Turbulence and energy dissipation from wave breaking

TL;DR

This work probes how broadband wave breaking generates turbulence and dissipates energy in the upper ocean using a novel multi-layer Navier–Stokes solver that resolves scales from ~0.5 m to ~1 km. By letting breaking emerge from the spectrum rather than being prescribed, the study reveals vorticity-rich, near-surface turbulence and a self-similar, near-surface dissipation profile that matches field observations when nondimensionalized by the significant wave height $H_s$ and depth-integrated dissipation $\\Psi$. An empirical shape $\\hat{\\epsilon} = 2(|\\hat{z}| + 1)^{-3}$ captures the vertical dissipation structure, with a near-surface slope of $\\epsilon \\propto z^{-1}$ transitioning to steeper decay deeper than roughly one $H_s$, and the global energy balance links to the fifth moment of the breaking-front distribution via Phillips-type dissipation $S_{ds}$. The results show that wave breaking can dominate near-surface turbulence production and provide a coherent framework to interpret observations and improve surface-boundary-layer modeling, while remaining an idealized study focused on breaking in the absence of wind forcing and other turbulence sources. These insights contribute to more accurate representations of upper-ocean mixing and energy pathways in ocean models.

Abstract

Wave breaking is a critical process in the upper ocean: an energy sink for the surface wave field and a source for turbulence in the ocean surface boundary layer. We apply a novel multi-layer numerical solver resolving upper-ocean dynamics over scales from O(50cm) to O(1km), including a broad-banded wave field and wave breaking. The present numerical study isolates the effect of wave breaking and allows us to study the surface layer in wave-influenced and wave-breaking-dominated regimes. Following our previous work showing wave breaking statistics in agreement with field observations, we extend the analysis to underwater breaking-induced turbulence and related dissipation (in freely decaying conditions). We observe a rich field of vorticity resulting from the turbulence generation by breaking waves. We discuss the vertical profiles of dissipation rate which are compared with field observations, and propose an empirical universal shape function. Good agreement is found, further demonstrating that wave breaking can dominate turbulence generation in the near-surface layer. We examine the dissipation from different angles: the global dissipation of the wave field computed from the decaying wave field, the spectral dissipation from the fifth moment of breaking front distribution, and a turbulence dissipation estimated from the underwater strain rate tensor. Finally, we consider how these different estimates can be understood as part of a coherent framework.

Figures (11)

• Figure 1: (a) Illustration of the layered discretization for $N_L=15$ (for only part of the domain). Color shows the horizontal velocity and the gray lines show the layers. (b) The geometric distribution of layer thickness. The blue marks are the uniformly-spaced interpolation points used in analysis in Figure \ref{['fig:aver_layer_interp']}.
• Figure 2: (a) Energy evolution of each case showing the system is slowly decaying without external forcing. Because equipartition of kinetic and potential energy is expected for linear waves, we show twice the kinetic energy, $2E_k$ (dashed lines); twice the potential energy, $2E_p$ (dotted lines); and the total energy, $E = E_k + E_p$ (solid lines). The sampling period is indicated with the gray shading. (b) The spectra of each case at the start of the sampling period $t=100$ s. For both plots, the color of lines indicate different initial spectrum $P$ value listed in Table \ref{['tab:cases']}. The bottom horizontal axis shows dimensional variables while the top horizontal axis shows non-dimensional variables normalized by peak wavenumber $k_p$ or peak frequency $\omega_p$.
• Figure 3: A 3D illustration of the simulation. The surface velocity field $u$ is shown in gray scale and offset by a distance from the surface vorticity field. The zoomed-in view shows one of the detached breaking fronts based on Gaussian curvature (see W23 for detail). The vorticity component $\Omega_z$ is shown in Blue-Red color. The blue frame indicates the surface; the purple frame indicates depth $z\approx - 3H_s$; the green framed indicate the $x-z$ plane. The same color notation for plane orientation is used in Figure \ref{['fig:vorticity']}.
• Figure 4: (a) Breaking distribution $\Lambda(c)$ for increasing wave intensity. (b) Pre-multiplied breaking distribution $c^5\Lambda(c)$. For both plots, the bottom horizontal axis shows dimensional variables while the top horizontal axis shows non-dimensional variables normalized by peak phase speed $c_p$.
• Figure 5: (a) An example of a slice of $x-z$ plane at $y=0$ for a moderately steep case P02 at $t=120$. (c) The surface layer. (e) The layer of average depth around $3H_s$. (b,d,f) Same as (a,c,e) but for the least steep case P008.