Skip to main content
SpringerLink
Log in
Book cover

Physics of Lakes pp 173–205Cite as

Subgrid-Scale Parameterization in Numerical Simulations of Lake Circulation

  • Chapter
  • First Online:

  • 914 Accesses

Abstract

Turbulence is ubiquitous in geophysical flows. There are hardly any situations in the dynamics of natural waters and the atmosphere that do not involve turbulent effects at some point, and only little insight can be gained in the dominant processes, if turbulence is not taken into account. Thus, the modelling of this phenomenon has attracted a great many researchers and more and more advanced models, suitable for the description of a large variety of geophysical flows, evolved over the last decades. A general model embracing all aspects of turbulence is still out of reach. Nevertheless, there has been an enormous progress in the understanding of turbulence in the past. More recently, Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES) provided data that were previously only available by high-precision laboratory setups (or not at all) and had a large impact on the development of new turbulence models. It seems likely that, especially in oceanography and meteorology, LES will take a position equitable to the ensemble averaged methods in the near future. At present, however, in spite of the development of larger and faster computers, LES is still too expensive for standard simulations of geophysical interest. In this chapter, we will mainly investigate a simple parameterization of the subgrid-scale turbulent closure which is based on the Reynolds Averaged Navier-Stokes Equations (RANS) and the subgrid-scale eddy viscosity parameterizations. We employ the three-dimensional, hydrodynamic, semi-implicit, finite difference model, developed by Song and Haidvogel and extended for the TVD treatment of the advection terms by Y. Wang. Subgrid-Scale Parameterization in Numerical Simulations of Lake Circulation are used and Smagorinsky’s formulation and Mellor–Yamada ’s level-2.5 model are used to parameterize the horizontal and vertical eddy viscosities. Numerical results in an assumed rectangular basin with constant depth and in Lake Constance are displayed and discussed. A comparison is made with prescribed constant eddy viscosities; it clearly shows the importance of subgrid-scale parameterizations in numerical simulations of lake circulation. We do not claim that such subgrid-scale parameterizations are the best choice of possible closure conditions, but only show that suitable parameterizations of the small-scale (subgrid-scale) turbulent motions are needed.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    An introduction to mathematical turbulence in continuous system is given in Vol. 1, Chap. 6 of this book series. For more details see e.g. Hutter and Jöhnk (2004) [17], Chaps. 10–12.

  2. 2.

    This is the shallow water approximation of

    $$\begin{aligned} R_i=\frac{g}{\rho }\frac{||\mathrm{g}rad \rho ||}{2 I\!\!I_{\varvec{D}}}, \end{aligned}$$

    where \(I\!\!I_{\varvec{D}}\) is the second invariant of the tensor \(\varvec{D}={\mathrm {sym}}(\mathrm{grad}\,\mathbf{v})\).

  3. 3.

    For a two-layer model with constant depth an estimate of the internal Rossby radius is \(\sqrt{gH_e\varepsilon }/f\) and yields here \(R_i\simeq 3-5~{\text {km}}\). A typical width of Lake Constance is app. 10 km. So a mid-lake signal of a Kelvin-type wave should be observable.

  4. 4.

    For a detailed analysis, see Vol. 2, Chap. 14.

References

  1. Appt, J., Imberger, J. and Kobus, H.: Basin-scale motion in stratified Upper Lake Constance. Limnol Oceanogr., 49(4), 919–933 (2004)

    Google Scholar 

  2. Bastiaans, R.J.M., Rindt, C.C.M. and Van, A.A.: Experimental analysis of a confined transitional plume with respect to subgrid-scale modelling. Int. J. Heat Mass Transfer, 41, 3989–4007 (1998)

    Google Scholar 

  3. Bäuerle, E.: Transverse baroclinic oscillations in Lake Überlingen. Aquatic Sciences, 56(2), 145–160 (1994)

    Google Scholar 

  4. Blumberg, A.F., and Mellor, G.L.: A Description of a three-dimensional coastal ocean circulation model. In: Mooers (Ed.), Three-dimensional Coastal Ocean Models, Coastal and Estuarine Sciences, 4, 1–16 (1987)

