# First AGM Meeting

**First AGM Meeting on Geometric Fluid Dynamics 2014-2015**

**16 May 2014, University of Surrey**

**[IMPORTANT! -- All non-speaker attendees are requested to register by writing to Cesare Tronci]**

This one-day meeting will cover various disciplines related to geometric fluid dynamics. Specific topics will include:

-- geometry of fluid flows,

-- geometric integration,

-- multiscale fluid models.

In addition, a mini-session on "Other Topics in Geometric Mechanics" will be held in the form of short PhD talks.

__Timetable:__

09.30 -- 10.15 Ian Roulstone (Surrey)

10.15 -- 11.00 Colin Cotter (Imperial)

11.00 -- 11.30 coffee break

11.30 -- 12.15 Darryl Holm (Imperial)

12.15 -- 13.00 Tom Bridges (Surrey)

13.00 -- 14.15 lunch

14.15 -- 14.45 François Gay-Balmaz (CNRS/ENS Paris)

14.45 -- 15.15 Claudia Wulff (Surrey)

15.15 -- 15.45 coffee break

15.45 -- 16.15 Dmitry Pavlov (Imperial)

16.15 -- 16.45 Hamid Alemi Ardakani (Surrey)

16.45 -- 17.00 coffee break

17.00 -- 17.30 Mini-session on *Other Topics in Geometric Mechanics*

Alexis Arnaudon (Imperial) & Esther Bonet Luz (Surrey)

17.30 -- 18.00 Cesare Tronci (Surrey)

*The meeting will take place in room 24AA04, AA building (more info *here*)*

__Titles and Abstracts__

**Hamid Alemi Ardakani**

*Sloshing dynamics with the Hamiltonian particle-mesh method*

The Hamiltonian Particle-Mesh (HPM) method of [2] is used to develop a symplectic integrator in Eulerian-Lagrangian coordinate systems for the problem of dynamic coupling between shallow-water sloshing and horizontal vehicle motion. A simple and fast numerical algorithm with excellent energy conservation over long times, based on the Störmer-Verlet method is implemented. Numerical simulations of the coupled dynamics are presented and compared to the results of [1] where the coupled fluid-vehicle problem is analysed in the pure Lagrangian Particle-Path (LPP) setting.

[1] H. Alemi Ardakani & T.J. Bridges. Dynamic coupling between shallow-water sloshing and horizontal vehicle motion, European Journal of Applied Mathematics 21 479–517 (2010).

[2] J. Frank, G. Gottwald & S. Reich. A Hamiltonian particle-mesh method for the rotating shallow-water equations, Meshfree Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, Springer 26 131–142 (2002).

**Thomas J. Bridges**

*Disintegration of relative equilibria and the emergence of modulation*

Relative equilibria are one of the most pervasive and interesting class of solutions in geometric mechanics. One is normally interested in non-degenerate RE. However, the degeneration of relative equilbria can lead to the generation of homoclinic orbits. One way to show this is to modulate relative equilbria. The modulation becomes more interesting when the relative equilbria are solutions of PDEs, in which case modulation can lead to the emergence of model PDEs with dispersion. It will be shown that this is the natural way that the KdV equation emergence.

TJB [2013] A universal form for the emergence of the Korteweg-de Vries equation, Proc. R. Soc. Lond. A 469 20120707.

**Colin J. Cotter**

*Jet particle solutions for an EPDiff equation*

I will present families of exact solutions of a regularised incompressible Euler equation that has unique C-infinity solutions for all time. These solutions are completely characterised by the positions of a finite set of Lagrangian particles, the first k derivatives of the velocity field at those positions, together with conjugate momenta for all of these coordinates. I will explain how to obtain the dynamics of these coordinates using a Clebsch variational principle, and then illustrate their behaviour through some numerical examples. A key aspect of the dynamics is that collisions between particles occur in the long time limit. Asymptotically these collisions produce (k+1)-jet particle solutions of the PDE out of k jet particle solutions. This provides a cartoon picture of a cascade to small scales.

This is joint work with Henry Jacobs (Imperial), Darryl Holm (Imperial) and David Meier (Brunel).

**François Gay-Balmaz**

*Flexible tubes conducting fluid: geometric nonlinear theory, stability and dynamics*

We derive a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. The theory also incorporates the change of the cross-section available to the fluid motion during the dynamics. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. We analyze the fully nonlinear behavior of the model when the axis of the tube remains straight. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the effects arising from the dynamical change of the cross-section. Finally, we derive and analyse several analytical, fully nonlinear solutions of traveling wave type in two dimensions.

