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Experimental study of particle trajectories below deep-water surface gravity wave groups

Published online by Cambridge University Press:  20 September 2019

T. S. van den Bremer*
Affiliation:
Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK
C. Whittaker
Affiliation:
Department of Civil and Environmental Engineering, University of Auckland, Auckland 1010, New Zealand
R. Calvert
Affiliation:
Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, UK
A. Raby
Affiliation:
School of Engineering, University of Plymouth, Plymouth PL4 8AA, UK
P. H. Taylor
Affiliation:
Faculty of Engineering and Mathematical Sciences, University of Western Australia, Crawley, WA 6009, Australia
*
Email address for correspondence: [email protected]

Abstract

Owing to the interplay between the forward Stokes drift and the backward wave-induced Eulerian return flow, Lagrangian particles underneath surface gravity wave groups can follow different trajectories depending on their initial depth below the surface. The motion of particles near the free surface is dominated by the waves and their Stokes drift, whereas particles at large depths follow horseshoe-shaped trajectories dominated by the Eulerian return flow. For unidirectional wave groups, a small net displacement in the direction of travel of the group results near the surface, and is accompanied by a net particle displacement in the opposite direction at depth. For deep-water waves, we study these trajectories experimentally by means of particle tracking velocimetry in a two-dimensional flume. In doing so, we provide visual illustration of Lagrangian trajectories under groups, including the contributions of both the Stokes drift and the Eulerian return flow to both the horizontal and the vertical Lagrangian displacements. We compare our experimental results to leading-order solutions of the irrotational water wave equations, finding good agreement.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

