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\DOI{10.5802/crmeca.247}
\datereceived{2021-12-17}
\daterevised{2024-02-19}
\dateaccepted{2024-03-04}
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\begin{noXML}

%\makeatletter
%\def\TITREspecial{\relax}
%\def\cdr@specialtitle@english{More than a half century of Computational Fluid Dynamics}
%\def\cdr@specialtitle@french{Plus d'un demi-si\`ecle de m\'ecanique des fluides num\'erique}
%\makeatother

\CDRsetmeta{articletype}{research-article}

\title{Climate models}

\alttitle{Mod\`{e}les de climat}

\author{\firstname{Aurore} \lastname{Voldoire}\CDRorcid{0000-0001-9585-7792}}
\address{CNRM, Universit\'e de Toulouse, M\'et\'eoFrance, CNRS, Toulouse, France}
\email{aurore.voldoire@meteo.fr}

\begin{abstract}
The first climate models have emerged in the 60s and have been
continuously developed since then. 
They have progressively included the representation of
all the climate system components: the atmosphere, the ocean, the
cryosphere and the biosphere. Inside each component, they have also
been enriched by the representation of more processes with the aim of
improving their realism. These models are used to make climate
projections over the coming century but they are above all a laboratory
tool to improve our understanding of the climate
system.
\end{abstract}

\begin{altabstract}
Les premiers mod\`{e}les climatiques sont apparus dans les ann\'{e}es
60 et n'ont cess\'{e} d'\^{e}tre d\'{e}velopp\'{e}s depuis. Ils ont
progressivement inclus la repr\'{e}sentation de toutes les composantes
du syst\`{e}me climatique : l'atmosph\`{e}re, l'oc\'{e}an, la
cryosph\`{e}re et la biosph\`{e}re. Au sein de chaque composante, ils
ont \'{e}galement \'{e}t\'{e} enrichis par la repr\'{e}sentation d'un
plus grand nombre de processus dans le but d'am\'{e}liorer leur
r\'{e}alisme. Ces mod\`{e}les sont utilis\'{e}s pour faire des
projections climatiques sur le si\`{e}cle \`{a} venir mais ils sont
surtout un outil de laboratoire pour am\'{e}liorer notre
compr\'{e}hension du syst\`{e}me climatique.
\end{altabstract}

\keywords{\kwd{Climate}\kwd{Ocean}\kwd{Atmosphere}\kwd{Biosphere}\kwd{Projection}}

\altkeywords{\kwd{Climat}\kwd{Oc\'ean}\kwd{Atmosph\`{e}re}\kwd{Biosph\`{e}re}\kwd{Projections}}

\maketitle

\end{noXML}


\section{Introduction}

Atmospheric and ocean models are based on computational fluid dynamics.
In the following, we will explain what are climate models and how they
are built. This article attempts to highlight the hypotheses and
inherent limitations of these models. The first climate models have
been developed based on weather forecasting models. The original
question raised in pioneering studies~\cite{1} was ``what happens if we
integrate atmospheric models over periods longer than a few days?''
Does the numerical atmosphere reach an equilibrium? And if yes does
this equilibrium look like reality? These atmospheric models called
General Circulation Models (GCM) were based on the Navier--Stokes
equations and were developed in the early 1950s~\cite{2} thanks to the
development of computers. These models were able to simulate the
large-scale circulation and the basic features of the atmospheric
climate and have then been used as a digital laboratory by Manabe and
Wetherald~\cite{3} to assess the effect of doubling the carbon
dioxide atmospheric concentration on the climate. It is remarkable that
theirs results have not been denied since then, the award of the 2021
Nobel Prize to S. Manabe is a recognition of these innovative\break studies.

