Simple Mathematical, Dynamical Stochastic Models Capturing the Observed Diversity of the El Niño Southern Oscillation (ENSO)

Simple Mathematical, Dynamical Stochastic Models Capturing the Observed Diversity of the El Niño Southern Oscillation (ENSO) Lecture 5: A Simple Stochastic Model for El Niño with Westerly Wind Bursts Andrew J. Majda, Nan Chen and Sulian Thual Center for Atmosphere Ocean Science Courant 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 the traditional 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 Conditions El Niño is a climate pattern that includes the interactions between 1. 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 Warm phase: El Niño Cold phase: La Niña 3 Nino 3.4 Index 2 1 Delaying... 0 1 2 Super El Nino Super El Nino Super El Nino 3 1980 1985 1990 1995 2000 2005 2010 2015 Years Eastern Pacific El Niño, including two types of super El Niño: 1) 1982-1983 and 1997-1998, 2) 2014-2016. Lecture 5 & 6. 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 ENSO The anomalous climate patterns in the equatorial Pacific affect global climate through teleconnections, which are atmospheric interactions between widely separated regions. 4 / 24

The Starting Model: Deterministic, Linear and Stable Atmosphere yv x θ = 0 yu y θ = 0 ( x u + y v) = E q/(1 Q) u, v : winds θ : temperature E q = αt : latent heat Ocean τ U c 1 YV + c 1 x H = c 1 τ x YU + Y H = 0 τ H + c 1 ( x U + Y V ) = 0 U, V : ocean current H : thermocline depth τ x = γu : wind stress SST τ T = c 1 ζe q + c 1 ηh T : sea surface temperature η : thermocline feedback (η stronger in eastern Pacific) Latent heat E q T Thermocline feedback ηh SST Atm Wind Stress τ x u Ocn 5 / 24

The Starting Model: Deterministic, Linear and Stable Atmosphere yv x θ = 0 yu y θ = 0 ( x u + y v) = E q/(1 Q) u, v : winds θ : temperature E q = αt : latent heat Ocean τ U c 1 YV + c 1 x H = c 1 τ x YU + Y H = 0 τ H + c 1 ( x U + Y V ) = 0 U, V : ocean current H : thermocline depth τ x = γu : wind stress SST τ T = c 1 ζe q + c 1 ηh T : sea surface temperature η : thermocline feedback (η stronger in eastern Pacific) 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 Pacific thermocline.) 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 different meridional axis y and Y due to different Rossby radius in atmosphere and ocean allowing a systematic meridional decomposition of the system into the well-known parabolic cylinder functions, keeping the system easily solvable (Majda; 2003) 6 / 24

Original Truncated to φ 0 (y) and ψ 0 (y) Atmosphere yv x θ = 0 yu y θ = 0 ( x u + y v) = E q/(1 Q) 1 x K A = χ A E q(2 2 Q) 1 x R A /3 = χ A E q(3 3 Q) (B.C.) K A (0, τ) = K A (L A, τ) (B.C.) R A (0, τ) = R A (L A, τ) Ocean τ U c 1 YV + c 1 x H = c 1 τ x YU + Y H = 0 τ H + c 1 ( x U + Y V ) = 0 τ K O + c 1 x K O = χ O c 1 τ x /2 τ R O (c 1 /3) x R O = χ O c 1 τ x /3 (B.C.) K O (0, τ) = r W R O (0, τ) (B.C.) R O (L O, τ) = r E K O (L O, τ) SST τ T = c 1 ζe q + c 1 ηh τ T = c 1 ζe q + c 1 η(k O + R O ) Reconstructed variables: Whole globe u = (K A R A )φ 0 + (R A / 2)φ 2 θ = (K A + R A )φ 0 (R A / 2)φ 2 U = (K O R O )ψ 0 + (R O / 2)ψ 2 H = (K O + R O )ψ 0 + (R O / 2)ψ 2 Rossby wave r W Reflected Kelvin wave Pacific Ocean 120 E Pacific Ocean Kelvin wave Reflected Rossby wave r E 80 W 7 / 24

6 u 6 U 6 H 6 T Atmosphere: yv x θ = 0 5 5 5 5 yu y θ = 0 4 4 4 4 ( x u + y v) = E q/(1 Q) Year 3 Year 3 Year 3 Year 3 Ocean: τ U c 1 YV + c 1 x H = c 1 τ x 2 2 2 2 YU + Y H = 0 1 1 1 1 τ H + c 1 ( x U + Y V ) = 0 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 SST: 2 0 2 0.2 0 0.2 20 0 20 2 0 2 τ T = c 1 ζe q + c 1 ηh Linear solution with N A = 64 and N O = 28, where the decay rate is set to be zero for illustration purpose. 8 / 24

