Geodynamics Lecture 7 Heat conduction and production

Geodynamics Lecture 7 Heat conduction and production Lecturer: David Whipp david.whipp@helsinki.fi 23.9.2014 Geodynamics www.helsinki.fi/yliopisto 1

Goals of this lecture Gain a conceptual and mathematical understanding of heat conduction Learn the concepts of radiogenic heat production in the Earth View the effects of heat conduction and production on the thermal field in the crust 2

Why talk about heat transfer? Grand Prismatic Spring, Yellowstone National Park, U.S. Image: http://en.wikipedia.org The dynamics of the Earth are dominantly controlled by gravitational and thermal processes. In fact, you could argue that thermal processes have the largest impact. Why? 3

Why now? www.usgs.gov Faulting Folding Many rock properties are largely a function of temperature. Faulting and brittle deformation may occur near the surface, but folding and ductile flow are dominant at depth. 4

The effects of partial melting on rock strength Partial melting dramatically decreases rock strength Only about 7% melt is required for to decrease rock strength by 80-90% Rosenberg et al., 2007 5

Heat transfer in the lithosphere Conduction: The diffusive transfer of heat by kinetic atomic or molecular interactions within the material. Also known as thermal diffusion. Advection: The transfer of heat by physical movement of molecules or atoms within a material. A type of convection, mostly applied to heat transfer in solid materials. Production: Not really a heat transfer process, but rather a source of heat. Sources in the lithosphere include radioactive decay, friction in deforming rock or chemical reactions such as phase transitions. 6

Heat transfer in the lithosphere Conduction: The diffusive transfer of heat by kinetic atomic or molecular interactions within the material. Also known as thermal diffusion. Focus in this lecture Advection: The transfer of heat by physical movement of molecules or atoms within a material. A type of convection, mostly applied to heat transfer in solid materials. Production: Not really a heat transfer process, but rather a source of heat. Sources in the lithosphere include radioactive decay, friction in deforming rock or chemical reactions such as phase transitions. 7

Basic ideas of heat conduction The conduction of heat in solids is a diffusion process, and well described by Fourier s laws, the basic mathematical relationships describing diffusion Fourier s first law states that the flux of heat in a material q is directly proportional to the temperature gradient What would this relationship look as an equation? 8

Fourier s first law In 1D, the mathematical translation of Heat flux q is directly proportional to the thermal gradient in a material is q = k dt dy Here, T represents temperature and y represents spatial position, depth in the Earth in this case Thus, dt/dy is the change in temperature with distance, the thermal gradient The proportionality constant k is known as the thermal conductivity 10

Fourier s first law In 1D, the mathematical translation of Heat flux q is directly proportional to the thermal gradient in a material is q = Why is there a negative sign? k dt dy 11

What is thermal conductivity? The mathematical translation of Heat flux q is directly proportional to the thermal gradient in a material is q = k dt dy Sandstone Thermal conductivity is a proportionality factor As you can easily see, rocks with a high thermal conductivity will produce a large heat flow, whereas rocks with a low thermal conductivity will have near zero heat flow Thermal conductivity of most crustal rocks is 2-3 W m - 1 K - 1 Salt Stüwe, 2007 12

Global heat flow map http://www.cbe.cornell.edu 13

Global heat flow map Global average: 87 mw m -2 Continents: 65 ± 1.6 mw m -2 Oceans: 101 ± 2.2 mw m -2 http://www.cbe.cornell.edu Here we can clearly see the connection between geodynamic setting and heat flow 14

Radiogenic heat production Radiogenic heat production, A or H, results from the decay of radioactive elements in the Earth, mainly 238 U, 235 U, 232 Th and 40 K. A is generally used for volumetric heat production and H for heat production by mass. These elements occur in the mantle, but are concentrated in the crust, where radiogenic heating can be significant The surface heat flow in continental regions is ~65 mw m -2 and ~37 mw m -2 is from radiogenic heat production (57%) Concentration Rock Type U (ppm) Th (ppm) K (%) Reference undepleted (fertile) mantle 0.031 0.124 0.031 Depleted peridotites 0.001 0.004 0.003 Tholeiitic basalt 0.07 0.19 0.088 Granite 4.7 20 4.2 Shale 3.7 12 2.7 Average continental crust 1.42 5.6 1.43 Chondritic meteorites 0.008 0.029 0.056 Turcotte and Schubert, 2014 15