The introduction of the ocean system in climate models has quickly
become a necessity since energy transport in the climate system is half
done by the ocean, which has a much larger heat capacity~\cite{4}.
Since then, models have evolved to include all the relevant components
to simulate the climate over a few centuries: atmosphere, ocean,
cryosphere and biosphere (Figure~\ref{fig1}) and each component representation
has been continuously improved. This increased of model complexity has
been possible thanks to the continuous increase in computing
capabilities. These models are now routinely used to make climate
projections for the next century, but also as a digital laboratory to
explore the complex processes that make up the climate system. The
original version of this article has been published in the encyclopedia
of environment~\cite{5}.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Main components of the climate system (atmosphere,
oceans, cryosphere, biosphere and lithosphere). {[}Source: Report by the
Climate Change Science Program, 2003, {\textcopyright}~US Climate
Change Program.{]}}
\end{figure}

\section{Climate model components}\label{sec1}

\subsection{Ocean and atmosphere}\label{sec1.1}

{\vspace*{2pt}}
Both ocean and atmosphere are fluids, therefore their dynamics follow
the Navier--Stokes equations. In climate models, several simplifications
are made to this set of equations, in particular the hydrostatic
approximation (allows for neglecting the vertical acceleration).
Actually, both components are modelled using the so-called primitive
equations system which consists of the equation of momentum, the
continuity equation for mass conservation, the thermodynamic equation
for energy conservation and the equation of state~\cite{6,7}.
Translated in numerical form, these equations are solved on a mesh
(Figure~\ref{fig2})~\cite{8} and the temporal evolution is done in
successive time steps. There are several numerical methods for
representing the fluid mechanics differential
equations. Among these, the most commonly used are the finite
difference and the spectral method, the latter being more efficient
computationally but also more dissipative. These equations are used to
determine the averaged state parameters (temperature, winds, humidity
of the atmosphere, temperature, currents and salinity of the ocean) on
each mesh cell at each time step. The type of mesh used in climate
models also differs in between models. The obvious way to discretize
the equations in spherical coordinates is to use a regular
latitude-longitude grid. However, such grids suffer from complications
associated with the convergence of meridians near the pole. To
alleviate this singularity, other types of grid have been used:
triangular grid for spectral models and more recently finite difference
models turn to icosahedric grids which have the advantage of
anisotropy~\cite{9}, some groups even turn to using unstructured
grids~\cite{10} which allow to adapt the local resolution to the local
processes.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Mesh structure of a climate model. 
[Source: {\textcopyright} Goose
et~al.~\cite{8}, licence CC-BY-NC.]}
\end{figure}

In climate simulations, the size of a mesh cell is of the order of a
hundred kilometres, and the time step is a few tens of minutes. Subgrid
processes, such as cloud or precipitation formation, cannot therefore
be represented directly by fluid dynamics equations. As these processes
must be taken into account: they are thus represented using
empirical~parameterizations that picture their effect on the state
variables at the scale resolved by the dynamics equations.

Atmospheric models~include parameterizations for~radiative~transfer,
turbulence, convection, cloud and precipitation formation, cloud
interactions with radiation, etc. Ocean models also contain
parameterizations to represent the solar~radiation penetration into
the~water, mixing related to smaller scale eddies, tide-induced mixing
and turbulence. Parameterizations are also used to
represent~transfers~of water, energy and momentum between the ocean and
the atmosphere. Such parameterizations are equations written
empirically from observations, but they intend to be universal in a
given model, in the sense that they are the same at any point on the
globe\footnote{Some parameterizations can however
include latitude as an input parameter~\cite{11}.}.

The atmospheric models used for climate simulations are thus
constructed in the same way as weather forecasting models. The
difference stems from the period over which these models are
integrated~and the way in which their results are interpreted (see Part
3). Additionally, global weather prediction models are operated using
cells of a few kilometres horizontally and a few hundred metres
vertically. To reduce their calculation cost, climate models use larger
cells (see Figure~\ref{fig2}), especially on the horizontal, which
allows integrations over several decades or even centuries. Note that
even if the cells are larger, the time-step of climate models remains
of a few tens of minutes, therefore, they can represent the succession
of meteorological phenomena such as rain, drought, storms, etc.


\subsection{Other components}\label{sec1.2}

In addition to the atmosphere and the ocean, the climate system has to
include a land surface component and a representation of the
cryosphere. Continental surface models aim to represent the exchange of
water and energy between the soil, the biosphere and the atmosphere.
They are based on energy and water conservation principles. For
example, an area of forest will capture more solar energy because it is
darker (its albedo is lower) than an area covered with pastures. On the
other hand, if the surface soil is dry, a tree will be able, thanks to
its deeper roots, to use water in the deep soil and produce a flow of
water to the atmosphere by evapotranspiration, while grass will not be
able to do so. These models also represent seasonal snow on the ground
as well as frost in the soil. As in atmospheric models, many processes
take place at scales smaller than the mesh size and need to be
parameterized.

