Solitary Waves In Stratified Fluids Research Articles

Numerical research on internal solitary waves in stratified fluid

Internal solitary waves are a common nonlinear large amplitude internal wave in nature, which can cause strong divergence and sudden strong currents in seawater during propagation, thus having great destructive effects on marine structures. The paper focuses on numerical wave generation based on three types of internal solitary wave theories. The waves are numerical simulated by RANS method in the paper. The numerical results of internal solitary wave theories are compared and analyzed. It shows that under the same numerical water tank scale and calculation conditions, the mKdV theory can obtain the largest wave amplitude, followed by the eKdV theory and the KdV theory. In addition, background currents will accelerate the propagation speed of internal solitary waves and increase their amplitude slightly.

Time-fractional generalized Boussinesq equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions

Construct fractional order model to describe Rossby solitary waves can provide more pronounced effects and deeper insight for comprehending generalization and evolution of Rossby solitary waves in stratified fluid. In the paper, from the quasi-geostrophic vorticity equation with dissipation effect and complete Coriolis force, based on the multi-scale analysis and perturbation method, a classical generalized Boussinesq equation is derived to describe the Rossby solitary waves in stratified fluid. Further, by employing the reduction perturbation method, the semi-inverse method, the Agrawal method, we derive the Euler–lagrangian equation of classical generalized Boussinesq equation and obtain the time-fractional generalized Boussinesq equation. Without dissipation effect, by using Lie group analysis method, the conservation laws of time-fractional Boussinesq equation are given. Finally, with the help of the improved (G′/G) expansion method, the exact solutions of the above equation are generated. Meanwhile, in order to consider the dissipation effect, we have to derive the approximate solutions by adopting the New Iterative Method. We remark that the fractional order model can open up a new window for better understanding the waves in fluid.

An Evans-function approach to spectral stability of internal solitary waves in stratified fluids

Frequently encountered in nature, internal solitary waves in stratified fluids have been investigated experimentally, theoretically, and numerically. Mathematically, these waves are exact solutions of the incompressible 2D Euler equations. Contrasting with a rich existence theory and the development of methods for their computation, their stability analysis has hardly received attention at a rigorous mathematical level.This paper proposes a new approach to the investigation of stability of internal solitary waves in a continuously stratified fluid and carries out the following four steps of this approach: (I) to formulate the eigenvalue problem as an infinite-dimensional spatial-dynamical system, (II) to introduce finite-dimensional truncations of the spatial-dynamics description, (III) to demonstrate that each truncation, of any order, permits a well-defined Evans function, (IV) to prove absence of small zeros of the Evans function in the small-amplitude limit. The latter notably implies the low-frequency spectral stability of small-amplitude waves to arbitrarily high truncation order.

Benjamin-Ono-Burgers-MKdV Equation for Algebraic Rossby Solitary Waves in Stratified Fluids and Conservation Laws

In the paper, by using multiple-scale method, the Benjamin-Ono-Burgers-MKdV (BO-B-MKdV) equation is obtained which governs algebraic Rossby solitary waves in stratified fluids. This equation is first derived for Rossby waves. By analysis and calculation, some conservation laws are derived from the BO-B-MKdV equation without dissipation. The results show that the mass, momentum, energy, and velocity of the center of gravity of algebraic Rossby waves are conserved and the presence of a small dissipation destroys these conservations.

The evolution equation for the second-order internal solitary waves in stratified fluids of finite depth

Abstract The evolution equation for the second-order internal solitary waves in stratified fluids, that are incomepressible inviscid ones of mixed stratification in which the density varies with depth in lower layer but keeps uniform in upper layer with a finite depth, is derived directly from the Euler equation by using the balance between the nonlinearity and the dispersion with the scaling μ= O(ɛ), and by introducing the Gardner-Morikawa transformation, asymptotic expansions and the matching of the solutions for the upper and lower layers of fluid via the computer algebraic operation. The evolution equation and its solution derived for the first-order wave amplitude are consistent with the classical ones, and the desired equation governing the second-order amplitude is reduced as follows ∂ f 2 ∂ τ - c 1 ∂ f 2 ∂ ξ + α ∂ ( f 1 f 2 ) ∂ ξ + β ∂ ∂ ξ ∫ - ∞ ∞ f 2 ( ξ ' ) g ( ξ - ξ ' ) d ξ ' = G 2 ( f 1 , c 1 , κ 1 ; c 0 , ϕ ) Where g ( ξ ) = - 1 2 π ∫ - ∞ ∞ k coth ( k l ) e j k ξ d k and G 2 ( f 1 , c 1 , κ 1 ; c 0 , ϕ ) is the inhomogeneous term, f 1 first-order wave amplitude.

