -g-Ic, of the buoyancy force, the latter would have to make good both ohmic and viscous losses, i.e. (C21)
+ would be replaced by 0'= (Qj Q")/T, but (C21a) would still be valid.
The estimation of turbulent momentum transport is not straightforward. Adopting the Reynolds analogy for

Downloaded by [Universite Laval] at 23:08 09 July 2014

one might write in a first approximation

where, by (C23),v$ = v$ and, if 7'1: is to vanish identically when ( V ) ' is solid body rotation, v;fk = v&. Arguments of the type used in elementary treatments of the kinetic theory of gases suggest that

- with similar results for other off-diagonal elements of the "viscosity tensor" v$.
Fluctuating turbulent magnetic fields, of strength b u, contribute terms of the same
v1iXy- - '. order of magnitude to (C24). The estimates (C24) and (C15a) imply that ic~,1Px/P, 3 x 10- m2s- Although this exceeds the molecular viscosity ofthe core, the transfer of mean momentum by Reynolds stresses on length scales of order L is negligible in the main body of the core. Nevertheless, because it acts on largescale motions, it is this turbulent viscosity that should be employed in the description of internal shear layers that may exist within the core, such as the shear layer surrounding the tangent cylinder that has recently been studied by Ruzmaikin (19931, Hollerbach (1994) and by Kleorin et al. (1995). It is therefore this turbulent velocity that should be used in computing the viscous dissipation, Q"', in that shear layer, which might conceivably be a significant part of the total viscous dissipation Q" in the core.
APPENDIX D: THERMODYNAMICS
This appendix has three objectives: (1) to provide a summary of thermodynamic relations needed in the main body of the paper, (2) to provide a synopsis of the derivation of (2.36)-(2.40), and (3) to set up a simple model of the core as a binary alloy.

GEODYNAMO CONVECTION

87

The starting point for the first objective is the internal energy per unit mass ~ ' ( pS,, 5)
for which
+ dC1=$ d p T d S + p d t , P

and from which it therefore follows that

Downloaded by [Universite Laval] at 23:08 09 July 2014

from which we have

It is sometimesconvenient however to use p , S and t as independent variables instead of
+ p, S and t. In this case the enthalpy, E~ = E' p / p , plays the role of ~ ' ( pS,, 5 ) and
(D5)-(D7) are replaced by

p-'dp =(pu~)-'dp-uSdS-aedt,
+ d T =( a s / p ) d p (T/c,)dS - (h'/c,)dt, + + d p =( a 5 / p ) d p (ht;/cp)dS p'dt,
from which it follows that

(D13) (D14) (D15)

(D16,17)

Note that all extensive quantities (E', S, etc.) are per unit mass (not per mole) and that
correspondingly 5 is the mass fraction (not the molar fraction) of the light constituent.

88

S. 1. BRAGINSKY A N D P. H. ROBERTS

A comparison of (D5)-(D7) with (D13)-(D15) suggests that, in an analogy with the names ‘thermal coefficient of volume expansion’ for a, ‘isothermal compositional expansion coefficient’ for a$, and ‘isothermal compositional derivative of chemical potential’ for p;, we might name as the ‘entropy coefficientof volume expansion’, a<the ‘adiabatic compositional expansion coefficient’, and p c the ‘adiabatic compositional derivative of chemical potential’. Central roles are played by a5 and as in the theory developed in this paper.
Three other useful quantities that arise are

Downloaded by [Universite Laval] at 23:08 09 July 2014

It follows in the usual way that
( 2 )=%a=’::, c

c p - c , = a 2 T u ~ , a$--,<=-. asThs

(D22,23,24)

All these results are readily generalized if the core is modeled by a multi-constituent alloy. The starting point that replaces (Dl)is

+ +K

ds’ = p d p T d S

pkdSk,

P

k= 1

the summation being over all K constituents of the alloy. By the definition of t kas
a mass fraction, it follows that

so that (Dla) may be rewritten as

K
1i k = 1,
k=l

Evidently our binary alloy, of a light constituent (L) and a heavy constituent (H), concerns K = 2. In this case (Dlb) coincides with (Dl)when we define p = p L - p H ,
( = tr.and t8 = 1 - tL.For other K , results (D2)-(D24) may be generalized in an
obvious way. We now turn to our second objective: a summary of the facts that we need from
diffusiontheory for the case K = 2. For a more complete discussion, see chapter VI of Landau & Lifshitz(1987).They derive linear relations for the fluxes of light constituent and entropy which, in a notation that differs from theirs, are

