Well now, that was from 1858, here is a summary from 2021;
Plain Language Summary
Increasing CO2 reduces the rate at which energy leaves Earth, causing a net energy gain at its surface. The resulting warming increases the rate that energy leaves the planet. The planet stops warming once it regains balance. Studies usually assume that doubling atmospheric CO2 always produces the same eventual global temperature rise (called the “equilibrium climate sensitivity”), whatever the starting CO2 level. We show, on the contrary, that in nearly all the computer climate models we have examined, the extra warming for each doubling goes up as the CO2 level increases. In most models, the warmer the climate becomes, the more it has to warm in order to balance a further CO2 doubling because warming becomes less effective at rebalancing the flow of energy. This effect increases projections of warming, especially for scenarios of greatest CO2 increase.
And then the math;
2 Equilibrium Warming
Let
T be the globally averaged surface temperature and Δ
T ≡
T −
T<em>pi</em> be the warming relative to the preindustrial period. We define Δ
T<em>eq</em>(
C) as the equilibrium warming caused by changing the CO2 concentration from its preindustrial value (
pCO2,<em>pi</em> ≈ 280ppm) to a new value (
pCO2), where
Cis the number of CO2 doublings relative to this preindustrial period,
(1)
Under preindustrial conditions,
C<em>pi</em> = 0; in an abrupt 2 × CO2 simulation,
C = 1; and so forth. Table
S1 is a glossary of all symbols used in this paper.
One condition for equilibrium is that the net top-of-atmosphere radiative flux
N (downwards positive) is zero, on average. If we assume that
N depends solely on
C and
T, then we can express a change in
N in an abrupt
n × CO2 simulation as an initial change due to
C and a subsequent change due to
T:
(2)
(3)
(4)
F is the
radiative forcing, the change in
N relative to a given initial condition (
C<em>i</em>,
T<em>i</em>) caused by
C doublings of CO2 while holding surface temperature fixed (
F(
C<em>i</em>,
T<em>i</em>,
C) ≡
N(
C<em>i</em> +
C,
T<em>i</em>) −
N(
C<em>i</em>,
T<em>i</em>)), and
λ is the
radiative feedback, the dependence of
N on
T (
λ(
C,
T) ≡
∂N(
C,
T)/
∂T), where the sign convention implies the feedback is typically negative. We can find Δ
T<em>eq</em>(
C) by setting
N(
C,
T) = 0:
(5)
where we assume
N(
C<em>pi</em>,
T<em>pi</em>) = 0, since the preindustrial climate was roughly in equilibrium.
Under preindustrial concentrations, the spectral line shape of CO2 absorption bands creates a logarithmic dependence of
N on changes in
pCO2, so that the
forcing per CO2
doubling (
) is often assumed to be constant (Myhre et al.,
1998). Our definition of radiative forcing also includes adjustments of the atmosphere, land, and ocean to CO2 changes that occur independently of subsequent changes in surface temperature (e.g., Kamae et al.,
2015; Sherwood et al.,
2014). This “effective radiative forcing” is also often assumed to be constant per CO2 doubling (Forster et al.,
2016), as is the radiative feedback (Gregory et al.,
2004; Hansen et al.,
1985). Substituting these constant terms into Equation
5, we can solve for Δ
T<em>eq</em>(
C):
(6)
Assuming a constant
and
λ is equivalent to approximating
N(
T,
C) with the linear Taylor expansion of
N around preindustrial values of
C<em>pi</em> and
T<em>pi</em> (i.e.,
, where
C = Δ
C because
C<em>pi</em> = 0). The linear approximation of Equation
6 is ubiquitous in climate science (e.g., Knutti et al.,
2017; Stocker et al.,
2013).
The linear approximation implies that the
equilibrium climate sensitivity (Δ
T2<em>x</em>), the equilibrium warming per CO2 doubling, is simply
, which, being a ratio of two constants, is itself a constant. It should therefore not matter how many CO2 doublings are used to estimate it since Δ
T2<em>x</em> = Δ
T<em>eq</em> (
C1)/
C1 = Δ
T<em>eq</em> (
C2)/
C2. Figure
1a shows instead that our estimates of Δ
T<em>eq</em>(
C)/
C increase with CO2 concentration for 13 of 14 models. Colored bars show estimates made by extrapolating regressions of years 21–150 of
N against Δ
T to equilibrium (
N = 0) for abrupt 2<em>C</em> × CO2 simulations (Gregory et al.,
2004, see also solid gray lines in Figure
S1). In these estimates,
N and Δ
T are anomalies: for LongRunMIP, we subtract the model's control simulation's mean value; for CMIP6, we subtract the linear fit of the control simulation after the branch point for the abrupt
n × CO2 simulations. We use only one ensemble member for each simulation.
Now that is a American Geophysical Union publication, and I am sure that old Westie thinks he is smarter than all the real scientists in the AGU.