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References

Andrews, D. G. & McIntyre, M. E. 1978 An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89, 609646.Google Scholar
Bagnold, R. A. 1947 Sand movement by waves: some small-scale experiments with sand of very low density. J. Inst. Civil Engng. 27 (4), 447469.Google Scholar
Belcher, S. E., Grant, A. L. M., Hanley, K. E., Fox-Kemper, B., Roekel, L. V., Sullivan, P. P., Large, W. G., Andy, A. B., Hines, A., Calvert, D. et al. 2012 A global perspective on Langmuir turbulence in the ocean surface boundary layer. Geophys. Res. Lett. 39 (18), L18605.Google Scholar
van den Bremer, T. S. & Breivik, Ø. 2017 Stokes drift. Phil. Trans. R. Soc. Lond. A 376, 20170104.Google Scholar
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), 27012722.Google Scholar
van den Bremer, T. S. & Taylor, P. H. 2016 Lagrangian transport for two-dimensional deep-water surface gravity wave groups. Proc. R. Soc. A 472, 20160159.Google Scholar
Bühler, O. 2014 Waves and Mean Flows, 2nd edn. Cambridge University Press.Google Scholar
Christensen, K. H. & Terrile, E. 2009 Drift and deformation of oil slicks due to surface waves. J. Fluid Mech. 620, 313332.10.1017/S0022112008004606Google Scholar
Craik, A. D. D. & Leibovich, S. 1976 A rational model for Langmuir circulations. J. Fluid Mech. 73 (3), 401426.Google Scholar
D’Asaro, E. A., Thomson, J., Shcherbina, A. Y., Harcourt, R. R., Cronin, M. F., Hemer, M. A. & Fox-Kemper, B. 2014 Quantifying upper ocean turbulence driven by surface waves. Geophys. Res. Lett. 41 (1), 102107.10.1002/2013GL058193Google Scholar
Deike, L., Pizzo, N. & Melville, W. K. 2017 Lagrangian transport by breaking surface waves. J. Fluid Mech. 829, 364391.Google Scholar
DiBenedetto, M. H. & Ouellette, N. T. 2018 Preferential orientation of spheroidal particles in wavy flow. J. Fluid Mech. 856, 850869.10.1017/jfm.2018.738Google Scholar
DiBenedetto, M. H., Ouellette, N. T. & Koseff, J. R. 2018 Transport of anisotropic particles under waves. J. Fluid Mech. 837, 320340.10.1017/jfm.2017.853Google 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), 977991.Google Scholar
Eames, I. 2008 Settling of particles beneath water waves. J. Phys. Oceanogr. 38, 28462853.Google Scholar
Groeneweg, J. & Klopman, G. 1998 Changes of the mean velocity profiles in the combined wave-current motion described in a GLM formulation. J. Fluid Mech. 370, 271296.10.1017/S0022112098002018Google Scholar
Grue, J. & Kolaas, J. 2017 Experimental particle paths and drift velocity in steep waves at finite water depth. J. Fluid Mech. 810, R1.Google Scholar
Haney, S., Fox-Kemper, B., Julien, K. & Webb, A. 2015 Symmetric and geostrophic instabilities in the wave-forced ocean mixed layer. J. Phys. Oceanogr. 45 (12), 30333056.Google Scholar
Herbers, T. H. C. & Janssen, T. T. 2016 Lagrangian surface wave motion and Stokes drift fluctuations. J. Phys. Oceaonogr. 46 (4), 10091021.Google Scholar
Jones, C. E., Dagestad, K., Breivik, Ø., Holt, B., Röhrshrs, J., Christensen, K., Espeseth, M., Brekke, C. & Skrunes, S. 2016 Measurement and modelling of oil slick transport. J. Geophys. Res. 121 (10), 77597775.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.Google Scholar
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245, 535581.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1962 Radiation stress and mass transport in gravity waves, with applications to ‘surf beats’. J. Fluid Mech. 13, 481504.10.1017/S0022112062000877Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stresses in water waves; a physical discussion, with applications. Deep-Sea Res. 2, 529562.Google 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, 131169.10.1017/jfm.2017.774Google Scholar
McIntyre, M. E. 1981 On the wave momentum myth. J. Fluid Mech. 106, 331347.Google Scholar
McIntyre, M. E. 1988 A note on the divergence effect and the Lagrangian-mean surface elevation in periodic water waves. J. Fluid Mech. 189, 235242.Google Scholar
McWilliams, J. C. 2016 Submesoscale currents in the ocean. Proc. R. Soc. Lond. A 472 (2189), 20160117.Google Scholar
McWilliams, J. C. & Restrepo, J. M. 1999 The wave-driven ocean circulation. J. Phys. Oceanogr. 29, 25232540.Google Scholar
McWilliams, J. N., Restrepo, J. M. & Lane, E. M. 2004 An asymptotic theory for the interaction of waves and currents in coastal waters. J. Fluid Mech. 511, 135178.10.1017/S0022112004009358Google Scholar
Mei, C. C., Liu, P. L. F. & Carter, T. G.1972 Mass transport in water waves. Tech. Rep. 146. MIT Rep. Ralph M. Parsons Lab. Water Resources Hydrodynamics.Google Scholar
Mellor, G. 2016 On theories dealing with the interaction of surface waves and ocean circulation. J. Geophys. Res. Oceans 121, 44744486.10.1002/2016JC011768Google Scholar
Melville, W. K. & Rapp, R. 1988 The surface velocity field in steep and breaking waves. J. Fluid Mech. 189, 122.10.1017/S0022112088000898Google Scholar
Monismith, S. G., Cowen, E. A., Nepf, H. M., Magnaudet, J. & Thais, L. 2007 Laboratory observations of mean flows under surface gravity waves. J. Fluid Mech. 573, 131147.Google Scholar
Nokes, R. 2014 Streams 2.02: System Theory and Design. University of Canterbury.Google Scholar
Paprota, M., Sulisz, W. & Reda, A. 2016 Experimental study of wave-induced mass transport. J. Hydraul Res. 54 (4), 423434.Google Scholar
Pizzo, N. E. 2017 Surfing surface gravity waves. J. Fluid Mech. 823, 316328.Google Scholar
Pizzo, N. & Melville, W. 2013 Vortex generation by deep-water breaking waves. J. Fluid Mech. 734, 198218.Google 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), 15191533.Google Scholar
Santamaria, F., Boffetta, F., Martins Afonso, M., Mazzino, A., Onorato, M. & Pugliese, D. 2013 Stokes drift for inertial particles transported by water waves. Europhys. Lett. 102 (1), 14003.Google Scholar
Schäffer, H. A. 1996 Second-order wavemaker theory for irregular waves. Ocean Engng 23, 4788.Google Scholar
Smith, J. A. 2006 Observed variability of ocean wave Stokes drift, and the Eulerian response to passing groups. J. Phys. Oceanogr. 36, 13811402.10.1175/JPO2910.1Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441455.Google Scholar
Sullivan, P. P. & McWilliams, J. C. 2010 Dynamics of winds and currents coupled to surface waves. Annu. Rev. Fluid Mech. 42, 1942.Google Scholar
Swan, C. 1990 Convection within an experimental wave flume. J. Hydraul Res. 28, 273282.Google Scholar
Swan, C. & Sleath, J. F. A. 1990 A second approximation to the time-mean Lagrangian drift beneath progressive gravity waves. Ocean Engng 1, 6579.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), 126138.Google Scholar
Umeyama, M. 2012 Eulerian-Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry. Phil. Trans. R. Soc. A 370 (1964), 16871702.10.1098/rsta.2011.0450Google Scholar
Van Dyke, M. 1982 An Album of Fluid Motion. Parabolic Press.Google Scholar
Weber, J. E. H. 2011 Do we observe Gerstner waves in wave tank experiments? Wave Motion 48 (4), 301309.Google Scholar
Whittaker, C. N., Fitzgerald, C. J., Raby, A. C., Taylor, P. H., Orszaghova, J. & Borthwick, A. G. L. 2017 Optimisation of focused wave group runup on a plane beach. Coast. Engng 121, 4455.Google Scholar