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The breaking of interfacial waves at a submerged bathymetric ridge

  • ERIN L. HULT (a1), CARY D. TROY (a2) and JEFFREY R. KOSEFF (a1)

The breaking of periodic progressive two-layer interfacial waves at a Gaussian ridge is investigated through laboratory experiments. Length scales of the incident wave and topography are used to parameterize when and how breaking occurs. Qualitative observations suggest both shear and convection play a role in the instability of waves breaking at the ridge. Simultaneous particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) measurements are used to calculate high resolution, two-dimensional velocity and density fields from which the local gradient Richardson number Rig is calculated. The transition to breaking occurred when 0.2 ≤ Rig ≤ 0.4. In these wave-ridge breaking events, the destabilizing effects of waves steepening in shallow layers may be responsible for breaking at higher Rig than for similar waves breaking through shear instability in deep water (Troy & Koseff, J. Fluid Mech., vol. 543, 2005b, p. 107). Due to the effects of unsteadiness, nonlinear shoaling and flow separation, the canonical Rig > 0.25 is not sufficient to predict the stability of interfacial waves. A simple model is developed to estimate Rig in waves between finite depth layers using scales of the incident wave scale and topography. The observed breaking transition corresponds with a constant estimated value of Rig from the model, suggesting that interfacial shear plays an important role in initial wave instability. For wave amplitudes above the initial breaking transition, convective breaking events are also observed.

