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| The large temperature excess in young planets leads to a large surface heat flow which is evacuated as much by intense volcanic activity as by thermal diffusion across a boundary layer. However, most fluid dynamic codes in Earth Science are implementing a zero vertical velocity at the surface of their models. This condition forbids any direct transport of material and therefore any advective extraction of heat from depth. We propose a new set of boundary conditions on the top surface for the momentum and energy equations. The vertical velocity is not imposed to zero but we compute the topography generated from this velocity. A diffusion term mimics the various processes that can redistribute the topography (mechanical and chemical erosion, gravity sliding, magma spreading on the surface...). The resulting topography affects the internal flow by imposing an equivalent vertical stress to the mantle. We show that with minimal approximations, the new condition can be implemented in numerical models. In the energy equation we impose a surface temperature only when the surface velocity is downward and a zero temperature gradient elsewhere. Our 2D and 3D numerical calculations of bottom and internally heated convection show that when increasing the Rayleigh number, the model evolves continuously from the typical pattern of convection and heat diffusion through a thermal boundary layer to a planform where all the heat is brought to the surface by large plumes. The extracted heat flux increases with the Rayleigh number from Ra^.3 at low Rayleigh number to Ra^.5 at high Rayleigh number where simultaneously, the temperature decreases from .5 to very low values. We discuss the implication of this model for the early evolution of the Earth and other solid planets, and the present state of Io. |
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