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Nonlinear Processes in Geophysics An interactive open-access journal of the European Geosciences Union

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Nonlin. Processes Geophys., 25, 387-412, 2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.
Research article
28 May 2018
Exploring the Lyapunov instability properties of high-dimensional atmospheric and climate models
Lesley De Cruz1, Sebastian Schubert2, Jonathan Demaeyer1, Valerio Lucarini2,3,4, and Stéphane Vannitsem1 1Royal Meteorological Institute of Belgium, Brussels, Belgium
2Meteorological Institute, CEN, University of Hamburg, Hamburg, Germany
3Department of Mathematics and Statistics, University of Reading, Reading, UK
4Centre for the Mathematics of Planet Earth, University of Reading, Reading, UK
Abstract. The stability properties of intermediate-order climate models are investigated by computing their Lyapunov exponents (LEs). The two models considered are PUMA (Portable University Model of the Atmosphere), a primitive-equation simple general circulation model, and MAOOAM (Modular Arbitrary-Order Ocean-Atmosphere Model), a quasi-geostrophic coupled ocean–atmosphere model on a β-plane. We wish to investigate the effect of the different levels of filtering on the instabilities and dynamics of the atmospheric flows. Moreover, we assess the impact of the oceanic coupling, the dissipation scheme, and the resolution on the spectra of LEs.

The PUMA Lyapunov spectrum is computed for two different values of the meridional temperature gradient defining the Newtonian forcing to the temperature field. The increase in the gradient gives rise to a higher baroclinicity and stronger instabilities, corresponding to a larger dimension of the unstable manifold and a larger first LE. The Kaplan–Yorke dimension of the attractor increases as well. The convergence rate of the rate function for the large deviation law of the finite-time Lyapunov exponents (FTLEs) is fast for all exponents, which can be interpreted as resulting from the absence of a clear-cut atmospheric timescale separation in such a model.

The MAOOAM spectra show that the dominant atmospheric instability is correctly represented even at low resolutions. However, the dynamics of the central manifold, which is mostly associated with the ocean dynamics, is not fully resolved because of its associated long timescales, even at intermediate orders. As expected, increasing the mechanical atmosphere–ocean coupling coefficient or introducing a turbulent diffusion parametrisation reduces the Kaplan–Yorke dimension and Kolmogorov–Sinai entropy. In all considered configurations, we are not yet in the regime in which one can robustly define large deviation laws describing the statistics of the FTLEs.

This paper highlights the need to investigate the natural variability of the atmosphere–ocean coupled dynamics by associating rate of growth and decay of perturbations with the physical modes described using the formalism of the covariant Lyapunov vectors and considering long integrations in order to disentangle the dynamical processes occurring at all timescales.

Citation: De Cruz, L., Schubert, S., Demaeyer, J., Lucarini, V., and Vannitsem, S.: Exploring the Lyapunov instability properties of high-dimensional atmospheric and climate models, Nonlin. Processes Geophys., 25, 387-412,, 2018.
Publications Copernicus
Short summary
The predictability of weather models is limited largely by the initial state error growth or decay rates. We have computed these rates for PUMA, a global model for the atmosphere, and MAOOAM, a more simplified, coupled model which includes the ocean. MAOOAM has processes at distinct timescales, whereas PUMA surprisingly does not. We propose a new programme to compute the natural directions along the flow that correspond to the growth or decay rates, to learn which components play a role.
The predictability of weather models is limited largely by the initial state error growth or...