Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Pad\'e approximation

Complex analytical structure of Stokes wave for two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth is analyzed. Stokes wave is the fully nonlinear periodic gravity wave propagating with the constant velocity. Simulations with the quadruple and variable precisions are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with $2\pi/3$ radians angle on the crest. A conformal map is used which maps a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half-plane. The Stokes wave is fully characterized by the complex singularities in the upper complex half-plane. These singularities are addressed by rational (Pad\'e) interpolation of Stokes wave in the complex plane. Convergence of Pad\'e approximation to the density of complex poles with the increase of the numerical precision and subsequent increase of the number of approximating poles reveals that the only singularities of Stokes wave are branch points connected by branch cuts. The converging densities are the jumps across the branch cuts. There is one branch cut per horizontal spatial period $\lambda$ of Stokes wave. Each branch cut extends strictly vertically above the corresponding crest of Stokes wave up to complex infinity. The lower end of branch cut is the square-root branch point located at the distance $v_c$ from the real line corresponding to the fluid surface in conformal variables. The limiting Stokes wave emerges as the singularity reaches the fluid surface. Tables of Pad\'e approximation for Stokes waves of different heights are provided. These tables allow to recover the Stokes wave with the relative accuracy of at least $10^{-26}$. The tables use from several poles to about hundred poles for highly nonlinear Stokes wave with $v_c/\lambda\sim 10^{-6}.$Comment: 38 pages, 9 figures, 4 tables, supplementary material

Beyond the random phase approximation: Stimulated Brillouin backscatter for finite laser coherence times

We develop a statistical theory of stimulated Brillouin backscatter (BSBS) of a spatially and temporally partially incoherent laser beam for laser fusion relevant plasma. We find a new collective regime of BSBS (CBSBS) with intensity threshold controlled by diffraction, an insensitive function of the laser coherence time, $T_c$, once light travel time during $T_c$ exceeds a laser speckle length. The BSBS spatial gain rate is approximately the sum of that due to CBSBS, and a part which is independent of diffraction and varies linearly with $T_c$. We find that the bandwidth of KrF-laser-based fusion systems would be large enough to allow additional suppression of BSBS.Comment: 8 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:1105.209

Weak Turbulent Kolmogorov Spectrum for Surface Gravity Waves

We study the long-time evolution of gravity waves on deep water exited by the stochastic external force concentrated in moderately small wave numbers. We numerically implement the primitive Euler equations for the potential flow of an ideal fluid with free surface written in canonical variables, using expansion of the Hamiltonian in powers of nonlinearity of up to fourth order terms. We show that due to nonlinear interaction processes a stationary energy spectrum close to $|k| \sim k^{-7/2}$ is formed. The observed spectrum can be interpreted as a weak-turbulent Kolmogorov spectrum for a direct cascade of energy.Comment: 4 pages, 5 figure

Coexistence of Weak and Strong Wave Turbulence in a Swell Propagation

By performing two parallel numerical experiments -- solving the dynamical Hamiltonian equations and solving the Hasselmann kinetic equation -- we examined the applicability of the theory of weak turbulence to the description of the time evolution of an ensemble of free surface waves (a swell) on deep water. We observed qualitative coincidence of the results. To achieve quantitative coincidence, we augmented the kinetic equation by an empirical dissipation term modelling the strongly nonlinear process of white-capping. Fitting the two experiments, we determined the dissipation function due to wave breaking and found that it depends very sharply on the parameter of nonlinearity (the surface steepness). The onset of white-capping can be compared to a second-order phase transition. This result corroborates with experimental observations by Banner, Babanin, Young.Comment: 5 pages, 5 figures, Submitted in Phys. Rev. Letter

Extraction processing of concentrated solutions of uranyl nitrate with high impurities content

Process flowsheet of recycling uranium concentrated solutions with its purification from insoluble impurities of iron, silicon, molybdenum, calcium oxides and hydroxides and soluble impurities with application of centrifugal extractors cascade has been developed and suggested for commercial introduction. The process was carried out at extractant saturation (30 % tributyl phosphate in hydrocarbon diluent) in extraction assembly lower than a limiting level (85...95 g/l) and in wash assembly - at limiting saturation (up to 120 g/l). As a result the waste uranium content in water-tail solutions 0,01...0,04 g/l and minimal content of impurities in re-extractors is provide

Показатели надежности схем городской электрической сети для питания потребителей второй и третьей категорий

