Fundamental Mathematics Group seminars

Using data to accurately and efficiently model turbulent flows: Data assimilation and parameter recovery

Speaker: Elizabeth Carlson

Date: Wednesday 30 March, 12-1pm

Venue: 39AA04

Abstract: One of the challenges of the accurate simulation of turbulent flows is that initial data is often incomplete. Data assimilation circumvents this issue by continually incorporating the observed data into the model. An emerging approach to data assimilation known as the Azouani-Olson-Titi (AOT) algorithm introduced a feedback control term to the 2D incompressible Navier-Stokes equations (NSE) in order to incorporate sparse measurements. The solution to the AOT algorithm applied to the 2D NSE was proven to converge exponentially to the true solution of the 2D NSE with respect to the given initial data.

In this talk, we present our tests on the robustness, improvements, and implementation of the AOT algorithm, as well as generate new ideas based off these investigations. First, we discuss the application of the AOT algorithm to the 2D NSE with an incorrect parameter and prove it still converges to the correct solution up to an error determined by the error in the parameters. This led to the development of a simple parameter recovery algorithm, whose convergence we recently proved in the setting of the Lorenz equations. It has now been proven by a co-author for the full 2D NSE, presenting new insights into the equation itself. We may also discuss applications to climate models.

Entirely out of character? Dispersive dynamics in the moving frame

Speaker: Daniel Ratliff (Loughborough)

Date: Wednesday 23 October 2019

Abstract: In the evolution of nonlinear waves, localised structures and defects can form and persist, and even stable waves can bifurcate to accommodate these. One way in which their formation can be understood in through the use of the Whitham Modulation equations (WMEs), a dispersionless set of quasilinear PDEs. However, a persistent problem is how to regularise this system via the inclusion of dispersive effects to prevent the emergence of multivalued wave quantities. Surprisingly, it transpires that such features already lurk within the WMEs whenever they are hyperbolic – one merely has to wait long enough in a suitable moving frame. This takes the form of the Korteweg – de Vries (KdV) equation, which is universal in the sense that its coefficients are tied to abstract properties of the original Lagrangian.This leads to a more general question – can the properties of the characteristics be used to infer the resulting dynamics? This is demonstrated to be true, and the connection between established concepts in hyperbolic systems (such as the Hamiltonian-Hopf bifurcation and linear degeneracy) and some well-known nonlinear dispersive equations, such as the Two-Way Boussinesq and modified KdV equations, are made.

Magnetofluid turbulence, broken symmetry and the dynamo problem

Speaker: John V. Shebalin (George Mason University, USA)

Date: Thursday 30 May 2019

Abstract: The problem of understanding if and how a magnetohydrodynamic (MHD) dynamo process could produce the Earth’s magnetic field was first proposed by Larmor in 1919. In 1995, Glatzmaier and Roberts demonstrated through numerical simulation that such an MHD dynamo process existed. There still remained the question of fundamental cause: What was it about MHD that allowed for the creation of a dominant, quasi-stationary geomagnetic dipole field? The magnetofluid in the Earth’s outer core is in a turbulent state and the answer as to fundamental cause appears to lie in the statistical mechanics of ideal MHD turbulence. Statistical theory makes the essential prediction that when a turbulent magnetofluid is in quasi-equilibrium, the magnetic energy in the dipole component is equal to the global magnetic helicity multiplied by the dipole wavenumber. Furthermore, theory predicts that the dipole magnetic field is quasi-stationary. Additional results indicate that a knowledge of the Earth's exterior magnetic dipole field allows for a robust estimate of the magnetic helicity and toroidal dipole field contained in the outer core. Numerical simulations indicate that these results also apply in the case of forced and dissipative, i.e., real MHD turbulence. Here, we describe the theoretical analysis and numerical simulations that lead to these conclusions. We also test an assertion that oblateness may be related to dipole alignment and find that while there is an effect, rotation appears to be far more important. Lastly, we discuss how inertial waves in a rotating magnetofluid might affect dipole alignment through dynamical symmetry breaking, though this is still an open question.

A survey of applications of ray phase-space eikonal methods in linear wave conversion

Speaker: Alain J. Brizard, Saint Michael’s College (USA)

Date: Tuesday 14 May  2019

Abstract: The problem of linear wave conversion involves the exchange of wave action between two waves of different types (i.e., satisfying different linear dispersion relations) as they interact in a nonuniform medium. Since uncoupled waves travel on a distinct dispersion manifold in ray phase space, whose coordinates (x, k) include the position x of a ray and its canonically-conjugate wavevector k, it is convenient to use eikonal methods to study the intersection (in ray phase space) of two rays of different types and construct a scattering matrix describing the transmission and conversion of wave action between the two waves. In particular, we demonstrate the modular nature of these ray phase-space eikonal methods.

Hydrodynamic pilot-waves: A wave-mediated non-linear non-smooth dynamical system that displays spontaneous pattern formation

Speaker: Carlos Rios

Date: Wednesday 27 March 2019

Abstract: In 2005, Yves Couder discovered that a millimetric droplet bouncing on the free surface of a vertically oscillating bath displays a robust droplet-wave association. The droplet does not merge with the underlying bath, yielding a system that resembles a ball bouncing on a trampoline. These droplets can become unstable to lateral perturbations, impacting the surface at a different point at each bounce, i.e. they start to “walk” along the surface. Each droplet impact triggers new surface waves and, in turn, the surface deflection at each impact location influences subsequent bounces. Moreover, waves are reflected of boundaries and generated by neighbouring droplets. Consequently, droplets move in response to a wave field that was created by their past trajectory and is influenced by the surrounding environment. The resulting system is formed by a collection of wave-guided wave sources, which gives rise to elaborate dynamic configurations and unique emerging properties, some of which have analogies to quantum-scale phenomena. In this talk, we will present the problem in detail, introduce mathematical models that capture the behaviour of such systems (including details of free-surface impacts such as moving contact lines) and we will conclude with an experimental demonstration of hydrodynamic pilot-waves.

A study of the stochastic resonance as a periodic random dynamical system

Speaker: Anna Maria Cherubini (Universita' del Salento)

Date: Thursday 14 June 2018

Abstract: We present a framework for random dynamical systems with nonautonomous deterministic forcing and we prove the existence of an attracting random periodic orbit for a class of one-dimensional systems with a time-periodic component. As  a particularly relevant example, we study  a standard model for the stochastic resonance and  we use properties of the attractor to derive an indicator for the resonant regime.

*The work is  in collaboration with J.S.W. Lamb, M. Rasmussen and Y. Sato.