C2

Partially molten materials resist shearing and compaction. This resistance

is described by a fourth-rank effective viscosity tensor. When the tensor

is isotropic, two scalars determine the resistance: an effective shear and

an effective bulk viscosity. In this seminar, calculations are presented of

the effective viscosity tensor during diffusion creep for a 3D tessellation of

tetrakaidecahedrons (truncated octahedrons). The geometry of the melt is

determined by assuming textural equilibrium.  Two parameters

control the effect of melt on the viscosity tensor: the porosity and the

dihedral angle. Calculations for both Nabarro-Herring (volume diffusion)

and Coble (surface diffusion) creep are presented. For Nabarro-Herring

creep the bulk viscosity becomes singular as the porosity vanishes. This

singularity is logarithmic, a weaker singularity than typically assumed in

geodynamic models. The presence of a small amount of melt (0.1% porosity)

causes the effective shear viscosity to approximately halve. For Coble creep,

previous modelling work has argued that a very small amount of melt may

lead to a substantial, factor of 5, drop in the shear viscosity. Here, a

much smaller, factor of 1.4, drop is obtained.

Ice streams are fast flowing regions of ice that generally slide over a layer of unconsolidated, water-saturated subglacial sediment known as till.  A striking feature that has been observed geophysically is that subglacial till has been found to accumulate distinctively into sedimentary wedges at the grounding zones (regions where ice sheets begin to detach from the bedrock to form freely floating ice shelves) of both past and present-day ice sheets. These grounding-zone wedges have important implications for ice-sheet stability against grounding zone retreat in response to rising sea levels, and their origins have remained a long-standing question. Using a combination of mathematical modelling, a series of laboratory experiments, field data and numerical simulations, we develop a fluid-mechanical model that explains the mechanism of the formation of these sedimentary wedges in terms of the loading and unloading of deformable till in three dynamical regimes. We also undertake a series of analogue laboratory experiments, which reveal that a similar wedge of underlying fluid accumulates spontaneously in experimental grounding zones, we formulate local conditions relating wedge slopes in each of the scenarios and compare them to available geophysical radargram data at the well lubricated, fast-flowing Whillans Ice Stream.

Many scientists, and in particular mathematicians, report difficulty in understanding thermodynamics. So why is thermodynamics so difficult? To attempt an answer, we begin by looking at the components in an exposition of a scientific theory. These include a mathematical core, a motivation for the choice of variables and equations, some historical remarks, some examples and a discussion of how variables, parameters, and functions (such as equations of state) can be inferred from experiments. There are other components too, such as an account of how a theory relates to other theories in the subject.

It will be suggested that theories of thermodynamics are hard to understand because (i) many expositions appear to argue from the particular to the general (ii) there are several different thermodynamic theories that have no obvious logical or mathematical equivalence (iii) each theory really is subtle and requires intense study (iv) in most expositions different theories are mixed up, and the different components of a scientific exposition are also mixed up. So, by presenting one theory at a time, and by making clear which component is being discussed, we might reduce the difficulty in understanding any individual thermodynamic theory. The key is perhaps separation of the mathematical core from the physical motivation. It is also useful to realise that a motivation is not generally the same as a proof, and that no theory is actually true.

By way of illustration we will attempt expositions of two of the simplest thermodynamic theories – reversible and then irreversible thermodynamics of homogeneous materials – where the mathematical core and the motivation are discussed separately. In conclusion we’ll relate these two simple theories to other, foundational and generalised, thermodynamic theories.

The accurate modelling of geophysical flows often requires information which is difficult to measure and therefore poorly quantified. Such information may relate to the fluid properties or an unknown boundary condition, for example. The premise of this talk is that when the flow is bounded by a free surface, the deformation of this free surface contains useful information which can be used to infer such unknown quantities. The increasing availability of free surface data through remote sensing using drones and satellites provides the impetus to develop new mathematical methods and numerical tools to interpret the signature embedded in the free surface deformation. This talk will explore two recent examples drawn from glaciology and inspired from volcanology for which free surface data was successfully used to reconstruct an unknown field.