The paper considers two schemes of the urban distributive electric network for supply of the second and third category consumers for whom indicators of reliability have been determined.As for a network scheme of the 2nd category it has been determined that while installing low-oil and vacuum switches at a distributive point the most remote transformer substation has the highest non-availability  factor which is higher of the corresponding non-availability  factor of the nearest transformer substation by 1,21–1,23 fold  accordingly.As for a network scheme of the 3rd category consumers it has been obtained the data that reliability of consumers receiving supply from bus-bars of transformer substations with 0,38 kW voltage changes  significantly (by 1,2–1,9-fold) while changing a breakpoint of the network with 10 kW voltage.Рассмотрены две схемы городской распределительной электрической сети для питания потребителей второй и третьей категорий, для которых были определены значения показателей надежности.Для схемы сети потребителей второй категории было получено, что при установке маломасляных и вакуумных  выключателей на распределительном пункте наиболее удаленная трансформаторная подстанция имеет самый высокий коэффициент неготовности, который больше соответствующего коэффициента неготовности ближайшей трансформаторной подстанции в 1,21–1,23 раза соответственно.Для схемы сети потребителей третьей категории было получено, что надежность потребителей, питающихся от шин трансформаторной подстанции напряжением 0,38 кВ, значительно (в 1,2–1,9 раза) изменяется при изменении точки разрыва сети напряжением 10 кВ

АНАЛИЗ СТРУКТУРНОЙ НАДЕЖНОСТИ ГЛАВНЫХ СХЕМ ЭЛЕКТРИЧЕСКИХ СОЕДИНЕНИЙ АТОМНЫХ ЭЛЕКТРОСТАНЦИЙ

The reliability of the main circuit of electrical connections at a nuclear electric power plant that has two units with a capacity of 1,200 MW each has been determined. Reliability, economical, maneuverable properties of the atomic power plant under study are largely determined by its main circuit, so the choice of the circuit for the design and its status in the process of operation occur to be critical objectives. Main electrical connection circuits in nuclear electric power plants are selected on the basis of the schematic networks of the energy system and the land attached to the plant. The circuit of the connection of a nuclear power plant to the grid in the original normal operating modes at all stages of the construction of such a plant should provide the outcome of the full added capacity of a nuclear power plant and the preservation of its stability in the power system without the influence of the emergency system automatics when any outgoing transmission line is disabled. When selecting the main circuit the individual capacity of the installed units and their number are taken into account as well as the order of development of the plant and power supply system; the voltage on which the power of a plant is delivered; a shortcircuit current for switchgear high voltage and the need for their limitation by circuit means; the most power that can be lost when damage to any switch. A model of reliability of the main circuit of electrical connections is designed to detect all types of accidents that are possible at the coincidence of failures of elements with the repair and operational modes that differs in composition and damageability of the equipment, as well as under conditions of the development of accidents due to failure of operation of devices of relay protection and automation.Определена надежность главной схемы электрических соединений атомной электрической станции, на которой установлено два блока мощностью 1200 МВт каждый. Надежность, экономичность, маневренные свойства атомной электрической станции во многом определяются ее главной схемой, поэтому выбор схемы при проектировании и ее состояние в процессе эксплуатации – важнейшие задачи. Главные схемы электрических соединений атомной электрической станции выбираются на основании схемы сетей энергосистемы и того участка, к которому присоединяется данная электростанция. Схема присоединения атомной электрической станции к энергосистеме в нормальных исходных режимах на всех стадиях сооружения такой станции должна обеспечивать выдачу полной введенной мощности атомной электростанции и сохранение устойчивости ее работы в энергосистеме без воздействия противоаварийной системной автоматики при отключении любой отходящей линии электропередачи. При выборе главной схемы учитываются: единичная мощность устанавливаемых агрегатов и их число; очередность развития станции и энергосистемы; напряжения, на которых выдается электроэнергия станции; токи короткого замыкания для распределительного устройства повышенного напряжения и необходимость их ограничения схемным путем; наибольшая мощность, которая может быть потеряна при повреждении любого выключателя. Модель надежности главной схемы электрических соединений призвана выявить все виды аварий, возможных при совпадении отказов элементов с ремонтными и эксплуатационными режимами, отличающимися составом и повреждаемостью оборудования, а также при развитии аварий из-за отказов срабатывания аппаратов и устройств релейной защиты и автоматики