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

Maderich, Vladimir; Brovchenko, Igor; Kovalets, Kateryna; Seo, Seongbong; Kim, Kyeong Ok

doi:https://doi.org/10.5194/egusphere-2025-491

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https://doi.org/10.5194/egusphere-2025-491

Preprints

21 Feb 2025

| 21 Feb 2025

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

Vladimir Maderich, Igor Brovchenko, Kateryna Kovalets, Seongbong Seo, and Kyeong Ok Kim

Abstract. A gravitational sinking of the particulate organic matter (POM) is a key mechanism of 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, changed under aggregation, fragmentation, bacterial decomposition which depends on the water temperature and oxygen concentration, particle sinking velocity, age of the organic particles, ballasting and other factors. In this paper, we developed simple Eulerian-Lagrangian approach to solve equations for sinking particulate matter when the influence of the size and age of particles, temperature and oxygen concentration on their dynamics and degradation processes were taken into account. The model considers feedback between degradation rate and particle sinking velocity. We rely on the known parameterizations, but our Eulerian-Lagrangian approach to solving the problem differs, allowing the algorithm to be incorporated into biogeochemical global ocean models with relative ease. Two novel analytical solutions of a 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 results to a significant change in POM and POM flux vertical profiles. The calculations are compared with the available measurement data for POM and POM flux for the latitude band of 20–30° N in the Atlantic and Pacific Oceans and 50–60° S 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 ocean’s upper layer and suppressing it in the deep layer of the ocean. The influence of oxygen concentration in all cases considered was insignificant compared to the distribution of temperature with depth.

Received: 02 Feb 2025 – Discussion started: 21 Feb 2025

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Journal article(s) based on this preprint

16 Oct 2025

Simple Eulerian–Lagrangian approach to solving equations for sinking particulate organic matter in the ocean

Vladimir Maderich, Igor Brovchenko, Kateryna Kovalets, Seongbong Seo, and Kyeong Ok Kim

Geosci. Model Dev., 18, 7373–7387, https://doi.org/10.5194/gmd-18-7373-2025,https://doi.org/10.5194/gmd-18-7373-2025, 2025

Short summary

Vladimir Maderich, Igor Brovchenko, Kateryna Kovalets, Seongbong Seo, and Kyeong Ok Kim

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-491', Anonymous Referee #1, 31 Mar 2025

This is an interesting paper, examining a model of the vertical POC flux

in the ocean below the euphotic zone. I found the paper hard to follow in

places, and this was in part because of the choice of English usage: I would

strongly urge the authors to seek out a native English speaker to clean

this up.
There have been quite a few papers published recently that take a

similar approach to the problem of modeling particle flux in the

ocean, and the authors cite all of these (Kriest and Oschlies, 2008;

Omand et al., 2020; DeVries et al., 2014). However, it is unclear what

this manuscript presents that is new when compared with these other

papers. Indeed, as far as I can see, there is no detailed comparison

of results (except to show that one of their analytical solution is

equivalent to that of DeVries et al.). I would like to see an analysis

of what new things we learn from this model.
The model contains many assumption (as stated by the authors), but there

is little to no analysis of the consequences of these assumptions. For

example, everything is assumed to be a power-law (the mass-size relationship,

the sinking velocity etc.) and while this makes things analytically

tractable, it is unclear what observational evidence there is for them.

For example, size distributions are often assumed to be power-law,

but in reality this assumption often holds over a relatively small size range.
The model is a steady state model, and it's unclear if such an assumption

is a reasonable one. For example. export fluxes out of the euphotic zone can

vary significantly over time periods of days. So whilst I'm not opposed to

the use of the steady state assumption, I do wonder about its validity.
Line 93, the mass loss is proportional to particle mass, not volume.

The relationship in Equation (4) makes the correspondence between

mass and volume unclear. For example, is the diameter the equivalent

spherical diameter, is the volume the conserved volume or the encased volume?
Line 109: I must be missing something here, because it's unclear to me

that, practically, z-prime can never be larger than the inverse of psi.

This follows from re-writing equation (10) and realizing that

the constants eta, gamma0, and zeta are all positive. What am I missing?
The authors also need to make their notation more consistent. For example,

in equation (15) we get the definition for C_{p,d}. But in equation (16)

this becomes C_{p,d,i}. Also, in equation (16), n_d becomes n. In equation

(17) we are apparently integrating with respect to a constant (d_0 having

been defined as the initial particle diameter in equation (8)). So, the

notation needs to be tidied up throughout the paper, not just in these places.

