Charlotte Werndl  

On defining climate and climate change 
 
Article (Accepted version) 
(Refereed) 
 
 

 
Original citation: Werndl, Charlotte (2016) On defining climate and climate change. The British 
Journal for the Philosophy of Science, 67 . pp. 337-364. ISSN 1464-3537 
DOI: 10.1093/bjps/axu048 
 
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On Defining Climate and Climate Change

Abstract

The aim of the paper is to provide a clear and thorough conceptual analysis
of the main candidates for a definition of climate and climate change. Five
desiderata on a definition of climate are presented: it should be empirically
applicable, it should correctly classify different climates, it should not depend
on our knowledge, is should be applicable to the past, present and future and
it should be mathematically well-defined. Then five definitions are discussed:
climate as distribution over time for constant external conditions, climate as
distribution over time when the external conditions vary as in reality, climate as
distribution over time relative to regimes of varying external conditions, climate
as the ensemble distribution for constant external conditions, and climate as the
ensemble distribution when the external conditions vary as in reality. The third
definition is novel and is introduced as a response to problems with existing
definitions. The conclusion is that most definitions encounter serious problems
and that the third definition is most promising.

1



Contents

1 Introduction 3

2 Climate Variables and a Simple Climate Model 4

3 Desiderata on a Definition of Climate 5

4 Climate as Distribution Over Time 7
4.1 Definition 1. Distribution Over Time for Constant External Conditions 7
4.2 Definition 2. Distribution Over Time when the External Conditions

Vary as in Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 Definition 3. Distribution Over Time for Regimes of Varying External

Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.4 Infinite Versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Climate as Ensemble Distribution 14
5.1 Definition 4. Ensemble Distribution for Constant External Conditions 14
5.2 Definition 5. Ensemble Distribution when the External Conditions

Vary as in Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.3 Infinite Versions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Conclusion 19

2



1 Introduction

How to define climate and climate change is nontrivial and contentious, as also ex-
pressed by Todorov ([1986], p. 259):

The question of climatic change is perhaps the most complex and con-
troversial in the entire science of meteorology. No strict criteria exist on
how many dry years should occur to justify the use of the words “climatic
change”. There is no unanimous opinion and agreement among clima-
tologists on the definition of the term climate, let alone climatic change,
climatic trend or fluctuation.

In both public and scientific discourse the notions of climate and climate change
are often loosely employed, and it remains unclear what exactly is understood by
them.1 This unclarity is problematic because it may lead to considerable confusion
regarding the existence and extent of global warming, for example. How to define
climate and climate change is conceptually interesting, but choosing good definitions
is also important for being able to make true statements about our climate system.
As we will see, adopting definitions with serious problems may imply that the climate
has nothing to do with the actual properties of the climate system, that different cli-
mates are not correctly classified or that there is no relation to observational records
such as past mean surface temperature values. All this implies that the issue of defin-
ing climate and climate change is also of crucial importance to climate policy (House
of Commons Science and Technology Committee [2014]).

Of course, different definitions of climate and climate change are discussed in
the climate science literature. However, what is missing is a clear and thorough
conceptual analysis of the different definitions and their benefits and problems.2 This
paper aims to contribute to filling this gap. After introducing climate variables and a
simple climate model (Section 2), five main desiderata on a definition of climate will
be presented (Section 3). Then five main candidates for a definition of climate (and
the derivative definitions of climate change) will be discussed. By referring to the
five desiderata, their benefits and problems will be analysed (Sections 4-5). Problems
with existing definitions lead me to propose a novel definition of climate (Definition
3) which has not been discussed in the literature before (the other four definitions are
among the most commonly endorsed definitions of climate). Finally, the conclusion
will summarise the discussion (Section 6).

1An example for this is Stott and Kettleborough ([2002]).
2The most detailed discussion so far seems to be Lorenz ([unpublished]). Still, a more thorough

analysis is needed. For instance, Lorenz’s discussion does not take into account recent developments
in defining climate and in using time-dependent dynamical systems theory (where the external
conditions are allowed to vary) to define climate.

3



2 Climate Variables and a Simple Climate Model

When talking about the climate of a certain region one is interested in the distribu-
tion of certain variables (called the climate variables) of that region.3 These include
the dynamic meteorological variables, i.e., the variables that describe the state of the
atmosphere, such as the surface air temperature or the surface pressure. In the liter-
ature one often finds the statement that climate is the expected weather (e.g. Allen
[2003]; Lorenz [unpublished]), which suggests that the only variables of interest are
the dynamic meteorological variables. However, this is not clear. When scientists
talk about climate they often refer not only to dynamic meteorological variables but
to a more extended set of variables. These usually include the variables describing
the state of the ocean (such as the sub-surface ocean temperature) and sometimes
also other variables such as those describing glaciers and ice sheets. In general, when
talking about climate, one is definitely interested in the dynamic meteorological vari-
ables, and there are some other variables one is definitely not interested in, such as
those which describe the flora and fauna on Earth in all its details. Apart from this,
there is a middle ground of other variables such as those describing glaciers and ice
sheets, which one might only include in the list of climate variables in certain con-
texts. It seems best to me to accept this middle ground. Depending on the purpose
at hand, it is better to include more or fewer variables—as climate scientists do in
practice.4

FIGURE 1 SHOULD BE HERE

When illustrating the different definitions of climate, in order for the discussion
to be as accessible as possible, it will be assumed that the following simple climate
model is the true model of the evolution of the climate variables : The only climate
variable is the temperature with possible values in [0, 30]. It evolves according to the
deterministic evolution equation

xt+1 = f(xt) =

{
atxt for 0 ≤ xt ≤ 15;
at(30 −xt) for 15 < xt ≤ 30.

(1)

where xt denotes the temperature at day t. For this assumed true simple climate

3Here ‘region’ is broadly understood. For instance, the region could be London (site-specific
climate) or the Earth (spatially aggregated global climate). These spatial aspects of defining climate
present further conceptual challenges, which are beyond the scope of this paper.

4The climate of a region is generated by the climate system, which is an interactive system
consisting of the atmosphere, the hydrosphere, the cryosphere, the land surface and the biosphere,
forced or influenced by various external forcing mechanisms such as solar radiation (Solomon et al.
[2007]). Note that many variables figuring in the climate system are not climate variables. For
instance, the climate system includes a detailed description of the flora and fauna on Earth, but
these details are not of concern when talking about the climate.

