The set-down and set-up of directionally spread and crossing surface gravity wave groups

Abstract

For sufficiently directionally spread surface gravity wave groups, the set-down of the wave-averaged free surface, first described by Longuet-Higgins and Stewart (J. Fluid Mech. vol. 13, 1962, pp. 481-504), can turn into a set-up. Using a multiple-scale expansion for two crossing wave groups, we examine the structure and magnitude of this wave-averaged set-up, which is part of a crossing wave pattern that behaves as a modulated partial standing wave: in space, it consists of a rapidly varying standing-wave pattern slowly modulated by the product of the envelopes of the two groups; in time, it grows and decays on the slow time scale associated with the translation of the groups. Whether this crossing wave pattern actually enhances the surface elevation at the point of focus depends on the phases of the linear wave groups, unlike the set-down, which is always negative and inherits the spatial structure of the underlying envelope(s). We present detailed laboratory measurements of the wave-averaged free surface, examining both single wave groups, varying the degree of spreading from small to very large, and the interaction between two wave groups, varying both the degree of spreading and the crossing angle between the groups. In both cases, we find good agreement between the experiments, our simple expressions for the set-down and set-up, and existing second-order theory based on the component-by-component interaction of individual waves with different frequencies and directions. We predict and observe a set-up for wave groups with a Gaussian angular amplitude distribution with standard deviations of above 30-40• (21-28 • for energy spectra), which is relatively large for realistic sea states, and for crossing sea states with angles of separation of 50-70• and above, which are known to occur in the ocean.

Laboratory study of the wave-induced mean flow and set-down in unidirectional surface gravity wave packets on finite water depth

“…We begin by carrying out a Stokes expansion using the steepness α ≡ k 0 a, where a is the maximum amplitude of the (linear) free surface A 0 and k 0 is the carrier wavenumber. Using a Taylor-series expansion of the free surface boundary conditions (3a,b) about z = 0, we obtain after some manipulation (see also [45,46]), utilizing the Laplace equation 1:…”

Laboratory recreation of the Draupner wave and the role of breaking in crossing seas

“…Although the widely used (Mai et al 2016;Zhao et al 2017;McAllister et al 2018) method of harmonic extraction (Baldock et al 1996;Fitzgerald et al 2014) we use relies on a generalised Stokes expansion and is thus not affected by cubic interactions, it cannot be fully relied on in the event of wave breaking. In our experiments, wave breaking occurs at the crest and not at the corresponding 180 • phase-inverted trough.…”

“…This can be theoretically predicted (Okihiro, Guza & Seymour 1992;Herbers, Elgar & Guza 1994;Toffoli, Onorato & Monbaliu 2006;Christou et al 2009) based on second-order interaction kernels (Hasselmann 1962;Sharma & Dean 1981;Dalzell 1999;Forristall 2000). A set-up has been observed in field data (Walker et al 2004;Toffoli et al 2007;Santo et al 2013) and recently in detailed laboratory experiments (McAllister et al 2018). Second, when modelled numerically using a fully nonlinear potential flow model (Yan & Ma 2010), a wave of comparable steepness could not be created without triggering breaking in non-crossing-sea states (Adcock et al 2011).…”

Experimental Observation of Modulational Instability in Crossing Surface Gravity Wavetrains

“…This can be theoretically predicted [35][36][37][38] based on second-order interaction kernels [39][40][41][42]. Set-up has been observed in field data [43][44][45] and recently in detailed laboratory experiments [46]. For the Draupner wave, recorded in the North Sea on the 1st of January 1995 [47], the observation of set-up can be seen as evidence for crossing [43,48,49].…”