Citation: https://doi.org/10.5194/egusphere-2025-491-RC1
- AC1: 'Reply on RC1', Kyeong Ok Kim, 21 May 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-491/egusphere-2025-491-AC1-supplement.zip
  
  Citation: https://doi.org/10.5194/egusphere-2025-491-AC1
- AC2: 'Reply on RC1', Kyeong Ok Kim, 21 May 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-491/egusphere-2025-491-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-491-AC2
RC2:
'Comment on egusphere-2025-491', Anonymous Referee #2, 20 Apr 2025

The paper addresses the depth distribution of particulate organic matter (POM) and its associated flux, introducing analytical solutions and a numerical approach to solve corresponding differential equations.
The equations of the analytical part are based on the assumption that mass and sinking speed of a particle is governed by its diameter, which varies with time due to the degradation processes.

Time-varying solutions of differential equation for POM concentration and flux are given for constant and time-varying degradation rates and are finally converted into corresponding depth-varying solutions.
As a reader who does not juggle with DGL solutions every day, I find the derivation of the formulas difficult to track.

Readers should be able to do so without the need of many calculations on an extra sheet of paper. E.g., you might want to write the integrals that convert sinking speed formulas (9)/(22) into the corresponding depth formulas (10)/(23).

I also would like to see some details about how the depth-varying solutions for sinking speed, particle diameter [and degradation rate] and finally the depth-dependent particle concentration and flux are calculated, i.e., how does (10) applied to (8) and (9) yield (11) and (12) (and finally (14)), and how does (23) applied to (20)-(22) yield (24)-(26) (and finally (28)).

Also, I do not see if formulas (31) (for constant degradation) and (32) (for time-varying degradation), which consider the special case of a constant particle sinking speed, are derived from formulas (14) and (28), respectively.

This would be good to see, because (31) and (32) are used to discuss the differences of solutions with respect to the corresponding assumptions.
The paper also provides a numerical solution algorithm for particle concentration and flux.

The Algorithm is verified w.r.t. the derived analytical solutions (Fig. 1) and applied under the additional assumptions that particle flux is also influenced by (i) temperature and (ii) oxygen concentration.

Results indicate a clear dependence on the temperature profile and on parameters with uncertain range, e.g., the exponent mu which relates particle diameter and sinking speed.

In the corresponding sections 4 and 5 I found some places where S_p was used instead of C_p to refer to POM concentration (Algorithm 3, Figure 5, line 269), please correct.
You may want to place the legend of figures 4-6 (which repeats in every single panel plot) only once to the right of the panels.

I would add the explanation that the three columns of panels in figures 4-6 correspond to the model without dependency of temperature and oxygen (panels a and d), additional temperature dependence (panels b and e), and both additional dependencies (panels c and f) to the figure captions in order to make the figures self-explaining.
In line 280, I interpreted "1/r < 1" and "r > 1" as defining assumptions at first glance (which makes actually no sense without the definition of r). I would define r=1.25 first and then derive p_min and p_max by the ratios 1/r and r, respectively (probably without extra stressing "1/r<1" and "r>1").

Citation: https://doi.org/10.5194/egusphere-2025-491-RC2
- AC3: 'Reply on RC2', Kyeong Ok Kim, 21 May 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-491/egusphere-2025-491-AC3-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-491-AC3
RC3:
'Comment on egusphere-2025-491', Anonymous Referee #3, 27 Apr 2025

This paper presents a combined Eulerian-Lagrangian formulation of the organic particles descent while taking into account their size and age among other things, resulting in a non-trivial relation among them, unlike suggested by previous studies. This is a well-written article with strong positioning regarding motivation, contribution and place among the previous literature. My detailed major and minor comments, mainly refraining to comment on the formulations, are organized below in the order of appearance in the manuscript.
While the contribution is clear around line 45, I would recommend highlighting the novelty by explaining the unique contribution in the context of limitations of previous attempts in the literature.
One major comment I have is that the modeling results section needs to be more specific in terms of presenting the results. The authors are showing and referring to the results but not describing and discussing them.
Same goes for the validations -- I found it hard to follow which parts agree with the measurements more than others. Both of the aspects need to be addressed in the results section for the paper to be strong.
When saying Fig 4-6, as in line 270, I'd recommend being specific when referring to figures and panels to show results.
“As follow” is a typo in line 272
Another major comment is that the paper is missing the significance in terms of closing the loop and relating back the findings back to the implementation in the earth system/biogeochemical models as the authors had described in the introduction section.
Please include significance and follow-up work recommendations before the conclusion. The authors have shown appreciations of the limitations of the work at various places in the text, I'd recommend synthesizing them at the end before conclusions.
In light of my comments above, I’d recommend minor revisions to the manuscript before it is published. Good luck and congratulations on this useful work.

