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Heat Flow 1 Heat Flow • Earth’s thermal budget controls: – Volcanoes and intrusives – Plate tectonic movement – Earthquakes – Mountain building – Metamorphism – Thermal structure of the Earth’s interior 2 Why do we care about heat flow? • It is the fundamental cause of plate motions • It is an excellent indicator of the age of the oceans • It is a decent indicator of the age of the continents • It is an indicator of tectonic activity • It is related to uplift and subsidence via heating (expansion) and cooling (contraction) • Heat is an inexhaustible source of energy but much new technology is needed (steam versus hot dry rock) 3 Mean Rate of Heat Radiation Loss on Earth Source Energy radiated Averaged over (Watts) Earth’s surface (Watts/m2) Solar 2 x 1017 W 400 W/m2 Internal Heat 4.4 x 1013 W 8.7 x 10-2 W/m2 Energy released from ~1011 W Earthquakes Heat loss from clothed 2000 W/m2 Human body on cold, (-30C) windy (10 m/s) day (note: frostbite occurs (wind chill = -47C) < 15 minutes) 4 GEOS 4320 4 Heat Flow Methods • Conduction: transfer of heat through a material by atomic or molecular interaction within the material. • Convection: transfer of heat by movement of molecules from one location to another within the material • Water pot on stove: heat transferred through pot by conduction, through water by convection. 5 GEOS 4320 5 Heat Flow Methods • Heat flow is faster via convection than conduction • Radiation: direct transfer of heat by electromagnetic radiation (minor factor within the earth) • Advection: heat physically lifted up with the rocks via tectonic events, erosion, and isostatic rebound. 6 GEOS 4320 6 Conductive Heat Flow • Heat flows from a hot body to a cold body • rate of heat conduction through a solid temperature gradient • High heat flow high temperature gradient 7 GEOS 4320 7 Conductive Heat Flow • Q(z) = -k[(T + dT – T)/dz] (7.3) • where: Q is the rate of flow per unit area down (positive z direction) (W m-2) k is the thermal conductivity of the material the heat is traveling through (W m-1 C-1) T is temperature (C) z is depth from surface (m) (positive downwards) 8 GEOS 4320 8 Conductive Heat Flow • If take the limit as dz → 0 of Eqn. 7.3, then: • Q(z) = -k[dT/dz] (7.4) • This is Fourier’s law of conduction: heat flow is proportional to the thermal gradient • positive temperature gradient = net flow of heat upward (out of Earth) • thus, positive temperature gradient means negative rate of flow of heat: Q 9 GEOS 4320 9 Measuring heat flow (Q) is simply a matter of dropping a thermometer down a drill hole and logging the change in temperature with depth Q = -k T/z k = thermal conductivity 10 Measuring Heat Flow Temperature readings are on the lithological boundaries Geothermal gradient and heat flow calculation values are placed at center-thickness of each unit 11 Measurements of temperature gradients and thermal conductivity in boreholes and mines provide estimates of the rate of loss of heat from the Earth 12 GEOS 4320 12 Thermal Conductivities Substance k (W m-1 C-1) Silver 418 Magnesium 159 Silica / a-Quartz 1.36 / 7.69 Calcite 3.57 Olivine (Fa – Fo) 3.16 – 5.06 “rock” 1.7 – 3.3 (→ 6) Glass 1.2 Wood 0.1 13 GEOS 4320 13 Conductive Heat Flow 14 GEOS 4320 14 k=l 15 Heat Conduction through a Volume For a small volume of height dz and area a, any change in temperature dT in time dt depends on: 1. the flow of heat across the volume’s surface (net flow is in or out) 2. the heat generated in the volume 3. the thermal capacity (specific heat) of the material 16 GEOS 4320 16 Heat Conduction through a Volume 17 GEOS 4320 17 Heat Conduction through a Volume • Specific Heat (cP) of a material is the amount of heat necessary to raise the temperature of 1kg of that material by 1C • Specific heat is measured in J kg-1 C-1 (J = Joules) • Q = mcPT • localized heat