In order to present a quantitative indicator for the onset of instability, in this paper, the critical points of a stratified gravitational flow on a slope are found and analyzed. These points are obtained by means of the solution of the two-dimensional Navier-Stokes equations via the standard Arakawa-C finite-difference method. Results show that in the marginal Richardson numbers, the critical points begin to originate. Also, the cyclic evolution in the temporal differenced density field in the vicinity of the critical points is used as a quantitative criterion of the onset of mixing. Therefore, it is possible to predict the beginning of the mixing phenomenon via analysis of only a limited number of critical points.