The global atmospheric circulation in moist isentropic coordinates. Part II:

Entropy transport and stratification

Olivier Pauluis*

Courant Institute of Mathematical Sciences, New York University, New York, New York

Arnaud Czaja

Department of Physics, Imperial College, London, United Kingdom

Robert Korty

Department of Atmospheric Sciences, Texas A&M University, College Station, Texas

Submitted to Journal of Climate

August 5, 2008

*Corresponding author address: Courant Institute of Mathematical Sciences, New York University,

Warren Weaver Hall, 251 Mercer St., New York, NY 10012-1185.

1

Abstract

By averaging the circulation on isentropic surfaces, part of the eddy transport can be

incorporated into the global mean circulation. However, the entropy of moist air cannot be

uniquely defined, and isentropic surfaces depends strongly on the choices made in this defi-

nition. A general expresion for the entropy of a parcel of moist air is derived. It is shown to

be partially undetermined due to the presence of integrations constants. Two particular def-

initions for the entropy are discussed in greater detail here: the moist entropy and the liquid

water entropy. Both definitions correspond to the thermodynamic entropy, match Clausius’

formulation of the Second Law, and are conserved for reversible adiabatic transformation.

The atmospheric circulation transports both moist entropy and liquid water entropy from

low to high latitudes. However, the transport of liquid water entropy is larger at low latitudes,

while the atmosphere transports more moist entropy in the midlatitudes. The difference be-

tween the two entropy transports is proportional to the water transported by the circulation.

Finally, the effective stratification, defined as the ratio of the entropy transport to the mass

transport, is estimated. It is found that the stratifications for liquid water entropy and moist

entropy are similar in the midlatitudes, which is interpreted here as indicating that the sub-

tropical warm, moist air feeding the midlatitudes stormtracks is close to being convectively

unstable.

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1. Introduction

The atmospheric circulation is dominated in the midlatitudes by turbulent eddies on the synoptic

scales. Because of the turbulent nature of the flow, any description of a ’mean’ circulation depends

on the averaging procedure. The Eulerian-mean circulation is obtained by averaging the merid-

ional velocity on either surfaces of constant height or constant pressure. This leads to the classic

description of the three-cell structures, with the Hadley cell in the equatorial regions, Ferrel cell

in midlatitudes, and a thermally direct polar cell at higher latitudes (Peixoto and Oort 1992). In

contrast, the circulation obtained by averaging the flow on isentropic surfaces exhibits a single cell

structure from the Equator to the poles (Townsend and Johnson 1985; Johnson 1989; Held and

Schneider 1999).

The synoptic eddies that dominate flow in midlatitudes have a typical life-time on the order of

a few days. When an air parcel is moved in such an eddy, its pressure and altitude of an air parcel

can change significantly. In contrast, the parcel’s entropy remains relatively constant as these

motions can be approximately viewed as adiabatic. Indeed, the typical radiative cooling rate in the

atmosphere is on the order of 1K per day, while vertical variations of potential temperature are on

the order of 40K. Taken together, these observations imply that the time-scale for the isentropic

circulation is on the order of 40 days. For this reason, it is usually argued that averaging the

circulation on isentropic surfaces better captures the average particle trajectories than the Eulerian-

mean circulation.

A key issue with any analysis of the circulation in isentropic coordinates arises from the fact

that isentropic surfaces in a moist atmosphere are not uniquely defined. Pauluis et al. (2008b,a)

3

compare the zonal-mean global circulation on dry and moist isentropes. Dry isentropes here refer

to surfaces of constant potential temperature, and moist isentropes to surfaces of constant equiva-

lent potential temperature. They show that the mass transport on moist isentropes is approximately

twice as large in the midlatitudes than the mass transport on dry isentropes. The additional mass

transport on moist isentropes is attributed to the presence of a low-level poleward flow of warm,

moist air in the subtropics that ascends to the upper troposphere within the stormtracks and ac-

counts for roughly half of the poleward mass flow in the midlatitudes.

