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Atmos. Chem. Phys., 13, 1039–1056, 2013

www.atmos-chem-phys.net/13/1039/2013/

doi:10.5194/acp-13-1039-2013

© Author(s) 2013. CC Attribution 3.0 License.

Atmospheric

Chemistry

and Physics

Where do winds come from? A new theory on how water vapor

condensation inﬂuences atmospheric pressure and dynamics

A. M. Makarieva1,2, V. G. Gorshkov1,2, D. Sheil3,4,5, A. D. Nobre6,7, and B.-L. Li2

1Theoretical Physics Division, Petersburg Nuclear Physics Institute, 188300, Gatchina, St. Petersburg, Russia

2XIEG-UCR International Center for Arid Land Ecology, University of California, Riverside, CA 92521, USA

3School of Environment, Science and Engineering, Southern Cross University, P.O. Box 157, Lismore, NSW 2480, Australia

4Institute of Tropical Forest Conservation, Mbarara University of Science and Technology, Kabale, Uganda

5Center for International Forestry Research, P.O. Box 0113 BOCBD, Bogor 16000, Indonesia

6Centro de Ciˆ

encia do Sistema Terrestre INPE, S˜

ao Jos´

e dos Campos SP 12227-010, Brazil

7Instituto Nacional de Pesquisas da Amazˆ

onia, Manaus AM 69060-001, Brazil

Correspondence to: A. M. Makarieva (ammakarieva@gmail.com) and D. Sheil (douglassheil@itfc.org)

Received: 5 August 2010 – Published in Atmos. Chem. Phys. Discuss.: 15 October 2010

Revised: 29 April 2011 – Accepted: 3 December 2012 – Published: 25 January 2013

Abstract. Phase transitions of atmospheric water play a

ubiquitous role in the Earth’s climate system, but their direct

impact on atmospheric dynamics has escaped wide attention.

Here we examine and advance a theory as to how conden-

sation inﬂuences atmospheric pressure through the mass re-

moval of water from the gas phase with a simultaneous ac-

count of the latent heat release. Building from fundamental

physical principles we show that condensation is associated

with a decline in air pressure in the lower atmosphere. This

decline occurs up to a certain height, which ranges from 3

to 4km for surface temperatures from 10 to 30◦C. We then

estimate the horizontal pressure differences associated with

water vapor condensation and ﬁnd that these are comparable

in magnitude with the pressure differences driving observed

circulation patterns. The water vapor delivered to the atmo-

sphere via evaporation represents a store of potential energy

available to accelerate air and thus drive winds. Our estimates

suggest that the global mean power at which this potential

energy is released by condensation is around one per cent of

the global solar power – this is similar to the known station-

ary dissipative power of general atmospheric circulation. We

conclude that condensation and evaporation merit attention

as major, if previously overlooked, factors in driving atmo-

spheric dynamics.

1 Introduction

Phase transitions of water are among the major physical pro-

cesses that shape the Earth’s climate. But such processes

have not been well characterized. This shortfall is recognized

both as a challenge and a prospect for advancing our un-

derstanding of atmospheric circulation (e.g., Lorenz, 1983;

Schneider, 2006). In A History of Prevailing Ideas about the

General Circulation of the Atmosphere Lorenz (1983) wrote:

“We may therefore pause and ask ourselves

whether this step will be completed in the man-

ner of the last three. Will the next decade see new

observational data that will disprove our present

ideas? It would be difﬁcult to show that this can-

not happen.

Our current knowledge of the role of the various

phases of water in the atmosphere is somewhat in-

complete: eventually it must encompass both ther-

modynamic and radiational effects. We do not fully

understand the interconnections between the trop-

ics, which contain the bulk of water, and the re-

maining latitudes .. . Perhaps near the end of the

20th century we shall suddenly discover that we

are beginning the ﬁfth step.”

Deluc (1812, p. 176) mentioned that conversion of water

vapor to rain creates a kind of “airfree” space that may cause

Published by Copernicus Publications on behalf of the European Geosciences Union.

1040 A. M. Makarieva et al.: Condensation-induced atmospheric dynamics

wind gusts. Lorenz (1967, Eq. 86), as well as several other

authors after him (Trenberth et al., 1987; Trenberth, 1991;

Gu and Qian, 1991; Ooyama, 2001; Schubert et al., 2001;

Wacker and Herbert, 2003; Wacker et al., 2006), recognized

that local pressure is reduced by precipitation and increased

by evaporation. Qiu et al. (1993) noted that “the mass de-

pletion due to precipitation tends to reduce surface pressure,

which may in turn enhance the low-level moisture conver-

gence and give a positive feedback to precipitation”. Van den

Dool and Saha (1993) labeled the effect as a physically dis-

tinct “water vapor forcing”. Lackmann and Yablonsky (2004)

investigated the precipitation mass sink for the case of Hur-

ricane Lili (2002) and made an important observation that

“the amount of atmospheric mass removed via precipitation

exceeded that needed to explain the model sea level pressure

decrease”.

Although the pressure changes associated with evapora-

tion and condensation have received some attention, the in-

vestigations have been limited: the effects remain poorly

characterized in both theory and observations. Previous in-

vestigations focused on temporal pressure changes not spa-

tial gradients. Even some very basic relationships remain

subject to confusion. For example, there is doubt as to

whether condensation leads to reduced or to increased atmo-

spheric pressure (P¨

oschl, 2009, p. S12436). Opining that the

status of the issue in the meteorological literature is unclear,

Haynes (2009) suggested that to justify the claim of pres-

sure reduction one would need to show that “the standard

approaches (e.g., set out in textbooks such as “Thermody-

namics of Atmospheres and Oceans” by Curry and Webster,

1999) imply a drop in pressure associated with condensa-

tion”.

Here we aim to clarify and describe, building from basic

and established physical principles, the pressure changes as-

sociated with condensation. We will argue that atmospheric

water vapor represents a store of potential energy that be-

comes available to accelerate air as the vapor condenses.

Evaporation, driven by the sun, continuously replenishes the

store of this energy in the atmosphere.

The paper is structured as follows. In Sect. 2 we analyze

the process of adiabatic condensation to show that it is al-

ways accompanied by a local decrease of air pressure. In

Sect. 3 we evaluate the effects of water mass removal and

lapse rate change upon condensation in a vertical air col-

umn in approximate hydrostatic equilibrium. In Sect. 4 we

estimate the horizontal pressure gradients induced by water

vapor condensation to show that these are sufﬁcient enough

to drive the major circulation patterns on Earth (Sect. 4.1).

We examine why the key relationships have remained un-

known until recently (Sects. 4.2 and 4.3). We evaluate the

mean global power available from condensation to drive the

general atmospheric circulation (Sect. 4.4). Finally, we dis-

cuss the interplay between evaporation and condensation and

the essentially different implications of their physics for at-

mospheric dynamics (Sect. 4.5). In the concluding section

we discuss the importance of condensation as compared to

differential heating as the major driver of atmospheric cir-

culation. Our theoretical investigations strongly suggest that

the phase transitions of water vapor play a far greater role in

driving atmospheric dynamics than is currently recognized.

2 Condensation in a local air volume

2.1 Adiabatic condensation

We will ﬁrst show that adiabatic condensation is always ac-

companied by a decrease of air pressure in the local volume

where it occurs. The ﬁrst law of thermodynamics for moist

air saturated with water vapor reads (e.g., Gill, 1982)

dQ=cVdT+pdV+Ldγ, (1)

γ≡pv

p1,dγ

γ=dpv

pv−dp

p.(2)

Here pvis partial pressure of saturated water vapor, pis

air pressure, Tis absolute temperature, Q(Jmol−1) is mo-

lar heat, V(m3mol−1) is molar volume, L≈45kJ mol−1

is the molar heat of vaporization, cV=5

2Ris molar heat

capacity of air at constant volume (Jmol−1K−1), R=

8.3Jmol−1K−1is the universal gas constant. The small

value of γ < 0.1 under terrestrial conditions allows us to ne-

glect the inﬂuence made by the heat capacity of liquid water

in Eq. (1).

