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

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. Such systems are characterized by a dynamics that takes place on diverse spatial and/or temporal scales, with interactions between different scales combined with the presence of nonlinear processes. The existence of a variety of scales makes it hard to approach such systems using direct numerical integrations, since the problem is stiff. Additionally, simplifications based on naïve scale analysis, where only a limited set of scales are deemed important and the others are outright ignored, might be misleading or lead to strongly biased results. The nonlinear interaction with scales outside the considered range may, indeed, be important as a result of (possibly slow) upward or downward cascades of energy and information.

A crucial contribution to the understanding of multiscale systems comes from
the now classic Mori–Zwanzig theory

The climate provides an excellent example of a multiscale system, with
dynamical processes taking place on a very large range of spatial and
temporal scales. The chaotic, forced and dissipative dynamics and the
nontrivial interactions between different scales represent a fundamental
challenge in predicting and understanding weather and climate. A fundamental
difficulty in the study of the multiscale nature of the climate system comes
from the lack of any spectral gap, namely, a clear and well-defined
separation of scales. The climatic variability covers a continuum of
frequencies

On the other side, there is a fundamental need to construct efficient and
accurate parametrizations for describing the impact of small scales on larger
ones in order to improve our ability to predict weather and provide a better
representation of climate dynamics. For some time it has been advocated that
such parametrizations should include stochastic terms

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. Lyapunov
exponents

While this information is gathered at the linearized level, one may
nevertheless conjecture that LEs and CLVs associated with the slowest timescales (i.e., the smallest LEs in absolute value) can capture relevant
information on the large-scale dynamics and its correlations with the faster
degrees of freedom. In a sense, one may conjecture that the small LEs and the
corresponding CLVs could be used to gain access to a nontrivial effective
large-scale dynamics. See, for instance,

The L96 model provides a simple yet prototypical representation of a
two-scale system where large-scale, synoptic variables are coupled to
small-scale, convective variables. The Lorenz 96 model was quickly established as
an important test bed for evaluating new methods of data assimilation

Our Lyapunov analysis reveals the existence of a nontrivial slow bundle in tangent space, formed by a set of CLVs – associated with the smallest LEs – that was the only one with a nonnegligible projection onto the slow variables. The number of these CLVs is considerably larger than the number of slow variables, and it is extensive in the number of slow and fast degrees of freedom. At the same time, the directions associated with highly expanding and contracting LEs are aligned almost exclusively along the fast, small-scale degrees of freedom. Moreover, we show that the LE corresponding to the first CLV of the slow bundle (i.e., the most expanding direction within this subspace) approaches the finite-size Lyapunov exponent in a large-perturbation range, where linearization is not generally expected to apply.

Altogether, it should be made clear that the timescale separation between the slow bundle and the fast degrees of freedom is large but finite and stays finite when the number of degrees of freedom is let to diverge (i.e., it is not a standard hydrodynamics component). Additionally, the stability is not absolutely weak in the sense of nearly vanishing Lyapunov exponents.

The paper is organized as follows. Section 2 introduces both the L96 model and the fundamental tools of the Lyapunov analysis used in this paper. Evidence for the existence of a slow bundle is presented in Sect. 3. In Sect. 4, we investigate how this slow structure arises from the superposition of the instabilities of the slow and fast dynamics. Section 5, on the other hand, is devoted to a comparison with results of finite-size analysis. Finally, in Sect. 6 we discuss our results, further commenting on their generality and proposing future developments and applications.

The L96 model is a simple example of an extended multiscale system such as the Earth atmosphere. Its dynamics is controlled by synoptic variables, characterized by a slow evolution over large scales, coupled to the so-called convective variables characterized by a faster dynamics over smaller scales.

The synoptic variables

We remark that in our configuration, following

The presence of the additional forcing term acting on the

Moreover, the parameter

The L96 model thus contains

Apart from helping to clarify these master–slave limiting cases, such a
reformulation of the model also allows us to better understand that, in
order to maintain a fixed amplitude of the coupling term, it is necessary to
keep

Given the more natural definition of the energy, when expressed in terms of
the

We will fix

As mentioned above, the right tools to quantify rigorously the rate of
divergence (or convergence) of nearby trajectories are the LEs and their associated covariant CLVs. We provide here a qualitative description of these objects.
For a more thorough discussion, the reader can look to

For definiteness, let us consider an

In the presence of

In the following, we always refer to CLVs assuming that they have been properly normalized. With the above-mentioned exception of degeneracies, CLVs constitute an intrinsic (they do not depend on the chosen norm) tangent-space decomposition into the stable and unstable directions associated with the different LEs. LEs themselves have units of inverse time so that the largest positive (in absolute value) exponents – and their associated CLVs – describe fast growing (or contracting) perturbations, while the smaller ones correspond to longer timescales.

