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Environmental Fluid Mechanics

, Volume 1, Issue 1, pp 71–106 | Cite as

A One-Equation Turbulence Model for Geophysical Applications: Comparison with Data and the k−ε Model

  • L.B. Axell
  • O. Liungman
Article

Abstract

A one-equation turbulence model is presented, in which the turbulent kinetic energy k is calculated with a transport equation whereas the turbulent length scale l is calculated with an algebraic expression. The value of l depends on the local stratification and reduces to the classical κ|z| scaling for unstratified flows near a wall, where |z| is the distance to the wall. The length scale decreases during stable stratification, and increases for unstable stratification compared to the neutral case. In the limit of strong stable stratification, the so-called buoyancy length scale proportional to k1/2N−1 is obtained, where N is the buoyancy frequency. The length scale formulation introduces a single model parameter which is calibrated against experimental data. The model is verified extensively against laboratory measurements and oceanic data, and comparisons are made with the two-equation k-ε model. It is shown that the performance of the proposed k model is almost identical to that of the k-ε model. In addition, the stability functions of Launder are revisited and adjusted to obtain better agreement with recent data.

Prandtl number stability functions turbulence model turbulent length scales 

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References

  1. 1.
    Arneborg, L.: 2000, Oceanographic Studies of Internal Waves and Diapycnal Mixing, Dept. of Oceanography, Earth Sciences Centre, Göteborg University, Box 460, SE-405 30Göteborg, Sweden.Google Scholar
  2. 2.
    Axell, L. B.: 1998, On the variability of Baltic Sea deepwater mixing, J. Geophys. Res. 103, 21667–21682.Google Scholar
  3. 3.
    Blackadar, A. K.: 1962, The vertical distribution of wind and turbulence exchange in a neutral atmosphere, J. Geophys. Res. 67, 3095–3102.Google Scholar
  4. 4.
    Blanke, B. and Delecluse, P.: 1993, Variability of the tropical Atlantic Ocean simulated by a General Circulation Model with two different mixed-layer physics, J. Phys. Oceanogr. 23, 1363–1388.Google Scholar
  5. 5.
    Burchard, H. and Baumert, H.: 1995, On the performance of a mixed-layer model based on the k — ɛ turbulence closure, J. Geophys. Res. 100, 8523–8540.Google Scholar
  6. 6.
    Burchard, H., Bolding, K. and Villarreal, M.R.: 1999, GOTM, a General Ocean Turbulence Model. Theory, Implementation and Test Cases, European Commission, Report EUR 18745, 103 pp.Google Scholar
  7. 7.
    Burchard, H. and Petersen, O.: 1999, Models of turbulence in the marine environment-A comparative study of two-equation turbulence models, J. Mar. Syst. 21, 29–53.Google Scholar
  8. 8.
    Burchard, H., Petersen, O. and Rippeth, T. P.: Comparing the performance of the Mellor-Yamada and the k — " two-equation turbulence models, J. Geophys. Res. 103, 10543–10554.Google Scholar
  9. 9.
    Businger, J. A., Wyngaard, J. C., Izumi, Y. and Bradley, E. F.: 1971, Flux-profile relationships in the atmospheric surface layer, J. Atmospheric Sci. 28, 181–189.Google Scholar
  10. 10.
    Champagne, F. H., Harris, V. G. and Corrsin, S.: 1970, Experiments on nearly homogeneous shear flow, J. Fluid Mech. 41, 81–139.Google Scholar
  11. 11.
    D'Alessio, S. J. D., Abdella, K. and McFarlane, N. A.: 1998, A new second-order turbulence closure scheme for modeling the oceanic mixed layer, J. Phys. Oceanogr. 28, 1624–1641.Google Scholar
  12. 12.
    Deardorff, J. W., Willis, G. E. and Lilly, D. K.: 1969, Laboratory investigation of non-steady penetrative convection, J. Fluid Mech. 35, 7–31.Google Scholar
  13. 13.
    Demirov, E., Eifler, W., Ouberdous, M. and Hibma, N.: 1998, ISPRAMIX-a Three-Dimensional Free Surface Model for Coastal Ocean Simulations and Satellite Data Assimilation on Parallel Computers, European Commission, 76 pp., EUR 18129EN.Google Scholar
  14. 14.
    Fung, I. Y., Harrison, D. E. and Lacis, A. A.: 1984, On the variability of the net longwave radiation at the ocean surface, Rev. Geophys. 22, 177–193.Google Scholar
  15. 15.
