scholarly journals The Effect of Random Wind Forcing in the Nonlinear Schrödinger Equation

Fluids ◽  
2019 ◽  
Vol 4 (3) ◽  
pp. 121
Author(s):  
Leo Dostal
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Solitary Waves ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Multiple Scales ◽  
Ocean Waves ◽  
Nonlinear Schrodinger Equation ◽  
Wind Forcing ◽  
Nonlinear Schrödinger

The influence of a strong and gusty wind field on ocean waves is investigated. How the random wind affects solitary waves is analyzed in order to obtain insights about wave generation by randomly time varying wind forcing. Using the Euler equations of fluid dynamics and the method of multiple scales, a random nonlinear Schrödinger equation and a random modified nonlinear Schrödinger equation are obtained for randomly wind forced nonlinear deep water waves. Miles theory is used for modeling the pressure variation at the wave surface resulting from the wind velocity field. The nonlinear Schrödinger equation and the modified nonlinear Schrödinger equation are computed using a relaxation pseudo spectral scheme. The results show that the influence of gusty wind on solitary waves leads to a randomly increasing ocean wave envelope. However, in a laboratory setup with much smaller wave amplitudes and higher wave frequencies, the influence of water viscosity is much higher. This leads to fluctuating solutions, which are sensitive to wind forcing.

scholarly journals Two-dimensional modulation instability of wind waves

2019 ◽  
Vol 5 (4) ◽  
pp. 413-417 ◽  
Author(s):  
Roger Grimshaw
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Modulation Instability ◽  
Wind Waves ◽  
Nonlinear Schrodinger Equation ◽  
Wind Forcing ◽  
Nonlinear Schrödinger ◽  
Peregrine Breather

Abstract It is widely known that deep-water waves are modulationally unstable and that this can be modelled by a nonlinear Schrödinger equation. In this paper, we extend the previous studies of the effect of wind forcing on this instability to water waves in finite depth and in two horizontal space dimensions. The principal finding is that the instability is enhanced and becomes super-exponential and that the domain of instability in the modulation wavenumber space is enlarged. Since the outcome of modulation instability is expected to be the generation of rogue waves, represented within the framework of the nonlinear Schrödinger equation as a Peregrine breather, we also examine the effect of wind forcing on a Peregrine breather. We find that the breather amplitude will grow at twice the rate of a linear instability.

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