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Symmetry
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Wave Processes in Fluids with Symmetric Density Stratification
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Wave Processes in Fluids with Symmetric Density Stratification

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 November 2022) | Viewed by 16000

Special Issue Editor

Prof. Dr. Oxana E. Kurkina

E-Mail Website
Guest Editor
Laboratory of Modeling of Natural and Anthropogenic Disasters, Nizhny Novgorod State Technical University, 603950, 24 Minin street, Nizhny Novgorod, Russia

Special Issue Information

Dear Colleagues,

Internal gravity waves are one of the most important components of wave motions in stratified media. They arise and propagate at the interfaces of layers of different densities in a stratified fluid. The dynamics of internal waves has been studied quite well for two-layer stratification, and a large number of research articles have recently appeared devoted to studies of wave dynamics in a three-layer fluid, where more complex and interesting dynamic regimes can be observed.

A special issue is devoted to the description of wave processes in stratified fluids with a symmetric in vertical (with respect to mid-depth) distribution of density and / or shear flow, both layered and with continuous laws of density and flow velocity variations in depth. Such fluid configurations are specific because when described within approximate asymptotic models, symmetry leads to degeneration of nonlinearities in certain orders of expansion, and, as a consequence, to the necessity of taking into account additional higher-order terms, and, therefore, to new effects in wave dynamics which are not present in the absence of symmetry. In particular, the simplest evolutionary equation here is the modified Korteweg-de Vries equation, not the more common classical Korteweg-de Vries equation.

Сontributions are invited covering a broad range of topics including: analytical methods for deriving nonlinear evolutionary models, numerical modeling, nonlinear wave dynamics, localized non-radiating solutions (solitons and breathers), stationary solutions, wave interactions, shear instability, Lagrangian transport, accounting the influence of additional factors (capillary effects, rotation, etc.) in the context of wave phenomena in inhomogeneous symmetric fluids.

Prof. Dr. Oxana E. Kurkina
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • internal waves
  • stratified flows
  • symmetric density stratification
  • layered fluid
  • solitary waves
  • breathers
  • nonlinear evolution equations
  • numerical modeling

Published Papers (9 papers)

