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The breaking of interfacial waves at a submerged bathymetric ridge

Published online by Cambridge University Press:  17 September 2009

ERIN L. HULT*
Affiliation:
Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305, USA
CARY D. TROY
Affiliation:
Department of Civil and Environmental Engineering, Purdue University, West Lafayette, IN 47907, USA
JEFFREY R. KOSEFF
Affiliation:
Department of Civil and Environmental Engineering, Stanford University, Stanford, CA 94305, USA
*
Email address for correspondence: ehult@stanford.edu

Abstract

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.

Type
Papers
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Ai, J., Law, A. W. K. & Yu, S. C. M. 2006 On Boussinesq and non-Boussinesq starting forced plumes. J. Fluid Mech. 558, 357386.CrossRefGoogle Scholar
Boegman, L. & Ivey, G. N. 2009 Flow separation and resuspension beneath shoaling nonlinear internal waves. J. Geophys. Res. 114, C02018.CrossRefGoogle Scholar
Boegman, L., Ivey, G. N. & Imberger, J. 2005 The degeneration of internal waves in lakes with sloping topography. Limnol. Oceanogr. 50 (5), 16201637.CrossRefGoogle Scholar
Bogucki, D., Dickey, T. & Redekopp, L. G. 1997 Sediment resuspension and mixing by resonantly generated internal solitary waves. J. Phys. Oceanogr. 27 (7), 11811196.2.0.CO;2>CrossRefGoogle Scholar
Cacchione, D. & Wunsch, C. 1974 Experimental study of internal waves over a slope. J. Fluid Mech. 66, 223239.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Crimaldi, J. P. 2008 Planar laser induced fluorescence in aqueous flows. Exp. Fluids 44 (6), 851863.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Daviero, G. J., Roberts, P. J. W. & Maile, K. 2001 Refractive index matching in large-scale stratified experiments. Exp. Fluids 31 (2), 119126.CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Emery, K. O. & Gunnerson, C. G. 1973 Internal swash and surf. Proc. Natl Acad. Sci. USA 70 (8), 23792380.CrossRefGoogle ScholarPubMed
Fringer, O. B. & Street, R. L. 2003 The dynamics of breaking progressive interfacial waves. J. Fluid Mech. 494, 319353.CrossRefGoogle Scholar
Fructus, D., Carr, M., Grue, J., Jensen, A. & Davies, P. A. 2009 Shear induced breaking of large internal waves. J. Fluid Mech. 620, 129.CrossRefGoogle Scholar
Grue, J., Jensen, A., Rusas, P. & Sveen, J. K. 2000 Breaking and broadening of internal solitary waves. J. Fluid Mech. 413, 181.CrossRefGoogle Scholar
Hazel, P. 1972 Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51, 3961.CrossRefGoogle Scholar
Helfrich, K. R. 1992 Internal solitary wave breaking and run-up on a uniform slope. J. Fluid Mech. 243, 133154.CrossRefGoogle Scholar
Helfrich, K. R. & Melville, W. K. 1986 On long nonlinear internal waves over slope-shelf topography. J. Fluid Mech. 167, 285308.CrossRefGoogle Scholar
Helfrich, K. R. & Melville, W. K. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395425.CrossRefGoogle Scholar
Holyer, J. Y. 1979 Large amplitude progressive interfacial waves. J. Fluid Mech. 93, 433448.CrossRefGoogle Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509512.CrossRefGoogle Scholar
Hult, E. L., Troy, C. D. & Koseff, J. R. 2006 Laboratory images of breaking internal waves. Phys. Fluids 18, 091107.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Kunze, E. & Sanford, T. B. 1996 Abyssal mixing: where it is not. J. Phys. Oceanogr. 26 (10), 22862296.2.0.CO;2>CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Linden, P. F. & Redondo, J. M. 1991 Molecular mixing in Rayleigh–Taylor instability. Part I. Global mixing. Phys. Fluids 3 (5), 12691277.CrossRefGoogle Scholar
Lueck, R. G. & Mudge, T. D. 1997 Topographically induced mixing around a shallow seamount. Science 276, 18311833.CrossRefGoogle Scholar
Michallet, H. & Ivey, G. N. 1999 Experiments on mixing due to internal solitary waves breaking on uniform slopes. J. Geophys. Res. 104 (C6), 1346713478.CrossRefGoogle Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496508.CrossRefGoogle Scholar
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), 20932112.2.0.CO;2>CrossRefGoogle Scholar
Nagashima, H. 1971 Reflection and breaking of internal waves on a sloping beach. J. Oceanogr. 27 (1), 16.Google Scholar
Orlanski, I. & Bryan, K. 1969 Formation of the thermocline step structure by large-amplitude internal gravity waves. J. Geophys. Res. 74 (28), 69756983.CrossRefGoogle Scholar
Pawlak, G. & Armi, L. 2000 Vortex dynamics in a spatially accelerating shear layer. J. Fluid Mech. 376, 135.CrossRefGoogle Scholar
Pedlosky, J. & Thomson, J. 2003 Baroclinic instability of time-dependent currents. J. Fluid Mech. 490, 189215.CrossRefGoogle Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Ann. Rev. Fluid Mech. 35 (1), 135167.CrossRefGoogle Scholar
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
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.CrossRefGoogle ScholarPubMed
Rehmann, C. R. 1995 Effects of stratification and molecular diffusivity on the mixing efficiency of decaying grid turbulence. PhD thesis, Stanford University, Stanford.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Taylor, J. R. 1993 Turbulence and mixing in the boundary layer generated by shoaling internal waves. Dyn. Atmos. Oceans 19, 233258.CrossRefGoogle Scholar
Thorpe, S. A. 1978 On the shape and breaking of finite amplitude internal gravity waves in a shear flow. J. Fluid Mech. 85, 731.CrossRefGoogle Scholar
Thorpe, S. A. 1987 On the reflection of a train of finite-amplitude internal waves from a uniform slope. J. Fluid Mech. 178, 279302.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Troy, C. D. & Koseff, J. R. 2005 a The generation and quantitative visualization of breaking internal waves. Exp. Fluids 38 (5), 549562.CrossRefGoogle Scholar
Troy, C. D. & Koseff, J. R. 2005 b The instability and breaking of long internal waves. J. Fluid Mech. 543, 107136.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Venayagamoorthy, S. K. & Fringer, O. B. 2006 Numerical simulations of the interaction of internal waves with a shelf break. Phys. Fluids 18 (7), 7660376603.CrossRefGoogle Scholar
Vlasenko, V. & Hutter, K. 2002 Numerical experiments on the breaking of solitary internal waves over a slope-shelf topography. J. Phys. Oceanogr. 32 (6), 17791793.2.0.CO;2>CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Wessels, F. & Hutter, K. 1996 Interaction of internal waves with a topographic sill in a two-layered fluid. J. Phys. Oceanogr. 26 (1), 520.2.0.CO;2>CrossRefGoogle Scholar
Westerweel, J. & Scarano, F. 2005 Universal outlier detection for PIV data. Exp. Fluids 39 (6), 10961100.CrossRefGoogle Scholar
Zhu, D. Z. & Lawrence, G. A. 2001 Holmboe's instability in exchange flows. J. Fluid Mech. 429, 391409.CrossRefGoogle Scholar
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