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Winter Is Coming: A Dynamical Systems Approach to Better El Ni˜

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Winter Is Coming: A Dynamical Systems Approach to Better El Ni˜
Winter Is Coming: A Dynamical Systems
Approach to Better El Niño Predictions
Andrew Roberts
Esther Widiasih, Chris Jones, Axel Timmerman
October 14, 2014
1/26
Outline
• What is ENSO?
• Why do we care?
• Conceptual ENSO Models
• Dynamical Systems
2/26
The Data
3/26
2014 Predictions
August:
September: No ENSO this year
October 9: El Niñ0 watch back on. Chances back over 65 % (NOAA).
4/26
Physical Cartoon
From: NOAA
5/26
Jin’s Model - 1997
Recharge Oscillator Model:
dTE
= RTE + γhW − en (hW + bTE )3
dt
dhW
= −rhW − αbTE .
dt
Variables:
• TE temperature anomaly in E Pacific
• hW thermocline depth anomaly in W Pacific
Processes:
• Relaxation to mean state
• Upwelling
• Thermocline adjustment to wind-stress
6/26
Results
7/26
Supercharged Recharge Oscillator
dT1
= −α(T1 − Tr ) − δµ(T2 − T1 )2
dt
dT2
= −α(T2 − Tr ) + ζµ(T2 − T1 )[T2 − Tsub (T1 , T2 , h1 )]
dt
dh1
bLµ
= −r h1 −
(T2 − T1 )
dt
2β
Modifications from Recharge Oscillator model:
• Variables are no longer anomalies
• Subscript 1 indicates West, 2 indicates East
• Includes advection
8/26
Simulation vs. Data
From: Timmerman, Jin, and Abshagen – 2003
9/26
MMOs
• Timmerman, Jin, Abshagen (2003): Shilnikov-type mechansim
(saddle-focus)
• Other methods require multiple time-scales!
Change of variables:
• S = T2 − T1
• T = T1 − Tr
• h = h1 − K
10/26
New System
dS
= −αS + δµS 2 + ζµS [S + T + Cf (S, T , h)]
dt
dT
= −αT − δµS 2
dt
dh
bLµ
= −r h − K +
S
dt
2β
We want to analyze as a fast/slow system → need to non-dimensionalize!
11/26
Dimensionless System
x3
x 0 = ε(x 2 − ax) + x x + y − nz + d − c x −
3
y 0 = −ε(ay + x 2 )
x
z0 = m k − z −
2
• x ∼S
• y ∼T
• z ∼h
• ε = δ/ζ
bLµ(Tr −Tr 0 )
2h∗ β
αbL
a = δβh∗
rbL
m = ζβh
∗
• c =
•
•
• n, k, d come from f
12/26
Dimensionless System
x3
x = ε(x − ax) + x x + y − nz + d − c x −
3
0
2
y 0 = −ε(ay + x 2 )
x
z0 = m k − z −
2
• If ε is small (how small?), we have a fast/slow system.
• If m is also small, we have 1-fast, 2-slow OR 3 time-scale system.
• If m = O(1), we have a 2-fast, 1-slow system.
x3
εẋ = ε(x 2 − ax) + x x + y − nz + d − c x −
3
ẏ = −(ay + x 2 )
x
εż = m k − z −
2
13/26
Fast and Slow Dynamics
If we take the limit as ε → 0, the two versions become: (1) The layer
problem
x3
0
x = x x + y − nz + d − c x −
3
x
0
z =m k −z −
2
y0 = 0
Or (2) The reduced problem
x3
0 = x x + y − nz + d − c x −
3
x
0=m k −z −
2
2
ẏ = −(ay + x )
14/26
The Critical Manifold (ε = 0)
Figure: a = 0.4, c = 4, d = 3.69, k = 1, m = 0.26, n = 2.69
15/26
The Critical Manifold (ε = 0)
Figure: a = 0.4, c = 4, d = 3.69, k = 1, m = 0.26, n = 2.69
16/26
The Critical Manifold (ε = 0)
Figure: a = 0.4, c = 4, d = 3.69, k = 1, m = 0.26, n = 2.69
17/26
The Critical Manifold (ε = 0)
Figure: a = 0.4, c = 4, d = 3.69, k = 1, m = 0.26, n = 2.69
18/26
The Critical Manifold (ε = 0)
Figure: a = 0.4, c = 4, d = 3.69, k = 1, m = 0.26, n = 2.69
19/26
The Critical Manifold (ε = 0)
Figure: a = 0.4, c = 4, d = 3.69, k = 1, m = 0.26, n = 2.69
20/26
The Critical Manifold (ε = 0)
Figure: a = 0.4, c = 4, d = 3.69, k = 1, m = 0.26, n = 2.69
21/26
The Critical Manifold (ε = 0)
Figure: a = 0.4, c = 4, d = 3.69, k = 1, m = 0.26, n = 2.69
22/26
MMO Orbit: ε = 0.001
23/26
MMO Orbit: ε = 0.1
24/26
Re-dimensionalized Model Output
Cubic Approximation−Dimensionalized
28
T1
T2
27
26
25
24
23
22
0
20
40
60
80
100
120
140
160
180
200
100
h
90
80
70
60
0
20
40
60
80
100
120
140
160
180
200
25/26
Consequences in the US
From: NOAA
26/26
Fly UP