Unfortunately, Eqs. (

The mutually orthogonal vectors, obtained as a by-product of this procedure,
constitute a basis in tangent space and are usually referred to as
Gram–Schmidt vectors (by the name of the algorithm used to perform the
QR decomposition) or backward Lyapunov vectors (BLVs; because they are
obtained by integrating the system forward until a given point in time, thus
spanning the past trajectory with respect to this point). Being forced to be
mutually orthogonal, BLVs allow only reconstructing the orientation of the
subspaces spanned by the most expanding directions. In this work, we
concentrate on the CLVs for the identification of the various
expanding and/or contracting directions. This is done by implementing a dynamical
algorithm based on a clever combination of both forward and backward
iterations of the tangent dynamics, introduced in

In practice, one first evolves the forward dynamics, following a phase-space
trajectory to compute the full LS

The key idea is then to project a generic tangent-space vector

Or, in the case of degenerate LEs, it converges to a vector belonging to the corresponding Oseledets subspace.

. In practice, this backward procedure can be performed by expressing the CLVs in the BLVs basis,The tangent-space dynamics of L96 can be readily obtained by linearizing the
phase-space evolution Eqs. (

In this paper, we numerically integrate Eqs. (

Spatially extended systems are known to typically exhibit an extensive
Lyapunov spectrum

Actually, it is
customary to define

Extensivity of chaos.

The single-scale L96 model (i.e., Eq.

In order to appreciate the different role of

The existence of a limit spectrum implies that the Kolmogorov–Sinai entropy

We conclude this section with a brief remark on the Lyapunov spectrum in the
zero-dissipation limit (

The invariant measure is absolutely continuous with respect to the Lebesgue one in the energy shell.

. Moreover, from Fig.We now come to the central result of this paper, namely the existence of a nontrivial subspace in tangent space associated with the slow dynamics of the L96 model.

The individual LEs

The norm of the (rescaled)

However, it should be noted that, although the CLVs are intrinsic vectors,
their mutual angles do depend on the relative scales used to represent the
single variables and, in particular, fast and slow ones. If we change the
units of measure used to quantify the fast

Given the strong temporal fluctuations of

We have first computed the projection norm

Note also that the CLV associated with the only null LE (in the following we
simply denote it as the 0-CLV) displays a sharp peak of the projection norm

CLV projection onto the slow variables.

We are interested in the dependence of this bundle on the number of slow and
fast variables. As discussed in the previous section, the L96 model is
extensive in both the slow and fast variables, provided that the ratio

We first set

Slow-bundle scaling for

We next focus on the scaling with

In order to accurately determine the width of the central band, i.e., the slow-bundle dimension, we fix a threshold for the

Before concluding this section, we would like to briefly discuss the
time-resolved projected norm

Time trace and probability distribution of CLV instantaneous
projection in the

In Fig.

In Fig.

In the previous section, we have identified a slow bundle in the tangent space of the L96 model – a central band centered around the 0-CLV – whose covariant vectors are characterized by a large projection over the slow degrees of freedom. It is natural to expect this band to be associated not only with long timescales (i.e., the inverse of the corresponding LEs) but also with large-scale instabilities.

We begin by discussing the pedagogical example of the uncoupled limit
(

Note also that, in the absence of coupling, the Jacobian matrix has a block diagonal
structure, with the CLVs either belonging to the

We now proceed to discuss the coupled case. When the coupling is switched on,
it has a double effect: (i) it modifies the overall dynamics, i.e., the
evolution in phase space, in Eqs. (

Thanks to this approximation, we can define two

The modifications induced by real-space coupling are more substantial.
They can be quantified by computing the root-mean-square differences

The main mechanism responsible for the reshuffling of the CLV orientation is
the (multifractal) fluctuations of finite-time LEs

Perfect tangencies may occur, but only for a set of zero-measure initial conditions, such as the homoclinic tangencies in low-dimensional chaos.

. Fluctuations are also responsible for the so-calledLet us be more quantitative and introduce the finite-time Lyapunov exponents

We are interested in the probability distribution

Given the two restricted spectra

Consistently, in Fig.

