Turbulent and wave kinetic energy budgets in the airflow over wind-generated surface waves. (English) Zbl 1503.76044
Summary: 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.
MSC:
| 76F25 | Turbulent transport, mixing |
| 76D33 | Waves for incompressible viscous fluids |
| 76U60 | Geophysical flows |
| 76-05 | Experimental work for problems pertaining to fluid mechanics |
| 86A05 | Hydrology, hydrography, oceanography |
| 86A10 | Meteorology and atmospheric physics |
Keywords:
wave-turbulence interaction; surface gravity wave; wind-wave interaction; viscous sublayer; Stokes sublayer; high-resolution laboratory measurementReferences:
| [1] | Abe, H. & Antonia, R.A.2016Relationship between the energy dissipation function and the skin friction law in a turbulent channel flow. J. Fluid Mech.798, 140-164. · Zbl 1422.76082 |
| [2] | Anis, A. & Moum, J.1995Surface wave-turbulence interactions. Scaling \(\varepsilon (z)\) near the sea surface. J. Phys. Oceanogr.25 (9), 2025-2045. |
| [3] | Antonia, R., Teitel, M., Kim, J. & Browne, L.1992Low-Reynolds-number effects in a fully developed turbulent channel flow. J. Fluid Mech.236, 579-605. |
| [4] | Belcher, S.E. & Hunt, J.C.R.1998Turbulent flow over hills and waves. Annu. Rev. Fluid Mech.30 (1), 507-538. · Zbl 1398.86006 |
| [5] | Breuer, M., Peller, N., Rapp, C. & Manhart, M.2009Flow over periodic hills – numerical and experimental study in a wide range of Reynolds numbers. Comput. Fluids38 (2), 433-457. · Zbl 1237.76026 |
| [6] | Buckley, M.P. & Veron, F.2017Airflow measurements at a wavy air-water interface using PIV and LIF. Exp. Fluids58 (11), 161. |
| [7] | Buckley, M.P. & Veron, F.2019The turbulent airflow over wind generated surface waves. Eur. J. Mech. (B/Fluids)73, 132-143. |
| [8] | Buckley, M.P., Veron, F. & Yousefi, K.2020Surface viscous stress over wind-driven waves with intermittent airflow separation. J. Fluid Mech.905, A31. · Zbl 1460.76087 |
| [9] | Calhoun, R.J. & Street, R.L.2001Turbulent flow over a wavy surface: neutral case. J. Geophys. Res.: Oceans106 (C5), 9277-9293. |
| [10] | Caulliez, G.2013Dissipation regimes for short wind waves. J. Geophys. Res.: Oceans118 (2), 672-684. |
| [11] | Chalikov, D. & Belevich, M.Y.1993One-dimensional theory of the wave boundary layer. Boundary-Layer Meteorol.63 (1-2), 65-96. |
| [12] | Cherukat, P., Na, Y., Hanratty, T. & Mclaughlin, J.1998Direct numerical simulation of a fully developed turbulent flow over a wavy wall. Theor. Comput. Fluid Dyn.11 (2), 109-134. · Zbl 0920.76066 |
| [13] | Cheung, T.K. & Street, R.L.1988The turbulent layer in the water at an air-water interface. J. Fluid Mech.194, 133-151. |
| [14] | Cifuentes-Lorenzen, A., Edson, J.B. & Zappa, C.J.2018Air-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. |
| [15] | Cimarelli, A., Leonforte, A., De Angelis, E., Crivellini, A. & Angeli, D.2019On negative turbulence production phenomena in the shear layer of separating and reattaching flows. Phys. Lett. A383 (10), 1019-1026. · Zbl 1480.76061 |
| [16] | De Angelis, V., Lombardi, P. & Banerjee, S.1997Direct numerical simulation of turbulent flow over a wavy wall. Phys. Fluids9 (8), 2429-2442. |
| [17] | Deike, L., Melville, W.K. & Popinet, S.2016Air entrainment and bubble statistics in breaking waves. J. Fluid Mech.801, 91-129. · Zbl 1462.76040 |
| [18] | Deike, L., Popinet, S. & Melville, W.K.2015Capillary effects on wave breaking. J. Fluid Mech.769, 541-569. · Zbl 1431.76031 |
| [19] | 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. |
| [20] | 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. |
| [21] | Donelan, M.A., Babanin, A.V., Young, I.R. & Banner, M.L.2006Wave-follower field measurements of the wind-input spectral function. Part II: parameterization of the wind input. J. Phys. Oceanogr.36 (8), 1672-1689. |