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Research

14 pages, 1184 KiB  
Article
Solitary Waves and Their Interactions in the Cylindrical Korteweg–De Vries Equation
by Wencheng Hu, Jingli Ren and Yury Stepanyants
Symmetry 2023, 15(2), 413; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15020413 - 03 Feb 2023
Cited by 5 | Viewed by 1090
Abstract
We consider approximate, exact, and numerical solutions to the cylindrical Korteweg–de Vries equation. We show that there are different types of solitary waves and obtain the dependence of their parameters on distance. Then, we study the interaction of solitary waves of different types. [...] Read more.
We consider approximate, exact, and numerical solutions to the cylindrical Korteweg–de Vries equation. We show that there are different types of solitary waves and obtain the dependence of their parameters on distance. Then, we study the interaction of solitary waves of different types. Full article
(This article belongs to the Special Issue Wave Processes in Fluids with Symmetric Density Stratification)
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14 pages, 4499 KiB  
Article
Nonlinear Transformation of Sine Wave within the Framework of Symmetric (2+4) KdV Equation
by Oxana Kurkina and Efim Pelinovsky
Symmetry 2022, 14(4), 668; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14040668 - 24 Mar 2022
Cited by 3 | Viewed by 1583
Abstract
This paper considers the transformation of a sine wave in the framework of the extended modified Korteweg–de Vries equation or (2+4) KdV, which includes a combination of cubic and quintic nonlinearities. It describes the internal waves in a medium with symmetric vertical density [...] Read more.
This paper considers the transformation of a sine wave in the framework of the extended modified Korteweg–de Vries equation or (2+4) KdV, which includes a combination of cubic and quintic nonlinearities. It describes the internal waves in a medium with symmetric vertical density stratification, and all the considerations in this study are produced for the reasonable combinations of the signs of the coefficients for nonlinear and dispersive terms, provided by this physical problem. The features of Riemann waves—solutions of the dispersionless limit of the model—are described in detail: The times and levels of breaking are derived in an implicit analytic form depending on the amplitude of the initial sine wave. It is demonstrated that the shock occurs at two (for small amplitudes) or four (for moderate and large amplitudes) levels per period of sine wave. Breaking at different levels occurs at different times. The symmetric (2+4) KdV equation is not integrable, but nevertheless it has stationary solutions in the form of traveling solitary waves of both polarities with a limiting amplitude. With the help of numerical calculations, the features of the processes of a sinusoidal wave evolution and formation of undular bores are demonstrated and analyzed. Qualitative features of multiple inelastic interactions of emerging soliton-like pulses are displayed. Full article
(This article belongs to the Special Issue Wave Processes in Fluids with Symmetric Density Stratification)
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17 pages, 578 KiB  
Article
Modulation Instability of Surface Waves in the Model with the Uniform Wind Profile
by Susam Boral, Trilochan Sahoo and Yury Stepanyants
Symmetry 2021, 13(4), 651; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13040651 - 12 Apr 2021
Cited by 5 | Viewed by 1645
Abstract
The modulation instability of surface capillary-gravity water waves is analysed in a shear flow model with a tangential discontinuity of velocity. It is assumed that air blows along the surface of the water with a uniform profile in the vertical direction. Such a [...] Read more.
The modulation instability of surface capillary-gravity water waves is analysed in a shear flow model with a tangential discontinuity of velocity. It is assumed that air blows along the surface of the water with a uniform profile in the vertical direction. Such a model, despite its simplicity, plays an important role in hydrodynamics as the reference model for investigating basic physical phenomena of wave–current interactions and acquiring insights into a series of complex phenomena. In certain cases where the wavelength of interfacial perturbations is much bigger than the width of the shear flow profile, the model with the tangential discontinuity in the velocity is adequate for describing physical phenomena at least within limited spatial and temporal frameworks. A detailed analysis of the air-flow conditions under which modulation instability sets in is presented. It is also shown that the interfacial waves are subject to dissipative or radiative instability when negative-energy waves appear at the interface. Full article
(This article belongs to the Special Issue Wave Processes in Fluids with Symmetric Density Stratification)
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9 pages, 1384 KiB  
Article
Generation of Internal Gravity Waves Far from Moving Non-Local Source
by Vitaly Bulatov and Yury Vladimirov
Symmetry 2020, 12(11), 1899; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12111899 - 19 Nov 2020
Cited by 12 | Viewed by 1494
Abstract
We consider analytical solutions describing the generation of internal gravity waves far from a non-local source of disturbances. We suppose that the source moves on the surface of stratified medium of a finite depth. A model distribution of the non-local source shape with [...] Read more.
We consider analytical solutions describing the generation of internal gravity waves far from a non-local source of disturbances. We suppose that the source moves on the surface of stratified medium of a finite depth. A model distribution of the non-local source shape with radial symmetry is used. This approximation correctly describes (qualitatively) the main spatiotemporal characteristics of natural sources of generation of internal gravity waves in the ocean. The resulting solution is the sum of wave modes. The solution is presented as a series of eigenfunctions of the spectral problem of internal gravity waves. The results of numerical calculations of internal gravity waves components at different depths are presented and discussed. Full article
(This article belongs to the Special Issue Wave Processes in Fluids with Symmetric Density Stratification)
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11 pages, 2432 KiB  
Article
Analytical Approximations of Dispersion Relations for Internal Gravity Waves Equation with Shear Flows
by Vitaly Bulatov and Yury Vladimirov
Symmetry 2020, 12(11), 1865; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12111865 - 13 Nov 2020
Cited by 10 | Viewed by 1528
Abstract
The problem of internal gravity waves fields in a stratified medium of finite depth is considered for model distributions of background shear currents. For the analytical solution of the problem, a constant distribution of the buoyancy frequency and various linear dependences of the [...] Read more.
