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Solitons, Tsunamis and Oceanographical Applications of

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Extreme Environmental Events

Article Outline

Glossary

Definition of the Subject

Introduction

Shallow Water Waves and KdV Type Equations

Deep Water Waves and NLS Type Equations

Tsunamis as Solitons

Internal Solitons

Rossby Solitons

Bore Solitons

Future Directions

Bibliography

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Abbreviations

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).

Shallow and deep water waves:

Considering surface gravity waves in an ocean of depth h, they are called shallow‐water waves if \({h\ll \lambda}\), where λ is the wavelength (or from a practical point of view if \({h < 0.07 \lambda}\)). 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 \({h > 0.28\lambda}\) and the corresponding wave speed is given by \({c=\sqrt{g/k}}\), \({k=\frac{2\pi}{\lambda}}\).

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.

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.

Bore solitons:

The classic bore (also called mascaret, poroca and aeger) 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 intertial and gravitational effects. Slower bores can take on oscillatory profile with a leading dispersive shockwave followed by a train of solitons.

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Author information

Authors and Affiliations

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

    M. Lakshmanan

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  1. M. Lakshmanan
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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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Lakshmanan, M. (2011). Solitons, Tsunamis and Oceanographical Applications of. In: Meyers, R. (eds) Extreme Environmental Events. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7695-6_47

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