GEODYNAMO CONVECTION

89

Onsager’s reciprocity principle implies that

see for example Landau & Lifshitz (1980)or deGroot & Mazur (1962).It is convenient to eliminate V p from (D26) and (2.34)by writing (D25)as

It follows that

vp = --I‘1

B
-

-

V’

T

.

a’ a‘

(D25a)

Downloaded by [Universite Laval] at 23:08 09 July 2014

The rate of entropy production, os,is given by [cf. Landau and Lifshitz (198O)l

+ ToS=-K(VT T)2 1

T

a

+ Q’ + Q” + QR,

where

It is clear from (D29)that positivity of osrequires that c1’ and the thermal conductivity
K T should be non-negative.
It is convenient to introduce a different notation that eliminates a’, B’ and 7’:

The three original transport coefficients (a’,/?’ and y’) are apparently replaced by five, namely K T ,18k,$, k: and p’, but p‘ - p$k$ is in fact a thermodynamic property of the fluid (h‘),as is k$. In this new notation, we may write (D25) and (D28) as

+ TI^ = p’F, I* = - K ~ TV.

(D36,37)

Equations (2.36)-(2.40) now follow from (2.33)and (2.34)in a straightforward way. The ICB is a surface on which the solid and liquid core are in phase equilibrium and
on which therefore

[ p i = 0, [E“ - p < ] = 0 on the ICB;

(D38,39)

90

S. I. BRAGINSKY AND P. H. ROBERTS

seeforexample Loper & Roberts (1978).In Section 6, we introduced two different latent heats (6.30a,b):

hL = E: - c:, hN = h , - p 2 t Z N .

(D40,41)

+ Since T is continuous and E~ = TS, it follows from (D38)-(D41) that (D40) is

equivalent to

h N = TZ(S2- SN).

(D42)

Finally, in setting up a simple model of the core, we should recognize at the outset that information about the physical chemistry of the core is largely non-existent. Even its composition is uncertain. It is generally agreed only that its principal constituent is iron (Fe) and that light admixtures are also present. Which element dominates those admixtures is not known, the competing merits of oxygen (0)s,ulfur (S),and silicon (Si) being vigorously but inconclusively argued. In view of these uncertainties, it seems justified to take a simple view of the core, in which the specific volume, p - l , simply depends on the total amounts of heavy (H) and light (L) constituent in the core, the volume that each occupies depending only on p and T :

Downloaded by [Universite Laval] at 23:08 09 July 2014

where p H is the density of heavy fluid (iron)and peL,is the effectivedensity of the light constituent, which (because of volume changes that occur when it is dissolved in iron) differs from its actual density, pAd,in the absence of iron. Equation (D43) can be rewritten as
(D44,45)
By (D17)and (D44)we have"
PS P H
We shall assume for simplicity that p H / p e Land hence 6,, are independent of p , T and
and are the same for solid and fluid phases. It then follows that, since 5 and S are
constant in the FOC, so are a<,p / p H and the mean molecular weight of the core, A,
which is related to those of the heavy and light elements by

An expression for the density jump Ap = p N-p 2 at the ICB r = R , can be obtained by applying (D44)both to the FOC (wherethe density of iron is p H and the admixture

' We ignore here the difference between a5and a$. Taking from Appendix E as = 6.28 x 10- kg J "K,

- - hc= -0.5 x lo7J kg- I, T2= 5,300"K and ac 0.6, we see from (D24) that ar,-a$ = - h h S / T

- 6 x

0.1 x ac at r = r 2 .

