where

a is the volume coefficient of thermal expansion, g the acceleration of gravity, L the thickness of the layer, Q the heat flow through the lower boundary, A the internal heat generation, k the thermal conductivity, k the thermal diffusivity and v the kinematic viscosity (kinematic viscosity = dynamic viscosity / density). The critical value of the Rayleigh number further depends on the boundary conditions:

1. For no shear stress on the upper and lower boundaries, the upper boundary held at a constant temperature and all heating from below (A = 0), Ra_c = 27 pi⁴/4 = 658. At this Rayleigh number the horizontal dimension of a cell is 2.8L.

2. For no slip on the boundaries, the upper boundary held at a constant temperature and all heating from below (A = 0), Ra_c = 1708. At this Rayleigh number the horizontal dimension of a cell is 2.0L.

3. For no shear stress on the boundaries, a constant heat flux across the upper boundary and all heating from within the fluid (Q = 0) Ra_c is 868. At this Rayleigh number the horizontal dimension of a cell is 3.5L.

4. For no slip on the boundaries, a constant heat flux across the upper boundary and all heating from within the fluid (Q = 0), Ra_c is 2772. At this Rayleigh number the horizontal dimension of a cell is 2.4L.
 

Thus, although the exact value of the critical Rayleigh number Ra_c depends on the shape of the fluid system, the boundary conditions and the details of heating, it is clear in all cases that Ra_c is of the order of 10³ and that the horizontal cell dimension at the critical Rayleigh number is about two to three times the thickness of the convecting layer. For convection to be vigorous with little heat transported by conduction, the Rayleigh number must be about 10⁵. If the Rayleigh number exceeds 10⁶, then convection is likely to become more irregular.