ABSTRACT. We develop theoretically a description of a possible subglacial drainage mechanism for glaciers and ice sheets moving over saturated, deformable till. The model is based on the plausible assumption that flow of water in a thin film at the ice-till interface is unstable to the formation of a channelized drainage system, and is restricted to the case in which meltwater cannot escape through the till to an underlying aquifer. In describing the physics of such channelized drainage, we have generalized and extended Rothlisberger's model of channels cut into basal ice to include "canals" cut into the till, paying particular attention to the role of sediment properties and the mechanics of sediment transport. We show that sediment-floored Rothlisberger (R) channels can exist for high effective pressures, and wide, shallow, ice-roofed canals cut into the till for low effective pressures. Canals should form a distributed, non-arborescent system, unlike R channels. For steep slopes typical of alpine glaciers, both drainage systems can exist, but with the water pressure lower in the R channels than in the canals; the canal drainage should therefore be unstable in the presence of channels. For small slopes typical of ice sheets, only canals can exist and we therefore predict that, if channelized meltwater flow occurs under ice sheets moving over deformable till, it takes the form of shallow, distributed canals at low effective pressure, similar to that measured at Ice Stream B in West Antarctica. Geologic evidence derived from land forms and deposits left by the Pleistocene ice sheets in North America and Europe is also consistent with predictions of the model.

ABSTRACT. We develop theoretically a description of a possible subglacial drainage mechanism for glaciers and ice sheets moving over saturated, deformable till. The model is based on the plausible assumption that flow of water in a thin film at the ice-till interface is unstable to the formation of a channelized drainage system, and is restricted to the case in which meltwater cannot escape through the till to an underlying aquifer. In describing the physics of such channelized drainage, we have generalized and extended Rothlisberger's model of channels cut into basal ice to include "canals" cut into the till, paying particular attention to the role of sediment properties and the mechanics of sediment transport. We show that sediment-floored Rothlisberger (R) channels can exist for high effective pressures, and wide, shallow, ice-roofed canals cut into the till for low effective pressures. Canals should form a distributed, non-arborescent system, unlike R channels. For steep slopes typical of alpine glaciers, both drainage systems can exist, but with the water pressure lower in the R channels than in the canals; the canal drainage should therefore be unstable in the presence of channels. For small slopes typical of ice sheets, only canals can exist and we therefore predict that, if channelized meltwater flow occurs under ice sheets moving over deformable till, it takes the form of shallow, distributed canals at low effective pressure, similar to that measured at Ice Stream B in West Antarctica. Geologic evidence derived from land forms and deposits left by the Pleistocene ice sheets in North America and Europe is also consistent with predictions of the model.

Bakaninbreen in Svalbard and Trapridge Glacier in Yukon Territory, Canada, are two prominent examples of surging glaciers which are thought to be controlled by their thermal regime. Both glaciers have developed large bulges which have propagated forward as travelling wave fronts, and which are thought to divide relatively stagnant downstream cold-based ice from faster-moving warm-based upstream ice. Additionally, both glaciers are underlain by a wet, metres thick layer of deforming till. We develop a simple model for the cyclic surging behaviour of these glaciers, which interrelates the motion of the ice and till through a description of the subglacial hydrology. We find that oscillations (surges) can occur if the subglacial hydrological transmissivity is sufficiently low and the till layer is sufficiently thin, and we suggest that these oscillations are associated with the development and propagation of a travelling wave front down the glacier. We therefore interpret the travelling wave fronts on both Trapridge Glacier and Bakaninbreen as manifestations of surges. In addition, we find that the violence of the surge in the model is associated with the resistance to ice flow offered by undulations in the bed, and the efficiency with which occasional hydrological events can release water accumulated at the glacier sole.

We present a model for the determination of a sliding law in the presence of subglacial cavitation. This law determines the basal stress at a clean ice‒bedrock interface in terms of the velocity and effective pressure. The method is based on an exact solution of the Nye—Kamb (linearly viscous) sliding problem with cavities, and uses ideas of Lliboutry (1979) to construct, via renormalization methods, an approximate law for general bedrock form. We show that, for a bedrock whose spectrum has a power‒law behaviour, one obtains a sliding law which gives the basal shear stress proportional to a power of the velocity, and to a power of the effective pressure.The effect of subglacial cavitation on the drainage system is examined, using recent ideas of Kamb. For sufficiently high velocities, drainage through a Röthlisberger tunnel system is unstable, and drainage takes place through the linked system of cavities. This leads to a reduction of the effective pressure, and by taking account of this, one can rewrite the sliding law in terms of stress and velocity only.This sliding law can be multi‒valued, and it is suggested that this underlies the dynamic phenomenon of surges.