The problem of internal gravity waves fields in a stratified medium of finite depth is considered for model distributions of background shear currents. For the analytical solution of the problem, a constant distribution of the buoyancy frequency and various linear dependences of the background shear current on depth were used. The dispersion dependences are obtained, which are expressed in terms of the modified Bessel function of the imaginary index. Under the Miles–Howard stability condition and large Richardson numbers, the Debye asymptotics of the modified Bessel function of the imaginary index were used to construct analytical solutions. The dispersion equation is solved using the proposed analytical approximation. The properties of the dispersion equation are studied and the main analytical characteristics of the dispersion curves are investigated depending on the parameters of background shear flows. Full article
(This article belongs to the Special Issue Wave Processes in Fluids with Symmetric Density Stratification)
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13 pages, 2024 KiB  
Article
The Asymptotic Approach to the Description of Two-Dimensional Symmetric Soliton Patterns
by Yury Stepanyants
Symmetry 2020, 12(10), 1586; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12101586 - 24 Sep 2020
Cited by 5 | Viewed by 1996
Abstract
The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe stationary moving symmetric wave patterns consisting of two plane solitary waves of equal amplitudes moving at an angle to each other. [...] Read more.
The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe stationary moving symmetric wave patterns consisting of two plane solitary waves of equal amplitudes moving at an angle to each other. The results obtained within the approximate asymptotic theory are validated by comparison with the exact two-soliton solution of the Kadomtsev–Petviashvili equation (KP2-equation). The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin–Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers. Full article
(This article belongs to the Special Issue Wave Processes in Fluids with Symmetric Density Stratification)
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8 pages, 1565 KiB  
Article
Soliton–Breather Interaction: The Modified Korteweg–de Vries Equation Framework
by Ekaterina Didenkulova and Efim Pelinovsky
Symmetry 2020, 12(9), 1445; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12091445 - 02 Sep 2020
Cited by 3 | Viewed by 2091
Abstract
Pairwise interactions of particle-like waves (such as solitons and breathers) are important elementary processes that play a key role in the formation of the rarefied soliton gas statistics. Such waves appear in different physical systems such as deep water, shallow water waves, internal [...] Read more.
Pairwise interactions of particle-like waves (such as solitons and breathers) are important elementary processes that play a key role in the formation of the rarefied soliton gas statistics. Such waves appear in different physical systems such as deep water, shallow water waves, internal waves in the stratified ocean, and optical fibers. We study the features of different regimes of collisions between a soliton and a breather in the framework of the focusing modified Korteweg–de Vries equation, where cubic nonlinearity is essential. The relative phase of these structures is an important parameter determining the dynamics of soliton–breather collisions. Two series of experiments with different values of the breather’s and soliton’s relative phases were conducted. The waves’ amplitudes resulting from the interaction of coherent structures depending on their relative phase at the moment of collision were analyzed. Wave field moments, which play a decisive role in the statistics of soliton gases, were determined. Full article
(This article belongs to the Special Issue Wave Processes in Fluids with Symmetric Density Stratification)
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23 pages, 12695 KiB  
Article
Formation of the Dynamic Energy Cascades in Quartic and Quintic Generalized KdV Equations
by Denys Dutykh and Elena Tobisch
Symmetry 2020, 12(8), 1254; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12081254 - 29 Jul 2020
Cited by 1 | Viewed by 1832
Abstract
In this study we investigate for the first time the formation of dynamical energy cascades in higher order KdV-type equations. In the beginning we recall what is known about the dynamic cascades for the classical KdV (quadratic) and mKdV (cubic) equations. Then, we [...] Read more.
In this study we investigate for the first time the formation of dynamical energy cascades in higher order KdV-type equations. In the beginning we recall what is known about the dynamic cascades for the classical KdV (quadratic) and mKdV (cubic) equations. Then, we investigate further the mKdV case by considering a richer set of initial perturbations in order to check the validity and persistence of various facts previously established for the narrow-banded perturbations. Afterwards we focus on higher order nonlinearities (quartic and quintic) which are found to be quite different in many respects from the mKdV equation. Throughout this study we consider both the direct and double energy cascades. It was found that the dynamic cascade is always formed, but its formation is not necessarily accompanied by the nonlinear stage of the modulational instability. The direct cascade structure remains invariant regardless of the size of the spectral domain. In contrast, the double cascade shape can depend on the size of the spectral domain, even if the total number of cascading modes remains invariant. The results obtained in this study can be potentially applied to plasmas, free surface and internal wave hydrodynamics. Full article
(This article belongs to the Special Issue Wave Processes in Fluids with Symmetric Density Stratification)
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12 pages, 6404 KiB  
Article
Breather’s Properties within the Framework of the Modified Korteweg–de Vries Equation
by Ekaterina Didenkulova and Efim Pelinovsky
Symmetry 2020, 12(4), 638; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12040638 - 17 Apr 2020
Cited by 5 | Viewed by 1942
Abstract
We study a breather’s properties within the framework of the modified Korteweg–de Vries (mKdV) model, where cubic nonlinearity is essential. Extrema, moments, and invariants of a breather with different parameters have been analyzed. The conditions in which a breather moves in one direction [...] Read more.
We study a breather’s properties within the framework of the modified Korteweg–de Vries (mKdV) model, where cubic nonlinearity is essential. Extrema, moments, and invariants of a breather with different parameters have been analyzed. The conditions in which a breather moves in one direction or another has been determined. Two limiting cases have been considered: when a breather has an N-wave shape and can be interpreted as two solitons with different polarities, and when a breather contains many oscillations and can be interpreted as an envelope soliton of the nonlinear Schrödinger equation (NLS). Full article
(This article belongs to the Special Issue Wave Processes in Fluids with Symmetric Density Stratification)
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