Past Mathematical Geoscience Seminar

In contemporary ecology and mathematical biology undergraduate courses, textbooks focus on competition and predation models despite it being accepted that most species on Earth are involved in mutualist relationships. Mutualism is usually discussed more briefly in texts, often from an observational perspective, and obligate mutualism mostly not at all. Part of the reason for this is the lack of a simple math model to successfully explain the observations. Traditionally, particular nonlinearities  are used, which produce a variety of apparently disparate models.

The failure of the traditional linear model to describe coexisting mutualists has been documented from May (1973) through Murray (2001) to Bronstein (2015). Here we argue that this could be because of the use of carrying capacity, and propose the use of a nutrient pool instead, which implies the need for an autotroph (e.g. a plant) that converts nutrients into living resources for higher trophic levels. We show that such a linear model can successfully explain the major features of obligate mutualism when simple expressions for obligated growth are included.

The last 500 million years of Earth’s history have been punctuated by numerous episodes of abrupt climate change, some of them coincident with mass extinction events. Many of these climate events have been associated with massive volcanism, occurring during the emplacement of so-called Large Igneous Provinces (LIPs). Because of the significant impact of small modern eruptions on the Earth’s climate, a link between LIP volcanism and past climate change has been strongly advocated. Geochemical investigations of the sedimentary records which record major climate changes can give a profound insight into the proposed interactions between volcanic activity and climate. Mercury is a trace-gas emitted by modern volcanoes, which are the main source of this metal to the atmosphere. Ultimately atmospheric mercury is deposited in sediments, thus if enrichments in mercury are observed in sediments of the same age across the globe, a volcanic cause of these enrichments might be inferred. Osmium isotopes can also be used as a fingerprint of volcanic activity, as primitive basalts are enriched in unradiogenic 188Os. However, the continental crust is enriched in radiogenic 187Os. Therefore, the 187Os/188Os ratio can change with either more volcanic activity, or increased continental weathering during climate change. Changes in sedimentary mercury content and osmium isotopes can thus be used as markers of volcanism or weathering during climate events. However, a possible future step would be to quantify the amount of volcanism and/or weathering on the basis of these sedimentary excursions. The final part of this talk will introduce some simple quantitative models which may represent a first step towards such quantification, with the aim of further elaborating these models in the future.

The existence of planetary and stellar magnetic fields is attributed to the dynamo instability, the mechanism by which a background turbulent flow spontaneously generates a magnetic field by the constructive refolding of magnetic field lines. Many efforts have been made by several experimental groups to reproduce the dynamo instability in the laboratory using liquid metals. However, so far, unconstrained dynamos driven by turbulent flows have not been achieved in the intrinsically low magnetic Prandtl number $P_m$ (i.e. $Pm = Rm/Re << 1$) laboratory experiments. In this seminar I will demonstrate that the critical magnetic Reynolds number $Rm_c$ for turbulent non-helical dynamos in the low $P_m$ limit can be significantly reduced if the flow is submitted to global rotation. Even for moderate rotation rates the required energy injection rate can be reduced by a factor more than 1000. Our finding thus points into a new paradigm for the design of new liquid metal dynamo experiments.

I will describe our research on numerical methods for atmospheric dynamical cores based on compatible finite element methods. These methods extend the properties of the Arakawa C-grid to finite element methods by using compatible finite element spaces that respect the elementary identities of vector-calculus. These identities are crucial in demonstrating basic stability properties that are necessary to prevent the spurious numerical degradation of geophysical balances that would otherwise make numerical discretisations unusable for weather and climate prediction without the introduction of undesirable numerical dissipation. The extension to finite element methods allow these properties to be enjoyed on non-orthogonal grids, unstructured multiresolution grids, and with higher-order discretisations. In addition to these linear properties, for the shallow water equations, the compatible finite element structure can also be used to build numerical discretisations that respect conservation of energy, potential vorticity and enstrophy; I will survey these properties. We are currently developing a discretisation of the 3D compressible Euler equations based on this framework in the UK Dynamical Core project (nicknamed "Gung Ho"). The challenge is to design discretisation of the nonlinear operators that remain stable and accurate within the compatible finite element framework. I will survey our progress on this work to date and present some numerical results.

Following several decades of development by applied mathematicians, models of ocean wave interactions with sea ice floes are now in high demand due to the rapid recent changes in the world’s sea ice cover. From a mathematical perspective, the models are of interest due to the thinness of the floes, leading to elastic responses of the floes to waves, and the vast number of floes that waves encounter. Existing models are typically based on linear theories, but the thinness of the floes leads to the unique and highly nonlinear phenomenon of overwash, where waves run over the floes, in doing so dissipating wave energy and impacting the floes thermodynamically. I will give an overview of methods developed for the wave-floe problem, and present a new, bespoke overwash model, along with supporting laboratory experiments and numerical CFD simulations.

Dedalus is a new open-source framework for solving general partial differential equations using spectral methods.  It is designed for maximum extensibility and incorporates features such as symbolic equation entry, custom domain construction, and automatic MPI parallelization.  I will briefly describe key algorithmic features of the code, including our sparse formulation and support for general tensor calculus in curvilinear domains.  I will then show examples of the code’s capabilities with various applications to astrophysical and geophysical fluid dynamics, including a compressible flow benchmark against a finite volume code, and direct numerical simulations of turbulent glacial melting

In my research I model three components of the Earth system: the ice sheets, the ocean, and the solid Earth. In the first half of this talk I will describe the traditional approach that is used to model the impact of ice sheet growth and decay on global sea-level change and solid Earth deformation. I will then go on to explain how collaboration across the fields of glaciology, geodynamics and seismology is providing exciting new insight into feedbacks between ice dynamics and solid Earth deformation.

