## Spring 2019 Seminars and Abstracts Spring 2019 Seminars and Abstracts

Seminars take place on Monday afternoons 14:00–15:00, unless specified otherwise. Everyone is welcome.

### Talks

February 11, (TEC 1.06), Dr Toby Wood (University of Newcastle), Superfluid vortices and fluxtubes in a neutron star

February 25, (JSC 2.02), Prof Giles Thomas (University College London), Slamming behavior of large high speed catamarans in large waves

March 4, (MED 2.02), Dr Anita Faul (University of Cambridge), The model is simple until proven otherwise

March 11, (TEC 1.06), Prof Robb MacDonald (University College London), Application of complex analysis to the geometry of river valleys and networks

March 18, (TEC 1.06), Prof Ted Johnson (University College London), The long-wave vorticity dynamics of rotating buoyant outflows

March 25, (SCI 1.20), Dr Matthew Moore (University of Oxford), Moving beyond Wagner: applying classical impact theory to droplet impact problems

### Abstracts

Dr Toby Wood (February 11):

Neutron stars contain the highest densities and strongest magnetic fields found anywhere in the Universe, so they present a fantastic opportunity to test our understanding of the laws of physics under extreme conditions.  Throughout most of the star, the neutrons and protons are condensed into superfluids, which are coupled through mutual entrainment and electromagnetic forces.  We present a microscale MHD model for these fluids, using the Gross-Pitaevskii/Ginzburg-Landau framework, and employing (quasi)periodic boundary conditions.  We study the onset of superconductivity near the critical temperature, and the transition between type-I and type-II superconductivity, demonstrating an intermediate "type-1.5" regime of fluxtubes bunches.

Prof Giles Thomas (February 25):

When large high-speed catamarans (such as the one pictured) operate in waves, they can be exposed to large wave loads, particularly from slamming events – an impact that occurs when the bow enters the water at high relative velocities. These loads can cause damage to the hull by generating large bending moments as well as producing a dynamic hydroelastic response, known as whipping, which can cause fatigue issues.  This presentation will outline work using full scale measurements, model experiments and numerical techniques to better understand the behaviour of these vessels in waves. The work has led to improvements in the design of the vessels to reduce the magnitude of the loads and better structural configurations to avoid damage occurrence.

Dr Anita Faul (March 4):

Machine Learning and AI have enjoyed an unprecedented rise in popularity. In academia as well as industry, they are often viewed as the future solution to all problems. However, systems have become so complex that it is no longer humanly comprehensibly, how an algorithm arrives at an answer. In some cases, companies refuse to disclose the proprietary algorithm. This has lead to controversies such as the COMPAS algorithm giving scores on the likelihood to reoffend. The organisation ProPublica claims that the software exhibits racial bias [1] which the company disputes [2]. Another example is Amazon’s gender bias recruitment tool [3]. Partly to blame is the data used to train algorithms. If the data is biased, then the algorithm will be. More seriously, it might exacerbate the bias, since algorithms distill the essential distinguishing features. If these are then highly correlated with black - white, male - female, we have a problem. While humans can also have bias, they are also capable of realizing their world view is too simplistic. The talk presents work in progress of increasing the complexity of a model, if the data suggests more features are necessary to model the data. This approach aides to understand the ”black magic” inside the ”black box”.

Prof Rob MacDonald (March 11):

Valleys and networks shaped by groundwater-fed streams exhibit remarkable geometric properties, two of which are considered in this talk: (i) the finger-like shape of an individual valley is modelled by a diffusive free boundary whose solution gives the 3D structure of the valley; (ii) the 2D bifurcation dynamics of evolving streams is modelled by analytically by Loewner's equation and numerically using Matlab's Schwarz-Christoffel toolbox. Comparison to field data is made in both (i) and (ii).