    Google Scholar 

  5. Businger, J.A., Wyngaard, J.C., Izumi, Y. and Bradley, E.F.: Flux profile relationships in the atmospheric surface layer. J. Atmos. Sci., 28, 181–189 (1971)

    Google Scholar 

  6. Deardorff, J.W.: A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J. Fluid Mech., 41, 453–480 (1970)

    Google Scholar 

  7. Deardorff, J.W.: On the magnitude of the subgrid scale eddy coefficient. J. Comput. Phys., 7, 120–133 (1971)

    Google Scholar 

  8. Domaradski, J.A., Liu, W., Brachet, M.E.: An analysis of subgrid-scale interactions in numerically simulated isotropic turbulence. Phys Fluids A, 5, 1749–1759 (1993)

    Google Scholar 

  9. Germano, M., Piomelli, U., Moin, P., and Cabot, W.H.: A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A, 3(7), 1760–1765 (1991)

    Google Scholar 

  10. Güting, P.M. and Hutter, K.: Modeling wind-induced circulation in the homogeneous Lake Constance using \(k\)-\(\varepsilon \) closure. Aquat. Sci., 60, 266–277 (1998)

    Google Scholar 

  11. Haidvogel, D.B., Wilkin, J.L. and Young, R.: A semi-spectral primitive equation ocean circulation model using vertical sigma and orthogonal curvilinear horizontal coordinates. J. Comput. Phys., 94, 151–185 (1991)

    Google Scholar 

  12. Heinz, G.: Strömungen im Bodensee. Ergebnisse einer Meßkampagne im Herbst 1993. Mitteilungen der VAW der ETH Zürich, 135, 237p (1995)

    Google Scholar 

  13. Higgins, C.W., Parlange, M.B. and Meneveau, C.: The heat flux and the temperature gradient in the lower atmosphere. Geophys. Res. Lett., 31, L22105 (2004)

    Google Scholar 

  14. Hollan, E.: Strömungsmessungen im Bodensee. Sechster Bericht der AWBR, 6, 112–187 (1974)

    Google Scholar 

  15. Horst, T. W., Kleissl, J., Lenschow, D.H., Meneveau, C., Moeng, C.-H., Parlange, M.B., Sullivan, P.P. and Weil, J.C.: HATS: Field observations to obtain spatially filtered turbulence fields from transverse arrays of sonic anemometers in the atmospheric surface layer. J. Atmos. Sci., 61, 1566–1581 (2004)

    Google Scholar 

  16. Hutter, K.: Mathematische Vorhersage von barotropen und baroklinen Prozessen im Zürich- und Luganersee. Vierteljahrschrift der Naturforschenden Gesellschaft in Zürich, 129, 51–92 (1984)

    Google Scholar 

  17. Hutter, K. and Jöhnk, K.: Continuum Methods of Physical Modeling. Springer Verlag, Berlin etc., 635 p. (2004)

    Google Scholar 

  18. Ip, J.T.C. and Lynch, D.R.: Three-dimensional shallow water hydrodynamics on finite elements: Nonlinear time-stepping prognostic model. Report NML-94-1, Numerical Methods Laboratory, Dartmouth College, Hanover NH, USA 03755 (1994)

    Google Scholar 

  19. Killworth, P.D., Stainforth, D., Webb, D.J. and Paterson, S.M.: The development of a free-surface Bryan-Cox-Semtner ocean model. J. Phys. Oceanogr., 21, 1211–1223 (1991)

    Google Scholar 

  20. Lewellen, W.S. and Teske, M.: Prediction of the Monin-Obukhov similarity functions from an invariant model of turbulence. J. Atoms. Sci., 30, 1340–1345 (1973)

    Google Scholar 

  21. Lilly, D.K.: The representation of small-scale turbulence in numerical simulation experiments. Proc. IBM Scientific Computing Symposium on Environmental Sciences, Thomas J. Watson Research Center, Yorktown Heights, NY, 195–210 (1967)

