Stochastic Modelling and ENSO David Kelly Mathematics Department - - PowerPoint PPT Presentation

▶
stochastic modelling and enso
SMART_READER_LITE
LIVE PREVIEW

Stochastic Modelling and ENSO David Kelly Mathematics Department - - PowerPoint PPT Presentation

Jan 25, 2024 344 likes •774 views

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

slide-1
SLIDE 1

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

slide-2
SLIDE 2

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

slide-3
SLIDE 3

What are the stochastic features of ENSO?

David Kelly (UNC) Stochastic Climate December 3, 2013 3 / 41

slide-4
SLIDE 4

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

slide-5
SLIDE 5

Seasonal locking

  • However, peaks tend to happen around December.

David Kelly (UNC) Stochastic Climate December 3, 2013 5 / 41

slide-6
SLIDE 6

Spectral Analysis

The spectral density indicates the dominant modes of a signal. For a random signal f (t), the spectral density is given by E|ˆ f (k)|2 where ˆ f is the Fourier transform of f . Eg.

David Kelly (UNC) Stochastic Climate December 3, 2013 6 / 41

slide-7
SLIDE 7

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

slide-8
SLIDE 8

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

slide-9
SLIDE 9

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

slide-10
SLIDE 10

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

slide-11
SLIDE 11

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

slide-12
SLIDE 12

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

slide-13
SLIDE 13

Linear stability analysis

Suppose that Ψ : R+ → RN is the interannual climate variables dΨ dt = F(Ψ) given by a spatially discretized version of the model. Let ψ be the anomaly, and suppose that d(Ψ + ψ) dt = F(Ψ + ψ) + f (t) where f is an undefined source of noise.

David Kelly (UNC) Stochastic Climate December 3, 2013 13 / 41

slide-14
SLIDE 14

Linear stability analysis

If we assume that ψ2 ≪ ψ then d(Ψ + ψ) dt = F(Ψ) + ∇F(Ψ)ψ + O(ψ2) + f (t) This gives the linear approximation dψ 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

slide-15
SLIDE 15

Discretization of the model

We work with the time discretization ψn+1 = ψn + Fnψn∆t + f n∆t . where Fn = ∇F(Ψn). We let f n = ∆t−1/2ξn, where ξ is a sequence of identically distributed Gaussian random variables, with Eξn = 0 and EξnξT

m = Dn,mC .

The number Dn,m measures temporal correlation and the matrix C measures spatial correlation.

David Kelly (UNC) Stochastic Climate December 3, 2013 15 / 41

slide-16
SLIDE 16

Writing down the solution

Since the model is linear, the solution is easy to write down. Let Rj,k be the semi-group for the linear part. That is, if un+1 = un + Anun∆t then uk = Rj,kuj. We will solve ψn+1 = ψn + Anψn∆t + ξn∆t1/2 using Duhamel’s principle.

David Kelly (UNC) Stochastic Climate December 3, 2013 16 / 41

slide-17
SLIDE 17

Writing down the solution

We have that ... ψn+1 = ψn + Anψn∆t + ξn∆t1/2 = (1 + An∆t)ψn + ξn∆t1/2 = Rn,n+1ψn + ξn∆t1/2 Repeating this ... ψn+1 = Rn,n+1(Rn−1,nψn−1 + ξn−1∆t1/2) + ξn∆t1/2 = Rn−1,n+1ψn−1 +

  • Rn−1,nξn−1∆t1/2 + Rn,nξn∆t1/2
  • And finally

ψn+1 = R0,n+1ψ0 +

n

  • k=0

Rk,nξk∆t1/2

David Kelly (UNC) Stochastic Climate December 3, 2013 17 / 41

slide-18
SLIDE 18

Measuring the variability of the anomaly

Given the solution, it is easy to write down the mean Eψn = E

  • R0,nψ0 +

n−1

  • k=0

Rk,n−1ξk∆t1/2

  • = R0,nEψ0 .

And the variance E

  • |ψn − Eψn|2
  • is given by

ER0,n(ψ0 − Eψ0), R0,n(ψ0 − Eψ0) +

n−1

  • j=0

n−1

  • k=0

ERk,n−1ξk, Rj,n−1ξj∆t

David Kelly (UNC) Stochastic Climate December 3, 2013 18 / 41

slide-19
SLIDE 19

Measuring the variability of the anomaly

The noise term is the important one. It simplifies to

n−1

  • j=0

n−1

  • k=0

ERk,n−1ξk, Rj,n−1ξj∆t =

n−1

  • j=0

n−1

  • k=0

ERT

j,n−1Rk,n−1ξk, ξj∆t

=

n−1

  • j=0

n−1

  • k=0

tr

  • RT

j,n−1Rk,n−1Dj,kC

  • ∆t = tr(ZC)

where Z =

n−1

  • j=0

n−1

  • k=0

RT

j,n−1Rk,n−1Dj,k∆t

David Kelly (UNC) Stochastic Climate December 3, 2013 19 / 41

slide-20
SLIDE 20

Measuring the variability of the anomaly

Alternatively, we might want to measure the anomaly of a particular feature of the model. Let P : RN → RM for some M ≤ N.

  • Eg. P could be the NINO3 average.

