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178
V.
Lucarini
and
K. Fraedrich
See Figs.
la)-lc)
for results for values
of
Ec
up
to 0.02. We then conclude that,
in
spite
of
introducing a second unstable direction, which is responsible for
mixing the phases
of
the waves, the inclusion
of
the impact
of
the viscous
dissipation on the thermal energy balance acts with continuity on the dynamical
indicators, by reducing the overall instability, increasing the predictability, and
by
confining the asymptotic dynamics to a more limited (in terms
of
dimensionality) set.
(a)
(b)
(c)
"'-~--~--~--~------~--~--
~--
~--~
0.9
1
--
____
_
0,8
0,7
0,6
0,5
0,4
0,3
0,2
O,'~===========~
-O
.l~
~
-0_2
0
0.002 0.004 0.0
06
0.008 ' 0.
01
0.
01
2 - 0.014 0.
016
0.
018
0.02
Be
0.95
0.9
0.85
0.8
~075
r
------------------------------------------
07r
0.65
0.6,
0.
55
1
°
4.2
1
-
4.19
41B
f
4.17
416
1
"
"t:: 4.15
4.14
413
1
4
.1
2
4-
11
0-
0.002 0.004 0.
006
0.008
~
12
0.014 0.016 0.
018
0.
02
Eo
~
--~--,
-~
--~
0.002 0.0
04
0,006 0.008
O
~
----O_014
0.016 0.0
18
0.
02
Ec
Figure I: Four largest Lyapunov exponents (a), metric entropy h (b) and Kaplan-Yorke d
KY
dimension (cl
as
a function
of
Ec. Note the continuity for Ec = 0
of
all parameters and the distinct
linear behavior - see specifically the dashed lines
in
(b) and (cl - for Ec < 0.008. The linear
behavior
of
the second and fourth Lyapunov exponents branching off zero
in
(a) extends throughout
Ec
'" 0.018. Details
in
the text.
Symmetry-Break in a Minimal Lorenz-Like System
179
7. Symmetry-break and low-frequency variability
We now focus on spectral properties
of
the system. We first observe that the
variables
AI'
A
2
,
B
I
,
B
2
,
Z2
feature mostly ultra-low frequency variability. In
order to provide some qualitative highlights on these variables,
in
Fig. 2
we
show, from top to bottom, the projection on
Al
of
three typical trajectories for
Ec
= 10-
3
,
Ec = 10-
4
, and Ec = 10-
5
, respectively. Going from the top via the
middle to the bottom panel, the time scale increases by a factor
of
10 and 100,
respectively. The striking geometric similarity confirms that the time scales
of
the dominating variability for the slow variables can be estimated as O(Ec-
l
)
As we take the Ec
~
0 limit, such a time scale goes to infinity, which agrees
with the fact that for Ec
= 0 these variables converge asymptotically to fixed
values. Moreover,
as
(A
2
)
=
(A12
+
An
~
b/8{r/2
-
8)
= 2 when
&.-1
-long time
averages are considered, Fig. I suggests that the system switches (on short time
scales) back and forth between
Al
- and
A2
-dominated dynamics. Same applies
for the
BI
and
B2
pair
of
variables.
On the other hand, the dynamics
of
XI'
X
2
,
~,
Y2'
ZI
is basically
controlled by the
0(1)
time scale
1/
~
«1/
~
, so that a clear-cut separation
between fast and slow variables can be justified. In Fig. 3
we
depict, from the
top to bottom panel, some trajectories
of
the fast variable XI for Ec = 10-
3
,
Ec = 10-
4
, and Ec = 10-
5
, respectively. Similarly to Fig. 2, the time scale
of
the
x-axis is scaled according to
Ec-
I
•
Since XI is a fast variable, the dominating
high-frequency variability component is only barely affected by changing
Ec,
because
Al
has a weak dependence on
Ec,
as discussed in the previous section
(see also Fig. la). Nevertheless, in Fig. 3
we
observe an amplitude modulation
occurring on a much slower 0(&·-1) time scale
of
the order
of
1/
~
. Note that,
as in Fig.
