the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Aggregation of Slightly Buoyant Microplastics in ThreeDimensional Vortex Flows
Abstract. Although the movement and aggregation of microplastics at the ocean surface has been well studied, less is known about the subsurface. Within the MaxeyRiley framework governing the movement of small spheres with high drag in fluid, aggregation of buoyant particles is encouraged in vorticitydominated regions. We explore this process in an idealized model of a threedimensional eddy with an azimuthal and overturning circulation. In the axially symmetric state, particles that do not accumulate at the top boundary are attracted to a closed contour consisting of periodic orbits. Such a contour exists when drag on the particle is sufficiently strong. For small slightlybuoyant particles, this contour is located close to the periodic fluid trajectory. If the symmetric flow is perturbed by a symmetrybreaking disturbance, additional attractors arise near periodic orbits of fluid particles within the resonance zones created by the disturbance. Disturbances with periodic time dependence produce even more attractors, with a shape and location that recurs periodically, and which are composed of quasiperiodic orbits of rigid particles. Not all such contours attract, and particles released in the vicinity may instead be attracted to a nearby attractor. Examples are presented along with mappings of the respective basins of attraction.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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Interactive discussion
Status: closed

CC1: 'Comment on egusphere20231624', Francisco BeronVera, 04 Aug 2023
Publisher’s note: the content of this comment was removed on 8 August 2023 since the comment was posted by mistake.
Citation: https://doi.org/10.5194/egusphere20231624CC1 
RC1: 'Comment on egusphere20231624', Francisco BeronVera, 08 Aug 2023
Dear Irina et al.,
0. 'No' above does not strictly mean 'No' but something in bewteen 'No' and 'Yes' and in many cases something closer to 'Yes' but the system doesn't allow me to express that. Also I have revealed my identity by posting a comment as CC by mistake; I don't mind. [It took me a while to get this review submitted through the regular(?) channel. A very confusing system; NPG used to be much easier in the past.]
1. My main concern with this paper is the lack of relevance of the flow considered for ocean dynamics. There's anecdotal reference to observed behavior. But this flow does not represent the solution, neither exact nor approximate, to any known dynamics.
2. Furthermore, the periodic time dependence considered and the ensuing emergence of invariant tori is largely of academic interest rather than a representation of observed behavior.
3. If this cylindrical vortex flow were to be of geophysical interest, then it should first be 'embedded in the ocean,' which will require one to appropriately redefine the Coriolis and centrifugal acceleration terms; cf eg Ripa (1987).
4. But if the authors do not wish to do 3) I believe the way the MR eq is modified to be written in a rotating frame  a seeming novelty?  can be substantively simplified. There appear to be too many Coriolis 'forces' involved. The MR reads (in the authors' notation except for the use of the Stokes time, \tau)
\dot v = 3R/2 Du/Dt + \tau^{1} (u  v) + (1  3R/2) g
(cf Haller & Sapsis, 2008). In a frame rotating with angular velocity \Omega one just must replace
\dot v > \dot v + 2\Omega\times v + \Omega\times \Omega \times x
and
Du/Dt > Du/Dt + 2\Omega\times u + \Omega\times \Omega \times x
These simple replacements lead to (3) after rearranging.
5. It should be \dot x = v_p(x,t) in (3); otherwise one doesn't have a closed system for (x,v_p). But there are x, x_p and r? What is, in particular, r (in the definition of g_r)?
6. BeronVera et al. do not ignore the centrifugal force. They work in a GFD setting in which the gravitational and centrifugal forces balance one another on a horizontal plane (tangent to the planet). The resulting gravity (which defines the vertical direction) is in general a function of longitude and latitude. As is common in GFD this dependence is ignored, assuming spherical geometry, but the fundamental balance above is kept.
7. Fountain et al. is not included in the reference list.
8. Eq (8) is a modest modification of H&S' eq (31).
9. Fluid particles cannot deform or split; thus are rigid. A different name should be used to denote inertial or finitesize particles, for example, inertial or finitesize particles!
