Anomalous wave statistics induced by abrupt depth change

Laboratory experiments reveal that variations in bottom topography can qualitatively alter the distribution of randomized surface waves. A normally distributed, unidirectional wave field becomes highly skewed and non-Gaussian upon encountering an abrupt depth change. A short distance downstream, wave statistics conform closely to a gamma distribution, affording simple estimates for skewness, kurtosis, and other statistical properties. Importantly, the exponential decay of the gamma distribution is much slower than Gaussian, signifying that extreme events occur more frequently. Under the conditions considered here, the probability of a rogue wave can increase by a factor of 50 or more. We also report on the spectral content of the waves produced in the experiments.

Figure 1

(a) Experimental schematic. (b),(c) Surface displacement measured at a representative upstream and downstream location. (d),(e) Corresponding histograms. (f) ${\eta }_{\text{std}}$ as it varies in space for six different driving amplitudes. (g),(h) Similar for skewness and kurtosis, showing an anomalous region at $x=15$ cm where deviation from Gaussian is most pronounced.

Figure 2

Histograms of $\eta$ sampled at the anomalous location ($x=15$ cm) for six different driving amplitudes, $\mathrm{\Delta }\theta$. For $\mathrm{\Delta }\theta >0.5^{\circ }$, behavior is non-Gaussian and the gamma distribution (blue dashed curve) provides a robust statistical description.

Figure 3

(a),(b) Variation of the parameters ${\eta }_{\gamma }$ and $\alpha$ with driving amplitude. Small amplitudes generate Gaussian statistics (green region), and larger amplitudes give rise to the gamma distribution (blue region), with ${\eta }_{\gamma }$ and $\alpha$ estimated for each. Both ${\eta }_{\gamma }$ and ${\alpha }^{-2}$ increase nearly linearly with amplitude. (c) The linear fit of ${\alpha }^{-2}$ leads to a power-law prediction for skewness, which compares well with direct measurements.

Figure 4

Power spectra of displacement (top) and surface slope (bottom) taken at representative upstream and downstream locations, along with estimated power laws.