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Turbulent and wave kinetic energy budgets in the airflow over wind-generated surface waves

Published online by Cambridge University Press:  14 June 2021

Kianoosh Yousefi Open the ORCID record for Kianoosh Yousefi [Opens in a new window] ,
Fabrice Veron Open the ORCID record for Fabrice Veron [Opens in a new window]  and
Marc P. Buckley Open the ORCID record for Marc P. Buckley [Opens in a new window]
Kianoosh Yousefi*
Affiliation:
Department of Mechanical Engineering, University of Delaware, Newark, DE19716, USA School of Marine Science and Policy, University of Delaware, Newark, DE19716, USA
Fabrice Veron
Affiliation:
School of Marine Science and Policy, University of Delaware, Newark, DE19716, USA
Marc P. Buckley
Affiliation:
Institute of Coastal Research, Helmholtz-Zentrum Geesthacht, 21502Geesthacht, Germany
*
†Email address for correspondence: [email protected]
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Abstract

The momentum and energy exchanges at the ocean surface are central factors determining the sea state, weather patterns and climate. To investigate the effects of surface waves on the air–sea energy exchanges, we analyse high-resolution laboratory measurements of the airflow velocity acquired above wind-generated surface waves using the particle image velocimetry technique. The velocity fields were further decomposed into the mean, wave-coherent and turbulent components, and the corresponding energy budgets were explored in detail. We specifically focused on the terms of the budget equations that represent turbulence production, wave production and wave–turbulence interactions. Over wind waves, the turbulent kinetic energy (TKE) production is positive at all heights with a sharp peak near the interface, indicating the transfer of energy from the mean shear to the turbulence. Away from the surface, however, the TKE production approaches zero. Similarly, the wave kinetic energy (WKE) production is positive in the lower portion of the wave boundary layer (WBL), representing the transfer of energy from the mean flow to the wave-coherent field. In the upper part of the WBL, WKE production becomes slightly negative, wherein the energy is transferred from the wave perturbation to the mean flow. The viscous and Stokes sublayer heights emerge as natural vertical scales for the TKE and WKE production terms, respectively. The interactions between the wave and turbulence perturbations show an energy transfer from the wave to the turbulence in the bulk of the WBL and from the turbulence to the wave in a thin layer near the interface.

JFM classification

Turbulent Flows: Wave-turbulence interactions Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wind-wave interactions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A33
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© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abe, H. & Antonia, R.A. 2016 Relationship between the energy dissipation function and the skin friction law in a turbulent channel flow. J. Fluid Mech. 798, 140–164.CrossRefGoogle Scholar
Anis, A. & Moum, J. 1995 Surface wave–turbulence interactions. Scaling $\varepsilon (z)$ near the sea surface. J. Phys. Oceanogr. 25 (9), 2025–2045.2.0.CO;2>CrossRefGoogle Scholar
