scholarly journals Nonlinear hydroelastic waves on a linear shear current at finite depth

2019 ◽  
Vol 876 ◽  
pp. 55-86 ◽  
Author(s):  
T. Gao ◽  
Z. Wang ◽  
P. A. Milewski
Keyword(s):  
Solitary Waves ◽  
Small Amplitude ◽  
Modulational Instability ◽  
Finite Depth ◽  
Time Dependent ◽  
Mapping Technique ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Constant Vorticity

This work is concerned with waves propagating on water of finite depth with a constant-vorticity current under a deformable flexible sheet. The pressure exerted by the sheet is modelled by using the Cosserat thin shell theory. By means of multi-scale analysis, small amplitude nonlinear modulation equations in several regimes are considered, including the nonlinear Schrödinger equation (NLS) which is used to predict the existence of small-amplitude wavepacket solitary waves in the full Euler equations and to study the modulational instability of quasi-monochromatic wavetrains. Guided by these weakly nonlinear results, fully nonlinear steady and time-dependent computations are performed by employing a conformal mapping technique. Bifurcation mechanisms and typical profiles of solitary waves for different underlying shear currents are presented in detail. It is shown that even when small-amplitude solitary waves are not predicted by the weakly nonlinear theory, we can numerically find large-amplitude solitary waves in the fully nonlinear equations. Time-dependent simulations are carried out to confirm the modulational stability results and illustrate possible outcomes of the nonlinear evolution in unstable cases.

scholarly journals Two-dimensional stability of finite-amplitude gravity waves on water of finite depth with constant vorticity

2017 ◽  
Vol 830 ◽  
pp. 631-659 ◽  
Author(s):  
M. Francius ◽  
C. Kharif
Keyword(s):  
Growth Rate ◽  
Modulational Instability ◽  
Finite Depth ◽  
Finite Amplitude ◽  
Numerical Results ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Constant Vorticity ◽  
Shear Current ◽  
Modulational Instabilities

A numerical investigation of normal-mode perturbations of a two-dimensional periodic finite-amplitude gravity wave propagating on a vertically sheared current of constant vorticity is considered. For this purpose, an extension of the method developed by Rienecker & Fenton (J. Fluid Mech., vol. 104, 1981, pp. 119–137) is used for the numerical computations of the finite-amplitude waves on a linear shear current. This method enables to compute accurately waves with or without critical layers and pressure anomalies. The numerical results of the linear stability analysis extend the weakly nonlinear analytical results of Thomas et al. (Phys. Fluids, vol. 24, 2012, 127102) to fully nonlinear waves. In particular, the restabilization of the Benjamin–Feir modulational instability, whatever the depth, for an opposite shear current is confirmed. For these sideband instabilities, the numerical results show some deviations with the weakly nonlinear theory as the wave steepness of the basic wave and vorticity are increased. Besides the modulational instabilities, new instability bands corresponding to quartet and quintet instabilities, which are not sideband disturbances, are discovered. The present numerical results show that with opposite shear currents, increasing the shear reduces the growth rate of the most unstable sideband instabilities but enhances the growth rate of these quartet instabilities, which eventually dominate the Benjamin–Feir modulational instabilities.

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 Experimental study of dispersion and modulational instability of surface gravity waves on constant vorticity currents

2020 ◽  
Author(s):  
Ton van den Bremer ◽  
James Steer ◽  
Dimitris Stagonas ◽  
Eugeny Buldakov ◽  
Alistair Borthwick
Keyword(s):  
Shear Rate ◽  
Numerical Solutions ◽  
Modulational Instability ◽  
Surface Current ◽  
Surface Velocity ◽  
Spectral Evolution ◽  
Weakly Nonlinear ◽  
Shear Rates ◽  
Constant Vorticity ◽  
Uniform Current

<p>We examine experimentally the dispersion and stability of weakly nonlinear waves on opposing linearly vertically sheared current profiles (with constant vorticity). Measurements are compared against predictions from the unidirectional  1D+1 constant vorticity nonlinear Schrödinger equation (the vor-NLSE) derived by Thomas et al. (Phys. Fluids, vol. 24, no. 12, 2012, 127102). The shear rate is negative in opposing currents when the magnitude of the current in the laboratory reference frame is negative (i.e. opposing the direction of wave propagation) and reduces with depth, as is most commonly encountered in nature. Compared to a uniform current with the same surface velocity, negative shear has the effect of increasing wavelength and enhancing stability. In experiments with a regular low-steepness wave, the dispersion relationship between wavelength and frequency is examined on five opposing current profiles with shear rates from 0 to -0,87 s<sup>-1</sup> For all current profiles, the linear constant vorticity dispersion relation predicts the wavenumber to within the 95% confidence bounds associated with estimates of shear rate and surface current velocity. The effect of shear on modulational instability was determined by the spectral evolution of a carrier wave seeded with spectral sidebands on opposing current profiles with shear rates between 0 and -0.48 s<sup>-1</sup>. Numerical solutions of the vor-NLSE are consistently found to predict sideband growth to within two standard deviations across repeated experiments, performing considerably better than its uniform-current NLSE counterpart. Similarly, the amplification of experimental wave envelopes is predicted well by numerical solutions of the vor-NLSE, and significantly over-predicted by the uniform-current NLSE.</p>

