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. 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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'(< 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 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> 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). 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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