ATMOSPHERIC DYNAMICS
Constrained work output of the moist
atmospheric heat engine in a
warming climate
F. Laliberté,1 J. Zika,2 L. Mudryk,3 P. J. Kushner,1 J. Kjellsson,3 K. Döös4
Incoming and outgoing solar radiation couple with heat exchange at Earth’s surface
to drive weather patterns that redistribute heat and moisture around the globe, creating
an atmospheric heat engine. Here, we investigate the engine’s work output using
thermodynamic diagrams computed from reanalyzed observations and from a climate
model simulation with anthropogenic forcing. We show that the work output is always
less than that of an equivalent Carnot cycle and that it is constrained by the power
necessary to maintain the hydrological cycle. In the climate simulation, the hydrological
cycle increases more rapidly than the equivalent Carnot cycle. We conclude that the
intensification of the hydrological cycle in warmer climates might limit the heat engine’s
ability to generate work.
As a reflection of the seminal work of Carnot, atmospheric motions have been described as an important component of the plane- tary heat engine (1). The concept of a heat engine is closely associated with the idea
of work: For two cycles that transport the same
amount of heat between the same two reservoirs,
the one that generates the least irreversible en-
tropy will produce the most work (2). We quan-
tify the atmosphere’s work output through a
budget of its entropy production. Previous attempts
at obtaining such a budget either resulted in gross
estimates (3, 4) or required highly specific data
from climate models for a precise analysis (5–8).
Some of these studies showed that the hydrolog-
ical cycle was an important contributor to the gen-
eration of irreversible entropy (6, 9, 10), suggesting
that moist processes, including the frictional
dissipation associated with falling hydrometeors
(11, 12), tend to limit the work output of the at-
mospheric heat engine. On a warming Earth, the
increase in precipitable water (13) has been iden-
tified as a reason for the tropical overturning to
slow down (14), and studies over a wide range of
climates suggest that global atmospheric motions
are reduced in extremely warm climates (15–17).
Models forced according to a climate change sce-
nario also exhibit this behavior in their tropical
circulation (18). Here, we employ a method that
uses high-frequency and high-resolution data
to obtain an atmospheric entropy budget from
climate models and reanalyses. This method does
not depend on specialized model output, making
the diagnostic applicable to the suite of models
produced for the Climate Model Intercomparison
Project phase 5 (CMIP5) and paving the way to
a systematic analysis of the entropy budget in
climate models, as proposed by some authors (8).
We base our analysis on the first law of thermodynamics describing moist air (19, 20).
Tds=dt ¼ dh=dt−adp=dt þ mdq T =dt:
The material derivatives of moist enthalpy h and
moist entropy s (19–21) are given by dh=dt and
ds=dt, respectively.
The equation of state used here provides a
comprehensive treatment of moist thermodynamics, including the effect of the latent heat of
fusion on h and s (19, 20). The specific ratio q T
represents the total mass of water divided by the
total mass of wet air (humid air plus water condensate). The work output −adp=dt is given by the
product of the specific volume a with the vertical
velocity in pressure coordinates −dp=dt. The chemical potential m quantifies the effect of adding or
removing moisture; it is equal to the sum of two
terms with different physical meanings (see the
supplementary text). The first of these terms
accounts for the moistening inefficiencies that
accompany the irreversible entropy production
associated with the addition of water vapor to
unsaturated air (9, 10). The second term accounts
for the enthalpy changes associated with the drying and moistening of air. For the atmospheric
thermodynamic cycle, we will show that when m
is positive, mdq T =dt primarily quantifies the moistening inefficiencies accounted for by the first
term, and when m is negative mdq T=dt primarily
quantifies how much power is associated with
combined moisture and dry air fluxes between
the surface and the precipitation level accounted
for by the second term.
Averaging the first law using a mass-weighted
annual and global spatial mean [denoted as {·}]
results in simplification. First, fdh=dtg equals the
difference between interior moist enthalpy sinks
and the moist enthalpy sources at Earth’s surface
stemming from diffusive fluxes. If we assume
that the atmospheric system is in steady state
(and therefore approximately yearly periodic),
the sinks cancel the sources and fdh=dtg van-
ishes. Moreover, under this averaging, Q•moist ¼
fmdq T=dtg is positive because it quantifies the
power necessary to maintain the hydrological
cycle and accounts for the moistening inefficien-
cies (10), and W ¼ f−adp=dtg is also positive be-
cause it is associated with the dissipation of
kinetic energy at the viscous scale (22). Writing
Q• total ¼ f Tds=dtg, then, the first law reads [equa-
tion 4 in (10)]
W ¼ Q• total − Q• moist
W is thus reduced by the moistening inefficien-
cies accounted for by Q•moist. In the following
sections, we obtain a diagnostic for Q•total and
Q• moist based on the area occupied by the atmo-
spheric thermodynamic cycle in a temperature-
entropy diagram (hereafter T − s diagram) and in
a specific humidity-chemical potential diagram
(hereafter q T − m diagram), respectively.
We analyze two different data sources. The
first source is a coupled climate model simulation using the Community Earth System Model
(CESM) version 1.0.2 (23). The time period 1981
to 2098 is simulated using a combination of historical radiative forcing estimates and the Representative Concentration Pathway 4.5 (24) future
scenario (hereafter historical RCP45). The second source is the period 1981 to 2012 of the
Modern-Era Retrospective Analysis for Research
and Applications (MERRA) reanalysis (25). For
these two data sets, we use a recently developed
method (20) to project the material derivative
ds=dt from eulerian space to s•ðs,TÞ, its representation in the T − s diagram. We use the same
method to project dq T=dt to q•T ðm,q TÞ, its representation in the q T − m diagram. Each of these
quantities is associated with a closed, uniquely
defined mass flux stream function in its respective coordinate system.
Ytotalðs, TÞ ¼ ∫∞ T s•ðs, T′Þd T′,
ðs•; T• Þ ¼ ð−∂T Ytotal,∂s Ytotal Þ,
Each stream function describes a separate aspect
of the large-scale atmospheric thermodynamic
cycle. This approach is similar to a method that
has been previously used to study the atmospheric
and oceanic circulations in thermodynamic coor-
dinates (26–29).
In a T − s diagram, the quantity Ytotal describes
a clockwise cycle (Fig. 1, A and B) with three
main branches. In the lower branch, a large
fraction of air is transported along the surface
saturation curve (1000 hPa, 100% relative humidity) and, as it moves toward warmer temperatures,
picks up heat through exchanges at Earth’s surface. In the tropical branch, air is transported
from warm temperatures to colder temperatures at almost constant moist entropy along
the zonal-mean tropical (15°S to 15°N) profile. The
zonal-mean tropical profile thus represents the
transformations that tropical air masses undergo
540 30 JANUARY 2015 • VOL 347 ISSUE 6221 sciencemag.org SCIENCE
1Department of Physics, University of Toronto, Ontario,
Canada. 2University of Southampton, National Oceanography
Centre, Southampton, UK. 3British Antarctic Survey,
Cambridge, UK. 4Department of Meteorology, Stockholm
University, Stockholm, Sweden.
*Corresponding author. E-mail: frederic.laliberte@utoronto.ca
