The full continuity equation eq.(3) may be expanded about the mean state density field by first writing it in the form:
and then using:
which leads to:
on neglecting second-order and higher terms.
We shall refer to eqs.(11), (12), (13) and (8), written as:
as the quasi-Boussinesq (QB) equations of motion. They represent a closed system for the four unknowns :
, , and .