Bin Shi

Symplectic integrators are the established standard for long-term simulations of nearly-integrable Hamiltonian systems due to their preservation of geometric structures. However, they suffer from an inherent limitation: secular phase-shift errors. While the qualitative ''shape'' of invariant tori is preserved, the numerical solution gradually drifts along the torus, leading to a phase-lag accumulation that degrades long-term positional accuracy. Inspired by the Craig-Wayne-Bourgain (CWB) scheme, originally developed as an analytical tool for infinite-dimensional systems, we introduce a numerical operator that incorporates frequency updates into a dimension-enlarged Newton iteration to compute quasi-periodic solutions. Unlike conventional time-stepping integrators, our alternating numerical procedure eliminates phase-lag accumulation by directly solving for instantaneous positions and phase angles. Theoretically, provided sufficient computational resources, the phase error can be reduced arbitrarily, remaining independent of the total integration time. Our algorithm translates the Nash-Moser iteration into a practical numerical framework, marking a significant departure from traditional Kolmogorov-Arnold-Moser (KAM) theory. While KAM provides rigorous existence proofs, its requirement for global Diophantine conditions and the total exclusion of resonant sets render it numerically inaccessible. By employing a ''step-by-step'' exclusion process and incrementally enlarging the dimension, our algorithm resolves irrationality conditions locally. This approach demonstrates that the ''numerical irrationality problem'' is not an intrinsic barrier to computation, offering a constructive, executable alternative to the non-executable nature of global KAM-based methods.

In this paper, we explore optimal disturbances of blockings in the equivalent barotropic atmosphere using the conditional nonlinear optimal perturbation (CNOP) approach. Considering the initial blocking amplitude, the optimal disturbance exhibits a solitary wave-like pattern. As the size increases incrementally, the spatial pattern becomes more concentrated, and the nonlinear evolution becomes more pronounced. During the evolution, it only focuses on gradually intensifying the blocking amplitude without any other influence. Additionally, based on the medium-range experiments, the time-delay optimal disturbance appears to lead to larger errors, making it more challenging to predict. Considering the preexisting synoptic-scale eddies, the optimal disturbance displays a sharply concentrated pattern, even more concentrated by increasing the size. However, it is worth noting that the nonlinear evolution undergoes significant changes, compared to disturbances of the initial blocking amplitude. Meanwhile, we find that the optimal disturbance not only strongly impacts the amplitude of blockings but also their shape, making eddy straining and wave breaking more chaotic and predominant, further influencing the development of weather extremes. This suggests that blockings are more sensitive to perturbations of preexisting synoptic-scale eddies than initial blocking amplitudes. Furthermore, the perturbations of the synoptic-scale eddies are more likely to lead to the development of weather extremes, making them less predictable. In medium-range experiments, it is also found that time-delay disturbances result in larger errors, particularly during the decay period. Finally, we discuss how the variations of westerly wind influence optimal disturbances in spatial patterns and nonlinear evolution as well as their relation to predictability.