sources / sinks: latent heat, shear heating, and endothermic/ exothermic chemical reactions 18 GEOS 4320 18 Specific Heats Substance cP (kJ kg-1 C-1) Silver Magnesium Silica / a-Quartz 0.70 / 0.698 Calcite .793 Olivine (Fa) 0.55 “rock” Glass Wood 19 GEOS 4320 19 Heat Conduction through a Volume • The net gain of heat per unit time is equal to [the heat entering across z] minus [the heat leaving across (z + dz)]: • aQ(z) – aQ(z + dz) • -adz[∂Q/∂z] (7.6) (applying the Taylor expansion series) • Internal heat generation (e.g., radioactivity) at a rate A per unit volume per unit time is given as • Aadz (7.7) 20 GEOS 4320 20 Heat Conduction through a Volume • Given that the material has a density r and specific heat cP, with a temperature increase dT in time dt (dT, dt → 0 ), then heat is gained at the rate: • cP adz r [∂T/∂t] (7.9) • combining and equating these expressions gives the rate at which heat is gained by the volume element: • cP adz r [∂T/∂t] = Aadz - adz[∂Q/∂z] 21 GEOS 4320 21 Heat Conduction through a Volume • cP adz r [∂T/∂t] = Aadz - adz[∂Q/∂z] • as adz is common to all three terms, it can be divided out, leaving: • cPr [∂T/∂t] = A - [∂Q/∂z] (7.10) • substituting for Q (heat flow / unit area) Eqn. 7.4, • cPr [∂T/∂t] = A + k[∂2T/∂z2] (7.12) 22 GEOS 4320 22 Heat Conduction through a Volume • cPr [∂T/∂t] = A + k[∂2T/∂z2] (7.12) • solving for ∂T/∂t gives the one-dimensional heat- conduction equation: • [∂T/∂t] = (k/cPr)[∂2T/∂z2] + A/cPr (7.13) • First derivatives indicate slope (temperature gradients) • second derivatives indicate minima and maxima (heat sinks and sources) 23 GEOS 4320 23 Heat Conduction through a Volume • Expanding the second derivative term to three dimensions involves the Laplacian operator: • { [∂2T/∂x2] + [∂2T/∂y2] + [∂2T/∂z2] } = 2T • Substituting 2T into Eqn. 7.13 gives the three- dimensional heat-conduction equation: • [∂T/∂t] = (k/cPr)2T + A/cPr (7.15) • where k/cPr represents the thermal diffusivity k, the ability of a material to lose heat by conduction. GEOS 4320 24 24 Heat Conduction through a Volume • In a steady-state situation, dT/dt =0, and there is no change in temperature with time, then: • 2T = -A/k (7.16) • In the special situation where there is no heat generation (A = 0): • [∂T/∂t] = k2T (7.17) • where k (= k/cPr) represents the thermal diffusivity, the ability of a material to lose heat by conduction. This is the diffusivity equation. 25 GEOS 4320 25 Heat Conduction through a Volume • Advection: heat physically lifted up with the rocks via tectonic events, erosion, and isostatic rebound. • If we consider how the temperature of a small volume changes with time if it is relative motion; a additional term has to be added to the one- dimensional heat conduction equation. • Let uz be the velocity of the moving volume element in the z direction; at any time t its depth is z + uzt. The expression uz(∂T/∂z) accounts for the affects of the volume motion through the thermal gradient. 26 GEOS 4320 26 Heat Conduction through a Volume • If we expand the velocity to three dimensions, then u = u(x, y, z) is the velocity vector of the material. The term u • 2T, the advective- transfer term, is added to the three-dimensional heat-conduction equation: • [∂T/∂t] = k2T + A/cPr - u • 2T (7.19) • Most commonly, uz is the rate at which erosion or deposition takes place; uz could also represent the rate of isostatic rebound. 