Heat in the Earth

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Heat in the Earth

• Volcanoes, magmatic intrusions, earthquakes,
  mountain building and metamorphism are all
  controlled by the generation and transfer of heat
  in the Earth.
• The Earths thermal budget controls the activity
  of the lithosphere and asthenosphere and the
  development of the basic structure of the Earth.

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• Heat arrives at the surface of the Earth from its
  interior and from the Sun.
• The heat arriving from the Sun is by far the
  greater of the two
• Heat from the Sun arriving at the Earth is 2x1017
  W
• Averaged over the surface this is 4x102 W/m2
• The heat from the interior is 4x1013 W and 8x10-2
  W/m2
• However, most of the heat from the Sun is
  radiated back into space. It is important because
  it drives the surface water cycle, rainfall, and
  hence erosion. The Sun and the biosphere keep the
  average surface temperature in the range of
  stability of liquid water.
• The heat from the interior of the Earth has
  governed the geological evolution of the Earth,
  controlling plate tectonics, igneous activity,
  metamorphism, the evolution of the core, and
  hence the Earths magnetic field.

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Heat Transfer Mechanisms

• Conduction
• Transfer of heat through a material by atomic or
  molecular interaction within the material
• Radiation
• Direct transfer of heat as electromagnetic
  radiation
• Convection
• Transfer of heat by the movement of the molecules
  themselves
• Advection is a special case of convection

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Conductive Heat Flow

• Heat flows from hot things to cold things.
• The rate at which heat flows is proportional to
  the temperature gradient in a material
• Large temperature gradient higher heat flow
• Small temperature gradient lower heat flow

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Imagine an infinitely wide and long solid plate
with thickness dz . Temperature above is T
dT Temperature below is T Heat flowing down is
proportional to The rate of flow of heat per
unit area up through the plate, Q, is
In the limit as dz goes to zero
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• Heat flow (or flux) Q is rate of flow of heat per
  unit area.
• The units are watts per meter squared, W m-2
• Watt is a unit of power (amount of work done per
  unit time)
• A watt is a joule per second
• Old heat flow units, 1 hfu 10-6 cal cm-2 s-1
• 1 hfu 4.2 x 10-2 W m-2
• Typical continental surface heat flow is 40-80 mW
  m-2
• Thermal conductivity k
• The units are watts per meter per degree
  centigrade, W m-1 C-1
• Old thermal conductivity units, cal cm-1 s-1 C-1
• 0.006 cal cm-1 s-1C-1 2.52 W m-1 C-1
• Typical conductivity values in W m-1 C-1
• Silver 420
• Magnesium 160
• Glass 1.2
• Rock 1.7-3.3
• Wood 0.1

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Lets derive a differential equation describing
the conductive flow of heat

• Consider a small volume element of height dz and
  area a.
• Any change in the temperature of this volume in
  time dt depends on
• Net flow of heat across the elements surface
  (can be in or out or both)
• Heat generated in the element
• Thermal capacity (specific heat) of the material

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The heat per unit time entering the element
across its face at z is aQ(z) . The heat per
unit time leaving the element across its face at
zdz is aQ(zdz) . Expand Q(zdz) as Taylor
series The terms in (dz)2 and above are small
and can be neglected
The net change in heat in the element is (heat
entering across z) minus (heat leaving across
Zdz)
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Suppose heat is generated in the volume element
at a rate A per unit volume per unit time. The
total amount of heat generated per unit time is
then A a dz Radioactivity is the prime
source of heat in rocks, but other possibilities
include shear heating, latent heat, and
endothermic/exothermic chemical
reactions. Combining this heating with the
heating due to changes in heat flow in and out of
the element gives us the total gain in heat per
unit time (to first order in dz as This tells
us how the amount of heat in the element changes,
but not how much the temperature of the element
changes.
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The specific heat cp of the material in the
element determines the temperature increase due
to a gain in heat. Specific heat is defined as
the amount of heat required to raise 1 kg of
material by 1?C. Specific heat is measured in
units of J kg-1 ?C-1 . If material has density ?
and specific heat cp, and undergoes a temperature
increase of dT in time dt, the rate at which heat
is gained is We can equate this to the rate at
which heat is gained by the element
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Simplifies to
In the limit as dt goes to zero
Several slides back we defined Q as
This is the one-dimensional heat conduction
equation.
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The term k/?cp is known as the thermal
diffusivity ?. The thermal diffusivity expresses
the ability of a material to diffuse heat by
conduction. The heat conduction equation can be
generalized to 3 dimensions
The symbol in the center is the gradient operator
squared, aka the Laplacian operator. It is the
dot product of the gradient with itself.
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This simplifies in many special situations. For
a steady-state situation, there is no change in
temperature with time. Therefore
In the absence of heat generation, A0
Scientists in many fields recognize this as the
classic diffusion equation.
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Talk at board about the qualitative behavior of
the Heat Conduction equation
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Equilibrium Geotherms

