The Earth's Heat and Temperature

Geophysics 221

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 ice-water 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 film cutting through an ice 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:

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 .

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 .

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

Pressure, thermal expansion, heat capacity, 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

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.

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:

Geotherm: T(z,t) = erf[z/2 (kt)]

What is the error function erf?

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.

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:

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

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

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

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

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!

Hot spots

From the USGS publications website

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