27 GEOS 4320 27 Radiogenic heat production (A) Isotope A, µW kg-1 H (W kg-1) τ½ (a) C (kg kg-1) 238U 9.46 x 10-5 4.47 x 109 30.8 x 10-9 235U 5.69 x 10-4 7.04 x 108 0.22 x 10-9 Total U 96.7 9.91 x 10-5 31.0 x 10-9 232Th 26.3 2.64 x 10-5 1.40 x 1010 124 x 10-9 40K 2.92 x 10-5 1.25 x 109 36.9 x 10-9 Total K 0.0035 4.48 x 10-9 31.0 x 10-5 Table of main parameters for heat generating isotopes in the Earth. A is the heat production of the element per unit weight; H is heat release of the isotope or element per unit weight for the concentration C; t½ is its half life in years; and C is the present mean mantle concentration of the heat producing elements [Modified from Table 4-2, page 137, D. L. Turcotte and G. Schubert, Geodynamics (2nd ed.), Cambridge 28 University Press, 456 pp., 2002.] Important Parameters for Heat Flow Analysis Property, Symbol Approximate Range Heat flow, Q 0 - 125 mW m-2 Vertical temperature gradient, dT/dz 10 to 80 °C/km Thermal conductivity, k . - marine sediments 0.6 - 1.2 W m-1 °C-1 - continental sediments 1 - 5 W m-1 °C-1 heat generation, A 0-8 10-6 W m-3 Specific heat, cP 0.85-1.25 kJ kg-1 °C-1 Density of crustal rocks and lithosphere, r 2200 to 3400 kg m-3 29 Heat Flow Data for Continents 30 Artemieva and Mooney,JGR, 2001 Global heat flow measurements Age crustal formation vs. Heat Flow Plot of heat flow vs. tectonothermal age. Data are plotted in the following age groups: C – Cenozoic; M – Mesozoic; LPa – Late Paleozoic; E – Early Paleozoic; LPr – Late Proterozoic; E – Early Proterozoic; A – Archean. On the heat flow scale the plus sign indicate the data mean with the box height indicating +/- one standard deviation of the data. On the age scale the plus sign indicate the middle of the age range and the box width indicates the age range. Numbers in parentheses indicates the number of data in each age group. [Reproduced from Morgan, P., The thermal structure and thermal evolution of the continental lithosphere, Phys. Chem. Earth, 15, 107-193, 1984]. 31 Craton-edge Heat Flow Measurements The Golden gate east of Clarens 32 Craton-edge Heat Flow Measurements 33 Craton-edge Heat Flow Measurements Why in the world is finding the edge of the craton important? 34 Kaapvaal Craton Geotherms The intersection of the geotherm with the adiabat indicates the thickness of the lithosphere 35 Kaapvaal Craton Geotherms and Kimberlites 36 Kaapvaal Craton Geotherms and Kimberlites Letseng-le-Terae Kimberlite The 603 ct Lesotho Promise Diamond 37 Calculation of Geotherms • Temperature in a column of rock is controlled by several parameters: – Internal: conductivity, specific heat, density, and radioactive heat generation – External: heat flow into the column, surface temperature, erosion/deposition at the top of the column • Equilibrium geotherm: column of rock (1-D) with no erosion/deposition and constant heat flow; column reaches a state of thermal equilibrium- temperature at any point is steady (∂T/∂t = 0). 38 Calculation of Geotherms • Equilibrium geotherm: As ∂T/∂t = 0, Eqn 7.16 applies: • ∂2T/∂z2 = -A/k (7.20) • Boundary conditions: i. T = 0 at z = 0 ii. surface heat flow Q = -k(∂T/∂z) = -Q0 at z = 0 • Integrating Eqn 7.20 once gives • ∂T/∂z = (-A/k)z + c1 (7.21) 39 Calculation of Geotherms • From the boundary condition ii, -k(∂T/∂z) = -Q0 at z = 0, (∂T/∂z) = Q0/k = c1 (Eqn 7.22). • Substituting 7.22 into 7.21 and integrating again: • T = (-A/k)(z2/2) + (Q0/k)z +c2 (7.23) • From the boundary condition i, T = 0 when z = 0, c2 must be zero. The temperature in the column is then given by: • T = (-Az2/2k) + Q0z/k (7.24) 40 Calculation of Geotherms • An alternate set of boundary conditions could be used to estimate the equilibrium crustal geotherms given d as the depth of the moho and Qd as the mantle heat flow into the base of the crust: – T = 0 at z = 0 – Q = -Qd at z = d • The temperature in the column 0 ≤ z ≤ d is given by: • T = (-Az2/2k) + [(Qd + Ad)/k]z (7.28) • Contributions to the surface heat flow come from both mantle (Qd) and internal crustal (Ad) sources; similarly, mantle heat flow contributes Qd/k to the temperature at depth z. 41 Calculation of Geotherms • Equilibrium geotherm: no change in temperature with time (steady-state); ∂T/∂t = 0 • Depth 0 ≤ z ≤ d • T = 0 at z = 0 • T = Td at z = d • No internal heat generation (A = 0) Eqn 7.20: ∂2T/∂z2 = -A/k = 0 ∂T/∂z = (Td – 0)/(d – 0) = Td/d