Aside from snow cover on land, the cryosphere is mainly composed of two
elements: sea ice and glaciers. Sea ice is the ice forming over the
ocean, whereas glaciers are formed over continents and include ice
sheets which are polar cap glaciers. Sea ice models represent the
exchanges of water, salt and energy between the sea ice and the ocean,
but also between the ice pack and the atmosphere. In these models, the
ice pack moves as a function of near-surface winds and ocean currents.
These movements lead to sea ice piling up in some areas and
divergence\footnote{Divergence: indicates that
dynamic tends to export ice to other regions. For example, if sea ice
is transported offshore along a coast by local winds, the ice will tend
to disappear or become thinner at the coast.} in other. Climate models
also include ice sheet models but these are generally very simplified
and are primarily intended to represent the effect of ice cover on
land, without any change in their extent. More complex ice sheet models
already exist, particularly for the Arctic and Antarctic, but they
require the atmospheric parameters to be represented with finer grids
than in current global climate models. Their coupling within climate
models is one of the current challenges of climate modelling and
becomes crucial for very long-term climate simulations (several
centuries).

In climate models, the different components are coupled: they evolve
jointly, interacting with each other. In general, continental surface
models and atmospheric models interact at each time step of the
atmospheric model (a few minutes). Exchanges between ocean models and
atmospheric models take place at least once a day or even hourly
depending on the model.

\section{Complexity and computational cost}\label{sec2}


To make realistic climate projections, climate models have to describe
the four components of the climate system: atmosphere, ocean, biosphere
and cryosphere. The complexity of a model is defined as the number of
climate processes taken into account. The first climate models
developed in the 1970s represented only the atmospheric component. By
the late 1990s, models evolved rapidly and managed to include all four
components. Since then, models have constantly improved by refining
existing parameterizations and adding the representation of new
processes. For instance, most current models represent the life cycle
of particles in suspension in the atmosphere, called aerosols. The
models thus represent their transport by winds, their interaction with
cloud formation processes and their leaching by rain. The most complex
models also represent the entire carbon cycle, and allow to estimate
carbon fluxes between the different components. The fluxes themselves
depend on climate, properties of seawater and the ability of vegetation
and marine biosphere to convert carbon into organic matter. In these
models, the carbon dioxide atmospheric and oceanic concentration
becomes a modelled variable and is no longer an external input.

There are also very simplified, one-dimensional models based on the
terrestrial radiation balance, which simulate only the global mean
surface temperature without any information on the regional
disparities. However, in order to represent the effect of an increase
in carbon dioxide concentration in these simple models, information
from the full models described above is needed. In between these highly
simplified energy balance models and the full models, there is a range
of models of increasing complexity (Figure~\ref{fig3}), developed for
specific applications.\looseness=-1


\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Level of model complexity as a function of the
components represented. {[}Source: {\textcopyright}
{Figure 1.2 in} Le Treut, H., R. Somerville,
U. Cubasch, Y. Ding, C. Mauritzen, A. Mokssit, T. Peterson and M.
Prather, 2007: Historical Overview of Climate Change. In: Climate
Change 2007: The Physical Science Basis. Contribution of Working Group
I to the Fourth Assessment Report of the Intergovernmental Panel on
Climate Change {[}Solomon, S., D. Qin, M. Manning, Z. Chen, M. Marquis,
K. B. Averyt, M. Tignor and H. L. Miller (eds.){]}. Cambridge University
Press, Cambridge, United Kingdom and New York, NY, USA.{]}}
\end{figure}



The most complex models are also the most expensive ones in terms of
computing time. However, depending on the study performed, it is not
always necessary to take all processes into account. In particular, the
inclusion of the full representation of the carbon cycle relies on
parameterizations that are still uncertain and which feedbacks may be
important. In order to study the impact of a given increase in
atmospheric carbon dioxide concentration on features such as El
Ni\~{n}o phenomena, it could be preferable to use a model without
representation of the carbon cycle, which results will be easier to
interpret. The results will then help to understand the projections of
the carbon cycle model. It is therefore sometimes necessary to decouple
the questions to improve our understanding of processes.

