(A.1)

since from the gas law and . Temperature tendency therefore is given by
 

(A.2)

(A.3)

Continuity and Thermodynamics lead to

(A.8)

Defining hydrostatic reference and perturbation,

(A.10)

which can be written as

(A.11)

This can be expanded as

(A.12)

In terms of ρ , this is

(A.13)

which is

(A.14)

This can be expressed in terms of temperature and pressure perturbations for the buoyancy term because
 

(A.15)

So

(A.19)

where and are the values of at the top and surface and .

(A.20)

Therefore

(A.21)

Expanding total derivatives noting that and and also that is hydrostatic