| [22] | Donelan, M.A., Babanin, A.V., Young, I.R., Banner, M.L. & Mccormick, C.2005Wave-follower field measurements of the wind-input spectral function. Part I: measurements and calibrations. J. Atmos. Ocean. Technol.22 (7), 799-813. |
| [23] | Donelan, M.A., Dobson, F.W., Smith, S.D. & Anderson, R.J.1993On the dependence of sea surface roughness on wave development. J. Phys. Oceanogr.23 (9), 2143-2149. |
| [24] | Donelan, M.A., Drennan, W.M. & Katsaros, K.B.1997The air-sea momentum flux in conditions of wind sea and swell. J. Phys. Oceanogr.27 (10), 2087-2099. |
| [25] | Drazen, D.A., Melville, W.K. & Lenain, L.2008Inertial scaling of dissipation in unsteady breaking waves. J. Fluid Mech.611, 307-332. · Zbl 1151.76387 |
| [26] | Druzhinin, O., Troitskaya, Y., Tsai, W.-T. & Chen, P.-C.2019The study of a turbulent air flow over capillary-gravity water surface waves by direct numerical simulation. Ocean Model.140, 101407. |
| [27] | Druzhinin, O.A., Troitskaya, Y.I. & Zilitinkevich, S.S.2016aStably stratified airflow over a waved water surface. Part 1: stationary turbulence regime. Q. J. R. Meteorol. Soc.142 (695), 759-772. |
| [28] | Druzhinin, O.A., Troitskaya, Y.I. & Zilitinkevich, S.S.2016bStably stratified airflow over a waved water surface. Part 2: wave-induced pre-turbulent motions. Q. J. R. Meteorol. Soc.142 (695), 773-780. |
| [29] | Einaudi, F. & Finnigan, J.1993Wave-turbulence dynamics in the stably stratified boundary layer. J. Atmos. Sci.50 (13), 1841-1864. |
| [30] | El Telbany, M.M.M. & Reynolds, A.J.1982The structure of turbulent plane Couette flow. Trans. ASME J. Fluids Engng104 (3), 367-372. |
| [31] | Fairall, C. & Larsen, S.E.1986Inertial-dissipation methods and turbulent fluxes at the air-ocean interface. Boundary-Layer Meteorol.34 (3), 287-301. |
| [32] | Fedorov, A.V. & Melville, W.K.1998Nonlinear gravity-capillary waves with forcing and dissipation. J. Fluid Mech.354, 1-42. · Zbl 0974.76017 |
| [33] | Fedorov, A.V., Melville, W.K. & Rozenberg, A.1998An experimental and numerical study of parasitic capillary waves. Phys. Fluids10 (6), 1315-1323. |
| [34] | Finnigan, J.J. & Einaudi, F.1981The 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. |
| [35] | Grachev, A.A. & Fairall, C.W.2001Upward momentum transfer in the marine boundary layer. J. Phys. Oceanogr.31 (7), 1698-1711. |
| [36] | Grare, L., Lenain, L. & Melville, W.K.2013Wave-coherent airflow and critical layers over ocean waves. J. Phys. Oceanogr.43 (10), 2156-2172. |
| [37] | Günther, A. & Von Rohr, P.R.2003Large-scale structures in a developed flow over a wavy wall. J. Fluid Mech.478, 257-285. · Zbl 1032.76501 |
| [38] | Hamed, A.M., Kamdar, A., Castillo, L. & Chamorro, L.P.2015Turbulent boundary layer over 2D and 3D large-scale wavy walls. Phys. Fluids27 (10), 106601. |
| [39] | Hara, T. & Belcher, S.E.2004Wind profile and drag coefficient over mature ocean surface wave spectra. J. Phys. Oceanogr.34 (11), 2345-2358. |
| [40] | Hara, T. & Sullivan, P.P.2015Wave boundary layer turbulence over surface waves in a strongly forced condition. J. Phys. Oceanogr.45 (3), 868-883. |
| [41] | Harris, J., Belcher, S. & Street, R.1996Linear dynamics of wind waves in coupled turbulent air-water flow. Part 2. Numerical model. J. Fluid Mech.308, 219-254. · Zbl 0875.76156 |
| [42] | Henn, D.S. & Sykes, R.I.1999Large-eddy simulation of flow over wavy surfaces. J. Fluid Mech.383, 75-112. · Zbl 0931.76035 |
| [43] | Högström, U., Sahlée, E., Smedman, A.-S., Rutgersson, A., Nilsson, E., Kahma, K.K. & Drennan, W.M.2015Surface stress over the ocean in swell-dominated conditions during moderate winds. J. Atmos. Sci.72 (12), 4777-4795. |
