Maximum Entropy Production in Climate Theory
RICHARD GOODY
Harvard University, Cambridge, Massachusetts
(Manuscript received 21 July 2006, in final form 7 November 2006)
ABSTRACT
R. D. Lorenz et al. claim that recent data on Mars and Titan show that planetary atmospheres are in
unconstrained states of maximum entropy production (MEP). Their model as it applies to Venus, Earth,
Mars, and Titan is reexamined, and it is shown that their claim is not justified. This does not necessarily
imply that MEP is incorrect, and inapplicable to atmospheres, but it does mean that the difficult and
unexplored problem of dynamical constraints on the MEP solution must be understood if it is to be of value
for climate research.
1. The MEP hypothesis
Paltridge (1975, 1978) first proposed that Earth’s cli-
mate structure might be explained from a hypothesis of
maximum entropy production (MEP). If correct, this
proposal would be of crucial importance to future cli-
mate research because it provides the hitherto missing
global constraint of the second law of thermodynamics.
Subsequent investigations have generally supported
Paltridge’s work, but not to the degree that MEP is
accepted as a useful principle in modern climate re-
search.
A recent review by Ozawa et al. (2003) summarizes
the literature since Paltridge’s first contribution, and
concludes with an optimistic assessment of the value of
the MEP hypothesis for climate systems. In my estima-
tion, in addition to Paltridge’s work and directly related
researches, the most important contributions to Ozawa
et al.’s conclusion are the following: first, a theoretical
paper by Dewar (2003) demonstrating the correctness
of MEP for an unconstrained system (i.e., constrained
only by energy and mass conservation); second, 50 yr of
theoretical investigations of laboratory flows, based on
MEP, which Ozawa et al. interpret as supportive of the
hypothesis; and third, a claim by Lorenz et al. (2001)
that the horizontal thermal structures of Mars and Ti-
tan support the MEP hypothesis.
Dewar’s is a statistical theory; his work has not been
challenged to my knowledge, although such theories
have been frequently and unsuccessfully discussed in
the past (Jaynes 1980). The proof is unconstrained,
apart from energy and mass conservation. Thus for any
atmospheric problem in which the fluid equations,
planetary rotation, gravitational constraints on vertical
structure, thermal inertia, etc. are known to be impor-
tant, unconstrained MEP is inapplicable. Since funda-
mental dynamical theory and numerical models indi-
cate that all of these are crucial parameters for climate,
it is not surprising that little advance has been made in
this field. This point was made by Rodgers (1976) in his
comments on Partridge’s first paper. It is, therefore,
disturbing that Dewar needs to claim successes with
climate theory as supportive of his work.
I have a different view from Ozawa et al. (2003) as to
the value of the evidence from investigations of labo-
ratory flows, and I shall comment further in section 4.
Finally, the investigation of Lorenz et al. (2001) is an
order-of-magnitude study using a two-box, uncon-
strained model with gray emission and absorption.
They demonstrate that MEP gives a better account of
the horizontal temperature structure for Mars and Ti-
tan than one particular scaling approximation. If we are
to claim verification for a fundamental thermodynamic
principle, however, quantitative agreement with ob-
served data must be demonstrated rather than improve-
ments over debatable approximations.
The primary objective of this paper is limited. In the
next two sections I shall examine whether the model of
Lorenz et al. (2001) does, in fact, agree with the best
data from Venus, Earth, Mars, and Titan. I find that it
Corresponding author address: Richard Goody, 101 Cumloden
Drive, Falmouth, MA 02540.
E-mail: goody@huarp.harvard.edu
JULY 2007 G O O D Y 2735
DOI: 10.1175/JAS3967.1
© 2007 American Meteorological Society
JAS3967

does not. In section 4 I discuss the imposition of con-
straints on an MEP calculation.
2. A gray, box model for MEP
Following Lorenz et al. (2001) I examine two latitude
zones: zone 1 is tropical (30° N– 30° S), and zone 2 is
extratropical (30°– 90° N and 30°– 90° S). Each is charac-
terized by a single value of the temperature, T, the
incoming solar radiation, S, and the outgoing thermal
radiation, E (assumed equal to
T 4 where
is Stefan’s
constant). Energy balance requires
S1  S2  E1  E2 . 1
The flux of energy between zones is
F  S1  E1  E2  S2 . 2
The rate of entropy production created by this flux is
s˙  F

