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The paper considers dynamo generated by a shallow fluid layer in a celestial body (planet or star). This dynamo is based on the extra invariant for interacting magnetic Rossby waves. The magnetohydrodynamics (MHD) is linearized on the background of strong toroidal magnetic field. The extra invariant is used to show that the background field is maintained., 12 pages, 2 figures

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 system of four wave packets (like light pulses in optical fibers) is shown to have an extra adiabatic-like invariant, in addition to the energy, momentum, and Manley–Rowe relations. A theoretical argument is suggested showing that the 4-wave interaction in one-dimensional systems can restrict spectral broadening.

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This paper reports a robust regime in the dynamics of three coupled modes when one mode is pumped, the second is dissipated, and the third is not subject to any external action. In this regime, the first and the second modes vanish, while the third mode becomes large. This is applied to the problem of plasma confinement by magnetic field, when the third mode represents zonal (poloidal) flow. This paper demonstrates that in order to generate strong zonal (poloidal) flow---a transport barrier---in magnetically confined fusion plasma, it is beneficial to have significant decrement in addition to increment of other modes.

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.

We consider the advection of a passive tracer when the velocity field is a superposition of random waves. Green's function for the turbulent transport (turbulent diffusion and turbulent drift) is derived. This Green's function is shown to imply sub-diffusive or super-diffusive behavior of the tracer. For the analysis we introduce the statistical near-identity transformation. The results are confirmed by numerical simulations.

We investigate the dynamics of a one dimensional mass-spring chain with non-monotone dependence of the spring force vs. spring elongation. For this strongly nonlinear system we find a family of exact solutions that represent nonlinear waves. We have found numerically that this system displays a dynamical phase transition from the stationary phase (when all masses are at rest) to the twinkling phase (when the masses oscillate in a wave motion). This transition has two fronts which propagate with different speeds. We study this phase transition analytically and derive relations between its quantitative characteristics.

We discuss dynamic processes in materials with non-monotonic constitutive relations. We introduce a model of a chain of masses joined by springs with a non-monotone strain–stress relation. Numerical experiments are conducted to find the dynamics of that chain under slow external excitation. We find that the dynamics leads either to a vibrating steady state (twinkling phase) with radiation of energy, or (if dissipation is introduced) to a hysteresis, rather than to an unique stress–strain dependence that would correspond to the energy minimization.

Zakharov discovered that wave kinetic equations for weak turbulence have exact power-law solutions which are similar to the Kolmogorov spectrum of hydrodynamic turbulence. We present a considerably simplified derivation of these solutions, i.e. Kolmogorov–Zakharov spectra. Even to a greater extent we simplify the derivation of certain “universal” corrections to these spectra, obtained by Kats and Kontorovich. The technique is utilized as well to derive the expressions for the so-called Mellin functions (that describe the behavior of weak turbulence in the vicinity of the Kolmogorov–Zakharov spectra). Using this general approach, we obtain Kolmogorov–Zakharov spectra for a class of weak turbulent media with dissipative nonlinearity. We also find a large family of anisotropic spectra, which are exact solutions of the corresponding kinetic equation. As a physical example, we consider the plasma turbulence when the main nonlinear process is the scattering of plasmons by electrons.

We consider the evolution of a passive scalar advected by a velocity field which is a superposition of random linear waves. An equation for the average concentration of the passive scalar is derived (in the limit of small molecular diffusion) using the weak turbulence methodology. In addition to the enhanced diffusion, this equation contains the correction to the (Stokes) drift. Both of these terms have the fourth order with respect to wave amplitudes. The formulas for the coefficients of turbulent diffusion and turbulent drift are derived.

We consider the phenomenon of suppression of short waves by a long wave, observed by Mitsuyasu in 1966. The recently proposed [1] essentially 3-D explanation of this phenomenon is reviewed and compared with more traditional 2-D explanations. Several physical implications of this 3-D explanation are suggested and the experimental verification is discussed.

A variational method is proposed for the description of the free surface dynamics in the strongly nonlinear regime. In the simplest case, when there is a single symmetric finger, this method gives a simple integrable Hamiltonian system with two degrees of freedom (which are, roughly speaking, the height and the width of the finger). The general solution of this system is found, and physical implications are discussed.

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.

We consider the ideal magnetohydrodynamics (MHD) of a shallow fluid layer on a rapidly rotating planet or star. The presence of a background toroidal magnetic field is assumed, and the "shallow water" beta-plane approximation is used. We derive a single equation for the slow large length scale dynamics. The range of validity of this equation fits the MHD of the lighter fluid at the top of Earth's outer core. The form of this equation is similar to the quasi-geostrophic (Q-G) equation (for usual ocean or atmosphere), but the parameters are essentially different. Our equation also implies the inverse cascade; but contrary to the usual Q-G situation, the energy cascades to smaller length scales, while the enstrophy cascades to the larger scales. We find the Kolmogorov-type spectrum for the inverse cascade. The spectrum indicates the energy accumulation in larger scales. In addition to the energy and enstrophy, the obtained equation possesses an extra (adiabatic-type) invariant. Its presence implies energy accumulation in the 30° sector around zonal direction. With some special energy input, the extra invariant can lead to the accumulation of energy in zonal magnetic field; this happens if the input of the extra invariant is small, while the energy input is considerable.

