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Internal hydraulic jumps in two-layer flows with upstream shear

Published online by Cambridge University Press:  15 January 2016

K. A. Ogden  and
Karl R. Helfrich
K. A. Ogden
Affiliation:
MIT/WHOI Joint Program in Oceanography, Cambridge, MA 02139, USA
Karl R. Helfrich*
Affiliation:
Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, MA 02536, USA
*
†Email address for correspondence: khelfrich@whoi.edu
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Abstract

Internal hydraulic jumps in flows with upstream shear are investigated using two-layer shock-joining theories and numerical solutions of the Navier–Stokes equations. The role of upstream shear has not previously been thoroughly investigated, although it is important in many oceanographic situations, including exchange flows. The full solution spaces of several two-layer theories, distinguished by how dissipation is distributed between the layers, with upstream shear are found, and the physically allowable solution space is identified. These two-layer theories are then evaluated using more realistic numerical simulations that have continuous density and velocity profiles and permit turbulence and mixing. Two-dimensional numerical simulations show that none of the two-layer theories reliably predicts the relation between jump height and speed over the full range of allowable solutions. The numerical simulations also show that different qualitative types of jumps can occur, including undular bores, energy-conserving conjugate state transitions, smooth-front jumps with trailing turbulence and overturning turbulent jumps. Simulation results are used to investigate mixing, which increases with jump height and upstream shear. A few three-dimensional simulations results were undertaken and are in quantitative agreement with the two-dimensional simulations.

JFM classification

Geophysical and Geological Flows: Hydraulic control Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 789 , 25 February 2016 , pp. 64 - 92
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Copyright
© 2016 Cambridge University Press 

