NCAR TOY - February 2008
4
Method of Multiple Scale --  Embid and Majda style.
To avoid secularity the second order term must be smaller than the leading order term.
Embid, P and Majda, A ÒLow Froude number limiting dynamics for stably stratified flow with small or finite Rossby number.Ó Geophysical and Astrophysical Fluid Dynamics, 87 pp 1-50
All about the method of multiple scales.
Andy and Pedro had already worked out the framework for looking at these dynamics,

They had already shown that the operator is skew-hermitian