The partial pressure of saturated water vapor obeys the

Clausius-Clapeyron equation:

dpv

pv=ξdT

T, ξ ≡L

RT ,(3)

pv(T ) =pv0 exp(ξ0−ξ ), (4)

where pv0 and ξ0correspond to some reference tempera-

ture T0. Below we use T0=303 K and pv0 =42hPa (Bolton,

1980) and neglect the dependence of Lon temperature.

We will also use the ideal gas law as the equation of state

for atmospheric air:

pV =RT , (5)

dp

p+dV

V=dT

T.(6)

Using Eq. (6) the ﬁrst two terms in Eq. (1) can be written

in the following form

cVdT+pdV=RT

µdT

T−µdp

p,

(7)

µ≡R

cp

=2

7=0.29, cp=cV+R.

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A. M. Makarieva et al.: Condensation-induced atmospheric dynamics 1041

Writing dγin Eq. (1) with use of Eqs. (2) and (3) as

dγ

γ=ξdT

T−dp

p(8)

and using the deﬁnition of ξ(Eq. 3) we arrive at the following

form for the ﬁrst law of thermodynamics (Eq. 1):

dQ=RT

µdT

T(1+µγ ξ 2)−µdp

p(1+γ ξ ).(9)

In adiabatic processes dQ=0, and the expression in

braces in Eq. (9) turns to zero, which implies:

dT

T=dp

pϕ(γ ,ξ ), ϕ(γ ,ξ ) ≡µ1+γ ξ

1+µγ ξ 2≡ϕ. (10)

Note that µ,γand ξare all dimensionless; γand ξare vari-

ables and µis a constant, ϕ(0,0)=µ. This is a general de-

pendence of temperature on pressure in an adiabatic atmo-

spheric process that involves phasetransitions of water vapor

(evaporation or condensation), i.e. change of γ. At the same

time γitself is a function of temperature as determined by

Eq. (8):

dγ

γ=ξdT

T−dp

p=dT

T

ξϕ −1

ϕ=(ξϕ −1)dp

p.(11)

One can see from Eqs. (10) and (11) that the adiabatic phase

transitions of water vapor are fully described by the relative

change of either pressure dp/p or temperature dT / T . For the

temperature range relevant for Earth we have ξ≡L/RT ≈18

so that

ξ µ −1≈4.3.(12)

Noting that µ,γ,ξare all positive, from Eqs. (10), (11)

and (12) we obtain

ξϕ −1=ξ µ 1+γ ξ

1+µγ ξ 2−1=ξµ −1

1+µγ ξ 2>0.(13)

Condensation of water vapor corresponds to a decrease of

γ, dγ < 0. It follows unambiguously from Eqs. (11) and (13)

that if dγis negative, then dpand dTare negative too. This

proves that water vapor condensation in any adiabatic pro-

cess is necessarily accompanied by reduced air pressure.

2.2 Adiabatic condensation cannot occur at constant

volume

Our previous result refutes the proposition that adiabatic con-

densation can lead to a pressure rise due to the release of

latent heat (P¨

oschl, 2009, p. S12436). Next, we show that

while such a pressure rise was implied by calculations as-

suming adiabatic condensation at constant volume, in fact

such a process is prohibited by the laws of thermodynamics

and thus cannot occur.

Using Eqs. (6), (10) and (8) we can express the relative

change of molar volume dV /V in terms of dγ /γ :

dV

V= − 1−ϕ

ϕξ −1dγ

γ.(14)

Putting dV=0 in Eq. (14) we obtain

(1−ϕ)dγ

(ξϕ −1)γ =0.(15)

The denominator in Eq. (15) is greater than zero, see

Eqs. (12) and (13). In the numerator we note from the def-

inition of ϕ(Eq. 10) that 1−ϕ=2γ

7+2γ ξ 2h5

2γ+ξ(ξ−1)i. The

expression in square brackets lacks real roots:

5

2γ+ξ2−ξ=0, ξ =1

2 1±is10−γ

γ!, γ ≤1.(16)

In consequence, Eq. (15) has a single solution dγ=0. This

proves that condensation cannot occur adiabatically at con-

stant volume.

2.3 Non-adiabatic condensation

To conclude this section, we show that for any process where

entropy increases, dS=dQ/T >0, water vapor condensation

(dγ <0) is accompanied by drop of air pressure (i.e., dp<0).

We write the ﬁrst law of thermodynamics Eqs. (9) and (11)

as

dS

R

µ

1+µγ ξ 2=dT

T−ϕdp

p,dT

T=1

ξdγ

γ+dp

p.(17)

Excluding dT /T from Eq. (17) we obtain

dp

p(ξϕ −1)=dγ

γ−ξµ

1+µγ ξ

dS

R.(18)

The term in round brackets in Eq. (18) is positive, see

Eq. (13), the multiplier at dSis also positive. Therefore, when

condensation occurs, i.e., when dγ /γ <0, and dS>0, the left-

hand side of Eq. (18) is negative. This means that dp/p<0,

i.e., air pressure decreases.

Condensation can be accompanied by a pressure increase

only if dS<0. This requires that work is performed on the gas

such as occurs if it is isothermally compressed. (We note too,

that if pure saturated water vapor is isothermally compressed

condensation occurs, but the Clausius-Clapeyron equation

(Eq. 3) shows that the vapor pressure remains unchanged be-

ing purely a function of temperature.)

3 Adiabatic condensation in the gravitational ﬁeld

3.1 Difference in the effects of mass removal and

temperature change on gas pressure in

hydrostatic equilibrium

We have shown that adiabatic condensation in any local vol-

ume is always accompanied by a drop of air pressure. We

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1042 A. M. Makarieva et al.: Condensation-induced atmospheric dynamics

will now explore the consequences of condensation for the

vertical air column.

Most circulation patterns on Earth are much wider than

they are high, with the ratio height/length being in the order

of 10−2for hurricanes and down to 10−3and below in larger

regional circulations. As a consequence of mass balance, ver-

tical velocity is smaller than horizontal velocities by a similar

ratio. Accordingly, the local pressure imbalances and result-

ing atmospheric accelerations are much smaller in the verti-

cal orientation than in the horizontal plane, the result being

an atmosphere in approximate hydrostatic equilibrium (Gill,

1982). Air pressure then conforms to the equation

−dp

dz=ρg, p(0)≡ps=g

∞

Z

0

ρ(z)dz. (19)

Applying the ideal gas equation of state (Eq. 5) we have from

Eq. (19)

dp

dz= − p

h, h ≡R T

Mg .(20)

This solves as

p(z) =psexp

−

z

Z

0

dz0

h(z0)

.(21)

Here Mis air molar mass (kg mol−1), which, as well as tem-

perature T (z), in the general case also depends on z.

The value of ps(Eq. 19), air pressure at the surface, ap-

pears as the constant of integration after Eq. (19) is integrated

over z. It is equal to the weight of air molecules in the at-

mospheric column. It is important to bear in mind that ps

does not depend on temperature, but only on the amount of

gas molecules in the column. It follows from this observa-

tion that any reduction of gas content in the column reduces

surface pressure.

