at which temperature decreases with height as an air parcel is lifted
adiabatically and without phase change. ($\sim 10$ K km-1). Above this level,
radiative cooling is not sufficient to cause convection, and only weak
overturning is present.

Fig. 4a shows a snapshot of the vertical velocity distribution at 4 km;
updrafts and downdrafts fill the domain, with magnitudes in the range 2-4 m
s-1. What sets the magnitude of these updrafts, and how does it depend on the
radiative cooling rate? The entropy budget provides a useful perspective.

The steady-state entropy budget for the dry RCE case may be written (section
III),

$\langle\dot{Q}_{\text{rad}}\rangle\left(\frac{1}{T_{a}}-\frac{1}{T_{s}}\right)=\langle\dot{S}^{\text{fric}}_{i}\rangle+\langle\dot{S}^{\text{heat}}_{i}\rangle,$

where
$\dot{Q}_{\text{rad}}=A^{-1}\int_{\Omega_{A}}\rho\dot{q}_{\text{rad}}\,dV$ is
the integrated radiative heating rate, expressed in units of W m-2 by dividing
by the area of the domain $A$, the angle brackets refer to a time average,
$T_{s}$ is the surface temperature, and $T_{a}$ is the effective temperature
at which the atmosphere is cooled by radiation, defined,

$\frac{\langle\dot{Q}_{\text{rad}}\rangle}{T_{a}}=\frac{1}{A}\int_{\Omega_{A}}\left\langle\frac{\rho\dot{q}_{\text{rad}}}{T}\right\rangle\,dV.$
(57)

In the atmosphere, we expect frictional dissipation to dominate over molecular
heat diffusion as an irreversible entropy source except in a molecular
boundary layer immediately adjacent to the surface [269, 331]. The entropy
production owing to molecular heat diffusion may therefore be neglected
entirely if we replace the surface temperature $T_{s}$ with the temperature of
the air above this molecular boundary layer, $T_{sa}$ [309]. The entropy
budget of the dry RCE state then becomes,

$\langle\dot{Q}_{\text{rad}}\rangle\left(\frac{1}{T_{a}}-\frac{1}{T_{sa}}\right)=\langle\dot{S}^{\text{fric}}_{i}\rangle.$
(58)

Identifying $T_{sa}$ as the input temperature, $T_{a}$ as the output
temperature, and $|\langle\dot{Q}_{\text{rad}}\rangle|$ as the heat input,
(58) may be used to define the Carnot efficiency (section II.2.2) and
mechanical efficiency (section II.2.3) of convection in RCE.

In a statistically steady state, we expect the rate of generation of kinetic
energy by the atmosphere to equal its dissipation. We may write this balance,

$\langle\dot{W}_{K}\rangle=T_{\text{fric}}\langle\dot{S}^{\text{fric}}_{i}\rangle,$
(59)

where $T_{\text{fric}}$ is the effective temperature of frictional dissipation
defined as in (21). The left-hand side of (59) gives the work done by the
pressure gradient force in producing kinetic energy of the winds, and the
right-hand side gives the frictional dissipation rate.

Combining the entropy budget (58) and the mechanical energy budget (59) allows
one to derive an estimate of the vigor of convection (as measured by its rate
of work) given the heat input $|\langle\dot{Q}_{\text{rad}}\rangle|$ and
estimates of the effective temperatures $T_{sa}$, $T_{a}$, and
$T_{\text{fric}}$ [63],

$\langle\dot{W}_{K}\rangle=\langle\dot{Q}_{\text{rad}}\rangle
T_{\text{fric}}\left(\frac{1}{T_{a}}-\frac{1}{T_{sa}}\right).$ (60)

The pressure-work $\dot{W}_{K}$ may be related, using hydrostatic balance, to
the upward buoyancy flux, which, in turn, may be used to derive a scale for
the vertical velocity [see e.g., 73]. For dry RCE, the mechanical efficiency
is close to its maximum value, and, for fixed values of the temperatures
$T_{sa}$, $T_{a}$, and $T_{\text{fric}}$, (60) gives that the rate of work
done by the convective heat engine scales linearly with the heat input
$|\langle\dot{Q}_{\text{rad}}\rangle|$.

Figure 4: Snapshot of vertical velocity at a height of 4 km and at day 90 in
simulations of (a) dry and (b) moist radiative-convective equilibrium. Black
contour in (b) shows regions within clouds (cloud water content greater than
0.01 g kg-1).

####  IV.1.2 Moist radiative-convective equilibrium

Let us now consider the case of moist RCE. The dry RCE simulation is rerun,
but with the lower boundary condition assumed to be a saturated surface of
water (Fig. 3c). The enthalpy flux from the surface now includes evaporative
fluxes of water vapor in addition to sensible heat, and the transport and
interaction of the three phases of water in the atmosphere is accounted for.
blueOnce steady state is reached, the precipitation rate through the lower
boundary is equal to the rate at which water vapor is evaporated into the
atmosphere.

The simulation highlights a number of important differences between dry and
moist RCE:

  * •

In moist RCE, rising air parcels rapidly become saturated, leading to
condensation and latent heating. As a result, the temperature lapse rate in
moist RCE is reduced from dry adiabatic to nearly moist adiabatic151515The
moist adiabatic lapse rate is the rate at which temperature decreases with
height for a saturated air parcel lifted adiabatically accounting for the
latent heat released by condensation. [Fig. 3b; 328].

  * •

The irreversible fallout of precipitation reduces the water content of air and
allows descending air parcels to be unsaturated. This leads to an asymmetry in
upward motion; moist convection favors narrow rapid updrafts and broad weak
descent [Fig. 4b; 20].

  * •

Turbulence in moist RCE is weaker than in the dry case [269]; in our
simulations, the mid-tropospheric vertical velocity distribution is
substantially narrower in moist RCE compared to dry RCE (Fig. 5), despite the
fact that the moist case has a larger heat input
$|\langle\dot{Q}_{\text{rad}}\rangle|$ (see Fig. 3).

  * •

Finally, Robe and Emanuel [303] showed that cloud updraft velocities in moist
RCE are virtually independent of the heat input
$|\langle\dot{Q}_{\text{rad}}\rangle|$ [see also 48], in contrast to the case
of dry RCE, for which updraft velocities increase with the heat input [73]. In
moist RCE, increased radiative cooling is balanced by an increased area
fraction of clouds rather than any change in their updraft velocities.

These differences suggest that the effects of moisture fundamentally change
the dynamics of moist convection compared to its dry counterpart. A theory of
moist convection must therefore account for these differences; it is the
search for such a theory toward which we now turn.

###  IV.2 Theories of moist convection

####  IV.2.1 The moist convective heat engine

Emanuel and Bister [72] and Rennó and Ingersoll [298] attempted to use the
entropy budget to derive a theory for moist convective updraft velocties. As
we shall see below, however, both theories are limited in their utility
because they do not properly account for the irreversible entropy production
associated with moist processes.

The theory of Emanuel and Bister [72] focuses on the integrated vertical
buoyancy flux associated with convection, given by,

$\langle F_{b}\rangle=\frac{1}{A}\int_{\Omega_{A}}\left\langle\rho
wb\right\rangle\,dV,$

where $w$ is the vertical velocity of the fluid, and

$b=-{g_{\earth}}\left(\frac{\rho-\overline{\rho}}{\overline{\rho}}\right)$

is the buoyancy, defined using the horizontal mean density $\overline{\rho}$.
If the pressure field is in approximate hydrostatic balance with the mean
density, the integrated buoyancy flux is approximately equal to the rate at
which the atmosphere performs work in order to generate the kinetic energy of
the winds $\langle F_{b}\rangle\approx\langle\dot{W}_{K}\rangle$ [e.g., 309].
Singh and O’Gorman [331] show that this is a very good approximation in
simulations of RCE.

Emanuel and Bister [72] assumed that the buoyancy flux at a given level could
be approximated by the buoyancy of an air parcel lifted adiabatically from the
subcloud layer multiplied by the total cloud mass flux. Integrating
vertically, this allows the kinetic energy generation rate to be written,

$\langle\dot{W}_{K}\rangle\approx\langle F_{b}\rangle\approx\langle
M_{c}\rangle(\text{CAPE}),$

where $M_{c}$ is the upward cloud mass flux from the subcloud layer, and CAPE
is the convective available potential energy, defined as the kinetic energy
produced by buoyancy forces as an air parcel of unit mass rises adiabatically
to the tropopause (the temperature of such a parcel is shown in the dotted
line on Fig. 3.). Further assuming that frictional dissipation is the dominant
source of irreversible entropy production in the atmosphere, the above
equation may be combined with the entropy budget to give a scaling relation
for the CAPE,

$\text{CAPE}=\frac{\langle\dot{Q}_{\text{rad}}\rangle T_{\text{fric}}}{\langle
M_{c}\rangle}\left(\frac{1}{T_{a}}-\frac{1}{T_{sa}}\right).$ (61)

This provides a velocity scaling for cloud updrafts given by
$w\sim\sqrt{2\text{CAPE}}$. A similar scaling was derived by Rennó and
Ingersoll [298] based on a heat engine analysis of an air parcel completing a
cycle rising through a cloud and descending through the environment. Since the
cloud mass flux $\langle M_{c}\rangle$ scales with the integrated radiative
cooling rate $\langle\dot{Q}_{\text{rad}}\rangle$ in RCE, (61) implies that
the cloud updraft velocity is only weakly dependent on
$\langle\dot{Q}_{\text{rad}}\rangle$, as seen in numerical simulations [303].

The flaw in the theories of Emanuel and Bister [72] and Rennó and Ingersoll
[298] is that they assume that the dominant irreversible entropy production
mechanism in the atmosphere is that of frictional dissipation, thereby
neglecting irreversible moist processes. As we show below, this is a poor
assumption for moist RCE, and it is almost certainly a poor assumption for
convection on Earth. A full theory for moist convection based on the entropy
budget requires careful consideration of the irreversible entropy production
associated with moist processes.

Figure 5: Empirical probability distribution functions (PDFs) of the vertical
velocity at a height of 4 km in simulations of dry (gray) and moist (black)
RCE. PDFs are constructed from hourly snapshots from days 90-100 of each
simulation.

####  IV.2.2 Moist irreversible processes in radiative-convective equilibrium

Pauluis and Held [269] were the first to provide a comprehensive estimate of
the material entropy budget of both dry and moist RCE. The authors used
simulations of RCE with a two-dimensional cloud-permitting model in which only
the liquid-vapor phase transition was considered to provide a detailed
evaluation of the relative importance of frictional dissipation compared to
other irreversible sources of entropy. Since then, Romps [309] and Singh and
O’Gorman [331] have confirmed these results using three-dimensional
simulations that include the ice phase.

Table 2 shows an example of the entropy budget in a simulation of moist RCE
over a liquid water surface held at a temperature of 301.5 K. The simulation
differs from the examples shown in Figs. 3-5 in that it allows for the
interactive effects of radiation by including an explicit parameterization of
radiative transfer. The simulation is identical to the control simulation in
Singh and O’Gorman [331] except that it is run with a horizontal grid spacing
of 2 km, and on a doubly periodic domain $288\times 288$ km2 in size. We
include interactive radiation and use a larger domain in order to allow the
simulation to undergo the phenomenon of “convective self-aggregation”, which
is described in the next subsection.

The key result of Pauluis and Held [269], also evident in table 2, is that the
entropy source associated with frictional dissipation
$\langle\dot{S}_{i}^{\text{fric}}\rangle$ is a relatively small component of
the irreversible entropy production of RCE, accounting for less than $\sim$
15% of the total. This is in stark contrast to dry RCE, in which frictional
dissipation is the main mechanism by which entropy is produced irreversibly.

| symbol | units | disaggregated | aggregated
---|---|---|---|---
entropy budget |  |  |  |
export | $-\langle\dot{S}^{\text{mat}}_{e}\rangle$ | mW m-2 K-1 | $37.4\pm 2.1$ | $36.5\pm 3.4$
irreversible source | $\langle\dot{S}^{\text{mat}}_{i}\rangle$ | mW m-2 K-1 | $37.7\pm 0.2$ | $34.9\pm 0.2$
heat diffusion | $\langle\dot{S}_{i}^{\text{heat}}\rangle$ | mW m-2 K-1 | $-0.24\pm 0.0$ | $-0.12\pm 0.0$
frictional dissipation | $\langle\dot{S}_{i}^{\text{fric}}\rangle$ | mW m-2 K-1 | $6.0\pm 0.1$ | $2.4\pm 0.1$
hydrometeor sedimentation | $\langle\dot{S}_{i}^{\text{sed}}\rangle$ | mW m-2 K-1 | $11.0\pm 0.1$ | $9.1\pm 0.1$
mixing & phase change | $\langle\dot{S}_{i}^{\text{mem}}\rangle$ | mW m-2 K-1 | $21.0\pm 0.1$ | $23.5\pm 0.1$
radiative cooling rate | $-\langle\dot{Q}_{\text{rad}}\rangle$ | W m-2 | $106.4\pm 0.5$ | $122.6\pm 0.8$
mechanical efficiency | $\eta_{M}$ | % | 1.5 | 0.5
Carnot efficiency | $\eta_{C}$ | % | 9.2 | 7.5
precipitation efficiency | $\epsilon$ | % | 25 | 49
Table 2: Entropy budget and other statistics calculated from a simulation of
moist radiative-convective equilibrium with interactive radiation. Simulation
is run following the configuration of Singh and O’Gorman [331], but on a
$288\times 288$ km horizontal domain and with a horizontal grid spacing of 2
km. Lower boundary condition is an ocean surface held fixed at 301.5 K, and
simulation is initialized from a state of rest. “Disaggregated” corresponds to
mean over day 20-40 of the simulation, when convection remains scattered, and
humidity variations across the domain are weak (Fig. 6a). “Aggregated”
corresponds to a mean over day 150-230 of the simulation, when the domain
consists of a single moist region containing convection surrounded by a dry
region devoid of clouds (Fig. 6b). Uncertainties associated with estimating
the steady-state budget using a finite timeseries are quantified using a
block-bootstrap method and given as the 90${}^{\text{th}}$ percentile
confidence interval.

