All seminars (unless otherwise stated) will take place on **Tuesdays at 3.00pm in ****Room 500 **which is located on the 5th floor of the Mathematics Department. See How to Find Us for further details. There will be tea afterwards in Maths Room 606. If you require any more information on the Applied seminars please contact Prof Slava Kurylev e-mail: y.kurylev AT ucl.ac.uk or tel: 020-7679-7896.

## 14 January 2014

NO SEMINAR

## 21 January 2014

Prof B M Brown - University of Cardiff

##### Title: Scattering and inverse scattering for a left-definite Sturm-Liouville problem

Abstract:

This talk reports on a scattering and an inverse scattering theory for the SturmLiouville equation with a weight that changes sign, but with non negative potential. The crucial ingredient of the approach is a generalized transform built on the Jost solutions of the problem and hence termed the Jost transform and the associated PaleyWiener theorem linking growth properties of transforms with support properties of functions. One motivation for this investigation comes from the CamassaHolm equation for which the solution of the Cauchy problem can be achieved by the inverse scattering transform of a certain Sturm-Liouville problem.

## 28 January 2014

Prof Ted Shepherd - University of Reading

##### Title: How Predictable is the effect of climate change on atmospheric circulation?

Abstract:

Although climate change is often characterised as global warming, the impact of climate change will vary greatly from region to region. Regional aspects of climate change are controlled by atmospheric circulation patterns, which moreover exhibit considerable chaotic variability. Model predictions of the atmospheric circulation response to climate change are in many cases highly uncertain, presumably because of systematic errors in the climate models (e.g. the location of the jet stream). The fact that these errors have stubbornly persisted despite increases in spatial resolution suggests that they are somehow linked to unresolved processes, whose effects need to be parameterised in the models. Thus, improving climate models requires a better understanding of multi-scale interactions. There are good reasons to believe that model bias, the divergence of model projections, and chaotic variability are somehow related. This talk will present some examples of these kinds of uncertainties and some potential ways forward.

## 04 February 2014

NO SEMINAR

## 11 February 2014

Dr Michael Patterson - University of Bristol

##### Title: Unusual features of rotating fluids

Abstract:

In this talk I will discuss the use of a multi-layer Particle Image Velocimetry (PIV) system that allows the measurement of 2D velocity fields at a range of fluid depths. I will then discuss application of this technique to different experiments where the fluid is subject to rotation and a variety of distinct boundary conditions including an evaporating upper boundary and an unusual half cone geometry that represents a pathological case for the classical linear theory of Greenspan & Howard (1963).

## 18 February 2014

**READING WEEK - NO SEMINAR**

## 25 February 2014

Prof Giovanni Vasconcelos - Imperial College and Federal University of Pernambuco (Brasil)

##### Title: Fluid dynamics in multiply connected domains

Abstract:

Many problems in two-dimensional fluid dynamics can be conveniently formulated as boundary-value problems for analytic functions in the complex plane. If the flow domain is simply or doubly connected, the problem can often be solved exactly by standard conformal mapping techniques. The situation is much more complicated, however, in the case of higher connectivity because conformal mappings for such domains are notoriously difficult to obtain. In this talk, I will describe a large class of conformal mappings from a bounded circular domain to multiple-slit domains which are relevant for several fluid systems. (The slit maps are written in terms of functions appropriately chosen from a family of Schottky-Klein prime functions associated with the circular domain which we have recently obtained.) As a first example, exact solutions for multiple steady bubbles in a Hele-Shaw cell will be constructed in closed form. Time-dependent solutions for multiple Hele-Shaw bubbles will also be discussed. Point vortex dynamics around obstacles can also be tackled with our formalism, and some examples will be presented. Applications to 2D Stokes flow and other vortex flows will be briefly discussed.

## 04 March 2014

Prof Alexandra Tzella - University of Birmingham

##### Title: Front propagation for fast reaction and small diffusivity in a chain of counterrotating vortices

Abstract:

We investigate the influence of steady cellular flows on the propagation of chemical fronts modeled by the Fisher-Kolmogorov nonlinearity. Previous theoretical investigations have shown that the long-time speed of propagation is determined by an eigenvalue problem whose solution is difficult to compute when the molecular diffusivity is small and the reactions are fast i.e., when the Peclet (Pe) and Damkohler (Da) numbers are large. Here, we employ a WKB approach to obtain an expression for the front speed in terms of a periodic path that minimizes a certain functional, which is valid for a broad range of Pe, Da ≫ 1. We show that for Da ≪ Pe, the periodic path closely follows the flow streamlines near the cell boundaries while for Da ≫ Pe, the path is nearly a straight line that sweeps across the centre of the cells. The theoretical results are compared with numerics obtained by evaluation of the eigenvalue problem. Their relation with previous results obtained using the G-equation is discussed.

