Submitted as: development and technical paper
24 Apr 2023
Submitted as: development and technical paper |  | 24 Apr 2023
Status: this preprint is currently under review for the journal GMD.

A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water column

Tor Nordam, Ruben Kristiansen, Raymond Nepstad, Erik van Sebille, and Andy M. Booth

Abstract. A common task in oceanography is to model the vertical movement of particles such as microplastics, nanoparticles, mineral particles, gas bubbles, oil droplets, fish eggs, plankton, or algae. In some cases, the distribution of vertical rise or settling velocities of the particles in question can span a wide range, covering several orders of magnitude, often due to a broad particle size distribution or differences in density. This requires numerical methods that are able to adequately resolve a wide and possibly multi-modal velocity distribution.

Lagrangian particle methods are commonly used for these applications. strength of such methods is that each particle can have its own rise or settling speed, which makes it easy to achieve a good representation of a continuous distribution of speeds. An alternative approach is to use Eulerian methods, where the partial differential equations describing the transport problem are solved directly with numerical methods. In Eulerian methods, different rise or settling speeds must be represented as discrete classes, and in practice only a limited number of classes can be included.

Here, we consider three different examples of applications for a water-column model: positively buoyant fish eggs, a mixture of positively and negatively buoyant microplastics, and positively buoyant oil droplets being entrained by waves. For each of the three cases we formulate a model for the vertical transport, based on the advection-diffusion equation with suitable boundary conditions and in one case a reaction term. We give a detailed description of an Eulerian and a Lagrangian implementation of these models, and we demonstrate that they give equivalent results for selected example cases. We also pay special attention to the convergence of the model results with increasing number of classes in the Eulerian scheme, and the number of particles in the Lagrangian scheme. For the Lagrangian scheme, we see the 1/√Np convergence as expected for a Monte Carlo method, while for the Eulerian implementation, we see a second order (1/N2k) convergence with the number of classes.

Tor Nordam et al.

Status: open (until 19 Jun 2023)

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on gmd-2023-49', Anonymous Referee #1, 23 May 2023 reply
    • AC1: 'Reply on RC1', Tor Nordam, 24 May 2023 reply
      • RC3: 'Reply on AC1', Anonymous Referee #1, 24 May 2023 reply
  • RC2: 'Comment on gmd-2023-49', Alethea Mountford, 23 May 2023 reply

Tor Nordam et al.

Tor Nordam et al.


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Short summary
We describe and compare two common methods used to simulate the vertical transport of material in the ocean, called Eulerian and Lagrangian models. They both solve the same transport problems, but use different approaches for how to represent the underlying equations on the computer. The main focus of our study on the numerical accuracy of the two approaches. Our results should be useful for other researchers creating or using these types of transport models.