GEODYNAMO CONVECTION

91

concentration is t2= t;,) and to the SIC (where they are p i and t N )A.fter some algebra,
we obtain

where Asp is the densityjump through solidification and Atp is the density jump arising from the difference in t; between liquidus and solidus:

=P N ( l -pH/&),

A<P =PN(PH/Pi)ur42W

(D48a, b)

- Here 1 - pH/j& l o - ’ is very small, and we shall therefore replace (D48b) by

< P -- P N @5t 2 N .

(D48c)

We now recall the rejectionfactor, so called because it quantifies the amount of light constituent rejected by the solid when the fluid freezes onto the SIC:

Downloaded by [Universite Laval] at 23:08 09 July 2014

Dividing (D48c) by p N u c t 2making use of (D44),(D46) and (D48), we find that

[It is not necessary to specify where the denominator in (D50) is to be evaluated, since p/pH is the same everywhere in the FOC in our model.] The right-hand side of (D5O)
does not depend on any special property of the alloy, such as,,S t,, or even its
composition! It is a potentially useful method of discriminating between the rival claims of Ad = 0,S and Si.If the phase diagrams of FeAd for these three elements could be measured at megabar pressures, it would be discovered which of the resulting values of rFs agreed best with the value deduced from (D50); see Appendix E. That element might then be considered to be the most likely light constituent predominating in the core.
To make the model defined by (D44) definite, we must estimate two of the three
unknowns 6,, t;, and p/p,, the third then following from (D44). We know p from
Earth models such as PREM and p H from shock wave experiments on iron; we can therefore find p/pH; see Appendix E. The final datum is, in the case of Si, obtained by estimating 6,, from laboratory experiments at normal pressures; in the case of S, it is
the 5, implied by a conjecture by Boness and Brown (1990);in the case of 0,we adopt
a value for 6,, derived by Loper (1978).
APPENDIX E: NUMERICAL VALUES
Many parameters play parts, of varying degrees of importance, in determining how the core behaves. These parameters can be conveniently thought of as falling into four .categories:geometrical, thermodynamic, physico-chemical and electromagnetic.

Downloaded by [Universite Laval] at 23:08 09 July 2014

92

S. I. BRAGINSKY AND P. H. ROBERTS

Those accessible to seismology are the best determined. They have already been listed in Table 1 in Section 3.
The temperature of the core is known with much less precision. It is constrained by
the condition T2= T,(p,,t,), which expresses the fact that the ICB is at the melting temperature corresponding to pressure p , and composition t,, but the functional relationship T, = T,(p, t) is unknown. This is hardly surprising while uncertainty
persists about which light element predominates amongst the light admixtures of the iron alloy, the competing merits of S, Si and 0 being variously, but inconclusively,
argued. There is even current uncertainty about what T, is for pure iron at core
pressures, different experimental techniques leading to significantly different T,. The present situation is described, and the existing contradictions are analyzed, by Anderson (1994, 1995). Direct static measurements of the melting point of iron have been performed by Boehler (1993)up to pressures of 2 Mbar. He also obtained, by extrapola-
tion to p , = 3.3 Mbar, an ICB temperature of T2= T,(p,)= 4850°K. In contrast,
shock wave measurements of T, at pressures near 3.3 Mbar have been made by many authors and have generally shown a markedly greater value of T,. According to Anderson (1994),the reason for the differencemay be the existence of a phase transition to a new, high pressure, fcc form of iron, with a triple point near 2 Mbar, and which deflectsthe melting curve, T = T,(p), upwards. Until uncertainty is removed, Anderson (1994) recommends that the value T,(p,)= 6000°K be adopted for pure iron. The theoretical calculations of Poirier & Shankland (1993)of the melting point of fcc iron at 3.3 Mbar gave 6060"K, and they further suggested a rather large depression of the melting point, namely 500-1000"K, through the presence of the alloying elements. Anderson (private communication) also recommended to us that we should suppose
that this depression is of order 700"K, so implying that T,= 5300°K. In contrast,
Boehler (1993),who measured T, for an Fe-0 alloy, found that the depression of the melting point due to oxygen is very small, at least up to pressures of 2 Mbar, implying
that T,=4850°K. Estimates of y, a, c p and other thermodynamic properties of Earth's core have
been made by many authors. Convenient tabulations have been provided by Stacey
(1992), who gave for example y1 =y(R,) = 1.27 and y2 =y(R,) = 1.44. In
later work (Stacey, 1994), he modified several of his estimates, and in particular took, as we shall, y1 = 1.27 and y, = 1.35. This smallness of the reduction ( - 6%) in y, underscores the robustness of this parameter in studies of Earth's interior. As (3.7e,f)show, the Gruneisen constant is in fact the only thermodynamic parameter needed when estimating the adiabatic temperature gradient, -V, Ta(r). Given the temperature T,= 5300°K of the CMB, T, follows throughout the core by integration in r. We performed this integration in a simple (and perhaps simplistic) way. By (3.7e,f) we have

