Past Mathematical Geoscience Seminar

Stick-slip behavior is a distinguishing characteristic of the flow of Whillans Ice Stream. Distinct from stick-slip on northern hemisphere glaciers, which is generally attributed to supraglacial melt, the behavior is thought be be controlled by fast processes at the bed and by tidally-induced stress. Modelling approaches to studying this phenomenon typically consider ice to be an elastically-deforming solid (e.g. Winberry et al, 2008; Sergienko et al, 2009). However, there remains a question of whether irreversible, i.e. viscous, deformation is important to the stick-slip process; and furthermore whether the details of stick-slip oscillations are important to ice stream evolution on longer time scales (years to decades).

To address this question I use two viscoelastic models of varying complexity. The first is a modification to the simple block-and-slider models traditionally used to examine earthquake processes on a very simplistic fashion. Results show that the role of viscosity in stick-slip depends on the dominant stress balance. These results are then considered in the context of a continuum description of a viscoelastic ice stream with a rate-weakening base capable of exhibiting stick-slip behavior. With the continuum model we examine the spatial and temporal aspects of stick-slip, their dependence on viscous effects, and how this behavior impacts the mean flow. Different models for the evolution of basal shear stress are examined in the experiments, with qualitatively similar results. A surprising outcome is that tidal effects, while greatly affecting the spectrum of the stick-slip cycle, may have relatively little effect on the mean flow.

Clouds play a key role in the climate system. Driven by radiation, clouds power the hydrological cycle and global atmospheric dynamics. In addition, clouds fundamentally affect the global radiation balance by reflecting solar radiation back to space and trapping longwave radiation. The response of clouds to global warming remains poorly understood and is strongly regime dependent. In addition, anthropogenic aerosols influence clouds, altering cloud microphysics, dynamics and radiative properties. In this presentation I will review progress and limitations of our current understanding of the role of clouds in climate change and discuss the state of the art of the representation of clouds and aerosol-cloud interactions in global climate models, from (slightly) better constrained stratiform clouds to new frontiers: the investigation of anthropogenic effects on convective clouds.

Many years ago, Mark Kac was consulted by biologist colleague Lamont Cole regarding field-based observations of animal populations that suggested the existence of 3-4 year cycles in going from peak to peak. Kac provided an elegant argument for how purely random sequences of numbers could yield a mean value of 3 years, thereby establishing the notion that pattern can seemingly emerge in random processes. (This does not, however, mean that there could be a largely deterministic cause of such population cycles.)

By extending Kac's argument, we show how the distribution of cycle length can be analytically established using methods derived from random graph theory, etc. We will examine how such distributions emerge in other natural settings, including large earthquakes as well as colored Brownian noise and other random models and, for amusement, the Standard & Poor's 500 index for percent daily change from 1928 to the present.

We then show how this random model could be relevant to a variety of spatially-dependent problems and the emergence of clusters, as well as to memory and the aphorism "bad news comes in threes." The derivation here is remarkably similar to the former and yields some intriguing closed-form results. Importantly, the centroids or "centers of mass" of these clusters also yields clusters and a hierarchy then emerges. Certain "universal" scalings appear to emerge and scaling factors reminiscent of Feigenbaum numbers. Finally, as one moves from one dimension to 2, 3, and 4 dimensions, the scaling behaviors undergo modest change leaving this scaling phenomena qualitatively intact.

Finally, we will show how that an adaptation of the Langevin equation from statistical physics provides not simply a null-hypothesis for matching the observation of 3-4 year cycles, but a remarkably simple model description for the behavior of animal populations.

Colloidal suspensions do not freeze uniformly; rather, the frozen phase (e.g. ice) becomes segregated, trapping bulk regions of the colloid within, which leads to a fascinating variety of patterns that impact both nature and technology. Yet, despite the central importance of ice segregation in several applications, the physics are poorly understood in concentrated systems and continuum models are available only in restricted cases. I will discuss a particular set of steady-state ice segregation patterns that were obtained during a series of directional solidification experiments on concentrated suspensions. As a case study, I will focus of one of these patterns, which is very reminiscent of ice lenses observed in freezing soils and rocks; a form of ice segregation which underlies frost heave and frost weathering. I will compare these observations against an extended version of a 'rigid-ice' model used in previous frost heave studies. The comparison between theory and experiment is qualitatively correct, but fails to quantitatively predict the ice-lensing pattern. This leaves open questions about the validity of the assumptions in 'rigid-ice' models. Moreover, 'rigid-ice' models are inapplicable to the study of other ice segregation patterns. I conclude this talk with some possibilities for a more general model of freezing colloidal suspensions.

