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1
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Is Weather Chaotic? Coexistence of Chaos and Order
Within a Generalized Lorenz Model
by
Bo-Wen Shen1*,Roger A. Pielke Sr.2,Xubin Zeng3, Jong-Jin Baik4,
Tiffany Reyes1, Sara Faghih-Naini5, Robert Atlas6, and Jialin Cui1
1San Diego State University, USA
2CIRES, University of Colorado at Boulder, USA
3The University of Arizona, USA
4Seoul National University, South Korea
5Friedrich-Alexander University Erlangen-Nuremberg, Germany
6AOML, National Oceanic and Atmospheric Administration, USA
*Email: bshen@sdsu.edu; Web: https://bwshen.sdsu.edu
Big Data, Data Assimilation and Uncertainty Quantification
Institut Henri Poincaré (IHP), Paris, France
12-15 November 2019
2
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Outline
vIntroduction
•30 day Predictions of African Easterly Waves (AEWs) and Hurricanes
•Goals and Approaches
vLorenz Models (Lorenz, 1963, 1969)
•Chaos and Two Kinds of Butterfly Effects (BE1 and BE2)
•The Lorenz 1963 Model and BE1/Chaos
•Three Types of Solutions (e.g., Steady-state, Chaotic, and Limit Cycle Orbits)
•The Lorenz 1969 Model and BE2/Instability
vMajor Features of Lorenz’s Butterfly
•Divergence, Boundedness, and Recurrence
vA Generalized Lorenz Model (Shen, 2019a)
•Slow and Fast Variables
•Aggregated Nonlinear Negative Feedback
•Two Kinds of Attractor Coexistence: Coexistence of Chaos and Order
vA Hypothetical Mechanism for Predictability of AEWs
vSummary and Outlook
4
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Track Forecast Intensity Forecast
OBS
OBS
model
model
•Shen, B.-W., W.-K. Tao, and M.-L. Wu, 2010b: African Easterly Waves and African Easterly Jet
in 30-day High-resolution Global Simulations. A Case Study during the 2006 NAMMA period.
Geophys. Res. Lett., L18803, doi:10.1029/2010GL044355.
(Hurricane Helene: 12-24 September, 2006)
•How can high-resolution global models have skill?
Simulations of Helene (2006) Between Day 22-30
5
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Goals and Approaches
To achieve our goals, we performed a comprehensive literature review and
derived a generalized Lorenz model (GLM) to:
1. understand butterfly effects (i.e., chaos theory),
2. reveal and detect the coexistence of chaotic and non-chaotic
processes,
3. emphasize the dual nature of chaos and order in weather, and
4. propose a hypothetical mechanism for the periodicity and predictability
(of multiple African easterly waves, AEWs)
Our goals include addressing the following questions:
• Can global models have skill for extended-range (15-30 day)
numerical weather prediction? Why?
• Is weather chaotic?
6
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Outline
vIntroduction
•30 day Predictions of African Easterly Waves (AEWs) and Hurricanes
•Goals and Approaches
vLorenz Models (Lorenz, 1963, 1969)
•Chaos and Two Kinds of Butterfly Effects (BE1 and BE2)
•The Lorenz 1963 Model and BE1/Chaos
•Three Types of Solutions (e.g., Steady-state, Chaotic, and Limit Cycle Orbits)
•The Lorenz 1969 Model and BE2/Instability
vMajor Features of Lorenz’s Butterfly
•Divergence, Boundedness, and Recurrence
vA Generalized Lorenz Model (Shen, 2019)
•Slow and Fast Variables
•Aggregated Nonlinear Negative Feedback
•Two Kinds of Attractor Coexistence: Coexistence of Chaos and Order
vA Hypothetical Mechanism for Predictability of AEWs
vSummary and Outlook
7
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Butterfly Effect of the First and Second Kind
Two kinds of butterfly effects can be identified as follows (Lorenz, 1963, 1972):
1. The butterfly effect of the first kind (BE1):
Indicating sensitive dependence on initial conditions (Lorenz, 1963).
•control run (blue):
!" #" $ % &'"("')
•parallel run (red):
!" #" $ % '"( * +" ' "
+ % (, -('.
continuous dependence
(within a time interval)
sensitive dependence
2. The butterfly effect of the second kind (BE2):
ametaphor (or symbol ) for indicating that small perturbations can create
a large-scale organized system (Lorenz, 1972/1969).
