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Core-Mantle Interactions It was shown in the preceding chapter that the heat output of the core is likely to be somewhere between 3 x 10~2 and 10~3 W. the lower figure being the one appropriate to a cooling core with no radioactive sources. We also saw in Chapter 3 that the interpretation of mantle layer D" as a thermal boundary layer with a steep gradient of 10/km requires a heat flow from the core Qc of about 9 x 10~2 W if the thermal conductivity of the lower mantle is 6 W/m deg (14 meal/cm deg s). Since it simplifies things a bit to assume a steady state, we shall henceforth assume that radioactivity of potassium is indeed the source of the core heat and assign it a value of 9 x 10~2 W. For a total surface heat flow Q0 = 4 x 10~3 W. heat generation in the mantle must amount to 3.1 x 10~3 W. corresponding to a volumetric average rate of 2.44 x 10-8 W/m3, which is not an impossible figure, considering that uranium alone, at the accepted concentration of 18 ppb (see Chapter 2) would generate about 8 x 10-9 W/m3. The question we now turn to is whether the heat flux Qc from the core has any discernible eject on the behavior of the mantle. There are several possible approaches to the problem, none very successful. The first to be examined is the "efficiency" of the mantle. EFFICIENCY OF THE MANTLE We saw in Chapter 1 that the mantle apparently does work on conti- nental crust, which is locally uplifted, deformed, and fractured. When 100

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Core-Mantle Interactions 101 two continents collide to form a mountain range, the mantle on which the continents are rafted must provide the necessary mechanical energy. This is mechanical work the mantle does on its surroundings. We want to see how this mechanical energy could be related to the heat input of the core. We consider a much simplified state in which the mantle receives heat from the core at the rate Qc and at the temperature Tc of the core-mantle boundary (CMB). The mantle contains radioactive heat sources with density ~ and discharges heat at its top at the rate QO and at temperature To; in addition it does work on the continental crust (not a part of the mantle) at rate W. In the steady state, a simple energy balance requires W = Qc + ~ edV-QO, v (5.1) where integration is over the volume V of the mantle. The entropy balance equation is T iVk(T) dV+lTdV, (5.2) where the two last terms on the right represent, respectively, the en- tropy generated irreversibly by heat conduction and by viscous dissipa- tion. Suppose that ~ is uniformly distributed so that r edV eV Jv T Tm where Tm, the "average" temperature of the mantle, is defined as 1 1 r dV = ~ -. Tm V Jv T Substituting in (5.1) to eliminate the unknown quantity eV, we get W Q (1 _ Tm) + Q (Tm 1) T J T where J- Ink (VTIT)2 dV and ~ additional assumption that rV /TdV (5.3) (5.4) Iv CHITS dV. If we make the 1/Tm IV ~dV = ~/Tm ~ where Tm

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102 ENERGETICS OF THE EARTH is defined by (5.4) and ~ is the total viscous dissipation, we obtain for W W Q (I_Tm +Q Tm_1 JT (5.6 Tc ) (To ) Equation (5.6) is rather remarkable in that it reveals the rather special role played by Qc Suppose indeed that we split Q0 into its two parts, Qo = Qc + Q', where Q' is the heat generated in the mantle. Then (5.6) can be written as (To T J ' ~ ( T -1 ~ -J Tm -1?, (5.7) and since TC ~ Tm' the factor modifying Qc is always larger than the factor affecting Q'. Heat from the core is more efficient than an equal amount of heat generated in the mantle. This is so because the rate of entropy production associated with core heat is less than that of mantle heat. Hence the reason for looking at mantle efficiency as a means of assessing the effect of core heat. We suppose that the oceanic crust and lithosphere form a part of the mantle, which is permissible since they consist mainly of mantle ma- terial that participates in the mantle flow. Thus we take Q0 - 4 x 10~3 W and To-300K. This is not quite correct since this value of Q0 includes the radioactive heat generated in continents outside the mantle, while the temperature at a continental Moho, or at the base of the continental lithosphere, is certainly greater than 300K. To calculate Tm and J. we must define the thermal structure of the mantle. Using the mantle temperatures estimated in Chapter 3, we divide the mantle into four layers, as follows: An upper boundary layer (UBL) with thickness H = 100 km and conductivity k = 4 W/m deg chosen such that the gradient ,30 at the surface is 20/km. The gradient decreases gradually with depth to give a temperature TH = 1300K atz = 100 km. 2. Layer 2, extending from 100 to 700 km, with an adiabatic tempera- ture prof~le. Because of the entropy changes associated with the phase changes that occur in this layer, the adiabatic gradient is steep. The temperature scale height h2 is taken to be 865 km, so that the tempera- ture T700 = 2600K. On this adiabat, the temperature at 400 km is 1838K or 1565C. In the layer, k = 4 W/m deg.

