# The climate system and the second law of thermodynamics

Martin S. Singh<EMAIL_ADDRESS>School of Earth, Atmosphere, and
Environment, Monash University, Victoria, Australia  Morgan E O’Neill
Department of Earth System Science, Stanford University, California, USA

###### Abstract

The second law of thermodynamics implies a relationship between the net
entropy export by the Earth and its internal irreversible entropy production.
The application of this constraint for the purpose of understanding Earth’s
climate is reviewed. Both radiative processes and material processes are
responsible for irreversible entropy production in the climate system.
Focusing on material processes, an entropy budget for the climate system is
derived which accounts for the multi-phase nature of the hydrological cycle.
The entropy budget facilitates a heat-engine perspective of atmospheric
circulations that has been used to propose theories for convective updraft
velocities, tropical cyclone intensity, and the atmospheric meridional heat
transport. Such theories can only be successful, however, if they properly
account for the irreversible entropy production associated with water in all
its phases in the atmosphere. Irreversibility associated with such moist
processes is particularly important in the context of global climate change,
for which the concentration of water vapor in the atmosphere is expected to
increase, and recent developments toward understanding the response of the
atmospheric heat engine to climate change are discussed. Finally, the
application of variational approaches to the climate and geophysical flows is
briefly reviewed, including the use of equilibrium statistical mechanics to
predict behavior of long-lived coherent structures, and the controversial
maximum entropy production principle.

###### Contents

  1. I Introduction
    1. I.1 Motivation
    2. I.2 Applications of the second law in climate research
    3. I.3 Structure of the review
  2. II The second law of thermodynamics applied to the climate system
    1. II.1 The boundaries of the climate system
      1. II.1.1 The planetary climate system
      2. II.1.2 The material climate system
      3. II.1.3 The transfer climate system
    2. II.2 The climate system as a heat engine
      1. II.2.1 Heat engines and irreversibility
      2. II.2.2 Carnot efficiency of the climate system
      3. II.2.3 Mechanical efficiency of the climate system
  3. III Irreversible processes in the climate system
    1. III.1 Derivation of the material entropy budget
      1. III.1.1 Single-component fluids
      2. III.1.2 Multi-component fluids
    2. III.2 Thermodynamics of a moist atmosphere
      1. III.2.1 Irreversible phase changes
      2. III.2.2 Diffusive mixing
      3. III.2.3 Irreversible sedimentation of precipitation
      4. III.2.4 The entropy budget of a moist atmosphere
      5. III.2.5 The role of latent heating
    3. III.3 Irreversible entropy production in the ocean
  4. IV The entropy budget of atmospheric convection
    1. IV.1 Radiative-convective equilibrium (RCE)
      1. IV.1.1 Dry radiative-convective equilibrium
      2. IV.1.2 Moist radiative-convective equilibrium
    2. IV.2 Theories of moist convection
      1. IV.2.1 The moist convective heat engine
      2. IV.2.2 Moist irreversible processes in radiative-convective equilibrium
      3. IV.2.3 The role of mixing and microphysics
    3. IV.3 Convective organization and the mechanical efficiency of moist convection
  5. V The thermodynamics of tropical cyclones
    1. V.1 Potential intensity theory
    2. V.2 Work and entropy budgets of TCs
    3. V.3 Open TCs
  6. VI The global circulation of the atmosphere
    1. VI.1 The material entropy budget of the global atmosphere
    2. VI.2 The global atmospheric heat engine
      1. VI.2.1 A thermodynamic perspective of the global atmospheric circulation
      2. VI.2.2 Theories for the global atmospheric heat engine
      3. VI.2.3 The atmospheric heat engine under climate change
    3. VI.3 Heat engines on other planets
      1. VI.3.1 Rocky planets
      2. VI.3.2 Giant planets
  7. VII Modeling the second law of thermodynamics
    1. VII.1 Cloud-permitting models
    2. VII.2 Global climate models
  8. VIII Variational approaches for climate and geophysical flows
    1. VIII.1 Entropic energies
      1. VIII.1.1 Available potential energy
      2. VIII.1.2 Exergetics
    2. VIII.2 Statistical mechanical approaches for steady flows
      1. VIII.2.1 Theoretical development
      2. VIII.2.2 Applications
    3. VIII.3 Maximum entropy production principle controversy
  9. IX Conclusions & perspectives

##  I Introduction

###  I.1 Motivation

The Earth is a highly irreversible thermodynamic system. It receives energy
and entropy from the sun and radiates energy and entropy to outer space. But
while the incoming and outgoing fluxes of energy are roughly in balance, the
Earth exports vastly more entropy than it receives [280, 337]. For a climate
whose statistics are stationary, the second law of thermodynamics requires
that this net export of entropy be balanced by irreversible production of
entropy within the climate system. The second law therefore provides a
fundamental steady-state constraint on the climate system, relating a measure
of its internal activity, the irreversible production of entropy, to fluxes of
entropy at its boundaries.

In fact, the Earth’s climate is not steady; it has undergone vast changes over
Earth’s history, from the icy cold of Snowball Earth episodes [122] to the
extreme warmth of the Late Cretaceous that allowed crocodile-like reptiles to
roam the Arctic [351]. Such climate variability occurs on a range of
timescales [97] and implies imbalances in the planetary energy and entropy
budgets. In the context of the climate change observed in recent decades and
projected over the next century, these energy imbalances are small relative to
the total incoming and outgoing fluxes [356], and the steady-state assumption
provides a useful framework for understanding the second law as applied to the
climate system.

A range of processes are involved in the irreversible production of entropy
within the climate system including the absorption and emission of radiation,
the frictional dissipation of winds and ocean currents, molecular diffusion of
heat and mass, and phase changes of water within the hydrological cycle.
Indeed, life itself is an irreversible process, although we will not discuss
the biotic generation of entropy in this review. But despite their ubiquity,
irreversible processes are often treated in simplified ways in studies of the
large-scale atmospheric circulation [63], while numerical models of the
climate system often treat irreversible processes in a physically inconsistent
way [e.g., 17], neglect certain irreversible processes altogether [e.g., 269,
262], or include spurious numerical sources of entropy [375]. Moreover,
interaction between the communities of climate scientists and physicists
developing tools for the understanding of irreversible processes remains
limited. Fostering collaborations between these communities has the potential
to reveal new methods for analyzing and understanding the climate system,
particularly in the context of a rapidly changing climate [196].

The purpose of this review is to provide an introduction to the application of
the second law of thermodynamics to the climate system suitable both for
scientists active within climate research and for a more general audience of
physicists. The research frontier in climate physics is rich with fascinating,
complex problems in their own right. Amidst a period of rapid anthropogenic
climate change [340], many of the most societally urgent problems, such as
predicting future freshwater availability, crop viability, or storm frequency,
are also the most difficult, requiring collaboration with public and private
decision-makers. A training in traditional physics is excellent preparation
for climate research, and researchers with expertise in areas including
statistical mechanics, fluid dynamics and physical chemistry have much to
offer as part of a vibrant, interdisciplinary climate science community [216,
370].

###  I.2 Applications of the second law in climate research

The second law has been applied in diverse ways within the broad field of
climate science. An important strand of research focuses on the quantification
of irreversibility within the climate system through analysis of its entropy
budget. The bulk of Earth’s irreversible entropy production occurs as a result
of radiative processes [337]. But applications of the second law often focus
on a subset of the climate system that includes only matter and considers
radiation as part of the system’s surroundings. This perspective allows for
the definition of the material entropy budget, in which radiation acts as an
external (and reversible) heat source, and irreversible radiative processes
play no role [101]. The steady-state material entropy budget requires a
balance between material sources of entropy, such as frictional dissipation,
heat and mass diffusion, and irreversible phase changes, and the net sink of
entropy owing to radiative heating at high temperature and radiative cooling
at low temperature [269].

The material entropy budget provides a framework for analyzing the climate
system as a heat engine. A number of studies have used a heat-engine based
perspective to bluederive theoretical constraints on the behavior of
atmospheric circulations of various scales, including convective clouds [72,
298], tropical cyclones [69, 366], and the global circulation [15]. Like a
heat engine, the climate system ingests heat in a warm region, transports it
to a cool region where it is expelled, and performs an amount of work in the
process. But unlike a traditional heat engine, the work performed by the
climate system must be dissipated within the system itself [147, 193]. This
cycle of kinetic energy production and dissipation may also be described
through the Lorenz energy cycle and the concept of available potential energy
[APE; see section VIII.1.1 and 188, 189]. The APE, and the related concept of
exergy [345], provide measures of the climate system’s ability to perform
work. Such concepts allow the second law, usually formulated in terms of
entropy, to be recast in terms of transformations between different energy
reservoirs.

A major challenge for heat-engine based theories applied to the atmosphere is
that they must properly account for the influence of “moist”
processes—processes associated with water in the atmosphere. Moist processes
are responsible for the bulk of the irreversible material entropy production
in Earth’s atmosphere, and this limits the efficiency with which the climate
system’s heat engine may generate kinetic energy in winds and ocean currents
[271, 269, 309]. The effects of irreversible moist processes are particularly
relevant in the context of global climate change [169, 331], as the
concentration of water vapor in the atmosphere is expected to increase with
warming roughly following the Clausius-Clapeyron relation [244].

The second law has also been applied in climate research in ways that go
beyond classical thermodynamics. In its most general form, the second law
governs the macroscale evolution of an isolated system with many degrees of
freedom toward a more probable state. In the field of statistical geophysical
fluid dynamics, the system is a two-dimensional ideal fluid, and its degrees
of freedom are the set of possible flow fields. Tools from statistical
mechanics may then be applied to find the equilibrium flow structures based on
the maximization of an entropy variable, subject to appropriate constraints.
This approach has provided a range of insights into nonequilibrium, steady-
state geophysical flows on Earth and other planets [e.g., 22, 200] in
situations where a heat engine analysis is not applicable.

An advantage of the statistical approach is that it fundamentally involves a
maximization problem, and it is therefore amenable to the powerful techniques
described by the calculus of variations. A disadvantage is that it is only
formally valid for equilibrium systems and cannot be applied to the climate
system as a whole. A generalization of the entropy maximum formalism to non-
equilibrium systems would therefore be of considerable value to climate
research. Such a generalization was proposed by Paltridge [254, 255], who
suggested that the climate system evolves to a state that maximizes its
entropy _production rate_. We briefly review the maximum entropy production
(MEP) principle, but we emphasize that there are a range of theoretical and
modeling issues that pre-empt its broad acceptance in the field (section
VIII.3).

###  I.3 Structure of the review

The bulk of this review is focused on Earth’s atmosphere, where most
irreversible entropy production within the (material) climate system occurs.
While this review primarily adopts a view of the second law focused on entropy
production, irreversibility in the climate system may also be framed in
energetic terms through the concepts of exergy [e.g., 9], and available
potential energy [188]. We briefly discuss these approaches in section VIII.1;
the reader is referred to Tailleux [345] for a more complete treatment.
Finally, we emphasize that this review covers only a small fraction of the
broader research field of climate dynamics; a thorough review of the physics
of climate change has been recently published in this journal by Ghil and
Lucarini [97]. The remainder of the review is structured as follows.

Section II introduces the basic thermodynamic properties of the climate
system. We discuss methods of defining the boundaries of the system, including
the planetary and material definitions used most commonly in the literature.
We also describe the climate system as a heat engine, and we show how
classical engineering concepts such as the work performed and the efficiency
may be meaningfully applied to the climate system.

Section III sketches a derivation of the entropy budget of the climate system.
We focus on the material entropy budget of the atmosphere, and we describe the
physical and mathematical origins of the main irreversible processes. We also
briefly discuss the oceanic entropy budget and recent work estimating
irreversible processes in the ocean.

Sections IV and V review applications of the second law of thermodynamics to
atmospheric convection and tropical cyclones, respectively. In particular, we
highlight how the irreversibility of moist processes fundamentally changes the
fluid dynamics of the atmosphere.

Section VI considers the global atmospheric circulation from a thermodynamic
perspective. We consider theories of the global atmospheric heat engine and we
discuss how it may change under climate change. We also review research
describing the heat engines of other planets and bodies in the Solar System
and beyond.

Section VII discusses some of the challenges faced in developing numerical
models of the climate system that accurately represent the second law of
thermodynamics. We describe practical and theoretical limitations of present
modeling frameworks, and we suggest strategies to aid future model
development.

Section VIII provides an introduction to variational approaches to
understanding geophysical fluid dynamics and the climate generally. We discuss
the application of such approaches to atmospheric energetics and turbulence in
large-scale geophysical flows. Here, both the classical thermodynamic
definition of entropy, as well as the Boltzmann entropy of statistical
mechanics, are employed. We also discuss the controversial maximum entropy
production (MEP) principle, which has motivated much research into the climate
system’s entropy budget.

Section IX concludes this review with a summary and discussion of outstanding
research questions. We particularly highlight those areas that are likely to
benefit from engagement with a broader community of physicists.

##  II The second law of thermodynamics applied to the climate system

The second law of thermodynamics is fundamentally concerned with
irreversibility; certain physical processes or transformations proceed
spontaneously in one direction, but not in the reverse direction. Common
everyday examples include the cooling of a cup of tea to room temperature when
it is left out or the evaporation of water from wet clothes hung out on a dry
day. We do not expect a cup of tea to extract heat from its surroundings and
spontaneously boil, and neither do we expect liquid water to condense out of
the air on already wet clothes. All real macroscopic physical processes
involve some degree of irreversibility, and the second law provides a
framework for understanding such irreversible processes.

A modern expression of the second law states that the entropy $S$ of an
isolated system must not decrease with time [51]:

$\frac{dS}{dt}\geq 0.\hskip 20.0pt\text{(isolated system)}$ (1)

Here, we refer to an isolated system as one that does not exchange mass or
energy with its environment. The entropy is a function of the state of the
system. If an isolated system’s entropy $S$ does not change, it is said to be
reversible, while irreversible processes cause an increase in $S$.

The entropy may be defined using statistical mechanics as a measure of the
number of microstates corresponding to a given macrostate, or in classical
thermodynamics by the relationship

$\frac{dS}{dt}=\frac{\dot{Q}_{\text{rev}}}{T},\hskip 20.0pt\text{(closed,
reversible system)}$ (2)

valid for a closed, reversible system. Here, a closed system may exchange
energy but not mass with its environment, $\dot{Q}_{\text{rev}}$ represents a
reversible heat transport from the surroundings to the system, and $T$ is the
temperature at which this heat is transported [e.g., 51, 135]. While (2) is
valid only for a closed, reversible system, as a state function, the entropy
$S$ remains well defined under both reversible and irreversible conditions.

The climate system exchanges energy with space in the form of radiation, and
it is therefore not isolated. The second law for a non-isolated system may be
written in the more general form,

$\frac{dS}{dt}=\dot{S}_{e}+\dot{S}_{i},$ (3)

where $\dot{S}_{e}$ is the net import of entropy from the surroundings and
$\dot{S}_{i}$ is the production of entropy within the system owing to
irreversible processes [51]. The second law of thermodynamics requires that
$\dot{S}_{i}\geq 0$.

