Strombolian volcanoes are thought to maintain their semi-permanent eruptive style by means of counter-current two-phase convective flow in the volcanic conduit leading from the magma chamber, driven by the buoyancy provided by exsolution of volatiles such as water vapour and carbon dioxide in the upwelling magma, due to pressure release. A model of bubbly two-phase flow is presented to describe this, but it is found that the solution breaks down before the vent at the surface is reached. We propose that the mathematical breakdown of the solution is associated with the physical breakdown of the two-phase flow regime from a bubbly flow to a churn-turbulent flow. We provide a second two-phase flow model to describe this regime, and we show that the solution can be realistically continued to the vent. The model is also in keeping with observations of eruptive style.

# Past Mathematical Geoscience Seminar

The vast majority of the World's documented meteorite specimens have been collected from Antarctica. This is due to Antarctica’s ice dynamics, which allows for the significant concentration of meteorites onto ice surfaces known as Meteorite Stranding Zones. However, meteorite collection data shows a significant anomaly exists: the proportion of iron-based meteorites are under-represented compared to those found in the rest of the World. Here I explain that englacial solar warming provides a plausible explanation for this shortfall: as meteorites are transported up towards the surface of the ice they become exposed to increasing amounts of solar radiation, meaning it is possible for meteorites with a high-enough thermal conductivity (such as iron) to reach a depth at which they melt their underlying ice and sink back downwards, offsetting the upwards transportation. An enticing consequence of this mechanism is that a sparse layer of meteorites lies just beneath the surface of these Meteorite Stranding Zones...

Inverse problems arise in many applications. One could solve them by adopting a Bayesian framework, to account for uncertainty which arises from our observations. The solution of an inverse problem is given by a probability distribution. Usually, efficient methods at hand to extract information from this probability distribution involves the solution of an optimization problem, where the objective function is highly nonconvex. In this talk, we explore a reformulation of inverse problems, which helps in convexifying the objective function. We also discuss a method to sample from this probability distribution.

Much of our understanding of the tropospheric dynamics relies on the concept of discrete internal modes. However, discrete modes are the signature of a finite system, while the atmosphere should be modeled as infinite and "is characterized by a single isolated eigenmode and a continuous spectrum" (Lindzen, JAS 2003). Is it then unphysical to use discrete modes? To resolve this issue we obtain an approximate radiation condition at the tropopause --- this yields an EBC. We then use this EBC to compute a new set of vertical modes: the leaky rigid lid modes. These modes decay, with decay time-scales for the first few modes ranging from an hour to a week. This suggests that the rate of energy loss through upwards propagating waves may be an important factor in setting the time scale for some atmospheric phenomena. The modes are not orthogonal, but they are complete, with a simple way to project initial conditions onto them.

The EBC formulation requires an extension of the dispersive wave theory. There it is shown that sinusoidal waves carry energy with the group speed c_g = d omega / dk, where both the frequency omega and wavenumber k are real. However, when there are losses, complex k's and omega's arise, and a more general theory is required. I will briefly comment on this theory, and on how the Laplace Transform can be used to implement generic EBC.

In this talk, I will present two different aspects of the ice flow modelling, including a theoretical part and an applied part. In the theoretical part, I will derive some "mechanical error estimators'', i.e. estimators that can measure the mechanical error between the most accurate ice flow model (Glen-Stokes) and some approximations based on shallowness assumption. To do so, I will follow residual techniques used to obtain a posteriori estimators of the numerical error in finite element methods for non-linear elliptic problems. In the applied part, I will present some simulations of the ice flow generated by the Rhone Glacier, Switzerland, during the last glacial maximum (~ 22 000 years ago), analyse the trajectories taken by erratic boulders of different origins, and compare these results to geomorphological observations. In particular, I will show that erratic boulders, whose origin is known, constitute valuable data to infer information about paleo-climate, which is the most uncertain input of any paleo ice sheet model.

