The Lagrange form of the nonlinear Schrödinger equation for low-vorticity waves in deep water: rogue wave aspect

The nonlinear Schrödinger equation (NLS equation) describing weakly rotational wave packets in an infinity-depth fluid in the Lagrangian coordinates is derived. The vorticity is assumed to be an arbitrary function of the Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is proved that the modulation instability criteria of the low-vorticity waves and deep water potential waves coincide. All the known solutions of the NLS equation for rogue waves are applicable to the low-vorticity waves. The effect of vorticity is manifested in a shift of the wave number in the carrier wave. In case of vorticity dependence on the vertical Lagrangian coordinate only (the Gouyon waves) this shift is constant. In a more general case, where the vorticity is dependent on both Lagrangian coordinates, the shift of the wave number is horizontally heterogeneous. There is a special case with the Gerstner waves where the vorticity is proportional to the square of the wave amplitude, and the resulting non-linearity disappears, thus making the equations of the dynamics of the Gerstner wave packet linear. It is shown that the NLS solution for weakly rotational waves in the Eulerian variables can be obtained from the Lagrangian solution by the ordinary change of the horizontal coordinates.