Introduction
Atmospheric motions are governed by a web of complex interactions among the
different components of the earth system . Solar
radiation and the earth's rotation are the primary large-scale drivers of the
chaotic atmospheric dynamics, while turbulent motions add a layer of
complexity at small scales. This picture is further complicated by the
presence of features such as ocean–land interactions, vegetation,
anthropocentric forcing, and the
hydrological cycle. Understanding both the transient (i.e. meteorological)
and mean (i.e. climatic) properties of this system is one of today's major
scientific challenges.
Since 's seminal work, dynamical systems
techniques have been widely applied to the study of the atmosphere. For
example, the use of tools such as the Lyapunov exponents or the
Kolmogorov–Sinai entropy has led to important advances in our understanding
of atmospheric predictability . An important result has
been to show that atmospheric motions are chaotic but not random: their
trajectories stay close to a high-dimensional object called an attractor
.
This object occupies only a fraction of the atmospheric phase space, meaning
that its dimension D is smaller than the number of variables used to
describe the system. D is an important quantity because it represents the
number of degrees of freedom of the system, namely the minimum number of
variables needed to represent the dynamics. The computation of D for
atmospheric attractors has posed a challenge to the dynamical systems
community for several decades. Whereas in the early 1980s several estimates
pointed to a low-dimensional D<10 attractor
, a later review of the numerical limitations
of the available techniques suggested that they tended to underestimate D
for complex systems . However, further estimates
of D were hardly attempted, because D>10 implies that low-dimensional
models should fail in describing the atmospheric dynamics.
D is a mean property, since it describes the dimension of the attractor for
the whole atmospheric trajectory. However, it is often more useful to
determine instantaneous dynamical systems metrics that describe transient
states ζ of complex attractors. A quantity that contains such
information is the local dimension d(ζ) .
The value of d is proportional to the active number of degrees of freedom
and provides information on how predictable the state ζ and its future
evolution are . By averaging d over all
possible ζ, one recovers the attractor dimension D. Unfortunately,
the computation of d has posed even greater challenges than that of D.
The original method developed by used box counting
techniques. First, a small portion of the phase space is partitioned in
hypercubes of different sizes. One then looks at the amount of space filled
up in each hypercube. The scaling of this quantity across different scales is
proportional to d. The complexity of this technique prevented computations
for high-dimensional systems such as atmospheric flows. Very recently, some
of the authors of this paper have contributed to developing an alternative
way to obtain d, based on the universal behaviour of Poincaré recurrences
in chaotic systems
. In a few
words (see “Methodology and data” for the details), the recurrences of a
state ζ of a chaotic dynamical system of arbitrary dimension have a
universal asymptotic distribution in the limit of infinite recurrences. The
parameters of this distribution are linked to the instantaneous dimension
d(ζ) and to another important dynamical quantity, namely the inverse of
the average persistence time of the trajectory around ζ
. Estimating these parameters via Poincaré
recurrences is easier than with the box counting algorithms because the
method avoids altogether computations in scale space. Since the asymptotic
distribution is known, one can numerically check that enough recurrences are
taken into account for the parameter estimates by performing standard
statistical tests (see ).
Having overcome the technical difficulties inherent to the calculation of
instantaneous dynamical systems metrics, the remaining step is the choice of
the states ζ of interest. Since it is impractical – not to say
impossible – to consider all atmospheric observables at once, we focus our
analysis on variables which are representative of events which affect human
welfare and society, namely sea-level pressure (slp; cyclones, windstorms
etc.), 2 m temperature (t2m; heat waves, cold spells), and precipitation
frequency (prp; droughts, floods). In dynamical systems terms, these fields
represent projections of the full phase-space dynamics onto specific
subspaces, called Poincaré sections. In and
we have shown that d and θ can be used
to characterize regional-scale atmospheric fields and further provide
information on the predictability linked to a given atmospheric state. It is
therefore important to investigate these indicators at different spatial
scales to fully understand the insights they can provide. In this study we
present a novel analysis based on d and θ computed for the whole
Northern Hemisphere (NH). We further investigate for the first time the
mutual correlations between the dynamical properties of different climate
variables.
The paper is organized as follows: in Sect. 2 we give an overview of the
dynamical indicators, the methodology to compute them, and the data used. In
Sect. 3 we present and discuss the dynamical properties of each of the three
atmospheric fields separately, while in Sect. 4 we analyse them jointly.
Finally, we discuss our results and summarize our conclusions in Sect. 5.