Antonia, R., Teitel, M., Kim, J. & Browne, L. 1992 Low-Reynolds-number effects in a fully developed turbulent channel flow. J. Fluid Mech. 236, 579–605.CrossRefGoogle Scholar
Belcher, S.E. & Hunt, J.C.R. 1998 Turbulent flow over hills and waves. Annu. Rev. Fluid Mech. 30 (1), 507–538.CrossRefGoogle Scholar
Breuer, M., Peller, N., Rapp, C. & Manhart, M. 2009 Flow over periodic hills – numerical and experimental study in a wide range of Reynolds numbers. Comput. Fluids 38 (2), 433–457.CrossRefGoogle Scholar
Buckley, M.P. & Veron, F. 2017 Airflow measurements at a wavy air–water interface using PIV and LIF. Exp. Fluids 58 (11), 161.CrossRefGoogle Scholar
Buckley, M.P. & Veron, F. 2019 The turbulent airflow over wind generated surface waves. Eur. J. Mech. (B/Fluids) 73, 132–143.CrossRefGoogle Scholar
Buckley, M.P., Veron, F. & Yousefi, K. 2020 Surface viscous stress over wind-driven waves with intermittent airflow separation. J. Fluid Mech. 905, A31.CrossRefGoogle Scholar
Calhoun, R.J. & Street, R.L. 2001 Turbulent flow over a wavy surface: neutral case. J. Geophys. Res.: Oceans 106 (C5), 9277–9293.CrossRefGoogle Scholar
Caulliez, G. 2013 Dissipation regimes for short wind waves. J. Geophys. Res.: Oceans 118 (2), 672–684.CrossRefGoogle Scholar
Chalikov, D. & Belevich, M.Y. 1993 One–dimensional theory of the wave boundary layer. Boundary-Layer Meteorol. 63 (1–2), 65–96.CrossRefGoogle Scholar
Cherukat, P., Na, Y., Hanratty, T. & McLaughlin, J. 1998 Direct numerical simulation of a fully developed turbulent flow over a wavy wall. Theor. Comput. Fluid Dyn. 11 (2), 109–134.CrossRefGoogle Scholar
Cheung, T.K. & Street, R.L. 1988 The turbulent layer in the water at an air-water interface. J. Fluid Mech. 194, 133–151.CrossRefGoogle Scholar
Cifuentes-Lorenzen, A., Edson, J.B. & Zappa, C.J. 2018 Air–sea interaction in the southern ocean: exploring the height of the wave boundary layer at the air–sea interface. Boundary-Layer Meteorol. 169 (3), 461–482.CrossRefGoogle Scholar
Cimarelli, A., Leonforte, A., De Angelis, E., Crivellini, A. & Angeli, D. 2019 On negative turbulence production phenomena in the shear layer of separating and reattaching flows. Phys. Lett. A 383 (10), 1019–1026.CrossRefGoogle Scholar
De Angelis, V., Lombardi, P. & Banerjee, S. 1997 Direct numerical simulation of turbulent flow over a wavy wall. Phys. Fluids 9 (8), 2429–2442.CrossRefGoogle Scholar
Deike, L., Melville, W.K. & Popinet, S. 2016 Air entrainment and bubble statistics in breaking waves. J. Fluid Mech. 801, 91–129.CrossRefGoogle Scholar
Deike, L., Popinet, S. & Melville, W.K. 2015 Capillary effects on wave breaking. J. Fluid Mech. 769, 541–569.CrossRefGoogle Scholar
Donelan, M.A. 1990 Air-sea interaction. In Ocean Engineering Science (ed. B. Le Méhauté & D.M. Hanes), The Sea, vol. 9, chap. 7, pp. 239–292. John Wiley & Sons.Google Scholar
Donelan, M.A. 1998 Air-water exchange processes. In Physical Processes in Lakes and Oceans (ed. J. Imberger), Coastal and Estuarine Studies, vol. 54, chap. 2, pp. 19–36. American Geophysical Union.CrossRefGoogle Scholar
Donelan, M.A., Babanin, A.V., Young, I.R. & Banner, M.L. 2006 Wave-follower field measurements of the wind-input spectral function. Part II: parameterization of the wind input. J. Phys. Oceanogr. 36 (8), 1672–1689.CrossRefGoogle Scholar
