Geophysics 221
The Earth's Heat and Temperature
Some definitions and Units
Temperature Scales
Temperature - a measure of the kinetic energy of the molecules
in a substance - for a perfect gas the average energy of a molecule
is equal to mv2/2 = kT where k = Boltzmann's constant = 1.38 X
10-23 joules/K.
Kelvin Scale - begins at absolute zero. Water triple point at
273.16 K.
Celsius - same units as Kelvin - Water triple point @ 0 C.
Heat and Energy
Calorie - amount of work needed to be done in order to raise the
temperature of 1 gram of water 1 degree C.
Joule - Amount of energy required to push with a force of 1 Newton for
1 meter. 1 Calorie = 4.1868 J.
Watt - Unit of power. 1 Watt = 1 Joule/second
Heat Flux (Flow): amount of energy flowing through an area in a given
time. In geophysics, typical SI unit = mW/m2. Old units are 'Heat Flow
Units' = HFU = 1 cal/cm2/sec = 41.9 mW/m2
Thermal Properties of Materials
Specific Heat (Heat Capacity) = cp (at constant pressure),
measure of the amount of heat required to increase the
temperature of materials. Q = change in heat content = cpmT,
m = mass of body.
Some Values in J/kg/K
Water
4180
Ice (-10 C)
2100
Iron
447
Glass
67
Olivine
815
Mantle
1260
Typical Rock
700
Periclase (MgO)
924
Volume Co-efficient of Thermal Expansion, measure of the change in
the volume with increase in temperature,
,
Note that the change in density is (T) = a(1 - (T - Ta))
Some values in units of 10-5 K-1
Granite
2.4
Gabbro
1.6
Peridotite
2.4
Mantle (Jarvis & Peltier)
1.4
Halite
13
Steel
1.1
Glass
0.1 - 1.3
Ice
5
Latent Heat of phase transformation. Exothermic - energy
released, endothermic - energy absorbed upon transformation.
Example, latent heat of melting of water - temperature of the icewater mixture stays buffered at the melting temperature until all of
the ice is consumed. Units in J/kg.
Some values in kJ/kg (most at atmospheric pressure)
Water to Ice (fusion)
335
Molten Fe to Solid Fe
275
Basalt (Lava Lakes, Hawaii)
400
Olivine to Spinel (400 km)
Exothermic? Occurs shallow
Spinel to Perovskite (670 km)
Endothermic? Occurs deeper.
Thermal Conductivity. Describes how easily heat can be
transported through a material. The higher the thermal
conductivity the greater the heat transported. Often related to
electrical conductivity.
Note that the heat flow Q = kT/z (conductivity times the
thermal gradient).
Some values of thermal conductivity:
Material
Thermal Conductivity (W/mC)
Diamond
~1600!
Diamond fi
crystal due to thermal conduction from the person's fingers
Silver
418
Magnesium
159
Glass
1.2
Sedimentary Rock
1.2 to 4.2 (Turcotte & Schubert)
Granite
2.4 to 3.8
Basalt
1.3 to 2.9 (Turcotte & Schubert)
Pyroxenite
4.1 to 5
Upper Mantle
6.7 (Jarvis & Peltier)
Lower Mantle
20 (Jarvis & Peltier)
Wood
0.1
Thermal Diffusivity: Describes the ability of a material to lose
heat by conduction - depends on conductivity, density, and heat
capacity by = k/Cp.
Note - has units of length2/time - if a temperature change occurs
with a time interval  then the changes will occur a distance on the
order of . Say for a regular crustal rock  ~ 1.5 X 10-6 m2/s
then:
Shows that conduction of heat in the earth is a very slow process!
Temperature Within the Earth
How is the temperature determined? Use constraints of:
1.
Outer core must be molten  at or above melting point
(liquidus)
Boehler (Diamond Anvil on Pure Fe) - 4000  K +- 200
K at CMB
Knittle and Jeanloz (Diamond Anvil of FeO) - 4800 C +500 C .
2.
Inner core solid  at or below solidification point (solidus)
Boehler (Diamond Anvil on Pure Fe) - 4850 K +- 200 K
.
