Lagrangian Measurement of Steep Directionally Spread Ocean Waves: Second-Order Motion of a Wave-Following Measurement Buoy

Abstract

The notion that wave-following buoys provide less accurate measurements of extreme waves than their Eulerian counterparts is a perception commonly held by oceanographers and engineers (Forristall 2000, J. Phys. Oceanogr., 30, 1931–1943, https://doi.org/10.1175/1520-0485(2000)030<1931:WCDOAS>2.0.CO;2 ). By performing a direct comparison between the two types of measurement under laboratory conditions, we examine one of the hypotheses underlying this perception and establish whether wave measurement buoys in extreme ocean waves correctly follow steep crests and behave in a purely Lagrangian manner. We present a direct comparison between Eulerian gauge and Lagrangian buoy measurements of steep directionally spread and crossing wave groups on deep water. Our experimental measurements are compared with exact (Herbers and Janssen 2016, J. Phys. Oceanogr., 46, 1009–1021, https://doi.org/10.1175/JPO-D-15-0129.1 ) and new approximate expressions for Lagrangian second-order theory derived herein. We derive simple closed-form expressions for the second-order contribution to crest height representative of extreme ocean waves—namely, for a single narrowly spread wave group, two narrowly spread crossing wave groups, and a single strongly spread wave group. In the limit of large spreading or head-on crossing, Eulerian and Lagrangian measurements become equivalent. For the range of conditions that we test, we find that our buoy behaves in a Lagrangian manner, and our experimental observations compare extremely well to predictions made using second-order theory. In general, Eulerian and Lagrangian measurements of crest height are not significantly different for all degrees of directional spreading and crossing. However, second-order bound-wave energy is redistributed from superharmonics in Eulerian measurements to subharmonics in Lagrangian measurement, which affects the “apparent” steepness inferred from time histories and poses a potential issue for wave buoys that measure acceleration.

Real-world rogue wave probabilities

“…Wave buoys are known to underestimate extreme crests through several mechanisms, such as lateral movements around the crest, being dragged through the crest, or linearization of the sea state due to their Lagrangian motion. Even though these effects were found to be of minor importance 40 , we cannot rule out that our conclusions are potentially biased by this. Therefore, our buoy data could underestimate the total number of rogue waves to some degree, and the influence of second-order effects on wave crests might be even higher if measured by a different sensor (this does not affect wave heights, though).…”

Estimating ocean wave directional spreading using wave following buoys: a comparison of experimental buoy and gauge data

“…In McAllister and van den Bremer (2019), it was shown that the mooring configuration used for the experiments analysed here had no observable effect on the vertical displacement η. However, lateral (x, y) buoy position, which Fig.…”

“…As only three quantities are used for directional estimates made using buoy measurements, bimodal directional distributions may present a challenge. Identifying crossing sea states is important, as they can be hazardous for seafaring vessels and potentially increase the likelihood of observing large wave crests (McAllister et al 2019).…”

Deterministic wave forecasting with the Zakharov equation

“…No matter the source, wave measurements inevitably suffer from a number of uncertainties. Wave buoy measurements exhibit deviations from the actual sea state due to the buoy response (McAllister & van den Bremer 2019), while radar is limited by wave shadowing (Plant & Farquharson 2012) (the inability to observe in the radar shadow of a wave crest). While an accurate measurement of the sea-surface height at a single instant, like that furnished by the WASS (wave acquisition stereo system) of Fedele et al (2013), would be ideal, radar measurements are further impacted by the rotation time of the radar antennae (typically several seconds (Al-Ani et al 2020)).…”