On the theory of solitary Rossby waves

1977 ◽  
Vol 82 (4) ◽  
pp. 725-745 ◽  
Author(s):  
L. G. Redekopp
Keyword(s):  
Wave Speed ◽  
Rossby Waves ◽  
Vries Equation ◽  
Finite Amplitude ◽  
Long Wave ◽  
Atmospheric Stratification ◽  
De Vries ◽  
Eddy Structures ◽  
Layer Region ◽  
Wave Limit

The evolution of long, finite amplitude Rossby waves in a horizontally sheared zonal current is studied. The wave evolution is described by the Korteweg–de Vries equation or the modified Korteweg-de Vries equation depending on the atmospheric stratification. In either case, the cross-stream modal structure of these waves is given by the long-wave limit of the neutral eigensolutions of the barotropic stability equation. Both non-singular and singular eigensolutions are considered and the appropriate analysis is developed to yield a uniformly valid description of the motion in the critical-layer region where the wave speed matches the flow velocity. The analysis demonstrates that coherent, propagating, eddy structures can exist in stable shear flows and that these eddies have peculiar interaction properties quite distinct from the traditional views of turbulent motion.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

Comments on the use of the Korteweg-de Vries equation in the study of anharmonic lattices

1974 ◽  
Vol 339 (1616) ◽  
pp. 119-126 ◽  
Keyword(s):  
Differential Equation ◽  
Lattice Dynamics ◽  
Continuum Limit ◽  
Vries Equation ◽  
Wave Dispersion ◽  
Order Partial Differential Equation ◽  
Lennard Jones ◽  
Long Wave ◽  
De Vries ◽  
The Continuum

The use of the Korteweg-de Vries equation as the continuum limit of the equations describing the anharmonic motion of atoms in a lattice is examined in the light of the periodic solutions recently constructed by Askar (1973). It is shown that the Korteweg-de Vries equation does not exactly represent the behaviour of the nonlinear lattice in the limit of long waves and that a sixth-order partial differential equation gives a more accurate description of the lattice dynamics. The relative accuracies of two long wave dispersion relations derived from the Korteweg-de Vries equation are discussed and numerical results are presented for Morse, Born-Meyer and Lennard-Jones potentials. Askar’s paper contains several misprints and errors and the corrected forms of his equations are presented in the appendix.

scholarly journals Topographically forced long waves on a sheared coastal current. Part 2. Finite amplitude waves

1997 ◽  
Vol 343 ◽  
pp. 153-168 ◽  
Author(s):  
S. R. CLARKE ◽  
E. R. JOHNSON
Keyword(s):  
Wave Equation ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Coastal Current ◽  
Constant Vorticity ◽  
Long Wave ◽  
Long Time ◽  
Wave Limit ◽  
Parameter Values ◽  
Upstream Flow

This paper analyses the finite-amplitude flow of a constant-vorticity current past coastal topography in the long-wave limit. A forced finite-amplitude long-wave equation is derived to describe the evolution of the vorticity interface. An analysis of this equation shows that three distinct near-critical regimes occur. In the first the upstream flow is restricted, with overturning of the vorticity interface for sufficiently large topography. In the second quasi-steady nonlinear waves form downstream of the topography with weak upstream influence. In the third regime the upstream rotational fluid is partially blocked. Blocking and overturning are enhanced at headlands with steep rear faces and decreased at headlands with steep forward faces. For certain parameter values both overturning and partially blocked solutions are possible and the long-time evolution is critically dependent on the initial conditions. The reduction of the problem to a one-dimensional nonlinear wave equation allows solutions to be followed to much longer times and parameter space to be explored more finely than in the related pioneering contour-dynamical integrations of Stern (1991).