A simplification to (3) may be made for systems close to steady state, where
$dS/dt\approx 0$. At a given instant, this assumption is likely to be poor for
the climte system; Huang and McElroy [129] computed observational estimates of
various measures of the atmosphere’s thermodynamic disequilibrium and found
substantial seasonal variation. On longer timescales, however, the magnitude
of the entropy tendency due to internally generated and forced climate
variability is likely to be a small fraction of the total irreversible
production of entropy by the climate system. Peixoto et al. [280] argued that
for time averages over periods longer than a year, $dS/dt\approx 0$, and the
second law of thermodynamics as applied to the climate system may be written,

$\langle\dot{S}_{i}\rangle=-\langle\dot{S}_{e}\rangle,$ (4)

where the angle brackets refer to a time average over a suitably long period.
According to (4), the time-mean irreversible entropy production rate of the
climate system is equal to the time-mean net rate of export of entropy to
space. For applications to the Earth, it will prove useful to measure entropy
exchanges per unit area of the Earth’s surface, giving $\dot{S}_{i}$ the units
of W m-2 K-1.

The steady-state entropy budget (4) states that, in order to maintain entropy
producing processes such as those associated with winds, ocean currents, and
the hydrological cycle, the climate system must export a greater quantity of
entropy than it receives. This is manifest in the relatively high entropy
contained in the radiation emitted from Earth to space compared to the lower
entropy of the solar beam. More generally, the entropy budget places a
fundamental constraint on the climate system by relating a measure of its
internal activity, the total irreversible entropy production, to fluxes at its
boundaries. One of the main purposes of studies of the climate’s entropy
budget is to leverage this constraint to better understand aspects of the
climate system’s behavior.

In the remainder of this section, we describe different methods of evaluating
the entropy fluxes into and out of the climate system depending on how the
system’s boundaries are defined (section II.1). We also introduce the concept
of the climate system as a heat engine, and we define the work done by the
climate system and its thermodynamic and mechanical efficiency (section II.2).

###  II.1 The boundaries of the climate system

Applying the second law to the climate system requires a proper definition of
the climate system’s boundaries; where does the Earth’s climate system end and
“the surroundings” begin? In most applications, the climate system is defined
to include the atmosphere, oceans, and the uppermost few meters of the land
surface. While this definition excludes the solid Earth, the smallness of the
geothermal heat flux indicates that irreversible processes in Earth’s interior
are likely to be weak compared to those in the atmosphere and ocean. The
irreversible entropy production in the climate system is therefore
approximately equal to that of the entire Earth system.

Defining the boundaries of the climate system also requires consideration of
the role of radiation within it. Like matter, radiation obeys the second law
of thermodynamics, and the interaction between radiation and matter may be
shown to be an irreversible source of entropy [e.g., 34, 33]. The extent to
which the irreversibility of radiative processes is included in (3) depends on
the extent to which radiation is included as part of the climate system or
excluded as part of the surroundings.

Bannon [12] summarizes a number of possible definitions of the climate system,
but here we limit our discussion to three common definitions used in studies
of the Earth’s entropy budget (Fig. 1):

  1. 1.

The planetary climate system: the Earth and its atmosphere is treated as a
control volume, and the climate system is defined as all substances, both
matter and radiation, within this volume [e.g., 12]. This is the most
expansive definition, and it leads to the largest value of the irreversible
entropy production $\dot{S}_{i}$.

  2. 2.

The material climate system: the climate system is defined to include only
matter within the Earth and atmosphere, and all photons are considered part of
the surroundings [e.g., 101].

  3. 3.

The transfer climate system: discussed in Bannon [12], and recently advocated
for by Gibbins and Haigh [99], the transfer climate system is defined to
include matter plus internal radiative transfer (photons that are emitted and
absorbed by matter within the system) but to exclude external radiative
transfer (photons that are incident from the sun or emitted directly to
space). Unlike the planetary climate system, the transfer climate system
cannot be defined using a control volume approach because it excludes some
photons present within the atmosphere (dashed black arrows on Fig. 1).

Figure 1: Schematic of different definitions of the climate system and its
boundary in the application of the second law of thermodynamics. The planetary
system is defined as a control volume including all particles (matter and
radiation) encompassed by a fictitious surface denoted “top of the atmosphere”
(TOA). The material system includes only matter (circles) and excludes all
radiation (arrows) as part of the surroundings. The transfer system [99]
includes internal radiative transfer (photons that are emitted and absorbed by
matter within the system; solid black arrows), but excludes external radiative
transfer (photons that are incident from the sun or emitted directly to outer
space; dashed black arrows).

Although each perspective provides a consistent description of the climate
system, the magnitude of the entropy export $-\dot{S}_{e}$ and the
irreversible entropy production $\dot{S}_{i}$ differ greatly between the
planetary, material, and transfer definitions, and previous authors have
disagreed on which perspective is most relevant for understanding climate
system behavior [e.g., 74, 75, 101]. Below we briefly outline the calculation
of the entropy export $-\dot{S}_{e}$ according to each perspective. Following
Goody [101] and a number of other authors [e.g. 252, 196, 262], we will argue
that the material climate system is most relevant to understanding the
dynamics of the atmosphere and ocean, and the material entropy budget will be
the focus of much of the later sections of this review.

####  II.1.1 The planetary climate system

The planetary climate system consists of a control volume bounded by a
fictitious surface beyond the atmosphere which we will refer to as the “top of
the atmosphere” (TOA; Fig. 1)111Since the density of the atmosphere decreases
exponentially with height, there is no precise dividing line between the
atmosphere and space. Conceptually, it is useful to consider the TOA to be at
roughly 80 km above sea level, where the gas density becomes so low that the
approximation of local thermodynamic equilibrium breaks down (see section
III.1.1).. Within this control volume exists all matter within the climate
system (and indeed the entire Earth system), as well as photons emitted by the
sun (shortwave radiation) and those emitted by the Earth and atmosphere
(longwave radiation)222The shortwave/longwave nomenclature is motivated by the
fact that the spectra of solar and terrestrial radiation have practically no
overlap. . A full account of the second law applied to the planetary climate
system must consider the entropy embodied in both matter and radiation; the
irreversible entropy production by the planetary climate system may then be
divided into a component $\dot{S}_{i}^{\text{mat}}$ associated with material
processes, and a component $\dot{S}_{i}^{\text{rad}}$ owing to the interaction
of matter with radiation.

The radiative component $\dot{S}_{i}^{\text{rad}}$ is a result of the
irreversibility of absorption, emission, and scattering processes that occur
within the climate system. In particular, the transformation of a focused beam
of shortwave radiation, with an effective emission temperature of $\sim$ 6000
K, into diffuse emission of longwave radiation, with an effective emission
temperature of $\sim 250$ K, is highly irreversible. Callies and Herbert [34]
provide a derivation of the equations governing the entropy of the radiation
field, showing how $\dot{S}_{i}^{\text{rad}}$ may be expressed in the classic
form of the product of a generalized thermodynamic flux and a generalized
thermodynamic force. The authors further demonstrate the irreversibility of
radiative interactions by showing that $\dot{S}_{i}^{\text{rad}}$ is positive
definite for the separate cases of absorption/emission and scattering. Here we
do not provide a detailed account of the various irreversible radiative
processes in the climate system [see e.g., 180, 104, 376, 281, 282, for more
detailed treatments]. Instead, we characterize the planetary entropy budget
through the time-mean net export of entropy out of the climate system
$-\langle\dot{S}_{e}\rangle$, given by the net flux of entropy across its
boundary.

For the planetary climate system, the relevant boundary is the TOA, and the
relevant fluxes are those carried by shortwave and longwave radiation.
Defining $\Omega$ as the volume of the climate system, we may write the net
export of entropy out of the system as,

$-\langle\dot{S}_{e}\rangle=\frac{1}{A}\int_{\partial\Omega}\left\langle
J_{SW}+J_{LW}\right\rangle\,\mathrm{d}A,$

where $\partial\Omega$ represents the boundary of $\Omega$, in this case the
TOA, the angle brackets represent a time mean, $A$ is the surface area of
Earth, $J$ is the radiant flux of entropy out of the climate system, and
subscripts $SW$ and $LW$ refer to shortwave and longwave radiation,
respectively. Here we follow the convention in atmospheric science to refer to
a flux as the transport of a quantity per unit area (also known as flux
density), and we divide the integral on the right-hand side by $A$ to express
$\langle\dot{S}_{e}\rangle$ per unit area of the Earth’s surface. For a system
approximately in steady state, we must also have a time-mean balance between
the shortwave and longwave radiant energy fluxes $F$,

$\frac{1}{A}\int_{\partial\Omega}\left\langle
F_{SW}+F_{LW}\right\rangle\,\mathrm{d}A=0.$

Previous authors have estimated the entropy fluxes $J_{LW}$ and $J_{SW}$ from
both observations [337, 151] and climate models [180, 262]. Before we discuss
these estimates, however, it is useful to consider the planetary entropy
budget for a simplified model of the climate system in order to build some
intuition of the magnitude and behavior of various components of
$\langle\dot{S}_{e}\rangle$.

Figure 2: A simple model for Earth’s climate and the associated vertical
energy and entropy fluxes. The climate system is assumed to be horizontally
homogeneous, and it includes a surface, with temperature $T_{s}$, and a
single-layer atmosphere, with temperature $T_{a}$. Arrows show energy fluxes
$F$ and their corresponding entropy fluxes $J$ expressed per unit area. The
subscripts $LW$ and $SW$ denote shortwave and longwave radiant fluxes,
respectively, and the subscripts $s$ and $a$ refer to the surface and
atmosphere, respectively. Shortwave radiation is divided into its upward and
downward components. The turbulent flux of enthalpy from the surface to the
atmosphere is made up of a sensible heat flux $F_{SH}$ and a latent heat flux
$F_{LH}$. Other terms in the equations are described in the text.

The simple model is described schematically in Fig. 2; it is similar to models
presented in Bannon [12] and Kato and Rose [151], and our discussion of the
different entropy production rates follows that of Gibbins and Haigh [99]. The
model is horizontally homogeneous, representing globally-averaged conditions,
and it consists of a surface and a single-layer atmosphere. Both the surface
and atmosphere are assumed to be completely opaque to longwave radiation and
to behave as blackbodies for radiation in the longwave portion of the
electromagnetic spectrum. The atmosphere is assumed to be transparent to
shortwave radiation, and the surface has a fixed shortwave albedo of $\alpha$,
reflecting a fraction $\alpha$ of the incoming solar radiation to space and
absorbing the rest. Energy and entropy transports in this model occur via
radiative fluxes between the surface, atmosphere, and space, and via turbulent
fluxes of latent333Latent heat refers to the energy embodied in water vapor
that is released upon condensation. and sensible heat between the surface and
the atmosphere.

Assuming steady state, we may write energy balance equations for the TOA,
atmosphere, and surface, respectively, given by Kato and Rose [151]:

$\displaystyle(1-\alpha)F^{\downarrow}_{SW}$ $\displaystyle=F_{LW}^{a},$ (5a)
$\displaystyle F_{LW}^{s}+F_{LH}+F_{SH}$ $\displaystyle=2F_{LW}^{a},$ (5b)
$\displaystyle(1-\alpha)F^{\downarrow}_{SW}{+F_{LW}^{a}}$
$\displaystyle=F_{LW}^{s}+F_{LH}+F_{SH},$ (5c)

where $F_{SW}^{\downarrow}$ is the downward solar energy flux input to the
Earth, $F^{s}_{LW}$ and $F^{a}_{LW}$ are the longwave energy fluxes from the
surface and atmosphere, respectively, and $F_{LH}$ and $F_{SH}$ are the latent
and sensible heat fluxes from the surface to the atmosphere, respectively
(Fig. 2). Since the model is time-invariant, we omit the time-averaging
operator in equations (5), but the fluxes should be interpreted as time means.

The longwave energy fluxes from the surface and atmosphere are given by the
well-known Stefan-Boltzmann law

$F=\sigma T^{4},$ (6)

where $F$ is the energy flux, $T$ is the temperature of the emitting body and
$\sigma$ is the Stefan-Boltzmann constant. Approximating the sun as a
blackbody, the downward solar energy flux at the TOA is given by,

$F^{\downarrow}_{SW}=\delta_{\text{sun}}\sigma T_{\text{sun}}^{4},$

where $\delta_{\text{sun}}=\frac{\Omega_{0}\cos\phi}{\pi}$, with
$\Omega_{0}=6.77\times 10^{-5}$ being the solid angle subtended by the sun’s
disk [337], and $\phi$ the zenith angle of the sun’s rays. Here we take a
global mean value of $\cos\phi=0.25$.

To apply the one-layer model to the Earth, we set the solar temperature to be
$T_{\text{sun}}=5777$ K [280], and we set $\alpha=0.3$ to roughly match the
planetary albedo of Earth [338]. Using the energy balance equations (5), this
constrains the atmospheric temperature, which for this model is equal to the
effective emission temperature of the planet, to be $T_{a}=255$ K. We are then
free to set either the sensible and latent heat fluxes from the surface or the
surface temperature $T_{s}$. On the basis that turbulent dynamics place a
strong constraint on the lapse rate in convecting atmospheres [73], we fix the
temperature difference between the surface and atmosphere by setting the
surface temperature to roughly match Earth’s global-mean surface temperature
$T_{s}=288$ K, and we allow the surface fluxes to adjust to satisfy the energy
balance equations.

We now evaluate the planetary entropy budget for the one-layer model. The
entropy fluxes associated with radiation of a given wavelength and angular
distribution may be derived from the fundamental statistical mechanics of a
Boson gas [311], or through a number of semi-classical methods [249, 376]. For
a blackbody, a formula for the entropy flux $J$ may be derived by integrating
the spectral entropy flux distribution over all frequencies to give [376],

$J=\frac{4}{3}\sigma T^{3}.$ (7)

Combining (7) and (6), the blackbody entropy flux may be expressed in terms of
the energy flux $F$ as,

$J=\frac{4}{3}\frac{F}{T}.$ (8)

The entropy flux emitted by a blackbody is larger, by a factor of $4/3$, than
the entropy loss of the emitting object $F/T$. This additional entropy
transport may be interpreted as the irreversible entropy production associated
with emitting radiation into a vacuum [80]444An elegant derivation of (8) is
presented by Feistel [80]. Consider two parallel plates held at fixed
temperatures exchanging energy through radiation. Assuming they are
blackbodies, the energy flux from each plate may be described by the Stefan-
Boltzmann law (6), and it will produce a transfer of heat from the hotter
plate to the colder plate. By the second law of thermodynamics, this energy
exchange must be associated with positive irreversible entropy production,
with the rate of irreversible entropy production tending to zero as the
temperature difference between the plates approaches zero. Feistel [80] showed
that the only functional form for the entropy flux associated with the
radiation from each plate consistent with this expectation is that described
by (8).. Eq. (7) may be used to evaluate the entropy fluxes from the surface
and atmosphere in the one-layer model.

