Abstract
We derive the order parameter equation which describes the evolution of spatio-temporal patterns close to the Bénard instability in a rotating large aspect ratio system for high Prandtl number fluids. Since this order parameter equation contains rather complicated nonlinear terms we present a model equation which can be obtained from the order parameter equation by suitable simplification of the nonlinearity. For this model equation we calculate the family of roll solutions and investigate their stability with respect to long scale instabilities and examine the onset of the Küppers-Lortz instability. Then we present spatiotemporal patterns which are obtained from a numerical evaluation of the model equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Busse, F.H.: Rep. Progr. Phys.41, 1929 (1978)
Busse, F.H.: In: Hydrodynamic Instabilities and the Transition to Turbulence. Topics in Applied Physics, Vol. 45, Swinney, H.L., Gollub, J.P. (eds.). Berlin, Heidelberg, New York: Springer 1981
Küppers, G., Lortz, D.: J. Fluid Mech.35, 609 (1969)
Küppers, G.: Phys. Lett. A32, 7 (1970)
Krishnamurti, R.: In: Eighth symposium on naval hydrodynamics. Washington: Office of Naval Research 1971
Busse, F.H.: Transition to turbulence via the statistical limit cycle route. In: Turbulence and chaotic phenomena in fluids. Tatsumi, T. (ed.). Amsterdam: Elsevier 1984
Busse, F.H., Heikes, K.E.: Science208, 173 (1980)
Busse, F.H., Clever, R.M.: In: Recent developments in theoretical and experimental fluid mechanics. Müller, U., Roesner, K.G., Schmidt, B. (eds.), p. 376. Berlin, Heidelberg, New York: Springer 1979
Clever, R.M., Busse, F.H.: J. Fluid Mech.94, 609 (1979)
Busse, F.H.: In: Chaos and Order in Nature (Springer Series in Synergetics, Vol. 11, Haken, H. (ed.). Berlin, Heidelberg, New York: Springer 1981
Niemela, J.J., Donnelly, R.J.: Phys. Rev. Lett.57, 2524 (1986)
Zhong, F., Ecke, R., Steinberg, V.: Physica D51, 596 (1991)
Bodenschatz, E., Cannell, D.S., Ecke, R., HU, Y.C., Lerman, K., Ahlers, G.: Experiments on three systems with nonvariational aspects (Preprint 1992)
Fantz, M., Friedrich, R., Bestehorn, M., Haken, H.: Evolution of patterns in rotating Bénard convection. In: From phase transitions to chaos. Györgyi, G., Kondor, I., Sasvári, L., Tél, T. (eds.). Singapore: World Scientific 1992
Fantz, M., Friedrich, R., Bestehorn, M., Haken, H.: Physica D61, 147 (1992)
Haken, H.: Synergetics. An introduction, 3. Auflage. Springer Series in Synergetics, Vol. 1. Berlin, Heidelberg, New York: Springer 1983
Haken, H.: Advanced synergetics, 2. Aufl. Springer Series in Synergetics, Vol. 20. Berlin, Heidelberg, New York: Springer 1987
Friedrich, R., Bestehorn, M., Haken, H.: Int. Mod. Phys. B4, 365 (1990)
Tu, Y., Cross, M.C.: Phys. Rev. Lett.69, 2515 (1992)
Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability. Oxford: Oxford University Press 1961
Gertsberg, V.I., Shivashinsky, G.I.: Progr. Theor. Phys.65, 1219 (1981)
Proctor, M.R.E.: J. Fluid Mech.113, 469 (1981)
Bestehorn, M., Haken, H.: Phys. Lett. A99, 265 (1983); Z. Phys. B57, 329 (1984)
Pomeau, Y., Manneville, J.: J. Phys. Lett.40, 609 (1979)
Cross, M.C., Newell, A.C.: Physica D10, 299 (1984)
Swift, J., Hohenberg, P.C.: Phys. Rev. A15, 319 (1977)
Busse, F.H.: J. Fluid Mech.96, 243 (1980)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Neufeld, M., Friedrich, R. & Haken, H. Order parameter equation and model equation for high Prandtl number. Rayleigh-Bénard convection in a rotating large aspect ratio system. Z. Physik B - Condensed Matter 92, 243–256 (1993). https://doi.org/10.1007/BF01312183
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01312183