Integrate ∂T = (Td/d) ∂z T = (Td/d)z 42 Calculation of Geotherms: One Layer Models • d = 50 km, k = 2.5 W m-1 °C-1, Qd = 21 x 10-3 W m-2 *, A = 1.25 mW m-3, from Eqn 7.27. • a: standard model; shallow gradient ~30 °C km-1, deep gradient ≤15 °C km-1. • b: k reduced to 1.7 W m-1 °C-1; shallow gradient raised to ~45 °C km-1. • c: A raised to 2.5 mW m-3; shallow gradient raised to >50 °C km-1. • d: Qd doubled to 42 x 10-3 W m-2; shallow gradient raised to ~40 °C km-1. • e: Qd halved to 10.5 x 10-3 W m-2; shallow gradient reduced to ~27 °C km-1. * page 281 erroneously reports this as 43 21 x 10-3 mW m-2 Calculation of Geotherms: Two Layer Models • Eqns 7.20 – 7.28 apply • each layer considered z1 separately z2 • temperature gradients must be matched across boundaries Boundary conditions: For the first layer 0 ≤ z < z1: i) T = 0 at z = 0 ∂2T/∂z2 = -A1/k ii) A = A1 = 4.2 for 0 ≤ z < z1 For the second layer z1 ≤ z < z2: iii) A = A2 = 0.8 for z1 ≤ z < z2 iv) Q = -Q2 = -63 at z = z2 ∂2T/∂z2 = -A2/k 44 Calculation of Geotherms: Two Layer Models The solution to these two differential equations is subject to: Boundary conditions Matching both temperature and temperature gradient at z = z1 is given as two equations: T = -A1z2/2k + {Q2/k + [A2/k(z2 – z1)] + A1z1/k}z for 0 ≤ z < z1 (7.31) T = -A2z2/2k + {Q2/k + A2z2/k}z + [(A1 - A2)/2k]z12 for 0 ≤ z < z1 (7.32) The calculated equilibrium geotherm from these equations for an Archaean crust is shown above. This indicates rather high crustal temperatures 45 existed during this time period (2.5 – 4.5 Ga) of the Earth. Comparison of Geotherms The Archaean gradient approaches the shape of curve d, which has double the heat flow Qd than the standard model, curve a, indicating higher heat flow in the Archaean 46 Typical Craton geotherm Conductive to ~200 km (~6°C/km) The geotherm beneath the Slave Province in Canada, as inferred from xenoliths from the Jericho pipe. Squares are unsheared xenoliths, triangles are sheared xenoliths, and asterisks are megacrystalline xenoliths. The conductive geotherm in the mantle extrapolates to the base of the crust at X and to the mantle adiabat at L. The shallow 47 geotherm G is from borehole measurements. From Sleep 2005 AREPS Timescale of Conductive Heat Flow • Consider model rock column with gradient curve a from Fig. 7.3 • Increase Qd from 21 x 10-3 W m-2 to 42 x 10-3 W m-2; temperature of rock column would increase until new equilibrium temperatures were attained along curve d. • This process occurs very slowly; consider a rock at z = 20 km, with initial temperature T0 = 567 °C. – After 20 Ma conduction would have raised the temperature to T20 = 580 °C. – After 100 Ma the temperature T100 > 700 °C. – The equilibrium temperature Te = 734 °C. 48 Timescale of Conductive Heat Flow • Theses values can be estimated using the thermal diffusion equation (7.17): • [∂T/∂t] = k2T • The characteristic time t = l2/k indicate the amount of time it takes for a temperature change to propagate a distance of l in a medium having thermal diffusivity k. • The characteristic thermal diffusion distance l = √(kt) estimates the distance in which changes in temperature propagate in a time t. 49 Timescale of Conductive Heat Flow • Example 1: thermal transfer from a subduction zone at z = 100 km via conduction >> 10s Ma • Melting and intrusion are important mechanisms for heat transfer above subduction zones 50 Timescale of Conductive Heat Flow • Example 2: metamorphic belt caused by a deep heated heat source is characterized by abundant intrusions, often mantle-derived material. • Magmatism occurs because large increases in deep heat flow cause large-scale melting at depth long before the heat can penetrate towards the surface by conduction. 