On the Boussinesq approximation and its role in the theory of internal waves

Long, R.L.

doi:10.1111/j.2153-3490.1965.tb00193.x

Cited by 45 publications

(46 citation statements)

References 5 publications

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“…The traditional justification for setting e = 0 may be stated in our notation as the fact that h 2 a «7T"2 for most stratifications of real interest. This widely used approximation has the advantage that it simplifies the analysis considerably; for instance, it makes the transformation ( However, some researchers [4,24] have concluded that the Boussinesq approximation can be delicate and can lead to significant inaccuracy when finite-amplitude solutions, especially solitary waves, are considered. Therefore, we shall proceed with our analysis and computation without this simplification, recognizing that the Boussinesq case may be included in our exact formulation merely by setting e = O.…”

Section: Semilinear Form Of Eigenvalue Problemmentioning

confidence: 99%

A Computational Method for Solitary Internal Waves in a Continuously Stratified Fluid

Turkington¹,

Eydeland²,

Wang³

1991

Stud Appl Math

125

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An iterative algorithm is presented to compute steady, translational nonlinear waves in an incompressible fluid with a stable density stratification. The method yields solitary internal waves as exact solutions of the governing Euler equations without the restrictions on wave amplitude or density profile involved in the various approximate models of weakly nonlinear long waves. Computed examples of large‐amplitude solutions in shallow and deep fluid regimes are given. A natural variational principle derived from the structure of the underlying dynamical equations forms the basis of the method. The construction of the numerical algorithm and the analysis of the properties of solutions are both developed from the same principle.

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Section: Semilinear Form Of Eigenvalue Problemmentioning

confidence: 99%

A Computational Method for Solitary Internal Waves in a Continuously Stratified Fluid

Turkington¹,

Eydeland²,

Wang³

1991

Stud Appl Math

125

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“…In the case of a homogeneous fluid, Eqs. (7) and (9) Finally, another phenomenon of stratified fluids (Long, 1965) that has application to the rotating homogeneous case is the solitary wave at rest in a basic flow wo{x o )' In the present case there is no wave when W o is constant because the governing equation (15)…”

Section: Homogeneous Liquidmentioning

confidence: 99%

Finite amplitude disturbances in a rotating fluid

Long

1971

Tellus A: Dynamic Meteorology and Oceanography

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A liquid is rotating with angular velocity vector along the z-axis. A uniform gravity field acts along the negative z-axis and the liquid is stratified with arbitrary density gradient. A steady disturbance exists with all properties independent of one of the horizontal coordinates, say, y. It is then shown that the primitive equations can be integrated once to yield a second-order vorticity equation. This equation is generally non-linear and complicated but greatly simplifies when the fluid is homogeneous. In one special and interesting case the problem becomes identical to two-dimensional flow of a certain stratified fluid in a uniform gravity field.

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“…The KdV model has been suggested for the studying of nonlinear internal waves in the mid 1960's, in a stationary variant of the KdV equation (Long, 1965;Benjamin, 1966). Between 1970 and, this model was developed and was adjusted to explain various atmospheric and ocean processes.…”

Section: Analytical Model Of Nonlinear Internal Gravity Wavesmentioning

confidence: 99%

“…Nevertheless, some mathematical research reveals that the passage from a full hydrodynamic model to the limit of a quasistatic model is absent (Long, 1965;Kshevetskii andLeble, 1985, 1988). The reason for this strange phenomenon is concealed in equation nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

Analytical and numerical investigation of nonlinear internal gravity waves

Кшевецкий

2001

Nonlin. Processes Geophys.

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Abstract. The propagation of long, weakly nonlinear internal waves in a stratified gas is studied. Hydrodynamic equations for an ideal fluid with the perfect gas law describe the atmospheric gas behaviour. If we neglect the term ρ dw/dt (product of the density and vertical acceleration), we come to a so-called quasistatic model, while we name the full hydrodynamic model as a nonquasistatic one. Both quasistatic and nonquasistatic models are used for wave simulation and the models are compared among themselves. It is shown that a smooth classical solution of a nonlinear quasistatic problem does not exist for all t because a gradient catastrophe of nonlinear internal waves occurs. To overcome this difficulty, we search for the solution of the quasistatic problem in terms of a generalised function theory as a limit of special regularised equations containing some additional dissipation term when the dissipation factor vanishes. It is shown that such solutions of the quasistatic problem qualitatively differ from solutions of a nonquasistatic nature. It is explained by the fact that in a nonquasistatic model the vertical acceleration term plays the role of a regularizator with respect to a quasistatic model, while the solution qualitatively depends on the regularizator used. The numerical models are compared with some analytical results. Within the framework of the analytical model, any internal wave is described as a system of wave modes; each wave mode interacts with others due to equation nonlinearity. In the principal order of a perturbation theory, each wave mode is described by some equation of a KdV type. The analytical model reveals that, in a nonquasistatic model, an internal wave should disintegrate into solitons. The time of wave disintegration into solitons, the scales and amount of solitons generated are important characteristics of the nonlinear process; they are found with the help of analytical and numerical investigations. Satisfactory coincidence of simulation outcomes with analytical ones is revealed and some examples of numerical simulations illustrating wave disintegration into solitons are given. The phenomenon of internal wave mixing is considered and is explained from the point of view of the results obtained. The numerical methods for internal wave simulation are examined. In particular, the influence of difference interval finiteness on a numerical solution is investigated. It is revealed that a numerical viscosity and numerical dispersion can play the role of regularizators to a nonlinear quasistatic problem. To avoid this effect, the grid steps should be taken less than some threshold values found theoretically.

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On the Boussinesq approximation and its role in the theory of internal waves

Cited by 45 publications

References 5 publications

A Computational Method for Solitary Internal Waves in a Continuously Stratified Fluid

A Computational Method for Solitary Internal Waves in a Continuously Stratified Fluid

Finite amplitude disturbances in a rotating fluid

Analytical and numerical investigation of nonlinear internal gravity waves

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