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Core-Mantle Interactions 103 3. Layer 3, from 700 to 2800 km, with temperature scale height ha = 9130 km, to give T2800 = 3300K. In layer 3, k = 6 W/m deg. 4. A lower boundary layer (LBL), extending to the core-mantle boundary, with an average gradient of 10/km. At the core boundary, TCA!B = 4300K and Qc, the heat flux from the core, is 9.15 x 1012 W. if k is 6 W/m deg. With the above numbers, the average temperature Tm comes out at 2050K, and J = 0.97 x 10~i W/deg. Substituting numbers in Equation (5.6) gives W = 3.9 x 10~3 - q) W. (5.8) The contribution to W + ~ of the core heat Qc [1 - (Tm/TC )] amounts to only 0.48 x 10~3 W. less than 15 percent of the total. There is no way of estimating accurately the viscous dissipation ~ or the viscous entropy production ~ = .{v (~/T) dV _ q>/Tm . Practically all that can be said about ~ is that the ratio 4~/QO is likely to be of the order of the ratio (d/h) of the thickness of the convecting layer to the tempera- ture scale height h (Hewitt et al., 1975~.* In our model, the ratio d/h is ().7 in layer 2 and 0.23 in layer 3, so that, very roughly, ~ 0.32 QO = 1.2 x 10~3 W and W is ~ 2.7 x 10~3 W. which is ample since W is estimated in Chapter 1 to be of the order of 10~2 W or less. Since Qc contributes only a small amount to W + A, it might be concluded that the mantle would function almost as well as a source of mechanical energy if Qc were zero. This, however, is not a foregone conclusion. In the first place, if Qc = 0, the LBE disappears, the temper- ature at the core boundary drops, and so does Tm' and so does W if Qo/To ~ J. Secondly, W is sensitive to J. which, in turn, is sensitive to the thermal structure of the upper boundary layer. The UBE contributes about 98 percent of the value of J because it is in this layer that temper- ature gradients are steepest and temperatures lowest. J is also sensitive to k, since for given QO, the surface gradient ~Bo varies as 1/k. A thin surface layer with much reduced conductivity k = 1 W/m deg might raise J suff~ciently to reduce W to zero, or at least to such a low value that the contribution from Qc would become important. Finally, the *More precisely, these authors find that {h = QO (dIh ) (1 - ~/2), where ~ is the ratio of ineernally generated heat Q' to total heat flux QO. Since Q' = Qo - Qc, this gives ~ = 0.61 QO (dIh).

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104 ENERGETICS OF THE EARTH contribution to J from horizontal temperature gradients that we have neglected may also decrease W below the value given by (5.8~. To conclude, if W-10~2 W is taken as an observed quantity, it should theoretically be possible to determine Qc from either (5.6) or (5.71. Exact calculations are, however, not feasible at the moment, mostly because of uncertainties affecting A, the viscous dissipation, end d, the rate of entropy production by conduction of heat, which depends only on thermal conductivity and temperature distribution. Alternatively, if both W and Qc are chosen, the equations could be used to determine d, and from it, possible temperature distributions in the mantle. CONVECTION PATTERNS IN THE MANTLE Assuming now that the heat output of the core is approximately 9 x 10~2 W. with a corresponding heat flux Qc into the base of the mantle of 5.9 x 10-2 W/m2, we inquire as to what the pattern of convection is most likely to be in the mantle. We note that the mantle also contains radio- active heat sources, generating on the average approximately 3.4 x 10-8 W/m3. We first examine the two heat sources separately, postpon- ing until later a discussion of their combined effects. CONVECTIVE PATTERN FOR DISTRIBUTED SOURCES OF HEAT The first question that arises is whether the radioactive sources are uniformly distributed or, on the contrary, concentrated in the upper mantle. The latter hypothesis is suggested by the fact that uranium, thorium, and potassium are more abundant in the crust than in the mantle. In many crustal plutons, the concentration of those elements must decrease exponentially with depth, as shown by Lachenbruch (1968) (see Chapter 24. It has also been argued that the close-packed structure of minerals in their high-pressure form makes them inhospi- table to ions as large as those of the radioactive elements, which would, so to speak, be squeezed out of the lower mantle by the high pressure prevailing there. While all this is true, it is not clear how these elements could have been effectively eliminated from the lower mantle. Concentration in the continental crust occurred presumably by complicated and often re- peated cycles of partial melting in the mantle and partial remelting in the crust. (No one seems to be very sure as to how the exponential dis- tribution in individual plutons comes about.) Transfer of radioactive elements from the lower to the upper mantle would presumably also require at least one stage of partial melting affecting the whole of the

