Simple Mathematical, Dynamical StochasticModels Capturing the Observed Diversity of

the El Niño Southern Oscillation (ENSO)

Lecture 5: A Simple Stochastic Model for El Niño withWesterly Wind Bursts

Andrew J. Majda, Nan Chen and Sulian Thual

Center for Atmosphere Ocean ScienceCourant Institute of Mathematical Sciences

New York University

October 05, 2017

Outline of this lecture

1. Reviewing the coupled ENSO dynamical model.

2. Incorporating a novel wind burst parameterization into the coupled model.

3. Showing the skill of capturing both the dynamical and statistical features of thetraditional El Niño in the eastern Pacific, including the super El Niño.

Sulian Thual, Andrew J. Majda, Nan Chen and Samuel N. Stechmann, A Simple Stochastic Model for El

Niño with Westerly Wind Bursts, PNAS, 113(37), pp. 10245-10250, 2016.

1 / 24

Equatorial Climate Patterns in Different ConditionsEl Niño is a climate pattern that includes the interactions between1. atmosphere, 2. ocean, 3. sea surface temperature (SST).

(Figures are from NOAA)

El Niño & La Niña are the opposite phases of El Niño-Southern Oscillation (ENSO).2 / 24

Remarkable Observational Phenomena of the ENSO

El Niño Southern Oscillation

I Warm phase: El Niño

I Cold phase: La Niña

1980 1985 1990 1995 2000 2005 2010 2015−3

−2

−1

0

1

2

3

Years

Nino 3.4 Index

Super El Nino Super El Nino Super El Nino

Delaying ...

I Eastern Pacific El Niño, including two types of super El Niño: 1) 1982-1983and 1997-1998, 2) 2014-2016. — Lecture 5 & 6.

I A series of moderate El Niño but little La Niña: 1990-1995, 2002-2006— years with central Pacific El Niño (El Niño Modoki) — Lecture 7.

Therefore, ENSO is more than a simple regular oscillator!3 / 24

Global Impact of ENSOThe anomalous climate patterns in the equatorial Pacific affect global climate throughteleconnections, which are atmospheric interactions between widely separated regions.

4 / 24

The Starting Model: Deterministic, Linear and Stable

Atmosphere

Ocean

SST

− yv − ∂xθ = 0

yu − ∂yθ = 0

− (∂x u + ∂y v) = Eq/(1− Q)

∂τU − c1YV + c1∂x H = c1τx

YU + ∂Y H = 0

∂τH + c1(∂x U + ∂Y V ) = 0

∂τT = −c1ζEq + c1ηH

u, v : winds

θ : temperature

Eq = αT : latent heat

U,V : ocean current

H : thermocline depth

τx = γu : wind stress

T : sea surface temperature

η : thermocline feedback

(η stronger in eastern Pacific)

SST

Atm OcnLa

tent h

eat

E q∝

TWind Stressτx ∝ u

Thermocline feedback

ηH

5 / 24

The Starting Model: Deterministic, Linear and Stable

Atmosphere

Ocean

SST

− yv − ∂xθ = 0

yu − ∂yθ = 0

− (∂x u + ∂y v) = Eq/(1− Q)

∂τU − c1YV + c1∂x H = c1τx

YU + ∂Y H = 0

∂τH + c1(∂x U + ∂Y V ) = 0

∂τT = −c1ζEq + c1ηH

u, v : winds

θ : temperature

Eq = αT : latent heat

U,V : ocean current

H : thermocline depth

τx = γu : wind stress

T : sea surface temperature

η : thermocline feedback

(η stronger in eastern Pacific)I fundamentally different from the Cane-Zebiak and other nonlinear models that use internal

instability to trigger the ENSO cycles. (plus, emphasis of CZ model: eastern Pacificthermocline.)

I non-dissipative atmosphere consistent with the skeleton model of Madden-Julian Oscillation(Majda and Stechmann 2009, 2011); suitable to describe the dynamics of the Walker circulation

I different meridional axis y and Y due to different Rossby radius in atmosphere and ocean

I allowing a systematic meridional decomposition of the system into the well-known paraboliccylinder functions, keeping the system easily solvable (Majda; 2003)

6 / 24

Original Truncated to φ0(y) and ψ0(y)

Atmosphere

Ocean

SST

− yv − ∂xθ = 0

yu − ∂yθ = 0

− (∂x u + ∂y v) = Eq/(1− Q)

∂τU − c1YV + c1∂x H = c1τx

YU + ∂Y H = 0

∂τH + c1(∂x U + ∂Y V ) = 0

∂τT = −c1ζEq + c1ηH

∂x KA = −χAEq(2− 2Q)−1

− ∂x RA/3 = −χAEq(3− 3Q)−1

(B.C.) KA(0, τ) = KA(LA, τ)

