On finite-size Lyapunov exponents in multiscale systems

Abstract

We study the effect of regime switches on finite size Lyapunov exponents (FSLEs) in determining the error growth rates and predictability of multiscale systems. We consider a dynamical system involving slow and fast regimes and switches between them. The surprising result is that due to the presence of regimes the error growth rate can be a non-monotonic function of initial error amplitude. In particular, troughs in the large scales of FSLE spectra is shown to be a signature of slow regimes, whereas fast regimes are shown to cause large peaks in the spectra where error growth rates far exceed those estimated from the maximal Lyapunov exponent. We present analytical results explaining these signatures and corroborate them with numerical simulations. We show further that these peaks disappear in stochastic parametrizations of the fast chaotic processes, and the associated FSLE spectra reveal that large scale predictability properties of the full deterministic model are well approximated whereas small scale features are not properly resolved.PACS numbers: 05.45.-a Keywords: Finite-size Lyapunov exponents, predictability, ensemble generation, regimes, metastabilityThe atmosphere and the climate system are inherently complex multiscale systems with processes spanning spatial scales from millimetres to thousands of kilometres, and temporal scales from seconds to millennia. It is a formidable challenge to find consistent and effective reduced dynamical equations for the "slow" and "large" degrees of freedom with predictive power. An important question is how to measure the radius of predictability in such a multiscale system. One such measure is the maximal Lyapunov exponent λ max . A generic situation is that the fast degrees are strongly chaotic, causing the Lyapunov exponent of the whole system to be large, indicating poor predictability. However, the large scale slow behaviour can still be forecast with reasonable accuracy for times much longer than O(1/λ max ); for example, weather can be forecast on time scales above those expected from small-scale instabilities such as convection 25 . This is usually due to the small-scale instabilities growing faster but becoming nonlinearly saturated at a much smaller level than large scale instabilities.

Finite size Lyapunov exponent: review on applications

“…ODEs). It is worth citing that non-monotonic behavior of λ(δ) similar to that of the Bernoulli map (figure 7(b)) has recently been found in a toy model for the climate [88]. In particular, there maxima of the FSLE have been connected to the switching between 'metastable states'.…”

Predictability in Rotating Turbulence: Insights from a Shell Model Study

“…The frozen stage in the error evolution is perhaps the manifestation of the globally imposed timescale by the Coriolis force term modifying the character of the slow and fast dynamics in the state space of the shell model. The dip and the subsequent peak in the FSLE plot are signatures of the slow and the fast dynamics in the inertial range, respectively [44]. We speculate that the enhancement of predictability can be attributed to the coherent dynamics in the columnar structures formed in real rotating turbulence as predicted in theory (Taylor-Proudman theorem) and observed in direct numerical simulations [10,45,46].…”

Nonlinear dynamics approach to the predictability of the Cane–Zebiak coupled ocean–atmosphere model

“…4), the error growth rate given by the FSLE can be a highly fluctuating function of initial state. In particular, troughs in the large scales of FSLE spectra are shown to be a signature of slow regimes (in the slow system -ocean), whereas regimes occurring in the fast system (atmosphere) are shown to cause large peaks in the FSLE spectra where error growth rates far exceed those estimated from the maximal Lyapunov exponent (Mitchell and Gottwald, 2012). Figure 9 shows the average error growing time T p (in months) as a function of the error tolerance level for a small uncertainty of fixed size δ = 0.05 • C to grow to a value = 1.5 • C, and is obtained by Eq.…”

“…(5). Since the FSLE depends on the initial state, one can consider the application of the FSLE for a particular season (ENSO spring predictability barrier) or regime of the system as in Mitchell and Gottwald (2012). In order to consider the FSLE seasonal or regime dependence, the algorithm can be modified to avoid the rescaling at finite δ n as follows.…”