Articles | Volume 13, issue 12
https://doi.org/10.5194/gmd-13-5935-2020
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/gmd-13-5935-2020
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Numerical integrators for Lagrangian oceanography
SINTEF Ocean, Trondheim, Norway
Department of Physics, Norwegian University of Science and Technology, Trondheim, Norway
Rodrigo Duran
National Energy Technology Laboratory, Albany, OR 97321, USA
Theiss Research, San Diego, CA 92037, USA
Related authors
Jenny Margareta Mørk, Tor Nordam, and Siren Rühs
EGUsphere, https://doi.org/10.5194/egusphere-2025-2109, https://doi.org/10.5194/egusphere-2025-2109, 2025
Short summary
Short summary
A common task in applied oceanography is to calculate the trajectories of floating objects in the ocean. We propose an alteration to some common numerical methods to improve their performance in such computations, and compare results with and without this alteration. This will help researchers to ensure they obtain a higher accuracy in their results without compromising on computer resources.
Tor Nordam, Ruben Kristiansen, Raymond Nepstad, Erik van Sebille, and Andy M. Booth
Geosci. Model Dev., 16, 5339–5363, https://doi.org/10.5194/gmd-16-5339-2023, https://doi.org/10.5194/gmd-16-5339-2023, 2023
Short summary
Short summary
We describe and compare two common methods, Eulerian and Lagrangian models, used to simulate the vertical transport of material in the ocean. They both solve the same transport problems but use different approaches for representing the underlying equations on the computer. The main focus of our study is on the numerical accuracy of the two approaches. Our results should be useful for other researchers creating or using these types of transport models.
Jenny Margareta Mørk, Tor Nordam, and Siren Rühs
EGUsphere, https://doi.org/10.5194/egusphere-2025-2109, https://doi.org/10.5194/egusphere-2025-2109, 2025
Short summary
Short summary
A common task in applied oceanography is to calculate the trajectories of floating objects in the ocean. We propose an alteration to some common numerical methods to improve their performance in such computations, and compare results with and without this alteration. This will help researchers to ensure they obtain a higher accuracy in their results without compromising on computer resources.
Luca Kunz, Alexa Griesel, Carsten Eden, Rodrigo Duran, and Bruno Sainte-Rose
Ocean Sci., 20, 1611–1630, https://doi.org/10.5194/os-20-1611-2024, https://doi.org/10.5194/os-20-1611-2024, 2024
Short summary
Short summary
Transient Attracting Profiles (TRAPs) indicate the most attracting regions of the flow and have the potential to facilitate offshore cleanups in the Great Pacific Garbage Patch. We study the characteristics of TRAPs and the prospects for predicting debris transport from a mesoscale-permitting dataset. Our findings show the relevance of TRAP lifetime estimations to an operational application, and our TRAP tracking algorithm may even benefit other challenges that are related to search at sea.
Tor Nordam, Ruben Kristiansen, Raymond Nepstad, Erik van Sebille, and Andy M. Booth
Geosci. Model Dev., 16, 5339–5363, https://doi.org/10.5194/gmd-16-5339-2023, https://doi.org/10.5194/gmd-16-5339-2023, 2023
Short summary
Short summary
We describe and compare two common methods, Eulerian and Lagrangian models, used to simulate the vertical transport of material in the ocean. They both solve the same transport problems but use different approaches for representing the underlying equations on the computer. The main focus of our study is on the numerical accuracy of the two approaches. Our results should be useful for other researchers creating or using these types of transport models.