Prof Ted Johnson (March 18):

This talk discusses the evolution of coastal currents by considering, relative to a rotating frame, the flow development when buoyant fluid is injected into a quiescent fluid bounded by a solid wall. The initial rapid response is determined by the Coriolis force–pressure gradient balance with a Kelvin wave propagating rapidly, at the long-wave speed, with the bounding wall to its right (for positive rotation). However fluid columns can stretch or squash on ejection from coastal outflows so that the ejected fluid gains positive or negative relative vorticity. Depending on its sign, the image in the solid wall of this vorticity can reinforce or oppose the zero potential vorticity-anomaly (PVa) current set up by the Kelvin wave (KW). This talk presents a simple, fully nonlinear, dispersive, quasi geostrophic model to discuss the form of coastal outflows as the relative strength of vortex to KW driving is varied. The model retains sufficient physics to capture both effects at finite amplitude and thus the essential nonlinearity of the flow, but is sufficiently simple so as to allow highly accurate numerical integration of the full problem and also explicit, fully nonlinear solutions for the evolution of a uniform PVa outflow in the hydraulic limit. Outflow evolutions are shown to depend strongly on the sign of the PVa of the expelled fluid, which determines whether the vortex and KW driving are reinforcing or opposing, and on the ratio of the internal Rossby radius to the vortex-source scale, $|V/D^2 \Pi|^{1/2}$ , of the flow (where $D$ measures the outflow depth, $\Pi$ the PVa of the outflow and $V$ the volume flux of the outflow), which measures the relative strengths of the two drivers. Comparison of the explicit hydraulic solutions with the numerical integrations shows that the analytical solutions predict the flow development well with differences ascribable to dispersive Rossby waves on the current boundary and changes in the source region captured by the full equations but not present in the hydraulic solutions. The analytical results are then  extended to Rossby numbers of order unity in the long-wave, reduced-gravity, shallow -water equations: the semi-geostrophic equations.

Dr Matthew Moore (March 25):

Impact problems have a wide range of applications throughout real-world phenomena and industrial processes ranging from ship-slamming to inkjet printing and coating processes. With recent advances in camera technology, visualisation techniques and high-powered computing, we are increasingly able to see more of the phenomena occurring at small length- and timescales in droplet impacts. Understanding these phenomena can be vital in both promoting and inhibiting splashing.

In this talk, we will discuss the role that classical Wagner theory, developed for applications to naval architecture in ship-slamming, can play in the much smaller scale problem of droplet impact. Wagner theory was developed to help predict pressure profiles on the hulls of ships and sea-planes, so much of the finer detail of the splash itself was ignored. However, it is this very splash that is of importance in many droplet impact scenarios. Since impact problems are highly nonlinear and complex, it is desirable to use modelling to help predict certain properties, such as the location of the root of the splash jet (or ejecta), enabling us to focus our numerical or experimental investigations on a subset of the full impact problem. Therefore, we will develop a droplet Wagner model and compare the predictions to direct numerical simulations in Gerris, highlighting what the theory does well and where it is lacking. In the latter cases, we will discuss possible extensions to the theory to improve the predictions.

## Autumn 2018 Seminars and Abstracts Autumn 2018 Seminars and Abstracts

Seminars take place on Monday afternoons 14:00–15:00, unlesss specified otherwise. Everyone is welcome.

### Talks

October 8, (MED 1.02), Dr Anna Kalogirou (UEA), Linear stability and nonlinear dynamics of two-layer flow in the presence of surfactants

October 15, (Queens 0.09), Dr Pavel Berloff (Imperial College London), Dynamically Consistent Parameterization of Mesoscale Eddies

October 22, (Queens 2.22), Dr Raphael Stuhlmeier (University of Plymouth), (In)stability and evolution of inhomogeneous, broad-banded seas

October 29, (Queens 1.04), Prof Grae Worster (University of Cambridge), The Dynamics of Marine Ice Sheets

November 2, 4pm, (SCI 1.20), Dr Hiromitsu Takeuchi (Osaka City University), Stability of a vortex with winding number two of the nonlinear Schrödinger equation for Bose-Einstein condensates

