Covariant Lyapunov Vectors of a Quasi-geostrophic Baroclinic Model: Analysis of Instabilities and Feedbacks

TL;DR: In this paper, a covariant Lyapunov vector (CLV) is proposed to describe the directions of exponential expansion and decay of perturbations to the non-linear trajectory of the flow.

Abstract: 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 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 directions 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 $\Delta T$. We explore three settings of $\Delta T$, representative of relatively weak turbulence, well-developed turbulence, and intermediate conditions. 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 sufficient condition of instability. Barotropic instability is present only for few very unstable CLVs for large values of $\Delta T$. 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.

Summary

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 methods allow for computing CLVs for high dimensional chaotic systems and have led to a renewed interest in the 2 related theory.
• In section 6 the main results are discussed.

2 Linear Stability Analysis and Lyapunov Vectors

• Before the authors present the model used in their study (see next section 3), they recapitulate the mathematical background of covariant Lyapunov vectors (CLVs).
• First, they are covariant, hence each element of the basis is a time-dependent solution to the tangent linear equation (see below equation 3).
• The meaning of the BLV can be understood by the following examples.
• Therefore, the cut of the subspaces V −j (t) and V +n−j+1(t) contains only vectors which have an asymptotic grow rate of λj on the interval [−∞,∞].
• The backward iteration leads to a vector which aligns with the fastest growing vector of the backward dynamics in the respective backward subspaces.

3 The Model

• This is the first time that CLVs are computed for a geophysical model and then used to characterize the properties of linear stability of its chaotic solutions.
• The model is a spectral version of the classical model introduced by Phillips (1956).
• With the boundary conditions and the adimensionalization, the stream function has the following form in spectral space.
• In their study, the authors are obtaining the CLVs for aperiodic background states over a much longer time period of ca. 25 years.
• The fields show baroclinically unstable eddies moving eastward which can be seen from the phase shift between the upper and lower layer.

4 Atmospheric Circulation and the Lorenz Energy Cycle

• The authors study such a decomposition for three different values of the forced meridional temperature gradient ∆T (dotted: 39.81K , dashed: 49.77K, solid: 66.36K).
• All setups feature an unstable stationary solution and an attractor corresponding to a turbulent solution.
• In the lower layer a small eastward jet emerges in the middle of the channel.
• The baroclinic conversion CZP→EP quantifies the exchange between the potential energies of the mean and the eddy fields.
• There are also sinks of energy related to the various terms of friction, diffusion and newtonian cooling.

5 The Lorenz Energy Cycle Of The CLVs

• Classical stability analysis interprets the growth of the normal modes by introducing a Lorenz energy cycle between the normal modes and the zonal background state (see, e.g. (Holton, 2004)).
• In their model, this role is played by the total energy Etot.
• The average growth rates of the CLVs measured in these or any other norms is equal to the Lyapunov exponents.
• After applying integration by parts, this term equals the negative correlation between the convergence of heat transport of the CLV (−∂y(v′Mψ′T ) − ∂x(u′Mψ′T )) and the temperature of the background state.

6 Physical Properties Of The CLVs

• In this section the authors present the actual features of the LEC and the associated transports for the CLVs.
• Additionally, the authors show the properties of chaoticity that can be derived from the Lyapunov spectrum.

6.2.1 Energy Conversion Terms and Sinks

• Now the authors can unravel the connection of the physics of the CLVs captured by the LEC to their stability properties.
• This includes all growing CLVs and part of the decaying CLVs.
• Hence, with increasing ∆T the barotropic conversion of all unstable CLVs turns from negative and to positive values for the fast growing CLVs.
• The energy loss rate by Ekman friction is large for very stable CLVs (low Lyapunov exponents).
• The authors compare the slow growing/decaying CLVs with the decomposition of the flow into zonal mean and the eddies (see section 4) in the magnified view on the right side of the figures 6 and 7.

