THE SECOND LAW OF THERMODYNAMICS AND THE
GLOBAL CLIMATE SYSTEM: A REVIEW OF THE MAXIMUM
ENTROPY PRODUCTION PRINCIPLE
Hisashi Ozawa,1 Atsumu Ohmura,2 Ralph D. Lorenz,3 and Toni Pujol4
Received 10 April 2002; revised 19 June 2003; accepted 8 July 2003; published 26 November 2003.
[1] The long-term mean properties of the global climate
system and those of turbulent fluid systems are reviewed
from a thermodynamic viewpoint. Two general expres-
sions are derived for a rate of entropy production due to
thermal and viscous dissipation (turbulent dissipation)
in a fluid system. It is shown with these expressions that
maximum entropy production in the Earth
s climate
system suggested by Paltridge, as well as maximum
transport properties of heat or momentum in a turbulent
system suggested by Malkus and Busse, correspond to a
state in which the rate of entropy production due to the
turbulent dissipation is at a maximum. Entropy produc-
tion due to absorption of solar radiation in the climate
system is found to be irrelevant to the maximized prop-
erties associated with turbulence. The hypothesis of
maximum entropy production also seems to be applica-
ble to the planetary atmospheres of Mars and Titan and
perhaps to mantle convection. Lorenz
s conjecture on
maximum generation of available potential energy is
shown to be akin to this hypothesis with a few minor
approximations. A possible mechanism by which turbu-
lent fluid systems adjust themselves to the states of
maximum entropy production is presented as a self-
feedback mechanism for the generation of available
potential energy. These results tend to support the hy-
pothesis of maximum entropy production that underlies
a wide variety of nonlinear fluid systems, including our
planet as well as other planets and stars. INDEX TERMS: 3220
Mathematical Geophysics: Nonlinear dynamics; 3309 Meteorology and
Atmospheric Dynamics: Climatology (1620); 3379 Meteorology and
Atmospheric Dynamics: Turbulence; 9820 General or Miscellaneous:
Techniques applicable in three or more fields; KEYWORDS: thermody-
namics, global climate, maximum entropy production, energetics
Citation: Ozawa, H., A. Ohmura, R. D. Lorenz, and T. Pujol, The
second law of thermodynamics and the global climate system: A review
of the maximum entropy production principle, Rev. Geophys., 41(4),
1018, doi:10.1029/2002RG000113, 2003.
We must attribute to heat the great movements that we observe
all about us on the Earth. Heat is the cause of currents in the
atmosphere, of the rising motion of clouds, of the falling of rain
and of other atmospheric phenomena ... .
Sadi Carnot (1824)
1. INTRODUCTION
[2] The opening words of Carnot
s original treatise on
thermodynamics provide a good starting point for this
review paper. We consider that Carnot
s view contains
invaluable insight into the subject, which seems to have
been lost from our contemporary view of the world.
Carnot regarded the Earth as a sort of heat engine, in
which a fluid like the atmosphere acts as working sub-
stance transporting heat from hot to cold places, thereby
producing the kinetic energy of the fluid itself. His
general conclusion about heat engines is that there is a
certain limit for the conversion rate of the heat energy
into the kinetic energy and that this limit is inevitable for
any natural systems including, among others, the Earth
s
atmosphere. His suggestion on the atmospheric heat
engine has been rather ignored. It is the purpose of this
paper to reexamine Carnot
s view, as far as possible, by
reviewing works so far published in the fields of fluid
dynamics, Earth sciences, and nonequilibrium thermo-
dynamics.
[3] Figure 1 shows a schematic of energy transport
processes in a planetary system composed of the Earth,
the Sun, and outer space. Shortwave radiation emitted
from the Sun with a brightness temperature of about
5800 K is absorbed by the Earth, mainly in the equatorial
region. This energy is transported poleward through
direct motions of the atmosphere and oceans (the gen-
1Institute for Global Change Research, Frontier Research
System for Global Change, Yokohama, Japan
2Institute for Atmospheric and Climate Science, Swiss Fed-
eral Institute of Technology, Zurich, Switzerland
3Lunar and Planetary Laboratory, University of Arizona,
Tucson, Arizona, USA
4Departament de Fı´sica, Universitat de Girona, Catalonia,
Spain
Copyright 2003 by the American Geophysical Union. Reviews of Geophysics, 41, 4 / 1018 2003
8755-1209/03/2002RG000113$15.00 doi:10.1029/2002RG000113
● 4-1 ●

eral circulation). The energy is finally reemitted to space
via longwave radiation. Thus there is a flow of energy
from the hot Sun to cold space through the Earth. In the
Earth
s system the energy is transported from the warm
equatorial region to the cool polar regions by the atmo-
sphere and oceans. Then, according to Carnot, a part of
the heat energy is converted into the potential energy
which is the source of the kinetic energy of the atmo-
sphere and oceans. In this respect, the Earth
s system
can be regarded as a heat engine operating between
thermal reservoirs with different temperatures (equator
and poles). The determination of the strength of the
circulation, and hence the rate of heat transport, consti-
tutes a fundamental problem in thermodynamics of the
general circulation [e.g., Lorenz, 1967].
