various other nonlinear processes, including

system (including the atmosphere, the ocean, the

many that can be represented by step

land surface, etc.), at a given time in full detail,

functions (such as condensation).

then there would not be room for statistical

uncertainty, nor a need for this book. Indeed, if we 1 We use the expression ˜phase space™ rather casually. It

repeat a run of a General Circulation Model, which is the space spanned by the state variables x of a system

d x = f (x). In the case of the climate system, the state

is supposedly a model of the real climate system, dt

variables consist of the collection of all climatic variables at

on the same computer with exactly the same code,

all geographic locations (latitude, longitude, height/depth). At

operating system, and initial conditions, we obtain

any given time, the state of the climate system is represented by

a second realization of the simulated climate that one point in this space; its development in time is represented

is identical to the ¬rst simulation. by a smooth curve (˜trajectory™).

This concept deviates from the classical mechanical de¬nition

Of course, there is a ˜but.™ We do not know where the phase space is the space of generalized coordinates.

all factors that control the trajectory of climate in Perhaps it would be better to use the term ˜state space.™

1

1: Introduction

2

• The dynamics include linearly unstable create their own unpredictability. These models

processes, such as the baroclinic instability in behave in such a way that a repeated run will

the midlatitude troposphere. diverge quickly from the original run even if only

minimal changes are introduced into the initial

• The dynamics of climate are dissipative. The conditions.

hydrodynamic processes transport energy

from large spatial scales to small spatial

scales, while molecular diffusion takes place 1.1.1 The Paradigms of the Chaotic and

at the smallest spatial scales. Energy is Stochastic Model of Climate. In the paradigm

dissipated through friction with the solid of the chaotic model of the climate, and

earth and by means of gravity wave drag at particularly the atmosphere, a small difference

introduced into the system at some initial time

larger spatial scales.2

causes the system to diverge from the trajectory it

The nonlinearities and the instabilities make would otherwise have travelled. This is the famous

the climate system unpredictable beyond certain Butter¬‚y Effect3 in which in¬nitesimally small

characteristic times. These characteristic time disturbances may provoke large reactions. In terms

scales are different for different subsystems, such of climate, however, there is not just one small

as the ocean, midlatitude troposphere, and tropical disturbance, but myriads of such disturbances at

troposphere. The nonlinear processes in the system all times. In the metaphor of the butter¬‚y: there

amplify minor disturbances, causing them to are millions of butter¬‚ies that ¬‚ap their wings all

evolve irregularly in a way that allows their the time. The paradigm of the stochastic climate

interpretation as ¬nite-amplitude noise. model is that this omnipresent noise causes the

In general, the dissipative character of the system to vary on all time and space scales,

system guarantees its ˜stationarity.™ That is, it does independently of the degree of nonlinearity of the

not ˜run away™ from the region of phase space that climate™s dynamics.

it currently occupies, an effect that can happen in

general nonlinear systems or in linearly unstable

systems. The two factors, noise and damping, 1.2 Some Typical Problems and

Concepts

are the elements required for the interpretation of

climate as a stationary stochastic system (see also

1.2.0 Introduction. The following examples,

Section 10.4).

Under what circumstances should the output which we have subjectively chosen as being

of climate models be considered stochastic? A typical of problems encountered in climate

major difference between the real climate and any research, illustrate the need for statistical analysis

climate model is the size of the phase space. The in atmospheric and climatic research. The order

phase space of a model is much smaller than that of of the examples is somewhat random and it is

the real climate system because the model™s phase certainly not a must to read all of them; the purpose

space is truncated in both space and time. That is, of this ˜potpourri™ is to offer a ¬‚avour of typical

the background noise, due to unknown factors, is questions, answers, and errors.

missing. Therefore a model run can be repeated

with identical results, provided that the computing 1.2.1 The Mean Climate State: Interpretation

environment is unchanged and the same initial and Estimation. From the point of view of

conditions are used. To make the climate model the climatologist, the most fundamental statistical

output realistic we need to make the model parameter is the mean state. This seemingly trivial

unpredictable. Most Ocean General Circulation animal in the statistical zoo has considerable

Models are strongly dissipative and behave almost complexity in the climatological context.

linearly. Explicit noise must therefore be added First, the computed mean is not entirely reliable

to the system as an explicit forcing term to as an estimate of the climate system™s true long-

create statistical variations in the simulated system term mean state. The computed mean will contain

(see, for instance [276] or [418]). In dynamical errors caused by taking observations over a limited

atmospheric models (as opposed to energy-balance observing period, at discrete times and a ¬nite

models) the nonlinearities are strong enough to number of locations. It may also be affected

by the presence of instrumental, recording, and

2 The gravity wave drag maintains an exchange of

momentum between the solid earth and the atmosphere, which

3 Inaudil et al. [194] claimed to have identi¬ed a Lausanne

is transported by means of vertically propagating gravity waves.

See McFarlane et al. [269] for details. butter¬‚y that caused a rainfall in Paris.

1.2: Some Typical Problems and Concepts 3

Figure 1.1: The 300 hPa geopotential height ¬elds in the Northern Hemisphere: the mean 1967“81

January ¬eld, the January 1971 ¬eld, which is closer to the mean ¬eld than most others, and the January

1981 ¬eld, which deviates signi¬cantly from the mean ¬eld. Units: 10 m [117].

transmission errors. In addition, reliability is not Some individual January mean ¬elds (e.g.,

likely to be uniform as a function of location. 1971) are similar to the long-term mean ¬eld.

There are differences in detail, but they share

Reliability may be compromised if the data has

the zonal wavenumber 2 pattern5 of the mean

been ˜analysed™, that is, interpolated to a regular

¬eld. The secondary ridges and troughs have

grid using techniques that make assumptions

different intensities and longitudinal phases. Other

about atmospheric dynamics. The interpolation is

Januaries (e.g., 1981) 300 hPa geopotential height

performed either subjectively by someone who

¬elds are very different from the mean state. They

has experience and knowledge of the shape of

are characterized by a zonal wavenumber 3 pattern

dynamical structures typically observed in the

rather than a zonal wavenumber 2 pattern.

atmosphere, or it is performed objectively using a

The long-term mean masks a great deal of

combination of atmospheric and statistical models.

interannual variability. For example, the minimum

Both kinds of analysis are apt to introduce biases

of the long-term mean ¬eld is larger than the

not present in the ˜raw™ station data, and errors

minima of all but one of the individual January

at one location in analysed data will likely be

states. Also, the spatial variability of each of the

correlated with those at another. (See Daley [98]

individual monthly means is larger than that of the

or Thi´ baux and Pedder [362] for comprehensive

e

long-term mean. Thus, the long-term mean ¬eld is

treatments of objective analysis.)

not a ˜typical™ ¬eld, as it is very unlikely to be

Second, the mean state is not a typical state.

observed as an individual monthly mean. In that

To demonstrate this we consider the January

sense, the long-term mean ¬eld is a rare event.

Northern Hemisphere 300 hPa geopotential height

¬eld4 (Figure 1.1). The mean January height ¬eld, Characterization of the ˜typical™ January re-

quires more than the long-term mean. Speci¬cally,

obtained by averaging monthly mean analyses for

it is necessary to describe the dominant patterns

each January between 1967 and 1981, has contours

of spatial variability about the long-term mean and

of equal height which are primarily circular with

to say something about the range of patterns one

minor irregularities. Two troughs are situated over

is likely to see in a ˜typical™ January. This can be

the eastern coasts of Siberia and North America.

accomplished to a limited extent through the use of

The Siberian trough extends slightly farther south