The equations are reduced to nonlinear operator equations of Hilbert space by using functional theory, based on each variable Hilbert space of the general system of atmospheric Navier-Stokes dynamics equations. Thereby, considered the equation for the overall characteristics on this basis, using necessary simplification, the underdetermined partial differential equations are reduced to nonlinear partial differential equations about generalized energy. Because of the nonlinear properties of the partial differential equations, the existence of weak solutions is attempted to consider. According to the analysis, in the turbulent closure process, if the nonlinear coefficient of the generalized energy equation has characteristics of strongly elliptic, according to the projection method of the continuous normal operator equations, that nonlinear partial differential equations of generalized energy has a projection determined, which approach the weak solution, while the external force is known or fixed. In last, the existence of weak solutions for generalized energy is obtained. On the movement of atmosphere, its energy conservation is important. Therefore, conserve-ation of generalized energy is discussed. And the conditions of reaching the conservation of general-ized energy is obtained in the last of this paper.