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# Why models run hot: results from an irreducibly simple climate model

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An irreducibly simple climate-sensitivity model is designed to empower even non-specialists to research the question how much global warming we may cause. In 1990, the First Assessment Report of the Intergovernmental Panel on Climate Change (IPCC) expressed “substantial confidence” that near-term global warming would occur twice as fast as subsequent observation. Given rising CO2 concentration, few models predicted no warming since 2001. Between the pre-final and published drafts of the Fifth Assessment Report, IPCC cut its near-term warming projection substantially, substituting “expert assessment” for models’ near-term predictions. Yet its long-range predictions remain unaltered. The model indicates that IPCC’s reduction of the feedback sum from 1.9 to 1.5 W m−2 K−1 mandates a reduction from 3.2 to 2.2 K in its central climate-sensitivity estimate; that, since feedbacks are likely to be net-negative, a better estimate is 1.0 K; that there is no unrealized global warming in the pipeline; that global warming this century will be IPCC in its Fourth and Fifth Assessment Reports that are highlighted in the present paper is vital. Once those discrepancies are taken into account, the impact of anthropogenic global warming over the next century, and even as far as equilibrium many millennia hence, may be no more than one-third to one-half of IPCC’s current projections.
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Article Earth Science
Why models run hot: results from an irreducibly
simple climate model
Christopher Monckton
Willie W.-H. Soon
David R. Legates
William M. Briggs
Received: 27 August 2014 / Accepted: 12 November 2014 / Published online: 8 January 2015
Ó Science China Press and Springer-Verlag Berlin Heidelberg 2015
Abstract An irreducibly simple climate-sensitivity
model is designed to empower even non-specialists to
research the question how much global warming we may
cause. In 1990, the First Assessment Report of the Inter-
governmental Panel on Climate Change (IPCC) expressed
‘substantial conﬁdence’ that near-term global warming
would occur twice as fast as subsequent observation. Given
rising CO
2
concentration, few models predicted no warm-
ing since 2001. Between the pre-ﬁnal and published drafts
of the Fifth Assessment Report, IPCC cut its near-term
warming projection substantially, substituting ‘expert
assessment’ for models’ near-term predictions. Yet its
long-range predictions remain unaltered. The model indi-
cates that IPCC’s reduction of the feedback sum from 1.9
to 1.5 W m
-2
K
-1
mandates a reduction from 3.2 to 2.2 K
in its central climate-sensitivity estimate; that, since feed-
backs are likely to be net-negative, a better estimate is
1.0 K; that there is no unrealized global warming in the
pipeline; that global warming this century will be \1K;
and that combustion of all recoverable fossil fuels will
cause \2.2 K global warming to equilibrium. Resolving
the discrepancies between the methodology adopted by
IPCC in its Fourth and Fifth Assessment Reports that are
highlighted in the present paper is vital. Once those dis-
crepancies are taken into account, the impact of anthro-
pogenic global warming over the next century, and even as
far as equilibrium many millennia hence, may be no more
than one-third to one-half of IPCC’s current projections.
Keywords Climate change Climate sensitivity
Climate models Global warming Temperature
feedbacks Dynamical systems
1 Introduction
Are global-warming predictions reliable? In the 25 years of
IPCC’s First to Fifth Assessment Reports [15], the
atmosphere has warmed at half the rate predicted in FAR
(Fig. 1); yet, Professor Ross Garnaut [6] has written, ‘The
outsider to climate science has no rational choice but to
accept that, on a balance of probabilities, the mainstream
science is right in pointing to high risks from unmitigated
climate change.’ However, as Sir Fred Hoyle put it,
‘Understanding the Earth’s greenhouse effect does not
require complex computer models in order to calculate
useful numbers for debating the issue. To raise a delicate
point, it really is not very sensible to make approximations
and then to perform a highly complicated computer
calculation, while claiming the arithmetical accuracy of the
computer as the standard for the whole investigation’ [7].
The present paper describes an irreducibly simple but
robustly calibrated climate-sensitivity model that fairly
represents the key determinants of climate sensitivity,
Electronic supplementary material The online version of this
article (doi:10.1007/s11434-014-0699-2) contains supplementary
material, which is available to authorized users.
C. Monckton (&)
Science and Public Policy Institute, Haymarket, VA 20169, USA
e-mail: monckton@mail.com
W. W.-H. Soon
Harvard-Smithsonian Center for Astrophysics, Cambridge,
MA 02138, USA
D. R. Legates
Department of Geography, University of Delaware, Newark,
DE 19716, USA
W. M. Briggs
New York, NY 10021, USA
123
Sci. Bull. (2015) 60(1):122–135 www.scibull.com
DOI 10.1007/s11434-014-0699-2 www.springer.com/scp
ﬂexibly encompasses all reasonably foreseeable outcomes,
and reliably determines how much global warming we may
cause both in the short term and in the long term. The
model investigates and identiﬁes possible reasons for the
widening discrepancy between prediction and observation.
Simpliﬁcation need not lead to error. It can expose
anomalies in more complex models that have caused them to
run hot. The simple climate model outlined here is not
intended as a substitute for the general-circulation models.
Its purpose is to investigate discrepancies between IPCC’s
Fourth (AR4) and Fifth (AR5) Assessment Reports and to
reach a clearer understanding of how the general-circulation
models arrive at their predictions, and, in particular, of how
the balance between forcings and feedbacks affects climate-
sensitivity estimates. Is the mainstream science settled? Or is
there more debate [8] than Professor Garnaut suggests? The
simple model provides a benchmark against which to mea-
sure the soundness of the more complex models’ predictions.
2 Empirical evidence of models running hot
How reliable are the general-circulation models the
authority of whose output Professor Garnaut invites us to
accept without question? In 1990, FAR predicted with
‘substantial conﬁdence’ that, in the 35 years 1991–2025,
global temperature would rise by 1.0 [0.7, 1.5] K, equiv-
alent to 2.8 [1.9, 4.2] K century
-1
. Yet 25 years after that
prediction the outturn, expressed as the trend on the mean
of the two satellite monthly global mean surface
temperature anomaly datasets [9, 10], is 0.34 °C, equiva-
lent to 1.4 °C century
-1
—half the central estimate in FAR
and beneath the lower bound of the then-projected warm-
ing interval (Fig. 1). Global temperature would have to rise
over the coming decade at a rate almost twice as high as the
greatest supra-decadal rate observed since the global
instrumental record began in 1850 to attain even the lower
bound of the predictions in FAR, and would have to rise at
more than thrice the previous record rate—i.e., at 0.67 K
over the decade—to correspond with the central prediction.
Since 1990, IPCC has all but halved its estimates both of
anthropogenic forcing since 1750 and of near-term warm-
ing. Though the pre-ﬁnal draft of AR5 had followed
models in projecting warming at 0.5 [0.3, 0.7] K over
30 years, equivalent to 2.3 [1.3, 3.3] °C century
-1
,
approximating the projections on the four RCP scenarios,
the ﬁnal draft cut the near-term projection to 1.7 [1.0,
2.3] °C century
-1
, little more than half the 1990 interval
and only marginally overlapping it (Fig. 2).
Empirically based reports of validation failure in complex
general-circulation models abound in the journals [1429].
Most recently, Zhang et al. [30] reported that some 93.4 % of
altocumulus clouds observed by collocated CALIPSO and
CloudSat satellites cannot be resolved by climate models
with a grid resolution [1° (110 km). Studies of paleo-veg-
etation and pollens in China during the mid-Holocene cli-
mate optimum 6,000 years ago ﬁnd January (i.e., winter
minimum) temperatures to have been 6–8 K warmer than
present. Yet, Jiang et al. [31] showed that all 36 models in the
Paleoclimate Modeling Intercomparison Project backcast
Fig. 1 Medium-term global temperature trend projections from FAR, extrapolated from January 1990 to October 2014 (shaded region), vs. observed
anomalies (dark blue) and trend (bright blue), as the mean of the RSS, UAH, NCDC, HadCRUT4 and GISS monthly global anomalies [913]
Sci. Bull. (2015) 60(1):122–135 123
123
winter temperatures for the mid-Holocene cooler than the
present. Also, all but one model incorrectly simulated
annual-mean mid-Holocene temperatures in China as cooler
than the present [31]. Suggestions that current models
accurately simulate the mid-Holocene climate optimum rely
on comparisons between projected and observed summer
warming only, overlooking models’ failure to represent
winter temperatures correctly, perhaps through undue sen-
sitivity to CO
2
-driven warming.