Then E|P(ψn − Eψn)|2 = tr(ZC) where Z =

n−1

  • j=0

n−1

  • k=0

RT

j,n−1PTPRk,n−1Dj,k∆t

  • NB. From now on we always use the NINO3 average for P

David Kelly (UNC) Stochastic Climate December 3, 2013 20 / 41

slide-21
SLIDE 21

Stochastic Optimals

Let {vk, λk} and {wk, µk} be the eigenvectors-eigenvalue pairs for Z and C respectively. Then tr(ZC) =

N

  • i,j=1

λiµj|vi, wj|2 . 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

slide-22
SLIDE 22

Stochastic Optimals: Wind stress

First stochastic optimal (wind stress component).

David Kelly (UNC) Stochastic Climate December 3, 2013 22 / 41

slide-23
SLIDE 23

We can compare the dominant stochastic optimal for wind stress, with the dominant eigenvector of the

  • bserved wind stress.

David Kelly (UNC) Stochastic Climate December 3, 2013 23 / 41

slide-24
SLIDE 24

Observed wind stress

1 - Get a data set of wind-stress observations {W1, . . . , WM}. 2 - Filter out the large time scales to obtain { ˜ W1, . . . , ˜ WM}. 3 - Compute the covariance matrix C = 1 N

M

  • j=1

( ˜ Wj − ¯ W )( ˜ Wj − ¯ W )T 4 - Find the eigenvectors (EOFs) of C.

David Kelly (UNC) Stochastic Climate December 3, 2013 24 / 41

slide-25
SLIDE 25

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

slide-26
SLIDE 26

Stochastic Optimals: Wind stress

First stochastic optimal (wind stress component).

David Kelly (UNC) Stochastic Climate December 3, 2013 26 / 41

slide-27
SLIDE 27

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

slide-28
SLIDE 28

The first stochastic optimal / EOF pair accounts for almost all of the anomaly.

David Kelly (UNC) Stochastic Climate December 3, 2013 28 / 41

slide-29
SLIDE 29

Importance of stochastic optimals

Truncated version of the variance tr(ZC) = N

i,j=1 λiµj|vi, wj|2

David Kelly (UNC) Stochastic Climate December 3, 2013 29 / 41

slide-30
SLIDE 30

What do simulations look like?

David Kelly (UNC) Stochastic Climate December 3, 2013 30 / 41

slide-31
SLIDE 31

Seasonal locking of extreme events

David Kelly (UNC) Stochastic Climate December 3, 2013 31 / 41

slide-32
SLIDE 32

Spectral analysis

David Kelly (UNC) Stochastic Climate December 3, 2013 32 / 41

slide-33
SLIDE 33

Chaotic forcing vs stochastic forcing.

David Kelly (UNC) Stochastic Climate December 3, 2013 33 / 41

slide-34
SLIDE 34

Slow-Fast system

Suppose the weather variables yε satisfy some chaotic dynamics dyε dt = ε−2g(yε) Note that yε(t) = y(ε−2t) where ˙ y = g(y). Suppose the climate variables x satisfy dxε dt = ε−1h(xε, yε) + f (xε, yε) 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

slide-35
SLIDE 35

Slow-Fast system

For mathematical convenience, we assume that h = h(y) and f = f (x), so that dxε dt = ε−1h(yε) + f (xε) Or in the integral form xε(t) = xε(0) + ε−1 t h(yε(s))ds + t f (xε(s))ds

David Kelly (UNC) Stochastic Climate December 3, 2013 35 / 41

slide-36
SLIDE 36

The fast dynamics

When ε ≪ 1, the fast component behaves very randomly - just like the Bernoulli shift. If we assume that the initial condition of yε is distributed randomly, then W ε(t) = ε−1 t h(yε(s))ds becomes a random variable. If we can classify the statistics of W ε in the limit, then perhaps we can do the same for xε.

David Kelly (UNC) Stochastic Climate December 3, 2013 36 / 41

slide-37
SLIDE 37

The fast dynamics

Since yε behaves randomly, the signal W ε is the sum of a sequence of decorrelated random variables. ε−1 t h(yε(s))ds =

⌊t/ε2⌋

  • j=0

ε−1 (j+1)ε2

jε2

h(yε(s))ds =

⌊t/ε2⌋

  • j=0

ε j+1

j

h(y(s))ds Recall that this is how to build Brownian motion. One can actually show that the statistics of W ε converge to the statistics

  • f (a multiple of) Brownian motion B as ε → 0.

David Kelly (UNC) Stochastic Climate December 3, 2013 37 / 41

slide-38
SLIDE 38

A continuous map

In general, the map Z → x defined by the solution to x(t) = x(0) + Z(t) + t f (x(s))ds is continuous in the sup-norm topology. This means that if convergence results for Z translate nicely to convergence results for x.

David Kelly (UNC) Stochastic Climate December 3, 2013 38 / 41

slide-39
SLIDE 39

Convergence to an SDE

So if W converges to B, then the solution of xε(t) = xε(0) + W ε(t) + t f (xε(s))ds converges to x(t) = x(0) + B(t) + t f (x(s))ds Or as an SDE dx = dB + F(x)dt

David Kelly (UNC) Stochastic Climate December 3, 2013 39 / 41

slide-40
SLIDE 40

General systems

The same type of result holds for the general slow-fast system dxε dt = ε−1h(xε, yε) + f (xε, yε) but the argument is a lot more complicated.

David Kelly (UNC) Stochastic Climate December 3, 2013 40 / 41

slide-41
SLIDE 41

The statistical behaviour of a deterministic, chaotic slow-fast system can be approximated by an SDE.

David Kelly (UNC) Stochastic Climate December 3, 2013 41 / 41