Munehiko Yamaguchi1

1. Rosenstiel School of Marine and Atmospheric Science, University of Miami

MPO672

ENSO Dynamics, Prediction and Predictability by Prof. Kirtman

14 Dec. 2009

Optimals in ENSO prediction

Why weather and climate prediction are not perfect?

Two factors which limit weather and climate prediction;

1. Numerical model is not perfect

2. Initial condition is not perfect

dx/dt = Px

x(t)=Mxtrue(t0)

P: Perfect model, M: Propagation operator

xtrue(t0) : Initial condition with zero analysis error

If we could have the perfect model and the perfect initial condition, the prediction is perfect forever!!!

Unfortunately we do not have P and xtrue

Unfortunately we do not have P and xtrue because;

1. Our limited understanding of the nature

2. Analysis error resulting from observation error and data assimilation scheme itself.

Create numerical model and initial condition as ”accurately” as possible.

Three approaches for ENSO prediction

1. dx/dt = Lx + N

2. dx/dt = Gx + N

3. dx/dt = FxComplicated

system

1. dx/dt = Lx + NENSO is damped and stochastically forced by weather noise.

The predictability is limited by the stochastic forcing exciting optimally growing modes or initial errors efficiently project onto the optimally growing modes.

Approach to improve prediction

•Force the system by accurate N

•Remove the component of growing perturbations (optimals) from the analysis error.

2. dx/dt = Gx + NENSO is self-sustained due to weak nonlinearity of the coupled ocean-atmosphere coupled system and is periodic, that is, perfectly predictable.

The irregularity of ENSO is due to weather noise (N) and the loss of predictability is primarily due to this stochastic noise.

Approach to improve prediction

•Add weather noise (N) with proper timing, amplitude.

3. dx/dt = FxENSO is intrinsically chaotic due to the nonlinear dynamics of the ocean-atmosphere coupled system.

The loss of predictability is primarily due to the uncertainty in the initial conditions.

Approach to improve prediction

•Increase the complexity of the dynamics and physics in numerical model (F) and also increase the model resolution.

•Use the state of the art data assimilation scheme to create initial condition as accurately as possible.

OptimalsOptimals are the fastest growing perturbations over a finite time interval.

Why stochastic optimals are important

•Optimals are used to represent the uncertainty in the initial conditions and to explain the error growth of weather system.

•Optimals are used to explore the predictability in coupled ocean-atmosphere models like dx/dt = Lx + N.

Various approaches to estimate the optimals

1. Singular vector (SV) method

2. Breeding of growing method (BGM)

3. Linear inverse models

4. …

Singular vector method

dx/dt = Lx; u(t)=Mv(t0)

Growth rate = λ

= (u(t), u(t))/(v(t0), v(t0))

= (Mv(t0), Mv(t0))/(v(t0), v(t0))

= (v(t0), M*Mv(t0))/(v(t0), v(t0))

L: Linearized dynamical operator

M: Tangent-linear operator

By solving the following eigenvalue problem, we can get a perturbation (singular vector), which has the largest growth rate in the dynamical system in a linear sense

Dynamical system

Let’s think of the growth rate of perturbation v(t0)

λx=Ax,

where A= M*M

Breeding of growing mode

Bred vectors are used as an alternative to the optimals (less expensive to compute compared to SVs)

The bred vectors are the finite time nonlinear extension of local Lyapunov vectors.

Breeding cycle

Normal or Non-normal growth

dx/dt = Lx; u(t)=Mv(t0)

Linearized dynamical system

1. Tangent linear operator, M, is most likely non-normal because of the complex weather and climate system. (the fact that M is normal is very strict because M has to satisfy M*M=MM*)

2. M has both normal modes (eigenvectors; the structure of perturbation does not change with time) and non-normal modes (singular vectors; the structure of perturbation changes with time).

3. In the tangent linear system, the normal modes grow forever while the non-normal modes show the transient growth.

4. Therefore, it might be expected that normal modes dominate the growth of perturbations for longer optimization time.

Two papers given by Prof. Kirtman for this project

1. Kleeman and Moore (1997), A theory for the limitation of ENSO predictability due to stochastic atmospheric transients.

• they estimated the optimals using SV method (the perturbation structure changes with time)

2. Cai et al. (2003), Bred vectors of the Zebiak-Cane model and their potential application to ENSO predictions.

• they estimated the optimals using BGM (the perturbation structure does not change with time; the perturbation looks like ENSO mode)

Kleeman and Moore (1997) said…

This is the fastest growing perturbation (causing the maximum variability in SST in NINO3)

Win

d St

ress

Hea

t flu

x

Cai et al. (2003) said…

This is the fast growing perturbation (causing the large variability in ocean energy)

130E 150E 180 150W 120W 90W

Arrow: wind perturbation

Contour: SST perturbation (shaded area with negative values)

The perturbation structure looks like ENSO itself

Pros and cons of the linear theory

Good points

• Easy to handle it

• Easy to analyze it (full set of eigenvectors and singular vectors)

•Applicable to the growth of perturbation, particularly as precursors to events

Not good points

•The system might be oversimplified.

•The system makes a linear growth assumption of perturbation (non-linearity might play a role to explain the variability of ENSO).

•The system assumes a clear separation of time-scales between weather system and climate system, but the assumption might deteriorate the prediction skill.

Example: Importance of non-linearity

Background

ENSO mature stageBred vector

The bred vectors are sensitive to the background state. The above bred vector with a mature ENSO stage does not grow with time, implying the importance of non-linearly growing perturbations.

Background

ENSO warm precursorBred vector

Seamless forecast

Seamless forecast handle both weather and climate system, using a fully coupled state of the art ocean-atmospheric model with high resolution.

The system is fully non-linear and do not assume the clear separation of time-scales between weather and climate system.

However, both ocean and atmosphere models are far from perfect (representation of MJO, for example), so all we need to do is just to keep doing research to understand the dynamics and improve the model!!!

Summary

ENSO dynamics falls into one of the following three categories:

1. Linear system driven by stochastic forcing

2. Self-sustained and periodic

3. Chaotic

Optimals in the linear system explain the variability of ENSO to some extent. The variability would be limited by;

1. Oversimplified dynamical system

2. Linear growth assumption of perturbation

3. Separation of time-scales between weather and climate system

Summary 2

Seamless forecast system is expected to offer some insight to ENSO prediction.

However, there would be many issues to be addressed even in the state of art ocean and atmosphere models and in the study of optimals in the non-linear dynamics.

Thank you for listening