4



model the external conditions (i.e. the phenomena not described by the climate vari-
ables) consist just of at. at represents the solar energy reaching the Earth at day t
and is assumed to be periodically fluctuating between the values 1 and 2 (and at = 1
on 1 July 1983). Figure 1 shows the evolution equation when at = 1 (left) and at = 2
(right). Realistic and state-of-the-art climate models are of course much more com-
plex deterministic models (Parker [2006]; see Appendix A for a general mathematical
definition of deterministic models).5 They usually arise from ordinary or partial dif-
ferential equations, implying that the time parameter t varies continuously (and not
in discrete steps as for the simple model). For the purposes of this paper these dif-
ferences are irrelevant as all the points made for the simple model carry over to the
more complex continuous-time climate models.6

3 Desiderata on a Definition of Climate

For many policy decisions and scientific questions what matters are not the specific
values of the climate variables at a certain point of time but the distribution over
the climate variables arising for a certain configuration of the climate system. For
instance, when policy advisors ask whether events of extreme heat under our current
greenhouse gas emissions will be roughly the same as under the greenhouse gas emis-
sions projected towards the end of the 21th century, they have in mind a comparison
of two temperature distributions. They are not concerned with the temperature at
certain days (such as whether there will be extreme heat on 1 July 2099).

Furthermore, climate scientists tend to think that distributions over climate vari-
ables are easier to predict than the actual path taken by the climate variables. Thus
both in scientific and policy discourse the notion of the distribution over the climate
variables arising for a certain configuration of the climate system is needed, and at an
intuitive level the climate simply amounts to this notion. The following five desiderata
present fairly weak requirements on a more rigorous definition of this notion (these
desiderata will play a crucial role in the analysis of the five definitions of climate in
Sections 4 and 5).

Desideratum 1. A definition of climate should be empirically applicable. In par-
ticular, (i) it should be possible to estimate the past and present climate from the

5For a discussion of how climate models are confirmed, see Steele and Werndl ([2013]).
6It should be noted that, for several reasons (e.g., incomplete knowledge or the inability to

resolve processes numerically at sub-grid level), stochastic models are sometimes used to describe
the evolution of climate variables (e.g. Hasselmann [1976]; Checkroun et al. [2011]). Deterministic
models are the norm and thus what follows focuses on deterministic models. However, all the
definitions of climate discussed in this paper have a stochastic counterpart.

5



time series of observations (if good records are available), and (ii) the future climate
should be about the future values of the climate variables. A definition that neither
fulfills (i) nor (ii) is called empirically void.

Desideratum 2. A definition of climate should correctly classify different climates
for time periods which are uncontroversially regarded as belonging to different cli-
mates. For instance, the climate which prevailed in the middle of the last ice age in
London is regarded as different from the climate in London in the past few years.
Therefore, a definition of climate should classify that these time periods belong to
different climates.

Desideratum 3. The climate should not depend on our knowledge. When, e.g., pol-
icy advisors ask what the past climate was and whether there will be climate change,
they do not refer to notions which would differ if we had better or worse knowledge.
How well we can predict the climate and climate change, of course, depends on our
knowledge, but what the climate is and whether there is climate change is indepen-
dent of our knowledge.

Desideratum 4. A definition of climate should be applicable to the past, present
and future. Scientists think that there were several climates in the past, that there
is a present climate and that there will be several climates in the future. Thus a
definition of climate should be applicable to all these cases. If, e.g., a definition were
only about the future climate, it would be regarded as incomplete and a definition
that can also be applied to the past and present would be demanded.

Desideratum 5. A definition of climate should be mathematically well-defined. In
particular, it cannot refer to a limit which does not exist.

As we will see, it will turn out that only one definition discussed in this paper
fulfills all the desiderata and hence is a promising definition of climate. Yet this
should not be taken to imply that, in principle, there can only be one definition of
climate. Different notions of the distribution over the climate variables arising for a
certain configuration of the climate system (i.e. what climate intuitively amounts to)
might be important for different purposes. Thus it is not excluded that there could
be several definitions of climate.

Let us now turn to the five definitions of climate (and the derivative definitions
of climate change).

6



4 Climate as Distribution Over Time

4.1 Definition 1. Distribution Over Time for Constant Ex-
ternal Conditions

In reality the external conditions (such as the amount of solar energy at for the as-
sumed true simple evolution of the temperature) are not constant. Suppose that the
external conditions take the form of small fluctuations around a mean value c over
a certain time period. Then, according to the first definition, the climate over this
time period is defined by the finite distribution over time which arises for the true
climate model under constant external conditions c. Different distributions over time
correspond to different climates. There is climate change when there are different
distributions for two successive time periods. This can be because of different exter-
nal conditions (external climate change) or because of different initial values under
constant external conditions (internal climate change). This definition is widely en-
dorsed (e.g. Dymnikov and Gritsoun [2001]; Lorenz [unpublished]).

FIGURE 2 SHOULD BE HERE

Let me illustrate this definition with the assumed true simple evolution of the
temperature (cf. Section 2). Here the amount of solar energy reaching the Earth
fluctuated around the mean value c = 1.5 for the thirty years period from 1 July
1983 to 30 June 2013. Suppose that the initial temperature on 1 July 1983 was
18.85 ◦C. Then the climate over this period is the distribution over time which arises
when the model (1) with at = c = 1.5 (for all t) and with initial value 18.85 is evolved
over thirty years. That is, the measure assigned to a subset A of [0, 30] is:7

The number of days under constant external conditions c with a temperature in A

The total number of modelled days (=10950 (30∗365))
.8

(2)
The distribution arising in this way is shown in Figure 2. For instance, the measure
assigned to [15, 20] (the proportion of temperature values between 15 ◦C and 20 ◦C)
is 0.517.

7The question arises how long the time period should be over which the climate distribution is
defined (cf. Subsection 4.3). Suppose that the relevant time period is thirty years. Then even if the
external conditions fluctuate around c for less than thirty years, say three years, then climate over
this time period is still defined as the distribution under constant external conditions over thirty
years. This makes intuitive sense: if, e.g., there are high temperatures over three years because of
the El Niño, then this is just a warm period under a certain climate.

8This paper paper ignores lap years.

7



There is a fundamental distinction between climate as a distribution or average of
actual properties of the climate system versus climate as a model-immanent notion
(which does not refer to distributions or averages of the actual properties of the
climate system). Because the external conditions are assumed to be constant but vary
in reality, Definition 1 is a model-immanent notion and not a distribution of the actual
properties of the climate system. This would not represent a problem if this definition
were empirically applicable in the following sense: When the external conditions are
small fluctuations around a mean value c, then the distributions over time of the
actual climate system are approximately equal to the distributions over time under
constant external conditions c. However, there are doubts about this. Let me illustrate
this with the assumed true simple climate model. The actual distribution (where the
solar energy at fluctuated between 1 and 2 with at = 1 on 1 July 1983) is given by
the actual evolution of the temperature over the thirty years period from 1 July 1983
to 30 June 2013 (with initial temperature 18.85). That is, the measure assigned to a
subset A of [0, 30] is:

The number of days of the actual climate system with a temperature in A

The total number of days (=10950 (30∗365))
. (3)

Figure 3 shows the distribution arising in this way. It is clear that the temperature
distribution over time for constant external conditions c = 1.5 (Figure 2) and the
actual temperature distribution over time (Figure 3) are completely different. For
instance, the measure assigned to [15, 20] (the proportion of temperature values be-
tween 15 ◦C and 20 ◦C) is 0.517 for the former distribution but 0.082 for the latter
distribution.

