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A model for internal bores in continuous stratification

Published online by Cambridge University Press:  21 November 2014

Brian L. White  and
Karl R. Helfrich
Brian L. White*
Affiliation:
Marine Sciences Department, University of North Carolina at Chapel Hill, Murray Hall, Chapel Hill, NC 27599-3300, USA
Karl R. Helfrich
Affiliation:
Department of Physical Oceanography, Woods Hole Oceanographic Institution, 266 Woods Hole Road, Woods Hole, MA 02543-1050, USA
*
†Email address for correspondence: [email protected]
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Abstract

We describe a model for the speed of an internal bore as a function of amplitude in continuous stratification of arbitrary form. The model is developed from the Dubreil-Jacotin–Long theory for nonlinear solitary waves in the conjugate flow limit, which represents an internal hydraulic jump, by allowing dissipation across the jump. The bore speeds predicted by the model are consistent in both the small- and large-amplitude limits with the waveguide intrinsic to the ambient stratification. The model therefore represents a significant advancement over previous theories limited to sharp two-layer stratification. The model shows good agreement with Navier–Stokes simulations of both undular and turbulent internal bores generated by dam break into a continuously stratified ambient with a finite pycnocline, predicting both the front speed as well as the velocity and density structure through the bore. A model is required for the structure of the energy dissipation, and we introduce a one-parameter closure that produces excellent agreement with numerical results, particularly in the parameter limit that maximizes the overall dissipation. By varying the dissipation parameter, the model reproduces previous two-layer theories in the thin-pycnocline limit, and suggests an improved two-layer front speed relationship. It is demonstrated that, even for the sharp two-layer limit, continuous stratification, and particularly the nonlinear waveguide, must be accounted for in order to accurately predict the bore speed and structure.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 761 , 25 December 2014 , pp. 282 - 304
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Copyright
© 2014 Cambridge University Press 

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References

Baines, P. G. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.Google Scholar
Bell, J. B. & Marcus, D. L. 1992 A second-order projection method for variable-density flows. J. Comput. Phys. 101, 334–348.Google Scholar
Benjamin, T. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.Google 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.Google Scholar
Borden, Z., Meiburg, E. & Constantinescu, G. 2012 Internal bores: an improved model via a detailed analysis of energy budget. J. Fluid Mech. 703, 279–314.Google Scholar
Brown, D. J. & Christie, D. R. 1998 Fully nonlinear solitary waves in continuously stratified incompressible Boussinesq fluids. Phys. Fluids 10, 2569–2586.Google Scholar
Chu, V. & Baddour, R.1977 Surges, waves and mixing in two-layer density stratified flow. In Proceedings of the 17th Congress of the International Association of Hydraulics Research, vol. 1, pp. 303–310.Google Scholar
Cummins, P., Vagle, S., Armi, L. & Farmer, D. 2003 Stratified flow over topography: upstream influence and generation of nonlinear internal waves. Proc. R. Soc. Lond. A 459, 1467–1487.CrossRefGoogle Scholar
Dubreil-Jacotin, M. 1934 Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie. J. Math. Pure Appl. 13, 217–291.Google Scholar
Esler, J. G. & Pearce, J. D. 2011 Dispersive dam-break and lock-exchange flows in a two-layer fluid. J. Fluid Mech. 667, 555–585.Google Scholar
Helfrich, K. & White, B. 2010 A model for large-amplitude internal solitary waves with trapped cores. Nonlinear Process. Geophys. 17, 303–318.CrossRefGoogle Scholar
King, S. E., Carr, M. & Dritschel, D. G. 2010 The steady-state form of large-amplitude internal solitary waves. J. Fluid Mech. 666, 477–505.Google Scholar
Klemp, J., Rotunno, R. & Skamarock, W. 1997 On the propagation of internal bores. J. Fluid Mech. 331, 81–106.Google Scholar
Lamb, K. 2000 Conjugate flows for a three-layer fluid. Phys. Fluids 12, 2169–2185.Google Scholar
Lamb, K. G. 2002 A numerical investigation of solitary internal waves with trapped cores formed via shoaling. J. Fluid Mech. 451, 109–144.Google Scholar
Lamb, K. G. & Wan, B. G. 1998 Conjugate flows and flat solitary waves for a continuously stratified fluid. Phys. Fluids 10, 2061–2079.CrossRefGoogle Scholar
Lamb, K. & Wilkie, K. P. 2004 Conjugate flows for waves with trapped cores. Phys. Fluids 16, 4685–4695.Google Scholar
Li, M. & Cummins, P. 1998 A note on hydraulic theory of internal bores. Dyn. Atmos. Oceans 28 (1), 1–7.Google Scholar
Long, R. 1953 Some aspects of the flow of stratified fluids, I. A theoretical investigation. Tellus 5, 42–57.Google Scholar
Marino, B., Thomas, L. & Linden, P. 2005 The front condition for gravity currents. J. Fluid Mech. 536, 49–78.Google Scholar
Pineda, J. 1999 Circulation and larval distribution in internal tidal bore warm fronts. Limnol. Oceanogr. 44, 1400–1414.Google Scholar
Pratt, L. J. & Whitehead, J. A. 2008 Rotating Hydraulics. Springer.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
Stastna, M. & Lamb, K. 2002 Large fully nonlinear internal solitary waves: the effect of background current. Phys. Fluids 14, 2987–2999.CrossRefGoogle Scholar
Walter, R., 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.Google Scholar
White, B. L. & Helfrich, K. R. 2008 Gravity currents and internal waves in a stratified fluid. J. Fluid Mech. 616, 327–356.Google Scholar
White, B. L. & Helfrich, K. R. 2012 A general description of a gravity current front propagating in a two-layer stratified fluid. J. Fluid Mech. 711, 127–167.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Wood, I. & Simpson, J. 1984 Jumps in layered miscible fluids. J. Fluid Mech. 140, 215–231.Google Scholar
Yih, C. S. & Guha, C. R. 1955 Hydraulic jumps in a fluid system of two layers. Tellus 7, 358–366.Google Scholar