Covariant Lyapunov vectors of a quasi-
geostrophic baroclinic model: analysis of
instabilities and feedbacks
Article
Accepted Version
Schubert, S. and Lucarini, V. (2015) Covariant Lyapunov
vectors of a quasi-geostrophic baroclinic model: analysis of
instabilities and feedbacks. Quarterly Journal of the Royal
Meteorological Society, 141 (693). pp. 3040-3055. ISSN 1477-
870X doi: https://doi.org/10.1002/qj.2588 Available at
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Covariant Lyapunov Vectors of a Quasi-geostrophic
Baroclinic Model
Sebastian Schub ert
1,2
and Valerio Lucarini
2,3
1
IMPRS - ESM, MPI f. Meteorology, University Of Hamburg, Hamburg, Germany,
Email: sebastian.schubert@mpimet.mpg.de
2
Meteorological Institute, CEN, University Of Hamburg, Hamburg, Germany
3
Department of Mathematics and Statistics, University of Reading, Reading,
United Kingdom
First Draft: October 2014
Second Draft: March 2015
Third Draft: June 2015
This Draft: August 2015
Abstract
The classical approach for studying atmospheric variability is based on defining a back-
ground state and studying the linear stability of the small fluctuations around such a state.
Weakly non-linear theories can be constructed using higher order expansions terms. While
these methods have undoubtedly great value for elucidating the relevant physical processes,
they are unable to follow the dynamics of a turbulent atmosphere. We provide a first example
of extension of the classical stability analysis to a non-linearly evolving quasi-geostrophic flow.
The so-called covariant Lyapunov vectors (CLVs) provide a covariant basis describing the di-
rections of exponential expansion and decay of perturbations to the non-linear trajectory of
the flow. We use such a formalism to re-examine the basic barotropic and baroclinic processes
of the atmosphere with a quasi-geostrophic beta-plane two-layer model in a periodic channel
driven by a forced meridional temperature gradient ∆T . We explore three settings of ∆T ,
representative of relatively weak turbulence, well-developed turbulence, and intermediate con-
ditions. We construct the Lorenz energy cycle for each CLV describing the energy exchanges
with the background state. A positive baroclinic conversion rate is a necessary but not suf-
ficient condition of instability. Barotropic instability is present only for few very unstable
CLVs for large values of ∆T . Slowly growing and decaying hydrodynamic Lyapunov modes
closely mirror the properties of the background flow. Following classical necessary conditions
for barotropic/baroclinic instability, we find a clear relationship between the properties of the
eddy fluxes of a CLV and its instability. CLVs with positive baroclinic conversion seem to
form a set of modes for constructing a reduced model of the atmosphere dynamics.
1 Introduction
A classical topic of dynamical meteorology and climate dynamics is the study of mid-latitude
atmospheric variability and the investigation of the unstable eddies responsible for the synoptic
weather. These unstable eddies affect predictability on time scales of the order of a few days and on
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spatial scales of the order of a few hundreds kilometers (Kalnay, 2003). They play a crucial climatic
role of transporting heat poleward, so that their accurate characterization is of utmost importance.
Classical attempts at understanding their properties are based on linearization of some basic state
and normal mode analysis, possibly extended to weakly non-linear regimes (Pedlosky, 1987), and
on the provision of simple climatic closures (Stone, 1978), with stochastic models trying to fill in the
gap (Farrell and Ioannou, 1993). Two types of energy conversion between the background state and
the fluctuations have been proposed. The barotropic instability converts energy between the kinetic
energy of the background state and the eddy field. As a result the momentum gradients in the
background profile are reduced by an unstable barotropic process (Kuo, 1949). The second type of
instability is related to the presence of a sufficient vertical shear in the background state (Charney,
1947; Eady, 1949; Kuo, 1952). The energetics of the so-called baroclinic instability is dominated by
the following processes. The available potential energy of the zonal flow is converted into available
potential energy of the eddy field, which is then converted into eddy kinetic energy. As a result of
these processes, the center of mass of the atmosphere is lowered and heat is transported against
the temperature gradient. Necessary instability conditions for the linear stability of generic zonal
symmetric states are given by the Charney-Stern theorem (Charney and Stern, 1962; Eliassen,
1983). The baroclinic and barotropic energy conversions between the zonal mean and the eddies
underpin the Lorenz energy cycle (LEC), thus providing the link between weather instabilities
and climate (Lorenz, 1955; Lucarini, 2009; Lucarini et al., 2014), seen as a non-equilibrium steady
state. Simple two layer quasi-geostrophic (QG) models (Pedlosky, 1964; Phillips, 1954) provide
a qualitative correct picture of the synoptic scale instabilities and energetics of the mid-latitude
dynamics (Oort, 1964; Li et al., 2007).
The classical approach for studying atmospheric variability is based on defining a background
state and studying the linear stability of the small fluctuations around such a state. Weakly non-
linear theories can be constructed using higher order expansions terms. While these approaches
provide useful insight into the mechanisms responsible for instabilities and the non-linear stabi-
lization, they miss the crucial point of allowing for the investigation of the actual properties of the
turbulent regimes, where the system evolves with time in a complex manner, and is far from being
in the idealized base state considered in the instability analysis (Speranza and Malguzzi, 1988;
Hussain, 1983).
