Abstract/Summary

The chaotic nature of atmospheric dynamics presents a central challenge to the accurate prediction of future weather. It is a well-known fact that the predictability of instantaneous weather is inherently limited to about two weeks, beyond which skilful prediction is impossible no matter how small the initial error is. This study seeks to advance the knowledge related to the limited predictability by addressing three theoretical topics. The first topic concerns the mathematical origins of the predictability barrier. In a simplified context, what appears to be a contradiction between the finite-time limit and the regularity of the governing equations is reconciled through understanding the practical role of the slope of the energy spectrum in the latter. The next topic explores the properties of error growth under the hybrid k −3 - k − 5 3 energy spectrum that approximates the atmosphere. With the aid of simplified turbulence models, the synoptic-scale k −3 range is found to substantially dampen the fast error growth characteristic of a k − 5 3 spectrum in the first decade of wavenumbers in the mesoscale range, so that the fast growth may only emerge when global numerical weather prediction models begin to resolve scales on the order of a few kilometres. The final topic focusses on the relationship between metrics that quantify error growth and predictability. Two popular metrics, namely the Continuous Ranked Probability Score and the root-mean-square error, are found to be mathematically related under certain conditions. Simulated results show that the relationship approximately holds in idealised turbulent environments despite the required conditions not being fully met. This study demonstrates that simple models can often be useful in identifying key mechanisms of error growth that lead to the limit of predictability. Future work involving simple models is encouraged to substantiate such understanding further.