Radiogenic heat production Rock type By mass, H [W kg Heat production By volume, A [W m By volume, A [µw m Reference undepleated (fer/le) mantle 7.39E-12 2.44E-08 0.024 "Depleated" perido/tes 3.08E-13 1.02E-09 0.001 Tholeii/c basalt 1.49E-11 4.41E-08 0.044 Granite 1.14E-09 3.01E-06 3.008 Shale 7.74E-10 1.86E-06 1.857 Average con/nental crust 3.37E-10 9.26E-07 0.927 Chondri/c meteorites 3.50E-12 1.15E-08 0.012 Calculated from Turcotte and Schubert, 2014 17

Radiogenic heat production Stüwe, 2007 H Crustal heat production over time Because radiogenic heat production is the result of radioactive decay, the parent isotope concentrations have continually decreased throughout Earth s history Today, the total amount of radiogenic heat is about half of what was produced in the Archean at ~3 Ga 19

LETTER doi:10.1038/nature13728 Spreading continents kick-started plate tectonics Patrice F. Rey 1, Nicolas Coltice 2,3 & Nicolas Flament 1 A a b c d e Continent Temperature (K) Stresses acting on cold, thick and negatively buoyant oceanic 0 1,000 lithosphere are thought to be crucial to the initiation of subduction 0 and tectonic impact of a thick and buoyant continent surrounded by a stag- 2,000 that of present-day tectonic forces driving orogenesis 1. To explore the the operation of plate tectonics 1,2, which characterizes the presentday geodynamics of the Earth. Because the Earth s interior 200 was 1,820 K nant lithospheric lid, we produced a series of two-dimensional thermomechanical numerical models of the top 700 km of the Earth, using 293 hotter K in the Archaean eon, the oceanic crust may have been thicker, thereby temperature-dependent densities and visco-plastic rheologies that depend making the oceanic lithosphere more buoyant than 400 at present 3, and on temperature, melt fraction and depletion, stress and strain rate (see 400 km whether subduction and plate tectonics occurred during this time is Methods). The initial temperature field is the horizontally averaged temperature profile of a stagnant-lid convection calculation for a mantle 600 ambiguous, both in the geological record and in geodynamic models 4. Here we show that because the oceanic crust was thick and buoyant 5,,200 K hotter than at present (Fig. 1A, a and Extended Data Fig. 2). The early continents may have produced intra-lithospheric gravitational absence ofba lateraldeviatoric temperature gradients stress (MPa) ensures that Reference no convective density stresses(g cm 3 ) stresses large enough to drive their gravitational spreading, to initiate subduction at their margins and to trigger episodes of subduc- A buoyant and stiff 0 continent 225 km thick (strongly 0 depleted mantle act on the lid, allowing100 us to isolate 300 the 500 dynamic effects2.8 of the3.0 continent. 3.2 3.4 Basaltic crust tion. Our model predicts the co-occurrence of deep to progressively root 170 km thick overlain by felsic crust 40 km thick; see Fig. 1B, a) is shallower mafic volcanics and arc magmatism within continents in inserted within the lid, on the left side of the domain to exploit 40 40 Crust the symmetry of the problem (Fig. 1A, a). A mafic crust 15 km thick covers the a self-consistent geodynamic framework, explaining the enigmatic multimodal volcanism and tectonic record of Archaean cratons 6.Moreover, our model predicts a petrological stratification and tectonic struc- thick greenstone80 covers on continents, as well as80 thick basaltic Strongly crust on whole system (Fig. 1A, a), consistent with the common occurrence of ture of the sub-continental lithospheric mantle, two predictions that the oceanic lid 3. depleted are consistent with xenolith 5 and seismic studies, respectively, and consistent with the existence of a mid-lithospheric seismic discontinuity 7. tinent imparts 120 a horizontal force large enough Our numerical solutions show that the presence oflithospheric a buoyant con- Depletion 120 to induce mantle a long period The slow gravitational collapse of early continents could have kickstarted transient episodes of plate tectonics until, as the Earth s inte- (Fig. 1 and Extended 160 Data Fig. (,50 150 Myr) of slow collapse of the whole continental lithosphere Strain rate rior cooled and oceanic lithosphere became heavier, plate tectonics 10 15 3), s 1 in agreement with the dynamics of spreading for gravity currents 19.Hence,acontinentoflargervolumeleads 225 became self-sustaining. to larger gravitational power and faster collapse. BecauseAsthenosphere of lateral spreading of thebcontinent, the100 adjacent 300lithospheric 500 lid is slowly 2.8pushed 3.0 under 3.2 3.4 Present-day plate tectonics is primarily driven by the negative buoyancy of cold subducting plates. Petrological and geochemical proxies of its margin (Fig. 1A, 0 b and Extended Data Fig. 3A, 0a). For gravitational Basaltic crust subduction preserved in early continents point to subduction-like processes stress lower than the yield stress of the oceanic lid, thickening of the margin already operating before 3 billion years (Gyr) ago 8,9 and perhaps of the lid is slow, and viscous drips (that is, Rayleigh Taylor instabilities) 40 40 as early as 4.1 Gyr ago 10. However, they are not unequivocal, and geodynamic modelling suggests that the thicker basaltic crust produced by typical of stagnant-lid convection 20, mitigate the thermal thickening mantleof detach from its base (Extended Data Fig. 3A, a and b). These Lithospheric instabilities, partial melting of a hotter Archaean or Hadean mantle would have had the lid. 80 80 increased lithospheric buoyancy and inhibited subduction 3,4.Mantleconvection When gravitational stresses overcome the yield stress of the lithospheric under a stagnant lid with extensive volcanism could therefore lid, subduction is initiated (Fig. 1A, b and c). Depending on the half-width have preceded the onset of subduction 11. In this scenario, it is classically of the continent 120 and its density contrast with the 120 adjacent oceanic lid (that assumed that the transition from stagnant-lid regime to mobile-lid regime is, its gravitational power) Strain three rate 10 15 situations s 160 1 can arise: first, Asthenosphere subduction and the onset of plate tectonics require that convective stresses overcame initiates and stalls (Extended Data Fig. 3b); second, 160the slab detaches and the strength of the stagnant lid 12 at some stage in the Archaean. the lid stabilizes (Fig. 1A, d and e and Extended Data Fig. 3c); or third, On the modern Earth, gravitational stresses due to continental buoyancy recurrent detachment of the slab continues until recycling of the oceanic can contribute to the initiation of subduction 2,13. The role of continental lid is completed, followed by stabilization (Extended Data Fig. 3d). When Depth (km) Depth (km) Depth (km) 21

