Numerical modeling of quasiplanar giant water waves

In this work we present a further analytical development and a numerical implementation of the recently suggested theoretical model for highly nonlinear potential long-crested water waves, where weak three-dimensional effects are included as small corrections to exact two-dimensional equations written in the conformal variables [V.P. Ruban, Phys. Rev. E 71, 055303(R) (2005)]. Numerical experiments based on this theory describe the spontaneous formation of a single weakly three-dimensional large-amplitude wave (alternatively called freak, killer, rogue or giant wave) on the deep water.Comment: revtex4, 8 pages, 7 figure

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

Free Surface in 2D Potential Flow: Singularities, Invariants and Virtual Fluid

We study a 2D potential flow of an ideal fluid with a free surface with decaying conditions at infinity. By using the conformal variables approach, we study a particular solution of Euler equations having a pair of square-root branch points in the conformal plane, and find that the analytic continuation of the fluid complex potential and conformal map define a flow in the entire complex plane, excluding a vertical cut between the branch points. The expanded domain is called the "virtual" fluid, and it contains a vortex sheet whose dynamics is equivalent to the equations of motion posed at the free surface. The equations of fluid motion are analytically continued to both sides of the vertical branch cut (the vortex sheet), and additional time-invariants associated with the topology of conformal plane and Kelvin's theorem for virtual fluid are explored. We called them "winding" and virtual circulation. This result can be generalized to a system of many cuts connecting many branch points, and resulting in a pair of invariants for each pair of branch points. We develop an asymptotic theory that shows how a solution originating from a single vertical cut forms a singularity at the free surface in infinite time, the rate of singularity approach is double-exponential, and supercedes the previous result of the short branch cut theory with finite time singularity formation. The present work offers a new look at fluid dynamics with free surface by unifying the problem of motion of vortex sheets, and the problem of 2D water waves. A particularly interesting question that arises in this context is whether instabilities of the virtual vortex sheet are related to breaking of steep ocean waves when gravity effects are included

On the Quantum Kinetic Equation in Weak Turbulence

The quantum kinetic equation used in the study of weak turbulence is reconsidered in the context of a theory with a generic quartic interaction. The expectation value of the time derivative of the mode number operators is computed in a perturbation expansion which places the large diagonal component of the quartic term in the unperturbed Hamiltonian. Although one is not perturbing around a free field theory, the calculation is easily tractable owing to the fact that the unperturbed Hamiltonian can be written solely in terms of the mode number operators.Comment: 12 pages, LATEX, no figures, to appear in Phys. Rev.

Studying the utilization techniques of ammonium hexafluorosilicate

The utilization techniques of ammonium hexafluorosilicate have been proposed and studied. Thermodynamic calculations of equilibrium gas phase compositions of topaz concentrate fluoridation reaction and reaction of (NH4)2SiF6 absorption by ammonium hydroxide were given. Experimental investigations in studying gas phase composition were carried out. The sublimation process of ammonium hexafluorosilicate as well as the process of its dissolving in ammonia water with silicon dioxide obtaining was studie

Topaz concentrate desiliconization with ammonium bifluoride

Fluoridizing agent for the process of topaz concentrate desiliconization has been selected, topaz thermodynamic potentials have been appraised, thermodynamic probabilities of fluorination reaction of topaz concentrate main constituents have been calculated. The results of studying the concentrate desiliconization process with ammonium bifluoride by the methods of thermogravimetry, RFA, microphotography are presente