Latent heat released when water condenses means that

more energy has to be removed from a given volume of satu-

rated air for a similar decline in temperature when compared

to dry air. This is why the moist adiabatic lapse rate is smaller

than the dry adiabatic lapse rate. Accordingly, given one and

the same surface temperature Tsin a column with rising air,

the temperature at some distance above the surface will be

on average higher in a column of moist saturated air than in

a dry one.

However, this does not mean that at a given height air pres-

sure in the warmer column is greater than air pressure in the

colder column (cf. Meesters et al., 2009; Makarieva and Gor-

shkov, 2009c), because air pressure p(z) (Eq. 21) depends

on two parameters, temperature T (z) and surface air pres-

sure (i.e., the total amount of air in the column). If the total

amount of air in the warmer column is smaller than in the

colder column, air pressure in the surface layer will be lower

in the warmer column despite its higher temperature.

In the following we estimate the cumulative effect of gas

content and lapse rate changes upon condensation.

3.2 Moist adiabatic temperature proﬁle

Relative water vapor content (Eq. 2) and temperature Tde-

pend on height z. From Eqs. (10), (11) and (20) we have

−dT

dz≡0=ϕT

h, ϕ ≡µ1+γ ξ

1+γ µξ 2,(22)

−1

γ

dγ

dz=ξϕ −1

h≡ξ µ −1

1+µγ ξ 21

h.(23)

Equation (22) represents the well-known formula for a moist

adiabatic gradient as given in Glickman (2000) for small

γ < 0.1. At γ=0 we have ϕ(γ , ξ) =µand 0d=Mdg/cp=

9.8Kkm−1, which is the dry adiabatic lapse rate that is in-

dependent of height z,Md=29gmol−1. For moist saturated

air the change of temperature Tand relative partial pressure

γof water vapor with height is determined by the system of

differential equations (Eqs. 22, 23).

Differentiating both parts of the Clapeyron-Clausius equa-

tion (Eq. 3) over zwe have, see Eq. (22):

dpv

dz= − pv

hv, hv≡RT 2

L0 =T

ξ0 =h

ξ ϕ ,

(24)

pv(z) =pvs exp

−

z

Z

0

dz0

hv

, pvs ≡pv(0).

The value of hvrepresents a fundamental scale height for the

vertical distribution of saturated water vapor. At Ts=300 K

this height hvis approximately 4.5km.

Differentiating both parts of Eq. (2) over zwith use of

Eqs. (20) and (24) and noticing that hv=h/(ξϕ) we have

−1

γ

dγ

dz=1

pv

dpv

dz−1

p

dp

dz=1

hv−1

h≡1

hγ

,

hγ≡hvh

h−hv.(25)

This equation is equivalent to Eq. (23) when Eqs. (22)

and (24) are taken into account. Height hγrepresents the

vertical scale of the condensation process. Height scales hv

(Eq. 24) and hγ(Eq. 25) depend on ϕ(γ , ξ ) (Eq. 22) and,

consequently, on γ. At Ts=300K height hγ≈9 km, in

close proximity to the water vapor scale height described by

Mapes (2001).

3.3 Pressure proﬁles in moist versus dry air columns

We start by considering two static vertically isothermal at-

mospheric columns of unit area, A and B, with temperature

T (z) =Tsindependent of height. Column A contains moist

air with water vapor saturated at the surface, column B con-

tains dry air only. Surface temperatures and surface pressures

Atmos. Chem. Phys., 13, 1039–1056, 2013 www.atmos-chem-phys.net/13/1039/2013/

A. M. Makarieva et al.: Condensation-induced atmospheric dynamics 1043

,km ,K ,km

h

n

HzL∆p

s

HT

s

Lp

A

HzL-p

B

HzL, hPa, hPa,km

hHzL,km

h

v

HzL,km T

s303 K

Ts293 K

Ts283 K

02468101214

z

0

2

4

6

8

10

12

14 HaL

260 270 280 290 300 310

Ts

0

10

20

30

40 HbL

02468

z

-40

-30

-20

-10

0

10

20

30

40 HcL

Fig. 1. (a) Scale height of saturated water vapor hv(z) (Eq. 24), hydrostatic scale height of water vapor hn(z) (Eq. 26), and scale height of

moist air h(z) (Eq. 20) in the column with moist adiabatic lapse rate (Eq. 22) for three values of surface temperature Ts;(b) condensation-

induced drop of air pressure at the surface (Eq. 27) as dependent on surface temperature Ts;(c) pressure difference versus altitude zbetween

atmospheric columns A and B with moist and dry adiabatic lapse rates, Eqs. (30) and (31), respectively, for three values of surface temperature

Ts. Height zcat which pA(zc)−pB(zc)=0 is 2.9, 3.4 and 4.1 km for 283, 293 and 303K, respectively. Due to condensation, at altitudes below

zcthe air pressure is lower in column A despite it being warmer than column B.

in the two columns are equal. In static air Eq. (19) is exact

and applies to each component of the gas mixture as well

as to the mixture as a whole. At equal surface pressures, the

total air mass and air weight are therefore the same in both

columns. Water vapor in column A is saturated at the surface

(i.e., at z=0) but non-saturated above it (at z > 0). The sat-

urated partial pressure of water vapor at the surface pv(Ts)

(Eq. 4) is determined by surface temperature and, as it is in

hydrostatic equilibrium, equals the weight of water vapor in

the static column.

We now introduce a non-zero lapse rate to both columns:

the moist adiabatic 0(Eq. 22) to column A and the dry adia-

batic 0din column B. (Now the columns cannot be static: the

adiabatic lapse rates are maintained by the adiabatically as-

cending air.) Due to the decrease of temperature with height,

some water vapor in column A undergoes condensation. Wa-

ter vapor becomes saturated everywhere in the column (i.e.,

at z≥0), with pressure pv(z) following Eq. (24) and density

ρv=pvMv/(RT )≡pv/(ghn)following

ρv(z) =ρv(Ts)hns

hn(z) exp

−

z

Z

0

dz0

hv(z0)

,

(26)

ρv(Ts)≡pv(Ts)

ghn(Ts), hn≡RT (z)

Mvg, T (z) =Ts−0z.

Here hn(z) is the scale height of the hydrostatic distribution

of water vapor in the isothermal atmosphere with T (z)=Ts.

The change in pressure δpsin column A due to water va-

por condensation is equal to the difference between the initial

weight of water vapor pv(Ts)and the weight of saturated wa-

ter vapor:

δps=pv(Ts)−g

∞

Z

0

ρv(z)dz≤pv(Ts)−ρv(Ts)ghv(Ts)

=pv(Ts)1−hvs

hns=pv(Ts)1−MvgTs

L0s.(27)

The inequality in Eq. (27) represents a conservative estimate

of δpsdue to the approximation hv(z)=hv(Ts)made while

integrating ρv(z) (26). As far as hv(z) declines with height

more rapidly than hn(z), Fig. 1a, the exact magnitude of

this integral is smaller, while the value of δpsis larger. The

physical meaning of estimate (Eq. 27) consists in the fact

that the drop of temperature with height compresses the wa-

ter vapor distribution hns/hvs-fold compared to the hydro-

static distribution (Makarieva and Gorshkov, 2007, 2009a;

Gorshkov et al., 2012).

The value of δps(Eq. 27) was calculated as the differ-

ence between the weight per unit surface area of vapor in

the isothermal hydrostatic column and the weight of water

vapor that condensed when a moist adiabatic lapse rate was

applied. This derivation can also be understood in terms of

the variable conventionally called the adiabatic liquid water

content (e.g., Curry and Webster, 1999, Eq. 6.41). We can

represent the total mixing ratio of moisture (by mass) as qt≡

qv+ql=(ρv+ρl)/ρ, where ρvis the mass of vapor and ρlis

the mass of liquid water per unit air volume; qt1. The total

adiabatic liquid water content in the column equals the inte-

gral of qlρover zat constant qt,qlρ=qtρ−qvρ=qtρ−ρv.