Stochastic Wind Bursts Parameterization Random atmospheric disturbances in the tropics, including westerly wind bursts, easterly wind bursts and the convective envelope of the MJO, are possible triggers to ENSO variability (Vecchi and Harrison, 2000; Tziperman and Yu, 2007; Hendon et al., 2007; Hu and Fedorov, 2015; Puy et al., 2016). All those atmospheric disturbances are usually more prominent 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 ) Total winds (140 180) Wind bursts Zonal Winds SST Thermocline depth 1998 1998 1998 1998 1998 Time (year) 1997 1997 1997 1997 1997 10 0 10 K m 20 0 20 10 0 10 4 2 0 2 4 100 0 100 9 / 24

Stochastic Wind Bursts Parameterization Random atmospheric disturbances in the tropics, including westerly wind bursts, easterly wind bursts and the convective envelope of the MJO, are possible triggers to ENSO variability (Vecchi and Harrison, 2000; Tziperman and Yu, 2007; Hendon et al., 2007; Hu and Fedorov, 2015; Puy et al., 2016). All those atmospheric disturbances are usually more prominent 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 ) Total winds (140 180) Wind bursts Zonal Winds SST Thermocline depth 1988 1988 1988 1988 1988 Time (year) 1987 1987 1987 1987 1987 1986 1986 1986 1986 1986 10 0 10 K m 20 0 20 10 0 10 4 2 0 2 4 100 0 100 9 / 24

Stochastic Wind Bursts Parameterization Random atmospheric disturbances in the tropics, including westerly wind bursts, easterly wind bursts and the convective envelope of the MJO, are possible triggers to ENSO variability (Vecchi and Harrison, 2000; Tziperman and Yu, 2007; Hendon et al., 2007; Hu and Fedorov, 2015; Puy et al., 2016). All those atmospheric disturbances are usually more prominent 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 ) Total winds (140 180) Wind bursts Zonal Winds SST Thermocline depth 2016 2016 2016 2016 2016 Time (year) 2015 2015 2015 2015 2015 2014 2014 2014 2014 2014 10 0 10 K m 20 0 20 10 0 10 4 2 0 2 4 100 0 100 9 / 24

SST v.s. WWB (from Chen et al, Nature Geoscience, 2015) contour: SST black 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 possible triggers 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 the warm pool SST (Fedorov et al., 2015; Tziperman and Yu, 2007; Hendon et al., 2007). 13 / 24

A summary of the observational facts. 1. Wind bursts, including WWB, EWB and MJO-related winds, are all possible triggers 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 the warm 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 τ x = γ(u + u p), u p = a p(τ)s p(x)φ 0 (y) u p : wind bursts, s p : spatial structure Evolution da p/dτ = d pa p + σ p(t W )Ẇ (τ), T W : western Pacific SST Markov Jump Process: stochastic dependency on warm pool SST Markov States States Switch { σp0 : quiescent σ p(t W ) = σ p1 : active P(quiescent active at t + t) = r 01 t + o( t) P(active quiescent at t + t) = r 10 t + o( t) 14 / 24

Stochastic Wind Bursts: in western Pacific depending on warm pool SST Total wind stress Wind burst τ x = γ(u + u p), u p = a p(τ)s p(x)φ 0 (y) u p : wind bursts, s p : spatial structure Evolution da p/dτ = d pa p + σ p(t W )Ẇ (τ), T W : western Pacific SST Markov Jump Process: stochastic dependency on warm pool SST Markov States States Switch { σp0 : quiescent σ p(t W ) = σ p1 : active P(quiescent active at t + t) = r 01 t + o( t) P(active quiescent at t + t) = r 10 t + o( t) Fundamentally different from Jin et al., 2007 that relies on the eastern Pacific SST and 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 yv x θ = 0 yu y θ = 0 ( x u + y v) = E q/(1 Q) u, v : winds θ : temperature E q = αt : latent heat Ocean τ U c 1 YV + c 1 x H = c 1 τ x YU + Y H = 0 τ H + c 1 ( x U + Y V ) = 0 U, V : ocean current H : thermocline depth τ x = γ(u+u p) : wind stress (including stochastic wind bursts) SST τ T = c 1 ζe q + c 1 ηh T : sea surface temperature η : thermocline feedback (η stronger in eastern Pacific) Total wind stress Wind burst Evolution τ x = γ(u + u p), u p = a p(τ)s p(x)φ 0 (y) da p/dτ = d pa p + σ p(t W )Ẇ (τ), u p : wind bursts, s p : spatial structure T W : western Pacific SST 15 / 24

