Variational Principles for Lagrangian Averaged Fluid Dynamics

The Lagrangian average (LA) of the ideal fluid equations preserves their transport structure. This transport structure is responsible for the Kelvin circulation theorem of the LA flow and, hence, for its convection of potential vorticity and its conservation of helicity. Lagrangian averaging also preserves the Euler-Poincar\'e (EP) variational framework that implies the LA fluid equations. This is expressed in the Lagrangian-averaged Euler-Poincar\'e (LAEP) theorem proven here and illustrated for the Lagrangian average Euler (LAE) equations.Comment: 23 pages, 3 figure

Zebra Mussels Affect Benthic Predator Foraging Success

The introduction of zebra mussels (Dreissena spp.) to North America has resulted in dramatic changes to the complexity of benthic habitats. Changes in habitat complexity may have profound effects on predator-prey interactions in aquatic communities. Increased habitat complexity may affect prey and predator dynamics by reducing encounter rates and foraging success. Zebra mussels form thick contiguous colonies on both hard and soft substrates. While the colonization of substrata by zebra mussels has generally resulted in an increase in both the abundance and diversity of benthic invertebrate communities, it is not well known how these changes affect the foraging efficiencies of predators that prey on benthic invertebrates. We examined the effect of zebra mussels on the foraging success of four benthic predators with diverse prey-detection modalities that commonly forage in soft substrates: slimy sculpin (Cottus cognatus), brown bullhead ( Ameirus nebulosus), log perch (Percina caprodes), and crayfish (Orconectes propinquus). We conducted laboratory experiments to assess the impact of zebra mussels on the foraging success of predators using a variety of prey species. We also examined habitat use by each predator over different time periods. Zebra mussel colonization of soft sediments significantly reduced the foraging efficiencies of all predators. However, the effect was dependent upon prey type. All four predators spent more time in zebra mussel habitat than in either gravel or bare sand. The overall effect of zebra mussels on benthic-feeding fishes is likely to involve a trade-off between the advantages of increased density of some prey types balanced against the reduction in foraging success resulting from potential refugia offered in the complex habitat created by zebra mussels

Explicit Lie-Poisson integration and the Euler equations

We give a wide class of Lie-Poisson systems for which explicit, Lie-Poisson integrators, preserving all Casimirs, can be constructed. The integrators are extremely simple. Examples are the rigid body, a moment truncation, and a new, fast algorithm for the sine-bracket truncation of the 2D Euler equations.Comment: 7 pages, compile with AMSTEX; 2 figures available from autho

Complete integrability versus symmetry

The purpose of this article is to show that on an open and dense set, complete integrability implies the existence of symmetry

Integrable discretizations of some cases of the rigid body dynamics

A heavy top with a fixed point and a rigid body in an ideal fluid are important examples of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(n)=so(n)\ltimes\mathbb R^n$. We give a Lagrangian derivation of the corresponding equations of motion, and introduce discrete time analogs of two integrable cases of these systems: the Lagrange top and the Clebsch case, respectively. The construction of discretizations is based on the discrete time Lagrangian mechanics on Lie groups, accompanied by the discrete time Lagrangian reduction. The resulting explicit maps on $e^*(n)$ are Poisson with respect to the Lie--Poisson bracket, and are also completely integrable. Lax representations of these maps are also found.Comment: arXiv version is already officia

Zeeman doppler imaging of the surface activity and magnetic fields of young solar-type stars

The cyclic magnetic activity of the modern-day Sun is generally considered to be powered by a self-regenerating interface-layer dynamo. However, Zeeman Doppler Imaging of the spots and magnetic fields of active young solar-type stars suggests that a distributed rather than an interface-layer dynamo is present. This paper outlines techniques we have used to map and study the spots and surface magnetic fields of a small sample of young active solar-type stars, the results obtained, and the implications for magnetic field generation in young cool stars

Discrete Variational Optimal Control

This paper develops numerical methods for optimal control of mechanical systems in the Lagrangian setting. It extends the theory of discrete mechanics to enable the solutions of optimal control problems through the discretization of variational principles. The key point is to solve the optimal control problem as a variational integrator of a specially constructed higher-dimensional system. The developed framework applies to systems on tangent bundles, Lie groups, underactuated and nonholonomic systems with symmetries, and can approximate either smooth or discontinuous control inputs. The resulting methods inherit the preservation properties of variational integrators and result in numerically robust and easily implementable algorithms. Several theoretical and a practical examples, e.g. the control of an underwater vehicle, will illustrate the application of the proposed approach.Comment: 30 pages, 6 figure

Geodesic Warps by Conformal Mappings

In recent years there has been considerable interest in methods for diffeomorphic warping of images, with applications e.g.\ in medical imaging and evolutionary biology. The original work generally cited is that of the evolutionary biologist D'Arcy Wentworth Thompson, who demonstrated warps to deform images of one species into another. However, unlike the deformations in modern methods, which are drawn from the full set of diffeomorphism, he deliberately chose lower-dimensional sets of transformations, such as planar conformal mappings. In this paper we study warps of such conformal mappings. The approach is to equip the infinite dimensional manifold of conformal embeddings with a Riemannian metric, and then use the corresponding geodesic equation in order to obtain diffeomorphic warps. After deriving the geodesic equation, a numerical discretisation method is developed. Several examples of geodesic warps are then given. We also show that the equation admits totally geodesic solutions corresponding to scaling and translation, but not to affine transformations

Research on the Geography of Agricultural Change: Redundant or Revitalized?

Future research directions for agricultural geography were the subject of debate in Area in the late 1980s. The subsequent application of political economy ideas undoubtedly revived interest in agricultural research. This paper argues that agricultural geography contains greater diversity than the dominant political economy discourse would suggest. It reviews ‘other’ areas of agricultural research on policy, post-productivism, people, culture and animals, presenting future suggestions for research. They should ensure that agricultural research continues revitalized rather than redundant into the next millennium

Symmetry Reduction by Lifting for Maps

We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincar\'{e} section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques.Comment: laTeX, 31 pages, 5 figure