Cited articles
Ali, S. and Shah, M.: A Lagrangian Particle Dynamics Approach for Crowd Flow Segmentation and Stability Analysis, in: 2007 IEEE Conference on Computer Vision and Pattern Recognition, Minneapolis, MN, USA, 17–22 June 2007, IEEE, 1–6, https://doi.org/10.1109/CVPR.2007.382977, 2007. a, b
Allshouse, M. R., Ivey, G. N., Lowe, R. J., Jones, N. L., Beegle-Krause, C., Xu, J., and Peacock, T.: Impact of windage on ocean surface Lagrangian coherent structures, Environ. Fluid Mech., 17, 473–483, 2017. a
Barkan, R., McWilliams, J. C., Molemaker, M. J., Choi, J., Srinivasan, K., Shchepetkin, A. F., and Bracco, A.: Submesoscale dynamics in the northern Gulf of Mexico. Part II: Temperature–salinity relations and cross-shelf transport processes, J. Phys. Oceanogr., 47, 2347–2360, 2017. a
Beron-Vera, F. J., Olascoaga, M. J., and Goni, G. J.: Oceanic mesoscale eddies as revealed by Lagrangian coherent structures, Geophys. Res. Lett., 35, L12603, https://doi.org/10.1029/2008GL033957, 2008. a
Beron-Vera, F. J., Olascoaga, M. J., Brown, M. G., Koçak, H., and Rypina, I. I.: Invariant-tori-like Lagrangian coherent structures in geophysical flows, Chaos, 20, 017514, https://doi.org/10.1063/1.3271342, 2010. a
Bogacki, P. and Shampine, L. F.: A 3(2) pair of Runge-Kutta formulas, Appl. Math. Lett., 2, 321–325, 1989. a
Breivik, Ø. and Allen, A. A.: An operational search and rescue model for the Norwegian Sea and the North Sea, J. Marine Syst., 69, 99–113, 2008. a
de Boor, C.: A practical guide to splines, Springer-Verlag, New York Berlin Heidelberg, 2001. a
De Dominicis, M., Pinardi, N., Zodiatis, G., and Lardner, R.: MEDSLIK-II, a Lagrangian marine surface oil spill model for short-term forecasting – Part 1: Theory, Geosci. Model Dev., 6, 1851–1869, https://doi.org/10.5194/gmd-6-1851-2013, 2013. a
Dieci, L. and Lopez, L.: A survey of numerical methods for IVPs of ODEs with discontinuous right-hand side, J. Comput. Appl. Math., 236, 3967–3991, https://doi.org/10.1016/j.cam.2012.02.011, 2012. a
Dormand, J. and Prince, P.: A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6, 19–26, 1980. a
Dormand, J. and Prince, P.: A reconsideration of some embedded Runge-Kutta formulae, J. Comput. Appl. Math., 15, 203–211, https://doi.org/10.1016/0377-0427(86)90027-0, 1986. a
Dugstad, J., Fer, I., LaCasce, J., Sanchez de La Lama, M., and Trodahl, M.: Lateral Heat Transport in the Lofoten Basin: Near-Surface Pathways and Subsurface Exchange, J. Geophys. Res.-Oceans,124, 2992–3006, https://doi.org/10.1029/2018JC014774, 2019. a
Duran, R., Beron-Vera, F. J., and Olascoaga, M. J.: Extracting quasi-steady Lagrangian transport patterns from the ocean circulation: An application to the Gulf of Mexico, Scientific Reports, 8, 5218, https://doi.org/10.1038/s41598-018-23121-y, 2018. a, b
Enright, W., Jackson, K., Nørsett, S., and Thomsen, P.: Effective solution of discontinuous IVPs using a Runge-Kutta formula pair with interpolants, Appl. Math. Comput., 27, 313–335, https://doi.org/10.1016/0096-3003(88)90030-6, 1988. a
Farazmand, M. and Haller, G.: Computing Lagrangian coherent structures from their variational theory, Chaos, 22, 013128, https://doi.org/10.1063/1.3690153, 2012. a
García-Martínez, R. and Flores-Tovar, H.: Computer modeling of oil spill trajectories with a high accuracy method, Spill Sci. Technol. B., 5, 323–330, 1999. a
Gladwell, I., Shampine, L., and Thompson, S.: Solving ODEs with MATLAB, Cambridge University Press, New York, NY, USA, 2003. a
Gräwe, U.: Implementation of high-order particle-tracking schemes in a water column model, Ocean Model., 36, 80–89, 2011. a
Griffiths, D. F. and Higham, D. J.: Numerical methods for ordinary differential equations, Springer-Verlag, London, 2010. a
Hairer, E. and Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, Heidelberg, 1996. a
Hairer, E., Nørsett, S. P., and Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, 1st edn., Springer-Verlag Berlin Heidelberg, ISBN 978-3-662-12609-7, https://doi.org/10.1007/978-3-662-12607-3, 1987. a, b, c
