Presentation on theme: "Heat in the Earth Volcanoes, magmatic intrusions, earthquakes, mountain building and metamorphism are all controlled by the generation and transfer of."— Presentation transcript:
1 Heat in the EarthVolcanoes, magmatic intrusions, earthquakes, mountain building and metamorphism are all controlled by the generation and transfer of heat in the Earth.The Earth’s thermal budget controls the activity of the lithosphere and asthenosphere and the development of the basic structure of the Earth.
2 The heat arriving from the Sun is by far the greater of the two 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 twoHeat from the Sun arriving at the Earth is 2x1017 WAveraged over the surface this is 4x102 W/m2The heat from the interior is 4x1013 W and 8x10-2 W/m2However, 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 Earth’s magnetic field.
3 Heat Transfer Mechanisms ConductionTransfer of heat through a material by atomic or molecular interaction within the materialRadiationDirect transfer of heat as electromagnetic radiationConvectionTransfer of heat by the movement of the molecules themselvesAdvection is a special case of convection
4 Conductive Heat Flow Heat flows from hot things to cold things. The rate at which heat flows is proportional to the temperature gradient in a materialLarge temperature gradient – higher heat flowSmall temperature gradient – lower heat flow
5 Imagine an infinitely wide and long solid plate with thickness δz . Temperature above is T + δTTemperature below is THeat 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:
6 Heat flow (or flux) Q is rate of flow of heat per unit area. The units are watts per meter squared, W m-2Watt is a unit of power (amount of work done per unit time)A watt is a joule per secondOld heat flow units, 1 hfu = 10-6 cal cm-2 s-11 hfu = 4.2 x 10-2 W m-2Typical continental surface heat flow is mW m-2Thermal conductivity kThe units are watts per meter per degree centigrade, W m-1 °C-1Old thermal conductivity units, cal cm-1 s-1 °C-10.006 cal cm-1 s-1°C-1 = 2.52 W m-1 °C-1Typical conductivity values in W m-1 °C-1 :Silver 420Magnesium 160Glass 1.2RockWood 0.1
7 Let’s derive a differential equation describing the conductive flow of heat Consider a small volume element of height δz and area a.Any change in the temperature of this volume in time δt depends on:Net flow of heat across the element’s surface (can be in or out or both)Heat generated in the elementThermal capacity (specific heat) of the material
8 Expand Q(z+δz) as Taylor series: 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 neglectedThe net change in heat in the element is (heat entering across z) minus (heat leaving across Z+δz):
9 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 thenA a δzRadioactivity 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.
10 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:
11 Simplifies to:In the limit as δt goes to zero:Several slides back we defined Q as:This is the one-dimensional heat conduction equation.
12 The term k/ρcp is known as the thermal diffusivity κ 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:The symbol in the center is the gradient operator squared, aka the Laplacian operator. It is the dot product of the gradient with itself.
13 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.
14 Talk at board about the qualitative behavior of the Heat Conduction equation
15 Equilibrium Geotherms The temperature vs. depth profile in the Earth is called the geotherm.An equilibrium geotherm is a steady state geotherm.Therefore:
16 Boundary conditionsSince this is a second order differential equation, we should expect to need 2 boundary conditions to obtain a solution.A possible pair of bc’s is:T=0 at z=0Q=Q0 at z=0Note: Q is being treated as positive upward and z is positive downward in this derivation.
17 Solution Integrate the differential equation once: Use the second bc to constrain c1Note: Q is being treated as positive upward and z is positive downward in this derivation.Substitute for c1:
18 Solution Integrate the differential equation again: Use the first bc to constrain c2Substitute for c2:Link to spreadsheet
19 Oceanic Heat FlowHeat flow is higher over young oceanic crustHeat flow is more scattered over young oceanic crustOceanic crust is formed by intrusion of basaltic magma from belowThe fresh basalt is very permeable and the heat drives water convectionOcean 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.
20 These data have been empirically modeled in two ways: d= t (0-70 my)andd=6.4 – 3.2e-t/ ( my)
21 Half Space ModelSpecified temperature at top boundary.No bottom boundary condition.Cooling and subsidence are predicted to follow square root of time.Plate ModelSpecified 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.
23 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.
24 Range of Continental and Oceanic Geotherms in the crust and upper mantle
27 Conductive Geotherm~10-20 C per kmAdiabatic Geotherm~ C per kmConvective GeothermAdiabatic “middle”Thermal boundary layerat top and bottom
28 Illustration of mantle melting during decompression Solid and liquid in the EarthIllustration of mantle melting during decompression
29 Rayleigh-Benard Convection Newtonian viscous fluid – stress is proportional to strain rateA tank of fluid is heated from below and cooled from aboveInitially heat is transported by conduction and there is no lateral variationFluid on the bottom warms and becomes less denseWhen density difference becomes large enough, lateral variations appear and convection beginsThe cells are 2-D cylinders that rotate about their horizontal axesWith more heating, these cells become unstable by themselves and a second, perpendicular set formsWith more heating this planform changes to a vertical hexagonal pattern with hot material rising in the center and cool material descending around the edgesFinally, with extreme heating, the pattern becomes irregular with hot material rising randomly and vigorously.
30 Rayleigh-Benard Convection The stages of convection have been modeled mathematically and are characterized by a “non-dimensional” number called the Rayleigh numbera is the volume coefficient of thermal expansiong gravityd the thickness of the layerQ heat flow through lower boundaryA, κ, k you known is kinematic viscosityThe critical value of Ra for gentle convection is about 103.The aspect ratio for R-B convection cells is about 2-3 to 1Ra above 105 will produce vigorous convectionRa above 106 will produce irregular convection
31 Reynold’s number – indicates whether flow is laminar or turbulent Ra for both the upper and lower mantle seems to be consistent with vigorous convectionWhile 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!Reynold’s number – indicates whether flow is laminar or turbulentAll mantle convection in the Earth is predicted to be laminarMantle convection movies from CaltechMore mantle convection moviesMore
32 Two-layer vs. Whole Mantle Convection 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 ConvectionModern tomographic images give a very different picture!
33 Illustration of slab pull and ridge push Plate Driving ForcesIllustration of slab pull and ridge push