The physical reason for the small value of
$\langle\dot{S}_{i}^{\text{fric}}\rangle$ is that moist RCE includes a number
of additional irreversible sources of entropy, including entropy production
associated with the sedimentation flux of precipitation
$\langle\dot{S}_{i}^{\text{sed}}\rangle$ [271] and that associated with the
mixing of water vapor and dry air $\langle\dot{S}_{i}^{\text{mix}}\rangle$ and
irreversible phase changes $\langle\dot{S}_{i}^{\text{evap}}\rangle$ and
$\langle\dot{S}_{i}^{\text{melt}}\rangle$ [269], which we have combined in
table 2 into a single term $\langle\dot{S}_{i}^{\text{mem}}\rangle$,

$\langle\dot{S}_{i}^{\text{mem}}\rangle=\langle\dot{S}_{i}^{\text{mix}}+\dot{S}_{i}^{\text{evap}}+\dot{S}_{i}^{\text{melt}}\rangle.$

The sum of all irreversible entropy sources must, in steady state, balance the
total radiative sink of entropy $-\langle\dot{S}_{e}^{\text{mat}}\rangle$. The
additional sources of entropy associated with phase change, mixing, and
precipitation sedimentation must therefore occur at the expense of entropy
production associated with frictional dissipation. In fact, the sources
$\langle\dot{S}_{i}^{\text{mem}}\rangle$ and
$\langle\dot{S}_{i}^{\text{sed}}\rangle$ are the two largest irreversible
entropy sources in RCE, accounting for the vast majority of the irreversible
entropy production (table 2).

Before discussing the implications of the above results, we briefly mention
some caveats regarding the interpretation of our numerically simulated entropy
budget. In most models, spurious numerical sources and sinks of entropy exist
which can induce errors in the budget. Moreover, irreversible molecular
processes that produce entropy in the atmosphere are not explicitly resolved
in cloud-permitting simulations, and the parameterizations that attempt to
account for their effects do not always faithfully represent the second law of
thermodynamics. For example, the entropy production associated with heat
diffusion is small but negative in our simulation of RCE, appearing to violate
the second law. We further discuss the reasons for such apparently unphysical
entropy sinks, and other issues relating to accurately modeling the entropy
budget, in section VII.

Despite the above caveats, the overall dominance of moist irreversible
processes and the small value of the entropy production owing to frictional
dissipation found in our simulation of RCE is consistent with previous
numerical [269, 309] and theoretical [269, 102] analyses and is therefore
likely to be robust. Since, in steady state, the frictional dissipation of
winds in the atmosphere must balance their generation by mechanical work, this
implies that moist convection has a low mechanical efficiency $\eta_{M}$
compared to dry convection and compared to the efficiency of an ideal Carnot
heat engine (table 2). Here we define the mechanical efficiency similarly to
(20), with
$\langle\dot{Q}_{\text{in}}\rangle=|\langle\dot{Q}_{\text{rad}}\rangle|$ the
radiative cooling rate, so that,

$\eta_{M}=\frac{\langle\dot{W}_{K}\rangle}{|\langle\dot{Q}_{\text{rad}}\rangle|}.$

Pauluis and Held [269] performed a nondimensional analysis of their RCE
simulations to show that this mechanical efficiency depends on the relative
importance of latent heat transport compared to sensible heat transport and
the relative importance of the work done by water vapor compared to the total
work performed by moist convection. In moist RCE at temperatures
characteristic of Earth’s tropics, latent heat transport is the dominant
vertical heat transport mechanism, and water vapor expansion accounts for a
substantial fraction of the total work. Both of these factors reduce the
efficiency of moist convection.

Romps [309] pointed out that irreversible melting and freezing further
contributes to the low mechanical efficiency of moist convection. He also
noted that the reduced lapse rate in moist convection compared to dry
convection (Fig. 3b) results in a smaller temperature difference between the
input and output temperatures given the same radiative cooling profile. For
given vertical profile of $\dot{q}_{\text{rad}}$, the moist convective heat
engine therefore has a lower Carnot efficiency than its dry convective
counterpart.

An alternative perspective on the reduced mechanical efficiency of moist
convection compared to dry convection is to consider the atmosphere as a
combination heat engine and steam cycle. As described by Pauluis [267], the
thermodynamic action of atmospheric convection is to transport heat vertically
in the atmosphere, but also to dehumidify the atmosphere. These two
thermodynamic operations are in competition with each other, such that the
dehumidification process reduces the work available for the atmospheric heat
engine. The magnitude of the reduction in available work associated with this
dehumidification depends on the relative importance of latent heat transport
compared to sensible heat transport, and on the relative humidity at which the
mixed-cycle engine operates.

The implications of the small mechanical efficiency of moist convection are
profound. Firstly, it provides an explanation for the reduced kinetic energy
of moist convection relative to dry convection as highlighted in the snapshots
in Fig. 4 and the vertical velocity distributions in Fig. 5. Furthermore, a
small mechanical efficiency is inconsistent with the theories of Emanuel and
Bister [72] and Rennó and Ingersoll [298]; such theories are predicated on the
dominance of entropy production associated with frictional dissipation within
the entropy budget, and they cannot account for the case in which the entropy
budget is dominated by moist processes. Finally, the presence of moist
irreversible sources of entropy means that the entropy bluesink
$-\langle\dot{S}_{e}^{\text{mat}}\rangle$ no longer places a bluedirect
constraint on the work done by atmospheric convection; a change in the heat
input $|\langle\dot{Q}_{\text{rad}}\rangle|$ or its effective temperature
$T_{a}$ may be balanced by changes in the entropy sources associated with
moisture, rather than those associated with frictional dissipation.

####  IV.2.3 The role of mixing and microphysics

Despite the challenges described above, the entropy budget may nevertheless
provide guidance toward a theory of moist convective intensity. In particular,
Pauluis and Held [269] argued that the importance of the moist processes for
the entropy budget implies that moist convective updraft velocities may depend
strongly on the effects of condensate on the buoyancy of air and ultimately on
the microphysical processes that determine cloud and precipitation formation.
Additionally, the importance of vapor diffusion and irreversible phase change
in the entropy budget potentially suggests that the mixing of cloudy and non-
cloudy air parcels may be of central importance to any theory for convective
vigor in moist RCE.

Indeed, current theories for moist convective updraft velocities suggest that
they are limited by the sedimentation velocity of precipitation [261], or that
they are determined by the efficiency of mixing between clouds and their
environment [328, 322, 330]. In the latter case, it is argued that the import
of subsaturated air from the environment allows the profile of temperature in
moist RCE to decrease with height faster than that of a moist adiabat, thereby
allowing for finite CAPE. At present, estimates of the rate at which this
import of subsaturated air occurs must rely on detailed simulations with high-
resolution models such as those reported above. Such simulations do not
resolve the molecular mixing processes directly, but rely on turbulence
closures whose validity in the vicinity of clouds is difficult to establish.
Whether the entropy budget may be used to constrain the rate at which cloudy
and non-cloudy air is mixed is a potentially important area of future work.

Figure 6: Snapshots of column relative humidity (colors) and contour of column
cloud water content of 0.5 kg m-2 (black) in a simulation of moist RCE with
interactive radiation at (a) day 30 and (b) day 230. Simulation is run
following the configuration of Singh and O’Gorman [331], but on a $288\times
288$ km horizontal domain and with a horizontal grid spacing of 2 km. Lower
boundary condition is an ocean surface held fixed at 301.5 K, and simulation
is initialized from a state of rest.

###  IV.3 Convective organization and the mechanical efficiency of moist
convection

Previous studies of the entropy budget of RCE have been limited to cases of
disorganized convection, characterized by a statistical equilibrium in which
convective clouds grow and decay quasi-randomly within the domain [Fig. 4;
269, 309, 331]. However, convection in Earth’s atmosphere is often organized
into larger-scale structures such as squall lines, mesoscale convective
complexes, or tropical cyclones [127]. While the heat engine characteristics
of tropical cyclones have been the subject of considerable research (see
section V), the broader characteristics of the entropy budget of organized
moist convection remain relatively unexplored. In this section, we provide a
preliminary analysis of the entropy budget of a particular type of organized
convection to highlight the potential for future research in this area.

Under certain conditions, simulations of RCE are known to spontaneously
develop organization in a process termed “self-aggregation”. In the aggregated
state, clouds and convective activity become confined to a small region that
remains moist, while the rest of the domain is characterized by a dry
troposphere and subsiding air [e.g., 24, 374, 68]. Studies of convective self-
aggregation have shown that it results from feedbacks between clouds, water
vapor, and radiation that lead to an instability of the disaggregated state to
perturbations in tropospheric humidity [68]. This instability is sensitive to
the imposed surface temperature [374] as well as other details of the model
formulation, including the resolution and domain size [235, 144]. Given its
sensitivity to model formulation, the idealization of the framework of RCE,
and the long timescale taken for convection to self-aggregate (see below), the
importance of self-aggregation as a mechanism for convective organization on
Earth remains debated [140]. It is nevertheless of interest to understand how
the thermodynamic characteristics of the aggregated state differ from those of
the disaggregated state. For example, does self-aggregation increase or
decrease the mechanical efficiency of moist convection?

The simulation of moist RCE described in the previous section initially
develops a state in which clouds occur randomly and relatively uniformly
throughout the domain (Fig. 6a). After roughly 50 days, the simulation begins
to aggregate, and after roughly 150 days, a new state in which clouds are
clustered into a small region is obtained (Fig. 6b).

We focus our analysis on two quasi-steady periods within this simulation in
which convection is disaggregated (days 20-40) and aggregated (days 150-230),
respectively. By comparing the entropy budget in these periods, table 2
presents, for the first time, an analysis of how the entropy budget is
affected by convective organization. In each period, the statistics of the
flow are relatively steady, and the entropy budget is roughly closed (table
2), while in the $\sim 100$ day period in between, the process of self
aggregation occurs, and we do not expect the statistics to be steady. The
initiation of the self-aggregation process therefore limits the length of the
period over which we analyze the disaggregated state. However, the
disaggregated state is less variable than the aggregated state, and the
uncertainty in the steady-state budget introduced by such variability is
relatively small in both the aggregated and disaggregated cases (table 2).

In our simulation, convective self-aggregation is associated with a large
fractional reduction in the entropy production associated with frictional
dissipation, implying a reduction in the rate of work performed by moist
convection. Indeed, the mechanical efficiency of moist convection decreases by
a factor of three between the aggregated and disaggregated state. The
reduction in $\langle\dot{S}_{i}^{\text{fric}}\rangle$ is balanced by an
increase in entropy production associated with mixing and phase change
$\langle\dot{S}_{i}^{\text{mem}}\rangle$ and a slight decrease in the total
irreversible entropy production of the atmosphere. The entropy source owing to
the sedimentation of precipitation $\langle\dot{S}_{i}^{\text{sed}}\rangle$
also decreases with aggregation, and this is associated with a doubling of the
precipitation efficiency, defined as the ratio of total condensation in the
atmosphere to the surface precipitation, in the aggregated state compared to
the disaggregated state.

The results above suggest that the RCE atmosphere is a less efficient heat
engine when convection is aggregated compared to when it is not, but that the
efficiency by which condensation in the atmosphere is converted to
precipitation at the surface increases under aggregation. The reasons for
these differences in efficiency are likely to be related to the large-scale
reorganization of the humidity distribution that occurs under aggregation. For
example, the humidity of the near-cloud environment is likely to be higher
when convection is aggregated, contributing to the higher precipitation
efficiency of the aggregated state. On the other hand, the variability of
humidity within the domain also increases under aggregation [374], and this
may lead to a larger entropy source associated with vapor diffusion,
contributing to the lower mechanical efficiency of aggregated convection.

The idealizations inherent in a simulation of RCE on a finite domain limit the
direct applicability of our results to Earth’s atmosphere. Previous studies of
aggregation in RCE have shown that domain geometry affects the number of
aggregated regions produced [373], while Jakob et al. [140] noted that a state
of RCE is only observed in the atmosphere on spatial scales substantially
larger than our domain. Moreover, a variety of additional mechanisms are
present in Earth’s atmosphere, including large-scale waves and other large-
scale circulations [e.g., 127], inhomogeneities in the surface [e.g., 300],
and background wind shear [e.g., 313] which may organize convection more
efficiently than the mechanisms that cause the $\sim 100$-day development of
aggregation in our simulation. Further work is required to determine if our
RCE results are relevant to these more general forms of convective
organization. Nevertheless, we note that the tendency for tropical convection
to aggregate has been hypothesized to increase in a warmer climate [see e.g.,
372]. Our results suggest that such an increase in aggregation may have
implications for the global atmospheric heat engine under future climate
change [see section VI.2.3 and 169].

##  V The thermodynamics of tropical cyclones

When planetary rotation is included, simulations of RCE spontaneously generate
one or more rapidly rotating storms analogous to tropical cyclones (TCs) on
Earth [240, 153, 38, 292]. TCs are stunning examples of organized deep
convection. They are variously called hurricanes, typhoons or cyclones,
depending on their intensity and the ocean basin in which they occur. TCs are
characterized by a primary circulation, consisting of a rapidly rotating
vortex around a low-pressure center, and a secondary circulation, consisting
of a (mostly) thermally direct overturning circulation. The small exception
comes from the enigmatic, dry TC eye, in which a thermally indirect flow is
produced as buoyant dry air is mechanically forced to descend. Outside the dry
central eye, high-entropy moist air ascends in the saturated eyewall until it
reaches the upper troposphere, where it spreads out laterally to large radii.
At intermediate radii, deep thunderstorms comprise one or more asymmetric
rainbands that spiral away from the TC center. Precipitation within the
eyewall and the rainbands is a rapid, irreversible sink of rising water mass
and occurs primarily within 200 km from the TC center, while radiative cooling
slowly removes energy and entropy from the air at larger radii.

