Nonlinear Stability Analyses of Problems in Patterned Ground Formation and Penetrative Convection

Applied mathematics, Geophysics

McKay, Geoffrey

University:University of Glasgow

关键词: Applied mathematics, Geophysics;   

Others  :  http://theses.gla.ac.uk/76416/1/13834196.pdf

来源: University of Glasgow

【 摘 要 】

In this thesis we present nonlinear energy stability analyses of a number of problems associated with the phenomenon of penetrative convection. In particular, we consider penetrative convection models which incorporate nonlinear buoyancy terms or internal heat generation (either together or separately). To begin we give a brief account of penetrative convection and situations in which it naturally occurs in geophysics. We also discuss the energy methods which we shall employ in later chapters. Our analysis begins by proving continuous dependence of the solution to the Boussinesq equations, both forward and backward in time, on a heat source and on the heat flux on the lower boundary. Linear theory and the energy method are then employed to study the effect of nonlinear density relations or a non uniform heat source on the onset of penetrative convection. We next introduce and describe patterned ground, a geological phenomenon whose formation is believed to involve penetrative convection in a saturated porous medium. We discuss the influence of a cubic density law and time-periodic solar radiation on the stability of the porous layer and on the size of the stone polygons. We then perform a linear analysis of a two layer problem which models the formation of patterned ground under water. Our predictions for patterned ground are compared with observations made by field workers and results from previous mathematical analyses. To conclude, we use a generalized energy to prove the stabilizing influence of rotation on a fluid layer, even when the layer is subject to internal heating.