Speaker
Prof.
Andrew Soward
Description
The onset of instability of a Boussinesq fluid within a
rapidly rotating spherical shell is considered, when a thin
unstable layer lies adjacent to the inner sphere boundary,
radius r_i . This layer, which is of radial extent O(ε^2 r_i
), sits beneath the remaining (stably stratified) fluid. The
variable stratification is effected by a combination of
differential heating between the inner and outer boundaries
and a uniform distribution of heat sinks within the fluid.
As in previous small Ekman number E studies, convection
takes on the familiar cartridge belt structure which, in
view of the differential heating, is localised within a thin
layer adjacent to (but outside) the tangent cylinder to the
inner sphere (Dormy et al., J. Fluid Mech., vol. 501, 2004,
pp. 43–70). We consider the situation when the domain of
convection sits entirely within the unstable layer which
requires that E ^{1/8} ≪ ε. In this case the axial extent of
the convecting column from the equatorial plane is small and
of size O(ε r_i ). We investigate the eigensolutions of the
ordinary differential equation governing the axial structure
both numerically for moderate (but small) values of the
stratification parameter ε and analytically for ε ≪ 1. At
the lowest order of the expansion in powers of ε, the
eigenmodes resemble those for the plane layer convective
models of the classical form described by Chandrasekhar
(Hydrodynamic and Hydromagnetic Stability, 1961). In
particular, the eigenmodes exhibit the features of exchange
of stabilities and over- stability although these overstable
modes only occur for Prandtl number P less than unity. The
onset of convection at large P is steady but for small P it
is oscillatory with frequency ±Ω. At the next order,
curvature effects remove any plane layer degeneracies.
Notably, the exchange of stabilities modes oscillate at low
frequency causing the short axial columns to propagate as a
wave with a small angular velocity (slow modes), while the
magnitudes of both the Rayleigh number and frequency of the
two overstable modes (fast modes) split. When P < 1 the slow
modes that exist at large azimuthal wavenumbers M make a
continuous transition to the preferred fast modes at small M
. At all values of P the critical Rayleigh number
corresponds to a mode exhibiting prograde propagation,
whether it be a fast or slow mode. This feature is shared by
the uniform classical convective shell models, as well as
Busse’s annulus model (J. Fluid Mech., vol. 44, 1970, pp.
441–460). Those models do not possess any stable
stratification and typically are prone to easily excitable
Rossby or inertial modes of convection at small P . By way
of contrast these structures can not exist in our model for
small ε due to the viscous damping in the outer thick stable
region.
