Approximation of the quasigeostrophic system with the primitive systems

In this paper we show that the quasigeostrophic system is well approximated by the primitive systems. More precisely, we prove that if the initial data are weakly well‐prepared then the maximal time existence of the regular solution of the primitive system goes to infinity and the regular solution goes to the solution of the quasigeostrophic system, strongly on an arbitrary time interval. By weakly well‐prepared initial data we mean that the initial data of the primitive systems is converging to an initial data with zero oscillating part, without any assumptions on the speed.