PROBLEMS OF GEOCOSMOS

Proceedings of the 7th International Conference "Problems of Geocosmos" (St. Petersburg, Russia, 26-30 May 2008)
ω cr ( Pr)
≅ const,
( ) 0 , Pr R Pr ℜ cr ≅
R 0 = const
in this larger Prandtl number case. This simplification becomes possible for two reasons. Firstly, the inner
rigid core is relatively small and, secondly, the change of the relative (integrated) convective efficiency is
also small for our b≤0.35 inspite of the sharp buoyancy source rise ~b -3 at the inner boundary.
The real dimensional value of compositional power flux Qi~b 2 strongly depends on the inner core
radius and rises sharply as b grows up to a certain critical radius that is slightly greater than one in the current
Earth value (see Starchenko 2003 for details). When the radius of the inner core is small (consequently Qi, is
small too), the Rayleigh number of flow is also small. Thus, it is hard to excite the compositional convection
when the inner core is relatively small because the critical Rayleigh numbers in Table III are too large.
Small thermal Prandtl numbers.
To describe thermal convection we neglect the non-adiabatic input heat flux and set Qi=0. Then for given
Qo>0, we may set Qo=1 in (4) as before as long as redefine the Rayleigh number to be
2 2 −1
R = 4π
Qo
( αgor
o ) TCP
/ κν with the original value of Qo.
The relative efficiency of such convection is determined by the integral IS from (8). In the case of
pure thermal convection we can safely stay in the small Prandtl number limit because the typical thermal
diffusion coefficient κ is larger than viscosity ν. See e.g. Braginsky and Roberts (1995) who have suggested
this molecular thermal Prandtl number Pr≈0.1 for the Earth’s outer liquid core.
We solve the complete system (20) with boundary conditions (21) and (22) for this small Prandtl
number case. For various Prandtl numbers, our local results for this system are listed in Table III. We can
conclude from Table III that the critical Rayleigh number for almost adiabatic thermal convection becomes
smaller as the radius of the inner core increases. Possibly this help to maintain thermal convection even
during the later stages of the planetary evolution when the internal radioactive heat sources become too
weak.
Table III. Critical parameters for thermal ( Pr