This thesis deals with the mathematical modelling of nutrient uptake by plant roots. It starts with the Nye-Tinker-Barber model for nutrient uptake by a single bare cylindrical root. The model is treated using matched asymptotic expansion and an analytic formula for the rate of nutrient uptake is derived for the first time. The basic model is then extended to include root hairs and mycorrhizae, which have been found experimentally to be very important for the uptake of immobile nutrients. Again, analytic expressions for nutrient uptake are derived. The simplicity and clarity of the analytical formulae for the solution of the single root models allows the extension of these models to more realistic branched roots. These models clearly show that the "volume averaging of branching structure" technique commonly used to extend the Nye-Tinker-Barber with experiments can lead to large errors. The same models also indicate that in the absence of large-scale water movement, due to rainfall, fertiliser fails to penetrate into the soil. This motivates us to build a model for water movement and uptake by branched root structures. This model considers the simultaneous flow of water in the soil, uptake by the roots, and flow within the root branching network to the stems of the plant. The water uptake model shows that the water saturation can develop pseudo-steadystate wet and dry zones in the rooting region of the soil. The dry zone is shown to stop the movement of nutrient from the top of the soil to the groundwater. Finally we present a model for the simultaneous movement and uptake of both nutrients and water. This is discussed as a new tool for interpreting available experimental results and designing future experiments. The parallels between evolution and mathematical optimisation are also discussed.

This paper presents a synopsis of some recent work, still in progress, aimed at elucidating a quantitative explanation of the processes by which flow chimneys form when certain types of alloys are directionally solidified. If (for example) light fluid is released at the liquid-solid "mushy" (dendrite) zone, and cooling is from below, then the intermediate fluid flow undergoes convection through the porous dendrite mass. This can lead to an "instability" of the form of the mushy zone, such that upwelling light fluid flows preferentially in channels within the dendrite mass. What we seek to develop here, is a mathematical basis by which this phenomenon may be properly understood. Accordingly, a mathematical model is developed, simplified, and partially analysed, and as a result we are able to make one specific prediction concerning a criterion for the onset of convection and freckling. This prediction is equivalent to the classical Rayleigh number condition for convective instability.

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Irregularities in observed population densities have traditionally been attributed to discretization of the underlying dynamics. We propose an alternative explanation by demonstrating the evolution of spatiotemporal chaos in reaction-diffusion models for predator-prey interactions. The chaos is generated naturally in the wake of invasive waves of predators. We discuss in detail the mechanism by which the chaos is generated. By considering a mathematical caricature ofthe predator-prey models, we go on to explain the dynamical origin of the irregular behavior and to justify our assertion that the behavior we present is a genuine example of spatiotemporal chaos.The widespread spatial and temporal irregularities in actual population densities are in marked contrast to the smooth predictions of early ecological models (1-3). Simple ordinary differential equation models for three or more interacting species have long been known to exhibit chaotic behavior (4-6), but when only two species are involved, such simple models cannot predict chaos. Standard explanations for observed irregularities in such systems rely on discretization of space, time, or population density (7-9); difference equations, coupled oscillators, and cellular automata frequently exhibit chaotic solutions in appropriate parameter regimes. This raises the question of whether ecological chaos arising from interactions between two species depends intrinsically on discretization of population behavior. We present results suggesting that this is not the case. We show that spatiotemporal chaos can arise naturally in a class of reaction-diffusion models of predator-prey interactions. Necessary ingredients for this are classical oscillatory predator-prey dynamics (10, 11) plus random (diffusive) movement of both predators and prey. With these ingredients, a spatiotemporal invasion of predators may have chaotic solutions in its wake.Chaotic solutions of reaction-diffusion equations are very rare and have previously been demonstrated, arising by quite different mechanisms, only in models of chemical reactions (12-16) and cardiac electrical activity (17, 18). Our work demonstrates chaos in ecological models of two interacting species without either discretization or delay effects. Waves of InvasionWe consider reaction-diffusion models for predator-prey interactions of the form ap/lt = DpA2p/x2 + fp(p, h)[la] ahlat = Dh a2h/aX2 + fh(p, h).[lb]Herep(x, t) and h(x t) are the population densities of predators and prey with diffusion coefficients Dp and Dh, respectively, and x and t denote space and time. Throughout, we restrict attention to one-dimensional spatial domains. Biologically realistic kinetic terms fp and fh will have two nontrivial equilibria, a "prey only" state, in which p = 0, h = ho, and a "coexistence" state, in which p = Ps, h = h,. Models of type 1 have been studied by many previous authors in the case when P = pSI h = hs is a stable equilibrium. In this case, one of the classical types of solution is a wave of invasion, that is, a travel...