| [44] | Högström, U., Smedman, A., Sahleé, E., Drennan, W., Kahma, K., Pettersson, H. & Zhang, F.2009The 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. |
| [45] | Hsu, C.-T. & Hsu, E.Y.1983On 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. |
| [46] | Hsu, C.-T., Hsu, E.Y. & Street, R.L.1981On 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. |
| [47] | Hudson, J.D., Dykhno, L. & Hanratty, T.J.1996Turbulence production in flow over a wavy wall. Exp. Fluids20 (4), 257-265. |
| [48] | Husain, N.T., Hara, T., Buckley, M.P., Yousefi, K., Veron, F. & Sullivan, P.P.2019Boundary layer turbulence over surface waves in a strongly forced condition: LES and observation. J. Phys. Oceanogr.49 (8), 1997-2015. |
| [49] | Hussain, A.F.1983Coherent structures – reality and myth. Phys. Fluids26 (10), 2816-2850. · Zbl 0524.76066 |
| [50] | Hussain, A.K.M.F. & Reynolds, W.C.1970The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech.41, 241-258. |
| [51] | Hussain, A.K.M.F. & Reynolds, W.C.1975Measurements in fully developed turbulent channel flow. Trans. ASME J. Fluids Engng97 (4), 568-578. |
| [52] | Iafrati, A.2011Energy dissipation mechanisms in wave breaking processes: spilling and highly aerated plunging breaking events. J. Geophys. Res.: Oceans116 (C7), C07024. |
| [53] | Iafrati, A., Babanin, A. & Onorato, M.2013Modulational instability, wave breaking, and formation of large-scale dipoles in the atmosphere. Phys. Rev. Lett.110 (18), 184504. · Zbl 1349.86003 |
| [54] | Janssen, P.2004The Interaction of Ocean Waves and Wind. Cambridge University Press. |
| [55] | Janssen, P.A.1999On the effect of ocean waves on the kinetic energy balance and consequences for the inertial dissipation technique. J. Phys. Oceanogr.29 (3), 530-534. |
| [56] | Kihara, N., Hanazaki, H., Mizuya, T. & Ueda, H.2007Relationship between airflow at the critical height and momentum transfer to the traveling waves. Phys. Fluids19 (1), 015102. · Zbl 1146.76440 |
| [57] | 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. |
| [58] | Kim, J., Moin, P. & Moser, R.1987Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech.177, 133-166. · Zbl 0616.76071 |
| [59] | 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. |
| [60] | Kitoh, O., Nakabyashi, K. & Nishimura, F.2005Experimental 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. · Zbl 1137.76305 |
| [61] | Komen, G.J., Cavaleri, L., Donelan, M., Hasselmann, K., Hasselmann, S. & Janssen, P.1996Dynamics and Modelling of Ocean Waves. Cambridge University Press. · Zbl 0816.76001 |
| [62] | Kreplin, H.-P. & Eckelmann, H.1979Behavior of the three fluctuating velocity components in the wall region of a turbulent channel flow. Phys. Fluids22 (7), 1233-1239. |
| [63] | Krogstad, P.-Å. & Antonia, R.1999Surface roughness effects in turbulent boundary layers. Exp. Fluids27 (5), 450-460. |
| [64] | Kruse, N., Günther, A. & Von Rohr, P.R.2003Dynamics of large-scale structures in turbulent flow over a wavy wall. J. Fluid Mech.485, 87-96. · Zbl 1103.76025 |
| [65] | Kruse, N., Kuhn, S. & Von Rohr, P.R.2006Wavy wall effects on turbulence production and large-scale modes. J. Turbul.7, N31. |
| [66] | Lamb, H.1932Hydrodynamics, 6th edn. Cambridge University Press. · JFM 58.1298.04 |
| [67] | Large, W. & Pond, S.1981Open ocean momentum flux measurements in moderate to strong winds. J. Phys. Oceanogr.11 (3), 324-336. |
| [68] | Liu, J. & Merkine, L.1976On 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. |
| [69] | Liu, S., Kermani, A., Shen, L. & Yue, D.K.2009Investigation of coupled air-water turbulent boundary layers using direct numerical simulations. Phys. Fluids21 (6), 062108. · Zbl 1183.76321 |
