A classic example is the Lorenz attractor, in the shape of
butterfly wings, which has a fractal structure (Fig. 25.1d).
This is the attractor of a very simple model, with only three
degrees of freedom, which was formulated by Lorenz (1963),
to illustrate the chaotic behavior of the atmosphere.
The result is that although the initial condition is effectively forgotten, it is nevertheless critical to the determination of the ‘true trajectory’ of the system. It is so critical that
it makes it absurd for ‘long-term’ simulations, that it, beyond
time t 1 , to focus on a ‘single real trajectory’ for the model
originating from a given initial state. Chaotic systems, like
the atmosphere, are in fact characterized by a high dependence on the initial conditions: a tiny difference between two
different starting points becomes exponentially greater. It
can be shown that they are also highly dependent on the
parameters used. For these systems, long-term deterministic
predictions simply do not make sense. It is therefore necessary to focus on a set of trajectories, not to define a single
result, but instead to assemble a set of possible results.
Indeed, only the ‘average’ trajectory is significant as it
represents a particular statistic of the attractor of the system.
In practice, it is therefore meaningless to try to calculate
what the weather will be like on a given day at a given
location, beyond a few weeks into the future. Only climate
magnitudes (averages, differences, etc.) have some meaning.
To calculate these climate averages, two solutions are possible. If we consider that the system is stationary, that is to
say that its statistical features do not change over time (this
would be the case, for example, in the absence of forcings),
then it would simply be a matter of averaging the model
results over several decades, as is done by geographers who
use an average of weather variables over thirty years to
define a ‘climate’. We consider that the trajectory followed
by the model represents the system that can be interpreted
statistically, as is the case for the real climate. However,
when the system is subjected to a forcing, such as the current
anthropogenic disturbance, the system can no longer be
considered ‘stationary’, and its statistical characteristics (i.e.
climate values) will evolve over time. A temporal average is
therefore no longer relevant and several simulations need to
be carried out, differing only in their initial condition. This
ensemble simulation establishes the range of different trajectories covering the range of possibilities.
Climate and Determinism
Before further describing (paleo)climate models, certain
paradoxes surrounding the idea of climate should be highlighted. As we have seen above, climatology conveys a
statistical approach, as opposed to meteorology, which has,
above all, a deterministic perspective. The fundamental
reason for this distinction stems mainly from the chaotic
nature of the atmosphere, which becomes inherently unpredictable in a short period of time. If all the boundary conditions of the atmosphere (and all parameters) are considered
to be either constant, or having a simple annual cyclical
forcing, then the system will be stationary, in other words
the statistical variables that define the climate will be stable
over time. Although the terms of the system of Eq. (25.1) are
never zero and the atmosphere changes endlessly, we refer to
the climate model as being in equilibrium. Conversely, the
climate will change only when the boundary conditions of
the atmosphere, or some of its parameters, change over a
time period of a decade or more, in other words, a timeframe
compatible with the concept of climate. In a way, although a
climate model aims to represent primarily atmospheric
variables, it is only the slower physical components, other
than the atmosphere, that cause the ‘climate’ of the model to
evolve, i.e. to change the statistical distribution of the results.
This is the case, for example, when there is a change in the
ocean, carbon cycle, ice caps. The evolution of the climate is
therefore only predictable if the ‘non-atmospheric’ components are predictable. The chaotic nature of the atmosphere
does not imply that climate is unpredictable.
The Framework of a Climate Model
Selecting Components of the Climate System:
Model and Boundary Conditions
In practice, it is not possible (or even desirable) to have a
mathematical model that simulates all the phenomena that
can interact with and modify the climate, including not only
the atmosphere and the oceans, but also the terrestrial and
marine biosphere, biogeochemical cycles, ice cap dynamics,
hydrology and continental erosion. The first difficulty is to
choose a relevant subsystem to define the variables of the
model, and that can be the object of a temporal evolution
given by a system of evolution equations, as above in (25.1).
The other factors have to be imposed, in other words fixed as
boundary conditions or forcings. For example, when coupled
ocean-atmosphere models are used to simulate the past, the
following will typically be imposed: (1) changes in coastlines, everything related to continental surfaces, particularly
changes in topography, ice sheet extent and height, changes
in ocean bathymetry; (2) everything related to atmospheric
concentrations, in particular greenhouse gases, but also, in
some cases, dust and other aerosols; (3) changes in the
orbital parameters of the Earth. We then examine the
response of the ocean-atmosphere system to these boundary
conditions and forcings, either taken together in order to
obtain a realistic simulation of the climate or taken separately to study the role of each of them individually. This is
referred to as a sensitivity experiment.