Like all storms, TCs are transient; they make an interesting thermodynamic
system in part due to their genesis and death. TCs are open systems,
continuously exchanging energy, mass and entropy laterally with the rest of
the atmosphere, as well as vertically via fluxes through the surface and TOA
(Fig. 7, right side). These characteristics contrast with the idealized RCE
simulations discussed above, in which transient effects and lateral fluxes
from a larger environment are not considered. In RCE, the atmosphere is an
open system only to the extent that water may enter and leave it through
surface evaporation and precipitation.

The idealization of TCs as being steady and closed to lateral fluxes (Fig. 7
left side) has nevertheless been a rewarding model for understanding their
basic physics, and we will consider an idealized steady-state TC in a closed
domain as a starting point for our discussion. First we review steady-state
theory for the potential intensity (highest-achievable surface wind speed, or
lowest-achievable central surface pressure deficit) of a TC given the
thermodynamic parameters of the environment (section V.1). We next review how
the secondary (overturning) circulation of TCs as a whole can be viewed as a
combination heat/steam engine, and we discuss recently developed techniques to
calculate the integrated work production of a TC (section V.2). Finally, we
consider TCs as open systems, capable of genesis and extinction, and mention
areas for future research (section V.3).

###  V.1 Potential intensity theory

It is generally agreed upon that the primary energy source of a TC is the flux
of enthalpy from the sea surface [31, 159, 202, 301, 302, 69]161616For a
translation of Kleinschmidt Jr. [159] see the appendix of Gray [107].. This
flux is driven by the thermodynamic disequilibrium between the sea surface and
the subsaturated air immediately above it. Acting like a heat engine, the TC
transports energy from the warm surface to the cooler troposphere, producing
potentially catastrophic winds in the process. Riehl [301] first identified
the enthalpy disequilibrium as being responsible for the energy flux from the
sea, and Kleinschmidt Jr. [159] provided the first estimate of the maximum
wind speed of a TC. That study and a later attempt by Malkus and Riehl [202]
laid the groundwork for quantifying TC thermodynamics, preceding the
celebrated modern-day potential intensity (PI) theory [69, 70, 18] for the
maximum attainable surface wind speed (or minimum central surface pressure) of
an inviscid, axisymmetric TC embedded in a given thermodynamic environment.

Emanuel [69] explicitly likened a TC to a Carnot heat engine, demonstrating
that the same analytic result for potential intensity can be derived from
either the equations of motion and the first law of thermodynamics, or from
consideration of an idealized thermodynamic cycle performed by an air parcel
within a TC. The thermodynamic cycle consists of four legs as follows (Fig.
7): Isothermal expansion: boundary layer air that converges toward the low
pressure center expands but stays relatively isothermal, heated by sensible
heat fluxes from the sea surface. Simultaneously, surface latent heat fluxes
dramatically increase the air’s moist entropy. Adiabatic expansion: frictional
convergence of boundary layer air near the surface is balanced by ascent in
the deep, cloudy eyewall of the TC. This process is roughly slantwise moist
neutral: saturated rising parcels do not ascend strictly vertically but rather
along sloping surfaces of constant angular momentum $M$. During sloping ascent
away from the TC core, parcels initially approximately conserve their moist
entropy $s$ (neglecting the mass loss due to precipitation, which is a small
fraction of the air mass), such that surfaces of constant $M$ and $s$ are
parallel. Isothermal compression: as air is exhausted farther radially over
time, it loses energy radiatively as it starts to descend, and finally
(adiabatic compression) the parcel slowly subsides back to the boundary layer
along a vortex line. These last two legs do not occur separately in real TCs;
rather, air parcels lose both entropy and energy as air descends at large
radii. However, the artificial separation of the two legs aids in mathematical
comparison to a Carnot cycle.

Figure 7: A schematic of two idealized tropical cyclone overturning
circulations (radial cross-section). Grey arrows indicate dry air mass fluxes.
Black arrows indicate boundary fluxes of heat $Q$, entropy $S$, water content
$r_{T}$, and momentum $M$ into and out of the domain, and arrows indicate the
characteristic direction of the fluxes in particular regions. (a) A closed
circulation bounded by an outer wall or doubly periodic boundary conditions,
with the idealized Carnot cycle legs indicated. (b) A TC in lateral contact
with the rest of the atmosphere.

The outer environment restores a parcel’s $M$ and $s$ back to environmental
conditions, ultimately through contact with the saturated, frictional sea
surface. With assumptions about the storm structure in the free troposphere
above the boundary layer, chiefly the assumption of slantwise moist
neutrality, one can determine the energetics of the storm needing only to
specify the TC’s boundary layer radial structure of $s$ as a function of $M$
and environmental factors like surface air temperature $T_{s}$ and outflow air
temperature $T_{o}$ [69]. Having specified these conditions, a theoretical
closed loop describing a parcel undergoing cyclic changes in pressure, volume,
temperature, and entropy that is closely analogous to a heat engine may be
defined. This parcel travels the outermost loop of the entire TC circulation,
allowing it to experience the largest temperature range and heating/cooling
range, and thus achieve the maximum wind speed $v_{\text{max}}$:

$v^{2}_{\text{max}}=\frac{T_{s}-T_{o}}{T_{s}}\frac{C_{k}}{C_{D}}(h^{*}_{s}-h_{BL}).$
(62)

This expression is the outcome of PI theory. Note that the first factor on the
right-hand side $(T_{s}-T_{o})/T_{s}$ is identical to the Carnot efficiency of
a heat engine operating between thermal reservoirs at temperatures $T_{s}$ and
$T_{o}$. The constants $C_{k}$ and $C_{D}$ are bulk surface exchange
coefficients for enthalpy and momentum, respectively. The atmospheric specific
enthalpy $h$ here is typically approximated as $h=q_{d}h_{d}+q_{v}h_{v}\approx
c_{pd}(T-T_{0})+L_{v}q_{v}$ for dry air heat capacity at constant pressure
$c_{pd}$, latent heat of vaporization $L_{v}$, mass fraction of water vapor
$q_{v}$, and reference temperature $T_{0}$. The relevant enthalpy
disequilibrium that drives flux from the ocean to the atmosphere is the
difference between the saturated enthalpy of the sea surface $h_{s}^{*}$ and
the enthalpy of the subsaturated boundary layer $h_{BL}$.

According to PI theory, the positive feedback responsible for TC
intensification is wind-induced surface heat exchange [WISHE, 69], whereby
stronger surface winds induce stronger sea-to-air energy fluxes, which
intensify the vortex and lead to stronger surface winds. This feedback is
eventually arrested and balanced by frictional drag at the surface. TC
structure and PI theory were reviewed by Emanuel [71], Camp and Montgomery
[35], Emanuel [64], Wang [367]; recently Emanuel [66] comprehensively reviewed
the full breadth of tropical cyclone research.

PI theory was further developed by Emanuel [70, 71] and a complementary
approach was provided by Holland [123]. In these theories, heating due to
kinetic energy dissipation was not considered, so the thermodynamic efficiency
in (62) includes the surface temperature $T_{s}$ in the denominator. Bister
and Emanuel [18] showed that, when frictional dissipation is taken into
account, the potential intensity is increased, with the outflow temperature
$T_{o}$ replacing $T_{s}$ in the denominator of (62). In numerical models that
don’t include this frictional heating term, the original formulation is still
appropriate [e.g., 49]. Recent work also shows that the outflow temperature
$T_{o}$ is not a constant [67, 65, 78], so the constant Carnot efficiency-like
term in (62) does not accurately reflect the range of temperatures at which
the atmosphere loses heat.

The application of heat engine theory to cyclones is even simpler for dry
fluids. Rennó et al. [299] undertook a heat engine analysis for nearly-dry
dust devils (for which planetary rotation is unimportant), where the driving
temperature difference was defined as that between the surface and the top of
the boundary layer. Mrowiec et al. [233] showed that the original Emanuel [69]
PI theory, formulated assuming a saturated core neutral to slantwise
convection, was also valid for purely dry TCs that only receive sensible heat
from the surface (provided that there is a sufficient WISHE-type feedback).
This works because PI theory incorporates the role of moisture only in the
definition of entropy.

Potential intensity theory is useful for bounding the upper limit of peak TC
wind speeds. However, the expression for $v_{\text{max}}^{2}$ is not
equivalent to a calculation of work produced by the extremely dissipative TC
heat engine [19]. Hakim [111] demonstrated that, even for the most extremal
closed path considered by PI theory, the Carnot cycle analogy is only
partially appropriate; while the isothermal expansion and adiabatic expansion
legs were decent approximations of the inflowing and ascending/outflowing air,
respectively, the return flow was neither isothermal nor adiabatic. And of
course, in a highly dissipative moist TC, $\oint Tds$ does not equal the work
produced in a cycle, as it would for a reversible Carnot cycle.

A recent development of PI theory considers a “differential Carnot cycle”
[314] in which the closed parcel path is taken as an infinitesimally wide
region bounding the extremal overturning streamline only for the inflowing and
ascending/outflowing branches of the circulation. This approach avoids the
somewhat unrealistic description of the TC descending branch as two separate
legs of adiabatic compression and isothermal cooling, and it better
illustrates that the thermodynamic efficiency of the TC’s overall overturning
circulation is unaccounted for in the calculation of $v_{\text{max}}$. For a
calculation of TC work and entropy production as a whole, the entire
circulation must considered. Renno [297] proposed that one could predict the
pressure drop anywhere within a realistic vortex by taking into account the
irreversibility of the vortical heat engine due to the presence of moisture or
dust. Very recent work has shown that, indeed, a TC’s core wind structure is
closely related to the sources and magnitude of entropy production available
in the system [365, 366].

###  V.2 Work and entropy budgets of TCs

Consider a steady-state TC, far from equilibrium with its environment. Let the
TC system be defined by a cylinder centered on the TC center, bounded below by
the ocean surface and above by the top of the atmosphere (Fig. 7). Though a
developing TC does grow in volume of dry air at the expense of the environment
(as defined perhaps by something like the size of the expanding outflow
region), we shall neglect the cyclogenesis/cyclolysis stages and consider a
steady-state mature TC with a fixed amount of dry air in RCE.

To evaluate the work performed by TCs in RCE, we apply a procedure of
isentropic averaging to analyze the overturning circulation of an axisymmetric
TC simulation described in O’Neill and Chavas [245] [see also 312, 276, 234].
To do so, we introduce the potential temperature

$\theta=T(p_{\text{ref}}/p)^{R_{d}/c_{pd}}$ (63)

where $p_{\text{ref}}=1000$ hPa is a reference pressure. The potential
temperature is conserved for a parcel of dry air that experiences adiabatic
changes in pressure, and it is related to the entropy of dry air by the
approximate equation $s_{d}\approx c_{pd}\ln(\theta/\theta_{0})$, for some
reference potential temperature $\theta_{0}$. We also introduce the equivalent
potential temperature $\theta_{e}$, which is a similar adiabatic invariant for
moist air. The equivalent potential temperature $\theta_{e}$ may be related to
the entropy by $s=c_{pd}\ln(\theta_{e}/\theta_{e0})$ for some reference
equivalent potential temperature $\theta_{e0}$ [273], blueprovided the liquid
water reference entropy $s_{l0}$ is taken to be zero. While other adiabatic
invariants may be defined that take into account the physical value of
$s_{l0}$ [211, 212], here we will proceed with the equivalent potential
temperature $\theta_{e}$ as it is commonly used in the tropical meteorology
literature.

Being thermally direct warm-core vortices, TCs are characterized by rising
high-$\theta_{e}$ air and subsiding low-$\theta_{e}$ air. Isentropic averaging
blue[where the word “isentropic” here is more historical and less accurate
[see 213]] allows one to recast spatial data into a spatial dimension and a
thermodynamic dimension. The vertical mass flux at each height is binned by a
discretized thermodynamic variable such as $\theta_{e}$. These isentropic mass
fluxes may then be used to define a streamfunction $\Psi_{e}(\theta_{e},z)$ as
a function of equivalent potential temperature and altitude that provides a
thermodynamic perspective on the overturning circulation. To ensure closure of
the streamlines, calculations of $\Psi_{e}$ are performed by first subtracting
the average vertical velocity at each vertical level of the analysis domain.
Pauluis and Mrowiec [276] first applied this technique to statistically-steady
disorganized convection in RCE. Mrowiec et al. [234] subsequently applied the
procedure to a TC, where the $\theta_{e}$ structure is a rather natural radial
coordinate (Fig. 8a). Mrowiec et al. [234] showed that the bulk of the upward
mass flux occurred in asymmetric convective regions that don’t appear in an
Eulerian streamfunction. As a result, the extremal isentropic streamfunction
mass flux is always higher than the extremal Eulerian streamfunction mass
flux, by a factor of three or so. This is also true when the TC is
axisymmetric, as a comparison of the peak mass flux in the Eulerian and
isentropic streamfunctions given in Fig. 8b and c shows. This demonstrates
that the TC upward mass flux does not exclusively occur in the central
eyewall.

Figure 8: Structure and overturning circulation of an axisymmetric tropical
cyclone simulation at steady state [data from a 30-day window of a TC in a
6,000 km radius domain at $40^{\circ}$ latitude as described in O’Neill and
Chavas [245]]. (a) Equivalent potential temperature ($\theta_{e}$ [K], darker
shades for higher values) and absolute angular momentum (dashed; contour value
increasing with radius). (b) Eulerian mass overturning streamfunction $\Psi$
[kg s-1] integrated radially outward to 800 km; each contour represents 10% of
the mass flux. (c) Isentropic mass overturning streamfunction $\Psi_{e}$ [kg
s-1] integrated radially outward to 800 km; each contour represents 10% of the
mass flux. (d) Temperature-entropy ($T$-$s$) diagram corresponding to parcel
paths in panel (c) using the MAFALDA technique [274], where data gaps present
in the contouring procedure in panel (c) have been filled via linear
interpolation. Numbers in panels (b) and (c) give maximum streamfunction
magnitude in units of $10^{9}$ kg s-1.