FIGURE 3 SHOULD BE HERE

Similar results can be found for several climate models. For instance, Daron
([2012]) numerically investigated the Lorenz equations where one parameter is subject
to fluctuations around a mean value. He found that the finite distributions over time
can differ significantly from the distributions arising when the parameters are held
fixed. A resonance effect, which can also arise for small fluctuations, is responsible
for these different distributions. Also, there is a body of work indicating that the
seasonal cycle of the sun leads to different distributions over time. It was found in
Goswami et al. ([2006]) for a model of the monsoon, Jin et al. ([1994]) for a model
of the El Niño, Lorenz ([1990]) for a simple general circulation model and Kurgansky
et al. ([1996]) for a baroclinic low-order model of the atmosphere that the finite
distributions over time are different when the seasonal cycle of the sun is included.
It could certainly be that similar results hold for the true climate model. Thus the
first definition of climate may be empirically void (thereby violating Desideratum
1). Note also that there is the problem that when the external conditions are not

8



small fluctuations around a mean value c but vary considerably, then this definition
is not applicable. All this shows a need to take the varying external conditions of the
climate system into account. Thus the question arises whether a similar definition
can be found where the external conditions are allowed to vary.

4.2 Definition 2. Distribution Over Time when the External
Conditions Vary as in Reality

Most directly this can be achieved by defining the climate over a certain time period
as the finite distribution over time for the actual evolution of the climate variables
(i.e. when the external conditions vary as in reality). Again, different climates cor-
respond to different distributions. There is climate change when there are different
distributions for succeeding time periods (and there can be external climate change
as well as internal climate change due to different initial values for the same external
conditions). To come back to the assumed true simple evolution of the temperature
(cf. Section 2): here the climate for the time period from 1 July 1983 to 30 June 2013
is the temperature distribution (3) as shown in Figure 3.

FIGURE 4 SHOULD BE HERE

This definition simply refers to a distribution of actual properties of the climate
system (so, unlike Definition 1, it is not a model-immanent notion). Because of this,
arguably, there could be no definition of climate that is easier to estimate from the
observations. For this reason, it is very popular and implicitly employed in many
climate studies (cf. Lorenz [unpublished]). The World Meteorological Organisation
publishes standard climate normals that are statistics taken over a period of thirty
years (most recently from 1961 to 1990). Thus the climate is commonly identified
with the actual distribution of the atmosphere over a period of thirty years (cf. Hulme
et al. [2009]).

However, there is a serious problem with this definition, which can best be illus-
trated with a simple hypothetical scenario. Suppose that the time period from 1 July
1000 to 30 June 1030 was marked by two different regimes because on 1 July 1015
the Earth was hit by a meteor and, as a consequence, became a colder place with
less temperature variation. More specifically, suppose again that the only climate
variable is the temperature and that before the hit of the meteor the evolution of the
temperature was given by equation (1) where at fluctuates periodically between 1.8
and 2 (with at = 1.8 and initial temperature 23.249 on 1 July 1000). After the hit of
the meteor the evolution was given by equation (1) where at fluctuates periodically
between 1 and 1.2 (with at = 1 on 1 July 1015).

9



FIGURE 5 SHOULD BE HERE

Suppose that for the second definition one decides to consider thirty years time
periods, and one focuses on the time period from 1 July 1000 to 30 June 1030. Then
the climate is the actual temperature distribution over the period from 1 July 1000 to
30 June 1030 as shown in Figure 4. However, Figure 5 reveals that this distribution is
composed of two distributions: the actual temperature distribution over time before
the hit of the meteor (from 1 July 1000 to 30 June 1015) and after the hit of the
meteor (from 1 July 1015 to 30 June 1030). These two distributions are very different,
e.g., the measure assigned to [25, 30] (the proportion of temperature values between
25 ◦C and 30 ◦C) is 0.142 for the former but zero for latter distribution. Because
a change in external conditions is responsible for these different distributions, one
would like to say that there is no single climate for the period from 1 July 1000 to
30 June 1030. Instead, one would like to say that there are two climates during this
period, one before the hit of the meteor, and one after the hit of the meteor. Yet, the
second definition does not imply this because it allows us to talk about the climate
over the time period from 1 July 1000 to 30 June 1030. More generally, when the
external conditions change drastically over a time period in a way that gives rise to
different distributions, one would like to say that there are different climates. Yet the
second definition does not imply this because it allows that the external conditions
change drastically for the time periods over which the climate distribution is defined.
Hence the second definition suffers from the shortcoming that it does not correctly
classify different climates for time periods which are uncontroversially regarded as
belonging to different climates (thereby violating Desideratum 2).

4.3 Definition 3. Distribution Over Time for Regimes of
Varying External Conditions

To avoid this problem, a promising possibility seems to be a slight modification of
the second definition. Namely, suppose that the actual external conditions over a
time period are subject to a certain regime of varying external conditions. Then
the climate over this time period is defined as the finite distribution over time which
arises under the regime of varying external conditions. Obviously, for this definition
to be adequate, more must be said about what counts as a regime of varying external
conditions, since of course, the period before and after the meteor hits is a regime
of varying external conditions. For instance, a reasonable requirement is that the
mean of the external conditions should at least be approximately constant. To my
knowledge, this definition is novel and has not been explicitly endorsed in the climate
literature. Again, different climates correspond to different distributions. There is

10



climate change when there are different climates for two successive time periods, and
there can be external climate change as well as internal climate change (due to dif-
ferent initial values).

Let me illustrate this definition with the assumed true simple evolution of the
temperature (cf. Section 2). A regime of varying external conditions is when the
solar energy fluctuates periodically between 1 and 2 (starting with 1), and the solar
energy was subject to this regime from 1 July 1983 to 30 June 2013. Thus the climate
over this period is the distribution over time under this regime of varying external
conditions (with initial temperature 18.85).9 That is, a subset A of [0, 30] is assigned
the measure:

The number of days under the external conditions regime with a temperature in A

The total number of modelled days (=10950 (30∗365))
.

(4)
Since the regime of varying external conditions coincides with the actual path taken
by the external conditions from 1 July 1983 to 30 June 2013, the third definition of
climate is simply given by the distribution shown in Figure 3.