2,
such a slow dynamics
of
the "coarse grained" XI variable for
various values
of
Ec is statistically similar when the time is scaled according to
Ec-
I
•
As in the previous case, the slow amplitude modulation determines the
changeovers between extended periods
of
XI - and X
2
-dominated dynamics,
thus ensuring the phase mixing and ergodicity
of
the system. In the Ec
~
0
limit, as discussed above, the relative strength
of
the two phase components
of
the
(l,
I) waves is determined by the initial conditions,
as
mixing requires
~
Ec-
I
time units. The same considerations apply for the
~
and Y
2
pair
of
variables.
Further insight can be obtained by looking at the spectral properties
of
the
fast and slow variables; some relevant examples
of
power spectra for Ec = 10-
3
,
180
V.
Lucarini
and
K.
Fraedrich
Ec =
10
-
4
,
and Ec =
10
-
5
are depicted in Fig. 4, where we have considered the
low frequency range by concentrating on time scales larger than
10/
A,
. The
white noise nature
of
the fast variable X 1 is apparent and so is the overall lack
of
sensitivity
of
its spectral properties with respect to
Ec.
Note that any
signature
of
the amplitude modulation, which is responsible for phase mixing
observed in Fig. 3, is virtually absent. This shows how crucial physical
processes are masked, due to their weakness, when using a specific metric, like
that provided by the power spectrum. Instead, the slow variable
Al
has a
distinct red power spectrum, which features a dominating
1-
3
/
2
scaling in the
low frequency range (black line) between the time scales
~
0.5 Ec-
I
and
~
10
Ec-
I
•
Such a scaling regime dominates the ultra-low frequency variability
described
in
Fig. 2 and
is
responsible for the long-term memory
of
the signal.
For higher frequencies, the spectral density
p(p) decreases - see the sharp
corner in the for
I""
O.SEc - as
1-4
, and then a classical red noise spectrum
DC
1-
2
is realized for very low values
of
the spectral density. Note also that, in
agreement with our visual perception
of
Fig.
2,
the spectral density p(p) scales
according to
Ec - I in a large range
of
time scales.
7
2,-
--,----,----
~--
_,--_,----,_--
_,----,_--,_--_,
~:~
"'"
0 1 2 3 4 5 6 7 8 9 10
X 10'
1 2
,---
,_
---,
--
--,-
--
,-
--
-,
----
,-
--
-,
--
-,--
--
,---
_,
~:~
"'"
0 1 2 3 4 5 6 7 8 9 10
X
10
5
~:~
"'"
-2
0
1 2 3 4 5 6 7 8 9 10
t X
10
6
Figure
2:
Impact
of
the viscous-thermal feedback on the time scales
of
the system. From top to
bottom: typical evolution
of
the variable A I for different values
of
the Eckert number (Ec =
10-
3
• Ec
=
10
-
4
,
Ec =
10-
5
,
respectively). Note that the time scale is magnified by a factor
of
I,
10
and 100
from top to bottom. Details
in
the text.
Symmetrv-Break in l.l Minimal Loreflz-Liki'
S:vstem
181
"
S
'i
~
~-2m---'---·c_----~-
----:~-----~----~---.-.-~---~~-----~-----.-
...
~
Figure
3:
Impact of the viscous-thennal feedback on the time scales
of
the system. From top to
bottom: typical evolution
of
the variable
Xl
for different values
of
the Eckert number (Ee
'"
1O-
J
,
Be
'"
10-.
1
,
Ee = 10-', respectively). Note that the ljme scale is magnified
by
a factor
of
1,
10
and 100
from top to bottom. Details in the text.
Figure
4:
Power spectrum
of
1'1
1
and
Xl
(multiplied times
10-
8
)
for Ee =
10',
Be
=
10-",
and Ee ",10-
5
in units
of
power per unit frequency. The black straight lines corre,pond
to
r
312
scalings. Details in
th(,
texL
8.
Summary
and
Conciusions
III
this work
we
have derived and thoroughly analyzed a IO-variable system
of
Lorenz-like ODEs obtained by a severe spectral truncation
of
the 2D x-z
convective equations for the streamfunction ljf and temperature
{}
in
Boussinesq approximation and subsequent nondimensionaIization. In
the
tmncation
we
retain the terms corresponding to the real and imaginary part
of
the modes
'P"w
(streamfunction) and e
m
.
n
(temperature) associated to the
x-z
182
V.