Due to the irreversibility associated with reflection, the upward and downward
shortwave entropy fluxes must be treated separately. The downward shortwave
flux of entropy is given simply by,

$J_{SW}^{\downarrow}=\frac{4}{3}\delta_{\text{sun}}\sigma T_{\text{sun}}^{3},$
(9)

representing the sun’s blackbody entropy flux reduced by the factor
$\delta_{\text{sun}}$. The same approximation cannot be used for the upward
flux because of the change of its angular distribution upon reflection.
Instead, we assume that the reflection is diffuse (Lambertian), so that the
radiance of the reflected beam is independent of direction. Stephens and
O’Brien [337] solved for this case, finding that the entropy flux of the
reflected radiation could be approximated as,

$J^{\uparrow}_{SW}=\frac{4}{3}\chi(\alpha\delta_{\text{sun}})\sigma
T_{\text{sun}}^{3},$ (10)

where $\chi(u)=u[a_{0}+a_{1}\ln(u)]$, and $a_{0}=0.9652$ and $a_{1}=-0.2777$
are empirical constants. Wu and Liu [376] provide a detailed evaluation of
this approximation and a number of other analytic formulae for entropy fluxes
associated with non-blackbody radiation.

| Model | $\langle J^{\text{TOA}}_{SW}\rangle$ | $\langle J^{\text{TOA}}_{LW}\rangle$ | $-\langle\dot{S}_{e}\rangle$ | $-\langle\dot{S}_{e}^{\text{trans}}\rangle$ | $-\langle\dot{S}_{e}^{\text{mat}}\rangle$
---|---|---|---|---|---|---
one-layer model (Fig. 2) |  | 31 | 1247 | 1279 | 109 | 39
Kato and Rose [151] | obs. | -55 | 1238 | 1183 | 76 | 49
Stephens and O’Brien [337] | obs. | 20 | 1230 | 1250 |  |
Peixoto et al. [280] | obs. | -41${}^{\text{a}}$ | 925${}^{\text{a}}$ | 884${}^{\text{a}}$ |  | 41
Lembo et al. [176] | CMIP6${}^{\text{b}}$ |  |  |  |  | 58
Pascale et al. [262] | HadCM3 |  |  | 911${}^{\text{a}}$ | 101 | 52
Fraedrich and Lunkeit [86] | PlanetSim |  |  | 880${}^{\text{a}}$ | 69 | 35
Goody [101] | GISS |  |  |  |  | 52
Table 1: Observational (obs.) and model-based estimates of the top of
atmosphere entropy fluxes of shortwave $\langle J_{SW}^{\text{TOA}}\rangle$
and longwave $\langle J^{\text{TOA}}_{LW}\rangle$ radiation and the planetary
$-\langle\dot{S}_{e}\rangle$, transfer
$-\langle\dot{S}^{\text{trans}}_{e}\rangle$ and material
$-\langle\dot{S}_{e}^{\text{mat}}\rangle$ entropy export rates of the climate
system (per unit area of Earth’s surface) from various prior studies and for
the one-layer model summarized in Fig. 2. Fluxes toward outer space are
defined as positive, and units are mW m-2 K-1.
${}^{\text{a}}$Entropy fluxes estimated using (11).
${}^{\text{b}}$Mean over 7 models participating in the sixth phase of the
Coupled Model Intercomparison Project (CMIP6) [76].

Eqs. (7), (9) and (10) may be used to evaluate the planetary entropy budget
for the one-layer model (table 1). According to this model, the TOA entropy
fluxes are dominated by longwave radiation; the longwave entropy flux at the
TOA is factor of 40 larger than the net shortwave entropy flux. Furthermore,
the net shortwave entropy flux at the TOA is directed upwards, despite the net
shortwave energy flux being downwards. This counter-intuitive result is
possible because the entropy associated with photons in the diffuse radiation
reflected from the Earth’s surface is much higher than the entropy of photons
in the beam of radiation incident on the Earth [337]. Summing the net
shortwave and longwave fluxes and using the steady-state entropy budget (4),
the one-layer model gives an estimate of the total irreversible entropy
production by the climate system $\langle\dot{S}_{i}\rangle=1279$ mW m-2 K-1.

Despite the simplicity of the one-layer model, its entropy budget is similar
to more detailed estimates of Earth’s planetary entropy budget based on
observations (table 1). For example, Stephens and O’Brien [337] used satellite
observations of TOA radiation to estimate the planetary entropy budget,
finding a similar dominance of the longwave fluxes and a value of the total
entropy export $-\langle\dot{S}_{e}\rangle$ roughly 2% smaller in magnitude
than that of the one-layer model. This difference is partially accounted for
by the one-layer model’s neglect of temperature variations within the
atmosphere. Lesins [177] showed that the outgoing longwave entropy flux is
maximized for an isothermal atmosphere, with meridional and vertical
temperature variations reducing the flux by a factor of the order of 1%.

Kato and Rose [151] also provided estimates of the entropy flux at the TOA
based on satellite observations. They applied the simple blackbody formula (8)
to both incoming and outgoing shortwave radiation. This neglects the
irreversible entropy production associated with diffuse reflection, resulting
in an underestimate of the entropy flux by reflected solar radiation. But the
overall planetary entropy budget is nevertheless broadly similar to that of
the one-layer atmosphere model.

On the other hand, the observational study of Peixoto et al. [280] and a
number of studies based on global climate models [86, 262] present planetary
entropy budgets that are inconsistent with the one-layer model, with estimates
of the entropy export $-\langle\dot{S}_{e}\rangle$ 30-40% smaller in
magnitude. The reason for this discrepancy is that these studies evaluate the
flux of entropy by radiation as

$J^{\prime}=\frac{F}{T},$ (11)

where $F$ is the radiative energy flux and $T$ is the temperature of the
emitting object. This definition is appealing, because it gives the radiant
entropy flux as being equal to the loss of entropy by the emitting object, but
it fails to account for the irreversible nature of spontaneous emission and
absorption represented by the $4/3$ factor in (8) [80]. For non-blackbody
radiation, (11) also neglects differences between the temperature of the
emitting object and the spectrally-varying emission temperature of the emitted
radiation. As a result, the use of (11) can imply irreversible entropy
production rates owing to radiative processes that are locally negative,
violating the second law [34].

Studies such as Peixoto et al. [280] therefore do not fully account for the
irreversible entropy production associated with radiative processes and they
underestimate the planetary entropy production rate [74, 75, 337]. Such
studies nevertheless remain relevant, because, as will be shown below, entropy
production associated with radiative transfer does not affect the material
entropy budget, and conclusions regarding material entropy production are
unaffected by whether one uses (11) or (8) to estimate radiant entropy fluxes.

####  II.1.2 The material climate system

To motivate study of the material entropy budget, consider a thought
experiment in which the heating and cooling owing to radiative absorption and
emission within the climate system is replaced by identical heating and
cooling rates produced by a reversible mechanism. Such a change would have no
effect on the matter within the climate system; the equations governing the
fluid dynamics of the atmosphere and oceans are only concerned with radiation
insofar as it heats or cools the fluid [101, 262]. The atmospheric
circulation, hydrological cycle, and ocean currents would behave exactly as
before. For this reason, Goody [101] advocated for a view of the entropy
budget that focuses exclusively on matter rather than radiation.

The material climate system includes all matter within the Earth, atmosphere,
and oceans but considers radiation as part of the surroundings [101, 12]. The
import of entropy into the material system $\dot{S}_{e}^{\text{mat}}$ occurs
through the heating and cooling of matter that absorbs and emits radiation,
and it may be written [101],

$\dot{S}^{\text{mat}}_{e}=\frac{1}{A}\int_{\Omega}\frac{\rho\dot{q}_{\text{rad}}}{T}\,dV,$
(12)

where $\dot{q}_{\text{rad}}$ is the net radiative heating rate per unit mass,
$\rho$ is the density, and the integral is over $\Omega$, representing the
entire climate system. Once again, we divide the integral on the right-hand
side by the surface area of the Earth $A$ in order to express
$\dot{S}_{e}^{\text{mat}}$ per unit area. We may also define the irreversible
material entropy production $\dot{S}_{i}^{\text{mat}}$, which by the steady-
state material entropy budget satisfies
$\langle\dot{S}_{i}^{\text{mat}}\rangle=-\langle\dot{S}_{e}^{\text{mat}}\rangle$.
The steady-state material entropy budget may then be written,

$\langle\dot{S}^{\text{mat}}_{i}\rangle=-\frac{1}{A}\int_{\Omega}\left\langle\frac{\rho\dot{q}_{\text{rad}}}{T}\right\rangle\,dV.$
(13)

The requirement that $\dot{S}_{i}^{\text{mat}}$ be positive is an application
of the second law known as the Clausius-Dunhem inequality [283]. As we shall
show below, for the simple one-layer model introduced in the previous
subsection, $\dot{S}_{i}^{\text{mat}}\geq 0$ if turbulent fluxes transport
heat down the temperature gradient.

Since the material entropy budget incorporates the effect of radiation as an
external (and reversible) heating, the material irreversible entropy
production rate $\dot{S}_{i}^{\text{mat}}$ accounts for only a portion of the
total irreversible entropy production of the climate system $\dot{S}_{i}$,
while the remainder $\dot{S}_{i}^{\text{rad}}$ is associated with irreversible
entropy production owing to radiation. As we will show in detail in section
III, the material entropy production includes irreversible processes such as
frictional dissipation, molecular heat diffusion, irreversible mixing, and
irreversible chemical reactions.

Consider again the simple one-layer model illustrated in Fig. 2. We may define
the time-mean net radiative heating rate of the atmosphere per unit area

$\langle\dot{Q}_{\text{rad}}\rangle=\frac{1}{A}\int_{\Omega_{A}}\left\langle\rho\dot{q}_{\text{rad}}\right\rangle\,dV,$

where $\Omega_{A}$ is the volume of the atmosphere. The heating rate
$\langle\dot{Q}_{\text{rad}}\rangle$ may be evaluated based on the radiative
energy fluxes shown in Fig. 2,

$\langle\dot{Q}_{\text{rad}}\rangle=\sigma T_{s}^{4}-2\sigma T_{a}^{4}.$ (14)

Further, using the energy balance equations for the one-layer model (5), one
may show that the net radiative heating of the surface is given by
$-\langle\dot{Q}_{\text{rad}}\rangle$, and that this is equal to the transport
of energy from the surface to the atmosphere via turbulent fluxes,

$-\langle\dot{Q}_{\text{rad}}\rangle=\langle F_{LH}+F_{SH}\rangle.$

Using the previous two equations and noting that the temperatures of the
atmosphere and surface are assumed to be constant, the steady-state material
entropy budget of the one-layer model may be expressed,

$\langle\dot{S}^{\text{mat}}_{i}\rangle=\left\langle
F_{LH}+F_{SH}\right\rangle\left(\frac{1}{T_{a}}-\frac{1}{T_{s}}\right).$

This equation demonstrates that, in steady state, the irreversible entropy
production $\langle\dot{S}_{i}^{\text{mat}}\rangle$ is positive provided the
surface fluxes move energy from high to low temperature. In Earth-like
climates, $T_{s}>T_{a}$, and the surface blueturbulent fluxes transport energy
from the surface to the atmosphere, so that
$\langle\dot{S}^{\text{mat}}_{i}\rangle$ is positive as required by the second
law. It is sometimes suggested that the greenhouse effect is incompatible with
the second law of thermodynamics. The one-layer model shows this is not true;
the greenhouse effect associated with the absorption of longwave radiation
allows the model to maintain a surface temperature higher than the effective
emission temperature of the planet while it maintains positive irreversible
entropy production.

For the parameters chosen, the one-layer model gives a material entropy
production rate of $\langle\dot{S}_{i}^{\text{mat}}\rangle=39$ mW m-2 K-1.
Given the assumptions of the model, this provides only a rough estimate of the
material entropy production in the climate system, but it highlights the small
magnitude of $\langle\dot{S}_{i}^{\text{mat}}\rangle$ compared to the total
irreversible production rate $\langle\dot{S}_{i}\rangle$, implying that the
bulk of the irreversible entropy production in the climate system occurs due
to radiative processes [74, 75, 180, 337, 101].

More detailed estimates of the material entropy budget of Earth’s climate
system confirm the picture above (table 1). Studies based on observations
[280, 151] as well as global climate model simulations [101, 86, 262, 176]
have estimated the material entropy export
$-\langle\dot{S}_{e}^{\text{mat}}\rangle$, finding values in the relatively
broad range of 35-60 mW m-2 K-1. The large range in such estimates is partly
due to methodological differences across studies. For example, Peixoto et al.
[280] considered only global- and annual-mean radiative fluxes and temperature
profiles in order to estimate $\langle\dot{S}_{e}^{\text{mat}}\rangle$, while
Kato and Rose [151] also took into account spatial and temporal variations.
But differences also arise because the spatial distribution of radiative
heating $\dot{q}_{\text{rad}}$ is strongly dependent on properties such as the
surface albedo and the distribution of clouds and water vapor in the
atmosphere, and it is therefore difficult to estimate accurately from
observations and dependent on uncertain parameterizations in models. As a
result, even among studies applying similar methodologies to climate model
output [86, 262], the estimated value of
$\langle\dot{S}_{e}^{\text{mat}}\rangle$ can vary substantially.

Under steady-state conditions, estimation of the net entropy export leads
directly to an estimate of the material entropy production of the climate
system. However, differences between estimates of
$-\langle\dot{S}_{e}^{\text{mat}}\rangle$ and
$\langle\dot{S}_{i}^{\text{mat}}\rangle$ of up to 30% have been reported in
the literature [176]. In principle, such differences may result from
imbalances in the entropy budget due to climate variation [101], but estimates
of the imbalance in the planetary energy budget [356] suggest that such
differences are likely to be on the order of a few mW m-2 K-1. Rather,
differences between estimates of the entropy export and estimates of material
entropy production reflect the difficulty of diagnosing irreversible processes
using available observations or using standard model outputs. We further
discuss the issues surrounding the estimation of irreversible processes in the
climate system in section VI.1 and in climate models in section VII.

####  II.1.3 The transfer climate system

While the majority of studies of Earth’s entropy budget adopt one of the two
definitions discussed above, Gibbins and Haigh [99] have recently advocated
for an alternate definition that is intermediate between the planetary and
material perspectives, which they refer to as the transfer climate system
[also discussed in 12, as their case “MS2”]. The transfer climate system is
similar to the material climate system, but it additionally includes radiation
that is “internal” to the climate system (Fig. 1). Internal radiation
corresponds to photons that transport energy between different material
elements of the climate system, in contrast to those photons that are incident
on the Earth from the sun or that are emitted to outer space. Since the
transfer approach includes some, but not all, radiation as part of the climate
system, it gives an irreversible entropy production rate
$\dot{S}_{i}^{\text{trans}}$ whose magnitude is between the planetary and
material values.

Gibbins and Haigh [99] argue that the transfer climate system provides an
entropy budget that is more robust to details of internal heat transport
mechanisms within the climate system. In particular, they note that, from the
perspective of the transfer entropy budget, heat transport from high to low
temperature is associated with the same amount of irreversible entropy
production whether it is caused by radiative fluxes or conductive fluxes. But
only in the latter case would this entropy production be included in the
material entropy budget.

