scholarly journals Two-dimensional generalized solitary waves and periodic waves under an ice sheet

2011 ◽  
Vol 369 (1947) ◽  
pp. 2957-2972 ◽  
Author(s):  
Jean-Marc Vanden-Broeck ◽  
Emilian I. Părău
Keyword(s):  
Solitary Waves ◽  
Gravity Waves ◽  
Ice Sheet ◽  
Periodic Waves ◽  
Two Dimensional ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Dimensional Gravity ◽  
Nonlinear Solutions

Two-dimensional gravity waves travelling under an ice sheet are studied. The flow is assumed to be potential. Weakly nonlinear solutions are derived and fully nonlinear solutions are calculated numerically. Periodic waves and generalized solitary waves are studied.

scholarly journals Stability and dynamics of two-dimensional fully nonlinear gravity–capillary solitary waves in deep water

2016 ◽  
Vol 809 ◽  
pp. 530-552 ◽  
Author(s):  
Z. Wang
Keyword(s):  
Solitary Waves ◽  
Deep Water ◽  
Wave Equations ◽  
Turning Point ◽  
Phase Speed ◽  
Amplitude Curve ◽  
Two Dimensional ◽  
Fully Nonlinear ◽  
Dimensional Gravity ◽  
The Stability

The stability and dynamics of two-dimensional gravity–capillary solitary waves in deep water within the fully nonlinear water-wave equations are numerically studied. It is well known that there are two families of symmetric gravity–capillary solitary waves – depression waves and elevation waves – bifurcating from infinitesimal periodic waves at the minimum of the phase speed. The stability of both branches was previously examined by Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) by means of a numerical spectral analysis. Their results show that the depression solitary waves with single-valued profiles are stable, while the elevation branch experiences a stability exchange at a turning point on the speed–amplitude curve. In the present paper, we provide numerical evidence that the depression solitary waves with an overhanging structure are also stable. On the other hand, Dias et al. (Eur. J. Mech. B, vol. 15, 1996, pp. 17–36) numerically traced the elevation branch and discovered that its speed–amplitude bifurcation curve features a ‘snake-like’ behaviour with many turning points, whereas Calvo & Akylas (J. Fluid Mech., vol. 452, 2002, pp. 123–143) only considered the stability exchange near the first turning point. Our results reveal that the stability exchange occurs again near the second turning point. A branch of asymmetric solitary waves is also considered and found to be unstable, even when the wave profile consists of a depression wave and a stable elevation one. The excitation of stable gravity–capillary solitary waves is carried out via direct numerical simulations. In particular, the stable elevation waves, which feature two troughs connected by a small dimple, can be excited by moving two fully localised, well-separated pressures on the free surface with the speed slightly below the phase speed minimum and removing the pressures simultaneously after a period of time.

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Anomalous width variation of rarefactive ion acoustic solitary waves in the context of auroral plasmas

2004 ◽  
Vol 11 (2) ◽  
pp. 219-228 ◽  
Author(s):  
S. S. Ghosh ◽  
G. S. Lakhina
Keyword(s):  
Solitary Waves ◽  
Nonlinear Theory ◽  
Acoustic Waves ◽  
Large Amplitude ◽  
Magnetized Plasma ◽  
Amplitude Variation ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Ion Acoustic Solitary Waves ◽  
Ion Acoustic

Abstract. The presence of dynamic, large amplitude solitary waves in the auroral regions of space is well known. Since their velocities are of the order of the ion acoustic speed, they may well be considered as being generated from the nonlinear evolution of ion acoustic waves. However, they do not show the expected width-amplitude correlation for K-dV solitons. Recent POLAR observations have actually revealed that the low altitude rarefactive ion acoustic solitary waves are associated with an increase in the width with increasing amplitude. This indicates that a weakly nonlinear theory is not appropriate to describe the solitary structures in the auroral regions. In the present work, a fully nonlinear analysis based on Sagdeev pseudopotential technique has been adopted for both parallel and oblique propagation of rarefactive solitary waves in a two electron temperature multi-ion plasma. The large amplitude solutions have consistently shown an increase in the width with increasing amplitude. The width-amplitude variation profile of obliquely propagating rarefactive solitary waves in a magnetized plasma have been compared with the recent POLAR observations. The width-amplitude variation pattern is found to fit well with the analytical results. It indicates that a fully nonlinear theory of ion acoustic solitary waves may well explain the observed anomalous width variations of large amplitude structures in the auroral region.

scholarly journals On the Dynamics of Two-Dimensional Capillary-Gravity Solitary Waves with a Linear Shear Current

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Dali Guo ◽  
Bo Tao ◽  
Xiaohui Zeng
Keyword(s):  
Solitary Waves ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Numerical Study ◽  
Two Dimensional ◽  
Shear Current ◽  
Linear Shear ◽  
The Stability ◽  
Depression Wave ◽  
Elevation Wave