3 An irreducibly simple climate-sensitivity model
An irreducibly simple climate-sensitivity model is now
described. It is intended to enable even non-specialists to
study why the models are running hot and to obtain reason-
able estimates of future anthropogenic temperature change.
The model is calibrated against the climate-sensitivity
interval projected by the CMIP3 suite of models and against
global warming since 1850. Its utility is demonstrated by its
application to the principal outputs of the CMIP5 models and
to other questions related to climate sensitivity.
The simple model, encapsulated in Eq. (1), determines
the temperature response DT
t
forcings and consequent temperature feedbacks over any
given period of years t:
DT
t
¼ q
1
t
DF
t
r
t
k
1
¼ q
1
t
DF
t
r
t
k
0
G
¼ q
1
t
DF
t
r
t
k
0
ð1 gÞ
1
¼ q
1
t
DF
t
r
t
k
0
ð1 k
0
f
t
Þ
1
¼ q
1
t
k ln
C
t
C
0

r
t
k
0
ð1 k
0
f
t
Þ
1
;
ð1Þ
where q
t
is the fraction of total anthropogenic forcing rep-
resented by CO
2
over t years, and its reciprocal allows for
non-CO
2
forcings as well as the CO
2
forcing; DF
t
is the
radiative forcing in response to a change in atmospheric CO
2
concentration over t years, which is the product of a constant
k and the proportionate change (C
t
/ C
0
)inCO
2
concentra-
tion over the period [3, 32]; r
t
is the transience fraction,
which is the fraction of equilibrium sensitivity expected to
be attained over t years; and k
?
is the equilibrium climate-
sensitivity parameter, which is the product of the Planck
sensitivity parameter k
0
[4] and the open-loop or system
gain G, which is itself the reciprocal of 1 minus the closed-
loop gain g, which is in turn the product of k
0
and the sum f
t
of all temperature feedbacks acting over the period.
This simple equation represents, in an elementary but
revealing fashion, the essential determinants of the tem-
perature response to any anthropogenic radiative pertur-
bation of the climate and permits even the non-specialist to
generate respectable approximate estimates of temperature
response over time. It is not, of course, intended to replace
the far more complex general-circulation models; rather, it
is intended to illuminate them.
4 Parameters of the simple model
The parameters of the simple model are now described.
4.1 The CO
2
fraction q
t
The principal direct anthropogenic radiative forcing is
CO
2
. Other inﬂuential greenhouse gases are CH
4
,N
2
O, and
tropospheric O
3
. In AR4, it was estimated that CO
2
would
contribute some 70 % of total net anthropogenic forcing
from 2001 to 2100, so that q
100
= 0.7. Likewise, AR5, on
1990 2000 2010 2020 2030 2040
+2.0
+1.5
+1.0
+0.5
0.0
–1.0
——— IPCC (1990): 0.19 — 0.43 ºC/decade
IPCC (2013): 0.10 — 0.23 ºC/decade
Temperature anomaly (ºC)
Year
Fig. 2 Near-term global warming projection intervals from FAR (red arrows) and AR5 (expert assessment: green arrows), overlaid on the
CMIP5 model projections based on the four RCP scenarios from AR5, which zeroed the models’ projections to observed temperature (black
curve) in 1990. Based on AR5 [5]
124 Sci. Bull. (2015) 60(1):122–135
123
projects that CO
2
concentration by 2100 will be 936 ppmv,
but that the inﬂuence of other greenhouse gases will raise
that value to 1,313 ppmv CO
2
equivalent (CO
2
e), again
implying a CO
2
fraction q
100
= 0.7. Note that the dis-
crepancy between ratios of forcings and of CO
2
concen-
trations is small over the relevant intervals.
However, AR5 concludes at p. 165 that CO
2
contributed
80 % of greenhouse-gas forcing from 2005 to 2011:
‘Based on updated in situ observations, this assessment
concludes that these trends resulted in a 7.5 % increase in
RF from GHGs from 2005 to 2011, with carbon dioxide
(CO
2
) contributing 80 %.’ Furthermore, models have
greatly exaggerated the growth of atmospheric CH
4
con-
centration. It is reasonable to suppose that CO
2
will rep-
resent not \83 % of total anthropogenic forcings over the
twenty-ﬁrst century: i.e., q
t
C 0.83. To retain compatibility
with IPCC’s practice of expressing the CO
2
fraction q
t
as a
percentage of total anthropogenic forcing, the convention
has been retained here. Accordingly, the total anthropo-
genic forcing may be derived by taking the reciprocal of
the CO
2
fraction; thus, q
t
C 0.83 ) q
t
-1
B 1.2. The CO
2
t
) is essentially being scaled by this
factor, as a measure of weighting the CO
2
.
4.2 The CO
2
t
The CO
2
radiative forcing is the product of a coefﬁcient k
and the proportionate change in CO
2
concentration [4];
thus, where C
0
is the unperturbed concentration,
DF
t
¼ klnðC
t
=C
0
Þ;
j
k ¼ 5:35: ð2Þ
The value of the coefﬁcient k was reduced by 15 %, from 6.3
in SAR to 5.35 in TAR. Thus, for instance, if CO
2
concen-
tration doubles, DF
t
will be 5.35 ln 2 = 3.708 W m
-2
. IPCC
now expresses ‘very high conﬁdence’ [5] in the greenhouse-
gas radiative forcings including that from CO
2
, which,
applying Eq. (2) to pre-industrial and 2011 forcings of
approximately 280 and 394 ppmv, respectively, is
1.82 W m
-2
, the value given in AR5. Therefore, the value of
k is here taken as constant. However, its current value 5.35
was obtained by intercomparison between three models [32].
It has not been convincingly derived empirically.
4.3 The Planck climate-sensitivity parameter k
0
To determine climate sensitivity where feedbacks are
absent or net-zero, a direct forcing is multiplied by the
Planck or instantaneous sensitivity parameter k
0
, denomi-
nated in Kelvin per Watt per square meter. Where feed-
backs are absent or net-zero, the equilibrium-sensitivity
parameter k
?
is equal to k
0
. At the characteristic-emission
altitude (CEA), at about the 300-mb pressure altitude,
where incoming and outgoing radiative ﬂuxes are by def-
inition equal, Eq. (3) gives incoming and hence, by deﬁ-
nition, outgoing radiative ﬂux F
E
:
F
E
¼
pr
2
4pr
2
Sð1 aÞ¼239:4Wm
2
; ð3Þ
where F
E
is the product of the ratio pr
2
/4pr
2
of the surface
area of the disk the Earth presents to the Sun to that of the
rotating sphere; total solar irradiance S = 1,368 W m
-2
;
and 1 a, where a = 0.3 is the Earth’s albedo. Then, since
F ¼ erT
4
; j Stefan-Boltzmann relation ð4Þ
mean CEA effective temperature T
E
is given by Eq. (5),
T
E
¼
F
E
er

1=4
¼
239:4
5:67 10
8

1=4
¼ 254:9K; ð5Þ
where emissivity e = 1 and the Stefan–Boltzmann constant
r = 5.67 9 10
-8
Wm
-2
K
-4
.
The CEA is *5 km above ground level. Since mean
surface temperature is 288 K and the mean tropospheric
lapse rate is about 6.5 K km
-1
, Earth’s effective radiating
temperature T
E
= 288 - 5(6.5) = 255 K, in agreement
with Eq. (5). Accordingly, a ﬁrst approximation of the
zero-feedback sensitivity parameter k
0
is DT
E
/DF
E
, thus
F
E
¼ erT
4
E
) k
0
¼
DT
E
DF
E
¼
1
4erT
3
E
¼
T
E
4erT
4
E
¼
T
E
4F
E
¼
254:9
4ð239:4Þ
¼ 0:27 KW
1
m
2
:
ð6Þ
However, [33], cited in AR4, pointed out that
‘[i]ntermodel differences in k
0
arise from different
spatial patterns of warming; models with greater high-
latitude warming, where the temperature is colder, have
smaller values of k
0.
’’
Accordingly [33], followed by AR4, gave
k
0
= 3.2
-1
= 0.3125 K W
-1
m
2
to allow for variation
with latitude (note, however, that AR4 expresses k
0
in
Wm
-2
K
-1
). Other values of k
0
in the literature are
0.29–0.30 [3437]. Though the value of k
0
may vary
somewhat over time, IPCC’s value 0.3125 K W
-1
m
2
may
safely be taken as constant at sub-millennial timescales.
