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Simulations of shear instabilities in interfacial gravity waves

Published online by Cambridge University Press:  11 February 2010

MICHAEL F. BARAD*
Affiliation:
Environmental Fluid Mechanics Laboratory, Stanford University, Stanford, CA 94305-4020, USA
OLIVER B. FRINGER
Affiliation:
Environmental Fluid Mechanics Laboratory, Stanford University, Stanford, CA 94305-4020, USA
*
Email address for correspondence: barad@stanford.edu

Abstract

An adaptive numerical method is employed to simulate shear instabilities in open-ocean internal solitary-like gravity waves. The method is second-order accurate, employs block-structured adaptive mesh refinement, solves the incompressible Navier–Stokes equations and allows for the simulation of all of the length scales of interest by dynamically tracking important regions with recursively-nested finer grids. Two-dimensional simulations are performed over a range of parameters, which allows us to assess the conditions under which the shear instabilities in the waves occur, including a method to evaluate the critical Richardson number for instability based on the bulk wave parameters. The results show that although the minimum Richardson number is well below the canonical value of 1/4 in all simulations, this value is not a sufficient condition for instability, but instead a much lower Richardson number of 0.1 is required. When the Richardson number falls below 0.1, shear instabilities develop and grow into two-dimensional billows of the Kelvin–Helmholtz type. A linear stability analysis with the Taylor–Goldstein equation indicates that an alternate criterion for instability is given by σiTw > 5, where σi is the growth rate of the instability averaged over Tw, the period in which parcels of fluid are subjected to a Richardson number of less than 1/4. A third criterion for instability requires that Lw/L > 0.86, where Lw is half the length of the region in which the Richardson number falls below 1/4 and L is the solitary wave half-width. An eigendecomposition of the rate-of-strain tensor is performed to show that the pycnocline thickness increases within the wave because of pycnocline-normal strain and not because of diffusion, which has important ramifications for stability. A three-dimensional simulation indicates that the primary instability is two-dimensional and that secondary, three-dimensional instabilities occur thereafter and lead to strong dissipation and mixing.