scholarly journals Solitary Waves and Undular Bores in a Mesosphere Duct

2015 ◽  
Vol 72 (11) ◽  
pp. 4412-4422 ◽  
Author(s):  
Roger Grimshaw ◽  
Dave Broutman ◽  
Brian Laughman ◽  
Stephen D. Eckermann
Keyword(s):  
Solitary Waves ◽  
Stokes Equations ◽  
Initial Conditions ◽  
Solitary Wave Solution ◽  
Internal Gravity Wave ◽  
Boussinesq Approximation ◽  
Navier Stokes ◽  
Wave Radiation ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

Abstract Mesospheric bores have been observed and measured in the mesopause region near 100-km altitude, where they propagate horizontally along a duct of relatively strong density stratification. Here, a weakly nonlinear theory is developed for the description of these mesospheric bores. It extends previous theories by allowing internal gravity wave radiation from the duct into the surrounding stratified regions, which are formally assumed to be weakly stratified. The radiation away from the duct is expected to be important for bore energetics. The theory is compared with a numerical simulation of the full Navier–Stokes equations in the Boussinesq approximation. Two initial conditions are considered. The first is a solitary wave solution that would propagate without change of form if the region outside the duct were unstratified. The second is a sinusoid that evolves into an undular bore. The main conclusion is that, while solitary waves and undular bores decay by radiation from the duct, they can survive as significant structures over sufficiently long periods (~100 min) to be observable.

scholarly journals Nonlinear Disintegration of the Internal Tide

2008 ◽  
Vol 38 (3) ◽  
pp. 686-701 ◽  
Author(s):  
Karl R. Helfrich ◽  
Roger H. J. Grimshaw
Keyword(s):  
Solitary Waves ◽  
Nonlinear Waves ◽  
Numerical Solutions ◽  
Internal Tide ◽  
Tidal Energy ◽  
Initial Amplitude ◽  
Governing Equations ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Hydrostatic Limit

Abstract The disintegration of a first-mode internal tide into shorter solitary-like waves is considered. Since observations frequently show both tides and waves with amplitudes beyond the restrictions of weakly nonlinear theory, the evolution is studied using a fully nonlinear, weakly nonhydrostatic two-layer theory that includes rotation. In the hydrostatic limit, the governing equations have periodic, nonlinear inertia–gravity solutions that are explored as models of the nonlinear internal tide. These long waves are shown to be robust to weak nonhydrostatic effects. Numerical solutions show that the disintegration of an initial sinusoidal linear internal tide is closely linked to the presence of these nonlinear waves. The initial tide steepens due to nonlinearity and sheds energy into short solitary waves. The disintegration is halted as the longwave part of the solution settles onto a state close to one of the nonlinear hydrostatic solutions, with the short solitary waves superimposed. The degree of disintegration is a function of initial amplitude of the tide and the properties of the underlying nonlinear hydrostatic solutions, which, depending on stratification and tidal frequency, exist only for a finite range of amplitudes (or energies). There is a lower threshold below which no short solitary waves are produced. However, for initial amplitudes above another threshold, given approximately by the energy of the limiting nonlinear hydrostatic inertia–gravity wave, most of the initial tidal energy goes into solitary waves. Recent observations in the South China Sea are briefly discussed.

scholarly journals A method for computing unsteady fully nonlinear interfacial waves

1997 ◽  
Vol 351 ◽  
pp. 223-252 ◽  
Author(s):  
JOHN GRUE ◽  
HELMER ANDRÉ FRIIS ◽  
ENOK PALM ◽  
PER OLAV RUSÅS
Keyword(s):  
Solitary Waves ◽  
Bottom Topography ◽  
The Body ◽  
Interfacial Waves ◽  
Nonlinear Method ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Taylor Series Expansions ◽  
Layer Flow ◽  
Fully Nonlinear Method

