Chaotic Hamiltonian dynamics of particle's horizontal motion in the atmosphere

Abstract

A non-separable, near-integrable, two degrees-of-freedom (d.o.f.) Hamiltonian system arising in the context of particle's dynamics on a geopotential of the atmosphere is studied. In the unperturbed system, homoclinic orbits to periodic motion in the longitude angle exist, giving rise to several types of homoclinic chaos in the perturbed system; a new type of homoclinic chaos is found, exhibiting chaos which is non-uniformly distributed in the longitude angle along the homoclinic loop while it is topologically uniformly distributed in this angle in a neighborhood of the hyperbolic periodic orbit, i.e. the return map to different values of the longitude angle are topologically conjugate only near the hyperbolic periodic orbit. The concept of colored energy surfaces and its corresponding energy-momentum map are developed as analytical tools for delineating the regions in the four-dimensional phase space where the various types of homoclinic chaos prevail. Geometrical interpretation of the Melnikov analysis is offered for detecting this non-uniformity in angle. Applying the results of the analysis to the planetary atmosphere with an infinite wavelength perturbation of amplitude ϵ, we find that for eastward going particles with initial velocities {(ifu 0, v 0) = (ū, 0)} the chaotic band occupies two narrow (O(ϵ)) bands on both sides of the equator nea latitudes φ = ± arccos(1 − 2 u) . By contrast, for westward going particles the chaotic zone is thicker ( O( ϵ; ) ) and is centered on the equator. Moreover, the dependence of the chaotic zone's thickness on the perturbation frequency is much more sensitive to the value of the initial speed for eastward going particles than for those going westward. Qualitative and quantitative differences between the distribution of the homoclinic chaotic motion in the longitude angle for westward and eastward travelling wave perturbations are predicted and numerically confirmed.