## 11 March 2014

Prof Laure Zanna - University of Oxford

##### Title: Towards a stochastic closure for ocean mesoscale eddies

Abstract:

The goal of this study is to construct a stochastic parameterization of ocean mesoscale eddies in order to account for the fluctuations in subgrid transport and to represent upscale turbulent cascades. Simulations are performed in a quasi- geostrophic (QG) model in a double-gyre configuration. The model equation and model output are coarse-grained, giving rise to an eddy source term which represents the eddy-eddy and eddy-mean flow interactions and associated Reynolds stresses. The eddy source term, its mean and fluctuations are analyzed as function of the resolved scales and external parameters.

A functional form of the resolved scales, based on a representation of turbulence as a Non-Newtonian viscoelastic medium acting on a mean flow field, is used to describe the eddy source term mean, variance and decorrelation timescale. Probability density functions (PDFs) of the eddy source term conditional on the resolved scales are then calculated, capturing the fluctuations associated with mesoscale eddies and their impact on the mean flow. Scalings for the mean, standard deviation, skewness, and kurtosis of the conditional PDFs are provided as function of the grid size, forcing, and stratification of the coarse-resolution model.

In light of these scalings, the implementation of a stochastic closure based on the conditional PDFs requires in principle very little tuning and no preliminary high-resolution (QG) model runs to diagnose the subgrid forcing. Some preliminary experiments will be discussed.

## 18 March 2014

Prof Ivan Graham - Bath University

##### Title: On shifted Laplace and related preconditioners for finite element approximations of the Helmholtz equation

Abstract:

As a model problem for high-frequency wave scattering, we study both the interior impedance problem for the Helmholtz equation and the truncated sound soft scattering problem. Finite element

approximations of this problem for high wavenumber are notoriously hard to solve. The analysis of solvers such as GMRES is also hard, since the corresponding system matrices are complex, non-Hermitian and usually highly non-normal and so information about spectra and condition numbers generally does not give much information about the convergence rate.

We consider preconditioners built from approximating the corresponding PDE problem with added absorption (the ``shifted Laplace problem''). Using techniques from PDE analysis we show how the

absorption should be chosen so as to obtain wavenmuber independent GMRES convergence for the preconditioned problem. We also present an analysis of additive Schwarz domain decomposition methods, proving estimates on the rate of convergence explicitly in terms of wavenumber and absorption. We give numerical illustrations of the performance of the solver for some constant and variable wavespeed problems.

The analysis is driven by estimates for the field of values of the preconditioned matrices. Some interesting special cases of trapping domains for which the problem posesses pseudomodes are

encountered along the way.

The talk includes joint work with Martin Gander, Euan Spence and Eero Vainikko.

## 25 March 2014

Dr Lyubov Chumakova - University of Edinburgh

##### Title: Leaky modes -- discrete versus continuous spectrum in the atmosphere

Abstract:

Much of our understanding of tropospheric dynamics is based on the concept of discrete internal modes. Internal gravity waves, such as those associated with convective systems, propagate at definite speeds, typically associated with the first to third baroclinic vertical modes. These waves are the dynamical backbone of the tropospheric dynamics, even though their nature and speed can be altered significantly by nonlinearity, moist convection, mean wind shear, etc. These discrete modes are a signature of systems of finite extent, and are derived in a case when the atmosphere is bounded above by a rigid lid. In reality, the atmosphere does not have a definite top, and, some argue, should be modeled as semi-infinite, leading to a continuous spectrum. Are the discrete rigid lid modes then just a fallacy of overly simplified theoretical models? In this talk I will present a correction to the rigid lid by using a boundary condition at the top of the troposphere, that allows for a fraction of waves to escape to the stratosphere. The new discrete "leaky" modes decay with characteristic time-scales, which are in the ballpark of many atmospheric phenomena. I will also address the mathematical question of why in seemingly identical physical situations of an unbounded atmosphere and its subsection with a radiation condition at the top give different spectral characteristics. This is joint work with R.R. Rosales (MIT) and E.G. Tabak (NYU).