Vol 62, No 11 (2017)

Extreme fluctuations are modeled by a point system of stochastic equations, in which power spectra inversely proportional to the frequency are produced under the effect of white noise. The distribution of extreme fluctuations corresponds to the maximum of statistical entropy, which points to their stability in nature. By calculating the spectral entropy of random processes, it becomes possible to investigate their stability directly from power spectra without the need to calculate the amplitude distribution functions. The spectral entropy as a function of white noise amplitude has a minimum. The position of the spectral entropy minimum corresponds to the critical state of the system in which the spectra of fluctuating quantities are inversely proportional to the frequency.

The mixing of toroidal plasma under the conditions of sawtooth oscillations is considered using the Kadomtsev model. A new mixing formula for the averaged distribution function of fast transit and trapped particles is proposed in the methodology of a kinetic equation averaged over drift trajectories. The proposed formula generalizes the known results for the case of non-circular magnetic surfaces, an arbitrary aspect ratio, and charged particle drift trajectories significantly deviating from the magnetic surfaces. The formula is applicable for a sufficiently wide class of instabilities. The 3D kinetic equation is numerically solved using the FPP- 3D computation code for parameters close to the ITER inductive scenario. The alpha particle distribution function and the power introduced by alpha particles in plasma when sawtooth oscillations occur are calculated. It is shown that such oscillations may change the energy input of a thermonuclear reaction in certain areas by several times.

The technique of searching for optimal control, which makes it possible to transfer a dynamic object from an arbitrary point of the phase space onto the surface of an arbitrary multidimensional ellipsoid in the shortest time, is developed.

At the pointed cusp of a two-dimensional plate, a tip of small length h is broken off. By means of asymptotic analysis, a new effect of “wandering” of eigenfrequencies of longitudinal vibrations of the plate with the blunted cusp is found: as h → +0, the frequencies prove to be almost periodic functions in the logarithmic scale ln h; i.e., when the fragment length decreases, they chaotically move at a high speed O(h⁻¹) along the real semi-axis (κ_†, +∞), while κ_† > 0 is the cutoff point of the continuous spectrum of the problem with an ideal cusp.

A linear set of equations is proposed for a strongly thickness-heterogeneous (in particular, multilayer) shallow shell. The model unifies the equations of the Mushtary−Donnell−Vlasov technical-theory and the Timoshenko−Reissner equations, which take into account transverse shear. The thickness-heterogeneous shell is replaced with an equivalent homogeneous transversally isotropic shell, the elasticity modula of which are chosen just as the previously determined elasticity modula for heterogeneous plates. In the test example for a multilayered cylindrical shell, the approximate solution according to the proposed model is compared with the exact solution of the three-dimensional problem. The model gives good results in accuracy for a reasonably wide level of inhomogeneity.

In recent decades, Moscow has experienced the impact of remarkable climate changes on a scale that has significantly exceeded the climate changes in most of the world’s populated regions. Analysis of operation of the Moscow energy system under these new conditions has allowed us to reveal that the climate changes have determined alleviation of energy supply requirements during the cold season, contributed to decreased overall energy consumption, and led to reduced seasonal irregularity of the annual power load schedule. The results of this study allow us to conclude that an increase in the annual-mean temperature by 3–4°C in temperate and cold climate zones for continental regions brings no apparent negative consequences for operation of the energy system.