Building a Maxey–Riley framework for surface ocean inertial particle dynamics

Abstract

A framework for the study of surface ocean inertial particle motion is built from the Maxey-Riley set. A new set is obtained by vertically averaging each term of the original set, adapted to account for Earth's rotation effects, across the extent of a sufficiently small spherical particle that floats at an assumed unperturbed airsea interface with unsteady nonuniform winds and ocean currents above and below, respectively. The inertial particle velocity is shown to exponentially decay in time to a velocity that lies close to an average of seawater and air velocities, weighted by a function of the seawater-to-particle density ratio. Such a weighted average velocity turns out to fortuitously be of the type commonly discussed in the search-and-rescue literature, which alone cannot explain the observed role of anticyclonic mesoscale eddies as traps for marine debris or the formation of great garbage patches in the subtropical gyres, phenomena dominated by finite-size effects. A heuristic extension of the theory is proposed to describe the motion of nonspherical particles by means of a simple shape factor correction, and recommendations are made for incorporating wave-induced Stokes drift, and allowing for inhomogeneities of the carrying fluid density. The new Maxey-Riley set outperforms an ocean adaptation that ignored wind drag effects and the first reported adaption that attempted to incorporate them.

Observation and quantification of inertial effects on the drift of floating objects at the ocean surface

“…Examples of relevant aspects include clustering at the center of the subtropical gyres 18,19 , phenomenon supported on measurements of plastic debris concentration 7 and the analysis of undrogued drifter trajectories 18,19 , or the role of mesoscale eddies as attractors or repellers of inertial particles depending on the polarity of the eddies and the buoyancy of the particles 19,27,28 despite the Lagrangian resilience of their boundaries [48][49][50][51] , which is also backed on observations 52 . The cited phenomena, which act on quite different timescales, all require both O(1) and O(τ ) terms in (12) for their description 18,19,27,28 consistent with the slow manifold M τ in (14), rather than the critical M 0 , controlling the time-asymptotic dynamics of the τ → 0 limit of the Maxey-Riley set (5).…”

“…Unlike stated in Beron-Vera, Olascoaga, and Miron 19 , the domain of applicability of the Maxey-Riley set is not extensible to all possible δ values, which nominally range in a very large interval bounded by 1 from below. Indeed, the fraction of submerged particle volume 19…”

“…which nominally ranges in [0, 1) and gives the emerged (resp., submerged) particle's projected (in the flow direction) area as πΨa 2 (resp., π(1 − Ψ)a 2 ). 19 A. The full set…”

“…Indeed, u is not given a priori as in the standard fluid mechanics setting 23 . Rather, it follows from vertically integrating the drag force 19 . In other words, inertial effects are felt by the particle even when τ = 0.…”

“…The differences in their trajectories are here explained as resulting from inertial effects, i.e., those due to the buoyancy and finite size of the drifters, which prevent them from instantaneously adjusting their velocities to changes in the carrying ocean current and wind fields. This is done by making use of a recently proposed framework for surface ocean inertial particle motion 19 , which is derived from the Maxey-Riley set 20 , the de-jure framework for the study of inertial particle dynamics in fluid mechanics [21][22][23] .…”