Donelan, M.A., Babanin, A.V., Young, I.R., Banner, M.L. & McCormick, C. 2005 Wave-follower field measurements of the wind-input spectral function. Part I: measurements and calibrations. J. Atmos. Ocean. Technol. 22 (7), 799–813.CrossRefGoogle Scholar
Donelan, M.A., Dobson, F.W., Smith, S.D. & Anderson, R.J. 1993 On the dependence of sea surface roughness on wave development. J. Phys. Oceanogr. 23 (9), 2143–2149.2.0.CO;2>CrossRefGoogle Scholar
Donelan, M.A., Drennan, W.M. & Katsaros, K.B. 1997 The air–sea momentum flux in conditions of wind sea and swell. J. Phys. Oceanogr. 27 (10), 2087–2099.2.0.CO;2>CrossRefGoogle Scholar
Drazen, D.A., Melville, W.K. & Lenain, L. 2008 Inertial scaling of dissipation in unsteady breaking waves. J. Fluid Mech. 611, 307–332.CrossRefGoogle Scholar
Druzhinin, O., Troitskaya, Y., Tsai, W.-t. & Chen, P.-c. 2019 The study of a turbulent air flow over capillary-gravity water surface waves by direct numerical simulation. Ocean Model. 140, 101407.CrossRefGoogle Scholar
Druzhinin, O.A., Troitskaya, Y.I. & Zilitinkevich, S.S. 2016 a Stably stratified airflow over a waved water surface. Part 1: stationary turbulence regime. Q. J. R. Meteorol. Soc. 142 (695), 759–772.CrossRefGoogle Scholar
Druzhinin, O.A., Troitskaya, Y.I. & Zilitinkevich, S.S. 2016 b Stably stratified airflow over a waved water surface. Part 2: wave-induced pre-turbulent motions. Q. J. R. Meteorol. Soc. 142 (695), 773–780.CrossRefGoogle Scholar
Einaudi, F. & Finnigan, J. 1993 Wave-turbulence dynamics in the stably stratified boundary layer. J. Atmos. Sci. 50 (13), 1841–1864.2.0.CO;2>CrossRefGoogle Scholar
El Telbany, M.M.M. & Reynolds, A.J. 1982 The structure of turbulent plane Couette flow. Trans. ASME J. Fluids Engng 104 (3), 367–372.CrossRefGoogle Scholar
Fairall, C. & Larsen, S.E. 1986 Inertial-dissipation methods and turbulent fluxes at the air-ocean interface. Boundary-Layer Meteorol. 34 (3), 287–301.CrossRefGoogle Scholar
Fedorov, A.V. & Melville, W.K. 1998 Nonlinear gravity–capillary waves with forcing and dissipation. J. Fluid Mech. 354, 1–42.CrossRefGoogle Scholar
Fedorov, A.V., Melville, W.K. & Rozenberg, A. 1998 An experimental and numerical study of parasitic capillary waves. Phys. Fluids 10 (6), 1315–1323.CrossRefGoogle Scholar
Finnigan, J.J. & Einaudi, F. 1981 The interaction between an internal gravity wave and the planetary boundary layer. Part II: effect of the wave on the turbulence structure. Q. J. R. Meteorol. Soc. 107 (454), 807–832.CrossRefGoogle Scholar
Grachev, A.A. & Fairall, C.W. 2001 Upward momentum transfer in the marine boundary layer. J. Phys. Oceanogr. 31 (7), 1698–1711.2.0.CO;2>CrossRefGoogle Scholar
Grare, L., Lenain, L. & Melville, W.K. 2013 Wave-coherent airflow and critical layers over ocean waves. J. Phys. Oceanogr. 43 (10), 2156–2172.CrossRefGoogle Scholar
Günther, A. & Von Rohr, P.R. 2003 Large-scale structures in a developed flow over a wavy wall. J. Fluid Mech. 478, 257–285.CrossRefGoogle Scholar
Hamed, A.M., Kamdar, A., Castillo, L. & Chamorro, L.P. 2015 Turbulent boundary layer over 2D and 3D large-scale wavy walls. Phys. Fluids 27 (10), 106601.CrossRefGoogle Scholar
Hara, T. & Belcher, S.E. 2004 Wind profile and drag coefficient over mature ocean surface wave spectra. J. Phys. Oceanogr. 34 (11), 2345–2358.CrossRefGoogle Scholar