8
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
The Lorenz (1963) Model (3DLM)
•Note that X, Y, and Z represent the amplitudes of Fourier modes for the
streamfunction and temperature.
•A phase space (or state space) is defined using the state variables X, Y
and Z as coordinates. The dimension of the phase space is determined by
the number of variables.
•A trajectory or orbit is defined by time varying components within the
phase space, also known as a solution.
•Two nonlinear terms form a nonlinear feedback loop (NFL).
•r –Rayleigh number: (Ra/Rc)
a dimensionless measure of the temperature
difference between the top and bottom surfaces of
a liquid; proportional to effective force on a fluid;
• σ –Prandtl number: (ν/κ)
the ratio of the kinetic viscosity (κ, momentum
diffusivity) to the thermal diffusivity (ν);
• b –Physical proportion: (4/(1+a2)), b = 8/3;
• a –a=l/m, the ratio of the vertical height, h, of the
fluid layer to the horizontal size of the convection
rolls. b = 8/3; l = aπ/H and m = π/H.
The classical Lorenz model (Lorenz, 1963) with three variables and
three parameters, referred to as the 3DLM, is written as follows:
9
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Three Attractors Within the 3DLM
A steady-state solution
with a small r
A chaotic solution
with a moderate r
A limit cycle
with a large r
Depending on the relative strength of dissipations, four types of solutions within
dissipative systems are:
a. Steady state solutions with a weak heating term (i.e., ! < !
#; !
#=24.74);
b. Chaotic solutions with a moderate heating term (i.e., !
#< r < *#; *#=313 );
c. Limit cycle solutions with a strong heating term (i.e., *#< !);
d. Coexistence of chaotic and steady-state solutions (24.06 <!<24.74).
control run in blue
parallel run in red
10
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Three Attractors Within the 3DLM
A steady-state solution
with a small r
A chaotic solution
with a moderate r
A limit cycle
with a large r
A point attractor A chaotic attractor A periodic attractor
(a spiral sink)
!
"< r < %"
11
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Limit Cycle: An Isolated Closed Orbit
•A limit cycle (black) is indicated by the
convergence of 200 orbits (color).
•A limit cycle (LC) is an isolated closed
orbit.
•Nearby trajectories spiral into it.
•LC orbits are determined by the
structure of the system itself. It has no
long term memory regarding ICs.
dependence of phases on ICs
oscillatory errors
color orbits: ! ∈ [1,10]
black orbit ! ∈ [9,10]r=350
12
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Impact of Initial Tiny Perturbations Within the 3DLM
•Steady state or nonlinear periodic solutions have no (long-term) memory
regarding their initial tiny perturbations
Øinitial tiny perturbations completely dissipate
•Chaotic solutions display a sensitive dependence on initial conditions
Øinitial tiny perturbations do not dissipate (before making a large
impact)
•3DLM: within the chaotic solutions, any tiny perturbation can cause large
impacts. Is this feature realistic?
•We may ask what kind of impact tiny perturbations may introduce in real
world models
13
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Concurrent Visualizations: Butterfly Effects?
•A selected frame from a global animation of the vertical velocity in pressure coordinates from a run
initialized at 0000 UTC 21 October 2005. The corresponding animation is available as a google
document: http://bit.ly/2GS2flD. The animation displays dissipation of the initial noise associated with an
imbalance between the model and the initial conditions (Shen, 2019b and references therein)
14
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Lorenz 1963 and 1969 Models
•Lorenz (1963) Model (3DLM): èBE1
•nonlinear and chaotic
•limited scale interactions (3 modes)
•Lyapunov exponent (LE) analysis
•KE and PE, PDE based (Rayleigh-
Benard Convection)
•Lorenz (1972/1969) Model: èBE2
•multiscale but linear (21 modes)
•growth rate analysis using a realistic
basic state
•KE,PDE based (a conservative
system with no forcing or dissipation)
•Lorenz (1996/2005) Model:
•nonlinear and chaotic with multiple
spatial scales
•equal weighting in dissipations
•KE, not PDE based
2D (x,z) flow
2D (x,y) flow
Also see Rotunno and Synder
(2008) and Durran and Gingrich
(2014)
No PDEs
15
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Comments on the Lorenz 1984 Model (Lorenz, 1990)
The above idealized system was proposed by Lorenz in 1984 for qualitatively
depicting atmospheric circulation, known as the Lorenz (1984) model. Due to the
following issues, results obtained using the Lorenz 1984 model should be
analyzed and interpreted with caution:
1. Detailed derivations of the Lorenz (1984) model were missing (e.g., Veen
2002a, b); it is difficult to trace the physical source of the forcing terms
(parameters “F” and “G” in Eqs. (1)-(3) of Lorenz 1984) in the model.