For the transfer climate system, the entropy import rate
$\dot{S}^{\text{trans}}_{e}$ is equal to the sum of the entropy tendencies
associated with the absorption of solar radiation and the emission of longwave
radiation directly to space. According to the one-layer model, the atmosphere
emits an amount of radiation $\sigma T_{a}^{4}$ directly to space, while the
surface absorbs an equal amount of radiation from the sun. Assuming steady-
state conditions, we may therefore write the time-mean irreversible entropy
production rate $\langle\dot{S}_{i}^{\text{trans}}\rangle$ for the one-layer
model as

$\langle\dot{S}^{\text{trans}}_{i}\rangle=\sigma
T_{a}^{4}\left(\frac{1}{T_{a}}-\frac{1}{T_{s}}\right).$

This may be evaluated with the parameters of the model to give
$\langle\dot{S}^{\text{trans}}_{i}\rangle=109$ mW m-2 K-1.

A disadvantage of using the transfer approach is that the transfer entropy
import rate $\dot{S}_{e}^{\text{trans}}$ depends on the origin and destination
of each photon that enters the climate system, rather than just the net
radiative heating rate or the TOA fluxes, making its estimation from
observations and models more involved. Nevertheless, Kato and Rose [151] have
recently estimated the transfer entropy budget from observations, and some of
its characteristics may be deduced from the results of previous studies using
global climate models [86, 262]. As expected, the magnitude of the time-mean
entropy export $-\langle\dot{S}_{e}^{\text{trans}}\rangle$ is between the
corresponding values for the planetary and material entropy budgets (table 1).
Compared to the observational and climate-model based estimates, the simple
one-layer model overestimates the transfer entropy production rate. This is
likely because of its neglect of solar absorption in the atmosphere, which
leads to an artificially high value of the solar absorption temperature [cf.
99].

But even detailed model-based estimates of the transfer entropy production
rate differ from each other considerably. This highlights our limited
knowledge of the transfer entropy production rate, which has only recently
been explicitly defined in the context of the climate system’s entropy budget
[12, 99]. Better quantification of the transfer entropy budget and further
understanding of its relationship to the thermodynamics of the climate system
present promising avenues for future research.

###  II.2 The climate system as a heat engine

The climate system is often described as a heat engine, transporting energy
from the warm tropical surface to the cold polar troposphere and producing
kinetic energy, in the form of atmospheric and oceanic circulations, in the
process [e.g., 27, 189, 15, 267, 12, 169]. But there are some important
differences between the climate system and the classic engineering account of
a heat engine. For example, the circulations produced by the climate engine
act on, and are dissipated within, the system itself [147]. This creates an
important negative feedback loop, in which the circulations themselves act to
reduce the temperature differences that are responsible for their existence
[e.g., 15]. In the following, we clarify how concepts such as the work output
and the thermodynamic efficiency of a heat engine may be meaningfully applied
to the climate system.

####  II.2.1 Heat engines and irreversibility

Consider a heat engine operating between two thermal reservoirs at different
temperatures. The engine ingests heat at a rate $\dot{Q}_{\text{in}}$ from the
warm reservoir at a temperature $T_{\text{in}}$, transporting it to the cool
reservoir at temperature $T_{\text{out}}$ where it is expelled at a rate
$\dot{Q}_{\text{out}}$. In the process, the engine is able to perform work at
the rate $\dot{W}_{\text{ext}}$. Here we include the subscript ext to
emphasize that this work is done on an external body. For instance, the engine
may be used to drive a piston that accelerates a locomotive. The eventual
dissipation of the locomotive’s kinetic energy occurs outside of the engine.

The action of a heat engine may be described by combining the first and second
laws of thermodynamics under steady-state conditions to form the Gouy-Stodola
theorem [e.g., 12],

$\eta_{C}\dot{Q}_{\text{in}}=\dot{W}_{\text{ext}}+\dot{S}^{\text{mat}}_{i}T_{\text{out}},$
(15)

where $\dot{S}^{\text{mat}}_{i}$ represents the irreversible entropy
production rate of the engine and

$\eta_{C}=\frac{T_{\text{in}}-T_{\text{out}}}{T_{\text{in}}}.$

If the engine is perfectly reversible, it produces work with an efficiency
given by $\eta_{C}$, equal to the Carnot efficiency. Irreversible processes
decrease the work output relative to this theoretical maximum. In the
engineering context, irreversible entropy production results in “lost work”,
and the aim of the engineer is to reduce $\dot{S}^{\text{mat}}_{i}$ as much as
possible.

The climate system does not have a warm or cold reservoir, and the heating
rates $\dot{Q}_{\text{in}}$ and $\dot{Q}_{\text{out}}$ and the associated
temperatures are more difficult to define. Moreover, the climate system as a
whole cannot perform work on any external body. For the climate system, we
therefore have that $\dot{W}_{\text{ext}}=0$ and any traditionally defined
efficiency is also zero. Nevertheless, previous authors have defined an
efficiency of the climate system in various ways. In the following, we detail
two such definitions. We first construct an equivalent Carnot efficiency of
the climate system that relates its irreversible entropy production to the
heat input through radiation. We then define a mechanical efficiency of the
climate system that relates the work performed in generating the atmospheric
and oceanic circulation to the heat input by radiation. This second definition
may be considered to be the analogue to the engineering concept of the
efficiency of a heat engine applied to the climate system.

####  II.2.2 Carnot efficiency of the climate system

To define the Carnot efficiency of the climate system, we take (15), set
$\dot{W}_{\text{ext}}=0$, and replace $\dot{S}_{i}^{\text{mat}}$ and
$\dot{Q}_{\text{in}}$ with their time-averaged values to give,

$\eta_{C}=\frac{\langle\dot{S}^{\text{mat}}_{i}\rangle
T_{\text{out}}}{\langle\dot{Q}_{\text{in}}\rangle}.\hskip
20.0pt(\dot{W}_{\text{ext}}=0)$

The numerator $\langle\dot{S}^{\text{mat}}_{i}\rangle T_{\text{out}}$ gives a
measure of the strength of irreversible processes within the climate system,
which was taken by Bannon [12] to be a measure of the activity of the
atmosphere and oceans. The task is then to determine the effective input and
output temperatures and heating rate $\langle\dot{Q}_{\text{in}}\rangle$.
Since there is no warm or cold reservoir, the heating rates must be defined in
an averaged sense. In particular, we may take the energy input as the sum over
all regions that experience net radiative heating [e.g., 12],

$\langle\dot{Q}_{\text{in}}\rangle=\frac{1}{A}\int_{\Omega}\left\langle\rho\dot{q}_{\text{rad}}^{+}\right\rangle\,dV,$
(16)

where $\dot{q}_{\text{rad}}^{+}$ is the net radiative heating rate when it is
positive and zero otherwise, and we have divided the integral on the right-
hand side by $A$, the surface area of the Earth, to express the heating rate
per unit area. We may similarly define the effective temperature of heat input
by,

$\frac{\langle\dot{Q}_{\text{in}}\rangle}{T_{\text{in}}}=\frac{1}{A}\int_{\Omega}\left\langle\frac{\rho\dot{q}_{\text{rad}}^{+}}{T}\right\rangle\,dV.$
(17)

Making similar definitions of the heat output and output temperature based on
the radiative cooling rate, (15) may be written

$\langle\dot{S}_{i}^{\text{mat}}\rangle=\langle\dot{Q}_{\text{in}}\rangle\left(\frac{1}{T_{\text{out}}}-\frac{1}{T_{\text{in}}}\right).$
(18)

Here we have assumed that $\dot{W}_{\text{ext}}=0$, and hence that
$\langle\dot{Q}_{\text{in}}\rangle$ = $\langle\dot{Q}_{\text{out}}\rangle$ for
the climate system. Comparing with (13), the heat engine relation above may be
seen to be equivalent to the steady-state material entropy budget.

For the simple one-layer atmosphere model described by Fig. 2, the heat input
$\langle\dot{Q}_{\text{in}}\rangle$ is given by
$\langle\dot{Q}_{\text{rad}}\rangle$ in (14), and the input and output
temperatures are those of the surface and the atmosphere, respectively. The
model therefore has a Carnot efficiency of $\eta_{C}=11\%$. Bannon [12] found
a slightly lower value of $\eta_{C}=8\%$ using a similar single-layer model of
the climate system which allows for atmospheric absorption of shortwave
radiation, while Gibbins and Haigh [99] found Carnot efficiencies in the range
6-12% also using highly simplified models of the climate system.

It is important to note that the heating rate
$\langle\dot{Q}_{\text{in}}\rangle$ is not an external parameter for a given
planet (as it would be in traditional heat engine analysis), but it depends on
features such as the surface albedo and the cloud and water vapor
distribution, which may vary with climate. Furthermore, the definitions given
above for the input and output heating rates are non-unique [12]. This non-
uniqueness has some parallels in the ambiguity of defining the boundaries of
the climate system in our discussion of planetary, material, and transfer
entropy production rates above. For example, taking
$\langle\dot{Q}_{\text{out}}\rangle$ as cooling by longwave emission directly
to space and $\langle Q_{\text{in}}\rangle$ as solar absorption gives an
efficiency based on the transfer climate system [99]. This perspective was
used in Bannon and Lee [13] to derive approximate upper bounds to the Carnot
efficiencies of Earth, Mars, Venus, and Titan (see also section VI.3).

Finally, we note that some authors define a Carnot efficiency for the climate
system based on a heating rate that includes additional terms such as the
heating owing to frictional dissipation and that due to latent and sensible
heat fluxes [e.g., 147]. This approach was used by Lucarini [193] to relate
the Carnot efficiency to the generation of kinetic energy by the climate
system. Here, we instead define a separate mechanical efficiency which relates
the generation of kinetic energy by the climate system to the radiative energy
input $\langle\dot{Q}_{\text{in}}\rangle$.

####  II.2.3 Mechanical efficiency of the climate system

While the climate system cannot perform work on an external body, the
atmosphere and ocean perform work on themselves and each other, and this work
drives the winds and ocean currents. More specifically, work represents a
conversion between kinetic energy and internal or potential energy within the
climate system [e.g., 193, 196]. This conversion may occur reversibly via
motions of the ocean and atmosphere, or irreversibly via dissipative
processes. In the latter case, kinetic energy may be transformed into internal
or potential energy555Dissipation can irreversibly increase potential energy
through the thermal expansion associated with frictional heating; see section
VIII.1.1., but the reverse transformation is prohibited by the second law of
thermodynamics.

In Earth’s climate system, kinetic energy dissipation occurs via two
processes:

  1. 1.

Frictional dissipation of the winds and ocean currents that occurs as a result
of the turbulent cascade of kinetic energy to scales small enough for
viscosity to act.

  2. 2.

Frictional dissipation in the microscopic shear zones surrounding particles
that have appreciable sedimentation velocities relative to the fluid (e.g.,
raindrops and snowflakes) [271, 269].

At steady state, the time-mean rate at which the climate system performs
reversible work $\langle\dot{W}_{\text{rev}}\rangle$ must be equal to the
total dissipation rate owing to these two processes. We may therefore write
the steady-state mechanical energy budget of the climate system as [269, 309],

$\langle\dot{W}_{\text{rev}}\rangle=\langle\dot{D}_{\text{fric}}\rangle+\langle\dot{D}_{\text{sed}}\rangle,$
(19)

where $\langle\dot{D}_{\text{fric}}\rangle$ is the time-mean rate of
frictional dissipation of the winds and ocean currents, and
$\langle\dot{D}_{\text{sed}}\rangle$ is the time-mean rate of dissipation
associated with the sedimentation of precipitation. We may also define the
time-mean rate of generation of kinetic energy associated with winds and ocean
currents by $\langle\dot{W}_{K}\rangle$. In steady state, we must have
$\langle\dot{W}_{K}\rangle=\langle\dot{D}_{\text{fric}}\rangle$, a balance
often expressed through the Lorenz energy cycle [e.g., 188], which describes
the atmospheric heat engine as a series of conversions between different
reservoirs of internal, potential and kinetic energy. We discuss the Lorenz
energy cycle in more detail in section VIII.1.1.

According to (19), $\langle\dot{W}_{K}\rangle$ accounts for only a portion of
the reversible work performed by the climate system; the remainder is used to
lift water upwards through the atmosphere to balance the downward irreversible
flux of water owing to precipitation [269]. Since $\langle\dot{W}_{K}\rangle$
represents the work responsible for powering the atmospheric and oceanic
circulation, it may be considered to be the “useful” component of the total
reversible work. This motivates the definition of the mechanical efficiency of
the climate system $\eta_{M}$ by

$\eta_{M}=\frac{\langle\dot{W}_{K}\rangle}{\langle\dot{Q}_{\text{in}}\rangle}.$
(20)

The mechanical efficiency refers to the efficiency with which the climate
system generates and dissipates kinetic energy of the winds and ocean currents
[269, 102]. It is similar to the classic concept of the efficiency of a heat
engine, except that the useful work $\langle\dot{W}_{K}\rangle$ is done on,
and dissipated within, the system itself.

To examine the factors affecting the mechanical efficiency, we use the fact
that the dissipation rates $\langle\dot{D}_{\text{fric}}\rangle$ and
$\langle\dot{D}_{\text{sed}}\rangle$ are associated with irreversible entropy
sources which we denote $\langle\dot{S}_{i}^{\text{fric}}\rangle$ and
$\langle\dot{S}_{i}^{\text{sed}}\rangle$, respectively. This allows the heat-
engine relation (18) to be written

$\langle\dot{Q}_{\text{in}}\rangle\left(\frac{1}{T_{\text{out}}}-\frac{1}{T_{\text{in}}}\right)=\langle\dot{S}^{\text{fric}}_{i}\rangle+\langle\dot{S}^{\text{sed}}_{i}\rangle+\langle\dot{S}^{\text{nm}}_{i}\rangle,$

where $\dot{S}^{\text{nm}}_{i}$ represents irreversible material entropy
production by non-mechanical processes such as heat diffusion, irreversible
mixing, and irreversible chemical reactions [102]. As we will show in more
detail in section III, the entropy source
$\langle\dot{S}_{i}^{\text{fric}}\rangle$ may be written in terms of the time-
mean dissipation rate $\langle\dot{D}_{\text{fric}}\rangle$ and an effective
temperature $T_{\text{fric}}$ so that,

$\langle\dot{S}_{i}^{\text{fric}}\rangle=\frac{\langle\dot{D}_{\text{fric}}\rangle}{T_{\text{fric}}}.$

Here, the effective temperature of frictional dissipation is defined to
satisfy

$\frac{\langle\dot{D}_{\text{fric}}\rangle}{T_{\text{fric}}}=\int_{\Omega}\frac{\rho\epsilon}{T}\,dV,$
(21)

where $\epsilon$ is the local rate of frictional dissipation per unit mass.
Generally, $T_{\text{fric}}$ is weighted near the warm lower boundary of the
atmosphere, implying $T_{\text{fric}}>T_{\text{out}}$.

