Simple Eulerian-Lagrangian approach to solve equations for sinking particulate organic matter in the ocean

A gravitational sinking of the particulate organic matter (POM) is a key mechanism of the vertical transport of carbon in the deep ocean and its subsequent sequestration. The size spectrum of these particles is formed in the euphotic layer by the primary production and various mechanisms including food web consumption. The mass of particles, as they descend, decreases under bacterial decomposition and the influence of grazing by filter feeders which depends on the water temperature and oxygen concentration, particle sinking velocity, age of the organic particles, ballasting and other factors. In this study, we consider the influence of the size and age of particles, temperature and oxygen concentration on their dynamics and degradation processes. The model takes into account feedback between the degradation rate and sinking velocity of particles.  We rely on the known parameterisations, but our Eulerian-Lagrangian approach to analytically and numerically solving the problem differs, allowing the model to be incorporated into biogeochemical global ocean models with relative ease. Two novel analytical solutions of the system of the one-dimensional Eulerian equation for POM concentration and Lagrangian equations for particle mass and position were obtained for constant and age-dependent degradation rates. At a constant rate of particle sinking, they correspond to exponential and power-law profiles of the POM concentration. It was found that feedback between degradation rate and sinking velocity significantly changes POM concentration and POM flux vertical profiles.  The calculations are compared with the available POM concentration and flux measurement data for the latitude band of 20-30^oN in the Atlantic and Pacific Oceans and 50-60^o in the Southern Ocean. The dependence of the degradation rate on temperature significantly affected the profiles of POM concentration enhancing the degradation of sinking particles in the upper layers of the oceans and suppressing it in the deep layers of the oceans. The influence of oxygen concentration in all cases considered was insignificant compared to the temperature distribution with depth.