    Google Scholar 

  22. Liu, S., Katz, J. and Meneveau, C.: Evolution and modeling of subgrid scales during rapid straining of turbulence. J. Fluid Mech., 387, 281–320 (1999)

    Google Scholar 

  23. Maiss, M., Ilmberger, J. and Münnich, K.O.: Vertical mixing in Überlingersee (Lake Constance) traced by a SF\(_6\) and heat. Aquatic Sciences, 56(4), 329–347 (1994)

    Google Scholar 

  24. Mansour, N.N., Moin, P., Reynolds, W.C. and Ferziger, J.H.: Improved methods for large-eddy simulations of turbulence. In: Durst, editor. Proceeding of the 1st Symposium on Turbulent Shear Flows. University Park, PA, USA. Berlin, Springer (1979)

    Google Scholar 

  25. Martin, P.J.: Simulation of the mixed layer at OWS November and Papa with several models. J. Geophys. Res.,90, 903–916 (1985)

    Google Scholar 

  26. Mason, P.J. and Callen, N.S.: On the magnitude of the subgrid-scale eddy coefficient in large-eddy simulations of turbulent channel flow. J. Fluid Mech., 162, 439–462 (1986)

    Google Scholar 

  27. Mellor, G.L.: Analytic prediction of the properties of stratified planetary surface layers. J. Atoms. Sci., 30, 1061–1069 (1973)

    Google Scholar 

  28. Mellor, G.L. and Yamada, T.: A hierarchy of turbulence closure models for planetary boundary layers. J. Atoms. Sci., 31, 1791–1806 (1974)

    Google Scholar 

  29. Mellor, G.L., and Yamada, T.: Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys. Space Phys., 20, 851–875 (1982)

    Google Scholar 

  30. Meneveau, C. and Katz, J.: Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech., 32, 1–32 (2000)

    Google Scholar 

  31. Moin P. and Kim, J.: Numerical investigation of turbulent channel flow. J. Fluid Mech., 118, 341–377 (1982)

    Google Scholar 

  32. O’Neil J, Meneveau C.: Subgrid-scale stresses and their modeling in a turbulent plane wake. J. Fluid Mech., 349, 253–293 (1997)

    Google Scholar 

  33. Peeters, F.: Horizontale Mischung in Seen. PhD thesis, ETH Zürich (1994)

    Google Scholar 

  34. Piomelli, U., Moin, P. and Perziger, J.H.: Model consistency in large eddy simulation of turbulent channel flows. Phys. Fluids, 31, 1884–1891 (1988)

    Google Scholar 

  35. Pohlmann, T.: A three dimensional circulation model of the South China Sea. In: Nihoul and Jamart (eds.) Three-Dimensional Models of Marine and Estuarine, Dynamics, 245–268 (1987)

    Google Scholar 

  36. Piomelli, U., Perziger, J.H. and Moin, P.: New approximate boundary conditions for large eddy simulations of wall-boundary flows. Phys. Fluids A, 1, 1061–1068 (1987)

    Google Scholar 

  37. Porté-Agel, F., Meneveau, C. and Parlange, M.B.: A scale-dependent dynamic model for large-eddy simulation: application to a neutral atmospheric boundary layer. J. Fluid Mech., 415, 261–284 (2000)

    Google Scholar 

  38. Porté-Agel, F, Parlange, M.B., Meneveau, C. and Eichinger, W.: A priori field study of the subgrid-scale heat fluxes and dissipation in the atmospheric surface layer. J. Atmospheric Sciences, 58, 2673–2698 (2001)

    Google Scholar 

  39. Porté-Agel, F., Pahlow, M., Meneveau, C. and Parlange, M.B.: Atmospheric stability effect on subgrid-scale physics for large-eddy simulation. Adv. in Water Resources, 24(9-10), 1085–1102 (2001)

    Google Scholar 

  40. Smagorinsky, J.S.: General circulation experiments with the primitive equations: I. The basic experiment. Mon. Weather Rev., 91, 99–164 (1963)

