We present a theory for the coupled flow of ice, subglacial water and subglacial sediment, which is designed to represent the processes which occur at the bed of an ice sheet. The ice is assumed to flow as a Newtonian viscous fluid, the water can flow between the till and the ice as a thin film, which may thicken to form streams or cavities, and the till is assumed to be transported, either through shearing by the ice, squeezing by pressure gradients in the till, or by fluvial sediment transport processes in streams or cavities. In previous studies, it was shown that the dependence of ice sliding velocity on effective pressure provided a mechanism for the generation of bedforms resembling ribbed moraine, while the dependence of fluvial sediment transport on water film depth provides a mechanism for the generation of bedforms resembling mega-scale glacial lineations. Here, we combine these two processes in a single model, and show that, depending largely on the granulometry of the till, instability can occur in a range of types which range from ribbed moraine through three-dimensional drumlins to mega-scale glacial lineations.
We provide and analyse a model for the growth of bacterial biofilms based on the concept of an extracellular polymeric substance as a polymer solution, whose viscoelastic rheology is described by the classical Flory-Huggins theory. We show that one-dimensional solutions exist, which take the form at large times of travelling waves, and we characterize their form and speed in terms of the describing parameters of the problem. Numerical solutions of the time-dependent problem converge to the travelling wave solutions.
We develop numerical solutions of a theoretical model which has been proposed to explain the formation of subglacial bedforms. The model has been shown to have the capability of producing bedforms in two dimensions, when they may be interpreted as ribbed moraine. However, these investigations have left unanswered the question of whether the theory is capable of producing fully three-dimensional bedforms such as drumlins. We show that, while the three-dimensional calculations show realistic quasi-three-dimensional features such as dislocations in the ribbing pattern, they do not produce genuine threedimensional drumlins. We suggest that this inadequacy is due to the treatment of subglacial drainage in the theory as a passive variable, and thus that the three-dimensional forms may be associated with conditions of sufficient subglacial water flux.
[1] This is a computational study of gravity-driven fingering instabilities in unsaturated porous media. The governing equations and corresponding numerical scheme are based on the work of Nieber et al. (2003) in which nonmonotonic saturation profiles are obtained by supplementing the Richards equation with a nonequilibrium capillary pressure-saturation relationship, as well as including hysteretic effects. The first part of the study takes an extensive look at the sensitivity of the finger solutions to certain key parameters in the model such as capillary shape parameter, initial saturation, and capillary relaxation coefficient. The second part is a comparison to published experimental results that demonstrates the ability of the model to capture realistic fingering behavior.Citation: Chapwanya, M., and J. M. Stockie (2010), Numerical simulations of gravity-driven fingering in unsaturated porous media using a nonequilibrium model, Water Resour. Res., 46, W09534,
We compare and investigate the performance of the exact scheme of the Michaelis-Menten (M-M) ordinary differential equation with several new non-standard finite difference (NSFD) schemes that we construct by using Mickens' rules. Furthermore, the exact scheme of the M-M equation is used to design several dynamically consistent NSFD schemes for related reactiondiffusion equations, advection-reaction equations and advection-reaction-diffusion equations. Numerical simulations that support the theory and demonstrate computationally the power of NSFD schemes are presented.
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