2nd year PhD student talks

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• James Cummins, Tom Daggitt, William Oxley, Alistair Hales, Jenny Dingwall, Matt Davison, Elvinas Ribinskas, James Mason, Joseph Webber
• Friday 27 May 2022, 14:45-17:01
• MR2, Centre for Mathematical Sciences, Wilberforce Road, Cambridge.

If you have a question about this talk, please contact Prof. Jerome Neufeld.

• 1445 – 1458 James Mason
• 1458 – 1511 Tom Daggitt
• 1511 – 1524 William Oxley
• 1524 – 1537 Alistair Hales
• 1537 – 1550 Jenny Dingwall

• 1550 – 1610 coffee

• 1610 – 1623 Matt Davison
• 1623 – 1636 Elvinas Ribinskas
• 1636 – 1649 James Cummins
• 1649 – 1701 Joseph Webber

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James Mason

Macroscopic behaviour in a two-species exclusion process via the method of matched asymptotics

We consider a two-species simple exclusion process on a periodic lattice. We use the method of matched asymptotics to derive evolution equations for the two population densities in the dilute regime, namely a cross-diffusion system of partial differential equations for the two species densities. First, our result captures non-trivial interaction terms neglected in the mean-field approach, including a non-diagonal mobility matrix with explicit density dependence. Second, it generalises the rigorous hydrodynamic limit of Quastel [Commun. Pure Appl. Math. 45(6), 623—679 (1992)], valid for species with equal jump rates and given in terms of a non-explicit self-diffusion coefficient, to the case of unequal rates in the dilute regime. In the equal-rates case, by combining matched asymptotic approximations in the low- and high-density limits, we obtain a cubic polynomial approximation of the self-diffusion coefficient that is numerically accurate for all densities. This cubic approximation agrees extremely well with numerical simulations. It also coincides with the Taylor expansion up to the second order in the density of the self-diffusion coefficient obtained using a rigorous recursive method.

Tom Daggitt

Variations in Observations of Geosynchronous Magnetopause and Last Closed Drift Shell Crossings with Magnetic Local Time

We present an analysis of events in which the number of electrons trapped in Earth’s outer radiation belts drops rapidly due to inward movement of the outer edge of Earth’s magnetic field. These observations are compared to models of the largest trapped electron orbits derived from models of Earth’s magnetic field and particle tracing models. These models of the largest trapped orbits agree well with the losses seen over the timescale of hours, but fail to reproduce more rapid decreases in the number of electrons measured on the timescale of minutes. We show that different satellites in geostationary orbit observe different trends in the trapped electron population on timescales of less than a day during geomagnetic storms due to their separation in longitude. These differences demonstrate that data from at least three satellites in geostationary orbit, ideally more, may be required for accurate, high time resolution forecasting and reconstruction of Earth’s radiation belts during geomagnetic storms.

William Oxley

Amplitude Thresholds for Zombie Vortex Instability   Rotating, stratified and sheared flows, where the shear is in a direction perpendicular to the (stable) vertical stratification, are found in a wide range of settings that include the atmosphere and oceans, as well as protoplanetary disks. The addition of stable stratification to rotating shear flows is of interest as it can have a destabilising effect. A number of studies of these types of flows, through numerical experiments, show vortices appearing in lattice like structures, after spreading across the domain due to an initial localised perturbation. This finite amplitude instability was first identified by Marcus et. al. 2013, who gave it the name ‘Zombie Vortex Instability’ (ZVI) due to its formation mechanism. The individual vortices result from the excitation of baroclinic critical layers, and each vortex then acts as a source for a new initial perturbation, which allows the process to repeat. Baroclinic critical layers are similar to their classical namesake, and take the form of sharp changes in perturbation variables around locations which are singular in the linear inviscid theory. In contrast to classical critical layers, which form where the perturbation phase speed matches the speed of the mean flow, baroclinic critical layers form at the locations where the phase speed of a perturbation (relative to the background shear) matches the characteristic gravity wave speed. This instability has been suggested by Marcus et. al. (2013) as a candidate to destabilize dead zones in protoplanetary discs, although that suggestion has been questions by other authors (e.g. Lesur & Latter 2016). Building on previous analytical and numerical studies, I have been investigating ZVI numerically to try and gain more insight into the formation mechanism, and in particular what are the conditions we must impose on the initial perturbation in order to produce ZVI . One of the key challenges to overcome is how to actually identify ZVI , and how to focus on one individual occurrence of the replication process. To do this, a carefully chosen initial condition is used and the problem is simulated using various different parameters.

