2. Heat Flow. Ge 163 3/30/15-

Transcription

1 2. Heat Flow Ge 163 3/30/15-

2 Outline 1. Diffusion equation 2. Unsteady solutions 3. Measurement of surface heat flow 4. Fluctuations in surface temperature 5. Overview of heat loss 6. Measurement of heat flow 7. Heat production 8. Chondritic coincidence 9. Simple geotherm 10. Constraining surface temperature changes (climate changes) 11. Heat flow on Faults

3 Fourier s Law q = kt,x q = heat flux (units: W/(m^2) k = thermal conductivity (units: W/(m K) T temperature x spatial coordinate

4 Derivation of the heat heat (diffusion of equation) Use conservation of energy T,t (x,t) = κt,xx (x,t) + F(x,t) multi-dimensional: T,t = κ 2 T + F κ= thermal diffusivity = k/(cρ) units m 2 /s c is the specific heat

5 Heuristic reasoning for diffusion & κ If temperatures change with a characteristic time τ, they will propagate a distance on the order of (κτ) 1/2 A time L /κ is required for temperatures to propagate a distance L.

6 Heat Flow with Arbitrary Initial Conditions The solution of this transient problem is helpful in understanding the basic physics of diffusion. [Use separation of variables, for example). T,t = κt,xx 0 < x <1 0 < t < BC's : (0 < t < ) T(0,t) = 0 T(1,t) = 0 IC : (0 x 1) T(x,0) = φ(x)

7 General solution T(x,t) = where A n = 2 n =1 A n e (nπ )2 κt sin(nπx) 1 φ(x)sin(nπx)dx 0 Two important points: (a) The only difference between the Fourier sine expansion of ϕ(x) and the solution is the insertion of the time factor e (nπ )2 κt (b) Terms further out in the series get small very fast

8 How do the temperature oscillations imposed onto the Earth s surface by diurnal, seasonal, and climatic effects diffuse into the Earth? T(t) Z=0 z Assume temperature at the Earth s surface varies as: T (0,t) = T o + ΔT cosωt ω = 2π f f =frequency;ω =circular frequency;τ =period τ = 1 f = 2π ω

9 Turcotte & Schubert use separation of variables to solve this equation T = T o + ΔTe z /d cos(ωt z d ) d = κτ π = ω 2k

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11 Can we drill deep (sample) enough? d ~ κτ π κ ~ 10 6 m 2 s 1 Period d, depth diurnal 24 hours 0.16 meter annual 365 days 10 meter climatic 10,000 yrs 1 km

12 Numerical solution of the thermal wave associated with T s pulse at the surface

13 Measurement of the surface heat flux requires measurement of the temperature gradient and via Fourier s Law: q=-kdt/dz q can be determined if k is measured in the laboratory: dt/dz ~ 20 to 30 C km -1 k ~ 2-3 W m -1 K -1 q ~ mw m -2 [An old unit in some papers: 1 heat flow unit [hfu=10-6 cal/ (cm 2 s) = mw/m 2 ) ]

14 The drilling process thermally perturbs the surrounding earth; Must wait for the system to return to equilibrium

15 4 recent heat flow Measurements from The Canadian shield. Temp measured with a thermister at every 10 m and k measured in lab from core cuttings For <k>=2.05 W m -1 K -1 dt/dz = 15.4 K km -1 q=27.8 mw m -2 [Guillou-Frottier, et al., 1995]

16 Smoothed Heat Flow from borehole measurements mw/m^2 SMU web site

17 Some rough estimates of average Heat flow out of the continents and Oceans [Turcotte & Schubert, 1982] q c = 56.3 mw m 2 q O = 78.2 mw m 2

18 What about the heat source term in the heat Equation? T,t = κ 2 T + F F is from the decay of Uranium, Thorium, and Potassium. For example: U Pb He + 6β + Q Q=47.4 MeV/atom Beta decay as the transformation of a neutron into a proton and an electron; Electron expelled from the nucleus as a negative beta particle