[4] Lorenz [1960] suspected that the Earth
s atmo-
sphere operates in such a manner as to generate avail-
able potential energy at a possible maximum rate. The
available potential energy is defined as the amount of
potential energy that can be converted into kinetic en-
ergy. Independently, Paltridge [1975, 1978] suggested
that the mean state of the present climate is reproduc-
ible as a state with a maximum rate of entropy produc-
tion due to horizontal heat transport in the atmosphere
and oceans. Figure 2 shows such an example [Paltridge,
1975]. Without considering the detailed dynamics of the
system, the predicted distributions (air temperature,
cloud amount, and meridional heat transport) show re-
markable agreement with observations. Later on, several
researchers investigated Paltridge
s work and obtained
essentially the same result [Grassl, 1981; Shutts, 1981;
Mobbs, 1982; Noda and Tokioka, 1983; Sohn and Smith,
1993, 1994; Ozawa and Ohmura, 1997; Pujol and Llebot,
1999a, 1999b]. His suggestion was criticized by Essex
[1984], however, since a predominant amount of entropy
production is due to direct absorption of solar radiation
at the Earth
s surface, which was a missing factor in
Paltridge
s work. Since then, the radiation problem has
been a central objection to Paltridge
s work [e.g., Lesins,
1990; Stephens and O
Brien, 1993; Li et al., 1994; Li and
Chylek, 1994]. As we shall discuss in section 3, the large
background radiative down-conversion of energy from
solar to terrestrial temperatures is essentially a linear
process which is irrelevant to the maximized process
related to nonlinear turbulence. In fact, Ozawa and
Ohmura [1997] applied the maximum condition specifi-
cally to the entropy production associated with the tur-
bulent heat transport in the atmosphere and reproduced
vertical distributions of air temperature and heat fluxes
that resemble those of the present Earth. Thus it is likely
that the global climate system is regulated at a state with
a maximum rate of entropy production by the turbulent
heat transport, regardless of the entropy production by
the absorption of solar radiation [Shimokawa and
Ozawa, 2001; Paltridge, 2001]. This result is also consis-
tent with a conjecture that entropy of a whole system
connected through a nonlinear system will increase
along a path of evolution, with a maximum rate of
entropy production among a manifold of possible paths
[Sawada, 1981]. We shall resolve this radiation problem
in this paper by providing a complete view of dissipation
processes in the climate system in the framework of an
entropy budget for the globe.
[5] The hypothesis of the maximum entropy produc-
tion (MEP) thus far seems to have been dismissed by
some as coincidence. The fact that the Earth
s climate
system transports heat to the same extent as a system in
a MEP state does not prove that the Earth
s climate
system is necessarily seeking such a state. However, the
coincidence argument has become harder to sustain now
that Lorenz et al. [2001] have shown that the same
condition can reproduce the observed distributions of
temperatures and meridional heat fluxes in the atmo-
spheres of Mars and Titan, two celestial bodies with
atmospheric conditions and radiative settings very dif-
ferent from those of the Earth. A popular account of this
work is given by Lorenz [2001a] and Lorenz [2003].
[6] Similar suggestions have been proposed in the
general field of fluid dynamics. For thermal convection
of a fluid layer heated from below (i.e., Be´nard [1901]
convection), Malkus [1954] suggested that the observed
mean state represents a state of maximum convective
heat transport. For turbulent flow of a fluid layer under
a simple shear, Malkus [1956] and Busse [1970] sug-
gested that the realized state corresponds to a state with
a maximum rate of momentum transport. Their ap-
proach is now called the “optimum theory” or “upper
bound theory” and is well known in the field [e.g.,
Howard, 1972; Busse, 1978]. Their suggestions were re-
cently shown to be unified into a single condition in
which the rate of entropy production by the turbulent
Figure 1. A schematic of energy transport processes in the
planetary system of the Earth, the Sun, and space. The Earth
receives the shortwave radiation from the hot Sun and emits
longwave radiation into space. The atmosphere and oceans act
as a fluid system that transports heat from the hot region to
cold regions via general circulation.
4-2 ● Ozawa et al.: THERMODYNAMICS OF CLIMATE 41, 4 / REVIEWS OF GEOPHYSICS