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Core-Mantle Interactions 105 lower mantle, not just a small blob or plume here and there.* Such partial melting and upward migration of the melt would necessarily lead to a gross chemical differentiation with respect to elements other than the radioactive ones, of which there is no sign, the whole mantle ap- pearing on the contrary to be chemically homogeneous. Partial melting of the lower mantle requires temperatures much in excess of the pres- ent ones; it implies therefore a thermal event of the first magnitude, of which there also is no sign. Even the gravitational energy released by separation of the core would be insufficient to account for it. Thus while it is possible that the rate of production of radiogenic heat in the lower mantle could be somewhat smaller than it is in the upper mantle, the likelihood of it being zero is not great. We prefer to proceed on the assumption that the rate is uniform. The Rayleigh number Ra appropriate for uniformly distributed sources iS aeGR6 Ra = Cp K2v (5-9) (Chandrasekhar, 1961), where R is the outer radius of the sphere or spherical shell, G is the gravitational constant, 6.67 x 10-~ m3/kg S2, K is the thermal diffusivity, v is the kinematic viscosity, and ~ is the radiogenic heat production rate needed to provide a total surface heat flow of 4 x 10~3 W; ~ = 3.4 x 10-8 W/m3. We take ~ = 2 x 10-5/deg, K = 1.2 x 10-6 m2/s, v = 2 x 10~7 m2/s, and cp = 1.25 x 103 J/kg deg. This gives Ra = 8.4 x 109. What happens at such high Rayleigh number is not precisely known. Chandrasekhar (1961) showed that the critical Rayleigh number for the onset of convection is about 2 x 103 for a sphere with a free surface and 5 x 103 for a fixed-surface boundary; in both cases the onset of in- stability is associated with the mode I = 1 (in spherical harmonic language) corresponding to a single cell with one uprising current and one descending current 180 away. The critical Rayleigh number for a spherical shell remains of the same order, but the preferred mode depends now on the ratio of the outer radius to the inner radius of the shell; in circumstances similar to those obtaining in the earth, the pre- ferred modes whould be harmonics 3 to 5, depending again on boundary conditions. (The harmonic I = 3 means that the convection pattern in *How pervasive the process must be to remove all radioactive elements from the lower mantle may be gauged by noticing that in spite of its long geological history of repeated local partial melting, the upper mantle still contains appreciable radiogenic heat sources.

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106 ENERGETICS OF THE EARTH the mantle breaks down into three cells, rather than just one as in the case I = 11. Hsui et al. (1972) have numerically pursued the problem for a sphere to higher Rayleigh numbers. For free-surface boundary conditions, the single-cell pattern is maintained up to a Rayleigh number of 5 x 106; for a fixed-surface boundary, a two-cell pattern appears at Ra = 5 x 105. What happens at Ra-10~ for a spherical shell like the mantle can only be guessed. Presumably, the high Rayleigh number and finite thickness of the shell will both tend to break the circulation into smaller cells, with harmonic number higher than 5. There is a vast literature on convection patterns in the plane geom- etry of an infinite layer, but most of it refers to the case of a fluid heated from below rather than to a fluid heated from within. The adap- tation to the geometry of a spherical shell is not straightforward. In particular, it is not clear how the elongated cylindrical rolls with hori- zontal axis characteristic of convection in a plane layer could survive transposition to a thick shell. Examination of this problem requires close attention to the nonlinear terms in the dynamic equations. Busse (1975a) finds that a spherical geometry leads to a qualitative difference between convective patterns of odd and even harmonic order 1; it effectively prohibits patterns of odd order.* This difference does not have a direct analogue in the case of a plane layer, for which solutions with different wave numbers differ only quantitatively. A possible pat- tern of even order is shown in Figure 5-1. Busse points out that his results are apparently not applicable to the mantle, since gravity and other geophysical data for the earth do not seem to show a preference for even orders, a fact he attributes tentatively to the inhomogeneous structure of the upper mantle and continents. We may also note in passing an interesting attempt by Walzer (1971) to determine possible convection patterns from a purely kinematic argument based on group theory. Assuming at the start that flow lines are circular, so as to minimize dissipation, he seeks the patterns of superposed tiers of con- vective cells that have maximum symmetry or regularity (in the sense of group theory) and that lead to a maximum rate of outward heat transport. The facts of plate tectonics strongly suggest that the convection pattern in the mantle depends on time; plates suddenly change direc- tion, oceanic ridges apparently migrate in longitude or latitude, or both, and large plates break up into smaller ones. Whereas it is generally *A spherical harmonic function is such that its value at opposite points on the sphere is equal and of the same sign for even orders, of opposite sign for odd orders.

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Core-Mantle Interactions 107 )~1 FIGURE ~1 A possible pattern in a spherical shell with internal heat sources. The motion is ascending or descending in the shaded area according to circumstances. Reproduced with permission from Busse (1975a). accepted that flow is steady at the onset of instability in fluids with high Prandtl number, there is experimental evidence (e.g., Krishnamurti, 1970) that the flow may become time dependent at Rayleigh numbers much greater than the critical Rc characteristic of the onset of instabil- ity. The onset of time dependency depends on both the Rayleigh numberRa and the Prandtl numbers = V/K, and, as Jones (1977) points out, there is no experimental evidence pertaining to fluids with very high Pr and small Ra/Pr ratio, as obtains in the mantle. The importance of the latter condition (Ra/Pr << 1) has been pointed out by G. M. Corcos (personal communication to Jones, 19771. If we nevertheless extrapolate Krishnamurti's results (Figure 5-2) to large Pr, it appears that in a fluid heated from below, the flow might become time depen- dent atRa > 105-106. Numerical calculations at infinite Pr (Busse, 1967) show that as the Rayleigh number is increased above its critical value, the pattern of flow changes from a two-dimensional (elongated rolls) to a three-dimensional, but steady, pattern that, at a still higher Rayleigh