Hara, T. & Sullivan, P.P. 2015 Wave boundary layer turbulence over surface waves in a strongly forced condition. J. Phys. Oceanogr. 45 (3), 868–883.CrossRefGoogle Scholar
Harris, J., Belcher, S. & Street, R. 1996 Linear dynamics of wind waves in coupled turbulent air–water flow. Part 2. Numerical model. J. Fluid Mech. 308, 219–254.CrossRefGoogle Scholar
Henn, D.S. & Sykes, R.I. 1999 Large-eddy simulation of flow over wavy surfaces. J. Fluid Mech. 383, 75–112.CrossRefGoogle Scholar
Högström, U., Sahlée, E., Smedman, A.-S., Rutgersson, A., Nilsson, E., Kahma, K.K. & Drennan, W.M. 2015 Surface stress over the ocean in swell-dominated conditions during moderate winds. J. Atmos. Sci. 72 (12), 4777–4795.CrossRefGoogle Scholar
Högström, U., Smedman, A., Sahleé, E., Drennan, W., Kahma, K., Pettersson, H. & Zhang, F. 2009 The atmospheric boundary layer during swell: a field study and interpretation of the turbulent kinetic energy budget for high wave ages. J. Atmos. Sci. 66 (9), 2764–2779.CrossRefGoogle Scholar
Hsu, C.-T. & Hsu, E.Y. 1983 On the structure of turbulent flow over a progressive water wave: theory and experiment in a transformed wave-following coordinate system. Part 2. J. Fluid Mech. 131, 123–153.CrossRefGoogle Scholar
Hsu, C.-T., Hsu, E.Y. & Street, R.L. 1981 On the structure of turbulent flow over a progressive water wave: theory and experiment in a transformed, wave-following co-ordinate system. J. Fluid Mech. 105, 87–117.CrossRefGoogle Scholar
Hudson, J.D., Dykhno, L. & Hanratty, T.J. 1996 Turbulence production in flow over a wavy wall. Exp. Fluids 20 (4), 257–265.CrossRefGoogle Scholar
Husain, N.T., Hara, T., Buckley, M.P., Yousefi, K., Veron, F. & Sullivan, P.P. 2019 Boundary layer turbulence over surface waves in a strongly forced condition: LES and observation. J. Phys. Oceanogr. 49 (8), 1997–2015.CrossRefGoogle Scholar
Hussain, A.F. 1983 Coherent structures – reality and myth. Phys. Fluids 26 (10), 2816–2850.CrossRefGoogle Scholar
Hussain, A.K.M.F. & Reynolds, W.C. 1970 The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech. 41, 241–258.CrossRefGoogle Scholar
Hussain, A.K.M.F. & Reynolds, W.C. 1975 Measurements in fully developed turbulent channel flow. Trans. ASME J. Fluids Engng 97 (4), 568–578.CrossRefGoogle Scholar
Iafrati, A. 2011 Energy dissipation mechanisms in wave breaking processes: spilling and highly aerated plunging breaking events. J. Geophys. Res.: Oceans 116 (C7), C07024.Google Scholar
Iafrati, A., Babanin, A. & Onorato, M. 2013 Modulational instability, wave breaking, and formation of large-scale dipoles in the atmosphere. Phys. Rev. Lett. 110 (18), 184504.CrossRefGoogle ScholarPubMed
Janssen, P. 2004 The Interaction of Ocean Waves and Wind. Cambridge University Press.CrossRefGoogle Scholar
Janssen, P.A. 1999 On the effect of ocean waves on the kinetic energy balance and consequences for the inertial dissipation technique. J. Phys. Oceanogr. 29 (3), 530–534.2.0.CO;2>CrossRefGoogle Scholar
Kihara, N., Hanazaki, H., Mizuya, T. & Ueda, H. 2007 Relationship between airflow at the critical height and momentum transfer to the traveling waves. Phys. Fluids 19 (1), 015102.CrossRefGoogle Scholar
