The deterministic Lorenz system is an important mathematical model for describing the law of atmospheric motion. It has a profound application background and has been extensively studied by many scholars. However, the climate environment is affected by abrupt factors, and the determined situation cannot fully explain the regularity of atmospheric motion. Based on this, in this paper, we study the progressive behavior of a stochastic Lorenz system driven by additive white noise. We firstly obtain the existence of the random attractor of the random Lorenz system by properly estimating the existence of random absorption set under the condition that the parameters are not constrained. We secondly prove that the stochastic Lorenz system converges to a deterministic system when the perturbation parameter tends to zero. Thirdly, by the theory of the upper semicontinuity, we obtain that random attractor converges to the global attractor in the sense of Hausdorff halfdistance. This indicates that the stability of the Lorenz system is not affected by environmental factors such as tsunamis, earthquakes, etc.