Predicting the occurrence of rogue waves in the presence of opposing currents with a high-order spectral method

We discuss the dynamics of unidirectional random wave fields that propagate against an opposing current through laboratory experiments and direct numerical simulations of the Euler equations solved with a high-order spectral method. Both approaches demonstrate that the presence of a negative horizontal velocity gradient increases the probability of the occurrence of extreme and rogue waves in the course of their propagation with the emergence of a rapid transition from weakly to strongly non-Gaussian properties. Numerical simulations capture quantitatively well the statistical properties of laboratory observations and substantiate that underlying physics are associated to quasiresonant nonlinear interactions triggered by the background current.

Figure 1

Schematic representation of the wave-current experimental facilities (not to scale).

Figure 2

(a) Vertical, (b) longitudinal, and (c) transverse profiles of the horizontal current velocity.

Figure 3

Configuration of the numerical solution.

Figure 4

Evolution of significant wave height $H_s$ as a function of the nondimensional distance $x/L_p$ from the wave maker for different current speeds: (a) without and (b) with wall friction dissipation. Marks indicate experimental observations, the solid lines refer to numerical simulations, and the dashed vertical lines indicate the start and end of the transition region from no current to regime flow.

Figure 5

Energy loss due to wave breaking.

Figure 6

Evolution of experimental (dashed red line) and numerical (solid blue line) wave spectrum at different distances from the wave maker and different current velocities; the input wave spectrum (dash-dotted yellow line) is presented as a benchmark.

Figure 7

Evolution of the peak frequency as a function of the distance from the wave maker. Comparison between experiments and numerical simulations.

Figure 8

Average wave energy transfer from numerical simulations: (a) without and (b) with friction and breaking dissipation terms.

Figure 9

Evolution of local steepness, $k_p{H}_s/2$, during wave propagation. Comparison between experiments and numerical simulations.

Figure 10

Spatial evolution of the skewness along the wave flume for different current velocities. Comparison between experiments and numerical simulations.

Figure 11

Spatial evolution of the kurtosis along the wave flume for different current velocities. Comparison between experiments and numerical simulations.

Figure 12

Maximum of kurtosis in the tank for different current velocities. Comparison between experiments and numerical simulations.

Figure 13

Plots of the exceedance probability of wave height at the location of maximum kurtosis for different current velocities. Comparison between experiments and numerical simulations.