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Discussion papers | Copyright
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.

Research article 10 Oct 2018

Research article | 10 Oct 2018

Review status
This discussion paper is a preprint. It is a manuscript under review for the journal Nonlinear Processes in Geophysics (NPG).

Lyapunov analysis of multiscale dynamics: The slow manifold of the two-scale Lorenz '96 model

Mallory Carlu1, Francesco Ginelli1, Valerio Lucarini2,3,4, and Antonio Politi1 Mallory Carlu et al.
  • 1SUPA, Institute for Complex Systems and Mathematical Biology, King's College, University of Aberdeen, Aberdeen, UK
  • 2Department of Mathematics and Statistics, University of Reading, Reading, UK
  • 3Centre for the Mathematics of Planet Earth, University of Reading, Reading, UK
  • 4CEN, University of Hamburg, Hamburg, Germany

Abstract. We investigate the geometrical structure of instabilities in the two-scales 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 manifold in tangent space, composed by a set of vectors with a significant projection on the slow degrees of freedom; they correspond to the smallest (in absolute sense) Lyapunov exponents and thereby to the longer time scales. We show that the dimension of this manifold 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 non-trivial subset of degrees of freedom. More precisely, we show that the slow manifold 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 these results may pave the way for future applications to ensemble forecasting and data assimilation in weather and climate models.

Mallory Carlu et al.
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Mallory Carlu et al.
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Publications Copernicus
Short summary
We explore the nature of instabilities in a well-known meteorological toy model, the Lorenz 96, to unravel key mechanisms of interaction between scales of different resolutions and time scales. To do so, we use a mathematical machinery known as Lyapunov analysis, allowing us to capture the degrees of chaoticity associated with fundamental directions of instability. We find a non-trivial group of such directions projecting significantly on slow variables, associated with long term dynamics.
We explore the nature of instabilities in a well-known meteorological toy model, the Lorenz 96,...