51 The Error Function • An integral that occurs often in the solution to physical problems: x • erf(x) = (2/√p) ∫ e -y 2 dy (7.35) 0 Error function table in Appendix 5, p.638-9 put equation 52 on board Instantaneous Cooling or Heating • A semi-infinite solid with upper surface z = 0 has no heat generation (A = 0) and an initial temperature throughout of T = T0 at t = 0. • Determine how the solid cools for t > 0 given that for z = 0, T = 0 for t > 0. • Use thermal diffusion equation (7.17): • [∂T/∂t] = k2T 53 Instantaneous Cooling or Heating • Solution to the thermal diffusion equation that satisfies the boundary conditions: • T = T0 erf[z/(2√(kt))] (7.34) • The time taken to reach a given temperature is proportional to z2 and inversely proportional to k. • The temperature gradient is given by differentiating (7.34) wrt z: 54 Instantaneous Cooling or Heating • ∂T/∂z = T0 ∙ (2/√p) ∙ 1/(2√(kt)) ∙ e-z/(4kt) • ∂T/∂z = T0/(√(pkt)) e-z/(4kt) (7.36) • The error-function to the heat-conduction equation can be applied to many geological situations. • Example: dike of 2w and infinite extent in y and z directions; no heat generation, ignore latent heat of solidification. Use: thermal diffusion equation. 55 Instantaneous Cooling or Heating • Initial conditions: • T = T0 at t = 0 for -w ≤ x ≤ w; • T = 0 at t = 0 for -w ≤ x ≤ w. • Solution to the thermal diffusion equation that satisfies the initial conditions (Eqn 7.37): • T = (T0/2){erf [(w – x)/(2√(kt))] + erf [(w + x)/(2√(kt))] 56 Instantaneous Cooling or Heating • For w = 1m, intrusion temperature of T0 = 1000 °C, and k = 10-6 m2 s-1, then the temperature at the center of the dike: – Tc = 640 °C after one week; – Tc = 340 °C after one month; and – Tc = 100 °C after one year. • Note that small dikes cool rapidly 57 Instantaneous Cooling or Heating • General case: temperature of the dike Tc – Tc = T0/2 when t = w2/k – Tc = T0/4 when t = 5w2/k • High temperatures outside the dike are confined to a narrow zone: – Distance w from dike edge Tmax = ~T0/4 – T0/2 reached about distance w/4 from dike edge • Contact metamorphism generally confined to a narrow zone 2w (dike width) on either side. 58 Dike Intrusion Contact Metamorphism Golden Gate National Park, Clarens, South Africa 59 Dike Intrusion Contact Metamorphism T0 + 590 °C at ts = ts 60 Instantaneous Cooling or Heating • Example: Periodic Variation of Surface Temperature • Earth’s surface temperature varies periodically (daily, annually, ice ages); we make temperature measure- ments deep enough to minimize these effects • Periodic contribution to surface temp modeled as T0eiwt, where: T0 is the maximum variation of mean surface temperature; w is 2p multiplied by the frequency of variation; and i is (-1). 61 Periodic Variation of Surface Temperature • The temperature T(z, t) is given by the diffusion equation with the following boundary conditions: – T(0, t) = T0eiwt, and – T(z, t) → 0 as z → ∞ • The separation of variables technique is used is used to solve this problem, as explained in the text, and the solution for T(z, t) is equation 7.45 (p.284) 62 Periodic Variation of Surface Temperature • Skin Depth L: the depth at which the amplitude of the periodic disturbance is 1/e of the amplitude at the surface. L is calculated by: • L = √(2k/(wrcP) (7.46) • Given a sandstone with k = 2.5 Wm-1C-1, cP = 103 J kg-1 C-1, and r = 2.3 kgm-3, and the daily variation w = 7.27 x 10-7 s-1), L is ~17cm; • For the annual variation w = 2 x 10-7 s-1, L = 3.3m • For an ice age (with period of the order of 100ka), L > 1km 63 Periodic Variation of Surface Temperature • Phase Difference f: difference between surface temperature variation and the variation at depth z is given by: • F = z√[(wrcP)/(2k)] (7.47) • At skin depth L, f = 1 radian • When f = p, temperature at depth z is exactly half a cycle out-of-phase with the surface temperature. 64 Problem Set #5: Heat Flow • Pages 321-323 and Additional below • Problems: A-1, 5, 22, 2, 14 – 20: evens, A-2, A-3 • Show all work! • Due 9 October 2009 65