Methodology and data
The attractor of a dynamical system is a geometrical object defined in the
space hosting all the possible states of the system (the so-called phase
space) . Each point on the attractor ζ can be
characterized by two dynamical quantities: (i) the local dimension
d(ζ), which provides the number of degrees of freedom active locally
around ζ, and (ii) the inverse persistence of the state ζ:
θ(ζ), which is a measure of the mean residence time of the system
around ζ.
Local dimensions
The term attractor dimension usually refers to a global measure
. D indicates the average number of
degrees of freedom of a dynamical system. Several methods to measure D were
developed in the 1980s .
These techniques have a certain number of adjustable parameters and require
the system to be embedded in a subspace of the phase space. They provide good
estimates of D only when the trajectories are sufficiently long to estimate
the embedding parameters. Such computations are therefore problematic in
systems with large numbers of degrees of freedom and give biased results when
applied to atmospheric flows
.
The technique we exploit here results from the application of extreme value
theory to Poincaré recurrences in dynamical systems
. In this approach, the
returns to points on chaotic attractors are fully characterized by extreme
value laws. In practice, one needs long trajectories x(t) that approximate
sequences of states on the attractor. One then fixes a point ζ on the
trajectory and computes the probability P that x(t) will return in a ball
of radius ϵ centered on point ζ. The
theorem, modified in
, states that logarithmic returns
g(x(t))=-log(dist(x(t),ζ)) are distributed as
P(g(x(t))>s(q),ζ)≃exp-x(t)-μ(ζ)σ(ζ).
Here s is a high threshold associated with a quantile q of the series
g(x(t)) itself, linked to the radius ϵ via s=g-1(ϵ).
In other words, requiring that the orbit falls within a ball of radius
ϵ around the point ζ is equivalent to asking that the series
g(x(t)) is over the threshold s. Here we adopt q=0.98 to determine s.
The resulting distribution is the exponential member of the generalized
Pareto distribution family. The parameters μ and σ depend on the
point ζ chosen on the attractor. σ(ζ) then provides the
local dimensions d(ζ) via the simple relation σ=1/d(ζ). This
result is very powerful because it provides a new way to compute local
dimensions on the attractor and to recover D as the average of d on all
the ζs without the need for embedding, as is required in most dimension
computation algorithms . We want to stress
that this procedure is not just a statistical fitting. The reason why it
provides good estimates of d that were impossible to obtain with previous
techniques derives from the universality of the extreme value statistics for
Poincaré recurrences: one knows a priori the statistics of such recurrences
and can then check that they are achieved for the numerical trajectory
examined.
Local persistence
The stability of the state ζ is measured by θ(ζ), namely the
inverse of the average residence time of trajectories around ζ. For
discrete maps θ is uniquely defined (see
, for details): if ζ is a fixed point of
the dynamics, θ(ζ)=0. For a point that leaves the neighbourhood of
ζ immediately, θ=1. For continuous flows, the definition of
θ depends on the Poincaré map chosen and precisely on the Δt
chosen to discretize the flow. Since θ is the inverse of the average
residence time, it is measured in units of 1/Δt. In general, the
higher the persistence of the point ζ, the longer the previous and
subsequent states of the system will resemble ζ. The residence time can
be computed by introducing a further parameter in the previous law. This
parameter, known as the extremal index, is such that
P(g(x(t))>s(q))≃exp-θx-μ(ζ)σ(ζ).
To estimate θ, we adopt the Süveges estimator
. For a fixed quantile q, the estimator is
defined as
θ=∑iNc(1-q)Si+N+Nc-∑iNc(1-q)Si+N+Nc2-8Nc∑iNc(1-q)Si2∑iNc(1-q)Si,
where N is the number of recurrences above the chosen quantile,
Nc the number of observations which form a cluster of at least
two consecutive recurrences, and Si is the length of each cluster i.
This length is the number of consecutive time steps during which the
trajectory remains within a radius ϵ of ζ. For further details
on the derivation of this estimator, the reader is referred to
.
Data
We use daily fields from the NCEP/NCAR reanalysis ,
with a horizontal resolution of 2∘. The analysis is carried out over
the whole Northern Hemisphere for all days of the year over the period
1948–2013. The observables of interest are sea-level pressure, 2 m
temperature, and precipitation frequency. A previous study
has shown that the results obtained are largely
independent of the dataset used and of its spatial resolution.
Anomalies are defined as deviations from the long-term daily mean. So, for
example, the anomaly of t2m at a given location on 5 December 2000 is
computed relative to the mean value of all 5 Decembers in the dataset at that
location.
The relevance of the composite anomaly maps for the different variables is
evaluated using a sign test. This identifies geographical areas where at
least 2/3 of the composite maps have the same-sign anomaly. Assuming a
binomial process with the same number of draws as the composite maps and
equal chances of positive or negative outcomes (binomial distribution with
success rate 0.5), a 2/3 threshold is beyond the 99.99th percentile of the
distribution. In Figs. , , and ,
such regions are marked by the thick black lines.
Statistics of local dimension d and persistence θ for daily
sea-level pressure (slp) data from the NCEP/NCAR reanalysis. Time series of
daily values of d (a) and θ (b). Autocorrelation
function ACF(d) (c) and ACF(θ) (d) for 3 years in
daily lags.
Dynamical properties of individual observables
Sea-level pressure (d,θ)
The local dimension, d, of the slp field shows a marked variability
throughout the analysis period, with values ranging from as low as 8.6 to as
high as 33.2 (Fig. a). The average dimension D, which in this case
is roughly 19.4, therefore provides incomplete information concerning the
field of interest, since the number of locally active degrees of freedom
(identified by d) can vary by a factor of almost 4. The autocorrelation
function (ACF) of d (Fig. c) highlights a robust variability
pattern which is not immediately evident from the raw time series. There is a
clear semi-annual cycle, with peak autocorrelation values in excess of 0.21.
Over a full year there are therefore two positive and two negative peaks in
autocorrelation, with the second positive peak typically displaying a larger
magnitude than the first. This is consistent with previous analyses which
have identified a strong seasonal dependence in d
. The presence of a
semi-annual cycle leads us to interpret the ACF as being modulated by the
four seasons, with the first positive peak corresponding to cross-season
correlation and the second, larger, peak corresponding to correlation between
the same seasons in successive years. This periodicity could be linked to
semi-annual slp variability features at the mid-latitudes
.
The inverse persistence, θ, shows a marked variability, with values
ranging from 0.28 to 0.65 (i.e. 1.6 to 3.6 days in terms of 1/θ)
(Fig. b). We note that these values should not be compared directly
to the persistence of the traditional weather regimes defined using
clustering algorithms as (i) here we consider a full hemispheric domain,
while weather regimes are typically computed for specific regions, and
(ii) the requirement that the flow does not leave the neighbourhood of the
state ζ is a more restrictive condition than continued permanence
within a given cluster. Indeed, if one considers the typical partition
of the atmospheric patterns over
the North Atlantic into four weather regimes, the probability of being in one
of them is of order 0.25, whereas the probability of being close to
ζ is set by the threshold s – in our
case 0.02 (see Sect. 2.1). Concerning the ACF (Fig. d), θ
shows a very different pattern to d. The year-to-year correlation between
the same seasons is still large and positive, but the semi-annual oscillation
seen in d is almost entirely absent. Indeed, the winter values appear to be
anticorrelated with those of the other three seasons.
Monthly average values (a, b) and standard deviation
(c, d) for the local dimension d (a, c) and inverse
persistence θ (b, d) of sea-level pressure (slp, blue), 2 m
temperature (t2m, red), and precipitation frequency (prp, black) data.
An analysis of monthly-mean values confirms the strong seasonal control on
the dynamical characteristics of the field. In the summer months, both the
magnitude (Fig. a, b) and variability of the two metrics reaches a
minimum (Fig. c, d). With the autumn season, the local dimension
and θ increase rapidly while the variability remains low. As winter
progresses the variability increases, θ remains roughly constant while
the local dimension shows a marked decrease, albeit remaining well above the
summertime values. In spring d grows back to values similar to those seen
in autumn, while its variability peaks and θ starts decreasing. This
picture is consistent with the ACFs described above. The annual cycle of
θ can be explained as follows: stability peaks in summer when the
mid-latitude storm tracks and wave activity are comparatively weak, decreases
in autumn and winter and starts increasing again during spring. For the
annual cycle of d, the maxima occurring in the intermediate season can be
explained as follows: assuming that there is a winter and a
summer attractor, the transitional seasons are more unstable because
the atmospheric flow can explore both the summer and winter configurations.
In dynamical systems terminology, the spring/autumn atmospheric flow sits on
a saddle-like point of the dynamics.
Occurrence of extremes (observations beyond the 0.02 (minima) and
0.98 (maxima) quantiles) of local dimension d and inverse persistence
θ in different seasons. (a–d) sea-level pressure (slp),
(e–h) 2 m temperature (t2m), and (i–l) precipitation
frequency (prp).
One can further look at the slp anomalies corresponding to extremes in d
(Fig. ), here defined as events beyond the 0.02 and 0.98
percentiles of the full distribution (see dashed lines in
Fig. a). The maxima of d occur primarily during the spring and
autumn months, while the minima are mainly found in summer and winter
(Fig. ). This mirrors the monthly-mean values discussed above.
The d maxima correspond to a complex anomaly pattern spanning the whole
hemisphere, but a sign test shows that in both the spring and autumn seasons
there is very little agreement between the individual events
(Fig. a). This suggests that there is no single, dominant
hemispheric-scale slp configuration leading to large dimensional extremes.
The wintertime d minima again display very limited sign agreement, with the
only significant feature being an intensification and eastward extension of
the climatological Aleutian low-pressure centre (Fig. b). Such
a result is very different from what has recently been observed in the North
Atlantic region, where both d minima and d maxima systematically
correspond to precise large-scale features . In
contrast, the summer minima display an extensive and significant region of
negative anomalies over the pole. The former pattern is an enhancement of the
relatively low climatological slp values seen over the Arctic basin during
the summer months. This results in a strengthened climatological meridional
gradient (and presumably a strengthened polar vortex and a reduced air-mass
exchange between the mid and high latitudes, although we recognize that slp
is not the optimal field to diagnose this), and we therefore hypothesize that
it matches a relatively predictable configuration.
We next analyse slp anomalies corresponding to extremes in θ. The
θ maxima occur predominantly during the autumn and winter months,
while the minima are mostly found in spring and summer
(Fig. c, d). The high mean persistence found in the summer
months therefore also corresponds to instantaneous maxima in this quantity.
The maxima of θ in both autumn and winter correspond to a
circum-hemispheric wave-like structure and show very little sign agreement
(Fig. c). Since θ maxima are by definition unstable
states, the lack of sign agreement might simply be due to the zonal
propagation of the wave-like anomalies in time, although we do not explore
this idea further here. The θ minima correspond to a mostly zonally
symmetric pattern with a significant positive slp anomaly over the pole and
locally significant negative anomalies throughout the mid and low latitudes
(Fig. d). Over the North Atlantic, this results in a negative
NAO-like dipole which is consistent with the anomaly pattern found for
regional persistence maxima in the Euro-Atlantic domain
. The reversal in the sign of the polar anomalies
relative to summertime d minima is difficult to interpret. However, we note
that 's regional analysis also showed
opposite-sign anomalies at the high latitudes for d and θ minima.
More generally, our results highlight that persistent, predictable states are
primarily associated with zonally symmetric slp anomalies and therefore
modulations of the zonal flow. Indeed, past studies have interpreted the
hemispheric circulation as being dominated by a zonal-flow attractor with
blocked or wavy states being associated with an unstable fixed point
. Similarly, an enhanced zonality of the
large-scale flow has been linked to increased downstream predictability on
regional scales .
(a) Scatter plot of local dimension d and inverse
persistence θ for slp data. Each point represents the value
corresponding to 1 day in the NCEP/NCAR reanalysis. The colour indicates the
month of the year the data point falls in. Blue dotted lines indicate the
0.02 and 0.98 percentiles of the d,θ distributions.
(b) Corresponding cross-correlation function between d and
θ.
Composite anomalies with respect to the seasonal cycle in sea-level
pressure (slp) for the four phase-space regions delimited by the blue dotted
lines in Fig. . Maxima of d (a), minima of d
(b), maxima of θ (c), and minima of θ
(d). Geographical composites are shown for seasons which account for
25 % or more of the extreme occurrences (Fig. ). Units:
hPa.
The seasonal control on the two dynamical systems metrics we discuss here can
be further investigated through a d–θ scatter plot
(Fig. a). This highlights how each season forms a distinct
diagonal band of relatively well-correlated d and θ values.
Figure b indeed confirms that the two metrics have high
cross-correlation values, with the lag-0 correlation approaching 0.7. The
lagged cross-correlation shows a semi-yearly cycle, with the peak correlation
values reflecting integer year shifts and peak anticorrelation values
reflecting a shift of approximately one season. This latter feature can be
easily understood in terms of the above analysis. Both d and θ peak
in autumn and spring. If we imagine shifting the d curve forwards or
backwards by one season, the autumn θ peak will now match a dimension
trough, thus leading to a negative correlation. The smaller positive and
negative peaks in the cross-correlation function correspond to shifts of one
and three seasons, respectively, such that the two cycles are roughly in
quadrature.