This paper discusses in greater detail why entropy cannot be uniquely defined and how the

different definition of entropy are related. We focus primarily on two specific definitions of en-

tropy: the moist entropy and the liquid water entropy. Both definitions are fully consistent with

the Second Law of thermodynamics. They both correspond to Clausius’ thermodynamic entropy

and are conserved in reversible adiabatic transformations. The key difference lies in that each uses

a different reference state for water. In the case of the liquid water entropy, the reference state for

water is water vapor at a given reference pressure and temperature. In contrast, for moist entropy,

the reference state is that of liquid water at the same reference temperature. The difference between

the liquid and moist entropy of a parcel of moist air is equal to the total water content multiplied

by a constant.

The liquid water entropy and the moist entropy are affected in very different manners by evapo-

ration or precipitation. Moist entropy increases when water vapor evaporates at the Earth’s surface,

but it is approximately conserved by condensation and precipitation. In contrast, liquid water en-

tropy is not significantly modified by evaporation at the Earth’s surface, but increases rapidly when

4

precipitation occurs. Overall, the behavior of liquid water entropy is very similar to that of potential

temperature, while moist entropy is closely related to the equivalent potential temperature.

Section 2 reviews the definition of entropy for a parcel of cloudy air. In section 3, we compare

the tendency equations for the different definitions of entropy. Section 4 discusses the entropy

transport and the gross stratification in the circulation. We show that the larger mass transport

on moist isentropes also corresponds to enhanced transport of moist entropy associated with the

poleward water vapor transport in midlatitudes. Interestingly, we find that in midlatitudes the

gross dry and moist stratifications are comparable. This situation stands in stark contrast to the

tropics where the potential temperature stratification is significantly larger than the gross moist

stratification. The Appendix discusses alternative definitions of entropy, and show that any valid

expression for the thermodynamic entropy can be written as a linear combination of the liquid

water entropy and the moist entropy.

2. Entropy of moist air

Cloudy air can be treated as a mixture of dry air, water vapor, and condensed water (which can be

either liquid water or ice). The specific humidity qv, the specific humidity for condensate water qc,

and the specific humidity for total water qT are respectively the mass of water vapor, condensed

water, and total water per unit mass of cloudy air. The mass of dry air per unit mass of cloudy air

is thus 1− qT . The specific entropy of moist air S is equal to the weighted average of the specific

5

entropies of dry air sd, water vapor sv and condensed water sc:

S = (1− qT )sd + qvsv + qcsc. (1)

The specific entropies of the individual components are

sd = Cpd lnT

T0

−Rd lnpdp0

+ sd0 (2a)

sv = Cpv lnT

T0

−Rv lne

e0

+ sv0 (2b)

sc = Cc lnT

T0

+ sc0. (2c)

Here, Cpd, Cpv and Cl are the specific heat capacities at constant pressure of dry air, water vapor

and liquid water, Rd and Rv are the ideal gas constants for dry air and water vapor, T is the air

temperature, and pd and e are the partial pressure of dry air and water vapor. These expressions are

obtained by assuming that both dry air and water vapor behave as an ideal gas and by neglecting

the specific volume of condensed water. The definitions of the specific entropies (2a-2c) include

several integration constants: the reference temperature T0; reference partial pressures for dry air

p0 and water vapor e0; and the reference entropies for dry air sd0, water vapor sv0, and condensed

water sc0. The expressions for the specific entropies here are such that they are equal to their refer-

ence values when the component is at the reference temperature and partial pressure. A common

practice is to choose some typical atmospheric values for the reference temperature T0 and partial

pressures pd0, and e0, and to set the reference value for the specific entropies of dry air and of either

liquid water or water vapor to 0.

The choice of these integration constants is not entirely arbitrary. Indeed, condensed water

and water vapor are not distinc chemical components but are different phases of the same fluid.

6

To properly account for phase transitions, the difference between the entropies of water vapor and

liquid water is equal to to the enthalpy difference divided by the absolute temperature when water

vapor and liquid water are in thermodynamic equilibrium:

sv − sc =LvT. (3)

For phase transition, thermodynamic equilibrium occurs when the partial pressure of water vapor is

equal to the saturation vapor pressure es(T ), which is itself function of the temperature. Expression

(3) is only valid at saturation e = es(T ): an additional term appears on the right-hand side of (3)

when the partial pressure of water vapor is not equal to the saturation value.