(B.C.) RA(0, τ) = RA(LA, τ)

∂τKO + c1∂x KO = χOc1τx/2

∂τRO − (c1/3)∂x RO = −χOc1τx/3

(B.C.) KO(0, τ) = rW RO(0, τ)

(B.C.) RO(LO , τ) = rE KO(LO , τ)

∂τT = −c1ζEq + c1η(KO + RO)

Reconstructed variables:

u = (KA − RA)φ0 + (RA/√

2)φ2

θ = −(KA + RA)φ0 − (RA/√

2)φ2

U = (KO − RO)ψ0 + (RO/√

2)ψ2

H = (KO + RO)ψ0 + (RO/√

2)ψ2

Whole globe

Pacific Ocean

80 W120 E Pacific Ocean

Reflected Rossby wave

ReflectedKelvin wave

rE

Rossby wave Kelvin wave

rW

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Yea

r

u

0 5 10 15

1

2

3

4

5

6

−2 0 2

Yea

r

U

0 5 10 15

1

2

3

4

5

6

−0.2 0 0.2

Yea

r

H

0 5 10 15

1

2

3

4

5

6

−20 0 20

Yea

r

T

0 5 10 15

1

2

3

4

5

6

−2 0 2

Atmosphere:

− yv − ∂xθ = 0

yu − ∂yθ = 0

− (∂x u + ∂y v) = Eq/(1 − Q)

Ocean:

∂τ U − c1YV + c1∂x H = c1τx

YU + ∂Y H = 0

∂τ H + c1(∂x U + ∂Y V ) = 0

SST:

∂τ T = −c1ζEq + c1ηH

Linear solution with NA = 64 and NO = 28, where the decay rate is set to be zero forillustration purpose.

8 / 24

Stochastic Wind Bursts ParameterizationI Random atmospheric disturbances in the tropics, including westerly wind bursts,

easterly wind bursts and the convective envelope of the MJO, are possibletriggers to ENSO variability (Vecchi and Harrison, 2000; Tziperman and Yu, 2007; Hendon et al.,

2007; Hu and Fedorov, 2015; Puy et al., 2016).

I All those atmospheric disturbances are usually moreprominent in the equatorial Pacific prior to El Niño events.

These wind bursts lie in ∼ 1 month time scale and are unresolved in the model.(westerly (−→): from west to east easterly (←−): from east to west

)

−10 0 10

1997

1998

Total winds (140−180)

m/s

Tim

e (y

ear)

150 200 250

1997

1998

Wind bursts

m/s

−20 0 20

150 200 250

1997

1998

Zonal Winds

m/s

−10 0 10

150 200 250

1997

1998

SST

K

−4 −2 0 2 4

150 200 250

1997

1998

Thermocline depth

m

−100 0 100 9 / 24

Stochastic Wind Bursts ParameterizationI Random atmospheric disturbances in the tropics, including westerly wind bursts,

easterly wind bursts and the convective envelope of the MJO, are possibletriggers to ENSO variability (Vecchi and Harrison, 2000; Tziperman and Yu, 2007; Hendon et al.,

2007; Hu and Fedorov, 2015; Puy et al., 2016).

I All those atmospheric disturbances are usually moreprominent in the equatorial Pacific prior to El Niño events.

These wind bursts lie in ∼ 1 month time scale and are unresolved in the model.(westerly (−→): from west to east easterly (←−): from east to west

)

−10 0 10

1986

1987

1988

Total winds (140−180)

m/s

Tim

e (y

ear)

150 200 250

1986

1987

1988

Wind bursts

m/s

−20 0 20

150 200 250

1986

1987

1988

Zonal Winds

m/s

−10 0 10

150 200 250

1986

1987

1988

SST

K

−4 −2 0 2 4

150 200 250

1986

1987

1988

Thermocline depth

m

−100 0 100 9 / 24

Stochastic Wind Bursts ParameterizationI Random atmospheric disturbances in the tropics, including westerly wind bursts,

easterly wind bursts and the convective envelope of the MJO, are possibletriggers to ENSO variability (Vecchi and Harrison, 2000; Tziperman and Yu, 2007; Hendon et al.,

2007; Hu and Fedorov, 2015; Puy et al., 2016).

I All those atmospheric disturbances are usually moreprominent in the equatorial Pacific prior to El Niño events.

