Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-25T06:18:22.156Z Has data issue: false hasContentIssue false
  • English
  • Français

Effects of truncation in modal representations of thermal convection

Published online by Cambridge University Press:  20 April 2006

Philip S. Marcus
Philip S. Marcus
Affiliation:
Present address: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass. Center for Radiophysics and Space Research, Cornell University
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine the Galerkin (including single-mode and Lorenz-type) equations for convection in a sphere to determine which physical processes are neglected when the equations of motion are truncated too severely. We test our conclusions by calculating solutions to the equations of motion for different values of the Rayleigh number and for different values of the limit of the horizontal spatial resolution. We show how the gross features of the flow such as the mean temperature gradient, central temperature, boundary-layer thickness, kinetic energy and temperature variance spectra, and energy production rates are affected by truncation in the horizontal direction. We find that the transitions from steady-state to periodic, and then to aperiodic convection depend not only on Rayleigh number but also very strongly on the horizontal resolution of the calculation. All of our models are well resolved in the vertical direction, so the transitions do not appear to be due to poorly resolved boundary layers. One of the effects of truncation is to enhance the high-wavenumber end of the kinetic energy and thermal variance spectra. Our numerical examples indicate that, as long as the kinetic energy spectrum decreases with wavenumber, a truncation gives a qualitatively correct solution.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 103 , February 1981 , pp. 241 - 255
Copyright
© 1981 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Clever, R. M. & Busse, F. H. 1974 Transition to time-dependent convection. J. Fluid Mech. 66, 67–79.Google Scholar
Curry, J. H. 1978 A generalized Lorenz system. Commun. Math. Phys. 60, 193–204.Google Scholar
Deardorff, J. W. & Willis, G. E. 1967 Investigation of turbulent thermal convection between horizontal plates. J. Fluid Mech. 28, 657–704.Google Scholar
Gollub, J. P. & Benson, S. V. 1980 Time-dependent instabilities and the transition to turbulent convection. J. Fluid Mech. 100, 449–470.Google Scholar
Latour, J., Spiegel, E. A., Toomre, J. & Zahn, J.-P. 1976 Stellar convection theory. I. The anelastic modal equations. Astrophys. J. 207, 233–243.Google Scholar
Lorenz, E. N. 1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar
McLaughlin, J. B. & Martin, P. C. 1975 Transition to turbulence in a statically stressed fluid system. Phys. Rev. A 12, 186–203.Google Scholar
Marcus, P. S. 1978 Nonlinear thermal convection in Boussinesq fluids and ideal gases with plane-parallel and spherical geometries. Ph.D. thesis. Ann Arbor, MI: University Microfilms.
Marcus, P. S. 1979 Stellar convection. I. Modal equations in spheres and spherical shells. Astrophys. J. 231, 176–192.Google Scholar
Marcus, P. S. 1980a Stellar convection. II. A multi-mode numeric solution for convection in spheres. Astrophys. J. 239, 622–639.Google Scholar
Marcus, P. S. 1980b Stellar convection. III. Convection at large Rayleigh numbers. Astrophys. J. 240, 203–217.Google Scholar
Ruelle, D. & Takens, F. 1971 On the nature of turbulence. Commun. Math. Phys. 20, 167–192.Google Scholar
Toomre, J., Gough, D. O. & Spiegel, E. A. 1977 Numerical solutions of single-mode convection equations. J. Fluid Mech. 79, 1–31.Google Scholar
Toomre, J., Zahn, J.-P., Latour, J. & Spiegel, E. A. 1976 Stellar convection theory. II. Single-mode study of the second convection zone in an A-type star. Astrophys. J. 207, 545–563.Google Scholar