is

x0,„ ≈ 2 Re(z 0,„ p „ ).

N

Then its future state δ time units later is given by

Figure 15.14: ENSO: Annual cycle of the variance

δ

of the cyclo-stationary POP-coef¬cients. (Solid:

r„ +s e’i·/n + noise.

„ +δ

x0,„ +δ ≈ 2 Re z 0,„ p

imaginary component; hatched: real component.)

s=1

This yields a typical evolution in time from x0,„

when the noise is set to zero.

by multiplying the patterns (Figure 15.13, right)

Figure 15.15 shows the typical evolution of

by 0.60 times the standard deviation of the POP

equatorial zonal 10 m wind (left panel) and

coef¬cients.

SST (right panel) when the initial state is given

Note that the variance of the POP coef¬cients by the imaginary part of the cyclo-stationary

has a marked annual cycle (Figure 15.14). The POP in January. The diagram illustrates that,

annual average is about ¬ve. Both components depending upon the sign, a La Ni˜ a or El Ni˜ o

n n

have maximum variability in northern autumn, but typically evolves from the January state depicted

they are not phased identically. Also, note that the in Figure 15.13 (top row).

15: POP Analysis

350

Figure 15.15: ENSO: ˜Typical™ evolution of SST and zonal wind from a prescribed initial state. The

horizontal axes represent the longitudinal position, and the vertical axis represents time over 24 months

(time increasing downwards). The imaginary component of the POP (see Figure 15.13, top row) in

January is the initial state.

15.5 State Space Models continuous) system equation for the k dynamical

variables Z = (Z1 , . . . , Zk )T ,

15.5.1 Overview. We have described POPs as

Zt+1 = F D (Zt , ±, t) + noise (15.31)

eigenmodes of an empirically determined system

matrix. However, POPs can be placed in a much and an observation equation for the observed

variables X = (X1 , . . . , Xm )T ,

more general setting as members of the class of

state space models. We will explain this concept

Xt = P T Zt + noise

in the next subsection, brie¬‚y describe its merits,

(15.32)

k

= j=1 Zt, j p j + noise.

and introduce the Principal Interaction Patterns

(PIPs).

Operator F D represents a class of models that

While the general idea is ubiquitous in climate

may be nonlinear in the dynamical variables Zt

research, speci¬c attempts to explicitly and

and depends on a set of free parameters ± =

objectively determine reduced phase spaces have

(±1 , ±2 , . . .).

been made only recently. So far, these attempts

Matrix P generally has many more columns

have dealt with simpli¬ed systems and have

(m) than rows (k). The system equations (15.31)

mostly addressed the complicated methodical and

therefore describe a dynamical system in a smaller

conceptual aspects of the problem; there is still a

phase space than the space that contains Xt . Ideally

way to go until these techniques will be applied

in applications, a reduced system governed by the

routinely by researchers trying to understand

same dynamics as the full system can be identi¬ed.

the dynamics of the real ocean and the real

The advantage of such low-order systems over

atmosphere. This ¬eld is certainly a frontier

the original high-dimensional system is, at least

of climate research, and we may expect new

in theory, that the low-order system is easier

developments in the future.

to ˜understand.™ Experience, however, suggests

that the system state vector must have very low

State Space Models. A complex dimension if the dynamics are to be analytically

15.5.2

dynamical system with an m-dimensional state tractable.

vector Xt can often be approximated as being

driven by a simpler dynamical system with a state 15.5.3 State Space Models as Conceptual Tools

vector Zt of dimension k < m. Mathematically, and as Numerical Approximations. One appli-

such processes can be approximated by a state cation of the state space models is the conceptu-

space model. These models consist of a discrete (or alization of hypotheses without determining the

15.5: State Space Models 351

unknown parameters ± and P. Indeed, almost all of physical reasoning. The number k might also

be speci¬ed a priori. The parameters ± and the

dynamical reasoning can be expressed as a state

patterns P are ¬tted simultaneously to a time series

space model. For example, the barotropic vorticity

by minimizing the mean square error [P; ±]

equation may be seen as a state space model in

of the approximation of the (discretized) time

which the system state vector evolves in a space

derivative of the observations X by the state space

that excludes a large class of waves. Time series

model:

models, such as the Box“Jenkins ARMA models

described in [10.5.5,6] can also be expressed in

[P; ±] = E Xt+1 ’ Xt ’

state space model form.14

In other applications, attempts are made to actu-

P(F[Zt , ±, t] ’ Zt ) .

2

(15.34)

ally determine the underlying dynamical variables

Zt and the unknown parameters ± for a given class

The patterns P that minimize (15.34) are called

of dynamical operators F. The Principal Interac-

Principal Interaction Patterns (PIPs) [167]. If only

tion Pattern ansatz proposed by Hasselmann [167]

a ¬nite time series of observations X is available,

is probably the most general formalization of this

the expectation E(·) is replaced by a summation

type (see [15.5.4] below).

over time.

The noise term in (15.31) is often disregarded

In general, minimization of (15.34) does not

in nonlinear dynamical analyses. However, dis-

result in a unique solution. In particular, if L

regarding the noise in low-order systems (k <

is any non-singular matrix, and if P minimizes

10) usually changes the dynamics of the system

(15.34), then the set of patterns P = PL will

signi¬cantly since the low-order system is a closed

also minimize (15.34) as long as the corresponding

system without noise. However, components of the

model F = L’1 F belongs to the a priori

climate system, such as the tropical troposphere or

speci¬ed class of models. This problem may be

the thermohaline circulation in the ocean, are never

solved by imposing a constraint. For example, one

closed; they continuously respond to ˜noise™ from

might require that the linear term in the Taylor

other parts of the climate system, hence the noise

expansion of F is a diagonal matrix.

term in (15.31). It is doubtful if the fundamental

Successful applications of the PIP idea to dy-