The effect of weak shear on finite-amplitude internal solitary waves

1999 ◽  
Vol 395 ◽  
pp. 125-159 ◽  
Author(s):  
S. R. CLARKE ◽  
R. H. J. GRIMSHAW
Keyword(s):  
Evolution Equation ◽  
Solitary Waves ◽  
Vries Equation ◽  
Physical Space ◽  
Finite Amplitude ◽  
Initial Condition ◽  
Weak Current ◽  
Inflection Points ◽  
Wide Range ◽  
De Vries

A finite-amplitude long-wave equation is derived to describe the effect of weak current shear on internal waves in a uniformly stratified fluid. This effect is manifested through the introduction of a nonlinear term into the amplitude evolution equation, representing a projection of the shear from physical space to amplitude space. For steadily propagating waves the evolution equation reduces to the steady version of the generalized Korteweg–de Vries equation. An analysis of this equation is presented for a wide range of possible shear profiles. The type of waves that occur is found to depend on the number and position of the inflection points of the representation of the shear profile in amplitude space. Up to three possible inflection points for this function are considered, resulting in solitary waves and kinks (dispersionless bores) which can have up to three characteristic lengthscales. The stability of these waves is generally found to decrease as the complexity of the waves increases. These solutions suggest that kinks and solitary waves with multiple lengthscales are only possible for shear profiles (in physical space) with a turning point, while instability is only possible if the shear profile has an inflection point. The unsteady evolution of a periodic initial condition is considered and again the solution is found to depend on the inflection points of the amplitude representation of the shear profile. Two characteristic types of solution occur, the first where the initial condition evolves into a train of rank-ordered solitary waves, analogous to those generated in the framework of the Korteweg–de Vries equation, and the second where two or more kinks connect regions of constant amplitude. The unsteady solutions demonstrate that finite-amplitude effects can act to halt the critical collapse of solitary waves which occurs in the context of the generalized Korteweg–de Vries equation. The two types of solution are then used to qualititatively relate previously reported observations of shock formation on the internal tide propagating onto the Australian North West Shelf to the observed background current shear.

scholarly journals Higher-order Korteweg-de Vries models for internal solitary waves in a stratified shear flow with a free surface

2002 ◽  
Vol 9 (3/4) ◽  
pp. 221-235 ◽  
Author(s):  
R. Grimshaw ◽  
E. Pelinovsky ◽  
O. Poloukhina
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Solitary Waves ◽  
Vries Equation ◽  
Wave Theory ◽  
Higher Order ◽  
Internal Solitary Waves ◽  
Long Wave ◽  
De Vries ◽  
Stratified Shear Flow

Abstract. A higher-order extension of the familiar Korteweg-de Vries equation is derived for internal solitary waves in a density- and current-stratified shear flow with a free surface. All coefficients of this extended Korteweg-de Vries equation are expressed in terms of integrals of the modal function for the linear long-wave theory. An illustrative example of a two-layer shear flow is considered, for which we discuss the parameter dependence of the coefficients in the extended Korteweg-de Vries equation.

scholarly journals Experiments on ion-acoustic rarefactive solitons in a multi-component plasma with negative ions

1985 ◽  
Vol 33 (2) ◽  
pp. 237-248 ◽  
Author(s):  
Y. Nakamura ◽  
J. L. Ferreira ◽  
G. O. Ludwig
Keyword(s):  
Vries Equation ◽  
Critical Concentration ◽  
Negative Ions ◽  
Finite Amplitude ◽  
Ion Temperature ◽  
Ion Acoustic ◽  
De Vries ◽  
Component Plasma ◽  
Ion Acoustic Solitons

Ion-acoustic solitons in a three-component plasma which consists of electrons and positive and negative ions have been investigated experimentally. When the concentration of negative ions is smaller than a certain value, positive or compressive solitons are observed. At the critical concentration, a broad pulse of small but finite amplitude propagates without changing its shape. When the concentration is larger than this value, negative or rarefactive solitons are excited. The velocity and the width of these solitons are measured and compared with predictions of the Korteweg-de Vries equation which takes the negative ions and the ion temperature into consideration. Head-on and overtaking collisions of the rarefactive solitons have been observed to show that the solitons are not affected by these collisions.

scholarly journals Short-Lived Large-Amplitude Pulses in the Nonlinear Long-Wave Model Described by the Modified Korteweg-De Vries Equation

2005 ◽  
Vol 114 (2) ◽  
pp. 189-210 ◽  
Author(s):  
Roger Grimshaw ◽  
Efim Pelinovsky ◽  
Tatiana Talipova ◽  
Michael Ruderman ◽  
Robert Erdelyi
Keyword(s):  
Large Amplitude ◽  
Vries Equation ◽  
Wave Model ◽  
Long Wave ◽  
De Vries

