Energetics of the Earth (1980) / Chapter Skim
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5 CORE-MANTLE INTERACTIONS
5 CORE-MANTLE INTERACTIONS
Pages 100-124
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From page 100... ...
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)
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From page 101... ...
. 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)
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From page 102... ...
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 temperature T700 = 2600°K.
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From page 103... ...
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 temperature gradients are steepest and temperatures lowest.
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From page 104... ...
We first examine the two heat sources separately, postponing 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.
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From page 105... ...
(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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From page 106... ...
The facts of plate tectonics strongly suggest that the convection pattern in the mantle depends on time; plates suddenly change direction, oceanic ridges apparently migrate in longitude or latitude, or both, and large plates break up into smaller ones. Whereas it is generally *
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From page 107... ...
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 dependent atRa > 105-106.
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From page 108... ...
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 boundary 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.
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From page 109... ...
, for a fluid heated from within; heating from below, as in (c) , produces a hot rising sheet narrower than the cold sinking sheet.
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From page 110... ...
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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From page 111... ...
The heat flux q, it will be remembered, is the heat transported, mostly by convection, toward the surface of the outer core; its value at a point on the CMB will depend on the pattern of core convection. The inner core plays a role here, as Busse has shown (see, for instance, Busse and Cuong, 19771.
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From page 112... ...
THE VON ZEIPEL INSTABILITY The van Zeipel instability arises from van Zeipel's theorem, which states that hydrostatic equilibrium can be achieved in a homogeneous, rotating, self-gravitating body with distributed internal heat sources only if ~ is distributed according to a very special law (Verhoogen, 19481. It is well known that hydrostatic equilibrium in a homogeneous body requires that density, pressure, and temperature be constant on surfaces of constant potential (gravitational plus centrifugal)
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From page 113... ...
implies that the ratio of the heat flux q to gravitational acceleration g must be constant throughout the body. When conditions (5.15)
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From page 114... ...
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 continents seem difficult to move around; they certainly move more slowly than purely oceanic plates.
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From page 115... ...
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
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From page 116... ...
The two-phase region may, to a first approximation, be treated as homogeneous, with an effective expansion c', up cz = catpAT 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
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From page 117... ...
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.
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From page 118... ...
As applied to the olivine-spinel transition, 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)
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From page 119... ...
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.
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From page 120... ...
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.
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From page 121... ...
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.
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From page 122... ...
, 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 decrease 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.
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From page 123... ...
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 structures, as are also horizontal temperature differences in the upper mantle, localization of volcanoes, or, more generally, any large-scale departure from homogeneity, uniformity, or isotropy.
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From page 124... ...
Perhaps it is not surprising that so much of this book should be concerned with interactions of thermal and gravitational fields.
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