For (3) to be true, the difference between the reference state for water vapor and liquid water

must verify

δs0 = sv0 − sc0 =Lv(T0)

T−Rv ln

e0

es(T0). (4)

Furthermore if the reference partial pressure e0 is chosen to be the saturation vapor pressure at the

reference temperature e0 = es(T0), then it is not possible to impose the conditions that sv0 and sc0

be zero simultaneously.

Equation (4) ensures that the entropy of phase transition matches (3) at the reference temper-

ature and pressure. It does not guarantee that (3) is true at all temperatures. This latter property

can be enforced by choosing a consistent definition for the saturation vapor pressure es(T ). To

do this, one can insert the definition of the specific entropies (2b-2c) in (3), which then yields an

expression for the saturation vapor pressure as function of temperature. This yields

lnes(T )

es(T0)=LvRv

(1

T0

− 1

T

). (5)

7

This procedure is equivalent to ensuring that the saturation vapor pressure obeys the Clausius-

Clapeyron relationship.

The entropy difference between liquid water and water vapor is dominated by the contribution

of the latent heat Lv

T. From a practical point of view, it is useful to choose a reference state such

that either the specific entropy of water vapor or of liquid water is small in comparison to the

contribution of the latent heat; i.e., either:

|sc| <<LvT0

or |sv| <<LvT0

.

The first alternative can be achieved by setting the integration constant sc0 to zero (as well as sd0).

In this case, the entropy of liquid water and water vapor are:

sv = Cl lnT

T0

+LvT−Rv ln

e

es(T )(6a)

sc = Cl lnT

T0

. (6b)

Using this definition for the entropy of the cloudy air yields the moist entropy Sm:

Sm =((1− qT )Cpd + qTCl) lnT

T0

− (1− qT )Rd lnpdp0

+ qvLvT− qvRv ln

e

es(T0).

(7)

The moist entropy is itself related to the equivalent potential temperature by

Sm = ((1− qT )Cpd + qTCl) lnθeT0

.

The second alternative is to set the integration constant sv0 to 0, so that the specific entropies

8

for water vapor and liquid water are:

sv = Cpv lnT

T0

−Rv lne

e0

(8a)

sc = Cpv lnT

T0

− LvT−Rv ln

es(T )

e0

. (8b)

Using these expressions for the entropy of cloudy air, one obtains the liquid water entropy Sl:

Sl =((1− qT )Cpd + qTCpv) lnT

T0

− (1− qT )Rd lnpdp0

− qvRv lne

e0

+ qlLvT− qlRv ln

es(T )

e0

.

(9)

The liquid water entropy is related to the liquid water potential temperature by

Sl = ((1− qT )Cpd + qTCpv) lnθlT0

.

These expressions for the liquid water entropy and moist entropy differ solely in the choice of

the reference state for the water. In particular, the difference between these two entropies is:

Sm − Sl = qT

(Lv(T0)

T0

−Rv lne0

es(T0)

)= qT δs0. (10)

The liquid water entropy and the moist entropy are two independent state variables. As the state

of a parcel of cloudy air under the assumption of thermodynamic equilibrium can be uniquely

determined by the knowledge of three state variables, it is thus possible to determine all the ther-

modynamic properties of a parcel of moist air from its liquid water entropy, moist entropy, and a

third state variable such as pressure.

