A model of laminated weak turbulence

A model of laminated weak turbulence (WT) is presented. This model consists of two co-existing layers one with continuous spectra, covered by Kolmogorov ́s theory and KAM tori, and one with discrete spectra, covered by discrete classes of waves and Clipping method. Some known laboratory experiments and numerical simulations are explained in the frame of this model. A few problems which might appear in numerical simulations of WT due to their principal discreteness are pointed out. 1 WT in infinite domains In [12] Kolmogorov presented energy spectrum of turbulence describing the distribution of the energy among turbulence vortices as function of vortex size and thus founded the field of mathematical analysis of turbulence. Kolmogorov regarded some inertial range of wave numbers, between viscosity and dissipation, and suggested that at this range, turbulence is (1) locally homogeneous (no dependence on position) and (2) locally isotropic (no dependence on direction) which can be summarized as follows: probability distribution for the relative velocities of two particles in the fluid only depends on the distance between particles. Using these suggestions and dimensional analysis, Kolmogorov deduced that energy distribution, called now Kolmogorov ́s spectrum, is proportional to k−5/3 for wave numbers k. Results of numerical simulations and real experiments carried out to prove this theory are somewhat contradictious. On the one hand, probably the most spectacular example of the validity of Kolmogorov ́s spectra is provided in [3] where measurements in tidal currents near Seymour Narrows north of Campbell River on Vancouver Island were described and −5/3 spectra appeared at the range of almost 10 (energy dissipation at a scale of a millimeter and energy input at 100 meters). On the other hand, Kolmogorov ́s spectra have been obtained under the assumptions opposite to Kolmogorov ́s [4] so that exponent −5/3 corresponds to both direct and inverse cascades. Zakharov-Filonenko spectrum [18] gives exponent γ = −19/4, not −5/3. This difference in exponents is principal for the whole Kolmogorov ́s theory because the exponent −5/3 was obtained as the only solution for energy spectrum of a form kγ . Some well-known physical phenomena are not allowed by Kolmogorov ́s theory, for instance, (1.1) Faraday instability or (1.2) zonal extended vertices (flows in latitudinal direction) in the atmosphere, organized structures spanning over many scales. (1.3) A very interesting phenomenon of ”frozen turbulence” was discovered in direct numerical simulation with dynamical equation in [17] which also ∗Author acknowledges support of the Austrian Science Foundation (FWF) under projects SFB F013/F1304.