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The dynamics of breaking internal solitary waves on slopes

  • Robert S. Arthur (a1) and Oliver B. Fringer (a1)


Using direct numerical simulations (DNS), we investigate the structure and energetics of breaking internal waves on slopes. We employ a Navier–Stokes code in an idealized three-dimensional domain where an internal solitary wave of depression impinges upon a sloping bottom. Seven cases with varying initial wave amplitude and bathymetric slope, but constant wave Reynolds number $\mathit{Re}_{w}$ are considered. Volume-integrated values of dissipation and irreversible mixing are related to the density and velocity structure of the wave throughout the breaking process. The majority of dissipation (63 %) occurs along the no-slip bottom boundary. Most of the remaining dissipation (35 %) and nearly all irreversible mixing occurs in the interior after breaking, when density overturns are present at the interface. Breaking introduces three-dimensionality to the flow field that is driven by the lateral breakdown of density overturns and the lobe–cleft instability typical of gravity currents. The resulting longitudinal rolls (streamwise vorticity) increase dissipation by roughly 8 % and decrease irreversible mixing by roughly 20 % when compared with a similar two-dimensional simulation. The bulk mixing efficiency is shown to increase for larger and smaller values of the internal Iribarren number ${\it\xi}$ , with a minimum for intermediate values of ${\it\xi}$ and a peak near ${\it\xi}=0.8$ for plunging breakers. This trend is explained by the degree of two-dimensionality in the flow, and agrees with previous results in the literature after accounting for Reynolds number effects. Local turbulence quantities are also calculated at ‘virtual moorings’, and a location upslope of the breakpoint but downslope of the intersection of the pycnocline and the bottom is shown to provide a signal that is most representative of the volume-integrated dissipation and mixing results.