This definition avoids the shortcoming of the second definition of not being able
to correctly classify different climates. For instance, the external conditions before
and after the hit of the meteor correspond to two different regimes of varying exter-
nal conditions (namely, the solar energy reaching the Earth fluctuating periodically
between 1.8 and 2 starting with 1.8 (first regime), and the solar energy fluctuating
periodically between 1 and 1.2, starting with 1 (second regime)). Thus, as desired,
the climate before and after the hit of the meteor differ, i.e. the two distributions
shown in Figure 5 correspond to different climates.

This definition of climate is a model-immanent notion and not a distribution of
actual properties of the climate system. This does not represent a problem because
the definition is empirically applicable: when the actual climate system is subject to
a certain regime of varying external conditions over a long-enough time period, then
the climate of this time period coincides with the distribution over time of the actual
evolution of the climate variables. Hence this definition is attractive because there is
an immediate link to the observations.

9The question arises how long the time period should be over which the climate distribution is
defined (cf. the comments at the end of this subsection). Suppose that the relevant time period is
thirty years. Then even if the external conditions are subject to a certain regime of varying external
conditions for less than thirty years, say three years, then climate over this time period is still
defined as the distribution under the regime over thirty years (e.g., if there are high temperatures
over three years because of the El Niño, then this is just a warm period under a certain climate).

11



Since this definition is promising, let me comment on a few issues (which also
arise for Definition 1 and 2). First, there is the question of over which finite time
period the distributions should be taken. It seems promising to adopt a pragmatic
approach as outlined by Lorenz ([unpublished]): The purpose of research will influ-
ence the choice of the time interval, e.g., if one is interested in inter-glacial climate the
time period will be relatively long. Also, the time period should be long enough so
that no specific predictions can be made (e.g., longer than the predictability horizon
given by the El Niño) but short enough to ensure that changes which are conceived
as climatic are subsumed under different climates. What is important is to provide
some motivation for the chosen time period (often thirty years is chosen without any
clear motivation, which is problematic). Second, it should be mentioned that there
will nearly always be climate change. Yet this does not constitute a problem: one is
usually only interested in major climate change where the distributions differ signif-
icantly (and not just slightly).

To sum up the discussion of climate as a finite distribution over time: Because of
the assumption of constant external conditions, Definition 1 may be empirically void
(thereby violating Desideratum 1). Definition 2 does not correctly classify different
climates and thus fails to meet Desideratum 2. In contrast, Definition 3 is promising
because it meets all the desiderata.

4.4 Infinite Versions

There are also infinite versions of the three definitions of climate as distribution over
time discussed above. According to these infinite versions, the climate is defined by
the infinite distribution over time when the time period in equation (2) (for Definition
1), equation (3) (for Definition 2) or equation (4) (for Definition 3) goes to infinity.
According to the infinite version of Definition 2, the climate is the actual distribution
of the climate system as time goes to infinity, which has the consequence that there
cannot be climate change. Because of this, it is hardly ever adopted. The infinite
versions of Definition 1 and Definition 3, however, are widely endorsed (Dymnikov
and Gritsoun [2001]; Lorenz [unpublished]; Palmer [1999]).

The main advantage of these infinite distributions is that they are mathematically
more tractable, implying that the tools of dynamical systems theory can be used to
analyse them. For instance, recall that climate as distribution over time is defined
relative to the initial values of the climate variables. For the infinite version of Defi-
nition 1 results of dynamical systems theory show that under certain conditions the
climate is the same for almost all initial values (no corresponding results are known
for Definition 2 and Definition 3). Namely, this is the case when the dynamics is
ergodic or when there is a physical measure (see Appendix B for a mathematical

12



statement of these results). These results are of limited relevance, however, because
it is unclear whether the true climate model under constant external conditions is
ergodic or has a physical measure. What is more, as I will now argue, the infinite
versions of Definitions 1, 2 and 3 suffer from serious problems (in addition to the
problems already outlined in the Subsections 4.1-4.3). Therefore, for defining the cli-
mate, finite distributions over time are preferable to infinite distributions over time
(and this is the reason why the presentation above focused on finite distributions).

The first problem with these infinite versions is that the relevant infinite limits
may not exist. More specifically, while infinite limits of equation (2) (Definition 1)
usually exist (Petersen [1983]), the infinite limits of equation (3) (Definition 2) and
equation (4) (Definition 3) often do not exist (Mancho et al. [forthcoming]). To the
best of my knowledge, it is an open question whether they exist for the true climate
model. Hence there is the problem that the climate may not be mathematically well-
defined (thereby violating Desideratum 5).10

Second, to make sure that the infinite distributions relate to the actual finite dis-
tributions of the climate system, one has to assume that the distributions over finite
time periods of interest are approximated by the infinite distributions. However,
there are doubts about this. For many climate models the convergence to the infinite
distributions is very slow. In particular, Lorenz ([1968]) introduced the notion of al-
most intransitivity to describe systems where distributions taken over very long but
finite time intervals differ from one interval to each other and thus from the infinite
distribution. Lorenz ([1968], [1970], [1976], [unpublished]) gave examples of simple
models that show almost intransitive behaviour. He concluded that the true climate
model may well be almost intransitive, particularly when the climate variables include
the ocean variables. Furthermore, there are several climate models that incorporate
some realism about the evolution of the climate variables where the distributions
taken over long finite time intervals differ considerably from the infinite distribu-
tions. For instance, Bhattacharya et al. ([1982]) for an energy-balance model, Daron
([2012]) and Daron and Stainforth ([2013]) for a coupled ocean-atmosphere model
and Sempf et al. ([2007]) for a model of the wintertime atmospheric circulation over
the northern hemisphere found considerable differences between distributions taken

10Furthermore, when the external conditions vary, i.e., for equation (3) (Definition 2) and equation
(4) (Definition 3), there are two infinite distributions, which usually differ. More specifically, when
the finite distributions range from time point t0 to time point t1, one infinite distribution arises
for t0 → −∞ and the other for t1 → ∞ (Kloeden and Rasmussen [2011]). Consequently, there is
the problem which of the infinite distributions should be identified with the climate. For the past
one can check which infinite distribution is approximated by the observed distributions. However,
if the concern is the future, it is unclear how to choose between the two distributions. If the wrong
distribution is chosen, the climate will be empirically void (thereby violating Desideratum 1).

13



over long finite intervals and distributions in the infinite limit. It is an open question
whether similar results hold for the true climate model, but there remains the worry
that the infinite distributions may not relate to the actual distributions of the climate
system and thus are empirically void (thereby violating Desideratum 1).

To sum up: climate as an infinite distribution over time is easier to analyse
mathematically. However, there are the additional problems that the relevant limits
may not exist (thereby violating Desideratum 5) and that the definitions may be
empirically void (thereby violating Desideratum 1).

