A direct numerical simulation of nonbreaking-surface-waves induced mixing

Nonbreaking surface waves (NBSWs) induce vertical mixing even under the windless condition (WLC).  Recent laboratory experiments (e.g., Dai et al. 2010) demonstrated this mixing clearly; stratified water was vertically mixed by the NBSWs under the WLC.  The estimated vertical diffusivity amounts to O(10^-5 m²/s), two orders of magnitude lager than the molecular diffusivity.  Yet, the mechanism of the mixing was not clarified in this laboratory experiment.   Recent numerical studies (e.g., Tsai et al. 2017; Fujiwara et al. 2020) on the other hand showed that the NBSWs under the WLC formed streamwise vortices  beneath the water surface through the CL2 mechanism like Langmuir circulations.  However, the intensity of the mixing was not evaluated in their numerical studies due to short integration time or artificially large eddy viscosity/diffusivity.  As a consequence, how the NBSWs under the WLC could induce the vertical mixing remains to be investigated.  In fact, local generation of turbulence by the wave orbital velocity is proposed as another mechanism of the NBSW-induced turbulence (e.g., Dai et al. 2010; Qiao et al. 2016).  Here, in order to investigate whether and how the NBSW alone could induce such the large vertical mixing, we performed a direct numerical simulation (DNS) of the NBSW under the WLC as in Dai et al. (2010).  The DNS with a sigma-coordinate free-surface nonhydrostatic model reveals that streamwise vortices like Langmuir circulations, developed exponentially at first, grow to be finite amplitude and keep slowly increasing in size and intensity.  At the finite-amplitude stage, the simulated water temperature was vertically mixed from near the surface.  The vertical eddy diffusivity was O(10^-5 m²/s) very near the surface, which is overall similar to the previous estimation (Dai et al.  2010), but its vertical profile was different.  Enstrophy analysys reveals that CL2 mechanism, the same as for Langmuir circulations, kept working even in the finite-amplitude stage to induce the intense mixing near the surface.