Since $\langle\dot{D}_{\text{fric}}\rangle=\langle\dot{W}_{K}\rangle$ in
steady state, we may combine the previous two equations to give

$\langle\dot{W}_{\text{max}}\rangle=\langle\dot{W}_{K}\rangle+T_{\text{fric}}\langle\dot{S}_{i}^{\text{sed}}+\dot{S}^{\text{nm}}_{i}\rangle,$
(22)

where

$\langle\dot{W}_{\text{max}}\rangle=\left(\frac{T_{\text{fric}}}{T_{\text{out}}}\right)\eta_{C}\langle\dot{Q}_{\text{in}}\rangle$

is the maximum rate at which work can be performed by the system for a given
$T_{\text{fric}}$, achieved when there are no other irreversible entropy
sources besides that associated with frictional dissipation of the winds and
ocean currents [269].

Note that, since we generally expect $T_{\text{fric}}>T_{\text{out}}$, the
mechanical efficiency implied by the performance of work at a rate
$\langle\dot{W}_{\text{max}}\rangle$ is higher than the Carnot efficiency
$\eta_{C}$. This is possible because the work is being performed on the system
itself, providing an additional dissipative heat source $\langle
D_{\text{fric}}\rangle$. The rate of work $\langle\dot{W}_{\text{max}}\rangle$
therefore does not represent the maximum work that can be done on an external
body, which is limited by Carnot’s theorem to not exceed that of an ideal
Carnot engine [298, 19, 120]. Rather, $\langle\dot{W}_{\text{max}}\rangle$
represents the maximum rate of work that would be performed by an ideal Carnot
engine in which the heat input and output is given by the combination of
radiative heating and cooling and dissipative heating experienced by the
climate system.

For a given value of $\langle\dot{W}_{\text{max}}\rangle$, the mechanical
efficiency of the climate system is determined by the amount of entropy
produced by precipitation sedimentation and non-mechanical irreversible
processes and the effective temperature of frictional dissipation. We shall
see in the following section that processes associated with water, including
diffusion of water vapor and irreversible phase change, are responsible for
most of the non-mechanical irreversible entropy production in the atmosphere.
These processes, coupled with the work required to lift water upward through
the atmosphere, reduce the mechanical efficiency of the atmosphere relative to
a hypothetical atmosphere that does not contain water, and they exert a strong
influence on the dynamics of the atmospheric circulation.

##  III Irreversible processes in the climate system

We now consider in detail the different processes that contribute to
irreversible entropy production in the climate system. Because of its clear
relationship to work and kinetic energy generation, we focus on material
entropy production in the atmosphere and oceans. We first sketch the
derivation of the material entropy budget for both a single-component and
multi-component fluid (section III.1). Readers familiar with the governing
equation for a fluid’s entropy (38) may proceed to the following sections
where we consider the application of these results to the atmosphere (section
III.2) and ocean (section III.3) more specifically. A number of previous
authors have provided more detailed treatments focused on the atmosphere
[e.g., 113, 265, 94] and ocean [e.g., 109], and in a more general context
[e.g., 51].

###  III.1 Derivation of the material entropy budget

####  III.1.1 Single-component fluids

For simplicity, we begin by considering the entropy budget of a fluid made of
a single chemical component, or equivalently, a mixture with a fixed
composition. The atmosphere and ocean both have variable composition, and they
must be treated as multi-component fluids; the more complex multi-component
case is discussed in section III.1.2 below.

Consider the second law of thermodynamics applied to a fluid element of unit
mass. If the fluid’s interactions are purely reversible, we have,

$\frac{ds}{dt}=\frac{\dot{q}_{\text{rev}}}{T},$

where $s$ is the entropy of the fluid element and $\dot{q}_{\text{rev}}$ is
the reversible heating rate, both expressed per unit mass. We also have the
first law of thermodynamics,

$\frac{du}{dt}=\dot{q}+\dot{w},$

where $u$ is the internal energy of the fluid element, $\dot{w}$ is rate of
work done on the fluid element by its environment, and $\dot{q}$ is the
heating rate. Assuming reversible conditions, $\dot{q}=\dot{q}_{\text{rev}}$,
and the work is given by $\dot{w}_{\text{rev}}=-p\frac{d\alpha}{dt}$, where
$p$ is the pressure and $\alpha=\rho^{-1}$ is the specific volume of the fluid
element. Combining the above equations, we have

$\frac{du}{dt}=T\frac{ds}{dt}-p\frac{d\alpha}{dt}.$ (23)

This is the fundamental thermodynamic relation linking entropy to other state
variables for any substance of fixed composition [see e.g., 51, 171].

While (23) was derived for reversible conditions and under the assumption of
thermodynamic equilibrium, it may be applied under more general circumstances
provided an approximation known as local thermodynamic equilibrium is valid
[e.g., 51]. This approximation allows for thermodynamic functions such as
temperature, pressure, and entropy, to be defined locally within a fluid as a
function of space and time. In the bulk of the atmosphere and ocean, local
thermodynamic equilibrium is a very good approximation. The exception is at
very high altitudes ($\gtrsim 80$ km), where the density of the gas becomes so
low that molecular collisions become infrequent [e.g., 126]. But this region
accounts for a trivial fraction of the atmosphere’s mass, and it is therefore
reasonable to assume (23) is valid when considering the entropy budget of the
atmosphere or climate system as a whole [e.g., 177, 193].

To use (23), we require a separate expression for the rate of change of the
fluid’s internal energy. The equation governing the specific internal energy
$u$ of a single-component fluid under the influence of radiation may be
written [51],

$\frac{\partial\rho u}{\partial
t}+\nabla\cdot\left(\rho\mathbf{v}u+\mathbf{D}_{u}\right)=-p\nabla\cdot{\mathbf{v}}+\rho\dot{q}_{\text{rad}}+\rho\epsilon,$
(24)

where $\rho$ is the fluid density, $\mathbf{v}$ is the fluid velocity,
$\mathbf{D}_{u}$ is the heat flux owing to molecular diffusion, and
$\dot{q}_{\text{rad}}$ and $\epsilon$ are the heating rates owing to radiation
and frictional dissipation, respectively, expressed per unit mass of the
fluid. The first term on the right-hand side gives the rate of work done on
the fluid by its surroundings. Eq. (24) is an Eulerian equation for the
internal energy as a function of space and time, while (23) is a Lagrangian
equation valid for a given element of fluid. These viewpoints may be related
to one another by expressing the Lagrangian derivative $d/dt$, representing
the rate of change following a given fluid element666The Lagrangian derivative
is sometimes given the special notation $\frac{D}{Dt}=\frac{\partial}{\partial
t}+\mathbf{v}\cdot\nabla$., in terms of its Eulerian counterpart,

$\frac{du}{dt}=\frac{\partial u}{\partial t}+\mathbf{v}\cdot\nabla u.$

The internal energy equation may then be written in Lagrangian form as,

$\rho\frac{du}{dt}=-\rho
p\frac{d\alpha}{dt}+\rho\dot{q}_{\text{rad}}-\nabla\cdot\mathbf{D}_{u}+\rho\epsilon,$
(25)

where we have used the equation for mass continuity,

$\frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho\mathbf{v}\right)=0,$
(26)

and we have rewritten the work term in terms of the Lagrangian rate of change
of specific volume $\alpha$.

Combining the internal energy equation (25) with the fundamental thermodynamic
relation (23) and using mass continuity, we may write an explicit equation
governing the fluid’s entropy,

$\frac{\partial\rho s}{\partial
t}+\nabla\cdot\left(\rho\mathbf{v}s\right)+\nabla\cdot\left(\frac{\mathbf{D}_{u}}{T}\right)-\frac{\rho\dot{q}_{\text{rad}}}{T}=\frac{\rho\epsilon}{T}-\frac{\mathbf{D}_{u}\cdot\nabla
T}{T^{2}}.$ (27)

This represents the local Eulerian entropy budget of a single-component fluid.
Terms on the left-hand side give, from left to right, the local rate of change
of entropy, the flux divergence of entropy by fluid motions, the flux
divergence of entropy owing to molecular heat diffusion, and the entropy
tendency due to radiative heating and cooling. The right-hand side contains
the entropy production due to irreversible processes.

The connection between the local Eulerian entropy budget and the material
entropy budget of the climate system may be readily seen by integrating (27)
in space and averaging in time. Since there are no advective and molecular
fluxes at the top of the atmosphere, the flux divergences on the left-hand
side vanish on integration over the entire climate system. Further considering
steady-state conditions, the time tendency also vanishes, and we have

$-\frac{1}{A}\int_{\Omega}\left\langle\frac{\rho\dot{q}_{\text{rad}}}{T}\right\rangle\,dV=\langle\dot{S}_{i}^{\text{fric}}\rangle+\langle\dot{S}_{i}^{\text{heat}}\rangle,$
(28)

where we have defined

$\displaystyle\dot{S}_{i}^{\text{fric}}$
$\displaystyle=\frac{1}{A}\int_{\Omega}\rho\dot{s}_{i}^{\text{fric}}\,dV=\frac{1}{A}\int_{\Omega}\frac{\rho\epsilon}{T}\,dV,$
(29) $\displaystyle\dot{S}_{i}^{\text{heat}}$
$\displaystyle=\frac{1}{A}\int_{\Omega}\rho\dot{s}_{i}^{\text{heat}}\,dV=-\frac{1}{A}\int_{\Omega}\frac{\mathbf{D}_{u}\cdot\nabla
T}{T^{2}}\,dV.$ (30)

Comparing (28) to the material entropy budget (13), we find that the material
entropy production is given by
$\dot{S}_{i}^{\text{mat}}=\dot{S}_{i}^{\text{fric}}+\dot{S}_{i}^{\text{heat}}$.
For a planetary atmosphere comprised of a single-component fluid, there are
two processes that lead to irreversible material entropy production: molecular
heat diffusion and viscous dissipation.

It is important to note that our derivation of (27) required only the internal
energy equation and the fundamental thermodynamic relation, which may be taken
as the relation defining entropy. Eq. (27) contains no additional information
about the flow that is not already contained in the energy budget [309].
Rather, the additional information provided by the second law is contained in
the requirement for the irreversible entropy production terms on the right-
hand side of (27) to be positive definite [94]. This puts constraints on the
form of the molecular heat flux $\mathbf{D}_{u}$ and the viscous dissipation
$\rho\epsilon$.

For example, it is easy to show that the second law is satisfied for the
simple case of Fickian diffusion of temperature, for which
$\mathbf{D}_{u}=-\kappa\nabla T$, for some $\kappa\geq 0$. The associated
irreversible entropy production due to heat diffusion is [280],

$\rho\dot{s}^{\text{heat}}_{i}=\kappa\left(\frac{|\nabla T|}{T}\right)^{2},$

which is positive definite as required by the second law. For more complex
heat diffusion laws (e.g., those applicable to anisotropic materials), the
requirement of positive definite entropy production may be used to constrain
the functional form of $\mathbf{D}_{u}$ and ensure it is consistent with the
second law.

####  III.1.2 Multi-component fluids

The budget equation (28) is valid for a fluid whose composition is invariant
in time and space. But the dynamics of Earth’s atmosphere and ocean are both
strongly influenced by their variable composition. In particular, irreversible
entropy production in the atmosphere is dominated by processes associated with
water in all its phases [269]. We therefore must consider Earth’s atmosphere
as a multi-component fluid when discussing its entropy budget.

We consider a fluid that is a mixture of $N$ species, and we denote the
density of species $x$ by $\rho_{x}$. The continuity equation for each species
may be written

$\frac{\partial\rho_{x}}{\partial
t}+\nabla\cdot\left(\rho_{x}\mathbf{v}+\mathbf{D}_{x}\right)=\rho\dot{\chi}_{x}.$

where the velocity $\mathbf{v}$ is the barycentric velocity, given by the
mass-weighted mean velocity over all species [51], and $\mathbf{D}_{x}$ is the
non-advective flux of species $x$, representing processes such as Brownian
motion of molecules and, in the atmosphere, sedimentation of hydrometeors such
as raindrops and snowflakes [94].

The quantity $\dot{\chi}_{x}$ represents the mass source of species $x$ per
unit mass of the mixture due to chemical reactions. Mass conservation requires
that

$\sum_{x}\dot{\chi}_{x}=0.$

For example, in the atmosphere, condensation represents a source of liquid
water and a sink of water vapor of equal magnitude. Since by definition the
barycentric velocity gives the velocity of the center of mass of an element of
the fluid mixture, mass conservation also requires that the non-advective mass
fluxes of all species sum to zero:

$\sum_{x}\mathbf{D}_{x}=0.$ (31)

Combining the previous three equations, it may be shown that density of the
mixture $\rho=\sum_{x}\rho_{x}$ satisfies the continuity equation (26).

It is useful to define the mass fraction of a species $q_{x}=\rho_{x}/\rho$.
The mass fraction of water vapor, $q_{v}$, is known as the specific humidity.
The mass continuity equation for each species may be rearranged into a
Lagrangian equation for $q_{x}$,

$\rho\frac{dq_{x}}{dt}=\rho\dot{\chi}_{x}-\nabla\cdot\mathbf{D}_{x}.$ (32)

We may also define the specific internal energy of the mixture by

$u=\sum_{x}q_{x}u_{x},$ (33)

where $u_{x}$ is the specific internal energy of species $x$. Finally, we may
write the fundamental thermodynamic relation for each species,

$\frac{du_{x}}{dt}=T\frac{ds_{x}}{dt}-p_{x}\frac{d\alpha_{x}}{dt},$ (34)

where $s_{x}$ is the specific entropy of species $x$, $p_{x}$ is the partial
pressure of species $x$ and $\alpha_{x}=\rho_{x}^{-1}$. Combining the previous
two equations, we may write a thermodynamic relation for the mixture given by
[267],

$\frac{du}{dt}=T\frac{ds}{dt}-p\frac{d\alpha}{dt}+\sum_{x}g_{x}\frac{dq_{x}}{dt}.$
(35)

Here, the entropy of the mixture $s$ is defined analogously to (33),
$\alpha=\rho^{-1}$, and the total pressure $p$ is the sum of the partial
pressures of each species. The quantity $g_{x}=u_{x}+p_{x}\alpha_{x}-Ts_{x}$
is the specific Gibbs free energy for each species $x$; it is equal to the
chemical potential divided by the molar mass.

Use of (35) involves some approximation that should be noted. In addition to
the assumption that the fundamental thermodynamic relation is valid for each
species, we have assumed that each species has the same temperature $T$.
Furthermore, by expressing the internal energy of the mixture as the mass-
weighted sum of the internal energy of each of species in (33), we have
neglected interfacial effects between the different species. Such effects are
important for understanding the formation of clouds and precipitation in the
atmosphere [290], but they are typically neglected when considering its bulk
thermodynamics. Note, however, that we have made no assumption of chemical
equilibrium between species; as we shall see, phase changes outside of
equilibrium are an important irreversible entropy source in the atmosphere.