5 Climate as Ensemble Distribution

5.1 Definition 4. Ensemble Distribution for Constant Exter-
nal Conditions

The first three definitions of climate are distributions over time. In stark contrast to
this, the remaining two definitions are ensemble distributions of the possible values of
the climate variables. Suppose that the concern is to make predictions at time t1 in
the future and that from the present t0 until t1 the external conditions take the form
of small fluctuations around a mean value c. Then, according to the fourth definition,
the climate at time t1 is the distribution of the possible values of the climate variables
at t1 under constant external conditions c conditional on our uncertainty in the initial
values at t0. As before, different distributions correspond to different climates. This
is a common definition (e.g., Lorenz [unpublished]; Stone and Knutti [2010]).

FIGURE 6 SHOULD BE HERE

Let me illustrate this definition with the assumed true simple evolution of the
temperature (cf. Section 2). Suppose that the temperature is measured to be between
29.77 ◦C and 30.00 ◦C degrees today, i.e. at t0 = 1 July 2013. Let pt0 be the uniform
probability density over [29.77, 30.00] representing this uncertainty in the initial
temperature. Further, suppose that the concern is to predict the temperature at
time t1 = 1 July 2040. The amount of solar energy reaching the Earth will fluctuate
around the mean value c = 1.5 from 1 July 2013 to 1 July 2040. Hence the climate
on 1 July 2040 is given by the distribution where the probability of a subset A of
[0, 30] is

Pct0,t1 (A), (5)

where Pct0,t1 (A) is the probability assigned to A by the density that arises when pt0
is evolved forward to 30 June 2040 under equation (1) with at = c = 1.5 (for all

14



t). Figure 6 shows the distribution arising in this way. For instance, the probability
assigned to [20, 25] (that the temperature will be between 20 ◦C and 25 ◦C) is 0.241.

Clearly, this definition is a model-immanent notion since it refers to an ensem-
ble of initial conditions that is evolved forward in time but actual initial conditions
are unique. Proponents of this definition think that this model-immanent notion is
predictively useful in the following sense: What one is interested in is the actual
ensemble distribution at time t1, i.e. the distribution over the possible values of the
climate variables given the uncertainty in the initial values and the actual path of
the external conditions taken by the climate system. So the assumption is made
that when the external conditions take the form of small fluctuations around a mean
value c from t0 to t1, then the ensemble distribution at time t1 under constant external
conditions c is approximately the same as the actual ensemble distribution at time t1.

However, there are doubts about this assumption. Let me illustrate this with the
assumed true simple climate model. The actual temperature ensemble distribution is
shown in Figure 7. It is the distribution where the probability of a subset A of [0, 30]
is

Pt0,t1 (A), (6)

where Pt0,t1 (A) is the probability assigned to A by the density that arises when pt0
is evolved forward to 1 July 2040 under equation (1) given the periodic fluctuations
of the solar energy at between 1 and 2 (with at = 1 on 1 July 2013). It is clear that
the ensemble distribution for constant external conditions c = 1.5 (Figure 6) and the
actual ensemble distribution (Figure 7) are very different. For instance, the proba-
bility assigned to [20, 25] (that the temperature will be between 20 ◦C and 25 ◦C) is
0.241 for the former but 0 for the latter distribution.11

FIGURE 7 SHOULD BE HERE

Similar results hold for several climate models. For instance, Daron ([2012])
numerically investigated the Lorenz equations where one parameter is subject to
fluctuations around a mean value. He found that the ensemble distributions can differ
significantly from the ensemble distributions when the parameters are held fixed. The
mechanism responsible for the different distributions is a resonance effect, which can
also arise for small fluctuations. Also, there is growing evidence that the inclusion

11Note that the actual ensemble distribution on 30 June 2040 is given by the uniform distribution
over [0, 30]. Thus the ensemble distribution under constant external conditions c = 1.5 (Figure 6)
is not the average of the ensemble distribution on 1 July 2040 (where at = 2) and the ensemble
distribution on 30 June 2040 (where at = 1). That is, it is not the average of the ensemble
distributions for the periodically fluctuating external conditions.

15



of the seasonal cycle of the sun may lead to different ensemble distributions: it was
found in Gowsami et al. ([2006]) for model of the monsoon, in Jin et al. ([1994]) for
a model of the El Niño and in Lorenz ([1990]) for a very simple general circulation
model that the ensemble distributions differ when the seasonal cycle of the sun is
included. It is certainly possible that similar results hold for the true climate model.
Hence there is the problem that this definition of climate may be empirically void
(thereby violating Desideratum 1). There is also the additional problem that when
the external conditions are not small fluctuations around a mean value, then the
definition is not applicable. All this shows the need to take the varying external
conditions into account.

5.2 Definition 5. Ensemble Distribution when the External
Conditions Vary as in Reality

Most directly this can be achieved by defining the climate as the actual ensemble
distribution. More specifically, suppose again that the concern is to make predictions
at time t1 in the future. According to the fifth definition, the climate at time t1 is
the distribution of the possible values of the climate variables at t1 given the actual
path taken by the external conditions and conditional on our uncertainty in the ini-
tial values at t0. As before, different distributions correspond to different climates.
This definition is widely endorsed (Daron [2012]; Daron and Stainforth [2013]; Smith
[2002]). To illustrate this definition with the assumed true simple evolution of the
temperature (cf. Section 2): The climate on 1 July 2040 conditional on our knowl-
edge that the temperature was between 29.77 ◦C and 30.00 ◦C on 1 July 2013 is the
ensemble distribution (6) as shown in Figure 7.

Definition 5 is again a model-immanent notion since it refers to an ensemble of
initial conditions that is evolved forward in time but actual initial conditions are
unique. This model-immanent notion is predictively very useful because it quantifies
the likelihood of the future possible properties of the actual climate system given our
present uncertainty in the initial values.

However, there are several problems (which also arise for Definition 4). First of
all, climate is defined relative to our present uncertainty about the initial values. This
implies that the climate and the derivative notion of climate change are dependent
on our knowledge and that Desideratum 3 is not met.

Second, this definition is always presented as defining the future climate. So the
question arises what the past and present climate amount to (which are also needed
to define climate change). While I have not found anything in print about this, it

16



seems natural to say that the climate on 1 July 2013 is the distribution that scien-
tists on, say, 1 July 1983 would have predicted as the climate of 1 July 2013 based on
their uncertainty about the initial values on 1 July 1983 and the actual path taken
by the external conditions.12 Climate change is then defined as the change between
the present and the future climate. Next to external climate change there is also
internal climate change (because of different uncertainties in the initial values or dif-
ferent prediction lead-times). In the example the climate of 1 July 2013 is defined
by choosing the reference point thirty years in the past (i.e. a prediction lead-time
of thirty years), but this choice seems arbitrary. One could argue that pragmatic
considerations as discussed in Subsection 4.3 will fix a suitable prediction lead-time,
say 30 years. Yet, then the same prediction lead-time should be used when defining
the future climate. However, this is undesirable because for the future climate the
prediction lead-time should only be determined by how many years scientists want
to predict in the future (and should not be constrained to, say, 30 years). So it is
difficult to see how the present and past climate could be defined. Hence Definition
4 is not applicable to the past, present and future and Desideratum 4 is violated.

