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

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.

The role of Lagrangian drift in the geometry, kinematics and dynamics of surface waves

“…We then presented the explicit connection between the Lagrangian mean flow and wave geometry, kinematics and dynamics for the scenario when the Lagrangian mean flow was weak. This included explicit expressions for the dependence of the Benjamin–Feir instability growth rate (Abrashkin & Pelinovsky 2017, 2018), as well as the range of unstable perturbation wavenumbers and frequencies, on the Lagrangian mean flow. To illustrate these points, we presented results from a recent laboratory demonstration that highlight the striking dependence of finite amplitude wave behaviour on the underlying shear flow.…”

“…The finite amplitude correction to the Stokes drift is shown to result in a Doppler-like shift to the frequency of these waves, which explains the curious conclusion of Stewart & Joy (1974). The finite amplitude Stokes correction to phase speed sets the growth rate of the Benjamin–Feir instability (BFI) (Benjamin & Feir 1967; Zakharov 1968) and hence in turn there is a connection between the Lagrangian mean flow (which sets the phase speed) and the strength of this instability, as was originally noted in Abrashkin & Pelinovsky (2017). Here, we explicitly present the growth rate of the most unstable mode and the range of unstable wavenumbers as a function of the Lagrangian mean flow.…”

“…Next we find the third-order expansions for Stokes waves by expanding the Gerstner waves solutions to include Lagrangian mean flows, (see also the related discussions in Abrashkin & Pelinovsky 2017, 2018). Going through the algebra, we find where and Note, the phase now has dependence on and is unknown at this stage.…”

“…The form of the phase velocity is enough to provide a heuristic derivation of the nonlinear Schrödinger equation (NLSE) on a weak Lagrangian shear flow (Abrashkin & Pelinovsky 2017), following the classical geometrical optics approach (Benjamin & Feir 1967; Yuen & Lake 1982; Abrashkin & Pelinovsky 2018). Whereas previous, Eulerian derivations of the NLSE in the presence of shear have favoured the special case of constant vorticity which permits potential theory (Baumstein 1998; Thomas, Kharif & Manna 2012); the Lagrangian framework handles arbitrary more readily.…”

An interfacial Gerstner-type trapped wave

“…Lagrangian perturbation solutions for surface waves with weak vorticity in a one-layer ocean of infinite depth have been discussed in [22] and [23]. For Gerstner type waves in a fluid layer of finite depth, an exact solution does not exist [6,7].…”