Since g is roughly proportional to P, we represented the right-hand side of (El)by the following simple interpolant:

GEODYNAMO CONVECTION

93

Downloaded by [Universite Laval] at 23:08 09 July 2014

where a, = 0.029619 and a, = 0.002207 are constants chosen so that (E2a) agrees with the q l and q 2implied by the values of y1 and y2 quoted above. From (El) and (E2a, b) it follows that

+ T = T,exp{ -(x2 - 1)[a, + $ a , ( x 2 l)]}.

(E3)

This gives Tl = 4000°K. The adiabaticgradient at the CMB is then 0.89"K km- I; at the ICB it is 0.276"K km-'. The 'average' temperature (6.25b) of the FOC implied by this model is To= 4590°K.
The shock Hugoniot for pure iron, as determined experimentally by Brown and McQueen (1986),intersects the pT-curve implied by (E3) and PREM, at a radius of approximately 2780 km within the FOC, where p = 204 GPa, T = 4520°K and the core density is p = 10.84 lo3 kgm-,. The shock data for this p and Tgives the corresponding pH as 11.98 lo3kg m-3, so that

pH/p = 1.1057,

which is the same throughout the core according to model (D44). We already noted in Appendix D that the density discontinuity, Vp at the ICB is
composed of two parts: Asp due to contraction on solidification, and Acp due to the difference in liquidus and solidus compositions, the former of which is comparatively small. Taking Asp=0.1103kgm-3 so that A g p = A p - A s p z 0 . 5 x 103kgm-3, we find from (D50) that the rejection coefficient, pFS,is approximately 0.41. Because the difference between the atomic radii of 0 and Fe is probably large under core conditions, we would expect that 0 would have a low solubility in solid Fe and that the rejection factor, rFS,and the density discontinuity, Ap, would be rather larger if 0 were the principal alloying constituent. Correspondingly, the small difference between the
- atomic radii of Fe, S, and Si (see below) suggests that, rF; and Ap should be small
(though not zero) for Ad = S or Si. Our estimate of rFs 41% leads us to favor Ad = S
and Si over Ad = 0.We now consider how 6, and 5, can be estimated for Ad =0,
S and Si. Boness and Brown (1990) noted that the (Wigner-Seitz) atomic radii of Fe and
S are close to one another when p is between 100 and 350Gpa and that these elements can therefore readily form solid-solution alloys. They used this fact to model FeS mixtures under core conditions. They found, from very detailed quantum mechanical calculations, that the dependence of p on p for Fe,S is (for core temperatures) rather close to that of the PREM model of Dziewonski and Anderson (1981). While conceding that it is somewhat speculative to do so, we adopt the Fe3S model of Boness and Brown (1990) as the basis for our estimates
+ for Ad = S, i.e. we take 5 = As/(& 3A,,) = 0.16 which by (D44) and (E4) implies
that,,a = 0.66.
Matassov (1977) measured the density, p ( 0 , FeSi at atmospheric pressure over
a range of 4 expected to cover core compositions. Although he found that the specific