Temperature (d,θ)
The local dimension d of the t2m field shows a marked variability
throughout the analysis period, with a range similar to that of d(slp):
8.9<d(t2m)<33.3 (Fig. a). The average dimension D is
roughly 17.6, slightly lower than D(slp). The ACF of d
(Fig. c) again displays a semi-annual cycle, albeit with larger ACF
values than those seen for the slp. We note that the ACF structure should not
be linked directly to the large seasonal cycle in temperature, since here we
are considering d(t2m), which is not necessarily linked to the absolute
value of the field. The inverse persistence θ spans a range
corresponding to periods between 1.9 and 6.3 days (0.16<θ<0.54,
Fig. b), indicating a higher persistence than slp. θ's ACF
(Fig. d) is again different to that seen for d. The inter-year
same-season correlation is still large and positive, but the semi-annual
oscillation seen in d is entirely absent. Indeed, the winter values appear
to be anticorrelated with those of the other three seasons, albeit with some
weak modulation in the negative correlation values on seasonal scales. This
difference is driven by the small offset between the seasonal cycles of d
and θ, as discussed below.
An analysis of monthly-mean values (Fig. ) confirms the strong
seasonal control on the dynamical characteristics of the field, but also
highlights a radically different picture from that seen for the slp. In the
summer months, d and its variability peak, while θ and its
variability display a local maximum. With the autumn season, both the local
dimension and θ reach a local minimum, only to increase again during
wintertime. During spring, both metrics display a second minimum before
returning to their high summertime values. The seasonal cycle in the
variability of both indicators roughly matches that of the indicators
themselves. The fact that the monthly-mean minima in d are broader and
occur with a 1-month shift relative to those in θ accounts for why the
semi-annual ACF cycle is only seen in the former variable. The general
picture is therefore consistent with the ACFs described above. We hypothesize
that the summertime and wintertime local maxima in d are associated with
the inherent difficulty in forecasting the onset and duration of warm and
cold spells . This is
presumably linked to a high-dimensional atmospheric configuration, namely one
with a large number of allowed preceding and future evolutions. The annual
cycle of θ also displays summertime and wintertime local maxima and
suggests that the winter (and to a lesser degree the summer) temperature
fields are comparatively unstable, while the transitional seasons have a more
sluggish dynamical evolution. This can be linked to the presence of
wintertime cold spells and summertime heat waves which are usually
non-stationary and locally short-lived, although notable exceptions can
occur. One can therefore picture the t2m dynamics as following a single
potential well configuration, with the extremes located in winter and summer.
The dynamics of slp and t2m are very different, and the interaction between
the two could be akin to a Langevin-like model .
The slp would be the variable pushed into the winter or
summer potential wells, while the temperature acts as a forcing
noise term with extremes in winter and summer.
Statistics of local dimension d and persistence θ for daily
2 m temperature (t2m) data from the NCEP/NCAR reanalysis. Time series of
daily values of d (a) and θ (b). Autocorrelation
function ACF(d) (c) and ACF(θ) (d) for 3 years in
daily lags.
One can further look at the t2m anomalies corresponding to extremes in d
and θ, again defined as events beyond the 0.02 and 0.98 percentiles of
the full distribution (see dashed lines in Fig. a). The d
maxima occur predominantly during summer, while the minima are mostly found
in autumn (Fig. e, f), consistently with the seasonal cycle
described above. The spatial anomalies corresponding to d maxima are weak
and display very little sign agreement (Fig. a), suggesting
that there is no single large-scale pattern matching these extremes. This is
consistent with the theory that the high dimensionality of the temperature
field may be associated with warm and cold spells, which are highly
non-stationary and therefore will not emerge in a composite plot.
Interestingly, the only region showing strong sign agreement is over eastern
Africa, and also emerges in the composite anomalies for the dynamical
extremes of prp (see below). The anomalies associated with minima in d are
stronger, but show a similarly low sign agreement (Fig. b). The
θ maxima and minima occur predominantly during winter and spring,
respectively (Fig. g, h), and again show low sign agreement
(Fig. c, d). While, as discussed above, the dynamical extremes
elucidate a number of features of the temperature's seasonality and
variability, they seem to afford relatively little insight concerning its
geographical nature.
In the d–θ scatter plot for t2m, the winter months form a cluster
corresponding to high θ, relatively high d values, while the spring
and autumn seasons form a low θ, low d cluster (Fig. a).
Summertime forms a continuation of the spring/autumn band, extending it
toward higher d and θ values. The relatively broad scatter of the
cloud points to a weak correspondence between d and θ. Indeed, the
two metrics show lower cross-correlation values than those seen for slp, with
a lag-0 correlation of just above 0.5 (Fig. b). The lagged
cross-correlation shows a semi-yearly cycle, with the peak correlation values
reflecting full and half-year shifts, in agreement with the synchronous
double peak in both metrics shown in Fig. . Similarly, the two
large negative peaks in the cross-correlation function correspond to shifts
of one and three seasons, respectively, leading to situations in which the
two yearly cycles are in anti-phase.