Lucarini
and
K.
Fraedrich
wave vector K =
(nan,nm)
with
(m,n)=
(1,1)
and
(m,n)
= (2,2)' and the
imaginary part
of
the x-symmetric modes
gO,2
and
gO,4'
As modes
characterized by a faster varying spatial structure and different parity are added
to the Lorenz spectral truncation, which describes the dynamics
of
the real part
of
P
1
,]
and
of
the imaginary part
of
g1,1
and
gO,2
only, the system
we
consider
in this work represents the thermal impact
of
viscous dissipation, controlled by
the Eckert number
Ec. The presence
of
a forth parameter - together with the
usual Lorenz parameters r (relative Rayleigh number),
(j
(Prandl number), and b
(geometric factor) - marks a crucial difference between this ODEs system and
other extensions
of
the Lorenz system proposed
in
the literature - see, e.g., [4-
6]. Moreover,
as
can be deduced following the argumentations
of
Nicolis [16],
this system specifically allows for the closed-form computation
of
the entropy
production,
as
the Eckert number enters into its evaluation. Up to our
knowledge, this
is
the first ODE system obtained
as
a truncation
of
the
convective equations presented in the scientific literature featuring all these
additional properties. The following results are worth mentioning:
1.
When the Eckert number is set to 0,
as
common in most applications, the
dynamical system is invariant with respect to the action
of
a specific
symmetry group, which basically shifts the phases
of
the streamfunction
and thermal waves with
m > 0 . At a physical level this implies that the
initial conditions set the relative strength
of
waves with the same spatial
structure but
n12
-shifted phases.
In
particular, it is found that the Lorenz
system is included
in
the ODEs system analyzed here when specific initial
conditions, which set the parity
of
the slower spatially varying modes, are
selected. The absence
of
phase-mixing implies a lack
of
ergodicity, and the
degeneracy
of
the dynamics is reflected also in the non-hyperbolicity
of
the system. When the classical parameters' values r =
28,
(j
=
10,
b =
8/3
are selected, three
of
the Lyapunov exponents vanish, one
corresponding to the direction
of
the flow, the other two accounting for the
toroidal symmetry. The value
of
the only positive exponent coincides with
that
of
the Lorenz system, and the value
of
one
of
the six negative
exponents agrees with that
of
the negative Lyapunov exponent
of
the
Lorenz system. Therefore, the Lorenz system contains already all the
interesting unstable dynamics described by this extended ODEs system,
and features exactly the same value for the metric entropy.
Correspondingly, while the five variables (fast) describing the modes Pl,l ,
gl,l , and
gO,2
have an erratic behavior, the other five variables (slow)
converge to fixed values.
2.
When Ec
'*
0,
the symmetry
of
the system is broken, and coupling occurs
between the fast and slow variable over a time scale O(Ec
-1).
This
is
clarified by adopting multi scale formalism.
If
we
select, as in the original
Lorenz system, r =
28,
(j
=
10,
b =
8/3,
the system is chaotic for
Symmetry-Break in a Minimal Lorenz-Like System
183
o < Ec
~
0.045 , whereas for higher values
of
Ec a quasi-periodic regime
is realized. In the chaotic regime, the symmetry-break is accompanied by
the establishment
of
a hyperbolic dynamics: two Lyapunov exponents
branch off from zero (one positive, one negative) linearly with
Ec.
Therefore, a second unstable direction with a second o (Ec-
I
)
time scale
11
~
»
11
AI
is established. The impact
of
the thermal-viscous feedback
is
stabilizing,
as
indicated by the metric entropy and the Kaplan-Yorke
attractor dimension monotonically decreasing with
Ec,
with a marked
linear behavior for
Ec
~
0.008.
3.
The coupling establishes dynamics on time scales
of
the order
of
Ec-
I
responsible for the changeovers between extended periods
of
dominance
of
waves
of
specific phase, both for the slow and for the fast
variables. Such processes, which result from small terms in the evolution
equations, correspond to the phase mixing
of
the waves and ensures the
ergodicity
of
the system. In particular, the slow variables have a non-trivial
time-evolution and are characterized by a dominating
1-
312
scaling in the
low frequency range for time scales between
0.5
Ec-
I
and
10
Ec-
I
.
