scholarly journals Propagation regimes of interfacial solitary waves in a three-layer fluid

2015 ◽  
Vol 22 (2) ◽  
pp. 117-132 ◽  
Author(s):  
O. E. Kurkina ◽  
A. A. Kurkin ◽  
E. A. Rouvinskaya ◽  
T. Soomere
Keyword(s):  
Internal Waves ◽  
Fluid Layer ◽  
Soliton Solutions ◽  
Boussinesq Approximation ◽  
Finite Amplitude ◽  
Stable Configuration ◽  
Weakly Nonlinear ◽  
Small Parameters ◽  
Specific Equation ◽  
Kdv Type Equations

Abstract. Long weakly nonlinear finite-amplitude internal waves in a fluid consisting of three inviscid layers of arbitrary thickness and constant densities (stable configuration, Boussinesq approximation) bounded by a horizontal rigid bottom from below and by a rigid lid at the surface are described up to the second order of perturbation theory in small parameters of nonlinearity and dispersion. First, a pair of alternatives of appropriate KdV-type equations with the coefficients depending on the parameters of the fluid (layer positions and thickness, density jumps) are derived for the displacements of both modes of internal waves and for each interface between the layers. These equations are integrable for a very limited set of coefficients and do not allow for proper description of several near-critical cases when certain coefficients vanish. A more specific equation allowing for a variety of solitonic solutions and capable of resolving most near-critical situations is derived by means of the introduction of another small parameter that describes the properties of the medium and rescaling of the ratio of small parameters. This procedure leads to a pair of implicitly interrelated alternatives of Gardner equations (KdV-type equations with combined nonlinearity) for the two interfaces. We present a detailed analysis of the relationships for the solutions for the disturbances at both interfaces and various regimes of the appearance and propagation properties of soliton solutions to these equations depending on the combinations of the parameters of the fluid. It is shown that both the quadratic and the cubic nonlinear terms vanish for several realistic configurations of such a fluid.

scholarly journals Propagation regimes of interfacial solitary waves in a three-layer fluid

2015 ◽  
Vol 2 (1) ◽  
pp. 1-41
Author(s):  
O. E. Kurkina ◽  
A. A. Kurkin ◽  
E. A. Rouvinskaya ◽  
T. Soomere
Keyword(s):  
Internal Waves ◽  
Fluid Layer ◽  
Soliton Solutions ◽  
Boussinesq Approximation ◽  
Finite Amplitude ◽  
Stable Configuration ◽  
Weakly Nonlinear ◽  
Small Parameters ◽  
Specific Equation ◽  
Kdv Type Equations

Abstract. Long weakly nonlinear finite-amplitude internal waves in a fluid consisting of three inviscid immiscible layers of arbitrary thickness and constant densities (stable configuration, Boussinesq approximation) bounded by a horizontal rigid bottom from below and by a rigid lid at the surface are described up to the second order of perturbation theory in small parameters of nonlinearity and dispersion. First, a pair of alternatives of appropriate KdV-type equations with the coefficients depending on the parameters of the fluid (layer positions and thickness, density jumps) are derived for the displacements of both modes of internal waves and for each interface between the layers. These equations are integrable for a very limited set of coefficients and do not allow for proper description of several near-critical cases when certain coefficients vanish. A more specific equation allowing for a variety of solitonic solutions and capable of resolving most of near-critical situations is derived by means of the introduction of another small parameter that describes the properties of the medium and rescaling of the ratio of small parameters. This procedure leads to a pair of implicitly interrelated alternatives of Gardner equation (KdV-type equations with combined nonlinearity) for the two interfaces. We present a detailed analysis of the relationships for the solutions for the disturbances at both interfaces and various regimes of the appearance and propagation properties of soliton solutions to these equations depending on the combinations of the parameters of the fluid. It is shown both the quadratic and the cubic nonlinear terms vanish for several realistic configurations of such a fluid.