Landslides are generally associated with a trigger, such as an earthquake, a rapid snowmelt or a large storm. The trigger event can generate a single landslide or many thousands. This paper examines: (i) The frequency-area statistics of several triggered landslide event inventories, which are characterized by a three-parameter inverse-gamma probability distribution (exponential for small landslide areas, power-law for medium and large areas). (ii) The use of proxies (newspapers) for compiling long-time series of landslide activity in a given region, done in the context of the Emilia-Romagna region, northern Italy. (iii) A stochastic model developed to evaluate the probability of landslides intersecting a simple road network during a landslide triggering event.

Nowadays there are a large number of satellite and airborne observations of the large ice sheet that covers Antarctica. These include maps of the surface elevation, ice thickness, surface velocity, the rate of snow accumulation, and the rate of change of surface elevation. Uncertainty in the possible rate of future sea level rise motivates using all of these observations and models of ice-sheet flow to project how the ice sheet will behave in future, but this is still a challenge. To make useful predictions, especially in the presence of potential dynamic instabilities, models will need accurate initial conditions, including flow velocity throughout the ice thickness. The ice sheet can be several kilometres thick, but most of the observations identify quantities at the upper surface of the ice sheet, not within its bulk. There is thus a question of how the subsurface flow can be inferred from surface observations. The key parameters that must be identified are the viscosity in the interior of the ice and the basal drag coefficient that relates the speed of sliding at the base of the ice sheet to the basal shear stress. Neither is characterised well by field or laboratory studies, but for incompressible flow governed by the Stokes equations they can be investigated by inverse methods analogous to those used in electric impedance tomography (which is governed by the Laplace equation). Similar methods can also be applied to recently developed 'hybrid' approximations to Stokes flow that are designed to model shallow ice sheets, fast-sliding ice streams, and floating ice shelves more efficiently. This talk will give a summary of progress towards model based projections of the size and shape of the Antarctic ice sheet that make use of the available satellite data. Some of the outstanding problems that will need to be tackled to improve the accuracy of these projections will also be discussed.

Convection in a porous medium plays an important role in many geophysical and industrial processes, and is of particular current interest due to its implications for the long-term security of geologically sequestered CO_2. I will discuss two different convective systems in porous media, with the aid of 2D direct numerical simulations: first, a Rayleigh-Benard cell at high Rayleigh number, which gives an accurate characterization both of the convective flux and of the remarkable dynamical structure of the flow; and second, the evolution and eventual `shut-down' of convection in a sealed porous domain with a source of buoyancy along only one boundary. The latter case is also studied using simple box models and laboratory experiments, and can be extended to consider convection across an interface that can move and deform, rather than across a rigid boundary. The relevance of these results in the context of CO_2 sequestration will be discussed.

Unstable dynamical systems can be stabilized, and hence the solution

recovered from noisy data, provided two conditions hold. First, observe

enough of the system: the unstable modes. Second, weight the observed

data sufficiently over the model. In this talk I will illustrate this for the

3DVAR filter applied to three dissipative dynamical systems of increasing

dimension: the Lorenz 1963 model, the Lorenz 1996 model, and the 2D

Navier-Stokes equation.

Thermomechanical processes observed in deformable solids under intensive dynamic or quasi-static loadings consist of coupled mechanical, thermal and fracturing stages. The fracturing processes involve formation, motion and interaction of defects in crystals, phase transitions, breaking of bonds between atoms, accumulation of micro-structural damages (pores, cracks), etc. Irreversible deformations, zones of adiabatic shear micro-fractures are caused by these processes. Dynamic fracturing is a complicated multistage process, which includes appearance, evolution and confluence of micro-defects and formation of embryonic micro-cracks and pores that can grow and lead to the breaking-up of bodies with formation of free surfaces. This results in a need to use more advanced mathematical and numerical techniques.

This talk presents modelling of irreversible deformation near the tip of a crack in a porous domain containing oil and gas during the hydraulic fracturing process. The governing equations for a porous domain containing oil and gas are based on constructing a mathematical model of thermo-visco-elasto-plastic media with micro-defects (micro-pores) filled with another phase (e.g., oil or/and gas). The micro-pores can change their size during the process of dynamical irreversible deformation. The existing pores can expand or collapse. The model was created by using fundamental thermodynamic principles and, therefore, it is a thermodynamically consistent model. All the processes (i.e., irreversible deformation, fracturing, micro-damaging, heat transfer) within a porous domain are strongly coupled. An explicit normalized-corrected meshless method is used to solve the resulting governing PDEs. The flexibility of the proposed technique allows efficient running using a great number of micro- and macro- fractures. The results are presented, discussed and future studies are outlined.

