Oberseminar Numerische Mathematik / Scientific Computing
Professor Andrew Majda
Courant Institute of Mathematical Sciences, New York
Systematic Strategies for Low Dimensional Stochastic Mode Reduction in Dynamical Systems with Many Degrees of Freedom
Abstract:
This lecture discusses systematic mathematical strategies for low- dimensional stochastic mode reduction for turbulent large dimensional dynamical systems and their application to modelling low frequency weather dynamics and climate change. A remarkable fact of Northern Hemisphere low frequency variability is that it can be efficiently described by only a few teleconnection patterns that explain most of the total variance. These few teleconnection patterns not only exert a strong influence on regional climate and weather, they are also related to climate change. These properties of teleconnection patterns make them an attractive choice as basis functions for climate models with a highly reduced number of degrees of freedom. The development of such reduced climate models involves the solution of two major issues: 1) how to properly account for the unresolved modes, also known as the closure problem; and 2) how to define a small set of basis functions that optimally represent the dynamics of the major teleconnection patterns.In this lecture examples of stochastic mode reduction are discussed ranging from an explicit solvable pedagogical example with three modes to a prototype atmospheric general circulation model with a thousand degrees of freedom where an effective reduced stochastic model with only ten low frequency modes captures the statistical dynamical behavior. A controversial topic in the recent climate modeling literature is the fashion in which metastable low-frequency regimes in the atmosphere occur despite nearly Gaussian statistics for these planetary waves. Here a simple 57-mode paradigm model for such metastable atmospheric regime behavior is introduced and analyzed through hidden Markov model (HMM) analysis of the time series of suitable low- frequency planetary waves. The analysis of this paradigm model elucidates how statistically significant metastable regime transitions between blocked and zonal statistical states occur despite nearly Gaussian behavior in the associated probability distribution function and without a significant role for the low-order truncated nonlinear dynamics alone; turbulent backscatter onto the three-dimensional subspace of low-frequency modes is responsible for these effects. It also is demonstrated that suitable stochastic mode reduction strategies, which include both augmented cubic nonlinearity and multiplicative noise, are also capable of capturing the metastable low-frequency regime behavior through a single stochastic differential equation compared with the full turbulent chaotic 57-mode model. This feature is attractive for issues such as long-term weather predictability. Research papers regarding most of the research here can be found on Majda's faculty website: http://www.math.nyu.edu/faculty/majda
| Datum: | 08.06.2007 | Zeit: | 14:15 Uhr | Ort: | FU Berlin, Institut für Mathematik II, Arnimallee 6, 14195 Berlin. | Raum: | 032 im Erdgeschoss |