Zhaohua Wu Zhaohua Wu
Center for Ocean-Land Atmosphere Studies Center for Ocean-Land Atmosphere Studies
Calverton, Maryland, USA Calverton, Maryland, USA
Annual Cycle and the Prediction of
Interannual Variability
MY CONFESSION
MY CONFESSION
OUTLINE OUTLINE
• Speculations (preliminary thoughts)
– Climate variability of different timescales are driven by different physical mechanisms
– Variability associated with one physical mechanism is more predictable than the total variability
• Justifications
• Annual cycle and anomaly
• Prediction in a new paradigm
TYPICAL DATA
TYPICAL DATA
AN ANALOGUE AN ANALOGUE
climate data
Random forcing
Low frequency oscillation
High frequency oscillation
DIFFICULTY IN PREDICTION
DIFFICULTY IN PREDICTION
ADVANTAGES OF SEPARATION
ADVANTAGES OF SEPARATION
ENSO MODELS ENSO MODELS
• Delayed Oscillator
cT t bT t eT A t
dt t dT
0
3
sin
Local instability
Delayed term related to Kelvin wave reflection at the eastern boundary
Nonlinear damping
Background
climatology
SOLUTIONS OF THE FORCED SOLUTIONS OF THE FORCED DELAYED OSCILLATION MODEL DELAYED OSCILLATION MODEL
0 2 4 6 8 10 12 14 16 18 20
-10 -5 0 5
10 Free and Annualy-Forced Delayed Oscillators
0 2 4 6 8 10 12 14 16 18 20
-2 -1 0 1
2 Modulated Annual Cycle in the Forced Delayed Oscillator
year
AM/FM SINGNAL AM/FM SINGNAL
40 42 44 46 48 50 52 54 56 58 60
-1.5 -1 -0.5 0 0.5 1 1.5
PREDICTION OF ANOMALY PREDICTION OF ANOMALY
• Anomaly
• Prediction model
N X
M
X n n
1
t X t A t
X
anomaly data annual cycle
The matrix M and the anomaly are dependent on the definition of annual cycle.
The matrix M is very sensitive to high frequency component
of quasi-periodic signal, leading to the difficulty of prediction
ANNUAL CYCLE IN ANOMALY ANNUAL CYCLE IN ANOMALY
1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965
0 5 10 15
climatological annual cycle
degree Celsius
time (year)
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
-2 -1 0 1
2 Anomaly wrt climatological annual cycle
degree Celsius
Magnified Anomaly
Note: After removing climatological annual cycle, we still see
locally in the anomaly a sort of annual cycle. Then, what
is the annual cycle?
FORCED LORENZ MODEL FORCED LORENZ MODEL
3 1 2
2
rx
1x x x
dt
dx
2 1
3
bx
3x x
dt
dx
T
a t x
x dt s
dx 2
2
sin
1 1
PROBLEMS with AMS DEFINITION PROBLEMS with AMS DEFINITION
• Numerous versions of annual cycle
• Assuming the climate system is stationary
• Implying the climate system is linear
• Implying the later evolution of climate system can change what has already happened
1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981
24 24.5 25 25.5 26 26.5 27 27.5 28 28.5
year
degree Celcius
CTI and its annual cycles CTI
Annual Cycle based on 1971-1976
Annual Cycle based on 1976-1981
Annual Cycle based on 1971-1981
DEFINITION OF MAC DEFINITION OF MAC
From data analysis perspective, AC is
“An adaptively and intrinsically determined
temporally local component of the climate data which contains frequency and amplitude
modulations and has quasi-annual period”
(Amplitude frequency modulated annual cycle, or MAC)
THE THE ULTIMATE ULTIMATE CONSTRAIN CONSTRAIN OF DATA ANALYSIS
OF DATA ANALYSIS
• Data analysis should be temporally local, for
– Later evolution can not change the past – What matter to a dynamic system’s future
evolution are its initial condition and boundary condition
A B C D
t
METHOD FOR EXTRACTING MAC METHOD FOR EXTRACTING MAC
• The Ensemble Empirical Mode Decomposition (EEMD) – Based on the Empirical Mode Decomposition, a
method based on the principles of adaptiveness and temporal locality
– A natural dyadic filter based on data characteristics – Mimicking a multiple observation scenario for singly
observed data by adding noise to the data and averaging out the added noise through ensemble approach
– A noise-assisted data analysis (NADA) method
EXTREMA & ENVELOPES EXTREMA & ENVELOPES
Norden E Huang
C hir p W av es
EMPIRICAL MODE DECOMPOSITION &
EMPIRICAL MODE DECOMPOSITION &
HILBERT-HUANG TRANSFORM HILBERT-HUANG TRANSFORM
receiver
signal source 1
signal source 2
Overall Signal
0 50 100 150 200 250 300
-2 0 2
0 50 100 150 200 250 300
-1 0 1
0 50 100 150 200 250 300
-2 0 2
EMPIRICAL MODE DECOMP.