The value of δps(Eq. 27) is equal to this integral (mass per

unit area) multiplied by the gravitational acceleration (giving

weight per unit area):

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1044 A. M. Makarieva et al.: Condensation-induced atmospheric dynamics

δps=g

∞

Z

0

qlρdz=g

∞

Z

0

qtρdz−

∞

Z

0

ρvdz

.(28)

The ﬁrst integral in braces gives the mass of vapor in the

considered atmospheric column if water vapor were a non-

condensable gas, qv=qt=const. This term is analagous to the

ﬁrst term, pv(Ts), in the right-hand side of Eq. (27), where

a static isothermal column was considered. The second term

is identical to the second term, gR∞

0ρvdz, in Eq. (27).

Using the deﬁnition of hv(Ts)(Eq. 24), hn(Ts)(Eq. 26)

and recalling that Mv/Md=0.62 and pv(Ts)=γsps, see

Eq. (4), we obtain the following expression for the δpses-

timate (Eq. 27), Fig. 1b:

δps

ps≈γs 1−0.62 1+γsµξ2

s

µξs+γsµξ2

s!.(29)

Note that δps/psis proportional to γsand increases exponen-

tially with the rise of temperature.

After an approximate hydrostatic equilibrium is estab-

lished, the vertical pressure proﬁles for columns A and B

become, cf. Eq. (21):

pA(z)=ps1−δps

psexp

−

z

Z

0

dz0

hA(z0)

, hA≡RT

Mg ;(30)

pB(z) =psexp

−

z

Z

0

dz0

hB(z0)

, hB≡RTd

Mdg.(31)

Here M(z)=Md(1−γ )+Mvγ;γ≡pv(z)/pA(z) and T (z)

obey Eqs. (22) and (23), Td(z)≡Ts−0dz.

In Fig. 1c the difference pA(z)−pB(z) is plotted for three

surface temperatures, Ts=10◦C, 20 ◦C and 30 ◦C. In all three

cases condensation has resulted in a lower air pressure in col-

umn A compared to column B everywhere below zc≈2.9, 3.4

and 4.1km, respectively. It is only above that height that the

difference in lapse rates makes pressure in the moist column

higher than in the dry column.

3.4 Comparing forces due to condensation and

buoyancy

Fig. 1c describes the relative contributions of latent heat re-

lease and the condensation vapor sink to the horizontal pres-

sure differences. This result can also be illustrated by com-

paring the vertical forces associated with phase transitions of

water vapor.

The buoyant force acting per unit moist air volume can be

written as

fB=ρpgρ

ρp−1=

ρpgT (z)

Td(z)

1

1−(Mv/Md)γ (z) −1.

Here ρpis the density of the air moist air parcel that ascends

in the environment with density ρ. (When fBis taken per unit

mass by dividing by density ρpand integrated over z, one

obtains the convective available potential energy (CAPE)

(Glickman, 2000), which represents work performed by the

buoyant force on the rising air parcel. As work of the buoy-

ant force on the air parcel that is descending dry adiabatically

is usually negative, total energy available for a buoyancy-

induced circulation can be close to zero even at large positive

CAPE (Gorshkov et al., 2012).)

Figure 2a shows the buoyant force acting on an air vol-

ume from column A that rises moist adiabatically in the

dry adiabatic environment of column B: ρp=pB(z)M/RT ,

ρ=pB(z)Md/RTd. Here pBis given by Eq. (31), Tdfol-

lows the dry adiabatic proﬁle Td(z) =Ts−0dz, where 0d=

9.8Kkm−1, while temperature T (z) and molar mass M(z) =

Md[1−(Mv/Md)γ (z)]of the rising air satisfy Eqs. (22)–

(23). The positive value of the buoyant force at the surface

is due to the lower molar density of the moist versus dry air.

The same ﬁgure shows the condensation pressure gradi-

ent force that acts in the column where moist saturated air

ascends adiabatically:

fC=pv

p

∂p

∂z −∂pv

∂z = −p∂γ

∂z .

Here pand γconform to Eqs. (22)–(23).

As Fig. 2a shows, the two forces have different spatial lo-

calization. The condensation force has a maximum in the

lower atmosphere where the amount of vapor is maximized.

The buoyant force grows with height following the accumu-

lating difference between the moist adiabatic and dry adia-

batic temperatures. At Ts=300km at z=8km the differ-

ence theoretically amounts to over 50K.

The buoyant force estimated in Fig. 2 represents a the-

oretical upper limit that assumes no heat transfer between

the ascending air and its environment. Maximum tempera-

ture differences observed in the horizontal direction in real

weather systems are typically much smaller than 50K at

any height. Indeed, even in the warm-core tropical storms

– i.e., in intense precipitation events – the horizontal temper-

ature difference between the core and the external environ-

ment rarely exceeds a few degrees Kelvin (e.g., Knaff et al.,

2000). In Fig. 2b the same forces are plotted, but for the

buoyant force estimated for an environment having a mean

tropospheric lapse rate of 6.5Kkm−1(rather than the dry

adiabatic lapse rate 9.8Kkm−1). As Fig. 2b shows, the

magnitude of the buoyant force drops rapidly with dimin-

ishing differences in temperature. Convective available po-

tential energy associated with the buoyant force shown in

Fig. 2a is R8km

0(fB/ρp)dz=8.5×103Jkg−1. This ﬁgure is

several times higher than the typical values calculated from

the lapse rate soundings of the atmospheric column below

12 km height in the most intense convection events like

thunderstorms and tornadoes (e.g., Thompson et al., 2003;

Kis and Straka, 2010). Furthermore, in a recent comparison

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A. M. Makarieva et al.: Condensation-induced atmospheric dynamics 1045

012345678

z ,km

0

1

2

3

4

5

6

7

8

9

10

mbarkm

HaL

012345678

z ,km

0

1

2

3

4

5

6

7

8

9

10

mbarkm

HbLTs303 K

Ts293 K

Ts283 K

Fig. 2. Condensation force fC(solid curves) and buoyant force fB(dashed) acting at height zon a moist air volume ascending in an

environment with dry adiabatic lapse rate 9.8K km−1(a) and mean tropospheric lapse rate 6.5 Kkm−1(b) for different values of surface

temperature Ts.

of nocturnal and diurnal tornadoes (Kis and Straka, 2010) it

was found that signiﬁcant tornadoes can form at both large

and very small CAPE values, pointing to the importance of

different mechanisms for the generation of intense circula-

tion systems.

The key message from Fig. 2 is that the condensation

force remains comparable in magnitude to the buoyant force

even when the latter is allowed (for the sake of argument) to

take unrealistically high values. Furthermore the condensa-

tion force dominates in the lower atmosphere with the buoy-

ant force more pronounced only in the upper atmosphere.

We note that both the buoyant and condensation forces are

vertically directed. But we emphasise that their action in the

atmosphere is manifested in the formation of horizontal pres-

sure gradients. This follows from the independent stipulation

that the atmosphere is vertically in approximate hydrostatic

equilibrium. In Sect. 4 we derive the horizontal pressure gra-

dients associated with the condensation force.