In July 2011, the observation of a massive phytoplankton bloom underneath a sea ice–covered region of the Chukchi Sea shifted the scientific consensus that regions of the Arctic Ocean covered by sea ice were inhospitable to photosynthetic life. Although the impact of widespread phytoplankton blooms under sea ice on Arctic Ocean ecology and carbon fixation is potentially marked, the prevalence of these events in the modern Arctic and in the recent past is, to date, unknown. We investigate the timing, frequency, and evolution of these events over the past 30 years. Although sea ice strongly attenuates solar radiation, it has thinned significantly over the past 30 years. The thinner summertime Arctic sea ice is increasingly covered in melt ponds, which permit more light penetration than bare or snow-covered ice. We develop a simple mathematical model to investigate these physical mechanisms. Our model results indicate that the recent thinning of Arctic sea ice is the main cause of a marked increase in the prevalence of light conditions conducive to sub-ice blooms. We find that as little as 20 years ago, the conditions required for sub-ice blooms may have been uncommon, but their frequency has increased to the point that nearly 30% of the ice-covered Arctic Ocean in July permits sub-ice blooms. Recent climate change may have markedly altered the ecology of the Arctic Ocean.

Many northern hemisphere climate records show a series of rapid climate changes - Dansgaard-Oesgher (D-O) cycles - that recurred on centennial to millennial timescales throughout most of the last glacial period.  They consist of sudden warming jumps of order 10°C, followed generally by a slow cooling lasting a few centuries, and then a rapid temperature drop into a cold period of similar length.  Most explanations for D-O events call on changes in the strength of the Atlantic meridional overturning circulation (AMOC), but the mechanism for triggering and pacing such changes is uncertain. Changes in freshwater delivery to the ocean are assumed to be important. 

Here, we investigate whether the proposed AMOC changes could have occurred as part of a natural relaxation oscillation, in which runoff from the northern hemisphere ice sheets varies in response to each warming and cooling event, and in turn provides the freshwater delivery that controls the ocean circulation.  In this mechanism the changes are buffered and paced by slow changes in salnity of the Arctic ocean.  We construct a simple model to investigate whether the timescales and magnitudes make this a viable mechanism.  

Radar data from both Greenland and Antarctica show folds and other disruptions to the stratigraphy of the deep ice. The mechanisms by which stratigraphy deforms are related to the interplay between ice flow and topography. Here we show that when ice flows across valleys or overdeepenings, viscous overturnings called Moffatt eddies can develop. At the base of a subglacial valley, the shear on the valley walls is transfered through the ice, forcing the ice to overturn. To understand the formation of these eddies, we numerically solve the non-Newtonian Stokes equations with a Glen's law rheology to determine the critical valley angle for the eddies to form. The decrease in ice viscosity with shear enhances shear localization and, therefore, Moffatt eddies form in smaller valley angles (steeper slopes) than in a fluid that does not localize shear, such as a Newtonian fluid. When temperature is incorporated into the ice rheology, the warmer basal ice is less viscous and eddies form in larger valley angles (shallower slopes) than in isothermal ice. We apply our simulations to the Gamburtsev Subglacial Mountains and solve for the ice flow over radar-determined topography. These simulations show Moffatt eddies on the order of 100 meters tall in the deep subglacial valleys.

The mathematical design of the table flood demonstrator Wetropolis will be presented. Wetropolis illustrates the concepts of extreme rainfall and flooding.

It shows how extreme rainfall events  can cause flooding of a city due to groundwater and river flood peaks. Rainfall is supplied randomly in space using four outcomes (in a reservoir, on a moor, at both places or nowhere) and randomly in time using four rainfall intensities (1s, 2s, 4s, or 9s during a 10s Wetropolis day), including one extreme event, via two skew-symmetric discrete probability distributions visualised by two Galton boards. Wetropolis can be used for both public outreach and as scientific testing environment for flood mitigation and data assimilation.

More information: https://www.facebook.com/resurging.flows

The spreading of a viscous fluid in between a rigid, horizontal substrate and an overlying elastic sheet is presented as a simplified model of the hydraulic fracturing process. In particular, the talk will focus on the case of a permeable substrate for which leak-off arrests the propagation of the fluid and permits the development of a steady state. The different regimes of  gravitationally-driven and elastically-driven flow will be explored, as will the cases of a stiff and flexible sheet, before a discussion of the influence that particles included in the fluid have on the fracture propagation. 

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.

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. 

It has been conjectured that marine ice sheets (those that

flow into the ocean) are unconditionally unstable when the underlying

bed-slope runs uphill in the direction of flow, as is typical in many

regions underneath the West Antarctic Ice Sheet. This conjecture is

supported by theoretical studies that assume a two-dimensional flow

idealization. However, if the floating section (the ice shelf) is

subject to three-dimensional stresses from the edges of the embayments

into which they flow, as is typical of many ice shelves in Antarctica,

then the ice shelf creates a buttress that supports the ice sheet.

This allows the ice sheet to remain stable under conditions that may

otherwise result in collapse of the ice sheet. This talk presents new

theoretical and experimental results relating to the effects of

three-dimensional stresses on the flow and structure of ice shelves,

and their potential to stabilize marine ice sheets.

There is wide interest in the oceanographic and engineering communities as to whether linear models are satisfactory for describing the largest and steepest waves in open ocean. This talk will give some background on the topic before describing some recent modelling. This concludes that non-linear physics produces only small increases in amplitude over that expected in a linear model — however, there are significant changes to the shape and structure of extreme wave-group caused by the non-linear physics.