Past Mathematical Geoscience Seminar

Across much of the ablation region of the western Greenland Ice Sheet, hydro-fracture events related to supraglacial lake drainages rapidly deliver large volumes of meltwater to the bed of the ice sheet. We investigate what triggers the rapid drainage of a large supraglacial lake using a Network Inversion Filter (NIF) to invert a dense local network of GPS observations over three summers (2011-2013). The NIF is used to determine the spatiotemporal variability in ice sheet behavior (1) prior to lake drainage, and in response to (2) vertical hydro-fracture crack propagation and closure, (3) the opening of a horizontal cavity at the ice-sheet bed that accommodates the rapid injection of melt-water, and (4) extra basal slip due to enhanced lubrication. We find that the opening and propagation of each summer’s lake-draining hydro-fracture is preceded by a local stress perturbation associated with ice sheet uplift and enhanced slip above pre-drainage background velocities. We hypothesize that these precursors are associated with the introduction of meltwater to the bed through neighboring moulin systems.

Studies of thermal convection in planetary interiors have largely focused on convection above the critical Rayleigh number. However, convection in planetary mantles and crusts can also occur under subcritical conditions. Subcritical convection exhibits phenomena which do not exist above the critical Rayleigh number. One such phenomenon is spatial localization characterized by the formation of stable, spatially isolated convective cells. Spatial localization occurs in a broad range of viscosity laws including temperature-dependent viscosity and power-law viscosity and may explain formation of some surface features observed on rocky and icy bodies in the Solar System.

Shear Thickening fluids such as cornstarch and water show remarkable response under impact, which allows, for example, a person to run on the surface of the suspension. We perform constant velocity impact experiments along with imaging and particle tracking in a shear thickening fluid at velocities lower than 500 mm/s and suspension heights of a few cm. In this regime, where inertial effects are insignificant, we find that a solid-like dynamically jammed region with a propagating front is generated under impact. The suspension is able to support large stresses like a solid only when the front reaches the opposite boundary. These impact-activated fronts are generated only above a critical velocity. We construct a model by taking into account that sufficiently large stresses are generated when this solid like region spans to the opposite boundary and the work necessary to deform this solid like material dissipates the kinetic energy of the impacting object. The model shows quantitative agreement of the measured penetration depth using high speed video of a person running on cornstarch and water suspensions.

The role of surface-water flooding in controlling fluxes of water and carbon between the land and the atmosphere is increasingly recognized in studies of the Earth system. Simultaneous advances in remote earth observation and large-scale land-surface and hydrological modeling promise improvements in our ability to understand these linkages, and suggest that improvements in prediction of river flow and inundation extents may result. Here we present an analysis of newly-available observational estimates of surface water inundation obtained through satellite Earth observation with results from simulations produced by using the Joint UK Land Environment Simulator (JULES) land-surface model operating at 0.5 degree resolution over the African continent. The model was forced with meteorological input from the WATCH Forcing Data for the period 1981-2001 and sensitivity to various model configurations and parameter settings were tested. Both the PDM and TOPMODEL sub-grid scale runoff generation schemes were tested for parameter sensitivities, with the evaluation focussing on simulated river discharge in sub-catchments of the Congo, Nile, Niger, Orange, Okavango and Zambezi rivers. It was found that whilst the water balance in each of the catchments can be simulated with acceptable accuracy, the individual responses of each river vary between model configurations so that there is no single runoff parameterization scheme or parameter values that yields optimal results across all catchments. We trace these differences to the model’s representation of sub-surface flow and make some suggestions to improve the performance of large-scale land-surface models for use in similar applications. Our findings also demonstrate links between episodes of extensive surface water flooding and large-scale climatic indices, although the pattern of correlations contains a level of spatial and temporal detail that warrants careful attention to the climatology of individual situations. These findings suggest that the use of Earth observation data together with improved models of large-scale hydrology have the potential to improve our ability to predict surface-water flooding and to develop our understanding of the role of flooding in driving components of the water and carbon cycles.

Contraction cracks form captivating patterns such as those seen in dried mud or the polygonal networks that cover the polar regions of Earth and Mars. These patterns can be controlled, for example in the artistic craquelure sometimes found in pottery glazes. More practically, a growing zoo of patterns, including parallel arrays of cracks, spiral cracks, wavy cracks, lenticular or en-passant cracks, etc., are known from simple experiments in thin films – essentially drying paint – and are finding application in surfaces with engineered properties. Through such work we are also learning how natural crack patterns can be interpreted, for example in the use of dried blood droplets for medical or forensic diagnosis, or to understand how scales develop on the heads of crocodiles.