4 Relevance of the condensation-induced pressure

changes for atmospheric processes

4.1 Horizontal pressure gradients associated with vapor

condensation

We have shown that condensation of water vapor produces

a drop of air pressure in the lower atmosphere up to an al-

titude of a few kilometers, Fig. 1c, in a moist saturated hy-

drostatically adjusted column. In the dynamic atmospheric

context the vapor condenses and latent heat is released dur-

ing the ascent of moist air. The vertical displacement of air

is inevitably accompanied by its horizontal displacement.

This translates much of the condensation-induced pressure

difference to a horizontal pressure gradient. Indeed, as the

upwelling air loses its water vapor, the surface pressure di-

minishes via hydrostatic adjustment producing a surface gra-

dient of total air pressure between the areas of ascent and

descent. The resulting horizontal pressure gradient is propor-

tional to the the ratio of vertical to horizontal velocity w/u

(Makarieva and Gorshkov, 2009b).

We will illustrate this point regarding the magnitude of the

resulting atmospheric pressure gradient for the case of a sta-

tionary ﬂow where the air moves horizontally along the x-

axis and vertically along the z-axis; there is no dependence

of the ﬂow on the y coordinate. The stationary continuity

equation for the mixture of condensable (vapor) and non-

condensable (dry air) gases can be written as

∂(Ndu)

∂x +∂(Ndw)

∂z =0;(32)

∂(Nvu)

∂x +∂(Nvw)

∂z =S;(33)

S≡w∂Nv

∂z −Nv

N

∂N

∂z =wN ∂γ

∂z , N =Nv+Nd.(34)

Here Ndand Nvare molar densities of dry air and satu-

rated water vapor, respectively; γ≡Nv/N, see Eq. (2), S

(Eq. 34) is the sink term describing the non-conservation of

the condensable component (water vapor). Saturated pres-

sure of water vapor depends on temperature alone. Assum-

ing that vapor is saturated at the isothermal surface we have

∂Nv/∂x=0, so Nvonly depends on z. (This condition ne-

cessitates either that there is an inﬂux of water vapor via

evaporation from the surface (if the circulation pattern is

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1046 A. M. Makarieva et al.: Condensation-induced atmospheric dynamics

immobile), or that the pressure ﬁeld moves as vapor is lo-

cally depleted. The second case occurs in compact circula-

tion patterns like hurricanes and tornadoes (Makarieva and

Gorshkov, 2011; Makarieva et al., 2011).) As the air ascends

with vertical velocity w, vapor molar density decreases due

to condensation and due to the expansion of the gas along

the vertical gradient of decreasing pressure. The latter ef-

fect equally inﬂuences all gases, both condensable and non-

condensable. Therefore, the volume-speciﬁc rate S(x, z) at

which vapor molecules are locally removed from the gaseous

phase is equal to w[∂Nv/∂z−(Nv/N)∂N/∂z], see Eqs. (1)

and (2). The second term describes the expansion of vapor at

a constant mixing ratio which would have occurred if vapor

were non-condensable as the other gases. (If vapor did not

condense, its density would decrease with height as a con-

stant proportion of the total molar density of moist air as

with any other atmospheric gas.) Further clariﬁcation of the

meaning of (Eq. 34) is provided in Sect. 4.2 below, and in

Appendix A which offers additional interpretation, see also

(Gorshkov et al., 2012).

The mass of dry air is conserved, Eq. (32). Using this fact,

Eq. (34) and ∂Nv/∂x=0 in Eq. (33) one can see that

N∂u

∂x +∂w

∂z +w∂N

∂z =0.(35)

Now expressing ∂N /∂ x=∂Nd/∂x+∂ Nv/∂ x from Eqs. (32)

and (33) with use of Eq. (35) we obtain

∂N

∂x =w

u∂Nv

∂z −Nv

N

∂N

∂z .(36)

Using the equation of state for moist air p=NRT and water

vapor pv=NvRT we obtain from Eqs. (36) and (25):

∂p

∂x =∂pv

∂z −pv

p

∂p

∂z w

u= − γ p

hγ

w

u.(37)

Here velocities wand urepresent vertical and horizontal

(along x-axis) velocities of the ascending air ﬂow, respec-

tively. Scale height hγis deﬁned in Eq. (25). A closely re-

lated formula for horizontal pressure gradient can be applied

to an axis-symmetric stationary ﬂow with ∂p/∂x replaced by

radial gradient ∂p/∂r (Makarieva and Gorshkov, 2009b).

Equation (37) shows that the difference between the scale

heights hvand h(Eq. 25) of the vertical pressure distribu-

tions for water vapor and moist air leads to the appearance

of a horizontal pressure gradient of moist air as a whole

(Makarieva and Gorshkov, 2007; Gorshkov et al., 2012).

This equation contains the ratio of vertical to horizontal ve-

locity. Estimating this ratio it is possible to evaluate, for

a given circulation, what sorts of horizontal pressure gradi-

ents are produced by condensation and whether these gradi-

ents are large enough to maintain the observed velocities via

the positive physical feedback described by Eq. (37).

For example, for Hadley cells at hγ=9km, γ=0.03 and

a typical ratio of w/u∼2×10−3(Rex, 1958) we obtain from

Eq. (37) a pressure gradient of 0.7Pakm−1. On a distance of

1500km such a gradient would correspond to a pressure dif-

ference of around 10hPa, which is close to the upper range

of the actually observed pressure differences in the region

(e.g., Murphree and Van den Dool, 1988, Fig. 1). Similar

pressure differences and gradients, also comparable in mag-

nitude to δps(Eq. 27) and ∂p/∂r are observed within cy-

clones, both tropical and extratropical, and persistent atmo-

spheric patterns in the low latitudes (Holland, 1980; Zhou

and Lau, 1998; Br¨

ummer et al., 2000; Nicholson, 2000; Sim-

monds et al., 2008). For example, the mean depth of Arctic

cyclones, 5hPa (Simmonds et al., 2008), is about ten times

smaller than the mean depth of a typical tropical cyclone

(Holland, 1980). This pattern agrees well with the Clausius-

Clapeyron dependence of δps, Fig. 1b, which would pre-

dict an 8 to 16-fold decrease with mean oceanic tempera-

ture dropping by 30–40 ◦C. The exact magnitude of the pres-

sure gradient and air velocities will depend on the horizon-

tal size of the circulation pattern, the magnitude of friction

and degree of the radial symmetry (Makarieva and Gorshkov,

2009a,b, 2011; Makarieva et al., 2011).

Our estimate of the horizontal pressure gradient in a

Hadley cell illustrates that our approach when coupled to

fundamental atmospheric parameters, yields horizontal pres-

sure gradients of magnitudes similar to those actually ob-

served in large-scale circulation patterns. If we had obtained

a much smaller magnitude from Eq. (34) we could conclude

that the impact of the vapor sink is negligible and cannot ex-

plain the observations. This did not happen. Rather the result

adds credibility to our proposal that the vapor sink is a major

cause of atmospheric pressure gradients.

Difﬁculties in the understanding of atmospheric circula-

tion relate to circumstances where uncertainty over the dy-

namics of water vapor play a role – even if the nature of that

role remains debatable. For example, modern global circula-

tion models do not satisfactorily account for the water cycle

of the Amazon River Basin, with the estimated moisture con-

vergence being half the actual amounts estimated from the

observed runoff values (Marengo, 2006). We note that cli-

mate science offers no quantitative theory of Hadley circula-

tion based on current theories and the effects of differential

heating alone (Held and Hou, 1980; Fang and Tung, 1999;

Schneider, 2006). Efforts to address this challenge are on-

going but progress is limited (e.g., Lindzen and Hou, 1988;

Robinson, 2006; Walker and Schneider, 2005, 2006). In one

recent review concerning theories of general circulation the

understanding of atmospheric moisture and its inﬂuences,

particularly, lack of relevant theoretical concepts, were iden-

tiﬁed as a persistent challenge (Schneider, 2006).

