Hostname: page-component-5b777bbd6c-j65dx Total loading time: 0 Render date: 2025-06-24T04:44:10.394Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite-amplitude thermal convection in a spherical shell

Published online by Cambridge University Press:  29 March 2006

Richard E. Young
Richard E. Young
Affiliation:
Advanced Study Program, National Center for Atmospheric Research, Boulder, Colorado 80302 Present address: Space Science Division, NASA, Ames Research Center, Moffett Field, California 94035.
Get access Rights & Permissions [Opens in a new window]

Abstract

The properties of finite-amplitude thermal convection for a Boussinesq fluid contained in a spherical shell are investigated. All nonlinear terms are retained in the equations, and both axisymmetric and non-axisymmetric solutions are studied. The velocity is expanded in terms of poloidal and toroidal vectors. Spherical surface harmonics resolve the horizontal structure of the flow, but finite differences are used in the vertical. With a few modifications, the transform method developed by Orszag (1970) is used to calculate the nonlinear terms, while Green's function techniques are applied to the poloidal equation and diffusion terms.

Axisymmetric solutions become unstable to non-axisymmetric perturbations at values of the Rayleigh number that depend on Prandtl number and shell thickness. However, even when stable, axisymmetric solutions are not a preferred solution to the full equations; steady non-axisymmetric solutions are obtained for the same parameter values. Initial conditions determine the characteristics of the finite-amplitude solutions, including, in the cases of non-axisymmetry, whether or not a steady state is achieved. Transitions in horizontal flow structure can occur, accompanied by a transition in functional dependence of heat flux on Rayleigh number. The dominant modes in the solutions are usually the modes most unstable to the onset of convection, but not always.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 63 , Issue 4 , 15 May 1974 , pp. 695 - 721
Copyright
© 1974 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.)

Article purchase

Temporarily unavailable

References

Batchelor, G. K. 1967 An Introduction to Fluid Dynumim. Cambridge University Press.
Benjamin, T. B. 1970 Upstream influence. J. Fluid Mech. 40, 49–79.Google Scholar
Flammer, C. 1957 Spheroidal Wave Functions. Stanford University Press.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Maxworthy, T. 1970 The flow created by a sphere moving along the axis of a rotating, slightly viscous fluid. J. Fluid Mech. 40, 453–480.Google Scholar
Mcintyre, M.E. 1972 On Long's hypothesis of no upstream influence in uniformly stratified or rotating flow. J. Fluid Mech. 52, 209–243.Google Scholar
Miles, J. W. 1968 Lee waves in stratified flow. Part 1. Thin barrier. J. Fluid Mech. 32, 549–568.Google Scholar
Miles, J. W. 1969a The lee-wave regime for a slender body in a rotating flow. J. Pluid Mech. 36, 261–288.Google Scholar
Miles, J. W. 1969b Transient motion of a dipole in a rotating flow. J. Fluid Mech. 39, 433–442.Google Scholar
Miles, J. W. 1970 The oseenlet as a model for separated flow in a rotating viscous liquid. J. Fluid Mech. 42, 207–217.Google Scholar
Miles, J. W. 1971 Boundary-layer separation in a rotating flow. J. Pluid Mech. 45, 513–526.Google Scholar
Miles, J. W. 1972 Upstream influence of a dipole in a rotating flow. J. Fluid Mech. to appear.Google Scholar
Orloff, K. & Bossel, H. 1971 Laser-Doppler velocity measurements of swirling flows with upstream influence. Bull. Am. Phyys. Soc. 16 (2), 1331 (abstract only).Google Scholar
Pritchard, W. 1969 The motion generated by an obstacle moving along the axis of a uniformly rotating fluid. J. Fluid Mech. 39, 443–464.Google Scholar
Stewartson, K. 1968 On inviscid flow of a rotating fluid past an axially symmetric body using Oseen's equations. Quart. J. Mech. Appl. Math. 21, 353–373.Google Scholar
Trustrum, K. 1971 An Oseen model of the two-dimensional flow of a stratified fluid over an obstacle. J. Fluid Mech. 50, 177–188.Google Scholar