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.

We find a Kolmogorov-type spectrum for the 2D turbulence of waves with the “sound-like” dispersion law Ω k =u|k| . The spectrum is proportional to P 3 5 , where P is the flux of energy through the scales. This provides an example of turbulence which is weaker than strong hydrodynamic turbulence, but stronger than the weak turbulence of dispersive waves. To analyze the localness of the turbulence we use the simplest closure (the so-called one-loop approximation). We apply the results to the turbulence of shallow water gravity waves, two-dimensional acoustic turbulence.

Can some extra quantities (besides energy and momentum) be conserved in 3-wave (resonance) interactions? The paper describes all 3-wave interactions in which infinitely many independent quantities are conserved. The results are obtained due to the application of web geometry. The correspondence between 3-wave interactions and 3-webs is established. All wave systems, related to the inverse scattering tranform method, correspond to the simplest (the so called coordinate) web. Each extra invariant of a 3-wave interaction defines new conservation law of the corresponding wave kinetic equation (which describes turbulence of the waves). The results are applied to the nonlinear theory of decay instability.

We show that rotating shallow water dynamics possesses an approximate (adiabatic-type) positive quadratic invariant, which exists not only at mid-latitudes (where its analogue in the quasigeostrophic equation has been previously investigated), but near the equator as well (where the quasigeostrophic equation is inapplicable). Deriving the extra invariant, we find "small denominators" of two kinds: (1) due to the triad resonances (as in the case of the quasigeostrophic equation) and (2) due to the equatorial limit, when the Rossby radius of deformation becomes infinite. We show that the "small denominators" of both kinds can be canceled. The presence of the extra invariant can lead to the generation of zonal jets. We find that this tendency should be especially pronounced near the equator. Similar invariant occurs in magnetically confined fusion plasmas and can lead to the emergence of zonal flows., Comment: 29 pages, 4 figures

We give a complete description of one-dimensional 4-wave resonance interactions in which some extra quantities (besides momentum, energy, number of quasiparticles) are conserved. In this way we obtain new consideration laws for the kinetic equations for waves. In particular, we consider waves in optical fibers, the system of four resonantly interacting wave packets, long wave interactions of annihilation-creation type, various wave systems with quadratic dispersion laws. The results can be important for various problems concerning nonlinear wave dynamics, e.g. for nonlinear optics of waveguides.

A new invariant for an arbitrary system of Rossby waves (or drift waves in plasma) is obtained.

A new invariant for drift wave (or Rossby wave) turbulence in two cases, (1) in zonal flow and (2) in the large scale range, is discovered. This invariant is proved to be a unique additional invariant (besides energy and momentum). Thus the first examples of wave systems with a finite number of additional invariants are obtained. A new Kolmogorov-type spectrum with the flux of the additional invariant through the scales is derived and the structure of fluxes in the k -space of invariants is analyzed.

Two new effects the drift turbulence can display are disclosed: (1) the turbulence spectrum in k -space separates into unconnected components of large and small scales, (2) the very presence of weak small-scale turbulence imposes rigid restrictions on powerful large-scale components.

It is shown that a train of long waves can suppress a short-wave field due to four-wave resonance interactions. These interactions lead to the diffusion (in Fourier space) of the wave action of the short-wave field, so that the wave action is transported to the regions of higher wavenumbers, where it dissipates more effectively. The diffusion equation is derived.

A Lagrangian for strongly nonlinear unsteady water waves (including overturning waves) is obtained. It is shown that the system of quadratic equations for the Stokes coefficients, which determine the shape of a steady wave (discovered by Longuet‐Higgins 100 years after Stokes derived his system of cubic equations) directly follows from the canonical system of Lagrange equations. Applications to the investigation of the stability of water waves and to the construction of numerical schemes are pointed out.

Perturbational approaches have a long history of application to dynamical systems. A special role in this subject belongs to Hamiltonian systems with a small parameter 6, which are integrable when 6 = 0. Expanding in powers of E is a common tool in celestial mechanics. When one tries to find the solution as a power series in 6, one encounters the classical problem of resonances. The same happens when one attempts to find additional invariants of motion. Attempts to overcome these difficulties stimulated the development of the theory of canonical transformations, simplifying the system by mapping it to some “normal form”. These and related issues were elaborated in “New Methods of Celestial Mechanics” by A.Poincare [1] and in “Dynamic Systems” by D.D.Birkhoff [2].

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.