**Darryl D. Holm**

*Multiscale turbulence models based on convected fluid microstructure*

The Euler-Poincaré approach to complex fluids is used to derive multiscale equations for computationally modelling Euler flows as a basis for modelling turbulence. The model is based on a *kinematic sweeping ansatz* (KSA) which assumes that the mean fluid flow serves as a Lagrangian frame of motion for the fluctuation dynamics. Thus, we regard the motion of a fluid parcel on the computationally resolvable length scales as a moving Lagrange coordinate for the fluctuating (zero-mean) motion of fluid parcels at the unresolved scales. Even in the simplest 2-scale version on which we concentrate here, the contributions of the fluctuating motion under the KSA to the mean motion yields a system of equations that extends known results and appears to be suitable for modelling nonlinear backscatter (energy transfer from smaller to larger scales) in turbulence using multiscale methods.

**Dmitry Pavlov**

*A matrix-based framework to structure-preserving discretization of continuum theories*

I will describe a framework for constructing discrete models of infinite-dimensional systems which preserve underlying geometry. These models lead to numerical methods that capture the dynamics of the system without energy or momenta loss and preserve momentum maps in discrete realm. This work starts with developing a matrix-based exterior calculus. This calculus extends the classical Discrete Exterior Calculus providing us with the notions of discrete Lie derivative, interior product etc., while preserving many properties of their continuous counterparts. This matrix-based exterior calculus has been used to create structure-preserving discretizations of various systems, such as fluid dynamics, magnetohydrodynamics, complex fluids etc. I will show how it can be used to construct a variational integrator for the Euler and EPDiff equations. I will also describe how these methods can be extended to create a new model of discrete differential geometry. This approach uses ideas of noncommutative geometry and can lead to a structure-preserving discretization of general relativity.

**Ian Roulstone**

*Differential geometry of the semi-geostrophic and Euler equations*

The role of contact and symplectic geometry of the semi-geostophic (SG) equations, in describing their Legendrian and Hamiltonian properties, will be reviewed. Using the geometry of 2-forms in 4 dimensions and the geometry of 3-forms in 6 dimensions, we show that the incompressible Euler equations in 2 and 3 dimensions admit geometric structures akin to the those present in the SG equations.

**Cesare Tronci**

*Recent results in hybrid kinetic-fluid models for plasmas*

Hybrid kinetic-fluid models govern the interaction of a hot plasma (governed by a kinetic theory) with a fluid bulk (governed by MHD). Different nonlinear hybrid models are reviewed, including the pressure-coupling scheme (PCS) used in modern hybrid simulations. This latter scheme suffers from not conserving total energy. Upon adopting the Vlasov description for the hot component, the conventional PCS model and a Hamiltonian variant are compared. A linear stability analysis shows that spurious instabilities appear at high frequencies in the conventional model. On the other hand, this instability is removed in the Hamiltonian version.

C. Tronci, E. Tassi, E. Camporeale, P.J. Morrison, *Hybrid Vlasov-MHD models: Hamiltonian vs. non-Hamiltonian*, arXiv:1403.2773

**Claudia Wulff**

*Symplectic time-semidiscretizations of semilinear Hamiltonian PDEs*

We consider semilinear Hamiltonian evolution equations, in this talk we specify to the semilinear wave equation. We show that its flow and an A-stable Runge Kutta time discretization of it (e.g. the implicit mid point rule) are smooth in time as maps from an open subset of a sufficiently high rung of a scale of Hilbert spaces into a low rung, and obtain full order convergence results for the time-discretization for sufficiently smooth initial data. Then we prove fractional order convergence results for the trajectory error of the time-semidiscretization in the case of non-smooth initial data, and show that the energy error, for nonsmooth initial data, has much higher order than the trajectory error. Finally we show that time semidiscretizations of analytic data conserve a modified energy up to an exponentially small error.

This is joint work with Marcel Oliver (Jacobs University) and Chris Evans (Surrey).

__Mini-session on Other Topics in Geometric Mechanics__

**Alexis Arnaudon**

*From the Lagrange top to new integrable wave equations*

In integrable systems theory, it is well-known that the classical R-matrix approach, associated with loop algebras, is one of the most powerful tool to prove complete integrability of Hamiltonian dynamical equations. An extension to this approach can be derived by adding a spatial parameter to the dynamical variables and thus obtain infinite dimensional integrable systems. This idea is very close to the recently developed G-Strands equations, where the dynamical variables depend on time and space with value in the Lie algebra of a Lie group G. In this talk I will present an application of the R-matrix theory when G=SO(3). In the case, where only the time is present, it proves the integrability of the Lagrange top system in a unusual way but when the space is added, it generates a new hierarchy of integrable 1+1 wave equations based on the geometry of the Lagrange top.

**Esther Bonet Luz**

*Geometry and symmetry of quantum variational principles*

The geometric setting of quantum variational principles is presented. Schrödinger's dynamics of pure states is obtained as the Euler-Lagrange equation associated to the well-known Dirac-Frenkel variational principle. Euler-Poincaré reduction theory is applied, motivated by the underlying unitary symmetries in quantum mechanics. This theory allows us to formulate new variational principles for various quantum descriptions such as Liouville-Von Neumann equation, Heisenberg dynamics, Dirac's interaction picture and Moyal-Wigner formulation on phase space.

*Contact*: