The Euler-Poincaré Equations in Geophysical Fluid Dynamics

Recent theoretical work has developed the Hamilton's-principle analog of Lie-Poisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: 1. Euler-Poincaré equations (the Lagrangian analog of Lie-Poisson Hamiltonian equations) are derived for a parameter dependent Lagrangian from a general variational principle of Lagrange d'Alembert type in which variations are constrained; 2. an abstract Kelvin-Noether theorem is derived for such systems. By imposing suitable constraints on the variations and by using invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we cast several standard Eulerian models of geophysical fluid dynamics (GFD) at various levels of approximation into Euler-Poincaré form and discuss their corresponding Kelvin-Noether theorems and potential vorticity conservation laws. The various levels of GFD approximation are related by substituting a sequence of velocity decompositions and asymptotic expansions into Hamilton's principle for the Euler equations of a rotating stratified ideal incompressible fluid. We emphasize that the shared properties of this sequence of approximate ideal GFD models follow directly from their Euler-Poincaré formulations. New modifications of the Euler-Boussinesq equations and primitive equations are also proposed in which nonlinear dispersion adaptively filters high wavenumbers and thereby enhances stability and regularity without compromising either low wavenumber behavior or geophysical balances

The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories

We study Euler–Poincaré systems (i.e., the Lagrangian analogue of Lie–Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d'Alembert type. Then we derive an abstract Kelvin–Noether theorem for these equations. We also explore their relation with the theory of Lie–Poisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler–Poincaré system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler–Poincaré systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional Camassa–Holm equations, which have many potentially interesting analytical properties. These equations are Euler–Poincaré equations for geodesics on diffeomorphism groups (in the sense of the Arnold program) but where the metric is H^1 rather thanL^2

A connection between the Camassa-Holm equations and turbulent flows in channels and pipes

In this paper we discuss recent progress in using the Camassa-Holm equations to model turbulent flows. The Camassa-Holm equations, given their special geometric and physical properties, appear particularly well suited for studying turbulent flows. We identify the steady solution of the Camassa-Holm equation with the mean flow of the Reynolds equation and compare the results with empirical data for turbulent flows in channels and pipes. The data suggests that the constant $\alpha$ version of the Camassa-Holm equations, derived under the assumptions that the fluctuation statistics are isotropic and homogeneous, holds to order $\alpha$ distance from the boundaries. Near a boundary, these assumptions are no longer valid and the length scale $\alpha$ is seen to depend on the distance to the nearest wall. Thus, a turbulent flow is divided into two regions: the constant $\alpha$ region away from boundaries, and the near wall region. In the near wall region, Reynolds number scaling conditions imply that $\alpha$ decreases as Reynolds number increases. Away from boundaries, these scaling conditions imply $\alpha$ is independent of Reynolds number. Given the agreement with empirical and numerical data, our current work indicates that the Camassa-Holm equations provide a promising theoretical framework from which to understand some turbulent flows.Comment: tex file, 29 pages, 4 figures, Physics of Fluids (in press

Lagrangian Reduction, the Euler--Poincar\'{e} Equations, and Semidirect Products

There is a well developed and useful theory of Hamiltonian reduction for semidirect products, which applies to examples such as the heavy top, compressible fluids and MHD, which are governed by Lie-Poisson type equations. In this paper we study the Lagrangian analogue of this process and link it with the general theory of Lagrangian reduction; that is the reduction of variational principles. These reduced variational principles are interesting in their own right since they involve constraints on the allowed variations, analogous to what one finds in the theory of nonholonomic systems with the Lagrange d'Alembert principle. In addition, the abstract theorems about circulation, what we call the Kelvin-Noether theorem, are given.Comment: To appear in the AMS Arnold Volume II, LATeX2e 30 pages, no figure

Emergent singular solutions of non-local density-magnetization equations in one dimension

We investigate the emergence of singular solutions in a non-local model for a magnetic system. We study a modified Gilbert-type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the non-local effects. We pass to a coupled density-magnetization model and perform a linear stability analysis, noting the effect of the length scales of non-locality on the system's stability properties. We carry out numerical simulations of the coupled system and find that singular solutions emerge from smooth initial data. The singular solutions represent a collection of interacting particles (clumpons). By restricting ourselves to the two-clumpon case, we are reduced to a two-dimensional dynamical system that is readily analyzed, and thus we classify the different clumpon interactions possible.Comment: 19 pages, 13 figures. Submitted to Phys. Rev.

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

Averaged Template Matching Equations

By exploiting an analogy with averaging procedures in fluid dynamics, we present a set of averaged template matching equations. These equations are analogs of the exact template matching equations that retain all the geometric properties associated with the diffeomorphismgrou p, and which are expected to average out small scale features and so should, as in hydrodynamics, be more computationally efficient for resolving the larger scale features. Froma geometric point of view, the new equations may be viewed as coming from a change in norm that is used to measure the distance between images. The results in this paper represent first steps in a longer termpro gram: what is here is only for binary images and an algorithm for numerical computation is not yet operational. Some suggestions for further steps to develop the results given in this paper are suggested