Third, there is the problem that climate, thus defined, does not have anything
to do with distributions over time of actual properties of the climate system (cf.
Schneider and Dickinson [2000]). This leaves us in the awkward situation that the
observational records of past temperatures etc. do not tell us anything about climate
and climate change. Therefore, Desideratum 1 is not met.

To sum up the discussion of climate as future ensemble distribution: Because of
the assumption of constant external conditions, Definition 4 may be empirically void
(thereby violating Desideratum 1). While Definition 5 avoids this shortcoming, there
are several other problems (which also arise for Definition 4). Namely, the climate
depends on our knowledge about the initial values (thereby violating Desideratum
3). Also, it is unclear how the past and present climate could be defined, and thus
Desideratum 4 is not met. Finally, there is no relation to the past observations of
the climate system, and thus Desideratum 1 is violated. Given all these problems, it
seems better to say that Definition 5 is not about the climate but just refers to the
expected future distribution of the climate variables (a distribution which is useful
to consider for predictive purposes).13

12This has also been suggested by the climate scientist David Stainforth (personal communica-
tion).

13One could also introduce an ensemble definition for regimes of varying external conditions (cor-
responding to Definition 3). The need for Definition 3 arose because it avoids a serious problem
encountered by Definition 2 (that different climates are not classified correctly). No problems are
avoided by introducing an ensemble definition for regimes of varying external conditions (compared
to Definition 5). Hence there is no need for such a definition.

17



5.3 Infinite Versions

There are also infinite versions of the two ensemble distributions of climate just dis-
cussed. According to these infinite versions, the climate is defined as the infinite
distribution which arises when the prediction lead-time in equation (5) (for Defini-
tion 4) or equation (6) (for Definition 5) goes to infinity. These infinite versions are
popular definitions (e.g. Checkroun et al. [2011]; Stone et al. [2009]).

The main advantage of these infinite versions is that they are mathematically more
tractable, and, as a consequence, dynamical systems theory can be used to analyse
them. Let me give two examples for this. First, recall that climate as the future en-
semble distribution is defined relative to the present uncertainty in the initial values.
For the infinite versions conditions are known under which the infinite ensemble dis-
tribution is the same for any arbitrary uncertainty in the initial values (then climate
does not depend on our uncertainty and Desideratum 3 is not violated). Namely, this
is the case when the dynamics is mixing or when there is a strong physical measure
(for the infinite version of Definition 4), or when there is a time-dependent strong
physical measure (for the infinite version of Definition 5) (see Appendix C for a math-
ematical statement of these results). As nice as these results are, their relevance is
unclear because it is unknown whether the true climate model under constant exter-
nal conditions is mixing or has a strong physical measure, or whether the true climate
dynamics has a time-dependent strong physical measure (cf. Daron [2012]).

Second, recall that the climate as the future ensemble distribution does not seem
to have anything to do with the time series of past observations. For the infinite
version of Definition 4 it can be shown that when the dynamics is mixing or has a
strong physical measure, then the infinite ensemble distribution and the infinite dis-
tribution over time are the same for almost all initial conditions (note that there are
no corresponding results for Definition 514). Hence if distributions over finite long
time periods approximate the distribution over an infinite time period, the infinite
ensemble distribution can be estimated from the observations and Desideratum 1 is
not violated (see Appendix D for a mathematical statement of these results). How-
ever, again, the relevance of these results is unclear because it is unknown whether
the true climate model under constant external conditions is mixing or has a strong
physical measure (and whether distributions over finite long time periods approxi-
mate the distributions over an infinite time period – cf. Subsection 4.4).

What speaks against the infinite versions of Definitions 4 and 5 is that they suf-

14There cannot be any corresponding results for Definition 5: Because the external conditions
vary, the infinite ensemble distributions vary with time but the infinite distributions over time do
not. Hence the infinite ensemble distributions cannot be equal to the infinite distributions over time.

18



fer from several shortcomings (in addition to the ones already outlined in Subsec-
tions 5.1-5.2). Consequently, for defining the climate finite ensemble distributions
are preferable to infinite ensemble distributions, and for this reason the discussion
above concentrated on finite distributions. More specifically, there is the problem
that the infinite limits defining the infinite ensemble distributions may not exist, im-
plying that Desideratum 5 may not be met (Lasota and Mackey [1985]; Provatas and
Mackey [1991]). To my knowledge, it is unknown whether these limits exist for the
true climate model.15

Furthermore, in order for the infinite ensemble distribution to be predictively use-
ful, one has to assume that the ensemble distribution at time point t1 (what is of
concern in practice) is approximated by the infinite ensemble distribution. However,
there are doubts about this. For several models that incorporate some realism about
the evolution of the climate variables (including models with constant and varying
external conditions) the finite ensemble distributions only converge very slowly to
the infinite ensemble distributions (cf. Smith [2002]). For instance, for the coupled
ocean-atmosphere model investigated by Daron ([2012]) and Daron and Stainforth
([2013]) there is convergence only after 100 model years. Further, if the true climate
model turned out to be almost intransitive, the convergence could be very slow (Bhat-
tacharya et al. [1982]; Sempf et al. [2007]). Whether for the true climate model the
finite ensemble distributions for predictions lead-times of interest are approximated
by the infinite ensemble distributions is unknown, but there remains the worry that
the infinite ensemble distribution may be empirically void (thereby violating Desider-
atum 1).

To conclude: defining climate as an infinite ensemble distribution makes the math-
ematical analysis of climate more tractable. However, there are the additional prob-
lems that the relevant limits may not exist (implying that Desideratum 5 may not be
met) and that the definitions may be empirically void (implying that Desideratum 1
may not be met).

6 Conclusion

The aim of the paper was to provide a clear and thorough analysis of the main candi-
dates for a definition of climate and climate change. Five definitions were discussed:

15For Definition 5 where the external conditions vary (i.e. for equation (6)) there is the additional
problem that there are two infinite limits: one for t0 → −∞ and one for t1 → ∞ (the latter limits
exist only very rarely, cf. Kloeden and Rasmussen 2011). If both limits exist, they usually differ.
Hence the question arises which of them should be identified with the climate, and it is unclear how
to answer this question.

19



climate as the finite distribution of the climate variables over time for constant exter-
nal conditions (Definition 1), climate as the finite distribution of the climate variables
over time when the external conditions vary as in reality (Definition 2), climate as the
finite distribution of the climate variables over time relative to a regime of varying
external conditions (Definition 3), climate as the ensemble distribution of the climate
variables for constant external conditions (Definition 4), and climate as the ensemble
distribution of the climate variables when the external conditions vary as in reality
(Definition 5). Definition 3 is a novel contribution of this paper and was proposed
as a response to problems with existing definitions. The other four definitions are
among the most commonly endorsed definitions of climate.

A definition of climate should be empirically applicable (Desideratum 1), it should
correctly classify different climates (Desideratum 2), it should not depend on our
knowledge (Desideratum 3), it should be applicable to the past, present and future
(Desideratum 4) and it should be mathematically well-defined (Desideratum 5). The
discussion has shown that only Definition 3 (where climate is the finite distribution
over time under a certain regime of varying external conditions) meets all the desider-
ata and hence is the most promising definition.

This can also be seen as follows. The Definition 2 is unattractive because it does
not classify different climates correctly (i.e., Desideratum 2 is not met). Definitions
of climate where the external conditions vary as in reality are preferable to definitions
where the external conditions are held constant because the latter may be empiri-
cally void (thereby violating Desideratum 1). Distributions over time are preferable
to ensemble distributions because for the latter the climate depends on our knowledge
(thereby violating Desideratum 3), it is unclear how to define the past and present
climate (thereby violating Desideratum 4) and there is no relation to the observa-
tional record (thereby violating Desideratum 1). Given this, Definition 3 comes out
as the most promising definition.

Infinite versions of Definitions 1-5 were also discussed. They were quickly dis-
missed since they suffer from the additional problems that they may be empirically
void (thereby not meeting Desideratum 1) and that the relevant limits may not exist
(thereby not meeting Desideratum 2).

Finally, we are now in the position to look at the characterisation of climate given
by the influential report of the IPCC (Solomon et al. [2007], p. 942):

Climate in a narrow sense is usually defined as the average weather, or
more rigorously, as the statistical description in terms of the mean and
variability of relevant quantities over a time period ranging from months

20



to thousands or millions of years. The classical period for averaging these
variables is 30 years, as defined by the World Meteorological Organiza-
tion. The relevant quantities are most often surface variables such as
temperature, precipitation and wind. Climate in a wider sense is the
state, including a statistical description, of the climate system.

This characterisation is vague and ambiguous. ‘Climate in a narrow sense’ may be
intended to refer to the distributions of a more limited set of climate variables and
‘climate in a wider sense’ to the distributions of a more extended set of climate vari-
ables (cf. Section 2). Apart from this, ‘climate in a narrow sense’ seems to refer to a
distribution over time, i.e. to Definition 1, 2 or 3. The most direct interpretation is
that it refers to the finite distribution over time for the actual path of the external
conditions (i.e. Definition 2, which, as argued, is unattractive). ‘Climate in a wider
sense’ is even more open to different interpretations and could in principle be inter-
preted as referring to any of the definitions discussed in this paper.

In any case, the vagueness may well be intended to subsume under one charac-
terisation the various different definitions of climate. Still, I hope that this paper
has raised awareness of the importance of choosing a good definition of climate. In
this way conceptual confusion and wrong statements about our climate system can
hopefully be avoided.

Appendix

A. Deterministic Models

Climate models are deterministic models (X, ΣX,T(x,t0, t)) of the evolution of the
climate variables. Here the set X represents all possible values of the climate vari-
ables, the σ-algebra ΣX represents all subsets of X of interest, and T(x,t0, t) :
X × Z × Z → X (the dynamics) is a measurable function such that T(x,t0, t0) = x
and T(x,t0, t + s) = T(T(x,t0, t), t,s) for all t0, t,s and x.

16 Intuitively speaking,
T(x,t0, t) gives one the value of the climate variables at time t when x is the value
at time t0. The solution when x is the initial value of the climate variables at t0
is the function Tx,t0 (t) : Z → X, Tx,t0 (t) = T(x,t0, t). These deterministic models
are time-dependent, and the recently developed theory of time-dependent dynamical
systems is needed to analyse them (cf. Kloeden and Rasmussen [2011]).

16If T(x,t0, t) is defined only on X×Z×Z∩[t0,∞), then the dynamics is non-invertible (as opposed
to the usual case in climate science when the dynamics is invertible, i.e. defined on X ×Z×Z). All
that is stated in the Appendix can be easily adapted to carry over to a non-invertible dynamics.

21



If the governing equations do not depend on time, the models are time-independent
and one is in the realm of classical dynamical systems theory. Then the reference
to t0 can be omitted because all that matters is the evolved time period t between
two values of the climate variables. That is, the solution through x is the function
Tx(t) = T(x,t), where T(x,t) gives the value of the climate variables that started in
x after t time steps. Suppose that the functions Tt : X → X, Tt(x) = T(x,t), are
bijective for all t and that there is a probability measure µ on X which is invariant,
i.e.

µ(T(A,t)) = µ(A) for all A ∈ ΣX and all t ∈ Z. (7)
Then (X, ΣX,T(x,t),µ) is a measure-preserving deterministic model (cf. Petersen
[1983]).

B. Independence from the Initial Conditions for the Infinite
Version of Definition 1

For the infinite version of Definition 1 there are two cases where the dependence on
the initial conditions is very weak (thus many argue that, pragmatically speaking,
there is no dependence – cf. Lorenz 1970, unpublished). First, a measure-preserving
deterministic model (X, ΣX,T(x,t),µ) is ergodic iff for all A ∈ ΣX

lim
k→∞

1

k

k−1∑
t=0

χA(Tt(x)) = µ(A) (8)

for any x ∈ B with µ(B) = 1 (cf. Petersen 1983).17 Ergodicity immediately implies
that the value assigned to any set A by the infinite version of Definition 1 is the same
for almost all x (i.e. those in B).

The second result is about attractors Ω which are invariant18 sets Ω ⊆ X ⊆ Rn
that attract all initial values in X, i.e., limt→∞ dist(Tt(x), Ω) = 0 for all x ∈ X. A
measure µΩ on Ω is called a physical measure iff for Lebesgue-almost all

19 x ∈ X

lim
k→∞

1

k

k−1∑
t=0

χA(Tt(x)) = µΩ(A), (9)

whenever µΩ(δA) = 0 where δA denotes the boundary of A (cf. Eckmann and Ruelle
[1985]; Ruelle [1976]). Hence for a physical measure the value assigned to any A ∈ ΣX
with µΩ(δA) = 0 by the infinite version of Definition 1 is the same for Lebesgue-almost
all initial values.

17χ(A) is the characteristic function, i.e. χ(x) = 1 for x ∈ A and 0 otherwise.
18That is, Tt(Ω) = Ω for all t ∈ Z.
19That is, for all x ∈ Y , Y ⊆ X, with λ(X \Y ) = 0, where λ is the Lebesgue measure.