The isentropic streamfunction may be considered to be an approximation of the
thermodynamic temperature-entropy ($T$-$s$) diagram; $T$ decreases
monotonically with altitude between the surface boundary layer and the
tropopause, and moist entropy increases with increasing $\theta_{e}$, so the
isentropic streamfunction is quite similar to a $T$-$s$ diagram if you flip it
upside-down (compare panels 8c and d). Pauluis [274] introduced a more formal
method to approximate thermodynamic cycles based on Eulerian data by assuming
that each closed circuit in the isentropic streamfunction is the path of a
parcel of air (named the Mean Air Flow As Lagrangian Dynamics Approximation,
MAFALDA). Isentropically-averaged variables of interest may then be
interpolated along each closed path. This is a strong idealization of real
parcel motion within a TC, which is not expected to exhibit closed parcel
paths in the vicinity of the eyewall. However it allows for an approximate
calculation of thermodynamic cycles experienced by the overturning circulation
(Fig. 8d), including the motions in randomly distributed convective towers.
The isentropic streamfunction suggests a complicated, distinctly not-Carnot-
like spectrum of closed paths in $T$-$s$ space, and the bulk of the TC
convection in fact occurs radially outward of the eyewall. This can be
confirmed using the MAFALDA procedure [Fig. 8d, and see Pauluis [274]].

The total work produced by an air parcel containing a unit mass of dry air
traversing a reversible Carnot cycle is equal to the net heating:
$\widetilde{W}_{\text{max}}=-\oint\alpha_{d}dp=\oint Tds$. Here $s$ is the
entropy expressed per unit mass of dry air, $\alpha_{d}=\rho_{d}^{-1}$ is the
specific volume of dry air, and the $\widetilde{()}$ indicates an integral
over a closed parcel path. We follow Pauluis [274] in considering integrals
along paths following the trajectory of dry air as the working fluid rather
than the barycentric flow as was done in previous sections. Furthermore,
unlike the definition of $\dot{Q}_{\text{in}}$ in (16), the net heating in the
following development includes that due to frictional dissipation and
diffusion, so the Carnot efficiency is an upper bound of the mechanical
efficiency. We denote this total positive heating as $Q_{\text{tot}}^{+}$.
Because the relevant system here is a single air parcel traversing a closed
cycle, frictional heating and diffusion can be considered as external heat
sources [277].

In an irreversible MAFALDA loop within a TC, the net heating is partitioned
into the “Gibbs penalty” and the reversible work $\widetilde{W}_{\text{rev}}$.
The Gibbs penalty [267, 274, 277] is due to the systematic removal of Gibbs
free energy from the TC system because water enters the open system via
evaporation in subsaturated conditions (low Gibbs free energy) and leaves via
precipitation under generally saturated conditions (high Gibbs free energy),
and it corresponds to a loss of work associated with irreversible entropy
production. The reversible work $\widetilde{W}_{\text{rev}}$ itself is
partitioned between the production of kinetic energy $\widetilde{W}_{K}$ and
the work used to increase the geopotential of water of any phase
$\widetilde{W}_{\text{H${}_{2}$O}}$,
$\widetilde{W}_{\text{rev}}=\widetilde{W}_{K}+\widetilde{W}_{\text{H${}_{2}$O}}$.
Thus $\widetilde{W}_{\text{max}}$ can be decomposed in the following way [267,
274, 277, 78]:

$\displaystyle\underbrace{\oint
Tds}_{\widetilde{W}_{\text{max}}}\,\,\,=\,\,\,\underbrace{-\oint\alpha_{d}dp}_{\widetilde{W}_{K}+\widetilde{W}_{\text{H${}_{2}$O}}}\,\,\,-\,\,\,{\underbrace{\oint\sum\limits_{x=v,l,s}g_{x}dr_{x}}_{-\widetilde{\Delta
g}}}$ (64)

where $g_{x}$ is the Gibbs free energy of water in phase $x$, expressed here
per unit mass of dry air, and $r_{x}$ is the mixing ratio, equal to the mass
of water in phase $x$ per unit mass of dry air. This is a closed line integral
of the fundamental thermodynamic relation (35), written in terms of enthalpy
and expressed per unit mass of dry air.

The most efficient parcel path (‘inner core path’) in the TC is the most
extremal one that travels from the bottom of the boundary layer, up the
eyewall and to the top of the outflow, experiencing the maximum gradient in
both temperature and entropy. The mechanical efficiency of a given closed
MAFALDA trajectory $\eta_{M}=\widetilde{W}_{K}/\widetilde{Q}_{\text{tot}}^{+}$
may be approximately expressed as [277, 78]:

$\eta_{M}=\eta_{C}-\frac{\widetilde{W}_{\text{H{\hbox{_2}}O}}}{\widetilde{Q}_{\text{tot}}^{+}}-\frac{\widetilde{\Delta
G}}{\widetilde{Q}_{\text{tot}}^{+}},$ (65)

where $\eta_{C}=(T_{\text{in}}-T_{\text{out}})/T_{\text{in}}$ is the Carnot
efficiency and

$\widetilde{\Delta
G}=-T_{\text{out}}\oint\sum\limits_{x=v,l,s}\frac{g_{x}}{T}dr_{x}.$

The temperatures $T_{\text{in}}$ and $T_{\text{out}}$ are the effective
temperatures experienced by the parcel during the heating and cooling legs of
the closed cycle, respectively. They are defined analogously to (17) but based
on the total heating rate $Q_{\text{tot}}$, and evaluated for a closed line
integral. Pauluis and Zhang [277] numerically simulated an idealized three-
dimensional TC and calculated a remarkable mechanical efficiency of
$\eta_{M}\approx 0.7\eta_{C}$ for the inner core parcel path. Similar values
of the mechanical efficiency as a fraction of the Carnot efficiency were
reported by Fang et al. [78] in a more realistic simulation of Hurricane
Edouard (2014). They showed that both $\widetilde{Q}_{\text{tot}}^{+}$ and
$\widetilde{W}_{K}$ of the inner core path increased sharply during a period
of strong intensification. The intensification occurred as the storm grew and
axisymmetrized, while $\widetilde{W}_{\text{H${}_{2}$O}}$ and
$\widetilde{\Delta G}$ remained relatively stationary. In the numerical TC
experiments of Pauluis and Zhang [277] and Fang et al. [78], MAFALDA
trajectories that extend to the upper troposphere have a high efficiency, with
$\widetilde{W}_{K}>\widetilde{W}_{H_{2}O},\widetilde{\Delta G}$. These results
highlight the potential utility of calculations of TC work and entropy budgets
for understanding TC dynamics. blueA limitation of the above framework is that
the magnitudes of $\widetilde{W}_{\text{max}}$ and $\widetilde{\Delta g}$ are
sensitive to the choice of the liquid water reference entropy $s_{l0}$ [213].
But whether this sensitivity matters for useful interpretation is a matter of
debate [see 275, 214].

The values found for TC mechanical efficiencies given above cannot be directly
compared with the efficiencies found in aggregated vs. disaggregated RCE
states in section IV, because the RCE calculations given in table 2 are
integrated over the domain, while the above TC efficiencies are only measured
for the parcel path with the highest efficiency. A comparison of the
efficiency of convection in rotating and non-rotating RCE has not yet been
carried out in the literature, but the MAFALDA procedure could easily be
employed in both cases for a direct comparison.

Wang and Lin [365, 366] compared simulations of a dry TC, a reversible moist
TC (no precipitation and attendant entropy production), and a highly
irreversible, realistic moist TC. The dry and reversible moist storms
displayed very similar structure with a deep lower inflow layer, a wide and
weak eyewall ascent region and no entropy minimum, in contrast to the
realistic TC. The realistic TC was also much smaller, even though it exhibited
much higher peak wind intensity. Wang and Lin [366] calculated full entropy
budgets for the same simulations, exploiting the heat engine nature of a TC to
develop an analytical expression that relates wind intensity and wind
structure. These results suggest that irreversibility due to water phase
changes and precipitation substantially reduces the overall mechanical
efficiency of realistic moist TCs and is responsible for their small, compact
core as compared to dry TC counterparts.

###  V.3 Open TCs

Though the isentropic averaging procedure produces a closed streamfunction
amenable to a work calculation, the TCs studied in the research described
above are all open to the farther environment. Dry air and water substance are
exchanged at the system’s lateral boundaries, causing the system to import or
export energy and entropy laterally. Note that the upper outflow of the TC in
Fig. 8b clearly flows beyond the right-hand boundary, and boundary layer air
enters from the outer edge, but the isentropic streamfunction in panel c still
appears as a closed circulation. In fact, it has been shown that most of the
condensed water in a TC comes from the lateral convergence of water vapor,
rather than locally from the sea surface [e.g., 168, 379]. But the MAFALDA
procedure can only operate on closed thermodynamic cycles. In order to close
an isentropic streamfunction under substantial lateral exchange with the
environment, the average vertical velocity is removed from the subdomain
containing the TC at each vertical level. The integrals are calculated out to
a rather limited radius away from the TC center (500-800 km) in order for the
signature of the mass flux in the eyewall to remain appreciable in spite of
its small volume. An isentropic average that integrates outward to the
deformation radius of the storm would be dominated by weaker convection
occurring far from the TC eye.

TCs have been treated as open systems in thermodynamic studies [e.g., 185,
347, 348, 149], with TC-environmental exchange considered alongside the
traditional vertical boundary sinks and sources. Juračić and Raymond [149]
calculated a moist entropy budget for TCs using dropsonde data interpolated to
a three-dimensional grid and included a calculation of lateral fluxes of
entropy between the tropical cyclone (in a $4^{\circ}\times 4^{\circ}$ storm-
following domain) and the environment. Irreversible entropy production was
estimated as instantaneously balancing the entropy sink due to radiative
cooling. This assumption could be inaccurate for TCs that are experiencing a
lot of environmental shear (and in general lateral exchange with the
environment), which can bring in low-entropy air [347] and lead to evaporation
underneath the cooling cirrus canopy. A constraint on the environmental
exchange with a highly dissipative TC is that the net entropy out of the TC
domain (to space as well as the broader atmosphere/ocean system) must be large
and positive in order to keep a TC at steady state. Lateral fluxes of entropy
out of the TC domain could be negative if radiative cooling is sufficiently
high; Juračić and Raymond [149] found that such fluxes could be of either sign
in their observed cases, and showed some dependence on the strengthening or
weaking status of the TC.

No study has yet attempted a heat-engine analysis of the entire lifecycle of a
TC, but Tang [349, 350] constructed a simplified framework that illuminates
the role of lateral entropy fluxes before and during tropical cyclogenesis.
However, their numerical model omits material production of entropy within the
TC domain. The intensification study of Fang et al. [78] is another promising
avenue. Heat engine concepts could be further brought to bear in the
literature seeking to understand the annual frequency of TCs globally [e.g.,
125, 128]. Use of MAFALDA to estimate closed trajectories, and the
thermodynamic analysis of Wang and Lin [365, 366], are among recent approaches
that can be leveraged to better understand how TCs evolve in a warming world.
In short, there are many exciting tools and approaches newly available to
probe TCs that exploit the second law of thermodynamics.

##  VI The global circulation of the atmosphere

In this section, we consider the global atmospheric circulation from a
thermodynamic perspective. We first describe the material entropy budget of
the global atmosphere and we compare it to the entropy budget of RCE (section
VI.1). Next, we consider the global atmospheric heat engine, and we review
theories for its meridional energy transport and its response to global
climate change VI.2). Finally, we broaden our perspective to consider the heat
engines of other planets in the Solar System and beyond (section VI.3).

###  VI.1 The material entropy budget of the global atmosphere

Estimating the entropy budget of the global atmosphere is challenging;
observational studies often employ relatively crude estimates of effective
temperatures [280] that limit the accuracy of the resultant estimates of
irreversible entropy production. Global climate models are able to provide
more detailed diagnostics than those available from observations, but they
present difficulties of their own. In particular, global climate models are
typically run at horizontal grid spacings of the order of 100 km and they are
therefore unable to resolve convective clouds. Irreversible entropy production
associated with moist convection, which was described in detail in the
previous section and is known to account for a large fraction of the total
irreversible entropy production in the atmosphere [262], must be wholly
parameterized within a global climate model. The extent to which
parameterizations of convection accurately represent this entropy production
remains unknown. But even assuming that a model’s parameterizations accurately
reflect the effect of subgrid processes on the model’s resolved grid, the low
resolution and use of simplified thermodynamic formulations [86, 262] within
global climate models imply that their entropy budgets differ from that of
Earth’s atmosphere (see section VII), and care must be taken to ensure that
comparisons across studies and models consider entropy sinks and sources at
similar spatial and temporal scales [194]. Finally, while detailed analysis of
individual models is possible [262], standard outputs from model
intercomparison projects only allow for the calculation of approximate entropy
budgets, and this can lead to difficulty closing the budget [176].

Despite the above challenges, there are a number of broad features of the
atmosphere’s material entropy budget that are known with some confidence. The
total material entropy production of the atmosphere makes up the vast bulk of
the material entropy production of the climate system (and indeed that of the
entire Earth system). In section II.1.2, the material entropy production of
the climate system was estimated to be in the range 35-60 mW m-2 K-1 [176].
The lower end of this range is only slightly higher than the estimates of the
entropy production of RCE given in table 2, suggesting that the bulk of the
irreversible entropy production in the atmosphere is produced by processes
acting vertically within each column. Indeed, Lucarini et al. [198] and
Lucarini and Pascale [194] found that roughly 90% of the irreversible entropy
production in the climate system could be associated with vertical heat
fluxes, with the remaining 10% associated with horizontal heat transport.

As in RCE, the atmosphere’s entropy budget is dominated by moist processes.
Lembo et al. [176] found that entropy production associated with the
hydrological cycle (terms $\langle\dot{S}_{i}^{\text{mem}}\rangle$ and
$\langle\dot{S}_{i}^{\text{sed}}\rangle$ in our formulation) accounted for
80-90% of the estimated total irreversible entropy production in an ensemble
of state-of-the-art global climate models, consistent with theoretical
expectations given the dominance of latent heat transport in the energy
exchange between the surface and atmosphere at a global level [269]. Of the
component associated with moist processes, the bulk is due to phase change and
vapor diffusion $\langle\dot{S}_{i}^{\text{mem}}\rangle$; Lembo et al. [176]
estimated that the entropy production associated with precipitation
sedimentation accounted for only roughly 4-6 mW m-2 K-1, consistent with an
observational estimate of the dissipation owing to precipitation sedimentation
given by Pauluis and Dias [268].

