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In response to the radiative forcing $\Delta $*Q*, the various variables characterising the state of the
climate system will change, modifying the radiative fluxes at the tropopause.
These modifications involve very complex mechanisms. However, insights into
the behaviour of the climate system can be gained by assuming that the changes
in the radiative fluxes at the tropopause can be estimated as a function of
the changes in global mean surface temperature,
$\Delta $T_{s}. If we denote
the imbalance in the radiative budget $\Delta $R, we can write:

where ${\lambda}_{f}$ is called the feedback parameter (expressed in W m

If the perturbation lasts a sufficiently long time, the system will eventually reach a new equilibrium, corresponding to $\Delta $R = 0. This can be used to compute the global mean equilibrium temperature change in response to $\Delta $Q as:

(-1/${\lambda}_{f}$) is a measure of the equilibrium climate sensitivity, i.e. the change in
the global mean temperature at equilibrium in response to a radiative forcing.
The equilibrium climate sensitivity is often determined using climate
model simulations. For practical reasons, it is thus generally defined as the global
mean surface temperature change after the climate system has reached a new
equilibrium in response to a doubling of the *CO*_{2} concentration in
the atmosphere. It is measured in °C and according to the most
recent estimates by the IPCC (Randall et al. 2007), its value is likely to be in the
range 2-4.5°C.

The changes in surface temperature *T*_{S} are associated with
modifications of many other variables that also affect the global heat budget. If we
consider an ensemble of *n* variables *x*_{i} that affect
*R*, neglecting the second order terms, we can express
${\lambda}_{f}$ as a function of those variables as:

$${\lambda}_{f}=\frac{\partial R}{\partial {T}_{S}}=\sum _{i=1,n}\frac{\partial R}{\partial {x}_{i}}\frac{\partial {x}_{i}}{\partial {T}_{S}}\phantom{\rule{0ex}{0ex}}$$ | (4.6) |

${\lambda}_{f}$ can thus be
represented by the sum of the feedback parameters associated with each
variable *x*_{i}. The most common analyses generally focus on the
variables that directly affect the balance at the tropopause through physical
processes. For simplicity, we will called them the direct physical feedbacks
(section 4.2). The standard decomposition of ${\lambda}_{f}$ then involves their separation into
feedbacks related to the temperature (${\lambda}_{T}$), the water vapour (${\lambda}_{w}$), the clouds (${\lambda}_{c}$) and the surface albedo
(${\lambda}_{\alpha}$). The temperature feedback is itself further split into ${\lambda}_{T}$ = ${\lambda}_{0}$ + ${\lambda}_{L}$. In evaluating ${\lambda}_{0}$, we assume that the
temperature changes are uniform throughout the troposphere, while ${\lambda}_{L}$ is called the lapse
rate feedback (see section 4.2.1), and is the feedback due to the non-uniformity
of temperature changes over the vertical.

$${\lambda}_{f}=\sum _{i}{\lambda}_{i}={\lambda}_{0}+{\lambda}_{L}+{\lambda}_{w}+{\lambda}_{c}+{\lambda}_{\alpha}\phantom{\rule{0ex}{0ex}}$$ | (4.7) |

Although indirect effects (such as changes in ocean or atmospheric dynamics and biogeochemical feedbacks) can also play an important role, they are excluded from this decomposition. Biogeochemical feedbacks will be discussed in section 4.3. Some indirect feedbacks, in which the dominant processes cannot be simply related to the heat balance at the tropopause will be mentioned briefly in Chapters 5 and 6, but a detailed analysis is left for more advanced courses.