I will discuss mud cracks, how they form, and their use as a simple laboratory analogue system. For flat mud layers I will show how sequential crack formation leads to a rectilinear crack network, with cracks meeting each other at roughly 90°. By allowing cracks to repeatedly form and heal, I will describe how this pattern evolves into a hexagonal pattern. This is the origin of several striking real-world systems: columnar joints in starch and lava; cracks in gypsum-cemented sand; and the polygonal terrain in permafrost. Finally, I will turn to look at crack patterns over uneven substrates, such as paint over the grain of wood, or on geophysical scales involving buried craters, and identify when crack patterns are expected to be dominated by what lies beneath them. In exploring all these different situations I will highlight the role of energy release in selecting the crack patterns that are seen.

Ice streams are narrow bands of rapidly sliding ice within an otherwise

slowly flowing continental ice sheet. Unlike the rest of the ice sheet,

which flows as a typical viscous gravity current, ice streams experience

weak friction at their base and behave more like viscous 'free films' or

membranes. The reason for the weak friction is the presence of liquid

water at high pressure at the base of the ice; the water is in turn

generated as a result of dissipation of heat by the flow of the ice

stream. I will explain briefly how this positive feedback can explain the

observed (or inferred, as the time scales are rather long) oscillatory

behaviour of ice streams as a relaxation oscillation. A key parameter in

simple models for such ice stream 'surges' is the width of an ice stream.

Relatively little is understood about what controls how the width of an

ice stream evolves in time. I will focus on this problem for most of the

talk, showing how intense heat dissipation in the margins of an ice stream

combined with large heat fluxes associated with a switch in thermal

boundary conditions may control the rate at which the margin of an ice

stream migrates. The relevant mathematics involves a somewhat non-standard

contact problem, in which a scalar parameter must be chosen to control the

location of the contact region. I will demonstrate how the problem can be

solved using the Wiener-Hopf method, and show recent extensions of this

work to more realistic physics using a finite element discretization.

A form of PDE-constrained inversion is today used as an engineering tool for seismic imaging. Today there are some successful studies and good workflows are available. However, mathematicians will find some important unanswered questions: (1) robustness of inversion with highly nonconvex objective functions; (2) scalable solution highly oscillatory problem; and (3) handling of uncertainties. We shall briefly illustrate these challenges, and mention some possible solutions.

One of the main obstacles to forecasting sea level rise over the coming centuries is the problem of predicting changes in the flow of ice sheets, and in particular their fast-flowing outlet glaciers. While numerical models of ice sheet flow exist, they are often hampered by a lack of input data, particularly concerning the bedrock topography beneath the ice. Measurements of this topography are relatively scarce, expensive to obtain, and often error-prone. In contrast, observations of surface elevations and velocities are widespread and accurate.

In an ideal world, we could combine surface observations with our understanding of ice flow to invert for the bed topography. However, this problem is ill-posed, and solutions are both unstable and non-unique. Conventionally, this problem is circumvented by the use of regularization terms in the inversion, but these are often arbitrary and the numerical methods are still somewhat unstable.

One philosophically appealing option is to apply a fully Bayesian framework to the problem. Although some success has been had in this area, the resulting distributions are extremely difficult to work with, both from an interpretive standpoint and a numerical one. In particular, certain forms of prior information, such as constraints on the bedrock slope and roughness, are extremely difficult to represent in this framework.

A more profitable avenue for exploration is a semi-Bayesian approach, whereby a classical inverse method is regularized using terms derived from a Bayesian model of the problem. This allows for the inclusion of quite sophisticated forms of prior information, while retaining the tractability of the classical inverse problem. In particular, we can account for the severely non-Gaussian error distribution of many of our measurements, which was previously impossible.

On calm clear nights a minimum in air temperature can occur just above the ground at heights of order 0.5m or less. This is contrary to the conventional belief that ground is the point of minimum. This feature is paradoxical as an apparent unstable layer (the height below the point of minimum) sustains itself for several hours. This was first reported from India by Ramdas and his coworkers in 1932 and was disbelieved initially and attributed to flawed thermometers. We trace its history, acceptance and present a mathematical model in the form of a PDE that simulates this phenomenon.

Marine-ice formation occurs on a vast range of length scales: from millimetre scale frazil crystals, to consolidated sea ice a metre thick, to deposits of marine ice under ice shelves that are hundreds of kilometres long. Scaling analyses is therefore an attractive and powerful technique to understand and predict phenomena associated with marine-ice formation, for example frazil crystal growth and the convective desalination of consolidated sea ice. However, there are a number of potential pitfalls arising from the assumptions implicit in the scaling analyses. In this talk, I tease out the assumptions relevant to these examples and test them, allowing me to derive simple conceptual models that capture the important geophysical mechanisms affecting marine-ice formation. 