Model-based estimates of the entropy production associated with frictional
dissipation of the winds $\langle\dot{S}_{i}^{\text{fric}}\rangle$ vary from
roughly 6 mW m-2 K-1 [176], similar to values found for disaggregated RCE, up
to twice this value [262], with correspondingly large ranges in estimates of
the rate of work performed by the atmospheric heat engine and its mechanical
efficiency. The reason for this wide range is likely due to difficulties in
estimating the frictional dissipation rate in global climate models [176];
such models often have multiple parameterizations that dissipate kinetic
energy, and they may or may not include frictional heating within their
thermodynamic formulation [262]. Additionally, kinetic energy that is both
generated and dissipated at scales smaller than the model grid is not included
in the model’s mechanical energy or entropy budgets. Frictional dissipation
estimated from global climate models should therefore be considered to be only
the portion of the dissipation that is associated with the large-scale flow;
it is unclear to what extent one should compare such estimates to those
derived from higher-resolution models such as presented in section IV.

In summary, while quantitative estimates remain uncertain, qualitatively, the
entropy budget of the global atmosphere shares a number of similarities with
the simpler case of RCE discussed in section IV. In particular, the dominance
of entropy production associated with moist processes limits the mechanical
efficiency of the global atmospheric heat engine, and it limits the rate at
which work is done by the pressure gradient force. As we shall see below, this
fact plays an important role in understanding the atmospheric heat engine’s
response to global climate change.

###  VI.2 The global atmospheric heat engine

####  VI.2.1 A thermodynamic perspective of the global atmospheric circulation

Despite the long history of research describing the atmosphere as a heat
engine [e.g., 27, section II.2], relatively few studies have expressed the
global atmospheric circulation in traditional thermodynamic coordinates [e.g.,
temperature-entropy ($T$-$s$) space; 169]. Rather, the global atmospheric
circulation is more commonly characterized in terms of the meridional mass
overturning streamfunction. This streamfunction may be constructed based on an
Eulerian average at constant height or pressure, in which case it quantifies
the average mass flow in the latitude-height plane. Alternatively, an
isentropic averaging technique similar to that described in section V.2 but
applied to the vertical dimension rather than a horizontal dimension may be
used to re-express the streamfunction as a function of latitude and an
entropy-based vertical coordinate (specifically, potential temperature). This
isentropic streamfunction provides a thermodynamic perspective on the global
atmospheric circulation, and it may be used to quantify the global atmospheric
heat engine.

Figure 9: Mean meridional mass overturning streamfunction estimated from the
ERA-Interim reanalysis [52] using 6-hourly snapshots for the years 1981-2000
and calculated using (a) pressure, (b) potential temperature, and (c)
equivalent potential temperature as a vertical coordinate using the method
described in Pauluis et al. [273]. Black contours represent clockwise motion
and gray contours represent anticlockwise motion. Contour interval is
$10\times 10^{9}$ kg s-1 with zero contour omitted. Numbers give maximum
streamfunction magnitude in each hemisphere in units of $10^{9}$ kg s-1.
Hadley, Ferrel, and Polar cells are labeled in (a). Thick maroon curves show
mean (b) potential temperature and (c) equivalent potential temperature near
the surface (at the pressure level $p=0.9988p_{s}$, where $p_{s}$ is the
surface pressure).

An estimate of the traditional Eulerian streamfunction reveals the three
circulation cells known to characterize each hemisphere of Earth’s annual-mean
circulation: the Hadley, Ferrel, and Polar cells (Fig. 9a; the Polar Cell in
the Northern Hemisphere is too weak to be displayed at the contour interval
shown). The Hadley and Polar cells are thermally direct; their mass fluxes
imply warm air rising, cool air sinking, and downgradient energy transport.
The Ferrel Cell, on the other hand, is thermally indirect; its implied energy
transport is toward the equator. Such a cell is able to exist because
eddies—motions in the atmosphere representing departures from a time or zonal
mean—produce a poleward energy transport at midlatitudes that more than
compensates for the equatorward energy transport implied by the mean
circulation. In spite of the presence of the Ferrel Cell, the total energy
transport by eddies plus the mean flow remains poleward across the
midlatitudes.

The Ferrel cell does not appear when the streamfunction is calculated using
isentropic averaging. The lower panels of Fig. 9 show estimates of the
streamfunction based on mass fluxes averaged at fixed potential temperature
$\theta$ (Fig. 9b) and fixed equivalent potential temperature $\theta_{e}$
(Fig. 9c) following Pauluis et al. [272, 273]. As discussed in section V.2,
the potential temperature $\theta$ is approximately proportional to the
logarithm of the entropy of dry air $s_{d}$. Isosurfaces of $\theta$ are
approximately parallel to isosurfaces of dry entropy, and we therefore refer
to such surfaces as dry isentropes. blueFor an air parcel with a fixed total
water content, the equivalent potential temperature $\theta_{e}$ is
proportional to the logarithm of the entropy $s$. Since $s$ includes the
entropy of both dry air and water within an air parcel, we refer to surfaces
of constant $\theta_{e}$ as moist isentropes.

Both the dry- and moist-isentropic mass streamfunctions are characterized by a
single, thermally direct overturning cell in each hemisphere. Physically, the
Eulerian and isentropic mass streamfunctions differ because there is a strong
tendency for poleward moving air to be warmer (and moister) than equatorward
moving air at the same pressure level. A latitudinal exchange of air at a
given pressure level contributes to the isentropic overturning but does not
contribute to the Eulerian overturning. The dry- and moist-isentropic
streamfunctions therefore provide a view of the global circulation of the
atmosphere that is more directly connected to its meridional energy transport
than the view provided by the Eulerian streamfunction.

The moist-isentropic streamfunction captures energy transport associated with
the latent heat content of moist air as well as its sensible heat content. As
a result, the meridional mass transport by the atmosphere is $\sim 1.5$ times
larger when viewed on moist isentropes compared to either the dry-isentropic
view or the Eulerian view [273]. The moist-isentropic streamfunction also
differs from its dry-isentropic counterpart in that it has a roughly
consistent “depth” in isentropic coordinates from the equator to the pole,
with no obvious demarcation between the tropics and extratropics. This is
consistent with the meridional energy transport of the atmosphere, which is
also “seamless” between the tropics and extratropics [355].

On the basis of the moist-isentropic streamfunction, Pauluis et al. [273]
argue for a revised view of the global atmospheric circulation that includes a
“moist branch” that involves bluewarm and moist air that is advected poleward
in the subtropics and rises through the warm sectors of extratropical
cyclones. Recent studies have highlighted the utility of the moist-isentropic
view for understanding the thermodynamics of atmospheric circulations such as
the monsoon [44] and tropical cyclones (see section V.2). Such analyses may be
useful for quantifying the contribution of circulations of different scales to
the global atmospheric heat engine [45].

####  VI.2.2 Theories for the global atmospheric heat engine

A key goal of climate dynamics research is the development of a theory for the
meridional heat transport in the atmosphere. One approach toward achieving
this goal is to relate the time-mean meridional heat transport $\langle
F_{H}\rangle$ to the meridional temperature gradient through an “eddy
diffusivity” $K$, such that,

$\langle F_{H}\rangle\propto K\frac{\Delta T}{\Delta y},$

where $\Delta T/\Delta y$ gives a measure of the gradient of temperature $T$
in the meridional direction $y$ averaged over a suitable latitude band. The
theoretical challenge is to understand the dependence of $K$ on the mean
thermodynamic state of the atmosphere.

Theories for the eddy diffusivity $K$ go back at least half a century [e.g.,
108, 341]. Of particular note for the present review is the study of Barry et
al. [15], in which a scaling for $K$ was developed by treating the atmospheric
circulation as a heat engine.

Barry et al. [15] assumed that the net atmospheric energy flux out of the
tropics $\langle F_{H}\rangle$ could be related to the frictional dissipation
rate associated with large-scale atmospheric circulations
$\langle\dot{D}_{LS}\rangle$ through an expression of the form,

$\langle\dot{D}_{LS}\rangle\propto\frac{\Delta T}{T_{0}}\langle F_{H}\rangle.$
(66)

Here, large-scale circulations are those that are significantly affected by
the Earth’s rotation and are well resolved by global climate models. As in the
previous equation, $\Delta T$ represents a characteristic temperature
difference across the midlatitude zone while $T_{0}$ is a characteristic
temperature, giving $\Delta T/T_{0}$ the form of a heat engine efficiency. The
theory is closed using a mixing length argument, which expresses the
diffusivity $K$ in terms of the dissipation rate and a characteristic length
scale over which fluid parcels are displaced by eddies. Barry et al. [15]
showed that, through the appropriate choice of mixing length, their expression
for the diffusivity was able to account for changes in atmospheric heat
transport in simulations with a comprehensive global climate model across a
wide range of parameters.

A number of other diffusive theories for the atmospheric heat transport have
been proposed [e.g., Green [108], Stone [341], Lapeyre and Held [172], Held
and Larichev [116], Gallet and Ferrari [87]. See also Held [115] for a recent
review]. Generally, such theories are developed on the basis of the budget of
available potential energy rather than entropy (see section VIII.1.1), but
recent work by Chang [40] casts both Barry et al. [15] and Held and Larichev
[116] in a common entropy budget-focused framework, showing that they are both
limiting cases of a more general theory for eddy diffusivity.

A common feature of many theories for the atmospheric eddy diffusivity,
including that of Barry et al. [15], is that they do not explicitly consider
the effect of moist processes. Indeed, the relation (66) may be compared to
similar relations used to develop theories of atmospheric convection discussed
in section IV.1.2, in which the rate of work performed by atmospheric
convection is related to a forcing parameter through a thermodynamic
efficiency. As we have seen, this approach fails for moist convection because
it neglects irreversible processes associated with water in all its phases,
which account for the bulk of the material entropy production in the
atmosphere. But such an approach, suitably adapted, may nevertheless have
relevance to the larger-scale circulations that contribute to
$\langle\dot{D}_{LS}\rangle$. Such circulations are primarily driven by
horizontal heating and temperature gradients, while, as noted above, the bulk
of the entropy production in the atmosphere is associated with vertical
heating and temperature gradients [194].

Indeed, a number of authors have sought to adapt theories of the atmospheric
eddy diffusivity to include moist thermodynamics [see e.g., 173, 243].
Recently, the use of diffusive closures based on the energy content of air
(including latent energy) rather than its temperature have been identified as
a promising direction [e.g., 82, 133, 6, 230]. However, theoretical
justification for this approach remains incomplete, and understanding
atmospheric heat transport in a moist atmosphere remains an area of active
research.

Accounting for moist processes is particularly important in the context of
global climate change: as the world warms, the concentration of water vapor in
the atmosphere is expected to increase by roughly 7% for each kelvin increase
in temperature, following the Clausius-Clapeyron equation. This rapid increase
in atmospheric humidity clearly must be taken into account in any theory for
the atmospheric heat engine in a warming climate.

####  VI.2.3 The atmospheric heat engine under climate change

In the last few decades, the climate science community has collectively
developed a large archive of simulation data containing projections of global
climate change that is freely available to researchers [76]. This archive
provides an opportunity to study how the atmospheric heat engine is affected
by climate change, at least in the context of global climate models. While
evaluating the entropy budget is challenging based only on the available
outputs [176], Laliberté et al. [169] recently developed a technique for
diagnosing the strength of the atmospheric heat engine using only standard
model outputs.

Consider the fundamental thermodynamic relation (35), applied to a parcel of
air and written in terms of enthalpy as,

$T\frac{ds}{dt}=\frac{dh}{dt}-\alpha\frac{dp}{dt}-\sum_{x}g_{x}\frac{dq_{x}}{dt}.$
(67)

Recall that thermodynamic equilibrium between phase $x$ and $y$ is defined by
$g_{x}=g_{y}$, so that the last term on the right-hand side sums to zero for
phase changes in equilibrium [267]. Under the condition that all phase changes
occur in thermodynamic equilibrium, the last term is therefore only non-zero
when water is either added or removed from the air, and (67) may be written,

$T\frac{ds}{dt}=\frac{dh}{dt}-\alpha\frac{dp}{dt}+(g_{d}-g_{v})\frac{dq_{T}}{dt},$
(68)

where $q_{T}$ is the total mass fraction of water. This equation applies
equally to the addition or removal of water vapor in unsaturated conditions
and the addition or removal of liquid (or solid) water in saturated
conditions, for which $g_{v}=g_{l}$ (or $g_{v}=g_{s}$).

At first glance, the assumption of phase equilibrium suggests that (68) is not
well suited for application to the atmosphere, which often experiences phase
changes far from equilibrium. But evaporation or sublimation at subsaturation
may be taken into account provided the parcel to which (68) is applied
includes only water substance that is in thermodynamic equilibrium with the
surrounding air. For example, by excluding falling precipitation from the
parcel definition, evaporation of precipitation at subsaturation may be recast
as the addition of water vapor to unsaturated air, which is included in the
last term on the right-hand side of (68). On the other hand, melting and
freezing that occurs outside of phase equilibrium cannot be treated this way,
and irreversibility associated with the melt/freeze cycle is neglected by this
framework.