4.4 The temperature-feedback sum f
t
The temperature change driven by a direct forcing may
itself engender temperature feedbacks—additional forcings
whose magnitude is dependent upon that of the temperature
change that triggered them. The direct forcing may be
ampliﬁed by positive feedbacks or attenuated by negative
feedbacks. Feedbacks are thus denominated in W m
-2
K
-1
of directly caused temperature change. The feedback sum
Sci. Bull. (2015) 60(1):122–135 125
123
f
t
=
P
i
f
i
, the sum of all temperature feedbacks acting on
the climate over some period t, is the prime determinant of
climate sensitivity in that, in IPCC’s understanding, it
doubles or triples a direct forcing. Yet its value is far from
settled. Indeed, uncertainty as to the magnitude of f
t
is the
greatest of the many uncertainties in the determination of
climate sensitivity. As Fig. 3 shows, IPCC’s interval 1.9
[1.5, 2.4] W m
-2
K
-1
in AR4 [cf. 33] was sharply cut to 1.5
[1.0, 2.2] W m
-2
K
-1
in AR5. Yet, the climate-sensitivity
interval [2.0, 4.5] K in the CMIP3 model ensemble [4] was
slightly increased to [2.1, 4.7] K in CMIP5 [5]. The user
may adopt any chosen value for the feedback sum.
4.5 The closed-loop gain g
t
and the open-loop
or system gain G
t
The effect of temperature feedbacks is to augment or
diminish the instantaneous temperature response DT
0
to a
direct forcing. The closed-loop gain g
t
is the product of the
instantaneous or Planck climate-sensitivity parameter k
0
and the feedback sum f
t
. The open-loop or system gain
factor G
t
is equal to (1 - g
t
)
-1
. Both g
t
and G
t
are unitless.
The equilibrium temperature response DT
?
is the product
of the instantaneous temperature response DT
0
and the
system gain factor G
t
.
4.6 The equilibrium climate-sensitivity parameter k
?
The equilibrium-sensitivity parameter k
?
,inKW
-1
m
2
,is
the product of the Planck parameter k
0
= 3.2
-1
KW
-1
m
2
and the system gain factor G
t
. Climate sensitivity DT
?
is
the product of k
?
and a given forcing DF
?
.
4.7 Derivation of G
?
, g
?
, and f
?
from DT
?
/DF
?
To ﬁnd the system gain G
?
, the loop gain g
?
and the
feedback sum f
?
, implicit in any given equilibrium-
response projection DT
?
, ﬁrst divide DT
?
by DF
?
to
obtain k
?
. Then, G
?
, g
?
, f
?
are all functions of k
?
and
k
0
, thus:
G
1
¼
k
1
k
0
; g
1
¼ 1
k
0
k
1
;
f
1
¼ k
1
0
k
1
1
Wm
2
K
1
:
ð7Þ
Table 1 shows the given feedback sums f
?
in AR4,
AR5, with the implicit central estimates of g
?
, G
?
, and
k
?
.
4.8 The transience fraction r
t
Not all temperature feedbacks operate instantaneously.
Instead, feedbacks act over varying timescales from decades
to millennia. Some, such as water vapor or sea ice, are short-
acting, and are thought to bring about approximately half of
the equilibrium warming in response to a given forcing over
a century. Thus, though approximately half of the equilib-
rium temperature response to be expected from a given
[Planck]
Water
vapor
Lapse
rate
Water
vapor
plus
lapse
rate
Cloud Albedo Total
+2
+1
0
–1
–2
AR5
AR4
W m
–2
K
–1
–3
Temperature feedback
Fig. 3 Individual climate-relevant temperature feedbacks and their
estimated values. Central estimates in AR4 are marked with
leftward-pointing blue arrows; in AR5 with rightward-pointing red
arrows. The feedback sum f (right-hand column) falls on 1.5 [1.0,
2.2] W m
–2
K
–1
for AR5, compared with 1.9 [1.5, 2.4] W m
–2
K
–1
for AR4. The Planck value shown as a ‘feedback’ is not a true
feedback, but a part of the climatic reference system. Diagram
Table 1 Derivation of the equilibrium-sensitivity parameter k
?
from the Planck parameter k
0
and the feedback sum f
?
, based on the lower,
central and upper estimates of f
?
in AR4 (left) and AR5 (right)
AR4 Derivation of k
?
AR5
f
?
g
?
G
?
k
?
f
?
g
?
G
?
k
?
Unampliﬁed
feedback sum
Closed-
loop
gain
System
gain
factor
Equilibrium-
sensitivity
parameter
k
0
= 3.2
-1
Unampliﬁed
feedback sum
Closed-
loop
gain
System
gain
factor
Equilibrium-
sensitivity
parameter
f
1
? f
2
?  ? f
n
k
0
f
?
(1–g
?
)
-1
k
0
G
?
Derivation f
1
? f
2
?  ? f
n
k
0
f
?
(1–g
?
)
-1
k
0
G
?
(W m
-2
K
-1
) Unitless Unitless (K W
-1
m
2
) Units (W m
-2
K
-1
) Unitless Unitless (K W
-1
m
2
)
1.5 0.469 1.882 0.588 Low est. 1.0 0.313 1.455 0.455
1.9 0.594 2.462 0.769 Best est. 1.5 0.469 1.882 0.588
2.4 0.750 4.000 1.250 High est. 2.2 0.688 3.200 1.000
126 Sci. Bull. (2015) 60(1):122–135
123
forcing will typically manifest itself within 100 years of the
forcing (Fig. 4), the equilibrium temperature response may
not be attained for several millennia [38, 39]. In Eq. (1), the
delay in the action of feedbacks and hence in surface tem-
perature response to a given forcing is accounted for by the
transience fraction r
t
. For instance, it has been suggested in
recent years that the long and unpredicted hiatus in global
warming may be caused by uptake of heat in the benthic
strata of the global ocean (for a fuller discussion of the cause
of the hiatus, see the supplementary matter). The construc-
tion of an appropriate response curve via variations over
time in the value of the transience fraction r
t
allows delays of
this kind in the emergence of global warming to be modeled
at the user’s will.
In [38], a simple climate model was used, comprising an
advective–diffusive ocean and an atmosphere with a Planck
sensitivity DT
0
= 1.2 K, the product of the direct radiative
forcing5.35ln2= 3.708 W m
-2
in response to a CO
2
dou-
bling and the zero-feedback climate-sensitivity parameter
k
0
= 3.2
-1
KW
-1
m
2
. The climate object thus deﬁned was
forced with a 4 W m
-2
pulse at t = 0, and the evolutionary
curve of climate sensitivity (Fig. 4) was determined. Equilib-
rium sensitivity was found to be 3.5 K, of which 1.95 K is
shown as occurring after 50 years, implying r
50
= 0.56. For
comparison, AR4 gave 3.26 K as its central estimate of equi-
librium climate sensitivity to a doubling of CO
2
concentration,
implying k
?
= 3.26/(5.35 ln 2) = 0.88 K W
-1
m
2
. The mean
of projected concentrations on the six SRES emissions sce-
narios in AR4, obtained by enlarging the graphs and overlaying
a precise grid on them and reading off and averaging the annual
values, is 713 ppmv in 2100 compared with 368 ppmv in 2000.
The central estimate of twenty-ﬁrst-century warming in
AR4 was 2.8 K, of which 0.6 K was committed warming
already in the pipeline. Of the remaining 2.2 K, some 70 %,
or 1.54 K, was CO
2
driven. AR4’s implicit centennial sen-
sitivity parameter k
100
was thus 1.54 K / [5.35 ln(713 / 368)
Wm
-2
], or 0.44 K W
-1
m
2
, which is half of the implicit
equilibrium-sensitivity parameter k
?
= 0.88 K W
-1
m
2
.
In AR4, the implicit centennial transience fraction r
100
is
thus 0.50, close to the 0.56 found in [38]. Table 2 gives
approximate values of r
t
corresponding to f
?
B 0 and
f
?
= 0.5, 1.3, 2.1, and 2.9. Where f
t
B 0.3, for all t, r
t
may
safely be taken as unity: at sufﬁciently small f
t
, there is little
difference between instantaneous and equilibrium response.
For f
?
on 2.1 [1.3, 2.9], r
t
is simply the fraction of equi-
librium sensitivity attained in year t, as shown in Fig. 4.
