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

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.

The dynamics of the atmosphere and the climate system is characterised by the
property of sensitivity to initial states

The property of sensitivity to initial conditions in deterministic dynamical
systems is often evaluated by computing the Lyapunov exponents that
correspond to the asymptotic rates of amplification or decay of
infinitesimally small perturbations, e.g.

In parallel to these investigations in the context of basic sciences, several
attempts to compute Lyapunov exponents in the context of meteorological and
climate models have been made (see

However, the atmosphere cannot be treated as an autonomous system, as it
interacts with other components of the climate system. These other components
are characterised by longer timescales of motions. They are typically less
intensely affected by some of the physical processes responsible for
atmospheric instabilities, most notably convective and baroclinic
instability. Moreover, the energetics of the atmosphere is mainly driven by
thermodynamic processes that are dominated by the inhomogeneous absorption of
solar radiation. The surface oceanic circulation, by contrast, is mostly
mechanically driven by atmospheric winds

This raises the question as to the impact of the coupling to other
sub-domains of the climate system: are the other sub-domains of the climate
system stabilising the atmosphere or not?

Yet the problem of the predictability (in terms of Lyapunov instability) of
the full-scale climate system including the different sub-domains is still
open. Recently a new coupled ocean–atmosphere model was developed that could
help answer key questions on the predictability properties of this type of
system

As originally envisioned by

Additionally, CLVs allow for understanding the properties of the tangent
space and assess the hyperbolicity of the system, through the analysis of the
statistics of the angles between the stable and unstable tangent manifolds
across the attractor. These angles should always be bounded away from 0 or

As is well known, geophysical fluid dynamical (GFD) systems are characterised
by relevant processes on multiple spatial and temporal scales of motion

The problem becomes particularly interesting when considering the coupling of
two sub-domains with vastly different timescales, as done in the case of a
low-order coupled ocean–atmosphere system in

We wish to provide some first steps of a wider research programme aimed at performing a systematic investigation of the properties of the tangent space of GFD systems in a turbulent regime of motion. A first objective is to gain a better understanding of the multiscale properties of the dynamics and of the energy exchanges occurring across such scales. Furthermore, this programme aims at understanding the relevance of violations to the uniform hyperbolicity conditions in terms of predictability on different timescales, including the response – in a statistical mechanical sense – of the system to static and time-dependent perturbations.

In the present paper, we explore for the first time the Lyapunov spectra of
the primitive-equation model, PUMA, and of intermediate-order configurations
of the coupled ocean–atmosphere system, MAOOAM. The first model is
characterised by the presence of multiple scales of motions resulting from
the fact that ageostrophic motions are

In Sect.

The Portable University Model of the Atmosphere (PUMA) was introduced by

Let us briefly summarise the equations of motion of PUMA and how the model is
integrated in time. For further details we refer the reader to the PUMA
User's Guide

PUMA is forced by Newtonian cooling which accounts in a crude yet effective
way for the emission and the absorption of long- and short-wave radiation and
for the heat convergence associated with convective processes (following

Symbols and variables in the PUMA equations.

PUMA uses spherical harmonics and grid-point fields of the prognostic variables. Utilising the Fourier transform along the zonal direction and a Legendre transformation, PUMA computes the linear terms in spectral space and the non-linear terms in grid-point space. The time-stepping scheme is a combination of a leap-frog scheme with the Robert–Asselin filter.

The PUMA User's Guide includes more details and a complete description of the
exact implementation and form of the various forcings

Although the atmospheric dynamics of both models are largely governed by the
same processes, MAOOAM differs in many respects from the stand-alone PUMA
model. Most importantly, the atmosphere of MAOOAM features both a mechanical
and a thermal coupling with a shallow-water ocean layer, which is absent in
PUMA. Furthermore, MAOOAM is a mid-latitude model which uses the
quasi-geostrophic approximation

The dynamics of MAOOAM's two-layer atmosphere is described by the
quasi-geostrophic vorticity equations, expressed in terms of the
streamfunction fields

Following

Variables and parameters in the MAOOAM equations.

The prognostic equations for the atmospheric and oceanic temperature fields,
using an energy balance scheme as in

The thermal wind relation allows one to link the atmospheric temperature
anomaly

The model equations are nondimensionalised, and the dynamical fields are
expanded in a configurable set of Fourier modes. The MAOOAM code computes the
coefficients for the resulting set of ordinary differential equations (ODEs)
as algebraic formulae of the wavenumbers. These ODEs are then integrated
using a fourth-order Runge–Kutta integration scheme. We refer the reader to

In what follows, we will use a shorthand notation that uses the maximum
wavenumbers

Let us write the evolution laws of the autonomous system presented in
Sect.

Let us consider a small perturbation along the trajectory,

If one or more LEs are positive, small errors on the initial conditions of
the system grow exponentially and the system is chaotic. In that case, the
time horizon of the system's predictability is proportional to the inverse of
the largest Lyapunov exponent,

The computation of the backward Lyapunov exponents follows the standard
algorithm of

An ensemble

At every time step

As the model is integrated forward from time

Every

Mean and variance of the local Lyapunov exponents are calculated.

