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Heat in the Earth Volcanoes, magmatic intrusions, earthquakes, mountain building and metamorphism are all controlled by the generation and transfer of heat in the Earth. Volcanoes, magmatic intrusions, earthquakes, mountain building and metamorphism are all controlled by the generation and transfer of heat in the Earth. The Earths thermal budget controls the activity of the lithosphere and asthenosphere and the development of the basic structure of the Earth. The Earths thermal budget controls the activity of the lithosphere and asthenosphere and the development of the basic structure of the Earth.

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Heat arrives at the surface of the Earth from its interior and from the Sun. Heat arrives at the surface of the Earth from its interior and from the Sun. The heat arriving from the Sun is by far the greater of the two The heat arriving from the Sun is by far the greater of the two Heat from the Sun arriving at the Earth is 2x10 17 W Heat from the Sun arriving at the Earth is 2x10 17 W Averaged over the surface this is 4x10 2 W/m 2 Averaged over the surface this is 4x10 2 W/m 2 The heat from the interior is 4x10 13 W and 8x10 -2 W/m 2 The heat from the interior is 4x10 13 W and 8x10 -2 W/m 2 However, most of the heat from the Sun is radiated back into space. It is important because it drives the surface water cycle, rainfall, and hence erosion. The Sun and the biosphere keep the average surface temperature in the range of stability of liquid water. However, most of the heat from the Sun is radiated back into space. It is important because it drives the surface water cycle, rainfall, and hence erosion. The Sun and the biosphere keep the average surface temperature in the range of stability of liquid water. The heat from the interior of the Earth has governed the geological evolution of the Earth, controlling plate tectonics, igneous activity, metamorphism, the evolution of the core, and hence the Earths magnetic field. The heat from the interior of the Earth has governed the geological evolution of the Earth, controlling plate tectonics, igneous activity, metamorphism, the evolution of the core, and hence the Earths magnetic field.

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Heat Transfer Mechanisms Conduction Conduction Transfer of heat through a material by atomic or molecular interaction within the material Transfer of heat through a material by atomic or molecular interaction within the material Radiation Radiation Direct transfer of heat as electromagnetic radiation Direct transfer of heat as electromagnetic radiation Convection Convection Transfer of heat by the movement of the molecules themselves Transfer of heat by the movement of the molecules themselves Advection is a special case of convection Advection is a special case of convection

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Conductive Heat Flow Heat flows from hot things to cold things. Heat flows from hot things to cold things. The rate at which heat flows is proportional to the temperature gradient in a material The rate at which heat flows is proportional to the temperature gradient in a material Large temperature gradient – higher heat flow Large temperature gradient – higher heat flow Small temperature gradient – lower heat flow Small temperature gradient – lower heat flow

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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: In the limit as δz goes to zero:

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Heat flow (or flux) Q is rate of flow of heat per unit area. Heat flow (or flux) Q is rate of flow of heat per unit area. The units are watts per meter squared, W m -2 The units are watts per meter squared, W m -2 Watt is a unit of power (amount of work done per unit time) Watt is a unit of power (amount of work done per unit time) A watt is a joule per second A watt is a joule per second Old heat flow units, 1 hfu = 10 -6 cal cm -2 s -1 Old heat flow units, 1 hfu = 10 -6 cal cm -2 s -1 1 hfu = 4.2 x 10 -2 W m -2 1 hfu = 4.2 x 10 -2 W m -2 Typical continental surface heat flow is 40-80 mW m -2 Typical continental surface heat flow is 40-80 mW m -2 Thermal conductivity k Thermal conductivity k The units are watts per meter per degree centigrade, W m -1 ° C -1 The units are watts per meter per degree centigrade, W m -1 ° C -1 Old thermal conductivity units, cal cm -1 s -1 ° C -1 Old thermal conductivity units, cal cm -1 s -1 ° C -1 0.006 cal cm -1 s -1° C -1 = 2.52 W m -1 ° C -1 0.006 cal cm -1 s -1° C -1 = 2.52 W m -1 ° C -1 Typical conductivity values in W m -1 ° C -1 : Typical conductivity values in W m -1 ° C -1 : Silver420 Silver420 Magnesium160 Magnesium160 Glass 1.2 Glass 1.2 Rock1.7-3.3 Rock1.7-3.3 Wood0.1 Wood0.1

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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 elements surface (can be in or out or both) 2.Heat generated in the element 3.Thermal capacity (specific heat) of the material Lets derive a differential equation describing the conductive flow of heat

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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 The net change in heat in the element is (heat entering across z) minus (heat leaving across Z+δz):

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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.

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The specific heat c p 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 1 C. Specific heat is measured in units of J kg -1 C -1. If material has density ρ and specific heat c p, 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:

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In the limit as δt goes to zero:Several slides back we defined Q as: This is the one-dimensional heat conduction equation. Simplifies to:

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The term k/ρc p 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: The symbol in the center is the gradient operator squared, aka the Laplacian operator. It is the dot product of the gradient with itself.

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This simplifies in many special situations. For a steady-state situation, there is no change in temperature with time. Therefore: In the absence of heat generation, A=0: Scientists in many fields recognize this as the classic diffusion equation.

