Calculation of the anomalous exponents in the rapid-change model of passive scalar advection to order ε3\varepsilon^{3}

The field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the Gaussian velocity field with zero mean and correlation function \propto\delta(t-t')/k^{d+\eps}. Inertial-range anomalous exponents, identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients, are calculated within the $\varepsilon$ expansion to order $\varepsilon^{3}$ (three-loop approximation), including the exponents in anisotropic sectors. The main goal of the paper is to give the complete derivation of this third-order result, and to present and explain in detail the corresponding calculational techniques. The character and convergence properties of the $\varepsilon$ expansion are discussed; the improved ``inverse'' $\varepsilon$ expansion is proposed and the comparison with the existing nonperturbative results is given.Comment: 34 pages, 5 figures, REVTe

Anomalous exponents in the rapid-change model of the passive scalar advection in the order ϵ3\epsilon^{3}

Field theoretic renormalization group is applied to the Kraichnan model of a passive scalar advected by the Gaussian velocity field with the covariance $- <{\bf v}(t,{\bf x}){\bf v}(t',{\bf x'})> \propto\delta(t-t')|{\bf x}-{\bf x'} |^{\epsilon}$. Inertial-range anomalous exponents, related to the scaling dimensions of tensor composite operators built of the scalar gradients, are calculated to the order $\epsilon^{3}$ of the $\epsilon$ expansion. The nature and the convergence of the $\epsilon$ expansion in the models of turbulence is are briefly discussed.Comment: 4 pages; REVTeX source with 3 postscript figure

Field theoretic renormalization group for a nonlinear diffusion equation

The paper is an attempt to relate two vast areas of the applicability of the renormalization group (RG): field theoretic models and partial differential equations. It is shown that the Green function of a nonlinear diffusion equation can be viewed as a correlation function in a field-theoretic model with an ultralocal term, concentrated at a spacetime point. This field theory is shown to be multiplicatively renormalizable, so that the RG equations can be derived in a standard fashion, and the RG functions (the $\beta$ function and anomalous dimensions) can be calculated within a controlled approximation. A direct calculation carried out in the two-loop approximation for the nonlinearity of the form $\phi^{\alpha}$, where $\alpha>1$ is not necessarily integer, confirms the validity and self-consistency of the approach. The explicit self-similar solution is obtained for the infrared asymptotic region, with exactly known exponents; its range of validity and relationship to previous treatments are briefly discussed.Comment: 8 pages, 2 figures, RevTe

Perspectives of the Reduction of Nutrient Export from River Watersheds through the Introduction of Best Available Technologies for Agricultural Production: Based on Modeling Results

© 2020, Pleiades Publishing, Ltd. Abstract: Estimate is made for the possible reduction of nutrient export from the drainage basins of tributaries of the Kuibyshev Reservoir through the introduction of the best available technologies of agricultural production. The activity of more than eight hundreds of agricultural enterprises in the reservoir drainage basin was analyzed. Different types of underlying surface were classified with the use of space photographs. Mathematical simulation was used to calculate first the distributed agricultural nitrogen and phosphorus load onto the drainage basin and next the export of nutrients with river runoff into the reservoir either under current conditions or after the introduction of the best available technologies (BAT) to agricultural production. The simulations show that, under the conditions of medium water abundance, the total export from the drainage basins of the Sviyaga, Kazanka, Mesha, and Bol’shoi Cheremshan will decrease by about 367 t N/year and 12 t P/year after the introduction of BAT into the agricultural practice. The application of nitrogen and phosphorus into the soil as components of mineral and organic fertilizers in most administrative territories of the region under consideration was found to be less than the nutrient demand of agricultural crops, resulting in the gradual depletion of such elements in soils and a decrease in the risk of their migration into water bodies

Paleolithic Man of Denisova Cave and Zoogeography of Pleistocene Mammals of Northwestern Altai

Mammal population of the northwestern Altai included residents, autochthonous species, cosmopolitans, and migrants. The last clearly indicate biogeographical relationships of the biota of the Altai Mountains in the Pleistocene. Most of them penetrated into the Altai from the south. The majority of ungulates and rodents migrated from Central Asia. Yak, red dog, and snow leopard came from the Himalayas, Pamir, and Tien Shan. The natural environment of the Altai Mountains in the Pleistocene enabled migrations of these mammals from the south to north. The same opportunity was true of the ancient man. It is possible to assume that humans migrated from southeastern Asia and Indochina along the eastern foothills of the Himalayas and Nan Shan Mountains to the northwest, to the Zaisan Depression and Altai. This resulted in inevitable exchange of gene material of Paleolithic human populations of southeastern Asia and the Altai