In light of
the schematic picture above,
we define the temperature at the bottom of the plate
to be T_{p},
the temperature of the fluid below the plate
to be T_{cp},
and the temperature of the internal fluid outside the plate
to be T_{c}.
Since the temperature of the fluid below the plate
T_{cp} is specified by


(5) 
it is sufficient to evaluate T_{p}
in order to obtain T_{cp}.
Using boundarylayer theory (Turcotte and Schubert 1982), the relation
between the Nusselt number and the Rayleigh number is given by


(6) 
where the prime denotes values for the convection under the plate,
and a is the horizontal size of a convection cell.
Since temperature and its gradient must be continuous at the boundary,
we have


(7) 
For the fluid beneath the plate,
the thickness of the layer is 1d
and temperature at the top of the fluid layer
is the temperature just beneath the plate T_{p},
Ra' is expressed in terms of the Rayleigh number
for the whole layer Ra as,


(8) 
From these equations, a relation between the temperature
under the plate T_{p}
and the Rayleigh number Ra can be straightforwardly obtained.


(9) 
From (9),
T_{p} can be calculated
for a given value of the Rayleigh number.
Further, the averaged temperature of the fluid under the plate
T_{cp}
= (T_{1}+T_{p})/2
and temperature difference induced by the plate
can also be estimated.