The onset of thermal convection
in rotating spherical shells
Emmanuel Dormy1
Andrew Soward2
Christopher Jones2
Dominique Jault3
Philippe cardin3
1Institut de Physique du Globe de Paris/CNRS, 4 place Jussieu, F-75252, Paris, France.
2Department of Mathematical Sciences, University of Exeter,
Exeter, EX4 4QE, UK
3LGIT/CNRS, Universit4e Joseph Fourier BP53, 38041 Grenoble
Cedex 9, France
J. Fluid Mech. (2004), vol. 501, pp. 4370.
Abstract.
The correct asymptotic theory for the linear onset of instability of a
Boussinesq
fluid rotating rapidly in a self-gravitating sphere containing a uniform
distribution
of heat sources was given recently by Jones et al. (2000). Their analysis
confirmed
the established picture that instability at small Ekman number E is
characterized by
quasi-geostrophic thermal Rossby waves, which vary slowly in the axial
direction on
the scale of the sphere radius ro and have short azimuthal length scale
O(E1/3ro).
They also confirmed the localization of the convection about some cylinder
radius
s =sM roughly ro/2. Their novel contribution concerned the implementation
of global
stability conditions to determine, for the first time, the correct Rayleigh
number,
frequency and azimuthal wavenumber. Their analysis also predicted the value
of the
finite tilt angle of the radially elongated convective rolls to the
meridional planes. In
this paper, we study small-Ekman-number convection in a spherical
shell. When the
inner sphere radius ri is small (certainly less than sM), the Jones et
al. (2000) asymptotic
theory continues to apply, as we illustrate with the thick shell ri =0.35
ro. For a large
inner core, convection is localized adjacent to, but outside, its tangent
cylinder, as
proposed by Busse & Cuong (1977). We develop the asymptotic theory for the
radial
structure in that convective layer on its relatively long length scale
O(E2/9ro). The
leading-order asymptotic results and first-order corrections for the case
of stress-free
boundaries are obtained for a relatively thin shell ri =0.65 ro and
compared with
numerical results for the solution of the complete PDEs that govern the
full problem
at Ekman numbers as small as 107. We undertook the corresponding asymptotic
analysis and numerical simulation for the case in which there are no
internal heat
sources, but instead a temperature difference is maintained between the
inner and
outer boundaries. Since the temperature gradient increases sharply with
decreasing
radius, the onset of instability always occurs on the tangent cylinder
irrespective
of the size of the inner core radius. We investigate the case ri =0.35
ro. In every
case mentioned, we also apply rigid boundary conditions and determine the
O(E1/6)
corrections due to Ekman suction at the outer boundary. All analytic
predictions
for both stress-free and rigid (no-slip) boundaries compare favourably with
our full
numerics (always with Prandtl number unity), despite the fact that very
small Ekman
numbers are needed to reach a true asymptotic regime.