Heat in the Earth Heat in the Earth Volcanoes, magmatic intrusions, Volcanoes, magmatic intrusions, earthquakes, mountain building earthquakes, mountain building and metamorphism are all and metamorphism are all controlled by the generation and controlled by the generation and transfer of heat in the Earth. transfer of heat in the Earth. The Earth’s thermal budget The Earth’s thermal budget controls the activity of the controls the activity of the lithosphere and asthenosphere lithosphere and asthenosphere and the development of the basic and the development of the basic structure of the Earth. structure of the Earth.
Transcript
Heat in the EarthHeat in the Earth
Volcanoes, magmatic intrusions, Volcanoes, magmatic intrusions, earthquakes, mountain building and earthquakes, mountain building and metamorphism are all controlled by metamorphism are all controlled by the generation and transfer of heat the generation and transfer of heat in the Earth.in the Earth.
The Earth’s thermal budget controls The Earth’s thermal budget controls the activity of the lithosphere and the activity of the lithosphere and asthenosphere and the development asthenosphere and the development of the basic structure of the Earth.of the basic structure of the Earth.
Heat arrives at the surface of the Earth from its interior and Heat arrives at the surface of the Earth from its interior and from the Sun.from the Sun.
The heat arriving from the Sun is by far the greater of the The heat arriving from the Sun is by far the greater of the twotwo
Heat from the Sun arriving at the Earth is 2x10Heat from the Sun arriving at the Earth is 2x101717 W W
Averaged over the surface this is 4x10Averaged over the surface this is 4x1022 W/m W/m22
The heat from the interior is 4x10The heat from the interior is 4x101313 W and 8x10 W and 8x10-2-2 W/m W/m22
However, most of the heat from the Sun is radiated back into However, most of the heat from the Sun is radiated back into space. It is important because it drives the surface water space. It is important because it drives the surface water cycle, rainfall, and hence erosion. The Sun and the biosphere cycle, rainfall, and hence erosion. The Sun and the biosphere keep the average surface temperature in the range of keep the average surface temperature in the range of stability of liquid water.stability of liquid water.
The heat from the interior of the Earth has governed the The heat from the interior of the Earth has governed the geological evolution of the Earth, controlling plate tectonics, geological evolution of the Earth, controlling plate tectonics, igneous activity, metamorphism, the evolution of the core, igneous activity, metamorphism, the evolution of the core, and hence the Earth’s magnetic field.and hence the Earth’s magnetic field.
Heat Transfer MechanismsHeat Transfer Mechanisms ConductionConduction
Transfer of heat through a material by atomic or Transfer of heat through a material by atomic or molecular interaction within the materialmolecular interaction within the material
RadiationRadiation Direct transfer of heat as electromagnetic radiationDirect transfer of heat as electromagnetic radiation
ConvectionConvection Transfer of heat by the movement of the molecules Transfer of heat by the movement of the molecules
themselvesthemselves
Advection is a special case of convectionAdvection is a special case of convection
Conductive Heat FlowConductive Heat Flow Heat flows from hot things to cold Heat flows from hot things to cold
things.things. The rate at which heat flows is The rate at which heat flows is
proportional to the temperature proportional to the temperature gradient in a materialgradient in a material Large temperature gradient – higher Large temperature gradient – higher
heat flowheat flow Small temperature gradient – lower heat Small temperature gradient – lower heat
flowflow
Imagine an infinitely wide and long solid plate with thickness δz .
