Alexander M. Balk

Dynamics of chains with non-monotone stress–strain relations. I. Model and numerical experiments

Abstract 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.

Dynamics of chains with non-monotone stress-strain relations. II. Nonlinear waves and waves of phase transition

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.

On the nonlocal turbulence of drift type waves

Abstract 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.

New invariant for drift turbulence

Abstract 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.

A Lagrangian for water waves

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.

A NEW INVARIANT FOR ROSSBY WAVE SYSTEMS

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

On the Kolmogorov-Zakharov spectra of weak turbulence

Abstract 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.

New Conservation Laws for the Interaction of Nonlinear Waves

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.

Rotating shallow water dynamics: extra invariant and the formation of zonal jets.

We show that rotating shallow water dynamics possesses an approximate (adiabatic-type) positive quadratic invariant, which exists not only at midlatitudes (where its analog in the quasigeostrophic equation has been previously investigated), but near the equator as well (where the quasigeostrophic equation is inapplicable). When deriving the extra invariant we find two kinds of small denominators: (i) those due to the triad resonances (as in the case of the quasigeostrophic equation) and (ii) those due to the equatorial limit, when the Rossby radius of deformation becomes infinite. We show that both kinds of small denominators 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. A similar invariant occurs in magnetically confined fusion plasmas and can lead to the emergence of zonal flows.

PASSIVE SCALAR IN A RANDOM WAVE FIELD : THE WEAK TURBULENCE APPROACH

Abstract 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.