The magma chamber - an underground vat of fluid magma that is tapped during volcanic eruptions - has been the foundation of models of volcanic eruptions for many decades and successfully explains many geological observations.  However, geophysics has failed to image the postulated large magma chambers, and the chemistry and ages of crystals in erupted magmas indicate a more complicated history.  New conceptual models depict subsurface magmatic systems as dominantly uneruptible crystalline networks with interstitial melt (mushes) extending deep into the Earth's crust to the mantle, containing lenses of potentially eruptible (low-crystallinity) magma.  These lenses would commonly be less dense than the overlying mush and so Rayleigh Taylor instabilities should develop leading to ascent of blobs of magma unless the growth rate is sufficiently slow that other processes (e.g. solidification) dominate.  The viscosity contrast between a buoyant layer and mush is typically extremely large; a consequence is that the horizontal dimension of a magma reservoir is commonly much less than the theoretical fastest growing wavelength assuming an infinite horizontal layer.  

I will present laboratory experiments and linear stability analysis for low Reynolds number, laterally confined Rayleigh Taylor instabilities involving one layer that is much thinner and much less viscous than the other.  I will then apply the results to magmatic systems, comparing timescales for development of the instability and the volumes of packets of rising melt generated, with the frequencies and sizes of volcanic eruptions.  I will then discuss limitations of this work and outstanding fluid dynamical problems in this new paradigm of trans-crustal magma mush systems.

Glacier flow is an example of a gravity driven non-linear viscous flow at low Reynolds numbers. As a glacier flows over an undulating bed, the surface topography is modified in response. Some information about bed conditions is therefore contained in the shape of the surface and the surface velocity field. I will present theoretical and numerical work on how basal conditions on glaciers affect ice flow, and how one can obtain information about basal conditions through surface-to-bed inversion. I’ll give an overview over inverse methodology currently used in glaciology, and how satellite data is now routinely used to invert for bed properties of the Greenland and the Antarctic Ice Sheets.

Plumes are a characteristic feature of convective flow through porous media. Their dynamics are an important part of numerous geological processes, ranging from mixing in magma chambers to the convective dissolution of sequestered carbon dioxide. In this talk, I will discuss models for the spread of convective plumes in a heterogeneous porous environment. I will focus particularly on the effect of thin, roughly horizontal, low-permeability barriers to flow, which provide a generic form of heterogeneity in geological settings, and are a particularly widespread feature of sedimentary formations. With the aid of high-resolution numerical simulations, I will explore how a plume spreads and flows in the presence of one or more of these layers, and will briefly consider the implications of these findings in physical settings.

For a reductive group $ G $, Steinberg established a map from the Weyl group to nilpotent $ G $-orbits using momentmaps on double flag varieties.  In particular, in the case of the general linear group, he re-interpreted the Robinson-Schensted correspondence between the permutations and pairs of standard tableaux of the same shape in terms of product of complete flags.

We generalize his theory to the case of symmetric pairs $ (G, K) $, and obtained two different maps.  In the case where $ (G, K) = (\GL_{2n}, \GL_n \times \GL_n) $, one of the maps is a generalized Steinberg map, which induces a generalization of the RS correspondence for degenerate permutations.  The other is an exotic moment map, which maps degenerate permutations to signed Young diagrams, i.e., $ K $-orbits in the Cartan space $ (\lie{g}/\lie{k})^* $.

We explain geometric background of the theory and combinatorial procedures which produces the above mentioned maps.

This is an on-going joint work with Lucas Fresse.
 

We present the linearized metrizability problem in the context of parabolic geometries and subriemannian geometry, generalizing the metrizability problem in projective geometry studied by R. Liouville in 1889. We give a general method for linearizability and a classification of all cases with irreducible defining distribution where this method applies. These tools lead to natural subriemannian metrics on generic distributions of interest in geometric control theory.

TBC