Kim, H.T., Kline, S.J. & Reynolds, W.C. 1968 An experimental study of turbulence production near a smooth wall in a turbulent boundary layer with zero pressure-gradient. Tech. Rep. MD-20. Department of Mechanical Engineering, Stanford University, Stanford, CA, USA.Google Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.CrossRefGoogle Scholar
Kitaigorodskii, S. & Donelan, M. 1984 Wind-wave effects on gas transfer. In Gas Transfer at Water Surfaces (ed. W. Brutsaert & G. H. Jirka), pp. 147–170. Springer.CrossRefGoogle Scholar
Kitoh, O., Nakabyashi, K. & Nishimura, F. 2005 Experimental study on mean velocity and turbulence characteristics of plane Couette flow: low-Reynolds-number effects and large longitudinal vortical structure. J. Fluid Mech. 539, 199–227.CrossRefGoogle Scholar
Komen, G.J., Cavaleri, L., Donelan, M., Hasselmann, K., Hasselmann, S. & Janssen, P. 1996 Dynamics and Modelling of Ocean Waves. Cambridge University Press.Google Scholar
Kreplin, H.-P. & Eckelmann, H. 1979 Behavior of the three fluctuating velocity components in the wall region of a turbulent channel flow. Phys. Fluids 22 (7), 1233–1239.CrossRefGoogle Scholar
Krogstad, P.-Å. & Antonia, R. 1999 Surface roughness effects in turbulent boundary layers. Exp. Fluids 27 (5), 450–460.CrossRefGoogle Scholar
Kruse, N., Günther, A. & Von Rohr, P.R. 2003 Dynamics of large-scale structures in turbulent flow over a wavy wall. J. Fluid Mech. 485, 87–96.CrossRefGoogle Scholar
Kruse, N., Kuhn, S. & von Rohr, P.R. 2006 Wavy wall effects on turbulence production and large-scale modes. J. Turbul. 7, N31.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Large, W. & Pond, S. 1981 Open ocean momentum flux measurements in moderate to strong winds. J. Phys. Oceanogr. 11 (3), 324–336.2.0.CO;2>CrossRefGoogle Scholar
Liu, J. & Merkine, L. 1976 On the interactions between large-scale structure and fine-grained turbulence in a free shear flow. I. The development of temporal interactions in the mean. Proc. R. Soc. Lond. A 352, 213–247.Google Scholar
Liu, S., Kermani, A., Shen, L. & Yue, D.K. 2009 Investigation of coupled air-water turbulent boundary layers using direct numerical simulations. Phys. Fluids 21 (6), 062108.CrossRefGoogle Scholar
Longuet-Higgins, M.S. 1969 Action of a variable stress at the surface of water waves. Phys. Fluids 12 (4), 737–740.CrossRefGoogle Scholar
Makin, V.K. & Kudryavtsev, V.N. 1999 Coupled sea surface–atmosphere model: 1. Wind over waves coupling. J. Geophys. Res.: Oceans 104 (C4), 7613–7623.CrossRefGoogle Scholar
Makin, V.K. & Mastenbroek, C. 1996 Impact of waves on air-sea exchange of sensible heat and momentum. Boundary-Layer Meteorol. 79 (3), 279–300.CrossRefGoogle Scholar
Melville, W. 1983 Wave modulation and breakdown. J. Fluid Mech. 128, 489–506.CrossRefGoogle Scholar
Melville, W.K. & Fedorov, A.V. 2015 The equilibrium dynamics and statistics of gravity–capillary waves. J. Fluid Mech. 767, 449–466.CrossRefGoogle Scholar
Moon, I.J., Hara, T., Ginis, I., Belcher, S.E. & Tolman, H.L. 2004 Effect of surface waves on air–sea momentum exchange. Part I: effect of mature and growing seas. J. Atmos. Sci. 61 (19), 2321–2333.2.0.CO;2>CrossRefGoogle Scholar
Pahlow, M., Parlange, M.B. & Porté-Agel, F. 2001 On Monin–Obukhov similarity in the stable atmospheric boundary layer. Boundary-Layer Meteorol. 99 (2), 225–248.CrossRefGoogle Scholar
