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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: m.haigh15@imperial.ac.uk
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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

REFERENCES

Abernathey, R., Ferreira, D. & Klocker., A. 2013 Diagnostics of isopycnal mixing in a circumpolar channel. Ocean Model. 72, 1–16.CrossRefGoogle Scholar
Abraham, E.R. & Bowen, M.M. 2002 Chaotic stirring by a mesoscale surface-ocean flow. Chaos 12, 373–381.CrossRefGoogle ScholarPubMed
Bachman, S.D., Fox-Kemper, B. & Bryan, F.O. 2020 A diagnosis of anisotropic eddy diffusion from a high-resolution global ocean model. J. Adv. Model. Earth Syst. 12 (2), e2019MS001904.CrossRefGoogle Scholar
Balarac, G., Le Sommer, J., Meunier, X. & Vollant., A. 2013 A dynamic regularized gradient model of the subgrid-scale scalar flux for large eddy simulations. Phys. Fluids 25 (7), 075107.CrossRefGoogle Scholar
Berloff, P.S., McWilliams, J.C. & Bracco, A. 2002 Material transport in oceanic gyres. Part I: phenomenology. J. Phys. Oceanogr. 32 (3), 764–796.2.0.CO;2>CrossRefGoogle Scholar
Dubos, T. & Babiano, A. 2002 Two-dimensional cascades and mixing: a physical space approach. J. Fluid Mech. 467, 81–100.CrossRefGoogle Scholar
Eyink, G.L. 2001 Dissipation in turbulent solutions of 2D Euler equations. Nonlinearity, 14 (4), 787–802.CrossRefGoogle Scholar
Fox-Kemper, B., Lumpkin, R. & Bryan, F.O. 2013 Chapter 8 – lateral transport in the ocean interior. In Ocean Circulation and Climate (International Geophysics) (ed. G. Siedler, S.M. Griffies, J. Gould & J.A. Church), vol. 103, pp. 185–209. Academic.CrossRefGoogle Scholar
Gent, P.R. & McWilliams, J.C. 1990 Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr. 20 (1), 150–155.2.0.CO;2>CrossRefGoogle Scholar
Gent, P.R., Willebrand, J., McDougall, T.J. & McWilliams, J.C. 1995 Parameterizing eddy-induced tracer transports in ocean circulation models. J. Phys. Oceanogr. 25 (4), 463–474.2.0.CO;2>CrossRefGoogle Scholar
Griffies, S.M. 1998 The Gent–McWilliams skew flux. J. Phys. Oceanogr. 28 (5), 831–841.2.0.CO;2>CrossRefGoogle Scholar
Haigh, M., Sun, L., Shevchenko, I. & Berloff, P. 2020 Tracer-based estimates of eddy-induced diffusivities. Deep-Sea Res. 160, 103264.CrossRefGoogle Scholar
Haigh, M., Sun, L., McWilliams, J.C. & Berloff, P. 2021 a On eddy transport in the ocean. Part I: the diffusion tensor. Ocean Model. 164, 101831.CrossRefGoogle Scholar
Haigh, M., Sun, L., McWilliams, J.C. & Berloff, P. 2021 b On eddy transport in the ocean. Part II: the advection tensor. Ocean Model. 165, 101845.CrossRefGoogle Scholar
Haigh, M. & Berloff, P. 2021 On co-existing diffusive and anti-diffusive tracer transport by oceanic mesoscale eddies. Ocean Model. 168, 101909.CrossRefGoogle Scholar
Kamenkovich, I., Rypina, I.I. & Berloff, P.S. 2015 Properties and origins of the anisotropic eddy-induced transport in the North Atlantic. J. Phys. Oceanogr. 45 (3), 778–791.CrossRefGoogle Scholar
Kamenkovich, I., Berloff, P., Haigh, M., Sun, L. & Lu, Y. 2021 Complexity of mesoscale eddy diffusivity in the ocean. Geophys. Res. Lett. 48 (5), e2020GL091719.CrossRefGoogle Scholar
Klocker, A., Ferrari, R., LaCasce, J. & Merrifield, S. 2012 Reconciling float-based and tracer-based estimates of eddy diffusivities in the Southern Ocean. J. Mar. Res. 70, 569–602.CrossRefGoogle Scholar
Ledwell, J.R., Watson, A.J. & Law, C.S. 1998 Mixing of a tracer in the pycnocline. J. Geophys. Res.: Oceans 103 (C10), 21499–21529.CrossRefGoogle Scholar
Marshall, J., Shuckburgh, E., Jones, H. & Hill, C. 2006 Estimates and implications of surface eddy diffusivity in the southern ocean derived from tracer transport. J. Phys. Oceanogr. 36 (9), 1806–1821.CrossRefGoogle Scholar
Meneveau, C. & Katz, J. 2000 Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech. 32 (1), 1–32.CrossRefGoogle Scholar
Nadiga, B.T. 2008 Orientation of eddy fluxes in geostrophic turbulence. Phil. Trans. R. Soc. Lond. A 366 (1875), 2489–2508.Google ScholarPubMed
Nencioli, F., d'Ovidio, F., Doglioli, A.M. & Petrenko, A.A. 2013 In situ estimates of submesoscale horizontal eddy diffusivity across an ocean front. J. Geophys. Res.: Oceans 118 (12), 7066–7080.CrossRefGoogle Scholar
Redi, M.H. 1982 Oceanic isopycnal mixing by coordinate rotation. J. Phys. Oceanogr. 12 (10), 1154–1158.2.0.CO;2>CrossRefGoogle Scholar
Rypina, I.I., Kamenkovich, I., Berloff, P.S. & Pratt, L.J. 2012 Eddy-induced particle dispersion in the near-surface North Atlantic. J. Phys. Oceanogr. 42 (12), 2206–2228.CrossRefGoogle Scholar
Smith, R.D. & Gent, P.R. 2004 Anisotropic Gent–McWilliams parameterization for ocean models. J. Phys. Oceanogr. 34 (11), 2541–2564.CrossRefGoogle Scholar
Smith, K.S. & Ferrari, R. 2009 The production and dissipation of compensated thermohaline variance by mesoscale stirring. J. Phys. Oceanogr. 39 (10), 2477–2501.CrossRefGoogle Scholar
Stanley, Z., Bachman, S.D. & Grooms, I. 2020 Vertical structure of ocean mesoscale eddies with implications for parameterizations of tracer transport. J. Adv. Model. Earth Syst. 12 (10), e2020MS002151.CrossRefGoogle Scholar
Starr, V. 1968 Physics of Negative Viscosity Phenomena, vol. 256. McGraw-Hill.Google Scholar
Young, W.R., Rhines, P.B. & Garrett, C.J.R. 1982 Shear-flow dispersion, internal waves and horizontal mixing in the ocean. J. Phys. Oceanogr. 12 (6), 515–527.2.0.CO;2>CrossRefGoogle Scholar