integration [see 10.4.2]. That is, short-term functions are available. Consequently, the study of

statistical forcing was not believed to affect the climate variability is reduced to the analysis of

dynamics of systems that respond slowly to such the structure of the forcing functions. The system

forcing. Hasselmann ([165]; see [10.4.3]) was (10.23) can generate many complicated modes of

apparently the ¬rst to recognize the inconsistency variation if it is nonlinear. To understand such

of this concept. He demonstrated that low- a system it is necessary to identify a subspace

frequency variability in systems like the climate of the full phase space that contains the relevant

could simply be the integrated response of a nonlinear dynamics.10

linear (or nonlinear) system forced by short-term

variations, such as those of the macroturbulent

atmospheric ¬‚ow at midlatitudes. The success 10.4.3 Stochastic Climate Models. Neither the

of this proposal is demonstrated in [10.4.3] and search for external forcing functions nor the search

possible generalizations are brie¬‚y mentioned in for nonlinear sub-systems has been convincingly

successful in explaining the observed variability in

[10.4.4].

the climate system. Hasselmann [165] suggested

a third mechanism for generating low-frequency

10.4.2 Statistical Dynamical Models. The variations in the system described by (10.22). This

purpose of Statistical Dynamical Models (SDM) concept, Stochastic Climate Modelling, is now

is to describe the behaviour of a ˜climate variable™ used widely.

yt that varies on time scales „Y and has dynamics Suppose the forcing f in (10.22) is zero and

that are described by a differential equation of the consider the evolution of the system from an initial

form value.

Early on, for 0 ¤ t < „Y , one may assume

dy

= V (y, x) + f. (10.22) that V (yt , xt ) ≈ V (y0 , xt ) so that V acts only in

dt

response to random variable Xt . During this time

Here xt is another climate variable that varies on a period

much shorter time scale „ X . Generally, V is some

dYt

= V (y0 , xt )

nonlinear function of yt and xt , and f represents (10.24)

dt

external forcing.

Now let A„ be an operator that averages a behaves as a stochastic process, say Zt . Since

climate variable over the time scale „ . Because Xt varies on time scales „ X „Y , the derived

— such that

„x „ y , there is a time scale „ 10

This is easier said than done. One possibility is to ¬t

Principal Interaction Patterns (see [15.1.6] and Hasselmann

A„ — (x) ≈ constant [167]) to observed or simulated data. Regardless of the method

d A„ — (y) dy used, the investigator must have a clear understanding of the

≈ . dynamics of the studied process.

dt dt

10: Time Series and Stochastic Processes

212

process Zt also varies on short time scales. After

discretization of (10.24) we ¬nd

Yt+1 = ±Yt + Zt (10.25)

with ± = 1. Equation (10.25) describes a random

walk when Zt is a white noise process [10.2.6].

Thus, the system gains energy and the excursions

grow, even if, in an ensemble sense, the mean

solution is constant.

Later, when t ≥ „Y , the operator V does depend

on Yt . Since the trajectories of the system are

bounded, a negative feedback mechanism must be

invoked. An approximation of the form

V (Yt , Xt ) ≈ ’βYt + Zt (10.26)

Figure 10.14: Result of an extended Ocean

General Circulation Model experiment forced with

is often suitable. This leaves (10.25) unchanged

except that ± = 1 ’ β. Equation (10.25) white noise freshwater ¬‚uxes.

Top: Net freshwater ¬‚ux into the Southern Ocean.

now describes an AR(1) process. The stationarity

condition ± < 1 is obtained for suf¬ciently small Bottom: Mass transport through the Drake

Passage.

time steps.