Citation: https://doi.org/10.5194/egusphere-2025-491-RC3
- AC4: 'Reply on RC3', Kyeong Ok Kim, 21 May 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-491/egusphere-2025-491-AC4-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-491-AC4

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-491', Anonymous Referee #1, 31 Mar 2025

This is an interesting paper, examining a model of the vertical POC flux

in the ocean below the euphotic zone. I found the paper hard to follow in

places, and this was in part because of the choice of English usage: I would

strongly urge the authors to seek out a native English speaker to clean

this up.
There have been quite a few papers published recently that take a

similar approach to the problem of modeling particle flux in the

ocean, and the authors cite all of these (Kriest and Oschlies, 2008;

Omand et al., 2020; DeVries et al., 2014). However, it is unclear what

this manuscript presents that is new when compared with these other

papers. Indeed, as far as I can see, there is no detailed comparison

of results (except to show that one of their analytical solution is

equivalent to that of DeVries et al.). I would like to see an analysis

of what new things we learn from this model.
The model contains many assumption (as stated by the authors), but there

is little to no analysis of the consequences of these assumptions. For

example, everything is assumed to be a power-law (the mass-size relationship,

the sinking velocity etc.) and while this makes things analytically

tractable, it is unclear what observational evidence there is for them.

For example, size distributions are often assumed to be power-law,

but in reality this assumption often holds over a relatively small size range.
The model is a steady state model, and it's unclear if such an assumption

is a reasonable one. For example. export fluxes out of the euphotic zone can

vary significantly over time periods of days. So whilst I'm not opposed to

the use of the steady state assumption, I do wonder about its validity.
Line 93, the mass loss is proportional to particle mass, not volume.

The relationship in Equation (4) makes the correspondence between

mass and volume unclear. For example, is the diameter the equivalent

spherical diameter, is the volume the conserved volume or the encased volume?
Line 109: I must be missing something here, because it's unclear to me

that, practically, z-prime can never be larger than the inverse of psi.

This follows from re-writing equation (10) and realizing that

the constants eta, gamma0, and zeta are all positive. What am I missing?
The authors also need to make their notation more consistent. For example,

in equation (15) we get the definition for C_{p,d}. But in equation (16)

this becomes C_{p,d,i}. Also, in equation (16), n_d becomes n. In equation

(17) we are apparently integrating with respect to a constant (d_0 having

been defined as the initial particle diameter in equation (8)). So, the

notation needs to be tidied up throughout the paper, not just in these places.

Citation: https://doi.org/10.5194/egusphere-2025-491-RC1
- AC1: 'Reply on RC1', Kyeong Ok Kim, 21 May 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-491/egusphere-2025-491-AC1-supplement.zip
  
  Citation: https://doi.org/10.5194/egusphere-2025-491-AC1
- AC2: 'Reply on RC1', Kyeong Ok Kim, 21 May 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-491/egusphere-2025-491-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-491-AC2
RC2:
'Comment on egusphere-2025-491', Anonymous Referee #2, 20 Apr 2025

The paper addresses the depth distribution of particulate organic matter (POM) and its associated flux, introducing analytical solutions and a numerical approach to solve corresponding differential equations.
The equations of the analytical part are based on the assumption that mass and sinking speed of a particle is governed by its diameter, which varies with time due to the degradation processes.

Time-varying solutions of differential equation for POM concentration and flux are given for constant and time-varying degradation rates and are finally converted into corresponding depth-varying solutions.
As a reader who does not juggle with DGL solutions every day, I find the derivation of the formulas difficult to track.

Readers should be able to do so without the need of many calculations on an extra sheet of paper. E.g., you might want to write the integrals that convert sinking speed formulas (9)/(22) into the corresponding depth formulas (10)/(23).

I also would like to see some details about how the depth-varying solutions for sinking speed, particle diameter [and degradation rate] and finally the depth-dependent particle concentration and flux are calculated, i.e., how does (10) applied to (8) and (9) yield (11) and (12) (and finally (14)), and how does (23) applied to (20)-(22) yield (24)-(26) (and finally (28)).

Also, I do not see if formulas (31) (for constant degradation) and (32) (for time-varying degradation), which consider the special case of a constant particle sinking speed, are derived from formulas (14) and (28), respectively.

This would be good to see, because (31) and (32) are used to discuss the differences of solutions with respect to the corresponding assumptions.
The paper also provides a numerical solution algorithm for particle concentration and flux.

The Algorithm is verified w.r.t. the derived analytical solutions (Fig. 1) and applied under the additional assumptions that particle flux is also influenced by (i) temperature and (ii) oxygen concentration.

Results indicate a clear dependence on the temperature profile and on parameters with uncertain range, e.g., the exponent mu which relates particle diameter and sinking speed.

In the corresponding sections 4 and 5 I found some places where S_p was used instead of C_p to refer to POM concentration (Algorithm 3, Figure 5, line 269), please correct.
You may want to place the legend of figures 4-6 (which repeats in every single panel plot) only once to the right of the panels.