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3. Entropy and the second law of thermodynamics

The second law of thermodynamics requires that the entropy of an air parcel is conserved for

closed, reversible adiabatic transformations. An adiabatic process is one where no external energy

source or sink is involved; a reversible process is one in which there is no entropy production;

and a closed process is one in which there is no mass exchanged (i.e., one where the total specific

humidity qT remains constant). In addition, for closed processes in the presence of either energy

exchange or irreversibility, the second law of thermodynamics can be written:

ρdS

dt=Q

T+ Sirr (11)

whereQ is the external heating rate per unit volume, and Sirr is the irreversible entropy production

by the atmospheric process, which is always positive. The entropy per unit of mass S can be

the moist entropy Sm, the liquid water entropy Sl or the entropy obtained for any other choice

of the integration constant discussed in section 2. Both the liquid water entropy Sl or the moist

entropy Sm are perfectly valid definitions of the entropy and can be used in the second law of

thermodynamics. As long as the total water is conserved along the parcel trajectories, the rates of

change for the liquid water entropy and moist entropy are identical and given by the second law of

thermodynamics (11).

In the atmosphere, many processes, such as including precipitation, evaporation, and diffusion,

involve exchange of water between air parcels or with the Earth’s surface. Such processes in which

water in either phase is added or removed from a parcel are here referred to as open transforma-

tions. If qv is the rate of change of specific humidity owing to diffusion, and qc is the rate of change

10

of the specific humidity for condensed water owing to precipitation, the entropy tendency becomes

ρdS

dt=Q

T+ Sirr + (sv − sd)ρqv + (sc − sd)ρqc. (12)

The tendencies qv and qc do not include phase transitions per se, as these correspond to internal

transformations and are already accounted for in (11). Equation (12) applies to both the liquid

water entropy Sl and moist entropy Sm and differs only by the value for the entropy of condensed

water sc and water vapor sv. One must use the definitions (6a-6b) for the moist entropy, and

(8a-8b) for the liquid water entropy. Because moist entropy and liquid water entropy rely on

different reference values for the specific entropies of water vapor and condensed water, they are

also affected differently by evaporation and precipitation.

In the definition of moist entropy, the reference state is chosen such that the specific entropy of

the condensed water is closed to that of moist air, but the entropy of water vapor is much larger due

to the term Lv

Tin (6a). In the entropy tendency equation, the term proportional to Lv

Tis much larger

than any of the other ones in sv− sd and in sc− sv. Hence, if we retain only the terms proportional

to the latent heat in the entropy tendency, we get

ρdSmdt≈ Q

T+ Sirr +

LvTρqv. (13)

Evaporation acts as a source of moist entropy that is approximately proportional to the surface

latent heat flux. By contrast, precipitation has little effect on moist entropy.

In the case of the liquid water entropy, the reference state for water is such that the entropy of

water vapor is close to that of dry air, while that of liquid water is much smaller due to the latent

heat contribution−Lv

Tin (8b). If we retain only the contribution from the terms proportional to the

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latent heat in (sv − sd) and (sc − sd), we can approximate the liquid water entropy tendency by:

ρdSldt≈ Q

T+ Sirr −

LvTρqc. (14)

While evaporation has little effect on the liquid water entropy, precipitation acts as a large source

of the liquid water entropy. The entropy increase is approximately proportional to the net latent

heat released by the condensation. (It should be stressed, however, that the entropy increase occurs

when condensed water is removed and not when water vapor condenses.) Thus, in contrast to the

moist entropy, the liquid water entropy cannot be viewed as almost conserved during precipitating

convection.

4. Entropy transport and gross stratification

The global atmospheric circulation described by Pauluis et al. (2008b,a) is closely tied to an

equator-to-pole entropy transport. The atmosphere receives more energy than it emits in equa-

torial latitudes, while the emission of infrared radiation exceeds the energy inputs at high latitude.

The second law of thermodynamics implies that these external energy sources and sinks also act as

entropy sources and sinks, with the net entropy input equal to the external energy source divided

by the temperature at which it occurs. In addition, irreversible processes within the atmosphere

produce a certain amount of entropy, as discussed by Pauluis and Held (2002). The ratio of the

internal production to the external entropy sources scales as ∆TT∼ 0.1, where ∆T is the tempera-

ture difference between the energy sources and sinks, and T is a typical atmospheric temperature.

To first approximation, the internal entropy production can be neglected when compared to the

12

external sources and sinks. The global atmospheric circulation acts to transport entropy from the

source regions at low latitudes to entropy sinks at higher latitudes, and is in many ways similar to

the global energy transport discussed in Czaja and Marshall (2006).