The full Lyapunov spectrum of a model allows us to compute some additional
interesting properties of its attractor. One of these is the Kaplan–Yorke or
Lyapunov dimension

The second important property of the attractor is the Kolmogorov–Sinai or
metric entropy

Since the Lyapunov exponents are obtained by considering limiting conditions
where the initial perturbations are very small and the time span over which
the growth or decay rate is very long, they cannot reasonably be used to
study predictability outside such conditions. FTLEs

In this paper, we focus on the FTLEs and their relation to the asymptotic
mean LEs. Hence, we are interested in averages

If

We choose a simple set-up of PUMA. In this spirit, we also switch off
orography. The system is forced via a constant temperature gradient between
the Equator and the respective poles, as detailed in
Sect.

The objective of our experiments with PUMA is to compute the backward
Lyapunov exponents. For this we perform spin-up simulations for 30 years from
random initial conditions. We then obtain the first 200 Lyapunov exponents
using the Benettin algorithm described in Sect.

Note that in order to compute the Lyapunov exponents, it is necessary to
construct the tangent linear of PUMA. We generated parts of the code using
the program TAF by

Table

All experiments are performed with the same integration parameters. The time
step of 0.2 nondimensional time units corresponds to 32.3 min in
dimensional units. Before calculating the Lyapunov spectrum, a transient run
of

Model parameter values that are identical across all MAOOAM configurations used in this study.

The experiments are performed for different resolutions as discussed in
Sect.

nodissip

This experiment corresponds to the set-up of

nodissip-reducedstress

For this “reduced-stress” experiment, the coupling parameter

dissipation

One of the physical processes that was not included in MAOOAM v1.0

dissipationx10

In this experiment,

dissipation-reducedstress

This experiment has the same parameters as the “dissipation” experiment,
except for the coupling parameter

Note that these idealised experiments do not take into account any dependence
of the eddy (or turbulent) viscosity on the truncation scale, as is usually
done in turbulence

Here we present the results for the two different experiments with PUMA,
described in Sect.

Figure

There are two very small exponents since the model set-up is zonally
symmetric, which in the limit of continuum creates an additional zero mode
(see

The 200 largest LEs of the Lyapunov spectra of PUMA for the two
different set-ups with

Nevertheless, the 50 K spectrum in comparison to the 60 K spectrum has a smaller slope where the LE are near zero and negative. This may suggest the presence a longer term regime switching behaviour. One potential source for such a regime change is the switching between blocked and non-blocked states of the mid-latitudes atmosphere.

We have computed the blocking rate employing the well-established
Tibaldi–Molteni Index

Next, the existence of a large deviation law for the FTLEs is verified, as
described in Sect.

In
Figs.

Our intent is to make at least a qualitative assessment of the convergence
rate for

Distributions and rate functions of

Distributions and rate functions of

Distributions and rate functions of

Distributions and rate functions of

We make the following observations. The graphs suggest a convergence of the rate function for all LEs. Also, the rate functions' shape is approximately parabolic and the estimates of the rate functions appear to converge to the asymptotic with a comparable speed regardless of the value of the corresponding LE.

Distributions and rate functions of

Distributions and rate functions of

Distributions and rate functions of

Distributions and rate functions of

We interpret these results as stemming from the lack of a clear-cut timescale
separation in a purely atmospheric model like PUMA. This is in opposition to
what was originally speculated in

We have shown that in a primitive-equation model with a high-dimensional
phase space of

The Lyapunov analysis is performed on the set of model configurations
described in Sect.

Furthermore, as we could expect, the predictability is enhanced for models
where the scale-dependent dissipation term is present. The decrease in

The largest Lyapunov exponent

Figures

The highly populated central manifold of MAOOAM is in stark contrast to the
few near-zero LEs in PUMA. Being a purely atmospheric model, PUMA's Lyapunov
spectrum does not exhibit the large timescale separation present in MAOOAM.
Indeed, the spectrum of PUMA bears more resemblance to that of the QG
two-layer model of

Lyapunov spectra of MAOOAM for the “nodissip” experiment, for
model configurations from atm. 2

Lyapunov spectra of MAOOAM for the “nodissip-reducedstress”
experiment. Colours and arrows as in Fig.

Lyapunov spectra of MAOOAM for the “dissipation” experiment.
Colours and arrows as in Fig.

Upon increasing the number of modes in the ocean and the atmosphere, the
number of positive Lyapunov exponents (indicated with a vertical arrow)
consistently increases, but not as much as the number of strongly negative
exponents. This suggests that most of the additional spatial scales that are
resolved by the higher-resolution models are highly dissipative, hence
increasing the number of strongly negative Lyapunov exponents. The additional
positive and near-zero exponents that are introduced at these scales
nevertheless indicate that the added resolution still resolves some scales
that are important for the description of the dynamics. This is in agreement
with the conclusion in

Figure

As the number of dimensions increases quadratically and not linearly for the
consecutive model resolutions, it is instructive to rescale

Lyapunov spectra of MAOOAM for the “dissipationx10” experiment.
Colours and arrows as in Fig.