6.2.2 Convergence of Heat and Momentum Transport and Vertical Velocity

• The previously discussed energy conversion (CBC , CBT and CPK) shown in fig.
• For lower ∆T the convergence of heat transport is largest near the center of the channel.
• Correspondingly, the momentum transports of the CLVs cause a convergence of momentum in the center of the channel, resulting a pointier jet (see fig. 8 a).
• This results into the fact that the authors do observe a pointy jet as mean state of the system (see also fig.
• The CLVs with positive baroclinic conversion feature upward and northward heat transport, while the opposite holds for the CLVs with negative baroclinic conversion rate.

7 Explaining the variability of the background flow

• So far the authors have studied the linear stability of their model by determining physical properties of CLVs (see section 6).
• Moreover, the Eady modes grow similar to cyclones due to a vertical westward tilt of the troughs and lows which induces a positive baroclinic energy transfer to the modes from the respective zonal background state and a northward eddy heat transport.
• The authors address first how these subspaces of CLVs are constructed.
• In this figure the authors also compare the Bsn cascade with a randomly chosen basis.

8 Summary & Conclusion

• The authors objective in this study was to determine the physical properties of the tangent linear space of a quasi-geostrophic model of the mid latitudes atmosphere.
• This understanding of the dynamics is linked to the decomposition of the atmospheric flow into a zonal mean state and an eddy field.
• The slowly growing and decaying CLVs exhibit similar properties as they all feature terms of a positive baroclinic conversion and a negative barotropic conversion including the associated momentum transport to the middle of the channel and the northward heat transport.
• This is due to the slow decorrelation of the slow growing/decaying CLVs with the background trajectory.
• Furthermore, the CLVs will be constructed for models featuring time-scale separations like the authors expect them in the primitive equations and coupled atmosphere/ocean models.

Figures

Figure 8: Left (a, c, e): The mean zonal profiles of Convergence of Momentum Transport. Right (b, d, f): Northward Heat Transport (b, d, f). The x axis indicates the jth CLV. In (a, c, e) the solid lines indicate the sign switch of the barotropic conversion CBT from positive to negative. In (b, d, f) the dash-dotted lines indicate the sign switch of the baroclinic conversion CBC from positive to negative. The black dotted lines show the CLV with smallest positive LE. The y axis shows the distribution in the meridional direction in 103km.

Table 3: Transports and Conversions - The conversion is positive if the correlation between gradient of the background state and eddy transport of the CLVs is negative. Note that for CPK the baroclinic stream function ψT is proportional to the vertical gradient of the stream function.

Figure 3: Flow Chart of the Lorenz Energy Cycle for three ∆T (Units of Conversions are 105m2/s3). The arrows indicate the average sign of the energy conversions, sinks and sources of the zonal and eddy energies. For every temperature gradient (∆T1 = 39.81K, ∆T2 = 49.77K and ∆T3 = 66.36K) the dominant source of energy is Newtonian cooling, which inputs energy to the zonal mean potential energy. The important conversions are the baroclinic conversion which is related to the northward heat transport (see fig. 2 c) and the barotropic conversion related to the center pointed momentum transport (see fig. 2 d). The main energy losses occour by converting the potential energy of the eddies into kinetic energy, where it is lost mainly due to kinetic diffusion and Ekman friction. We observe an intensification of the cycle for a larger meridional temperature gradient ∆T .

Figure 10: The panels show the average correlation (solid lines) of the subspaces spanned by the n fastest growing CLVs (a) and the n fastest decaying CLVs (b) with the considered trajectories of ∆T (dotted: 39.81K , dashed: 49.77K, solid: 66.36K). The parameter n is indicated on the x axis. The average is done over the mean correlation for different reference points of the CLVs (41 reference points, equally distributed over a 12 years period). The corresponding σ area is indicated by the shaded regions. The vertical dashed dotted lines indicate where the expansion includes exactly all baroclinically unstable CLVs (Panel a) or all baroclinically stable CLVs (Panel b). Panel b) also shows the expansion into a randomly chosen basis (almost diagonal dashed lines). The comparison of both panels shows the higher explanatory power of CLVs with a higher LE.