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108 ENERGETICS OF THE EARTH 106~ ~ 105 D - ._ C) A 104 TV of Turbulent flow / / Time dependent / / . a . Dimensional / ~ ~ lot E, Flow | _ _ . 1 ,~; 3-dimensional' Flow lll t . ~ Steady Dimensional F1OW 1 1 '' ~ Pr ~ . 03 l No motion 0.I 1 10 lot 103 104 1 Prandtl number FIGURE ~2 Flow regimes in a layer of fluid heated from below. The diagram illus- trates the dependence on Raylei~ and Prandtl numbers of the onset of time-dependent flow. Reproduced with permission from Krishnamurti (1970). number, becomes time dependent. This time-dependent flow is quite different from the oscillatory motion that appears in fluids (e.g., mercury) with very low Prandtl number, and that can be explained with linear theory. The time dependence we are concerned with here, and which results from the nonlinear terms in the dynamic and heat equa- tions consists of a slow change in pattern, well illustrated by the numerical experiments of McKenzie et al. (19741. In one particular experiment (see their Figure 17), a fluid heated from within is enclosed in a rectangular box twice as wide as it is deep. The Rayleigh number is 1.4 x 106. The motion, which initially consists of a single cell or roll, is unstable and gradually breaks down into two rolls with both ascend- ing plumes at the vertical boundaries of the box. This is followed by formation of four rolls, two of which are larger than the others. The larger rolls grow at the expense of the smaller ones until the two-roll pattern is approximately restored. A new instability in the upper bound- ary layer develops, leading again to the four-roll pattern, and the cycle is repeated on a time scale that, for the mantle, would be on the order of 107-108 yr. The authors have reason to believe that this unsteady flow is a feature of the solution of the nonlinear equations rather than an artifact of the numerical scheme used in the calculations.

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Core-Mantle Interactions 109 We need to mention one more feature of the convection pattern in a fluid heated from within to which McKenzie has drawn attention (McKenzie et al., 19741. In contrast to fluids heated from below, in which upwellings and downwellings are localized and symmetrical, the pattern in fluids heated *om within tends to consist of localized down- wellings separated by regions of diffuse and slow upwelling (Figure 5-3~. This occurs because, if heat is generated everywhere, the flow must bring all parts of the fluid close to the upper surface to permit them to lose heat by conduction. Also, the buoyancy force is distributed : ~:) I_ ~ 'A - l FIGURE ~3 Computer simulation of convection cells. The two left cells, (a) and (c), are for liquid heated from below; those on the right, (b) and (d), have internal heat sources. The two top patterns, (a) and (b), are for constant viscosity; in (c) and (d) the viscosity decreases with increasing temperature. In case (a), the pattern is nearly symmetrical, whereas in (b) there is a relatively narrow sink- ing sheet with upwelling everywhere else. Variable viscosity does not much alter the pattern, (d), for a fluid heated from within; heating from below, as in (c), produces a hot rising sheet narrower than the cold sinking sheet. Deep black is hot; gray is cold. Reproduced, with permission from D. P. McKenzie and F. Richter, Scientific Ameri- can, 235, 82, 1976. Copyright(~) Scientific American.

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110 ENERGETICS OF THE EARTH throughout the fluid rather than concentrated in the lowermost layer as in the case of a fluid heated from below. This raises the question as to whether localized upwelling such as is believed to occur at oceanic ridges or in deep-mantle plumes could ever develop in a fluid with only internal heat sources (McKenzie and Weiss, 19751. HEAT FROM THE CORE We now examine what happens to the mantle when it receives from the core, through its lower boundary, a heat flux q = Qc/S = 5.9 x 0-2 W/m2 (S is the area of the core-mantle boundary). The appropriate Rayleigh number based on q rather than on an as- sumed constant temperature difference between top and bottom of the mantle is (Foster, 1971) ag q d4 R = K up Cp (5. 10) where d is the thickness of the layer. Using the same numbers as before, we get Rq = 5.7 x 108. For an incompressible fluid with infinite Prandtl number, in the Boussinesq approximation, Foster shows that forRq > 107 the flow is intermittent with a mean period of intermittence given by and a horizontal wave length vpcp (5.11) agq 2 7r {K2 up Cp: ,/4 a 0.13 aGq (5.12) The mechanism that leads to intermittence may be the one illustrated by Howard (1966), and further explored by Jones (1977) and by Elsasser et al. (19 791. At the start, a thermal conductive boundary layer forms at the CMB to carry the heat q into the mantle. Initially it is so thin that the local Rayleigh number, based on the temperature difference across the boundary layer and its thickness 3, is less than about 103 and too small to induce convection; but as the thickness of the boundary layer increases it becomes unstable and ejects matter, which detaches itself from the CMB and rises by buoyancy. It is then replaced by a cold