It is not possible to provide a similar table for values of
f
?
given in AR4 or AR5, since IPCC provides no evolu-
tionary curve similar to that in Fig. 4. Nevertheless,
Table 2, derived from [38], allows approximate values of r
t
to be estimated.
5 How does the model represent different conditions?
The simple model has only ﬁve tunable parameters: the CO
2
fraction q
t
, dependent on projected CO
2
concentration
change; the CO
2
t
; the transience frac-
tion r
t
; the Planck sensitivity parameter k
0
, on which the
instantaneous temperature response DT
0
and the system gain
2
4
6
8
10
Mean and 95 % bounds
12
1.3
2.1
2.9
f
Years
250 500 1000 10000
(K)
Fig. 4 The time evolution of the probability distribution of future
climate states, generated by a simple climate model forced by a step-
function climate forcing DF
?
=4Wm
–2
at t = 0. The climate
model considers a range of different feedback strengths and has a
reference sensitivity DT
0
= 1.2 K. The black curve shows the time
evolution of the state with the mean sensitivity, ﬂanked by the 95 %
conﬁdence interval (blue region). Right panel: equilibrium [i.e. t =
?] probability distribution. Higher-sensitivity climates have a larger
response time and take longer to equilibrate. Note the switch to a log
time axis after 500 years. The equilibrium sensitivity interval DT
?
on
3.5 [2.0, 12.7] K shown in the graph corresponds to a loop gain
g
?
= 0.7 [0.4, 0.9] and a feedback-sum f
?
= 2.1 [1.3, 2.9] W m
–2
K
–1
.
Table 2 Approximate values of r
t
at values f
?
B 0 and f
?
= 0.5, 1.3, 2.1, and 2.9 over periods t = 25–300 years, derived from [38]
Approximate values of r
t
Years t 25 50 75 100 125 150 175 200 225 250 275 300
f
?
B 0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
f
?
= 0.5 0.65 0.70 0.74 0.77 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.85
f
?
= 1.3 0.55 0.63 0.65 0.68 0.70 0.71 0.72 0.73 0.74 0.75 0.75 0.76
f
?
= 2.1 0.40 0.49 0.53 0.56 0.57 0.59 0.60 0.61 0.62 0.63 0.64 0.64
f
?
= 2.9 0.15 0.19 0.22 0.23 0.25 0.26 0.27 0.28 0.29 0.30 0.30 0.30
Sci. Bull. (2015) 60(1):122–135 127
123
G
t
are separately dependent; and the feedback sum f
t
,of
which the equilibrium-sensitivity parameter k
?
is a function.
These ﬁve parameters permit representation of any
combination of anthropogenic forcings; of expected
warming at any stage from inception to equilibrium after
perturbation by forcings of any magnitude or sign; and of
any combination of feedbacks, positive or negative, linear
or nonlinear. The model makes explicit the relative con-
tributions of forcings and feedbacks to projected anthro-
pogenic global warming. Feedbacks, mentioned [1,000
times in AR5, are the greatest source of uncertainty in
predicting anthropogenic temperature change.
6 Calibration against climate-sensitivity projections
in AR4
To establish that the model generates climate sensitivities
sufﬁciently close to IPCC’s values, its output is compared
to the equilibrium Charney climate-sensitivity interval 3.26
[2.0, 4.5] K in response to a CO
2
doubling (AR4). Here, the
equilibrium value of the transience fraction r
?
in (1)is
unity by deﬁnition;, since CO
2
alone is the focus of equi-
librium-sensitivity studies, q
t
-1
is likewise unity. Thus,
DT
?
becomes simply the product of k
?
and DF
t
(Table 3).
The chief reason why the central estimate in AR4 is 14 %
greater than the model’s central estimate is that IPCC’s
central estimate is close to the mean of the upper and lower
bounds, while the model’s central estimate is closer to the
lower than to the upper bound because it is derived from
AR4’s central estimate of the feedback sum. This asymmetry
is inherent in Eq. (1), but is not reﬂected in AR4’s central
estimate. The sensitivity interval 2.9 [2.2, 4.6] K found by the
simple model is accordingly close enough to the interval 3.26
[2.0, 4.5] K in AR4, Box 10.2, to calibrate the model.
7 Calibration against observed temperature change
since 1850
The HadCRUT global surface temperature dataset [12]
shows global warming of 0.8 K from January 1850 to April
2014. CO
2
concentration in 1850 was *285 ppmv against
393 ppmv in 2011, so that DF
t
= 5.35 ln(393 / 285) =
1.72 W m
-2
. Total radiative forcing from 1750 to 2011
was 2.29 W m
-2
(AR5). Taking forcing from 1750 to 1850
as approximately 0.1 W m
-2
, forcing from 1850 to 2011
was about 2.19 W m
-2
, so that q
t
-1
= 2.19 / 1.72 = 1.27.
Using these inputs, warming since 1850 is determined by
the model and compared with observation in Table 4.
Assuming that all global warming since 1850 was
anthropogenic, the model fairly reproduces the change in
global temperature since then, suggesting that the 0.6 K
committed but unrealized warming mentioned in AR4,
AR5 is non-existent. If some global warming was natural,
then a fortiori the likelihood of committed but unrealized
warming is small.
8 Application of the model to global-warming
projections in AR5
8.1 The climate-sensitivity interval
In FAR, the implicit central estimate of k
?
was
0.769 K W
-1
m
2
, giving an equilibrium climate sensitiv-
ity 2.9 K in response to a CO
2
doubling. The CMIP3
model ensemble in AR4, p. 798, box 10.2 gave as its
central estimate an equilibrium sensitivity of 3.26 K,
implying that k
?
= 0.879 K W
-1
m
2
and consequently
that f = 2.063 W m
-2
K
-1
, somewhat above the
1.9 W m
-2
K
-1
given in [32].
In AR5, however, the reduction in the central estimate
of f to 1.5 W m
-2
K
-1
cut IPCC’s implicit central estimate
of k
?
to 0.588 K W
-1
m
2
, halving the feedback compo-
nent k
?
- k
0
in k
?
from 0.566 to 0.275 K W
-1
m
2
.
IPCC, by reducing the feedback sum enough to halve the
contribution of feedbacks to equilibrium sensitivity in
AR5, had in effect cut its central estimate of climate sen-
sitivity by one-third, from 3.3 to 2.2 K. Yet, for the ﬁrst
time, the panel decided that no central estimate of climate
sensitivity would be published. The Summary for Policy-
makers in AR5 says
Table 3 Comparison of the Charney-sensitivity interval 2.9 [2.2, 4.6] K generated by the model on the basis of the feedback-sum interval f on
1.9 [1.5, 2.4] (AR5) with the CMIP3 sensitivity interval 3.26 [2.0, 4.5] K (AR4)
AR4 f k
?
DF
2x
DT
2x
DT
2x
Variance
2x CO
2
AR4 Table 1 5.35 ln 2 Model (k
?
DF
2x
) AR4 Box 10.2 AR4-model
model
(W m
-2
K
-1
)(KW
-1
m
2
)(Wm
-2
) (K) (K) (%)
Lowest 1.5 0.588 2.20 2.00 -9
Best 1.9 0.769 3.708 2.85 3.26 14
Highest 2.4 1.250 4.60 4.50 -2
128 Sci. Bull. (2015) 60(1):122–135
123
No best estimate for equilibrium climate sensitivity
can now be given because of a lack of agreement on
values across assessed lines of evidence and studies.
The simple model indicates that, as a result of the fall in
the interval of estimates of f from 1.9 [1.5, 2.4]
Wm
-2
K
-1
in AR4 to 1.5 [1.0, 2.2] W m
-2
K
-1
in AR5,
the Charney-sensitivity interval in response to a CO
2
doubling should have been reduced from 3.26 [2.0. 4.5] K
to 2.2 [1.7, 2.7] K. Yet, the CMIP5 climate-sensitivity
interval given in AR5 is 3.2 [2.1, 4.7] K (AR5).
The central estimate is near half as high again as it
would have been if the method in AR4 had been followed
(Table 5). The simple model suggests that the CMIP5
Charney-sensitivity estimates published in AR5 are unduly
high and that the central estimate has apparently been
overstated by almost half.
8.2 Projected warming in the RCP forcing scenarios
In AR5, IPCC introduces four new forcing scenarios, based
on net anthropogenic forcings of 2.6, 4.5, 6.0, and
8.5 W m
-2
over 1750–2100, of which approximately
2.3 W m
-2
is shown as having occurred by 2011. There
has also been global warming of approximately 0.9 K
since 1750.