We derive a time-stepping method for unsteady fully nonlinear two-dimensional motion of a two-layer fluid. Essential parts of the method are: use of Taylor series expansions of the prognostic equations, application of spatial finite difference formulae of high order, and application of Cauchy's theorem to solve the Laplace equation, where the latter is found to be advantageous in avoiding instability. The method is computationally very efficient. The model is applied to investigate unsteady trans-critical two-layer flow over a bottom topography. We are able to simulate a set of laboratory experiments on this problem described by Melville & Helfrich (1987), finding a very good agreement between the fully nonlinear model and the experiments, where they reported bad agreement with weakly nonlinear Korteweg–de Vries theories for interfacial waves. The unsteady transcritical regime is identified. In this regime, we find that an upstream undular bore is generated when the speed of the body is less than a certain value, which somewhat exceeds the critical speed. In the remaining regime, a train of solitary waves is generated upstream. In both cases a corresponding constant level of the interface behind the body is developed. We also perform a detailed investigation of upstream generation of solitary waves by a moving body, finding that wave trains with amplitude comparable to the thickness of the thinner layer are generated. The results indicate that weakly nonlinear theories in many cases have quite limited applications in modelling unsteady transcritical two-layer flows, and that a fully nonlinear method in general is required.

A note on waveless subcritical flow past symmetric bottom topography

2016 ◽  
Vol 28 (4) ◽  
pp. 562-575
Author(s):  
R. J. HOLMES ◽  
G. C. HOCKING
Keyword(s):  
Nonlinear Analysis ◽  
Nonlinear Problem ◽  
Vries Equation ◽  
Bottom Topography ◽  
Finite Depth ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Subcritical Flow ◽  
Weakly Nonlinear Analysis ◽  
Subcritical Solutions

This paper re-examines the problem of the flow of a fluid of finite depth over two Gaussian-shaped obstructions on the stream bed. A weakly nonlinear analysis in the form of the Korteweg–de Vries equation is used to compare with the results of the fully nonlinear problem. The main focus is to find waveless subcritical solutions, and contours showing the obstruction height and separation values that result in waveless solutions are found for different Froude numbers and different obstruction widths.

scholarly journals Spectral methods for internal waves: indistinguishable density profiles and double-humped solitary waves

2011 ◽  
Vol 18 (3) ◽  
pp. 351-358 ◽  
Author(s):  
M. Dunphy ◽  
C. Subich ◽  
M. Stastna
Keyword(s):  
Euler Equations ◽  
Solitary Waves ◽  
Dynamic Simulations ◽  
Large Lakes ◽  
Full Spectrum ◽  
Density Profiles ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
First Order ◽  
Spectral And Pseudospectral Methods

Abstract. Internal solitary waves are widely observed in both the oceans and large lakes. They can be described by a variety of mathematical theories, covering the full spectrum from first order asymptotic theory (i.e. Korteweg-de Vries, or KdV, theory), through higher order extensions of weakly nonlinear-weakly nonhydrostatic theory, to fully nonlinear-weakly nonhydrostatic theories and finally exact theory based on the Dubreil-Jacotin-Long (DJL) equation that is formally equivalent to the full set of Euler equations. We discuss how spectral and pseudospectral methods allow for the computation of novel phenomena in both approximate and exact theories. In particular we construct markedly different density profiles for which the coefficients in the KdV theory are very nearly identical. These two density profiles yield qualitatively different behaviour for both exact, or fully nonlinear, waves computed using the DJL equation and in dynamic simulations of the time dependent Euler equations. For exact, DJL, theory we compute exact solitary waves with two-scales, or so-called double-humped waves.

scholarly journals Solitary waves on a ferrofluid jet

2014 ◽  
Vol 750 ◽  
pp. 401-420 ◽  
Author(s):  
M. G. Blyth ◽  
E. I. Părău
Keyword(s):  
Magnetic Field ◽  
Solitary Wave ◽  
Solitary Waves ◽  
Phase Speed ◽  
Bond Number ◽  
Azimuthal Magnetic Field ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
Limiting Profile ◽  
Moving Jet

AbstractThe propagation of axisymmetric solitary waves on the surface of an otherwise cylindrical ferrofluid jet subjected to a magnetic field is investigated. An azimuthal magnetic field is generated by an electric current flowing along a stationary metal rod which is mounted along the axis of the moving jet. A numerical method is used to compute fully nonlinear travelling solitary waves, and the predictions of elevation waves and depression waves made by Rannacher and Engel (New J. Phys., vol. 8, 2006, pp. 108–123) using a weakly nonlinear theory are confirmed in the appropriate ranges of the magnetic Bond number. New nonlinear branches of solitary wave solutions are identified. As the Bond number is varied, the solitary wave profiles may approach a limiting configuration with a trapped toroidal-shaped bubble, or they may approach a static wave (i.e. one with zero phase speed). For a sufficiently large axial rod, the limiting profile may exhibit a cusp.

Sign in / Sign up

Export Citation Format

Share Document