Model Simulations: Hovmollers x-t Zonal winds u 120 120 Currents U Thermocline Depth H 120 120 SST T Wind Bursts Amplitude SST Indices a p 120 120 120 Markov States 118 118 118 118 118 118 118 116 116 116 116 116 116 116 114 114 114 114 114 114 114 112 112 112 112 112 112 112 Year 110 110 110 110 110 110 110 108 108 108 108 108 108 108 106 106 106 106 106 106 106 104 104 104 104 104 104 104 102 102 102 102 102 102 102 100 100 100 m 100 K 100 10 0 10 100 4 2 0 2 4 K 100 0 0.5 1 T West 5 0 5 1 0 1 50 0 50 5 0 5 T East The coupled model succeeds in capturing quasi-regular moderate El Niño 16 / 24

Model Simulations: Hovmollers x-t Zonal winds u 310 310 Currents U Thermocline Depth H 310 310 SST T Wind Bursts Amplitude SST Indices a p 310 310 310 Markov States 308 308 308 308 308 308 308 306 306 306 306 306 306 306 304 304 304 304 304 304 304 302 302 302 302 302 302 302 Year 300 300 300 300 300 300 300 298 298 298 298 298 298 298 296 296 296 296 296 296 296 294 294 294 294 294 294 294 292 292 292 292 292 292 292 290 290 290 m 290 K 290 10 0 10 290 4 2 0 2 4 K 290 0 0.5 1 5 0 5 1 0 1 50 0 50 5 0 5 T West T East The coupled model succeeds in capturing quasi-regular moderate El Niño super El Niño as that during 1997-1998 16 / 24

Model Simulations: Hovmollers x-t Zonal winds u 160 160 Currents U Thermocline Depth H 160 160 SST T Wind Bursts Amplitude SST Indices a p 160 160 160 Markov States 158 158 158 158 158 158 158 156 156 156 156 156 156 156 154 154 154 154 154 154 154 152 152 152 152 152 152 152 Year 150 150 150 150 150 150 Year 150 148 148 148 148 148 148 148 146 146 146 146 146 146 146 144 144 144 144 144 144 144 142 142 142 142 142 142 142 140 140 140 m 140 K 140 10 0 10 140 4 2 0 2 4 K 140 0 0.5 1 5 0 5 1 0 1 50 0 50 5 0 5 T West T East The coupled model succeeds in capturing quasi-regular moderate El Niño super El Niño as that during 1997-1998 super El Niño as that during 2014-2016 16 / 24

Model Simulations: Time series 17 / 24

Model Simulations: PDF and Power Spectrum Moderate and Extreme El Niño events in the eastern Pacific, frequency 2-7 years. 18 / 24

Model Simulations: PDF and Power Spectrum Moderate 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-dependent noise in the stochastic wind bursts. 18 / 24

Mechanism of the overall ENSO formation 19 / 24

Mechanism of 1997-1998 El Niño Observations 1997-1998 Model Simulation Mimicking 1997-1998 20 / 24

Mechanism of 2014-2016 El Niño Observations 2014-2016 Model Simulation Mimicking 2014-2016 21 / 24

Mechanism: Ocean Kelvin and Rossby waves τ K O + c 1 x K O = χ O c 1 τ x /2 τ R O (c 1 /3) x R O = χ O c 1 τ x /3 H = (K O + R O )ψ 0 + (R O / 2)ψ 2 τ T = c 1 ζe q + c 1 ηh 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 and easterly winds is coupled to the simple ocean-atmosphere system. 3. The coupled model succeeds in simulating traditional El Niño and capturing the observational record in the eastern Pacific. 4. The coupled model is able to distinguish the two types of super El Niño. (More details will be discussed in Lecture 6 by Sulian Thual) 5. With more physics in the model (such as nonlinear advection and mean trade wind anomaly), the simple modeling framework allows the study of central Pacific El 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 dynamical models. Thank you 24 / 24