Editorial Type:
Article
Article Type:
Research Article

Convective Spherical Shell: II. With Rotation

B. Durney High Altitude Observatory, Boulder, Colo.

Search for other papers by B. Durney in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Sep 1968
DOI:
https://doi.org/10.1175/1520-0469(1968)025<0771:CSSIWR>2.0.CO;2
Page(s):
771–778
Restricted access

We are aware of a technical issue preventing figures and tables from showing in some newly published articles in the full-text HTML view.
While we are resolving the problem, please use the online PDF version of these articles to view figures and tables.

Abstract

The problem of a convective, rotating spherical shell is considered in Herring's approximation. The temperature is expanded in spherical harmonics, YLm(θ,Φ), and the velocity field in basic poloidal pLm(r,t) and toroidal tLm(r,t) vectors. The scalar pLm(r,t) [tLm(r,t)], together with YLm(θ,Φ), defines a basic poloidal [toroidal] vector. The equations for pLm(r,t) and tLm(r,t) with different L's are coupled by the Taylor number and two types of solutions are possible: symmetric or antisymmetric about the equator.

For the case of axial symmetry and for a Rayleigh number equal to 1500, we calculate the convective steady-state solution with rotation by successively increasing the Taylor number from zero, its value for no rotation. Using free surface boundary conditions, the relevant equations determine the radial and time-dependent parts of the temperature and velocity field, with the exception of t10(r), the lowest toroidal component of the axisymmetric solution having equatorial symmetry. The conservation of the total angular momentum in the direction of the axis of rotation then determines t10(r). The stabilizing effect of rotation on axisymmetric convection is specially important at the equator. For Taylor numbers larger than ∼502, axisymmetric convection is completely inhibited and the spherical shell rotates as a solid body.

Abstract

The problem of a convective, rotating spherical shell is considered in Herring's approximation. The temperature is expanded in spherical harmonics, YLm(θ,Φ), and the velocity field in basic poloidal pLm(r,t) and toroidal tLm(r,t) vectors. The scalar pLm(r,t) [tLm(r,t)], together with YLm(θ,Φ), defines a basic poloidal [toroidal] vector. The equations for pLm(r,t) and tLm(r,t) with different L's are coupled by the Taylor number and two types of solutions are possible: symmetric or antisymmetric about the equator.

For the case of axial symmetry and for a Rayleigh number equal to 1500, we calculate the convective steady-state solution with rotation by successively increasing the Taylor number from zero, its value for no rotation. Using free surface boundary conditions, the relevant equations determine the radial and time-dependent parts of the temperature and velocity field, with the exception of t10(r), the lowest toroidal component of the axisymmetric solution having equatorial symmetry. The conservation of the total angular momentum in the direction of the axis of rotation then determines t10(r). The stabilizing effect of rotation on axisymmetric convection is specially important at the equator. For Taylor numbers larger than ∼502, axisymmetric convection is completely inhibited and the spherical shell rotates as a solid body.

Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 342 105 31
PDF Downloads 43 14 0