Maximum entropy production, cloud feedback, and climate change
Garth W. Paltridge,1,2 Graham D. Farquhar,1 and Matthias Cuntz1
Received 7 March 2007; revised 7 May 2007; accepted 2 July 2007; published 26 July 2007.
[1] A steady-state energy-balance climate model based on
a global constraint of maximum entropy production is used
to examine cloud feedback and the response of surface
temperature T to doubled atmospheric CO2. The constraint
ensures that change in zonal cloud amount q necessarily
involves change in the convergence KX of meridional
energy flow. Without other feedbacks, the changes in q, KX
and T range from about 2%, 2 Wm
2 and 1.5 K respectively
at the equator to
2%,
2 Wm
2 and 0.5 K at the poles.
Global-average cloud effectively remains unchanged with
increasing CO2 and has little effect on global-average
temperature. Global-average cloud decreases with
increasing water vapour and amplifies the positive
feedback of water vapour and lapse rate. The net result is
less cloud at all latitudes and a rise in T of the order of 3 K at
the equator and 1 K at the poles. Ice-albedo and solar
absorption feedbacks are not considered. Citation: Paltridge,
G. W., G. D. Farquhar, and M. Cuntz (2007), Maximum entropy
production, cloud feedback, and climate change, Geophys. Res.
Lett., 34, L14708, doi:10.1029/2007GL029925.
1. Introduction
[2] Dewar [2003, 2005] recently published a proof of the
concept of maximum entropy production (MEP) as it
applies to non-linear systems. The proof revived interest
in the MEP-based climate model of one of us [Paltridge,
1975, 1978] and the physics and mathematical treatment of
the model were examined and updated as described briefly
in the Appendix. The MEP constraint allows the model to
calculate the broad steady-state distribution of cloud and
surface temperature without the need for detailed consider-
ation of the internal dynamics of the system. In principle
therefore the model inherently includes cloud feedback,
which is perhaps the most arguable (and potentially the
most significant) of the various feedbacks built into large-
scale general circulation climate models. Despite the fact
that cloud feedback was formally identified in the early days
of the World Climate Research Program as one of the most
significant scientific problems restricting climate research,
even the sign of cloud feedback is still not known for certain
today [Randall et al., 2003].
[3] This paper examines, albeit at the basic level of an
energy balance climate model, what the MEP principle
suggests with regard to cloud feedback. Radiative forcing
information from various sources is used to calculate
changes in the infrared input parameters of the MEP model
caused by a doubling of CO2 and by the subsequent
feedback involving atmospheric water vapour and lapse
rate (WV/LR feedback). Sensitivities of cloud amount q
and surface temperature T to doubled CO2 are established
from model trials with and without the changes of the input
parameters.
2. Long-Wave Parameters of the MEP Model
[4] The model assumes a two-band radiating atmosphere
in the long-wave part of the spectrum. All the atmospheric
emission or absorption takes place in one of the bands
which is regarded as 100% opaque. The other represents the
8 to 14 micron atmospheric window and the various smaller
windows in the absorption spectrum of the atmosphere, and
is regarded as 100% clear. The absorbing gases of the
atmosphere are regarded as a blanket, the top of which in
clear skies corresponds to the height from which the
radiation of the opaque band is emitted upward. The bottom
of the blanket radiates downwards from a radiative temper-
ature very close to ground temperature, so that the net
exchange of radiation between ground and atmosphere
within the opaque band is effectively zero.
[5] The emissivity ea of the atmosphere is determined by
the ratio of the width of the opaque band relative to that of
the total black-body spectrum, but weighted appropriately
by the shape of the spectrum. Transmission of radiation
between ground and space in clear skies (or between ground
and cloud base in cloudy skies) can take place only through
the atmospheric window whose width is 1-ea. Upward
emission of radiation Ru from the top of the radiating
blanket in clear skies is given by easTbt4, where s is the
Stefan Boltzmann constant and Tbt is the temperature at the
top of the radiating blanket. Tbt is defined in terms of
the black-body radiation at that temperature, but expressed
as a fraction of the black-body radiation at the temperature
of the Earth’s surface. That is, it is expressed in terms of a
parameter F = sTbt4 /sT 4 such that Ru = FeasT 4.
[6] The upward flux Ru
C of radiation to space from cloud
and from the atmosphere above cloud is envisaged as black-
body radiation emanating from an effective cloud top that is
slightly higher and colder than real cloud top. It takes into
account the presence above real cloud top of a blanket of
CO2 whose emissivity and thickness are much less than
those of the clear-sky blanket. The effective cloud top
temperature is again defined in terms of black-body radia-
tion at that temperature. It is expressed in terms of param-
eters f and f 0 such that Ru
C = f. f 0sT 4. Here f 0sT 4 is the
black-body radiation at the temperature of cloud base, and f
is essentially a fractional measure of the temperature differ-
ence between cloud base and effective cloud top.
[7] Increasing the concentration of atmospheric CO2 has
the immediate effect of slightly closing the atmospheric
window (increasing the emissivity ea), slightly thickening
GEOPHYSICAL RESEARCH LETTERS, VOL. 34, L14708, doi:10.1029/2007GL029925, 2007
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1Research School of Biological Sciences, Australian National University,
Canberra, ACT, Australia.
2Also at Institute of Antarctic and Southern Ocean Studies, University
of Tasmania, Hobart, Tasmania, Australia.
Copyright 2007 by the American Geophysical Union.