1
T2  1
T1. 3
From (2) and the definition of E(
T 4)
s˙  T 2
3

S2
T2
 T 1
3

S1
T1
. 4
For maximum entropy production
ds˙
dT1

dT2
dT1


3T 2
2

S2
T 2
2  3T 1
2

S1
T 1
2  0. 5
From (1) it can be shown that
dT2
dT1
 


T1
T2

3
. 6
Consequently,


E1
E2

5
4



S1
S2


1  3E1
S1
1  3E2
S2

. 7
From (1),
S1
E1

1  E1
E2
1  S1
S2
, 8
and similarly for the ratio S2/E2. Substituting (8) into (7),


E1
E2

1
4



S1
S2


4  E2
E1  3S2
S1
4  E1
E2  3S1
S2

. 9
Equation (9) is an algebraic equation for E1/E2 in
terms of S1/S2. The results shown in Table 1 have been
obtained from a numerical solution.
This solution is identical to that of Rodgers (1976),
except that Rodgers’ solution is continuous and re-
quires use of the calculus of variations. Rodgers also
conjectured that the solution would not be greatly
changed if E were to replace T(E1/4) in (3). With the
two-box model this yields exactly
E1
E2



S1
S2

1
2
, 10
the same result as obtained by Rodgers. In the param-
eter range of solar ratio that occurs in Table 1, there is
no significant difference between results from (10) and
(9).
Lorenz et al. (2001) linearize both F and E with re-
spect to temperature. They obtain an approximate re-
lationship between the linearity coefficients. In the lin-
ear limit E1 → E2 and S1 → S2, it can be shown that the
solution of Lorenz et al. is the same as (10).
3. Data for Venus, Earth, Mars, and Titan
Table 1 shows the incoming solar flux and outgoing
thermal flux for Earth, Mars, Venus, and Titan for
three different situations: radiative equilibrium (Eequ 
S); MEP (Emep); and observed (Eobs). The following
procedure was used. The best mean values of S and E
were chosen from a study of all of the available data on
both incoming and outgoing radiances. Since the two
means must be equal [Eq. (1)] the most reliable was
TABLE 1. Solar and thermal emissions for Venus, Earth, Mars,
and Titan. The numbers in parentheses are probable errors for
measured quantities. Where no errors are indicated, no data were
available.
Fluxes (W m2)
Eequ Emep Eobs
Venus Mean 157.3 (2.2) 157.3 157.3 (3.8)
Zone 1 191.9 (2.7) 174.8 161.4 (3.8)
Zone 2 122.8 (1.7) 139.8 152.9 (3.8)
Zone 1
Zone 2
1.56 (0.03) 1.25 (0.01) 1.06 (0.04)
Earth Mean 237.0 (5.9) 237.0 237.0 (5.9)
Zone 1 300.4 (7.5) 269.7 258.5 (6.5)
Zone 2 173.6 (4.3) 204.3 215.5 (5.4)
Zone 1
Zone 2
1.73 (0.06) 1.32 (0.03) 1.20 (0.04)
Mars Mean 116.7 116.7 116.7
Zone 1 140.2 128.7 139.0
Zone 2 93.2 104.7 94.1
Zone 1
Zone 2
1.51 1.23 1.48
Titan Mean 2.81 2.81 2.81 (0.012)
Zone 1 3.43 3.12 2.96 (0.012)
Zone 2 2.20 2.50 2.67 (0.012)
Zone 1
Zone 2
1.56 1.25 1.11 (0.06)
2736 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 64