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the total heat loss of the mantle. Attempts to
reconstruct the thermal history of the Earth
from a geophysical point of view have, since the
time of Lord Kelvin, been in apparent disagree-
ment with geochemical and geological observa-
tions. The geophysical approach uses simplified
parameterized models of mantle cool-
ing , usually involving whole-mantle convection.
Effects of time-dependence, sphericity, pressure,
plates and continents are generally ignored. The
rate of cooling in early Earth history obtained
in these models is generally too rapid to allow
a sufficient present-day secular cooling rate.
Geochemical estimates of radioactive element
concentrations in the mantle appear to be too
low to explain the observed present mantle heat
loss. With present estimates of radioactive heat
production in the mantle, simple parameterized
models of whole mantle convection lead to a cool-
ing of the Earth at the beginning of its history
thatistoofastandtheyareunabletoexplain
the present heat loss of the Earth. This is one of
the heat flow paradoxes of geophysics and it is
related to the age of the Earth paradox of Lord
Kelvin's time. There are several paradoxes associ-
ated with U, Th and K and their daughters such
as heat, helium, lead and argon.
The balance between internal heat produc-
tion and efficiency of heat transfer for a mantle
with a temperature-dependent viscosity is such
that the system is not sensitive to the initial con-
ditions and self-regulates at each time step. A
local increase in heat production or temperature
results in a lower viscosity and more vigorous
convection that carries away the excess heat.
Mantle convection may not be self-regulating.
Cooling may be regulated by plate tectonics --
involving the sizes and stiffnesses of the plates
and the distribution of continents -- leading to a
weak dependence of the heat flow on the man-
tle viscosity and temperature. A hotter mantle
results in a deeper onset of melting, more exten-
sive melting, and a thicker buoyant crust and a
dryer, stiffer lithosphere, which can reduce the
total heat loss. Mantle viscosity and plate rhe-
ology depend on water content as well as tem-
perature. These effects may be more important
than the lowering of mantle viscosity by high
temperature; a hotter mantle can actually result
inlowerheatflow.
It is common to assume the existence of a
hidden mantle reservoir -- a stealth or phan-
tom reservoir -- enriched in radiogenic elements
and sequestered from global mantle circulation,
to explain the present-day global heat budget.
A deep, invisible undegassed mantle reservoir
enriched in heat-producing elements has been
the traditional explanation for the various heat
flow and U--Pb--He paradoxes. But it is difficult to
store heat-producing elements in the deep man-
tle. It is more likely that heat flow is variable with
time, that plate tectonics controls the cooling
rate of the mantle, and that the depleted MORB-
reservoir does not occupy the whole of the upper
mantle. There are various missing element para-
doxes in geochemical mass-balance calculations
and there is good evidence that the deeper man-
tle may be irreversibly stratified. But the inacces-
sible regions are more likely to be refractory and
depleted. It is the assumption that depleted mid-
ocean ridge basalts and their depleted residues
represent the entire upper mantle that is respon-
sible for the idea that there are missing heat-
producing elements. Recycled and delaminated
crust, not currently at the Earth's surface, and
ultra-enriched magmas, such as kimberlites, can
readily make up the perceived He, Th and K
deficits.
Wrap up
One-dimensional and homogenous mantle mod-
els, or models with a downward increase in
radioactive heating have dominated the attention
of convection modelers. Paradoxes such as the
Archean catastrophe, overheating of the lower
mantle, persistence of cold continental keels and
the missing heat-source 'problem' can be traced
to these non-realistic assumptions and initial and
boundary conditions.
A variety of evidence indicates that high
temperatures and efficient gravitational differ-
entiation determined the initial conditions of
the Earth. On an Earth-sized body the effects
of accretional energy and high pressure in the
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