19 Rock H (W kg -1 ) ref Ref undepleated mantle Some heat production values F=H/C (F being the dt/dt used In previous equations 6.3 x TS basalt 2.6 x TS granite 9.6 x TS Chondritic meterorites 5.1 x TS Upper mantle 1.0 x Davies book

20 Assume perfect insulation dt dt ~ H c Assume that c~10 3 J kg -1 K -1 For granite dt/dt ~ 3 x 10-5 K yr -1 or 30 K after 1 Myr For upper mantle dt/dt ~ 3 x 10-8 K yr -1 or 0.03 K after 1 Myr

21 The decay of unstable isotopes is time-dependent, if there are N unstable isotopes in a rock, then dn dt = λn λ Is the decay constant N = N o e λt t 1/2 = half life = ln2 λ isotope t 1/2 (yr) 238 U 4.47x U 7.04x Th 1.40x K 1.25x10 9

22 Chondrites From U, Th, K H=5.1x10-12 W kg -1 Q (Chondritic Earth) = HM =(5.1x10-12 Wkg -1 )(5.93x10 24 kg) = 3.0x10 13 W Q Earth = q c A c + q o A o Q Earth =(56.3 mw m -2 )(2x10 8 km 2 ) + (78.2 mw m -2 )(3.1x10 8 km 2 ) = 3.55 x W =35 TW (Tera Watts) Chondritic Coincidence [Wasserburg et al., 1964] Ur = Urey ratio = present heat generation present heat loss

23 Let us look at a simple geotherm [i.e. T(z)]: q s T s Z=0 T,t = κt,zz + F T,t = 0 (Steady-state) T(z = 0) = T s Z q(z = 0) = q s solution T = T s + q s k z F 2κ z2

24 For the geotherm, take T s = 0 C q s = 70 mw m -2 ρ = 3300 kg m -3 H = 6.2 x K kg -1 ( reference undepleted mantle ) k = 4 W m -1 K -1 κ = 10-6 m 2 s T(ºC) 3000 Z (km) 200 Basalt solidus

25 Additional application: Constraining surface temperature changes (environmental changes) Lachenbruch, A. H., and Marshall, B. V., Changing climate: Geothermal evidence from permafrost in the Alaskan Arctic, Science 234, , 1986.

26 Heat flow measurements in the Permafrost of the North Slope of Alaska. Lachenbruch & Marshall [1986]

27 Inference is for a rise in temperature, although the estimate is Non-unique. Lachenbruch & Marshall [1986]

28 Inference of Temperature histories (either forward or inverse) use the part of the geotherm thought to be transient. Pollack & Huang, Means and 1 SD for 34 & 51 boreholes

29 1 deg C increase over 500 Yr, but mostly in last century Reconstructed Temp from Boreholes 5-year running mean of the globally averaged instrumental record of surface air temperature since 1860 Pollack, H. N. and Huang, S.: Climate reconstruction from subsurface temperatures, Ann. Rev. Earth Planetary Sci., 28, , 2000.

30 Pollack & Huang, Extracting Regional Variations from boreholes

31 Frictional Heating on Faults Dilemma between stress on seismogeneic predicted By rock friction experiments (>200 MPa) and the values inferred from seismic studies (such as stress drop from earthquakes, ~3 Mpa) Can this be resolved from heat flow studies: u = velocity τ = shear stress q = uτ (heat production on fault)

32 Lachenbruch & Sash [1988]

33 Heat signature on the Chelungpu fault associated with the 1999 Chi Chi, Taiwan earthquake Downhole temperature measurements were made in Sept., 2005 Kano et al. [2006]

34 Drilling after Tohoku 2011 eq IODP Expedition 343 Lowered a temperature observatory in a boreholein July 2012, 855 mbsf (7,000 m water depth)

35 Fulton et al., Science, 2013

36 Magnified view From previous slide Modeled Heat flow Anomaly Fault friction < 0.08 Fulton et al., Science, 2013