Structural sensitivities of soft and steep nonlinear global modes in spatially developing media

Structural sensitivities of soft and steep nonlinear global modes arising in a complex Ginzburg–Landau model are investigated by calculating the leading-order variation of their amplitude and frequency to an open-loop forcing and a closed-loop perturbation. The soft global mode is found to be sensitive to both the open-loop and closed perturbations in the region of linear absolute instability where its amplitude is not negligible. In particular, the frequency of the soft global mode exhibits large sensitivity at the location where the frequency of soft global mode was shown to be determined in the previous WKBJ analysis. On the other hand, the steep global mode shows a large response in the amplitude and the frequency to the open-loop perturbation located far upstream. To the closed-loop perturbation, the steep global mode is most sensitive at the location where the stationary front is located, consistent with the previous WKBJ analysis. Finally, the sensitivities analyzed for the fully nonlinear global mode are compared to those obtained from a weakly nonlinear analysis. It shows that the weakly nonlinear analysis fails to capture the sensitivity behavior obtained from the fully nonlinear global mode particularly under the strong advection yielding steep global mode.