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114 ENERGETICS OF THE EARTH the geographic orientation of erogenic belts and midoceanic ridges, which show complete disregard for the rotational axis, in contrast to the axisymmetric Eddington currents and their north-south (or south- north) surface velocities. No one will argue that von Zeipel's effect is the only cause of convec- tion in the mantle, any more than one would argue that the small climatic difference in temperature between the poles and equator on the surface of the earth has important mechanical effects on mantle circu- lation. It is important to remember, though, that this effect amounts to an inherent instability at zero Rayleigh number that could well trigger other near-marginal instabilities and help select among modes with nearly the same critical Rayleigh number. EFFECTS OF SURFACE PLATES Suboceanic upper mantle differs from subcontinental upper mantle with respect to distribution of radioactivity and temperature and with respect to theological and mechanical properties. Plates carrying con- tinents seem difficult to move around; they certainly move more slowly than purely oceanic plates. Because of their buoyancy, continents can- not be subducted; when they collide to form mountains, kinetic energy of mantle flow is transformed to strain and potential energy outside the mantle, and consequently the mantle flow must slow down. It is clear that constraints on the motion of plates will have an effect on mantle flow, extending perhaps throughout the mantle (Davies, 19771. Much of the time-dependent behavior of plates may be due to such surficial causes rather than to intrinsic properties of the convecting system. If this is true, the plate motions at any time depend on what has happened before; to account for the present-day pattern may well require know- ledge of the events of the past billion years. EFFECTS OF PHASE TRANSITIONS IN THE MANTLE The problem is to find in what way the existence of phase transitions in the mantle, notably near 400 km and 65~km depth' could affect the pattern of convection. Can a rising or sinking current go through such a transition, or will the flow be restricted to homogeneous layers lying between phase discontinuities? In spite of much work (Verhoogen, 1965; Schubert and Turcotte, 1971; Richter, 1973; Schubert et al., 1975) the question has not been completely answered yet. Schubert and Turcotte performed a stability analysis for a fluid layer heated from below in which a horizontal

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Core-Mantle Interactions 115 boundary z = 0 separates the two phases. The phase transition is assumed to be univariant, i.e., at any pressure the equilibrium tempera- ture Te is determined and falls on a Clapeyron curve with slope dTe/dP = /` V/li S = Te ~ VIEW, where ~ V and I`S are respectively the volume and entropy changes associated with the phase transformation and AH = Te ~ S is the latent heat of the transformation. Schubert and Turcotte show that the critical Rayleigh number for convection through the phase boundary depends now on two additional numbers, S and RQ, the first of which is a measure of the density jump at the transition, the second of which measures the latent heat l`H. They find that for given RQ, the critical Rayleigh number decreases with increasing S; for given S. the critical Rayleigh number increases with increasing RQ. Their analysis is, however, marred by their choice of an erroneous boundary condition, namely, that at the phase boundary z = 0 the vertical compo- nent of the velocity is continuous, we = w2. Conservation of mass requires, of course, that paws = p2w2, so that if pi 7& P2 (as must necessarily be the case if there is a phase transition), we cannot be equal to w2. In 1975, Schubert et al. extended the analysis to the case, more relevant to the mantle, of a divariant phase transition such as the olivine-spinel transition. The transition is said to be divariant because the system now has two components (e.g., Mg2SiO4 and Fe2SiO4~. There are now four variables (not three, as Schubert et al. assert), namely, P. T. and the mole fractions of one of the two components in both phases. The effect of this is that the equilibrium temperature (or pressure) is no longer determined by the pressure (or temperature) as it would be in an univariant equilibrium. If pressure is raised at constant temperature, the transformation of olivine into spinet starts at a lower pressure, Pi, and ends at a higher pressure, P2; within the range AP = P2 - Pi both phases coexist, with different and varying compositions. It is important to remember that the composition of these phases also changes as a function of pressure. As the pressure rises, the yet un- transformed olivine becomes progressively richer in magnesium, as does the growing spinet, which, at the beginning of the transformation, is richer in iron than the original olivine. Similarly, at constant pressure the transition occurs with rising temperature in the sense spinet olivine, and is spread over a temperature range l`T. On a P-T plot (Figure 5~) for a given initial gross composition, the beginning and end of the transformation are indicated by the curves 1 and 2; for any point lying between these curves, the two phases coexist. The first step taken by Schubert et al. (1975) is to determine the adiabatic (or better, isentropic) gradient within the divariant zone, i.e.,

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116 ENERGETICS OF THE EARTH A ., ~ mob if/ 1 TIC I~ depth ~ FIGURE ~4 Convection through a divariant phase transforma- tion. Curves (1) and (2) represent, respectively, the beginning and end of the transformation from phase 1 to phase 2. AC is the tem- perature distribution prior to the onset of convection, AB is an adiabatic path, starting at A, through the divariant zone. to define the slope of the curve AB in Figure 5 - on which the entropy at any point is equal to the entropy at the starting point A. Because the compositions of both phases vary from point to point along the curve, an exact calculation is difficult. Schubert et al. make simplifications that amount to reducing to zero the width of the divariant zone. The general trend of their analysis is, nevertheless, correct. The two-phase region may, to a first approximation, be treated as homogeneous, with an effective expansion c', up cz = cat- pAT where ~ is the ordinary thermal expansion of olivine or spinet (assumed to be equal); the second term represents the effect of the transition with total density jump Ap taking place over a temperature interval ~ T. at constant pressure, and is very much larger, perhaps 100 times larger, than c', which can be safely dropped. (Note that for the olivine-spinel