Alistair Hales

Reduction of Leading-Edge Noise Using Tailored Turbulence Anisotropy

In this talk I give an outline of a new mathematical model to approximate the leading edge noise from the scattering of anisotropic flow off a rigid aerofoil. Thin aerofoil theory is used to model an aerofoil as a semi-infinite plate and the scattering of incoming turbulence is solved via the application of the Wiener-Hopf technique. This theoretical is integrated over a wavenumber-frequency spectrum to account for general incoming turbulence which is obtained using the method of Gaussian decomposition, accounting for and modelling anisotropy in the incoming turbulence.

Jenny Dingwall

Modelling the accumulation of buoyant particles under wind-driven and convective turbulence using large-eddy simulations

Buoyant material such as microplastics tend to accumulate near the ocean surface in regions with convergent surface currents where they can be harmful to marine life. We investigate the accumulation of buoyant tracers and Lagrangian surface particles by small-scale turbulence in the ocean mixed layer under combined wind and convective forcing using large-eddy simulations. Surface cooling drives convection, and under this regime persistent convective vortices form which trap buoyant material, leading to large concentrations. For sufficiently weak winds, convective vortices survive but become less effective at clustering material as the wind stress increases. Under strong wind forcing, convective vortices are no longer visible, but some particle clustering occurs in downwelling regions associated with longitudinal wind rolls.

Matt Davison

Reaction-Diffusion Dynamics of the Tropical Atmosphere   The Madden-Julian Oscillation (MJO) is a precipitating disturbance that propagates eastward across the Indian and Pacific oceans every 30-60 days. There is no single accepted theoretical explanation for the MJO and in climate models the MJO is often poorly represented. It has been suggested that there is a relation between the MJO and the phenomenon of convective aggregation observed in ‘convection-resolving’ numerical simulations. I will describe an extension of a previous theoretical model of convective aggregation, based solely on reaction-diffusion, to include large scale dynamics. The new model consists of the shallow water equations and a moisture equation coupled through the effects of moisture on radiation and latent heat release. When the latter are chosen such that there are two stable states alongside the initial unstable radiative-convective equilibrium, convective aggregation is observed. Since this model has dynamics we can include rotation, and on an equatorial beta-plane the dynamical model exhibits coherent, moist, eastward propagating disturbances, and therefore appears to represent the mechanism behind the MJO .

Elvinas Ribinskas

Reduced modelling of ice sheet response to climate perturbations

Ice sheet melting associated to climate change poses the risk of significantly increasing the sea level. Modelling how ice sheets respond to climate perturbations allows us predict this change. Reduced models in particular can be used to also infer how the underlying physical processes influence ice sheet response to external forcings. I will present a reduced model of a land-terminating ice sheet that is forced by a prescribed ablation-accumulation law. The model consists of a simplified ice flow law, basic meltwater network and a basal boundary condition that couples ice flow to the basal meltwater pressure. The steady-state solutions of the model are found to be non-unique. Also, we learn that longitudinal stresses have to be included into the model for it to represent a realistic ice sheet. In the end, I will present preliminary results of the improved model and possible directions for future work.

James Cummins

Unconventional computing & the n-queens problem

Traditional computers can not keep up with the increasing speed and power expected of them. Unconventional computing architectures overcome this by using gain-dissipative systems to improve computational performance. These platforms may comprise of lasers, superconducting qubits, polariton condensates, or photon condensates. In this talk we explore how such systems can solve one of the oldest problems in mathematics: the n-queens problem.

Joseph Webber

Dynamics of super-absorbent hydrogels

Hydrogels are soft materials formed from a hydrophilic polymer scaffold surrounded by adsorbed water molecules, and may comprise over 99% water by volume in their fully-swollen state. As such, when allowed to swell and dry, their volumes may change to an extreme degree, rendering linear poroelastic models invalid owing to the large magnitudes of the strains involved. Models proposed in the literature to describe this extreme swelling and drying often rely on a complex molecular-scale understanding of polymer-water interactions and formulate gel dynamics in terms of a free energy density. In this talk, I will summarise a new formulation for the dynamics of hydrogels which treats them as instantaneously incompressible linear-elastic materials, whilst allowing for nonlinearities in the isotropic strains corresponding to swelling and drying. Key features of this model include a complete description of any gel using only three material parameters, a nonlinear diffusion equation governing swelling and drying, and an associated equation to describe the displacement field, from which the shape of the hydrogel can be deduced. Applications of this model to a number of different problems will then be discussed briefly.

This talk is part of the Fluid Mechanics (DAMTP) series.

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