Palmer, J.A., Mejia-Alvarez, R., Best, J.L. & Christensen, K.T. 2012 Particle-image velocimetry measurements of flow over interacting barchan dunes. Exp. Fluids 52 (3), 809–829.CrossRefGoogle Scholar
Panofsky, H.A. & Dutton, J.A. 1984 Atmospheric Turbulence: Models and Methods for Engineering Applications, 1st edn. Wiley.Google Scholar
Papadimitrakis, Y.A., Hsu, E.Y. & Street, R.L. 1986 The role of wave-induced pressure fluctuations in the transfer processes across an air–water interface. J. Fluid Mech. 170, 113–137.CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Reynolds, W.C. & Hussain, A.K.M.F. 1972 The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech. 54, 263–288.CrossRefGoogle Scholar
Rutgersson, A. & Sullivan, P.P. 2005 The effect of idealized water waves on the turbulence structure and kinetic energy budgets in the overlying airflow. Dyn. Atmos. Oceans 38 (3–4), 147–171.CrossRefGoogle Scholar
Sauer, J.A., Muñoz-Esparza, D., Canfield, J.M., Costigan, K.R., Linn, R.R. & Kim, Y.-J. 2016 A large-eddy simulation study of atmospheric boundary layer influence on stratified flows over terrain. J. Atmos. Sci. 73 (7), 2615–2632.CrossRefGoogle Scholar
Shaikh, N. & Siddiqui, K. 2010 An experimental investigation of the near surface flow over air-water and air-solid interfaces. Phys. Fluids 22 (2), 025103.CrossRefGoogle Scholar
Shaikh, N. & Siddiqui, K. 2011 Near-surface flow structure over wind-generated water waves, Part II: characteristics of separated and non-separated flows. Ocean Dyn. 61 (1), 143–154.CrossRefGoogle Scholar
Shen, L., Zhang, X., Yue, D.K.P. & Triantafyllou, M.S. 2003 Turbulent flow over a flexible wall undergoing a streamwise travelling wave motion. J. Fluid Mech. 484, 197–221.CrossRefGoogle Scholar
Siddiqui, K. & Loewen, M.R. 2010 Phase-averaged flow properties beneath microscale breaking waves. Boundary-Layer Meteorol. 134 (3), 499–523.CrossRefGoogle Scholar
Sjöblom, A. & Smedman, A.-S. 2002 The turbulent kinetic energy budget in the marine atmospheric surface layer. J. Geophys. Res.: Oceans 107 (C10), 6–18.Google Scholar
Smedman, A., Högström, U., Bergström, H., Rutgersson, A., Kahma, K. & Pettersson, H. 1999 A case study of air-sea interaction during swell conditions. J. Geophys. Res.: Oceans 104 (C11), 25833–25851.CrossRefGoogle Scholar
Smedman, A., Högström, U., Sahlée, E., Drennan, W., Kahma, K., Pettersson, H. & Zhang, F. 2009 Observational study of marine atmospheric boundary layer characteristics during swell. J. Atmos. Sci. 66 (9), 2747–2763.CrossRefGoogle Scholar
Smedman, A., Tjernström, M. & Högström, U. 1994 The near-neutral marine atmospheric boundary layer with no surface shearing stress: a case study. J. Atmos. Sci. 51 (23), 3399–3411.2.0.CO;2>CrossRefGoogle Scholar
Smedman, A.-S. 1988 Observations of a multi-level turbulence structure in a very stable atmospheric boundary layer. Boundary-Layer Meteorol. 44 (3), 231–253.CrossRefGoogle Scholar
Spalart, P.R. 1988 Direct simulation of a turbulent boundary layer up to $R_{\theta }= 1410$. J. Fluid Mech. 187, 61–98.CrossRefGoogle Scholar
Sullivan, P.P., Banner, M.L., Morison, R.P. & Peirson, W.L. 2018 Turbulent flow over steep steady and unsteady waves under strong wind forcing. J. Phys. Oceanogr. 48 (1), 3–27.CrossRefGoogle Scholar