    Galperin, B., Kantha, L. H., Hassid, S. and Rosati, A.: 1988, A quasi-equilibrium turbulent energy model for geophysical flows, J. Atmospheric Sci. 45, 55–62.Google Scholar
  16. 16.
    Galperin, B., Rosati, A., Kantha, L. H. and Mellor, G. L.: 1989, Modeling rotating stratified turbulent flows with application to oceanic mixed layers, J. Phys. Oceanogr. 18, 901–916.Google Scholar
  17. 17.
    Gargett, A. E.: 1984, Vertical eddy diffusivity in the ocean interior, J. Mar. Res. 42, 359–393.Google Scholar
  18. 18.
    Gargett, A. E.: 1990, Do we really know how to scale the turbulent kinetic energy dissipation rate ɛ due to breaking of oceanic internal waves?, J. Geophys. Res. 95, 15971–15974.Google Scholar
  19. 19.
    Gargett, A. E. and Holloway, G.: 1984, Dissipation and diffusion by internal wave breaking, J. Mar. Res. 42, 15–27.Google Scholar
  20. 20.
    Gaspar, P., Grégoris, Y. and Lefevre, J.-M.: 1990, A simple eddy kinetic energy model for simulations of the vertical mixing: tests at station Papa and Long-Term Upper Ocean Study site, J. Geophys. Res. 95, 16179–16193.Google Scholar
  21. 21.
    Goosse, H., Deleersnijder, E. and Fichefet, T.: 1999, Sensitivity of a global coupled ocean-sea ice model to the parameterization of vertical mixing, J. Geophys. Res. 104, 13681–13695.Google Scholar
  22. 22.
    Gregg, M. C.: 1989, Scaling turbulent dissipation in the thermocline, J. Geophys. Res. 94, 9686–9698.Google Scholar
  23. 23.
    Högström, U.: 1996, Review of some basic characteristics of the atmospheric surface layer, Boundary-Layer Meteorol. 78, 215–246.Google Scholar
  24. 24.
    Itsweire, E. C., Koseff, J. R., Briggs, D. A. and Ferziger, J. H.: 1993, Turbulence in stratified shear flows: Implications for interpreting shear-induced mixing in the ocean, J. Phys. Oceanogr. 23, 1508–1522.Google Scholar
  25. 25.
    Ivey, G.N. and Imberger, J.: 1991, On the nature of turbulence in a stratified fluid. Part I: The energetics of mixing, J. Phys. Oceanogr. 21, 650–658.Google Scholar
  26. 26.
    Kantha, L. H. and Clayson, A. C.: 1994, An improved mixed layer model for geophysical applications, J. Geophys. Res. 99, 25235–25266.Google Scholar
  27. 27.
    Kato, H. and Phillips, O. M.: 1969, On the penetration of a turbulent layer into a stratified fluid, J. Fluid Mech. 37, 643–655.Google Scholar
  28. 28.
    Kundu, P. K.: 1990, Fluid Mechanics, Academic Press, Inc., New York, 638 pp.Google Scholar
  29. 29.
    Large, W. G., McWilliams, J. C. and Doney, S. C.: 1994, Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization, Rev. Geophys. 32, 363–403.Google Scholar
  30. 30.
    Launder, B. E.: 1975, On the effects of a gravitational field on the turbulent transport of heat and momentum, J. Fluid Mech. 67, 569–581.Google Scholar
  31. 31.
    Liungman, O.: 2000, Tidally forced internal wave mixing in a k — ɛmodel framework applied to fjord basins, J. Phys. Oceanogr. 30, 352–368.Google Scholar
  32. 32.
    Luyten, P. J., Deleersnijder, E., Ozer, J. and Ruddick, K. G.: 1996, Presentation of a family of turbulence closure models for stratified shallow water flows and preliminary application to the Rhine outflow region, Cont. Shelf Res. 16, 101–130.Google Scholar
  33. 33.
    Martin, P. J.: 1985, Simulation of the mixed layer at OWS November and Papa with several models, J. Geophys. Res. 90, 903–916.Google Scholar
  34. 34.
    McPhee, M. G.: 1994, On the turbulent mixing length in the oceanic boundary layer, J. Phys. Oceanogr. 24, 2014–2031.Google Scholar
  35. 35.
    Mellor, G. L.: 1989, Retrospect on oceanic boundary layer modeling and second moment closure, Proc. Fifth Aha Huliko'a Hawaiian Winter Workshop, Hawaii Institute of Geophysics, Honolulu, HI, pp. 251–272.Google Scholar
  36. 36.
    Mellor, G. L. and Yamada, T.: 1974, A hierarchy of turbulence closure models for planetary boundary layers, J. Atmospheric Sci. 31, 1791–1806.Google Scholar
  37. 37.
    Mellor, G. L. and Yamada, T.: 1982, Development of a turbulence closure model for geophysical problems, Rev. Geophys. 20, 851–875.Google Scholar
  38. 38.
    Moum, J. N., Caldwell, D. R. and Paulson, C. A.: 1989, Mixing in the equatorial surface layer and thermocline, J. Geophys. Res. 94, 2005–2021.Google Scholar
  39. 39.