Hairer, E., Wanner, G., and Lubich, C.: Geometric Numerical Integration, in:
Structure-Preserving Algorithms for Ordinary Differential Equations, Springer-Verlag, Berlin, Heidelberg, https://doi.org/10.1007/3-540-30666-8, 2006. a
Isaacson, E. and Keller, H. B.: Analysis of Numerical Methods, Dover Publications, New York, USA, 1994. a
Kloeden, P. E. and Platen, E.: Numerical Solution of Stochastic Differential
Equations, Springer-Verlag, Berlin, Heidelberg, 1992. a
Kress, R.: Numerical Analysis, in: Graduate Texts in Mathematics, Springer-Verlag, New York, https://doi.org/10.1007/978-1-4612-0599-9, 1998. a
Lekien, F. and Marsden, J.: Tricubic interpolation in three dimensions, Int. J. Numer. Meth. Eng., 63, 455–471, https://doi.org/10.1002/nme.1296, 2005. a, b
Lévy, M., Resplandy, L., Klein, P., Capet, X., Iovino, D., and Éthé, C.: Grid degradation of submesoscale resolving ocean models: Benefits for offline passive tracer transport, Ocean Model., 48, 1–9, 2012. a
Maslo, A., de Souza, J. M. A. C., Andrade-Canto, F., and Outerelo, J. R.: Connectivity of deep waters in the Gulf of Mexico, J. Marine
Syst., 203, 103267, https://doi.org/10.1016/j.jmarsys.2019.103267, 2020. a
Narváez, D. A., Klinck, J. M., Powell, E. N., Hofmann, E. E., Wilkin, J., and Haidvogel, D. B.: Modeling the dispersal of eastern oyster (Crassostrea virginica) larvae in Delaware Bay, J. Mar. Res., 70, 381–409, 2012. a
Nieto, R. and Gimeno, L.: A database of optimal integration times for Lagrangian studies of atmospheric moisture sources and sinks, Scientific Data, 6, 59, https://doi.org/10.1038/s41597-019-0068-8, 2019. a
Nordam, T.: nordam/ODE-integrators-for-Lagrangian-particles 0.9, Version 0.9, Zenodo, https://doi.org/10.5281/zenodo.4041979, 2020. a, b
Nordam, T., Brønner, U., Skancke, J., Nepstad, R., Rønningen, P., and Alver, M. O.: Numerical integration and interpolation in marine pollutant transport modelling, in: Proceedings of the 40th AMOP Technical Seminar, Calgary, AB, Canada, 3–5 October 2017, Environment and Climate Change Canada, Ottawa, 586–609, https://hdl.handle.net/11250/2652834, 2017. a
North, E. W., Adams, E. E., Schlag, Z., Sherwood, C. R., He, R., Hyun, K. H., and Socolofsky, S. A.: Simulating Oil Droplet Dispersal From the Deepwater Horizon Spill With a Lagrangian Approach, in: Monitoring and Modeling the Deepwater Horizon Oil Spill: A Record Breaking Enterprise, Wiley, 195, 217–226, 2011. a
Onink, V., Wichmann, D., Delandmeter, P., and van Sebille, E.: The Role of Ekman Currents, Geostrophy, and Stokes Drift in the Accumulation of Floating Microplastic, J. Geophys. Res.-Oceans, 124, 1474–1490, https://doi.org/10.1029/2018JC014547, 2019. a
Onu, K., Huhn, F., and Haller, G.: LCS Tool: A computational platform for Lagrangian coherent structures, J. Computat. Sci., 7, 26–36, 2015. a
Peng, J. and Dabiri, J. O.: Transport of inertial particles by Lagrangian coherent structures: application to predator–prey interaction in jellyfish feeding, J. Fluid Mech., 623, 75–84, https://doi.org/10.1017/S0022112008005089, 2009. a
Povinec, P., Gera, M., Holý, K., Hirose, K., Lujaniené, G., Nakano, M., Plastino, W., Sýkora, I., Bartok, J., and Gažák, M.: Dispersion of Fukushima radionuclides in the global atmosphere and the ocean, Appl. Radiat. Isotopes, 81, 383–392, https://doi.org/10.1016/j.apradiso.2013.03.058, 2013. a
Riuttanen, L., Hulkkonen, M., Dal Maso, M., Junninen, H., and Kulmala, M.: Trajectory analysis of atmospheric transport of fine particles, SO2, NOx and O3 to the SMEAR II station in Finland in 1996–2008, Atmos. Chem. Phys., 13, 2153–2164, https://doi.org/10.5194/acp-13-2153-2013, 2013. a
Rivas, D. and Samelson, R. M.: A Numerical Modeling Study of the Upwelling Source Waters along the Oregon Coast during 2005, J. Phys. Oceanogr., 41, 88–112, https://doi.org/10.1175/2010JPO4327.1, 2011. a, b
