On the spatial statistics of free-surface turbulence and the complementarity of 'dimples' and 'scars'
The air--water interface governs the exchange of heat and gas between natural water bodies and their surrounding environment. Turbulence beneath the free surface imprints characteristic features: near-circular depressions (`dimples') and elongated indentations (`scars'). Recent studies have shown that these are linked to sub-surface flow features in a temporal sense. For instance, rapid increases in mean-square surface divergence due to upwelling events, precede dimple count surges. Yet the spatial structure of these connections remains unquantified. We employ spatial statistics to consider the spatial and temporal correlations between dimples and scars and two key velocity-derived fields, surface divergence $β=\partial_x u+\partial_y v$ and vertical vorticity $ω=\partial_x v-\partial_y u$. The dimples and scars are modelled as inhomogeneous Poisson point-processes, with intensity fields driven by the local variance of $β$ and $ω$. Parameters, including spatial support radius $r$ and time lag $τ$, are estimated by maximum likelihood against six DNS datasets, quantifying the spatial and temporal connection between dimples, scars, surface divergence and vorticity. Our results demonstrate a clear complementarity: Dimples show strong local connection to the vertical vorticity field but has weak spatial connection with surface divergence and a spatially ``global'' model is required for dimples to work as estimators of surface divergence; scars, in a similar but opposite manner, couple locally to surface divergence but globally to the vertical vorticity. The complementarity sheds new light on the way dimples and scars may be used to infer fluxes across the surface, e.g., in remote sensing contexts.