Furthermore, many climate researchers readily acknowl-

edge that the current incomplete understanding of the gen-

eral circulation precludes a theory-based analysis, from fun-

damental physical principles, of the role of latitudinal at-

mospheric mixing in stabilizing the Earth’s thermal regime

important – this is not a minor and thus neglected detail

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A. M. Makarieva et al.: Condensation-induced atmospheric dynamics 1047

but is central in debates concerning climate sensitivity (e.g.,

Lindzen and Choi, 2009; Trenberth et al., 2010). It would

seem to many that new ideas are needed. If these ideas were

obvious, and followed directly from current paradigms, they

would have already been identiﬁed and accepted – thus we

should not be surprised that the new ideas we all seek may

challenge conventional perspectives. We conclude that our

approaches are a promising new avenue for further examina-

tion.

4.2 Condensation rate and hydrostatic equilibrium

Let us dwell in greater detail on the physical meaning of

Eq. (34) that speciﬁes condensation rate in a unit volume.

The second term in brackets, (Nv/N)∂N/∂z, describes how

the molar density of vapor would change during adiabatic

ascent if the water vapor were non-condensable and there

would be no condensation in the column. This reference

term is needed to discriminate the density change caused by

condensation from the density change due to gravitational

expansion. As we presume that moist air as a whole is in

hydrostatic equilibrium, see Eq. (20), it is total molar den-

sity Nthat must be used as such a reference. Indeed, total

molar density remains in hydrostatic equilibrium in the ab-

sence of condensation as well as in its presence. In the limit

Nv→NEq. (34) gives a physically meaningful result, S=0.

Indeed, when atmosphere consists of water vapor only and

is in hydrostatic equilibrium, no condensation takes place.

Condensation occurs only when water vapor distribution is

non-equilibrium.

When condensation is absent, dry air is in hydrostatic equi-

librium. But when water vapor condenses and its distribution

is compressed several-fold compared to the hydrostatic dis-

tribution, the dry air must be “stretched” compared to its

hydrostatic distribution. Only in this case, when the non-

equilibrium deﬁcit of vapor in the upper atmosphere is com-

pensated by the non-equilibrium excess of dry air, the moist

air as a whole will remain in equilibrium. The distribution of

Ndis non-equilibrium and cannot be used instead of Nin the

reference term in Eq. (34).

The horizontal pressure gradient produced by condensa-

tion is therefore a direct consequence of hydrostatic adjust-

ment. The air expands upwards to compensate for vapor

deﬁcit, thus its pressure at the surface diminishes in the re-

gion of ascent. If no hydrostatic adjustment took place, the

dry air would remain in hydrostatic equilibrium (while moist

air as a whole would not). In this case dry air molar den-

sity Ndcould be used in the reference term in Eq. (34).

Putting Ndinstead of Nin Eq. (34), i.e., replacing Sby

Sd≡∂Nv/∂z−(Nv/Nd)∂Nd/∂z in Eq. (33), and performing

all the derivations in Sect. 4.1, one obtains ∂p/∂x=0. This

result is obvious: in the absence of hydrostatic adjustment,

the dry air distribution is not affected by condensation and

remains in equilibrium both in horizontal and vertical dimen-

sions. The non-equilibrium gradient of total air pressure re-

mains located in the vertical dimension and is not translated

onto horizontal dimension. Such a situation could take place

in an atmosphere that would be much higher than it is wide.

In the real atmosphere which is effectively very thin, most

part of the non-equilibrium pressure gradient is transferred to

the horizontal plane via rapid hydrostatic adjustment. Note

that Sd≡S/(1−γ ) and S≡Sd/(1+γd),γd≡Nv/Nd. The ex-

pressions for condensation rates in situations with or without

hydrostatic adjustment differ, respectively, by the absence or

presence of the multiplier 1/(1−γ ) in Eq. (34).

We emphasize that whether the hydrostatic adjustment

takes place or not, the disequilibrium gradient of total air

pressure persists, being located, respectively, either in the

horizontal or in the vertical dimension. Note that if S=Sd

then S≡Sd≡0, condensation is absent and atmospheric

pressure is in equilibrium in all directions (see Appendix A).

When asking for feedback on earlier versions of this text

several readers assumed that Eq. (34) for condensation rate is

an approximate form of the exact expression (Eq. 33). Here

we address this misunderstanding, see also Appendix A for

more details. Equation (33) represents a general continuity

(mass balance) equation for water vapor. It does not contain

any information about condensation – indeed, it is equally

valid for condensation S<0, evaporation S>0 or absence of

phase transitions altogether, S=0. Also, it is equally valid

for any dependence of Son spatial coordinates, velocities,

temperature, pressure or any other variables. In other words,

the continuity equation universally applies to all circulation

events. In the meantime, our task here is to study only those

circulation patterns that are induced by condensation asso-

ciated with adiabatic ascent. To do so, we need to specify

term Sin Eq. (33) so we can use this equation for the deter-

mination of condensation-induced pressure gradients. This

is done by means of Eq. (34), which says that: (1) in the

considered volume the only source of phase transitions is

condensation; (2) this condensation is caused by the adia-

batic ascent of moist saturated air (no condensation occurs

if the air moves horizontally because of isothermal surface)

and (3) that the moist saturated air is in hydrostatic equilib-

rium. We stress that none of these speciﬁc assumptions are

contained in the universal continuity equation (Eq. 33). (In

contrast to the generally applicable Eq. (33), Eq. (34) would

not be valid, for example, for the case of adiabatic descent,

or for a horizontal motion along a non-isothermal surface.)

We emphasize that S(Eq. 34) is based on speciﬁc physical

considerations, not on formal mathematical analogies.

4.3 Regarding previous oversight of the effect

For many readers a major barrier to acceptance of our propo-

sitions may be to understand how such a fundamental physi-

cal mechanism has been overlooked until now. Why has this

theory come to light only now in what is widely regarded as

a mature ﬁeld? We can offer a few thoughts based on our

readings and discussions with colleagues.

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1048 A. M. Makarieva et al.: Condensation-induced atmospheric dynamics

The condensation-induced pressure gradients that we have

been examining are associated with density gradients that

have been conventionally considered as minor and thus ig-

nored in the continuity equation (e.g., Sabato, 2008). For

example, a typical 1p=50 hPa pressure difference observed

along the horizontally isothermal surface between the outer

environment and the hurricane center (e.g., Holland, 1980)

is associated with a density difference of only around 5%.

This density difference can be safely neglected when esti-

mating the resulting air velocity ufrom the known pres-

sure differences 1p. Here the basic scale relation is given

by Bernoulli’s equation, ρu2/2=1p. The point is that a 5 %

change in ρdoes not signiﬁcantly impact the magnitude of

the estimated air velocity at a given 1p. But, as we have

shown in the previous section, for the determination of the

pressure gradient (Eq. 37) the density difference and gradi-

ent (Eq. 36) are key.

Considering the equation of state (Eq. 5) for the

horizontally isothermal surface we have p=Cρ, where

C≡RT /M =const. Irrespective of why the considered

pressure difference arises, from Bernoulli’s equation

we know that u2=21p/ρ=2C 1ρ/ρ,1ρ=ρ0−ρ. Thus,

if one puts 1ρ/ρ=1p/p equal to zero, no veloc-

ity forms and there is no circulation. Indeed, we

have u2=21p/ρ=2C 1ρ/ρ=2C(1ρ/ρ0)(1+1ρ/ρ0+. . .).

