A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water column
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)
RC1: 'Comment on gmd-2023-49', Anonymous Referee #1, 23 May 2023
AC1: 'Reply on RC1', Tor Nordam, 24 May 2023
- RC3: 'Reply on AC1', Anonymous Referee #1, 24 May 2023 reply
- AC1: 'Reply on RC1', Tor Nordam, 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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The work of Nordam et al. compares the Eulerian and Lagrangian formulations of the vertical transport equation of buoyant (or negatively buoyant) particles in the seawater column. Specifically, three types of particles were analyzed, namely: buoyant fish eggs, positive/negative buoyant microplastic, and buoyant oil drops, each one characterized by specific boundary conditions and rising/settling velocity. The study shows the accuracy and the convergence of the two approaches by varying several parameters; among those, the most relevant is the number of classes describing the particle population (Eulerian approach) and the number of particles (Lagrangian approach). The results indicate that both the Eulerian and Lagrangian schemes predict the same distribution with an error that reduces with the square of the class number and the square root of the particle number, respectively.
In general, I found the analysis rigorous, but the draft organization could be improved. For instance, there are many subsections and continuous cross-references to other sections in the text. The overall result is often a redundancy of notions and a not fluent reading in some parts. I suggest defining and developing the methods and the results separately.
Finally, The relevance of the research questions and the outcomes of the present work are not evident. The fact that Eulerian and Lagrangian Schemes give similar results is not surprising and trivially expected. The implementation of the boundary conditions adopted is also well-established. Finally, how can one transpose the present results, which are based on a 1D scheme, to more realistic 3D scenarios where the dynamics of the particles are more complex than those implemented here? What is the best approach in the latter cases?
Below I report some specific comments.
- L210. w instead of v within the square brackets?
- L270-279. This long paragraph seems more suited for an Introduction.
- L283. Why do you use the reflection of the particle displacement? Which is the physical meaning? The free surface should dampen the particle velocity independently of the mechanism involved (advection or scattering by diffusivity).
- Fig.1. The GOTM simulation predicts null diffusivity at the water interface. This seems consistent we the null diffusivity flux at the boundary. Is much more demanding to use the output from GOTM instead of Equation (15) in the transport equation?
- L416. Why this difference? Is it maybe owing to a substantial variation of the velocity distribution for the two classes?
- Fig.4. The graduation of the two y-axes overlaps.
- Eq. (19). Which is the value of H_s?