Finite-amplitude internal wavepacket dispersion and breaking

2001 ◽  
Vol 429 ◽  
pp. 343-380 ◽  
Author(s):  
BRUCE R. SUTHERLAND
Keyword(s):  
Numerical Simulations ◽  
Internal Waves ◽  
Large Amplitude ◽  
Finite Amplitude ◽  
Critical Amplitude ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
The Stability ◽  
The Waves ◽  
Horizontal Wavelength

The evolution and stability of two-dimensional, large-amplitude, non-hydrostatic internal wavepackets are examined analytically and by numerical simulations. The weakly nonlinear dispersion relation for horizontally periodic, vertically compact internal waves is derived and the results are applied to assess the stability of weakly nonlinear wavepackets to vertical modulations. In terms of Θ, the angle that lines of constant phase make with the vertical, the wavepackets are predicted to be unstable if [mid ]Θ[mid ] < Θc, where Θc = cos−1 (2/3)1/2 ≃ 35.3° is the angle corresponding to internal waves with the fastest vertical group velocity. Fully nonlinear numerical simulations of finite-amplitude wavepackets confirm this prediction: the amplitude of wavepackets with [mid ]Θ[mid ] > Θc decreases over time; the amplitude of wavepackets with [mid ]Θ[mid ] < Θc increases initially, but then decreases as the wavepacket subdivides into a wave train, following the well-known Fermi–Pasta–Ulam recurrence phenomenon.If the initial wavepacket is of sufficiently large amplitude, it becomes unstable in the sense that eventually it convectively overturns. Two new analytic conditions for the stability of quasi-plane large-amplitude internal waves are proposed. These are qualitatively and quantitatively different from the parametric instability of plane periodic internal waves. The ‘breaking condition’ requires not only that the wave is statically unstable but that the convective instability growth rate is greater than the frequency of the waves. The critical amplitude for breaking to occur is found to be ACV = cot Θ (1 + cos2 Θ)/2π, where ACV is the ratio of the maximum vertical displacement of the wave to its horizontal wavelength. A second instability condition proposes that a statically stable wavepacket may evolve so that it becomes convectively unstable due to resonant interactions between the waves and the wave-induced mean flow. This hypothesis is based on the assumption that the resonant long wave–short wave interaction, which Grimshaw (1977) has shown amplifies the waves linearly in time, continues to amplify the waves in the fully nonlinear regime. Using linear theory estimates, the critical amplitude for instability is ASA = sin 2Θ/(8π2)1/2. The results of numerical simulations of horizontally periodic, vertically compact wavepackets show excellent agreement with this latter stability condition. However, for wavepackets with horizontal extent comparable with the horizontal wavelength, the wavepacket is found to be stable at larger amplitudes than predicted if Θ [lsim ] 45°. It is proposed that these results may explain why internal waves generated by turbulence in laboratory experiments are often observed to be excited within a narrow frequency band corresponding to Θ less than approximately 45°.

Finite-Amplitude Baroclinic Instability of Time-Varying Abyssal Currents

2006 ◽  
Vol 36 (1) ◽  
pp. 122-139 ◽  
Author(s):  
Seung-Ji Ha ◽  
Gordon E. Swaters
Keyword(s):  
Baroclinic Instability ◽  
Soliton Solutions ◽  
Zonal Flow ◽  
Finite Amplitude ◽  
Marginal Stability ◽  
Time Variability ◽  
Time Varying ◽  
Weakly Nonlinear ◽  
Abyssal Current ◽  
The Stability

Abstract The weakly nonlinear baroclinic instability characteristics of time-varying grounded abyssal flow on sloping topography with dissipation are described. Specifically, the finite-amplitude evolution of marginally unstable or stable abyssal flow both at and removed from the point of marginal stability (i.e., the minimum shear required for instability) is determined. The equations governing the evolution of time-varying dissipative abyssal flow not at the point of marginal stability are identical to those previously obtained for the Phillips model for zonal flow on a β plane. The stability problem at the point of marginally stability is fully nonlinear at leading order. A wave packet model is introduced to examine the role of dissipation and time variability in the background abyssal current. This model is a generalization of one introduced for the baroclinic instability of zonal flow on a β plane. A spectral decomposition and truncation leads, in the absence of time variability in the background flow and dissipation, to the sine–Gordon solitary wave equation that has grounded abyssal soliton solutions. The modulation characteristics of the soliton are determined when the underlying abyssal current is marginally stable or unstable and possesses time variability and/or dissipation. The theory is illustrated with examples.