10. The derivation of the reduced eq on the slow manifold lacks a prove of the global attractivity of the latter. Fenichel does this (with the modifications introduced in H&S to the nonautonomous case).
11. In the Abstract and elsewhere, 'closed contour' should be replaced by 'loop' or 'torus' as contour is more suggestive of a level set of a scalar.
13. In l.~79 dX_b suggests random variable when everything is deterministic in this ms.
14. The comment on divergence in l.~178 seems interesting; however, I couldn't find the counterexample.
15. Finally, all the results relating behavior near (resonance) tori of the flow may be fine, theoretically, but they lack a connection with ocean dynamics. I urge the authors' to tone down the claimed relevance of their results to oceanography in a revised version of the paper.
Javier
Citation: https://doi.org/10.5194/egusphere20231624RC1 
RC2: 'Comment on egusphere20231624', Anonymous Referee #2, 28 Aug 2023
Review of "Aggregation of Slightly Buoyant Microplastics in ThreeDimensional Vortex Flows" by Rypina et al
In this manuscript, the authors analyse the trajectories of idealised but nontrivial particles in a complex analytical flow field, to find attractors in the phasespace.
This is a sophisticated and careful analysis of an interesting dynamical system, and I enjoyed reading it a lot. However, like the other reviewer I agree that the connection to actual microplastics in the ocean is not very clear.
While there are some general references to some literature in the introduction section, there is no attempt to reconcile in the discussion section what the results and conclusions mean for microplastic in the real ocean. This really dimities the potential impact of the manuscript, and I thus strongly encourage the authors to reflect on the extent to which the results can be generalised to more realistic scenarios. For example, how would the results change if
a. the eddy is also translating (advecting) and/or decaying;
b. the particles experience biofouling (see also the highly relevant work at https://aslopubs.onlinelibrary.wiley.com/doi/full/10.1002/lno.11879);
c. the particles are nonspherical;
d. the particles are negatively buoyant?
Furthermore, I have the following minor comments
 line 41: is it also important to state that these particles are spherical?
 line 5964: these statements here need more references. For example, are concentrations really largest at the sea surface?
 line 72: Is Froyland et al (2014) the most appropriate reference for this statement?
 line 84: E.g. https://www.nature.com/articles/s41598020728984 also uses an Eulerian approach
 line 100: is there being an attractor a sufficient condition for aggregation to occur? Or do other conditions also need to be fulfilled?
 line 103: added mas is not introduced yet, at this point
 line 104: how is drag different from inertia?
 line 129: a motivation for why the lift force, the Basset history force and the Faxen corrections are omitted is missing. This should be motivated
 line 162: known in advance of what?
 line 174: could the surface then also been seen as an attractor, possible one 'external' to the fluid?
 line 176: 'similar' instead of 'the like'?
 line 198: give examples of these cases?
 line 234: which phase space? Physical space?
 line 246: give references for these 'numerous authors'?
 The manuscript is very sloppy with Equation references. Very often, the word 'Eq.' Misses before an equation is references
 line 283: It would be helpful to start a new subsection here, detailing with fluid trajectories
 line 285: is there an estimate hat 'a sufficient length of time' is?
 line 433: How are the particles integrated? What timestepping scheme? What integration scheme? Which boundary conditions?
 line 474: If these are nondimensional units, they seem like enormous(?) particles, at 0.1% of the size of the eddy?? Or how does the scaling of $d$ work?
 In general, it would be helpful if the authors could also provide animations of their simulations
 The manuscript is also somewhat sloppy with math font in text. On some occasions (e.g. line 233, 367, 391), short equations seem to have been written in normal (italic) font, rather than math. This is especially a problem with negative numbers, where the minus kind of disappears.Citation: https://doi.org/10.5194/egusphere20231624RC2 
AC1: 'Comment on egusphere20231624', Irina I. Rypina, 29 Sep 2023
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere20231624/egusphere20231624AC1supplement.pdf

