PDE-Based Transport Maps for Inertial Particles in Dynamic Marine Environments

Assuming purely Lagrangian particle transport in the ocean may be justified at a macro scale, but can significantly differ from reality at smaller scales. First, the transported particles can experience significant buoyant forces due to varying density gradients in the ocean. Further, the inertia of particles implies that the particle velocity differs from the underlying ocean currents. In this work, we present an efficient and accurate approach to compute transport maps for inertial particles in dynamic ocean environments. Our work utilizes the Maxey-Riley equation and embeds this dynamic in a higher dimensional space to yield an advective transport equation that utilizes a modified velocity field. The solution of this equation when projected on the physical space yields a global transport map that accounts for buoyancy and inertial effects. The PDE at hand is solved efficiently using the method of composition to yield the transport maps for the inertial particles, without compounding of numerical errors. Our work is in part motivated by the need to better understand the transport of the sediment plumes generated from deep sea mining operations, thus we showcase the applications of the developed theory for this context in analytical and realistic flows in 2 and 3 dimensions. We also illustrate results for other dispersed particles including plastic debris.