The initial-value problem for the Korteweg-de Vries equation

1975 ◽  
Vol 278 (1287) ◽  
pp. 555-601 ◽  
Keyword(s):  
Initial Data ◽  
Initial Value Problem ◽  
Model Equation ◽  
Vries Equation ◽  
Initial Value ◽  
Exact Relation ◽  
De Vries ◽  
Periodic Initial Data ◽  
Wave Limit ◽  
Continuous Dependence Results

For the Korteweg-de Vries equation u t + u x + u u x + u x x x = 0 , existence, uniqueness, regularity and continuous dependence results are established for both the pure initial-value problem (posed on -∞< x <∞) and the periodic initial-value problem (posed on 0 ⩽ x ⩽ l with periodic initial data). The results are sharper than those obtained previously in that the solutions provided have the same number of L 2 -derivatives as the initial data and these derivatives depend continuously on time, as elements of L 2 . The same equation with dissipative and forcing terms added is also examined. A by-product of the methods used is an exact relation between solutions of the Korteweg-de Vries equation and solutions of an alternative model equation recently studied by Benjamin, Bona & Mahony (1972). It is proven that in the long-wave limit under which these equations are generally derived, the solutions of the two models posed for the same initial data are the same. In the process of carrying out this programme, new results are obtained for the latter model equation.

The long-wave limit in the asymptotic theory of hypersonic boundary-layer stability

1993 ◽  
Vol 246 ◽  
pp. 381-395 ◽  
Author(s):  
S. E. Grubin ◽  
V. N. Trigub
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Asymptotic Theory ◽  
Boundary Layer Stability ◽  
Hypersonic Boundary Layer ◽  
Similarity Parameter ◽  
Long Wave ◽  
Layer Region ◽  
Wave Limit ◽  
Parallel Approximation

This paper discusses the long-wave limit of the asymptotic theory of hypersonic boundary-layer stability for a gas with the Prandtl number ½ < σ < 1 and with the viscosity–temperature law being a power function. The investigation is confined to the local-parallel approximation.In the long-wave limit the vorticity mode starts to interact with the acoustic disturbances in the boundary-layer region. The general solution of the linear problem in the boundary-layer inner region is analysed numerically and analytically. This solution is matched with the long-wave vorticity-mode solution near the transition layer. As a result, the inviscid instability problem for a hypersonic boundary layer is formulated. The analytical solution of this problem is found and analysed. The different limits of the solution are considered and the universal forms of the dependence are obtained. A similarity parameter is found which is a function of the Prandtl number and the power in the viscosity–temperature law. A significant change of the solution behaviour is noticed when this parameter passes a critical value. The asymptotic structure of the amplification rate, as a function of the wavenumber, is described and discussed.

The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography

1990 ◽  
Vol 221 ◽  
pp. 263-287 ◽  
Author(s):  
T. R. Marchant ◽  
N. F. Smyth
Keyword(s):  
Water Wave ◽  
Wave Equations ◽  
Vries Equation ◽  
Phase Speed ◽  
Simple Wave ◽  
Flow Speed ◽  
Higher Order ◽  
Modulation Equations ◽  
Long Wave ◽  
De Vries

The extended Korteweg-de Vries equation which includes nonlinear and dispersive terms cubic in the wave amplitude is derived from the water-wave equations and the Lagrangian for the water-wave equations. For the special case in which only the higher-order nonlinear term is retained, the extended Korteweg-de Vries equation is transformed into the Korteweg-de Vries equation. Modulation equations for this equation are then derived from the modulation equations for the Korteweg-de Vries equation and the undular bore solution of the extended Korteweg-de Vries equation is found as a simple wave solution of these modulation equations. The modulation equations are also used to extend the solution for the resonant flow of a fluid over topography. This resonant flow occurs when, in the weakly nonlinear, long-wave limit, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. In addition to the effect of higher-order terms, the effect of boundary-layer viscosity is also considered. These solutions (with and without viscosity) are compared with recent experimental and numerical results.

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