Mathematical modelling of geophysical melt drainage - Hewitt, I (University of Oxford;Mathematical Institute, 2009)
Fluid flows involving transport of a liquid phase in close proximity with its solid phase involve continuous transfer of mass and heat, which can influence the nature of the drainage that occurs. We consider mathematical models for two such situations; magma flow in the mantle and water flow beneath glaciers. In part I, we derive a model for porous flow within a partially molten column of mantle undergoing decompression melting. By ignoring composition effects, and by scaling the equations appropriately, approximate analytical solutions can be found for one-dimensional upwelling, which allow the region and extent of melting to be determined. We study the dynamics of open channels of melt flow in the same situation, and find that such channels would have low pressure compared to the surrounding porous flow, and therefore draw in melt from a region of the size of a compaction length. We suggest that such channels could form through the unstable effects of melting caused by heat transfer by the upwelling melt. We emphasise the similarity with channels of meltwater that are known to exist beneath ice. In part II we pose a generalised model for subglacial water flow, which is described as an effective porous medium, the pore space being determined from an evolution equation. This is used to investigate the flow into a channel, which is found to be drawn from a surrounding region whose size, we suggest, determines the spacing between major drainage channels beneath ice sheets. These are compared to the observed spacing of eskers. A critical condition on the discharge necessary to sustain a channel is found, which may provide a criteria to decide where and when channelised drainage occurs. Lastly, a simple drainage model is used to explain seasonal variations in the velocity of a valley glacier.