November 5, (Queens 2.22), Prof Robin Cleveland (University of Oxford), Planning ultrasound surgery with 3D patient specific models

November 12, (MED 1.02), Dr Helen Burgess (University of St. Andrews), Long frontal waves and dynamic scaling in freely evolving equivalent barotropic flow

November 19, (Queens 0.08), Prof Kostas Bellibasakis (National Technical University of Athens), Wave propagation in nearshore/coastal environment by coupled-mode models (problems and applications)

November 19, 4pm, Prof Larry Forbes (University of Tasmania, Australia), Unstable Interfaces and Plumes

November 26, (ZICER 0.02), Dr David Lloyd (University of Surrey), Localised patterns and invasion fronts on the surface of a ferrofluid

December 10, (Queens 0.08), Dr Giovanni Barontini (University of Birmingham), Multi-component Bose-Einstein condensates: from neutron stars to ultracold engines

### Abstracts

Dr Anna Kalogirou (October 8):

A two-fluid shear flow in the presence of surfactants is considered. The flow configuration comprises two superposed layers of viscous and immiscible fluids confined in a long horizontal channel. The two fluids can have in general different densities, viscosities and thicknesses. The surfactants can be insoluble, i.e. located at the interface between the two fluids only, or soluble in the lower fluid. A primary aim of this study is to investigate the effect of surfactants on the stability of the interface, and in particular surfactants in high concentrations and above the critical micelle concentration (cmc). An asymptotic model valid in the approximation of a thin fluid layer is also derived, comprising a set of nonlinear PDEs to describe the evolution of the film thickness and surfactant concentration. Interfacial instabilities are induced due to the acting forces of gravity and inertia, as well as the action of Marangoni forces generated as a result of the dependence of surface tension on the interfacial surfactant concentration. The underlying physical mechanism responsible for the formation of interfacial waves will be discussed, together with the complex flow dynamics (typical nonlinear phenomena associated with thin-film flows include travelling waves, solitary pulses, quasi-periodic and chaotic dynamics).

Dr Pavel Berloff (October 15):

This work aims at developing new approach for parameterizing mesoscale eddy effects for use in non-eddy-resolving ocean circulation models. These effects are often modelled as some diffusion process or a stochastic forcing, and the proposed approach is implicitly related to the latter category. The idea is to approximate transient eddy flux divergence in a simple way, to find its actual dynamical footprints by solving a simplified but dynamically relevant problem, and to relate the ensemble of footprints to the large-scale flow properties.

Dr Raphael Stuhlmeier (October 22):

Nonlinear interaction, along with wind input and dissipation, is one of the three mechanisms which drive wave evolution, and is included in every modern wave-forecast model. The mechanism behind the nonlinear interaction terms in such models is based on the kinetic equation for wave spectra derived by Hasselmann. This does not allow, for example, for statistically inhomogeneous wave fields, nor for the modulational instability which depends on such inhomogeneity, and which has been implicated in the appearance of exceptionally high rogue waves.

Beginning with the basics of third-order wave theory, we sketch the derivation of a discretized equation for the evolution of random, inhomogeneous surface wave fields on deep water from Zakharov's equation, along lines first laid out by Crawford, Saffman, and Yuen. This allows for a general treatment of the stability and long-time behaviour of broad-banded sea states. It is investigated for the simple case of degenerate four-wave interaction, and the instability of statistically homogeneous states to small inhomogeneous disturbances is demonstrated. Furthermore, the long-time evolution is studied for several cases and shown to lead to a complex spatio-temporal energy distribution. The possible impact of this evolution on the statistics of rogue wave occurrence is explored within the framework of this simplified example.