Rye, H., Reed, M., and Ekrol, N.: The ParTrack model for calculation of the spreading and deposition of drilling mud, chemicals and drill cuttings, Environ. Modell. Softw., 13, 431–441, 1998. a
Serra, M., Sathe, P., Rypina, I., Kirincich, A., Ross, S. D., Lermusiaux, P., Allen, A., Peacock, T., and Haller, G.: Search and rescue at sea aided by hidden flow structures, Nat. Commun., 11, 2525, https://doi.org/10.1038/s41467-020-16281-x, 2020. a
Shadden, S. C. and Taylor, C. A.: Characterization of Coherent Structures in the Cardiovascular System, Ann. Biomed. Eng., 36, 1152–1162, https://doi.org/10.1007/s10439-008-9502-3, 2008. a
Shadden, S. C., Astorino, M., and Gerbeau, J.-F.: Computational analysis of an aortic valve jet with Lagrangian coherent structures, Chaos: An Interdisciplinary J. Nonlinear Sci., 20, 017512, https://doi.org/10.1063/1.3272780, 2010. a
Siegel, D., Kinlan, B., Gaylord, B., and Gaines, S.: Lagrangian descriptions of marine larval dispersion, Mar. Ecol. Prog. Ser., 260, 83–96, 2003. a
Sirois, A. and Bottenheim, J. W.: Use of backward trajectories to interpret the 5-year record of PAN and O3 ambient air concentrations at Kejimkujik National Park, Nova Scotia, J. Geophys. Res., 100, 2867–2881, https://doi.org/10.1029/94JD02951, 1995. a
Spivakovskaya, D., Heemink, A. W., and Deleersnijder, E.: Lagrangian modelling of multi-dimensional advection-diffusion with space-varying diffusivities: theory and idealized test cases, Ocean Dynam., 57, 189–203, 2007. a
van Sebille, E., Scussolini, P., Durgadoo, J. V., Peeters, F. J., Biastoch, A., Weijer, W., Turney, C., Paris, C. B., and Zahn, R.: Ocean currents generate large footprints in marine palaeoclimate proxies, Nat. Commun., 6, 6521, https://doi.org/10.1038/ncomms7521, 2015. a
van Sebille, E., Griffies, S. M., Abernathey, R., Adams, T. P., Berloff, P., Biastoch, A., Blanke, B., Chassignet, E. P., Cheng, Y., Cotter, C. J., Deleersnijder, E., Döös, K., Drake, H. F., Drijfhout, S., Gary, S. F., Heemink, A. W., Kjellsson, J., Koszalka, I. M., Lange, M., Lique, C., MacGilchrist, G. A., Marsh, R., Mayorga Adame, C. G., McAdam, R., Nencioli, F., Paris, C. B., Piggott, M. D., Polton, J. A., Rühs, S., Shah, S. H., Thomas, M. D., Wang, J., Wolfram, P. J., Zanna, L., and Zika, J. D.: Lagrangian ocean analysis: Fundamentals and practices, Ocean Model., 121, 49–75, 2018. a
Visser, A. W.: Lagrangian modelling of plankton motion: From deceptively simple random walks to Fokker-Planck and back again, J. Marine Syst., 70, 287–299, 2008. a
Williams, J.: Bspline-Fortran: Multidimensional B-Spline Interpolation of Data on a Regular Grid (Version 5.4.0), Zenodo, https://doi.org/10.5281/zenodo.1215290, 2018. a
Woods, J.: The Lagrangian Ensemble metamodel for simulating plankton ecosystems, Prog. Oceanogr., 67, 84–159, 2005. a
Yang, Y., He, G.-W., and Wang, L.-P.: Effects of subgrid-scale modeling on Lagrangian statistics in large-eddy simulation, J. Turbul., 9, 1–24, https://doi.org/10.1080/14685240801905360, 2008. a
Zelenke, B., O'Connor, C., Barker, C., Beegle-Krause, C. J., and Eclipse, L. (Eds.): General NOAA Operational Modeling Environment (GNOME) Technical Documentation, U.S. Dept. of Commerce, NOAA Technical Memorandum NOS OR&R 40, Emergency Response Division, Seattle, WA, NOAA, 105 pp., available at: https://response.restoration.noaa.gov/sites/default/files/GNOME_Tech_Doc.pdf (last access: 17 November 2020), 2012. a
Short summary
In applied oceanography, a common task is to calculate the trajectory of objects floating at the sea surface or submerged in the water. We have investigated different numerical methods for doing such calculations and discuss the benefits and challenges of some common methods. We then propose a small change to some common methods that make them more efficient for this particular application. This will allow researchers to obtain more accurate answers with fewer computer resources.
In applied oceanography, a common task is to calculate the trajectory of objects floating at the...