First AGM Meeting

Titles and Abstracts

Aythami Bethencourt De Leon

Arnold stability of a slice model with applications to front formation

We present an atmospheric slice model developed in a paper by Darryl Holm and Colin Cotter which was originally intended, among other things, to help understand the process of frontogensis (the equations in the Euler-Boussinesq case are similar to the ones postulated by Hoskins and Bretherton for front generation). This slice model arises naturally in a Lagrangian framework where the configuration space is a semidirect product Lie group, and it enjoys a Hamiltonian formulation as well (to be more exact, the slice equations in Lagrangian form can be rewritten as Lie-Poisson equations in the dual of a semidirect product Lie algebra). We study Arnold stability conditions in both the incompressible and compressible Euler-Boussinesq cases (which yield some interesting physical restrictions) and mention their possible relation to front formation.

Thomas J. Bridges

Geometry of loop space, Whitham modulation theory, and beyond*

The talk will start with an introduction to classical Whitham modulation theory (WMT), as presented in textbooks. A new viewpoint will also be given. As an example of why WMT is important, the instability of periodic travelling waves will be discussed. An interesting special case is near the stability transition, and a study of this case has led to some new interesting developments. Then a step back will be taken, and WMT looked at afresh. WMT is really all about relative equilibria on the loop space (with periodic orbits and periodic travelling waves a special case). When viewed from this perspective WMT emphasises RE on abelian groups only. Open question: what about WMT for Lagrangians on Lie groups? In principle, just consider Lagrangians on Lie groups, look for RE, and modulate. What are the resulting modulation PDEs? Sounds like the natural setting for this is the Euler-Poincaré framework....

* This is joint work with Daniel Ratliff.

Bin Cheng

Existence of global weak solutions to a hybrid kinetic-MHD model for plasma dynamics*

In plasma physics, it is very popular to study hybrid models. Various such models are introduced in recent literature that capture the nonlinear coupling of a kinetic equation for the energetic but rare particles of one species and magneto-hydrodynamics model that describe the combined effect of charged ions and electrons of the bulk fluid. The model we study is based on the so called "current coupling scheme" and possesses Hamiltonian structure, and hence respects energy conservation. This facilitates our proof of the existence of large data global weak solutions. We also prove that the weak solutions satisfy the physically relevant properties. The proof relies on a single approximation scheme that is carefully devised to preserve the total physical energy. It will be interesting to see if alternatively we regularize the Hamiltonian itself, what type of regularized PDE can arise.

* This is joint work with Endre Süli (Oxford) and Cesare Tronci (Surrey).

Paul Dellar

Lattice Boltzmann methods for models of suspensions of elongated particles

Jeffery's equation describes the rotation of the symmetry axis of a rigid spheroid immersed in a slow, viscous, incompressible Stokes flow with a uniform velocity gradient far from the particle. It forms the basis for modelling dilute suspensions of such particles. Brownian effects are negligible for sufficiently large particles, so one may simply treat the particles' orientations as a material vector field.The evolution equation for this vector field in a compressible fluid coincides with the ideal magnetohydrodynamic (MHD) induction equation,plus additional non-conservative terms involving the the fluid strain tensor and the divergence of the vector field. Both are available from the non-equilibrium parts of the orientational and hydrodynamic distribution functions in a discrete kinetic formulation, so a lattice Boltzmann algorithm for MHD may be adapted to simulate such suspensions.

François Gay-Balmaz

A variational formalism for the nonequilibrium thermodynamics of discrete and continuum systems

We present a Lagrangian variational formulation for nonequilibrium thermodynamics. This formulation extends the Hamilton principle of classical mechanics to include irreversible processes in both discrete and continuum systems. The irreversibility is encoded into a nonlinear nonholonomic constraint given by the expression of entropy production associated to the irreversible processes involved. The introduction of the concept of thermodynamic displacement allows the definition of a corresponding variational constraint. We also present the geometric structures underlying nonequilibrium thermodynamics. We illustrate our theory with both finite and infinite dimensional examples, including mechanical systems with friction, chemical reactions, electric circuits, and reacting fluids.

Claudia Wulff

Exponential estimates of symplectic slow manifolds*

In this talk consider analytic Hamiltonian slow-fast systems with finitely many slow degrees of freedom. We allow for infinitely many fast degrees of freedom. We present a result on the existence of an almost invariant symplectic slow manifold for which the error field is exponentially small. The method we use is motivated by a paper of MacKay from 2004. The method does not notice resonances, and therefore we do not pose any restrictions on the motion normal to the slow manifold other than it being fast and analytic. We also present a stability result and obtain a generalization of a result of Gelfreich and Lerman on an invariant slow manifold to (finitely) many fast degrees of freedom.

* This is joint work with Kristian Kristiansen (Copenhagen)