Yoo and Ahrens (Shock Wave on Pure Fe) - 7000 K
Shankland (Theory on Melting of Pure Fe) - 6160 K +250 K .
3.
4.
5.
6.
Mantle and Crust propagate S waves  solid  above
melting temperature
Asthenosphere has low rigidity  close to melting
Phase transitions at 400 km and 670 km depth  can
constrain from laboratory experiments.
Calculate assuming an adiabatic temperature gradient (i.e.
what would the temperature be if you simply compressed
the materials to the pressures expected) - Need knowledge
of
o
Pressure, thermal
expansion, heat capacity,
o
1.
density, and bulk
modulus.
Provides a lower bound
for the temperature
Calculate assuming a melting point temperature gradient
(i.e. what is the minimum temperature at which the mantle
will melt?) - Need knowledge of
o
o
Latent heat of melting,
gravity, solid and melt
densities
Provides an upper bound
for temperature in the
mantle.
Temperatures in the Crust and the Concept of the
Lithosphere
Lithosphere - rigid outer section of the earth above the
asthenosphere. It is the part of crust and uppermost mantle which
is sufficiently strong to move as a unit and define the plates. Two
definitions of the lithosphere:
Thermal Lithosphere - that portion of the earth in
which heat is transferred primarily by conduction
Mechanical lithosphere - that portion of the earth
which has sufficient strength so that it acts a a brittle
or rigid solid - cannot convect.
As such, because temperature will control the
mechanical properties of the rocks the thickness of
the lithosphere is dependent on the heat flow and
geothermal gradient. Worth looking at global heat
flow at this point.
Ridges - most of heat flux from the planet.
Subduction zones - low heat flux
Continents - variable - depends on age to some
degree - younger continental regions have higher
heat flow.
http://www.windows.umich.edu/cgi-bin/redirect.cgi/earth/interior/lithospheric_motion.html
Oceanic Lithosphere
Age Dependent - Simplest model assumes generation at
temperature Ta ridge with no heat generation. Must solve a
differential equation (see Fowler page 239) with the main results
that:
1.
Geotherm: T(z,t) = erf[z/2 (kt)]
What is the error function erf?
2.
3.
Heat Flow: Q(t) = -kTa/sqrt (t)
Lithospheric Thickness: Assume temperature in the
asthenosphere at the ridge axis Ta= 1300 C and the base of
the lithosphere defined by T = 1100 C. In this case one can
say that L = 2.016sqrt(t).
If  = 10-6 m2s-1 then L = 11sqrt(t) in kilometres and t in
Ma then will have the following curve but only good to
about 70 Ma. After this time the lithospheric thickness (and
heat flow and ocean depth) has stabilized and come into
equilibrium with the convecting asthenosphere. Oceanic
plate stays more or less the same.
Supported by studies of surface wave dispersion.
4.
Depth of the Ocean Floor with age
Age from MidAtlantic Ridge Topography of North Atlantic
Seafloor.
Depth stabilizes at about 70 Ma to ~5 km then increases
very slowly past this time. Effect due to increase in density
from thermal contraction as plate cools  density
increases with time  lithosphere must find new isostatic
equilibrium.
For t < 70 Ma, d = 2.5 km + 0.35t1/2 (t in Ma) (follows Airy)
for t 70 Ma, d = 6.4 km - 3.2exp(-t/62.8) (Lithosphere contstant
- follows Pratt).
Continental Lithosphere
Additional complication - continental crust contains
substantial radioactive elements - high internal heat
generation.
Heat transmitted primarily by conduction
Simplest model - plate of thickness D in which
radioactive materials are presumed to be uniformly
concentrated and producing heat at a rate A overlying the
rest of the earth which inputs a constant basal heat flux Qr
At equilibrium, T(z) = -Az2/2k + Qoz/k + Tsurface where
Qo is the observed surface heat flow.
Further Qo = Qr + AD (see Fowler, page 244)
Standard Model: A = 1.25 W/m3, k = 2.5 W/mC, and
Qr = 21 mW/m2 for a 50 km thickness (model after that
given by Fowler page 228).