We are interested in finding the Probability Density Function (PDF) of high dimensional chaotic systems such as a global atmospheric circulation model. The key difficulty stems from the so called “curse of dimensionality”. Representing anything numerically in a high dimensional space seems to be just too computationally expensive. Methods applied to dodge this problem include representing the PDF analytically or applying a (typically linear) transformation to a low dimensional space. For chaotic systems these approaches often seem extremely ad-hoc with the main motivation being that we don't know what else to do.

The Lorenz 95 system is one of the simplest systems we could come up with that is both chaotic and high dimensional. So it seems like a good candidate for initial investigation. We look at two attempts to approximate the PDF of this system to an arbitrary level of accuracy, firstly using a simple Monte-Carlo method and secondly using the Fokker-Planck equation. We also describe some of the (sometimes surprising) difficulties encountered along the way.

In many places, the Antarctic and Greenland ice sheets are fringed by tongues of ice floating on the ocean, called ice shelves. Recent observations and modelling suggest that melting and disintegration of the floating ice shelves can impact ice sheet flow, and hence have consequences for sea level rise. Of particular interest are observations of channels and undulations in the ice shelf base, for which the conditions for genesis remain unclear. To build insight into the potential for melting-driven instability of the ice shelf base, this talk will consider a free boundary problem with melting at the ice-ocean interface coupled to a buoyant plume of meltwater confined below a stationary ice shelf. An asymptotic model of turbulent heat transfer in the meltwater plume reveals that melting rates depend on ice-shelf basal slope, with potentially shocking consequences for the evolving ice-shelf geometry

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.

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.

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.

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.

Respiration is a redox reaction in which oxidation of a substrate (often organic) is coupled to the reduction of a terminal electron acceptor (TEA) such as oxygen. Iron oxides in various mineral forms are abundant in sediments and sedimentary rocks, and many subsurface microbes have the ability to respire using Fe(III) as the TEA in anoxic conditions. This process is environmentally important in the degradation of organic substrates and in the redox-cycling of iron. But low mineral solubility limits the bioavailability of Fe(III), which microbes access primarily through reductive dissolution. For aqueous nutrients, expressions for microbial growth and nutrient uptake rates are standardly based on Monod kinetics. We address the question of what equivalent description is appropriate when solid phase Fe(III) is the electron acceptor.

This talk will consist of two parts. In the first part we will present a motivating application from oil reservoir simulation, namely finding the location and trajectory of an oil producing well which maximises oil production. We will show how such a problem can be tackled through the use of radial basis function (RBF) approximation (also known as Kriging or Gaussian process regression) and a branch and bound global optimization algorithm.

In the second part of the talk we will show how one can improve the branch and bound algorithm through the use of Lipschitz continuity of the RBF approximation. This leads to an entirely new global optimization algorithm for twice differentiable functions with Lipschitz continuous Hessian. The algorithm makes use of recent cubic regularisation techniques from local optimization to obtain the necessary bounds within the branch and bound algorithm.

Inverse methods are frequently used in geosciences to estimate model parameters from indirect measurements. A common inverse problem encountered when modelling the flow of large ice masses such as the Greenland and the Antarctic ice sheets is the determination of basal conditions from surface data. I will present an overview over some of the inverse methods currently used to tackle this problem and in particular discuss the use of Bayesian inverse methods in this context. Examples of the use of adjoint methods for large-scale optimisation problems that arise, for example, in flow modelling of West-Antarctica will be given.

Data assimilation in highly nonlinear and high dimensional systems is a hard

problem. We do have efficient data-assimilation methods for high-dimensional

weakly nonlinear systems, exploited in e.g. numerical weather forecasting.

And we have good methods for low-dimensional (

Probability does not exist. At least no more so than "mass" "spin" or "charm" exist. Yet probability forecasts are common, and there are fine reasons for deprecating point forecasts, as they require an unscientific certainty in exactly what the future holds. What roles do our physical understanding and laws of physics play in the construction of probability forecasts to support of decision making and science-based policy? Will probability forecasting more likely accelerate or retard the advancement of our scientific understanding?

Model-based probability forecasts can vary significantly with alterations in the method of data assimilation, ensemble formation, ensemble interpretation, and forecast evaluation, not to mention questions of model structure, parameter selection and the available forecast-outcome archive. The role of each of these aspects of forecasting, in the context of interpreting the forecast as a real-world probability, is considered and contrasted in the cases of weather forecasting, climate forecasting, and economic forecasting. The notion of what makes a probability forecast "good" will be discussed, including the goals of "sharpness given calibration" and "value".

