Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-06-02T13:37:32.853Z Has data issue: false hasContentIssue false
  • English
  • Français

The effects of vertical mixing on nonlinear Kelvin waves

Published online by Cambridge University Press:  28 September 2020

Matthew N. Crowe Open the ORCID record for Matthew N. Crowe [Opens in a new window]  and
Edward R. Johnson Open the ORCID record for Edward R. Johnson [Opens in a new window]
Matthew N. Crowe*
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK
Edward R. Johnson
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK
*
†Email address for correspondence: m.crowe@ucl.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Submesoscale processes along coastal boundaries provide a potential mechanism for the dissipation of mesoscale kinetic energy in the ocean. Since these processes occur on scales not generally resolved by global ocean models, a physically motivated parametrisation is required to accurately describe their effects. Submesoscale dynamics is characterised by strong turbulent mixing, nonlinearity and topographic effects; all of which significantly modify the flow. A major component of the submesoscale boundary response to mesoscale forcing is the Kelvin – or coastally trapped – wave field, which has been shown to transport energy over large distances. This paper thus examines the influence of vertical mixing, nonlinearity and steep-slope topography on baroclinic Kelvin waves with the aim of assessing the importance of these effects. We consider the limit of a steep coastal boundary, weak mixing and weak nonlinearity and perform an asymptotic analysis to determine the modification of the classical Kelvin wave solution by these effects. Linear and nonlinear solutions are given and different mixing limits are discussed and compared with previous work. We find that vertical mixing acts to damp slowly propagating Kelvin waves while nonlinearity can cause wave breaking which may be important for fast waves. Steep-slope topography acts to modify the wave speed and structure consistent with previous work.

JFM classification

Geophysical and Geological Flows: Ocean processes Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A22
Check for updates
Copyright
© The Author(s), 2020. 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

