In this manuscript, it is analysed the onset and evolution of natural
convection of an incompressible fluid between spherical shells. The shells are
kept at a fixed temperature difference and aspect ratio, and the
Rayleigh-Benard convection is driven by different radial gravity profiles. The
analysis has been carried out by using a finite difference scheme to solve the
three-dimensional Navier-Stokes equations in spherical coordinates. Numerical
results are compared with theoretical predictions from linear and non-linear
stability analysis, and differ from the expected critical Rayleigh number Ra_c
= 1708 by less than 1 percent. In the range of Prandlt numbers Pr studied, and
for all the different gravity profiles analysed, the system presents a
dependence on its starting condition and flow history. Even in the region just
above the onset of convection, two stable states are observed, with qualitative
and quantitative differences, and exploring higher values of Ra introduces new
modes and time dependency phenomena in the flow. These results are corroborated
by spectral analysis