For a probability forecast to be decision-relevant as such, it must be reasonably interpreted as a basis for rational action through the reflection of the probability of the outcomes forecast. This rather obvious sounding requirement proves to be the source of major discomfort as the distinct roles of uncertainty (imprecision) and error (structural mathematical "misspecification") are clarified. Probabilistic forecasts can be of value to decision makers even when it is irrational to interpret them as probability forecasts. A similar statement, of course, can be said for point forecasts, or for spin. In this context we explore the question: do decision-relevant probability forecasts exist?

The part of the West Antarctic Ice Sheet that drains into the Amundsen Sea is currently thinning at such a rate that it contributes nearly 10 percent of the observed rise in global mean sea level. Acceleration of the outlet glaciers means that the sea level contribution has grown over the past decades, while the likely future contribution remains a key unknown. The synchronous response of several independent glaciers, coupled with the observation that thinning is most rapid at their downstream ends, where the ice goes afloat, hints at an oceanic driver. The general assumption is that the changes are a response to an increase in submarine melting of the floating ice shelves that has been driven in turn by an increase in the transport of ocean heat towards the ice sheet. Understanding the causes of these changes and their relationship with climate variability is imperative if we are to make quantitative estimates of sea level into the future.

Observations made since the mid‐1990s on the Amundsen Sea continental shelf have revealed that the seabed troughs carved by previous glacial advances guide seawater around 3‐4°C above the freezing point from the deep ocean to the ice sheet margin, fuelling rapid melting of the floating ice. This talk summarises the results of several pieces of work that investigate the chain of processes linking large‐scale atmospheric processes with ocean circulation over the continental shelf and beneath the floating ice shelves and the eventual transfer of heat to the ice. While our understanding of the processes is far from complete, the pieces of the jigsaw that have been put into place give us insight into the potential causes of variability in ice shelf melting, and allow us to at least formulate some key questions that still need to be answered in order to make reliable projections of future ice sheet evolution in West Antarctica.

Ocean climate models are unlikely routinely to have sufficient

resolution to resolve the turbulent ocean eddy field. The need for the

development of improved mesoscale eddy parameterisation schemes

therefore remains an important task. The current dominant mesoscale eddy

closure is the Gent and McWilliams scheme, which enforces the

down-gradient mixing of buoyancy. While motivated by the action of

baroclinic instability on the mean flow, this closure neglects the

horizontal fluxes of horizontal momentum. The down-gradient mixing of

potential vorticity is frequently discussed as an alternative

parameterisation paradigm. However, such a scheme, without careful

treatment, violates fundamental conservation principles, and in

particular violates conservation of momentum.

A new parameterisation framework is presented which preserves

conservation of momentum by construction, and further allows for

conservation of energy. The framework has one dimensional parameter, the

total eddy energy, and five dimensionless and bounded geometric

parameters. The popular Gent and McWilliams scheme exists as a limiting

case of this framework. Hence the new framework enables for the

extension of the Gent and McWilliams scheme, in a manner consistent with

key physical conservations.

Data assimilation aims to correct a forecast of a physical system, such as the atmosphere or ocean, using observations of that system, in order to provide a best estimate of the current system state. Since it is not possible to observe the whole state it is important to know how errors in different variables of the forecast are related to each other, so that all fields may be corrected consistently. In the first part of this talk we consider how we may impose constraints between different physical variables in data assimilation. We examine how we can use our knowledge of atmospheric physics to pose the assimilation problem in variables that are assumed to be uncorrelated. Using a shallow-water model we demonstrate that this is best achieved by using potential vorticity rather than vorticity to capture the balanced part of the flow. The second part of the talk will consider a further transformation of variables to represent spatial correlations. We show how the accuracy and efficiency with which we can solve the transformed assimilation problem (as measured by the condition number) is affected by the observation distribution and accuracy and by the assumed correlation lengthscales. Theoretical results will be illustrated using the Met Office variational data assimilation scheme.

There is much current concern over the future evolution of climate under conditions of increased atmospheric carbon. Much of the focus is on a bottom-up approach in which weather/climate models of severe complexity are solved and extrapolated beyond their presently validated parameter ranges. An alternative view takes a top-down approach, in which the past Earth itself is used as a laboratory; in this view, ice-core records show a strong association of carbon with atmospheric temperature throughout the Pleistocene ice ages. This suggests that carbon variations drove the ice ages. In this talk I build the simplest model which can accommodate this observation, and I show that it is reasonably able to explain the observations. The model can then be extrapolated to offer commentary on the cooling of the planet since the Eocene, and the likely evolution of climate under the current industrial production of atmospheric carbon.