As for a single-component fluid, we may derive an equation for the entropy
tendency of a multi-component fluid by substituting the thermodynamic relation
(35) into the equation governing the internal energy, which for a multi-
component fluid may be written [e.g., 51, 94],

$\rho\frac{du}{dt}+\nabla\cdot\left(\mathbf{D}_{u}+\sum_{x}\mathbf{D}_{x}h_{x}\right)=-\rho
p\frac{d\alpha}{dt}+\rho\dot{q}_{\text{rad}}+\rho\epsilon.$ (36)

This equation is identical to the single-component case (25) but for the
appearance of the flux divergence of enthalpy $h_{x}=u_{x}+p_{x}\alpha_{x}$
owing to molecular motions on the left-hand side777The heat flux
$\mathbf{D}_{u}$ is sometimes defined to include these molecular enthalpy
fluxes [51].. Substituting (35) into (36), using (32), and rearranging, one
may write an equation for the Lagrangian rate of change of entropy, given by

$\displaystyle\rho\frac{ds}{dt}+$
$\displaystyle\nabla\cdot\left(\frac{\mathbf{D}_{u}}{T}\right)+\sum_{x}\left[\tfrac{\nabla\cdot\left(\mathbf{D}_{x}h_{x}\right)}{T}-\tfrac{g_{x}\nabla\cdot{\mathbf{D}_{x}}}{T}\right]-\frac{\rho\dot{q}_{\text{rad}}}{T}=$
$\displaystyle-\frac{\mathbf{D}_{u}\cdot\nabla T}{T^{2}}-\sum_{x}\frac{\rho
g_{x}\dot{\chi}_{x}}{T}+\frac{\rho\epsilon}{T}.$

Using the fact that $h_{x}-g_{x}=Ts_{x}$, the terms involving the molecular
flux of species mass may be written,

$\tfrac{\nabla\cdot\left(\mathbf{D}_{x}h_{x}\right)}{T}-\tfrac{g_{x}\nabla\cdot{\mathbf{D}_{x}}}{T}=\nabla\cdot\left(\mathbf{D}_{x}s_{x}\right)+\mathbf{D}_{x}\cdot\left(\tfrac{1}{T}\nabla
h_{x}-\nabla s_{x}\right).$ (37)

Finally, using mass conservation (26), we may write a local Eulerian budget
equation for the entropy, given by

$\displaystyle\frac{\partial\rho s}{\partial
t}+\nabla\cdot\left(\rho\mathbf{v}s\right)+\nabla\cdot\left(\frac{\mathbf{D}_{u}}{T}+\sum_{x}\mathbf{D}_{x}s_{x}\right)-\frac{\rho\dot{q}_{\text{rad}}}{T}=$
$\displaystyle\rho\dot{s}_{i}^{\text{fric}}+\rho\dot{s}_{i}^{\text{heat}}+\rho\dot{s}_{i}^{\text{diff}}+\rho\dot{s}_{i}^{\text{chem}},$
(38)

where

$\displaystyle\rho\dot{s}_{i}^{\text{diff}}$
$\displaystyle=-\sum_{x}\mathbf{D}_{x}\cdot\left(\tfrac{1}{T}\nabla
h_{x}-\nabla s_{x}\right),$ $\displaystyle\rho\dot{s}_{i}^{\text{chem}}$
$\displaystyle=-\sum_{x}\frac{\rho g_{x}\dot{\chi}_{x}}{T}.$

The budget (38) is similar to the entropy budget derived for a single
component fluid, but it has an additional term associated with entropy
transport owing to the non-advective flux of mass $\mathbf{D}_{x}$ on the
left-hand side, and additional irreversible sources
$\rho\dot{s}_{i}^{\text{diff}}$ and $\rho\dot{s}_{i}^{\text{chem}}$ associated
with non-advective transport of mass and chemical reactions, respectively, on
the right-hand side [e.g., 51].

The entropy production associated with non-advective transport
$\rho\dot{s}_{i}^{\text{diff}}$ may be simplified further using the
fundamental thermodynamic relation written in terms of the enthalpy $h_{x}$,

$dh_{x}=Tds_{x}+\frac{1}{\rho_{x}}dp_{x},$ (39)

which allows one to write,

$\rho\dot{s}_{i}^{\text{diff}}=-\sum_{x}\frac{\mathbf{D}_{x}\cdot\nabla
p_{x}}{\rho_{x}T}.$ (40)

Molecular diffusion of a species $x$ induces positive irreversible entropy
production when it transports the species from high partial pressure to low
partial pressure. This corresponds to the entropy production associated with
the molecular mixing of the species within the fluid.

###  III.2 Thermodynamics of a moist atmosphere

We now discuss more specifically the irreversible entropy sources in the
atmosphere. To do so, we introduce an approximate thermodynamic treatment of a
moist atmosphere following Gassmann and Herzog [93, 94] and consistent with
other treatments in the literature [e.g., 113, 8]. We note that while some
numerical models of the atmosphere employ similar thermodynamic treatments
[e.g., 309, 28], many models employ simplified equation sets that neglect some
of the irreversible processes discussed. Moreover, most atmospheric models
include numerical sources and sinks of entropy in addition to the irreversible
physical sources discussed here, and diagnosing the entropy budget in such
models must be done with care. We further discuss the issue of numerical
production of entropy in models of the climate system in section VII.

The atmosphere is taken to be a mixture of water vapor ($v$), liquid water
($l$), solid water ($s$) and “dry” air $(d)$ containing all the well-mixed
gases. In Earth’s atmosphere, dry air is by far the most abundant component;
typical mass fractions of water vapor $q_{v}$ are no larger than 3-4%, while
typical mass fractions of condensed water species are orders of magnitude
smaller. Dry air and water vapor are assumed to be ideal gases governed by an
equation of state of the form,

$p_{x}=\rho_{x}R_{x}T,$

with $R_{x}=R^{*}/m_{x}$ being the gas constant for species $x$ expressed in
terms of the molar mass $m_{x}$ and the universal gas constant $R^{*}$.
Condensed water species are assumed to be incompressible, with negligible
specific volume. The specific entropy $s$ is then given by
$s=\sum_{x}q_{x}s_{x}$ where $x=(d,v,l,s)$ and the specific entropy of each
constituent is defined [113]

$\displaystyle s_{d}$
$\displaystyle=c_{pd}\ln(T/T_{0})-R_{d}\ln(p_{d}/p_{0})+s_{d0},$ (41a)
$\displaystyle s_{v}$
$\displaystyle=c_{pv}\ln(T/T_{0})-R_{v}\ln(p_{v}/p_{0})+s_{v0},$ (41b)
$\displaystyle s_{l}$ $\displaystyle=c_{pl}\ln(T/T_{0})+s_{l0},$ (41c)
$\displaystyle s_{s}$ $\displaystyle=c_{ps}\ln(T/T_{0})+s_{s0}.$ (41d)

Here $c_{px}$ is the isobaric specific heat capacity of each constituent,
which we take to be constant, and $T_{0}$, $p_{0}$, and $s_{x0}$ are the
temperature, pressure, and entropy of each constituent, respectively, at a
reference point which we take to be the triple point of water. The reference
entropies of water phases are related by,

$\displaystyle s_{v0}$ $\displaystyle=s_{l0}+\frac{L_{v0}}{T_{0}},$
$\displaystyle s_{s0}$ $\displaystyle=s_{l0}-\frac{L_{f0}}{T_{0}},$

where $L_{v0}$ and $L_{f0}$ are the latent heats of vaporization and freezing,
respectively, at the triple point of water. Specification of the entropy is
completed by setting the value of blue$s_{d0}$ and $s_{l0}$. blueBecause we
will not consider chemical reactions between dry air and water substance, the
values of these parameters do not affect the calculation of irreversible
entropy production in the atmosphere, and many authors take $s_{l0}=s_{d0}=0$
[e.g., 309]. But care must be taken to account for the effects of this choice
when interpreting entropy changes of open systems [213]. As noted previously,
this formulation also neglects interfacial effects between species, and the
resultant entropy budget therefore neglects certain irreversible processes
associated with the spatial distribution of different phases within an air
parcel (e.g., the irreversibility of cloud droplets coalescing together).

With the above caveats in mind, we now consider the irreversible entropy
sources in a moist atmosphere. Like a single-component fluid, entropy
production in the atmosphere includes that due to frictional dissipation
$\dot{s}_{i}^{\text{fric}}$ and heat diffusion $\dot{s}_{i}^{\text{heat}}$. In
subsections III.2.1-3 below, we focus on the remaining entropy production
terms $\dot{s}_{i}^{\text{chem}}$ and $\dot{s}_{i}^{\text{diff}}$ that owe
their existence to the multi-component nature of a moist atmosphere.

####  III.2.1 Irreversible phase changes

The irreversibility of phase changes is included within the entropy source
$\dot{s}_{i}^{\text{chem}}$, which we divide into components associated with
evaporation/condensation $\dot{s}_{i}^{\text{evap}}$ and melting/freezing
$\dot{s}_{i}^{\text{melt}}$. Sublimation may be considered to be a combination
of melting and evaporation.

Consider first evaporation and condensation of a liquid droplet in the air.
Denoting the evaporation rate per unit mass by $e$, we have that
$e=\dot{\chi}_{v}=-\dot{\chi}_{l}$, and hence,

$\dot{s}^{\text{evap}}_{i}=\frac{e(g_{l}-g_{v})}{T}.$

Positive irreversible entropy production requires $g_{v}<g_{l}$ for net
evaporation ($e>0$) and $g_{l}<g_{v}$ for net condensation ($e<0$), and the
phase change is reversible if $g_{v}=g_{l}$, a condition known as saturation.
Defining $g^{*l}_{v}$ as the value of the Gibbs free energy of water vapor at
saturation with respect to liquid, we have

$\dot{s}^{\text{evap}}_{i}=\frac{e(g_{v}^{*l}-g_{v})}{T}.$ (42)

Using the definitions of the entropy of water vapor (41b) and the enthalpy of
water vapor $h_{v}=c_{pv}(T-T_{0})+T_{0}s_{v0}$, this may be written [e.g.,
269],

$\dot{s}^{\text{evap}}_{i}=-eR_{v}\ln(\mathcal{R}),$ (43)

where $\mathcal{R}=p_{v}/p^{*l}_{v}$ is the relative humidity and
$p^{*l}_{v}(T)$ is the saturation vapor pressure over a liquid surface and is
only a function of temperature. When the relative humidity is less than one,
the air is subsaturated with respect to liquid water, and evaporation is
irreversible. When the relative humidity is greater than one, the air is
supersaturated, and condensation is irreversible.

The above discussion neglects the effects of surface tension and impurities on
phase equilibrium. Both of these effects are important for determining the
conditions under which cloud droplets may form and grow [290]. Were it not for
the abundance of aerosols that act as nuclei for the formation of cloud
droplets, supersaturation may be a common occurrence in the atmosphere. In
actual fact, substantial supersaturation with respect to liquid water is rare,
and condensation generally occurs close to phase equilibrium. Evaporation, on
the other hand, often occurs at relative humidities well below 100% and is an
important source of irreversible entropy production in the climate system.
Contributors to $\dot{s}_{i}^{\text{evap}}$ include the evaporation of
precipitation falling through subsaturated air, and evaporation from the
Earth’s surface, particularly over water bodies. Surface evaporation is driven
by the thermodynamic disequilibrium between the surface and the atmosphere,
but it is strongly modulated by kinetic effects; empirically, the evaporation
rate from a saturated surface is found to scale with the square of the
windspeed. This modulation gives rise to interesting feedbacks that can
amplify atmospheric circulations such as tropical cyclones (see section V).

A similar derivation to that given above may be performed for the solid/liquid
phase transition leading to [309]

$\dot{s}^{\text{melt}}_{i}=-mR_{v}\ln\left(\frac{p_{v}^{*l}}{p_{v}^{*s}}\right),$
(44)

where $m$ is the melting rate per unit mass and $p_{v}^{*s}$ is the saturation
vapor pressure over a solid ice surface. Note that both $p_{v}^{*l}$ and
$p_{v}^{*s}$ are functions of temperature only, and they coincide at the
freezing point888By neglecting the specific volumes of liquid and solid water,
we have assumed that the freezing point is independent of pressure and equal
to the triple point temperature. This is a very good approximation except at
very high pressures not experienced in Earth’s atmosphere. $T_{0}$. For
temperatures below $T_{0}$, $p_{v}^{*l}>p_{v}^{*s}$ and freezing is
irreversible, while for temperatures above $T_{0}$, $p_{v}^{*l}<p_{v}^{*s}$
and melting is irreversible. In the atmosphere, melting and freezing can occur
tens of kelvins from the freezing point, leading to a substantial irreversible
production of entropy. Furthermore, in regions of the atmosphere with
temperatures below freezing, the relative humidity with respect to ice
$p_{v}/p_{v}^{*s}$ ranges from a few percent to values approaching 200% [96].
Both sublimation at subsaturation and deposition at supersaturation may
therefore be important irreversible sources of entropy in the atmosphere,
contributing to both $\dot{s}_{i}^{\text{evap}}$ and
$\dot{s}_{i}^{\text{melt}}$.

####  III.2.2 Diffusive mixing

In the atmosphere, the non-advective flux of species mass occurs as a result
of two processes: 1) diffusive molecular mixing as a result of the random
Brownian motions of molecules of each species and 2) sedimentation of
condensed water particles massive enough to have appreciable terminal
velocities. We consider entropy production by molecular mixing
$\dot{s}_{i}^{\text{mix}}$ here and return to the sedimentation flux in the
following subsection.

According to (40), the entropy source due to molecular mixing is proportional
to the specific volume of each constituent. The specific volume of condensed
water species has been assumed to be negligible, and so we must only consider
diffusive fluxes of gaseous species. Consider the diffusive mixing of dry air
and water vapor. The mass conservation condition (31) requires that such
mixing involves equal and opposite mass fluxes of dry air and water vapor. But
in the atmosphere, fractional gradients in the partial pressure of water vapor
can be orders of magnitude larger than those of dry air, and the dominant
component of $s_{i}^{\text{mix}}$ is that due to the diffusion of water vapor
down its partial pressure gradient, given by [269, 309],

$\rho\dot{s}_{i}^{\text{mix}}\approx
R_{v}\mathbf{D}_{v}\cdot\nabla\ln\left(\frac{p_{v}}{p_{0}}\right).$ (45)

Diffusive mixing of water vapor and dry air is particularly strong at the
boundaries between clouds and the clear-air environment. Here, the combination
of diffusive mixing and evaporation of cloud and precipitation particles
produces a transport of water vapor into the environment that plays an
important role in governing the tropical relative humidity [310, 332].

In fact, the entropy source owing to the mixing of dry air and water vapor
$\dot{s}_{i}^{\text{mix}}$ has a close connection to that of evaporation
$\dot{s}_{i}^{\text{evap}}$. Consider a cloud droplet suspended in air with a
relative humidity $\mathcal{R}$. Evaporation from this droplet into the
subsaturated environment results in an irreversible entropy source given by
(43). Alternatively, suppose the evaporation from the droplet occurs
reversibly in a molecular boundary layer surrounding the droplet that is at
saturation. Diffusive mixing then transports this water vapor from the droplet
to the far-field which has relative humidity $\mathcal{R}$. Neglecting the
small contribution owing to the diffusion of dry air, this process results in
an irreversible source of entropy given by (45). As pointed out by Pauluis and
Held [269], the entropy production in these two cases is the same. Evaporating
water and transporting it from the droplet to its environment results in the
same irreversible entropy production regardless of the microscopic details of
the transport process.

