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OCR for page 100
CoreMantle
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 108 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 109 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
OCR for page 100
CoreMantle 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
coremantle 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 + ~ edVQO,
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 To300°K. 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 300°K.
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 = 1300°K 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 = 2600°K. On this adiabat, the temperature at 400 km is
1838°K or 1565°C. In the layer, k = 4 W/m deg.
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CoreMantle Interactions 103
3. Layer 3, from 700 to 2800 km, with temperature scale height ha =
9130 km, to give T2800 = 3300°K. In layer 3, k = 6 W/m deg.
4. A lower boundary layer (LBL), extending to the coremantle
boundary, with an average gradient of 10°/km. At the core boundary,
TCA!B = 4300°K 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
2050°K, 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 W10~2 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 102 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
108 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 closepacked
structure of minerals in their highpressure 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
OCR for page 100
CoreMantle 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
(59)
(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 108 W/m3. We take ~ = 2 x 105/deg,
K = 1.2 x 106 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 fixedsurface 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.
OCR for page 100
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 freesurface boundary conditions, the
singlecell pattern is maintained up to a Rayleigh number of 5 x 106; for
a fixedsurface boundary, a twocell pattern appears at Ra = 5 x 105.
What happens at Ra10~° 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 51. 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.
OCR for page 100
CoreMantle 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 52) to large Pr, it appears
that in a fluid heated from below, the flow might become time depen
dent atRa > 105106. 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 twodimensional (elongated rolls) to
a threedimensional, but steady, pattern that, at a still higher Rayleigh
OCR for page 100
108 ENERGETICS OF THE EARTH
106~
~ 105
D

._
C)
A
104
TV
of
Turbulent flow /
/ Time
dependent
/
/
.
a
. Dimensional
/ ~ ~ lot E, Flow 
_ _ .
1
,~; ·3dimensional' 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 timedependent
flow. Reproduced with permission from Krishnamurti (1970).
number, becomes time dependent. This timedependent 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 tworoll
pattern is approximately restored. A new instability in the upper bound
ary layer develops, leading again to the fourroll pattern, and the cycle
is repeated on a time scale that, for the mantle, would be on the order
of 107108 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.
OCR for page 100
CoreMantle 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
53~. 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.
OCR for page 100
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 deepmantle 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
02 W/m2 (S is the area of the coremantle 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
OCR for page 100
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 northsouth (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 nearmarginal 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 timedependent 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 presentday 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
OCR for page 100
CoreMantle 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
olivinespinel 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 PT 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.,
OCR for page 100
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 twophase
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 olivinespinel
OCR for page 100
CoreMantle 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 olivinespinel 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
twophase layer of thickness 2d; the total thickness of the three layers
is 2D. A parameter R is defined as
(X (163a13a ~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 singlephase 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
twophase 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 HT106107 m, but it may not be
met in a twophase region where HT ~ 104105 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 olivinespinel transi
tion, their calculations seem to show that for a mantle viscosity less
than 3 x 10221 x 1023 cm2/s, doublecell convection requires a smaller
overall superadiabatic gradient than singlecell 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 olivinespinel 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 singlephase 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 10i 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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CoreMantle Interactions 119
sumably a complicated, threedimensional, timedependent 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 timedependent character of mantle convection, which could easily
be explained on the FosterJones 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 longperiod (>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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CoreMantle 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 socalled 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 21; the highpotassium 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 earthmoon 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 lowpotassium 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 steadystate models, e.g., the effi
ciency of a steadystate mantle or a steadystate 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 selfregulating property of a convecting planet
adds credibility to the steadystate hypothesis.
The highpotassium model is essentially a steadystate 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 quasisteady, since radioactive sources necessarily decrease
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CoreMantle 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 selfregulating mechanism. The highpotassium 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 presentday 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 lowpotassium 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 211; 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 lowpotassium model demands more gravi
tational energy than the highpotassium 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 largescale
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.