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Papers
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Almgren, A. S., Bell, J. B., Colella, P., Howell, L. H. & Welcome, M. L. 1998 A conservative adaptive projection method for the variable density incompressible Navier–Stokes equations. J. Comput. Phys. 142, 146.CrossRefGoogle Scholar
Armi, L. & Farmer, D. M. 1988 The flow of mediterranean water through the Strait of Gibraltar. Prog. Oceanogr. 21 (1), 1.Google Scholar
Barad, M. F. & Colella, P. 2005 A fourth-order accurate adaptive mesh refinement method for Poisson's equation. J. Comput. Phys. 209 (1), 118.CrossRefGoogle Scholar
Barad, M. F., Colella, P. & Schladow, S. G. 2009 An adaptive cut-cell method for environmental fluid mechanics. Intl J. Numer. Meth. Fluids 60 (5), 473514.CrossRefGoogle Scholar
Barad, M. F. & Fringer, O. B. 2007 Numerical simulations of shear instabilities in open-ocean internal gravity waves. In Proceedings of the Fifth International Symposium on Environmental Hydraulics, Perth, 722–72.Google Scholar
Bell, J., Berger, M., Saltzman, J. & Welcome, M. 1994 Three-dimensional adaptive mesh refinement for hyperbolic conservation laws. SIAM J. Sci. Comput. 15 (1), 127138.CrossRefGoogle Scholar
Bell, J. B., Colella, P. & Glaz, H. M. 1989 A second-order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 85, 257283.CrossRefGoogle Scholar
Bell, J. B., Day, M. S., Shepherd, I. G., Johnson, M. R., Cheng, R. K., Grcar, J. F., Beckner, V. E. & Lijewski, M. J. 2005 From the cover: numerical simulation of a laboratory-scale turbulent V-flame. Proc. Natl Acad. Sci. 102 (29), 1000610011.CrossRefGoogle Scholar
Berger, M. J. & Colella, P. 1989 Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82 (1), 6484.CrossRefGoogle Scholar
Berger, M. J. & Leveque, R. J. 1989 An adaptive Cartesian mesh algorithm for the euler equations in arbitrary geometries. AIAA Paper 89-1930CP. AIAA, pp. 1–7.Google Scholar
Berger, M. J. & Oliger, J. E. 1983 Adaptive mesh refinement for hyperbolic partial differential equations. Tech Rep. NA-M-83-02 Stanford University.Google Scholar
Bogucki, D. & Garrett, C. 1993 A simple model for the shear induced decay of an internal solitary wave. J. Phys. Oceanogr. 23 (8), 17671776.2.0.CO;2>CrossRefGoogle Scholar
Cacchione, D. & Wunsch, C. 1974 Experimental study of internal waves over a slope. J. Fluid Mech. 66, 223239.CrossRefGoogle Scholar
Carter, G. S., Gregg, M. C. & Lien, R. C. 2005 Internal waves, solitary-like waves, and mixing on the Monterey Bay shelf. Cont. Shelf Res. 25 (12–13), 14991520.CrossRefGoogle Scholar
Caulfield, C. P. & Peltier, W. R. 2000 The anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech. 413, 147.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, 784812.CrossRefGoogle Scholar
Egbert, G. D. & Ray, R. D. 2000 Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data. Nature 405, 775778.CrossRefGoogle ScholarPubMed
Fjortoft, R. 1950 Application of integral theorems in deriving criteria of stability of laminar flow and for the baroclinic circular vortex. Geofys. Publ. 17, 152.Google Scholar
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 solitary waves. J. Fluid Mech. 620, 129.CrossRefGoogle Scholar
Fructus, D. & Grue, J. 2004 Fully nonlinear solitary waves in a layered stratified fluid. J. Fluid Mech. 505, 323347.CrossRefGoogle Scholar
Gossard, E. E. 1990 Radar in Meteorology (ed. Atlas, D.), pp. 477527. American Meteorological Society.CrossRefGoogle Scholar
Grue, J., Jensen, A., Rusas, P. O. & Sveen, J. K. 2000 Breaking and broadening of internal solitary waves. J. Fluid Mech. 413, 181217.CrossRefGoogle Scholar
Haigh, S. P. 1995 Non-symmetric Holmboe waves. PhD thesis, University of British Columbia, Vancouver, BC, Canada.Google Scholar
Hazel, 1972 Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51, 3961.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
Hogg, A. M. & Ivey, G. N. 2003 The Kelvin–Helmholtz to Holmboe instability transition in stratified exchange flows. J. Fluid Mech. 477, 339362.CrossRefGoogle Scholar
Hosegood, P., Bonnin, J. & van Haren, H. 2004 Solibore-induced sediment resuspension in the Faeroe–Shetland channel. Geophys. Res. Lett. 31 (9), L09301.CrossRefGoogle Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509512.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. & Silva, I. P. D. De 2000 Turbulent mixing in a sloping benthic boundary layer energized by internal waves. J. Fluid Mech. 418, 5976.CrossRefGoogle Scholar