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Aghsaee, P., Boegman, L., Diamessis, P. J. & Lamb, K. G. 2012 Boundary-layer-separation-driven vortex shedding beneath internal solitary waves of depression. J. Fluid Mech. 690, 321344.
Aghsaee, P., Boegman, L. & Lamb, K. G. 2010 Breaking of shoaling internal solitary waves. J. Fluid Mech. 659, 289317.
Aucan, J., Merrifield, M. A., Luther, D. S. & Flament, P. 2006 Tidal mixing events on the deep flanks of Kaena Ridge, Hawaii. J. Phys. Oceanogr. 36 (6), 12021219.
Boegman, L., Ivey, G. N. & Imberger, J. 2005 The degeneration of internal waves in lakes with sloping topography. Limnol. Oceanogr. 50 (5), 16201637.
Bouffard, D. & Boegman, L. 2013 A diapycnal diffusivity model for stratified environmental flows. Dyn. Atmos. Oceans 61, 1434.
Bourgault, D., Blokhina, M. D., Mirshak, R. & Kelley, D. E. 2007a Evolution of a shoaling internal solitary wavetrain. Geophys. Res. Lett. 34, L03601.
Bourgault, D. & Kelley, D. E. 2007b On the reflectance of uniform slopes for normally incident interfacial solitary waves. J. Phys. Oceanogr. 37 (5), 11561162.
Bourgault, D., Morsilli, M., Richards, C., Neumeier, U. & Kelley, D. E. 2014 Sediment resuspension and nepheloid layers induced by long internal solitary waves shoaling orthogonally on uniform slopes. Cont. Shelf Res. 72, 2133.
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.
Chou, Y. J. & Fringer, O. B. 2010 A model for the simulation of coupled flow-bed form evolution in turbulent flows. J. Geophys. Res. 115, C10041.
Cui, A.1999 On the parallel computation of turbulent rotating stratified flows. PhD thesis, Stanford University.
Davis, K. A. & Monismith, S. G. 2011 The modification of bottom boundary layer turbulence and mixing by internal waves shoaling on a barrier reef. J. Phys. Oceanogr. 41 (11), 22232241.
Dörnbrack, A. 1998 Turbulent mixing by breaking gravity waves. J. Fluid Mech. 375, 113141.
Fringer, O. B.2003 Numerical simulations of breaking interfacial waves. PhD thesis, Stanford University.
Fringer, O. B. & Street, R. L. 2003 The dynamics of breaking progressive interfacial waves. J. Fluid Mech. 494, 319353.
Gayen, B. & Sarkar, S. 2010 Turbulence during the generation of internal tide on a critical slope. Phys. Rev. Lett. 104 (21), 218502.
Gayen, B. & Sarkar, S. 2011 Boundary mixing by density overturns in an internal tidal beam. Geophys. Res. Lett. 38 (14), L14608.
Härtel, C., Carlsson, F. & Thunblom, M. 2000 Analysis and direct numerical simulation of the flow at a gravity-current head. Part 2. The lobe-and-cleft instability. J. Fluid Mech. 418, 213229.
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. 2006 Long nonlinear internal waves. Annu. Rev. Fluid Mech. 38, 395425.
Hult, E. L., Troy, C. D. & Koseff, J. R. 2011 The mixing efficiency of interfacial waves breaking at a ridge: 2. Local mixing processes. J. Geophys. Res. 116, C02004.
Klymak, J. M. & Moum, J. N. 2003 Internal solitary waves of elevation advancing on a shoaling shelf. Geophys. Res. Lett. 30 (20), 2045.
Koltakov, S. & Fringer, O. B. 2012 Moving grid method for numerical simulation of stratified flows. Intl J. Numer. Meth. Fluids 71 (12), 15241545.
Lamb, K. G. 2002 A numerical investigation of solitary internal waves with trapped cores formed via shoaling. J. Fluid Mech. 451, 109144.
Leichter, J. J., Wing, S. R., Miller, S. L. & Denny, M. W. 1996 Pulsed delivery of subthermocline water to Conch Reef (Florida Keys) by internal tidal bores. Limnol. Oceanogr. 41 (7), 14901501.
Li, X., Lu, P., Schaeffer, J., Shillington, J., Wong, P. S. & Shi, H. 1993 On the versatility of parallel sorting by regular sampling. Parallel Comput. 19 (10), 10791103.
Michallet, H. & Ivey, G. N. 1999 Experiments on mixing due to internal solitary waves breaking on uniform slopes. J. Geophys. Res. 104 (C6), 1346713477.
Munk, W. & Wunsch, C. 1998 Abyssal recipes II: energetics of tidal and wind mixing. Deep-Sea Res. 45 (12), 19772010.
Nam, S. H. & Send, U. 2011 Direct evidence of deep water intrusions onto the continental shelf via surging internal tides. J. Geophys. Res. 116, C05004.
Omand, M. M., Leichter, J. J., Franks, P. J., Guza, R. T., Lucas, A. J. & Feddersen, F. 2011 Physical and biological processes underlying the sudden appearance of a red-tide surface patch in the nearshore. Limnol. Oceanogr. 56 (3), 787801.
Pineda, J. 1994 Internal tidal bores in the nearshore: warm-water fronts, seaward gravity currents and the onshore transport of neustonic larvae. J. Mar. Res. 52 (3), 427458.
Scotti, A. & Pineda, J. 2004 Observation of very large and steep internal waves of elevation near the Massachusetts coast. Geophys. Res. Lett. 31 (22), L22307.
Scotti, A. & White, B. 2014 Diagnosing mixing in stratified turbulent flows with a locally defined available potential energy. J. Fluid Mech. 740, 114135.
Shih, L. H., Koseff, J. R., Ivey, G. N. & Ferziger, J. H. 2005 Parameterization of turbulent fluxes and scales using homogeneous sheared stably stratified turbulence simulations. J. Fluid Mech. 525, 193214.
Shroyer, E. L., Moum, J. N. & Nash, J. D. 2009 Observations of polarity reversal in shoaling nonlinear internal waves. J. Phys. Oceanogr. 39 (3), 691701.
Simpson, J. E. 1972 Effects of the lower boundary on the head of a gravity current. J. Fluid Mech. 53 (4), 759768.
Smyth, W. D. & Winters, K. B. 2003 Turbulence and mixing in Holmboe waves. J. Phys. Oceanogr. 33 (4), 694711.
Troy, C. D. & Koseff, J. R. 2005 The instability and breaking of long internal waves. J. Fluid Mech. 543, 107136.
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.
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.
Vlasenko, V. & Stashchuk, N. 2007 Three-dimensional shoaling of large-amplitude internal waves. J. Geophys. Res. 112, C11018.
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.
Walter, R. K.2014 Nonlinear internal waves, internal bores, and turbulent mixing in the nearshore coastal environment. PhD thesis, Stanford University.
Walter, R. K., Woodson, C. B., Arthur, R. S., Fringer, O. B. & Monismith, S. G. 2012 Nearshore internal bores and turbulent mixing in southern Monterey Bay. J. Geophys. Res. 117, C07017.
Walter, R. K., Woodson, C. B., Leary, P. R. & Monismith, S. G. 2014 Connecting wind-driven upwelling and offshore stratification to nearshore internal bores and oxygen variability. J. Geophys. Res. 116 (6), 35173534.
Winters, K. B. & D’Asaro, E. A. 1994 Three-dimensional wave instability near a critical level. J. Fluid Mech. 272, 255284.
Winters, K. B., Lombard, P. N., Riley, J. J. & D’Asaro, E. A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115128.
Zang, Y., Street, R. L. & Koseff, J. R. 1993 A dynamic mixed subgrid-scale model and its application to turbulent recirculating flows. Phys. Fluids A 5 (12), 31863196.
Zang, Y., Street, R. L. & Koseff, J. R. 1994 A non-staggered grid, fractional step method for time-dependent incompressible Navier–Stokes equations in curvilinear coordinates. J. Comput. Phys. 114, 1833.
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The dynamics of breaking internal solitary waves on slopes

  • Robert S. Arthur (a1) and Oliver B. Fringer (a1)


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