Explicit equations for two-dimensional water waves with constant vorticity

Governing equations for two-dimensional inviscid free-surface flows with constant vorticity over arbitrary non-uniform bottom profile are presented in exact and compact form using conformal variables. An efficient and very accurate numerical method for this problem is developed.Comment: 4 pages, 6 figures, published in 200

Anomalous wave as a result of the collision of two wave groups on sea surface

The numerical simulation of the nonlinear dynamics of the sea surface has shown that the collision of two groups of relatively low waves with close but noncollinear wave vectors (two or three waves in each group with a steepness of about 0.2) can result in the appearance of an individual anomalous wave whose height is noticeably larger than that in the linear theory. Since such collisions quite often occur on the ocean surface, this scenario of the formation of rogue waves is apparently most typical under natural conditions.Comment: 5 pages, 9 figure

Thermal vacancies in random alloys in the single-site mean-field approximation

A formalism for the vacancy formation energies in random alloys within the single-site mean-filed approximation, where vacancy-vacancy interaction is neglected, is outlined. It is shown that the alloy configurational entropy can substantially reduce the concentration of vacancies at high temperatures. The energetics of vacancies in random Cu-0.5Ni-0.5 alloy is considered as a numerical example illustrating the developed formalism. It is shown that the effective formation energy is increases with temperature, however, in this particular system it is still below the mean value of the vacancy formation energy which would correspond to the vacancy formation energy in a homogeneous model of a random alloy, such as given by the coherent potential approximation.Comment: 5 pages, 3 figure

Ideal hydrodynamics inside as well as outside non-rotating black hole: Hamiltonian description in the Painlev{\'e}-Gullstrand coordinates

It is demonstrated that with using Painlev{\'e}-Gullstrand coordinates in their quasi-Cartesian variant, the Hamiltonian functional for relativistic perfect fluid hydrodynamics near a non-rotating black hole differs from the corresponding flat-spacetime Hamiltonian just by a simple term. Moreover, the internal region of the black hole is then described uniformly together with the external region, because in Painlev{\'e}-Gullstrand coordinates there is no singularity at the event horizon. An exact solution is presented which describes stationary accretion of an ultra-hard matter ($\varepsilon\propto n^2$) onto a moving black hole until reaching the central singularity. Equation of motion for a thin vortex filament on such accretion background is derived in the local induction approximation. The Hamiltonian for a fluid having ultra-relativistic equation of state $\varepsilon\propto n^{4/3}$ is calculated in explicit form, and the problem of centrally-symmetric stationary flow of such matter is solved analytically.Comment: revtex4, 7 pages, no figure

On the nonlinear Schr\"odinger equation for waves on a nonuniform current

A nonlinear Schr\"odinger equation with variable coefficients for surface waves on a large-scale steady nonuniform current has been derived without the assumption of a relative smallness of the velocity of the current. This equation can describe with good accuracy the loss of modulation stability of a wave coming to a counter current, leading to the formation of so called rogue waves. Some theoretical estimates are compared to the numerical simulation with the exact equations for a two-dimensional potential motion of an ideal fluid with a free boundary over a nonuniform bottom at a nonzero average horizontal velocity.Comment: 6 pages, 10 figures, published in 201

Dynamics of quantum vortices in a quasi-two-dimensional Bose-Einstein condensate with two "holes"

The dynamics of interacting quantum vortices in a quasi-two-dimensional spatially inhomogeneous Bose-Einstein condensate, whose equilibrium density vanishes at two points of the plane with a possible presence of an immobile vortex with a few circulation quanta at each point, has been considered in a hydrodynamic approximation. A special class of density profiles has been chosen, so that it proves possible to calculate analytically the velocity field produced by point vortices. The equations of motion have been given in a noncanonical Hamiltonian form. The theory has been generalized to the case where the condensate forms a curved quasi-two-dimensional shell in the three-dimensional space.Comment: 6 pages, 8 figures, in English, published versio

Internal waves in a compressible two-layer atmospheric model: The Hamiltonian description

Slow flows of an ideal compressible fluid (gas) in the gravity field in the presence of two isentropic layers are considered, with a small difference of specific entropy between them. Assuming irrotational flows in each layer [that is ${\bf v}_{1,2}=\nabla\phi_{1,2}$], and neglecting acoustic degrees of freedom by means of the conditions ${div}(\bar\rho(z)\nabla\phi_{1,2})\approx0$, where $\bar\rho(z)$ is a mean equilibrium density, we derive equations of motion for the interface in terms of the boundary shape $z=\eta(x,y,t)$ and the difference of the two boundary values of the velocity potentials: $\psi(x,y,t)=\psi_1-\psi_2$. A Hamiltonian structure of the obtained equations is proved, which is determined by the Lagrangian of the form ${\cal L}=\int \bar\rho(\eta)\eta_t\psi \,dx dy -{\cal H}\{\eta,\psi\}$. The idealized system under consideration is the most simple theoretical model for studying internal waves in a sharply stratified atmosphere, where the decrease of equilibrium gas density with the altitude due to compressibility is essentially taken into account. For planar flows, a generalization is made to the case when in each layer there is a constant potential vorticity. Investigated in more details is the system with a model density profile $\bar\rho(z)\propto \exp(-2\alpha z)$, for which the Hamiltonian ${\cal H}\{\eta,\psi\}$ can be expressed explicitly. A long-wave regime is considered, and an approximate weakly nonlinear equation of the form $u_t+auu_x-b[-\hat\partial_x^2+\alpha^2]^{1/2}u_x=0$ (known as Smith's equation) is derived for evolution of a unidirectional wave.Comment: revtex4, 8 pages, submitted to JETP, information about Eq.(44) adde

Water waves over a strongly undulating bottom

Two-dimensional free-surface potential flows of an ideal fluid over a strongly inhomogeneous bottom are investigated with the help of conformal mappings. Weakly-nonlinear and exact nonlinear equations of motion are derived by the variational method for arbitrary seabed shape parameterized by an analytical function. As applications of this theory, band structure of linear waves over periodic bottoms is calculated and evolution of a strong solitary wave running from a deep region to a shallow region is numerically simulated.Comment: revtex4, 11 pages, 17 figures, extended version with numerical result

Enhanced rise of rogue waves in slant wave groups

Numerical simulations of fully nonlinear equations of motion for long-crested waves at deep water demonstrate that in elongate wave groups the formation of extreme waves occurs most intensively if in an initial state the wave fronts are oriented obliquely to the direction of the group. An "optimal" angle, resulting in the highest rogue waves, depends on initial wave amplitude and group width, and it is about 18-28 degrees in a practically important range of parameters.Comment: revtex4, 4 pages, 8 figures, minor corrections, additional numerical results include

An integrable localized approximation for interaction of two nearly anti-parallel sheets of the generalized vorticity in 2D ideal electron-magnetohydrodynamic flows

The formalism of frozen-in vortex lines for two-dimensional (2D) flows in ideal incompressible electron magnetohydrodynamics (EMHD) is formulated. A localized approximation for nonlinear dynamics of two close sheets of the generalized vorticity is suggested and its integrability by the hodograph method is demonstrated.Comment: 4 pages, revtex4, 1 eps figur