Stochastic Modelling and ENSO David Kelly Mathematics Department UNC Chapel Hill dtbkelly@gmail.com December 3, 2013 David Kelly (UNC) Stochastic Climate December 3, 2013 1 / 41
Outline 1 - Why is ENSO stochastic ? 2 - Variability can be explained by stochastic noise (Kleeman + Moore paper) 3 - Chaotic models vs stochastic models David Kelly (UNC) Stochastic Climate December 3, 2013 2 / 41
What are the stochastic features of ENSO? David Kelly (UNC) Stochastic Climate December 3, 2013 3 / 41
Variability of Amplitude and Period • No two events are the same in magnitude. • Is it really an oscillation? 2 − 10 yr periods. David Kelly (UNC) Stochastic Climate December 3, 2013 4 / 41
Seasonal locking • However, peaks tend to happen around December. David Kelly (UNC) Stochastic Climate December 3, 2013 5 / 41
Spectral Analysis The spectral density indicates the dominant modes of a signal. f ( k ) | 2 where ˆ For a random signal f ( t ), the spectral density is given by E | ˆ f is the Fourier transform of f . Eg . David Kelly (UNC) Stochastic Climate December 3, 2013 6 / 41
Spectral Analysis • 4 year period is significant ... But is surrounded by a lot of “noise”. • The spectrum decays like k − 2 ... Signature of red noise (Brownian motion). David Kelly (UNC) Stochastic Climate December 3, 2013 7 / 41
Why is ENSO stochastic The ENSO phenomenon features variability in • amplitude of events • frequency of events and has a spectral signature reminiscent of noise . This suggests noise is present ... But we would like to know if the noise is actually “ causing ” excitations. David Kelly (UNC) Stochastic Climate December 3, 2013 8 / 41
Evidence that ENSO events arise due to noise. Kleeman and Moore. A Theory of the Limitation of ENSO Predictability Due to Stochastic Atmospheric Transients . Journal of Atmospheric Science (1997). Kleeman and Moore. Stochastic Forcing of ENSO by the Intraseasonal Oscillation . Journal of Climate (1999). David Kelly (UNC) Stochastic Climate December 3, 2013 9 / 41
Idea behind paper 1 - Fix the model dynamics Ψ 2 - Propose a noisy perturbation ψ 3 - Find the type of noise that excites variability 4 - Show that this agrees with natural sources of noise David Kelly (UNC) Stochastic Climate December 3, 2013 10 / 41
The model The authors used a coupled atmosphere-ocean model ( ⋆ ) (variant of the ZC model in Dijkstra). ( ⋆ ) - Kleeman (1991,1993). David Kelly (UNC) Stochastic Climate December 3, 2013 11 / 41
The model The noise is assumed to enter the model through forcing in the wind components . This is reasonable given short term, unpredictable wind events, like Madden-Julian oscillation (MJO). David Kelly (UNC) Stochastic Climate December 3, 2013 12 / 41
Linear stability analysis Suppose that Ψ : R + → R N is the interannual climate variables d Ψ dt = F (Ψ) given by a spatially discretized version of the model. Let ψ be the anomaly, and suppose that d (Ψ + ψ ) = F (Ψ + ψ ) + f ( t ) dt where f is an undefined source of noise. David Kelly (UNC) Stochastic Climate December 3, 2013 13 / 41
Linear stability analysis If we assume that ψ 2 ≪ ψ then d (Ψ + ψ ) = F (Ψ) + ∇ F (Ψ) ψ + O ( ψ 2 ) + f ( t ) dt This gives the linear approximation d ψ dt = ∇ F (Ψ) ψ + f ( t ) So we have a linear model for the anomaly ψ that depends on the state of the unperturbed model Ψ. David Kelly (UNC) Stochastic Climate December 3, 2013 14 / 41
Discretization of the model We work with the time discretization ψ n +1 = ψ n + F n ψ n ∆ t + f n ∆ t . where F n = ∇ F (Ψ n ). We let f n = ∆ t − 1 / 2 ξ n , where ξ is a sequence of identically distributed Gaussian random variables, with E ξ n ξ T E ξ n = 0 and m = D n , m C . The number D n , m measures temporal correlation and the matrix C measures spatial correlation. David Kelly (UNC) Stochastic Climate December 3, 2013 15 / 41
Writing down the solution Since the model is linear, the solution is easy to write down . Let R j , k be the semi-group for the linear part. That is, if u n +1 = u n + A n u n ∆ t then u k = R j , k u j . We will solve ψ n +1 = ψ n + A n ψ n ∆ t + ξ n ∆ t 1 / 2 using Duhamel’s principle . David Kelly (UNC) Stochastic Climate December 3, 2013 16 / 41