In Table 6, IPCC’s projected intervals of warming from
1750–2100 on each of the four scenarios in AR5 are
compared with the output of the model. On all four sce-
narios, IPCC’s projected values for twenty-ﬁrst-century
warming are greatly in excess of the simple model’s pro-
jections. One reason for the discrepancy is that IPCC bases
its projections not on the period 2014–2100 but on the
difference between the means of two 20-year intervals
1986–2005 and 2081–2100, separated by 95 years. IPCC’s
method thus takes no account of the absence of global
warming in the past two decades.
8.3 An observationally based estimate of global
warming to 2100
The simple model may be deployed to obtain observa-
tionally based best estimates of global warming to 2100,
for instance, by adopting realistic values of the CO
2
forc-
ing DF
t
, the feedback sum f, the CO
2
fraction q
t
, and the
transience fraction r
t
.
8.3.1 The CO
2
forcing DF
t
RCP 8.5 is the ‘business-as-usual’ scenario in AR5.
However, the assumptions underlying it are unrealistic (see
Discussion). In the more realistic RCP 6.0 scenario,
Table 4 Modeled and observed global warming, January 1850 to April 2014
1850–2014 CO
2
(1850) CO
2
(2014) fq
t
-1
r
t
k
?
DF
t
DT
2x
(Model) DT
2x
(Obs.) Variance
Basis cf. 278 (1850) NOAA (2014) AR5 ﬁg. 9.43
2.19
1.72
Table 2 Table 1 5.35 ln (393/285) q
t
-1
r
t
k
?
DF
t
model
Units (ppmv) (ppmv) (W m
-2
K
-1
)(KW
-1
m
2
)(Wm
-2
) (K) (K) (%)
1.0 0.7 0.455 0.7
285 400 1.5 1.27 0.6 0.588 1.72 0.8 0.8 0
2.2 0.5 1.000 1.1
Sci. Bull. (2015) 60(1):122–135 129
123
atmospheric CO
2
concentration, currently 400 ppmv, is
projected to reach 670 ppmv by 2100, so that DF
t
from
2015 to 2100 will be 5.35 ln(670/400), or 2.760 W m
-2
.
8.3.2 The feedback sum f
A plausible upper bound to f may be found by recalling that
absolute surface temperature has varied by only 1 % or 3 K
either side of the 810,000-year mean [40, 41]. This robust
thermostasis [42, 43], notwithstanding Milankovich and
other forcings, suggests the absence of strongly net-posi-
tive temperature feedbacks acting on the climate.
In Fig. 5, a regime of temperature stability is represented
by g
?
B?0.1, the maximum value allowed by process
engineers designing electronic circuits intended not to
oscillate under any operating conditions. Thus, assuming
g
?
C 0.5, values of f
?
fall on [-1.6, ?0.3], giving k
?
on
[0.21, 0.35]. Where f
?
is thus at most barely net-positive, the
corresponding equilibrium-sensitivity interval is well con-
strained, falling on [0.8, 1.3] K. Of course, other assumptions
might be made; however, in a near-perfectly thermostatic
system, net-negative feedback is plausible, indicating that the
climate—far from amplifying any temperature changes
caused by a direct forcing—dampens them instead. Indeed,
this damping should be expected, since temperature change is
not merely a bare output, as voltage change is in an electronic
circuit: temperature change is also the instrument of self-
equilibration in the system, since radiative balance following
a forcing is restored by the prevalence of a higher tempera-
ture. Also, in electronic circuits, the singularity at g
?
=?1,
where the voltage transits from the positive to the negative
rail, has a physical meaning: in the climate, it has none. A
damping term absent in the models is thus required in Eq. (7)
and may be represented in Eq. (1) by a reduction of k
?
.
8.3.3 The CO
2
fraction q
t
IPCC’s implicit value for q
t
falls on [0.71, 0.89], the higher
values corresponding to the lower projected total anthro-
pogenic forcings. A reasonable interval for q
t
correspond-
ing to low values of f
t
is thus [0.8, 0.9], so that q
t
-1
falls on
[1.10, 1.25]. For comparison, on RCP 6.0 in AR5, the
implicit value for q
t
-1
is 1.194 (Table 6).
8.3.4 The transience fraction r
t
Where f
t
B 0.3, little error will arise if, for all t, r
t
is taken
as unity: For at sufﬁciently small f
t
, there is little difference
between instantaneous and equilibrium response.
8.3.5 Projected global warming from 2014 to 2100
From the values of f
t
,q
t
-1
, and r
t
thus determined, the model
projects global warming to 2100 (Table 7). On the assump-
tions that DF
t
= 2.760, r
t
= 1, f falls on [-1.6,
?0.32] W m
-2
K
-1
, and q
-1
falls on [1.10, 1.25], model-
projected warming DT
t
falls on 0.8 [0.6, 1.2] K. The narrow
response interval is a consequence of the temperature sta-
bility where g
t
falls on [-0.5, ?0.1] (Fig. 5). This stability is
consistent with the observed near-thermostasis over the past
810,000 years [40], with which IPCC’s implicit loop-gain
interval g
t
on [?0.23, ?0.74] seems inconsistent. For com-
parison, the projection in AR5 on RCP 6.0 is 2.2 [1.4, 3.1] K
and on RCP 8.5 is 3.7 [2.6, 4.8] K.
8.4 How much post-1850 global warming
was anthropogenic?
Assuming 285 ppmv CO
2
in 1850 and 400 ppmv in 2014,
and applying the observationally derived values of f
t
,
holding r
t
at unity, and taking q
t
-1
= 2.29/1.813 = 1.263
to allow for the greater fraction of past warming attribut-
able to CH
4
, the simple model determines the approximate
fraction of the 0.8 K observed global warming since 1850
that was anthropogenic as 78 % [62 %, 104 %].
If it is assumed that g
t
\ ?0.1, warming is already at
equilibrium, since r
t
? 1 for the implicit values
f
t
B?0.3 W m
-2
K
-1
, on this scenario there is probably
no committed but unrealized global warming. If AR4 is
correct in its estimate that 0.6 K warming is in the pipeline,
then \0.2 K anthropogenic warming has occurred since
1850, indicating that warming realized since then is sub-
stantially natural.
Table 5 Comparison of the Charney-sensitivity interval 2.2 [1.7, 3.7] K generated by the model on the basis of the feedback-sum interval f on
1.0 [1.5, 2.2] W m
-2
K
-1
(AR5) with IPCC’s published climate-sensitivity interval 3.2 [2.1, 4.7] K (AR5)
AR5 f
?
k
?
DF
2x
DT
2x
DT
2x
Variance
2x CO
2
AR5 ﬁg. 9.43 Table 1 5.35 ln 2 Model (k
?
DF
2x
) AR5 (SPM) AR5-model
model
(W m
-2
K
-1
)(KW
-1
m
2
)(Wm
-2
) (K) (K) (%)
Lowest 1.0 0.455 1.7 2.1 24
Best 1.5 0.588 3.708 2.2 3.2 46
Highest 2.2 1.000 3.9 4.7 21
130 Sci. Bull. (2015) 60(1):122–135
123
8.5 An observationally based estimate of Charney
sensitivity
With the observationally derived values of f, the model
provides a new estimate of the Charney sensitivity
(Table 7). If temperature feedbacks are at most weakly net-
positive, with loop gain g on [-0.5, ?0.1] as Fig. 5 and
810,000 years of thermostasis suggest, Charney sensitivity
may fall on 1.0 [0.8, 1.3] K. The model’s central estimate
is one-third of the 3.2 K central estimate from the CMIP5
model ensemble in AR5, or of the 3.26 K central estimate
from the CMIP3 model ensemble in AR4.
For comparison, in [44], g is found to fall on [-1.5,
?0.7], so that, assuming the forcing at CO
2
doubling is
4Wm
-2
, a little above the 3.71 W m
-2
that IPCC cur-
rently regards as canonical, the equilibrium Charney sen-
sitivity DT
29
falls on [0.5, 4.2] K. The model’s climate-
sensitivity interval is better constrained than the CMIP
models’ intervals because across a broad interval of weakly
positive to net-negative feedbacks there is little change in
the temperature response.
8.5.1 Charney sensitivity: summary of results
Table 8 summarizes the Charney climate-sensitivity
intervals in IPCC’s ﬁve successive Assessment Reports
FAR, SAR, TAR, AR4, and AR5 as amended in the light
of the simple model’s results and as found by the model
itself.