Prof Grae Worster (October 29):

Most of the West Antarctic Ice Sheet (WAIS) sits on bedrock that is one to two kilometres below sea level. Its weight causes the ice sheet to flow outwards towards the ocean, thinning as it goes until it is thin enough to float on the ocean as an ice shelf before it ultimately breaks up into ice bergs. Some areas of the WAIS have been accelerating in recent years, as the point at which the sheet begins to float recedes, and this contributes to the rise in global sea level. I shall describe some recent analogue laboratory experiments and associated mathematical models that describe and quantify the fundamental dynamical controls on ice sheets that terminate in the ocean, focusing particularly on the role that floating ice shelves play in buttressing the ice sheet against collapse.

Dr Hiromitsu Takeuchi (November 2):

The stability of doubly quantized vortex (DQV), a vortex with winding number two, in uniform system is a crucial problem in low temperature physics. If a DQV could be stable, a lot of literatures in the long history of research on superfluid system should be re-examined since they assumed a multiply quantized vortex should be unstable there. In this work, we revisit this fundamental problem of the stability of a DQV in uniform single-component Bose-Einstein condensates at zero temperature [1]. To reveal the stability, the system-size dependence of the excitation frequency of the system with a DQV was analyzed through large-scale simulations of the Bogoliubov-de Gennes equation. We found that the system remains dynamically unstable even in an infinite-system-size limit. The system-size dependence is characterized by introducing the perturbation theory, based on the theory of Hamiltonian dynamical systems [2] and the semi-classical theories based on the WKB approximation, extended to the case with complex eigen-energy.

[1] Hiromitsu Takeuchi, Michikazu Kobayashi, and Kenichi Kasamatsu, Is a Doubly Quantized Vortex Dynamically Unstable in Uniform

Superfluids?, Journal of the Physical Society of Japan 87, 023601 (2018); arXiv:1710.10810

[2] R. S. MacKay, in Hamiltonian Dynamical Systems, ed. R. S. MacKay and J. D. Meiss (Adam Hilger, Bristol, U.K., 1987) p. 137.

Prof Robin Cleveland (November 5):

High intensity focused ultrasound has been used clinically to thermally ablate tissue, for example destroying cancer tumours, and to mechanically fractionating tissue, for example enlargening the urethra in the prostate. In order to focus the ultrasound it is normally assumed that the sound speed in soft-tissue is uniform and so it is mostly limited to targets with soft-tissue paths. In reality tissue has a range of different sound speeds, with fat typically 100 m/s slower than other soft-tissue. This can affect the ability to focus accurately. Here two applications are considered using realistic 3D patient models derived from CT data. The first is thermal ablation of the kidney where it is shown that fat layers in the path can result in fragmentation of the focus. It is shown that if the phase aberration can be accounted for then it is possible to recover a tight focus. In the second application it is shown that be using an array it is possible to focus ultrasound to the centre of a vertebral disc, despite the presence of the bone structures. A paradigm for mechanically fractionating tissue in the disc is described employing cavitation nucleation agents. These examples, demonstrate how patient specific models can be employed to improve the performance of high intensity focused ultrasound.

Dr Helen Burgess (November 12):

We present a scaling theory that links the frequency of long frontal waves to the decay

rate of kinetic energy and inverse transfer of potential energy in freely evolving equivalent barotropic turbulence. The flow is initialised with a potential vorticity field whose characteristic length scale is LD, the Rossby radius of deformation. As the turbulence evolves fronts of width O(LD) emerge, bounding large vortices within which potential vorticity is well-mixed and arranged into a staircase structure. The jets collocated with the fronts support long wave undulations with wavelengths >> LD and time scales fast relative to the time scale of the flow evolution. These undulations facilitate collisions and mergers between the vortices, implicating the frontal dynamics in the growth of potential-energy-containing flow features. Mergers generate disturbances with radius of curvature O(LD), which then propagate along the jets, causing them to shed filaments of kinetic energy and smooth out. A decay law for the total frontal length LF(t)~t^(-1/3) follows from assuming self-similar vortex growth and using the dispersion relation for long frontal waves [1]. High resolution simulations show that kinetic energy, potential enstrophy, and enstrophy, which are concentrated along the fronts and proportional to LD*LF(t), decay like t^(-1/3). Interestingly this is the same decay law followed by enstrophy in the vortex populations of freely evolving barotropic turbulence [2, 3].