AC2: 'Comment on egusphere20231624', Irina I. Rypina, 29 Sep 2023
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere20231624/egusphere20231624AC2supplement.pdf
 AC3: 'Comment on egusphere20231624', Irina I. Rypina, 29 Sep 2023
 AC4: 'Comment on egusphere20231624', Irina I. Rypina, 29 Sep 2023
Interactive discussion
Status: closed

CC1: 'Comment on egusphere20231624', Francisco BeronVera, 04 Aug 2023
Publisher’s note: the content of this comment was removed on 8 August 2023 since the comment was posted by mistake.
Citation: https://doi.org/10.5194/egusphere20231624CC1 
RC1: 'Comment on egusphere20231624', Francisco BeronVera, 08 Aug 2023
Dear Irina et al.,
0. 'No' above does not strictly mean 'No' but something in bewteen 'No' and 'Yes' and in many cases something closer to 'Yes' but the system doesn't allow me to express that. Also I have revealed my identity by posting a comment as CC by mistake; I don't mind. [It took me a while to get this review submitted through the regular(?) channel. A very confusing system; NPG used to be much easier in the past.]
1. My main concern with this paper is the lack of relevance of the flow considered for ocean dynamics. There's anecdotal reference to observed behavior. But this flow does not represent the solution, neither exact nor approximate, to any known dynamics.
2. Furthermore, the periodic time dependence considered and the ensuing emergence of invariant tori is largely of academic interest rather than a representation of observed behavior.
3. If this cylindrical vortex flow were to be of geophysical interest, then it should first be 'embedded in the ocean,' which will require one to appropriately redefine the Coriolis and centrifugal acceleration terms; cf eg Ripa (1987).
4. But if the authors do not wish to do 3) I believe the way the MR eq is modified to be written in a rotating frame  a seeming novelty?  can be substantively simplified. There appear to be too many Coriolis 'forces' involved. The MR reads (in the authors' notation except for the use of the Stokes time, \tau)
\dot v = 3R/2 Du/Dt + \tau^{1} (u  v) + (1  3R/2) g
(cf Haller & Sapsis, 2008). In a frame rotating with angular velocity \Omega one just must replace
\dot v > \dot v + 2\Omega\times v + \Omega\times \Omega \times x
and
Du/Dt > Du/Dt + 2\Omega\times u + \Omega\times \Omega \times x
These simple replacements lead to (3) after rearranging.
5. It should be \dot x = v_p(x,t) in (3); otherwise one doesn't have a closed system for (x,v_p). But there are x, x_p and r? What is, in particular, r (in the definition of g_r)?
6. BeronVera et al. do not ignore the centrifugal force. They work in a GFD setting in which the gravitational and centrifugal forces balance one another on a horizontal plane (tangent to the planet). The resulting gravity (which defines the vertical direction) is in general a function of longitude and latitude. As is common in GFD this dependence is ignored, assuming spherical geometry, but the fundamental balance above is kept.
7. Fountain et al. is not included in the reference list.
8. Eq (8) is a modest modification of H&S' eq (31).
9. Fluid particles cannot deform or split; thus are rigid. A different name should be used to denote inertial or finitesize particles, for example, inertial or finitesize particles!
10. The derivation of the reduced eq on the slow manifold lacks a prove of the global attractivity of the latter. Fenichel does this (with the modifications introduced in H&S to the nonautonomous case).
11. In the Abstract and elsewhere, 'closed contour' should be replaced by 'loop' or 'torus' as contour is more suggestive of a level set of a scalar.
13. In l.~79 dX_b suggests random variable when everything is deterministic in this ms.
14. The comment on divergence in l.~178 seems interesting; however, I couldn't find the counterexample.
15. Finally, all the results relating behavior near (resonance) tori of the flow may be fine, theoretically, but they lack a connection with ocean dynamics. I urge the authors' to tone down the claimed relevance of their results to oceanography in a revised version of the paper.
Javier
Citation: https://doi.org/10.5194/egusphere20231624RC1 
RC2: 'Comment on egusphere20231624', Anonymous Referee #2, 28 Aug 2023
Review of "Aggregation of Slightly Buoyant Microplastics in ThreeDimensional Vortex Flows" by Rypina et al
In this manuscript, the authors analyse the trajectories of idealised but nontrivial particles in a complex analytical flow field, to find attractors in the phasespace.
This is a sophisticated and careful analysis of an interesting dynamical system, and I enjoyed reading it a lot. However, like the other reviewer I agree that the connection to actual microplastics in the ocean is not very clear.
While there are some general references to some literature in the introduction section, there is no attempt to reconcile in the discussion section what the results and conclusions mean for microplastic in the real ocean. This really dimities the potential impact of the manuscript, and I thus strongly encourage the authors to reflect on the extent to which the results can be generalised to more realistic scenarios. For example, how would the results change if
a. the eddy is also translating (advecting) and/or decaying;
b. the particles experience biofouling (see also the highly relevant work at https://aslopubs.onlinelibrary.wiley.com/doi/full/10.1002/lno.11879);
c. the particles are nonspherical;
d. the particles are negatively buoyant?
Furthermore, I have the following minor comments
 line 41: is it also important to state that these particles are spherical?
 line 5964: these statements here need more references. For example, are concentrations really largest at the sea surface?
 line 72: Is Froyland et al (2014) the most appropriate reference for this statement?
 line 84: E.g. https://www.nature.com/articles/s41598020728984 also uses an Eulerian approach
 line 100: is there being an attractor a sufficient condition for aggregation to occur? Or do other conditions also need to be fulfilled?
 line 103: added mas is not introduced yet, at this point
 line 104: how is drag different from inertia?
 line 129: a motivation for why the lift force, the Basset history force and the Faxen corrections are omitted is missing. This should be motivated
 line 162: known in advance of what?
 line 174: could the surface then also been seen as an attractor, possible one 'external' to the fluid?
 line 176: 'similar' instead of 'the like'?
 line 198: give examples of these cases?
 line 234: which phase space? Physical space?
 line 246: give references for these 'numerous authors'?
 The manuscript is very sloppy with Equation references. Very often, the word 'Eq.' Misses before an equation is references
 line 283: It would be helpful to start a new subsection here, detailing with fluid trajectories
 line 285: is there an estimate hat 'a sufficient length of time' is?
 line 433: How are the particles integrated? What timestepping scheme? What integration scheme? Which boundary conditions?
 line 474: If these are nondimensional units, they seem like enormous(?) particles, at 0.1% of the size of the eddy?? Or how does the scaling of $d$ work?
 In general, it would be helpful if the authors could also provide animations of their simulations
 The manuscript is also somewhat sloppy with math font in text. On some occasions (e.g. line 233, 367, 391), short equations seem to have been written in normal (italic) font, rather than math. This is especially a problem with negative numbers, where the minus kind of disappears.Citation: https://doi.org/10.5194/egusphere20231624RC2 
AC1: 'Comment on egusphere20231624', Irina I. Rypina, 29 Sep 2023
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere20231624/egusphere20231624AC1supplement.pdf

AC2: 'Comment on egusphere20231624', Irina I. Rypina, 29 Sep 2023
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2023/egusphere20231624/egusphere20231624AC2supplement.pdf
 AC3: 'Comment on egusphere20231624', Irina I. Rypina, 29 Sep 2023
 AC4: 'Comment on egusphere20231624', Irina I. Rypina, 29 Sep 2023
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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
 Preprint
(1897 KB)  Metadata XML

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