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| Titel |
The problem of the highest wave in a group: fully nonlinear simulations of water wave breathers vs weakly nonlinear theory |
| VerfasserIn |
A. Slunyaev, V. Shrira |
| Konferenz |
EGU General Assembly 2012
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 14 (2012) |
| Datensatznummer |
250059454
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| Zusammenfassung |
| Deep water waves often propagate in groups due to the Benjamin – Feir instability. The
weakly nonlinear theory for weakly modulated unidirectional waves within the framework of
the integrable nonlinear Schrödinger equation (NLS) predicts that for an initially uniform
wave train independently of its amplitude the highest wave in the group may be up to 3 times
the uniform wave amplitude, while the most modulationaly unstable wave results in
about 2.4 times wave enhancement. What are the highest waves in reality is not
known, the very notion is not well defined. We examine the problem of the highest
wave in a modulated wave train by direct numerical simulations of exact Euler
equations. In this case strong nonlinearity (including wave breaking) and strong linear
and nonlinear dispersion may become crucial. The main purpose of the study is to
re-define the notion of the highest wave and to establish the correspondence between
the weakly nonlinear analytic NLS theory and simulations of strongly nonlinear
analogues of the Akhmediev breathers (periodic in space modulations of a uniform
wave) within the framework of the primitive hydrodynamic equations. The initial
conditions specifying initial small perturbation of a uniform wave are chosen in
such a way that only one predefined unstable mode (one breather) is excited. The
simulations are performed for the range of initial steepness ka = 0.04-¦0.25 and
the modulation length of from 2 to 20 carrier wave lengths. The simulations are
focused on the final stage of the Benjamin – Feir instability, when large waves
emerge out of weakly modulated wave trains. It is shown that in the non-breaking
regimes wave crests may be amplified more than 4 times. Maximum wave trough
amplification is shown to be more than 3 times; though the maximal wave height
amplification in all non-breaking cases did not exceed 3. The moderation of wave height
amplification is due to the short length of emerging wave groups: the wave height differs
considerably from the doubled wave amplitude (strong dispersion). In the fully
nonlinear model the focusing time and the threshold of modulation growth are
greater than in the weakly nonlinear limit. The role of wave breaking is twofold: on
the one hand, the already intense waves cannot be amplified significantly as they
quickly break; on the other hand, close to the wave breaking onset the amplification
factors of weaker waves increase sharply, so that near-breaking and breaking waves
seem to represent the most dangerous case of wave dynamics. The results provide
an important element for developing of deterministic forecasting of rogue waves. |
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