2. As compared to fully dissipative systems where the time change rate of
volume of the solutions is negative, the volume of the solution within the 1984
model does not necessarily shrink to zero (e.g., p. 380 of Lorenz 1990).
•The variable X represents the strength of
a large scale westerly-wind current, and
also the geostrophically equivalent large-
scale poleward temperature gradient;
•Y and Z are the strengths of the cosine
and sine phases of a chain of superposed
waves, respectively.
16
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Outline
vIntroduction
•30 day Predictions of African Easterly Waves (AEWs) and Hurricanes
•Goals and Approaches
vLorenz Models (Lorenz, 1963, 1969)
•Chaos and Two Kinds of Butterfly Effects (BE1 and BE2)
•The Lorenz 1963 Model and BE1/Chaos
•Three Types of Solutions (e.g., Steady-state, Chaotic, and Limit Cycle Orbits)
•The Lorenz 1969 Model and BE2/Instability
vMajor Features of Lorenz’s Butterfly
•Divergence, Boundedness, and Recurrence
vA Generalized Lorenz Model (Shen, 2019a)
•Slow and Fast Variables
•Aggregated Nonlinear Negative Feedback
•Two Kinds of Attractor Coexistence: Coexistence of Chaos and Order
vA Hypothetical Mechanism for Predictability of AEWs
vSummary and Outlook
17
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
What Lorenz’s Butterfly Really Reveals
2. Boundedness
•No “blow-up” solutions
! " = $%& cos *" + , sin *"
oscillatory
grow or decay at an exponential rate
The statement of ``Orbits initially diverge and then curve back’’ includes the
following major features of butterfly solutions:
3. Recurrence/Folding
•Complex eigenvalues, / = 0 + ,*:real
part leads to a growing or decaying
solution; imaginary part gives the
oscillatory component.
1. Divergence of Trajectories
5. Ergodicity (Hilborn, 2000)
•Time averages are the same as state
space averages.
4. Error Saturations
•Max errors determined by the “size” of
the butterfly’s wings
18
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Based on the previous discussions, we may ask whether the following folklore
is an “accurate” analogy of the butterfly effect (Gleick, 1987; Drazin, 1992):
“For want of a nail, the shoe was lost.
For want of a shoe, the horse was lost.
For want of a horse, the rider was lost.
For want of a rider, the battle was lost.
For want of a battle, the kingdom was lost.
And all for the want of a horseshoe nail.”
However, Lorenz (2008) made the following comments:
1. Let me say right now that I do not feel that this verse is describing true
chaos, but better illustrates the simpler phenomenon of instability.
2. The implication is that subsequent small events will not reverse the outcome.
Chaos and the Butterfly Effect
Lorenz’s comments support the view that the verse neither describes (local)
time-varying convergence of trajectories nor indicates recurrence.
Do we all agree on the above? Prof. Lorenz expressed his concerns in 2008.
21
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Outline
vIntroduction
•30 day Predictions of African Easterly Waves (AEWs) and Hurricanes
•Goals and Approaches
vLorenz Models (Lorenz, 1963, 1969)
•Chaos and Two Kinds of Butterfly Effects (BE1 and BE2)
•The Lorenz 1963 Model and BE1/Chaos
•Three Types of Solutions (e.g., Steady-state, Chaotic, and Limit Cycle Orbits)
•The Lorenz 1969 Model and BE2/Instability
vMajor Features of Lorenz’s Butterfly
•Divergence, Boundedness, and Recurrence
vA Generalized Lorenz Model (Shen, 2019a)
•Slow and Fast Variables
•Aggregated Nonlinear Negative Feedback
•Two Kinds of Attractor Coexistence: Coexistence of Chaos and Order
vA Hypothetical Mechanism for Predictability of AEWs
vSummary and Outlook
22
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
A Generalized Lorenz Model (GLM)
As discussed in Shen (2019a) and Shen et al. (2019), the GLM with many
M modes possesses the following features:
(1) any odd number of M greater than three; a conservative system in the
dissipationless limit;
(2) three types of solutions (that also appear within the 3DLM);
(3) energy transfer across scales by the nonlinear feedback loop (NFL);
(4) slow and fast variables across various scales;
(5) aggregated negative feedback;
(6) increased temporal complexities of solutions associated with
additional (incommensurate) frequencies that are introduced by the
extension of the NFL (i.e., spatial mode-mode interactions);
(7) hierarchical scale dependence;
(8) two kinds of attractor coexistence;
•The 1st kind of Coexistence for Chaotic and Steady-state Solutions,
•The 2nd kind of Coexistence for Limit Cycle and Steady-state Solutions.
