A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water column
<p>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 the vertical rise or settling velocities of the particles in que...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Copernicus Publications
2023-09-01
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| Series: | Geoscientific Model Development |
| Online Access: | https://gmd.copernicus.org/articles/16/5339/2023/gmd-16-5339-2023.pdf |
| _version_ | 1797681427063504896 |
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| author | T. Nordam T. Nordam R. Kristiansen R. Nepstad E. van Sebille A. M. Booth |
| author_facet | T. Nordam T. Nordam R. Kristiansen R. Nepstad E. van Sebille A. M. Booth |
| author_sort | T. Nordam |
| collection | DOAJ |
| description | <p>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 the
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.</p>
<p>Lagrangian particle methods are commonly used for these applications. A
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.</p>
<p>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 an increasing number of classes in the Eulerian scheme and with the
number of particles in the Lagrangian scheme. For the Lagrangian scheme, we
see the <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M1" display="inline" overflow="scroll" dspmath="mathml"><mrow><mn mathvariant="normal">1</mn><mo>/</mo><msqrt><mrow><msub><mi>N</mi><mi>p</mi></msub></mrow></msqrt></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="39pt" height="22pt" class="svg-formula" dspmath="mathimg" md5hash="d9ca94831737cc2bfbb9f2192576cbc2"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gmd-16-5339-2023-ie00001.svg" width="39pt" height="22pt" src="gmd-16-5339-2023-ie00001.png"/></svg:svg></span></span> convergence, as expected for a Monte Carlo method,
while for the Eulerian implementation, we see a second-order (<span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M2" display="inline" overflow="scroll" dspmath="mathml"><mrow><mn mathvariant="normal">1</mn><mo>/</mo><msubsup><mi>N</mi><mi>k</mi><mn mathvariant="normal">2</mn></msubsup></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="28pt" height="16pt" class="svg-formula" dspmath="mathimg" md5hash="65ed4bae386134bc32043ea4a232773e"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gmd-16-5339-2023-ie00002.svg" width="28pt" height="16pt" src="gmd-16-5339-2023-ie00002.png"/></svg:svg></span></span>)
convergence with the number of classes.</p> |
| first_indexed | 2024-03-11T23:44:40Z |
| format | Article |
| id | doaj.art-1cd1e18ada9645249604026ccf8a2001 |
| institution | Directory Open Access Journal |
| issn | 1991-959X 1991-9603 |
| language | English |
| last_indexed | 2024-03-11T23:44:40Z |
| publishDate | 2023-09-01 |
| publisher | Copernicus Publications |
| record_format | Article |
| series | Geoscientific Model Development |
| spelling | doaj.art-1cd1e18ada9645249604026ccf8a20012023-09-19T09:55:17ZengCopernicus PublicationsGeoscientific Model Development1991-959X1991-96032023-09-01165339536310.5194/gmd-16-5339-2023A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water columnT. Nordam0T. Nordam1R. Kristiansen2R. Nepstad3E. van Sebille4A. M. Booth5SINTEF Ocean AS, Trondheim, NorwayDepartment of Physics, Norwegian University of Science and Technology, Trondheim, NorwayDepartment of Physics, Norwegian University of Science and Technology, Trondheim, NorwaySINTEF Ocean AS, Trondheim, NorwayInstitute for Marine and Atmospheric Research, Utrecht University, Utrecht, the NetherlandsSINTEF Ocean AS, Trondheim, Norway<p>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 the 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.</p> <p>Lagrangian particle methods are commonly used for these applications. A 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.</p> <p>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 an increasing number of classes in the Eulerian scheme and with the number of particles in the Lagrangian scheme. For the Lagrangian scheme, we see the <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M1" display="inline" overflow="scroll" dspmath="mathml"><mrow><mn mathvariant="normal">1</mn><mo>/</mo><msqrt><mrow><msub><mi>N</mi><mi>p</mi></msub></mrow></msqrt></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="39pt" height="22pt" class="svg-formula" dspmath="mathimg" md5hash="d9ca94831737cc2bfbb9f2192576cbc2"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gmd-16-5339-2023-ie00001.svg" width="39pt" height="22pt" src="gmd-16-5339-2023-ie00001.png"/></svg:svg></span></span> convergence, as expected for a Monte Carlo method, while for the Eulerian implementation, we see a second-order (<span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M2" display="inline" overflow="scroll" dspmath="mathml"><mrow><mn mathvariant="normal">1</mn><mo>/</mo><msubsup><mi>N</mi><mi>k</mi><mn mathvariant="normal">2</mn></msubsup></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="28pt" height="16pt" class="svg-formula" dspmath="mathimg" md5hash="65ed4bae386134bc32043ea4a232773e"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="gmd-16-5339-2023-ie00002.svg" width="28pt" height="16pt" src="gmd-16-5339-2023-ie00002.png"/></svg:svg></span></span>) convergence with the number of classes.</p>https://gmd.copernicus.org/articles/16/5339/2023/gmd-16-5339-2023.pdf |
| spellingShingle | T. Nordam T. Nordam R. Kristiansen R. Nepstad E. van Sebille A. M. Booth A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water column Geoscientific Model Development |
| title | A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water column |
| title_full | A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water column |
| title_fullStr | A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water column |
| title_full_unstemmed | A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water column |
| title_short | A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water column |
| title_sort | comparison of eulerian and lagrangian methods for vertical particle transport in the water column |
| url | https://gmd.copernicus.org/articles/16/5339/2023/gmd-16-5339-2023.pdf |
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