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Whole Mantle Dynamics November 30, 2009 lecture What equations are needed to construct a theory of planetary convection? •
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Fluid Flow (Navier-­‐Stokes) equation Heat Conduction (Fourier’s Law) Heat Production LaPlace’s (gravity) equation Constituitive (elastic-­‐plastic-­‐brittle) equation Equation of state Continuity equation Put above into numerical calculation, use boundary and initial conditions. Predict the nature of terrestrial convection. Difficulties include: •
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Uncertainties in material properties and heat production – assumptions may not be correct. Highly non-­‐linear (potentially unstable) numerical problems Rather than try to do a completely “realistic” model, numerical simulations are often used to explore how results depend on assumptions. Here is a 3-­‐D calculation for a spherical Earth that is isoviscous (clearly an incorrect assumption): Shown are surface temperatures with red being hot and blue being coldest. Material is subducting in the blue areas and upwelling at the red. On this “planet” there are many convection cells that have a spreading center point surrounded by the down-­‐
wellings. This is completely unlike convection within Earth’s mantle. However, by making viscosity depend on temperature and starting the calculation with surface plates that are rigid, one can get the following result: Here the number of convection cells is reduced and the overlying plates organize the flow. Up-­‐welling occurs along linear features (unlike the spot up-­‐welling in the previous figure). These two calculations are used to argue that the nature of plate tectonics on Earth is strongly influenced by the plates. A “Tale of Two Planets” Although similar to Earth in size and bulk composition, the nature of tectonics on Venus is entirely different. The image above (with large vertical exaggeration) is based on satellite altimetry and shows a surface dominated by shield volcanoes and lava flows. Based on the Russian lander, we know that the surface is basaltic. There is no evidence of plate tectonic features (spreading centers and subduction zones) yet the surface is relatively young. The leading hypothesis is that Venus is a “one plate” planet that has catastrophic “resurfacing events” on the 100 million year time scale. Why is Venus so different? Why not spreading and subduction on Venus? One leading idea is that Venus’s lithosphere is too strong. This seems counter intuitive since the surface temperature is about 600°C and rock has reduced strength at higher temperatures. However, the one thing Earth has that Venus does not is water. Water, incorporated into rocks as hydrated minerals, has an even larger effect on the ductility of rock. Perhaps water-­‐weakened rocks are the reason plates can break and subduct on Earth. Why is convection “inevitable” for planets? The Rayleigh Number must be considered. The two modes of heat transport in planets are conduction and convection. Conduction (Fourier’s Law) gets heat to the surface through rigid rock. Convection transports heat in the interior where higher temperatures generate greater ductility (lower viscosity). A very small planet will not convect if it is too cold to have ductile rock and if the internal heat can be transported along a conductive temperature profile. For a larger planet, the conductive temperature gradient leads to temperatures high enough to soften or melt the material at depth. At that point convection becomes the dominant mechanism of heat transport. This raises the question of how to predict whether a planet does or does not have internal convection. The answer to this is provided by analysis of the fluid flow equation (Navier-­‐Stokes). In short, a system will convect if the buoyancy forces are larger than the frictional forces (viscous forces) trying to impede motion. The ratio of buoyancy to frictional forces is called the Rayleigh number: R = Fbouyancy/Ffriction Using an analysis of the Navier-­‐Stokes equation, the Rayleigh number is a combination of material properties and properties of the system: R = αgρ ΔT h3/κν Where α is thermal expansion, g is gravitational acceleration, ρ is density, ΔT is the temperature difference from the top to the bottom, h is the height of the convecting system, κ is thermal diffusivity, and ν is viscosity. Note that increasing quantities on the top increases the Rayleigh number while increasing the quantities on the bottom decreases R. Note also that the size of the convecting system comes in as its cube. Thus, larger systems have much larger Rayleigh number (with all other variables held constant). Lab work and theory shows that there is a critical Raleigh number (Rc) that separates convecting from non-­‐convecting systems. For R > Rc, the system convects. The value of Rc is somewhat dependent on the geometry of the system and on how the material is being heated (from below or from within), but is usually found to be in a range around 1000. What is the Rayleigh number for Earth? Although some of the parameters in the equation given above are uncertain (viscosity at great depth in Earth is uncertain by several orders of magnitude), the estimates for Earth lie in the range from a million to a billion – much larger than the critical Rayleigh number. This strongly supports the idea that Earth’s mantle must convect.