Integrating (68) with mass weighting over the atmosphere $\Omega_{A}$ and
taking a time mean over a statistically steady state, the time derivative of
the enthalpy vanishes, and the above equation may be transformed into a budget
for work done by the atmosphere,

$\langle\dot{W}_{\text{max}}\rangle=\langle\dot{W}_{K}\rangle+\langle\Delta\dot{G}\rangle.$
(69)

where

$\langle\dot{W}_{K}\rangle=-\int_{\Omega_{A}}\left\langle\frac{dp}{dt}\right\rangle\,dV$
(70)

is the rate at which the atmosphere performs work to generate the kinetic
energy of the winds and

$\displaystyle\langle\dot{W}_{\text{max}}\rangle$
$\displaystyle=\int_{\Omega_{A}}\left\langle\rho
T\frac{ds}{dt}\right\rangle\,dV,$ $\displaystyle\langle\Delta\dot{G}\rangle$
$\displaystyle=\int_{\Omega_{A}}\left\langle\rho(g_{d}-g_{v})\frac{dq_{T}}{dt}\right\rangle\,dV.$

Here the term $\langle\dot{W}_{\text{max}}\rangle$ represents a measure of the
maximum rate of work that could be performed by the atmosphere bluein the
absence of moist processes (all else being equal). The term
$\langle\Delta\dot{G}\rangle$ represents a “Gibbs penalty” related to the
effects of moisture, and it primarily represents the power required to
maintain the hydrological cycle [267]. This budget is similar to (64) in
section V.2, but, because we consider the fluid velocity to be the barycentric
velocity of the mixture of air and condensed water rather than the velocity of
air, the work does not include the work required to lift water (see section
III.2.3). The work required to lift water is included in the Gibbs penalty
term $\langle\Delta\dot{G}\rangle$.

Laliberté et al. [169] devised a method for applying (70) to climate model
output in order to estimate the strength of the global atmospheric heat engine
and its changes under global warming. While the method allows for accurate
closure of the budget, it does not distinguish between physical processes and
the numerical production of entropy (see section VII). In principle, the
latter entropy source may be climate dependent and affect the results.
blueMoreover, (69) suffers from the same limitation as (64), in that the terms
$\langle\dot{W}_{\text{max}}\rangle$ and $\langle\Delta\dot{G}\rangle$ are
sensitive to the specification of the reference entropies for dry air and
liquid water in the definition of $s$. Notwithstanding these caveats, the
authors found that, while the maximum rate of work
$\langle\dot{W}_{\text{max}}\rangle$ increased under warming, this was offset
by an even larger increase in the power required to maintain the hydrological
cycle $\langle\Delta\dot{G}\rangle$ such that the rate of work done by the
atmospheric heat engine in generating winds $\langle\dot{W}_{K}\rangle$
decreased with warming. These results are consistent with a recent analysis of
changes in the Lorenz energy cycle and entropy budget in a suite of global
climate model simulations of future warming conducted by Lembo et al. [176].

Laliberté et al. [169] argued that the rapid increase in the power required to
maintain the hydrological cycle $\langle\Delta\dot{G}\rangle$ under warming
could be related to the rapid increase in the moisture content of the
atmosphere following the Clausius-Clapeyron relation. On the other hand, the
maximum rate of work $\langle\dot{W}_{\text{max}}\rangle$ is governed by the
radiative cooling rate of the atmosphere, and this is known to increase at a
more modest rate under global warming [e.g., 4]. The work performed by the
atmosphere to generate winds $\langle\dot{W}_{K}\rangle$ must then decrease
with warming in order to balance (69).

A similar reduction in kinetic energy generation and an increased dominance of
moist irreversible processes was found in idealized simulations of climate
warming induced by increased greenhouse gas concentrations [196], increased
solar irradiance [197], and increased ocean heat transport [160]. In these
studies, a framework based on a variant of the dry-entropy budget (see section
III.2.5) originally developed by Johnson [147] is applied to define an
efficiency $\eta$ relating the work $\langle\dot{W}_{K}\rangle$ to the total
heating owing to the combination of frictional dissipation, radiation, and
latent and sensible heat fluxes within the climate system. Lucarini [193]
further defined an irreversibility parameter as the ratio of the irreversible
entropy production owing to frictional dissipation to that associated with
down-gradient heat transport. The heat transport may be defined to include
sensible and latent heat fluxes [160], or to additionally include radiative
fluxes within the climate system [193]. According to [196], the increased
importance of latent heat fluxes in a warming climate is associated with a
decrease in efficiency $\eta$ and an increase in the irreversibility of the
climate system.

Figure 10: Entropy budget as a function of sea-surface temperature (SST) in
simulations of radiative-convective equilibrium taken from Singh and O’Gorman
[331] and plotted with log scale. Material entropy export
$\langle\dot{S}_{e}^{\text{mat}}\rangle$ (black circles), total irreversible
entropy production $\langle\dot{S}_{i}^{\text{mat}}\rangle$ (black line), and
irreversible entropy production owing to frictional dissipation
$\langle\dot{S}_{i}^{\text{fric}}\rangle$ (red triangles), precipitation
sedimentation $\langle\dot{S}_{i}^{\text{sed}}\rangle$ (blue squares) and
irreversible phase change & mixing $\langle\dot{S}_{i}^{\text{mem}}\rangle$
(green pluses). Dotted gray lines show Clausius-Clapeyron scaling, increasing
in proportion to the saturation vapor pressure at the sea surface. Adapted
from Singh and O’Gorman [331].

The results of Laliberté et al. [169] and the other studies cited above
contrast with a recent estimate of trends in the generation and dissipation of
kinetic energy in the atmosphere based on global climate models constrained by
satellite and in-situ observations, in which it was found that both the
kinetic energy and its generation/dissipation rate increased over the period
1979-2013 [260]. If accurate, this result suggests that recent warming has
been associated with an intensification of the atmospheric heat engine. It
should be noted, however, that the technique of estimating the atmospheric
state using global climate models constrained by historical observations
(known as reanalysis) is not well suited to evaluating climate trends [e.g.,
352], and further work is needed to confirm the results of Pan et al. [260]
using other methods.

The decrease in $\langle\dot{W}_{K}\rangle$ with warming seen in global
climate simulations also contrasts with cloud-permitting simulations of RCE,
in which the rate of work done by the atmospheric heat engine has been found
to increase with surface temperature [309, 331]. This is despite the fact that
the moisture content and radiative cooling rates vary similarly with
temperature in both RCE and global climate simulations [145].

Singh and O’Gorman [331] examined the entropy budget in a series of
simulations of RCE over a wide range of surface temperatures. Their results,
reproduced in Fig. 10, show that the magnitude of the irreversible entropy
production terms associated with moist processes roughly scale with the total
radiative entropy sink $\langle\dot{S}_{e}^{\text{mat}}\rangle$, rather than
with the Clausius Clapeyron equation. This allows the entropy production
associated with frictional dissipation
$\langle\dot{S}_{i}^{\text{fric}}\rangle$ to also increase with warming.

The contrasting response of the atmospheric heat engine to warming in global
climate models compared to high-resolution models run in RCE is puzzling and
points to fundamental gaps in our understanding of the atmosphere’s entropy
budget. A major difference between the two types of studies is one of spatial
scale. Global climate models simulate the entire atmosphere, including the
large-scale circulations that act to transport heat from the tropics to the
polar regions. But such models are generally run with horizontal resolutions
too coarse to explicitly represent cloud-scale circulations, and the work
$\langle\dot{W}_{K}\rangle$ includes only the work required to generate
circulations that are resolved by the model grid (see section VII).
Simulations of RCE, on the other hand, are typically run on domains too small
to contain large-scale circulations, and the work performed is entirely used
to generate moist convection. Future studies applying the technique of
Laliberté et al. [169] to a wider range of model types, including both
realistic and idealized configurations, is clearly needed to better understand
how circulations at different scales are affected by climate warming, and how
this is manifest in changes to the atmospheric heat engine. A further
outstanding question is whether the differences in mechanical efficiency
associated with convective organization highlighted in section IV.3 may
contribute to changes in the strength of the atmospheric heat engine in its
response to climate change.

###  VI.3 Heat engines on other planets

The rapidly growing zoo of detected exoplanets, in addition to our quirky
companions in the Solar System, continues to astound the imagination. The
assumptions that lead us to simple models in Earth’s troposphere can be
completely invalid or irrelevant on other planets. Nevertheless, heat engine
concepts that consider entropy budgets, appropriately tailored, have been
applied to rocky planets in our Solar System [192, 191, 103, 353, 320, 13],
Jupiter [191, 371], tidally locked rocky exoplanets [161] and hot Jupiters
[296, 162].

An important complication in describing the characteristics of planetary heat
engines is that, unlike for a traditional heat engine operating between two
thermal reservoirs, the Carnot efficiency of a planetary heat engine is not an
external parameter. The input temperature $T_{\text{in}}$ and output
temperature $T_{\text{out}}$ are both functions of the climate (see section
II.2.2). Even the effective emission temperature $T_{e}^{*}$ of a planet,
defined as the temperature of a blackbody if it were to emit the same amount
of radiant energy as the planet, is a function of its planetary albedo, which
is climate dependent.

As a planet’s climate changes, both its mechanical efficiency and Carnot
efficiency may simultaneously change, perhaps substantially. The history of
Venus presents a possible example of such behavior; in its early history,
Venus has been hypothesized to be water-rich before a runaway greenhouse
effect occurred [364]. If there had been an active hydrological cycle, we can
hypothesize that the mechanical efficiency of the Venusian heat engine
(potentially near the Carnot efficiency at present) would be much lower in the
past to account for the attendant irreversible entropy production associated
with moist processes. However, the Carnot efficiency itself may have been
lower and more Earth-like if the presence of water clouds and precipitation
allowed for clear-sky patches to cool off the lower atmosphere to space. Thus
mechanical efficiency as a fraction of the Carnot efficiency is a moving
target that may obscure dramatic changes in climate.

####  VI.3.1 Rocky planets

Schubert and Mitchell [320] and Bannon and Lee [13] estimated the Carnot
efficiency of the rocky planets with substantial atmospheres (Venus, Earth,
Mars, Saturn’s moon Titan). They found that Venus has a much higher Carnot
efficiency than the other bodies considered, at around 24%. Like the better-
observed Earth, Mars and Titan are likely to have sedimentation-related
material entropy production sources that significantly reduce the mechanical
efficiency of the climate.

Bannon and Lee [13] used a creative approach to establish an upper bound on
the Carnot efficiency and to study the importance of spatial variations of
absorption and emission temperatures for the planetary heat engine. Using a
variant of the transfer climate system definition (see section II.1), they
defined the entropic absorption temperature $T_{\text{abs}}$ by,

$\frac{\langle\dot{Q}_{\text{abs}}\rangle}{T_{\text{abs}}}=\frac{1}{A}\int_{\Omega}\left\langle\frac{\rho\dot{q}_{\text{abs}}}{T}\right\rangle
dV,$

where $\dot{q}_{\text{abs}}$ is the heating of the climate system by shortwave
radiation, with

$\langle\dot{Q}_{\text{abs}}\rangle=\frac{1}{A}\int_{\Omega}\langle\rho\dot{q}_{\text{abs}}\rangle\,dV.$
(71)

The entropic emission temperature $T_{e}$ was defined analogously as a
function of $\langle\dot{Q}_{e}\rangle$, based on the temperature at which
longwave radiation is emitted directly to space. Unlike (16), (71) does not
include net heating by longwave radiation due to radiative exchanges within
the climate system; it is purely defined based on absorption of photons
emitted by the sun. Bannon and Lee [13] showed that $T_{e}$ must be equal to
or greater than the corresponding “effective” emission temperature $T_{e}^{*}$
necessary for a corresponding blackbody planet of equivalent albedo to be in
radiative equilibrium with its star. Given a fixed stellar flux, orbital
radius and planetary size, $T_{e}^{*}$ is only a function of the planetary
albedo. Employing a thought experiment of the entropy balance of two idealized
planets, Bannon and Lee [13] demonstrated that the maximum entropy production
of a planet occurs when $T_{e}=T_{e}^{*}$, which can be understood as the
lowest temperature possible for the heat sink to space. When $T_{e}$ instead
varies across the planet, entropy production decreases. In contrast, a more
variable absorption temperature $T_{\text{abs}}$, or more variable albedo,
increases entropy production. For a fixed $T_{\text{abs}}$, maximum entropy
production occurs when $T_{e}=(3/4)T_{\text{abs}}$. Bannon and Lee [13]
further scaled these temperatures by a truly external temperature
$T_{\text{bb}}$, which is the temperature the planet would have if it were a
blackbody with zero albedo. The corresponding material entropy production can
be scaled similarly. This permits a nondimensional comparison of entropy
production between planets in a two-dimensional phase space of
$T_{e}/T_{\text{bb}}$ and $T_{\text{abs}}/T_{\text{bb}}$, and it neatly
demonstrates that Venus is indeed unique for its high Carnot efficiency, which
is close to the upper bound.

Venus has a high albedo (around 76%) and thick sulfuric acid cloud cover, such
that little shortwave radiation reaches the surface. Venus’ thin thermally
direct overturning circulation likely does not penetrate to the stable
atmospheric layer closest to the surface—it largely occurs aloft in the cloud
region, 50-55 km, away from a solid frictional surface. Evidence suggests a
possibility of convective overshooting in Venus’ atmosphere, which would
indicate a Venusian counterpart to Earth’s hydrologic cycle [e.g., 7, 220] and
would suggest that the mechanical efficiency of the Venusian atmosphere is
substantially lower than the Carnot efficiency. But dissipation due to
friction around hydrometeors was estimated to be unimportant by Lorenz and
Rennó [191]. The dominant dissipation mechanism that removes mechanical energy
is hypothesized to be breaking internal gravity waves [138]. Taken together,
the studies of Lorenz and Rennó [191], Schubert et al. [321], Bannon and Lee
[13] suggest that Venus has a very high absolute mechanical efficiency, but
the role of moist convection is too poorly understood to have high confidence.

Mars, uniquely in the Solar System, has sporadic global dust storms that
nearly shield the surface from the sun. This dust absorbs both shortwave and
longwave radiation, adding to the longwave absorption from the CO2 atmosphere.
As the dust sediments back to the surface, it is also likely to be an
important source of dissipation analogous to the dissipation of falling
hydrometeors discussed in section III.2.3. Given the small difference between
Mars’ estimated average absorption temperature and emission temperature in the
vertical, more work may potentially be produced during horizontal energy
transport than vertical transport, which the existence of global dust storms
and small dust devils seems to support [320, 139]. The dominance of Earth’s
vertical production of irreversible entropy cannot be assumed of other
planets. In particular, planets with shallow atmospheres and small surface-
emission layer temperature differences may still have a strong lateral
temperature gradient. This can lead to increased entropy loss to space,
coincident with increased work production and frictional dissipation in the
atmosphere. Koll and Abbot [161] showed that the temperature difference
relevant to the heat engine model of a tidally locked rocky exoplanet is the
permanent horizontal day-night temperature gradient and not the local vertical
lapse rate.