Arbic, B. K., Shriver, J. F., Hogan, P. J., Hurlburt, H. E., McClean, J. L., Metzger, E. J., Scott, R. B., Sen, A., Smedstad, O. M. & Wallcraft, A. J. 2009 Estimates of bottom flows and bottom boundary layer dissipation of the oceanic general circulation from global high-resolution models. J. Geophys. Res. Oceans 114 (C2).CrossRefGoogle Scholar
Bachman, S. D. 2019 The GM+E closure: a framework for coupling backscatter with the Gent and McWilliams parameterization. Ocean Model. 136, 85–106.CrossRefGoogle Scholar
Boyd, J. P. 1980 The nonlinear equatorial Kelvin wave. J. Phys. Oceanogr. 10 (1), 1–11.2.0.CO;2>CrossRefGoogle Scholar
Callies, J., Ferrari, R., Klymak, J. M. & Gula, J. 2015 Seasonality in submesoscale turbulence. Nat. Commun. 6, 6862.CrossRefGoogle ScholarPubMed
Callies, J., Flierl, G., Ferrari, R. & Fox-Kemper, B. 2016 The role of mixed-layer instabilities in submesoscale turbulence. J. Fluid Mech. 788, 5–41.CrossRefGoogle Scholar
Capet, X., McWilliams, J. C., Molemaker, M. J. & Shchepetkin, A. F. 2008 a Mesoscale to submesoscale transition in the California current system. Part I: flow structure, eddy flux, and observational tests. J. Phys. Oceanogr. 38 (1), 29–43.CrossRefGoogle Scholar
Capet, X., McWilliams, J. C., Molemaker, M. J. & Shchepetkin, A. F. 2008 b Mesoscale to submesoscale transition in the California current system. Part II: frontal processes. J. Phys. Oceanogr. 38, 44–64.CrossRefGoogle Scholar
Cavalcanti, M. M., Domingos Cavalcanti, V. N., Faminskii, A. & Natali, F. 2012 Decay of solutions to damped Korteweg–de Vries type equation. Appl. Maths Optim. 65 (2), 221–251.CrossRefGoogle Scholar
Crowe, M. N. & Taylor, J. R. 2018 The evolution of a front in turbulent thermal wind balance. Part 1. Theory. J. Fluid Mech. 850, 179–211.CrossRefGoogle Scholar
Crowe, M. N. & Taylor, J. R. 2019 a Baroclinic instability with a simple model for vertical mixing. J. Phys. Oceanogr. 49, 3273–3300.CrossRefGoogle Scholar
Crowe, M. N. & Taylor, J. R. 2019 b The evolution of a front in turbulent thermal wind balance. Part 2. Numerical simulations. J. Fluid Mech. 880, 326–352.CrossRefGoogle Scholar
D'Asaro, E., Lee, C., Rainville, L., Harcourt, R. & Thomas, L. 2011 Enhanced turbulence and energy dissipation at ocean fronts. Science 332 (6027), 318–322.CrossRefGoogle ScholarPubMed
Davey, M. K., Hsieh, W. W. & Wajsowicz, R. C. 1983 The free Kelvin wave with lateral and vertical viscosity. J. Phys. Oceanogr. 13 (12), 2182–2191.2.0.CO;2>CrossRefGoogle Scholar
Deremble, B., Johnson, E. R. & Dewar, W. K. 2017 A coupled model of interior balanced and boundary flow. Ocean Model. 119, 1–12.CrossRefGoogle Scholar
Dewar, W. K., Berloff, P. & Hogg, A. McC. 2011 Submesoscale generation by boundaries. J. Mar. Res. 69 (4–5), 501–522.CrossRefGoogle Scholar
Dewar, W. K. & Hogg, A. McC. 2010 Topographic inviscid dissipation of balanced flow. Ocean Model. 32 (1), 1–13.CrossRefGoogle Scholar
Duhaut, T. H. A. & Straub, D. N. 2006 Wind stress dependence on ocean surface velocity: implications for mechanical energy input to ocean circulation. J. Phys. Oceanogr. 36 (2), 202–211.CrossRefGoogle Scholar
Ferrari, R. & Wunsch, C. 2009 Ocean circulation kinetic energy: reservoirs, sources, and sinks. Annu. Rev. Fluid Mech. 41 (1), 253–282.CrossRefGoogle Scholar
Fox-Kemper, B., Ferrari, R. & Hallberg, R. 2008 Parameterization of mixed layer eddies. Part I: theory and diagnosis. J. Phys. Oceanogr. 38 (6), 1145–1165.CrossRefGoogle Scholar
Grimshaw, R. 1981 Evolution equations for long, nonlinear internal waves in stratified shear flows. Stud. Appl. Maths 65 (2), 159–188.CrossRefGoogle Scholar
Gula, J., Molemaker, M. J. & McWilliams, J. C. 2014 Submesoscale cold filaments in the Gulf Stream. J. Phys. Oceanogr. 44, 2617–2643.CrossRefGoogle Scholar
Gula, J., Molemaker, M. J. & McWilliams, J. C. 2016 Topographic generation of submesoscale centrifugal instability and energy dissipation. Nat. Commun. 7, 12811.CrossRefGoogle ScholarPubMed
Hogg, A. McC., Dewar, W. K., Berloff, P. & Ward, M. L. 2011 Kelvin wave hydraulic control induced by interactions between vortices and topography. J. Fluid Mech. 687, 194–208.CrossRefGoogle Scholar
Jansen, M. F., Adcroft, A., Khani, S. & Kong, H. 2019 Toward an energetically consistent, resolution aware parameterization of ocean mesoscale eddies. J. Adv. Model. Earth Syst. 11 (8), 2844–2860.CrossRefGoogle Scholar
Johnson, E. R. 1991 The scattering at low frequencies of coastally trapped waves. J. Phys. Oceanogr. 21 (7), 913–932.2.0.CO;2>CrossRefGoogle Scholar
Johnson, E. R. & Rodney, J. T. 2011 Spectral methods for coastal-trapped waves. Cont. Shelf Res. 31 (14), 1481–1489.CrossRefGoogle Scholar