The northward transports of liquid water entropy FSland moist entropy FSm are given by:

FSl(φ) =

∫ psurf

0

∫ 2π

0

vSla cosφdλdp

g, (15a)

FSm(φ) =

∫ psurf

0

∫ 2π

0

vSla cosφdλdp

g, (15b)

with φ the latitude, psurf is the surface pressure, a is the Earth radius, λ is the longitude, and g

is the gravitationall acceleration. Figure 1 shows the time averaged entropy fluxes. Qualitatively,

the atmosphere transports both the liquid water entropy and the moist entropy poleward, which is

broadly consistent the Earth’s radiative forcing. They differ quantitatively: the liquid water entropy

flux is larger than the moist entropy flux in low latitudes, and smaller in midlatitudes. As discussed

above, both the liquid water entropy and moist entropy are valid expression for the entropy of moist

air and obey the second law of thermodynamics. In particular, their rates of change are identical

for closed transformations (i.e., transformations in which neither water vapor nor condensed water

is added to the air parcels). In the absence of precipitation and evaporation, the transports of

liquid water entropy and of moist entropy would be identical. However, removing water vapor or

condensed water has a different impact on moist entropy than and liquid water entropy, as indicated

by the tendency equation (13) and (14). On the one hand, precipitation acts as a large source of

liquid water entropy, but this has little effect on moist entropy. The intense precipitation in the

equatorial regions shows up as a large divergence in the liquid water entropy flux. The enhanced

precipitation over the stormtracks is also a source of liquid water entropy and its effect is most

13

visible in the Northern Hemisphere during winter. On the other hand, evaporation has little effect

on the liquid water entropy, but it is a source of moist entropy. The moist entropy source at the

Earth’s surface is proportional to the total energy flux, i.e., the sum of the sensible and latent heat

flux. (By contrast, the liquid water entropy flux is solely attributable to the sensible heat flux.) The

moist entropy flux is divergent at low latitudes, where the atmosphere receives more energy than

it loses, and it is divergent at high latitudes, where the atmospheric cooling exceeds the surface

energy flux.

As detailed in Section 2, the difference between moist entropy and liquid water entropy is

proportional to the liquid water content:

Sm − Sl = δs0qT =

[Lv(T0)

T−Rv ln

e0

es(T0)

]qT ,

where Lv0 is the latent heat of vaporization at the reference temperature T0. This implies that

the difference between the moist and liquid water entropy fluxes is proportional to the total water

transport:

FSm − FSl= δs0Fq =

[Lv(T0)

T−Rv ln

e0

es(T0)

]Fq, (16)

with Fq =∫ ∫

ρvqTa cosφdλdz the meridional transport of total water. The difference between

the liquid water entropy and moist entropy fluxes in Figure 1 is proportional to the total water

transport. The Hadley circulation transports water vapor toward the equatorial regions, in the op-

posite direction to the global energy transport. Hence, at low latitudes, the liquid water entropy

flux is larger than the moist entropy flux. In contrast, in middle latitudes baroclinic eddies trans-

port water vapor poleward and the latent heat transport accounts for roughly half the total energy

transport. Similarly, the water contribution accounts for about half the moist entropy transport in

14

midltatitudes.

The relationship between mass and entropy transport can be assessed by defining an effective

stratification as the ratio of the entropy flux to the total mass transport:

∆Sl =FSl

∆Ψθl

, (17a)

∆Sm =FSm

∆Ψθe

. (17b)

Here, ∆Ψθland ∆Ψθe are the mass transport on dry and moist isentropes, as defined in Pauluis et al.

(2008a). The effective stratification measures the ability of the circulation to transport entropy and

are shown in Figure 2. To understand the relationship between these two stratifications, one can

first consider the vertical variations of moist entropy:

∂Sm∂z

=∂Sl∂z

+ δs0∂qT∂z

. (18)

Liquid water entropy usually increases with height, but the water content decreases sharply. These

two effects partially compensate each other. As a result, moist entropy often exhibits a mid-

tropospheric minimum in the tropical regions. In addition, one would expect the effective stratifi-

cation for moist entropy to be smaller than the gross stratification for liquid water entropy.

