The Earth's Heat and Temperature
Some definitions and Units
Kelvin Scale
- begins at absolute zero. Water triple point at 273.16 K.
Celsius - same
units as Kelvin - Water triple point @ 0 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 mcal/cm2/sec = 41.9 mW/m2
Some Values in J/kg/K
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Note that the
change in density is r(T)
= ra(1 - a(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 |
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. |
Note that the heat flow Q = kDT/Dz (conductivity times the thermal gradient).
Some values of thermal conductivity:
Material | Thermal Conductivity (W/mC) |
Diamond | ~1600! ![]() |
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 |
Note - has units of length2/time - if a temperature change occurs with a time interval t then the changes will occur a distance on the order of Ökt. Say for a regular crustal rock k ~ 1.5 X 10-6 m2/s then:
Shows that conduction of heat in the earth is a very slow process!
Boehler (Diamond
Anvil on Pure Fe) - 4000 ; K +- 200 K at CMB
Knittle and
Jeanloz (Diamond Anvil of FeO) - 4800 C +- 500 C .
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 .
Temperatures in the Crust and the Concept of the Lithosphere
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.
What is the error function erf?
If k = 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.
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 Þ density increases with time Þ 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 mW/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
- n2),
h = thickness, E = Young's modulus, n
= 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), rmis the density of the mantle, riis 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(rm - ri).
Complex solution in elasticity but result is:
Flexure at
Subduction Zones
In this case
w(x) = (a2/2D)exp(-x/a)[-Msin(x/a)
+ (Va + M)cos(x/a)]
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
Heat in the Crust of Alberta
Image by
Dr. Walter Jones and colleagues, U of Alberta - report on geothermal potential
in Alberta
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
Ra = (graDTD3)/kh
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°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