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Core-Mantle Interactions 117 transition Ap is 0.) Similarly, the effective heat capacity cp is defined as AH TAS cp=cp + AT =cp + AT . The adiabatic pressure coefficient of temperature is CITY orT -Up Opts pcp p2(cpAT+T A5) (5. 17) If ~ T ~ O. as for a univariant transformation, the right side of (5. 17) reduces to l`V/AS (where V = lip); the adiabatic slope is then equal to the slope of the Clapeyron curve. For the olivine-spinel transition, T AS is of the same order as, or somewhat larger than, cp /` T. so that the isentropic slope will be approximately one half to one fourth of the slope of the curves bounding the divariant region. Schubert et al. (1975) then proceed to make a stability analysis of a system consisting of two superposed layers separated by a divariant two-phase layer of thickness 2d; the total thickness of the three layers is 2D. A parameter R is defined as (X (163a-13a ~gD4 KV where ~ and `(3a are, respectively, the thermal expansion coefficient and the adiabatic gradient in the divariant region, while /3a is the adiabatic gradient in the single-phase layers. The critical Rayleigh number Rc now depends on the ratios d/D and cY/c' and on R; it rises rapidly when R increases from 102 to 106 (for d/D = 0.05, cY/c' = 1001. The analysis, however, is carried out on the assumption that the Boussinesq approxi- mation is valid in all three layers, including the divariant one. This is a surprising assumption to make, considering that the very essence of the two-phase region is its marked density gradient; a particle of fluid moving through it sustains over a short vertical distance a change in density of the order of 10 percent of the density itself. It is generally recognized (Hewitt et al., 1975) that the Boussinesq approximation is valid only if the ratio d/HT of the thickness of the convecting layer to the temperature scale height HT = cp/ga is ~ 1.0. The condition is generally met in the mantle, where HT-106-107 m, but it may not be met in a two-phase region where HT ~ 104-105 m.

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118 ENERGETICS OF THE EARTH However that may be, the main result of Schubert et al. (1975) is that the stability of the mantle is not changed much if the phase change is divariant rather than univariant. As applied to the olivine-spinel transi- tion, their calculations seem to show that for a mantle viscosity less than 3 x 1022-1 x 1023 cm2/s, double-cell convection requires a smaller overall superadiabatic gradient than single-cell convection through the phase change region ("double cell" rpeans convection confined to the two layers above and below the transition zone, with no flow across it). For 1023 < v < 1024, convection through the phase transition is the preferred mode. On the other hand, Richter (1973) finds, again by taking the olivine-spinel transition as a model, that (1) the vertical scale of motion is the entire depth of the fluid, (2) the horizontal scale is not significantly changed from the case- of a single-phase fluid, (3) the amplitude of the motion is not significantly changed, as the buoyancy effects are largely balanced by the effect of latent heat, and (4) the phase boundary varies in depth by as much as 30 km, for vertical velocities of the order of 10-i cm/yr. Nothing can be said of the effect on convection of the phase transi- tion near 650 km since its mineralogical nature is not definitely known; even the sign of the entropy change AS is unknown. McKenzie and Weiss (1975), however, argue that this phase change is important be- cause of differences in mechanical properties of the two phases, or phase assemblages. These differences will lead, according to the au- thors, to formation of a mechanical boundary layer across which heat is transferred by conduction; whether this boundary layer breaks up into plates or not depends on whether the convective forces exerted on it from below are sufficient to overcome its strength. There is, of course, as yet no compelling evidence for notable differences in me- chanical properties of the upper and lower mantles. Thus McKenzie and Weiss's argument appears too speculative to conclude from it that upper and lower mantles convect as two separate units, with no flow across the boundary. INFLUENCE OF THE CORE ON CONVECTION IN THE MANTLE From the lengthy discussion of the preceding sections, it must be clear that the convectional pattern of the mantle is presently unknown. Neither observations nor numerical experiments suffice as yet to de- scribe accurately what may be happening. The only point that seems relatively clear is that convection must affect the whole mantle; be- cause of relatively high Rayleigh numbers, the pattern of flow is pre

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Core-Mantle Interactions 119 sumably a complicated, three-dimensional, time-dependent one pro- ceeding simultaneously on several time, length, and height scales. A few years ago, the author was inclined to think that the core exerts a major influence on the mantle, that without core heat, convection in the mantle would be less vigorous than it is and possibly restricted to its upper part. This is not so clear anymore, for it is now certain, from the energy balance, that a large fraction of the earth's heat must come from the mantle. Since removal of all radioactivity from the lower mantle would require a major episode of melting and would produce a marked petrological differentiation that is not observed, it must be assumed that, at least to a first approximation, radioactivity is more or less evenly distributed in the mantle at a concentration corresponding to a supercritical Rayleigh number. The mantle is unstable, with or without the core. It will thus be difficult to pin down, in the largely unknown pattern of mantle convection, what may or may not be spe- cif~cally related to an influx of core heat. There are several ways by which, as discussed above, it might be possible to detect and single out the effects of core heat. The first was mentioned in the first section of this chapter, which dealt with the amount of mechanical work the mantle can do on continental plates. But as we saw, no conclusion can be drawn as yet, mostly because of the large uncertainties affecting the magnitude of the viscous dissipa- tion function and the temperature distribution. A second criterion for input of heat from the core was thought to be the time-dependent character of mantle convection, which could easily be explained on the Foster-Jones model of a mantle vigorously heated from below. It now seems likely that the Rayleigh number for radio- genic sources alone would be sufficiently high to induce time depen- dency. Finally, it was shown that, because of rotation, heat in the core will be preferentially transported to equatorial and polar regions, with a minimum in midlatitudes. No such effect is clearly recognizable on the earth's surface. But it is a long way from the core to the surface, and the amplitude of any horizontal temperature variation on the CMB iS likely to be markedly reduced, by conduction and convection, before it reaches the surface. Since the heat flow from the core represents, at most, less than one third of the surface heat flow, it is not likely that the effect could be readily detected on the surface. There is, however, one feature that may help to decide if core heat plays a role in shaping the convection pattern in the mantle. We noted earlier that, as pointed out by McKenzie et al. (1974), when heat sources are distributed, upwellings tend to be diffuse and spread out