23
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
A Generalized Lorenz Model (GLM)
The GLM is derived based on extensions of the NFL that can provide negative
nonlinear feedback to stabilize solutions. The GLM is written as follows:
3DLM
smaller
scale
modes
primary
scale
modes
•The “backbone” of the linearized NFL is analogous to the spring of the above system.
•A new pair of high wavenumber modes ("
#, %
#)that extends the NFL creates an
additional frequency in a new subsystem with a different spring constant.
(2)
(4)
(1)
(3)
(5)
(6)
24
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Note that the GLM is coupled by the extension of the nonlinear feedback
loop and the coefficients of the terms on the right-hand sides continuously
increase in association with increasing inclusion of high wavenumber
modes, as shown in Eq. (4).
!"
!# $ %&' ()& % " *+,
-
.
!"
/
!# $ &'
/01 %. ( -
.&'
/%!/01
."
/. 2 3 4 . 2 -5 6 *7,
The following two equations suggest that "
/is a fast variable while
"is a slow variable when -8. is small (i.e., .is large).
Slow and Fast Variables Within the GLM
(2)
(4)
26
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
An Indicator of Aggregated Negative Feedback
model rcheating
terms
solutions references
3DLM 24.74 rX
steady
, chaotic, or LC
Lorenz
(1963)
3D-NLM n/a rX
periodic
Shen
(2018)
5DLM 42.9 rX
steady, chaotic, or
LC/LT
Shen
(2014a,2015a,b)
5D-NLM n/a rX
quasi
-periodic
Faghih
-Naini
and Shen (2018)
6DLM 41.1 rX, rX1
steady
or chaotic
Shen
(2015a,b)
7DLM 116.9 rX
steady
,chaotic or LC/LT
Shen (2016, 2017)
7D-NLM n/a rX
quasi
-periodic
Shen and
Faghih-Naini
(2017)
8DLM 103.4 rX, rX1
steady
or chaotic
Shen (2017)
9DLM 102.9
rX, rX
1, rX2
steady
or chaotic
Shen (2017)
9DLMr 679.8 rX
steady
, chaotic, or LC/LT
Shen (2019a)
rc: a critical value of the Raleigh parameter for the onset of chaos; LC: limit cycle; LT: limit torus
A comparison of the 3D, 5D, 7D and 9D LMs, requiring larger heating parameters
for the onset of chaos in higher-dimensional LMs, indicates aggregated negative
feedback by small scale modes.
27
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
In Relation to Non-autonomous Systems
ØAn extension of the NFL by an additional pair of small scale modes can
introduce one frequency that is incommensurate with existing frequencies.
•The impact may be partially “emulated” by the inclusion of a periodic forcing
into the original system, yielding a non-autonomous system, as discussed
below. [Note that the introduced frequency is not necessarily
incommensurate.]
ØThe negative feedback associated with an additional pair of small scale modes
can be “parameterized” into the original system. If the system allows the
parameterized impact to be on or off as time proceeds. It is a non-autonomous
system.
!"
!# $ %" & '()*+#,
!-
!# $+./
!"
!# $ %" & -
!.
!# $ 0+-
A high dimensional
autonomous system
A low dimensional
system + a forcing term
1 $ %2 34+- 5 $ 5/ . 5 $ 0+
28
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
The First Kind of Attractor Coexistence Within the 9DLM
For the 1st kind of attractor coexistence within the 9DLM, the appearance of
a steady state solution (left and middle) or a chaotic solution (right)
depends on the initial conditions. This indicates final state sensitivity to ICs.
non-chaotic orbits chaotic orbit
r=680
29
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Figure: Time evolution of 2,048 orbits in the X-Y3-Z3 space using the 9DLM, showing spiral sinks
and a limit cycle/torus solution. The animation is available from https://goo.gl/sMhoUb.
