Applied and Computational Mathematics Seminars 2018/2019

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Title: Energy flux enhancement, intermittency and turbulence via Fourier triad phase dynamics in 1D Burgers equation

Speakers: Dr. Miguel Bustamante (UCD)

Date: Monday 17th September 2018

Time: 1pm

Location: Room 1.25, Science Centre North

Abstract:
We present a theoretical and numerical study of Fourier space triad phase dynamics in one-dimensional stochastically forced Burgers equation at Reynolds number Re ≈ 2.7×10^4. We demonstrate that Fourier triad phases over the inertial range display a collective behaviour characterised by intermittent periods of synchronisation and alignment, reminiscent of Kuramoto model (Kuramoto 1984) and directly related to collisions of shocks in physical space. These periods of synchronisation favour efficient energy fluxes across the inertial range towards small scales, resulting in strong bursts of dissipation and enhanced coherence of Fourier energy spectrum. The fast time scale of the onset of synchronisation relegates energy dynamics to a passive role: this is further examined using a reduced system with the Fourier amplitudes fixed in time -- a phase-only model. We show that intermittent triad phase dynamics persists without amplitude evolution and we broadly recover many of the characteristics of the full Burgers system. In addition, for both full Burgers and phase-only systems the physical space velocity statistics reveals that triad phase alignment is directly related to the non-Gaussian statistics typically associated with structure-function intermittency in turbulent systems.
This work was done in collaboration with Dr. Brendan P. Murray. It is published in Journal of Fluid Mechanics 850, 624-645 (2018). You can find an arXiv preprint here: https://arxiv.org/abs/1705.08960.

Title: Stochastic Backlund transformations

Speakers: Prof. Neil O’Connell (UCD)

Date: Monday 24th September 2018

Time: 1pm

Location: Room 1.25, Science Centre North


Abstract:
How does one introduce randomness into a classical Hamiltonian system in order to produce something which is related to the ‘corresponding’ quantum system? I will discuss this question from a probabilistic point of view, in the context of a one-dimensional example with exponential potential. I will also give some background and motivation.

Coffee and tea will be available in the School Common Room from 12.40pm

 


Title:        Gluing methods for Vortex dynamics in Euler flows

Speakers:    Manuel del Pino,(University of Bath and University of Chile)

Date:        Monday 3rd December 2018

Time:     1pm

Location:    Room 1.25, Science Centre North


Abstract:    
We consider the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. We construct smooth solutions with concentrated vorticities around k points which evolve according to the Hamiltonian system for the Kirchhoff-Routh energy, using an outer-inner solution gluing approach.

The asymptotically singular profile around each point resembles a scaled finite mass solution of Liouville's equation. We also discuss the vortex filament conjecture for the three-dimensional case.

This is joint work with Juan Dávila, Monica Musso and Juncheng Wei.


Coffee and tea will be available in the School Common Room from 12.40pm.





 


Title:        Storm surge forecasting and other operational ocean modelling at the Met Office

Speakers:    Clare O’Neill (Met. Office, UK)

Date:        Monday 26th November 2018

Time:     1pm

Location:    Room 1.25, Science Centre North


Abstract:    


The Met Office runs a wide variety of operational marine models, from waves to storm surge to ecosystems. These generate products for a variety of customers and partners including other national meteorological services, environment agencies, the UK navy, and the Copernicus Marine Environment Monitoring Service. This presentation will give an overview of the activities and outputs of the Met Office's Ocean Forecasting Research and Development group. It will then focus on the storm surge forecast system in particular, describing the current system as well as research activities aimed at improving forecasts in the future.


Coffee and tea will be available in the School Common Room from 12.40pm.





Title:        Revisiting black-hole perturbation theory: the hyperboloidal slice approach

Speakers:    Rodrigo Macedo (Queen Mary, University of London)

Date:        Monday 19th November 2018

Time:     1pm

Location:    Room 1.25, Science Centre North

Abstract:  

 
After reviewing the well-stablished notion of black-hole perturbation theory and the concept of quasinormal modes, we present a spectral representation of solutions to relativistic wave equations based on a geometrical approach in which the constant-time surfaces extend until future null infinity. Here, we restrict ourselves to an asymptotically flat, spherical symmetric spacetime (with focus on the Reisnner-Nordstrom solution). With the help of a Laplace transformation on the wave equation in question, we provide a geometrical interpretation to known algorithms (i.e. Leaver’s approach) apart from deriving an algorithm for obtaining all ingredients of the desired spectral decomposition, including quasi-normal modes, quasi-normal mode amplitudes as well as the jump of the Laplace-transform along the branch cut. The work explains extensively this procedure and includes detailed discussions of relevant aspects the contribution of infinity frequencies modes to the early time response of the black hole and its relation to the QNM-amplitudes grows rate.

Coffee and tea will be available in the School Common Room afterwards


Title:        Buckling and wrinkling of thin viscous and elastic sheets

Speakers:    Doireann O'Keily (University of Oxford)

Date:        Monday 22nd October 2018

Time:     1pm

Location:    Room 1.25, Science Centre North


Abstract:
Thin structures typically buckle out of plane when compressed, giving rise to a myriad of shapes. These range from classic Euler buckling to honey coiling to microscale wrinkling patterns, and in some cases correspond to failure, but in others may be exploited for applications such as flexible electronics. In this talk I will present two examples of compression-driven out-of-plane deformation in thin sheets. The first is a failure mode in the manufacture of glass sheets, and the second concerns the dynamic wrinkling of thin elastic sheets under confinement. This talk will involve a combination of experimental results and mathematical modelling.


