Applied Mathematics Seminar
University of Leeds
Welcome to the Leeds Applied Mathematics Seminar
Spring 2019
Monday 11 February 2019
Steven Boeing & Leif Denby (University of Leeds)
Cumulus clouds and the coherent structures that trigger them
In the below-cloud boundary layer of the Earth's atmosphere, coherent structures of air are formed through interaction with the Earth's surface. These structures trigger the formation of convective clouds. Due to their small size relative to the current achievable computational resolution these drivers of convection must be parameterised in contemporary weather and climate models (which have horizontal grid spacings on the order of 1-4km and 10-100km respectively). However, the growth, organisation and physical properties of these structures is currently poorly understood, making the parameterisations inaccurate. This is thought to be one of the main causes of poor weather forecasts and uncertainty in climate predictions.
Using techniques developed in mathematics (cumulants and Minkowski functionals), analysis tools have been developed during the GENESIS project to identify and quantify the properties of these coherent boundary layer structures. In this talk, we will show results of these techniques applied to archetypal simulations of shallow convection with and without ambient wind shear, delineating a path to a more physically-based representation of the formation of convective clouds.
We will also discuss ongoing efforts to model the clouds as rising thermals. Again, it appears that capturing the behaviour of the thermals near cloud base is key to understanding their behaviour. A Lagrangian method for simulating moist convection may help us generate new insights into the behaviour of the clouds as they start to form. Finally, we will show a new and exciting way of visualising cloud formation which uses a pyramid-shaped projector located at the School of Earth and Environment.
Monday 25 February 2019
Daniel Read (University of Leeds)
Polydisperse polymers: predicting fluctuations, flow and crystallisation
All industrial polymeric materials are polydisperse – they contain molecules with a wide range of lengths (and also, commonly, branches too). There exists a good deal of understanding of how simple liquids containing ideal monodisperse polymers (all the same length) behave when subjected to flow. However, there is a significant challenge in generalising these models to more messy, polydisperse materials.
In this first half of the talk I will attempt to give a broad introduction to common rheological (flow) measurements, and simple models for the dynamics of entangled polymers. Then (depending on time) I will try to give an overview of recent results for bidisperse polymers (two chain lengths), how these can be generalised to polydiperse materials in order to make quantitative predictions which are helpful to industry. I’ll also show how all this can be packaged up in useful software. And (if time) I might get chance to mention some very recent ongoing work on predicting how polymer flow affects nucleation barriers for crystallization.
Monday 25 March 2019
Rebecca Hoyle (University of Southampton)
Personality and social influence in strategic decision-making
Humans are highly social and cooperative. But how do our social ties sustain cooperation in the face of the temptation to pursue our own self-interest? And how do our individual personalities influence cooperation in groups?
Social dilemmas that pit the interest of the individual against that of their wider social group are often modelled using strategy games such as the prisoner's dilemma and the public goods game, where cooperation favours the group while the individual's immediate economic interest is best served by withholding it.
I will present results from simulations and experiments that investigate the effects of social influence, social network structure and personality on decision-making in these economic games, including a multiplex network model that suggests social behaviour dominates economic incentives in maintaining cooperation, an experiment investigating how the personality composition of groups affects cooperation, and an individual-based simulation probing whether selfish risk-taking can favour community survival.
Monday 29 April 2019
Priya Subramanian (University of Leeds)
Formation and spatial localization of phase field quasicrystals
The dynamics of many physical systems often evolve to asymptotic states that exhibit spatial and temporal variations in their properties such as density, temperature, etc. Regular patterns such as graph paper and honeycombs look the same when moved by a basic unit and/or rotated by certain special angles. They possess both translational and rotational symmetries giving rise to discrete spatial Fourier transforms. In contrast, an aperiodic crystal displays long range order but no periodicity.
Recently, quasicrystals which are related to aperiodic crystals have been observed to form in diverse physical systems such as metallic alloys (atomic scale) and dendritic-, star-, and block co-polymers (molecular scale). Such quasicrystals lack the lattice symmetries of regular crystals, yet have discrete Fourier spectra. We look to understand the minimal mechanism which promotes the formation of such quasicrystalline structures using a phase field crystal model. Direct numerical simulations combined with weakly nonlinear analysis highlight the parameter values where the quasicrystals are the global minimum energy state and help determine the phase diagram.
By locating parameter values where multiple patterned states possess the same free energy (Maxwell points), we obtain states where a patch of one type of pattern (for example, a quasicrystal) is present in the background of another (for example, the homogeneous liquid state) in the form of spatially localized dodecagonal (in 2D) and icosahedral (in 3D) quasicrystals. In two dimensions, we compute several families of spatially localized quasicrystals with dodecagonal structure and investigate their properties as a function of the system parameters. The presence of such metastable localized quasicrystals is significant as they affect the dynamics of the soft matter crystallization process.