Abstract: The objects of this study are to propose the dynamical mechanism of the growth of the initial error in numerical weather prediction.The equations of error growth are obtained from the spectral primitive equation model. The results have shown that the error growth rate is determined by the tangent linear equations along the orbit of exact solutions.The nonlinear advective process of error itself has no effect on the error growth rate in the mean square error sense a:d only redistributes error over the spectral space.The trajectory instability is the essential cause to error growth. A method to compute the instability growth rate is developed, which could be used to study the instability of the time dependent state. Then trajectory instability of the Lorenz system is analyzed and the theoretical results are verified.