    Nordblom, O.: 1997, Numerical Simulation of the Atmospheric Surface Layer, Department of Environmental Engineering, Division of Water Resources Engineering, Luleå University of Technology, Sweden.Google Scholar
  40. 40.
    Oakey, N. S.: 1982, Determination of the rate of dissipation of turbulent energy from simultaneous temperature and velocity shear microstructure measurements, J. Phys. Oceanogr. 12, 256–271.Google Scholar
  41. 41.
    Omstedt, A.: 1990, Modelling the Baltic Sea as thirteen sub-basins with vertical resolution, Tellus bf 42A, 286–301.Google Scholar
  42. 42.
    Omstedt, A. and Axell, L. B.: 1998, Modeling the seasonal, interannual, and long-term variations of salinity and temperature in the Baltic proper, Tellus 50A, 637–652.Google Scholar
  43. 43.
    Osborn, T. R.: 1980, Estimates of the local rate of vertical diffusion from dissipation measurements, J. Phys. Oceanogr. 10, 83–89.Google Scholar
  44. 44.
    Price, J. F.: 1979, On the scaling of stress-driven entrainment experiments, J. Fluid Mech. 90, 509–529.Google Scholar
  45. 45.
    Rodi, W.: 1980, Turbulence Models and Their Application in Hydraulics-A State-of-the-Art Review, International Association for Hydraulic Research, Rotterdamseweg 185, P.O. Box 177, 2600 MH Delft, The Netherlands.Google Scholar
  46. 46.
    Rodi, W.: 1987, Examples of calculation methods for flow and mixing in stratified flows, J. Geophys. Res. 92, 5305–5328.Google Scholar
  47. 47.
    Rosati, A. and Miyakoda, K.: 1988, A General Circulation Model for upper ocean simulation, J. Phys. Oceanogr. 18, 1601–1626.Google Scholar
  48. 48.
    Simpson, J. H. and Bowers, D. G.: 1984, The role of tidal stirring in controlling the seasonal heat cycle in shelf seas, Ann. Geophys. 2, 411–416.Google Scholar
  49. 49.
    Simpson, J. H., Crawford, W. R., Rippeth, T. P., Campbell, A. R. and Cheok, J. V. S.: 1996, The vertical structure of turbulent dissipation in shelf seas, J. Phys. Oceanogr. 26, 1579–1590.Google Scholar
  50. 50.
    Skyllingstad, E. D., Smyth, W. D., Moum, J. N. and Wijesekera, H.: 1999, Upper-ocean turbulence during a westerly wind burst: a comparison of large-eddy simulation results and microstructure measurements, J. Phys. Oceanogr. 29, 5–28.Google Scholar
  51. 51.
    Squires, K. D. and Yamazaki, H.: 1995, Preferential concentration of marine particles in isotropic turbulence, Deep Sea Res. 42, 1989–2004.Google Scholar
  52. 52.
    Stigebrandt, A.: 1987, A model for the vertical circulation of the Baltic deep water, J. Phys. Oceanogr. 17, 1772–1785.Google Scholar
  53. 53.
    Svensson, U.: 1978, A Mathematical Model for the Seasonal Thermocline, Report No. 1002, Dept. of Water Res. Eng., Lund Inst. of Technology, Lund, Sweden.Google Scholar
  54. 54.
    Svensson, U.: 1998, PROBE. Program for Boundary Layers in the Environment. System Description and Manual, Report Oceanography No. 24, Swedish Meteorological and Hydrological Institute, SE-601 76Norrköping, Sweden.Google Scholar
  55. 55.
    Svensson, U. and Sahlberg, J.: 1989, Formulae for pressure gradients in one-dimensional lake models, J. Geophys. Res. 94, 4939–4946.Google Scholar
  56. 56.
    Tennekes, H. and Lumley, J. L.: 1972, A First Course in Turbulence, The MIT Press, New York, 300 pp.Google Scholar
  57. 57.
    Turner, J. S.: 1973, Buoyancy Effects in Fluids, Cambridge University Press, London, 367 pp.Google Scholar
  58. 58.
    Webster, C. A. G.: 1964, An experimental study of turbulence in a density stratified shear flow, J. Fluid Mech. 19, 221–245.Google Scholar
  59. 59.
    Wieringa, J.: 1980, A reevaluation of the Kansas mast influence on measurements of stress and cup anemometer overspeeding, Boundary-Layer Meteorol. 18, 411–430.Google Scholar
  60. 60.
    Xing, J. and Davies, A. M.: 1995, Application of three dimensional turbulence energy models to the determination of tidal mixing and currents in a shallow sea, Prog. Oceanogr. 35, 153–205.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • L.B. Axell
    • 1
  • O. Liungman
    • 2
  1. 1.Swedish Meteorological and Hydrological InstituteNorrköpingSweden
  2. 2.Department of Oceanography, Earth Sciences CentreGöteborg UniversityGöteborgSweden

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