EMPIRICAL MODE DECOMP.
0 50 100 150 200 250 300
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
EMD SEPARATION
EMD SEPARATION
LENGTH-OF-DAY
LENGTH-OF-DAY
SOME KNOWN CYCLES
SOME KNOWN CYCLES
CHANGE OF OUR UNDERSTANDING CHANGE OF OUR UNDERSTANDING
• The phase of ENSO Interannual variability does not locked to annual cycle
• ENSO phase locking to annual cycle may be only a result of phase locking of the
“residual annual cycle” contained in the
traditional anomaly to annual cycle itself
PHASE LOCKING PHASE LOCKING
“Warm ENSO events tend to peaking in winter season” ?
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000
-2 -1 0 1 2
3 Traditional CTI (Nino3.4 Indec)
degree Celsius
year
1956 1956.5 1957 1957.5 1958 1958.5 1959 -1
-0.5 0 0.5 1 1.5
year
degree Celsius
1991 1991.5 1992 1992.5 1993 1993.5 1994 -1
-0.5 0 0.5 1 1.5
year
degree Celsius
A B A B
Note: From traditional anomaly, the phase locking can not be
determined directly; rather, it is determined indirectly through examining the standard deviation of individual month: if the interannual variability tends to peak in a particular month, the standard deviation of the
anomaly at that particular month is larger.
EVIDENCE OF LOCKING EVIDENCE OF LOCKING
Feb Apr Jun Aug Oct Dec
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
0.9standard deviation of traditional anomalies in each month
month
degree Celsius
ANOMALIES wrt TAC AND MAC ANOMALIES wrt TAC AND MAC
19700 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 2
4 6 8 10 12 14 16 18
20 CTI and its decompositions
year
scale (K)
PHASE LOCKING PHASE LOCKING ? ?
Feb Apr Jun Aug Oct Dec
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.9standard deviation of various anomalies in each month
month
degree Celsius
Anomaly w.r.t. TAC
Low-Fre. w.r.t. TAC
MAC - TAC
High-Fre. w.r.t. TAC
SCATTERED ENSO EVENTS SCATTERED ENSO EVENTS
Feb Apr Jun Aug Oct Dec
-2 -1.5 -1 -0.5 0 0.5 1 1.5
2 extremas and their appearance months
Kelvin
ENSO MODELS ENSO MODELS
• Models with intermediate complexity
– Specified climatology has an annual forcing term ~ almost always peaking in a particular season
– When perpetual monthly (e.g., jan, jul) climatology specified ~ not peaking at a particular season
• Delayed Oscillator
cT t bT t eT A t
dt t dT
0
3
sin
Local instability
Delayed term related to Kelvin wave reflection at the eastern boundary
Nonlinear damping
Background
climatology
SOLUTIONS OF THE FORCED SOLUTIONS OF THE FORCED DELAYED OSCILLATION MODEL DELAYED OSCILLATION MODEL
0 2 4 6 8 10 12 14 16 18 20
-10 -5 0 5
10 Free and Annualy-Forced Delayed Oscillators
0 2 4 6 8 10 12 14 16 18 20
-2 -1 0 1
2 Modulated Annual Cycle in the Forced Delayed Oscillator
year
“ “ SPRING BARRIER” SPRING BARRIER”
• The “spring barrier” appeared in the prediction of traditional anomaly .
• When anomaly is defined with respect to
MAC, “spring barrier” disappear
SPRING BARRIER PROBLEM SPRING BARRIER PROBLEM
2 4 6 8 10 12 14 16 18
1 2 3 4 5 6 7 8 9 10 11 12
Autocorrelations of Traditional Anomaly (wrt TAC)
starting month
month of lag
SPRING BARRIER REDUCED SPRING BARRIER REDUCED
2 4 6 8 10 12 14 16 18
1 2 3 4 5 6 7 8 9 10 11 12
Autocorrelations of Revised Anomaly (wrt MAC)
starting month
month of lag
MAC & ITS FREQUENCY
MAC & ITS FREQUENCY
ALTERNATIVE PREDICTION SCHEME ALTERNATIVE PREDICTION SCHEME
t
t
i n
i
i n
i
i
t A t t dt
C t
T
0
cos
1 1
n i
i i
predicted i
n i
i
t t
t A
t C t
T
1
, 1
