Abstract
Abstract. We investigate the geometrical structure of instabilities in the two-scale Lorenz 96 model through the prism of Lyapunov analysis. Our detailed study of the full spectrum of covariant Lyapunov vectors reveals the presence of a slow bundle in tangent space, composed by a set of vectors with a significant projection onto the slow degrees of freedom; they correspond to the smallest (in absolute value) Lyapunov exponents and thereby to the longer timescales. We show that the dimension of the slow bundle is extensive in the number of both slow and fast degrees of freedom and discuss its relationship with the results of a finite-size analysis of instabilities, supporting the conjecture that the slow-variable behavior is effectively determined by a nontrivial subset of degrees of freedom. More precisely, we show that the slow bundle corresponds to the Lyapunov spectrum region where fast and slow instability rates overlap, “mixing” their evolution into a set of vectors which simultaneously carry information on both scales. We suggest that these results may pave the way for future applications to ensemble forecasting and data assimilations in weather and climate models.
Highlights
Understanding the dynamics of multiscale systems is one of the great challenges in contemporary science, both for the theoretical aspects and the applications in many areas of interests for the society and the private sectors
The enthusiasm one may have for the Mori–Zwanzig formalism is partly counterbalanced by the fact that the effective coarse-grained dynamics is written in an implicit form so that it is of limited direct use
We have identified a slow bundle in the tangent space of the Lorenz 96 (L96) model – a central band centered around the 0-covariant Lyapunov vectors (CLVs) – whose covariant vectors are characterized by a large projection over the slow degrees of freedom
Summary
Understanding the dynamics of multiscale systems is one of the great challenges in contemporary science, both for the theoretical aspects and the applications in many areas of interests for the society and the private sectors. An extremely relevant possible advantage of using theory-based methods is the possibility of constructing scale-adaptive parametrizations (see discussion in Vissio and Lucarini, 2017, ;) Another angle on multiscale systems deals with the study of the scale–scale interactions, which are key in understanding instabilities and dissipative processes and the associated predictability and error dynamics. The amplitude of the perturbations of the fastest variables start saturating, while those affecting the slowest degrees of freedom grow at a pace mostly controlled by the (typically weaker) instabilities characteristic of the slower degrees of freedom While nonlinear tools, such as finite-size Lyapunov exponents (Aurell et al, 1997), are able to capture the rate of this multiscale growth, they lack the mathematical rigor of infinitesimal analysis.
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