Continental Lithosphere
Rigidity of the Lithosphere
Plates have the ability to bend - the static loading by local
masses will have an effect on bending and topography. This
is quantified by the
flexural rigidity D = Eh3/12(1 - 2),
h = thickness, E = Young's modulus,  = Poisson's ratio
(see seismic lectures for definitions of these elastic
parameters). Units of D are Nm = bending moment. D is a
fundamental parameter describing the plate - it tells how
much it will resist bending.
Some examples of this include:
Loading by Oceanic
Islands
Big island of Hawaii from space
Note zone of deepening near big Island - also increased topography
leading up to it due to increased heat in mantle (see hot spots a
little later).
Principal Idea shown in the cartoon below
L is the load (in terms of Force/area), mis the density of the mantle, iis
the density of the material infilling the depression.
L pushes down but is balanced by elastic deformation of the plate and bouyancy
according to Archimedes of gw( m - i).
Complex solution in elasticity but result is:
1.
2.
3.
w(x) = woexp(-x/)[cos(x/) + sin(x/)] where
wo = L3/8D is the maximum deflection beneath the load L
 = [4D/(m - i)g]1/4 is called the flexural parameter.
Note forebulge at about 200 km from the load. L = 1, D =
9.6e22 Nm, h = 24 km, mantle density = 3.3, fill (water)
density = 1.0, g = 10 ms^-2, E = 70 GPa,
Poisson's ratio = 0.25
Flexure at Subduction Zones
In this case
w(x) = (2/2D)exp(-x/)[-Msin(x/) + (V +
M)cos(x/)]
Also gives a slight bulge before the plate descends.
Formation of Sedimentary Basins
Called Flexural basins - include the Western
Canadian Sedimentary Basin.
Load at the surface produces a foreland basin.
Width of basin depends on lithospheric thickness.
The Rockies
o
o
Thrusting and emplacement form the west (starting
about140 Ma to about 40 Ma).
From about 35 Ma large amount of erosion (also
isostatic uplift) - almost 10 km of material has been
removed from the centre
Annual variations no more than ~100 m. Glacial
influences ~1 to 2 km.
Heat in the Crust of Alberta
Image by Dr. Walter Jones and colleagues, U of
Alberta - report on geothermal potential in Alberta
Cooling of the rest of the earth
Mantle Convection - Temperature increases when adiabatic temperature is
slightly exceeded.
Balance of forces - when a material heat up it expands becomes less dense bouyant force F = Volume time density times gravity times expansion
coefficient times temperature above adiabatic.
Balanced by
1.
2.
3.
Thermal conduction - if heat moves out to fast
the material cools to the adiabat and is stable depends on thermal diffusivity 
Resitive drag due to viscosity 
Total resistance proportional to 
Balance described by Rayleigh Number:
Ra = (gTD3)/
Convection will occur once Ra exceeds a critical value (not easy to find).
In a flat layer convection begins once Ra > 658
Once Ra ~10000 then all heat transport by convection.
(see http://cass.jsc.nasa.gov/science/kiefer/)
Thermal Boundary Layer at 670 km discontinuity
This animation shows the deformation in a slab of subducted oceanic lithosphere as it
impinges on a density interface at a depth of 670 km. The slab is 80 km thick and is
subducted at 8 cm/yr for a period of 12 Myr in this animation. The slab is denser than its
surroundings in the upper mantle, but it deforms and buckles when it meets resistance at
the 670 km interface
http://www.earth.monash.edu.au/Department/greg_houseman.html
Convection
The hot rock (yellow) rises slowly as the denser cold rock (blue) sinks. The layer is at least 700 km thick,
and could be as thick as 2900 km. The rock is at temperatures of order 1000 to 2000&deg;C and creeps like
a very viscous fluid. Its viscosity is about 20 orders of magnitude greater than that of water so velocity is
only centimeters per year, and the time interval of this animation is of order 10 million years.
Mantle Avalances!
http://www.npaci.edu/envision/v14.2/tackley.html
Hot spots
http://volcano.und.nodak.edu/vwdocs/vwlessons/hot_spots/introduction.html
From the USGS publications website
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