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References

Alford, M. H., Girton, J., Voet, G., Carter, G., Mickett, J. & Klymak, J. 2013 Turbulent mixing and hydraulic control of abyssal water in the Samoan Passage. Geophys. Res. Lett. 40, 4668–4674.Google Scholar
Almgren, A., Bell, J., Colella, P., Howell, L. & Welcome, M. 1998 A conservative adaptive projection method for the variable density incompressible Navier–Stokes equations. J. Comput. Phys. 142, 1–46.Google Scholar
Aspden, A., Nikiforakis, N., Dalziel, S. & Bell, J. 2008 Analysis of implicit LES methods. Commun. Appl. Maths Comput. Sci. 3 (1), 103–126.CrossRefGoogle Scholar
Baines, P. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.Google Scholar
Benjamin, T. 1968 Gravity currents and related phenomenon. J. Fluid Mech. 31, 209–248.CrossRefGoogle Scholar
Borden, Z. & Meiburg, E. 2013a Circulation based models for Boussinesq gravity currents. Phys. Fluids 25, 101301.Google Scholar
Borden, Z. & Meiburg, E. 2013b Circulation-based models for Boussinesq internal bores. J. Fluid Mech. 726, R1.CrossRefGoogle Scholar
Borden, Z., Meiburg, E. & Constantinescu, G. 2012 Internal bores: an improved model via a detailed analysis of the energy budget. J. Fluid Mech. 703, 279–314.Google Scholar
Carr, M., King, S. & Dritschel, D. 2012 Instability in internal solitary waves with trapped cores. Phys. Fluids 24, 016601.CrossRefGoogle Scholar
Choi, W. 2006 The effect of a background shear current on large amplitude internal solitary waves. Phys. Fluids 18, 036601.Google Scholar
Farmer, D. & Armi, L. 1999 Stratified flow over topography: the role of small-scale entrainment and mixing in flow establishment. Proc. R. Soc. Lond. A 455, 3221–3258.Google Scholar
Gregg, M. & Özsoy, E. 2002 Flow, water mass changes, and hydraulics in the Bosphorus. J. Geophys. Res. 107, C3.CrossRefGoogle Scholar
Grimshaw, R., Pelinovsky, E., Talipova, T. & Kurkin, A. 2004 Simulation of the transformation of internal solitary waves on oceanic shelves. J. Phys. Oceanogr. 34, 2774–2791.CrossRefGoogle Scholar
Grinstein, F., Margolin, L. & Rider, W.(Eds) 2007 Implicit Large Eddy Simulation: Computing Turbulent Fluid Dynamics. Cambridge University Press.Google Scholar
Helfrich, K. 1995 Time-dependent two-layer hydraulic exchange flows. J. Phys. Oceanogr. 25, 359–373.Google Scholar
Helfrich, K. R. & White, B. L. 2010 A model for large-amplitude internal solitary waves with trapped cores. Nonlinear Process. Geophys. 17 (4), 303–318.Google Scholar
Hickel, S., Adams, N. & Mansour, N. 2007 Implicit subgrid-scale modeling for large-eddy simulation of passive-scalar mixing. Phys. Fluids 19, 095102.Google Scholar
Holland, D., Rosales, R., Stefanica, D. & Tabak, E. 2002 Internal hydraulic jumps and mixing in two-layer flows. J. Fluid Mech. 470, 63–83.Google Scholar
Jacobson, T., Milewski, P. A. & Tabak, E. 2008 Mixing closures for conservation laws in stratified flows. Stud. Appl. Maths 121, 89–116.Google Scholar
Klemp, J., Rotunno, R. & Skamarock, W. 1997 On the propagation of internal bores. J. Fluid Mech. 331, 81–106.Google Scholar
Klymak, J. & Gregg, M. 2003 The role of upstream waves and a downstream density pool in the growth of lee waves: stratified flow over the Knight Inlet Sill. J. Phys. Oceanogr. 33, 1446–1461.Google Scholar
Klymak, J. & Gregg, M. 2004 Tidally generated turbulence over the Knight Inlet Sill. J. Phys. Oceanogr. 34 (5), 1135–1151.Google Scholar
Lamb, K. 2000 Conjugate slows for a three-layer fluid. Phys. Fluids 12 (9), 2169–2185.Google Scholar
Li, M. & Cummins, P. 1998 A note on hydraulic theory of internal bores. Dyn. Atmos. Oceans 28, 1–7.Google Scholar
Luzzatto-Fegiz, P. & Helfrich, K. R. 2014 Laboratory experiments and simulations for solitary internal waves with trapped cores. J. Fluid Mech. 757, 354–380.Google Scholar
Margolin, L., Rider, W. & Grinstein, F. 2006 River plumes as a soumodeling turbulent flow with implicit LES. J. Turbul. 7 (15), 1–27.Google Scholar
Milewski, P. A. & Tabak, E. G. 2015 Conservation law modelling of entrainment in layered hydrostatic flows. J. Fluid Mech. 772, 272–294.Google Scholar
Nash, J. & Moum, J. 2005 River plumes as a source of large-amplitude internal waves in the coastal ocean. Nature 437, 400–403.Google Scholar
Polzin, K., Speer, K., Toole, J. & Schmitt, R. 1996 Intense mixing of Antarctic Bottom Water in the equatorial Atlantic Ocean. Nature 380 (7), 54–57.Google Scholar
Remmler, S. & Hickel, S. 2012 Direct and large eddy simulation of stratified turbulence. Intl J. Heat Fluid Flow 35, 13–24.Google Scholar
Rottman, J. & Simpson, J. 1989 The formation of internal bores in the atmosphere: a laboratory model. Q. J. R. Meteorol. Soc. 115, 941–963.Google Scholar
Sánchez-Garrido, J., Sannino, G., Liberti, L., Lafuente, J. G. & Pratt, L. 2011 Numerical modeling three-dimensional stratified tidal flow over Camarinal Sill, Strait of Gibraltar. J. Geophys. Res. 116, C12026.Google Scholar
Stastna, M. & Lamb, K. 2002 Large fully nonlinear internal solitary waves: the effect of background current. Phys. Fluids 14 (9), 2987–2999.CrossRefGoogle Scholar
St Laurent, L. & Thurnherr, A. 2007 Intense mixing of lower thermocline water on the crest of the Mid-Atlantic Ridge. Nature 448 (9), 680–683.Google Scholar
Waite, M. & Smolarkiewicz, P. 2008 Instability and breakdown of a vertical vortex pair in a strongly stratified fluid. J. Fluid Mech. 606, 239–273.Google Scholar
White, B. & Helfrich, K. 2013 Rapid gravitational adjustment of horizontal shear flows. J. Fluid Mech. 721, 86–117.Google Scholar
White, B. & Helfrich, K. 2014 A model for internal bores in continuous stratification. J. Fluid Mech. 761, 282–304.Google Scholar
Wood, I. & Simpson, J. 1984 Jumps in layered miscible fluids. J. Fluid Mech. 140, 329–342.Google Scholar
Zhou, Y., Grinstein, F., Wachtor, A. & Haines, B. 2014 Estimating the effective Reynolds number in implicit large-eddy simulation. Phys. Rev. E 89, 013303.Google ScholarPubMed