Bibliography

[1] J. Nycander, D. G. Dritschel, and G. G. Sutyrin. The dynamics of long frontal waves in the shallow-water equations. Phys. Fluids A, 5:1089-1091, 1993.

[2] D. G. Dritschel, R. K. Scott, C. Macaskill, G. A. Gottwald, and C. V. Tran. Unifying scaling theory for vortex dynamics in two-dimensional turbulence. Phys. Rev. Lett., 101:094501, 2008.

[3] B. H. Burgess, D. G. Dritschel, and R. K. Scott. Extended scale invariance in the vortices of freely evolving two-dimensional turbulence. Phys. Rev. Fluids, 2:114702, 2017.

Prof Kostas Bellibasakis (November 19):

Results from the development and application of coupled-mode models to predict propagation of water waves in nearshore/coastal environment with variable bathymetry and other inhomogeneities are presented, including interaction with marine structures. This method models reflection, refraction, diffraction and dispersion phenomena, without introducing mild-slope type assumptions. The theory is based on an improved representation of the field in a series of local vertical modes, enhanced by appropriate terms to satisfy boundary conditions on the free surface and the sloping bottom. The additional modes significantly accelerate the convergence of the modal expansion and make the method suitable for horizontally large-scale applications. Next, with the aid of variational principles, the problem of propagation and interaction in non-homogeneous environment is reformulated as a system of partial differential equations on the horizontal plane, having the property to reduce to mild-slope type models in subregions where bathymetry and other parameters are slowly varying, saving computational cost. Various examples are presented and discussed demonstrating the applicability of the present method, including effects of waves on marine floating or fixed structures in an environment characterized by variable bottom topography.

Prof Larry Forbes (November 19):

In many applications involving waves at the surface of a fluid, it is often sufficient just to consider steady-state situations, where the wave pattern does not change noticeably with time.  Waves behind moving ships are one such example.  There is an enormous literature on such situations, and at least for two-dimensional flow, steady waves can be computed reasonably accurately.

As computers have developed, it has now become possible to look at 2D and even 3D unsteady problems, where the fluid interface evolves with time.  These unsteady flows have some surprising behaviour, both analytically and numerically.  If fluid viscosity is ignored, it is now known that classical flows such as the Rayleigh-Taylor Instability fail within finite time, when the curvature of the free surface becomes infinite at certain points.  This appears to be a common feature in unsteady inviscid flows.  In other geometries, such as the initially spherical outflow from a source, unsteady effects can lead to the surprising result that the lowest mode is the most unstable, so that a one-sided outflow jet evolves.

This talk will consider some examples of unsteady fluid flow, characterized by the presence of an unstable interface.  We will discuss how fluid viscosity and interface thickness change the singular behaviour predicted by (non-linear) inviscid theory, in some surprisingly subtle ways.

Dr David Lloyd (November 26):

In this talk, I will give an overview of work (both mathematical and experimental) on localised pattern formation seen on the surface of a magnetisable fluid seen in experiments. I will also present on-going work looking at proving the existence of radial spots and invasion fronts where cellular hexagon spikes are left behind in the wake of the front.

Dr Giovanni Barontini (December 20):

I will present our latest results on the thermodynamics of spin-1 polar Bose-Einstein Condensates, including evidence of the observation of 2-photon Feshbach resonances, and our progress towards the realization of a “quantum printer” that will allow us to simulate high-energy physics phenomena. In particular, I will present our recent work aiming at unveiling the Andreev-Bashkin effect in superfluid mixtures. In the final part of the talk I will present the progress of our project for the realization of quantum engines using ultracold atomic mixtures.