Abstract
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.
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Acknowledgements
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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Gräwe, U., Deleersnijder, E., Shah, S.H.A.M. et al. Why the Euler scheme in particle tracking is not enough: the shallow-sea pycnocline test case. Ocean Dynamics 62, 501–514 (2012). https://doi.org/10.1007/s10236-012-0523-y
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DOI: https://doi.org/10.1007/s10236-012-0523-y