22



C. Independence from the Uncertainty in the Initial Values
for the Infinite Versions of Definitions 4 and 5

The independence result for the infinite version of Definition 4 holds under two con-
ditions. First, a measure-preserving deterministic model (X, ΣX,T(x,t),µ) is mixing
iff for all probability densities pt0 (relative to µ) and all sets A ∈ ΣX (Petersen [1983];
Werndl [2009a], Werndl [2009b]):20

lim
t1→∞

Pct0,t1 (A) = µ(A). (10)

Mixing immediately implies that the measure assigned to a set by the infinite version
of Definition 4 is the same for all initial densities pt0 and hence there is no dependence
on the uncertainty.21

Second, a measure µΩ on the attractor Ω is called strongly physical iff for any
density pt0 (relative to the Lebesgue measure λ) on X and for any set A ∈ ΣX :22

lim
t1→∞

Pct0,t1 (A) = µ
Ω(A), (11)

whenever µΩ(δA) = 0 where δA denotes the boundary of A (cf. Ruelle [1976]; Tasaki
et al. [1998]). Here the dependence on the uncertainty in the initial values is negligi-
ble in the sense that the measure assigned to any A with µΩ(δA) = 0 by the infinite
version of Definition 4 is the same for all initial uncertainties.

The independence result for the infinite version of Definition 5 holds when t1 →
∞23 and when there are time-dependent strong physical measures. To define them,
the following definition is needed. A pullback attractor Ω ⊆ Z × Rn is an invariant
set24 where for all initial values x ∈ X

lim
t0→−∞

dist(T(x,t0, t), Ω(t)) = 0. (12)

20Since all that matters is the evolved time period, the limit could equally be taken for t0 →−∞.
21One might ask when the stronger condition holds that any initial probability density pt0 con-

verges to the measure µ in the sense that limt1→∞
∫
X
|1 −pct0,t1|dµ = 0, where p

c
t0,t1

is the density
that arises when pt0 is evolved forward to t1. Most climate models are invertible (cf. footnote 16).
In this case the stronger condition cannot hold because the measure is invariant. For non-invertible
models (X, ΣX,T(x,t),µ) this condition holds iff they are exact, i.e. when limt→∞ µ(T(A,t)) = 1
for all A ∈ ΣX with µ(A) > 0 (Berger [2001]; Lasota and Mackey [1985]). Exactness is a stronger
condition than mixing. It is unclear whether realistic non-invertible climate models are exact.

22Since all that matters is the evolved time period, the limit could equally be taken for t0 →−∞.
23Recall that there are two possible infinite limits for the infinite version of Definition 5 (cf.

footnote 15).
24That is, Ω(t) = T(Ω(t0), t0, t) for all t,t0 ∈ Z, where Ω(t) := {x ∈ Rn | (t,x) ∈ Ω}.

23



Time-dependent strong physical measures µΩt defined on Ω(t), t ∈ Z, where Ω is a
pullback attractor are defined by the condition that for any t and t0, any initial
density pt0 (relative to the Lebesgue measure λ) on X and for any set A ∈ ΣX (cf.
Buzzi [1999]):

lim
t0→−∞

Pt0,t1 (A) = µ
Ω
t (A), (13)

whenever µΩt (δA) = 0 (δA denotes the boundary of A). Hence for time-dependent
strong physical measures the dependence on the uncertainty is negligible because the
measure assigned to a subset A with µΩt (δA) = 0 by the infinite version of Definition
5 is the same for all initial uncertainties.

D. Relation to Infinite Distributions Over Time for the Infi-
nite Version of Definition 4

For the infinite version of Definition 4 two conditions are known which relate infi-
nite ensemble distributions to infinite distributions over time. First, if the measure-
preserving deterministic system (X, ΣX,T(x,t),µ) is mixing, equations (10) and (8)
imply that for any initial probability density pt0 and any set A ∈ ΣX

lim
t1→∞

Pct0,t1 (A) = µ(A) = limk→∞

1

k

k−1∑
t=0

χA(Tt(x)) (14)

for all initial values x ∈ B with µ(B) = 1. Hence the measure assigned to a set A by
the infinite ensemble distribution and the infinite distribution over time is the same
for almost all initial values (i.e. those in B).

Second, for an attractor Ω with a strong physical measure µΩ equations (11) and
(9) imply that for any initial density pt0 , any set A with µ

Ω(δA) = 0 and Lebesgue-
almost all initial values x ∈ X:

lim
t1→∞

Pct0,t1 (A) = µ
Ω(A) = lim

k→∞

1

k

k−1∑
t=0

χA(Tt(x)). (15)

Hence the measure assigned to a set A with µΩ(δA) = 0 by the infinite ensemble
distribution and the infinite distribution over time is the same for Lebesgue-almost
all initial values.

Funding

Arts and Humanities Research Council, UK (AH/J006033/1), Economic and Social
Research Council, UK (ES/K006576/1).

24



Acknowledgements

Many thanks to Mickael Checkroun, Stefaan Willem Conradie, Joseph Daron, Ro-
man Frigg, Gary Froyland, Casey Helgeson, Wendy Parker, Leonard Smith, David
Stainforth, Erica Thompson and audiences at the CHESS Seminar at the University
of Durham and the “Managing Severe Uncertainty”-Meetings at the London School
of Economics for valuable discussions and suggestions.

Charlotte Werndl, c.s.werndl@lse.ac.uk
Department of Philosophy, Logic and Scientific Method
London School of Economics and Political Science
Houghton Street
London WC2A 2AE, UK
c.s.werndl@lse.ac.uk

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Figure 1 caption: The assumed true simple evolution of the temperature when at = 1
(left) and at = 2 (right).

Figure 2 caption: The temperature distribution over time for the simple climate
model with constant c = 1.5 and with initial value 18.85 from 1 July 1983 to 30 June
2013.

Figure 3 caption: The actual temperature distribution over time for the assumed true
simple climate model with initial value 18.85 from 1 July 1983 to 30 June 2013.

Figure 4 caption: The actual temperature distribution over time for the meteor sce-
nario from 1 July 1000 to 30 June 1030.

Figure 5 caption: The actual temperature distribution over time before (from 1 July
1000 to 30 June 1015, left) and after the hit of the meteor (from 1 July 1015 to 30
June 1030, right).

Figure 6 caption: The temperature ensemble distribution on 1 July 2040 for the
simple climate model under constant c = 1.5 given the uncertainty about the initial
temperature on 1 July 2013.

Figure 7 caption: The actual temperature ensemble distribution on 1 July 2040 for
the simple climate model given the uncertainty about the initial temperature on 1
July 2013.

29


	New Journal Cover(accepted version refereed)
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