These wind bursts lie in ∼ 1 month time scale and are unresolved in the model.(westerly (−→): from west to east easterly (←−): from east to west

)

−10 0 10

2014

2015

2016

Total winds (140−180)

m/s

Tim

e (y

ear)

150 200 250

2014

2015

2016

Wind bursts

m/s

−20 0 20

150 200 250

2014

2015

2016

Zonal Winds

m/s

−10 0 10

150 200 250

2014

2015

2016

SST

K

−4 −2 0 2 4

150 200 250

2014

2015

2016

Thermocline depth

m

−100 0 100 9 / 24

SST v.s. WWB (from Chen et al, Nature Geoscience, 2015)

contour: SSTblack line: WWB

10 / 24

WWB and EWB in 1998 and 2014 events.(Time series are taken from Hu and Fedorov, PNAS 2016)

(westerly (−→): from west to east easterly (←−): from east to west

)

11 / 24

MJO-related wind stress signal in 1998 events.(Column 1-4 are taken from Puy et al, Climate Dynamics 2016)

12 / 24

A summary of the observational facts.

1. Wind bursts, including WWB, EWB and MJO-related winds, are all possibletriggers to ENSO variability.

2. Wind bursts occur in a much faster time scale than the ENSO cycle.

3. Wind bursts are mostly in the western Pacific are their strength is affected by thewarm pool SST (Fedorov et al., 2015; Tziperman and Yu, 2007; Hendon et al., 2007).

Next: Develop a wind burst model that includes all these observational facts.

13 / 24

A summary of the observational facts.

1. Wind bursts, including WWB, EWB and MJO-related winds, are all possibletriggers to ENSO variability.

2. Wind bursts occur in a much faster time scale than the ENSO cycle.

3. Wind bursts are mostly in the western Pacific are their strength is affected by thewarm pool SST (Fedorov et al., 2015; Tziperman and Yu, 2007; Hendon et al., 2007).

Next: Develop a wind burst model that includes all these observational facts.

13 / 24

Stochastic Wind Bursts: in western Pacific depending on warm pool SST

Total wind stress

Wind burst

Evolution

τx = γ(u + up),

up = ap(τ)sp(x)φ0(y)

dap/dτ = −dpap + σp(TW )W (τ),

up : wind bursts,

sp : spatial structure

TW : western Pacific SST

Markov Jump Process: stochastic dependency on warm pool SST

Markov States

States Switch

σp(TW ) =

{σp0 : quiescent

σp1 : active

P(quiescent→ active at t + ∆t) = r01∆t + o(∆t)

P(active→ quiescent at t + ∆t) = r10∆t + o(∆t)

Fundamentally different from Jin et al., 2007 that relies on the eastern Pacific SSTand D. Chen et al., 2015 that requires ad hoc prescription of wind burst thresholds.

14 / 24

Stochastic Wind Bursts: in western Pacific depending on warm pool SST

Total wind stress

Wind burst

Evolution

τx = γ(u + up),

up = ap(τ)sp(x)φ0(y)

dap/dτ = −dpap + σp(TW )W (τ),

up : wind bursts,

sp : spatial structure

TW : western Pacific SST

Markov Jump Process: stochastic dependency on warm pool SST

Markov States

States Switch

σp(TW ) =

{σp0 : quiescent

σp1 : active

P(quiescent→ active at t + ∆t) = r01∆t + o(∆t)

P(active→ quiescent at t + ∆t) = r10∆t + o(∆t)

Fundamentally different from Jin et al., 2007 that relies on the eastern Pacific SSTand D. Chen et al., 2015 that requires ad hoc prescription of wind burst thresholds. 14 / 24

The Coupled ENSO Model with Stochastic Wind Bursts

Atmosphere

Ocean

SST

− yv − ∂xθ = 0

yu − ∂yθ = 0

− (∂x u + ∂y v) = Eq/(1− Q)

∂τU − c1YV + c1∂x H = c1τx

YU + ∂Y H = 0

∂τH + c1(∂x U + ∂Y V ) = 0

∂τT = −c1ζEq + c1ηH

u, v : winds

θ : temperature

Eq = αT : latent heat

U,V : ocean current

H : thermocline depth

τx = γ(u+up) : wind stress

(including stochastic wind bursts)

T : sea surface temperature

η : thermocline feedback

(η stronger in eastern Pacific)

