Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model

Mallory Carlu; Francesco Ginelli; Valerio Lucarini; Antonio Politi

doi:10.5194/npg-26-73-2019

Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model

Mallory Carlu^* (Corresponding Author), Francesco Ginelli, Valerio Lucarini, Antonio Politi

^*Corresponding author for this work

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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.

Original language	English
Pages (from-to)	73-89
Number of pages	17
Journal	Nonlinear Processes in Geophysics
Volume	26
Issue number	2
DOIs	https://doi.org/10.5194/npg-26-73-2019
Publication status	Published - 7 May 2019

Bibliographical note

Data availability: All data have been generated by numerically integrating the model equations mentioned in the paper. While all algorithmic details are duly given in our paper, in case of future needs, the authors are willing to provide their numerical codes on request.

Acknowledgements.
Francesco Ginelli warmly thanks Massimo Cencini for truly invaluable early discussions. We acknowledge support from EU Marie Skłodowska-Curie ITN grant no. 642563 (COSMOS). Mallory Carlu acknowledges financial support from the Scottish Universities Physics Alliance (SUPA) as well as Sebastian Schubert and the Meteorological Institute of the University of Hamburg for the warm welcome and the stimulating discussions. Valerio Lucarini acknowledges the support received from the DFG Sfb/Transregion TRR181 project and the EU Horizon 2020 projects Blue-Action (grant agreement number 727852) and CRESCENDO (grant agreement number 641816).

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Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model
© Author(s) 2019. This work is distributed under the Creative Commons Attribution 4.0 License. https://creativecommons.org/licenses/by/4.0/
Final published version, 2.6 MBLicence: CC BY

https://arxiv.org/abs/1809.05065v3Licence: Unspecified

Cite this

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title = "Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model",

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.",

author = "Mallory Carlu and Francesco Ginelli and Valerio Lucarini and Antonio Politi",

note = "Data availability: All data have been generated by numerically integrating the model equations mentioned in the paper. While all algorithmic details are duly given in our paper, in case of future needs, the authors are willing to provide their numerical codes on request. Acknowledgements. Francesco Ginelli warmly thanks Massimo Cencini for truly invaluable early discussions. We acknowledge support from EU Marie Sk{\l}odowska-Curie ITN grant no. 642563 (COSMOS). Mallory Carlu acknowledges financial support from the Scottish Universities Physics Alliance (SUPA) as well as Sebastian Schubert and the Meteorological Institute of the University of Hamburg for the warm welcome and the stimulating discussions. Valerio Lucarini acknowledges the support received from the DFG Sfb/Transregion TRR181 project and the EU Horizon 2020 projects Blue-Action (grant agreement number 727852) and CRESCENDO (grant agreement number 641816).",

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N1 - Data availability: All data have been generated by numerically integrating the model equations mentioned in the paper. While all algorithmic details are duly given in our paper, in case of future needs, the authors are willing to provide their numerical codes on request. Acknowledgements. Francesco Ginelli warmly thanks Massimo Cencini for truly invaluable early discussions. We acknowledge support from EU Marie Skłodowska-Curie ITN grant no. 642563 (COSMOS). Mallory Carlu acknowledges financial support from the Scottish Universities Physics Alliance (SUPA) as well as Sebastian Schubert and the Meteorological Institute of the University of Hamburg for the warm welcome and the stimulating discussions. Valerio Lucarini acknowledges the support received from the DFG Sfb/Transregion TRR181 project and the EU Horizon 2020 projects Blue-Action (grant agreement number 727852) and CRESCENDO (grant agreement number 641816).

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N2 - 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.

AB - 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.

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Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model

Abstract

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