Figure 2 shows that indeed the effective stratification for liquid water entropy in the equatorial

regions is on the order of 100 JK−1kg−1 and is significantly larger than the gross stratification for

the moist entropy, which is often less than 50 JK−1kg−1. The Hadley circulation is characterized

by a low-level equatorward and an upper-level outflow. It also transports water vapor toward

the Equator, in the opposite direction to the transport of liquid water entropy, resulting in a small

transport of moist entropy. Due to the low effective stratification for moist entropy, the atmospheric

15

circulation is very inefficient at exporting (moist) entropy or energy out of the equatorial regions.

Czaja and Marshall (2006) argue that because the atmosphere is inefficient at transporting energy

or entropy out of the equatorial regions, most of the energy transport must take place in the ocean.

In midlatitudes, however, the two stratifications are very close to each other. The argument

for a lower moist stratification is based on the vertical variations of moisture (18) and assumes

implicitly that the circulation is composed of an upper level flow in one direction and a return flow

in the opposite direction at a lower level. However, the moist stratification can be higher than the

liquid water stratification if a flow of moist air is balanced by a return flow of drier air at the same

temperature and pressure. In this case, the transport of moist entropy can be much larger than the

transport of liquid water entropy. The effective stratifications in midlatitudes do not correspond to

either a direct circulation with upper level poleward flow and low level return flow, which would

imply a low moist stratification, or to a turbulent transport by horizontal eddies, which implies a

larger moist stratification. Rather, that the two effective stratifications are comparable indicates

that the midlatitudes flow is a mixture between direct and eddy-dominated circulations.

To explain why the effective stratifications for moist entropy and liquid water entropy are al-

most equal in the midlatitudes, we consider an idealized flow described in Figure 3, which shows

an idealized mass transport on isentropic filaments, similar to Figures 3 and 4 of Pauluis et al.

(2008a). We assume that the total circulation on dry isentropes is ∆Ψθl= Mθ. On dry isen-

tropes, the poleward flux takes place in the upper troposphere at θl = θup, and the return flow

occurs in the lower troposphere at θl = θlow.The additional mass transport on moist isentropes

∆M = ∆Ψθe −∆Ψθlmust take place in the lower troposphere on a dry isentrope with θl ∼ θlow,

16

with the additional return flow taking place on the same isentrope. In the lower troposphere, the

equivalent potential temperature of a parcel is always larger than its potential temperature. It must

also be smaller than the equivalent potential temperature in the upper troposphere:

θlow ≤ θe ≤ θup

Indeed, if it this were not the case, the parcel would be convectively unstable. This means that

on Figure 3, the entire circulation must take place within the triangle delineated by the diagonal

θl = θe, the vertical line θl = θlow and the horizontal line θe = θup.

For the stratifications of liquid water entropy and moist entropy to be equal, the differences

of potential temperature and equivalent temperature between the poleward and equatorward flows

must also be equal. When the circulation on dry isentropes takes place in the upper and lower

troposphere—i.e., if θup and θlow correspond respectively to the potential temperature near the

tropopause and near the surface—the dry and moist stratification will be equal only if the additional

near-surface poleward flow takes place at high value of θe similar to that of the upper troposphere,

i.e., θe ≈ θup. In addition, the equivalent potential in the return flow must be close to the potenital

temperature in the lower troposphere, i.e., θl ≈ θlow, which implies that the return flow has a

low humidity content. Hence, all of the equatorward flux Mθ takes place near the surface, at

low potential and equivalent potential temperature θl ≈ θe ≈ θlow, while the return flow has a

high potential temperature, with θe ≈ θup, but is split between an upper tropospheric branch with

θl ≈ θup and a near-surface branch with θl ≈ θlow. The fact that the dry and moist stratifications are

almost equal in midlatitudes is tied to the fact that the near-surface poleward flow is composed of

air parcels with high equivalent potential temperatures that rise into the upper troposphere within

17

the stormtracks.