Instead, in the case
of
fast variables, the phase mixing appears
as
a slow
amplitude modulation occurring on time scales
of
Ec - I which
superimposes on the fast dynamics controlled by the
0(1)
time scale
11
AI
.
The system introduced
in
this paper features very rich dynamics and,
therefore, may have prototypical value for phenomena generic to complex
systems, such as the interaction between slow and fast variables and the
presence
of
long term memory. Moreover, analysis shows how, neglecting the
coupling
of
slow and fast variables only on the basis
of
scale analysis -
as
usually done when discarding the Eckert number - can be catastrophic. In fact,
this leads to spurious invariances that affect essential dynamical properties
(ergodicity, hyperbolicity) and that cause the model losing its ability
to
describe
intrinsically multiscale processes. We have shown that a standard multiscale
approach allows for understanding the role
of
the small parameter controlling
the coupling. This may suggest that a careful re-examination
of
the scaling
procedures commonly adopted for defining simplified models, especially in the
climate science community, may be fruitful in the development
of
more efficient
modeling strategies. Note that in a recent study [21]
we
have tested that re-
feeding the kinetic energy lost to dissipation as positive thermal forcing to the
fluid (which corresponds to considering
Ec >
0)
brings the long-term global
energy budget
of
the system closer to zero by an order
of
magnitude. While the
numerical values presented in this paper refer to a specific choice for the
parameters
r,
(J,
b,
analogous properties - like the symmetry break and the
ensuing emergence
of
ergodicity and multiscale processes, the linearity
of
the
Lyapunov exponents with respect
to
Ec-
I
,
the low frequency variability are
expected for all values
of
r,
(J
, b leading to a chaotic dynamics.
184
V.
Lucarini
and
K.
Fraedrich
Bibliography
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E.N. Lorenz,
J.
Atmos. Sci.
20,130
(1963)
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Chaotic Evolution and Strange Attractors, Cambridge University Press, Cambridge,
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Y.
Peicai, Adv. Atmos. Sci.
3,
289 (1984)
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31 (2007) 747
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C. Lainscsek, F. Schtirrer, C. Letellier, and
J.
Maquet,
J.
Phys. A 31,7121 (1998)
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C. Sparrow, The Lorenz Equations. B!furcations, Chaos. and Strange Attractors, Springer, New
York, 1984
[9]
c.
Bonatti, L.J. Diaz, and M. Viana, Dynamics Beyond Un!form Hyperbolicity: A Global
Geometric and Probabilistic Perspective,
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l.
Stat. Phys. 131,543 (2008)
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D. Ruelle, Phys. Lett. A 245,
220
(1998)
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11,
5 (1998)
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J.
Stat. Phys. 134,381 (2009)
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R.
Metero!. Soc. 125,2855 (1999)
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Am. Met. Soc. 74,
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(1993)
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C. Nicolis, Quart.
J.
Roy. Meteor. Soc. 125, 1859 (1999)
[17] Z.-M.
Chen
and W.G. Price, Chaos, Solitons and Fractals 28 571 (2006)
[18] V. Oseledec,
Moscow
Math. Soc.19, 197 (1968)
[19]
G.
Benettin,
L.
Galgani,
A.
Giorgilli and J .M. Stre1cyn, Meccanica 15, 9 (1980)
[20] J.L. Kaplan and l.A. Yorke, Chaotic Behavior
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H.-O. Walter, (Springer, Berlin, 1979)
[21] V. Lucarini,
K. Fraedrich, F. Lunkeit, Thermodynamic Analysis
of
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l.
Roy. Met. Soc.
(2009); also at arXiv:0905.3669vl [physics.ao-ph]
Sub-Critical
'Transitions
in
Coupled
Map
Lattices
Hydrodynamics
Laboratory
Ecole
Poly
technique
91128
Palaiseau,
France
P. Manneville
(e-mail:
paul.manneville@polytechnique.edu)
185
Abstract:
Sub-critical
transitions
between
two
collective regimes
with
different
periods
have
been
observed
in a
4-dimensional
hypercubic
lattice
of
coupled
logistic
maps.