Total wind stress

Wind burst

Evolution

τx = γ(u + up),

up = ap(τ)sp(x)φ0(y)

dap/dτ = −dpap + σp(TW )W (τ),

up : wind bursts,

sp : spatial structure

TW : western Pacific SST

15 / 24

Model Simulations: Hovmollers x-t

m/s

Y

ear

Zonal windsu

150 200 250100

102

104

106

108

110

112

114

116

118

120

−5 0 5

m/s

CurrentsU

150 200 250100

102

104

106

108

110

112

114

116

118

120

−1 0 1

m

Thermocline DepthH

150 200 250100

102

104

106

108

110

112

114

116

118

120

−50 0 50

K

SSTT

150 200 250100

102

104

106

108

110

112

114

116

118

120

−5 0 5

−10 0 10100

102

104

106

108

110

112

114

116

118

120

m/s

Wind Bursts Amplitudea

p

−4 −2 0 2 4100

102

104

106

108

110

112

114

116

118

120

K

SST Indices

T West

T East

0 0.5 1100

102

104

106

108

110

112

114

116

118

120

Markov States

The coupled model succeeds in capturingI quasi-regular moderate El Niño

I super El Niño as that during 1997-1998I super El Niño as that during 2014-2016

16 / 24

Model Simulations: Hovmollers x-t

Yea

r

Zonal windsu

m/s150 200 250

290

292

294

296

298

300

302

304

306

308

310

−5 0 5

0 0.5 1290

292

294

296

298

300

302

304

306

308

310

Markov States

CurrentsU

m/s150 200 250

290

292

294

296

298

300

302

304

306

308

310

−1 0 1

Thermocline DepthH

m150 200 250

290

292

294

296

298

300

302

304

306

308

310

−50 0 50

SSTT

K150 200 250

290

292

294

296

298

300

302

304

306

308

310

−5 0 5

−10 0 10290

292

294

296

298

300

302

304

306

308

310

Wind Bursts Amplitudea

p

m/s−4 −2 0 2 4

290

292

294

296

298

300

302

304

306

308

310

SST Indices

K

T WestT East

The coupled model succeeds in capturingI quasi-regular moderate El NiñoI super El Niño as that during 1997-1998

I super El Niño as that during 2014-2016

16 / 24

Model Simulations: Hovmollers x-t

Yea

r

m/s

Zonal windsu

150 200 250140

142

144

146

148

150

152

154

156

158

160

−5 0 5

m/s

CurrentsU

150 200 250140

142

144

146

148

150

152

154

156

158

160

−1 0 1

Thermocline DepthH

m150 200 250

140

142

144

146

148

150

152

154

156

158

160

−50 0 50

K

SSTT

150 200 250140

142

144

146

148

150

152

154

156

158

160

−5 0 5

−10 0 10140

142

144

146

148

150

152

154

156

158

160

m/s

Wind Bursts Amplitudea

p

−4 −2 0 2 4140

142

144

146

148

150

152

154

156

158

160

K

SST Indices

T WestT East

0 0.5 1140

142

144

146

148

150

152

154

156

158

160

Yea

r

Markov States

The coupled model succeeds in capturingI quasi-regular moderate El NiñoI super El Niño as that during 1997-1998I super El Niño as that during 2014-2016

16 / 24

Model Simulations: Time series

17 / 24

Model Simulations: PDF and Power SpectrumModerate and Extreme El Niño events in the eastern Pacific, frequency ≈ 2-7 years.

The fat-tailed non-Gaussian PDF in the model is due to the state-dependentnoise in the stochastic wind bursts.

18 / 24

Model Simulations: PDF and Power SpectrumModerate and Extreme El Niño events in the eastern Pacific, frequency ≈ 2-7 years.

The fat-tailed non-Gaussian PDF in the model is due to the state-dependentnoise in the stochastic wind bursts. 18 / 24

Mechanism of the overall ENSO formation

19 / 24

Mechanism of 1997-1998 El Niño

Observations1997-1998

Model SimulationMimicking 1997-1998

20 / 24

Mechanism of 2014-2016 El Niño

Observations2014-2016

Model SimulationMimicking 2014-2016

21 / 24

Mechanism: Ocean Kelvin and Rossby waves

∂τKO + c1∂x KO = χOc1τx/2

∂τRO − (c1/3)∂x RO = −χOc1τx/3

⇓

H = (KO + RO)ψ0 + (RO/√

2)ψ2

∂τT = −c1ζEq + c1ηH

22 / 24

Summary

A simple modeling framework is developed for the ENSO.

1. The starting model is a coupled ocean-atmosphere model that is deterministic,linear and stable.

2. A stochastic parameterization of the wind bursts including both westerly andeasterly winds is coupled to the simple ocean-atmosphere system.

3. The coupled model succeeds in simulating traditional El Niño and capturing theobservational record in the eastern Pacific.

4. The coupled model is able to distinguish the two types of super El Niño. (Moredetails will be discussed in Lecture 6 by Sulian Thual)

5. With more physics in the model (such as nonlinear advection and mean tradewind anomaly), the simple modeling framework allows the study of central PacificEl Niño and therefore the El Niño diversity (Lecture 7, 8, 9).

23 / 24

Next week by Sulian Thual:Mechanisms of the 2014-2016 delayed super El Niño capturing by simple dynamicalmodels.

Thank you

24 / 24