The use of adaptive optics entails the design of a controller. This requires the development of a model of the plant to be controlled, which, in this case, consists of the atmosphere through which light is traveling. In optics, Zernike polynomials are used as a basis set for the expansion of wavefront phase distortions. Due to the turbulence induced stochastic nature of the underlying process involved, the spatial-temporal correlation functions of the Zernike polynomial phase expansion coefficients must be evaluated if a proper stochastic model of the plant is to be developed and adaptive optics is to be employed. In this work, these correlation functions are developed using a layered atmospheric model which takes into account wind effects and anisoplanatism. Calculations are provided for the first few low-order Zernike modes. Using these correlation functions, an underlying linear, stochastic, dynamical system, which represents the atmosphere and is adequate for control synthesis, is identified. Within an acceptable error bound, the correlation functions of this system are representative of the calculated functions. The deformable mirror is also modeled, output equations are specified, and the complete system is constructed.