Nonlinear dynamics of trapped waves on jet currents and rogue waves

Nonlinear dynamics of surface gravity waves trapped by an opposing jet current is studied analytically and numerically. For wave fields narrow band in frequency but not necessarily with narrow angular distributions the developed asymptotic weakly nonlinear theory based on the modal approach of Shrira and Slunyaev [J. Fluid. Mech. 738, 65 (2014)] leads to the one-dimensional modified nonlinear Schrödinger equation of self-focusing type for a single mode. Its solutions such as envelope solitons and breathers are considered to be prototypes of rogue waves; these solutions, in contrast to waves in the absence of currents, are robust with respect to transverse perturbations, which suggests a potentially higher probability of rogue waves. Robustness of the long-lived analytical solutions describing modulated trapped waves and solitary wave groups is verified by direct numerical simulations of potential Euler equations.

Figure 1

Simulation of propagation of one trapped mode with $k=0.1$ rad/m and $kH/2=0.15$ with an initial 5% longitudinal modulation. Surface elevation at time $t=829.4$ s; the maximal wave is characterized by $kH/2\approx 0.32$. The longitudinal cross section through the peak of the maximal wave is shown by lines above the current maximum and above the $x$ axis. The profile of the current is shown above the $y$ axis.

Figure 2

Simulation of a solitary group of trapped waves belonging to the fifth mode (the surface at $t\approx 868$ s is shown; see videoclip [13]). The snapshot longitudinal cross section is shown by red solid lines above the current maximum and above the $x$ axis; the corresponding sections of the initial condition are shown by thin black curves. The snapshot transverse cross section is shown in front of the surface above the $y$ axis.