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120 ENERGETICS OF THE EARTH over most of the convective volume. Upwelling in the mantle, on the contrary, seems concentrated in narrow sheets (ridges) or thin plumes, a feature more commonly observed in numerical experiments on fluids heated from below. The argument is, however, not decisive, because it has not been shown that the observed pattern of high heat flow at narrow oceanic ridges could not also be accounted for, in the whole or in part, by other factors such as the mechanical properties of the plates, the temperature dependence of viscosity, and so forth. INFLUENCE OF THE MANTLE ON CONVECTION IN THE CORE Imagine a core entirely devoid of heat sources so that it cannot convect. Imagine also a lower mantle with radiogenic sources sufficient to cause convection in it. This convection would presumably lead to the forma- tion, on the CMB, of cold spots corresponding to localized downwellings in the mantle. Could the horizontal temperature variations on the CMB cause convection in the core sufficient to generate a magnetic field? The question has not, to the writer's knowledge, been examined in detail; but a preliminary answer seems to be no (Jones, 1977), as the motion is likely to be restricted to a very thin zone at the very top of the core. There is no likelihood, in particular, of the development of the differential rotation necessary for the growth of a powerful toroidal field. If, however, the core is unstable, while the mantle convects intermit- tently, it seems likely that the pattern of convection in the core will reflect the temperature variations at the CMB. These variations include an overall decrease in temperature at the CMB during the convective part of the mantle cycle; hence, presumably, the Rayleigh number for the core will increase, and the core convective pattern will intensify and change. Should reversals of the magnetic field be caused by changes in the latitude of the zone of cyclonic circulation, as suggested by Levy (1972), then possibly a change in temperature on the CMB might induce a change in the polarity of the field, or in the mean frequency of reversals. This is what prompts Jones (1977) to argue that the estimated period of intermittence of mantle convection could be related to long- period changes in the average rate of magnetic reversals. On the whole, it does not seem impossible that long-period (>107 yr) fluctuations of the magnetic field might correlate with changes occur- ring in the mantle and at the CMB. It is difficult, however, to prove such a correlation because of the long time delay (~108 yr?) between events in the core and their manifestation at the surface. A few years ago, Irving and Park (1972) called attention to the apparent paleomagnetic

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Core-Mantle Interactions 121 polar wander path for North America, which occasionally shows strik- ing changes in direction. Since the apparent polar wander path reflects mostly changes in the motion of the plate from which it is determined, these "hairpins" in the polar wander path imply sudden changes in the motion of the plate, which, so to speak, stops and then starts back nearly in the direction from which it came. It is a curious fact that the two latest hairpins happened during the two latest periods (Cretaceous and Permian) in which the field remained in the same polarity for a time on the order of 30 million years. But this can hardly be more than a coincidence, since it is hard to see how a change in plate motion could affect instantaneously the convection pattern of the core, or vice versa. Thus, it would seem that at this stage of the game there is not much definite evidence for interaction between mantle and core, with one minor exception. This exception refers to the so-called irregular fluc- tuations in the length of the day, which almost certainly arise from the electromagnetic torque exerted by the core on the mantle. But this is not a geophysical phenomenon of the first magnitude, as far as energy is concerned. The very existence of the magnetic field implies, as we have seen, that the core must be losing heat to the mantle at a rate of between one tenth and one quarter of the total heat loss from the earth; yet there is no way of proving in the present state of the art that the core heat plays a significant role in the energetic economy of the mantle, or that without it plate tectonics would not be operational or would operate in a very different mode. I, for one, suspect that core heat is significant, but I must admit that I cannot prove it. SOME FINAL REMARKS Let us now return to some of the broad questions raised in the introduc- tion to Chapter 1 and summarize some of the evidence. We consider first the matter of possible energy sources, the two contenders being radioactivity and gravitational energy. With regard to radioactivity, we distinguish between the "low potassium" and "high potassium" models. The first assumes the abundance of radioactive elements to be that given in Table 2-1; the high-potassium model assumes that the abundance of potassium in the earth is close to its solar abundance or to its abundance in chondritic meteorites, with much of the terrestrial potassium buried in the core. The latter model seems preferable, if only because it removes the obligation of explaining why the earth should be more depleted in potassium than chondritic meteorites. The low- potassium model produces today somewhat less heat than now escapes at the surface; it implies therefore a cooling earth. The magnitude of the gravitational energy source is much less cer

