Geometry, Mechanics and Fluids

Symmetric Hamiltonian systems: stability

The simplest invariant sets of dynamical systems with symmetry are equilibria and periodic orbits, and orbits which become equilibria or periodic after symmetry reduction - the so-called relative equilibria (REs) and relative periodic orbits (RPOs). The stability theory for REs and RPOs is well understood for general dynamical systems with symmetry, but not for Hamiltonian systems, especially when the symmetry group in not compact. We have explored the interpretation of 'stability' in this context and have developed a theory for relative equilibria which generalises the Energy-Casimir and Energy-Momentum methods. The ideas and results have been illustrated by applications to models of the dynamics of rigid bodies in fluids.

Kähler Geometry and Fluid Mechanics

From slowly-evolving large-scale fluid flows, such as those observed 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 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. The use of different sets of dependent and independent variables in geophysical models of cyclones and fronts, has facilitated some remarkable simplifications of otherwise hopelessly difficult nonlinear problems. Quaternionic and hyper-Kähler structures emerge in models of nearly geostrophic flows in atmosphere and ocean dynamics, and it has also been shown that the three-dimensional Euler equations has a quaternionic structure in the dependent variables. Our current research is focused on Kähler, and generalised Calabi-Yau, structures in the incompressible Navier--Stokes equations in two and three dimensions.

Astrodynamics

Geometric methods are currently being used to model and analyse the dynamics and control of spacecraft in collaboration with the Astrodynamics group of the Surrey Space Centre. Projects include:

• Geometric studies of the singularities of attitude control by momentum exchange devices.
• New approaches to the design of attitude control algorithms.
• Trajectory design for solar-sail spacecraft.
• The dynamics and control of spacecraft with tethers.

  We also coordinate the European Research Training Network 'Astronet', the aims of which include the use of natural dynamics to optimize trajectories and control and thereby minimize fuel use and extend mission ranges.

Criticality in Shallow Water Hydrodynamics

Criticality, uniform flows and bulk quantities such as mass flux, total head and the momentum flux are at the heart of the subject of open-channel hydraulics. However, attempts to generalize criticality to non-trivial flows and unsteady flows have been largely unsuccessful. Recent research at Surrey has led to a new characterization of criticality which generalizes easily to nontrivial flows. The idea is that criticality is all about degeneracy of relative equilibria (RE). A surprising variety of well-known flows are in fact RE. An example is the classic Stokes waves in shallow water. A new nonlinear theory for degenerate RE has been developed and shows that criticality creates solitary waves, with the nature of the solitary wave determined by the nature of the underlying RE. An application to shallow water hydrodynamics shows that "secondary criticality" of Stokes waves signals a bifurcation to a new class of steady dark solitary waves which are biasymptotic to a Stokes wave with a phase jump in between, and synchronized with the Stokes wave. Other examples include criticality of two-layer flow with differing densities. For steady criticality the theory uses the Hamiltonian formulation of inviscid fluids, whereas for unsteady criticality the multi-symplectic Hamiltonian formulation is the basis for the theory.

Wave Interactions

Research at Surrey has been looking at new ways of studying the interaction between waves, particularly the stability of such interactions. An example is the stability problem for short-crested Stokes waves. One observes that an understanding of the linear stability of short-crested waves (SCWs) is closely associated with an understanding of the stability of the oblique non-resonant interaction between two waves. The proposed approach is to embed the SCWs in a six-parameter family of oblique non-resonant interactions. A variational framework is developed for the existence and stability of this general two-wave interaction. It is argued that the resonant SCW limit makes sense a posteriori, and leads to a new stability theory for both weakly-nonlinear and finite-amplitude SCWs. Surprisingly, even in the weakly nonlinear case one finds new results. The theory is developed in some generality for multi-symplectic Hamiltonian PDEs. This theory involves the interaction between two waves. Current research is studying the 3-wave, 4-wave and higher interactions.

Lagrangian Fluid Dynamics

Lagrangian fluid dynamics (LFD), where the dynamics of fluids is formulated in terms of trajectories of fluid parcels, presents an alternative to the more common Eulerian description and has its own merits and advantages. LFD is getting increasing attention in fluid dynamics, oceanography and related areas as Lagrangian codes and experimental techniques are developed with the Lagrangian point of view as the guide. Some topics of interest at Surrey are: Lagrangian versus Eulerian instability; the Lagrangian particle path formulation of water waves and its implications; LFD and numerical weather prediction; symmetry and symplecticity in LFD; Lagrangian oceanography; the geometry of Lagrangian drift.

Symmetric Hamiltonian systems: bifurcation theory and numerics

For the analysis of the long-time behaviour of mechanical systems with symmetry it is crucial to understand the bifurcations of REs and RPOs as internal parameters such as energy and other conserved quantities are varied. Whereas the theory of generic symmetry breaking bifurcations of such invariant sets is well-developed for general (non-Hamiltonian) systems, there are many fewer results on the corresponding theory for symmetric mechanical systems. This is due to the various conservation laws of mechanical systems with symmetry which change the generic behaviour of a dynamical system drastically and therefore have to be taken into account. So far a systematic numerical bifurcation analysis only exists for equilibria and periodic orbits of non-symmetric systems. We aim at the parallel development of theoretical and numerical methods for symmetry breaking bifurcations of simple invariant sets of symmetric mechanical systems using the code SYMPERCON. The results will be applied to various examples of mechanical systems from the areas mentioned above.

Data Assimilation in Numerical Weather Prediction

Data assimilation for weather forecasting involves incorporating observations into a numerical weather prediction model to produce the best estimate of the state of the atmosphere for the next forecast. Observations alone are insufficient to determine the state of the atmosphere because many regions of the atmosphere cannot be observed directly, and therefore data assimilation schemes must be designed to fill the data gaps using other information. Typically, such information is contained in our knowledge of covariances between model variables, which may be deduced either statistically or via the equations of motion and thermodynamics. At Surrey, we are conducting research into the use of variational methods to find optimal fits between dynamical models and observations, and in particular we are interested in the Hamiltonian properties of such schemes. Geometric features of Hamiltonian systems imply that there are important constraints that data assimilation schemes should respect, and we are currently studying these issues in systems with stochastic forcing.

Analysis of Perturbed Hamiltonian Systems

Autonomous Hamiltonian systems with one degree of freedom are completely understood, mainly because their solution curves in phase space are Hamiltonian contours. When the Hamiltonian depends explicitly on time, the dynamics becomes far more complicated, and a general analysis of the motion does not exist. One outstanding question in the theory of dynamical systems is the following: does a periodic or quasiperiodic solution of a dissipative system persist in the limit as dissipation goes to zero? For example in celestial mechanics, it is possible that in the process of the creation of solar systems similar to our own, some special orbits self-select because they are in a sense more attracting than others. Another fundamental problem of dissipative systems is that of understanding all their attractors and associated basins of attraction.