Effective scaling for the onset of thermal convection in rotating planetary cores

Corrections to the leading order asymptotics should not be ignored for the onset of thermal convection in a spherical shell driven by differential heating.

The critical Rayleigh number Rac for the onset of thermal convection in a rotating sphere or spherical shell scales with the Ekman number E as:

Rac = Rc * E^-4/3

However, several corrections should be applied to this formula, as shown by Dormy et al 2004 (J. Fluid Mech.):

For full spheres with internal heating (formula 4.1a from Dormy et al 2004): 
Rac = Rc * E^-4/3 + R1 * E^-7/6 + R2 * E^-1

and for spherical shells with differential heating (convection starts at the tangent cylinder, formula 3.25a from Dormy et al 2004):
Rac = Rc * E^-4/3 + R1 * E^-7/6 + R2 * E^-10/9

This last expression, in conjunction with a large R2 prefactor, makes the effective exponent for Rac converge slowly towards -4/3

This figure shows this effective exponent as a function of Ekman number for the spherical shell with differential heating and the full sphere with internal heating.
Prandtl number is 1.
The values are obtained using the formula 3.25 with values 5.4 and formula 4.1a with values 4.4 from Dormy et al (2004).
For differential heating in the spherical shell, a significant error is made if the corrections are ignored, even for Ekman number as low as 1e-8