I would add the explanation that the three columns of panels in figures 4-6 correspond to the model without dependency of temperature and oxygen (panels a and d), additional temperature dependence (panels b and e), and both additional dependencies (panels c and f) to the figure captions in order to make the figures self-explaining.
In line 280, I interpreted "1/r < 1" and "r > 1" as defining assumptions at first glance (which makes actually no sense without the definition of r). I would define r=1.25 first and then derive p_min and p_max by the ratios 1/r and r, respectively (probably without extra stressing "1/r<1" and "r>1").

Citation: https://doi.org/10.5194/egusphere-2025-491-RC2
- AC3: 'Reply on RC2', Kyeong Ok Kim, 21 May 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-491/egusphere-2025-491-AC3-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-491-AC3
RC3:
'Comment on egusphere-2025-491', Anonymous Referee #3, 27 Apr 2025

This paper presents a combined Eulerian-Lagrangian formulation of the organic particles descent while taking into account their size and age among other things, resulting in a non-trivial relation among them, unlike suggested by previous studies. This is a well-written article with strong positioning regarding motivation, contribution and place among the previous literature. My detailed major and minor comments, mainly refraining to comment on the formulations, are organized below in the order of appearance in the manuscript.
While the contribution is clear around line 45, I would recommend highlighting the novelty by explaining the unique contribution in the context of limitations of previous attempts in the literature.
One major comment I have is that the modeling results section needs to be more specific in terms of presenting the results. The authors are showing and referring to the results but not describing and discussing them.
Same goes for the validations -- I found it hard to follow which parts agree with the measurements more than others. Both of the aspects need to be addressed in the results section for the paper to be strong.
When saying Fig 4-6, as in line 270, I'd recommend being specific when referring to figures and panels to show results.
“As follow” is a typo in line 272
Another major comment is that the paper is missing the significance in terms of closing the loop and relating back the findings back to the implementation in the earth system/biogeochemical models as the authors had described in the introduction section.
Please include significance and follow-up work recommendations before the conclusion. The authors have shown appreciations of the limitations of the work at various places in the text, I'd recommend synthesizing them at the end before conclusions.
In light of my comments above, I’d recommend minor revisions to the manuscript before it is published. Good luck and congratulations on this useful work.

Citation: https://doi.org/10.5194/egusphere-2025-491-RC3
- AC4: 'Reply on RC3', Kyeong Ok Kim, 21 May 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-491/egusphere-2025-491-AC4-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-491-AC4

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

AR by Kyeong Ok Kim on behalf of the Authors (21 May 2025) Author's tracked changes

EF by Polina Shvedko (27 May 2025) Manuscript

EF by Polina Shvedko (27 May 2025) Author's response

ED: Referee Nomination & Report Request started (13 Jun 2025) by Pearse Buchanan

RR by Anonymous Referee #2 (13 Jun 2025)

ED: Publish subject to minor revisions (review by editor) (21 Jul 2025) by Pearse Buchanan

AR by Kyeong Ok Kim on behalf of the Authors (28 Jul 2025) Author's response Author's tracked changes Manuscript

ED: Publish as is (04 Aug 2025) by Pearse Buchanan

AR by Kyeong Ok Kim on behalf of the Authors (04 Aug 2025)

Journal article(s) based on this preprint

16 Oct 2025

Simple Eulerian–Lagrangian approach to solving equations for sinking particulate organic matter in the ocean

Vladimir Maderich, Igor Brovchenko, Kateryna Kovalets, Seongbong Seo, and Kyeong Ok Kim

Geosci. Model Dev., 18, 7373–7387, https://doi.org/10.5194/gmd-18-7373-2025,https://doi.org/10.5194/gmd-18-7373-2025, 2025

Short summary

Vladimir Maderich, Igor Brovchenko, Kateryna Kovalets, Seongbong Seo, and Kyeong Ok Kim

Supplement

https://doi.org/10.5194/egusphere-2025-491-supplement

Data sets

KKovalets/EuLag_DataSet: EuLag_DataSet (v0.0.0) K. Kovalets et al. https://doi.org/10.5281/zenodo.14782095

Model code and software

KKovalets/EuLag: EuLag (v0.2.0) K. Kovalets et al. https://doi.org/10.5281/zenodo.14782046

Vladimir Maderich, Igor Brovchenko, Kateryna Kovalets, Seongbong Seo, and Kyeong Ok Kim

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Short summary

We have developed a new simple Eulerian-Lagrangian approach to solve equations for sinking particulate organic matter in the ocean. We rely on the known parameterizations, but our approach to solving the problem differs, allowing the algorithm to be incorporated into biogeochemical global ocean models with relative ease. New analytical and numerical solutions confirmed that feedback between degradation rate and sinking velocity significantly changes particulate matter fluxes.


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