The feedback parameter ${\lambda}_{0}$ can be evaluated relatively easily because it simply represents the dependence of the outgoing longwave radiation on temperature through the Stefan-Boltzmann law. Using the integrated balance at the top of the atmosphere (see section 2.1.1):

and assuming the temperature changes to be homogenous in the troposphere

we obtain

$${\lambda}_{0}=\frac{\partial R}{\partial T}\frac{\partial T}{\partial T}=-4\sigma {T}_{e}^{3}\phantom{\rule{0ex}{0ex}}$$ | (4.10) |

This provides a value of
${\lambda}_{0}$
$\sim $-3.8
Wm^{-2}K^{-1}. More precise estimates using climate
models gives a value of around -3.2 Wm^{-2}K^{-1}.

We can then compute the equilibrium temperature change in response to a perturbation if this feedback was the only one active as:

For a radiative forcing due to a doubling of the CO_{2} concentration in the atmosphere
corresponding to about 3.8 Wm^{-2} (see Chapter 6), Eq. 4.11 leads to a climate sensitivity
that would be slightly greater than 1°C.

If now, we include all the feedbacks, we can write:

$$\Delta {T}_{S}=-\frac{\Delta Q}{\sum _{i}{\lambda}_{i}}=-\frac{\Delta Q}{{\lambda}_{0}+{\lambda}_{L}+{\lambda}_{w}+{\lambda}_{c}+{\lambda}_{\alpha}}\phantom{\rule{0ex}{0ex}}$$ | (4.12) |

All these feedbacks are often compared to the blackbody response of the system, represented by ${\lambda}_{0}$ , of :

$$\begin{array}{rcl}\Delta {T}_{S}& =& -\frac{1}{\left(1+\frac{{\lambda}_{L}}{{\lambda}_{0}}+\frac{{\lambda}_{w}}{{\lambda}_{0}}+\frac{{\lambda}_{c}}{{\lambda}_{0}}+\frac{{\lambda}_{\alpha}}{{\lambda}_{0}}\right)}\left(\frac{\Delta Q}{{\lambda}_{0}}\right)\\ & =& {\frac{1}{\left(1+\frac{{\lambda}_{L}}{{\lambda}_{0}}+\frac{{\lambda}_{w}}{{\lambda}_{0}}+\frac{{\lambda}_{c}}{{\lambda}_{0}}+\frac{{\lambda}_{\alpha}}{{\lambda}_{0}}\right)}}^{}\Delta {T}_{S,0}\\ & =& {f}_{f}\Delta {T}_{S,0}\end{array}$$ | (4.13) |

Here *f*_{f} is called the feedback factor. If *f*_{f} is larger than one, it
means that the equilibrium temperature response of the system is larger than the
response of a blackbody. As
${\lambda}_{0}$
is negative, Eq. 4.13 also shows that,
if a feedback parameter
${\lambda}_{i}$
is positive, the corresponding feedback amplifies the temperature changes
(positive feedback). Besides, if
${\lambda}_{i}$
is negative, it damps down the changes (negative feedback).

The concepts of radiative forcing, climate feedback and climate sensitivity are very
useful in providing a general overview of the behaviour of the system. However,
when using them, we must bear in mind that the framework described above represents a
greatly simplified version of a complex three-dimensional system. First, it does not
provide explicit information on many important climate variables such as the spatial
distribution of the changes or the probability of extreme events such as storms or
hurricanes. Second, the magnitude of the climate feedbacks and the climate
sensitivity depends on the forcing applied. Climate sensitivity is usually
defined through the response of the system to an increase in *CO*_{2}
concentration but some types of forcing are more 'efficient' than others for the same radiative
forcing, meaning that they induce larger responses. Third, the feedbacks
depend on the mean state of the climate system. For instance, it is pretty obvious
that feedbacks related to the cryosphere (see section 4.2.3) play a larger role in
relatively cold periods, where large amounts of ice are present at the surface, than in warmer
periods. Fourth, non-linearities in the climate system lead to large modifications
when a threshold is crossed as a response to the perturbation (see for instance
section 4.3). In such cases, the climate change is mainly due to the internal dynamics of
the system and is only weakly related to the magnitude of the forcing.
Consequently, the assumptions leading to Eq. 4.4 are no longer valid.