94

S. I. BRAGINSKY A N D P. H. ROBERTS

volume depends nonlinearly on 5, we found that his data is reasonably well fitted by (D44) with S, = 0.68 and p,, = 7.87 x lo3k g m - 3, the density of iron at NTP. This value of S, was self-consistently determined so that the implied p falls onto the
relevant segment of the (p-plot shown in Figure 4.10 of Matassov (1977)which it did at
+ ( = 15.75%. Also p:, = p i " ' S,,) = 4.68 x lo3kgm-3. This is significantly greater
than the density of solid crystalline silicon, which is pgi = 2.42 x lo3kgm-3. This underscores the importance of allowing for the effects of chemical interactions when
modeling alloys. For Ad = 0,Loper (1978)gave p,", = 4.4 x lo3k g m - 3 and p i = 8.57 x lo3kgm-3.
The former was obtained on the basis of laboratory data; the latter resulted from correcting the density p H = 7.87 x lo3kgrnp3quoted above to allow for close packing
at high pressure. It follows from (D45) that S, = 0.95, and then (D44) and (E4) give
[, = 14.1%. Our values for 6,, for Ad = S, Si and 0 led via (D46),D47) and (D49) to the values
f o r d , A and t2- shown in Table El. Our estimates of the mean molecular weight
Ad = Si and for Ad = S are close to the value, A = 48.1, to which Stacey (1994)was led
from other considerations, while that for 0 is somewhat smaller. Despite being
hampered by a serious lack of information about the effects of high pressure on the phase diagrams of FeAd where Ad = S, Si or 0,we are favorably impressed by the consistency the Ad = Si and Ad = S models present. While conceding that S is, according to the arguments given here, an equally plausible candidate for Ad, we shall nevertheless concentrate on an FeSi core below.

Table E l Compositional Parameters

Downloaded by [Universite Laval] at 23:08 09 July 2014

S

32.07

0.66

0.60

16.0

6.6

49.9

Si

28.09

0.68

0.61

15.8

6.5

48.3

0

16.00

0.95

0.84

14.1

5.8

41.3

Stacey (1994)recently revised his earlier estimates (Stacey 1992)of the specific heats.
' He argued that the electronic contribution to these had previously been under-
estimated, and he proposed that c, % 4.5 R per mole, where R = 8314Jmole-'"K- iS
the gas constant. For iron (AFe= 55.85),we therefore have cre= 670 J kg-'"K-', and
more generally for a material of mean atomic weight 2,we have

=CFe=--

AF' -

-

670

z AF' J

kg-

'OK.

u

v

A

A

Since

we have, after using (3.7f) to eliminate CI from (E6),
-Fe
L.

GEODYNAMO CONVECTION

95

so that, substituting values from Table 1and adopting the values of y and T a t the CMB and ICB given earlier, we obtain

6705 kg-'"K-'
"' = (j/AF,) - 0.0751'

670 J kg-'"K-l cp2= (A-/AF,) -0.0534'

(E7a, b)

Choosing 2 = 48.3 (see above), we find the values of cpl and cp2shown in Table E2.
These may be compared with the recent values of Stacey (1994):cpl = 845 J kg- 'OK-', cp2= 826J kg-'"K-'. Tfwe took S instead of Si, we would have cpl = 819 J kg-'"K-' and cp2= 798 Jkg-'"K-', whereas 0 would give cpl = 1006Jkg-'"K-' and cp2= 974 J kg-'"K-'. Since y is so robust, any change in c pimplies, according to (3.7f), a corresponding revision in the coefficient of volume expansion, u. We obtain (for
2 =48.3) the values shown in Table E2, which are close to those of Stacey (1994):
u1 = 1.68 x 10-50K-', u2 = 1.00 x 10-50K-'; Anderson (1994) gives u1 = 1.62 x I O - S O K - ~ , u2 = 0.83 x ~ o - ~ o K - ' .

Table E2 Values of some Thermodynamic Parameters of the Core

Tl =4000"K Tz =5300"K To =4590"K AT12 T2 - T , = 1300°K y1 = 1.35 y z = 1.27
a, = 1.76 x 10-5"K-1 az =0.98 x 10-50K-1
cPl =848Jkg-''K-' cP2 =826 J kg- '"K ~ as =8.30x 10-5kgJ-'"K,
as =6.28x 10-5kgJ-'oK,

Temperature of the CMB, Temperature of the ICB, Average temperature of the FOC, Temperature contrast across the FOC, Giuneisen constant at the CMB, Giuneisen constant at the ICB, Thermal coefficient of volume expansion at the CMB. Thermal coefficient of volume expansion at the ICB, Specificheat at constant pressure at the CMB, Specificheat at constant pressure at the ICB, Entropy coefficient of volume expansion at CMB, Entropy coefficient of volume expansion at ICB,

