Papers by Anatoly Abrashkin
Gerstner waves and Stokes Drift
EGU General Assembly Conference Abstracts, Apr 1, 2017
Radiophysics and Quantum Electronics, 2016
We show that the discrete frequency spectrum of a plane hydrodynamic flow of ideal incompressible... more We show that the discrete frequency spectrum of a plane hydrodynamic flow of ideal incompressible liquid with localized trajectories of the liquid particles can contain only one, two, or an infinite number of harmonics.
Инварианты Коши и точные решения нелинейных уравнений гидродинамики
Teoreticheskaya i Matematicheskaya Fizika
Дан обзор точных решений для гравитационных волн на глубокой воде. Все решения получены в рамках ... more Дан обзор точных решений для гравитационных волн на глубокой воде. Все решения получены в рамках лагранжева описания и являются обобщениями волны Герстнера (в случае неоднородного давления на свободной поверхности и с учетом вращения жидкости). Для каждого типа волн найден вид инвариантов Коши.
The paper describes a new derivation of the NLS equation, based on a Lagrangian coordinates appro... more The paper describes a new derivation of the NLS equation, based on a Lagrangian coordinates approach, in the presence of weak vorticity. First, an introduction presents several previously existing derivations of the NLS equation, and offers an interesting review of recent developments designed to take vorticity into account. Then, the Lagrange coordinates, and associated general equations are presented in section 2, while the new NLS equation related to this framework is derived in section 3. Several results are presented at the end of section 3, and in section 4 (only those related to envelope soliton solutions), and summarized in section 5. The paper is relatively well structured, even if several typos remain. Globally, several new results can be found in the manuscript, and for all these reasons, I recommend publication, after some modifi-
Nonlinear Schrödinger equation for low-vorticity waves in deep water
Reviewer 2: First, it suffers a lack of illustration. Indeed, a single figure appears, and intend... more Reviewer 2: First, it suffers a lack of illustration. Indeed, a single figure appears, and intends to show the full geometry of the problem. For instance, from the figure, I cannot understand what is this “average current” (average in time, in ‘a’ coordinate? In ‘b’ coordinate?) Neither can I see a weak vorticity. Thus, the definition of vorticity is confusing. Another way to say the same thing is that the Euler to Lagrange coordinate transform is not clear. Is a background vertical flow included? Or do we only consider the vertical flow induced by the waves?
Nonlinear Gouyon waves
Journal of Physics A: Mathematical and Theoretical, 2021
To study stationary periodic weakly vortical waves on water (the Gouyon waves), the method of the... more To study stationary periodic weakly vortical waves on water (the Gouyon waves), the method of the modified Lagrangian variables is suggested. The wave vorticity Ω is specified as a series in the small steepness parameter ε in the form: Ω=∑n=1∞εn⋅Ωnb , where Ω n are arbitrary functions of the vertical Lagrangian coordinate b. Earlier Gouyon (1958) studied waves in linear and quadratic approximations. The present paper provides a complete problem solution in a cubic approximation. Explicit expressions are obtained for the coordinates of the liquid particle trajectories and pressure. The quadratic correction to the wave velocity is determined. The comparison with the Stokes and Gerstner waves is made. The class of quadratically vortical Gouyon waves is distinguished, for which Ω1 = 0. On the basis of the general dispersion theory method for waves of this class, the nonlinear Schrödinger equation is written. The criterion, when the equation is focusing and rogue Gouyon waves can be formed, is indicated.
Izvestiya, Atmospheric and Oceanic Physics, 2018
⎯A nonlinear Schrödinger equation (NSE) describing packets of weakly nonlinear waves in an inhomo... more ⎯A nonlinear Schrödinger equation (NSE) describing packets of weakly nonlinear waves in an inhomogeneously vortical infinitely deep fluid has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is shown that the modulational instability criteria for the weakly vortical waves and potential Stokes waves on deep water coincide. The effect of vorticity manifests itself in a shift of the wavenumber of high-frequency filling. A special case of Gerstner waves with a zero coefficient at the nonlinear term in the NSE is noted.
Nonlinear Processes in Geophysics Discussions, 2016
The nonlinear Schrödinger equation (NLS equation) describing weakly rotational wave packets in an... more The nonlinear Schrödinger equation (NLS equation) describing weakly rotational wave packets in an infinity-depth fluid in the Lagrangian coordinates is derived. The vorticity is assumed to be an arbitrary function of the Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is proved that the modulation instability criteria of the low-vorticity waves and deep water potential waves coincide. All the known solutions of the NLS equation for rogue waves are applicable to the low-vorticity waves. The effect of vorticity is manifested in a shift of the wave number in the carrier wave. In case of vorticity dependence on the vertical Lagrangian coordinate only (the Gouyon waves) this shift is constant. In a more general case, where the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally heterogeneous. There is a special case with the Gerstner waves where the vorticity is proportional to the square o...
Izvestiya VUZ. Applied Nonlinear Dynamics, 2016
Предложена новая этимология основных русских числительных. Происхождение названий первых четырех ... more Предложена новая этимология основных русских числительных. Происхождение названий первых четырех чисел связывается с древнейшей эротической культурной традицией. Рождение названий чисел от пяти до десяти и числа двенадцать объясняется влиянием славянского алфавита. Даны интерпретации слов сорок, девяносто, сто и тысяча.
Extreme Vortical Waves Under External Pressure Action
Physics of Wave Phenomena, 2010
The problems of the asymptotic theory of weakly nonlinear surface waves in viscous fluid are disc... more The problems of the asymptotic theory of weakly nonlinear surface waves in viscous fluid are discussed. For standing waves on deep water, the solutions obtained in the first-and second-order approximations in a small parameter-wave steepness-are analyzed. The evolution equation for the amplitude of wave packet envelope is obtained where the inverse Reynolds number is equal to the squared steepness. It is shown that this is a nonlinear Schr¨odinger equation with linear dissipation.
Physical Review Letters, 2013
A vortical model for freak wave formation in water is presented. The wind action is simulated by ... more A vortical model for freak wave formation in water is presented. The wind action is simulated by nonuniform pressure on the free surface. The motion of the fluid is described by an exact solution of 2D hydrodynamic equations for ideal inviscid fluid in Lagrangian variables. Fluid particles rotate in circles of different radius. The model describes the appearance of a freak wave in the field of the Gerstner wave. The physical parameters of the wave and feasibility of the proposed scenario are discussed.
Fluid Dynamics, 2013
Plane periodic oscillations of an infinitely deep fluid are studied in the case of a nonuniform p... more Plane periodic oscillations of an infinitely deep fluid are studied in the case of a nonuniform pressure distribution over its free surface. The fluid flow is governed by an exact solution of the Euler equations in the Lagrangian variables. The dynamics of an oscillating standing soliton are described, together with the scenario of the soliton evolution and the birth of a wave of an anomalously large amplitude against the background of the homogeneous Gerstner undulation (freak wave model). All the flows are nonuniformly vortical.
Gouyon waves in water of finite depth
Monatshefte für Mathematik, 2022
Large Oceanic Gyres: Lagrangian Description
Journal of Mathematical Fluid Mechanics
Wind generated equatorial Gerstner-type waves
Discrete & Continuous Dynamical Systems - A
Generalization of Cauchy invariants for equatorial β- plane flows
Deep Sea Research Part II: Topical Studies in Oceanography
Unsteady Gerstner waves
Chaos, Solitons & Fractals
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Papers by Anatoly Abrashkin