Statistics of bubble plumes generated by breaking surface waves

Abstract

We examine the dependence of the penetration depth and fractional surface area (e.g., whitecap coverage) of bubble plumes generated by breaking surface waves on various wind and wave parameters over a wide range of sea state conditions in the North Pacific Ocean, including storms with sustained winds up to 22 ms$^{-1}$ and significant wave heights up to 10 m. Observations include arrays of freely drifting SWIFT buoys together with shipboard wind and optical video systems, which enabled concurrent high-resolution measurements of wind, waves, bubble plumes, and turbulence. We estimate bubble plume penetration depth from echograms that extend to more than 30 m depth in a surface-following reference frame collected by downward-looking echosounders integrated onboard the buoys. Our observations indicate that the mean and maximum bubble plume penetration depths exceed 10 m and 30 m beneath the surface at high winds, respectively, with a plume residence time of many wave periods. Bubble plume depths are well correlated with wind speeds, spectral wave steepness, and whitecap coverage. Plume depths scaled by total significant wave height are strongly linearly correlated with the inverse of wave age.Plume depths scaled by either wind sea or total significant wave height vary non-monotonically with increasing wind speeds. Dependencies of the combined observations on various non-dimensional predictors used for whitecap coverage estimation are also explored. This study provides first field evidence of a direct relation between bubble plume penetration depth and whitecap coverage, suggesting that the volume of bubble plumes could be estimated by remote sensing.

Scaling the Bubble Penetration Depth in the Ocean

“…with the expectation that through the wave-affected layer (in terms of the TKE dissipation rate enhancement) Γ » 1 with the wave breaking driven enhancement constraint to this layer (e.g., Drennan et al, 1992Drennan et al, , 1996. Previous work suggests that stronger forcing leads to deeper penetration depths (e.g., Derakhti et al, 2023;Graham et al, 2004;Strand et al, 2020). However, evaluating the behavior of observed penetration depths as a function of wind speed and wind energy input alone exhibits a low index of agreement.…”

“…Black solid line is a parameterization proposed by Derakhti et al. (2023): $${\overline{D}}_{bp}=0.092{\left({U}_{10N}\right)}^{1.58}$$. Black dashed line is a parameterization proposed by Derakhti et al.…”

“…Black dashed line is a parameterization proposed by Derakhti et al. (2023) based on a different thresholding method for detection of bubbles: $${\overline{D}}_{bp}=0.55{\left({U}_{10N}\right)}^{0.84}$$. Black dotted line is the parameterization presented by Vagle et al.…”

“…It is relevant to notice that solely on dimensional grounds the following is a reasonable expectation of the observations, 𝐴𝐴 |𝑝𝑝𝑑𝑑| = 𝑓𝑓 ( 𝑢𝑢 * ∕𝑐𝑐, 𝐿𝐿) . Then, introducing a simple linear relationship of the form, 𝐴𝐴 |𝑝𝑝 𝑑𝑑 | ∕𝐿𝐿 ∝ ( 𝐴𝐴 𝑢𝑢 * ∕𝑐𝑐 ) leads to penetration depths directly proportional to the dimensional expression, 𝐴𝐴 𝐴𝐴( 𝑢𝑢 * ∕𝑐𝑐) L, with a a constant coefficient and L a relevant length scale for the extent of the bubble transport and or injection (e.g., Derakhti et al, 2023) Definition of a dimensional 1:1 scaling followed by introducing two constants, 𝐴𝐴 𝐴𝐴Γ −1 (2𝜋𝜋∕𝑘𝑘𝑤𝑤) and BΓ −1 (H s ). The constant coefficients A and B were experimentally defined for the conditions presented here.…”

“…It is relevant to notice that solely on dimensional grounds the following is a reasonable expectation of the observations, $$\vert {p}_{d}\vert =f\left(\sfrac{{u}_{\ast }}{c},L\right)$$. Then, introducing a simple linear relationship of the form, $$\sfrac{\vert {p}_{d}\vert }{L}\propto $$ ($$\sfrac{{u}_{\ast }}{c}$$) leads to penetration depths directly proportional to the dimensional expression, $$a\,\left(\sfrac{{u}_{\ast }}{c}\right)$$ L , with a a constant coefficient and L a relevant length scale for the extent of the bubble transport and or injection (e.g., Derakhti et al., 2023). From this perspective, $$\sfrac{\vert {p}_{d}\vert }{L}\propto \left(\frac{{u}_{\ast }}{{c}_{w}}\right)\mathrm{ln}\left(\sfrac{L}{{z}^{\prime }}\right)$$ is dimensionally consistent and directly proportional to the forcing expressed as the inverse wave age.…”