Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T19:54:58.375Z Has data issue: false hasContentIssue false
  • English
  • Français

Grid-based calculation of the Lagrangian mean

Published online by Cambridge University Press:  08 April 2022

Hossein A. Kafiabad Open the ORCID record for Hossein A. Kafiabad [Opens in a new window]
Hossein A. Kafiabad*
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, EdinburghEH9 3FD, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Lagrangian averaging has been shown to be more effective than the Eulerian mean in separating waves from slow dynamics in two time scale flows. It also appears in many reduced models that capture the wave feedback on the slow flow. Its calculation, however, requires tracking particles in time, which imposes several difficulties in grid-based numerical simulations or estimation from fixed-point measurements. To circumvent these difficulties, we propose a grid-based iterative method to calculate the Lagrangian mean without tracking particles in time, which also reduces computation, memory footprint and communication between processors in parallelised numerical models. To assess the accuracy of this method several examples are examined and discussed. We also explore an application of this method in the context of shallow-water equations by quantifying the validity of wave-averaged geostrophic balance – a modified form of geostrophic balance accounting for the effect of strong waves on slow dynamics.

JFM classification

Geophysical and Geological Flows: Waves in rotating fluids Mathematical Foundations: Computational methods Geophysical and Geological Flows: Ocean circulation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 940 , 10 June 2022 , A21
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andrews, D.G. & McIntyre, M.E. 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609–646.CrossRefGoogle Scholar
Asselin, O. & Young, W.R. 2020 Penetration of wind-generated near-inertial waves into a turbulent ocean. J. Phys. Oceanogr. 50 (6), 1699–1716.CrossRefGoogle Scholar
Bachman, S.D., Shakespeare, C.J., Kleypas, J., Castruccio, F.S. & Curchitser, E. 2020 Particle-based lagrangian filtering for locating wave-generated thermal refugia for coral reefs. J. Geophys. Res.: Oceans 125 (7), e2020JC016106.CrossRefGoogle Scholar
Barkan, R., Winters, K.B. & McWilliams, J.C. 2017 Stimulated imbalance and the enhancement of eddy kinetic energy dissipation by internal waves. J. Phys. Oceanogr. 47 (1), 181–198.CrossRefGoogle Scholar
Bretherton, F.P. 1971 The general linearized theory of wave propagation. In Mathematical Problems in the Geophysical Sciences (ed. W.H. Reid), Lectures in Applied Mathematics, vol. 13, pp. 61–102. American Mathematical Society.Google Scholar
Bühler, O. 2009 Waves and Mean Flows. Cambridge University Press.CrossRefGoogle Scholar
Bühler, O. & McIntyre, M.E. 1998 On non-dissipative wave–mean interactions in the atmosphere or oceans. J. Fluid Mech. 354, 301–343.CrossRefGoogle Scholar
Charney, J.G. 1971 Geostrophic turbulence. J. Atmos. Sci. 28 (6), 1087–1095.2.0.CO;2>CrossRefGoogle Scholar
Gilbert, A.D. & Vanneste, J. 2018 Geometric generalised Lagrangian-mean theories. J. Fluid Mech. 839, 95–134.CrossRefGoogle Scholar
Grimshaw, R. 1975 Nonlinear internal gravity waves in a rotating fluid. J. Fluid Mech. 71 (3), 497–512.CrossRefGoogle Scholar
Kafiabad, H.A. & Bartello, P. 2018 Spontaneous imbalance in the non-hydrostatic Boussinesq equations. J. Fluid Mech. 847, 614–643.CrossRefGoogle Scholar
Kafiabad, H.A., Vanneste, J. & Young, W.R. 2021 Wave-averaged balance: a simple example. J. Fluid Mech. 911, R1.CrossRefGoogle Scholar
