Ocean Dynamics

, Volume 62, Issue 4, pp 501–514 | Cite as

Why the Euler scheme in particle tracking is not enough: the shallow-sea pycnocline test case

  • Ulf GräweEmail author
  • Eric Deleersnijder
  • Syed Hyder Ali Muttaqi Shah
  • Arnold Willem Heemink


During the last decades, the Euler scheme was the common “workhorse” in particle tracking, although it is the lowest-order approximation of the underlying stochastic differential equation. To convince the modelling community of the need for better methods, we have constructed a new test case that will show the shortcomings of the Euler scheme. We use an idealised shallow-water diffusivity profile that mimics the presence of a sharp pycnocline and thus a quasi-impermeable barrier to vertical diffusion. In this context, we study the transport of passive particles with or without negative buoyancy. A semi-analytic solutions is used to assess the performance of various numerical particle-tracking schemes (first- and second-order accuracy), to treat the variations in the diffusivity profile properly. We show that the commonly used Euler scheme exhibits a poor performance and that widely used particle-tracking codes shall be updated to either the Milstein scheme or second-order schemes. It is further seen that the order of convergence is not the only relevant factor, the absolute value of the error also is.


Particle tracking Milstein scheme Pycnocline Diffusion Residence time 



Eric Deleersnijder was a research associate with the Belgian Fund for Scientific Research (F.R.S.-FNRS). His contribution to the present study was achieved in the framework of the Interuniversity Attraction Pole TIMOTHY, which is funded by BELSPO under the contract IAP6.13, and the ARC 10/15-028 (Communauté Française de Belgique). Ulf Gräwe was funded by the Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie of Germany (BMBF) through grant number 01LR0807B.


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Ulf Gräwe
    • 1
    Email author
  • Eric Deleersnijder
    • 2
    • 3
  • Syed Hyder Ali Muttaqi Shah
    • 4
  • Arnold Willem Heemink
    • 4
  1. 1.Leibniz Institute for Baltic Sea Research (IOW)WarnemündeGermany
  2. 2.Institute of Mechanics, Materials and Civil Engineering (IMMC)Université catholique de LouvainLouvain-la-NeuveBelgium
  3. 3.Earth and Life Institute (ELI), G. Lemaître Centre for Earth and Climate Research (TECLIM)Université catholique de LouvainLouvain-la-NeuveBelgium
  4. 4.Delft Institute of Applied Mathematics (DIAM)Delft University of TechnologyDelftthe Netherlands

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