Thus, the most complex model is not necessarily the most suitable for
all uses. With an increased level of complexity, more degrees of
freedom are also introduced into the system. This complicates its
calibration and validation, sometimes at the expense of model
robustness. The interpretation of the results is also more delicate.
Additionally, the more complex the model, the more powerful the
computers required and the longer the computation time. Note also that
it is often necessary to carry out several tests before reaching
reliable conclusions on a given study. It is therefore important that
the calculation cost is reasonable in order to be able to perform
several tests.

The cost of running a climate model increases with its complexity and
with its resolution, i.e. the number of grid cells. The calculation
cost also depends on the numerical methods used to represent the model
equations. Thus, to simulate one year, current climate model needs from
a few dozen days to a thousand if only one processor was used. This is
why these models are implemented on supercomputers: by using a large
number of processors simultaneously, the simulation of a year of
climate can then be done within a few hours. Therefore, it takes
several weeks to make a 100-year climate projection simulation. The
volume of data produced is also very large, of the order of a few
hundred Gigabytes (GB) per simulation year, or a few dozen Terabytes
(TB) for a 100-year simulation.

A climate model is a numerical laboratory for testing hypotheses, as
done in biology with test tubes. In biology, the same treatment is
often repeated may times to verify its effectiveness. Similarly in
climate, it is often appropriate to carry out the simulations in the
form of ensembles. As the climate system is chaotic, it is sometimes
difficult to separate the climate changes that are due to imposed
forcing, for example an increase in greenhouse gases, from the internal
variability of the climate system. Simulation sets subject to the same
forcings but with slightly different starting states are then performed
to determine the robust impacts of the applied forcings. \break\break The less
numerically expensive the model, the larger the size of the ensembles
that can be performed and the more confidently the results can be
interpreted.

The calculation cost also determines the mesh size of the climate
models. Currently, these meshes are generally close to 100 km ${\times}$ 100~km,
although some models manage to reach a few tens of km$^{2}$. This
spatial resolution makes it possible to represent many phenomena such
as the succession of perturbations in the mid-latitudes, or tropical
monsoons. On the other hand, these global models do not allow the
representation of more local phenomena, such as cyclones and regional
circulations directly linked to the topography. Thus, in a climate
model with a resolution of 100~km, the topography is too smooth for the
Rh\^{o}ne valley to be represented distinctly, and this prevents
Mistral episodes from being represented in the model. 

To carry out studies on finer resolution
phenomena, a regional climate model can be used, as is done in weather
forecasting. The finer-grid regional model will need to be constrained
at the boundaries of its domain by a global climate model
(Figure~\ref{fig4}). Regional models are built on the same principles
as global models (similar algorithms, same processes) and can
themselves be more or less complex. Some represent only the atmospheric
component, but there are also coupled regional ocean-atmosphere models.
Such models make it possible to study smaller-scale phenomena, but the
results depend on both the regional model and global model used to
constrain at the boundary. It is therefore important to take account of
this double uncertainty, for example by carrying out a set of regional
simulations forced by several global models.


\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Orography and bathymetry of a global climate model
at 100~km resolution and of a regional model over Europe at 12~km
resolution. {[}Source: {\textcopyright} CNRM/Realisation Pierre Nabat
and Aurore Voldoire.{]}}
\end{figure}

\section{Validation of a climate model}\label{sec3}

\subsection{Input data}\label{sec3.1}

When simulating with a climate model, there are two types of input
data: initial state and forcings. While in weather forecasting,
the initial state is a key input, in climate modelling, forcings are
the most important input. The main forcings are: greenhouse gas
concentrations (except water vapour, simulated by the atmospheric
component), aerosol load, but also land cover, volcanic eruptions and
solar energy received at the top of the atmosphere. Because the climate
system is chaotic, information from the initial state is quickly lost.
The initial conditions are therefore of little importance for climate
simulations over several decades.

