  In this article we study the lifespan and asymptotics (in the large rotation
and stratification regime) for the Primitive system for highly ill-prepared
initial data in critical spaces. Compared to our previous works, we simplified
the proof and made it adaptable to the Rotating fluids system with highly
ill-prepared initial data decomposed as a sum of 2D horizontal part and a very
large 3D part. We also provide explicit convergence rates.