•A limit cycle (LC) is an isolated
closed orbit.
•Nearby trajectories spiral into it.
•LC orbits are determined by the
structure of the system itself.
§The total simulation time is !=
3.5.
§Transient orbits are only kept for
the last 0.25 time units, i.e. for the
time interval of [max (0, T-0.25),
T] at a given time T.
§The zoom-in of the domain starts
at != 0.25 and ends at != 0:45,
leading to a smooth domain
change from (X, Y3, Z3) = (-
1300,1200) x (-1100, 1100) x (-
1000,1700) to (-300;200) x (-
100,100) x (0,700).
The Second Kind of Attractor Coexistence Within the 9DLM
r=1600
30
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Two Kinds of Dependence on ICs
The 9DLM with attractor coexistence reveals:
• final state sensitivity to ICs, i.e., ICs determine whether
solutions are chaotic or steady state;
• sensitive dependence on ICs for chaotic solutions; and
• no long term memory regarding ICs for steady solutions.
An initial tiny perturbation:
• may be important or unimportant within the attractor
coexistence of the 9DLM, depending on various kinds of
orbits (i.e., various basins of attraction);
• is always important within the chaotic solutions of the 3DLM.
Any tiny perturbation can cause large impacts.
Shen, B.-W., 2019a: Aggregated Negative Feedback in a Generalized Lorenz Model. International
Journal of Bifurcation and Chaos Vol. 29, No. 3 (2019) 1950037 (20 pages).
https://doi.org/10.1142/S0218127419500378
32
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Outline
vIntroduction
•30 day Predictions of African Easterly Waves (AEWs) and Hurricanes
•Goals and Approaches
vLorenz Models (Lorenz, 1963, 1969)
•Chaos and Two Kinds of Butterfly Effects (BE1 and BE2)
•The Lorenz 1963 Model and BE1/Chaos
•Three Types of Solutions (e.g., Steady-state, Chaotic, and Limit Cycle Orbits)
•The Lorenz 1969 Model and BE2/Instability
vMajor Features of Lorenz’s Butterfly
•Divergence, Boundedness, and Recurrence
vA Generalized Lorenz Model (Shen, 2019a)
•Slow and Fast Variables
•Aggregated Nonlinear Negative Feedback
•Two Kinds of Attractor Coexistence: Coexistence of Chaos and Order
vA Hypothetical Mechanism for Predictability of AEWs
vSummary and Outlook
33
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Oscillatory Forecast Scores in the 30 Day Run: Why?
0.65
0.75
Aug 22 Sep 21
Correlation
Coefficients (CCs)
Is the forecast score a
monotonically
decreasing function of
time?
Note that (1) a limit cycle
is oscillatory; and (2) two
chaotic orbits may
produce time varying
convergence and
divergence
34
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
A Hypothetical Mechanism for the Predictability
at Extended-Range Scales
•Shen, B.-W., 2019b: On the Predictability of 30-day Global Mesoscale Simulations of Multiple African Easterly Waves during Summer 2006: A
View with a Generalized Lorenz Model. Geosciences 2019, 9(7), 281; https://doi.org/10.3390/geosciences9070281
Limit Cycle African Easterly Waves (AEWs)
system conditions
strong heating + nonlinearity
during summer (JAS)
features periodic recurrent, 27 AEWs per year
errors oscillatory oscillatory CCs
The realistic simulation of Hurricane Helene (2006) from Day 22 to 30
became possible as a result of the realistic simulation of the
1. periodicity (or recurrence) of AEWs and
2. downscaling process of the 4th AEW.
3 days
36
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Additional Support for Oscillatory Components
! " # $ % & '( )*+,
•Lorenz (1990) applied the Lorenz (1984) model to reveal “chaotic winter
and non-chaotic summer” (in the bottom left figure)
•Using the NCAR WACCM3 (Whole Atmosphere Community Climate Model
Model), Liu et al. (2009) documented oscillatory root mean square (RMS)
errors (i.e., with no error saturation) (in the bottom right figure).