However, the memory of the climate system is longer than that of a
weather prediction model, because in particular of the thermal inertia
of the oceans. This property is the ground base to make forecasts on a
seasonal scale. Indeed, seasonal forecast models are climate models in
which the atmosphere and ocean components are initialized based on
observations. They take advantage of the fact that the climate system
has significant inertia in the ocean but also on land to forecast
trends over the coming months. These systems are much less reliable
than weather forecasting models but provide relevant information on the
evolution of phenomena such as El Ni\~{n}o and on tropical variability
in general. Such forecasting models are run over past periods as in
forecast mode (ignoring the observation available after the
initialisation is done), this is called hindcasting. Hindcasting over a
large number of past years allows a seasonal forecasting system to be
tested and is a form of model validation. However, this type of
validation is limited to simulations of a few months, which is still
short compared to the climatic time-scales.

The use of climate models to  make forecasts at longer time horizons of
a few years is a current research
topic~\cite{12}. Some studies consider
predictability\footnote{Predictability: Predictability measures the
ability to predict with respect to the target time frames under the
assumption that the initial state is well known.} to be possible for up
to one or two years. Beyond a few years, initial conditions are of
little importance and applied forcings become dominant.

\subsection{Validation}\label{sec3.2}

Aside from being used in seasonal hindcasts, a climate model is
validated in successive stages. Each component of the model is
developed independently of the others by applying observations as input
variables. Within this constrained framework, it is verified that the
model produces realistic results. Thus, a land surface model will be
used alone by providing meteorological observations as input variables
and it will be assessed that the temperature, soil moisture and
vegetation evolve in accordance with the observations.

Each component is thus adjusted and validated separately from the
others. Each component is also used for studies specific to its domain.
In this step, a few model parameters can be considered to be
unconstrained by available observations and the range of acceptable
values for these parameters will be assessed. These parameters can
eventually be adjusted within this range once the complete model is
assembled. During the steps constrained by observations, the simulation
results can be directly compared with observations in terms of time
evolution.

For the full climate system, the chaotic nature of the climate system
implies that they do not follow closely the evolution of the real
system. A simulation over several decades, even when forced by observed
forcings, will not follow the chronology of observations due to this
chaotic nature (and model imperfections). Even if the model were
perfect, it would not be able to follow the sequence of weather events
as they occur. It will therefore not be possible to compare the
evolution of the temperature over a country directly with observations.
On the other hand, the model will be able to represent the
climatological characteristics of the climate
(Figure~\ref{fig5})~\cite{13}: the seasonal cycles of temperature and
precipitation averaged over 30 years can be compared with the (average)
climatology of the observations. Similarly, long-term trends can be
compared with those of the observations. For example, the models show
the reduction in spring snow cover at mid-latitudes during the 20{th}
century~\cite{14}.


\begin{figure}
\includegraphics{fig05}
\caption{\label{fig5}Map of the 30-year (1985--2014) averaged top of the
atmosphere outgoing long-wave radiation simulated by a set of models in
the CMIP6 database for the ``historical'' simulation. The map in the
lower right-hand corner represents the reference estimate over the same
period obtained from satellite measurements (CDR monthly product
``Outgoing Longwave Radiation''). {[}Source: {\textcopyright} 
CNRM/Aurore Voldoire production.{]}}
\end{figure}


Validation is not only possible on the long-term climate
characteristics. It will also assess the model’s ability to
adequately represent the daily variability of precipitation, i.e. its
ability to represent the number of dry days and very wet days in a
given season at a given location around the globe. The model’s
ability to represent extreme events, such as storms, can also be
assessed.

The validation step determines the limits of these models. For example,
they are often able to represent the distribution of different types of
clouds around the globe, but for the most part, they currently fail to
adequately represent the thin stratocumulus-type clouds on the eastern
edges of the tropical oceans, such as off Chile, California or
Angola~\cite{15}. When simulations of future climate are analysed,
these shortcomings will have to be taken into account as a limitation
of model reliability in these regions. The validation step also
indicates the processes that need to be improved in models to make them
more reliable.

Climate models produce information on all time scales
(Figure~\ref{fig1}), from the diurnal cycle to trends at the century
scale. They are therefore evaluated on their ability to represent many
processes at all these temporal and spatial scales. Our confidence in
these models emerges from the physical consistency of all the processes
represented. Downstream of the modelling activity, the work of climate
researchers also consists of processing large masses of data produced
by models using appropriate statistical methods.

