Dynamical analysis of blocking events:
spatial and temporal fluctuations of
covariant Lyapunov vectors
Article
Accepted Version
Schubert, S. and Lucarini, V. (2016) Dynamical analysis of
blocking events: spatial and temporal fluctuations of covariant
Lyapunov vectors. Quarterly Journal of the Royal
Meteorological Society, 142 (698). pp. 2143-2158. ISSN 1477-
870X doi: https://doi.org/10.1002/qj.2808 Available at
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Dynamical Analysis of Blocking Events: Spatial and Temporal
Fluctuations of Covariant Lyapunov Vectors
Sebastian Schub ert
1,2
and Valerio Lucarini
2,3,4
1
IMPRS - ESM, MPI f¨ur Meteorologie, University of Hamburg, Hamburg, Germany,
Email: sebastian.schub ert@mpimet.mpg.de
2
Meteorological Institute, CEN, University Of Hamburg, Hamburg, Germany
3
Department of Mathematics and Statistics, University of Reading, Reading, United
Kingdom
4
Walker Institute for Climate System Research, University of Reading, Reading,
United Kingdom
First Draft: November 2015
This Draft: January 2016
Abstract
One of the most relevant weather regimes in the mid-latitudes atmosphere is the persistent
deviation from the approximately zonally symmetric jet stream to the emergence of so-called
blocking patterns. Such configurations are usually connected to exceptional local stability prop-
erties of the flow which come along with an improved local forecast skills during the phenomenon.
It is instead extremely hard to predict onset and decay of blockings. Covariant Lyapunov Vectors
(CLVs) offer a suitable characterization of the linear stability of a chaotic flow, since they rep-
resent the full tangent linear dynamics by a covariant basis which explores linear perturbations
at all time scales. Therefore, we assess whether CLVs feature a signature of the blockings. As a
first step, we examine the CLVs for a quasi-geostrophic beta-plane two-layer model in a periodic
channel baroclinically driven by a meridional temperature gradient ∆T . An orographic forcing
enhances the emergence of localized blocked regimes. We detect the blocking events of the channel
flow with a Tibaldi-Molteni scheme adapted to the periodic channel. When blocking occurs, the
global growth rates of the fastest growing CLVs are significantly higher. Hence, against intuition,
the circulation is globally more unstable in blocked phases. Such an increase in the finite time
Lyapunov exponents with respect to the long term average is attributed to stronger barotropic
and baroclinic conversion in the case of high temperature gradients, while for low values of ∆T ,
the effect is only due to stronger barotropic instability. In order to determine the localization of
the CLVs we compare the meridionally averaged variance of the CLVs during blocked and un-
blocked phases. We find that on average the variance of the CLVs is clustered around the center of
blocking. These results show that the blocked flow affects all time scales and processes described
by the CLVs.
1 Introduction
The study of weather regimes in the atmosphere is a key topic in meteorology and geosciences. In
particular, blocking highs have been early on identified as persistent, large scale deviations from
the zonally symmetric general circulation [Rex, 1950, Baur, 1947]. Traditionally, the detection and
description of these events employs objective indicators based on pressure anomalies in the atmosphere
obtained from observational data or output of general circulation models [Lejen¨as and {\O}kland,
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1983, Tibaldi and Molteni, 1990, Schalge et al., 2011]. Such blocking events and related large scale
weather regimes provide an important contribution to the low frequency variability of the atmosphere.
In particular, one can interpret the mid-latitude atmosphere as jumping between a zonal regime
and a blocked regime, or, more in general, a regime where long waves are strongly enhanced [Benzi
et al., 1986, Sutera, 1986, Molteni et al., 1988, Ruti et al., 2006]. One needs to remark that the so-
called bimodality theory and the analyses which have confirmed - at least partially - its validity have
been criticized in the literature, see e.g. Nitsche et al. [1994] and Ambaum [2008]. In Charney and
DeVore [1979], Charney and Straus [1980], it was speculated that the existence of multiple stationary
equilibria in simple models of the atmospheric circulation is the root cause for weather regimes. In
their investigation of a highly truncated quasi-geostrophic (QG) models, several stationary states exist
due to an orographic forcing. Different weather regimes are then associated with the neighborhood
of the various stationary states. Contrary to this theory of multiple equilibria, it was found that in
less severely truncated models, which adopted realistic forcings, stationary states are far away from
the attractor and/or only one stationary state exists [Reinhold and Pierrehumbert, 1982, Tung and
Rosenthal, 1985, Speranza and Malguzzi, 1988]. In a recent contribution by Faranda et al. [2015], a
different paradigm is instead proposed: blocking events are seen as close returns to an unstable fixed
point in a suitably defined reduced space describing the large scale dynamics of the atmosphere.
When considering high-dimensional chaotic dynamics, we have to look at the problem of the pos-
sible existence of weather regimes by looking at the properties of the invariant measure supported on
the attractor of the system. A possible way to revisit the idea of transitions between atmospheric
regimes is based on looking at the switching between the neighborhood of unstable periodic orbits
[Gritsun, 2013]. We remind that unstable periodic orbits provide an alternative way to reconstruct the
properties of the attractor of a chaotic dynamical systems (Cvitanovic and Eckhard 1991). Also, het-
eroclinic connections between unstable stationary states were found in a highly truncated barotropic
model [Crommelin, 2003]. In models with higher complexity leftovers of these structures are found
and correlate with transitions between different weather regimes [Kondrashov et al., 2004, Sempf et al.,
2007]. In a reduced model phase space, this allows for identifying different dynamically stable weather
regimes and less stable transitions paths between them [Tantet et al., 2015].
