DH 3rd floor SR

The Preisach model of hysteresis has a long history, a convenient algorithmic form and "nice" mathematical properties (for a given value of nice) that make it suitable for use in differential equations and other dynamical systems. The difficulty lies in the fact that the "parameter" for the Preisach model is infinite dimensional—in full generality it is a measure on the half-plane. Applications of the Preisach model (two interesting examples are magnetostrictive materials and vadose zone hydrology) require methods to specify a measure based on experimental or observed data. Current approaches largely rely on direct measurements of experimental samples, however in some cases these might not be sufficient or direct measurements may not be practical. I will present the Preisach model in all its glory, along with some history and applications, and introduce an open inverse problem of fiendish difficulty.

Ice-dammed lakes form next to, on the surface of, and beneath glaciers

and ice sheets. Some lakes are known to drain catastrophically,

creating hazards, wasting water resources and modulating the flow of

the adjacent ice. My work aims to increase our understanding of such

drainage. Here I will focus on lakes that form next to glaciers and

drain subglacially (between ice and bedrock) through a channel. I will

describe how such a system can be modelled and present results from

model simulations of a lake that fills due to an input of meltwater

and drains through a channel that receives a supply of meltwater along

its length. Simulations yield repeating cycles of lake filling and

drainage and reveal how increasing meltwater input to the system

affects these cycles: enlarging or attenuating them depending on how

the meltwater is apportioned between the lake and the channel. When

inputs are varied with time, simulating seasonal meteorological

cycles, the model simulates either regularly repeating cycles or

irregular cycles that never repeat. Irregular cycles demonstrate

sensitivity to initial conditions, a high density of periodic orbits

and topological mixing. I will discuss how these results enhance our

understanding of the mechanisms behind observed variability in these

systems.

Turbidity currents - submarine flows of sediment - are capable of transporting particulate material over large distance. However direct observations of them are extremely rare and much is inferred from the deposits they leave behind, even though the characteristics of their source are often not known. The submarine flows of volcanic ash from the Soufriere Hills Volcano, Monsterrat provide a unique opportunity to study a particle-driven flow and the deposit it forms, because the details of the source are relatively well constrained and through ocean drilling, the deposit is well sampled.

We have formed simple mathematical models of this motion that capture ash transport and deposit. Our description brings out two dynamical features that strongly influence the motion and which have previously often been neglected, namely mixing between the particulate flow and the oceanic water and the distribution of sizes suspended by the flow. We show how, in even simple situations, these processes alter our views of how these currents propagate.

Mesocosm experiments provide a major test bed for models of plankton, greenhouse gas export to the atmosphere, and changes to ocean acidity, nitrogen and oxygen levels. A simple model of a mesocosm plankton ecology is given in terms of a set of explicit natural population dynamics rules that exactly conserve a key nutrient. These rules include many traditional population dynamics models ranging from Lotka-Volterra systems to those with more competitors and more trophic levels coupled by nonlinear processes. The rules allow a definition of an ecospace and an analysis of its behaviour in terms of equilibrium points on the ecospace boundary.

Ecological issues such as extinctions, plankton bloom succession, and system resilience can then be analytically studied. These issues are understood from an alternative view point to the usual search for interior equilibrium points and their classification, coupled with intensive computer simulations. Our approach explains why quadratic mortality usually stabilises large scale simulation, but needs to be considered carefully when developing the next generation of Earth System computer models. The ‘Paradox of the Plankton’ and ‘Invasion Theory’ both have alternative, yet straightforward explanations within these rules.

4D-Var is a widely used data assimilation method, particularly in the field of Numerical Weather Prediction. However, it is highly sequential: integrations of a numerical model are nested within the loops of an inner-outer minimisation algorithm. Moreover, the numerical model typically has a low spatial resolution, limiting the number of processors that can be employed in a purely spatial parallel decomposition. As computers become ever more parallel, it will be necessary to find new dimensions over which to parallelize 4D-Var. In this talk, I consider the possibility of parallelizing 4D-Var in the temporal dimension. I analyse different formulations of weak-constraint 4D-Var from the point of view of parallelization in time. Some formulations are shown to be inherently sequential, whereas another can be made parallel but is numerically ill-conditioned. Finally, I present a saddlepoint formulation of 4D-Var that is both parallel in time and amenable to efficient preconditioning. Numerical results, using a simple two-level quasi-geotrophic model, will be presented.