Statistical
findings
can
be
interpreted
in
terms
of
macroscopic
quantities
(measuring
global
properties)
escaping
from
potential
wells in
the
presence
of
noise
induced
by
chaos
in
the
microscopic
dynamics
(local
variables).
The
relevance
of
these
results
to
a
"thermodynamic"
understanding
of
the
dynamics
of
distributed
complex
systems
is briefly discussed.
Keywords:
coupled
map
lattices,
sub-critical
transitions,
escape
from
attractors.
1
Introduction
Large
assemblies
of
interacting
nonlinear
units
give
examples
of
complex sys-
tems
in which collective
behaviour
emerges,
characterised
by
the
ordered
evolution
of
averaged
quantities
in
spite
of
strong
chaos
persisting
at
the
scale
of
the
individual
units.
The
acronym
NTCB,
meaning
non-trivial
col-
lective behaviour,
has
been
coined
to
label
this
dynamics,
emphasising
the
fact
that
conventional
thermodynamic
arguments
seem
to
rule
them
out.
In
the
present
study,
we consider
coupled
map
systems
[1]
in
the
form:
XHI
-
_1_
" f
(xt)
.i
-
2d
+ 1
~
r
J'
,
j'EVj
where
t
denotes
(discrete)
time
and
Vj
the
neighbourhoods
of
sites j
at
nodes
of
ad-dimensional
hypercubic
simple
lattice.
Here,
Vj
is
limited
to
nearest
neighbours
and
the
local
evolution
is governed
by
the
logistic
map
fr(X)
=
rX(l-X).
Such
models have
already
been
studied
in various
space
dimensions
[2].
They
are
known
to
display
NTCB
for r
between
the
accumu-
lation
point
of
the
period-doubling
cascade
of
the
isolated
map,
roo
~
3.57,
and
its
crisis
point
rmax
= 4.
Thc
nontrivial
features arise from
the
fact
that
the
mechanisms
at
the
origin of
the
different macroscopic
states
and
the
transitions
from one
to
another
are
not
known.
In
particular,
the
mean-field
approximation
obtained
by
replacing variables by
mean
values
and
neglecting
correlations
is
unable
to
give
hints
on
the
emergence
of
a collective
behaviour
apparently
unrelated
to
the
properties
of
the
isolated
map.
186
P.
Manneville
Bifurcations
between
different
NTCB
has
been
observed
upon
varying
r.
Supercritical
period-doubling
cascades for d = 2
and
3
and
the
Hopf
bifur-
cation
toward
quasi-periodic
behaviour
for d = 5
are
well
documented
[3,2].
The
size
of
the
system
as
measured
by
the
number
of sites N = n
d
,
where
n is
number
of
sites along a
space
direction, is
an
important
parameter
and,
remarkably,
NTCB
is
better
and
better
defined
as
N increases
(the
so-called
thermodynamic limit). Accordingly,
it
is believed
that
the
asymptotic
evo-
lution
in
the
infinite-time
limit
is
accounted
for
by
a genuine
attractor
in
the
infinite-size limit.
Whereas
size effects
on
continuous
transitions,
espe-
cially
period-doubling
bifurcations,
can
be
understood
within
appropriately
adapted
standard
approaches
[4],
the
role
of
the
noise
that
they
introduce
in
discontinuous
transitions
needs
further
study. We focus
here
on
the
sub-
critical
bifurcation
between
two
NTBC
periodic regimes
T2
and
T4,
with
respective
periods
2
and
4, coexisting in
the
range
3.88 < r
::;
4
[2].
Only
preliminary
results
are
presented;
a
more
complete
analysis is in progress.
2
Results
NTCB
is identified from
the
evolution
of
the
mean
value
of
the
Xj over
the
lattice
at
time
t, X
t
=
N-
1
~j
Xj.
Size effects
alluded
to
above
are
easily observable
at
r = 4 as seen in Fig. 1 which displays
time
series of X
t
for
systems
with
n = 16, 19, 23, 27,
and
helical
boundary
conditions,!
the
numbers
of
sites increasing
by
factors
of
2 from
6.510
4
to
5.310
5
.