LETTER doi:10.1038/nature13728 Spreading continents kick-started plate tectonics Patrice F. Rey 1, Nicolas Coltice 2,3 & Nicolas Flament 1 A a Continent Temperature (K) 0 1,000 2,000 0 400 km Depth (km) 200 400 600 293 K 1,820 K b Continent weakens, spreads laterally Ba Deviatoric stress (MPa) 100 300 500 0 40 Reference density (g cm 3 ) 2.8 3.0 3.2 3.4 0 Basaltic crust 40 Crust c d Depletion Larger horizontal stresses force subduction of buoyant oceanic lithosphere Depth (km) b 80 120 160 0 Strain rate 10 15 s 1 100 300 500 80 120 225 0 Strongly depleted lithospheric mantle Asthenosphere 2.8 3.0 3.2 3.4 Basaltic crust e Slab detaches, margin stabilizes Depth (km) 40 80 120 160 Strain rate 10 15 s 1 40 80 120 160 Lithospheric mantle Asthenosphere 22

1D steady-state heat conduction + production Consider the heat flux across a slab of thickness δy Fig. 4.5, Turcotte and Schubert, 2014 The net heat flux is simply the heat flux out minus the heat flux in, or q(y + y) q(y) Using Taylor series expansion and Fourier s first law, it can be shown that q(y + y) q(y) = y dq dy = y d apple dt k dy dy apple d 2 T = y k dy 2 for constant thermal conductivity k 23

1D steady-state heat conduction + production If we assume the only source of changing the heat flux in the slab is radiogenic heat, we can say apple d 2 T H y = y k dy 2 or k d2 T + H =0 dy2 where ρ is rock density Fig. 4.5, Turcotte and Schubert, 2014 Assuming T = T0 and q = q0 at y = 0, the solution is T = T 0 + q 0 k y H 2k y2 With this equation, we can plot a geotherm, or change in temperature with depth 24

Heat conduction in the mantle? T = T 0 + q 0 k y H 2k y2 70 mw m -2 is a reasonable average heat flow value observed at the surface If we assume a surface temperature T0 = 0 C, rock density ρ = 3300 kg m -3, H = 7.38 10-12 W kg -1 and k = 4 W m -1 K -1, what is the predicted temperature at 100 km depth? Does this seem reasonable? What about at 200 km depth? 25

Heat conduction in the mantle? Fig. 4.8, Turcotte and Schubert, 2014 Clearly, heat flow in the mantle is not strictly conductive 26

Geotherms in the continental crust With a 40-km-thick crust, you can see that heat production can increase the Moho temperature considerably Note, that this model has a constant heat flow basal boundary condition k = 2.75 W m - 1 K - 1 qm = 20 mw m - 2 ymax = 40 km 27

