Physics

1
Physics Chemistry of the EarthHeat Flow II

• Jesse Fisher Lawrence

2
Where Does Earths Heat Come From?
Planetary Accretion
Differentiation

• Accretion of the Earth
• Kinetic energy from impacts is transferred into
  heat.
• Early Heat Source
• Differentiation of the Earth
• Potential Energy is converted into heat through
  differentiation of the Earth. (Especially the
  formation of the core).
• Early Heat Source
• Tidal Heating?
• Very small amount of heating
• Decay of radiogenic isotopes,
• Current Possibly Large Heat Source

Tidal Heating
Earth
Moon
Sun
Decay Rate
3
Radiogenic Isotope Decay
Planetary Accretion

• Radiogenic Isotope Decay
• Some particles are only created by solar
  processes
• So, when the Earth formed, the particles were
  trapped inside.
• Some isotopes decay too fast to maintain Earths
  Heat today.
• 11Be Half-life 14 seconds
• Others decay too slow to keep Earth warm
• 147Sm ? 143Nd t1/2 106 Gyr
• 87Rb ? 87Sr t1/2 48 Gyr
• Ideal Candidates
• 232Th ? 208Pb t1/2 14 Gyr
• 238U ? 206Pb t1/2 4.5 Gyr
• 40K ? 40Ar t1/2 1.25 Gyr
• 235U ? 207Pb t1/2 0.704 Gyr

• The contribution of each isotope depends on the
  intrinsic heat generation and initial abundance

4
Heat Production
Planetary Accretion

• The contribution of each isotope depends on the
  intrinsic heat production per unit mass and
  initial abundance

5
Vector Calculus Review

• Divergence Theorem
• The flow rate within the volume can be described
  by the flow rate at the surface of the volume.
• If q is heat flux (W m-2) and there is a
  volumetric heat source or production Q (W m-3)
  like heat released from radioactive decay, then
• We can integrate both sides.
• Use the divergence theorem on the left side

RC
Q
RE
q
If the whole Earth has the same volumetric heat
source or radioactive decay, then Or
q 82 mW m-2 QH 8.88X10-12Wkg-1
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The Earth is not Homogeneous

• The Earth is not homogeneous
• The core likely provides 10 of the heat to the
  mantle (Qcore .1 Qmantle)
• Plumes rise from the core-mantle boundary
• This heat source helps drive convection.
• The continental crust has more radiogenic
  material than the rest of the mantle
• Qcontinents 56 mW m-2
• Continents make up 87 of the Earths crustal
  volume.
• Qcrust 23 mW m-2
• The Earths temperature profile has changed over
  time
• The current temperature is a cumulative effect of
  past heat loss and heat production.
• Convection models suggest that about 20 of the
  heat comes from the cooling history of the Earth.

q 82 mW m-2 Qmantle 5.56X10-12 W kg-1
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Steady State Solutions

• The easiest case is the steady state solution,
  where there is not time derivative
• Temperature derivative w.r.t. time zero.
• Velocity of material is zero.
• This satisfies LaPlaces equation
• or
• Given
• At zero depth,
• So,

V 0
T0
L
TL
qL
Z

• Now add a constant heat source in the layer

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Continental Heat Flow

• A good approximation of the heat source in
  continental crust is
• Where hf is the e-folding depth for decrease in
  heat production.
• The temperature profile with depth variable heat
  production becomes

T0
L
TL
qL
Z

• The heat flow becomes
• where Q0 0, qs qL.

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Time Dependant Heat Flow I

• Start with the 1D heat diffusion equation
• Dimensionalize the 1D diffusion equation
• where L is the length scale, and ? is the time
  scale.
• So dimensionally,
• For an average rock (? 10-6 m-2 s-1) the time
  scale can easily be converted to length scale per
  year (about ? X 107 s yr-1) with

This is the time scale it takes to cool a slab
(subducted lithosphere, a batholith, etc ) by
1/e.
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Time Dependant Heat Flow II

• Starting with a half-space cooling,
• The time it takes, ?, the material to cool from
  the surface to some depth, L, produces a total of
  qT Joules,
• where L is the length scale, and ? is the time
  scale.
• The amount of energy removed must also be

Equating the two sides This result is
the same as the previous result.