Nature is always more complicated than the models we use to describe it. Ocean waves, for example, are constantly evolving. They grow when wind transfers kinetic energy to the water’s surface. They interact with each other in highly nonlinear ways. And once the wind dies down, their internal viscosity makes them dissipate away to nothing, leaving behind a surface as flat as glass.
This process is familiar to anyone who’s ever been to the seaside, yet physicists’ go-to models of water moving through the ocean essentially ignore it. Instead, the standard theory of wave transport, known as classical Stokes drift theory, concentrates on describing the flow of steady waves with finite amplitude. And while extensions of the theory incorporate complications such as nonlinear wave shapes, uneven seabeds and the inertia of particles carried along by the waves, they don’t account for waves that grow and fade over time.
For Marco Edoardo Rosti, a physicist at Japan’s Okinawa Institute of Science and Technology (OIST), this shortcoming has real-world consequences. “There is increasing interest in understanding near-surface transport processes because they are directly connected to problems such as pollutant dispersion, microplastic accumulation, sediment transport, and air-sea exchange,” he explains. “Since these processes depend sensitively on particle trajectories, even subtle modifications of the drift mechanism may become important over long times.”
In a study published in the journal EPL, Rosti and his colleagues Alessandro Chiarini, Tatsuo Izawa and Giulio Foggi Rota explored ways of remedying this deficiency. They focused on the simplest type of wave evolution: a freely decaying, single-frequency gravity wave, one where the force of gravity acts to restore equilibrium and smooth out the surface of the water. Physics World spoke with the team about the findings.
What is the most important advance in this work?
Classical Stokes drift theory predicts that on average, particles will get transported through the water horizontally, in the direction of the wave. But by combining fully nonlinear two-phase simulations with a perturbative analytical model, we demonstrate that when you allow those waves to decay, additional transport mechanisms appear.
Specifically, we identified a net vertical particle migration that arises solely from the temporal decay of the wave field, due to the interplay between fluid inertia and viscous dissipation. This migration persists even for fluids with relatively weak viscosity. Also, because the drift velocity depends on depth, even small vertical displacements alter particle trajectories and mixing properties.
More broadly, our results show that unsteadiness in the wave amplitude is not a minor correction to classical Stokes drift. It’s something that can qualitatively change how an individual parcel of fluid moves through space and time – what we call Lagrangian transport. This has important implications for understanding how floating and suspended materials, including sediments, aerosols and microplastics, get redistributed near the ocean surface.
Can you say more about those implications?
Because gravity waves at the ocean surface span a broad range of wavelengths, from centimetres to hundreds of meters, even a modest fractional vertical displacement per wave cycle can significantly change the depth of a drifting particle. And because the drift velocity beneath a wave decays exponentially with depth, these small vertical excursions can substantially alter how much horizontal drift a particle accumulates. That could influence whether microplastics remain near the surface (where they are most bioavailable), or whether they instead migrate to greater depths, where different biological and physical processes govern their fate.
Why have previous studies not seen this vertical drift?
The drift mechanism we identify accumulates gradually as the wave field decays. That means it only becomes apparent when “tracer” particles in the water are exposed to the same wave as its amplitude decreases over many periods. Most wave-tank experiments and field measurements instead follow tracers as they are swept through a passing wave packet, which is a fundamentally different configuration.
In open-sea conditions, similar arguments apply, and the presence of currents, wind forcing and turbulence all complicate the picture. These effects mask a signal that only shows up unambiguously in clean, unforced conditions.
What was the most challenging aspect of this work?
The most challenging part was to bridge the limits of classical wave theories with the fully nonlinear, two-phase simulations of an air-water interface. The classical Landau wave-decay theory, for example, assumes an isolated, weakly viscous fluid (water). However, our high-resolution simulations demonstrate (as previously reported in other studies) a deviation from these idealized assumptions. In fact, the dominant viscous energy dissipation occurs in a thin layer of air just above the water surface, rather than in the bulk.
The search for the missing plastic
To investigate how continuous wave energy decay affects Lagrangian particle transport, we developed a perturbative analytical model based on the classical Landau framework. This was a difficult step because the framework is inherently blind to the two-phase dissipative system. As a result, its asymptotic predictions progressively diverged from simulated flow fields at long times, likely due to intrinsic limitations of the Landau decay-wave model.
What do you plan to do next?
The drift mechanism we discuss in this study applies to a very idealized form of wave decay. In the future, we would like to test whether and how this mechanism propagates in more complex but realistic wave conditions. In particular, we want to work out how to measure such mechanisms in wave-tank experiments. To do so, we will study a different setup, one in which waves decay spatially instead of temporally, since this is closer to what observed in wave tanks. We would also like to study how our mechanism affects drift in waves that vary over time, with controlled unsteadiness, rather than simply decaying.