####  III.2.3 Irreversible sedimentation of precipitation

Consider an air parcel containing a mass fraction of precipitation $q_{p}$
with a sedimentation velocity $-v_{t}\mathbf{k}$ relative to air. Here
$\mathbf{k}$ is a unit vector pointing upwards (antiparallel to the
gravitational vector). Generally, it is reasonable to assume that the fall
speed $v_{t}$ is equal to the terminal velocity of precipitation set by a
balance between the downward gravitational force on hydrometeors
(precipitation particles) and the upward drag force on hydrometeors owing to
friction with the surrounding air [290].

The barycentric velocity of the air-precipitation mixture may be written,

$\mathbf{v}=(1-q_{p})\mathbf{v}_{a}+q_{p}\left(\mathbf{v}_{a}-v_{t}\mathbf{k}\right),$
(46)

where $\mathbf{v}_{a}$ is the velocity of air. Since
$\mathbf{v}_{a}\neq\mathbf{v}$, sedimentation of precipitation is coupled with
a compensating upward non-advective transport of air with a velocity,

$\mathbf{v}_{a}-\mathbf{v}=q_{p}v_{t}\mathbf{k}.$

The above equation implies an upward non-advective flux of dry air
$\mathbf{D}_{d}=\rho q_{d}q_{p}v_{t}\mathbf{k}$ and water vapor
$\mathbf{D}_{v}=\rho q_{v}q_{p}v_{t}\mathbf{k}$. On substitution into (40),
these non-advective fluxes give an irreversible entropy source totalling [94],

$\rho\dot{s}_{i}^{\text{sed}}=-\frac{q_{p}v_{t}}{T}\frac{\partial p}{\partial
z}.$

Assuming hydrostatic balance, this may be written,

$\rho\dot{s}_{i}^{\text{sed}}\approx\frac{\rho{g_{\earth}}q_{p}v_{t}}{T},$
(47)

where ${g_{\earth}}$ is Earth’s gravitational acceleration. The entropy source
$\dot{s}_{i}^{\text{sed}}$ is therefore positive when precipitation falls to
the surface ($v_{t}>0$).

The term $\dot{d}_{\text{sed}}={g_{\earth}}q_{p}v_{t}$ that appears on the
right-hand side of (47) represents a dissipation rate associated with the loss
of gravitational potential energy by falling precipitation. Physically, this
manifests as frictional dissipation due to the upward drag force acting on
precipitation particles sedimenting relative to the air [271, 269]. In our
derivation, however, we do not consider in detail the interface between air
and condensed water species, and $\rho\dot{d}_{\text{sed}}$ appears instead as
the hydrostatic approximation to an irreversible pressure work [94].

As discussed in section II.2.3, the integrated dissipation
$\langle\dot{D}_{\text{sed}}\rangle=\int_{\Omega}\langle\rho\dot{d}_{\text{sed}}\rangle\,dV$
is a sink in the time-averaged mechanical energy budget. This may be seen
explicitly by considering the time-averaged rate of work performed by the
barycentric flow, which may be written,

$\langle\dot{W}_{K}\rangle=-\int_{\Omega}\langle\mathbf{v}\cdot\nabla
p\rangle\,dV,$ (48)

where the angle brackets refer to a time mean. Rearranging the barycentric
velocity (46) into a reversible component
$\mathbf{v}_{\text{rev}}=\mathbf{v}_{a}$, associated with fluid motions, and
an irreversible component $\mathbf{v}_{\text{irr}}=-q_{p}v_{t}\mathbf{k}$,
associated with the sedimentation of precipitation, we may write,

$\displaystyle\langle\dot{W}_{K}\rangle=\underbrace{-\int_{\Omega}\langle\mathbf{v}_{\text{rev}}\cdot\nabla
p\rangle\,dV}_{\langle\dot{W}_{\text{rev}}\rangle}\,\,\,-\,\,\,\underbrace{\int_{\Omega}\langle\rho{g_{\earth}}q_{p}v_{t}\rangle\,dV}_{\langle\dot{D}_{\text{sed}}\rangle}.$

Here we have used hydrostatic balance to express the second term on the right-
hand side in terms of the gravitational acceleration. The first term on the
right-hand side gives the rate of work performed by reversible fluid motions
$\langle\dot{W}_{\text{rev}}\rangle$, and the second term gives the
precipitation-induced dissipation rate $\langle\dot{D}_{\text{sed}}\rangle$.
Comparing this equation to (19), we see that, in steady state,
$\langle\dot{W}_{K}\rangle=\langle\dot{D}_{\text{fric}}\rangle$. We may
therefore identify the rate of work performed by the barycentric flow as being
the rate of generation of kinetic energy of fluid motions. The total rate at
which reversible work is performed $\langle\dot{W}_{\text{rev}}\rangle$ is
larger and represents the work required not just to generate kinetic energy of
fluid motions, but also to lift water against the Earth’s gravitational field
[271, 269].

In general it is the work $\langle\dot{W}_{K}\rangle$ that is of most interest
in studies of the climate system, as it represents the work performed in
driving the atmospheric and oceanic circulations. In particular,
$\langle\dot{W}_{K}\rangle$ represents the source of kinetic energy in
computations of the transformations between internal, potential, and kinetic
energy that make up the Lorenz energy cycle [see section VIII.1.1 and 188].
However, most discussions of the Lorenz energy cycle to date treat moist
processes in a simplified way, and they do not explicitly consider the work
required to lift water through the Earth’s gravitational field [266].

####  III.2.4 The entropy budget of a moist atmosphere

Combining the results from the previous subsections, we may write the
approximate local Eulerian material entropy budget (38) for a moist
atmosphere,

$\frac{\partial\rho s}{\partial
t}+\nabla\cdot\left(\rho\mathbf{v}s\right)+\nabla\cdot\left(\frac{\mathbf{D}_{u}}{T}+\sum_{x}\mathbf{D}_{x}s_{x}\right)-\frac{\rho\dot{q}_{\text{rad}}}{T}=\rho\dot{s}_{i}^{\text{mat}}$
(49)

where the material irreversible entropy production is given by,

$\dot{s}_{i}^{\text{mat}}=\dot{s}_{i}^{\text{fric}}+\dot{s}_{i}^{\text{heat}}+\dot{s}_{i}^{\text{evap}}+\dot{s}_{i}^{\text{melt}}+\dot{s}_{i}^{\text{mix}}+\dot{s}_{i}^{\text{sed}}.$
(50)

The terms on the right-hand side represent, from left to right, frictional
dissipation of the winds (29), molecular heat diffusion (30), irreversible
evaporation and condensation (43), irreversible melting and freezing (44),
irreversible molecular mixing (45), and dissipation associated with the
sedimentation of precipitation (47).

Integrating the above equation in space, dividing by the Earth’s surface area
$A$, and averaging in time, one obtains the steady-state material entropy
budget for the entire atmosphere [309],

$\displaystyle\frac{1}{A}\int_{\partial\Omega_{A}}\left\langle\frac{\mathbf{D}_{u}}{T}+\sum_{x}\mathbf{D}_{x}s_{x}\right\rangle\cdot
d\mathbf{A}-$
$\displaystyle\frac{1}{A}\int_{\Omega_{A}}\left\langle\frac{\rho\dot{q}_{\text{rad}}}{T}\right\rangle\,dV$
$\displaystyle=\langle\dot{S}_{i}^{\text{mat}}\rangle,$ (51)

where $\Omega_{A}$ represents the volume of the atmosphere,
$\partial\Omega_{A}$ represents its boundary (the top of the atmosphere and
the surface) and $d\mathbf{A}$ is a surface element oriented with outward
pointing normal. The first integral on the left-hand side gives the boundary
fluxes of entropy owing to molecular heat and water transport. Specifically,
$\int_{\partial\Omega}\tfrac{\mathbf{D}_{u}}{T}\cdot d\mathbf{A}$ may be
interpreted as the flux of entropy out of the atmosphere carried by the
molecular flux of heat, while $\int_{\partial\Omega}\mathbf{D}_{x}s_{x}\cdot
d\mathbf{A}$ may be interpreted as the flux of entropy out of the atmosphere
carried by the molecular flux of species $x$. The right hand side
$\langle\dot{S}_{i}^{\text{mat}}\rangle$ gives the total material irreversible
entropy production in the atmosphere defined

$\langle\dot{S}_{i}^{\text{mat}}\rangle=\frac{1}{A}\int_{\Omega_{A}}\left\langle\rho\dot{s}_{i}^{\text{mat}}\right\rangle\,dV,$

and expressed per unit of the Earth’s surface area $A$. The total entropy
production rate associated with each process given in (50) may be defined
analogously.

Consider the atmosphere over a saturated liquid surface (e.g., the
ocean)999For a surface with relative humidity less than one, the surface
evaporative flux of vapor also involves an irreversible source of entropy that
should be added to $\dot{S}_{i}^{\text{mat}}$ on the right-hand side.. Using
the relationship $s_{v}^{*l}-s_{l}=L_{v}/T$, where $s_{v}^{*l}$ is the
saturation entropy of water vapor over a liquid surface and $L_{v}$ is the
latent heat of vaporization, we may write (51) as

$-\frac{1}{A}\int_{\partial\Omega_{A}}\left\langle\frac{F_{LH}+F_{SH}}{T}\right\rangle
dA-\frac{1}{A}\int_{\Omega_{A}}\left\langle\frac{\rho\dot{q}_{\text{rad}}}{T}\right\rangle\,dV=\langle\dot{S}_{i}^{\text{mat}}\rangle$
(52)

where $F_{LH}$ and $F_{SH}$ are the surface latent and sensible heat fluxes
from the surface to the atmosphere, and $dA=|d\mathbf{A}|$. In steady state,
conservation of total energy in the atmosphere requires the sensible and
latent heat flux to balance the net radiative cooling of the atmosphere. We
may therefore write the above equation in a simpler form [e.g., 269, 331],

$-\langle\dot{Q}_{\text{rad}}\rangle\left(\frac{1}{T_{a}}-\frac{1}{T_{s}}\right)=\langle\dot{S}_{i}^{\text{mat}}\rangle,$
(53)

where
$\dot{Q}_{\text{rad}}=\frac{1}{A}\int_{\Omega_{A}}\rho\dot{q}_{\text{rad}}\,dV$
is the net radiative heating of the atmosphere, and we have defined the
characteristic temperatures

$\frac{1}{T_{s}}=\frac{\int_{\partial\Omega_{A}}\langle\left(\frac{1}{T}\right)(F_{LH}+F_{SH})\rangle
dA}{\int_{\partial\Omega_{A}}\left\langle F_{LH}+F_{SH}\right\rangle dA}$

and

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

Here $T_{s}$ is a characteristic surface temperature representing the
temperature at which the atmosphere is heated by surface fluxes, and $T_{a}$
is a characteristic atmospheric temperature representing the temperature at
which the atmosphere is cooled by radiation. With these definitions, (53) is
identical to the material entropy budget of the one-layer atmosphere model
presented in section II.1.2.

Eq. (53) provides an intuitive perspective on the second law as applied to the
atmosphere. The atmosphere is heated by surface fluxes at the relatively warm
surface and cooled by radiation in the relatively cold troposphere, thus
creating an entropy sink that, in steady state, is balanced by irreversible
processes101010If one instead defines $T_{s}$ as the characteristic
temperature at which the surface is heated by radiation, (53) becomes the
entropy budget for the entire climate system. Since the ocean transports
energy from warm regions to cooler regions, this definition results in a
higher $T_{s}$, and a larger entropy sink. In steady state, this larger sink
is balanced by the additional irreversible entropy production that occurs in
the ocean (see section III.3).. A key task is then to determine relative
importance of the different processes contributing to
$\dot{S}_{i}^{\text{mat}}$. As highlighted by Pauluis and Held [269, 270], the
irreversible entropy production in the atmosphere is dominated by moist
processes, including the molecular mixing of water vapor
$\dot{S}_{i}^{\text{mix}}$, irreversible phase change
$\dot{S}_{i}^{\text{cond}}$ & $\dot{S}_{i}^{\text{melt}}$, and dissipation
associated with precipitation sedimentation $\dot{S}_{i}^{\text{sed}}$. In the
next three sections, we will discuss in detail the role played by irreversible
moist processes in the dynamics of convective clouds (section IV), tropical
cyclones (section V) and the general circulation (section VI).

An important limitation of the derivation leading to (53) is the assumption
that all chemical species within the atmosphere have the same temperature.
This is a good approximation for dry air, water vapor, and clouds, but it is
not accurate for hydrometeors with appreciable sedimentation velocities, which
often differ in temperature from their surroundings by several kelvins. While
the assumption of uniform temperature is made commonly in studies of the
atmosphere’s entropy budget, it does not allow for consideration of
irreversible entropy production associated with heat diffusion between
precipitation and the surrounding atmosphere. Bannon [8] and Goody [102]
derived order-of-magnitude estimates to suggest that such heat diffusion may
contribute significantly to the entropy budget of moist convection. Bannon [8]
and Raymond [295] derived equation sets suitable for numerical modeling that
include the relevant irreversible production terms, but, to our knowledge,
there are no detailed modeling studies of the entropy budget that include
entropy production associated with heat diffusion between hydrometeors and
their surroundings. The magnitude of the errors induced by neglecting this
entropy source, and whether it is larger than other errors in simulated
entropy budgets due to, for example, numerical truncation [see e.g., 375, and
section VII], remains unknown.

####  III.2.5 The role of latent heating

Latent heat release does not appear explicitly in the atmospheric entropy
budget. This is because phase changes that occur at equilibrium do not cause a
change in entropy. Rather, the effects of latent heat in (51) are included
through the entropy input into the atmosphere owing to molecular fluxes of
water species. Nevertheless, many studies of the entropy budget make explicit
mention of entropy production by latent heat release and find that it is a
major contributor to the global atmospheric budget [e.g., 262, 280, 86]. This
is because such studies consider an “external” view of the effects of
moisture, in which latent heating is treated as an additional external heat
source, similar to radiative heating, but the atmosphere is otherwise treated
largely as a single-component fluid (see section III.1.1). Our derivation
above, in contrast, considers an “internal” view, in which the atmosphere is
treated as a true multi-component fluid, with phase changes being cast as mass
exchanges between the different components. As we discuss below, both of these
perspectives are mathematically valid, but only the internal approach
blueprovides a direct expression of the second law of thermodynamics applied
to the atmosphere.

