The free rigid body dynamics: generalized versus classic

In this paper we analyze the normal forms of a general quadratic Hamiltonian system defined on the dual of the Lie algebra $\mathfrak{o}(K)$ of real $K$ - skew - symmetric matrices, where $K$ is an arbitrary $3\times 3$ real symmetric matrix. A consequence of the main results is that any first-order autonomous three-dimensional differential equation possessing two independent quadratic constants of motion which admits a positive/negative definite linear combination, is affinely equivalent to the classical "relaxed" free rigid body dynamics with linear controls.Comment: 12 page

Two-component {CH} system: Inverse Scattering, Peakons and Geometry

An inverse scattering transform method corresponding to a Riemann-Hilbert problem is formulated for CH2, the two-component generalization of the Camassa-Holm (CH) equation. As an illustration of the method, the multi - soliton solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data for CH2.Comment: 22 pages, 3 figures, draft, please send comment

Elliptic instability in the Lagrangian-averaged Euler-Boussinesq-alpha equations

We examine the effects of turbulence on elliptic instability of rotating stratified incompressible flows, in the context of the Lagragian-averaged Euler-Boussinesq-alpha, or \laeba, model of turbulence. We find that the \laeba model alters the instability in a variety of ways for fixed Rossby number and Brunt-V\"ais\"al\"a frequency. First, it alters the location of the instability domains in the $(\gamma,\cos\theta)-$parameter plane, where $\theta$ is the angle of incidence the Kelvin wave makes with the axis of rotation and $\gamma$ is the eccentricity of the elliptic flow, as well as the size of the associated Lyapunov exponent. Second, the model shrinks the width of one instability band while simultaneously increasing another. Third, the model introduces bands of unstable eccentric flows when the Kelvin wave is two-dimensional. We introduce two similarity variables--one is a ratio of the Brunt-V\"ais\"al\"a frequency to the model parameter $\Upsilon_0 = 1+\alpha^2\beta^2$, and the other is the ratio of the adjusted inverse Rossby number to the same model parameter. Here, $\alpha$ is the turbulence correlation length, and $\beta$ is the Kelvin wave number. We show that by adjusting the Rossby number and Brunt-V\"ais\"al\"a frequency so that the similarity variables remain constant for a given value of $\Upsilon_0$, turbulence has little effect on elliptic instability for small eccentricities $(\gamma \ll 1)$. For moderate and large eccentricities, however, we see drastic changes of the unstable Arnold tongues due to the \laeba model.Comment: 23 pages (sigle spaced w/figure at the end), 9 figures--coarse quality, accepted by Phys. Fluid

K\'arm\'an--Howarth Theorem for the Lagrangian averaged Navier-Stokes alpha model

The K\'arm\'an--Howarth theorem is derived for the Lagrangian averaged Navier-Stokes alpha (LANS$-\alpha$) model of turbulence. Thus, the LANS$-\alpha$ model's preservation of the fundamental transport structure of the Navier-Stokes equations also includes preservation of the transport relations for the velocity autocorrelation functions. This result implies that the alpha-filtering in the LANS$-\alpha$ model of turbulence does not suppress the intermittency of its solutions at separation distances large compared to alpha.Comment: 11 pages, no figures. Includes an important remark by G. L. Eyink in the conclusion

Variational Principles for Stochastic Fluid Dynamics

This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The Legendre transform of the Lagrangian formulation of these SPDEs yields their Lie-Poisson Hamiltonian form. The paper proceeds by: taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their It\^o representation; and then investigating the properties of these stochastic fluid models in comparison with each other, and with the corresponding deterministic fluid models. The circulation properties of the stochastic Stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models. As with deterministic ideal flows, motion along the stochastic Stratonovich paths also preserves the helicity of the vortex field lines in incompressible stochastic flows. However, these Stratonovich properties are not apparent in the equivalent It\^o representation, because they are disguised by the quadratic covariation drift term arising in the Stratonovich to It\^o transformation. This term is a geometric generalisation of the quadratic covariation drift term already found for scalar densities in Stratonovich's famous 1966 paper. The paper also derives motion equations for two examples of stochastic geophysical fluid dynamics (SGFD); namely, the Euler-Boussinesq and quasigeostropic approximations.Comment: 19 pages, no figures, 2nd version. To appear in Proc Roy Soc A. Comments to author are still welcome