In this dissertation, the filamentation time in spherical coordinates on an isentropic surface has been derived and used as a diagnostic tool to analyze both midlatitude and tropical trough thinning processes, wave breaking of Antarctic polar vortex, and mei-yu frontogenesis events. Filamentation time is defined as the e-folding time for the vorticity gradient growth rate. The diagnostic is an accurate measure of stirring processes under the condition of slowly varying velocity gradients. For the trough events, the results indicate that the filamentation time for the tropical trough thinning event is generally longer than those for the midlatitude trough events. From the mei-yu frontogenesis study, the results suggest filamentation time diagnosis matches well with the deformation part of the frontogenetical function. In addition to the effects of stretching and shearing deformation, the filamentation time diagnostic contains the effects of divergence. For the calculation of filamentation time on isentropic surfaces in spherical coordinates, it is acceptable to ignore the curvature effects in the tropics, but in both the midlatitudes and the tropics, isentropic divergence effects should be retained for improved accuracy. The modified filamentation time formula which includes the diabatic heating effect is derived and analyzed with trough thinning events and mei-yu frontogenesis event. In the trough thinning events, the results suggest the filamentation time with diabatic heating effect could capture additional local filamentation zones caused by the local convection, but for the whole cutoff process, the diabatic heating effect seems to play secondary role in the filamentation time diagnosis. In the mei-yu frontogenesis event, including the diabatic heating effect is beneficial to match better with the frontogenetical function analysis. Barotropic simulation is presented for comparing the forcing time scale and the filamentation time scale. For high resolution simulation, Chebyshev spectral method with parallel domain decomposition method is developed. In conclusion of this dissertation, the results suggest that the filamentation time diagnostic can serve as a useful aid in the analysis of observed synoptic banded flows.