McIntyre, M.E. & Norton, W.A. 2000 Potential vorticity inversion on a hemisphere. J. Atmos. Sci. 57, 1214–1235, corrigendum: 58, 2001, 949–949.2.0.CO;2>CrossRefGoogle Scholar
Mohebalhojeh, A.R. & Dritschel, D.G. 2001 Hierarchies of balance conditions for the $f$-plane shallow water equations. J. Atmos. Sci. 58, 2411–2426.2.0.CO;2>CrossRefGoogle Scholar
Moore, D. 1970 The mass transport velocity induced by free oscillations at a single frequency. Geophys. Fluid Dyn. 1, 237–247.CrossRefGoogle Scholar
Nagai, T., Tandon, A., Kunze, E. & Mahadevan, A. 2015 Spontaneous generation of near-inertial waves by the Kuroshio front. J. Phys. Oceanogr. 45 (9), 2381–2406.CrossRefGoogle Scholar
Rocha, C.B., Wagner, G.L. & Young, W.R. 2018 Stimulated generation: extraction of energy from balanced flow by near-inertial waves. J. Fluid Mech. 847, 417–451.CrossRefGoogle Scholar
Shakespeare, C.J., Gibson, A.H., Hogg, A.M., Keating, S.R., Bachman, S.D. & Velzeboer, N. 2021 Lagrangian filtering: a novel method for separating internal waves from non-wave flows in high-resolution simulations. In Earth and Space Science Open Archive, https://doi.org/10.1002/essoar.10507016.1.CrossRefGoogle Scholar
Shakespeare, C.J. & Hogg, A.M. 2017 Spontaneous surface generation and interior amplification of internal waves in a regional-scale ocean model. J. Phys. Oceanogr. 47 (4), 811–826.CrossRefGoogle Scholar
Shakespeare, C.J. & Hogg, A.M. 2018 The life cycle of spontaneously generated internal waves. J. Phys. Oceanogr. 48 (2), 343–359.CrossRefGoogle Scholar
Shakespeare, C.J. & Hogg, A.M. 2019 On the momentum flux of internal tides. J. Phys. Oceanogr. 49 (4), 993–1013.CrossRefGoogle Scholar
Skamarock, W.C. 2004 Evaluating mesoscale NWP models using kinetic energy spectra. Mon. Weath. Rev. 132 (12), 3019–3032.CrossRefGoogle Scholar
Soward, A.M. & Roberts, P.H. 2010 The hybrid Euler–Lagrange procedure using an extension of Moffatt's method. J. Fluid Mech. 661, 45–72.CrossRefGoogle Scholar
Sutherland, B.R., Achatz, U., Colm-cille, P.C. & Klymak, J.M. 2019 Recent progress in modeling imbalance in the atmosphere and ocean. Phys. Rev. Fluids 4 (1), 010501.CrossRefGoogle Scholar
Taylor, S. & Straub, D. 2020 Effects of adding forced near-inertial motion to a wind-driven channel flow. J. Phys. Oceanogr. 50 (10), 2983–2996.CrossRefGoogle Scholar
Thomas, J., Bühler, O. & Smith, K.S. 2018 Wave-induced mean flows in rotating shallow water with uniform potential vorticity. J. Fluid Mech. 839, 408–429.CrossRefGoogle Scholar
Vallis, G.K. 2017 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Vanneste, J. 2013 Balance and spontaneous wave generation in geophysical flows. Annu. Rev. Fluid Mech. 45, 147–172.CrossRefGoogle Scholar
Wagner, G.L. & Young, W.R. 2015 Available potential vorticity and wave-averaged quasi-geostrophic flow. J. Fluid Mech. 785, 401–424.CrossRefGoogle Scholar
Wagner, G.L. & Young, W.R. 2016 A three-component model for the coupled evolution of near-inertial waves, quasi-geostrophic flow and the near-inertial second harmonic. J. Fluid Mech. 802, 806–837.CrossRefGoogle Scholar
Waite, M.L. & Snyder, C. 2009 The mesoscale kinetic energy spectrum of a baroclinic life cycle. J. Atmos. Sci. 66 (4), 883–901.CrossRefGoogle Scholar
Warn, T. 1997 Nonlinear balance and quasi-geostrophic sets. Atmos. Ocean 35 (2), 135–145.CrossRefGoogle Scholar
Xie, J.-H. 2020 Downscale transfer of quasigeostrophic energy catalyzed by near-inertial waves. J. Fluid Mech. 904, A40.CrossRefGoogle Scholar
Xie, J.-H. & Vanneste, J. 2015 A generalised-Lagrangian-mean model of the interactions between near-inertial waves and mean flow. J. Fluid Mech. 774, 143–169.CrossRefGoogle Scholar