Explosive volcanic eruptions often produce large amounts of ash that is transported high into the atmosphere in a turbulent buoyant plume.  The ash can be spread widely and is hazardous to aircraft causing major disruption to air traffic.  Recent events, such as the eruption of Eyjafjallajokull, Iceland, in 2010 have demonstrated the need for forecasts of ash transport to manage airspace.  However, the ash dispersion forecasts require boundary conditions to specify the rate at which ash is delivered into the atmosphere.

Models of volcanic plumes can be used to describe the transport of ash from the vent into the atmosphere.  I will show how models of volcanic plumes can be developed, building on classical fluid mechanical descriptions of turbulent plumes developed by Morton, Taylor and Turner (1956), and how these are used to determine the volcanic source conditions.  I will demonstrate the strong atmospheric controls on the buoyant plume rise.  Typically steady models are used as solutions can be obtained rapidly, but unsteadiness in the volcanic source can be important.  I'll discuss very recent work that has developed unsteady models of volcanic plumes, highlighting the mathematical analysis required to produce a well-posed mathematical description.

One of the main obstacles to forecasting sea level rise over the coming centuries is the problem of predicting changes in the flow of ice sheets, and in particular their fast-flowing outlet glaciers. While numerical models of ice sheet flow exist, they are often hampered by a lack of input data, particularly concerning the bedrock topography beneath the ice. Measurements of this topography are relatively scarce, expensive to obtain, and often error-prone. In contrast, observations of surface elevations and velocities are widespread and accurate.

In an ideal world, we could combine surface observations with our understanding of ice flow to invert for the bed topography. However, this problem is ill-posed, and solutions are both unstable and non-unique. Conventionally, this problem is circumvented by the use of regularization terms in the inversion, but these are often arbitrary and the numerical methods are still somewhat unstable.

One philosophically appealing option is to apply a fully Bayesian framework to the problem. Although some success has been had in this area, the resulting distributions are extremely difficult to work with, both from an interpretive standpoint and a numerical one. In particular, certain forms of prior information, such as constraints on the bedrock slope and roughness, are extremely difficult to represent in this framework.

A more profitable avenue for exploration is a semi-Bayesian approach, whereby a classical inverse method is regularized using terms derived from a Bayesian model of the problem. This allows for the inclusion of quite sophisticated forms of prior information, while retaining the tractability of the classical inverse problem. In particular, we can account for the severely non-Gaussian error distribution of many of our measurements, which was previously impossible.

We will present results from studies of the impact of the non-slow (typically fast) components of a rotating, stratified flow on its slow dynamics. We work in the framework of fast singular limits that derives from the work of Bogoliubov and Mitropolsky [1961], Klainerman and Majda [1981], Shochet [1994], Embid and Ma- jda [1996] and others.

In order to understand how the flow approaches and interacts with the slow dynamics we decompose the full solution, where u is a vector of all the unknowns, as

u = u α + u ′α where α represents the Ro → 0, F r → 0 or the simultaneous limit of both (QG for

quasi-geostrophy), with

P α u α = u α    P α u ′α = 0 ,

and where Pαu represents the projection of the full solution onto the null space of the fast operator. We use this decomposition to find evolution equations for the components of the flow (and the corresponding energy) on and off the slow manifold.

Numerical simulations indicate that for the geometry considered (triply periodic) and the type of forcing applied, the fast waves act as a conduit, moving energy onto the slow manifold. This decomposition clarifies how the energy is exchanged when either the stratification or the rotation is weak. In the quasi-geostrophic limit the energetics are less clear, however it is observed that the energy off the slow manifold equilibrates to a quasi-steady value.

We will also discuss generalizations of the method of cancellations of oscillations of Schochet for two distinct fast time scales, i.e. which fast time scale is fastest? We will give an example for the quasi-geostrophic limit of the Boussinesq equations.

At the end we will briefly discuss how understanding the role of oscillations has allowed us to develop convergent algorithms for parallel-in-time methods.

Beth A. Wingate - University of Exeter

Jared Whitehead - Brigham Young University

Terry Haut - Los Alamos National Laboratory

Seismology currently undergoes rapid and exciting advances fueled by a simultaneous surge in recorded data (in both quality and quantity), realistic wave-propagation algorithms, and supercomputing capabilities. This enables us to sample parameter spaces of relevance for imaging the Earth's interior 3D structure with fully numerical techniques. Seismic imaging is the prime approach to illuminate and understand global processes such as mantle convection, plate tectonics, geodynamo, the vigorous interior of the Sun, and delivers crucial constraints on our grasp of volcanism, the carbon cycle and seismicity. At local scales, seismic Earth models are inevitable for hydrocarbon exploration, monitoring of flow processes, and natural hazard assessment.