Coffee and tea will be available in the School Common Room afterwards


Title:        Gravitational self-interactions of cosmic string loops

Speakers:    Jeremy Wachter (University of the Basque Country)

Date:        Monday 5th November 2018

Time:     1pm

Location:    Room 1.25, Science Centre North

Abstract:
Gravitational wave signals from cosmic strings are strongly influenced by the presence and character of generic features on string loops known as kinks and cusps. We find analytically the leading-order effect of gravitational self-interactions on strings near kinks and cusps, and discuss how these effects might influence loop evolution. We show the results of numerically evolving particular kinds of loops undergoing self-interactions, and comment on how the gravitational wave spectrum from loops might be affected.

Coffee and tea will be available in the School Common Room afterwards

Title:        Layers, instabilities and relaminarisation in horizontally shearing stratified flows

Speakers:    Dan Lucas (Keele University)

Date:        Monday 1st October 2018

Time:     1pm

Location:    Room 1.25, Science Centre North

Abstract:
In this talk we survey several new results for turbulent flows driven by a shear which is perpendicular to a stable stratification. First we show how in a body forced system layer formation can be associated to a stratified linear instability via nonlinear unstable steady solutions. We also show how mixing processes can be encapsulated by unstable periodic orbits embedded in the turbulence. Next we explore the differences when we instead drive the flow by moving boundaries in a plane Couette flow configuration. Here we find layers confined near the walls which are seemingly disconnected from any linear mechanism. However the layer scale provides insight into the way turbulence is shut-off when stratification becomes very large. Finally we investigate the effect of linear instabilities on the transition to turbulence in this system at very large Reynolds numbers and stratifications.

Coffee and tea will be available in the School Common Room from 12.40pm

 

Title:        Travelling-wave spatially periodic forcing of asymmetric binary mixtures

Speakers:    Lennon O'Naraigh (UCD)

Date:        Monday 8th October 2018

Time:     1pm

Location:    Room 1.25, Science Centre North

Abstract:
We study travelling-wave spatially periodic solutions of a forced Cahn-Hilliard equation. This is a model for phase separation of a binary mixture, subject to external forcing. We look at arbitrary values of the mean mixture concentration, corresponding to asymmetric mixtures (previous studies have only considered the symmetric case). We characterize in depth one particular solution which consists of an oscillation around the mean concentration level, using a range of techniques, both numerical and analytical. We determine the stability of this solution to small-amplitude perturbations. Next, we use methods developed elsewhere in the context of shallow-water waves to uncover a (possibly infinite) family of multiple-spike solutions for the concentration profile, which linear stability analysis demonstrates to be unstable. Throughout the work, we perform thorough parametric studies to outline for which parameter values the different solution types occur.

Coffee and tea will be available in the School Common Room afterwards


Title:        Time Domain Method for the Green Function in Schwarzschild Spacetime

Speakers:        Conor O’Toole (UCD)

Date:        Monday 15th October 2018

Time:     1pm

Location:    Room 1.25, Science Centre North

Abstract:


We discuss a method for modelling stellar-mass bodies inspiralling into massive (10^6 solar masses and up) black holes. This problem has significance for the European Space Agency's LISA mission, the first space-based gravitational wave observatory, planned for launch in 2034. The approach taken is to calculate the Green Function, in particular for an object moving around a Schwarzschild (non-rotating) black hole. This can then be used to calculate the so-called "self-force", orbital evolution and resulting gravitational radiation. By rewriting the wave equation in this background spacetime, we show how this problem can be posed as an initial value problem in effectively flat, 2D spacetime, allowing for a relatively straightforward numerical calculation of the Green's Function. We will outline this algorithm, the necessary initial conditions, and present results of this approach.


Title:        On the convergence of the normal form transformation in discrete wave turbulence theory for the Charney-Hasegawa-Mima (CHM) equation

Speakers:        Shane Walsh (UCD)

Abstract:

A crucial problem in discrete wave turbulence theory concerns extending the validity of the normal form
transformation beyond the weakly nonlinear limit. The main difficulty is that even if the transformation converges in a given domain around the origin, there is no assurance that all orbits starting in the domain will remain there at all times. Therefore a situation could arise whereby the original system exhibits behavior that is not captured by the normal form system evolution, regardless of the order of the transformation.

We demonstrate this for the CHM equation, Galerkin-truncated to 4 Fourier modes. By calculating the
transformation to 7th order (keeping all resonances up to 8-wave), we perform numerical simulations of both the original and mapped equations to find that the problems occur precisely when the initial conditions lead to precession resonance, a finite-amplitude phenomenon characterized by strong energy transfers across Fourier modes [1].

We use the dynamical systems approach to extend this result to complex wave-turbulent regimes in the CHM equation, leading to a working definition of convergence radius for normal transformations in terms of invariant manifolds.


Coffee and tea will be available in the School Common Room afterwards