Heat Transfer in the Earth

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Heat Transfer in the Earth

• What are the 3 types of heat transfer ?

1. Conduction 2. Convection 3. Radioactive heating

• Where are each dominant
• in the Earth ?

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Heat Transfer in the Earth

• Conduction
• - Oceanic Lithosphere
• - Some conduction occurs
• everywhere a temperature
• gradient exists
• - Inner core (?)

• Convection
• - Ocean water
• - Mantle interior
• - Outer Core
• - Inner core (?)

• Radioactive heating
• - Mantle interior
• - Continental crust

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Radioactive Element Abundance in Continental Crust

• The major heat producing elements in the crust
  are
• 40K , 238U, 235U,
  232Th.

• These elements have a half-life of about 1-10
  Ga.

• Heat production from elements in the continental
  crust
• is 0.6 pW/Kg and can account for nearly ½
  the observed
• surface heat flow

For example A heat production value of 2.5
mW/m3 through a 10 km depth slice produces 25
mW/m2 surface heat flux.
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The Mantle Heat Budget Puzzle

• The observed surface heat flux is 60-100 mW/m2.

• Total crust 10

• Upper mantle 3
• (3 nW/m3 to 650 km)

• Full mantle 20 -50
• ( extend to 3000 km)

• TOTAL 65 max

• What other factors may contribute to surface
  heat flow ?

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The Mantle Heat Budget Puzzle

• The observed surface heat flux is 60-100 mW/m2.

• Convecting mantle plumes 10
• Lower mantle may have higher radiogenic
  concentration
• - Reservoirs of primitive mantle
• - Accumulation of subducted oceanic crust

• This still may leave a discrepancy of at least
  15-20

• Heat from the outer core could contribute can
  this be calculated ?

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The Mantle Heat Budget Puzzle

• What kind of convective behavior will a heat
  source
• at the base of a box produce ?

• Can the number and wavelength of plumes be
  calculated ?

• We can study convection with a combination of
• internal heat sources and base heating and
  study
• style and even number of plumes produced...

• We can compare these
• predictions to what we
• know about plumes in the
• Earth's mantle from surface
• observations (volcanism,
• seismic tomography, etc.)

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Convective Heat Transport

• Convection is fluid flow driven
• by internal buoyancy and gravity

• Buoyancy is driven by horizontal
• density gradients
• Buoyancy can be positive or
• negative and occurs when a
• boundary layer becomes unstable.
• Mantle convection in the Earth
• occurs by solid state deformation
• and creep mechanisms
• (the mantle is NOT a fluid) over
• millions of years.

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Convective Heat Transport

• There is an intimate relationship
• between interior convection
• and the surface topography
• that it produces.
• Most convecting systems are
• described by two thermal boundary layers (at the
  top and
• bottom). Some by only one TBL.

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Fluid Mechanics and Mantle Flow

• The Earth's interior deforms by creep mechanisms
  over
• long periods of time geologic time
• We approximate movement of solid rocks as a
  viscous material
• We use fluid mechanical laws to understand
  mantle flow over
• geologic time scales

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Fluid Mechanics and Mantle Flow

• First we consider the governing conservation
  equations
• Conservation of Mass
• Conservation of Momentum
• Conservation of Energy

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See Class notes on development of Navier-Stokes
Equation
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Buoyancy the Thermal Expansion

• In the lower mantle thermal properties may be
  pressure-dependent
• The density contrast in the upper mantle for a
  ?? of 1000
• is about 3.
• In the lower mantle with thermal expansion
  reduced by only a
• factor of 3, the density contrast is only 1.

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Buoyancy in the Earth

• What other areas of the Earth has density
  differences ?

• Oceanic crust (due to mineralogy composition
• The contrast between oceanic crust (2.9 g/cm3)
  and
• the mantle is 12!
• The density contrast across the Mantle
  Transition Zone is 15.
• (Due to phase changes, so not a buoyancy
  source).
• The density contrast between the upper and lower
  mantle is small.