The evolution equation for the second-order internal solitary waves in stratified fluids of finite depth

The evolution equation for the second-order internal solitary waves in stratified fluids of finite depth

Stability of Small-amplitude Internal Solitary Waves in Stratified Fluids

Stability of Small-amplitude Internal Solitary Waves in Stratified Fluids

POINCARE-LIGHTHILL-KUO METHOD AND SYMBOLIC COMPUTATION

This paper elucidates the effectiveness of combining the Poincare-Lighthill-Kuo method (PLK method, for short) and symbolic computation. Firstly, the idea and history of the PLK method are briefly introduced. Then, the difficulty of intermediate expression swell, often encountered in symbolic computation, is outlined. For overcoming the difficulty, a semi-inverse algorithm was proposed by the author, with which the lengthy parts of intermediate expressions are first frozen in the form of symbols till the final stage of seeking perturbation solutions. To discuss the applications of the above algorithm, the related work of the author and his research group on nonlinear oscillations and waves is concisely reviewed. The computer-extended perturbation solution of the Duffing equation shows that the asymptotic solution obtained with the PLK method possesses the convergence radius of 1 and thus the range of validity of the solution is considerably enlarged. The studies on internal solitary waves in stratified fluid and on the head-on collision between two solitary waves in a hyperelastic rod indicate that by means of the presented methods, very complicated manipulation, unconceivable in hand calculation, can be conducted and thus result in higher-order evolution equations and asymptotic solutions. The examples illustrate that the algorithm helps to realize the symbolic computation on micro-commputers. Finally, it is concluded that with the aid of symbolic computation, the vitality of the PLK method is greatly strengthened and at least for the solutions to conservative systems of oscillations and waves, it is a powerful tool.

Numerical simulations of internal solitary waves with vortex cores

This paper deals with the numerical verification of the theory developed by Derzho and Grimshaw (DG) (1997, Phys. Fluids 9(11), 3378–3385) regarding solitary waves in stratified fluids with recirculation regions. The Boussinesq approximation is made and the stratification is chosen such that the Brunt-Väisälä frequency differs only slightly from uniform stratification. To establish the consistency of the numerical scheme the usual KdV and mKdV solutions are tested first and then the solutions obtained by DG are considered. It is found that these waves remain of permanent form and are stationary when viewed at their corresponding phase speed. The recirculation region remains stagnant to first order as predicted by DG.

Theoretical analysis and symbolic computation of internal solitary waves in stratified fluids

Theoretical analysis and symbolic computation of internal solitary waves in stratified fluids

The evolution equation for second-order internal solitary waves in stratified fluids of great depth

By using perturbation methods, the evolution equation is derived for the second-order internal solitary waves in stratified fluids of great depth, which is a kind of inhomogeneous linearized Benjamin-Ono equation.

Solitary waves in stratified fluids and their interaction

A systematic procedure is proposed for obtaining solutions for solitary waves in stratified fluids. The stratification of the fluid is assumed to be exponential or linear. Its comparison with existing results for an exponentially stratified fluid shows agreement, and it is found that for the odd series of solutions the direction of displacement of the streamlines from their asymptotic levels is reversed when the stratification is changed from exponential to linear. Finally the interaction of solitary waves is considered, and the Korteweg-de Vries equation and the Boussinesq equation are derived. Thus the known solutions of these equations can be relied upon to provide the answers to the interaction problem.

A solitary wave application of an iterative method for nonlinear elliptic eigenvalue problems

A basic problem is that of solving nonlinear elliptic eigenvalue problems in the plane. Problems of this type arise in many branches of physics and engineering. The particular problem considered here, internal solitary waves in stratified fluids, occurs in fluid dynamics. An iterative technique (based on a transformation of the objective functional in an equivalent variational formulation of the problem) is used to numerically solve for eigenvalues and eigenfunctions associated with the fluid system. The choice of implementation involves discretizing the problem using finite Fourier sine series in conjunction with fast Fourier transforms (FFT). Results from preliminary test problems indicate that this numerical method can be used to verify the range of validity of existing (analytic) asymptotic expansion methods, and that new large amplitude solutions (strongly nonlinear waves) that cannot be obtained from the existing asymptotic expansion methods can be obtained using this numerical method.

Algebraic Solitary Waves in Stratified Fluids

By using nonlinear perturbation method a time-dependent equation is derived which governs internal waves in stratified fluids of great depth. The resultant equation takes the following form :