\section{Climate projections}\label{sec4}

To make climate projections, modellers must prescribe the evolution of
forcings over the coming century, as input data for climate models.

For solar radiation and volcanoes, it is difficult to predict their
level of future activity. So simple assumptions are made: for solar
radiation, the last observed 11-year cycle is reproduced periodically
over the next century\footnote{Cyclicity simulated
from reconstructions over 9400 years for CMIP6 (Matthes \etal, 2017),
available at \url{https://solarisheppa.geomar.de/cmip6}.}, for volcanoes, only
their average effect is considered in general for the future without
variations (but more specific studies are carried out to address this
issue).               

For the other forcings, namely greenhouse
gases, aerosols and land use, their evolution is highly dependent on
the evolution of human activities, and are referred to as anthropogenic
forcings. Economists provide different scenarios (Figure~\ref{fig6}) of
how these forcings will change based on demographic and economic
assumptions. For this purpose, they use specific models known as
``integrated impact models~\cite{16}''.

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig6}Evolution of (a)~CO$_2$ concentration and (b)
sulphate aerosol emissions, observed (black line) and according to the
4 main scenarios (colored lines) for the 21st century considered in
AR5. {[}Source: {\textcopyright} CNRM / R\'{e}alisation Pierre
Nabat.{]}}
\end{figure}

Several types of evolution are envisaged, depending on whether or not
the population becomes aware of the risk and adapts its practices,
depending on demographic changes and whether or not regions are turning
in on themselves. As the economic projections remain uncertain,
economists produce several types of scenarios for changes in greenhouse
gas concentrations. Climate models are then integrated over the 21{st}
century by varying forcings under these economic scenarios. The initial
conditions are derived from simulations of the 20{th} century and
intrinsically incorporate the effect of the forcings over past periods.

Depending on the level of complexity of the model, the type of forcing
is adapted. For example, for models that do not represent the carbon
cycle, carbone dioxide concentrations will be prescribed to the model.
Conversely, if the model represents the carbon cycle, the model will
apply anthropogenic carbon emission forcings, with natural emissions
being dynamically represented in the model. Similarly, for models that
do not include a life-cycle representation of aerosols, concentrations
of these particles, which are themselves derived from specific models
forced by the chosen economic scenario, will be prescribed.

\section{IPCC and model intercomparison exercises}\label{sec5}

The Intergovernmental Panel on Climate Change (IPCC) regularly produces
reports that aim to summarize the state of scientific knowledge on
climate, its evolution, impacts and ways to adapt to projected changes.
In order to coordinate the work of the various teams implementing
climate models around the world, a model intercomparison exercise
(Coupled Model Intercomparison Project---CMIP) is carried out upstream
of the IPCC reports. The aim of these exercises is to propose common
experimental protocols to the modelling centres that will make it
possible to study climate change and characterise the uncertainty
associated with climate modelling in a rigorous manner.

The CMIP project also provides modelling centres with storage space and
tools to share the data produced by the models. These databases are
public\footnote{Earth System Grid Federation.} and
can be used by researchers around the world to conduct climate change
studies. For the last IPCC report published in 2021, 31 teams have
published data from 64 different models\footnote{As
of 22/01/2024.}. Many centres operate several versions of their models
by modifying the complexity of the model and/or its spatial resolution.

For the last CMIP exercise (6{th} phase), 4 scenarios of changes in
greenhouse gases and main aerosols were used to force the models
(Figure~\ref{fig6}). The lowest scenario, called ssp126, plans very
limited concentrations which requires a reduction in emissions from the
middle of the century and negative emissions from 2080 onwards. On the
opposite, the highest scenario, called ssp585, projects an increase in
emissions at the current rate which forecasts carbone dioxide
concentrations of 1135 ppm in 2100. In between these two extremes, the
ssp245 scenario represents an intermediate future where societies
partially adapt, and the ssp370 scenario, also plans an intermediate
future but with a higher concentration in carbone dioxide and a lower
reduction in aerosol emissions.