25 Modeling and Paleoclimatology
323
butterfly wings, which has a fractal structure (Fig. 25.1d).
This is the attractor of a very simple model, with only three
degrees of freedom, which was formulated by Lorenz (1963),
to illustrate the chaotic behavior of the atmosphere.
The result is that although the initial condition is effectively forgotten, it is nevertheless critical to the determination of the ‘true trajectory’ of the system. It is so critical that
it makes it absurd for ‘long-term’ simulations, that it, beyond
time t 1 , to focus on a ‘single real trajectory’ for the model
originating from a given initial state. Chaotic systems, like
the atmosphere, are in fact characterized by a high dependence on the initial conditions: a tiny difference between two
different starting points becomes exponentially greater. It
can be shown that they are also highly dependent on the
parameters used. For these systems, long-term deterministic
predictions simply do not make sense. It is therefore necessary to focus on a set of trajectories, not to define a single
result, but instead to assemble a set of possible results.
Indeed, only the ‘average’ trajectory is significant as it
represents a particular statistic of the attractor of the system.
In practice, it is therefore meaningless to try to calculate
what the weather will be like on a given day at a given
location, beyond a few weeks into the future. Only climate
magnitudes (averages, differences, etc.) have some meaning.
To calculate these climate averages, two solutions are possible. If we consider that the system is stationary, that is to
say that its statistical features do not change over time (this
would be the case, for example, in the absence of forcings),
then it would simply be a matter of averaging the model
results over several decades, as is done by geographers who
use an average of weather variables over thirty years to
define a ‘climate’. We consider that the trajectory followed
by the model represents the system that can be interpreted
statistically, as is the case for the real climate. However,
when the system is subjected to a forcing, such as the current
anthropogenic disturbance, the system can no longer be
considered ‘stationary’, and its statistical characteristics (i.e.
climate values) will evolve over time. A temporal average is
therefore no longer relevant and several simulations need to
be carried out, differing only in their initial condition. This
ensemble simulation establishes the range of different trajectories covering the range of possibilities.
Climate and Determinism
Before further describing (paleo)climate models, certain
paradoxes surrounding the idea of climate should be highlighted. As we have seen above, climatology conveys a
statistical approach, as opposed to meteorology, which has,
above all, a deterministic perspective. The fundamental
reason for this distinction stems mainly from the chaotic
nature of the atmosphere, which becomes inherently unpredictable in a short period of time. If all the boundary conditions of the atmosphere (and all parameters) are considered
to be either constant, or having a simple annual cyclical
forcing, then the system will be stationary, in other words
the statistical variables that define the climate will be stable
over time. Although the terms of the system of Eq. (25.1) are
never zero and the atmosphere changes endlessly, we refer to
the climate model as being in equilibrium. Conversely, the
climate will change only when the boundary conditions of
the atmosphere, or some of its parameters, change over a
time period of a decade or more, in other words, a timeframe
compatible with the concept of climate. In a way, although a
climate model aims to represent primarily atmospheric
variables, it is only the slower physical components, other
than the atmosphere, that cause the ‘climate’ of the model to
evolve, i.e. to change the statistical distribution of the results.
This is the case, for example, when there is a change in the
ocean, carbon cycle, ice caps. The evolution of the climate is
therefore only predictable if the ‘non-atmospheric’ components are predictable. The chaotic nature of the atmosphere
does not imply that climate is unpredictable.
The Framework of a Climate Model
Selecting Components of the Climate System:
Model and Boundary Conditions
In practice, it is not possible (or even desirable) to have a
mathematical model that simulates all the phenomena that
can interact with and modify the climate, including not only
the atmosphere and the oceans, but also the terrestrial and
marine biosphere, biogeochemical cycles, ice cap dynamics,
hydrology and continental erosion. The first difficulty is to
choose a relevant subsystem to define the variables of the
model, and that can be the object of a temporal evolution
given by a system of evolution equations, as above in (25.1).
The other factors have to be imposed, in other words fixed as
boundary conditions or forcings. For example, when coupled
ocean-atmosphere models are used to simulate the past, the
following will typically be imposed: (1) changes in coastlines, everything related to continental surfaces, particularly
changes in topography, ice sheet extent and height, changes
in ocean bathymetry; (2) everything related to atmospheric
concentrations, in particular greenhouse gases, but also, in
some cases, dust and other aerosols; (3) changes in the
orbital parameters of the Earth. We then examine the
response of the ocean-atmosphere system to these boundary
conditions and forcings, either taken together in order to
obtain a realistic simulation of the climate or taken separately to study the role of each of them individually. This is
referred to as a sensitivity experiment.
25 Modeling and Paleoclimatology
323