• The temperature vs. depth profile in the Earth is
  called the geotherm.
• An equilibrium geotherm is a steady state
  geotherm.
• Therefore

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Boundary conditions

• Since this is a second order differential
  equation, we should expect to need 2 boundary
  conditions to obtain a solution.
• A possible pair of bcs is
• T0 at z0
• QQ0 at z0
• Note Q is being treated as positive upward and z
  is positive downward in this derivation.

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Solution

• Integrate the differential equation once
• Use the second bc to constrain c1
• Note Q is being treated as positive upward and z
  is positive downward in this derivation.
• Substitute for c1

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Solution

• Integrate the differential equation again
• Use the first bc to constrain c2
• Substitute for c2
• Link to spreadsheet

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Oceanic Heat Flow Heat flow is higher over young
oceanic crust Heat flow is more scattered over
young oceanic crust Oceanic crust is formed by
intrusion of basaltic magma from below The fresh
basalt is very permeable and the heat drives
water convection Ocean crust is gradually covered
by impermeable sediment and water convection
ceases. Ocean crust ages as it moves away from
the spreading center. It cools and it contracts.
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These data have been empirically modeled in two
ways d2.5 0.35t2 (0-70 my) and d6.4
3.2e-t/62.8 (35-200 my)
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Half Space Model Specified temperature at top
boundary. No bottom boundary condition. Cooling
and subsidence are predicted to follow square
root of time.
Plate Model Specified temperature at top and
bottom boundaries. Cooling and subsidence are
predicted to follow an exponential function of
time. Roughly matches Half Space Model for first
70 my.
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The model of plate cooling with age generally
works for continental lithosphere, but is not
very useful. Variations in heat flow in
continents is controlled largely by changes in
the distribution of heat generating elements and
recent tectonic activity.
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Range of Continental and Oceanic Geotherms in the
crust and upper mantle
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Convection
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Conductive Geotherm 10-20 ?C per km Adiabatic
Geotherm 0.5-1.0 ?C per km Convective
Geotherm Adiabatic middle Thermal boundary
layer at top and bottom
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Solid and liquid in the Earth
Illustration of mantle melting during
decompression
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Rayleigh-Benard Convection

• Newtonian viscous fluid stress is proportional
  to strain rate
• A tank of fluid is heated from below and cooled
  from above
• Initially heat is transported by conduction and
  there is no lateral variation
• Fluid on the bottom warms and becomes less dense
• When density difference becomes large enough,
  lateral variations appear and convection begins
• The cells are 2-D cylinders that rotate about
  their horizontal axes
• With more heating, these cells become unstable by
  themselves and a second, perpendicular set forms
• With more heating this planform changes to a
  vertical hexagonal pattern with hot material
  rising in the center and cool material descending
  around the edges
• Finally, with extreme heating, the pattern
  becomes irregular with hot material rising
  randomly and vigorously.

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Rayleigh-Benard Convection

• The stages of convection have been modeled
  mathematically and are characterized by a
  non-dimensional number called the Rayleigh
  number
• a is the volume coefficient of thermal expansion
• g gravity
• d the thickness of the layer
• Q heat flow through lower boundary
• A, ?, k you know
• n is kinematic viscosity

The critical value of Ra for gentle convection is
about 103. The aspect ratio for R-B convection
cells is about 2-3 to 1 Ra above 105 will produce
vigorous convection Ra above 106 will produce
irregular convection
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• Ra for both the upper and lower mantle seems to
  be consistent with vigorous convection
• While R-B convection models are very useful, they
  do not approximate the Earth very well. The
  biggest problem is that they model uniform
  viscosity materials. The mantle is not uniform
  viscosity!
• Reynolds number indicates whether flow is
  laminar or turbulent
• All mantle convection in the Earth is predicted
  to be laminar
• Mantle convection movies from Caltech
• More mantle convection movies
• More
• More

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Studies like you did in lab, seemed to show that
subduction stopped at about 670 km depth. This
was interpreted to mean there was mantle
convection operating in the upper mantle that was
separate from convection in the lower mantle.
Two-layer vs. Whole Mantle Convection
Modern tomographic images give a very different
picture!
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Plate Driving Forces
Illustration of slab pull and ridge push