The connection between the internal and external viewpoints was first
elucidated by Pauluis and Held [270], who showed that the total irreversible
entropy production associated with phase change and water vapor mixing could
be approximately written as the sum of two terms, one related to the latent
heating rate, and the other related to the work performed by the expansion of
water vapor in the atmosphere. Using this relationship, the entropy budget
(51) may be written approximately as,

$\displaystyle\int_{\partial\Omega_{A}}\left\langle\frac{\mathbf{D}_{u}}{T}\right\rangle\cdot
d\mathbf{S}-\int_{\Omega_{A}}\left\langle\frac{\rho(\dot{q}_{\text{rad}}+\dot{q}_{\text{lat}})}{T}\right\rangle\,dV\approx$
$\displaystyle\langle\dot{S}_{i}^{\text{fric}}\rangle+\langle\dot{S}_{i}^{\text{heat}}\rangle+\langle\dot{S}_{i}^{\text{sed}}\rangle+\int_{\Omega_{A}}\left\langle\frac{1}{T}\frac{dp_{v}}{dt}\right\rangle\,dV,$
(54)

where $\dot{q}_{\text{lat}}$ is the net heating rate due to phase
changes111111This relationship neglects the temperature dependence of the
latent heats of vaporization and freezing [270, 309].. This equation is
similar to the single-component entropy budget [cf. (28)] with the latent
heating rate included as an external heating in addition to that due to
radiation, but it includes the entropy production associated with
sedimentation of hydrometeors blue$\langle\dot{S}_{i}^{\text{sed}}\rangle$ and
a term related to the work performed by water vapor expansion (last term on
the right-hand side).

Romps [309] explicitly showed that (54) is an approximation of the budget for
a quantity he referred to as the “dry” entropy, the entropy of a single-
component gas with the same pressure, temperature, heat capacity, and gas
constant as the moist atmosphere. A number of authors have considered budgets
similar to (54) but neglected the terms associated with hydrometeor
sedimentation and water vapor work [280, 86, 262]. The neglect of these terms
is consistent with the treatment of moist thermodynamics in many climate
models, which often neglect the effects of water substance on the specific
heat capacity and density of air, except in the calculation of buoyancy [269].
Indeed, Pascale et al. [262] obtained accurate closure of a budget similar to
(54) in a comprehensive global climate model without considering precipitation
sedimentation or water vapor work.

The dry-entropy budget and its variants provide a view of atmospheric
thermodynamics in which the latent heating and cooling associated with phase
changes is treated as an external heat source to an otherwise dry fluid. Such
a view has proven useful as a means of estimating terms within the entropy
budget [269, 270], or as an analysis tool in and of itself [280, 86, 262,
309]. But in contrast to the entropy budget, the dry-entropy budget includes
source terms that may be locally negative and do not represent irreversible
processes, and its connection to the second law of thermodynamics is less
direct.

###  III.3 Irreversible entropy production in the ocean

Comparatively few studies have investigated the entropy budget of the ocean
relative to that of the atmosphere, but the formalism developed in section
III.1.2 is equally applicable to the ocean. The oceanic entropy budget
includes irreversible entropy sources owing to frictional dissipation
$\dot{S}_{i}^{\text{fric}}$, heat diffusion $\dot{S}_{i}^{\text{heat}}$,
diffusion of mass $\dot{S}_{i}^{\text{diff}}$, and phase changes
$\dot{S}_{i}^{\text{chem}}$. In the ocean, $\dot{S}_{i}^{\text{diff}}$
accounts for irreversible entropy production owing to molecular mixing between
regions of high and low salinity, while $\dot{S}_{i}^{\text{chem}}$ accounts
for irreversible entropy production associated with the melting and freezing
of sea ice outside of phase equilibrium. Estimates of the entropy production
by these processes indicate that, while salt diffusion can be a significant
source of entropy in certain regions [109], both salt diffusion [326] and sea-
ice melt [14] contribute only a small portion of the total irreversible
entropy production of the ocean as a whole. In contrast to the atmosphere, the
ocean’s entropy budget may be approximated by that of a single-component
fluid, with irreversible entropy production occurring primarily through heat
diffusion and frictional dissipation.

A further difference between the atmosphere and the ocean is that the ocean is
heated and cooled almost exclusively at its upper surface (the exception is
the heating owing to the geothermal heat flux, which on Earth is
quantitatively small). Heat transport through sensible and latent heat fluxes
occurs at the air-sea interface, while the penetration depth of shortwave and
longwave radiation through ocean water is no more than 100 m and a few mm,
respectively. The temperature at which the ocean is heated or cooled by
radiation and turbulent fluxes is therefore very nearly equal to the surface
temperature. The steady-state material entropy budget of the ocean may
therefore be written [346, 14],

$\frac{1}{A_{O}}\int_{\partial\Omega_{O}}\left\langle\frac{F_{\text{rad}}+F_{LH}+F_{SH}}{T_{s}}\right\rangle\,dA=\langle\dot{S}_{i}^{\text{mat}}\rangle,$
(55)

where $F_{\text{rad}}$ is the upward radiant energy flux, $F_{SH}$ and
$F_{LH}$ are the sensible and latent heat fluxes from the ocean to the
atmosphere, the integral is over the ocean surface $A_{O}$, and the angle
brackets refer to a time mean. For simplicity, we have neglected the
geothermal heat flux into the ocean and we have assumed that the difference
between the effective temperature of precipitation and runoff and the
effective temperature of evaporation is negligible; Bannon and Najjar [14]
show that making these approximations has a relatively minor effect on the
ocean’s entropy budget.

Building on the work of Tailleux [346], Bannon and Najjar [14] derived an
observational estimate of the left-hand side of (55), finding a rate of
entropy export $-\langle\dot{S}_{e}\rangle=1.7$ mW m-2 K-1 averaged over the
ocean surface area. This estimate is somewhat higher than the material entropy
production rate of the ocean of 1.2-1.4 W m-2 K-1 per unit area of the ocean
surface found in the modeling study of Pascale et al. [262]. However, it
should be noted that the model’s thermodynamic formulation did not include
frictional heating, and so this was left out of the estimate of irreversible
entropy production and may influence the result. Recall that the total
material entropy production of the climate system is of the order 35-60 mW m-2
K-1 averaged over the Earth’s surface; irreversible entropy production by the
ocean is a very small fraction of this total, implying that the atmosphere
accounts for the bulk of the irreversble entropy production in the climate
system.

Bannon and Najjar [14] also derived an independent estimate of the material
entropy production in the ocean using estimates of the thermal diffusivity and
temperature structure of the global ocean. According to this estimate,
$\langle\dot{S}_{i}^{\text{heat}}\rangle\approx 0.86$ mW m-2 K-1 and
$\langle\dot{S}_{i}^{\text{fric}}\rangle\approx 0.64$ mW m-2 K-1 with small
contributions from salt diffusion and ice melt. Given the difficulty in
measuring diffusivities in the ocean, the resultant estimate of
$\langle\dot{S}_{i}^{\text{mat}}\rangle$ is in rather remarkable agreement
with the estimate of the entropy export $-\langle\dot{S}_{e}\rangle$ given
above.

The ocean, unlike the atmosphere, is forced both thermodynamically, through
surface heat and freshwater fluxes (so-called buoyancy fluxes), and
mechanically, through the work done on the ocean by the winds and
tides121212The ocean also performs work on the atmosphere. but this a
negligible term in the atmosphere’s mechanical energy and entropy budgets..
Together, these two forcings produce an overturning circulation that spans the
depth of the ocean and governs the exchange of heat and chemical species
between the atmosphere and ocean on timescales from hundreds to thousands of
years [e.g., 39]. An application of the oceanic entropy budget is in
understanding the role of each type of forcing in determining the ocean’s deep
overturning circulation.

It is well known that the circulation produced in a fluid heated and cooled at
the same geometric level is substantially weaker than if the heating occurs
below the cooling Sandström [318]131313See Kuhlbrodt [167] for an English
translation.. Mechanically-induced turbulent mixing may therefore play an
important role in amplifying the ocean’s overturning circulation by allowing
the surface buoyancy flux from the atmosphere to penetrate a finite depth into
the ocean, thereby allowing the ocean to be heated and cooled at different
levels. Indeed, Munk and Wunsch [236] and Wunsch and Ferrari [377] argued that
mechanically-induced mixing is critical to the observed ocean stratification,
and that mechanical forcing plays a dominant role in governing the ocean’s
overturning circulation. But more recent work notes that buoyancy forcing is
an important source of available potential energy for the global ocean [e.g.,
132, 344, 345, see also section VIII.1.1], and numerical evidence suggests
that the ocean’s overturning circulation responds both to changes in buoyancy
fluxes and changes in the atmospheric wind field [e.g., 231].

A key question is how buoyancy fluxes act to alter the kinetic energy
generation rate of the ocean. The estimate of the entropy source owing to
frictional dissipation $\langle\dot{S}_{i}^{\text{fric}}\rangle$ provided by
Bannon and Najjar [14] is based on the assumption that the frictional
dissipation rate is equal to the work input by the winds and tides. To the
extent that this budget is closed, it is therefore consistent with the notion
that buoyancy forcing results in no net increase in kinetic energy generation
of the oceanic circulation. We note, however, that the estimates of entropy
production given in Bannon and Najjar [14] are necessarily crude due to the
lack of detailed observations, and they are not sufficiently accurate to
strongly constrain the kinetic energy dissipation rate. Moreover, as pointed
out by Hughes et al. [132], buoyancy forcing may facilitate a release of
kinetic energy at large scales even if does not provide a net increase in the
total kinetic energy generation rate of the ocean. Constraining the magnitude
of such energy transfers between scales using either the entropy or mechanical
energy budgets remains an observational challenge.

##  IV The entropy budget of atmospheric convection

In the field of meteorology, convection refers to fluid motions that transport
heat in the vertical direction. This is primarily accomplished through clouds
and their associated circulations, from shallow boundary-layer clouds over the
subtropical ocean, to explosive continental convection that spans the depth of
the troposphere and can potentially produce lightning, hail and other severe
weather. Despite its ubiquity in the atmosphere, our fundamental understanding
of atmospheric convection and its interaction with planetary-scale flows
remains limited. Basic questions such as what physical factors determine cloud
updraft velocities are still not completely resolved. A key reason for this is
the importance of moist processes, including evaporation, latent heating, and
precipitation, in atmospheric convection. The complex interaction of moist
processes with atmospheric fluid dynamics presents a challenging theoretical
problem.

In this section, we describe how analysis of the second law of thermodynamics
has provided a range of insights into the dynamics of moist convection. We
begin by introducing a common idealized conceptual and modeling framework for
studying moist convection known as radiative-convective equilibrium (section
IV.1), before reviewing theories of moist convective updraft velocities
developed based on analysis of the second law (section IV.2). Finally, we
introduce some new results concerning the effect of the “organization” of
moist convection on its mechanical efficiency with the aim of motivating
further work in this area (section IV.3).

###  IV.1 Radiative-convective equilibrium (RCE)

Radiative-convective equilibrium (RCE) describes a hypothetical state in which
a surface of infinite extent, usually assumed to consist of liquid water, is
held at a fixed temperature, and the atmosphere above is allowed to cool under
the influence of radiation. The cooling destabilizes the atmosphere,
eventually leading to convection. “Equilibrium” is achieved when the
convective heat flux from the surface balances the integrated radiative
cooling rate [303]. Note that, while we retain the term RCE for consistency
with the corresponding literature, it is somewhat of a misnomer, as the state
of RCE is far from thermodynamic equilibrium. Rather, RCE is a non-
equilibrium, statistically steady, but turbulent, state, involving the
continuous irreversible production of entropy. RCE is similar to the canonical
fluid mechanics problem of Rayleigh-Bénard convection between two plates, but
the upper plate is absent and replaced by bulk cooling of the fluid.

RCE provides a starting point for thinking about vertical heat transport in an
atmosphere without horizontal variations. The first studies to calculate RCE
solutions used it as a model for the tropical-mean or global-mean climate
[205, 206]. While recent studies have constructed analytical approximations
for mean temperature and humidity profiles in RCE [310], solutions to the full
turbulent cloud field are only accessible through numerical models. But with
increased availability of computational resources, RCE has become a popular
numerical and theoretical framework for studying the dynamics of moist
convection [e.g., 117, 24, 374, 329, 354], and in particular, for developing
and testing theories of moist convective updraft velocities [e.g., 303, 269,
328, 322, 330]. Recent work has also considered the problem of rotating RCE,
in which the atmosphere is assumed to exist on a planet with a finite rotation
rate [e.g., 240, 153]. In this section, we will consider only nonrotating RCE,
but the rotating case will be relevant to the discussion of tropical cyclones
in section V.

A disadvantage of RCE is that it does not exist on Earth, and so there are no
direct observations with which to compare numerical solutions. Our discussion
in this section will therefore remain theoretical and modeling based, but we
will discuss estimates of the entropy budget of Earth’s atmosphere in section
VI.

Figure 3: Schematic of (a) dry and (c) moist radiative-convective equilibrium.
Horizontal black lines denote the tropopause in each case and vertical arrows
show the mean sensible (gray) and latent (black) heat fluxes into the
atmosphere in W m-2. Panel (b) shows mean temperature profiles in the dry
(gray) and moist (black) cases, and temperature of a parcel initialized with
the mean temperature and humidity of the lowest model level of the moist
simulation and lifted adiabatically while assuming all condensate is
immediately removed from the parcel by precipitation (dotted). Mean quantities
are calculated as horizontal averages over the domain and over days 90-100 in
each simulation.

####  IV.1.1 Dry radiative-convective equilibrium

While our primary interest is in understanding moist convection, it is
instructive to begin by considering the simpler case of “dry” convection, in
which the phase change of water plays no role in the dynamics, latent heating
is absent, and the enthalpy transport from the surface is entirely composed of
sensible heat fluxes. We further simplify the problem by assuming that the
radiative cooling rate may be expressed as a function of the temperature,

$\dot{q}_{\text{rad}}=-\frac{c_{pd}(T-T_{0})}{\tau},$ (56)

where $c_{pd}$ is the specific heat capacity of dry air, $T_{0}=200$ K, and
$\tau=40$ days [269].

To demonstrate the RCE state, we ran a simulation of the above configuration
using a “cloud-permitting” model of the atmosphere. Cloud-permitting models
are numerical models with grid spacings of $\mathcal{O}$(1 km), allowing them
to explicitly represent convective clouds on the model grid. Though more
highly resolved than global climate models, cloud-permitting models remain too
coarse to properly _resolve_ all but the largest cloud systems, and they must
rely on parameterizations for turbulence and other subgrid-scale processes. We
discuss the challenges in modeling the second law using cloud-permitting
models further in section VII.1.

Our RCE simulation was run with the Bryan Cloud Model [CM1 version 13; 28]
with the lower boundary fixed at a temperature of 301.5 K and without
moisture. The domain was $200\times 200$ km2 horizontally, with periodic
boundary conditions in both $x$ and $y$ dimensions, and with a horizontal grid
spacing of 500 m and 64 vertical levels. The model was run for 100 days, and
statistics were accumulated over days 90-100 representing a statistically
steady state. Further details of the model configuration are as specified in
Singh and O’Gorman [328].

A schematic of the dry RCE simulation is presented in Fig. 3a. The radiative
cooling (56) produces a troposphere with a depth of roughly 10 km in which
vigorous convection occurs and the domain-mean temperature decreases with
height at a rate close to the dry-adiabatic lapse rate141414This is the rate