The accumulation of surface meltwater on ice shelves can lead to the formation of melt lakes. These structures have been implicated in crevasse propagation and ice-shelf collapse; the Larsen B ice shelf was observed to have a large amount of melt lakes present on its surface just before its collapse in 2002. Through modelling the transport of heat through the surface of the Larsen C ice shelf, where melt lakes have also been observed, this work aims to provide new insights into the ways in which melt lakes are forming and the effect that meltwater filling crevasses on the ice shelf will have. This will enable an assessment of the role of meltwater in triggering ice-shelf collapse. The Antarctic Peninsula, where Larsen C is situated, has warmed several times the global average over the last century and this ice shelf has been suggested as a candidate for becoming fully saturated with meltwater by the end of the current century. Here we present results of a 1-D mathematical model of heat transfer through an idealized ice shelf. When forced with automatic weather station data from Larsen C, surface melting and the subsequent meltwater accumulation, melt lake development and refreezing are demonstrated through the modelled results. Furthermore, the effect of lateral meltwater transport upon melt lakes and the effect of the lakes upon the surface energy balance are examined. Investigating the role of meltwater in ice-shelf stability is key as collapse can affect ocean circulation and temperature, and cause a loss of habitat. Additionally, it can cause a loss of the buttressing effect that ice shelves can have on their tributary glaciers, thus allowing the glaciers to accelerate, contributing to sea-level rise.

Accurate methods for the first-order advection equation, used for example in tracking contaminants in fluids, usually exploit the theory of characteristics. Such methods are described and contrasted with methods that do not make use of characteristics.

Then the second-order wave equation, in the form of a first-order system, is considered. A review of the one-dimensional theory using solutions of various Riemann problems will be provided. In the special case that the medium has the ‘Goupillaud’ property, that waves take the same time to travel through each layer, one can derive exact solutions even when the medium is spatially heterogeneous. The extension of this method to two-dimensional problems will then be discussed. In two-dimensions it is not apparent that exact solutions can be found, however by exploiting a generalised Goupillaud property, it is possible to calculate approximate solutions of high accuracy, perhaps sufficient to be of benchmark quality. Some two-dimensional simulations, using exact one-dimensional solutions and operator splitting, will be described and a numerical evaluation of accuracy will be given.

In this talk I will review mathematical models used to describe the dynamics of ice sheets, and highlight some current areas of active research. Melting of glaciers and ice sheets causes an increase in global sea level, and provides many other feedbacks on isostatic adjustment, the dynamics of the ocean, and broader climate patterns. The rate of melting has increased in recent years, but there is still considerable uncertainty over why this is, and whether the increase will continue. Central to these questions is understanding the physics of how the ice intereacts with the atmosphere, the ground on which it rests, and with the ocean at its margins. I will given an overview of the fluid mechanical problems involved and the current state of mathematical/computational modelling. I will focus particularly on the issue of changing lubrication due to water flowing underneath the ice, and discuss how we can use models to rationalise observations of ice speed-up and slow-down.

Humpback whales are iconic mammals at the top of the Antarctic food chain. Their large reserves of lipid-rich tissues such as blubber predispose them to accumulation of lipophilic contaminants throughout their lifetime. Changes in the volume and distribution of lipids in humpback whales, particularly during migration, could play an important role in the pharmacokinetics of lipophilic contaminants such as the organochlorine pesticide hexachlorobenzene (HCB). Previous models have examined constant feeding and nonmigratory scenarios. In the present study, the authors develop a novel heuristic model to investigate HCB dynamics in a humpback whale and its environment by coupling an ecosystem nutrient-phytoplankton-zooplankton-detritus (NPZD) model, a dynamic energy budget (DEB) model, and a physiologically based pharmacokinetic (PBPK) model. The model takes into account the seasonal feeding pattern of whales, their energy requirements, and fluctuating contaminant burdens in the supporting plankton food chain. It is applied to a male whale from weaning to maturity, spanning 20 migration and feeding cycles. The model is initialized with environmental HCB burdens similar to those measured in the Southern Ocean and predicts blubber HCB concentrations consistent with empirical concentrations observed in a southern hemisphere population of male, migrating humpback whales.