Geometry, Dynamics and Fluids

Geometrical Fluid Mechanics
Darryl Holm (Imperial College)

When Hamilton's principle is formulated on a manifold, two classes of momentum maps may arise from the two possible actions -- left and right action -- of Lie groups on the manifold. These ideas apply, for example, to constrained Euler-Poincare variational principles for continuum dynamics. In that case, the manifold consists of the Lie group of diffeomorphisms (smooth invertible maps whose inverses are smooth), plus the set of vector spaces on which the Lie group acts. For fluids and other continuum dynamics, these additional vector spaces contain the properties such as mass and heat that are carried by the material when it deforms by the smooth maps.

The familiar Clebsch momentum map for continua arises from the cotangent lift of the right action, and the more exotic singular momentum maps arise from the cotangent lift of the left action. These actions yield conservation laws when they are symmetries of the Lagrangian. For example, the right action is a symmetry which leaves the Eulerian fluid velocity invariant under particle relabeling. The right action also generates the motion of the properties that are carried along with the fluid particles. The Eulerian fluid velocity is not invariant under the left action. However, this is no cause for regret. Coadjoint left action generates evolution of exotic singular, or weak solutions. These solutions are embedded manifolds on which the momentum is supported as a distribution.

The method for deriving these momentum maps is very easy to apply in Hamilton's principle and lots of examples will be discussed. For background information see:

D. D. Holm and J. E. Marsden, Momentum Maps and Measure-Valued Solutions (Peakons, Filaments and Sheets) for the EPDiff Equation. In The Breadth of Symplectic and Poisson Geometry, A Festshrift for Alan Weinstein, 203-235, Progr. Math. 232, J.E. Marsden and T.S. Ratiu (Editors), Birkhäuser Boston, Boston, MA, 2004. Also at http://arxiv.org/abs/nlin.CD/0312048

Dynamics from Geometry
Mark Roberts (University of Surrey)

Many dynamical systems arising from physical systems respect some kind of geometrical structure on their phase spaces. These structures strongly influence the behaviour of the systems, including stability, bifurcations and more complex dynamics. These lectures will provide an overview of the types of geometry that may occur, the relationships between these, and the influence they have on dynamics. Topics to be covered should include:

• classical mechanics, Hamiltonian systems, symplectic geometry;
• symmetry, conserved quantities, phase space reduction;
• Poisson geometry and dynamics;
• stability of Poisson equilibria and Hamiltonian relative equilibria;
• non-holonomic and controlled systems, the geometry of distributions;
• almost-Poisson systems and more exotic 'bracket dynamics'.

Kähler Geometry in Fluids
Ian Roulstone (University of Surrey)

From slowly-evolving large-scale fluid flows, such as we observe in the atmosphere and oceans, to rapidly changing and turbulent flows, fluid mechanics is believed to be described accurately by the classical Navier--Stokes-based equations of motion. Detailed computations of the three-dimensional incompressible Navier--Stokes equations vividly illustrate the fact that vorticity has a tendency to accumulate on quasi one-dimensional tubes or filaments and on quasi two-dimensional sheets. On larger scales (such as in the atmosphere and oceans) and in the asymptotic regimes that are most relevant for weather and climate forecasting, it can be shown that the solutions of the fluid equations stay close over finite, but useful, time intervals to the solutions of much simpler dynamical systems. These approximate models seek to describe flows in which there is a dominant balance between the Coriolis, buoyancy and pressure-gradient forces on fluid particles, which can be described very succinctly using vortex dynamics.

Recent research (Roubtsov and Roulstone (2001); McIntyre and Roulstone (2002); Roulstone et al. (2005)) suggests that ideas from Kähler geometry may be important in understanding the principles that govern the vortex dynamics of both the incompressible Navier--Stokes equations and the equations that govern those regimes most important to weather and climate. In these lectures, we shall first explain how the complex manifold structures emerge from the underlying partial differential equations, and then describe some of the implications for the fluid mechanics. We shall exploit ideas that will be presented in the other lectures on this course.

M.E. McIntyre and I. Roulstone (2002), Are there higher-accuracy analogues of semi-geostrophic theory? In Large-scale atmosphere---ocean dynamics, Vol. II: Geometric methods and models. J. Norbury and I. Roulstone (eds.), Cambridge: University Press.

V.N. Roubtsov and I. Roulstone (2001), Holomorphic structures in hydrodynamical models of nearly geostrophic flow. Proc. R. Soc. Lond., A 457 , 1519-1531.

I. Roulstone, B. Banos, J.D. Gibbon and V.N. Roubtsov (2005), Kähler geometry and the Navier-Stokes Equations. Arxiv: http//arxiv.org/abs/nlin.SI/0509023

Duality and Geometry in Mechanics
Mike Sewell (University of Reading)

This course will be based on the article Some Applications of Transformation Theory in Mechanics by the lecturer in Volume 2 of Large-Scale Atmosphere-Ocean Dynamics edited by J. Norbury and I. Roulstone and published by Cambridge University Press in 2002. The lectures will be selected from the following topics to be found in it.

Many aspects of transformation theory are reviewed, including Legendre duality and other types, lift transformations and canonical transformations. An example is the singularities which arise from convexifications of multivalued Legendre dual functions, such as the swallowtail, where a typical singular surface is reminiscent of a weather front. The review draws upon earlier work in, for example, plasticity theory, gas dynamics, shallow water theory, catastrophe theory, hamiltonian mass-point mechanics, and the theory of maximum and minimum principles. There is an intimate relation between lift transformations and hamiltonian structures. Properties of constitutive surfaces whose presence underlies gas dynamics and shallow water theory are indicated. Properties of semi-geostrophic central orbit theory are described.

Further information may be found in the book entitled Maximum and Minimum Principles by M.J.Sewell, published by Cambridge University Press in 1987.