Writing down the solution We have that ... ψ n +1 = ψ n + A n ψ n ∆ t + ξ n ∆ t 1 / 2 = (1 + A n ∆ t ) ψ n + ξ n ∆ t 1 / 2 = R n , n +1 ψ n + ξ n ∆ t 1 / 2 Repeating this ... ψ n +1 = R n , n +1 ( R n − 1 , n ψ n − 1 + ξ n − 1 ∆ t 1 / 2 ) + ξ n ∆ t 1 / 2 � � R n − 1 , n ξ n − 1 ∆ t 1 / 2 + R n , n ξ n ∆ t 1 / 2 = R n − 1 , n +1 ψ n − 1 + And finally n � R k , n ξ k ∆ t 1 / 2 ψ n +1 = R 0 , n +1 ψ 0 + k =0 David Kelly (UNC) Stochastic Climate December 3, 2013 17 / 41
Measuring the variability of the anomaly Given the solution, it is easy to write down the mean n − 1 � � � R k , n − 1 ξ k ∆ t 1 / 2 E ψ n = E R 0 , n ψ 0 + = R 0 , n E ψ 0 . k =0 � � | ψ n − E ψ n | 2 And the variance E is given by n − 1 n − 1 � � E � R 0 , n ( ψ 0 − E ψ 0 ) , R 0 , n ( ψ 0 − E ψ 0 ) � + E � R k , n − 1 ξ k , R j , n − 1 ξ j � ∆ t j =0 k =0 David Kelly (UNC) Stochastic Climate December 3, 2013 18 / 41
Measuring the variability of the anomaly The noise term is the important one. It simplifies to n − 1 n − 1 n − 1 n − 1 � � � � E � R T E � R k , n − 1 ξ k , R j , n − 1 ξ j � ∆ t = j , n − 1 R k , n − 1 ξ k , ξ j � ∆ t j =0 k =0 j =0 k =0 n − 1 n − 1 � � � � R T = tr j , n − 1 R k , n − 1 D j , k C ∆ t = tr ( ZC ) j =0 k =0 where n − 1 n − 1 � � R T Z = j , n − 1 R k , n − 1 D j , k ∆ t j =0 k =0 David Kelly (UNC) Stochastic Climate December 3, 2013 19 / 41
Measuring the variability of the anomaly Alternatively, we might want to measure the anomaly of a particular feature of the model. Let P : R N → R M for some M ≤ N . Eg . P could be the NINO3 average. Then E | P ( ψ n − E ψ n ) | 2 = tr ( ZC ) where n − 1 n − 1 � � R T j , n − 1 P T PR k , n − 1 D j , k ∆ t Z = j =0 k =0 NB . From now on we always use the NINO3 average for P David Kelly (UNC) Stochastic Climate December 3, 2013 20 / 41
Stochastic Optimals Let { v k , λ k } and { w k , µ k } be the eigenvectors-eigenvalue pairs for Z and C respectively. Then N λ i µ j |� v i , w j �| 2 . � tr ( ZC ) = i , j =1 The eigenvectors of Z are called the stochastic optimals . The eigenvectors of C are called the empirical orthogonal functions (EOFs). The anomaly is activated when the dominant stochastic optimal lines up with the dominant EOF. David Kelly (UNC) Stochastic Climate December 3, 2013 21 / 41
Stochastic Optimals: Wind stress First stochastic optimal (wind stress component). David Kelly (UNC) Stochastic Climate December 3, 2013 22 / 41
We can compare the dominant stochastic optimal for wind stress, with the dominant eigenvector of the observed wind stress. David Kelly (UNC) Stochastic Climate December 3, 2013 23 / 41
Observed wind stress 1 - Get a data set of wind-stress observations { W 1 , . . . , W M } . 2 - Filter out the large time scales to obtain { ˜ W 1 , . . . , ˜ W M } . 3 - Compute the covariance matrix M C = 1 � ( ˜ W j − ¯ W )( ˜ W j − ¯ W ) T N j =1 4 - Find the eigenvectors (EOFs) of C . David Kelly (UNC) Stochastic Climate December 3, 2013 24 / 41
Observed wind stress First eigenvector of the covariance. • Similar global pattern (up to plus-minus). David Kelly (UNC) Stochastic Climate December 3, 2013 25 / 41
Stochastic Optimals: Wind stress First stochastic optimal (wind stress component). David Kelly (UNC) Stochastic Climate December 3, 2013 26 / 41
Stochastic Optimals: Heat flux • The dipole structure indicates the importance of coupling in ENSO events. • Dipole structures agrees with MJO heat flux map. David Kelly (UNC) Stochastic Climate December 3, 2013 27 / 41
The first stochastic optimal / EOF pair accounts for almost all of the anomaly. David Kelly (UNC) Stochastic Climate December 3, 2013 28 / 41
Importance of stochastic optimals Truncated version of the variance tr ( ZC ) = � N i , j =1 λ i µ j |� v i , w j �| 2 David Kelly (UNC) Stochastic Climate December 3, 2013 29 / 41
What do simulations look like? David Kelly (UNC) Stochastic Climate December 3, 2013 30 / 41
Seasonal locking of extreme events David Kelly (UNC) Stochastic Climate December 3, 2013 31 / 41
Spectral analysis David Kelly (UNC) Stochastic Climate December 3, 2013 32 / 41
Chaotic forcing vs stochastic forcing. David Kelly (UNC) Stochastic Climate December 3, 2013 33 / 41
Slow-Fast system Suppose the weather variables y ε satisfy some chaotic dynamics dy ε dt = ε − 2 g ( y ε ) Note that y ε ( t ) = y ( ε − 2 t ) where ˙ y = g ( y ). Suppose the climate variables x satisfy dx ε dt = ε − 1 h ( x ε , y ε ) + f ( x ε , y ε ) This is a natural set-up in climate models. Eg . Barotropic flow (Majda et al 1999). David Kelly (UNC) Stochastic Climate December 3, 2013 34 / 41
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