As Table 8 shows, correcting the output of the CMIP5
models to determine the central estimate of temperature
response from the central estimate of the feedback sum and
to determine the entire sensitivity interval from the revised
feedback-sum interval given in AR5 reduces the sensitivity
interval from 3.2 [2.1, 4.7] K to 2.2 [1.7, 3.9] K, bringing the
Table 6 Comparison of projected warming under the RCP 2.6, 4.5, 6.0, and 8.5 radiative forcing scenarios, 2014–2100, as generated by the simple model and as given in AR5
RCP CO
2
(2100) CO
2
e (2100) r
t
q
t
-1
k
?
DF
t
DT
2x
(Model) DT
2x
(AR5) Variance
Basis Box SPM.1 Box SPM.1 Table 1
CO
2
e
CO
2
Table 2 5.35 ln (CO
2
/400) q
t
-1
r
t
k
?
DF
t
AR5 SPM RCP-model
model
(ppmv) (ppmv) (K W
-1
m
2
)(Wm
-2
) (K) (K) (%)
RCP 2.6 421 475 0.7 0.455 0.10 0.3 ?200
0.6 1.128 0.588 0.274 0.11 1.0 ?809
0.5 1.000 0.15 1.7 ?1,033
RCP 4.5 538 630 0.7 0.455 0.59 1.1 ?86
0.6 1.171 0.588 1.586 0.66 1.8 73
0.5 1.000 0.93 2.6 ?180
RCP 6.0 670 800 0.7 0.455 1.05 1.4 ?33
0.6 1.194 0.588 2.760 1.16 2.2 ?90
0.5 1.000 1.65 3.1 ?88
RCP 8.5 936 1,313 0.7 0.455 2.00 2.6 ?30
0.6 1.402 0.588 4.548 2.25 3.7 ?39
0.5 1.000 3.19 4.8 ?50
–1.0 –0.5
+0.1
+0.5 +1.0
+1.5
+2.0
+25
+20
+15
+10
+5
0
–5
–10
–15
–20
–25
Closed-loop gain γ
Equilibrium climate sensitivity ΔT
to 2x CO
2
(K)
Process
engineers’
design limit
γ
+0.1
Unstable
response
Stable
response
Unphysical response
Fig. 5 Climate sensitivity DT
?
at CO
2
doubling against closed-loop
gains g
?
on [-1, ?2]
Sci. Bull. (2015) 60(1):122–135 131
123
CMIP5 feedback-sum interval into line with IPCC’s interval.
If, however, the loop gain g is indeed below the process
engineers’ limit for stability, namely ?0.1, compatible with
the results in [21, 23], then the simple model’s output giving a
climate-sensitivity interval 1.0 [0.8, 1.3] K may be preferable.
9 How skillful is the model?
Remarkably, though the model is very simple, its output
proves to be broadly consistent with observation, while the
now-realized projections of the general-circulation models
have proven to be relentlessly exaggerated. If, for instance,
the observed temperature trend of recent decades were
extrapolated several decades into the future, the model’s
output would coincident with the observations thus
extrapolated (Fig. 6).
10 Discussion
The irreducibly simple model presented here aims speciﬁ-
cally to study climate sensitivity. Though it is capable of
representing in a rough and ready fashion all the forcings
and feedbacks discussed in AR5, the question arises whether
extreme simplicity renders such models altogether valueless
in contrast to the more complex general-circulation models.
Recently, it was explained in [45] that although the
complex models cover many physical, chemical and bio-
logical processes in their representation of the Earth’s cli-
mate system, the added complexity has naturally led to
great difﬁculty in identifying the chains of causality in the
climate object—what the authors call ‘the processes most
responsible for a certain effect.’
Two recent examples of the substantial uncertainty in
representing climate by complex models indicate that greater
complexity does not necessarily entail improved perfor-
mance, despite myriad improvements and intense scrutiny.
The ﬁrst example: It was recently reported [46] that
increased spatial resolution had led to improvements in
simulations of sea-level pressure, surface temperatures,
etc., in GISS’ latest model, E2, but that simultaneously,
‘some degradations are seen in precipitation and cloud
metrics.’’ Increased spatial resolution in a model, therefore,
does not automatically lead to improvement.
Table 7 Comparison of the climate-sensitivity interval 1.0 [0.8, 1.3] K generated by the model with IPCC’s climate-sensitivity interval 3.2 [2.1,
4.7] K [AR5, SPM]
fgk
?
DF
2x
DT
2x
DT
2x
Variance
Figure 5 Figure 5 k
0
(1-g)
-1
5.35 ln 2 Model (k
?
DF
2x
) CMIP5 AR5-model
model
(W m
-2
K
-1
)(KW
-1
m
2
)(Wm
-2
) (K) (K) (%)
-1.60 -0.5 0.208 0.77 2.1 ?172
-0.64 -0.2 0.260 3.708 0.96 3.2 ?233
?0.32 ?0.1 0.347 1.29 4.7 ?264
Table 8 Charney-sensitivity estimates from all ﬁve IPCC Assessment
Reports and, in bold face, from the simple model
Climate-sensitivity estimates Central
(K)
Lower
(K)
Upper
(K)
FAR (Models) 4.0 1.9 5.2
FAR (SPM) 2.5 1.5 4.5
SAR (SPM) 2.5 1.5 4.5
TAR (Models) 3.0 1.7 4.2
TAR (SPM) None 1.5 4.5
AR4 (CMIP3 models) 3.26 2.0 4.5
AR4 (SPM) 3.0 2.0 4.5
AR5 (SPM) None 1.5 4.5
AR5 (CMIP5 models) 3.2 2.1 4.7
AR5 (CMIP5: central estimate rebased
to mean feedback sumf)
2.9
AR5: adjusted for AR5 feedback sum
f on 1.5 [1.0, 2.2]
2.2 K 1.7 3.9
Simple model: f on -0.64 [-1.6, ?0.32] 1.0 K 0.8 1.3
2000 2010 2020 2030 20402000 2010 2020 2030 2040
Hansen (1988, scen. A)
IPCC (2007)
IPCC (1990)
IPCC (2013 2
nd
draft)
IPCC (2013 final draft)
OBS: HadCRUT4, 63 yr
OBS: RSS, 17 yr
Hansen (1988, scen. A)
IPCC (2007)
IPCC (1990)
IPCC (2013 2
nd
draft)
IPCC (2013 final draft)
OBS: HadCRUT4, 63 yr
OBS: RSS, 17 yr
0.50 K0.50 K
0.380.38
0.300.30
0.230.23
0.130.13
–1
–1
Year
2.0
1.0
0.5
2.5
1.5
(K)
Fig. 6 Near-term global warming projections (brick-red region)on
[0.13, 0.50] K decade
–1
, compared with observations (green region)
that fall on [0.0, 0.11] K decade
–1
, and the simple model’s 21st-
century warming projections (yellow arrow), falling on 0.09 [0.06,
–1
132 Sci. Bull. (2015) 60(1):122–135
123
The second example: The IPSL-CM5A modeling
group’s recent study [47] of the skill of horizontal and
vertical atmospheric grid conﬁguration in representing the
observed climate reported that, when the number of
atmospheric layers was increased from 19 to 39 to improve
stratospheric resolution, a substantial global energy
imbalance requiring retuning of model parameters resulted,
but that, paradoxically, these signiﬁcant impacts of the
model’s grid resolution had not led to any signiﬁcant
changes in projected climate sensitivity.
It is not necessarily true, therefore, that improvements in
the resolution of a model will reﬁne the determination of
climate sensitivity. By the same token, a reduction in
complexity—even an irreducible reduction—does not
necessarily entail a reduction in the reliability with which
climate sensitivity is determined.
On the other hand, it would be inappropriate to claim
that the simple model is preferable to the complex general-
circulation models. Its purpose is more limited than theirs,
being narrowly focused on determining the transient and
equilibrium responses of global temperature to speciﬁed
radiative forcings and feedbacks in a simpliﬁed fashion.
The simple model is not a replacement for the general-
circulation models, but it is capable of illuminating their
performance. It also puts climate-sensitivity modeling
within the reach of those who have no access to or famil-
iarity with the general-circulation models. In effect, this
paper is the user manual for the simple model, bringing it
within the reach of all who have a working knowledge of
elementary mathematics and physics.