    Google Scholar 

  41. Song, Y. and Haidvogel, D.B.: A semi-implicit ocean circulation model using a generalized topography-following coordinate. J. Comp. Phys., 115, 228–244 (1994)

    Google Scholar 

  42. Sullivan, P.P., Horst, T.W., Lenschow, D.H., Moeng, C.H. and Weil, J.C.: Structure of subfilter-scale fluxes in the atmospheric surface layer with application to large-eddy simulation modeling. J. Fluid Mech., 482, 101–139 (2003)

    Google Scholar 

  43. Tong, C., Wyngaard, J.C. and Brasseur. J.G.: Experimental study of the subgrid-scale stresses in the atmospheric surface layer. J. Atmos. Sci., 56, 2277–2292 (1999)

    Google Scholar 

  44. Uberoi, M.S.: Effect of wind-tunnel contraction on free stream turbulence. J. Aeronaut Sci., 23, 754–764 (1956)

    Google Scholar 

  45. Umlauf, L. and Burchard, H.: A generic length-scale equation for geophysical turbulence models. J. Marine Res., 61, 235–265 (2003)

    Google Scholar 

  46. Umlauf, L., Burchard, H. and Hutter, K.: Extending the \(k\)-\(\omega \) turbulent model towards oceanic applications. Ocean Modelling, 5, 195–218 (2003)

    Google Scholar 

  47. Walker, S.J.: A 3-dimensional non-linear variable density hydrodynamic model with curvilinear coordinates. Tech. Rep. OMR-60/00, CSIRO Division of Oceanography (1995)

    Google Scholar 

  48. Wang, Y., Hutter, K. and Bäuerle, E.: Wind-induced baroclinic response of Lake Constance. Annales Geophysicae, 18, 1488–1501 (2000)

    Google Scholar 

  49. Wang, Y.: Comparing different numerical treatments of advection terms for wind-induced circulations in Lake Constance. In: Straughan, B., Greve, R., Ehrentraut, H., and Wang, Y. (Eds.). Continuum Mechanics and Applications in Geophysics and the Environment. Springer, Heidelberg-New York, 368–393 (2001)

    Google Scholar 

  50. Wang, Y.: Importance of subgrid-scale parameterization in numerical simulations of lake circulation. Adv. Water Res., 26(3), 277–294 (2003)

    Google Scholar 

  51. Warner, J.C., Sherwood, C.R., Arango, H.G. and Signell, R.P.: Performance of four Turbulence Closure Methods Implemented using a Generic Length Scale Method. Ocean Modelling, 8, 81–113 (2005)

    Google Scholar 

  52. Wilcox, D.C.: Turbulence Modeling for CFD. DCW Industries, Inc., La Ca\(\tilde{\rm {n}}\)ada, CA., p. 540 (1998)

    Google Scholar 

  53. Zenger, A., Ilmberger, J., Heinz, G., Schimmele, M. and Münnich, K.O.: Untersuchungen zur Struktur der internen Seiches des Bodensees. Wasserwirtschaft, 79(12), 616–624 (1989)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. c/o Laboratory of Hydraulics, Hydrology and Glaciology at ETH, ETH Hönggerberg, HIT F43, 8093, Zürich, Switzerland

    Kolumban Hutter

  2. Chair of Fluid Dynamics, Department of Mechanical Engineering, TU Darmstadt, Otto-Berndt-Strasse 2, 64287, Darmstadt, Germany

    Yongqi Wang

  3. P.P. Shirshov Institute of Oceanology, Russian Academy of Sciences, prospect Mira 1, Kaliningrad, 236022, Russia

    Irina P. Chubarenko

Authors
  1. Kolumban Hutter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongqi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Irina P. Chubarenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kolumban Hutter .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Hutter, K., Wang, Y., Chubarenko, I.P. (2014). Subgrid-Scale Parameterization in Numerical Simulations of Lake Circulation. In: Physics of Lakes. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-00473-0_27

Download citation

Publish with us

Policies and ethics

Search

Navigation