Table 2: Properties of the attractor

Figure 5: Lyapunov Exponents [1/day] for three meridional temperature gradients (dotted: 39.81K , dashed: 49.77K, solid: 66.36K)

Table 1: Parameters and Variables used in this model and the respective adimensionalization scheme. Not that the scales for time and length are t = 104s = 1/f0 and L = 107 π m

Figure 6: Left Side: The three figures show the dependence of the inputs and the conversion of the Lorenz energy cycle on the corresponding Lyapunov exponent for each of the three meridional temperature gradients (dotted: 39.81K , dashed: 49.77K, solid: 66.36K). The magnified view (right side) shows the CLVs with near zero growth rate including the corresponding average eddy observables from the classical Lorenz energy cycle (gray horizontal lines). The y axis units are in 1/day.

Figure 2: The mean state of the trajectory (averaged over a period of 25 years) for the three forced meridional temperature gradients ∆T (dotted: 39.81K , dashed: 49.77K, solid: 66.36K). The superscript E indicates the eddy terms.

Figure 4: The graphs show the modulus of the average correlation <ψ B ,ψ′> ||ψB ||||ψ′|| between the background state and the CLVs, where the bilinear product < ·, · > is defined via the kinetic, potential and total energy. The grey shaded areas are the 3 σ confidence intervals. We estimated the effective number of degrees of freedom by dividing the time series into blocks corresponding to the e folding time of the autocorrelation function (Leith, 1973). The x axis indexes the CLVs. Similar results have been obtained for non zonal stationary states (Niehaus, 1981)

Figure 1: These snapshots show the streamfunction fields for the upper and lower layer. The upper is on the left side and the lower layer is found on the right side. Note, that in order to use a grey scale, the scale of the colorbars is different in every panel. The flow shows a clearly baroclinically unstable and barotropically stable configuration (see section 4)

Figure 9: The mean zonal profiles of the conversion of heat S? for the three meridional temperature gradients plotted for every CLV ((a) ∆T = 39.81 K, (b) ∆T = 49.77 K and (c) ∆T = 66.36 K). The black vertical dashdotted lines indicate the change of sign from positive to negative of the baroclinic conversion CBC. The dashed lines indicate the change of sign in the conversion from potential to kinetic energy CPK. The black dotted lines show the CLV with the smallest positive LE. The y-axis shows the distribution in the meridional direction in units of 103 km. The x-axis indicates the jth CLV. 25

Figure 7: Left Side: The four figures show the dependence of the different sinks of the Lorenz energy cycle on the corresponding Lyapunov exponent for each of the three meridional temperature gradients (dotted: 39.81K , dashed: 49.77K, solid: 66.36K). The magnified view (right side) shows the CLVs with near zero growth rate including the corresponding average eddy observables from the classical Lorenz energy cycle (gray horizontal lines). The y axis units are in 1/day.

Full text

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
1

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

Frequently asked questions

Q1. What are the contributions in "Covariant lyapunov vectors of a quasi- geostrophic baroclinic model: analysis of instabilities and feedbacks" ?

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. While these methods have undoubtedly great value for elucidating the relevant physical processes, they are unable to follow the dynamics of a turbulent atmosphere. The authors 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 directions of exponential expansion and decay of perturbations to the non-linear trajectory of the flow. The authors 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. The authors explore three settings of ∆T, representative of relatively weak turbulence, well-developed turbulence, and intermediate conditions. 

Q2. What are the future works mentioned in the paper "Covariant lyapunov vectors of a quasi- geostrophic baroclinic model: analysis of instabilities and feedbacks" ?

Future work will deal with two layer QG dynamics with different imposed boundary conditions. It is also promising to study higher resolutions given the discovery of wave-dynamical and damped-advective Floquet vectors by Wolfe and Samelson in a high resolution QG two layer model ( Wolfe and Samelson, 2006, 2008 ). This will include on the one hand orography acting on the lower layer ( Speranza et al., 1985 ; Charney and DeVore, 1979 ) and on the other hand potential vorticity anomalies imposed on the upper layer. The authors will also test the hypothesis whether the CLVs can be used for the construction of a reduced model of the atmospheric circulation.