Past Mathematical Geoscience Seminar

8 May 2015
14:15
Abstract
Alternating, fast cloud level zonal winds on Jupiter have been accurately measured for several decades but their depth of penetration into the Jovian interior, which is closely associated with the origin of the winds, still remains highly controversial. The Juno spacecraft, now on its way to Jupiter and will arrive there in 2016, will probe the depth of penetration of the zonal winds by accurately measuring their effects on the high-order zonal gravitational coefficients at unprecedentedly high precision. Interpretation of these gravitational measurements requires an accurate description of the shape, density structure and internal wind profile. We shall discuss the mathematical theory and accurate numerical simulation for the gravitational field of rapidly rotating, non-spherical gaseous Jupiter.
  • Mathematical Geoscience Seminar
22 April 2015
14:00
Lucas Goehring
Abstract
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.
  • Mathematical Geoscience Seminar
13 March 2015
14:15
Abstract
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.
  • Mathematical Geoscience Seminar
13 February 2015
14:15
Abstract

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.

  • Mathematical Geoscience Seminar
30 January 2015
14:15
Martin O'Leary
Abstract

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.

  • Mathematical Geoscience Seminar
12 December 2014
14:15
Abstract

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.

  • Mathematical Geoscience Seminar
5 December 2014
14:15
David Rees-Jones
Abstract

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. 

  • Mathematical Geoscience Seminar
21 November 2014
14:15
Mark Woodhouse
Abstract

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.

  • Mathematical Geoscience Seminar
7 November 2014
14:15
Martin O'Leary
Abstract

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.

  • Mathematical Geoscience Seminar
24 October 2014
14:15
Beth Wingate
Abstract

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

  • Mathematical Geoscience Seminar

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