The increased drift of steep focusing surface gravity waves

TL;DR

The paper investigates how mean Lagrangian drift in irrotational surface gravity waves is locally enhanced during focusing events, challenging the conventional additive assumption for multi-component wave fields. By formulating the drift in a fully Lagrangian framework and introducing a local mean pseudomomentum constraint, the authors derive a higher-order drift expression for narrow-banded packets governed by the NLSE and validate it with laboratory measurements and nonlinear simulations. They demonstrate that near-surface transport can be significantly amplified in focusing regions, with enhancements up to about $30\%$ relative to linear theory, especially for small bandwidths, and show good agreement between theory, simulations, and experiments. The results emphasize that local wave-field steepness, not just the sum of component steepness, governs transport and suggest a more local interpretation of wave-induced drift with potential implications for Langmuir-like circulations and upper-ocean mixing parameterizations.

Abstract

Irrotational and monochromatic surface gravity waves possess a mean Lagrangian drift which transports mass and enhances mixing in the upper ocean. In the ocean, where many surface waves are present, it is commonly assumed that the mean Lagrangian drift can be computed independently for each wave component and summed. Here we show, using laboratory measurements and fully nonlinear simulations of steep focusing wave packets, that this assumption underpredicts the average transport in regions of wave focusing by up to 30%. To explain these enhancements, we derive a new exact method for constraining the local mean Lagrangian drift in general flows by working in the Lagrangian reference frame. From this method, we derive a higher-order expression for the local mean Lagrangian drift in narrow-banded wave fields governed by the nonlinear Schrödinger equation (NLSE) that predicts near-surface enhancements when waves focus and steepen. The theoretical predictions of the local transport agree with the experiments, particularly for smaller bandwidth packets where the NLSE approximation is most valid. These findings highlight that it is the local steepness of the wave field, not just the sum of the steepnesses of the linear (non-interacting) wave components, which sets the strength of these enhancements.

Figures (5)

• Figure 1: Surface particle trajectories in a focusing wave packet with $\Delta = 0.8$ and $S = 0.27$. In panel $(a)$, the vertical elevation of particles is shown in time as a function of their initial location from the linear prediction of focusing $x_f$, normalized by the central wavenumber $k_c$. The colored lines represent particles downstream of focusing (blue), at focusing (red), and upstream of focusing (green). Likewise, on the right, panels $(b,c,d)$ show the physical particle trajectories of these downstream, at focusing, and upstream particles respectively, normalized by $k_c$. Note that the total transport during focusing (red) is much greater than that away from focusing, contrary to linear theory \ref{['lineartransport']} (dashed line) which states that all particles should experience the same transport.
• Figure 2: The total Lagrangian transport $\delta x$ of surface particles as a function of their initial distance from the linear prediction of maximum focusing $(x_0 - x_f)$, normalized by the central wavenumber $k_c$ for the same simulation as in figure (\ref{['fig:trajectories']}), $\Delta = 0.8$ and $S = 0.27$. The normalized linear prediction of the total transport $k_c \delta x_\text{lin}$\ref{['lineartransport']}, constant in space, is shown in red.
• Figure 3: Mean surface transport $\langle \delta x \rangle$ as a function of the linear prediction of maximum wave slope $S$. Panel $(a)$shows the mean transport normalized by the central wavenumber $k_c$ and linear bandwidth dependence $f(\Delta)$ so that the prediction of linear theory \ref{['lineartransport']} (red) collapses to a single curve for both the simulation and laboratory parameters. A polynomial fit of the discrete simulation points is shown in green. Panel $(b)$ shows the same data plotted as a percentage increase from linear theory.
• Figure 4: Percentage increases of the maximum $(a)$ and mean $(b)$ surface Lagrangian transport relative to linear theory \ref{['lineartransport']} for numerically simulated focusing wave packets as a function of parameter space $(S,\Delta)$. Discrete simulation runs are shown via colored markers, with interpolated values in between. Note the two distinct colorbar scalings for panels $(a,b)$. The red line outlining the parameter space represents the breaking slope threshold numerically determined by pizzo2021b which we found to be consistent with our simulations.
• Figure 5: The mean surface transport $\langle \delta x \rangle$ within the focusing region as a function of $S$ computed both directly from simulation (circles) and using our higher-order theory \ref{['theorytransport']} (lines) for each simulation. In $(a)$, $\langle \delta x \rangle$ is normalized by the central wavenumber $k_c$, and each line represents the theoretical prediction of mean transport for each bandwidth value. In $(b)$, $\langle \delta x \rangle$ is also normalized by the linear bandwidth dependence $f(\Delta)$\ref{['transportapprox']} which collapses the results. The theory performs best at lower values of $\Delta$ where the narrow-banded envelope assumption is most valid.