Semi-Circle Theorem of Unstable Spectrum Distribution of Heterotropic Perturbation and the Upper Bound Estimation of Its Growth Rate

Based on the theoretical analysis on the unstable spectrum distribution of heterotropic perturbation, the semi-circle theorem of such spectrum distribution is obtained -- the spectrum distributed within an upper semi-circle domain with radius R0 and taking origin as its circle center in the complex plane, and the upper bound estimation of growth rate is given simultaneously. It is found that the smaller the horizontal scale of perturbation is, the higher the model top located, the greater the estimated value of upper bound of such growth rate is. Also the increase of vertical shear of wind and latitude have the positive contribution to the increase of such growth rate. Finally when stratified stability decreases, the relative maximal growth rate increases, while the maximal growth rate decreases instead.