In this paper, we take inspiration from the classical point of view on the dynamics of blocking,
which focuses on the analysis of the linear instabilities of low-order models, but here we we consider
more Earth-like - at least, qualitatively - background turbulent atmospheric conditions. While the
attractors we consider are strange geometrical objects, we follow a mathematical approach such that
we are able to stick to the investigation of linear stability properties, which allows for a relatively
easy interpretation of the underlying physical mechanisms. Ever since Lorenz [1963], it is clear that
linear stability is a measure of predictability of the atmosphere. Therefore, the difficulty of predicting
- in time - the onset and decay of weather regimes and their persistence should be reflected in local
stability properties. The analysis of optimal linear perturbations indicated that the leading optimal
perturbation localizes where blocking occurs [Buizza and Molteni, 1996]. In a study by Frederiksen
[1997] normal modes for a time varying basic state were investigated. Naoe and Matsuda [2002] found
that - in contrast to the baroclinic instability - the emergence of blocking events can not be explained
by linear perturbations of fixed states of the atmosphere, instead non-linear processes have to be
included.
Our approach to this problem will be based on investigating blocking events using Covariant
Lyapunov Vectors (CLVs). These vectors form a norm-independent and covariant basis of the tangent
linear space [Ruelle, 1979, Eckmann and Ruelle, 1985, Trevisan and Pancotti, 1998, Ginelli et al.,
2007]. The long-time average of the growth rates of the CLVs give the Lyapunov exponents (LEs),
see discussion in Froyland et al. [2013] and Vannitsem and Lucarini [2015]. Note that by spanning
the tangent space of the attractor, CLVs allow in principle a precise calculation of the response
operator to an arbitrary perturbation of a dynamical system [Lucarini et al., 2014, Lucarini and
Sarno, 2011, Ruelle, 2009]. CLVs provide a powerful method for characterizing the properties of
weather regimes. First, they are a first order representation of the dynamics around a fully non-
linearly evolving background state, so that no simplifying hypotheses are made on the dynamics.
Second, they are a generalization of the normal mode instabilities of basic states of the atmosphere,
2
so that it is still possible to use all the machinery of linear ordinary differential equations. Taking these
points into consideration, it is suggestive to consider CLVs as a superior choice over other orthogonal,
hence norm-dependent Lyapunov vectors [Legras and Vautard, 1996].
Previously, we have investigated CLVs in a quasi geostrophic two layer model in a periodic channel
[Schubert and Lucarini, 2015]. In that work, we addressed how the average energy and momentum
transports of the CLVs are related to their growth and decay in respect to the background state and
how they explain the variance of the background state. Moreover, we provided a bridge between
the growth rate of the CLVs and the physical mechanisms responsible for the variability of the quasi-
geostrophic flow, namely the barotropic and baroclinic conversion, by a detailed analysis of the Lorenz
Energy cycle of each CLV. We note that our focus was exclusively on the long-term properties of the
flow, of its CLVs, and of the corresponding LEs.
In this paper, we are concerned with weather regimes in the background state, hence we will
study the fluctuations of the CLVs and of the finite-time LEs. The rationale of our study is then the
following. Using the classical Tibaldi-Molteni scheme blocking detection, we will determine when the
flow is unblocked and when/where the flow switches to a blocked state [Tibaldi and Molteni, 1990].
We will then address two questions.
1. Is there a systematic signature of blocked phases in the growth rates of the linear perturbations?
Is there a systematic change in the energetics of the flow?
2. Is the occurrence of blocked phases linked to the presence of specific patterns for the CLVs and
to their localization in the physical space?
Note that the second question is different from the average localization of the CLVs investigated in
Szendro et al. [2008]. In order to address these questions, we have extended the model of our previous
study with an orographic forcing, following [Charney and Straus, 1980]. As in our previous study,
the model is baroclinically driven by introducing a relaxation meridional temperatrue gradient ∆T
and dissipates energy via Ekman pumping, which parameterizes the effect of the planetary boundary
layer. The orography in our investigation is a Gaussian bump in the middle of the domain with
horizonal scale of O(1000) km. We explore the sensitivity of the problem by considering multiple
setups featuring different heights of the gaussian bump and different values of ∆T . The various
setups all exhibit chaotic conditions with many positive LEs.
We find that blocking increases with a higher meridional temperature gradient ∆T and is addition-
ally enhanced by orography, which, by breaking the zonal symmetry, contributes as a catalyst to the
process of a phase lock mechanism that allows standing perturbations to grow, as envisioned in Benzi
et al. [1986]. The spatial variance of the CLVs is dominantly located around the region where blocking
occurs when compared to the average variance during unblocked phases. Furthermore, the growth
rates of the fastest growing CLVs are higher during blocked phases, pointing at the fact that the
system has globally a lower predictability during blocked phases, possibly as a result of the difficulty
of predicting when the onset and decay of the blocking events. The observed increased instability
suggests also that the local dimension of the attractor is higher during blocking. We explain the
changed growth behavior by using a generalization of the Lorenz energy cycle between the CLVs and
the background state introduced in Schubert and Lucarini [2015]. We find that for high values of ∆T
the increased instability is dominantly caused by an increased input of energy to the CLVs by baro-
clinic and barotropic conversions, while for weakly baroclinic flows the intensification of barotropic
instability is the only active mechanism. We speculate that these results hint at a possible definition
of blocking by taking into account the properties of all CLVs at a particular time.
The structure of the paper can be summarized as follows. In section 2, we describe our experimental
set-up, by sketching the formulation of the QG model, the basic properties of CLVs, and the blocking
detection algorithm. In section 3, we present our results on the properties of blocked vs unblocked
phases, discussing the properties of the fluctuations of LEs and CLVs and investigating the sensitivity
of our results to changes in the forcing and in the orography. Finally, in section 4, we give an outlook
and summary and point the reader towards future work on these topics.
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