Javam, A., Imberger, J. & Armfield, S. W. 1999 Numerical study of internal wave reflection from sloping boundaries. J. Fluid Mech. 396, 183201.CrossRefGoogle Scholar
Klymak, J. M. & Moum, J. N. 2003 Internal solitary waves of elevation advancing on a shoaling shelf. Geophys. Res. Lett. 30 (20), 2045.CrossRefGoogle Scholar
Kundu, P. K. 2002 Fluid Mechanics, 2nd edn. Academic.Google Scholar
Lamb, K. G. 2002 A numerical investigation of solitary internal waves with trapped cores formed via shoaling. J. Fluid Mech. 451, 109144.CrossRefGoogle Scholar
Lamb, K. G. 2003 Shoaling solitary internal waves: on a criterion for the formation of waves with trapped cores. J. Fluid Mech. 478, 81100.CrossRefGoogle Scholar
Legg, S. & Adcroft, A. 2003 Internal wave breaking at concave and convex continental slopes. J. Phys. Oceanogr. 33, 22242246.2.0.CO;2>CrossRefGoogle Scholar
Long, R. R. 1956 Solitary waves in one- and two-fluid systems. Tellus 8, 460471.CrossRefGoogle Scholar
Marmorino, G. O. 1990 ‘Turbulent mixing’ in a salt finger staircase. J. Geophys. Res. 95, 1298312994.CrossRefGoogle Scholar
Maslowe, S. A. & Redekopp, L. G. 1980 Long nonlinear waves in stratified shear flows. J. Fluid Mech. 101, 321348.CrossRefGoogle Scholar
Michallet, H. & Ivey, G. N. 1999 Experiments on mixing due to internal solitary waves breaking on uniform slopes. J. Geophys. Res. 104, 1346713477.CrossRefGoogle Scholar
Miles, J. W. 1963 On the stability of heterogeneous shear flows. Part 2. J. Fluid Mech. 16, 209227.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, 20932112.2.0.CO;2>CrossRefGoogle Scholar
Munk, W. & Wunsch, C. 1998 Abyssal recipes. Part II. Energetics of tidal and wind mixing. Deep-Sea Res. 45, 19772010.CrossRefGoogle Scholar
Pereira, N. R. & Redekopp, L. G. 1980 Radiation damping of long, finite amplitude internal waves. Phys. Fluids 23, 21822183.CrossRefGoogle Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Rayleigh, J. W. S. 1880 On the stability or instability of certain fluid motions. Proc. Lond. Math. Soc. 9, 5770.Google Scholar
Scotti, A. & Pineda, J. 2004 Observations of very large and steep internal waves of elevation near the Massachusetts coast. Geophys. Res. Lett. 31, l22307.CrossRefGoogle Scholar
Skamarock, W., Oliger, J. & Street, R. L. 1989 Adaptive grid refinement for numerical weather prediction. J. Comput. Phys. 80, 2760.CrossRefGoogle Scholar
Skamarock, W. C. & Klemp, J. B. 1993 Adaptive grid refinement for two-dimensional and three-dimensional nonhydrostatic atmospheric flow. Month. Weath. Rev. 121, 788804.2.0.CO;2>CrossRefGoogle Scholar
Slinn, D. N. & Riley, J. J. 1998 Turbulent dynamics of a critically reflecting internal gravity wave. Theoret. Comput. Fluid Dyn. 11, 281303.CrossRefGoogle Scholar
Smyth, W. D. & Moum, J. N. 2000 Anisotropy of turbulence in stably stratified mixing layers. Phys. Fluids 12, 13431362.CrossRefGoogle Scholar
Smyth, W. D. & Peltier, W. R. 1990 Three-dimensional primary instabilities of a stratified, dissipative, parallel flow. Geophys. Astrophys. Fluid Dyn. 52, 249261.CrossRefGoogle Scholar
Squire, H. B. 1933 On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. R. Soc. Lond. Proc. A 142, 621628.CrossRefGoogle Scholar
Thorpe, S. A. 2004 Recent developments in the study of ocean turbulence. Annu. Rev. Earth Planet. Sci. 32, 91109.CrossRefGoogle Scholar
Troy, C. D. & Koseff, J. R. 2005 The instability and breaking of long internal waves. J. Fluid Mech. 543, 107136.CrossRefGoogle Scholar
Venayagamoorthy, S. K. & Fringer, O. B. 2007 On the formation and propagation of nonlinear internal boluses across a shelf break. J. Fluid Mech. 577, 137159.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, 17791793.2.0.CO;2>CrossRefGoogle Scholar
Werne, J. & Fritts, D. C. 1999 Stratified shear turbulence: evolution and statistics. Geophys. Res. Lett. 26, 439442.CrossRefGoogle Scholar
Wolfsberg, A. V. & Freyberg, D. L. 1994 Efficient simulation of single species and multispecies transport in groundwater with local adaptive grid refinement. Water Resour. Res. 30, 29792992.CrossRefGoogle Scholar
Woods, J. D. 1968 Wave-induced shear instability in the summer thermocline. J. Fluid Mech. 32, 791800.CrossRefGoogle Scholar
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