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Ai J., Law A. W. K. & Yu S. C. M. 2006 On Boussinesq and non-Boussinesq starting forced plumes. J. Fluid Mech. 558, 357386.
Boegman L. & Ivey G. N. 2009 Flow separation and resuspension beneath shoaling nonlinear internal waves. J. Geophys. Res. 114, C02018.
Boegman L., Ivey G. N. & Imberger J. 2005 The degeneration of internal waves in lakes with sloping topography. Limnol. Oceanogr. 50 (5), 16201637.
Bogucki D., Dickey T. & Redekopp L. G. 1997 Sediment resuspension and mixing by resonantly generated internal solitary waves. J. Phys. Oceanogr. 27 (7), 11811196.
Cacchione D. & Wunsch C. 1974 Experimental study of internal waves over a slope. J. Fluid Mech. 66, 223239.
Carr M. & Davies P. A. 2006 The motion of an internal solitary wave of depression over a fixed bottom boundary in a shallow, two-layer fluid. Phys. Fluids 18, 016601.
Carr M., Davies P. A. & Shivaram P. 2008 Experimental evidence of internal solitary wave-induced global instability in shallow water benthic boundary layers. Phys. Fluids 20 (6), 066603066603.
Chen C. Y., Hsu J. R. C., Cheng M. H. & Chen C. W. 2008 Experiments on mixing and dissipation in internal solitary waves over two triangular obstacles. Environ. Fluid Mech. 8 (3), 199214.
Crimaldi J. P. 2008 Planar laser induced fluorescence in aqueous flows. Exp. Fluids 44 (6), 851863.
Crimaldi J. P. & Koseff J. R. 2001 High-resolution measurements of the spatial and temporal scalar structure of a turbulent plume. Exp. Fluids 31 (1), 90102.
Dalziel S. B., Carr M., Sveen J. K. & Davies P. A. 2007 Simultaneous synthetic schlieren and PIV measurements for internal solitary waves. Meas. Sci. Technol. 18 (3), 533547.
Daviero G. J., Roberts P. J. W. & Maile K. 2001 Refractive index matching in large-scale stratified experiments. Exp. Fluids 31 (2), 119126.
Dean R. G. & Dalrymple R. A. 1984 Water wave mechanics for engineers and scientists. In River Edge. World Scientific. Advanced Series on Ocean Engineering, Vol. 2, 335.
De Silva I. P. D., Fernando H. J. S., Eaton F. & Hebert D. 1996 Evolution of Kelvin–Helmholtz billows in nature and laboratory. Earth Planet. Sci. Lett. 143 (1), 217231.
Diamessis P. J. & Redekopp L. G. 2006 Numerical investigation of solitary internal wave-induced global instability in shallow water benthic boundary layers. J. Phys. Oceanogr. 36 (5), 784812.
Diez F. J., Bernal L. P. & Faeth G. M. 2005 PLIF and PIV measurements of the self-preserving structure of steady round buoyant turbulent plumes in crossflow. Intl J. Heat Fluid Flow 26 (6), 873882.
Emery K. O. & Gunnerson C. G. 1973 Internal swash and surf. Proc. Natl Acad. Sci. USA 70 (8), 23792380.
Fringer O. B. & Street R. L. 2003 The dynamics of breaking progressive interfacial waves. J. Fluid Mech. 494, 319353.
Fructus D., Carr M., Grue J., Jensen A. & Davies P. A. 2009 Shear induced breaking of large internal waves. J. Fluid Mech. 620, 129.
Grue J., Jensen A., Rusas P. & Sveen J. K. 2000 Breaking and broadening of internal solitary waves. J. Fluid Mech. 413, 181.
Hazel P. 1972 Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51, 3961.
Helfrich K. R. 1992 Internal solitary wave breaking and run-up on a uniform slope. J. Fluid Mech. 243, 133154.
Helfrich K. R. & Melville W. K. 1986 On long nonlinear internal waves over slope-shelf topography. J. Fluid Mech. 167, 285308.
Helfrich K. R. & Melville W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395425.
Holyer J. Y. 1979 Large amplitude progressive interfacial waves. J. Fluid Mech. 93, 433448.
Howard L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509512.
Hult E. L., Troy C. D. & Koseff J. R. 2006 Laboratory images of breaking internal waves. Phys. Fluids 18, 091107.
Ivey G. N. & Nokes R. I. 1989 Vertical mixing due to the breaking of critical internal waves on sloping boundaries. J. Fluid Mech. 204, 479500.
Ivey G. N., Winters K. B. & De Silva I. P. D. 2000 Turbulent mixing in a sloping benthic boundary layer energized by internal waves. J. Fluid Mech. 418, 5976.
Kao T. W., Pan F. S. & Renouard D. 1985 Internal solitons on the pycnocline: generation, propagation, and shoaling and breaking over a slope. J. Fluid Mech. 159, 1953.
Kunze E. & Sanford T. B. 1996 Abyssal mixing: where it is not. J. Phys. Oceanogr. 26 (10), 22862296.
Law A. W. K., Wang H. & Herlina 2003 Combined particle image velocimetry/planar laser induced fluorescence for integral modelling of buoyant jets. J. Engng Mech. 129 (10), 11891196.
Ledwell J. R., Montgomery E. T., Polzin K. L., St. Laurent L. C., Schmitt R. W. & Toole J. M. 2000 Evidence for enhanced mixing over rough topography in the abyssal ocean. Nature 403 (6766), 179182.
Linden P. F. & Redondo J. M. 1991 Molecular mixing in Rayleigh–Taylor instability. Part I. Global mixing. Phys. Fluids 3 (5), 12691277.
Lueck R. G. & Mudge T. D. 1997 Topographically induced mixing around a shallow seamount. Science 276, 18311833.
Michallet H. & Ivey G. N. 1999 Experiments on mixing due to internal solitary waves breaking on uniform slopes. J. Geophys. Res. 104 (C6), 1346713478.
Miles J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496508.
Moum J. N., Farmer D. M., Smyth W. D., Armi L. & Vagle S. 2003 Structure and generation of turbulence at interfaces strained by internal solitary waves propagating shoreward over the continental shelf. J. Phys. Oceanogr. 33 (10), 20932112.
Nagashima H. 1971 Reflection and breaking of internal waves on a sloping beach. J. Oceanogr. 27 (1), 16.
Orlanski I. & Bryan K. 1969 Formation of the thermocline step structure by large-amplitude internal gravity waves. J. Geophys. Res. 74 (28), 69756983.
Pawlak G. & Armi L. 2000 Vortex dynamics in a spatially accelerating shear layer. J. Fluid Mech. 376, 135.
Pedlosky J. & Thomson J. 2003 Baroclinic instability of time-dependent currents. J. Fluid Mech. 490, 189215.
Peltier W. R. & Caulfield C. P. 2003 Mixing efficiency in stratified shear flows. Ann. Rev. Fluid Mech. 35 (1), 135167.
Phillips O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.
Polzin K. L., Toole J. M., Ledwell J. R. & Schmitt R. W. 1997 Spatial variability of turbulent mixing in the abyssal ocean. Science 276 (5309), 9396.
Rehmann C. R. 1995 Effects of stratification and molecular diffusivity on the mixing efficiency of decaying grid turbulence. PhD thesis, Stanford University, Stanford.
Shavit U., Lowe R. L. & Steinbuck J. V. 2007 Intensity capping: a simple method to improve cross-correlation PIV results. Exp. Fluids 42 (2), 225240.
Sveen J. K., Guo Y., Davies P. A. & Grue J. 2002 On the breaking of internal solitary waves at a ridge. J. Fluid Mech. 469, 161188.
Taylor J. R. 1993 Turbulence and mixing in the boundary layer generated by shoaling internal waves. Dyn. Atmos. Oceans 19, 233258.
Thorpe S. A. 1978 On the shape and breaking of finite amplitude internal gravity waves in a shear flow. J. Fluid Mech. 85, 731.
Thorpe S. A. 1987 On the reflection of a train of finite-amplitude internal waves from a uniform slope. J. Fluid Mech. 178, 279302.
Thorpe S. A. 1998 Some dynamical effects of internal waves and the sloping sides of lakes. Phys. Processes Lakes Oceans, Coast. Estuar. Stud. 54, 441460.
Toole J. M., Schmitt R. W., Polzin K. L. & Kunze E. 1997 Near-boundary mixing above the flanks of a midlatitude seamount. J. Geophys. Res. 102, 947959.
Troy C. D. & Koseff J. R. 2005 a The generation and quantitative visualization of breaking internal waves. Exp. Fluids 38 (5), 549562.
Troy C. D. & Koseff J. R. 2005 b The instability and breaking of long internal waves. J. Fluid Mech. 543, 107136.
Variano E. A. & Cowen E. A. 2007 Quantitative imaging of CO2 transfer at an unsheared free surface. In Transport at the Air-Sea Interface (ed. Garbe C. S., Handler R. A. & Jähne B., pp. 4357. Springer.
Venayagamoorthy S. K. & Fringer O. B. 2006 Numerical simulations of the interaction of internal waves with a shelf break. Phys. Fluids 18 (7), 7660376603.
Vlasenko V. & Hutter K. 2002 Numerical experiments on the breaking of solitary internal waves over a slope-shelf topography. J. Phys. Oceanogr. 32 (6), 17791793.
Wallace B. C. & Wilkinson D. L. 1988 Run-up of internal waves on a gentle slope in a two-layered system. J. Fluid Mech. 191, 419442.
Wessels F. & Hutter K. 1996 Interaction of internal waves with a topographic sill in a two-layered fluid. J. Phys. Oceanogr. 26 (1), 520.
Westerweel J. & Scarano F. 2005 Universal outlier detection for PIV data. Exp. Fluids 39 (6), 10961100.
Zhu D. Z. & Lawrence G. A. 2001 Holmboe's instability in exchange flows. J. Fluid Mech. 429, 391409.
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Journal of Fluid Mechanics
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