| [70] | Longuet-Higgins, M.S.1969Action of a variable stress at the surface of water waves. Phys. Fluids12 (4), 737-740. |
| [71] | Makin, V.K. & Kudryavtsev, V.N.1999Coupled sea surface-atmosphere model: 1. Wind over waves coupling. J. Geophys. Res.: Oceans104 (C4), 7613-7623. |
| [72] | Makin, V.K. & Mastenbroek, C.1996Impact of waves on air-sea exchange of sensible heat and momentum. Boundary-Layer Meteorol.79 (3), 279-300. |
| [73] | Melville, W.1983Wave modulation and breakdown. J. Fluid Mech.128, 489-506. |
| [74] | Melville, W.K. & Fedorov, A.V.2015The equilibrium dynamics and statistics of gravity-capillary waves. J. Fluid Mech.767, 449-466. |
| [75] | Moon, I.J., Hara, T., Ginis, I., Belcher, S.E. & Tolman, H.L.2004Effect of surface waves on air-sea momentum exchange. Part I: effect of mature and growing seas. J. Atmos. Sci.61 (19), 2321-2333. |
| [76] | Pahlow, M., Parlange, M.B. & Porté-Agel, F.2001On Monin-Obukhov similarity in the stable atmospheric boundary layer. Boundary-Layer Meteorol.99 (2), 225-248. |
| [77] | Palmer, J.A., Mejia-Alvarez, R., Best, J.L. & Christensen, K.T.2012Particle-image velocimetry measurements of flow over interacting barchan dunes. Exp. Fluids52 (3), 809-829. |
| [78] | Panofsky, H.A. & Dutton, J.A.1984Atmospheric Turbulence: Models and Methods for Engineering Applications, 1st edn. Wiley. |
| [79] | Papadimitrakis, Y.A., Hsu, E.Y. & Street, R.L.1986The role of wave-induced pressure fluctuations in the transfer processes across an air-water interface. J. Fluid Mech.170, 113-137. |
| [80] | Pope, S.B.2000Turbulent Flows. Cambridge University Press. · Zbl 0966.76002 |
| [81] | Reynolds, W.C. & Hussain, A.K.M.F.1972The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech.54, 263-288. |
| [82] | Rutgersson, A. & Sullivan, P.P.2005The effect of idealized water waves on the turbulence structure and kinetic energy budgets in the overlying airflow. Dyn. Atmos. Oceans38 (3-4), 147-171. |
| [83] | Sauer, J.A., Muñoz-Esparza, D., Canfield, J.M., Costigan, K.R., Linn, R.R. & Kim, Y.-J.2016A large-eddy simulation study of atmospheric boundary layer influence on stratified flows over terrain. J. Atmos. Sci.73 (7), 2615-2632. |
| [84] | Shaikh, N. & Siddiqui, K.2010An experimental investigation of the near surface flow over air-water and air-solid interfaces. Phys. Fluids22 (2), 025103. · Zbl 1183.76468 |
| [85] | Shaikh, N. & Siddiqui, K.2011Near-surface flow structure over wind-generated water waves, Part II: characteristics of separated and non-separated flows. Ocean Dyn.61 (1), 143-154. |
| [86] | Shen, L., Zhang, X., Yue, D.K.P. & Triantafyllou, M.S.2003Turbulent flow over a flexible wall undergoing a streamwise travelling wave motion. J. Fluid Mech.484, 197-221. · Zbl 1058.76032 |
| [87] | Siddiqui, K. & Loewen, M.R.2010Phase-averaged flow properties beneath microscale breaking waves. Boundary-Layer Meteorol.134 (3), 499-523. |
| [88] | Sjöblom, A. & Smedman, A.-S.2002The turbulent kinetic energy budget in the marine atmospheric surface layer. J. Geophys. Res.: Oceans107 (C10), 6-18. |
| [89] | Smedman, A., Högström, U., Bergström, H., Rutgersson, A., Kahma, K. & Pettersson, H.1999A case study of air-sea interaction during swell conditions. J. Geophys. Res.: Oceans104 (C11), 25833-25851. |
| [90] | Smedman, A., Högström, U., Sahlée, E., Drennan, W., Kahma, K., Pettersson, H. & Zhang, F.2009Observational study of marine atmospheric boundary layer characteristics during swell. J. Atmos. Sci.66 (9), 2747-2763. |
| [91] | Smedman, A., Tjernström, M. & Högström, U.1994The near-neutral marine atmospheric boundary layer with no surface shearing stress: a case study. J. Atmos. Sci.51 (23), 3399-3411. |