The
T4
regime
corresponds
to
4
bands
and
the
T2
regime
to
2
bands
only
with
the
't =
4k
+
3'
and'
4( k +
1)'
series
partly
masking
the'
4k
+
l'
and'
4k
+
3'
series.
For
n = 16, switchings
between
T4
and
T2
are
frequent
and
it
may
happen
that
a
phase
change
occurs
upon
return
to
the
T4
regime, e.g.
after
the
T2
episodes
at
t
:::::;
1.310
5
,
:::::;
1.4210
5
and:::::;
1.8210
5
.
For
n = 19, switchings
are
less frequent, even less for n = 23,
and
for n = 27 regime
T2
seems
sustained
(notice
the
change
of
time
scales for n = 23, 27).
This
situation
no
longer
changes as
n is increased
further.
These
observations
helps us
understanding
that
bifurcation
diagrams
strongly
depend
on
the
experimental
protocol. In
[2],
regimes
T2
and
T4
were found
to
exist in
the
range
r E [3.90,3.97]
with
n = 27 in sweep-
up/sweep-down
adiabatic
simulations
with
2,000
iterations
and
detection
of
the
state
obtained
after
elimination
of
1,500
iterations.
Considering
instead
10
4
iterations,
after
elimination
of
8 10
3
transients,
regime
T2
is
obtained
in
sweep-up
simulations
up
to
r
:::::;
3.94
where
a
transition
to
T4
is observed;
regime
T4
is
then
kept
up
to
rmax
= 4.
On
the
other
hand,
in
sweep-down
1
The
lattice
is
arranged
as
a
one-dimensional
array
of
size N = n
4
with
periodic
boundary
conditions.
The
coupling
then
reads
~
(Yj +
Yj+l
+ Yj
-1
+ Yj+n +
Yj-n
+
YJ+n2
+YJ-n
2
+YJ+n3
+YJ-
n
3)
for
j =
1,
...
,n
4
,
where
Yj =
Jr(Xj).
Other
kinds
of
boundary
conditions,
e.g.
periodic,
give
the
same
results
provided
that
the
system
is
large
enough.
Sub-Critical Transitions in Coupled Map Lattices 187
0.9
~----
~-
-~-
-
-----,
o'
r:"-<
;;-';-
"
;'
'1
0.8
0
.7
n=27
::
~
••
lli"_~~
2
6
10
Fig.
1.
Time
series
of
X
t
at
r = 4 for
systems
of
increasing sizes. Color indicates
time
modulo
4;
blue:
4k
+ 1, red:
4k
+ 2, green:
4k
+ 3, magenta:
4(k
+ 1).
simulations
T2
is
obtain
ed from T =
rmax
down
to
r
~
3.98,
next
T4
in
the
range [3.90,3.98]'
and
T2 again for T < 3.90. Turning
to
n = 54 (presu
mably
closer
to
the
thermodynamic
lilnit) using
the
same protocol,
the
sweep-up
T2---T4 bifurcation is
not
observed
and
the
T2
regime
obtained
for r = 3.88
is kept
up
to
r'
=
4.
In
sweep-down experiments
starting
frorH
an
initial con-
dition belonging
to
regime
T2,
no spontaneous
T2
---T4
transition
is observed
so
that
it
is
kept all
the
way down.
If
the
initial condition belongs
to
regime
T4,
it
persists down
to
r ~ 3.90 where
the
T4---T2
transition
takes place.
These
ap
parentl
y ill-m
atc
h
ed
results
can
in fact
be
rationalised by
taking
into acco
unt
the
role of
the
microscopic noise induc
ed
by
the
local chaotic
evolution within
a
few
assumptions of
thermodynamic
flavour as analysed
below.
3
Interpretation
and
discussion
The
findings described above
can
be quantit.atively
understood
by
taking
a
stoc
hastic
approac
h in
terms
of master equation
[5]
and
assuming
that
,
at
the
macroscopic level, we
are
dealing
with
a two-state system
T2
and
T4
characterised
by
two variables P
z
and
P
4
= 1 - P
2
which
are
the
prob
ab
ilities
for
system
of being found in s
tate
T2
or T4, respectively,
and
two transition