Precipitation frequency (d,θ)
We construct a daily precipitation frequency variable as follows: we assign a
value of 1 to each grid point and time step for non-zero precipitation rates
and a value of 0 otherwise. For this variable the statistical fit of the
recurrences to the expected distribution is better than for the precipitation
rate itself (not shown). Another motivation to use the precipitation
frequency data can be found in the multifractal analysis performed by
, where the authors warn against using the
precipitation data directly to measure dimensionality. The local dimension
d of the precipitation frequency (prp) shows a large variability throughout
the analysis period, with markedly higher values than those of the previous
variables: 48<d(prp)<132 (Fig. a). This is also reflected in
the average dimension D=83.1. These high values are consistent with the
very scattered, noisy nature of the precipitation field. The autocorrelation
function of d displays a semi-annual cycle with large autocorrelation
values at full-year lags and near-zero values at 6-month lags
(Fig. c). This is very different from the positive 6-month
autocorrelation values seen for d(slp) and d(t2m). The inverse
persistence θ (Fig. b) spans a range corresponding to periods
between 1.0 and 2.1 days (0.3<θ<0.7), indicating a lower persistence
than the previous variables, compatible with the precipitation's noisy
nature. θ's autocorrelation function displays features similar to
those discussed in Sects. 3.1 and 3.2 (Fig. d), namely a yearly ACF
peak with negative autocorrelations at intermediate lags.
(a) Scatter plot of local dimension d and inverse
persistence θ for 2 m temperature (t2m) data. Each point represents
the value corresponding to 1 day in the NCEP/NCAR reanalysis. The colour
indicates the month of the year the data point falls in. Blue dotted lines
indicate the 0.02 and 0.98 percentiles of the d,θ distributions.
(b) Corresponding cross-correlation function between d and
θ.
An analysis of monthly-mean values (Fig. ) reveals a marked
semi-seasonal cycle in d, whose absolute values and variability both peak
during spring and late summer/early autumn. These are seasons with enhanced
convective precipitation at mid-latitudes. θ displays a similar
variability behaviour, while the magnitude has a minimum in summer and an
extended period of higher values from autumn into early spring. During the
summer months, the mature phase of NH monsoon systems provides comparatively
persistent and predictable precipitation patterns. The high persistence might
also be favoured by the predominantly dry summers in the Mediterranean and
other mid-latitude regions, with long dry spells at regional scale being the
norm. This picture is consistent with the autocorrelation functions described
above. In particular, the low absolute ACF values of d seen at lags of 2 to
10 months can be linked to the asymmetry in the positive and negative peaks.
For example, for a lag of 6 months the first peak will roughly match the
second peak, but the first trough will not match the second trough.
Composite anomalies with respect to the seasonal cycle in 2 m
temperature (t2m) for the four phase-space regions delimited by the blue
dotted lines in Fig. . Maxima of d (a), minima of
d (b), maxima of θ (c), and minima of
θ (d). Geographical composites are shown for seasons which
account for 25 % or more of the extreme occurrences
(Fig. ). Units: ∘C.
Statistics of local dimension d and persistence θ for
precipitation frequency (prp) data from the NCEP/NCAR reanalysis. Time series
of daily values of d (a) and θ (b).
Autocorrelation function ACF(d) (c) and ACF(θ) (d)
for 3 years in daily lags.
(a) Scatter plot of local dimension d and inverse
persistence θ for precipitation frequency (prp) data. Each point
represents the value corresponding to 1 day in the NCEP/NCAR reanalysis. The
colour indicates the month of the year the data point falls in. Blue dotted
lines indicate the 0.02 and 0.98 percentiles of the d,θ distributions.
(b) Corresponding cross-correlation function between d and
θ.
Composite anomalies with respect to the seasonal cycle in
precipitation frequency (prp) for the four phase-space regions delimited by
the blue dotted lines in Fig. . Maxima of d (a),
minima of d (b), maxima of θ (c), and minima of
θ (d). Geographical composites are shown for seasons which
account for 25 % or more of the extreme occurrences (Fig. ).
Units: dimensionless quantity.
