Finite Dimensional Behaviors of the Primitive Equations Under Small Depth Assumption

In this article, we study the asymptotic degrees of freedom for solutions to the primitive equation (PEs for brevity). More precisely, we will prove that the long-time behavior of solutions to PEs is determined by the set of either finite Fourier modes, line elements, or volume elements. Our results show that the long-time behavior of the PEs is determined by the baro-tropic flows that are independent of the vertical direction in ?³. This study builds upon the previous article by the author concerning the existence and uniqueness of strong solutions to the PEs in thin domains.