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Non-linear Internal Waves

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

The objective of the present work is to study the generation and propagation of nonlinear internal waves in the frame of the shallow watertheory. These waves are generated inside a stratified fluid occupying a semi infinite channel of finite and constant depth by a wave makersituated in motion at the finite extremity of the channel. A distortion process is carried out to the variables and the nonlinear equations of theproblem using a certain small parameter characterizing the motion of the wave maker and double series representations for the unknown functions isintroduced. This procedure leads to a solution of the problem including a secular term, vanishing at the position of the wave maker. Thisinconvenient result is remedied using a multiple scale transformation of variables and it is shown that the free surface and the interface elevationssatisfy the well known KdV equation. The initial conditions necessary for the solution of the KdV equations are obtained...

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Abbreviations

Nonlinear waves :

Nonlinear waves are such waves which arise as solutions of the nonlinear mathematical models simulating physical phenomena in fluids.

Shallow water :

Shallow water means water waves for which the ratio between the amplitude and the wave length is relatively small. The linear theory of motion is inadequate for the description of shallow water waves.

Internal waves :

Internal waves are gravity waves that oscillate within a fluid medium. They arise from perturbations to hydrostatic equilibrium, where balance is maintained between the force of gravity and the buoyant restoring force. A simple example is a wave propagating on the interface between two fluids of different densities, such as oil and water. Internal waves typically have much lower frequencies and higher amplitudes than surface gravity waves because the density differences (and therefore the restoring forces) within a fluid are usually much smaller than the density of the fluid itself.

Pycnocline :

Pycnocline is a rapid change in water density with depth. In freshwater environments, such as lakes, this density change is primarily caused by water temperature, while in seawater environments such as oceans the density change may be caused by changes in water temperature and/or salinity.

Solitary waves :

Solitary waves are localized traveling waves, which asymptotically tend to zero at large distances.

Solitons :

Solitons are waves which appear as a result of a balance between a weakly nonlinear convection and a linear dispersion. The solitons are localized highly stable waves that retain their identity (shape and speed) upon interaction, and resemble particle like behavior. In the case of a collision, solitons undergo a phase shift.

Baroclinic fluid :

Baroclinic fluid is such a fluid for which the density depends on both the temperature and the pressure.

In atmospheric terms, the baroclinic areas are generally found in the mid‐latitude/polar regions.

Barotropic fluid :

Barotropic fluid is such a fluid for which the density depends only on the pressure.

In atmospheric terms, the barotropic zones of the Earth are generally found in the central latitudes, or tropics.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

    Moustafa S. Abou-Dina & Mohamed A. Helal

Authors
  1. Moustafa S. Abou-Dina
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  2. Mohamed A. Helal
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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© 2009 Springer-Verlag

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Abou-Dina, M.S., Helal, M.A. (2009). Non-linear Internal Waves. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_363

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