### Radiative forcing

Radiative forcing determines how much the CO_{2} in the atmosphere
increases temperatures. It is a measure of the difference between incoming
energy from the sun and the energy radiated back out to space from the
earth. Any differ results in temperatures increasing. You can choose
how much a given amount of CO_{2} in the atmosphere increases
temperatures by choosing the value of the climate sensitivity. The
default value is 3.2, which means that a doubling of atmospheric
CO_{2} produces a 3.2°C temperature increase.
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Radiative forcing quantifies the change in the earth’s energy balance
from the addition of greenhouse gasses and other climate forcers to
the atmosphere. DICE links the amount of carbon dioxide added to the
atmosphere since 1750 to radiative forcing. Radiative forcing is then
related to temperature change. This relationship is governed by climate
sensitivity. This parameter is a way of representing the equilibrium
temperature from doubling the concentration of CO_{2}. The default
setting of the model specifies a 3.2°C rise in temperature for a
doubling of CO_{2}. The user can set this between 1°C and 5°C.

The relationship between greenhouse gases and radiative forcing $F(t)$
is given by
\[
F(t)=\eta\frac{\ln\left(M_{AT}(t)\right)-\ln\left(M_{AT}^{1750}\right)}{\ln(2)}+F_{EX}(t)
\]
where $F_{EX}(t)$ represents forcing from gases other than CO_{2}
in time period $t$ and $\eta$ represents the increase in forcing from
the doubling of CO_{2} in the atmosphere. These other forcings are
estimated in a given period by a linear function of the difference
between forcings from non-CO_{2} today and an estimate of those
forcings in the year 2100:
\[
F_{EX}(t)=F_{EX}(0)+0.1\left(F_{EX}(2100)-F_{EX}(0)\right)\cdot t.
\]

Radiative forcing leads the warming in the atmosphere, which then warms the upper ocean, gradually warming the deep ocean. The model is: \begin{align*} T(t) & =T(t-1)+\xi_{1}\left[F(t)-\lambda T(t-1)-\xi_{2}\left(T(t-1)-T_{LO}(t-1)\right)\right]\\ T_{LO}(t) & =T_{LO}(t-1)+\xi_{3}\left(T(t-1)-T_{LO}(t-1)\right) \end{align*} where the $\xi_{i}$ are the transfer coefficients reflecting the rates of flow and thermal capacities of the sinks. In particular, $1/\xi_{1}$ is the thermal capacity of the atmosphere and the upper oceans, $1/\xi_{3}$ is the transfer rate from the upper ocean to the deep ocean, and $\xi_{2}$ is the ratio of the thermal capacity of the deep oceans to the transfer rate from the shallow to deep ocean.

The key parameter is $\lambda$, or climate sensitivity, is a way
of representing the equilibrium temperature from doubling the
concentration of CO_{2}. If we solve the temperature equation
for a constant temperature (i.e. equilibrium), we get
$\Delta T/\Delta F=1/\lambda$. If $T_{2xCO_{2}}$ is the equilibrium
impact of a doubling of CO_{2} concentrations, we get
$T_{2xCO_{2}}=\Delta F_{2xCO_{2}}/\lambda$ where
$\Delta F_{2xCO_{2}}$ is the change in radiative forcing from a
doubling of CO_{2}. Therefore, setting $\lambda$ allows us to
set the climate sensitivity.