Resolvent model-based analyses of coherent structures in Langmuir turbulence

TL;DR

This work extends resolvent analysis to turbulence–wave interactions in Langmuir turbulence by basing the linearised perturbation system on LES-derived mean states and a vertically varying eddy viscosity, with wave forcing represented by Stokes drift. The transfer operator framework identifies dominant forcing–response pairs and shows strong amplification for elongated, streamwise-structured motions, predicting both large-scale Langmuir cells and near-surface three-dimensional vortices as linear responses to nonlinear forcing. The results reveal a low-rank structure across scales, enabling credible reproduction of LES vertical-velocity spectra and spanwise length-scale trends, while also highlighting limitations in capturing the deepest-scale energy without improved forcing modelling. Overall, the resolvent approach provides a coherent, physics-based mechanism—linear amplification of sustained nonlinear forcing—for the formation and sustenance of multiscale Langmuir structures, offering a path toward efficient predictive tools for turbulence–wave interactions in the ocean surface boundary layer.

Abstract

We present an analysis of the coherent structures in Langmuir turbulence, a state of the ocean surface boundary layer driven by the interactions between water waves and wind-induced shear, via a resolvent framework. Langmuir turbulence is characterised by multiscale vortical structures, notably counter-rotating roll pairs known as Langmuir circulations. While classic linear stability analyses of the Craik-Leibovich equations have revealed key instability mechanisms underlying Langmuir circulations, the vortical rolls characteristic of Langmuir turbulence, the present work incorporates the turbulent mean state and varying eddy viscosity using data from large-eddy simulation (LES) to investigate the turbulence dynamics of fully developed Langmuir turbulence. Scale-dependent resolvent analyses reveal a new formation mechanism of two-dimensional circulating rolls and three-dimensional turbulent coherent vortices through linear amplification of sustained harmonic forcing. Moreover, the integrated energy spectra predicted by the principal resolvent modes in response to broadband harmonic forcing capture the dominant spanwise length scales that are consistent with the LES data. These results demonstrate the feasibility of resolvent analyses in capturing key features of multiscale turbulence-wave interactions in the statistical stationary state of Langmuir turbulence.

Figures (13)

• Figure 1: Convergence of the discretised resolvent system for case ${La}_t=0.2$ with $k_x H = 10\pi$, $k_z H=20\pi$ and $\omega= k_x U^L(y=-0.08H) = 459 u_*/H$, as indicated by the singular values, $\sigma_j$ (------), and the cumulative squared singular values, $\sum_{i=1}^{j} \sigma^2_i / \sum_{i=1}^{max} \sigma^2_i$ (-- -- --), for different numbers of Chebyshev collocation points: $N=64$ ($\bullet$), $N=128$ ( ) and $N=256$ ($\blacktriangledown$). The cumulative squared singular values are plotted against the right vertical axis.
• Figure 2: Profiles of (a) the Lagrangian mean velocity $U^L$ and (b) the eddy viscosity $\nu_t$ extracted from the simulations of cases with ${La}_t=0.2$ ($\bullet$) and ${La}_t=0.3$ ($\blacksquare$).
• Figure 3: Contours of the maximum energy amplification $G_{max}$\ref{['eq:Gmax']} at different streamwise and spanwise wavelengths for cases with (a) ${La}_t=0.2$ and (b) ${La}_t=0.3$. The dashed line indicates $\lambda_x = \lambda_z$.
• Figure 4: Structures of the optimal (a) velocity response $\hbox{\boldmath $u$}=(u,v,w)$ and (c) input forcing $\hbox{\boldmath $d$}=(d_x,d_y,d_z)$ for case ${La}_t=0.2$ at $(k_x, k_z, \omega)=(0, 2\pi/H, 0)$. In (a) and (c), the contours represent the streamwise component, $u$ and $d_x$, respectively; the vectors represent the cross-stream components, $(w,v)$ and $(d_z, d_y)$, respectively. The vertical variations in the mean squared response velocity and forcing components are plotted in (b) and (d), respectively: streamwise component ($\overline{u^2}$ or $\overline{d_x^2}$, $\bullet$), vertical component ($\overline{v^2}$ and $\overline{d_y^2}$, ) and spanwise component ($\overline{w^2}$ and $\overline{d_z^2}$, $\blacktriangle$).
• Figure 5: Vortex structures of the principal response mode for case ${La}_t=0.2$ with $k_x H=4$, $k_z H = 16.5$ and $\omega=k_x U^L(y=-0.12H)=44.9 u_*/H$. The vortex structures are elucidated using the iso-surfaces of the $Q$-criterion ($10\%$ of the maximum value), with red and blue indicating positive and negative streamwise vorticity, respectively.