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Encyclopedia of Complexity and Systems Science pp 1–25Cite as

Solitons, Tsunamis and Oceanographical Applications of

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Definition of the Subject

Surface and internal gravity waves arising in various oceanographic conditions are natural sources where one can identify/observe the generation, formation, and propagation of solitary waves and solitons. Unlike the standard progressive waves of linear dispersive type, solitary waves are localized structures with long wavelengths and finite energies and propagate without change of speed or form and are patently nonlinear entities. The earliest scientifically recorded observation of a solitary wave was made by John Scott Russel in August 1834 in the Union Canal connecting the Scottish cities of Glasgow and Edinburgh. The theoretical formulation of the underlying phenomenon was provided by Korteweg and de Vries in 1895 who deduced the now-famous Korteweg-de Vries (KdV) equation admitting solitary wave solutions. With the insightful numerical and analytical investigations of Martin Kruskal and coworkers in the 1960s, the KdV solitary waves have been shown to...

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Abbreviations

Bore solitons:

The classic bore (also called mascaret, poroca, and aegir) arises generally in funnel-shaped estuaries that amplify incoming tides, tsunamis, or storm surges, the rapid rise propagating upstream against the flow of the river feeding the estuary. The profile depends on the Froude number, a dimensionless ratio of inertial and gravitational effects. Slower bores can take on oscillatory profile with a leading dispersive shock wave followed by a train of solitons.

Internal solitons:

Gravity waves can exist not only as surface waves but also as waves at the interface between two fluids of different density. While solitons were first recognized on the surface of water, the commonest ones in oceans actually happen underneath as internal oceanic waves propagating on the pycnocline (the interface between density layers). Such waves occur in many seas around the globe, prominent among them being the Andaman and Sulu seas.

Rossby solitons:

Rossby waves are typical examples of quasigeostrophic dynamical response of rotating fluid systems, where long waves between layers of the atmosphere as in the case of the Great Red Spot of Jupiter or in the barotropic atmosphere are formed and may be associated with solitonic structures.

Shallow and deep water waves:

Considering surface gravity waves in an ocean of depth h, they are called shallow water waves if h ≪ λ, where λ is the wavelength (or from a practical point of view if h < 0.07λ). In the linearized case, for shallow water waves, the phase speed \( c=\sqrt{ gh} \), where g is the acceleration due to gravity. Water waves are classified as deep (practically) if h > 0.28λ, and the corresponding wave speed is given by \( c=\sqrt{g/k} \), \( k=\frac{2\pi }{\lambda } \).

Soliton:

A class of nonlinear dispersive wave equations in (1 + 1) dimensions having a delicate balance between dispersion and nonlinearity admit localized solitary waves which under interaction retain their shapes and speeds asymptotically. Such waves are called solitons because of their particle-like elastic collision property. The systems include Korteweg-de Vries, nonlinear Schrödinger, sine-Gordon, and other nonlinear evolution equations. Certain (2 + 1)-dimensional generalizations of these systems also admit soliton solutions of different types (plane solitons, algebraically decaying lump solitons, and exponentially decaying dromions).

Tsunami:

Tsunami is essentially a long-wavelength water wave train, or a series of waves, generated in a body of water (mostly in oceans) that vertically displaces the water column. Earthquakes, landslides, volcanic eruptions, nuclear explosions, and impact of cosmic bodies can generate tsunamis. Propagation of tsunamis is in many cases in the form of shallow water waves and sometimes can be of the form of solitary waves/solitons. Tsunamis as they approach coastlines can rise enormously and savagely attack and inundate to cause devastating damage to life and property.

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Authors and Affiliations

  1. Center for Nonlinear Dynamics, Bharathidasan University, Tiruchirapalli, India

    M. Lakshmanan

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    Robert A. Meyers

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Lakshmanan, M. (2013). Solitons, Tsunamis and Oceanographical Applications of. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_509-3

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  • DOI: https://doi.org/10.1007/978-3-642-27737-5_509-3

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  1. Latest

    Applications of Solitons, Tsunamis and Oceanographical
    Published:
    15 August 2022

    DOI: https://doi.org/10.1007/978-3-642-27737-5_509-4

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    Solitons, Tsunamis and Oceanographical Applications of
    Published:
    28 April 2014

    DOI: https://doi.org/10.1007/978-3-642-27737-5_509-3

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