We now return to (10.22) with f = 0, From Mikolajewicz and Maier-Reimer [276].

except we consider a system that varies around

an equilibrium state. If we assume that the

Mikolajewicz and Maier-Reimer [276] provide

disturbances are small, then the nonlinear operator

a particularly convincing example without explic-

V can be linearized as

itly ¬tting a simple stochastic climate model. They

V (x, y) = vx x + v y y (10.27) ran an Ocean General Circulation Model with up-

per boundary forcing consisting of constant wind

so that we again arrive at (10.25) with Zt = vx Xt . stress, and heat and freshwater ¬‚uxes. Additional

In both of these cases, the full nonlinear system freshwater ¬‚ux anomalies with characteristic time

can be approximated by a stationary AR process „x ∼ 1 were also added (Figure 10.14, top).

as long as there is negative feedback. Section 10.3 These additional anomalies were white in time

shows that such systems possess substantial and almost white in space. The ˜response,™ char-

low-frequency variations that are not related to acterized by the mass transport through the Drake

(deterministic) internal nonlinear dynamics or to Passage, is dominated by low-frequency variations

(also deterministic) external forcing. Instead, the with typical times „ y > 100 years (Figure 10.14,

system is fully random: it is entirely driven by the bottom).11 Subsequent research has shown that

short-term ¬‚uctuating noise Xt . this result is at least partly an artifact of the model

and its boundary conditions. None the less, this ex-

10.4.4 Examples. Frankignoul, in two reviews ample effectively demonstrates that the dynamics

[129, 131], summarizes a number of applications of a physical system can turn short-term stochastic

in which dynamical systems have been modelled forcing into low-frequency climate variability.

explicitly as stochastic climate models. Such Stochastic Climate Models can not be used to

systems include the sea-surface temperature at reproduce a physical system in detail. Neverthe-

midlatitudes, and Arctic and Antarctic sea ice and less, they are instrumental in the understanding

soil moisture. of the dynamics that prevail in complex general

For the midlatitude sea-surface temperature circulation models or observations.

(SST) the variable y is the SST and the variables

x that vary on short time scales are the air“sea 10.4.5 Generalizations. The main purpose of

heat ¬‚ux and the wind stress (Frankignoul and the stochastic climate model is to explain fun-

Hasselmann [133]). The characteristic times are damental dynamics from a zero-order approxi-

„ SST ≈ 6 months „x ≈ 8 days. Similarly, mation. Examples from various aspects of the

for Arctic sea ice extent (Lemke [251]), the low- climate system support the general concept that

frequency variable is the sea-ice extent and the

11 See also Weaver and Hughes [418].

short time scale variable represents weather noise.

10.5: Moving Average Processes 213

10.5.3 In¬nite Moving Averages and Auto-

short-term variations are a signi¬cant source of

Regressions. It is useful, for technical reasons,

low-frequency variability, although, of course, the

to be able to discuss in¬nite moving averages. A

dynamics may be more complicated. The operator

process Xt is said to be an in¬nite moving average

V may have preferred time scales, nonlinearities,

complex feedbacks and resonances, requiring ap- process if

proximations other than (10.26) or (10.27). How-

∞

ever, the principle will still be valid. Also, mul-

Xt = µ X + Zt + βl Zt’l (10.29)

tivariate systems may be considered”we present

l=1

various examples of multivariate systems that are

successfully represented by multivariate AR(1) where

processes when we introduce the Principal Oscil-

1 µ X is the mean of the process,

lation Patterns in Chapter 15.

2 {β j : j = 1, 2, . . .} is a sequence of

coef¬cients such that ∞ |β j | < ∞, and

10.5 Moving Average Processes and

j=1

Regime-dependent AR Processes

3 {Zt : t ∈ Z} is a white noise process.

10.5.1 Overview. This section deals with some

In¬nite auto-regressions are de¬ned similarly. A

topics that, up to now, have been only marginally

process Xt is said to be an in¬nite auto-regressive

relevant to climate research applications. Some

process if

readers might ¬nd it convenient to skip directly to

Chapter 11.

∞

Auto-regressive processes are part of a larger

Xt = ±0 + ±k Xt + Zt (10.30)

class of processes known as auto-regressive k=1