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On the stability of tracer simulations with opposite-signed diffusivities

Published online by Cambridge University Press:  28 February 2022

Michael Haigh Open the ORCID record for Michael Haigh [Opens in a new window]  and
Pavel Berloff Open the ORCID record for Pavel Berloff [Opens in a new window]
Michael Haigh*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Pavel Berloff
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina 8, 119333 Moscow, Russia
*
†Email address for correspondence: [email protected]
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Abstract

Many recent studies have diagnosed opposite-signed diffusion eigenvalues to be a prevalent feature of the transfer tensor for diffusive tracer transport by oceanic mesoscale eddies. This diagnosed tensor, which we refer to as the diffusion tensor, therefore accounts for tracer filamentation effects. The preferential orientation of this filamentation is quantified by the principal axis of the diffusion tensor, namely the diffusion axis. Parameterisations of eddy diffusion commonly invoke a diffusion tensor, typically one with non-negative eigenvalues to avoid numerical issues. Motivated by the need to parameterise tracer filamentation, in this study we examine diffusion of a Gaussian tracer patch with imposed opposite-signed diffusion eigenvalues, and in particular we focus on the time scale for the onset of instability. For a fixed diffusion axis, numerical instability is an inevitable consequence of persistent up-gradient fluxes associated with the negative eigenvalue. For typical oceanic scales and diffusion magnitudes, this time scale is of the order of $100$ days, but is shorter for larger negative eigenvalues or for finer grid resolutions. We show that imposing a time-dependent diffusion axis can lead to simulations with no onset of instability after 100 000 days of tracer evolution. Although motivated by oceanographic fluid dynamics, our results have much broader applications since diffusive processes are present in a wide range of fluid flows.

JFM classification

Geophysical and Geological Flows: Ocean processes Mixing: Turbulent mixing
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 937 , 25 April 2022 , R3
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

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