Numerical modeling of quasiplanar giant water waves

Abstract

In this work we present a further analytical development and a numerical implementation of the recently suggested theoretical model for highly nonlinear potential long-crested water waves, where weak three-dimensional effects are included as small corrections to exact two-dimensional equations written in the conformal variables [V. P. Ruban, Phys. Rev. E 71, 055303(R) (2005)]. Numerical experiments based on this theory describe the spontaneous formation of a single weakly three-dimensional large-amplitude wave (alternatively called freak, killer, rogue, or giant wave) on the deep water.

Breathing rogue wave observed in numerical experiment

“…[25,26]). Besides that, the linear dispersion relation resulting fromK is correct in the entire Fourier plane (it should be noted that in Ref.…”

“…Their detailed derivation and discussion can be found in Refs. [25,26]. We use Cartesian coordinates x, q, y, with y axis up-directed.…”

“…For such weakly 3D flows, the advantages of the conformal variables are almost recovered, as it has been explained in Refs. [25,26]. In the cited papers, the noncanonical Hamiltonian description in terms of the conformal variables has been generalized from 2D to 3D potential inviscid flows with a free surface, and the asymptotic expansion of the Hamiltonian functional on the small parameter ǫ = (l x /l q ) 2 has been suggested, where l x is a typical wave length, and l q is a large transversal scale along the wave crests.…”

“…In Ref. [26] some numerical results have been presented, for non-random initial conditions and typical dimensionless wave numbers about 20.…”

“…Therefore a relatively high spatial resolution (16384 × 256 in Ref. [26]) for the corresponding numerical algorithm has been possible due to the large number of local operations that result from the Fourier diagonalization. In Ref.…”