0094-8276/07/2007GL029925$05.00
L14708 1 of 6

the clear-sky radiative blanket and raising its top to a lower
temperature (decreasing the value of F ) and, by the same
mechanism, slightly raising the effective cloud-top to a
lower temperature (decreasing the value of f ). Feedback
via surface temperature raises the concentration of atmo-
spheric water vapour and further increases ea and decreases
F and f. It is assumed that lapse-rate feedback derives
primarily from latent heat release associated with the
formation of rain in the upper levels of the cloudy atmo-
sphere, thereby increasing the temperature of effective
cloud-top relative to the ground and hence increasing f
without affecting ea and F.
3. Calculations
[8] In terms of the above description, the downward
long-wave flux Rd at the ground and the upward long-wave
flux Ru at the top-of-the-atmosphere (TOA) are given by
Rd ¼ easT4 1
qð Þ þ f 0easT4q ð1Þ
and
Ru ¼ 1
eað ÞsT4 þ FeasT4


1
qð Þ þ f 0f sT4q ð2Þ
where q is the fractional cloud amount. These equations can
be differentiated with respect to CO2 concentration C and
(separately) to surface temperature T both for clear-sky
conditions (q = 0) and for fully clouded conditions (q = 1).
The differentiated equations can then be manipulated so as
to yield expressions for the partial derivatives @ea /@C, @F /
@C and @f /@C (each of them constant with respect to water
vapour and lapse rate) and @ea /@T, @F/@T and @f /@T (each
of them constant with respect to C) as a function of the
corresponding partial derivatives of the radiative fluxes Rd
and Ru.
[9] Global-average (or mid-latitude) values of the radia-
tive flux partial derivatives and their source references are
quoted in Table 1 together with the derivatives of the
infrared parameters calculated from them. The derivatives
with respect to C assume the present-day concentration of
CO2 to be one unit, so that numerically they are equal to the
changes from doubled CO2 – this on the further assumption
that the radiative response is linear over the range. The
derivatives with respect to T reflect the changes in water
vapour and lapse rate brought about by a rise in T of 1 K,
and among other things involve an assumption of constant
relative humidity at all levels of the atmosphere during
the changes of T. @f/@T is the sum of the changes due to the
separate water-vapour and lapse-rate feedbacks (see the
appropriately subscripted derivatives in Table 1).
[10] The parameters ea, F and f are fixed inputs for any
particular trial of the MEP model. Each trial yields a
distribution with latitude of T, q and the meridional energy
flux convergence KX. Here we simply re-run the model with
changes to the parameters as indicated by the derivatives of
the right hand column of Table 1. It is re-run first with the
changes corresponding only to doubled CO2. That is, ea, F
and f at each latitude are changed to ea + (@ea /@C)DC, F +
(@F/@C)DC and f + (@f /@C)DC where DC = 1. The zonal-
average increases DT in surface temperature so derived are
then used as the starting point of an iterative set of trials
taking WV/LR feedback into account. (@ea /@T)DT, (@F/
@T)DT and (@f/@T)DT are added to the relevant parameters
at each trial, with the DT at each latitude deriving from the
output of the preceding two trials. The iteration stops when
all the DT are infinitesimally small.
[11] The process assumes that CO2 is a well-mixed gas
and therefore that the parameter changes for doubled CO2
are the same at all latitudes. A similar and far more drastic
assumption is made with regard to water vapour and lapse
rate in that the sensitivities to T (the @ea/@T etc. of Table 1)
are also set to be the same at all latitudes.
[12] There are problems of stability in the optimisation
process (see the Appendix) associated with numerical
application of the MEP constraint when the non-linear
WV/LR feedback on temperature is large. The results here
are derived by repeating the WV/LR feedback calculations
over three equal steps (DT/3) of the ‘starting point’ tem-
perature change DT. This is a considerable approximation
that partially linearises (and reduces) the feedback.
4. Results and Discussion
[13] Case A of Figure 1 is the no-feedback distribution of
temperature change calculated when cloud, meridional
energy convergence, water vapour and lapse rate of each
latitude zone are held constant while the changes @ea /@C,
Table 1. Values of Flux Derivatives and Derived Infrared Parametersa
Flux
Derivatives
Value for
Clear Sky
Value for
Mean Cloud
Value for
q = 100%
Derived
Parameters Value
@Rd/@C 1.8 (1) @ea/@C .0045
@Ru/@C
4.5 (2) @F/@C
.0127
@Ru/@C
3.7 (2)
3.2 @f/@C
.0097
(@Rd/@T)wv 2.2 (3) (@ea/@T)wv .0056
(@Ru/@T)wv
1.8 (4) (@F/@T)wv
.0027
(@Ru/@T)wv
1.2 (4)
0.8 (@f/@T)wv
.0024
(@Ru/@T)lr 0 0.8 (5) 1.5 (@f/@T)lr .0048
aParenthesized numbers refer to sources as follows: (1) Fels et al. [1991]; (2) Intergovernmental Panel on Climate Change
[2001]; (3) derived from experimental data of Swinbank [1963]; (4) M. Dix (personal communication, 2006) involving
calculations with the radiation package of the CSIRO Mark 3 Climate Model; and (5) Bony et al. [2006]. Subscripts wv and lr
refer to feedback of water vapour and lapse rate respectively. The 100% cloud values are calculated from the clear-sky and
mean-cloud values assuming a mean cloud cover of 60%. (@Rd/@T)wv and (@Ru /@T)wv are the differences between fluxes
calculated for an atmosphere of constant relative humidity and one of constant specific humidity when the temperature at all
levels is raised by 1 K. They are therefore the changes due only to the change in water vapour when the temperature is raised.
Units of the @R/@C in columns 2, 3, and 4 are Wm
2, and of the @R/@T are Wm
2 K
1.
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