Sullivan, P.P., Edson, J.B., Hristov, T. & McWilliams, J.C. 2008 Large-eddy simulations and observations of atmospheric marine boundary layers above nonequilibrium surface waves. J. Atmos. Sci. 65 (4), 1225–1245.CrossRefGoogle Scholar
Sullivan, P.P. & McWilliams, J.C. 2002 Turbulent flow over water waves in the presence of stratification. Phys. Fluids 14 (3), 1182–1195.CrossRefGoogle Scholar
Sullivan, P.P. & McWilliams, J.C. 2010 Dynamics of winds and currents coupled to surface waves. Annu. Rev. Fluid Mech. 42 (1), 19–42.CrossRefGoogle Scholar
Sullivan, P.P., McWilliams, J.C. & Patton, E.G. 2014 Large-eddy simulation of marine atmospheric boundary layers above a spectrum of moving waves. J. Atmos. Sci. 71 (11), 4001–4027.CrossRefGoogle Scholar
Sun, Z., Zhu, Y., Hu, Y. & Zhang, S. 2018 Direct numerical simulation of a fully developed compressible wall turbulence over a wavy wall. J. Turbul. 19 (1), 72–105.CrossRefGoogle Scholar
Thais, L. & Magnaudet, J. 1996 Turbulent structure beneath surface gravity waves sheared by the wind. J. Fluid Mech. 328, 313–344.CrossRefGoogle Scholar
Tsai, W.-t., Chen, S.-m. & Lu, G.-h. 2015 Numerical evidence of turbulence generated by nonbreaking surface waves. J. Phys. Oceanogr. 45 (1), 174–180.CrossRefGoogle Scholar
Tsai, W.-t. & Hung, L.-p. 2010 Enhanced energy dissipation by parasitic capillaries on short gravity–capillary waves. J. Phys. Oceanogr. 40 (11), 2435–2450.CrossRefGoogle Scholar
Vollestad, P., Ayati, A.A. & Jensen, A. 2019 Experimental investigation of intermittent airflow separation and microscale wave breaking in wavy two-phase pipe flow. J. Fluid Mech. 878, 796–819.CrossRefGoogle Scholar
Wei, T. & Willmarth, W. 1989 Reynolds-number effects on the structure of a turbulent channel flow. J. Fluid Mech. 204, 57–95.CrossRefGoogle Scholar
Yang, D. & Shen, L. 2009 Characteristics of coherent vortical structures in turbulent flows over progressive surface waves. Phys. Fluids 21 (12), 125106.CrossRefGoogle Scholar
Yang, D. & Shen, L. 2010 Direct-simulation-based study of turbulent flow over various waving boundaries. J. Fluid Mech. 650, 131–180.CrossRefGoogle Scholar
Yang, D. & Shen, L. 2017 Direct numerical simulation of scalar transport in turbulent flows over progressive surface waves. J. Fluid Mech. 819, 58–103.CrossRefGoogle Scholar
Yousefi, K. 2020 Turbulence in the atmospheric wave boundary layer. Doctoral dissertation, University of Delaware, Newark, Delaware, USA.Google Scholar
Yousefi, K. & Veron, F. 2020 Boundary layer formulations in orthogonal curvilinear coordinates for flow over wind-generated surface waves. J. Fluid Mech. 888, A11.CrossRefGoogle Scholar
Yousefi, K., Veron, F. & Buckley, M.P. 2020 a Momentum flux measurements in the airflow over wind-generated surface waves. J. Fluid Mech. 895, A15.CrossRefGoogle Scholar
Yousefi, K., Veron, F. & Buckley, M.P. 2020 b Measurements of airside shear- and wave-induced viscous stresses over strongly forced wind waves. In Recent Advances in the Study of Oceanic Whitecaps (ed. P. Vlahos & E.C. Monahan), chap. 6, pp. 77–94. Springer.CrossRefGoogle Scholar
Zhang, X. 1995 Capillary–gravity and capillary waves generated in a wind wave tank: observations and theories. J. Fluid Mech. 289, 51–82.CrossRefGoogle Scholar
Zhang, X. 2002 Enhanced dissipation of short gravity and gravity capillary waves due to parasitic capillaries. Phys. Fluids 14 (11), L81–L84.CrossRefGoogle Scholar