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Ocean Dynamics

, Volume 58, Issue 3–4, pp 237–246 | Cite as

On the mathematical stability of stratified flow models with local turbulence closure schemes

  • Eric Deleersnijder
  • Emmanuel HanertEmail author
  • Hans Burchard
  • Henk A. Dijkstra
Article

Abstract

Occasionally, numerical simulations using local turbulence closure schemes to estimate vertical turbulent fluxes exhibit small-scale oscillations in space, causing the eddy coefficients to vary over several orders of magnitude on short distances. Theoretical developments suggest that these spurious oscillations are essentially due to the way the eddy coefficients depend on the vertical gradient of the model’s variables. An instability criterion is derived based on the assumptions that the artefacts under study are due to the development of small-amplitude, small time- and space-scale perturbations of a smooth solution. The relevance of this criterion is demonstrated by applying it to a series a closure schemes, ranging from the Pacanowski–Philander formulas to the Mellor–Yamada level 2.5 model.

Keywords

Marine modelling Vertical mixing Turbulence closure Stability 

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References

  1. Blanke B, Delecluse P (1993) Variability of tropical atlantic ocean simulated by a general circulation model with two different mixed-layer physics. J Phys Oceanogr 23:1363–1388CrossRefGoogle Scholar
  2. Brown PS, Pandolfo JP (1982) A numerical predictability problem in solution of the nonlinear diffusion equation. Mon Weather Rev 110:1214–1223CrossRefGoogle Scholar
  3. Burchard H (2002a) Applied turbulence modelling in marine waters. No. 100 in Lecture Notes in Earth Science. Springer, HeidelbergGoogle Scholar
  4. Burchard H (2002b) Energy-conserving discretization of turbulent shear and buoyancy production. Ocean Model 4:347–361CrossRefGoogle Scholar
  5. Burchard H, Deleersnijder E (2001) Stability of algebraic non-equilibrium second-order closure models. Ocean Model 3:33–50CrossRefGoogle Scholar
  6. Canuto VM, Howard A, Cheng Y, Dubovikov MS (2001) Ocean turbulence. Part I: one point closure model. Momentum and heat vertical diffusivities. J Phys Oceanogr 31:1413–1426CrossRefGoogle Scholar
  7. Davies AM, Jones JE (1991) On the numerical solution of the turbulence energy equations for wave and tidal flows. Int J Numer Methods Fluids 12:17–41CrossRefGoogle Scholar
  8. Davies HC (1983) The stability of some planetary boundary layer diffusion equations. Mon Weather Rev 111:2140–2143CrossRefGoogle Scholar
  9. Deleersnijder E, Burchard H (2003) Reply to Mellor’s comments on “Stability of algebraic non-equilibrium second-order closure models” by H. Burchard and E. Deleersnijder (Ocean Model 3:33–50 (2001)). Ocean Model 5:291–293CrossRefGoogle Scholar
  10. Deleersnijder E, Luyten P (1994) On the practical advantages of the quasi-equilibrium version of the Mellor and Yamada level 2.5 turbulence closure applied to marine modelling. Appl Math Model 18:281–287CrossRefGoogle Scholar
  11. Galperin B, Kantha LH, Hassid S, Rosati S (1988) A quasi-equilibrium turbulent energy model for geophysical flows. J Atmos Sci 45:55–62CrossRefGoogle Scholar
  12. Girard C, Delage Y (1990) Stable schemes for nonlinear vertical diffusion in atmospheric circulation models. Mon Weather Rev 118:737–745CrossRefGoogle Scholar
  13. Goosse H, Deleersnijder E, Fichefet T, England MH (1999) Sensitivity of a global coupled ocean-sea ice model to the parametrisation of vertical mixing. J Geophys Res 104:13681–13695CrossRefGoogle Scholar
  14. Hassid S, Galperin B (1983) A turbulent energy model for geophysical flows. Bound Layer Meteorol 26:397–412CrossRefGoogle Scholar
  15. Kato H, Phillips OM (1969) On the penetration of a turbulent layer into stratified fluid. J Fluid Mech 37:643–655CrossRefGoogle Scholar
  16. Kranenburg C (1980) On the stability of turbulent density-stratified shear flows. J Phys Oceanogr 10:1131–1133CrossRefGoogle Scholar
  17. Kranenburg C (1982) Stability conditions for gradient-transport models of turbulent density-stratified shear flows. Geophys Astrophys Fluid Dyn 19:93–104CrossRefGoogle Scholar
  18. Mellor GL (2003) Comments on “Stability of algebraic non-equilibrium second-order closure models” by H. Burchard and E. Deleersnijder (Ocean Model 3:33–50 (2001)). Ocean Model 5:193–194CrossRefGoogle Scholar
  19. Mellor GL, Yamada T (1974) A hierarchy of turbulence closure models for planetary boundary layers. J Atmos Sci 31:1791–1806CrossRefGoogle Scholar
  20. Mellor GL, Yamada T (1982) Development of a turbulence closure model for geophysical fluids problems. Rev Geophys Space Phys 20:851–875CrossRefGoogle Scholar
  21. Munk WH, Anderson ER (1948) Notes on a theory of the thermocline. J Mar Res 3:276–295Google Scholar
  22. Pacanowski RC, Philander SGH (1981) Parametrization of vertical mixing in numerical models of tropical oceans. J Phys Oceanogr 11:1443–1451CrossRefGoogle Scholar
  23. Phillips OM (1972) Turbulence in a strongly stratified fluid—is it unstable? Deep Sea Res 19:79–81Google Scholar
  24. Rodi W (1987) Examples of calculation methods for flow and mixing in stratified fluids. J Geophys Res 92(C5):5305–5328CrossRefGoogle Scholar
  25. Ruddick KG, Deleersnijder E, Luyten PJ, Ozer J (1995) Haline stratification in the Rhine-Meuse freshwater plume: a three-dimensional model sensitivity analysis. Cont Shelf Res 15:1597–1630CrossRefGoogle Scholar
  26. Yamada T (1977) A numerical experiment on pollutant dispersion in a horizontally-homogeneous atmospheric boundary layer. Atmos Environ 11:1015–1024CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Eric Deleersnijder
    • 1
  • Emmanuel Hanert
    • 2
    Email author
  • Hans Burchard
    • 3
  • Henk A. Dijkstra
    • 4
  1. 1.Centre for Systems Engineering and Applied Mechanics, Louvain School of EngineeringUniversité catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Department of MeteorologyUniversity of ReadingReadingUK
  3. 3.Leibniz Institute for Baltic Sea Research WarnemündeWarnemündeGermany
  4. 4.Institute for Marine and Atmospheric research UtrechtUtrecht UniversityUtrechtThe Netherlands

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