Downloaded by [Universite Laval] at 23:08 09 July 2014

Equations (D8)-(D12) and (D16)-(D24) contains a number of further physico-chemical parameters whose values depend on pressure and on the specific admixture involved. They are unknown, and can be estimated only very roughly. The value
- - h 5 - - lo7Jkg-' was given by Gubbins et al. (1979) for FeS. According to Kubas-
chewski & Alcock (1979), h5 -3 x lo6J kg- for Si at NTP, while for 0 it exceeds lo7Jkg-'. We have adopted h5 -5 x lo6J kg-'. We should emphasize that this
- value is very uncertain, and its dependence on pressure is unknown. The situation is
scarcely better for h, and h,. We assume that h, h,. We follow Gubbins et al. (1979) by taking h, = lo6J kg-'. This value falls into the interval of uncertainty, 0.8 x lo6 Jkg-' < h, < 1.5 x lo6Jkg-', given by Anderson & Young (1988) for pure iron.
The kinetic coefficient in which we are most interested is the magnetic diffusivity, u.
Its value is important for dynamo theory. The electrical resistivity of metals arises from the scattering of the conduction electrons by thermal oscillations of the ions (phonons)
and impurities (admixtures). The first of these processes introduces a linear increase of q with T,although increasingp tends to offset this. These dependencies are considerably complicated by phase transitions and by structural and chemical changes.
Measurements of electrical conductivity at high pressure were made by Keeler & Mitchell (1986)for Fe and by Matassov (1977)for FeSi. Matassov also includes the

96

S. 1. BRAGINSKY AND P. H. ROBERTS

Downloaded by [Universite Laval] at 23:08 09 July 2014

results Keeler and Mitchell for pure iron. The measurements were made by shock wave

techniques, so that an increase in p is accompanied by the increase in T along the

Hugoniot. Matassov (1977) found that an increase in the Si content decreases the

conductivity. He also found that, for a geophysically relevant concentration of Si,
namely 25% atomic or 4 = 14.4% (which is rather close to our estimate; see Table El),

the change in conductivity produced by an increase in pressure from 521 kbar to

2518 kbar and by an increase in temperature from 672°K to 2518°K nearly compen-

sated each other. For p = 1422kbar, which is scarcely more than p l , the conductivity
was found to be 8.7 x lo5Sm- '. To adapt this to core conditions, we should make

a reduction because of the greater temperature, T , e 4000"K, of the CMB, and also

decrease it further because the experimental specimens were solid, not fluid like the

core; melting always reduces the conductivity. While these corrections are somewhat

uncertain, a conductivity of 4 x lo5S m-' seems quite plausible, corresponding to
u] = 2 m2s- '. Because the increases in pressure and temperature with depth in the core

tend to change the conductivity in opposing ways, we anticipate that u] does not vary

strongly across the FOC. It may be anticipated however that the conductivity of the

SIC will be greater than that of the FOC for three reasons: solids conduct better than

fluids, p increases with depth in the SIC but the increase in T is slight, and the SIC

contains smaller amounts of impurities (the admixtures) that reduce its conductivity.

According to the data of Keeler and Mitchell, as cited by Matassov (1977),the electrical

conductivity of pure Fe is, at the same p and T,approximately twice that of FeSi at 14.4%.

Having no other experimental data available, we shall suppose that this result its typical,

and shall assume that u] = 2 m2s in the F O C but that q N = 1.5m's- in the SIC. ~

The thermal conductivity of the core can be estimated through its electrical

conductivity by using the Wiedemann-Franz law, K = k, Tlu], where k, =

0.02 W m s - 'OK - 2 is the Lorentz constant (appropriately modified because u] has been

used, instead of the more usual electrical conductivity, in the expression for K T ) .Taking

q=2rn2s-' and T , = 4 x 103"K, we find that KT=40Wm-'"K-' and

KT = 5.7 x

m2s-'. This led to the estimate of 9: = 5.4 x 1OI2 W for the heat flux

out of the core down the adiabat. This is about 8 times less than the heat flux through

- the surface of Earth, namely 42 x l o L 2W, according to Pollack et at. (1993). The measured viscosity of liquid iron at atmospheric pressure is v lop6m2 s-'.