Multi-Soliton and Rational Solutions for the Extended Fifth-Order KdV Equation in Fluids

2015 ◽  
Vol 70 (7) ◽  
pp. 559-566 ◽  
Author(s):  
Gao-Qing Meng ◽  
Yi-Tian Gao ◽  
Da-Wei Zuo ◽  
Yu-Jia Shen ◽  
Yu-Hao Sun ◽  
...  
Keyword(s):  
Kdv Equation ◽  
Soliton Solutions ◽  
Second Order ◽  
Wave Shape ◽  
Rational Solutions ◽  
Weakly Nonlinear ◽  
First Order ◽  
Propagation Properties ◽  
Kdv Type Equations ◽  
Fifth Order

AbstractKorteweg–de Vries (KdV)-type equations are used as approximate models governing weakly nonlinear long waves in fluids, where the first-order nonlinear and dispersive terms are retained and in balance. The retained second-order terms can result in the extended fifth-order KdV equation. Through the Darboux transformation (DT), multi-soliton solutions for the extended fifth-order KdV equation with coefficient constraints are constructed. Soliton propagation properties and interactions are studied: except for the velocity, the amplitude and width of the soliton are not influenced by the coefficient of the original equation; the amplitude, velocity, and wave shape of each soltion remain unchanged after the interaction. By virtue of the generalised DT and Taylor expansion of the solutions for the corresponding Lax pair, the first- and second-order rational solutions of the equation are obtained.

Rolls versus squares in thermal convection of fluids with temperature-dependent viscosity

1987 ◽  
Vol 178 ◽  
pp. 491-506 ◽  
Author(s):  
D. R. Jenkins
Keyword(s):  
Thermal Convection ◽  
Fluid Layer ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Finite Amplitude ◽  
Temperature Dependent Viscosity ◽  
Weakly Nonlinear ◽  
Expansion Procedure ◽  
The Stability ◽  
Dependent Viscosity

We consider finite-amplitude thermal convection, in a horizontal fluid layer. The viscosity of the fluid is dependent upon its temperature. Using a weakly nonlinear expansion procedure, we examine the stability of two-dimensional roll and three-dimensional square planforms, in order to determine which should be preferred in convection experiments. The analysis shows that the roll planform is preferred for low values of the ratio of the viscosities at the top and bottom boundaries, but the square planform is preferred for larger values of the ratio. At still larger values, subcritical convection is predicted. We also include the effects of boundaries having finite thermal conductivity, which enables favourable comparison to be made with experimental studies. A discrepancy between the present work and a previous study of this problem (Busse & Frick 1985) is discussed.

On the shape of progressive internal waves

1968 ◽  
Vol 263 (1145) ◽  
pp. 563-614 ◽  
Keyword(s):  
Experimental Evidence ◽  
Internal Waves ◽  
Gravity Waves ◽  
Boussinesq Approximation ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Fluid System ◽  
Expansion Technique ◽  
Theoretical Predictions ◽  
Two Fluid

An expansion technique, analogous to that of Stokes in the study of surface waves, is used to investigate the effects of finite amplitude on a progressive train of internal gravity waves. The paper is divided into two main parts, a study of interfacial waves in a two-fluid system and an examination of internal waves in a continuously stratified fluid. Experimental evidence is presented which confirms some of the theoretical predictions. The validity of the Boussinesq approximation is examined and particular examples are taken to illustrate the general results.

On the generation of shelves by long nonlinear waves in stratified flows

1997 ◽  
Vol 346 ◽  
pp. 345-362 ◽  
Author(s):  
DILIP PRASAD ◽  
T. R. AKYLAS
Keyword(s):  
Fluid Layer ◽  
Outer Region ◽  
Finite Depth ◽  
Finite Amplitude ◽  
Stratified Flows ◽  
Wave Disturbance ◽  
Necessary Condition ◽  
Matched Asymptotics ◽  
Weakly Nonlinear ◽  
Boussinesq Fluid

The phenomenon of shelf generation by long nonlinear internal waves in stratified flows is investigated. The problem of primary interest is the case of a uniformly stratified Boussinesq fluid of finite depth. In analysing the transient evolution of a finite-amplitude long-wave disturbance, the expansion procedure of Grimshaw & Yi (1991) breaks down far downstream, and it proves expedient to follow a matched-asymptotics procedure: the main disturbance is governed by the nonlinear theory of Grimshaw & Yi (1991) in the ‘inner’ region, while the ‘outer’ region comprises multiple small-amplitude fronts, or shelves, that propagate downstream and carry O(1) mass. This picture is consistent with numerical simulations of uniformly stratified flow past an obstacle (Lamb 1994). The case of weakly nonlinear long waves in a fluid layer with general stratification is also examined, where it is found that shelves of fourth order in wave amplitude are generated. Moreover, these shelves may extend both upstream and downstream in general, and could thus lead to an upstream influence of a type that has not been previously considered. In all cases, transience of the main nonlinear wave disturbance is a necessary condition for the formation of shelves.

