Phenomenology of decaying turbulence beneath surface waves

TL;DR

The paper addresses decaying turbulence beneath surface waves and demonstrates that wave-averaged dynamics produce coherent vortices oriented perpendicular to wave propagation, depth-alternating jets, and suppressed kinetic-energy dissipation. By drawing an analogy to rotating turbulence, the curl of the Stokes drift is treated as a background vorticity that catalyzes an inverse energy cascade, a perspective that is supported by a Bardina-type two-equation model that captures the evolution of $k(t)$ and the dissipation rate $\,ε$. The work shows that this wave-modulated turbulence behaves similarly to beta-plane turbulence, offering a framework to incorporate wave effects into turbulence models and suggesting a non-dimensional pseudovorticity number that connects to the Langmuir number. The results have implications for ocean-surface boundary-layer mixing and motivate refined parameterizations in two-equation turbulence closures for the presence of surface waves.

Abstract

This paper explores decaying turbulence beneath surface waves that is initially isotropic and shear-free. We start by presenting phenomenology revealed by wave-averaged numerical simulations: an accumulation of angular momentum in coherent vortices perpendicular to the direction of wave propagation, suppression of kinetic energy dissipation, and the development of depth-alternating jets. We interpret these features through an analogy with rotating turbulence (Holm 1996), wherein the curl of the Stokes drift, $\boldsymbol{\nabla}\times\boldsymbol{u}^S$, takes on the role of the background vorticity (for example, $(f_0 + βy) \hat{\boldsymbol{z}}$ on the beta plane). We pursue this thread further by showing that a two-equation model proposed by Bardina et al. (1985) for rotating turbulence reproduces the simulated evolution of volume-integrated kinetic energy. This success of the two-equation model -- which explicitly parametrizes wave-driven suppression of kinetic energy dissipation -- carries implications for modeling turbulent mixing in the ocean surface boundary layer. We conclude with a discussion about a wave-averaged analogue of the Rossby number appearing in the two-equation model, which we term the "pseudovorticity number" after the pseudovorticity $\boldsymbol{\nabla}\times\boldsymbol{u}^S$. The pseudovorticity number is related to the Langmuir number in an integral sense.

Figures (4)

• Figure 1: Vorticity in simulations decaying turbulence with $512^3$ finite volume cells in a unit cube after $1000$ time units. (a) Vertical relative vorticity $\zeta = \space\boldsymbol{\hat{z}} \boldsymbol{\cdot} \left ( \boldsymbol{\nabla} \times \boldsymbol u \right )$ in rotating turbulence with Coriolis parameter $f = 1/4$, (b) Horizontal relative vorticity $\eta = \space\boldsymbol{\hat{y}} \boldsymbol{\cdot} \left ( \boldsymbol{\nabla} \times \boldsymbol u \right )$ in turbulence beneath surface waves with Stokes shear $\partial_z u^\mathrm{S} = - z / 2$, and (c) $\eta$ in isotropic turbulence.
• Figure 2: As figure \ref{['decaying-turbulence']} but showing (a) $\zeta$ for rotating turbulence in the $xy$-plane; (b) $\eta$ for turbulence beneath surface waves in the $xz$-plane, (c) $\eta$ for isotropic turbulence in the $xz$-plane.
• Figure 3: The evolution of cross-wave momentum $v$ in the $xz$-plane (6 panels on the left) and horizontally-averaged along-wave-momentum $u$ (2 panels on the right) at $t = 40, 400$ and for three wave fields: "deep" (left panel, red lines), "medium" (middle panel, orange lines), and "weak" (right panel, blue lines). Dashed lines in the top right plot show the Stokes drift profile (normalized) for each case. The "medium" waves case uses the background vorticity $\boldsymbol{\Omega}_\text{waves}$ in \ref{['intro-stokes-shear']}. Both the $u$ profiles and the light gray "zero lines" are spaced apart by the scale $\delta u = 10^{-2}$. The $u$-profiles at right show the development of depth-alternating jets, including a counter-wave surface jet for all cases. The $v$-slices are similar between the three cases at early times, but at later times exhibit strong, localized wave-impacts in their respective regions of significant Stokes shear. Note that $v$ plays the role that vertical velocity plays in rotating turbulence. Note that the Rhines scale may be estimated as $\ell_R = 2 \pi \sqrt{U / | \partial_z^2 u^\mathrm{S} |}$. With $|\partial_z^2 u^\mathrm{S} | = \tfrac{1}{2}$ and $U \approx 10^{-2}$ for the medium waves case, for example, we estimate $\ell_R \approx 0.8$, consistent with the jet spacing apparent on the right panels.
• Figure 4: The decay of kinetic energy $k(t)$ in isotropic, rotating, and surface-wave-modulated turbulence. Solid lines show kinetic energy normalized by it's initial value, $k / k_0$, computed from large eddy simulations. Dashed lines show $k / k_0$ given by \ref{['vorticity-solution']}, which solves the phenomenological two-equation system in \ref{['kinetic-energy-eqn']}--\ref{['dissipation']}. The initial dissipation rate $\epsilon_0$ in \ref{['vorticity-solution']} is estimated from the numerical solution; specifically we evaluate \ref{['isotropic-solution']} at $\Delta t$, substitute $k(t=\Delta t)$ from the numerical solution, and solve for $\epsilon_0$. We use $a = 11/6$ and estimate $b$ from \ref{['beta-estimate']}. For the rotating case, $\Omega = 1/4$ yields $b = 0.033$. For the four surface-wave-modulated cases we use $b = 0.036$, where $\Omega \stackrel{\mathrm{def}}{=} \int \partial_z u^\mathrm{S} {\, \mathrm{d}} z = \left (1/4, 1/8, 1/16, 1/32 \right )$. Note that using $a = 1.75$ along with commensurate adjustments to $b$ matches the simulation data even more closely.