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Talk at board about the qualitative behavior of the Heat Conduction equation

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Equilibrium Geotherms The temperature vs. depth profile in the Earth is called the geotherm. The temperature vs. depth profile in the Earth is called the geotherm. An equilibrium geotherm is a steady state geotherm. An equilibrium geotherm is a steady state geotherm. Therefore: Therefore:

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Boundary conditions Since this is a second order differential equation, we should expect to need 2 boundary conditions to obtain a solution. Since this is a second order differential equation, we should expect to need 2 boundary conditions to obtain a solution. A possible pair of bcs is: A possible pair of bcs is: T=0 at z=0 T=0 at z=0 Q=Q 0 at z=0 Q=Q 0 at z=0 Note: Q is being treated as positive upward and z is positive downward in this derivation. Note: Q is being treated as positive upward and z is positive downward in this derivation.

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Solution Integrate the differential equation once: Integrate the differential equation once: Use the second bc to constrain c 1 Use the second bc to constrain c 1 Note: Q is being treated as positive upward and z is positive downward in this derivation. Note: Q is being treated as positive upward and z is positive downward in this derivation. Substitute for c 1 : Substitute for c 1 :

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Solution Integrate the differential equation again: Integrate the differential equation again: Use the first bc to constrain c 2 Use the first bc to constrain c 2 Substitute for c 2 : Substitute for c 2 : Link to spreadsheet Link to spreadsheet Link to spreadsheet Link to spreadsheet

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Oceanic Heat Flow Heat flow is higher over young oceanic crust Heat flow is more scattered over young oceanic crust Oceanic crust is formed by intrusion of basaltic magma from below The fresh basalt is very permeable and the heat drives water convection Ocean crust is gradually covered by impermeable sediment and water convection ceases. Ocean crust ages as it moves away from the spreading center. It cools and it contracts.

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These data have been empirically modeled in two ways: d=2.5 + 0.35t 2 (0-70 my) and d=6.4 – 3.2e -t/62.8 (35-200 my)

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Half Space Model Specified temperature at top boundary. No bottom boundary condition. Cooling and subsidence are predicted to follow square root of time. Plate Model Specified temperature at top and bottom boundaries. Cooling and subsidence are predicted to follow an exponential function of time. Roughly matches Half Space Model for first 70 my.

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The model of plate cooling with age generally works for continental lithosphere, but is not very useful. Variations in heat flow in continents is controlled largely by changes in the distribution of heat generating elements and recent tectonic activity.

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Range of Continental and Oceanic Geotherms in the crust and upper mantle

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Convection

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Conductive Geotherm ~10-20 C per km Adiabatic Geotherm ~0.5-1.0 C per km Convective Geotherm Adiabatic middle Thermal boundary layer at top and bottom

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Illustration of mantle melting during decompression Solid and liquid in the Earth

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Rayleigh-Benard Convection Newtonian viscous fluid – stress is proportional to strain rate Newtonian viscous fluid – stress is proportional to strain rate A tank of fluid is heated from below and cooled from above A tank of fluid is heated from below and cooled from above Initially heat is transported by conduction and there is no lateral variation Initially heat is transported by conduction and there is no lateral variation Fluid on the bottom warms and becomes less dense Fluid on the bottom warms and becomes less dense When density difference becomes large enough, lateral variations appear and convection begins When density difference becomes large enough, lateral variations appear and convection begins The cells are 2-D cylinders that rotate about their horizontal axes The cells are 2-D cylinders that rotate about their horizontal axes With more heating, these cells become unstable by themselves and a second, perpendicular set forms With more heating, these cells become unstable by themselves and a second, perpendicular set forms With more heating this planform changes to a vertical hexagonal pattern with hot material rising in the center and cool material descending around the edges With more heating this planform changes to a vertical hexagonal pattern with hot material rising in the center and cool material descending around the edges Finally, with extreme heating, the pattern becomes irregular with hot material rising randomly and vigorously. Finally, with extreme heating, the pattern becomes irregular with hot material rising randomly and vigorously.

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Rayleigh-Benard Convection The stages of convection have been modeled mathematically and are characterized by a non-dimensional number called the Rayleigh number The stages of convection have been modeled mathematically and are characterized by a non-dimensional number called the Rayleigh number is the volume coefficient of thermal expansion is the volume coefficient of thermal expansion g gravity g gravity d the thickness of the layer d the thickness of the layer Q heat flow through lower boundary Q heat flow through lower boundary A, κ, k you know A, κ, k you know is kinematic viscosity is kinematic viscosity The critical value of Ra for gentle convection is about 10 3. The aspect ratio for R-B convection cells is about 2-3 to 1 Ra above 10 5 will produce vigorous convection Ra above 10 6 will produce irregular convection

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Ra for both the upper and lower mantle seems to be consistent with vigorous convection Ra for both the upper and lower mantle seems to be consistent with vigorous convection While R-B convection models are very useful, they do not approximate the Earth very well. The biggest problem is that they model uniform viscosity materials. The mantle is not uniform viscosity! While R-B convection models are very useful, they do not approximate the Earth very well. The biggest problem is that they model uniform viscosity materials. The mantle is not uniform viscosity! Reynolds number – indicates whether flow is laminar or turbulent Reynolds number – indicates whether flow is laminar or turbulent All mantle convection in the Earth is predicted to be laminar All mantle convection in the Earth is predicted to be laminar Mantle convection movies from Caltech Mantle convection movies from Caltech Mantle convection movies from Caltech Mantle convection movies from Caltech More mantle convection movies More mantle convection movies More mantle convection movies More mantle convection movies More More More More More More

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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. Two-layer vs. Whole Mantle Convection

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Illustration of slab pull and ridge push Plate Driving Forces

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