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The non-linear system of Rossby waves or plasma drift waves is known to have a unique adiabatic-like extra invariant in addition to the energy and enstrophy. This invariant is physically significant because its presence implies the generation of zonal flow. The latter is known to slow down the anomalous transport of temperature and particles in nuclear fusion with magnetic confinement. However, the derivation of the extra invariant — unlike the energy and enstrophy — is based on the continuum of resonances, while in numerical simulations there are only finite number of resonances. We show that precisely the same invariant takes place in the Galerkin approximations (even of low order, with a few ODEs). To show this we make variation of boundary conditions, when the solution is periodic in different directions. We also simplify the derivation of the extra conservation.

The purpose of this research is to investigate the formation of zonal flows that can lead to the enhanced confinement of plasma in tokamaks. We show that zonal flows can be effectively formed by resonance triad interactions in the process of the inverse cascade. We discuss what energy sources are more effective for the formation of zonal flows.

It was found out in 1991 that the Fourier space dynamics of Rossby waves possesses an extra positive-definite quadratic invariant , in addition to the energy and enstrophy. This invariant is similar to the adiabatic invariants in the theory of dynamical systems. For many years, it was unclear if this invariant—known only in the Fourier representation—is physically meaningful at all, and if it is, in what sense it is conserved. Does the extra conservation hold only for a class of solutions satisfying certain constraints (like the conservation in the Kadomtsev–Petviashvili equation)? The extra invariant is especially important because this invariant (provided it is meaningful) has been connected to the formation of zonal jets (like Jupiter’s stripes). In the present paper, we find an explicit expression of the extra invariant in the physical (or coordinate) space and show that the invariant is indeed physically meaningful for any fluid flow. In particular, no constraints are needed. The explicit form also enables us to note several properties of the extra invariant.

We consider the dynamics of a system of Rossby waves with nonlinear interaction. It has been shown that such a system–besides the energy and momentum (enstrophy)–has an extra invariant, which is conserved approximately in the limit of small wave amplitudes. This invariant implies the anisotropy of the inverse cascade, when the energy is transferred from small scale eddies to the large scale zonal jets. The invariant has a quadratic (in wave amplitudes) term, which has a universal form, and a cubic term, whose form depends on the form of nonlinear interaction between Rossby waves. In this paper, we show that it is impossible to extend the extra invariant by higher order nonlinear terms (fourth order and higher) to obtain an exact conservation law. This fact holds irrespective of the form of nonlinearity (including the three-wave interaction). The extra invariant is similar to the adiabatic invariants in the theory of dynamical systems. We also show that on a “long time scale”, the cubic part of the invariant can be dropped (without sacrificing the accuracy of conservation), so that the extra invariant is given by the universal quadratic part. Finally, we show that the extra invariant mostly represents large scale modes, even to a greater extent than the energy. The presence of a small dissipation destroys the conservation of the enstrophy, which is based on small scale modes. At the same time, the extra invariant, along with the energy, remains almost conserved.

It is shown that the extra invariant for systems of the Rossby waves—which was discovered in 1991—provides the missing theoretical argument for the anisotropic accumulation of the Rossby wave energy in the large-scale zonal flow.

We consider the turbulence of potential deep-water surface gravity waves, which have dispersion law Ω k = g k (g is the gravity constant, k=| k | ). It is well known that there are no three wave resonance interaction for such waves, and the lowest-order resonance involves four waves. We show the importance of almost resonance three wave interactions; in particular, they determine the life-span of these waves, provided the range of scales is large enough.

We consider the behaviour of a passive tracer in multiscale velocity field, when there is no separation of scales; the energy spectrum of the velocity field extends into the region of long waves and even can be singular there. We suppose that the velocity field is a superposition of random waves. The turbulence of various ocean or atmospheric waves provides examples. We find anomalous diffusion (sub- and super-diffusion), anomalous drift (super-drift), and anomalous spreading of a passive tracer cloud. For the latter we find the existence of two regimes: (i) ‘close’ passive tracer particles diverge sub- or supper-exponentially in time, and (ii) a ‘large’ passive tracer cloud spreads as a power-law in time. The exponents, as well as the corresponding pre-factors, are found. The theory is confirmed by numerical simulations.

This is a story about an old problem of the outstanding French geometer J.D.Gergonne. It is about the answer to this problem which was widely accepted as correct (and rigorously proved) for many decades, and cited as such in highly respected encyclopedias, but is in reality absolutely erroneous — the right answer is just the opposite. It is a story about two other problems closely related to that of Gergonne, about the possibility of finding highly plausible answers to them via computer experimentation.

Recently, unexpected conservation laws have been discovered for various nonlinear wave systems. Among these systems is the system of Rossby waves, which describes the global dynamics of the atmosphere and the ocean. It turns out that these conservation laws are intimately related to a geometric theory---the geometry of webs---that originated more than 60 years earlier. This relation helped discover how many conservation laws a nonlinear wave system can have.

Many studies consider media with microstructure, which has variations on some microscale, while the macroproperties are under investigation. Sometimes the medium has several microscales, all of them being much smaller than the macroscale. Sometimes the variations on the macroscale are also included, which are taken into account by some procedures, like WKB or geometric optics. What if the medium has variations on all scales from microscale to macroscale? This situation occurs in several practical problems. The talk is about such situations, in particular, passive tracer in a random velocity field, wave propagation in a random medium, Schrodinger equation with random potential. To treat such problems we have developed the statistical near-identity transformation. We find anomalous attenuation of the pulse propagating in a multiscale medium.

Turbulent diffusion of a passive tracer caused by a random wavefield is believed to be quadratic with respect to the energy spectrum ek of the velocity field (i.e. proportional to 4, where is the order of the wave amplitudes). So, the wave turbulent diffusion (say, on the ocean surface or in the air) is often believed to be dominated by the turbulent diffusion due to the incompressible flow. In this paper, we show that the wave turbulent diffusion can be associated with the Doppler shift and find that the wave turbulent diffusion can be more significant than previously thought. This mechanism works if the velocity field is compressible and statistically anisotropic, with the result that the wave system has a significant Stokes drift. The contribution of this mechanism has a lower order in . We confirm our results with numerical simulations. To derive these results, we develop the statistical near-identity transformation.