## SPRING 2017 SEMINARS AND ABSTRACTS SPRING 2017 SEMINARS AND ABSTRACTS

### Hydrodynamic Interactions between Flagellar Filaments

Date: Monday 16th January, 2pm, (Queens 1.04)
Speaker: Dr Eric Lauga (University of Cambridge)
Abstract: Many small organisms possess flagella, slender whiplike appendages which are actuated in a periodic fashion in fluids and allow the cells to self-propel. In particular, most motile bacteria are equipped with multiple helical rotating flagella which interact hydrodynamically, synchronise, and can form a tight helical bundle behind a swimming cell. We consider here the problem of bundling and unbundling of these flagellar filaments. Most past theoretical work has approached the problem of bundling using numerical computations. Here, we present an asymptotic treatment of the interactions between elastic rotating filaments. We first show how to asymptotically compute the hydrodynamic kernels governing hydrodynamic interactions in the case of long filaments, and we then use these results to derive the nonlocal, nonlinear, equations of motions of each filament. We finally apply our results to a few simple configurations.

### Electron-MHD Evolution in Strongly Magnetised Neutron Stars

Date: Monday 23rd January, 2pm, (Queens 2.22)
Speaker: Dr Kostas Gourgouliatos (University of Leeds)
Abstract: Neutron stars host the strongest magnetic fields in the Universe that we know of, reaching 10^15 G in magnetars. However the origin of these magnetic fields and the way they power magnetar bursting activity is still under debate. In this seminar, I will present the Electron Magnetohydrodynamic formulation for the evolution of the magnetic field in neutron star crusts. I will discuss the results of plane parallel, axially symmetric and finally 3-D simulations, demonstrating that the combined effect of instabilities and secular evolution leads to the formation of very strong localised magnetic fields. These magnetic fields facilitate the observed bursts and the quiescent emission of magnetars. In particular instabilities can create pockets where the magnetic field is at least one order of magnitude stronger than the average magnetic field leading to a more economical magnetar theory.

### Vortex Reconnections and Rebounds in Trapped Atomic Bose-Einstein Condensates

Date: Monday 30th January, 2pm, (JSC 2.02)
Speaker: Dr Luca Galantucci (Newcastle University)
Abstract: Reconnections and interactions of filamentary coherent structures play a fundamental role in the dynamics of classical and quantum fluids, plasmas and nematic liquid crystals. In quantum fluids vorticity is concentrated into discrete (quantised) vortex lines (unlike ordinary fluids where vorticity is a continuous field), turning vortex reconnections into isolated events, conceptually easier to study.
In order to investigate the impact of non-homogeneous density fields on the dynamics of quantum reconnections, we perform a numerical study of two-vortex interactions in magnetically trapped elongated Bose--Einstein condensates in the T=0 limit. We observe different vortex interactions regimes depending on the vortex orientations and their relative velocity: unperturbed orbiting, bounce dynamics, single and double reconnection events [1]. The key ingredients driving the dynamics are the anti-parallel preferred alignment of the vortices and the impact of density gradients arising from the inhomogeneity of the trapping potential.
The results are confirmed by ongoing experiments in Trento performed employing an innovative non--destrutive real--time imaging technique capable of determining the temporal evolution of the axial dynamics [2] and the orientation of the vortices.

[1] S. Serafini, L. Galantucci et al. , arXiv: 1611.01691 (2016)

[2] S. Serafini, M. Barbiero et al., Phys. Rev. Lett., 115, 170402 (2015)

Date: Monday 6th February, 2pm, (Queens 2.22)
Speaker: Dr Matthew Turner (University of Surrey)

Date: Monday 13th February, 2pm, (JSC 2.02)

Date: Monday 20th February, 2pm, (SCI 1.20)
Speaker: Dr Robert Ferdman (UEA)

Date: Monday 27th February, 2pm, (JSC 2.02)

Date: Monday 6th March, 2pm, (JSC 1.02)
Speaker: Prof. Daniele Faccio (Heriot-Watt University)

Date: Monday 13th March, 2pm, (SCI 1.20)

Date: Monday 20th March, 2pm, (SCI 1.20)
Speaker: Dr Konstantin Ilin (University of York)

