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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union
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Volume 24, issue 2
Nonlin. Processes Geophys., 24, 255–264, 2017
https://doi.org/10.5194/npg-24-255-2017
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.
Nonlin. Processes Geophys., 24, 255–264, 2017
https://doi.org/10.5194/npg-24-255-2017
© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.

Research article 06 Jun 2017

Research article | 06 Jun 2017

Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water

Anatoly Abrashkin1 and Efim Pelinovsky2,3 Anatoly Abrashkin and Efim Pelinovsky
  • 1National Research University Higher School of Economics (HSE), 25/12 Bolshaya Pecherskaya Str., 603155 Nizhny Novgorod, Russia
  • 2Institute of Applied Physics RAS, 46 Ulyanov Str., 603950 Nizhny Novgorod, Russia
  • 3Nizhny Novgorod State Technical University n.a. R. Alekseev, 24 Minina Str., 603950 Nizhny Novgorod, Russia

Abstract. The nonlinear Schrödinger (NLS) equation describing the propagation of weakly rotational wave packets in an infinitely deep fluid in Lagrangian coordinates has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. The vorticity effects manifest themselves in a shift of the wave number in the carrier wave and in variation in the coefficient multiplying the nonlinear term. In the case of vorticity dependence on the vertical Lagrangian coordinate only (Gouyon waves), the shift of the wave number and the respective coefficient are constant. When the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally inhomogeneous. There are special cases (e.g., Gerstner waves) in which the vorticity is proportional to the squared wave amplitude and nonlinearity disappears, thus making the equations for wave packet dynamics linear. It is shown that the NLS solution for weakly rotational waves in the Eulerian variables may be obtained from the Lagrangian solution by simply changing the horizontal coordinates.

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The nonlinear Schrödinger equation describing weakly rotational wave packets in a fluid in the Lagrangian coordinates is derived. Rogue effects are possible in low-vorticity waves, and the effect of vorticity is manifested in a shift of the wave number in the carrier wave. Special attention is paid to Gouyon and Gerstner waves. It is shown that this equation in the Eulerian variables can be obtained from the Lagrangian solution with an ordinary change in the horizontal coordinates.
The nonlinear Schrödinger equation describing weakly rotational wave packets in a fluid in the...
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