Temperature above is T + δT
Temperature below is T
Heat flowing down is proportional to:
The rate of flow of heat per unit area up through the plate, Q, is:
z
TTT
)(
z
TkzQ
z
TTTkQ
)(
z
TkzQ
)(
In the limit as δz goes to zero:
Heat flow (or flux) Q is rate of flow of heat per unit Heat flow (or flux) Q is rate of flow of heat per unit area.area. The units are watts per meter squared, W mThe units are watts per meter squared, W m-2-2
Watt is a unit of power (amount of work done per unit Watt is a unit of power (amount of work done per unit time)time)
A watt is a joule per secondA watt is a joule per second Old heat flow units, 1 hfu = 10Old heat flow units, 1 hfu = 10-6-6 cal cm cal cm-2-2 s s-1-1
1 hfu = 4.2 x 101 hfu = 4.2 x 10-2-2 W m W m-2-2
Typical continental surface heat flow is 40-80 mW mTypical continental surface heat flow is 40-80 mW m-2-2
Thermal conductivity k Thermal conductivity k The units are watts per meter per degree centigrade, W mThe units are watts per meter per degree centigrade, W m--
1 °1 °CC-1-1
Old thermal conductivity units, cal cmOld thermal conductivity units, cal cm-1 -1 ss-1 °-1 °CC-1-1
0.006 cal cm0.006 cal cm-1 -1 ss-1°-1°CC-1 -1 = 2.52= 2.52 W mW m-1 °-1 °CC-1-1
Typical conductivity values in W mTypical conductivity values in W m-1 °-1 °CC-1-1 : : SilverSilver 420420 MagnesiumMagnesium 160160 Glass Glass 1.21.2 RockRock 1.7-3.31.7-3.3 WoodWood 0.10.1
Consider a small volume element of height δz and area a.
Any change in the temperature of this volume in time δt depends on:
1. Net flow of heat across the element’s surface (can be in or out or both)
2. Heat generated in the element
3. Thermal capacity (specific heat) of the material
Let’s derive a differential equation describing the conductive flow of heat
The heat per unit time entering the element across its face at z is aQ(z) .
The heat per unit time leaving the element across its face at z+δz is aQ(z+δz) .
Expand Q(z+δz) as Taylor series:
The terms in (δz)2 and above are small and can be neglected
...
!3!2)()(
3
33
2
22
z
Qz
z
Qz
z
QzzQzzQ
The net change in heat in the element is (heat entering across z) minus (heat leaving across Z+δz):
z
Qza
zzaQzaQ
)()(
Suppose heat is generated in the volume element at a rate A per unit volume per unit time. The total amount of heat generated per unit time is then
A a δz
Radioactivity is the prime source of heat in rocks, but other possibilities include shear heating, latent heat, and endothermic/exothermic chemical reactions.
Combining this heating with the heating due to changes in heat flow in and out of the element gives us the total gain in heat per unit time (to first order in δz as:
This tells us how the amount of heat in the element changes, but not how much the temperature of the element changes.
z
QzazAa
The specific heat cp of the material in the element determines the temperature increase due to a gain in heat.
Specific heat is defined as the amount of heat required to raise 1 kg of material by 1C.
Specific heat is measured in units of J kg-1 C-1 .
If material has density ρ and specific heat cp, and undergoes a temperature increase of δT in time δt, the rate at which heat is gained is:
We can equate this to the rate at which heat is gained by the element:
t
Tzacp
z
QzazAa
t
Tzacp
z
QzazAa
t
Tzacp
z
QA
t
Tcp
z
QA
t
Tcp
In the limit as δt goes to zero:
z
TkzQ
)(
Several slides back we defined Q as:
2
2
z
TkA
t
Tcp
pp c
A
z
T
c
k
t
T
2
2
This is the one-dimensional heat conduction equation.
Simplifies to:
The term k/ρcp is known as the thermal diffusivity κ. The thermal diffusivity expresses the ability of a material to diffuse heat by conduction.
The heat conduction equation can be generalized to 3 dimensions:
pp c
A
z
T
y
T
x
T
c
k
t
T
2
2
2
2
2
2
pp c
AT
c
k
t
T
2
The symbol in the center is the gradient operator squared, aka the Laplacian operator. It is the dot product of the gradient with itself.
z
T
y
T
x
TT
zyx
,,
,,
2
2
2
2
2
22
2
2
2
2
2
22
z
T
y
T
x
TT
zyx
pp c
AT
c
k
t
T
2
This simplifies in many special situations.
For a steady-state situation, there is no change in temperature with time. Therefore:
k
AT 2
In the absence of heat generation, A=0:
Tc
k
t
T
p
2
Scientists in many fields recognize this as the classic “diffusion” equation.
Talk at board about the Talk at board about the qualitative behavior of the qualitative behavior of the Heat Conduction equationHeat Conduction equation
Equilibrium GeothermsEquilibrium Geotherms
The temperature vs. depth profile in The temperature vs. depth profile in the Earth is called the geotherm.the Earth is called the geotherm.
An equilibrium geotherm is a steady An equilibrium geotherm is a steady state geotherm.state geotherm.