Date: Monday 24th April, 2pm, (SCI 1.20)

### Total Positivity: A Concept at the Interface Between Algebra, Analysis and Combinatorics

UEA Mathematics Colloquium (Joint Pure and Applied Research Seminar)
Date: Monday 8th May 2pm, (LT3)
Speaker: Prof. Alan Sokal (University College London and New York University

Date: Monday 15th May, 2pm, (JSC 1.02)
Speaker: Dr James Sprittles (University of Warwick)

## Autumn 2016 Seminars and Abstracts Autumn 2016 Seminars and Abstracts

### Unsteady Free-Surface Flow Cased by a Moving Circular Cylinder

Date: Monday 24th October, 3pm, (SCI 3.05)
Speaker: Dr Vasil Kostikov (Novosibirsk State University, Russia)
Abstract: A problem on non-stationary free surface flow of an infinitely deep ideal fluid generated due to the motion of a submerged body is considered. The initial formulation of the problem is reduced to an integral-differential system of equations for the functions defining the free surface shape, the normal and tangential components of velocity on the free boundary. Small- time asymptotic solution is constructed for the case of circular cylinder that moves with a constant acceleration from rest. The role of non-linearity is clarified by the analysis of this solution in the context of formation mechanism of added mass layers, splash jets and finite amplitude surface waves

### Challenges for Climate and Weather Prediction in the Era of Exascale Computer Architectures: Oscillatory Stiffness, Time-Parallelism and the Role of Long-Time Dynamics

Date: Monday 7th November, 2pm, (LT4)
Speaker: Prof. Beth Wingate (University of Exeter)
Abstract: For weather or climate models to achieve exascale performance on next-generation heterogeneous computer architectures they will be required to exploit on the order of hundred-million-way parallelism. This degree of parallelism far exceeds anything possible in today's models even though they are highly optimized. In this talk I will discuss one of the mathematical issues that leads to the limitations in space- and time-parallelism for climate and weather prediction models oscillatory stiffness in the PDE. Many PDEs used in weather and climate simulations have the form: PDF File where the linear operator L has pure imaginary eigenvalues, the nonlinear term $N(u,u)$ is of polynomial type, the operator $D$ represents dissipation, and $\epsilon$ is a small non-dimensional parameter. The operator $\epsilon^{-1}L$ results in time oscillations on an order $O(\epsilon)$ time scale, and generally necessitates small time steps if standard explicit numerical integrators are used. Even implicit integrators need to use small time steps if accuracy is required. I will discuss the role of resonances in the PDEs in formulating new numerical algorithms with the potential to go beyond the strong- and weak-scaling limitations that presently exist.

### Modulation Instability of Finite Amplitude Periodic Travelling Waves: Theme and Variations

Date: Monday 14th November, 2pm, (LT4)
Speaker: Prof. Tom Bridges (University of Surrey)
Abstract: The most well known and well understood example of modulation instability is the Benjamin-Feir instability of weakly nonlinear Stokes waves.  On the other hand, modulation instability for finite-amplitude periodic travelling waves is much less understood, with the only nonlinear theory being Whitham modulation theory which is dispersionless and produces shocks or blowup.
However, at the transition from stability to instability more precise results can be obtained including nonlinearity and dispersion, as well as saturation or enhancement. The talk will first give an overview of the weakly nonlinear case, including the work of Johnson which derives a higher order NLS equation at the transition point.  Then a new theory will be presented for nonlinear behaviour near finite amplitude modulational instability transition. Examples and implications for the modulational transition of finite-amplitude Stokes waves, as well as simplified examples where the complete theory can be worked out in detail. This talk is based on joint work with Daniel Ratliff.

### Mean Flows in 2D Turbulence

Date: Monday 21st November, 2pm, (SCI 1.20)
Speaker: Dr Jason Laurie (Aston University)
Abstract: An inverse turbulent cascade in a restricted two-dimensional periodic domain creates a large-scale condensate or mean flow--for a square aspect ratio this is a pair of coherent system-size vortices. We present a new theoretical analysis based on momentum and energy exchanges between the mean flow and the underlying turbulence and show that the mean velocity profile has an universal internal structure independent on the mechanisms of small-scale dissipation and small-scale forcing. We verify the theoretical predictions through extensive numerical simulations of the two-dimensional Navier-Stokes equations. We begin our analysis by investigating the square geometry before studying larger aspect ratios and predictions for zonal mean flows.

