The role of dissipation in the evolution of ocean swell

Abstract

[1] Dissipation of ocean swell, inferred from published oceanographic data, is investigated to determine if laboratory results on the dissipative stabilization of narrow-banded wave trains are applicable to ocean swell. Three issues are addressed. (i) Dimensional decay rates of ocean swell are about a million times smaller than typical decay rates of laboratory waves. Nevertheless, when decay rates are nondimensionalized using scales of dispersive and nonlinear effects, the dimensionless decay rates of ocean swell are comparable to those of laboratory waves, indicating that dissipation and nonlinear effects can influence ocean swell on the same time scale. (ii) The stability of ocean swell to small perturbations is examined within the theoretical framework of nonlinear Schrödinger-type models that either do or do not include dissipation. As in laboratory experiments, for swell with small enough nonlinearity, dissipation can inhibit and eventually stop the growth of small perturbations before nonlinearity becomes important. And as in laboratory experiments, we document herein an example of ocean swell with stronger nonlinearity that exhibits frequency downshifting, which is not predicted by any nonlinear Schrödinger-type model, including higher-order models, with or without dissipation. (iii) Given that dissipation can influence the evolution of ocean swell, we compare the predicted decay rates of four (published) dissipative models with observed decay rates, both in the ocean and in a laboratory wave tank. The model that presupposes an inextensible film on the free surface agrees best with measured rates of dissipation of ocean swell. Statement of the Problem[2] Ocean ''swell'' refers to slowly varying wave trains of surface water waves, which are typically generated by an oceanic storm, and which are observed to propagate over thousands of km without additional forcing [Snodgrass et al., 1966]. The standard mathematical model of the dynamics of surface water waves, first posed by Stokes [1847], is an energy-conserving system, with no dissipation. But water and air are both viscous fluids, so the energy of ocean swell must dissipate as the swell propagates. The dissipation rate of ocean swell is measurable [cf. Collard et al., 2009] but weak enough that Snodgrass et al. [1966] pronounced it ''negligible'' in their landmark paper. It appears that they meant that dissipation is negligible in the sense that ocean swell can propagate the entire length of the Pacific Ocean, more than 1/3 of the distance around the world, while still maintaining measurable wave amplitudes and significant coherence.[3] At about the same time as the important work of Snodgrass et al. [1966], several scientists around the world discovered what is now called either the ''modulational instability'' or the ''Benjamin-Feir instability'' [Lighthill, 1965;Benjamin and Feir, 1967; Benney and Newell, 1967;Ostrovsky, 1967;Whitham, 1967;Zakharov, 1967Zakharov, , 1968. The instability occurs in energy-conserving systems with dispersive waves ...

Superregular Breathers in Optics and Hydrodynamics: Omnipresent Modulation Instability beyond Simple Periodicity

“…The agreement with theoretical predictions is again excellent, thus demonstrating the existence of superregular breathers in hydrodynamics, in particular, for describing the nonlinear modulation instability in the framework of localized perturbations. Note that amplitudes are slightly lower in the experiments than theoretical prediction, in particular, due to the dissipation naturally existing when performing experiments in water-wave facilities [45].…”

Spatiotemporal measurement of surfactant distribution on gravity–capillary waves

“…While this ansatz captures most of the total pattern, there is a residual mode in the FI data (see figure 4(g)) that generates peak-to-mean variations in Γ of 0.03 and 0.01 Γ c for experiments 1 and 2, respectively. This higher-order mode probably arises through a nonlinear interaction between the meniscus and Faraday waves such as those that give rise to the Benjamin-Feir instability (Segur et al 2005;Henderson et al 2010;Akers 2012;Henderson & Segur 2013). Although this mode has a Bessel mode shape like the Faraday waves (F(r, θ ) ∼ J m (kr) cos(nθ + φ)) with m = 2n (twice the symmetry number of the fundamental mode), the distance between neighbouring extrema is comparable to that of the meniscus waves.…”

“…In our experiments, Ak ≈ 0.01-0.1, which means that the O((Ak) 3 ) nonlinearity is a very small effect. Furthermore, the damping effect of the surfactant both suppresses the Benjamin-Feir instability and affects the relative phase (Segur et al 2005;Henderson, Segur & Carter 2010;Akers 2012;Henderson & Segur 2013).…”

Role of the basin boundary conditions in gravity wave turbulence

“…This dissipation comes from the presence of surfactants and/or contaminants at the interface, which leads to an inextensible surface where fluid tangential velocity should be cancelled at the interface. This type of dissipation is known to strongly affect the stability of large-scale gravity waves in the ocean (Henderson & Segur 2013). For all frequencies, τ d is found to be much larger than τ theo d except at the forcing frequencies ∼1 Hz where the two curves intersect.…”