V. P. Ruban

Fluid description of trains of stationary mirror structures in a magnetized plasma

A simple magnetohydrodynamic model is used to interpret the large-scale features of trains of steady nonlinear magnetic structures, anticorrelated with density, that are commonly observed in anisotropic space plasmas. For this purpose, an equation of state is derived in the quasi-static regime, and stable solutions are obtained, by minimizing the potential energy under the constraint of particle conservation and frozen-in magnetic field. In agreement with satellite observations, these coherent structures take the form of magnetic holes at moderate β or humps when β is larger, provided the propagation angle and the anisotropy are sufficiently large. A phenomenon of bistability is also observed, associated with the existence of stable nonlinear structures in a regime where the plasma is stable with respect to the mirror instability.

Nonlinear stage of the benjamin-feir instability : Three-dimensional coherent structures and rogue waves

A specific, genuinely three-dimensional mechanism of rogue wave formation, in a late stage of the modulational instability of a perturbed Stokes deep-water wave, is recognized through numerical experiments. The simulations are based on fully nonlinear equations describing weakly three-dimensional potential flows of an ideal fluid with a free surface in terms of conformal variables. Spontaneous formation of zigzag patterns for wave amplitude is observed in a nonlinear stage of the instability. If initial wave steepness is sufficiently high (ka>0.06), these coherent structures produce rogue waves. The most tall waves appear in turns of the zigzags. For ka<0.06, the structures decay typically without formation of steep waves.

Hamiltonian dynamics of vortex lines in hydrodynamic-type systems

It is shown that the degeneracy of the noncanonical Poisson bracket operating on the space of solenoidal vector fields that arises due to the freezing-in of the curl of the velocity [E. A. Kuznetsov and A. V. Mikhailov, Phys. Lett. A 77, 37 (1980)] is lifted when the vorticity Ω is represented in terms of vortex lines. This representation makes it possible to integrate the equation of motion of the vorticity for a system with the Hamiltonian H=∫∣Ω∣dr.

On the nonlinear Schrödinger equation for waves on a nonuniform current

A nonlinear Schrödinger 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.

Breathing rogue wave observed in numerical experiment

Numerical simulations of the recently derived fully nonlinear equations of motion for long-crested water waves [V. P. Ruban, Phys. Rev. E 71, 055303(R) (2005)] with quasirandom initial conditions are reported, which show the spontaneous formation of a single extreme wave on deep water. This rogue wave behaves in an oscillating manner and exists for a relatively long time (many wave periods) without significant change of its maximal amplitude.

Collapse of vortex lines in hydrodynamics

A new mechanism is proposed for collapse in hydrodynamics associated with the “breaking” of vortex lines. The collapse results in the formation of point singularities of the vorticity field, i.e., a generalized momentum curl. At the point of collapse the vorticity |Ω| increases as ((t0 − t)−1 and its spatial distribution for t → t0 approaches quasi-two-dimensional: in the “soft” direction contraction obeys the law l1 → (t0 − t)3/2 whereas in the other two “hard” directions it obeys l2 → (t0 − t)1/2. It has been shown that this collapse scenario takes place in the general case for three-dimensional integrable hydrodynamics with the Hamiltonian ℋ = ∫|Ω| dr.

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 an arbitrary seabed shape parametrized by an analytical function. As applications of this theory, the band structure of linear waves over periodic bottoms is calculated and the evolution of strong solitary waves running from a deep region to a shallow region is numerically simulated.

Rogue waves at low Benjamin-Feir indices: Numerical study of the role of nonlinearity

The nonlinear interaction between waves in incoherent sea states is weaker than their dispersion. In this situation, random space-time focusing is the main mechanism of the formation of rogue waves. The numerical simulation has indicated that nonlinearity becomes important at the final stage of focusing and can significantly change predictions of the so-called second-order theory concerning the parameters of rogue waves. The elongation of the crest of a rogue wave as compared to that predicted by the second-order theory is an important effect promoting the “weighting of the tails” of the distribution function of the vertical deviation of the free surfaces.

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.

Giant waves in weakly crossing sea states

The formation of rogue waves in sea states with two close spectral maxima near the wave vectors k0 ± Δk/2 in the Fourier plane is studied through numerical simulations using a completely nonlinear model for long-crested surface waves [24]. Depending on the angle θ between the vectors k0 and Δk, which specifies a typical orientation of the interference stripes in the physical plane, the emerging extreme waves have a different spatial structure. If θ ≲ arctan(1/√2), then typical giant waves are relatively long fragments of essentially two-dimensional ridges separated by wide valleys and composed of alternating oblique crests and troughs. For nearly perpendicular vectors k0 and Δk, the interference minima develop into coherent structures similar to the dark solitons of the defocusing nonlinear Schroedinger equation and a two-dimensional killer wave looks much like a one-dimensional giant wave bounded in the transverse direction by two such dark solitons.