  Buoyant, finite-size or inertial particle motion is fundamentally unlike
neutrally buoyant, infinitesimally small or Lagrangian particle motion. The
de-jure fluid mechanics framework for the description of inertial particle
dynamics is provided by the Maxey-Riley equation. Derived from first principles
- a result of over a century of research since the pioneering work by Sir
George Stokes - the Maxey-Riley equation is a Newton-type-law with several
forces including (mainly) flow, added mass, shear-induced lift, and drag
forces. In this paper we present an overview of recent efforts to port the
Maxey-Riley framework to oceanography. These involved: 1) including the
Coriolis force, which was found to explain behavior of submerged floats near
mesoscale eddies; 2) accounting for the combined effects of ocean current and
wind drag on inertial particles floating at the air-sea interface, which helped
understand the formation of great garbage patches and the role of anticyclonic
eddies as plastic debris traps; and 3) incorporating elastic forces, which are
needed to simulate the drift of pelagic Sargassum. Insight on the nonlinear
dynamics of inertial particles in every case was possible to be achieved by
investigating long-time asymptotic behavior in the various Maxey-Riley equation
forms, which represent singular perturbation problems involving slow and fast
variables.
