Elena Kartashova

Model of intraseasonal oscillations in earth's atmosphere.

We suggest a way of rationalizing intraseasonal oscillations of Earths atmospheric flow as four meteorologically relevant triads of interacting planetary waves, isolated from the system of all of the rest of the planetary waves. Our model is independent of the topography (mountains, etc.) and gives a natural explanation of intraseasonal oscillations in both the Northern and the Southern Hemispheres. Spherical planetary waves are an example of a wave mesoscopic system obeying discrete resonances that also appears in other areas of physics.

Partitioning of ensembles of weakly interacting dispersing waves in resonators into disjoint classes

Abstract Weakly non-linear interactions of a finite number of waves in a bounded domain are considered. It is evident that the synchronism conditions in this case turn into equations in integers due to boundedness of the domain. A theorem about the partitioning of the set of vectors satisfying synchronism conditions into non-intersecting classes is proved. Cases are found in which for a fixed vector k with integer coordinates there exist no vectors constituting a solution together with it. Examples are given of situations in which the said classes are infinite, finite, or empty. Methods of algebraic number theory have been used.

Exact and quasiresonances in discrete water wave turbulence.

The structure of discrete resonances in water wave turbulence is studied. It is shown that the number of exact 4-wave resonances is huge (hundreds of millions) even in a comparatively small spectral domain when both scale and angle energy transport is taken into account. It is also shown that angle transport can contribute inexplicitly to scale transport. Restrictions for quasiresonance to start are written out. The general approach can be applied directly to mesoscopic systems encountered in condensed matter (quantum dots), medical physics, etc.

Discrete wave turbulence

In this letter we propose discrete wave turbulence (DWT) as a counterpart of classical statistical wave turbulence (SWT). DWT is characterized by resonance clustering, not by the size of clusters, i.e. it includes, but is not reduced to, the study of low-dimensional systems. Clusters with integrable and chaotic dynamics co-exist in different sub-spaces of the k-space. NR-diagrams are introduced, a graphical representation of an arbitrary resonance cluster allowing to reconstruct uniquely dynamical system describing the cluster. DWT is shown to be a novel research field in nonlinear science, with its own methods, achievements and application areas.

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.

Constructively factoring linear partial differential operators in two variables

We study conditions under which a partial differential operator of arbitrary order

On properties of weakly nonlinear wave interactions in resonators

n

Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves

in two variables or ordinary linear differential operator admits a factorization with a first-order factor on the left. The factorization process consists of solving, recursively, systems of linear equations, subject to certain differential compatibility conditions. In the generic case of partial differential operators one does not have to solve a differential equation. In special degenerate cases, such as ordinary differential, the problem is finally reduced to the solution of some Riccati equation(s). The conditions of factorization are given explicitly for second- and, and an outline is given for the higher-order case.

Dynamics of nonlinear resonances in Hamiltonian systems

Abstract Weakly nonlinear wave interactions in resonators are considered. Qualitative differences from the continuous case are found: prevalence of waves not taking part in any interactions at all; dependence of the number of interacting waves on the basin form; existence of basins where interactions are totally impossible; locality of interactions in the spectral space. Consideration of equations defining the resonance surface in finite fields as well as methods of number theory are used. Some aspects of relations between the continuous and discrete cases are discussed.

Laminated wave turbulence: Generic algorithms iii

The influence of an underlying current on 3-wave interactions of capillary water waves is studied. The fact that in irrotational flow resonant 3-wave interactions are not possible can be invalidated by the presence of an underlying current of constant non-zero vorticity. We show that: 1) wave trains in flows with constant non-zero vorticity are possible only for two-dimensional flows; 2) only positive constant vorticities can trigger the appearance of three-wave resonances; 3) the number of positive constant vorticities which do trigger a resonance is countable; 4) the magnitude of a positive constant vorticity triggering a resonance can not be too small.