MIT OpenCourseWarehttp://ocw.mit.edu 12.002 Physics and Chemistry of the Earth and Terrestrial Planets Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.32 We can use this function to calculate heat loss through the surface of a planet if the planet simply cools statically with time (no convection or plate tectonics). The surface heat flow that would be predicted from the error function can be computed from: ! q(z,t) = K"T"zwhere K is the thermal conductivity, and substituting the error function for T(z,t) gives the heatflow q: ! q(z,t) =2K(Tm" Ts)2#$t% & ' ( ) * e"z2$t+ , - . / 0 2Evaluating at the surface, z=0: ! q(0,t) =K(Tm" Ts)#$t% & ' ( ) * This implies that surface heat flow should decrease as the inverse of the square root of time in a conductively cooling half-space (or a very thick layer such that the base of the layer is not cooling yet). We can apply this, or see that it works, by looking at the oceanic lithosphere of the Earth. Oceanic Subsidence and Heat Flow The theory of sea-floor spreading says that new oceanic crust and lithosphere is created at mid-ocean ridges and that the new lithosphere moves away from the ridge crest with time. If this is correct, then the properties of the oceanic lithosphere, as a function of time, should be governed by the laws of heat conduction as this outer shell of the Earth cools through time. Understanding how this results in physical observables is important not just for our understanding of the Earth, but also for understanding how lithospheric age and cooling may be displayed on other planets.33 We have just shown that surface heat flow for a cooling lithosphere goes as one over the square root of time. Let’s look at the data from the oceans: Plotted against the observed heat flow data from the oceans the agreement between observed heat flow and square-root of time relationship looks pretty good for appropriate values of K, κ, etc.: http://www.earth.northwestern.edu/people/seth/107/Ridges/depthage.htm However, as you can see from the map below, there is actually a huge scatter in heat flow observations, because heat flow measurements easily perturbed by water circulation in the sediments on the sea floor as well as through cracks in the underlying basement. Courtesy of Seth Stein, Northwestern University. Used with permission.34 We can’t measure the temperatures down in the lithosphere below a few kilometers depth, but we can measure average temperature through the effects of thermal expansion and contraction of the cooling mantle. First, let’s calculate how much a column of mantle (and a little crust on top) contracts as it cools. (Because of conservation of mass, the amount of material stays the same, but the vertical, or radial, dimension of the material will decrease according to its temperature and the coefficient of thermal expansion: Let’s suppose, as before, that the initial temperature of the mantle is a uniform Tm and that later the temperature of the mantle is T(z,t). The coefficient of thermal expansion, α, is defined as: ! "l ="lo1+#(T $ To)[ ]35 where l! "o is the thickness of a column of rock at temperature To, and ! "l is height of the same column of rock at temperature T. Rearranging, we can see that the height of a column of rock changes by an amount: ! ("l #"lo) ="lo$(T # To)as the temperature changes from To to T. If we want to apply this to to a column that goes from the surface to great depth within an infinite half space, we just need to integrate: ! "l =#(T(z) $ To)0%&dzThere are multiple ways to proceed, but the easiest is to relate this to relate temperature to heat. Remembering that the change in temperature is related to the change in heat per unit volume by: ! "Q =#Cp"TWe can now substitute in change in heat per unit volume for temperature, to find: ! "l =#$Cp"Q(z)0%&dzThe integral just gives the change in heat in a column with unit surface area extending from the surface downwards to infinity. If we take the time derivative of this with respect to time, we will have the rate at which heat is changing in a column per unit time per unit surface area. This has to be equal to the rate at which heat is leaving the column through the surface at z=0, or in otherwords has to be equal to the surface heat flow, qs. ! "#l"t=$%Cp""t#Q(z)0&'dz( ) * + , - =$%Cpqs(t)This is easy because it just says that the rate of change of thickness of the material is linearly proportional to the surface heat flow for36 a conductively cooling half space. We have already calculated the surface heat flow, so we can just substitute that in: ! "#l"t=$%CpK(Tm& Ts)'(t) * + , - . Or, we can easily integrate this over time to find: ! "l =2#K(Tm$ Ts)%Cp&'( ) * * + , - - tFinally, remembering that K=κρCp: ! "l =#$2%(Tm& Ts) tThis means that the amount of “shinkage” during cooling should simply be proportional to the square root of time. It would be the amount of subsidence that we would expect to see in cooling oceanic lithosphere but, because there is water on top of the ocean floor, there is an amplification effect (the water adds a load) that we will learn how to calculate in a few weeks. For now we can just stick it in to get: ! "l =#$2%(Tm& Ts)'m('m&'w)( ) * + , - twhere ρm and ρw are the densities of the mantle and seawater, respectively. We can verify how well this works by looking at the depth of sea-floor of different ages. The figure above shows the square-root-time depth-age relationship is excellent out to about 60 m.y. or so. After that it doesn’t work so well and the ocean floor does not subside as fast as the error function model suggests. This is usually attributed to processes that limit the cooling at depths between 100 and 125 km depth beneath the oceanic lithosphere, but they are not well understood. This could be quite important in terms of the evolution of other planetary lithospheres where the only way to keep the planet loosing heat rapidly, if no plate tectonics or planetary37 resurfacing, is by keeping the lithosphere thin by some process that acts on the base of the lithosphere and does not let it thicken beyond some point. Concepts for Heat Convection Density and Temperature Mantle heat convection is driven by the presence of hotter rocks at great depth than are present at shallow depth. Of course, it is not the temperature itself, but rather the density of the rocks, that is important for convection. Density