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

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

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

Water wave collapses over quasi-one-dimensional non-uniformly periodic bed profiles

Nonlinear water waves interacting with quasi-one-dimensional, non-uniformly periodic bed profiles are studied numerically in the deep-water regime with the help of approximate equations for envelopes of the forward and backward waves. Spontaneous formation of localized two-dimensional wave structures is observed in the numerical experiments, which looks essentially as a wave collapse.Comment: 4 pages, 4 figure

Breaking of vortex lines - a new mechanism of collapse in hydrodynamics

A new mechanism of the collapse in hydrodynamics is suggested, due to breaking of continuously distributed vortex lines. Collapse results in formation of the point singularities of the vorticity field $|{\bf\Omega}|$. At the collapse point, the value of the vorticity blows up as $(t_0-t)^{-1}$ where $t_0$ is a collapse time. The spatial structure of the collapsing distribution approaches a pancake form: contraction occurs by the law $l_1\sim(t_0-t)^{3/2}$ along the "soft" direction, the characteristic scales vanish like $l_2\sim(t_0-t)^{1/2}$ along two other ("hard") directions. This scenario of the collapse is shown to take place in the integrable three-dimensional hydrodynamics with the Hamiltonian ${\cal H}=\int|{\bf\Omega}|d{\bf r}$. Most numerical studies of collapse in the Euler equation are in a good agreement with the proposed theory.Comment: revtex, 9 pages, no figure

Local approximation for contour dynamics in effectively two-dimensional ideal electron-magnetohydrodynamic flows

The evolution of piecewise constant distributions of a conserved quantity related to the frozen-in canonical vorticity in effectively two-dimensional incompressible ideal EMHD flows is analytically investigated by the Hamiltonian method. The study includes the case of axisymmetric flows with zero azimuthal velocity component and also the case of flows with the helical symmetry of vortex lines. For sufficiently large size of such a patch of the conserved quantity, a local approximation in the dynamics of the patch boundary is suggested, based on the possibility to represent the total energy as the sum of area and boundary terms. Only the boundary energy produces deformation of the shape with time. Stationary moving configurations are described.Comment: REVTEX4, 7 pages, 6 EPS-figures. Extended versio