We would like to approach the problem of studying the instabilities of the atmosphere in a
turbulent regime, taking advantage of some recent tools of dynamical systems theory and statis-
tical mechanics, namely the Covariant Lyapunov Vectors (CLVs)(Ginelli et al., 2007; Wolfe and
Samelson, 2007). These allow for studying linear perturbations of chaotic atmospheric flows and
investigating the dynamics of the tangent space. In the past Lyapunov vectors were proposed as
bases to study the growth and decay of linear perturbations and to associate such features to the
predictability of the flow and use them in data assimilation, see (Legras and Vautard, 1996; Kalnay,
2003). Ruelle (Ruelle, 1979) proposed first the idea of a covariant splitting of the tangent linear
space (see also (Trevisan and Pancotti, 1998). The covariance of this basis is the critical property
for a linear stability analysis, since the basis vectors can be seen as actual trajectories of linear
perturbations. The average growth rate of each CLV equals one of the Lyapunov exponents (LE).
The LEs describe the asymptotic expansion and decay rates of infinitesimal small perturbations
of a chaotic trajectory (Eckmann and Ruelle, 1985). The CLVs provide explicit information about
the directions of asymptotic growth and decay in the tangent linear space. For stationary states
the CLVs reduce to the normal modes. In the case of periodic orbits the CLVs coincide with the
Floquet vectors which for example have been obtained for the weakly unstable Pedlosky model
(Samelson, 2001a). Samelson also extended this analysis to unstable periodic orbits (Samelson,
2001b).
Recently, new methods to compute CLVs for arbitrary chaotic trajectories have been developed
by Ginelli et al. (2007), and Wolfe and Samelson (2006, 2007, 2008). These methods allow for
computing CLVs for high dimensional chaotic systems and have led to a renewed interest in the
2
related theory. For a comprehensive introduction we refer to Kuptsov and Parlitz (2012). CLVs
have been successfully obtained for one and two dimensional systems (Yang et al., 2009; Yang
and Radons, 2010; Takeuchi et al., 2011). Moreover, they have been studied for simple models of
geophysical relevance (Paz´o et al., 2010; Herrera et al., 2011) elucidating the potential benefits of
CLVs in Ensemble Prediction Systems over bred vectors and orthogonal Lyapunov vectors. In this
paper, we construct CLVs for the simple two layer QG model introduced by Phillips (1956) and
consider three values of the equator-to-pole relaxation temperature difference ∆T , corresponding
to low, medium, and high baroclinic forcing, and correspondingly developed turbulence. This
is intended as a first step in the direction of studying a hierarchy of more complex models of
geophysical flows.
It is of great interest to link the mathematical properties of the various CLVs to their energetics.
Thanks to covariance, we are able to construct the Lorenz energy cycle for each CLV, and then
deduce the rate of barotropic and baroclinic energy conversion, as well as of frictional dissipation.
In this way, we are able to associate the overall asymptotically growing or decaying property of
each CLV to specific physical processes.
The main results obtained in this way are the following. We observe that CLVs with higher
LEs gain energy via the baroclinic conversion, while energy is mainly lost by friction and diffusion.
This is accompanied by a northward heat transport, while warm air rises in the south and cold air
sinks in the north of the channel. For the lower negative LEs these processes are inverted. These
qualitative features do not depend on ∆T . As for the barotropic conversion, the fastest growing
CLVs (for the two largest values of ∆T ) gain energy by transporting momentum away from the
baroclinic jet of the background trajectory. For slow growing CLVs and all decaying CLVs the
barotropic conversion is always negative.
Empirical Orthogonal Functions (EOFs) (Peixto and Oort, 1992) can be used to construct
models of reduced complexity (Selten, 1995; Franzke et al., 2005). Unfortunately, the modes
constructed using this approach are based on correlations and are not related to the actual dynamics
of the flow. As a preliminary idea we take a first step towards a comparable concept employing
CLVs. We investigate how much variance of the background trajectories can be explained by the
CLVs. We find that CLVs with higher LEs explain the trajectory’s variance better than CLVs with
lower LEs. In particular, CLVs with a positive baroclinic conversion can be used as a meaningful
reduced basis of the background trajectory for all values of ∆T .
The structure of the paper is the following. After giving a short overview of the CLVs in section
2, we explain in section 3 the technical details of the model. In section 4 we present a brief summary
of the most important aspects of the LEC for our model. In section 5 we construct the LEC for
each of the CLVs and discuss its relation to the transports of heat and momentum. In section 6 the
main results are discussed. First the properties of the Lyapunov spectrum are evaluated then the
LEC of the CLVs is evaluated and connections are drawn between the background state and the
stable and unstable processes described by the CLVs. Finally, we report on the reconstruction of
the variance of the background trajectories using CLVs, comparing the efficiency of using unstable
vs. stable modes (section 7). We conclude the study with a short summary, conclusion and an
outlook for future studies.
2 Linear Stability Analysis and Lyapunov Vectors
Before we present the model used in our study (see next section 3), we recapitulate the mathemat-
ical background of covariant Lyapunov vectors (CLVs). They form basis of all linear perturbations
to a given background trajectory of a dynamical system and fulfill two important properties. First,
they are covariant, hence each element of the basis is a time-dependent solution to the tangent
linear equation (see below equation 3). This means the basis vectors are linearized approximations
of nearby evolving trajectories and we can identify physical processes between them and the back-
3
![Figure 5: Lyapunov Exponents [1/day] for three meridional temperature gradients (dotted: 39.81K , dashed: 49.77K, solid: 66.36K)](/figures/figure-5-lyapunov-exponents-1-day-for-three-meridional-3dgh142r.webp)