We have verified this to be the generic behavior, as expected due to the nonhyperbolic nature of the L96 model. The near tangencies between different
CLVs within the slow bundle provide strong numerical evidence of the
mixing between slow and fast degrees of freedom and are perfectly
consistent with the nonnegligible projection onto the slow

The intermittent nature of the instantaneous

Finally, we return to the restricted LEs to see whether – as implied by the
above conjecture – their knowledge can help to identify the slow-bundle
boundaries. In practice, we have first identified the borders of the region
covered by both slow LEs. They are given by the indices (within the
reconstructed spectrum) of the largest and smallest restricted slow LE,
labeled, respectively, as

Altogether, our analysis suggests that coupling in real space induces a sort of “short-range” interaction within tangent space: each LE (and the corresponding CLV) tends to affect and be affected by exponents with a similar magnitude and thereby characterizes a similar degree of instability in a sort of resonance phenomenon.

Note finally that the spectral band where the slow and fast restricted
Lyapunov spectra superimpose covers all the slow restricted LEs and a

So far we have studied the geometry of the L96 model, dealing exclusively with infinitesimal perturbations. A legitimate question is whether we can learn something more by looking at finite perturbations.

Finite-size analysis has been implemented in the L96 model since its
introduction

Here, we follow the excellent review

In this section we repeat this analysis in our setup, comparing the behavior
of the FSLE with the analysis of the tangent-space slow bundle. In the
following we use our standard parameters (

As we expect the FSLE to depend on the norm, we have decided to transform
this weakness into an advantage by studying the behavior of an entire family
of Euclidean norms, thereby extracting useful information from the
dependence on the chosen norm. More precisely, we introduce the

FSLE analysis.

In the inset of Fig.

The FSLEs obtained for different coupling parameters

For each coupling

Estimated slow dynamics finite-size instability

In practice, by interpreting the height of each plateau as a suitable
instability rate within the full Lyapunov spectrum, one can thereby extract
the corresponding index

Altogether, the slow-variable (large-scale) instability

It is remarkable that the analysis of a single pair of trajectories allows
for extracting information about (at least) two different Lyapunov exponents. We
conjecture that the linearly controlled growth of small, finite perturbations
stops as soon as the fast components saturate because of nonlinearities.
Afterwards, fast variables act as a sort of noise on the slow ones, whose
dynamics is still in the linear regime. Finally, in view of the above-mentioned closeness between the restricted and fully coupled LS, it is reasonable
to conjecture that, since coupling does not play a crucial role in tangent
space, the growth rate corresponds, in this second regime, to the maximal LE
of the slow variables, as indeed observed. Our result supports an earlier
conjecture of

Our analysis of the tangent-space structure of the L96 model has identified a slow bundle within the full tangent space. It is composed of the set of covariant Lyapunov vectors characterized by a nonnegligible projection over the slow degrees of freedom. Vectors in this set are associated with the smallest (in absolute value) LEs and thus with the longest timescales. We have verified that the number of such vectors increases linearly with the total number of degrees of freedom so that the slow-bundle dimension is an extensive quantity.

The upper and lower boundaries of the slow bundle are better defined for a
weak coupling

In order to clarify the origin of the slow bundle, we have introduced the notion of restricted Lyapunov spectra and argued that the central region, where the CLVs retain a significative projection over both slow and fast variables, corresponds to the range where the restricted spectra overlap with one another. In this region, fluctuations of the finite-time LEs much larger than the typical separation between consecutive LEs lead inevitably to frequent “near tangencies” between CLVs, thereby mixing slow and fast degrees of freedom into a nontrivial set of vectors which carries information on both sets of variables.

Besides, we have found that coupling in tangent space weakly influences the actual LEs, provided that it is accounted for in real space. This is one of the reasons for the finite-size analysis being able to give information about the instability of the slow bundle (i.e., the correspondence between the second plateau displayed by the FSLE and the upper boundary of the slow bundle). Further investigations are necessary to put our consideration on firmer ground.

So far, we have discussed the slow bundle in a setup where the fast degrees
of freedom are forced by a strong external drive

As already mentioned, the slow bundle is identified as the set of CLVs with a
nonnegligible projection onto the slow degrees of freedom. One might argue
that the average projection

Altogether, we conjecture that (i) the fast stable directions lying beyond
the slow-bundle central region are basically slaved degrees of freedom, which
do not contribute to the overall dynamical complexity, and (ii) the fast unstable
directions act as a noise generator for the

The mechanism discussed in Sect.

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.

AP and FG devised the research, MC is responsible for the simulations, and all authors analyzed the data and wrote the paper.

The authors declare that they have no conflict of interest.

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).

This paper was edited by Amit Apte and reviewed by two anonymous referees.