By comparing the projections of all models under a given scenario,
modelling uncertainty (distinct from scenario uncertainty) can be
assessed. This is about 2~{\textdegree}C for ssp370 scenario, which
projects a global mean temperature change of 4~{\textdegree}C as a
multi-model average compared to preindustrial levels
(Figure~\ref{fig7}a). It is well understood that the total uncertainty
in climate projections results both from uncertainty in the future of
demographics and the economy (scenario) and from uncertainty in the
climate models themselves. The level and source of uncertainty depends
on the climate variable considered. For instance, for September sea ice
extent (Figure~\ref{fig7}b), the uncertainty is dominated by the model
uncertainty whereas for the ocean acidity (Figure~\ref{fig7}c), the
scenario is the main source of uncertainty.

\begin{figure}
{\vspace*{-2pt}}
\includegraphics{fig07}
{\vspace*{-2pt}}
\caption{\label{fig7}\fontsize{9.5}{11.4}\selectfont
Scenarios of (a) global average temperature change
(in~{\textdegree}C), (b) September sea ice extent and (c) global ocean
surface pH based on the 5 greenhouse gas concentration trajectories.
The zero reference is the average over the end of the 19th century
(1850--1900). The color shading indicates the likely range for ssp370
and ssp126 scenarios. {[}Source: Figure SPM.8 in IPCC, 2021: Summary
for Policymakers. In: Climate Change 2021: The Physical Science Basis.
Contribution of Working Group I to the Sixth Assessment Report of the
Intergovernmental Panel on Climate Change {[}Masson-Delmotte, V., P.
Zhai, A. Pirani, S. L. Connors, C. P\'{e}an, S. Berger, N. Caud, Y.
Chen, L. Goldfarb, M. I. Gomis, M. Huang, K. Leitzell, E. Lonnoy, J. B. R.
Matthews, T. K. Maycock, T. Waterfield, O. Yelek\c{c}i, R. Yu, and B.
Zhou (eds.){]}. Cambridge University Press, Cambridge, UK and New York,
NY, USA, pp. 3--32,
\url{https://doi.org./10.1017/9781009157896.001}.{]}}
\end{figure}


In order to better compare the models with each other, climate
modellers have defined a more theoretical metric, called climate
sensitivity, which corresponds to the increase in surface temperature
after a doubling of carbone dioxide concentration compared to the
pre-industrial era. This is the change in temperature at equilibrium,
which implies that the model must be integrated long enough for the
global surface temperature to stabilize. This theoretical quantity
allows us to compare the magnitude of the feedbacks that appear in the
climate system following an increase in carbone dioxide concentration.
Indeed, the first impact of carbone dioxide is to warm the air locally,
but this warming has multiple consequences, called feedbacks. For
example, if the air warms up, the Clausius--Clapeyron relationship
indicates that it will be able to hold more water vapour. This can have
impacts on cloud formation, cloud positioning, etc. It is the relative
importance of these different feedbacks that is uncertain and the study
of several models allows to better understand them.

For the last intercomparison (CMIP6), the uncertainty between models
has rather increased~\cite{17}; and the CMIP6 models ensemble has a wider
range of climate sensitivity than the CMIP5 models ensemble. The higher
CMIP6 climate sensitivity values compared to CMIP5 can be traced to an
amplifying cloud feedback that is larger in CMIP6 by about 20\%. The
reasons for this increase in sensitivity in several models are the
subject of intense research. Several studies also attempt to use recent
observations to constrain sensitivity estimates. It is indeed crucial
to determine whether such strong values are plausible.

Climate models are thus laboratory tools to better understand processes
within the climate system and to advance knowledge. However, it should
not be forgotten that these models are imperfect. It is necessary to
compare them with observations in order to understand their limitations
and continuously improve them.

\section{Conclusion}

As recently highlighted by the Nobel Prize Academy, climate models are
physically based on the well-known system of equations of
Navier--Stokes. They are a numerical representation of the climate
system and have grown in complexity over the last 50 years. Now,
varying complexity climate models exist and the level of complexity for
each specific study is chosen according to its scientific objective.
Indeed, climate models are used to make projections of the future
climate under a scenario of economic and demographic evolution. But
they are also a very valuable laboratory tool for advancing our
understanding of the climate system.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization
that could benefit from this article, and have
declared no affiliations other than their research
organizations.

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