NOt MIne :)

1. 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 ?

2. 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

3. Radioactive Element Abundance in Continental Crust
 The major heat producing elements in the crust are
40
K ,
238
U,
235
U,
232
Th.
 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.

4. 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 ?

5. 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 ?

6. 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.)

7. 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.

8. 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.

13. Fluid Mechanics and Mantle Flow
 First we consider the governing conservation equations
 Conservation of Mass
 Conservation of Momentum
 Conservation of Energy

14. 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

15. See Class notes on development of Navier-Stokes Equation

20. 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%.

21. 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.

22. 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
o
C
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!

23. 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.

24. Analytical Calculations of Convecti
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.

25. Subduction, Mantle Viscosity, and P
 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

28. Subduction, Mantle Viscosity, and P
 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

33. 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).

36. Scaling Fluid Dynamic Models to Earth Systems
 To obtain:
(D/ )
3
= g T D
3
/ 4
 This is written in a general form which is often used
to describe a non-dimensional number, the Rayleigh number.
Ra = g T D
3
/ 
 What is a non-dimensional number ?

40. Non-Dimensional Numbers
True compatability requires both dynamic and th
 Prandlt number: is a property of the fluid
Pr =  / 
(ratio: diffusion of momentum and vorticity
In the Earth where viscosities are high, Pr ~
 Reynolds number: is a property of fluid flow
Re = VL / 
(ratio: of inertial forces / viscous forces)
In the Earth, Re ~ 10
-12

41. Non-Dimensional Numbers
The Nusselt and Rayleigh numbers give thermal sim
 Nusselt number: describes thermal properties
Nu = LFheat / 
(ratio: of total heat flux / conductive heat
 Rayleigh number: describes thermal and dyna

43. Non-Dimensional Numbers
Length scale = / D
Velocity scale: V = / D
Characteristic time: t = D
2
/ 
Other relevant scaling parameters:
Can you use any of these non-dimensional parameters
in your class projects ?

44. Boundary Layer Theory
Boundary layers are everywhere!
Airplane wing: note particles
in boundary layer surrounding
wing geometry
Wind Chill Factor: wind
that is strong enough to blow
away the warm thermal boundary
layer surrounding your skin.