Increasing T decreases v, while increases in p tend to increase v because larger p

- makes the relative displacement of atoms more difficult. The estimate of core vis-

cosity most often cited, namely v

m2s-' (Gans, 1972),stems from a statement

made in the modern theory of fluids: fluid viscosity does not change along the

melting curve, T= T,,,(p). And the whole fluid core is near (though above) the

melting temperature. Poirier (1988) gives v1 = 3 x 1 0 - 7 m 2 s - 1 and v2 = 6 x

m 2s- l . These values, which we adopt here, should nevertheless be used with caution,

No experimental measurements of v have yet been made at core pressures, and the

theory of fluids, from which the constancy of v on the melting curve was inferred,

contains a number of strongly simplfying assumptions. It is not impossible that, at the

megabar pressures prevailing in the FOC, a large increase in v with depth occurs.

The viscosity of a fluid is related to its compositional diffusivity by a relation of the form

GEODYNAMO CONVECTION

97

where k , = 1.38 x

JOK-’ is Boltzmann’s constant and a is of the order of a few

inter-atomic distances (see Frenkel, 1958).If v is the same as at atmospheric pressure,

then according to (E8) laboratory measurements of IC< are applicable to the core. On

this basis and using laboratory measurements of the diffusion of S and Si, Loper
- & Roberts (1983) made the estimate 3 x 10-9m2s-’. Poirier (1988) gave
- ’. IC< 6 x m2s- This diffusivity might be even smaller if the kinematic viscosity is

greater than 10-6m2s-1. We will suppose that lies between 10-9m2s-1 and
10- * rn’s-’, but both of these values are so small that molecular diffusion of composi-

tion may be safely ignored except over very small distances. For example, taking

K < - 3 x 10-9m2s-’, we find that the characteristic diffusion distance, (rC%)1’2 over

a time-interval of t = lo3yr is only about 10m. This weak molecular diffusion is

completely insignificant, compared with turbulent mixing.

We conclude this Appendix with the argument that let us to adopt 0.05as an estimate

of A2. Referring to (6.37d) and using values given in Tables 1 and E2, we see that

g,y2 R,/u; z 0.064. According to Lindemann’s law,

where K , is the isothermal incompressibility which may, with an error of less than lo%, be taken to be the adiabatic incompressibility, K , = p 11:. This, by (3.7e,f), is also y T(dp/dT),,<. It follows that

Downloaded by [Universite Laval] at 23:08 09 July 2014

but we should recognize that, depending as it does on the difference between two
- gradients, estimates made on the basis of (E10) are unlikely to be robust. Taking
yz = 1.27 (see Table E2), we find from (6.37d) that Amu 0.03. The estimation of
’, A,{ from (6.41)is even more uncertain. The pre-factor, 3v-zPN/y12PO, is 0.159. Taking
t2, = 0.06 and h5 = - 5 x lo6J kg- we find that the first term in square brackets,
~,h~/c,T,,is approximately -0.067. The second term depends on the unknown
- - physical chemistry of the core, and is very hard to estimate reliably. Taking
(dTm/d(a)p - AT,,,/&, where AT, 700°K is the depression of the melting tempera-
- - ture through the alloying elements, we find that the second term is about -0.05. In
total A,< 0.02 and, by (6.40),A, 0.05.
Note added in Proof: As stated below (2.Q we have not considered the effects of variable rotation and have
ignored the Poincare force, -ph x r, that strictly should be present on the right-hand
side of (2.1). We wish to draw attention here to a recently published review (Malkus, 1994) that discusses the dynamical implications of a varying R , especially for the luni-solar precession, and that considers the consequences for core energetics. Malkus, W.V.R., “Energy sources for planetary dynamos,” in: Lectures on Solar and Planetary Dynamos, (Eds. M.R.E. Proctor and A.D.Gilbert). Cambridge UK: University Press, pp. 161-179 (1994).