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Buoyancy in the Earth

• The buoyancy force (FB) of a ball bearing is
  -0.02 N
• FB for a plume head of 1000 km diameter and ??
  300 oC
• is a buoyancy of 2 x 1020 N.
• Subducting lithosphere to 600 km depth exerts a
  negative
• buoyancy of -40 x 1012 N per meter of trench.

Are plumes more dominant ? - Consider the length
of oceanic trenches...over 30,000 km!
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Buoyancy in the Earth

• Oceanic crust undergoes different phase
  transformations than
• the lithospheric mantle during subduction, so
  may be more
• or less dense than surrounding mantle at
  different times...
• Crustal weight will be more important in young
  lithosphere
• which is thinner (or earlier in the Earth's
  history...).
• The large range of magnitudes (10-20 orders of
  magnitude!)
• in buoyancy for Earth processes emphasize
  that fact that we
• must consider the structural volumes and not
  just density
• anomalies alone.

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• ACTIVITY
• Consider the force of a subducting plate entering
  into the mantle
• The oceanic plate has a negative buoyancy and
  sinks of its own
• weight because it is more dense.
• As it sinks it is surrounded by viscous mantle
  which resists
• the plate motion by viscous shear.
• The viscous stresses influence the plate
  velocity,
• slowing it down.
• The plate velocity adjusts until an equilibrium
  (force balance)
• is reached between the opposing forces of
  buoyancy and
• viscous stress.

Analytical Calculations of Convection
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Subduction, Mantle Viscosity, and Plate Velocity

• The buoyancy of the descending lithosphere is
  given by
• (see handout for diagram)

FB- -g L? ? ? ?T
?? is the average Temperature difference between
the slab and mantle and is approximated by -T/2
FB- -g L? ? ? T/2
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Subduction, Mantle Viscosity, and Plate Velocity

• Once plate velocity adjusts to the viscous shear
  in the mantle
• the forces are balanced,

Buoyancy Force Shear Force
FB ?
-g L? ? ? T/2 ? 2V

• Solve for V to get the resultant plate velocity

V -g L? ? ? T/4?
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Scaling Fluid Dynamic Models to Earth Systems

• The theory we just developed from assumptions of
• buoyancy forces and shear forces also tell us how
• various physical properties scale with each
  other.
• For example in the equation for fluid velocity

V L g ? ? T (sqrt(?)) /4? 2/3
If viscosity was 10 times lower then how would
the velocity change..... ?
the velocity would then increase by 10 2/3 (
4.6 times greater).
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Scaling Fluid Dynamic Models to Earth Systems

• To obtain

(D/? )3 ???g ??T D3 / 4???

• This is written in a general form which is often
  used
• to describe a non-dimensional number, the
  Rayleigh number.

Ra ??g ??T D3 / ???

• What is a non-dimensional number ?

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Non-Dimensional Numbers

• True compatability requires both dynamic
  and thermal similarity
• Prandlt number is a property of the fluid
• Pr ?? / ??
• (ratio diffusion of momentum and vorticity
  / diffusion of heat)
• In the Earth where viscosities are high, Pr
  1026 !
• Reynolds number is a property of fluid flow
• Re ??VL / ??
• (ratio of inertial forces / viscous
  forces)
• In the Earth, Re 10-12

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Non-Dimensional Numbers

• The Nusselt and Rayleigh numbers give thermal
  similarity
• Nusselt number describes thermal properties
• Nu LFheat / ????
• (ratio of total heat flux / conductive
  heat flux)
• Rayleigh number describes thermal and dynamic
  properties

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Non-Dimensional Numbers
Other relevant scaling parameters

• Length scale ??/ D
• Velocity scale V ??/ D
• Characteristic time t D2/ ?

Can you use any of these non-dimensional
parameters in your class projects ?
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Boundary Layer Theory
Boundary layers are everywhere!

Wind Chill Factor wind that is strong enough to
blow away the warm thermal boundary layer
surrounding your skin.
Airplane wing note particles in boundary layer
surrounding wing geometry
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