As one can see, discarding 1ρ compared to ρdoes indeed

correspond to discarding the higher order term of the small-

ness parameter 1ρ/ρ. But with respect to the pressure gradi-

ent, the main effect is proportional to the smallness parameter

1ρ/ρ0itself. If the latter is assumed to be zero, the effect is

overlooked. We suggest that this dual aspect of the magnitude

of condensation-related density changes has not been recog-

nized and this has contributed to the neglect of condensation-

associated pressure gradients in the Earth’s atmosphere.

Furthermore, the consideration of air ﬂows associated with

phase transitions of water vapor has been conventionally re-

duced to the consideration of the net ﬂuxes of matter ignoring

the associated pressure gradients. Suppose we have a linear

circulation pattern divided into the ascending and descend-

ing parts, with similar evaporation rates E(kgH2O m−2s−1)

in both regions. In the region of ascent the water vapor pre-

cipitates at a rate P. This creates a mass sink E−P, which

has to be balanced by water vapor import from the region of

descent. Approximating the two regions as boxes of height

h, length land width d, the horizontal velocity utassociated

with this mass transport can be estimated from the mass bal-

ance equation

ld (P −E) =utρhd , ut=(P −E )

ρ

l

h.(38)

Equation (38) says that the depletion of air mass in the

region of ascent at a total rate of ld(P −E) is com-

pensated for by the horizontal air inﬂux from the re-

gion of descent that goes with velocity utvia vertical

cross-section of area hd. For typical values in the trop-

ics with P−E∼5mmd−1=5.8×10−5kgH2Om−2s−1and

l/ h∼2×103we obtain ut∼1cms−1. For regions where pre-

cipitation and evaporation are smaller, the value of utwill be

smaller too. For example, Lorenz (1967) estimated utto be

∼0.3cms−1for the air ﬂow across latitude 40◦S.

With ρ≈ρdthe value of utcan be understood as the mass-

weighted horizontal velocity of the dry air+water vapor mix-

ture, which is the so-called barycentric velocity, see, e.g.,

(Wacker and Herbert, 2003; Wacker et al., 2006). There is no

net ﬂux of dry air between the regions of ascent and descent,

but there is a net ﬂux of water vapor from the region of de-

scent to the region of ascent. This leads to the appearance of

a non-zero horizontal velocity utdirected towards the region

of ascent. Similarly, vertical barycentric velocity at the sur-

face is wt≈(E −P )/ρ (Wacker and Herbert, 2003), which

reﬂects the fact that there is no net ﬂux of dry air via the

Earth’s surface, while water vapor is added via evaporation

or removed through precipitation. The absolute magnitude

of vertical barycentric velocity wtfor the calculated tropical

means is vanishingly small, wt∼+0.05mm s−1.

We speculate that the low magnitude of barycentric ve-

locities has contributed to the judgement that water’s phase

transitions cannot be a major driver of atmospheric dynamics.

However, barycentric velocities should not be confused with

the actual air velocities (e.g., Meesters et al., 2009). Unlike

the former, the latter cannot be estimated without considering

atmospheric pressure gradients (Makarieva and Gorshkov,

2009c). For example, in the absence of friction, the maxi-

mum linear velocity ucthat could be produced by conden-

sation in a linear circulation pattern in the tropics constitutes

uc=p21p/ρ ∼40 m s−1ut.(39)

Here 1p was taken equal to 10hPa as estimated from

Eq. (37) for Hadley cell in Sect. 4.1. As one can see, uc

(Eq. 39) is much greater than ut(Eq. 38). As some part of

potential energy associated with the condensation-induced

pressure gradient is lost to friction (Makarieva and Gorshkov,

2009a), real air velocities observed in large-scale circulation

are an order of magnitude smaller than uc, but still nearly

three orders of magnitude greater than ut.

4.4 The dynamic efﬁciency of the atmosphere

We will now present another line of evidence for the im-

portance of condensation-induced dynamics: we shall show

that it offers an improved understanding of the efﬁciency

with which the Earth’s atmosphere can convert solar energy

into kinetic energy of air circulation. While the Earth on

average absorbs about I≈2.4×02Wm−2of solar radiation

(Raval and Ramanathan, 1989), only a minor part η∼10−2

of this energy is converted to the kinetic power of atmo-

spheric and oceanic movement. Lorenz (1967, p. 97) notes,

“the determination and explanation of efﬁciency ηconstitute

the fundamental observational and theoretical problems of

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A. M. Makarieva et al.: Condensation-induced atmospheric dynamics 1049

atmospheric energetics”. Here the condensation-induced dy-

namics yields a relationship that is quantitative in nature and

can be estimated directly from fundamental atmospheric pa-

rameters.

A pressure gradient is associated with a store of poten-

tial energy. The physical dimension of pressure gradient co-

incides with the dimension of force per unit air volume,

i.e. 1Pam−1=1 N m−3. When an air parcel moves along

the pressure gradient, the potential energy of the pressure

ﬁeld is converted to the kinetic energy. The dimension of

pressure is identical to the dimension of energy density:

1Pa=1N m−2=1 J m−3. As the moist air in the lower part of

the atmospheric column rises to height hγwhere most part

of its water vapor condenses, the potential energy released

amounts to approximately δps(Eq. 27). The potential energy

released πvper unit mass of water vapor condensed, dimen-

sion J(kgH2O)−1, thus becomes

πv(Ts)=δps

ρv=RTs

Mv1−MvgTs

L0s.(40)

The global mean precipitation rate is

P∼103kgH2Om−2yr−1(L’vovitch, 1979), global mean

surface temperature is Ts=288K and the observed mean

tropospheric lapse rate 0o=6.5Kkm−1(Glickman, 2000).

Using these values and putting 0oinstead of the moist adi-

abatic lapse rate 0sin Eq. (40), we can estimate the global

mean rate 5v=P πvat which the condensation-related po-

tential energy is available for conversion into kinetic energy.

At the same time we also estimate the efﬁciency η=5v/I

of atmospheric circulation that can be generated by solar

energy via the condensation-induced pressure gradients:

5v=P πv∼3.5Wm−2, η ∼0.015.(41)

Thus, the proposed approach not only clariﬁes the dynamics

of solar energy conversion to the kinetic power of air move-

ment (solar power spent on evaporation →condensation-

related release of potential power →kinetic power gener-

ation), but it does so in a quantiatively tractable manner,

explaining the magnitude of the dissipative power associ-

ated with maintaining the kinetic energy of the Earth’s at-

mosphere.

Our estimate of atmospheric efﬁciency differs fundamen-

tally from a thermodynamic approach based on calculating

the entropy budgets under the assumption that the atmo-

sphere works as a heat engine, e.g. Pauluis et al. (2000);

Pauluis and Held (2002a,b), see also Makarieva et al. (2010).

The principal limitation of the entropy-budget approach is

that while the upper bounds on the amount of work that could

be produced are clariﬁed, there is no indication regarding the

degree to which such work is actually performed. In other

words, the presence of an atmospheric temperature gradient

is insufﬁcient to guarantee that mechanical work is produced.

In contrast, our estimate (Eq. 41) is based on an explicit cal-

culation of mechanical work derived from a deﬁned atmo-

spheric pressure gradient. It is, to our knowledge, the only

available estimate of efﬁciency ηmade from the basic phys-

ical parameters that characterize the atmosphere.

4.5 Evaporation and condensation

While condensation releases the potential energy of atmo-

spheric water vapor, evaporation, conversely, replenishes it.