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122 ENERGETICS OF THE EARTH lain. We include in it (1) that part of the gravitational energy released during accretion that was not immediately radiated away (what we first called "original heat," (2) the gravitational energy released by separa- tion of the core, (3) the gravitational energy released by separation of the inner core, (4) the tidal dissipation that may have been important during the early days of the earth-moon system, and (5) the kinetic energy of impacting meteorites of large size traveling at orbital veloci- ties. Only two of these quantities (separation of core and separation of inner core) are known, if only rather approximately. The other three items (original heat, tidal dissipation, and meteorite impact) are very uncertain and highly speculative. Separation of the inner core is not an important item in the earth's energy budget. It implies a cooling earth and is therefore compatible only with the low-potassium model. Sepa- ration of core from mantle, if it ever took place, must have been an energetic event of the first magnitude (~103i J), and so must have been the first item in the above list. All told, it would not take much stretch- ing of the imagination to suggest that all gravitational sources combined could indeed provide all the energy needed to run the earth. Since we know for certain that the earth also contains some radioactive ele- ments, it seems that there is no problem in finding adequate sources; most probably radiogenic and gravitational sources are both involved. But in what proportion? Radioactive and gravitational sources are both time dependent, the gravitational ones strikingly so. It may then be surprising that much of our previous discussion dealt with steady-state models, e.g., the effi- ciency of a steady-state mantle or a steady-state core. This emphasis on the steady state is not purely for the sake of mathematical simplicity. It also stems from an analysis by Tozer (1972, 1977), who argues that in a convecting planetary body in which viscosity depends strongly on temperature, the temperature distribution is in the long run nearly independent of the intensity of the heat sources. This happens because an increase in heat generation that will raise locally the temperature will also lower the viscosity, thus allowing for a more vigorous convection that will rapidly carry the excess heat away; if the heat sources de- crease in intensity, the viscosity will rise, convection will slow down, less heat will be carried away, and the temperature will rise to near its previous value. This self-regulating property of a convecting planet adds credibility to the steady-state hypothesis. The high-potassium model is essentially a steady-state model. It pro- vides, as we have seen, all the heat now escaping from the earth, and is therefore incompatible with cooling. It should perhaps better be called quasi-steady, since radioactive sources necessarily decrease

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Core-Mantle Interactions 123 with time; yet a rough equality of surface heat flow and radioactive heat generation may have prevailed throughout most of the earth's history, while mantle temperatures remained nearly constant by the operation of Tozer's self-regulating mechanism. The high-potassium model does not, however, exclude early contributions from gravitational sources. As noted in Chapter 3, present temperatures in the lower mantle and core seem to be several thousand degrees higher than the original temperatures for instance, those calculated from accretion theory by Hanks and Anderson (1969~. Even if the core had retained all the heat generated in it by its 0.1 percent potassium, its temperature could not have risen by more than some 700 in 4 billion years. Thus it seems that a gravitational source (e.g., separation of the core) may be needed to account for present-day temperatures. We recall that separation of the core would release about 103~ J. enough to heat the whole core by 7500, or the whole earth by 1500. The low-potassium model, on the other hand, is incompatible with any kind of steady state since it implies that the mantle must be cooling at a rate of approximately 100 per 109 yr in order to provide for the difference between present heat flow and radiogenic heat production (see Table 2-11; the core must also be cooling to provide the gravita- tional energy that drives the dynamo. It is not certain that separation of the core could provide enough energy to heat up both the mantle and core to the somewhat higher temperatures required in this model; pos- sibly a large tidal event, or a higher initial accretion temperature, will be needed. At any rate, the low-potassium model demands more gravi- tational energy than the high-potassium model. Significant as the release of gravitational energy may be, sight should not be lost of the part played by the gravitational field in the develop- ment of the "structures" alluded to in the introduction to Chapter 1. As mentioned, our problem is not only to find adequate energy sources; we must also account for observed structures. Regional variations in heat flow as a function of distance from an oceanic ridge, or as a function of geological age of a continental province, are examples of such struc- tures, as are also horizontal temperature differences in the upper man- tle, localization of volcanoes, or, more generally, any large-scale departure from homogeneity, uniformity, or isotropy. Starting from a grossly homogeneous earth, with uniformly distributed isotropic heat sources, either radiogenic or gravitational, how does one go about producing such ordered structures as the magnetic field? The answer to this question is, of course, convection. Convection is a prime example of a nonequilibrium state in which highly disorganized heat sources and random fluctuations somehow contrive to produce

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124 ENERGETICS OF THE EARTH coherent, organized flow and an ordered thermal structure (Prigogine, 19781. That convection is a major geophysical process can no longer be doubted. No theory of the earth based on transfer of heat only by conduction has been able to account satisfactorily for the production of basaltic magma, which follows almost automatically from the convec- tion hypothesis (Verhoogen, 19541. Convection is the main heat carrier in the mantle, and the prime mover of plates. But there can be no natural convection without gravity. Gravity and, in the core, rotation of the earth are the factors through which struc- tures develop. Perhaps it is not surprising that so much of this book should be concerned with interactions of thermal and gravitational fields.