Notwithstanding the local and noisy nature of precipitation, a number of
coherent features emerge from the geographical composites corresponding to
extremes in the two dynamical systems metrics, again defined as events beyond
the 0.02 and 0.98 percentiles of the full distribution (see dashed lines in
Fig. a). The bulk of the d and θ maxima occur during
spring, while the minima are primarily associated with the summertime
(Fig. i–l). d maxima show predominantly negative
precipitation anomalies (Fig. a). Significant features include
decreases over the East Asian Monsoon region, the Indian Ocean, and
continental and eastern Europe. Weak positive anomalies are mainly found over
the central Pacific Ocean, parts of the USA, and the Canadian Arctic
Archipelago. The anomalies associated with θ maxima closely track the
d maxima throughout the Northern Hemisphere, albeit with a stronger
preference for negative anomalies (Fig. c). Since the positive
extremes in both metrics occur predominantly during spring, we hypothesize
that the anomalies over East Asia could be linked to the shift between the
northerly flow associated with the winter monsoon and the southerly flow
associated with the summer monsoon. It is reasonable to expect that this
shift between very different large-scale flow configurations could contribute
to the high local dimensions. Similarly, the anomalies over the Indian Ocean
could be associated with the onset of the Indian Monsoon's large-scale flow.
d minima (Fig. b) show strong negative anomalies over the
continental USA and southern Indochina and the strong positive anomalies over
eastern Africa and the mid-latitude Pacific. θ minima again track
closely the patterns seen for the d minima (Fig. d). The
eastern African positive anomalies, roughly corresponding to northern
Ethiopia, might indicate a modulation of the late-summer rainfall peak in the
region . Similarly, the widespread negative
anomalies across eastern continental North America suggest a modulation of
the wet season over the Great Plains and northern Mexico.
The d–θ scatter plot (Fig. a) shows two clouds of
points and a clear separation between the late spring/summer and
autumn/winter seasons. While both clusters span a wide range of d and
θ, there is a clear vertical and horizontal offset between the two,
with the former seasons corresponding to lower values than the latter. The
two prp metrics show high cross-correlation features (Fig. b),
with the lag-0 correlation exceeding 0.6. The lagged cross-correlation shows
a yearly cycle, a negative correlation for lags of 2–4 months, and a local
maximum for lags of around 7 months.
Cross-analysis of the dynamical properties
We next address the co-variability of the dynamical indicators of the
different variables. In physical space, there is an obvious link between
anomalies in the large-scale slp and 2 m temperature fields. A similarly
close link can be found between precipitation and temperature or slp
anomalies. There are therefore strong grounds to expect some systematic
relationships to emerge.
We begin by analysing the cross-correlation functions between the slp and t2m
(Fig. a, b). d(slp) and d(t2m) are anticorrelated at
zero lag, as might be expected by their contrasting seasonal cycles described
above. The lagged cross-correlations display a roughly regular semi-yearly
cycle, which derives from the fact that both local dimensions display a
double peak, albeit in different seasons. The cross-correlation between
θ(slp) and θ(t2m) is more nuanced, owing to the fact that
θ(slp) displays high values throughout the autumn and winter while
θ(t2m) displays two well-separated peaks, one of which partially
overlaps the months of high θ(slp) values. The lag-0 correlations
are positive, albeit low, and peak negative cross-correlations are achieved
at lags of approximately 6–7 months. The lag-0 anticorrelation of the local
dimensions points to the fact that it is rare to find co-occurring slp and
t2m fields both displaying high predictability. This is compounded by the
fact that positive correlations between the θ are generally weak,
suggesting that persistent slp configurations do not necessarily match
equally persistent t2m patterns. An example of this are wintertime cold
spells at the mid-latitudes: while the large-scale circulation anomalies are
often very persistent, the temperature can evolve rapidly with a build-up of
cold t2m masses leading to a rapidly cooling region which then relaxes back
to near-climatological values as soon as the anomalous circulation pattern
weakens .
Cross-correlation functions for the local dimensions d (a, c, e) and inverse persistences θ (b, d, f) of sea-level
pressure (slp), 2 m temperature (t2m), and precipitation frequency (prp)
data. Positive lags indicate that the first variable precedes the second.
Scatter plots of the local dimensions d (a, c, e) and
inverse persistences θ (b, d, f) of sea-level pressure (slp),
2 m temperature (t2m), and precipitation frequency (prp) data.
The cross-correlation between d(slp) and d(prp) is shown in
Fig. c, d. At lag-0, the two variables have a moderate positive
correlation, with peak positive values being reached for negative shifts of
1–2 months (i.e. slp leading prp). Indeed, d(prp) has a broad peak
during the spring, then decreases rapidly through the summer season, and
peaks again in early autumn. These three features precede by roughly 1 month
the corresponding ones in the d(slp) signal. However, we note that peak
cross-correlation values are lower than those seen between d(slp) and
d(t2m). The two persistence metrics, by contrast, display peak correlation
at lag-0, since they both display a spring maximum and a summer minimum. The
large-scale circulation changes associated with the onset of the monsoonal
precipitation over Asia and Africa and the high summertime persistence in the
precipitation field therefore have a clear correspondence in d(slp) and
θ(slp), albeit with a small temporal lag in the case of the first.