scholarly journals The Complex Ginzburg Landau Model for an Oscillatory Convection in a Rotating Fluid Layer

2020 ◽  
Vol 25 (1) ◽  
pp. 75-91
Author(s):  
S.H. Manjula ◽  
P. Kiran ◽  
P. Raj Reddy ◽  
B.S. Bhadauria
Keyword(s):  
Heat Transfer ◽  
Thermal Instability ◽  
Fluid Layer ◽  
Landau Equation ◽  
Finite Amplitude ◽  
Oscillatory Mode ◽  
Stationary Mode ◽  
Ginzburg Landau ◽  
Weakly Nonlinear ◽  
Speed Modulation

AbstractA weakly nonlinear thermal instability is investigated under rotation speed modulation. Using the perturbation analysis, a nonlinear physical model is simplified to determine the convective amplitude for oscillatory mode. A non-autonomous complex Ginzburg-Landau equation for the finite amplitude of convection is derived based on a small perturbed parameter. The effect of rotation is found either to stabilize or destabilize the system. The Nusselt number is obtained numerically to present the results of heat transfer. It is found that modulation has a significant effect on heat transport for lower values of ωf while no effect for higher values. It is also found that modulation can be used alternately to control the heat transfer in the system. Further, oscillatory mode enhances heat transfer rather than stationary mode.

Internal waves of finite amplitude and permanent form

1966 ◽  
Vol 25 (2) ◽  
pp. 241-270 ◽  
Author(s):  
T. Brooke Benjamin
Keyword(s):  
Internal Waves ◽  
Present Theory ◽  
Boussinesq Approximation ◽  
Finite Amplitude ◽  
Physical Models ◽  
Physical Conditions ◽  
Explicit Formulae ◽  
The Waves ◽  
Wave Phenomena ◽  
Permanent Form

A theory is derived for the class of long two-dimensional waves, comprising solitary and periodic cnoidal waves, that can propagate with unchanging form in heterogeneous fluids. The treatment is generalized to the extent that the waves are supposed to arise on a horizontal stream of incompressible fluid whose density and velocity are arbitrary functions of height, and the upper surface of the fluid is allowed either to be free or to be fixed in a horizontal plane. Explicit formulae for the wave properties and a general interpretation of the physical conditions for the occurrence of the waves are achieved without need to specify particular physical models; but in a later part of the paper, §4, the results are applied to three examples that have been worked out by other means and so provide checks on the present theory. These general results are also shown to accord nicely with the principle of ‘conjugate-flow pairs’ which was explained by Benjamin (1962b) with reference to swirling flows along cylindrical ducts, but which is known to apply equally well to flow systems of the kind in question here.The theory reveals certain physical peculiarities of a type of flow model often used in theoretical studies of internal-wave phenomena, being specified so as to make the equation for the stream-function linear. In an appendix, some observations are also made regarding the ‘Boussinesq approximation’, which too is often used as a simplifying assumption in this field. It is shown, adding to a recent discussion by Long (1965), that finite internal waves may depend crucially on small effects neglected in this approximation.

Stability of solitary waves on deep water with constant vorticity

2021 ◽  
Author(s):  
Alexander Dosaev ◽  
Maria Shishina ◽  
Yuliya Troitskaya
Keyword(s):  
Solitary Waves ◽  
Deep Water ◽  
Equations Of Motion ◽  
Soliton Solutions ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Constant Vorticity ◽  
Rfbr Grant ◽  
Stability Of Solitary Waves ◽  
Wave Limit

&lt;p&gt;Waves on deep water with constant vorticity propagating in the direction of the shear are known to be weakly dispersive in the long wave limit. Weakly-nonlinear evolution of such waves can be described by the Benjamin-Ono equation, which is integrable and has stable soliton solutions. In the present study we investigate behaviour of finite-amplitude counterparts of Benjamin-Ono solitons by modelling their dynamics within exact equations of motion (Euler equations). Due to the solitons having a near-Lorentzian shape with slowly decaying tails, we need to approach them by examining periodic waves, whose crests, indeed, become more and more localised as the period increases. We perform a parameter space study and analyse how stability of very long waves depends on their amplitude and period. We show that large-amplitude solitary waves are unstable.&lt;br&gt;This research was supported by RFBR (grant No. 16-05-00839) and by the President of Russian Federation (grant No. MK-2041.2017.5). Numerical experiments were supported by RSF grant No. 14-17-00667, data processing was supported by RSF grant No. 15-17-20009.&lt;/p&gt;

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