Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-28T08:10:26.624Z Has data issue: false hasContentIssue false
  • English
  • Français

Lagrangian transport by deep-water surface gravity wavepackets: effects of directional spreading and stratification

Published online by Cambridge University Press:  28 November 2019

C. Higgins Open the ORCID record for C. Higgins [Opens in a new window] ,
T. S. van den Bremer Open the ORCID record for T. S. van den Bremer [Opens in a new window]  and
J. Vanneste Open the ORCID record for J. Vanneste [Opens in a new window]
C. Higgins*
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK
T. S. van den Bremer
Affiliation:
Department of Engineering Science, University of Oxford, OxfordOX1 3PJ, UK
J. Vanneste
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, EdinburghEH9 3FD, UK
*
†Email address for correspondence: christopher.higgins@keble.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The Lagrangian mass transport by non-dissipating surface gravity wavepackets consists of the Stokes drift and the wave-induced return flow. We examine how directional spreading and density stratification affect this mass transport for an isolated non-dissipating wavepacket in deep water using a perturbation expansion. For an unstratified ocean, we show that the net displacement by the return flow is finite, negative, the same at all vertical levels and inversely proportional to the depth for spanwise-infinite packets representing unidirectional (two-dimensional) seas, but zero for spanwise-localised packets representing directionally spread seas (three-dimensional). We resolve this difference by demonstrating that a transition between two-dimensional-like (finite) and three-dimensional-like (zero) displacement occurs on a time scale inversely proportional to the degree of directional spreading. For a stratified ocean, we show that in two dimensions the net displacement profile by the return flow oscillates slowly with depth, with a wavelength dependent on the ratio of buoyancy frequency to the surface wave group velocity, and infinite displacements are predicted when the surface wavepacket resonantly excites internal waves. In three dimensions, the net displacement remains zero in the presence of stratification, but finite-time displacements may be appreciably altered.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A42
Copyright
© 2019 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

van den Bremer, T. S. & Taylor, P. H. 2015 Estimates of Lagrangian transport by surface gravity wave groups: the effects of finite depth and directionality. J. Geophys. Res. 120 (4), 2701–2722.CrossRefGoogle Scholar
van den Bremer, T. S. & Taylor, P. H. 2016 Lagrangian transport for two-dimensional deep-water surface gravity wave groups. Proc. R. Soc. Lond. A 472 (2192), 20160159.CrossRefGoogle Scholar
Bühler, O. 2014 Waves and Mean Flows, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Christensen, K. H. & Terrile, E. 2009 Drift and deformation of oil slicks due to surface waves. J. Fluid Mech. 620, 313–332.CrossRefGoogle Scholar
Drivdal, M., Broström, G. & Christensen, K. H. 2014 Wave-induced mixing and transport of buoyant particles: application to the Statfjord A oil spill. Ocean Sci. 10 (6), 977–991.CrossRefGoogle Scholar
Ewans, K. 2002 Directional spreading in ocean swell. In Fourth International Symposium on Ocean Wave Analysis and Measurement, pp. 517–529. Ocean Wave Measurement and Analysis.Google Scholar
Haney, S. & Young, W. R. 2017 Radiation of internal waves from groups of surface gravity waves. J. Fluid Mech. 829, 280–303.CrossRefGoogle Scholar
Hasselmann, K. 1970 Wave-driven inertial oscillations. Geophys. Fluid Dyn. 1, 463–502.CrossRefGoogle Scholar
Herbers, T. H. C. & Janssen, T. T. 2016 Lagrangian surface wave motion and Stokes drift fluctuations. J. Phys. Oceanogr. 46 (4), 1009–1021.CrossRefGoogle Scholar
Jones, C. E., Dagestad, K., Breivik, Ø., Holt, B., Röhrs, J., Christensen, K., Espeseth, M., Brekke, C. & Skrunes, S. 2016 Measurement and modelling of oil slick transport. J. Geophys. Res. 121 (10), 7759–7775.CrossRefGoogle Scholar
Kinsman, B. 2002 Wind Waves: Their Generation and Propagation on the Ocean Surface. Dover.Google Scholar
Lebreton, L., Slat, B., Ferrari, F., Sainte-Rose, B., Aitken, J., Marthouse, R., Hajbane, S., Cunsolo, S., Schwarz, A., Levivier, A. et al. 2018 Evidence that the Great Pacific Garbage Patch is rapidly accumulating plastic. Sci. Rep. 8 (4666), 1–15.CrossRefGoogle ScholarPubMed
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245 (903), 535–581.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1984 Statistical properties of wave groups in a random sea state. Phil. Trans. R. Soc. Lond. A 312 (1521), 219–250.CrossRefGoogle Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1962 Radiation stress and mass transport in gravity waves, with application to ‘surf beats’. J. Fluid Mech. 13 (4), 481–504.CrossRefGoogle Scholar
McAllister, M. L., Adcock, T. A. A., Taylor, P. H. & van den Bremer, T. S. 2018 The set-down and set-up of directionally spread and crossing surface gravity wave groups. J. Fluid Mech. 835, 131–169.CrossRefGoogle Scholar
McIntyre, M. E. 1980 An introduction to the generalized Lagrangian-mean description of wave, mean-flow interaction. Pure Appl. Geophys. 118 (1), 152–176.CrossRefGoogle Scholar
Phillips, O. M. 1977 The Dynamics of The Upper Ocean. Cambridge University Press.Google Scholar
Rayleigh, Lord 1883 The form of standing waves on the surface of running water. Proc. Lond. Math. Soc. s1–15 (1), 69–78.CrossRefGoogle Scholar
Röhrs, J., Christensen, K. H., Hole, L. R., Broström, G., Drivdal, M. & Sundby, S. 2012 Observation-based evaluation of surface wave effects on currents and trajectory forecasts. Ocean Dyn. 62 (10), 1519–1533.CrossRefGoogle Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Toffoli, A. & Bitner-Gregersen, E. M. 2017 Types of Ocean Surface Waves, Wave Classification. pp. 1–8. John Wiley & Sons, Ltd.Google Scholar
Trinanes, J. A., Olascoaga, M. J., Goni, G. J., Maximenko, N. A., Griffin, D. A. & Hafner, J. 2016 Analysis of flight MH370 potential debris trajectories using ocean observations and numerical model results. J. Oper. Oceanogr. 9 (2), 126–138.Google Scholar
Ursell, F. 1950 On the theoretical form of ocean swell on a rotating earth. Mon. Not. R. Astron. Soc., Geophys. Suppl. 6 (s1), 1–8.CrossRefGoogle Scholar
Xu, Z. & Bowen, A. J. 1994 Wave - and wind-driven flow in water of finite depth. J. Phys. Oceanogr. 24 (9), 1850–1866.2.0.CO;2>CrossRefGoogle Scholar