Irreducible simplicity is the chief innovation embodied in
the simple model. While it is rooted in the mainstream
mathematics and physics of climate sensitivity and is capable
of reﬂecting no less wide a range of scenarios than the gen-
eral-circulation models, it allows a rapid but not unreliable
determination of climate sensitivity by anyone even at
undergraduate level, providing insights not only into the
relevant physics but also into the extent to which the more
complex models are adequately reﬂecting the physics.
The complex general-circulation models have been run-
ning hot for a quarter of a century. The simple model con-
ﬁrms the hot running and exposes several of the reasons for it.
Firstly, application of the simple model reveals that the
central climate-sensitivity estimate in the CMIP5 ensemble
is somewhat too high because IPCC has taken its mid-
range climate-sensitivity estimate as the mean of its upper-
and lower-bound estimates rather than determining it from
the mean feedback sum f
?
. By contrast, in [38] the central
climate-sensitivity estimate was perhaps more correctly
derived from the central feedback-sum estimate
f
?
= 2.1 W m
-2
K
-1
, the exact mean of the lower and
upper bounds f
?
on [1.3, 2.9] W m
-2
K
-1
. Accordingly,
in [38] the central climate-sensitivity estimate 3.5 K is
signiﬁcantly closer to the lower-bound estimate 2.0 K than
to the upper-bound estimate 12.7 K. The rapidly increasing
slope of climate sensitivity against loop gain g
?
as the
value of g
?
approaches unity (the singularity in the Bode
feedback-ampliﬁcation equation [48]), is the reason for this
asymmetry (Fig. 5), and is also the reason for the extre-
mely high-sensitivity estimates sometimes presented in the
journals. Implicitly, f
?
in the CMIP5 ensemble falls on
1.923 [1.434, 2.411] W m
-2
K
-1
. The mean of these two
values is 1.923 W m
-2
K
-1
. Based on the mean feedback
sum f
?
= 1.923 W m
-2
K
-1
, the CMIP5 central estimate
of climate sensitivity should have been 2.9 K, not 3.2 K.
Secondly, the simple model reveals that the climate
sensitivity 3.3 [2.0, 4.5] K in AR4 should have fallen
sharply to 2.2 [1.7, 3.7] K in AR5 commensurately with the
reduction of the feedback-sum interval between the two
reports (Fig. 3). For the variance between the CMIP3 and
CMIP5 projections of climate sensitivity is inferentially
conﬁned to the feedback-sum interval. If the CMIP5
models took account of signiﬁcant net-positive feedbacks
not included in AR5, Fig. 9.43, in the chart of climate-
relevant feedbacks (Fig. 3), it is not clear why that chart
was not updated to include them. The sharp reduction of
the feedback-sum interval in CMIP5 and hence in AR5
compared with the interval in CMIP3 and hence in AR4
mandates a sharp reduction in the climate-sensitivity
interval, which, however, was instead increased somewhat.
Thirdly, the simple model shows that even the reduced
feedback-sum interval in CMIP5 and hence in AR5 seems
implausibly high when set against the thermostasis over
geological timescales shown in [40]. In Fig. 5, g B?0.1 is
consistent with the inferred thermostasis. Charney sensi-
tivity would then be 1.3 K or less—below even the lower
bound of the climate-sensitivity interval [1.5, 3] K in AR5.
Fourthly, the simple model demonstrates that, in AR5,
the estimates of global warming to 2100 under the four
RCP scenarios (Table 5) project much more warming over
the twenty-ﬁrst century than they should. For instance,
under the RCP 2.6 scenario, it is expected that there will be
no more than 2.6 W m
-2
radiative forcing to 2100, of
which some 2.3 W m
-2
Even adding IPCC’s estimate of 0.6 K committed but
unrealized warming to the small warming yet to be gen-
erated by the 0.3 W m
-2
forcing still to come by 2100
under this scenario, it is not easy to understand why IPCC’s
upper-bound warming estimate on RCP 2.6 is as high as
1.7 K.
Fifthly, application of the simple model raises the
question why AR5 adopted the extreme RCP 8.5 scenario
at all. On that scenario, atmospheric CO
2
concentration is
projected to reach 936 ppmv by 2100 on the basis of two
implausible assumptions: ﬁrst, that global population will
be 12 billion by 2100, though the UN predicts that
Sci. Bull. (2015) 60(1):122–135 133
123
population will peak at little more than 10 billion by not
later than 2070 and will fall steeply thereafter; and sec-
ondly, that coal will contribute as much as 50 % of total
energy supply, though gas is rapidly replacing coal in many
countries, a process that will accelerate as shale gas comes
on stream. Furthermore, the observed increase in CH
4
concentration at a mean rate of 3 ppbv year
-1
from 1990 to
2011, taken with the history of very substantial over-pre-
diction of the CH
4
growth rate, does not seem to justify
IPCC in projecting that, on the RCP 8.5 scenario, the mean
rate of increase in CH
4
concentration from 2015 to 2100
will be 21 ppbv year
-1
, seven times the observed rate of
increase over recent decades.
The utility of the simple model lies in identifying dis-
crepancies such as those enumerated above. It should not
be seen as a substitute for the more complex models, but as
a simple benchmark against which the plausibility of their
outputs may be examined.
11 Conclusion
Resolving the discrepancies between the methodology
adopted by IPCC in AR4 and AR5 is vital. Once those
discrepancies are corrected for, it appears that the impact of
anthropogenic global warming over the next century, and
even as far as equilibrium many millennia hence, may be
no more than one-third to one-half of IPCC’s current
projections.
Suppose, for instance, that the equilibrium response to a
CO
2
doubling is, as the simple model credibly suggests it is,
\1 K. Suppose also that the long-run CO
2
fraction proves
to be as high as 0.9. Again, this possibility is credible.
Finally, suppose that remaining affordably recoverable
reserves of fossil fuels are as much as thrice those that have
been recovered and consumed so far. Then, the total
warming we shall cause by consuming all remaining
recoverable reserves will be little more than 2.2 K, and not
the 12 K imagined by IPCC on the RCP 8.5 scenario. If so,
the case for any intervention to mitigate CO
2
emissions has
not necessarily been made: for the 2.2 K equilibrium
warming we project would take place only over many
hundreds of years. Also, the disbeneﬁts of more extreme
heat may well be at least matched by the beneﬁts of less
extreme cold. It is no accident that 90 % of the world’s
living species thrive in the warm, wet tropics, while only
1 % live at the cold, dry poles. As a benchmark, AR5
estimates that adaptation to the 2–3 K global warming it
expects by 2100 will cost 0.2 %–2.0 % of global GDP,
broadly in line with the cost estimate of 0–3 % of GDP in
Lord Stern’s report for the UK Government on the eco-
nomics of climate change in 2006. However, the reviewed
journals of economics generally report that the cost of
mitigation today would be likely to exceed these low costs
of adaptation to projected global warming, perhaps by as
much as one or two orders of magnitude.
Under different assumptions, the simple model is of course
capable of reaching conclusions more alarming (but arguably
less reasonable) than those that have been sketched here. Be
that as it may, the utility of the model lies in making acces-
sible for the ﬁrst time the distinction between the relative
contributions of forcings and feedbacks; in exposing anom-
alies requiring clariﬁcation in the outputs of the general-cir-
culation models, which seem to agree ever more closely with
each other while departing ever farther from observation
(Fig. 1); and, above all, in facilitating the rapid and simple
estimation of both transient and equilibrium climate sensi-
tivity under a wide range of assumptions and without the need
either for climatological expertise or for access to the world’s
most powerful computers and complex models. The simple
model has its limitations, but it has its uses too.
Conﬂict of interest The authors declare that they have no conﬂict
of interest.
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... Lewis and Curry [25] calculated an ECS median of 1.50°C (with 5-95% range: 1.05-2.45°C). Bates [26] and Monckton et al. [27] evaluated a climate sensitivity in the neighborhood of 1°C. Kluft et al. [28] found an ECS range of 2.09-2.40°C ...
... Low ECS values agree with some alternative studies that have found serious disagreements between observations and the CMIP6 model predictions [17,18,30]. In fact, several observational-based and Earth thermal radiation studies suggest low ECS values between 0.5 and 2.5°C with a median around 1.5°C [4,5,[24][25][26][27][28][29]. These estimates are significantly lower than most of the GCM ECS values. ...
... The above reasoning is very approximate, but the result would agree well with several studies [4,5,[24][25][26][27][28][29]. For example, Scafetta [5] proposed an interpretation of the dynamics observed in the global surface temperature records based on natural oscillations since 1850 and concluded that the ECS central estimate had to be around 1.5°C; however, the proposed ECS value could also be lower (that is between 1.0 and 1.5°C) because the observed 0.9-1.0°C ...
