Comparing surface and deep horizontal distributions of depth-keeping particles in shallow fluid layers

TL;DR

This paper investigates whether surface dispersion of depth-keeping particles in shallow forced flows can serve as a quantitative proxy for horizontal transport at depth. The authors perform direct numerical simulations with Lagrangian particle tracking across multiple depths, examining how surface and subsurface flows differ as a function of the control parameter $Re_Fδ^2$. They identify four regimes in the $(z/δ, Re_Fδ^2)$ space, showing that surface patterns reflect upper-layer structures but lose correspondence with depth beneath the top quarter due to vertical shear and bottom boundary-layer dynamics. The findings provide a practical framework for inferring shallow subsurface transport from surface observations in coastal and lake environments, while emphasizing that deeper transport requires explicit knowledge of the vertical velocity profile. The study also outlines future work to incorporate rotation, time-dependent forcing, and higher levels of turbulence to broaden applicability.

Abstract

This study examines whether the dispersion of passive particles at the free surface of a generic (nonturbulent) shallow flow can reliably represent the behavior of depth-keeping particles below the surface. A shallow configuration characterize many aquatic environments, such as coastal regions and lakes, where horizontal scales far exceed vertical ones, large-scale flow structures dominate, and observations are sometimes limited to the surface. We compare surface and subsurface horizontal velocities in both direction and magnitude, identifying distinct behaviors depending on the parameter $Re_Fδ^2$, where $Re_F$ is the Reynolds number based on forcing, and $δ$ is the aspect ratio between the fluid layer depth and the horizontal forcing scale. At low $Re_Fδ^2$, deep flows match the surface flow in direction throughout the layer, but not in magnitude. At high $Re_Fδ^2$, the magnitude matches (outside the bottom boundary layer), but not always the direction. Despite these differences, for all $Re_Fδ^2$, surface particle patterns correlate with those in the upper quarter of the fluid layer. Filamentary structures caused by horizontal flow convergence remain spatially aligned within this region. Below it, at intermediate $Re_Fδ^2$, deep filaments become diffuse and eventually vanish. At high $Re_Fδ^2$, filaments persist at depth, but become spatially misaligned with surface filaments. These findings suggest that in shallow environments, surface observations can quantitatively infer subsurface transport processes in the upper quarter of the fluid layer. For the deeper part, knowledge of the vertical profiles of the mean flow yields insights into the horizontal transport processes.

Figures (16)

• Figure 1: Computational domain for the 3D simulations. The no-slip bottom wall and stress-free (rigid) upper boundary are colored in gray. The cyan lines sketch how the forcing is acting on the fluid. The red dot indicates the origin $O$ of our reference frame.
• Figure 2: Instantaneous fields of the speed ratio $s$ at $z=0.9\delta$, $0.5\delta$ and $0.1\delta$ (from left to right) for flows with $Re_F=70$, $325$, and $1060$ (from top to bottom), as indicated by the label in each panel. For all flows, $\delta=0.3$. The arrows indicate the horizontal velocity fields at the corresponding $z/\delta$ planes. For simplicity, only one quarter of the horizontal domain is shown. Snapshots of the vertical velocity field in the full (horizontal) domain for the three cases shown in panels (b), (e) and (h) can be found in Fig. 7 of Ref. FloresRamirez2025AsymmetricFlows.
• Figure 3: Vertical profiles of the average speed ratio $\langle\!\langle s \rangle\!\rangle$ for all simulations. Each subfigure corresponds to a different $\delta$ value as indicated by their titles. The arrow in the panels indicates increasing values of $Re_F$.
• Figure 4: Normalized boundary layer thickness $z_b/\delta$ versus $Re_F\delta^2$. The dashed line corresponds to the function $z_b/\delta=0.7[1+(Re_F\delta^2/b)^2]^{-1/4}$, see Eq. (\ref{['eq:zbd']}), with $b=11.1\pm 0.8$ a fitting parameter.
• Figure 5: Instantaneous fields of $\cos\theta$ at $z=0.9\delta$, $0.5\delta$ and $0.1\delta$ (from left to right) for flows with $Re_F=70$, $325$, and $1060$ (from top to bottom), as indicated by the label in each panel. For all flows, $\delta=0.3$. The arrows indicate the horizontal velocity fields at the corresponding $z/\delta$ planes. For simplicity, only one quarter of the domain is shown.