Cluster Dynamics of Planetary Waves

The dynamics of nonlinear atmospheric planetary waves is determined by a small number of independent wave clusters consisting of a few connected resonant triads. We classified the different types of connections between neighboring triads that determine the general dynamics of a cluster. Each connection type corresponds to substantially different scenarios of energy flux among the modes. The general approach can be applied directly to various mesoscopic systems with 3-mode interactions, encountered in hydrodynamics, astronomy, plasma physics, chemistry, medicine, etc.Comment: 6 pages, 3 figs, EPL, publishe

Laminated Wave Turbulence: Generic Algorithms II

The model of laminated wave turbulence puts forth a novel computational problem - construction of fast algorithms for finding exact solutions of Diophantine equations in integers of order $10^{12}$ and more. The equations to be solved in integers are resonant conditions for nonlinearly interacting waves and their form is defined by the wave dispersion. It is established that for the most common dispersion as an arbitrary function of a wave-vector length two different generic algorithms are necessary: (1) one-class-case algorithm for waves interacting through scales, and (2) two-class-case algorithm for waves interacting through phases. In our previous paper we described the one-class-case generic algorithm and in our present paper we present the two-class-case generic algorithm.Comment: to appear in J. "Communications in Computational Physics" (2006

A Model of Intra-seasonal Oscillations in the Earth atmosphere

We suggest a way of rationalizing an intra-seasonal oscillations (IOs) of the Earth atmospheric flow as four meteorological relevant triads of interacting planetary waves, isolated from the system of all the rest planetary waves. Our model is independent of the topography (mountains, etc.) and gives a natural explanation of IOs both in the North and South Hemispheres. Spherical planetary waves are an example of a wave mesoscopic system obeying discrete resonances that also appears in other areas of physics.Comment: 4 pages, 2 figs, Submitted to PR

Hierarchy of general invariants for bivariate LPDOs

We study invariants under gauge transformations of linear partial differential operators on two variables. Using results of BK-factorization, we construct hierarchy of general invariants for operators of an arbitrary order. Properties of general invariants are studied and some examples are presented. We also show that classical Laplace invariants correspond to some particular cases of general invariants.Comment: to appear in J. "Theor.Math.Phys." in May 200

Quadratic invariants for discrete clusters of weakly interacting waves

We consider discrete clusters of quasi-resonant triads arising from a Hamiltonian three-wave equation. A cluster consists of N modes forming a total of M connected triads. We investigate the problem of constructing a functionally independent set of quadratic constants of motion. We show that this problem is equivalent to an underlying basic linear problem, consisting of finding the null space of a rectangular M × N matrix with entries 1, −1 and 0. In particular, we prove that the number of independent quadratic invariants is equal to J ≡ N − M* ≥ N − M, where M* is the number of linearly independent rows in Thus, the problem of finding all independent quadratic invariants is reduced to a linear algebra problem in the Hamiltonian case. We establish that the properties of the quadratic invariants (e.g., locality) are related to the topological properties of the clusters (e.g., types of linkage). To do so, we formulate an algorithm for decomposing large clusters into smaller ones and show how various invariants are related to certain parts of a cluster, including the basic structures leading to M* < M. We illustrate our findings by presenting examples from the Charney–Hasegawa–Mima wave model, and by showing a classification of small (up to three-triad) clusters

Fourier analysis of wave turbulence in a thin elastic plate

The spatio-temporal dynamics of the deformation of a vibrated plate is measured by a high speed Fourier transform profilometry technique. The space-time Fourier spectrum is analyzed. It displays a behavior consistent with the premises of the Weak Turbulence theory. A isotropic continuous spectrum of waves is excited with a non linear dispersion relation slightly shifted from the linear dispersion relation. The spectral width of the dispersion relation is also measured. The non linearity of this system is weak as expected from the theory. Finite size effects are discussed. Despite a qualitative agreement with the theory, a quantitative mismatch is observed which origin may be due to the dissipation that ultimately absorbs the energy flux of the Kolmogorov-Zakharov casade.Comment: accepted for publication in European Physical Journal B see http://www.epj.or