The numerical study of the dynamics of two-dimensional capillary-gravity solitary waves on a linear shear current is presented in this paper. The numerical method is based on the time-dependent conformal mapping. The stability of different kinds of solitary waves is considered. Both depression wave and large amplitude elevation wave are found to be stable, while small amplitude elevation wave is unstable to the small perturbation, and it finally evolves to be a depression wave with tails, which is similar to the irrotational capillary-gravity waves.

scholarly journals Transverse instability of gravity–capillary solitary waves on deep water in the presence of constant vorticity

2019 ◽  
Vol 871 ◽  
pp. 1028-1043
Author(s):  
M. Abid ◽  
C. Kharif ◽  
H.-C. Hsu ◽  
Y.-Y. Chen
Keyword(s):  
Growth Rate ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Deep Water ◽  
Capillary Waves ◽  
Two Dimensional ◽  
Transverse Instability ◽  
Constant Vorticity ◽  
Dimensional Gravity ◽  
Wave Properties

The bifurcation of two-dimensional gravity–capillary waves into solitary waves when the phase velocity and group velocity are nearly equal is investigated in the presence of constant vorticity. We found that gravity–capillary solitary waves with decaying oscillatory tails exist in deep water in the presence of vorticity. Furthermore we found that the presence of vorticity influences strongly (i) the solitary wave properties and (ii) the growth rate of unstable transverse perturbations. The growth rate and bandwidth instability are given numerically and analytically as a function of the vorticity.

scholarly journals Dynamics of steep two-dimensional gravity–capillary solitary waves

2010 ◽  
Vol 664 ◽  
pp. 466-477 ◽  
Author(s):  
PAUL A. MILEWSKI ◽  
J.-M. VANDEN-BROECK ◽  
ZHAN WANG
Keyword(s):  
Solitary Waves ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Amplitude Wave ◽  
Linear Dispersion ◽  
Infinite Depth ◽  
Linear Dispersion Relation ◽  
Dimensional Gravity ◽  
Nonlinear Free Surface ◽  
The Waves

In this paper, the unsteady evolution of two-dimensional fully nonlinear free-surface gravity–capillary solitary waves is computed numerically in infinite depth. Gravity–capillary wavepacket-type solitary waves were found previously for the full Euler equations, bifurcating from the minimum of the linear dispersion relation. Small and moderate amplitude elevation solitary waves, which were known to be linearly unstable, are shown to evolve into stable depression solitary waves, together with a radiated wave field. Depression waves and certain large amplitude elevation waves were found to be robust to numerical perturbations. Two kinds of collisions are computed: head-on collisions whereby the waves are almost unchanged, and overtaking collisions which are either almost elastic if the wave amplitudes are both large or destroy the smaller wave in the case of a small amplitude wave overtaking a large one.

On the free-surface flow of very steep forced solitary waves

2013 ◽  
Vol 739 ◽  
pp. 1-21 ◽  
Author(s):  
Stephen L. Wade ◽  
Benjamin J. Binder ◽  
Trent W. Mattner ◽  
James P. Denier
Keyword(s):  
Free Surface ◽  
Solitary Waves ◽  
Flow Problem ◽  
Nonlinear Approximation ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Boundary Integral ◽  
Weakly Nonlinear ◽  
Nonlinear Solutions

AbstractThe free-surface flow of very steep forced and unforced solitary waves is considered. The forcing is due to a distribution of pressure on the free surface. Four types of forced solution are identified which all approach the Stokes-limiting configuration of an included angle of $12{0}^{\circ } $ and a stagnation point at the wave crests. For each type of forced solution the almost-highest wave does not contain the most energy, nor is it the fastest, similar to what has been observed previously in the unforced case. Nonlinear solutions are obtained by deriving and solving numerically a boundary integral equation. A weakly nonlinear approximation to the flow problem helps with the identification and classification of the forced types of solution, and their stability.

scholarly journals The effect of disturbances on the flows under a sluice gate and past an inclined plate

2007 ◽  
Vol 576 ◽  
pp. 475-490 ◽  
Author(s):  
B. J. BINDER ◽  
J.-M. VANDEN-BROECK
Keyword(s):  
Free Surface ◽  
Boundary Integral ◽  
Sluice Gate ◽  
Inclined Plate ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Potential Flows ◽  
Integral Equation Methods ◽  
Different Types ◽  
Nonlinear Solutions

Free surface potential flows past disturbances in a channel are considered. Three different types of disturbance are studied: (i) a submerged obstacle on the bottom of a channel; (ii) a pressure distribution on the free surface; and (iii) an obstruction in the free surface (e.g. a sluice gate or a flat plate). Surface tension is neglected, but gravity is included in the dynamic boundary condition. Fully nonlinear solutions are computed by boundary integral equation methods. In addition, weakly nonlinear solutions are derived. New solutions are found when several disturbances are present simultaneously. They are discovered through the weakly nonlinear analysis and confirmed by numerical computations for the fully nonlinear problem.

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