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... This is even the case for some working within Paradigm 1, e.g., Schwartz (2008) [236]. However, within Paradigm 1, the debate is considered non-trivial, e.g., see the debate over the Schwartz (2007) [237] study between Knutti et al. (2008) [238] and Schwartz (2008) [236], or that over Monckton et al. (2015a) [216] between Richardson et al. (2015) [239] and Monckton et al. (2015b) [240]. ...
... Therefore, rather than considering just one value for the climate sensitivity, for the rest of our analysis, we will consider a range of six different values for TCR: 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 °C This covers the IPCC's current "likely" range of 1.0-2.5 °C, but also considers a lower value of 0.5°C, recognizing that several recent studies have argued that the TCR could be less than 1.0 °C, e.g., refs. [152,207,208,215,216,219,221,227] as well as a higher value of 3.0°C. Similarly, we consider a range of six different values for ECS (1, 2, 3, 4, 5, and 6 °C), which encompasses the IPCC's current "likely" range of 1.5-4.5 °C, but also considers the possibility that the ECS might be lower than 1.5 °C, e.g., refs. ...
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... 0.5-2.5°C (Bates, 2016;Christy and McNider, 2017;van Wijngaarden and Happer, 2020;Kluft et al., 2019;Lewis and Curry, 2018;Lindzen and Choi, 2011;McKitrick and Christy, 2020;Monckton et al., 2015;Smirnov and Zhilyaev, 2021;Stefani, 2021). Indeed, decadal and millennial climatic oscillations (Alley, 2004;Christiansen and Ljungqvist, 2012;Esper, 2012;Kutschera et al., 2017;Ljungqvist, 2010;Matskovsky and Helama, 2014;Moberg, 2005;Scafetta, 2014Scafetta, , 2020a and additional solar/astronomical forcings are still debated (Connolly et al., 2021;Scafetta et al., 2004;Scafetta and West, 2006;Scafetta, 2012Scafetta, , 2013Scafetta, , 2021aScafetta et al., 2020b;Svensmark et al., 2017). ...
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... 0.5-2.5°C (Bates, 2016;Christy and McNider, 2017;van Wijngaarden and Happer, 2020;Kluft et al., 2019;Lewis and Curry, 2018;Lindzen and Choi, 2011;McKitrick and Christy, 2020;Monckton et al., 2015;Smirnov and Zhilyaev, 2021;Stefani, 2021). Indeed, decadal and millennial climatic oscillations (Alley, 2004;Christiansen and Ljungqvist, 2012;Esper, 2012;Kutschera et al., 2017;Ljungqvist, 2010;Matskovsky and Helama, 2014;Moberg, 2005;Scafetta, 2014Scafetta, , 2020a and additional solar/astronomical forcings are still debated (Connolly et al., 2021;Scafetta et al., 2004;Scafetta and West, 2006;Scafetta, 2012Scafetta, , 2013Scafetta, , 2021aScafetta et al., 2020b;Svensmark et al., 2017). ...
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Plain Language Summary The last‐generation Coupled Model Intercomparison Projects (CMIP6) global circulation models (GCMs) are used by scientists and policymakers to interpret past and future climatic changes and to determine appropriate (adaptation or mitigation) policies to optimally address scenario‐related climate‐change hazards. However, these models are affected by large uncertainties. For example, their equilibrium climate sensitivity (ECS) varies from 1.83°C to 5.67°C, which makes their 21st‐century predicted warming levels very uncertain. This issue is here addressed by testing the GCMs' global and local performance in predicting the 1980–2021 warming rates against the ERA5‐T2m records and by grouping them into three equilibrium climate sensitivity (ECS) classes (low‐ECS, 1.80–3.00°C; medium‐ECS, 3.01–4.50°C; high‐ECS, 4.51–6.00°C). We found that: (a) all models with ECS > 3.0°C overestimate the observed global surface warming; (b) Student t‐tests show model failure over 60% (low‐ECS) to 81% (high‐ECS) of the Earth's surface. Thus, the high and medium‐ECS GCMs do not appear to be consistent with the observations and should not be used for implementing policies based on their scenario forecasts. The low‐ECS GCMs perform better, although not optimally; however, they are also found unalarming because for the next decades they predict moderate warming: ΔTpreindustrial→2050 ≲ 2°C.
... These 9 facts suggest CO2's greenhouse-warming potential, logarithmically falling & now "well into the saturation regime" (32), is canceled by underestimated negative feedbacks, e.g. more aerosol emitted by faster forest growth due to warming & CO2 fertilization (33, 34, 35). Thus climate models run too hot (36). IPCC says effects of "aerosol-cloud interactions ... carry large uncertainties" (37; NB large error bars in 30-figSPM.5). ...
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Has the IPCC made the most expensive (trillions) scientific blunder of all time by portraying life-giving CO2 as a 'pollutant'? Yes. This 500-word 5-minute abstract, for a 2021 conference, is the culmination of 5 years of independent literature research of the entire scientific literature relevant to climate change (not just 'climate science', which is fatally biased by its dependence (for funding) on public belief in a 'climate emergency' due to 'man-made' warming). I'm a professional geologist. My hope is that other scientists, upon reading this abstract, will quickly appreciate CO2's obvious innocence, and spread the truth to colleagues. friends and family. For the well-being of society and your children, grandchildren, etc., it's up to us, until now the 'silent majority', to deny the fake 'consensus' (among CLIMATE scientists only), and stop the BBC and other media organisations peddling their ignorance-based misinformation to the trusting public, e.g. … https://www.bbc.co.uk/news/av/science-environment-52926683 FOR MORE CO2 TRUTHS (including feedbacks) in 5 minutes each, see my … https://www.researchgate.net/publication/341622566 … https://www.researchgate.net/publication/348369922 … https://www.researchgate.net/publication/348689944 Please consider pressing the 'Recommend' button (blue arrow, on the right). The more scientists denouncing the preposterous 'man-made warming' fallacy (a mere belief, like any other religion), the more 'civilians' will take notice and start thinking for themselves at last; maybe even politicians and opinion-moulding journalists.
... Comprehensive SCMs are also used to simulate the climate or carbon cycle (Friedlingstein et al., 2014;Joos et al., 1999;Knutti et al., 2008), explore responses to anthropogenic perturbations (Geoffroy et al., 2013;Hope, 2006;Meinshausen et al., 2009;Rogelj et al., 2014), or address model spread in the various model intercomparison projects (MIPs) (Knutti and Sedláček, 2012;Monckton et al., 2015;Rogelj et al., 2012). These analyses often include comparisons to more complex models . ...
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... A 2011 Asia-Pacific Journal of Atmospheric Science study using observational data rather than computer climate models concluded that "the models are exaggerating climate sensitivity" and overestimate how fast the earth will warm as CO2 levels increase (Lindzen & Choi, 2009). Two other studies using observational data found that IPCC projections of future global warming are too high (Monckton, et al., 2015), (Lewis & Curry, 2016). In a 2014 article, climatologist and former NASA scientist Roy Spencer, PhD, concluded that 95% of climate models have "over-forecast the warming trend since 1979." ...
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... With a quite different approach based on the mainly rural temperature trend of the Northern Hemisphere and considering an TSI increase as described by Scafetta & Willson [12], Soon and Connelli even deduce from their calculated residual temperature trends a climate sensitivity of only 0.44 ∘ C [15]. From an irreducibly simple model and only critically referring to published data from AR4 and AR5, Monckton et al. [78] also conclude that an ECS smaller than 1 ∘ C seems quite likely. ...
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Global Physical Climatology, Second Edition, provides an introduction to the science of climate and climate change. It begins with a basic introduction to the climate system, and then introduces the physics of the climate system, including the principles and processes that determine the structure and climate of the atmosphere, ocean, and land surface. This basic knowledge is then applied to understanding natural variability of the climate in both the present and past, the sensitivity of climate to external forcing, explanations for the ice ages, and the science of human-induced climate change. The physical principles and computer models necessary for understanding past climate and predicting future climate are introduced. Covers a great range of information on the Earth's climate system and how it works Includes a basic introduction to the physics of climate suitable for physical science majors Provides an overview of the central themes of modern research on climate change suitable for beginning researchers Incorporates problem sets to aid learning Offers an authoritative, clearly written, well-illustrated text with up-to-date data and modeling results.
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