Large, W. G., McWilliams, J. C. & Doney, S. C. 2011 Oceanic vertical mixing: a review and a model with a nonlocal boundary layer parameterization. Rev. Geophys. 32, 363–403.CrossRefGoogle Scholar
Marshall, D. P. & Adcroft, A. J. 2010 Parameterization of ocean eddies: potential vorticity mixing, energetics and arnold's first stability theorem. Ocean Model. 32 (3), 188–204.CrossRefGoogle Scholar
Martinsen, E. A. & Weber, J. E. 1981 Frictional influence on internal Kelvin waves. Tellus 33 (4), 402–410.CrossRefGoogle Scholar
Mashayek, A., Ferrari, R., Merrifield, S., Ledwell, J. R., St. Laurent, L. & Naveira Garabato, A. 2017 Topographic enhancement of vertical turbulent mixing in the Southern Ocean. Nat. Commun. 8, 14197.CrossRefGoogle ScholarPubMed
McCreary, J. P. & Lighthill, J. 1981 A linear stratified ocean model of the equatorial undercurrent. Phil. Trans. R. Soc. Lond. A 298 (1444), 603–635.Google Scholar
McWilliams, J. C. 2016 Submesoscale currents in the ocean. Proc. R. Soc. A 472, 20160117.CrossRefGoogle ScholarPubMed
McWilliams, J. C. 2017 Submesoscale surface fronts and filaments: secondary circulation, buoyancy flux, and frontogenesis. J. Fluid Mech. 823, 391–432.CrossRefGoogle Scholar
Nikurashin, M. & Ferrari, R. 2010 Radiation and dissipation of internal waves generated by geostrophic motions impinging on small-scale topography: application to the Southern Ocean. J. Phys. Oceanogr. 40 (9), 2025–2042.CrossRefGoogle Scholar
Penduff, T., Juza, M., Barnier, B., Zika, J., Dewar, W. K., Treguier, A.-M., Molines, J.-M. & Audiffren, N. 2011 Sea level expression of intrinsic and forced ocean variabilities at interannual time scales. J. Clim. 24 (21), 5652–5670.CrossRefGoogle Scholar
Poulsen, M. B., Jochum, M., Maddison, J. R., Marshall, D. P. & Nuterman, R. 2019 A geometric interpretation of Southern Ocean eddy form stress. J. Phys. Oceanogr. 49 (10), 2553–2570.CrossRefGoogle Scholar
Rodney, J. T. & Johnson, E. R. 2012 Localisation of coastal trapped waves by longshore variations in bottom topography. Cont. Shelf Res. 32, 130–137.CrossRefGoogle Scholar
Rodney, J. T. & Johnson, E. R. 2014 Meanders and eddies from topographic transformation of coastal-trapped waves. J. Phys. Oceanogr. 44 (4), 1133–1150.CrossRefGoogle Scholar
Rodney, J. T. & Johnson, E. R. 2015 Localised continental shelf waves: geometric effects and resonant forcing. J. Fluid Mech. 785, 54–77.CrossRefGoogle Scholar
Romea, R. D. & Allen, J. S. 1984 The effect of friction and topography on coastal internal kelvin waves at low latitudes. Tellus 36A (4), 384–400.CrossRefGoogle Scholar
Sen, A., Scott, R. B. & Arbic, B. K. 2008 Global energy dissipation rate of deep-ocean low-frequency flows by quadratic bottom boundary layer drag: computations from current-meter data. Geophys. Res. Lett. 35 (9).CrossRefGoogle Scholar
Shakespeare, C. J. & Hogg, A. McC. 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
Skyllingstad, E. D. & Samelson, R. M. 2012 Baroclinic frontal instabilities and turbulent mixing in the surface boundary layer. Part I: unforced simulations. J. Phys. Oceanogr. 42 (10), 1701–1716.CrossRefGoogle Scholar
Smith, R. 1972 Nonlinear Kelvin and continental-shelf waves. J. Fluid Mech. 52 (2), 379–391.CrossRefGoogle Scholar
Stocker, T. F., Qin, D., Plattner, G.-K., Tignor, M., Allen, S. K., Boschung, J., Nauels, A., Xia, Y., Bex, V. & Midgley, P. M. (Eds.) 2013 Contribution of working group I to the fifth assessment report of the intergovernmental panel on climate change. In Climate Change 2013: The Physical Science Basis. Cambridge University Press.Google Scholar
Thomas, L. N., Tandon, A. & Mahadevan, A. 2008 Submesoscale processes and dynamics. In Ocean Modeling in an Eddying Regime, vol. 177, pp. 17–38. American Geophysical Union.CrossRefGoogle Scholar
Vanneste, J. & Yavneh, I. 2004 Exponentially small inertia–gravity waves and the breakdown of quasigeostrophic balance. J. Atmos. Sci. 61 (2), 211–223.2.0.CO;2>CrossRefGoogle Scholar
Vassilicos, J. C. 2015 Dissipation in turbulent flows. Annu. Rev. Fluid Mech. 47 (1), 95–114.CrossRefGoogle Scholar
Venayagamoorthy, S. K. & Stretch, D. D. 2010 On the turbulent Prandtl number in homogeneous stably stratified turbulence. J. Fluid Mech. 644, 359–369.CrossRefGoogle Scholar
Wang, D.-P. & Mooers, C. N. K. 1976 Coastal-trapped waves in a continuously stratified ocean. J. Phys. Oceanogr. 6 (6), 853–863.2.0.CO;2>CrossRefGoogle Scholar
Young, W. R. & Chen, L. 1995 Baroclinic instability and thermohaline gradient alignment in the mixed layer. J. Phys. Oceanogr. 25, 3172–3185.2.0.CO;2>CrossRefGoogle Scholar