5. Conclusion

The entropy of a parcel of moist air is not uniquely defined. In two companion papers (Pauluis

et al. 2008b,a), we show that the global circulation on isentropic surfaces depends significantly on

the definition for the entropy. In particular, the mass transport on moist isentropes is approximately

twice as large as that on dry isentropes. In the present paper, we discuss why the entropy cannot be

uniquely defined and the implications that this fact has on both the second law of thermodynamics,

the entropy transport by the atmospheric circulation and its gross stratification.

The specific entropy of water can be defined only up to an additive constant. As the amount

of water present in the atmosphere varies, the entropy of moist air can be known only up to this

integration constant multiplied by the total water concentration. Here, we focus our discussion

on two particular choices for the entropy of moist air: the moist entropy Sm and the liquid water

entropy Sl. These particular choices are directly related to the equivalent potential temperature θe

and the liquid water potential temperature θl in use in atmospheric sciences.

Both the liquid water entropy and moist entropy correspond to the thermodynamics entropy.

In particular, both are conserved for closed, reversible and adiabatic transformations. Their rate of

change is identical as long as the total water content of an air parcel is conserved. However, they

are affected differently by the addition or removal of water. Evaporation greatly increases moist

entropy, but has little effect on the liquid water entropy. Conversely, precipitation increases the liq-

18

uid water entropy but leaves moist entropy almost unchanged. Hence, while both entropies exhibit

the same sources and sinks from radiation, surface sensible heat flux and internal production, the

entropy sources associated with the hydrological cycle differ markedly between moist entropy and

liquid water entropy.

The global circulation transports both moist entropy and liquid water entropy from the equator

to the poles. The two entropy transports are not equal, with circulation transporting more liquid

water entropy in the equatorial regions, and more moist entropy in the midlatitudes. The difference

between the two entropy transport is proportional to the transports of water vapor by the circulation.

This situation is analogous to the relationship between the transports of dry static energy, moist

static energy and latent heat. The relative importance of the transport of liquid water entropy and

moist entropy can thus be related to the fact that the atmosphere transports water vapor equatorward

in the subtropics and poleward in the midlatitudes.

We also define an effective stratification as the ratio between the total entropy transport to the

total mass transport in isentropic coordinates. One of the more surprising aspects of the midlati-

tude circulation lies in that the effective stratifications for moist entropy and liquid water entropy

- defined as the ratio of the total entropy transport to the total mass transport - are approximately

equal, with the exception of the Northern Hemisphere summer. This finding is in apparent con-

tradiction with the fact that the vertical variations of equivalent potential temperature are always

smaller then the vertical variation of potential temperature. It does however indicates that horizon-

tal fluctuations of equivalent potential temperature are comparable to the vertical fluctuations of

potential temperature. This also requires the equivalent potential temperature in the moist branch

19

of the circulation described in Pauluis et al. (2008b,a) to be characteristics of air parcels that are

almost convectively unstable and ready to ascend in the upper troposphere.

Appendix: Alternative choice for the entropy and the global cir-

culation

Liquid water entropy and moist entropy are not the only possible expression for entropy. Any

set of integration constants in Section 2 yields a valid definition of the entropy of moist air, say

Sa. However, such entropy would differs from either Sm and Sl by a constant multiplied by qT .

Because of (10), it can be expressed as a linear combination of Sm and Sl:

Sa = (1− a)Sm + aSl,

with a and arbitrary constant. The streamfunction on surface of constant value of Sa would be

different from the streamfunction on either dry and moist isentropes. This raises the question of

what aspect of the isentropic circulation does not depend on the integration constants.