•Based on dishpan experiments (e.g., Hide 1953), Lorenz (1993) suggested three
types of solutions, including: (1) steady state solutions, (2) chaotic solutions, and
(3) vacillation. “Amplitude vacillation” is defined as a solution whose amplitude
grows and periodically decays in a regular cycle (Lorenz 1963c; Ghil and
Childress 1987; Ghil et al. 2010). The amplitude vacillation can be viewed as a
limit cycle solution (e.g., Pedloksy, 1972; Smith and Reilly, 1977)
#'## )-./0,
•A theoretical study suggested that 40-day intra-seasonal oscillations may arise
from a Hopf bifurcation off the blocking flow. The corresponding limit cycle has
a period of 40 days (Ghil and Robertson, 2002)
38
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Concluding Remarks
1. Two kinds of butterfly effects in Lorenz studies can be defined as follows:
•The BE1: the sensitive dependence of solutions on initial conditions;
•The BE2: ametaphor for indicating the enabling role of a tiny
perturbation in producing an organized large-scale system.
2. The GLM possesses the following features:
a) Three types of attractors; b) Two kinds of attractor coexistence;
c) Aggregated negative feedback; d) Hierarchical scale dependence.
A higher-dimensional LM ≈a lower-dimensional LM +forcing terms.
3. Chaotic solutions only appear within the finite range of the Rayleigh
parameters. Chaotic and non-chaotic orbits may coexist, displaying two
kinds of data dependence. The BE1 does not always appear.
4. “As with Poincare and Birkhoff, everything centers around periodic
solutions,” Lorenz and chaos advocates focused on the existence of non-
periodic solutions and their complexities.
5. We propose that the entirety of weather possesses a dual nature of chaos
and order. The duality should be taken into consideration to revisit the
predictability problem.
39
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Future Tasks
ØDetect (Nonlinear) Oscillatory Signals by
• applying the Parallel Ensemble Empirical Mode Decomposition
(PEEMD) to decompose data into oscillatory modes and non-
oscillatory trend mode (Shen et al. 2017; Wu and Shen, 2016);
• performing the Recurrence Analysis (Reyes and Shen, 2019a,b);
• performing the Kernel Principle Component Analysis for
classification of solutions (Cui and Shen, 2019, under revision).
• [Apply the above to analyze MJO signals and compare results with
those using the analysis of Real-time Multivariate MJO (RMM) Index]
ØImprove the Simulations of Nonlinear Oscillatory Signals (e.g., limit
cycle or quasi-periodic orbits) by
• reducing numerical dissipations to avoid computational chaos (e.g.,
the Logistic equation vs. the Logistic map; Lorenz, 1989)
• examining the potential impact of increased resolutions and newly
added components on the generation of new incommensurate or
commensurate frequencies, leading to quasi-periodic solutions.
40
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
vThe entirety of weather possesses a dual nature of chaos and order.
• The above refined view is neither too optimistic nor pessimistic as
compared to the Laplacian view of deterministic predictability and
the Lorenz view of deterministic chaos.
v``there is no reason that the limit of predictability is a fixed number’’ as
suggested by Prof. Arakawa (Lewis, 2005, MWR).
• In some cases, we obtained realistic predictions with a
predictability of over two weeks.
Takeaway Messages
41
A Dual Nature of Chaos and Order in Weather IHP, Paris, France, 12 Nov. 2019
Acknowledgments and References
We thank Drs. R. Anthes, B. Bailey, J. Buchmann, D. Durran, M. Ghil, B. Mapes,
Z. Musielak, T. Krishnamurti (Deceased), G C Layek, C.-D. Lin, J. Rosenfeld, R.
Rotunno, I. A. Santos, C.-L. Shie, S. Vannitsem, and F. Zhang (Deceased) for
valuable comments and discussions.
Selected References:
1. Shen, B.-W.*, R. A. Pielke Sr., X. Zeng, J.-J. Baik, T.A.L. Reyes#, S. Faghih-Naini#, R.
Atlas, and J. Cui#, 2019: Is Weather Chaotic? Coexistence of Chaos and Order within a
Generalized Lorenz Model (to be submitted; available from ResearchGate:
http://doi.org/10.13140/RG.2.2.21811.07204)
2. Shen, B.-W.*, 2019a: Aggregated Negative Feedback in a Generalized Lorenz Model.
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