Abstract
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.
Highlights
The nonlinear Schrödinger (NLS) equation was first derived by Zakharov in 1967 (English edition; Zakharov, 1968), who used the Hamiltonian formalism for a description of wave propagation in deep water; see Benney and Newell (1967). Hasimoto and Ono (1972) and Davey (1972) obtained the same result independently
We have derived the vortex-modified nonlinear Schrödinger equation using the method of multiple-scale expansions in the Lagrangian variables
The nonlinear evolution equation for the wave packet in the form of the nonlinear Schrödinger equation has been derived as well. The novelty of this equation is related to the emergence of a new term proportional to the envelope amplitude and the variance of the coefficient of the nonlinear term
Summary
The nonlinear Schrödinger (NLS) equation was first derived by Zakharov in 1967 (English edition; Zakharov, 1968), who used the Hamiltonian formalism for a description of wave propagation in deep water; see Benney and Newell (1967). Hasimoto and Ono (1972) and Davey (1972) obtained the same result independently. Using the method of multiple scales, Johnson (1976) examined the slow modulation of a harmonic wave moving at the surface of an arbitrary shear flow with a velocity profile U (y), where y is the vertical coordinate He derived the NLS equation with coefficients that depend in a complicated way on a shear flow (Johnson, 1976). The effect of low vorticity (ε2 order of magnitude) in the paper by Hjelmervik and Trulsen (2009) is reflected in the NLS equation This fact, like the NLS nonlinear term for plane potential waves, may be attributed to the presence of an average current nonuniform over the fluid depth.
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