The answer to this question lies in that rather than isentropic surfaces, one should think in terms

of isentropic filaments defined as regions where both entropies Sm and Sl are constant. As Sm and

Sl are constant along such isentropic filament, so are all possible definition of the entropy of moist

air. Contrarily to the concept of isentropic surface, these isentropic filaments do not depend on any

specific choices in the definition of entropy. The mass transport on isentropic filament introduced

20

in Pauluis et al. (2008b,a) is defined as

Mml(Sm0, Sl0, φ) =1

τ

∫ τ

0

∫ 2π

0

∫ psfc

0

[vδ(Sm − Sm0)δ(Sl − Sl0)a cos θ]dp

gdλdt (19)

where τ is the time period over which the data are averaged, φ is the latitude, g the gravitational

acceleration, and psfc the surface pressure. This is the same definition as in Pauluis et al. (2008a),

except that we use here the entropies Sm and Sl rather than the potential temperature θe and θl.

If the isentropic filaments are defined using different definition of entropy, for example Sa =

(1 − a)Sm + aSl and Sb = (1 − b)Sm + bSl, with a 6= b, the mass transport on the isentropic

filament remain unchanged, and we have

Mab(Sa, Sb, φ)dSadSb = Mml(Sm, Sl, φ)dSmdSl

→Mab(Sa, Sb, φ) = |a− b|Mml(Sm, Sl, φ).

Contrarily to the streamfunction, the mass transport on isentropic filament is a robust feature of

the circulation that does not depend on any specific definitions for the entropy of moist air. In

Pauluis et al. (2008b,a), it is used to identify the thermodynamic properties of the poleward and

equatorward moving air, and separate between the lower and upper branch of the circulations in

the midlatitudes.

Acknowledgement

This work was supported by the NSF grants No. ATM-0545047 and PHY05-51164.

21

References

Czaja, A. and J. Marshall, 2006: The Partitioning of Poleward Heat Transport between the Atmo-

sphere and Ocean. 63, 1498–1511.

Held, I. M. and T. Schneider, 1999: The surface branch of the zonally averaged mass transport

circulation in the troposphere. J. Atmos. Sci., 56, 1688–1697.

Johnson, D. R., 1989: The forcing and maintenance of global monsoonal circulations: An isen-

tropic analysis. Advances in Geophysics, 31, 43–304.

Pauluis, O., A. Czaja, and R. Korty, 2008a: The global atmospheric circulation in moist isentropic

coordinates. Part I: Mass transport. submitted to J. Climate..

— 2008b: The global atmospheric circulation on moist isentropes. To be published in Science..

Pauluis, O. and I. M. Held, 2002: Entropy budget of an atmosphere in radiative-convective equi-

librium. Part I: Maximum work and frictional dissipation . J. Atmos. Sci., 59, 125–139.

Peixoto, J. P. and A. H. Oort, 1992: Physics of Climate. AIP Press, 520 pp.

Townsend, R. D. and D. R. Johnson, 1985: A diagnostic study of the isentropic zonally averaged

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22

List of Figures

1 Meridional entropy flux: moist entropy flux (solid line), liquid water entropy flux

(dashed line), and contribution of the total water transport to the moist entropy flux

(dash-dotted line). The upper panel shows the annual mean flux, the middle panel

the mean for June-July-August and the low panel the mean for December-January-

February. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Effective stratification in moist entropy isentropy (solid lines) and liquid water en-

tropy (dashed lines) for the annual mean (upper panel), June-July-August (middle

panel), and December-January-February (lower panel) circulation. . . . . . . . . . 25

3 Schematic representation of the distribution of the mass transport on isentropic

filaments in the midlatitudes. See text for details. . . . . . . . . . . . . . . . . . . 26

23

Figure 1: Meridional entropy flux: moist entropy flux (solid line), liquid water entropy flux (dashed

line), and contribution of the total water transport to the moist entropy flux (dash-dotted line). The

upper panel shows the annual mean flux, the middle panel the mean for June-July-August and the

low panel the mean for December-January-February.

24

Figure 2: Effective stratification in moist entropy isentropy (solid lines) and liquid water entropy

(dashed lines) for the annual mean (upper panel), June-July-August (middle panel), and December-

January-February (lower panel) circulation.

25

Figure 3: Schematic representation of the distribution of the mass transport on isentropic filaments

in the midlatitudes. See text for details.

26