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We derive a new completely integrable dispersive shallow water equation that is biHamiltonian and thus possesses an infinite number of conservation laws in invo-lution. The equation is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler's equations in the shallow water regime. The soliton solution for this equation has a limiting… (More)

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… (More)

We show here the global, in time, regularity of the three dimensional viscous Camassa–Holm (Navier–Stokes-alpha) (NS-a) equations. We also provide estimates , in terms of the physical parameters of the equations, for the Hausdorff and fractal dimensions of their global attractor. In analogy with the Kolmogorov theory of turbulence, we define a small spatial… (More)

In this paper we introduce and study a new model for three-dimensional turbulence, the Leray-α model. This model is inspired by the Lagrangian averaged Navier–Stokes-α model of turbulence (also known Navier–Stokes-α model or the viscous Camassa– Holm equations). As in the case of the Lagrangian averaged Navier–Stokes-α model, the Leray-α model compares… (More)

- Darryl D. Holm, Martin F. Staley
- SIAM J. Applied Dynamical Systems
- 2003

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… (More)

- H R Dullin, G A Gottwald, D D Holm
- Physical review letters
- 2001

We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic… (More)

Based on recent advances in the theory of Euler–Poincaré (EP) equations with ad-vected parameters and using the methods of Hamilton's principle asymptotics and averaged Lagrangians, we propose a new class of models for ideal incompressible fluids in three dimensions, including stratification and rotation for GFD applications. In these models, the amplitude… (More)

- Darryl D. Holm
- 2000

Lagrangian reduction by stages is used to derive the Euler-Poincaré equations for the nondis-sipative coupled motion and micromotion of complex fluids. We mainly treat perfect complex fluids (PCFs) whose order parameters are continuous material variables. These order parameters may be regarded geometrically either as objects in a vector space, or as coset… (More)

We review the properties of the nonlinearly dispersive Navier-Stokes-alpha (NS−α) model of incompressible fluid turbulence – also called the viscous Camassa-Holm equations in the literature. We first re-derive the NS−α model by filtering the velocity of the fluid loop in Kelvin's circulation theorem for the Navier-Stokes equations. Then we show that this… (More)