The most Earth-like planet in terms of entropy sources and sinks is likely
Saturn’s moon Titan. It has an active methane cycle with some resemblance to
Earth’s hydrological cycle, which is a source of irreversible entropy
generation due to drag around ‘rain’-drops. Similar to Mars however, there is
a small difference between the average absorption and emission temperatures,
rendering a rather low estimated Carnot efficiency of 4.1% [320]. It is
possible that the general circulation is able to produce more work than this
Carnot efficiency would suggest if Titan has substantial seasonal cross-
hemispheric heat transport, which was argued using energy balance and
numerical modeling by Mitchell [226].

####  VI.3.2 Giant planets

Rocky planets can be generally assumed to be in energy balance with their
star. They are much smaller on average than the fluid planets and are either
geologically ‘dead’ with a cold core, or the climate system is shielded from
an active core by an insulating rocky mantle, as in the case of Earth. Either
way, the geothermal heat flux on the rocky planets is typically an
insignificant fraction of the energy received from the sun. Their well-defined
solid surfaces bound the climate system on timescales shorter than the
evolution of the lithosphere. These traits make single- and multi-layer energy
balance models tractable and useful. Giant planets are much trickier systems.
Gas and ice giants, like Jupiter, Saturn, Uranus and Neptune, are likely to
lack any kind of solid lower boundary that could provide a large frictional
drag on winds, and modeling efforts do not always carefully consider the need
for a physically motivated dissipation mechanism [100]. In the absence of
solid frictional surfaces, leading dissipation mechanisms to balance
mechanical energy generation by thermally direct flows include turbulence and
fluid instabilities, shocks [58, 179], Ohmic dissipation [16] and magnetic
drag [285].

The idealization of the atmospheric circulation as a thermally direct
circulation similar to a Carnot engine is a reasonably good model for
atmospheric layers with approximately adiabatic lapse rates, such that
convective motions can move air quasi-adiabatically in the vertical as they
advect heat downgradient. This is certainly a good model for the Earth’s
troposphere, but the Earth’s stratosphere—more generally any atmosphere in
approximate radiative equilibrium—does not exhibit thermally direct quasi-
adiabatic motions [162]. The Brewer-Dobson overturning circulation in the
stably stratified stratosphere, for example, is mechanically forced [114], and
it is thought that regions within the giant planets, as well as brown dwarf
planets, should exhibit thermally indirect, wave-driven overturning
circulations [327], where the waves may be excited by convective (thermally
direct overturning) activity in an adjacent atmospheric layer.

Even in regions of the atmosphere that are characterized by thermally direct
circulations, the Carnot cycle is not necessarily a good model, particularly
for heavily irradiated giant exoplanets. These so-called ‘hot Jupiters’,
believed to be in abundance throughout the universe, have extremely short and
rapid orbits around their stars. The thermally direct overturning circulation
occurs approximately within an isothermal layer column-wise, with a strong
day-night gradient (where the day side is permanently irradiated because it is
tidally locked to always face the star). Koll and Komacek [162] modeled the
heat engine of hot Jupiters instead as an Ericsson cycle. Like the ideal
Carnot cycle, an Ericsson cycle is heated and cooled during isothermal
processes, but the other two legs are isobaric instead of isentropic. An ideal
Ericsson cycle has the same efficiency $\eta_{C}$, but the mechanical
efficiency of hot Jupiters could be larger or smaller depending on the unknown
potential role of precipitation during the cycle (see section II.2.3). Hot
Jupiters are likely to host multiple layers of cloud decks (including silicate
and titanium dioxide clouds) as well as hydrocarbon hazes [88].

Lastly, planets need not even be close to energy balance. This is observed for
Jupiter, Saturn and Neptune, which emit around 80%-160% more energy than they
receive from the sun [46, 134, 182]. These planets are still cooling,
shrinking and stratifying from their formation [131]. Given $\langle
Q_{\text{out}}\rangle>\langle Q_{\text{in}}\rangle$, and assuming that average
$T_{\text{in}}>T_{\text{out}}$ as on Earth, we could assume that these planets
are in entropy balance and determine that they must be producing additional
entropy irreversibly to balance the enhanced entropy export $\langle
Q_{\text{out}}\rangle/T_{\text{out}}$ to space. But why should we assume
entropy balance under these conditions? More likely, these planets are
secularly ‘ordering’ (stratifying by density) as well as cooling, and
accordingly losing net entropy to space. If the planets are indeed out of
entropy balance, how could we measure that remotely? On a fluid planet, can we
distinguish between a plausibly fast atmospheric adjustment to entropy balance
and a core region that loses entropy over time? The fluid behaviors (including
magnetohydrodynamics at depth) of the giant planets are still extremely
unconstrained, so this is an area ripe for further observational missions.

Lucarini [193] and Bannon and Lee [13] call for future climate models to
routinely calculate and provide emission and absorption temperatures, in order
to make possible quantitative heat engine analyses between simulations. As
global climate model capabilities improve in realism as well as flexibility
(i.e., models capable of simulating Mars, Jupiter, hot Jupiters, etc.), there
is now potential for comparative planetary climatology to tackle the evolving
Carnot and mechanical efficiencies of ancient and alien worlds. This approach
may help bound the possible range in climates of exoplanets, which for the
foreseeable future can be studied observationally only as point sources of
light.

##  VII Modeling the second law of thermodynamics

Given the impracticality of conducting controlled experiments on the climate
system and the sparseness of our networks for observing the atmosphere, ocean,
and land surface, numerical models represent an essential component of the
climate researcher’s toolkit. Climate models are used as numerical
laboratories to test hypotheses about how the climate system operates, as
state estimation tools to study aspects of the climate system that go beyond
those accessible to observations, and as tools for projecting future climate
change171717Numerical weather prediction models, closely related to global
climate models, are also used for weather forecasting.. Such models implement
numerical approximations to physical laws including conservation of energy,
mass, and momentum in order to solve for the evolution of the system. As
discussed in section III, these conservation laws, combined with a suitable
definition of entropy, are sufficient to specify the entropy budget. But
developing models of the climate that produce a realistic entropy budget that
satisfies the second law of thermodynamics remains challenging. In this
section, we discuss some of the issues raised when attempting to model the
second law in the context of two types of climate models: cloud-permitting
models (section VII.1) and global climate models (section VII.2).

###  VII.1 Cloud-permitting models

Cloud-permitting models are numerical models of the atmosphere with horizontal
grid spacing $\lesssim 2$ km, giving them sufficiently high resolution to
represent at least the largest convective cloud systems explicitly. This
contrasts with global climate models, discussed in the next subsection, for
which moist convection cannot be resolved on the model grid. Because of their
large computational cost, cloud-permitting models are usually run on regional
domains or idealized domains that do not encompass the whole Earth181818Recent
advances in computing technology are beginning to allow for global-scale
weather and climate models that approach cloud-permitting resolutions [see
e.g., 339, 368].. For example, section IV presents the results of idealized
simulations of RCE using a cloud-permitting model on a domain roughly
$200\times 200$ km2 in size.

Although their horizontal grid spacing is much smaller than grid spacings
typical for global climate models, cloud-permitting models remain too coarse
to properly resolve individual clouds and their associated turbulence.
Explicit resolution of atmospheric turbulence down to the inertial subrange is
estimated to require grid spacings of 100 meters or less [29], which is still
many orders larger than that required for direct numerical simulation (DNS) of
the atmosphere191919DNS has been applied to understand the detailed dynamics
of cloud entrainment [e.g., 223] and microphysics [e.g., 359], but
computational constraints currently limit the accessible Reynolds numbers far
below the requirements for even a single cloud life-cycle, let alone the
global atmosphere.. As a result, cloud-permitting models do not explicitly
resolve the diffusive molecular fluxes of heat, water, and momentum that are
involved in irreversible entropy production in the atmosphere. Rather, the
effects of these molecular fluxes must be approximated by the model’s subgrid-
scale turbulence parameterizations. Such parameterizations assume that the
turbulent cascade of kinetic energy toward the molecular scale may be
expressed in terms of the model’s resolved-scale flow. If this assumption is
satisfied, the frictional dissipation rate, and the associated irreversible
entropy production, implied by the parameterized subgrid-scale momentum
transports provide a good approximation to the dissipation rate and
irreversible entropy source owing to viscosity in the atmosphere [309]. In
other words, we expect subgrid-scale momentum transport to produce entropy
irreversibly, thereby satisfying the second law of thermodynamics [92].

In our simulation of RCE described in section IV, subgrid-scale fluxes of
momentum are indeed associated with positive frictional dissipation and
positive irreversible entropy production (see table 2). Subgrid-scale fluxes
of heat, however, are associated with a small but systematic sink of entropy.
As has been noted by previous authors [101, 309, 94], parameterized turbulent
heat transport does not necessarily produce an entropy source of the same sign
as molecular diffusion of heat in the atmosphere. The reason for this is that
the parameterization must account for heat transport by turbulent air motions
on scales smaller than the grid in addition to diffusion of heat at the
molecular scale. In particular, vertical turbulent heat transport is
associated with the exchange of air parcels at different height levels. But in
stably stratified conditions, air does not spontaneously move vertically;
rather, work must be done against the stratification to produce vertical
motion. In the atmosphere, and at resolved scales within numerical models, the
energy required for this work is provided by the kinetic energy associated
with turbulence. But for scales smaller than a model’s grid, turbulence is not
resolved, and the required work is supplied by the internal energy of the air
itself. Such an energy conversion from internal energy to work corresponds to
a sink of entropy, and if it were to occur spontaneously it would violate the
second law of thermodynamics [94]202020Traditional parameterizations of
subgrid-scale water vapor transport can also lead to local entropy sinks
because of the work required to diffuse water vapor vertically [94]..

In numerical models, parameterized turbulent heat transport does not occur in
isolation; rather, it is associated with turbulent transports of momentum and
mass. Previous authors have argued that the negative entropy production owing
to parameterized turbulent heat transport may be reconciled with the second
law by recognizing that turbulent heat transport and turbulent dissipation of
kinetic energy are different aspects of the same turbulent cascade [3, 287].
According to this view, the second law is satisfied provided that the total
entropy production associated with all turbulent transports is positive. In
shear-driven turbulence layers, it may be shown that this condition is
guaranteed if the Richardson number (a nondimensional ratio representing the
relative importance of buoyancy compared to shear) is below a critical value.
In many turbulence parameterizations, the critical Richardson number is taken
to represent the onset of shear-driven turbulence [e.g., 184], ensuring that
subgrid-scale turbulent heat and momentum transports only occur when they
would result in a net positive irreversible entropy production as required by
the second law.

Gassmann and Herzog [94] and Gassmann [91] have recently argued against the
above view, suggesting instead that the second law requires positive entropy
production for both parameterized turbulent heat transport and parameterized
turbulent momentum transport individually. The authors develop a formulation
of turbulent heat transport that satisfies this constraint by including a
subgrid work term in the mechanical energy equation. Effectively, this
formulation shifts the energy source of work done by turbulence against the
stratification from the internal energy of the fluid to the resolved-scale
motion. Gassmann [91] provides evidence that this formulation allows for a
more realistic simulation of a dry gravity wave, but it is at present not
widely adopted within the field. Clearly, continued research is needed to
further clarify the requirements placed by the second law of thermodynamics on
the formulation of turbulence parameterizations used in cloud-permitting
models [see e.g., 92].

###  VII.2 Global climate models

Global climate models, also known as general circulation models, are numerical
models of the atmosphere, land, and ocean, that are used for both weather
prediction and climate projection. Because they must cover the entire planet,
global climate models are typically run at lower resolution than cloud-
permitting models discussed above, and their horizontal grid spacing within
the atmosphere ($\gtrsim 20$ km) is too coarse to resolve convective clouds.
Since clouds and their associated circulations are responsible for a large
portion of the irreversible entropy production in the atmosphere, evaluating
and interpreting the entropy budget of a global climate model presents a
particular challenge.

Johnson [146] was one of the first to explicitly link the entropy budget of
global climate models to biases in their simulation of the atmosphere. The
author argued that numerical dissipation in climate models leads to an
artificial source of entropy that spuriously increases the material entropy
production of the simulated atmosphere. In order to maintain a steady state,
an opposing error in the simulated entropy import must also be present.
Johnson [146] suggested that this could occur via a cold bias in the model’s
temperature field, providing an explanation for “the general coldness of
climate models”. As pointed out by Lucarini and Ragone [195], however, a bias
in the entropy import to the atmosphere is just as likely to be associated
with a bias in the radiation field as in the temperature field. Moreover, the
cold bias referred to by Johnson [146] is much reduced in more recent
generations of global climate models [83].

Woollings and Thuburn [375] investigated numerical entropy generation in
climate-model simulations of a dry atmosphere in the absence of radiative heat
transport or heat exchange with the surface—effectively a thermodynamically
isolated atmosphere. The authors found both positive and negative numerical
entropy sources, contradicting Johnson’s assumption that numerical entropy
generation acts solely to increase the total internal entropy production of a
simulated atmosphere. Moreover, numerical entropy sinks within a fluid that is
otherwise isolated correspond to a local violation of the second law of
thermodynamics. To prevent such occurrences, Liu and Liu [186] suggested an
ad-hoc procedure in which the diabatic heating rate within a model is altered
to ensure consistency with the second law.

A more rigorous solution is to employ numerical formulations of the governing
equations that maintain their Poisson-bracket structure upon discretization
[93], thereby reproducing exact conservation of energy and entropy (in the
absence of non-conservative terms), and avoiding the problem of artificial
numerical sources of entropy [92]. But while such numerical formulations are
beginning to be used in cloud-permitting models [90], they are not typically
used in global climate models, which still suffer from numerical errors in
their energy and entropy budgets [195, 136]. Similar numerical errors cause
imbalances in the water budgets of global climate models [183], with follow on
effects for the energy and entropy budgets due to the latent heat carried by
water substance. Many global climate models also employ simplified
thermodynamic formulations that neglect processes such as the heating owing to
frictional dissipation [262]. Care must be taken to evaluate the entropy
budget of such models so that it may be compared meaningfully to that of the
Earth [269].