Geotherms in the continental crust Why might this matter? The approximate depth at which crustal rocks transition from brittle deformation to ductile deformation corresponds roughly to the 300 C isotherm in the crust Based on the models here, that depth could vary from ~15-40 km (), which has serious implications for how the crust deforms k = 2.75 W m - 1 K - 1 qm = 20 mw m - 2 ymax = 40 km 28

Distribution of heat producing elements Rock type By mass, H [W kg Heat production By volume, A [W m By volume, A [µw m Reference undepleated (fer/le) mantle 7.39E-12 2.44E-08 0.024 "Depleated" perido/tes 3.08E-13 1.02E-09 0.001 Tholeii/c basalt 1.49E-11 4.41E-08 0.044 Granite 1.14E-09 3.01E-06 3.008 Shale 7.74E-10 1.86E-06 1.857 Average con/nental crust 3.37E-10 9.26E-07 0.927 Chondri/c meteorites 3.50E-12 1.15E-08 0.012 As we ve seen, however, the concentration of heat-producing elements is not constant in the crust, but decreases with depth 29

Geotherms in the continental crust As before, changing the concentration of heatproducing elements shows us something important about how we might expect the crust to deform Here, we might predict partial (to complete) melting of the crust and very weak rocks if we don t consider decreasing heat production A = 2.5 μw m - 3 k = 2.75 W m - 1 K - 1 Onset of partial melting of crust qm = 20 mw m - 2 ymax = 40 km 30

Geotherms in thick continental crust The crust beneath the Tibetan Plateau is thought to be 70-80 km thick resulting from extensive shortening during orogenesis, and it is likely that this over-thickened crust is partially molten at depth as a result of radiogenic heat How might this affect the plateau topography? A = 2.5 μw m - 3 k = 2.75 W m - 1 K - 1 Onset of partial melting of crust qm = 20 mw m - 2 ymax = 80 km 31

Geodynamic implications We ve now see that radiogenic heat production can significantly influence the subsurface thermal field, particularly in the crust This may influence geodynamic behavior of the deforming crust by affecting the depth of the transition from brittle to ductile deformation, or the potential for partial melting of the crust Another important influence on the crustal thermal field is topography at the Earth s surface Why might topography matter? 32

Thermal field beneath periodic topography In our calculations of temperatures in the Earth thus far, we ve assumed a known surface temperature T0 at y = 0 Atmospheric temperatures decrease with elevation following an atmospheric lapse rate β (typical value: β = 6.5 C km -1 ) Combined, this suggests surface temperatures should vary with elevation in mountainous regions, an effect we can simulate Importantly, this is clearly a 2D problem 33

Thermal field beneath periodic topography To do this, we compensate for the temperature variations with elevation by scaling the temperature by ΔT at y = 0, yielding T (x, y) =T 0 + q my k + H 0h 2 r 1 e y/h r + T cos 2 x e 2 y/ k with q m H 0 h r T = h 0 k k where T0 is the surface temperature a sea level in nature, qm is the mantle heat flux, k is thermal conductivity, ρ is density, H0 is the surface heat production value, hr is the e-folding depth for heat production, λ is the wavelength of the 34 topography and h0 is the amplitude

Thermal field beneath periodic topography A = 2.5 μw m - 3 hr = 10 km k = 2.75 W m - 1 K - 1 qm = 20 mw m - 2 h0 = 4 km λ =??? km β = 6.5 C km - 1 35

Thermal field beneath periodic topography A = 2.5 μw m - 3 hr = 10 km k = 2.75 W m - 1 K - 1 qm = 20 mw m - 2 h0 = 4 km λ =??? km β = 6.5 C km - 1 36

Thermal field beneath topography Topography can influence the geometry of the thermal field beneath it, particularly as the relief and wavelength grow The depth of the perturbation is related to the wavelength, with longer wavelengths affecting deeper crustal levels This has important implications for data that depend on the thermal structure of mountainous regions, such as lowtemperature thermochronology data 37

Recap Heat conduction, or diffusion, is the process by which molecular or elemental interactions transfer energy to neighboring particles Heat conduction is dominantly controlled by rock material properties, such as the thermal conductivity or radiogenic heat production The thermal field in the crust plays an important role in determining how the crust deforms 38

References Rey, P. F., Coltice, N., & Flament, N. (2015). Spreading continents kick-started plate tectonics. Nature, 513(7518), 405 408. doi:10.1038/nature13728 Rosenberg, C. L., Medvedev, S. & Handy, M. (2007) Effects of melting on faulting and continental deformation. In: The Dynamics of Fault Zones, edited by M. R. Handy, G. Hirth, N. Hovius. Dahlem Workshop Report 95, The MIT Press, Cambridge, Mass., USA, 357-401. 39