| [92] | Smedman, A.-S.1988Observations of a multi-level turbulence structure in a very stable atmospheric boundary layer. Boundary-Layer Meteorol.44 (3), 231-253. |
| [93] | Spalart, P.R.1988Direct simulation of a turbulent boundary layer up to \(R_{\theta }= 1410\). J. Fluid Mech.187, 61-98. · Zbl 0641.76050 |
| [94] | Sullivan, P.P., Banner, M.L., Morison, R.P. & Peirson, W.L.2018Turbulent flow over steep steady and unsteady waves under strong wind forcing. J. Phys. Oceanogr.48 (1), 3-27. |
| [95] | Sullivan, P.P., Edson, J.B., Hristov, T. & Mcwilliams, J.C.2008Large-eddy simulations and observations of atmospheric marine boundary layers above nonequilibrium surface waves. J. Atmos. Sci.65 (4), 1225-1245. |
| [96] | Sullivan, P.P. & Mcwilliams, J.C.2002Turbulent flow over water waves in the presence of stratification. Phys. Fluids14 (3), 1182-1195. · Zbl 1185.76729 |
| [97] | Sullivan, P.P. & Mcwilliams, J.C.2010Dynamics of winds and currents coupled to surface waves. Annu. Rev. Fluid Mech.42 (1), 19-42. · Zbl 1345.76032 |
| [98] | Sullivan, P.P., Mcwilliams, J.C. & Patton, E.G.2014Large-eddy simulation of marine atmospheric boundary layers above a spectrum of moving waves. J. Atmos. Sci.71 (11), 4001-4027. |
| [99] | Sun, Z., Zhu, Y., Hu, Y. & Zhang, S.2018Direct numerical simulation of a fully developed compressible wall turbulence over a wavy wall. J. Turbul.19 (1), 72-105. |
| [100] | Thais, L. & Magnaudet, J.1996Turbulent structure beneath surface gravity waves sheared by the wind. J. Fluid Mech.328, 313-344. |
| [101] | Tsai, W.-T., Chen, S.-M. & Lu, G.-H.2015Numerical evidence of turbulence generated by nonbreaking surface waves. J. Phys. Oceanogr.45 (1), 174-180. |
| [102] | Tsai, W.-T. & Hung, L.-P.2010Enhanced energy dissipation by parasitic capillaries on short gravity-capillary waves. J. Phys. Oceanogr.40 (11), 2435-2450. |
| [103] | Vollestad, P., Ayati, A.A. & Jensen, A.2019Experimental investigation of intermittent airflow separation and microscale wave breaking in wavy two-phase pipe flow. J. Fluid Mech.878, 796-819. |
| [104] | Wei, T. & Willmarth, W.1989Reynolds-number effects on the structure of a turbulent channel flow. J. Fluid Mech.204, 57-95. |
| [105] | Yang, D. & Shen, L.2009Characteristics of coherent vortical structures in turbulent flows over progressive surface waves. Phys. Fluids21 (12), 125106. · Zbl 1183.76579 |
| [106] | Yang, D. & Shen, L.2010Direct-simulation-based study of turbulent flow over various waving boundaries. J. Fluid Mech.650, 131-180. · Zbl 1189.76267 |
| [107] | Yang, D. & Shen, L.2017Direct numerical simulation of scalar transport in turbulent flows over progressive surface waves. J. Fluid Mech.819, 58-103. · Zbl 1383.76253 |
| [108] | Yousefi, K.2020 Turbulence in the atmospheric wave boundary layer. Doctoral dissertation, University of Delaware, Newark, Delaware, USA. |
| [109] | Yousefi, K. & Veron, F.2020Boundary layer formulations in orthogonal curvilinear coordinates for flow over wind-generated surface waves. J. Fluid Mech.888, A11. · Zbl 1460.76113 |
| [110] | Yousefi, K., Veron, F. & Buckley, M.P.2020aMomentum flux measurements in the airflow over wind-generated surface waves. J. Fluid Mech.895, A15. · Zbl 1460.76114 |
| [111] | Yousefi, K., Veron, F. & Buckley, M.P.2020b 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. |
| [112] | Zhang, X.1995Capillary-gravity and capillary waves generated in a wind wave tank: observations and theories. J. Fluid Mech.289, 51-82. |
| [113] | Zhang, X.2002Enhanced dissipation of short gravity and gravity capillary waves due to parasitic capillaries. Phys. Fluids14 (11), L81-L84. · Zbl 1185.76420 |
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