As in the case of cloud-permitting models, irreversible entropy production in
global climate models is not modeled explicitly; it occurs within
parameterizations that calculate the effect of processes that occur at subgrid
scales. Because of their low resolution, global climate models require
parameterizations for processes such as ocean mesoscale eddies and atmospheric
convection that are not required by higher-resolution models. Developing
accurate parameterizations for these processes remains an ongoing challenge.
For example, errors in moist convection parameterizations have been argued to
be responsible for long-standing biases in the tropical precipitation
distribution simulated by global climate models [e.g., 250]. Evaluating the
ability of parameterizations of moist convection to accurately represent the
second law of thermodynamics therefore provides a potential pathway toward
their improvement. However, because convection parameterizations represent
both reversible processes (e.g., the generation of kinetic energy by cloud
motions), and irreversible processes (e.g., vapor mixing and irreversible
evaporation at the cloud edge) within the atmosphere, evaluating their
compliance with the second law remains a nontrivial theoretical challenge
[94].

The limitations in the representation of the second law in global climate
models, and in cloud-permitting models as discussed in the previous
subsection, lead to errors in their simulation of the atmospheric and oceanic
flow. Such errors are likely to be quantitatively small, but if they are
systematic, they may nevertheless be consequential for the mean climate and
its statistics. For instance, Singh and O’Gorman [331] reported that the
irreversible entropy source due to vapor diffusion in simulations of RCE was
strongly sensitive to vertical resolution, and this resulted in resolution
dependence of the simulated mechanical efficiency of moist convection. Given
these potential sensitivities, it is our view that a firm theoretical
foundation for the representation of the second law of thermodynamics in
climate models should be a goal of model developers. Increased availability
and use of thermodynamic diagnostics for the evaluation of climate models, as
recently advocated by Laliberté et al. [169] and Lembo et al. [176] provides
one possible step toward this goal. Promisingly, such diagnostics are
beginning to be included in standard analysis packages for the evaluation of
global climate models [e.g., the Earth System Model Evaluation Tool; 77].

##  VIII Variational approaches for climate and geophysical flows

The second law of thermodynamics implies that an isolated system evolves
toward a state of maximum entropy. The eventual state of such a system may
therefore be solved through an extremization procedure using variational
methods. While Earth’s climate is not isolated and exists far from
equilibrium, similar variational approaches have nonetheless found a range of
applications in the literature.

In particular, variational methods have been used to define measures of the
amount of energy “available” to do work on the climate system. This literature
involves an extremization of a particular energy reservoir under the
constraint that the total energy of the (presumed isolated) climate system is
constant. We discuss two examples of such measures in section VIII.1.

Furthermore, certain long-lived coherent structures in planetary fluids exist
in an “inertial” regime in which both forcing and dissipation are weak. Such
flows are amenable to analysis through statistical mechanics techniques where
a quasi-steady solution is determined through the maximization of an entropy
variable, despite the forced-dissipative nature of the broader climate system.
We discuss statistical mechanical approaches to geophysical fluid dynamics
(GFD) in section VIII.2.

Lastly, a controversial hypothesis due to Paltridge [254] extends the idea of
entropy maximization to forced-dissipative systems by arguing that such
systems tend to maximize their entropy production rate: this is the Maximum
Entropy Production (MEP) principle. We critically examine the MEP principle in
section VIII.3, concluding that its physical basis remains unclear, and its
application to the climate system remains speculative.

###  VIII.1 Entropic energies

The first law does not distinguish between heat and work, but the second law
breaks that symmetry; according to the second law, work can be completely
converted to heat but heat cannot be completely converted into work. Thus the
second law of thermodynamics indicates that the universe is irreversibly and
monotonically transforming energy from other forms into unusable internal
energy. In previous sections, this principle was expressed in terms of
entropy, but it may also be expressed in energetic terms by defining a measure
of the energy available to drive motions in a fluid. In this section, we
explore two parallel threads of research that seek to provide a definition of
such an energy measure.

Section VIII.1.1 discusses a common approach in atmospheric science and
physical oceanography, that of quantifying the Available Potential Energy
(APE) of the climate system as a source of kinetic energy. Section VIII.1.2
then describes exergy as an alternative and more formal measure of departure
from thermodynamic equilibrium and briefly reviews some applications to the
climate system. The reader is referred to Tailleux [345] for a detailed review
of APE, exergy and related concepts.

####  VIII.1.1 Available potential energy

Margules [208]212121As translated by Abbe [1]. and Lorenz [188] pioneered the
quantification of an atmosphere’s ability to drive motion [222]. Lorenz
defined the Available Potential Energy (APE) of the atmosphere $A$ as the
component of the total potential energy $P$ that may be converted to kinetic
energy of the general circulation222222Convective available potential energy
(CAPE), introduced in section IV.2, has some similarities to the concept of
APE, but it is defined as the energy available to an infinitesimal parcel on
an ascent through the atmosphere rather than for the fluid as a whole..
Evaluating the APE for Earth’s atmosphere, Lorenz [188] found it to be a very
small fraction of the total potential energy $P$.

The concept of APE is easy to illustrate. Imagine a water glass containing hot
water above cold water with a tilted interface between the two water masses
(Fig. 11a). At an initial time $t_{0}$, the water is at rest and in
hydrostatic balance. If the system is allowed to spontaneously evolve, one may
intuit that there will be rapid turbulent motion as the water masses reduce
the interface slope to zero. Thus, even though each column individually was in
hydrostatic balance initially, the horizontal gradient of density provides a
reservoir of potential energy that can be spontaneously converted to kinetic
energy: it is “available”. After sufficient time has passed, the turbulent
motions cease and both the hot water and cold water masses become slightly
warmer due to the frictional heating associated with the dissipation of the
kinetic energy. But the potential energy is lower than that of the initial
state232323In the Boussinesq limit, there is no internal energy reservoir, and
instead frictional dissipation of motion returns energy to the potential
energy reservoir. Thus if the fluid considered were Boussinesq, the potential
energy of the initial state and final state would be the same.. Had the glass
been stably stratified from the beginning, without horizontal gradients, it
would have remained motionless, maintaining its initial potential energy. APE
provides us with a mathematical tool to determine when and how much potential
energy may be released by spontaneous fluid motion.

Figure 11: Schematic of release of available potential energy. (a) Motionless
initial water glass with a non-zero interface slope between stably stratified
water masses. (b) Motionless final state after available potential energy $A$
was released, converted to kinetic energy $K$, and then converted to internal
energy $IE$ and non-available potential energy through frictional heating and
the associated thermal expansion.

Lorenz’s APE theory assumes an initially stable vertical profile, so energy
cannot be converted to work from purely vertical rearrangement, and it
requires the definition of a reference state that minimizes the total
potential energy $P$ subject to some constraints. Lorenz [188, 189] proposed
that the appropriate reference state is one in which the atmosphere is
reversibly and adiabatically re-arranged (holding mass constant) to a static
state of minimum potential energy $P_{r}$, leaving the residual potential
energy available to drive motion. Such a reference state is in mechanical
equilibrium (motionless and in hydrostatic balance), but not in thermal
equilibrium, given there remains a vertical temperature gradient associated
with vertical stratification. The APE is then defined over the entire
atmosphere [279]:

$A=\int_{\Omega_{A}}(P-P_{r})\rho dV.$ (72)

Letting $C(X,Y)$ indicate a rate of energy conversion from reservoir $X$ to
reservoir $Y$, the Lorenz energy cycle is:

$\langle\dot{A}\rangle=\langle\dot{G}\rangle-\langle C(A,K)\rangle\\\ $ (73)
$\langle\dot{K}\rangle=\langle C(A,K)\rangle-\langle\dot{D}\rangle.$ (74)

In steady state (indicated by the time-averaging brackets), APE is generated
at rate $\langle\dot{G}\rangle$ and converted to kinetic energy at rate
$\langle\dot{K}\rangle$. Kinetic energy is ultimately removed by frictional
dissipation at rate $\langle\dot{D}\rangle$.

The original APE of Lorenz [188] was only defined for a global integral over
small-amplitude perturbations from the resting state. The integrand in (72)
may be locally positive or negative, but the APE is positive definite upon
integration.

Van Mieghem [360] quickly pointed out limitations in the assumptions of the
reference state, remarking: “the hydrostatic hypothesis and the assumption of
incompressibility which are commonly used in atmospheric dynamics are far more
dangerous to introduce in energy studies”. Pearce [278] formulated a more
complete APE that included the impact of energy available upon adiabatic
rearrangement in the vertical direction and that is valid without assuming
that the atmosphere is in hydrostatic balance.

The need to define a positive-definite local APE (an APE density) was
addressed by Holliday and Mcintyre [124] for a stratified, incompressible
fluid. That same year, Andrews [5] developed a theory for local APE density
valid for nonhydrostatic and compressible flows. He identified an additional
energy reservoir in compressible atmospheres termed the available elastic
energy. Lorenz’s APE was shown to always be less than or equal to a volume
integral of the APE density [5]. Shepherd [325] derived a quantity equivalent
to the APE density using Hamiltonian methods and called it a “pseudoenergy”
(we will briefly review Hamiltonian GFD in section VIII.2).

The studies discussed above generally consider only the “dry” component of
APE, neglecting the potential energy associated with the presence of water in
the atmosphere. However, when the effects of moist processes are included, the
estimated APE production rate is far greater than the observed or estimated
frictional dissipation rate [266], indicating that frictional dissipation
$\dot{D}$ is not the only (nor even primary) sink of “moist” APE. Pauluis
[266] formulated the first budget for APE that includes sources, sinks, and
diffusion of water content in a moist atmosphere. He showed that heat and
water diffusion, precipitation, and re-evaporation can each change APE. But
despite the conceptual similarity between irreversible entropy generation and
APE destruction, there is no direct mapping between the sign of the change in
APE and the occurrence of irreversible moist processes; the sign of the APE
change is also a function of the vertical position of the air parcels in which
these processes occur relative to the vertical position of the same air
parcels in the adiabatically-rearranged minimum-$P$ reference state. In a
moist atmosphere, the vertical rearrangement of air parcels towards the
reference state can be complex. For example, a moist parcel of air may be in a
statically stable environment within the unsaturated boundary layer, but if it
is lifted adiabatically until the water vapor begins to condense and release
latent heat, the parcel may nevertheless acquire a higher altitude in the
minimum-$P$ reference state, even if no horizontal gradients are present.

Evaluating the APE involves applying a sorting algorithm to find the
appropriate reference state, and this can be computationally intensive,
particularly if moisture is considered [190, 293, 121, 336, 342].
Nevertheless, APE and the related Lorenz energy cycle are widely applied in
the atmospheric (usually using a dry formulation) and oceanic literature [see
e.g., 176, 132, 181, 363]. The zonal mean of APE has also been shown to scale
with the eddy kinetic energy or ‘storminess’ of the midlatitude storm tracks
[319], and this relationship has been used to help explain future changes in
storminess projected by global climate models [242].

####  VIII.1.2 Exergetics

Lorenz’s and subsequent approaches that minimize potential energy to define a
reference state may be characterized as a ‘mechanical’ perspective [129], in
which the minimum potential energy reference state is not necessarily a state
to which the atmosphere spontaneously tends. In contrast, the concept of
exergy facilitates a second-law based thermodynamic perspective on the
availability of energy to do work in a fluid which has been developed in
parallel with APE theory [e.g. 152, 60, 59, 187, 150, 209, 210, 165, 85, 9,
10, 284, 129, 215]. Some physics curricula and most engineering programs teach
the concept of exergy [coined by 294], which is used widely in the energy
industry [119]. In the climate literature it has also been called ‘static
entropic energy’ [59], ‘static exergy’ [150], ‘availability’ [11], ‘available
energy’ [9, 10], or ‘available enthalpy’ [209], with small variations in
formulation and assumptions. The subfield of study generally may be called
exergetics [150]. The relationship between APE and exergy of the atmosphere
has been discussed in Dutton [59], Marquet [209], Kucharski [165], Fortak
[85], Tailleux [345].

Exergy $B$ is the amount of energy in an out-of-equilibrium system that can be
converted to useful work upon moving to a reference state that is in
thermodynamic equilibrium with its environment. The reference state has lower
total energy than the original state. For an open system such as a power
plant, we may imagine the system exporting energy reversibly until it reaches
the reference state; the amount of energy exported is equal to the exergy.
Hermann [119] estimated the exergy available in various climate system
components, disregarding practicality. For example, if we were theoretically
able to extract the global gravitational and chemical exergy available in
freshwater precipitation back to the salty ocean surface, that would yield 44
TW of power. Wind energy is currently extracted by surface-based wind turbines
at a global rate of 743 GW [174], with a theoretical (and undesirable) upper
bound of blueat least 400 TW [218].

For exergy calculations of the climate system as a whole, defining the
“environment” becomes problematic. In this case, one may instead assume that
the climate system is isolated in order to define an exergy that measures the
portion of energy within the system that can perform work on the system
itself. As for the open-system case, the reference state has lower total
energy than the initial state, and the difference is the exergy. But in the
isolated-system case, the system cannot evolve to the reference state because
it cannot export energy.

To define both APE and exergy, the reference state is reached assuming total
mass is held constant. However instead of minimizing total potential energy,
exergy is defined by minimizing the Gibbs free energy of the reference state
at a reference temperature $T_{r}$. The reference state is then both static
(mechanical equilibrium) and isothermal at $T_{r}$ (thermal equilibrium), with
reference profiles in the vertical of pressure $p_{r}(z)$ and specific entropy
$s_{r}(z)$. The difference in the Gibbs free energy between the reference
state and the initial state is the exergy. The static exergy $B$ of a single
fluid may therefore be defined [e.g., 85, 9]:

$\displaystyle B=$ $\displaystyle
h(s,p)-h(s_{r},p_{r})-\alpha(p-p_{r})-T_{r}(s-s_{r})$ $\displaystyle=$
$\displaystyle[h(s,p)-h(s,p_{r})-\alpha(p-p_{r})]$