The t2m–prp pair is analysed in Fig. e, f. In this case, the
lag-0 cross-correlation between the local dimensions is very small, and peaks
for prp leading temperature by roughly 2 months. A similar picture is seen
for the persistence metrics, since the summer peak in θ(t2m) is out
of phase with the summer minimum in θ(prp). From a dynamical systems
perspective there therefore seems to be a significant lag between changes in
the monthly-mean properties of the large-scale temperature and precipitation
signals.
The cross-correlations consider the time series of the different metrics as a
whole, but provide little insight into the correlation between dynamical
extremes. We conclude our analysis by looking at the d–θ scatter
plots for the local dimensions and persistences of the three observables
(Fig. ). The negative lag-0 correlation found for d(slp)
and d(t2m) is evident (Fig. a), while the other two d
scatter plots (Fig. c, e) show a more diffuse distribution,
consistent with the low correlation values previously discussed. The
strongest match between both positive and negative d extremes is found for
the slp–prp pair. θ shows generally higher co-occurrences of extremes
across all pairs, with the most frequent match being for the late springtime
low θ extremes of the prp–t2m pair (Fig. b, d, f).
This indicates that (i) rapidly shifting slp patterns can lead to equally
rapid shifts in the large-scale temperature and precipitation fields, and
vice versa for persistent configurations; and that (ii) persistent 2 m
temperature and precipitation configurations show a systematic co-occurrence
during the spring months.
Conclusions
In the present study we have applied recent advances in dynamical systems
theory to estimate the local dimension and inverse persistence of
instantaneous atmospheric fields over the Northern Hemisphere. Persistence is
a very intuitive metric, which quantifies the average residence time of the
system's trajectory in phase space within the neighbourhood of the point of
interest. Local dimension is a proxy for the number of locally active degrees
of freedom in the system, and can thus be directly linked to the number of
possible configurations preceding and following the instantaneous field being
analysed. We have specifically focused on three observables: sea-level
pressure, 2 m temperature, and precipitation frequency. Despite the high
dimensionality of atmospheric dynamics, we find that the Northern Hemisphere
sea-level pressure and low-level temperature fields can on average be
described by roughly 15–20 degrees of freedom, while the noisier
precipitation field has an average dimension of over 80. We further note that
the dimension of the instantaneous fields can vary by almost a factor of 4
for a given observable. The links between the local dimension and persistence
of a given variable can be complex. While the two generally show a positive
lag-0 correlation, they can display very different seasonal cycles.
This study further analyses dynamical extremes, namely the instances where
one – or both – dynamical systems metrics are at the positive or negative
edge of their respective distributions. The dynamical extremes in d and
θ of a given variable occur independently for 2 m temperature, but
they coincide almost always for precipitation frequency and sea-level
pressure. Both d and θ are linked to atmospheric predictability,
since a persistent, low-dimensional state is intrinsically easier to forecast
than a rapidly shifting, high-dimensional situation. Fields where the
co-occurrence of d and θ extremes is more frequent – such as is the
case for slp – therefore provide more highly predictable (or unpredictable)
configurations than those where the two occurrences are rarer.
We further identify a number of robust correlations between the dynamical
properties of the different variables. For example, low-persistence cases in
slp often indicate a similar low persistence in t2m and prp. This is an
intuitive relationship since rapidly shifting slp patterns can lead to
equally rapid shifts in the large-scale temperature and precipitation fields.
Similarly, persistent prp and t2m configurations often co-occur. Other links
do not have a similarly straightforward physical interpretation. For example,
the local dimensions of prp and t2m seem to be mostly uncorrelated,
suggesting that predictable large-scale precipitation patterns do not
directly affect the predictability of the t2m field, unlike what is seen for
persistence.
Our results do not always match those obtained in
for the North Atlantic region. Indeed, there is a strong dependence on the
region chosen and the dynamics of the Northern Hemisphere include degrees of
freedom other than the North Atlantic dynamics. This is consistent with the
increase in the average dimension found. Atmospheric predictability is therefore overall different for the hemisphere than for regional
mid-latitude dynamics. We conclude that the dynamical systems metrics we
adopt here provide a wealth of information concerning the large-scale
atmospheric processes and dynamics. We are convinced that this analysis
framework will find applications in a wide number of climate studies.