Professor Bo-Wen Shen

Department of Mathematics and Statistics, San Diego State University

Web: https://bwshen.sdsu.edu

A view that weather is chaotic was proposed and is recognized based on the pioneering work of Prof. Lorenz who first introduced the concept of deterministic chaos. Chaos is defined as the sensitive dependence of solutions on initial conditions, also known as the butterfly effect. The appearance of deterministic chaos suggests finite predictability, in contrast to the Laplacian view of deterministic predictability. After a follow-up study in 1972, the butterfly effect has come to be known as a metaphor for indicating that a tiny perturbation such as a butterfly’s flap may ultimately cause a large impact, such as the creation of a tornado. The two studies discussed above, as well as Lorenz’s 1969 study, laid the foundation for chaos theory that is viewed as one of the three scientific achievements of the 20th century, inspiring numerous studies in multiple fields.

Our daily experiences with weather predictions largely support Lorenz’s view of a finite predictability. On the other hand, under some conditions, better predictability, as compared to the predictability limit documented within the scientific literature, has also been observed. For example, while some studies have suggested that a theoretical limit of predictability is two weeks, recent advances in supercomputing and high-resolution modeling technology (Shen et al., 2006; 2013) have yielded promising 30-day simulations for high impact weather systems (Shen et al., 2010). In a brief report highlighting remarkable simulations in 2011, Dr. Richard Anthes, President Emeritus of the University Corporation for Atmospheric Research (UCAR), proposed a visioning question: is the atmosphere more predictable than we assume (Anthes, 2011)? Since that time, high-dimensional Lorenz models (LM) and a generalized LM have been developed in order to refine the current view of weather being chaotic (Shen, 2014-2019; Faghih-Naini and Shen, 2018; Shen et al., 2018).

In this study, we provide a report to: (1) Illustrate two kinds of attractor coexistence within Lorenz models, including coexisting chaotic and non-chaotic attractors. (2) Propose that the entirety of weather possesses dual nature of chaos and order associated with the coexistence of chaotic and non-chaotic processes. Specific weather systems may appear chaotic or non-chaotic within their finite lifetime. The refined view on the nature of weather is neither too optimistic nor pessimistic as compared to the Laplacian view of deterministic predictability and the Lorenz view of deterministic chaos, respectively. For stable non-chaotic processes, their intrinsic predictability is deterministic. To this end, if we can identify non-chaotic processes in advance, we may obtain longer predictability or better estimates on predictability. To achieve this goal, I will present recent results obtained using the methods of the recurrence analysis and kernel principal component analysis (Reyes and Shen, 2019; Cui and Shen, 2019).

References:

Anthes, R., 2011: Turning the tables on chaos: is the atmosphere more predictable than we assume? UCAR Magazine, spring/summer, available at: https://news.ucar.edu/4505/turningtables-chaos-atmosphere-more-predictable-we-assume (last access: 26 November 2018).
Cui, J. and B.-W. Shen, 2019: Revealing the Co-existence of High-dimensional Chaotic and NonChaotic Orbits using Kernel PCA Method: A Case Study with a Generalized Lorenz Model. (under revisions, May 20, 2019). Available from RG https://doi.org/10.13140/RG.2.2.21764.17288.
Faghih-Naini, S. and B.-W. Shen, 2018: Quasi-periodic orbits in the five-dimensional nondissipative Lorenz model: the role of the extended nonlinear feedback loop. International Journal of Bifurcation and Chaos, Vol. 28, No. 6 (2018) 1850072 (20 pages). DOI: 10.1142/S0218127418500724.
Reyes, T. and B.-W. Shen, 2019: A Recurrence Analysis of Chaotic and Non-Chaotic Solutions within a Generalized Lorenz Model. Chaos, Solitons & Fractals. 125 (2019), 1-12. https://doi.org/10.1016/j.chaos.2019.05.003
Shen, B.-W., 2019: Aggregated Negative Feedback in a Generalized Lorenz Model. International Journal of Bifurcation and Chaos, Vol. 29, No. 3 (2019) 1950037 (20 pages). https://doi.org/10.1142/S0218127419500378
Shen, B.-W., 2018: On periodic solutions in the non-dissipative Lorenz model: the role of the nonlinear feedback loop. Tellus A: 2018, 70, 1471912, https://doi.org/10.1080/16000870.2018.1471912.
Shen, B.-W., 2017: On an extension of the nonlinear feedback loop in a nine-dimensional Lorenz model. Chaotic Modeling and Simulation (CMSIM), 2: 147–157, 2017.
Shen, B.-W., 2016: Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model. Nonlin. Processes Geophys., 23, 189-203, doi:10.5194/npg-23-189-2016, 2016.
Shen, B.-W., 2015: Nonlinear Feedback in a Six-dimensional Lorenz Model. Impact of an additional heating term. Nonlin. Processes Geophys., 22, 749-764, doi:10.5194/npg-22-749- 2015, 2015.
Shen, B.-W., 2014: Nonlinear Feedback in a Five-dimensional Lorenz Model. J. of Atmos. Sci., 71, 1701–1723. doi:http://dx.doi.org/10.1175/JAS-D-13-0223.1
Shen, B.-W., T. Reyes, and S. Faghih-Naini, 2019: Coexistence of Chaotic and Non-Chaotic Orbits in a New Nine-Dimensional Lorenz Model. 11th Chaotic Modeling and Simulation International Conference (Editors: C. H. Skiadas and I. Lubashevsky), Springer Proceedings in Complexity, https://www.springer.com/us/book/9783030152963
Shen, B.W., M. DeMaria, J.-L. F. Li, and S. Cheung, 2013: Genesis of Hurricane Sandy (2012) Simulated with a Global Mesoscale Model. Geophys. Res. Lett. 40. 2013, DOI: 10.1002/grl.50934.
Shen, B.-W., W.-K. Tao, and M.-L. Wu, 2010: African Easterly Waves in 30-day High-resolution Global Simulations: A Case Study during the 2006 NAMMA Period. Geophys. Res. Lett., 37, L18803, doi:10.1029/2010GL044355.
Shen, B.-W., R. Atlas, O. Oreale, S.-J Lin, J.-D. Chern, J. Chang, C. Henze, and J.-L. Li, 2006: Hurricane Forecasts with a Global Mesoscale-Resolving Model: Preliminary Results with Hurricane Katrina(2005). Geophys. Res. Lett., L13813, doi:10.1029/2006GL026143.