Kaplan–Yorke or Lyapunov dimension

Kaplan–Yorke or Lyapunov dimension

Figure

An additional experiment is performed by increasing the resolution of the
ocean and of the atmosphere separately starting from a specific symmetric
configuration “6

Kolmogorov–Sinai entropy

Lyapunov spectra of MAOOAM for the “dissipation” experiment, for
different model configurations starting from atm. 6

As a final analysis, we have given a preliminary look at whether large
deviation laws can be established for the long-term statistics of the FTLEs.
In what follows, we consider the “9

Kaplan–Yorke or Lyapunov dimension

In contrast to what was presented in

Estimate of the rate function describing the large deviation law of
the 351st FTLE for MAOOAM with no dissipation

Estimate of the rate function describing the large deviation law of
the first FTLE for MAOOAM with no dissipation

In brief, these results indicate that the dominant instabilities of the
coupled ocean–atmosphere system are well captured by MAOOAM, even at low
resolutions. However, the increase in the Lyapunov dimension with the
resolution implies that the relevant dynamics of the system are not yet fully
resolved, in agreement with

The chaotic nature of the atmosphere and of the climate system has been investigated in the present work in the context of a primitive-equation atmospheric model and a coupled ocean–atmosphere model. Both systems suggest that high-dimensional dynamical processes are at play with very interesting distinct specificities.

The Lyapunov spectra of the two models considered here have rather different
qualitative features, as a result of their structural differences, which have
profound impacts on the type of possible instability mechanisms. Following

MAOOAM is a coupled quasi-geostrophic atmosphere–ocean model, which, by definition, features a large timescale separation between ocean and atmosphere, and lacks a satisfactory representation of mesoscale and sub-mesoscale processes. PUMA is an atmospheric-only primitive-equation model, which can represent the faster, smaller-scale instabilities associated with processes occurring well below the Rossby deformation radius. On the other side, the lack of an active ocean component removes the presence of very slow scales and does not allow for a built-in scale separation in the dynamics.

We summarise here some findings.

In PUMA the spectrum of Lyapunov exponents changes in accordance with the
paradigm that stronger baroclinic forcing leads to a more unstable
atmosphere, as already observed in

For MAOOAM the Lyapunov spectrum is shaped considerably by the presence of
the ocean, with a large portion of exponents close to zero. The subspace
associated with these exponents corresponds to the central manifold as in the
theory of partially hyperbolic systems, and presents features analogous to
what was observed in

One can also conjecture that the set of physical modes, as defined by

The analysis of the FTLEs of MAOOAM reveals some interesting insight into the dynamics. Surprisingly, it is hard to find convergence for the rate functions of the FTLEs, even for those associated with the positive LEs. This may point to the presence of nontrivial ocean influence on the (mostly) atmospheric instabilities.

In the programme we want to develop starting from this investigation, we will employ CLVs in high-dimensional models to tackle various open problems. CLVs allow us to associate growth and decay rates with time-dependent physical modes, and provide a geographical portrait of where instability or damping develops.

First, what is the minimal but sufficient resolution? This is a crucial
question, in particular in view of the current computer power needed to
perform long-term numerical integrations. A possible way to quantify where
this threshold might be, is by means of the different modes identified by

Second, we want to understand multiscale instabilities better and find out
what the driving processes are behind their growth. Here, the covariance of
the CLVs with the tangent linear equation is the key for understanding
instabilities and their properties far away from an equilibrium.
Traditionally, even in a chaotic setting such an analysis relied on classic
normal mode instability of fixed stationary states

The PUMA model is a part of PLASIM, for which the source
code can be downloaded at

The Lyapunov spectra of the different PUMA and MAOOAM configurations, which were computed using the Benettin algorithm, are available as a Supplement.

We have performed an additional analysis that focuses on the convergence of
the standard deviation

The scaling of

Standard deviation

The decorrelation time of the near-zero FTLEs in MAOOAM is extremely long.
Accordingly, the time intervals

The metric

Standard deviation

The metric

The supplement related to this article is available online at:

VL and SS performed the analysis of PUMA. SS wrote the code to compute the LEs for PUMA. LDC and SV performed the analysis of MAOOAM. JD and SS wrote the code to compute the LEs for MAOOAM. LDC, SV and JD wrote MAOOAM. VL analysed the FTLEs in MAOOAM. All authors contributed to the writing of the manuscript.

Stéphane Vannitsem and Valerio Lucarini are members of the editorial board of the journal. The other authors declare that they have no conflict of interest.

This article is part of the special issue “Numerical modeling, predictability and data assimilation in weather, ocean and climate: A special issue honoring the legacy of Anna Trevisan (1946–2016)”. It is not associated with a conference.

The authors would like to thank Stephen G. Penny and an anonymous reviewer for improving the manuscript with their constructive comments.

Lesley De Cruz was supported by the BELSPO under contract BR/165/A2/Mass2Ant. Jonathan Demaeyer was supported by BELSPO under contract BR/121/A2/STOCHCLIM. Valerio Lucarini and Sebastian Schubert were supported by the DFG through the contract SFB/Transregio TRR181. Edited by: Juan Manuel Lopez Reviewed by: Stephen G. Penny and one anonymous referee