Heat Transfer II: Time-Dependent Heatflow and Evolution of the Oceanic Lithosphere

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1 Heat Transfer II: Time-Dependent Heatflow and Evolution of the Oceanic Lithosphere Learning objectives: I can explain the solution for the instantaneous heating or cooling of a semi-infinite half-space. I can use this solution for various problems in geophysics I can explain the plate-cooling model as well as how and why it differs from the half-space cooling model for the oceanic lithosphere and subsidence of the seafloor. I can use the plate cooling model to predict the thermal evolution and surface heat flow of oceanic lithosphere. For Mon 4/2 Read Turcotte & Schubert Sections Verify that the boundary conditions given by Eq equate to those given by Eq T 2T Section 4.15 shows that the partial differential equation can be solved by changing the variables t t y2 and y in the equation so that they compose the similarity variable. This produces an ordinary differential equation, which is easier to solve. What is that ordinary differential equation? Do problem Derive an expression for the thickness of the thermal boundary layer if we define it to be the distance to where = What is a geotherm? What is an isotherm? Be prepared to discuss and work on problems 1-3. For Wed 4/4 Read Turcotte & Schubert Sections 4.17 and 4.23 How do the initial conditions and boundary conditions of the plate cooling model compare with the half-space cooling model for the oceanic lithosphere? Examine Figure How does the half-space cooling model differ from the two plate cooling models and why? Be prepared to discuss and work on the problem set. Bring laptops 1

2 Problem Set 11 Due Monday 9 April, 12:30 pm 1) Problem If the surface temperature is increased suddenly by 10 K, how long is it before the temperature increases 2 K at a depth of 1 m ( = 1 mm 2 /s). 2) Problem Displacements along faults can bring rock masses with different temperatures... 3) The Hawaiian island chain sits on a broad bathymetric swell on the sea floor. A popular explanation is that a hot and buoyant mantle plume is pushing up on the base of the lithosphere, and lithosphere moves away from the mantle plume, it drags a layer of hot plume material with it. With time, the plume layer thins and cools. To test this explanation, you propose to do a heatflow survey someplace along the Hawaiian hotspot chain. First, you must determine where to do the survey. You know that it will take time for excess heat from the plume to diffuse up into the lithosphere and change the surface temperature gradient, and hence surface heat flow. You must therefore estimate how much time, and thus distance of Pacific plate motion (moving at 80 km/myr), it will take for a surface heat flow anomaly to be detectable. To construct your thermal model, assume that at time t = 0 Myr the seafloor was directly over where the lithosphere first encountered the plume (now below Kilauea). At this point, the lithosphere has a thickness of b = 80 km and a normal temperature so the temperature anomaly T=0 C everywhere in the lithosphere. Below the lithosphere, the plume introduces an excess temperature T = Tmax. You are only concerned with changes introduced by the plume, therefore you can neglect the background heat flow and normal thermal structure of the lithosphere cooling with age. You are to compute how the excess temperature T evolves with time t and depth y, and compute how the excess heat flow q at the surface (y = 0) evolves. Show solutions for Tmax= 100 C and 200 C, b= 80 km, and = 31 km 2 /Myr. The accuracy of your heatflow probes are such that the minimum heatflow anomaly you can possibly detect is ~2 mw/m 3. (a) Plot T vs. y at three different distances (x = t x 80 km/myr) from Kilauea (b) Plot q at the surface versus distance along the chain compared to a horizontal line at q = 2 mw/m 2. (c) Is the Hawaiian plume likely to be detectable with your heat flow probes? If so, at what distance from Kilauea should you place your probes? 2

3 Here are two ways to solving the problem. Method 1: Use the solution for T&S problem A rather crude approximation is to assume the plume layer is much thicker than the lithosphere (i.e., it extends to infinite depth). The initial condition is that the temperature anomaly T = 0 C in the lithosphere (i.e., T=T0=T1 = 0 C, 0 < y <b) and the temperature anomaly is maximum just below the base of the lithosphere (i.e., T = T2 = Tmax, y > b). The boundary conditions are no temperature anomaly at the surface ( T =T0 = 0 C at y = 0) and temperature anomaly is maximum deep in the plume layer ( T = T2 = Tmax at y = ). Method 2 (worth 50% extra credit): The better way to solve the problem is to model a plume layer of finite initial thickness. By superposing error function solutions, or alternatively, by using Fourier transforms, you can simulate the temperature structure shown to the right. The initial condition is T = 0 C (y b) and T = Tmax (b < y < c). The boundary conditions are T = 0 C at y = 0 and y =. Assume b = c = 80 km. y=0 y=b y=c 3

4 4) The figure to the right shows seafloor depths adjacent to the islands as a function of age along the chain. The original explanations for the rapid swell uplift and subsidence is that hot mantle rapidly thins the oceanic lithosphere such that its depth and subsequent cooling is like that of younger lithosphere [Detrick and Crough, 1978]. Essentially, the hotspot resets the thermal age of the Hawaiian lithosphere. At the shallowest part of the swell (i.e. around Big Island, which is shown at age = 0 in the plot below) the adjacent seafloor is ~90 Myr old. The shown curve is a fit to the Hawaiian swell depths assuming the adjacent seafloor was not 90 Myr, but instead has an age of treset, thus making the lithosphere behave as it were (90- treset) Myr younger. Try for yourself to bound the dots with two seafloor subsidence curves of different ages (treset) using the plate cooling model (Eq ). The data are in the file Hw11_data.txt and contain age of adjacent seafloor and depth. Assume a zero-age (t = 0) seafloor depth of 2.6 km. You will have to adjust the asthenospheric temperature (T1-T0) and/or plate thickness yl0 to fit the two ends of the plot. Reasonable ranges to try are T1-T0 ~ C and yl0 ~ km. **For reference, also plot a normal seafloor curve with t = 90 m.y. beneath Big Island.** 4

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