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With a slight focus on global-scale applications, I will present the underlying physical model of realistic wave propagation, its numerical discretization and link such forward modeling to updating Earth models by means of inverse modeling. The associated computational burden to solve high-resolution statistical inverse problems with precise numerical techniques is however entirely out of reach for decades to come. Consequently, seismologists need to take approximations in resolution, physics, data and/or inverse methodology. I will scan a number of such end-member approximations, and focus on our own approach to simultaneously treat wave physics realistically across the frequency band while maximizing data usage and allow for uncertainty quantification. This approach is motivated by decisive approximations on the model space for typical Earth structures and linearized inverse theory.

Ice sheets are among the key controls on global climate and sea-level change. A detailed understanding of ice sheet dynamics is crucial so to make accurate predictions of their mass balance into the future. Ice streams are the dominant negative component in this balance, accounting for up to 90$\%$ of the Antarctic ice flux into ice shelves and ultimately into the sea. Despite their importance, our understanding of ice-stream dynamics is far from complete.

A range of observations associate ice streams with meltwater. Meltwater lubricates the ice at its bed, allowing it to slide with less internal deformation. It is believed that ice streams may appear due to a localization feedback between ice flow, basal melting and water pressure in the underlying sediments. I will present a model of subglacial water flow below ice sheets, and particularly below ice streams. This hydrologic model is coupled to a model for ice flow. I show that under some conditions this coupled system gives rise to ice streams by instability of the internal dynamics.

The classical separate treatments of competition and predation, and an inability to provide a sensible theoretical basis for mutualism, attests to the inability of traditional models to provide a synthesising framework to study trophic interactions, a fundamental component of ecology. Recent approaches to food web modelling have focused on consumer-resource interactions. We develop this approach to explicitly represent finite resources for each population and construct a rigorous unifying theoretical framework with Lotka-Volterra Conservative Normal (LVCN) systems. We show that mixotrophy, a ubiquitous trophic interaction in marine plankton, provides the key to developing a synthesis of the various ways of making a living. The LVCN framework also facilitates an explicit redefinition of facultative mutualism, illuminating the over-simplification of the traditional definition.

We demonstrate a continuum between trophic interactions and show that populations can continuously and smoothly evolve through most population interactions without losing stable coexistence. This provides a theoretical basis consistent with the evolution of trophic interactions from autotrophy through mixotrophy/mutualism to heterotrophy.

Numerical models provide valuable tools for integrating understanding of riverine processes and morphology. Moreover, they have considerable potential for use in investigating river responses to environmental change and catchment management, and for aiding the interpretation of alluvial deposits and landforms. For this potential to be realised fully, such models must be capable of representing diverse river styles, and the spatial and temporal transitions between styles that can be driven by environmental forcing. However, while numerical modelling of rivers has advanced significantly over the past few decades, this has been accomplished largely by developing separate approaches to modelling different styles of river (e.g., meanders and braided networks). In addition, there has been considerable debate about what should constitute the ‘basic ingredients’ of river models, and the degree to which the environmental processes governing river evolution can be simplified in such models. This seminar aims to examine these unresolved issues, with particular reference to the simulation of large rivers and their floodplains.

Microbial biofilms grow on most rock and stone surfaces and may play critical roles in weathering. With climate change and improving air quality in many cities in Europe biofilms are growing rapidly on many historic stone buildings and posing practical problems for heritage conservation. With many new field and lab techniques available it is now possible to identify the microbes present and start to clarify their roles. We now need help modelling microbial biofilm growth and impacts in order to provide better advice for conservators.

Hazardous geophysical mass flows, such as snow avalanches, debris-flows and pyroclastic flows, often spontaneously develop large particle rich levees that channelize the flow and enhance their run-out. Measurements of the surface velocity near an advancing flow front have been made at the United States Geological Survey (USGS) debris-flow flume, where 10m^3 of water saturated sand and gravel are allowed to flow down an 80m chute onto a run-out pad. In the run-out phase the flow front is approximately invariant in shape and advances at almost constant speed. By tracking the motion of surface tracers and using a simple kinematic model, it was possible to infer bulk motion as incoming material is sheared towards the front, over-run and shouldered to the side. At the heart of the levee formation process is a subtle segregation-mobility feedback effect. Simple models for particle segregation and the depth-averaged motion of granular avalanches are described and one of the first attempts is made to couple these two types of models together. This process proves to be non-trivial, yielding considerable complexity as well as pathologies that require additional physics to be included.

Jakobshavn Isbrae and many other fast flowing outlet glaciers of present

and past ice sheets lie in deep troughs which often have several

overdeepened sections. To make their fast flow possible their bed needs

to be slippery which in turn means high basal water pressures. I will

present a model of subglacial water flow and its application to

Jakobshavn. I find that, somewhat surprisingly, the reason for

Jakobshavn's fast flow might be the pressure dependence of the melting

point of ice. The model itself describes the unusual fluid dynamics occurring underneath the ice; it has an interesting mathematical structure that presents computational challenges.