Therefore:Therefore: 2
20,
T T Aand
t z k
Boundary conditionsBoundary conditions
Since this is a second order Since this is a second order differential equation, we should differential equation, we should expect to need 2 boundary expect to need 2 boundary conditions to obtain a solution.conditions to obtain a solution.
A possible pair of bc’s is:A possible pair of bc’s is: T=0 at z=0T=0 at z=0 Q=QQ=Q00 at z=0 at z=0 Note: Q is being treated as positive upward and z is positive downward in this Note: Q is being treated as positive upward and z is positive downward in this
derivation.derivation.
SolutionSolution
Integrate the Integrate the differential differential equation once:equation once:
Use the second bc Use the second bc to constrain cto constrain c11
Note: Q is being treated as positive Note: Q is being treated as positive upward and z is positive downward in upward and z is positive downward in this derivation.this derivation.
Substitute for cSubstitute for c11::
1
T Azc
z k
01
Qc
k
0QT Az
z k k
SolutionSolution
Integrate the Integrate the differential equation differential equation again:again:
Use the first bc to Use the first bc to constrain cconstrain c22
Rayleigh-Benard ConvectionRayleigh-Benard Convection Newtonian viscous fluid – stress is proportional to strain Newtonian viscous fluid – stress is proportional to strain
raterate A tank of fluid is heated from below and cooled from aboveA tank of fluid is heated from below and cooled from above
Initially heat is transported by conduction and there is no Initially heat is transported by conduction and there is no lateral variationlateral variation
Fluid on the bottom warms and becomes less denseFluid on the bottom warms and becomes less dense When density difference becomes large enough, lateral When density difference becomes large enough, lateral
variations appear and convection beginsvariations appear and convection begins The cells are 2-D cylinders that rotate about their horizontal The cells are 2-D cylinders that rotate about their horizontal
axesaxes With more heating, these cells become unstable by themselves With more heating, these cells become unstable by themselves
and a second, perpendicular set formsand a second, perpendicular set forms With more heating this planform changes to a vertical With more heating this planform changes to a vertical
hexagonal pattern with hot material rising in the center and hexagonal pattern with hot material rising in the center and cool material descending around the edgescool material descending around the edges
Finally, with extreme heating, the pattern becomes irregular Finally, with extreme heating, the pattern becomes irregular with hot material rising randomly and vigorously.with hot material rising randomly and vigorously.
Rayleigh-Benard ConvectionRayleigh-Benard Convection The stages of convection The stages of convection
have been modeled have been modeled mathematically and are mathematically and are characterized by a “non-characterized by a “non-dimensional” number dimensional” number called the Rayleigh numbercalled the Rayleigh number
isis the volume coefficient the volume coefficient of thermal expansionof thermal expansion
g gravityg gravity d the thickness of the layerd the thickness of the layer Q heat flow through lower Q heat flow through lower
boundaryboundary A, A, κκ, k you know, k you know is kinematic viscosityis kinematic viscosity
4gd Q AdRa
k
The critical value of Ra for gentle convection is about 103.
The aspect ratio for R-B convection cells is about 2-3 to 1
Ra above 105 will produce vigorous convection
Ra above 106 will produce irregular convection
Ra for both the upper and lower mantle seems to Ra for both the upper and lower mantle seems to be consistent with vigorous convectionbe consistent with vigorous convection
While R-B convection models are very useful, they While R-B convection models are very useful, they do not approximate the Earth very well. The do not approximate the Earth very well. The biggest problem is that they model “uniform biggest problem is that they model “uniform viscosity” materials. The mantle is not uniform viscosity” materials. The mantle is not uniform viscosity!viscosity!
Reynold’s number – indicates whether flow is Reynold’s number – indicates whether flow is laminar or turbulentlaminar or turbulent All mantle convection in the Earth is predicted to be All mantle convection in the Earth is predicted to be
laminarlaminar
Mantle convection movies from CaltechMantle convection movies from Caltech More mantle convection moviesMore mantle convection movies MoreMore MoreMore
Modern tomographic images give a very different picture!
Studies like you did in lab, seemed to show that subduction stopped at about 670 km depth. This was interpreted to mean there was mantle convection operating in the upper mantle that was separate from convection in the lower mantle.
