Rogue Waves

Burkard Baschek and Jennifer Imai

Introduction

Rogue waves are commonly defined as waves with a trough-to-crest height exceeding the significant wave height by a factor of two or more and are therefore statistically extreme ocean gravity waves. While their absolute wave height may not always be large, they become a major hazard for vessels and offshore structures in rough seas.

We have investigated a wave buoy data set collected off the U.S. West coast with 7,157 rogue waves observed over a total of 80 years. It yields comprehensive statistics of the likelihood of rogue wave occurrence in the open ocean, coastal ocean, and in shallow water. The highest recorded rogue wave had a trough-to-crest height of 18.95 m. The average likelihood of occurrence is 63 per year in coastal waters and 101 per year in the open ocean. An extrapolation to conditions in the world ocean yields an average likelihood of encountering rogue waves along the main shipping routes in the North Atlantic of 0.8-1.2% per day for rogue waves exceeding 11 m in height. The results can be used to test rogue wave forecasting models and will help to improve the forecasting of hazardous ocean conditions.

Wave Data

In this study, we use wave amplitude data collected between 1993 and 2010 at 16 different Datawell Directional Waverider Buoys (type Mk-II and Mk-III) off the U.S. West coast (Figure 1) with a total of 80 years of data. The buoy locations are representative of open ocean conditions in deep water, shallow water, and for the coastal ocean (Table 1). The data were furnished by the Coastal Data Information Program (CDIP), Integrative Oceanography Division, operated by the Scripps Institution of Oceanography, San Diego, CA. Transmission related errors and data spikes were removed.

	Figure 1: Map of the U.S. West coast with locations of the CDIP buoys used in this study. The text shows the mooring number and the text colour indicates open ocean moorings (red), coastal moorings (blue), and shallow water moorings (black). The average number of rogue waves per year for a ratio of wave height to significant wave height H/Hs ≥ 2.0 is given by the coloured dots, the amount of available data by their size.

Rogue Wave Statistics

We define rogue waves as waves with a trough-to-crest wave height H exceeding the significant wave height Hs by at least a factor of 2, or a crest height with ξ/Hs > 1.25. Hs is calculated over a 30 minute interval as Hs = 4 std(a), with a: surface elevation, std: standard deviation.

69 waves of more than 11 m in height were recorded. The largest rogue wave occurred at buoy 71 (Harvest, CA) on February 24, 2008, 21:19 UTC (Figure 2). It is the second of two rogue waves immediately following each other. The trough-to-crest heights are 16.7 m and 18.95 m, the zero-crossing wave period is 15.5 s, and the H/Hs-ratios are 2.4 and 2.3. The horizontal displacements are 23.0 m and 16.8 m.

	Figure 2: Two rogue waves at mooring 71 (Harvest, CA) on February 24, 2008, 21:19 UTC. (a) Wave amplitude (solid line) and magnitude of horizontal displacement in wave direction (dashed line). Locations (1) to (5) correspond to panel (b). The trough-to-crest wave height of waves (2) and (3) is 16.7 m and 18.95 m and the zero-crossing period is 15.5 s. (b) Horizontal and vertical displacements indicate rotational motion. (c) Directional wave spectrum showing the wave energy density of the background wave field as function of direction [°] and frequency [Hz]. The direction and frequency of the rogue waves is marked with the red dot.

In 80.8 years of data and 6,1 ∙ 10^8 waves at all moorings, 7,157 rogue waves were observed, averaging 88.6 per year, or about one in 84,500 waves. In the open ocean, 100 rogue waves occur per year (one in 129,300 waves or 7.7 ∙ 10^-6, return period 5.75 d), and 63 per year in the coastal ocean (1.4 ∙10^-5, return period 3.6 d). The maximum observed H/Hs ratio was 2.57 for a 4.1 m wave (Figure 3).

	Figure 3: Exceedence probability P plotted as log(–log(P)) and as a function of normalized wave height H/Hs for the coastal ocean (dotted curves) and open ocean (dashed curves). The distributions by Naess (1985) is plotted for comparison. Values above those lines indicate a lower probability.

The average likelihood of encountering a rogue wave in a 24-h period is shown as function of Hs (solid lines, Figure 4), calculated from 5 open ocean moorings (28.3 years). The likelihood is almost independent of Hs and is predominantly a function of H/Hs, with a slight increase at large H/Hs ratios. In order to determine the probability P of encountering a rogue wave in a 24-hour period in the open ocean, the data have been approximated with P = 535exp(6.5)(1-H/Hs) for H/Hs ≤ 1.75 and P = 17,000exp(11)(1-H/Hs) for H/Hs > 1.75 for H/Hs–increments of 0.1 (dashed lines).

	Figure 4: Number of rogue waves per day for significant wave height increments of 1.0 m and H/Hs-increments of 0.1 calculated from the open ocean moorings (solid lines) and from the linear approximation as described in the text (dashed lines).

With this, the average likelihood of encountering a rogue wave of a certain size in the world ocean is estimated (Figure 5). Hs is calculated for each location as function of wind speed (level-3 QuikSCAT wind speed data) (Sverdrup and Munk, 1947). Rogue waves with H ≥ 5 m are about twice as common in the Southern Ocean as in the North Atlantic and North Pacific (up to 9%/day), while rogue waves with H ≥ 11 m have similar likelihoods in all three oceans.

	Figure 5: Map showing the average likelihood of encountering a rogue wave within a 24-h period in the open ocean. (a) Likelihood for waves with H/Hs ≥ 2.0 and H ≥ 5.0 m. (b) H/Hs ≥ 2.0, H ≥ 11.0 m.

References

Garrett, C., J. Gemmrich, and B. Baschek, 2011: Rogue Waves. In McGraw-Hill Yearbook of Science & Technology 2011. Accepted.

Baschek, B., and J. Imai, 2011: Rogue Wave Observations off the U.S. West Coast. Oceanography 24(2):158–165, doi:10.5670/oceanog.2011.35.