Here the nondimensional

The boundary conditions are to be applied on the boundary at and . Note that Rayleigh number is positive when and is the ratio of buoyancy force and viscous force. On the other hand, Prandtl number depends on the properties of the fluid: ratio of molecular diffusion due to momentum and heat.

We now eliminate all the dependent variables except . The curl of equation (11) gives

where we is the vorticity of the flow. In the above equation we have used the vector identity

In particular for the vertical velocity we have

where is the horizontal laplacian. This equation can be written is

If we eliminate then we get same equation (17) satisfied by .

From the vertical component of we have

from which we can determine the vertical vorticity component. From the definition of and the equation of continuity, we get

and these equation can be used to find and .