I know, I know; A woman trapped underwater during a diving expedition Groundbreaking! And yes, this might sound like a familiar plot. However, this time around there are no sharks. Time is their only real enemy.
The cinematographer on Breaking Surface was Anna Patarakina who also worked on the zombie horror-drama Zoo (2020). The underwater cinematographer was Eric Börjeson who also worked on the Oscar-nominated Kon-Tiki (2012) and the award-winning New Nordic horror movie Let the Right One In (2008, Låt den rätte komma in) which also has a US remake.
Various measures of the turbulence dissipation rate εsgs for the breaking event E3 of the case T1. (a) Phase-resolved horizontal-averaged turbulence dissipation rate εˇsgs in the surface following reference frame, (b) temporal variation of the vertically integrated εˇsgs, (c),(e) phase-resolved εsgs sampled by the two virtual drifters shown in Fig. 2, and (d),(f) their corresponding time-averaged vertical profiles εsgs (thick black lines) as well as the results for all available virtual drifters released at before the beginning of the breaking event T1-E3 (thin gray lines). The dashed lines in (d) and (f) demonstrate that εsgs does not follow the law-of-the-wall vertical scaling (1/z).
Model results of the vertical profile of the ensemble-time-averaged turbulence dissipation rates ε in (a) the surface-following reference frame and (b) the fixed reference frame. Here Heq the equilibrium wave height, WA the active whitecap coverage, εL, and εE are defined in Eqs. (7), (9), (10), and (11), respectively.
Variation of the total wave-breaking-induced TKE dissipation rates with the rate of wind energy input F. Vertical line segments represent the sensitivity of F values with respect to 2 < ce < 3. Definitions of the rest of symbols and lines are the same as in Fig. 10.
The present work is motivated by the study of Thomson et al. (2016), in which turbulence dissipation rates were estimated using Doppler velocity profiles within the upper meter of the wave-following surface. That study concluded that strong turbulence is isolated to a very thin layer (
Here, we revisit the topics of Thomson et al. (2016) by sampling a high-fidelity numerical model in the Lagrangian mode of the surface-following observations. We use a polydisperse two-fluid model (Derakhti and Kirby 2014a) with large-eddy simulation (LES) resolution and volume-of-fluid surface reconstruction (VOF) to simulate the generation and evolution of turbulence and bubbles beneath 3D short-crested wave breaking events in deep water (section 2). We first scale the model domain to match the observed whitecap coverage values, and we scale the model wave heights to match the wind-wave (i.e., equilibrium) portion of the observed spectrum (i.e., neglecting swell). We then determine the effects of sparse sampling and intermittent breaking, as well as the effects of data occlusion by bubbles and limitations in the vertical extent of the observed profiles (section 3). In section 4, we comment on the apparent discrepancy between the observed wind-input energy fluxes and total turbulence dissipation rates reported by Thomson et al. (2016). Examination of potential Lagrangian sampling bias related to a partially trapped drifter in convergence zones in the turbulence observations is left for future work.
In this section, we first present the model governing equations for continuity of mass and momentum of liquid and gas phases of a polydisperse two-fluid mixture, as described in Derakhti and Kirby (2014a). The model setup including details of the incident wave conditions and the scaling of the model domain to match observations of whitecap coverage are then described. Finally, we explain our methodology to convert the model results to surface following virtual drifters.
Demonstrations of model convergence and performance, including detailed comparisons of free surface evolution, bubble void fraction, integral properties of the bubble plume, organized and turbulent velocity fields and total wave-breaking-induced energy dissipation, for various deep- and shallow-breaking waves may be found in Derakhti and Kirby (2014a,b, 2016) and Derakhti et al. (2018, 2019, manuscript submitted to J. Geophys. Res. Oceans).
Our numerical experiments are carried out in a virtual wave tank of unperturbed constant depth h, extending a length Lx in the x direction, and Ly/2 in the transverse y direction. The vertical direction z in the fixed reference frame is positive upward and measured from the still water level. The virtual wave tank is sufficiently deep to avoid any depth-limited wave breaking, such that the experiments remain focused on whitecaps.
We need to choose a number of well-defined parameters to present both the wave breaking forcing and model results in a nondimensional form, such that they can be appropriately scaled to field conditions. Here our goal is to have the wave spectrum E and the fractional area of breaking crests of the simulated cases as consistent as possible with those observed in the field. The latter is usually referred to as the active part of the whitecap coverage W of visible breaking crests, hereafter referred to as WA, which is a space- and time-averaged quantity calculated over a given domain. There is a growing body of literature documenting a direct relationship between WA and the total wave breaking energy dissipation in the upper ocean (Callaghan et al. 2016, 2017; Callaghan 2018; Anguelova and Hwang 2016). We also need a characteristic breaking wave height to scale the vertical profiles of wave-breaking-related dynamical measures, such as the turbulence dissipation rates.
In this paper, our main goal is to examine potential sampling biases and convergence of statistics of the field observations of intermittent wave breaking turbulence collected by surface following platforms (e.g., SWIFT drifters) using our high resolution numerical simulations. To do this, we need to sample our model results, which are available at fixed Eulerian grid points, in a manner which is similar to how a physical drifter (Fig. 1f) obtains samples in the field.
The horizontal location of each virtual drifter is updated using the vertical average of the water horizontal velocity components over the surface layer of depth 0.2Heq. Figures 1g and 1h show the corresponding horizontal displacements of some of the virtual drifters released in a uniform grid and during the events E3 and E4 of the case T1 respectively. Figure 1i shows an example of the horizontal displacement of a SWIFT drifter in the field. In these frames, each color segment represents the horizontal displacement during a fixed time, equal to Ts in the model results and Tp/2 in the observations. Both simulated and observed results indicate that a drifter trapped in an active breaking crest may experience horizontal displacements that are significantly greater than when it is riding on a nonbreaking crest. This is consistent with the recent work of Deike et al. (2017b) and Pizzo et al. (2019).
As summarized in section 1, in most practical applications the long-time average (e.g., over many wave periods) of TKE dissipation rates over a relatively large surface area, O(100Lp 100Lp), is of interest. In this section, we first examine how the Eulerian averages of εsgs compare with those obtained from surface following virtual drifters. Then we comment on the convergence of statistics obtained from the virtual drifters. Last, we examine the effect of incomplete sampling of εsgs by the virtual drifters due to limited vertical field of view and occlusion due to the entrained bubbles. 59ce067264