### Variational Modelling of Water Waves and Their Impact on Moving Ships

Date: Monday 28th November, 2pm, (EFRY 1.01)
Speaker: Dr Anna Kalogirou (University of Leeds)
Abstract: The study of water waves has been an important area of research for years; their significance becomes obvious when looking at ocean and offshore engineering or naval architecture. Local weather and sea conditions can often lead to extreme wave phenomena, e.g. waves with irregular height. Waves with anomalously high amplitudes relative to the ambient waves are called rogue waves and can appear either at the coast or in the open ocean. The aim of this study is to investigate mathematically the generation and interaction of such waves and their impact on wave-energy devices and moving ships. The modelling is demonstrated by analysing variational methods asymptotically and numerically.
A reduced potential flow water-wave model is derived, based on the assumptions of waves with small amplitude and large wavelength. This model consists of a set of modified Benney-Luke equations describing the deviation from the still water surface and the velocity potential at the bottom of the domain. A novel feature in our model is that the dynamics are non-autonomous due to the explicit dependence of the equations on time. Numerical results obtained using a (dis)continuous Galerkin finite element method (DGFEM) are compared to a soliton splash experiment in a long water channel with a contraction at its end, resulting after a sluice gate is removed at a finite time. The removal of the sluice gate is included in the variational principle through a time-dependent gravitational potential.
The Benney-Luke approximation for water waves is also adapted to accommodate nonlinear ship dynamics. The new model consists of the classical water-wave equations, coupled to a set of equations describing the dynamics of the ship. We will first investigate the dynamics of the coupled system linearised around a rest state. For simplicity, we also consider a simple ship structure consisting of V-shaped cross-sections. The model is solved numerically using a DGFEM and the numerical results are compared to observations from experiments in wave tanks that employ geometric wave amplification to create nonlinear rogue-wave effects.

### Shock Waves in Non-Convex Dispersive Hydrodynamics

Date: Monday 12th December, 2pm, (LT3)
Speaker: Dr Gennady El (Loughborough University)
Abstract: Dispersive shock waves (DSWs) are unsteady nonlinear wavetrains replacing traditional shocks in media whose dominant wavebreaking regularisation mechanism is dispersion. DSWs are as ubiquitous as solitons and have been observed in a broad range of physical media including water waves and optical fibres. When the governing system represents a convex hyperbolic conservation law modified by weak dispersion,  as in the KdV equation, the resulting DSW has a well-defined structure  that includes a modulated periodic wave train, led by a solitary wave. The mathematical description of such DSWs in the framework of the Whitham modulation theory  is now well understood [1].
The DSW structure is also seen in non-convex dispersive-hydrodynamic systems, the modified KdV (mKdV) equation being probably the simplest example. Non-convexity of the hyperbolic flux in mKdV leads  to a rich set of unsteady solutions not observed in the convex case. Despite the fundamental nature of the problem such solutions have been obtained only recently. They are of direct relevance to the propagation of internal undular bores in the ocean and to the nonlinear dynamics of polarisation waves in two-component Bose-Einstein condensates.  In my talk I will describe a complete set of solutions to the Riemann initial value problem for the mKdV equation and show their (nontrivial) connections with the known solutions of the mKdV-Burgers equation such as undercompressive shocks and shock-rarefactions [2].
If time permits, I will also describe  expansion shock solutions of the Benjamin-Bona-Mahony (BBM) equation, a new type of shocks owing their existence to the particular type of non-convex linear dispersion.  An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition [3].

[1] G.A. El and M.A. Hoefer, Dispersive shock waves and modulation theory, Physica D 33, 5 - 67 (2016).

[2] G.A. El, M.A. Hoefer and M. Shearer, Dispersive and diffusive-dispersive shocks for non-convex conservation laws, SIAM Review 59, (2017) in press.

[3] G.A. El, M.A. Hoefer and M. Shearer, Expansion shock waves in regularised shallow water theory, Proc. Roy. Soc.  A, 472, 20160141 (2016).