Here we brieﬂy dwell on some salient differences between

evaporation and condensation to complete our picture regard-

ing how the phase transitions of water vapor generate pres-

sure gradients.

Evaporation requires an input of energy to overcome the

intermolecular forces of attraction in the liquid water to free

the water molecule to the gaseous phase, as well as to com-

press the air. That is, work is performed against local atmo-

spheric pressure to make space for vapor molecules that are

being added to the atmosphere via evaporation. This work,

associated with evaporation, is the source of potential energy

for the condensation-induced air circulation. Upon conden-

sation, two distinct forms of potential energy arise. One is

associated with the potential energy of raised liquid drops –

this potential energy dissipates to friction as the drops fall.

The second form of potential energy is associated with the

formation of a non-equilibrium pressure gradient, as the re-

moval of vapor from the gas phase creates a pressure short-

age of moist air aloft. This pressure gradient produces air

movement. In the stationary case total frictional dissipation

in the resulting circulation is balanced by the fraction of solar

power spent on the work associated with evaporation.

Evaporation is, fundamentally, a surface-speciﬁc process

because it represents a ﬂux of water molecules via the sur-

face of liquid. In contrast, condensation is a volume-speciﬁc

process that affects vapor molecules distributed in a certain

volume. The balance between condensation and evaporation

demands that to compensate for the amount of moisture con-

densed in a certain volume vapor must be transported to

that local volume via its borders. Adding more gas to a gas

volume where condensation has occurred is associated with

compression of the gas in the volume and, hence, with per-

forming work on the gas.

In the stationary case, as long there is a supply of en-

ergy and the relative humidity is less than unity, evaporation

from the planetary surface is adding water vapor to the atmo-

spheric column without changing its temperature. The rate of

evaporation is affected by turbulent mixing and is usually re-

lated to the horizontal wind speed at the surface. The global

mean power of evaporation cannot exceed the power of solar

radiation.

The primary cause of condensation is the cooling of air

masses as the moist air ascends and its temperature drops.

Provided there is enough water vapor in the ascending air, at

a local and short-term scale, condensation is not governed by

solar power but by stored energy and can occur at an arbitrar-

ily high rate dictated by the vertical velocity of the ascending

ﬂow, see Eq. (34).

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1050 A. M. Makarieva et al.: Condensation-induced atmospheric dynamics

Any circulation pattern includes areas of lower pressure

where air ascends, as well as higher pressure areas where it

descends. Condensation rates are non-uniform across these

areas – being greater in areas of ascent. Importantly, in such

areas of ascent condensation involves water vapor that is lo-

cally evaporated along with often substantial amounts of ad-

ditional water vapor transported from elsewhere. Therefore,

the mean rate of condensation in the ascending region of

any circulation pattern is always higher than the local rate

of evaporation. This inherent spatial non-uniformity of the

condensation process determines horizontal pressure gradi-

ents.

Consider a large-scale stationary circulation where the re-

gions of ascent and descent are of comparable size. A rel-

evant example would be the annually averaged circulation

between the Amazon River Basin (the area of ascent) and

the region of Atlantic ocean where the air returns from the

Amazon to descend depleted of moisture. Assuming that

the relative humidity at the surface, horizontal wind speed

and solar power are approximately the same in the two re-

gions, mean evaporation rates should be roughly similar as

well (i.e., coincide at least in the order of magnitude). How-

ever, the condensation (and precipitation) rates in the two re-

gions will be consistently different. In accordance with the

picture outlined above, the average precipitation rate Pain

the area of ascent should be approximately double the av-

erage value of regional evaporation rate Ea. The pressure

drop caused by condensation cannot be compensated by lo-

cal evaporation so as to produce a net zero effect on air pres-

sure. This is because in the region of ascent both the local

water vapor evaporated from the forest canopy of the Ama-

zon forest at a rate Ea∼Edas well as imported water vapor

evaporated from the ocean surface at a rate Edprecipitate,

Pa=Ed+Ea. This is conﬁrmed by observations: precipita-

tion in the Amazon river basin is approximately double the

regional evaporation, Pa≈2Ea(Marengo, 2004). The differ-

ence between regional rates of precipitation and evaporation

on land, R=Pa−Ea∼Ea, is equal to regional runoff – aerial

or liquid. In the region of descent the runoff thus deﬁned is

negative and corresponds to the ﬂux of water vapor that is

exported away from the region with the air ﬂow. Where Ris

positive, it represents the ﬂux of imported atmospheric water

vapor and the equal ﬂux of liquid water that leaves the region

of ascent to the ocean.

The fact that the climatological means of evaporation and

precipitation are seldom observed to be equal has been rec-

ognized in the literature (e.g., Wacker and Herbert, 2003), as

has the fact that local mean precipitation values are consis-

tently larger than those for evaporation (e.g., Trenberth et al.,

2003).

The inherent spatial non-uniformity of the condensation

process explains why it is condensation that principally de-

termines the pressure gradients associated with water vapor.

So, while evaporation is adding vapor to the atmosphere

and thus increasing local air pressure, while condensation

in contrast decreases it, the evaporation process is signiﬁ-

cantly more even and uniform spatially than is condensation.

Roughly speaking, in the considered example evaporation in-

creases pressure near equally in the regions of ascent and

descent, while condensation decreases pressure only in the

region of ascent. Moreover, as discussed above, the rate at

which the air pressure is decreased by condensation in the

region of ascent is always higher than the rate at which lo-

cal evaporation would increase air pressure. The difference

between the two rates is particularly marked in heavily pre-

cipitating systems like hurricanes, where precipitation rates

associated with strong updrafts can exceed local evaporation

rates by more than an order of magnitude (e.g., Trenberth and

Fasullo, 2007).

We have so far discussed the magnitude of pressure gradi-

ents that are produced and maintained by condensation in the

regions where the moist air ascends. This analysis is applica-

ble to observed condensation processes that occur on differ-

ent spatial scales, as we illustrated on the example of Hadley

cell. We emphasize that to determine where the ascending

air ﬂow and condensation predominantly occur is a separate

physical problem. For example, why the updrafts are located

over the Amazon and the downdrafts are located over the

Atlantic ocean and not vice versa. Here regional evapora-

tion patterns play a crucial role. In Sect. 4.1 we have shown

that constant relative humidity associated with surface evap-

oration, which ensures that ∂Nv/∂x=0, is necessary for the

condensation to take place. Using the deﬁnition of γ(Eq. 2)

Eq. (37) can be re-written as follows:

∂lnγ

∂x = − w

u

∂γ

∂z .(42)

This equation shows that the decrease of γwith height and,

hence, condensation is only possible when γgrows in the

horizontal direction, ∂lnγ /∂ x>0. Indeed, surface pressure

is lower in the region of ascent. As the air moves towards the

region of low pressure, it expands. In the absence of evapo-

ration, this expansion would make the water vapor contained

in the converging air unsaturated. Condensation at a given

height would stop.

Evaporation adds water vapor to the moving air to keep

water vapor saturated and sustain condensation. The higher

the rate of evaporation, the larger the ratio w/u at a given

∂ γ /∂z and, hence, the larger the pressure gradient (Eq. 37)

that can be maintained between the regions of ascent and

descent. A small, but persistent difference in mean evapo-

ration 1E<E between two adjacent regions, determines the

predominant direction of the air ﬂow. This explains the role

of the high leaf area index of the natural forests in keep-

ing evaporation higher than evaporation from the open wa-

ter surface of the ocean, for the forests to become the re-

gions of low pressure to draw moist air from the oceans

and not vice versa (Makarieva and Gorshkov, 2007, 2010;

Makarieva et al., 2013). Where the surface is homogeneous

with respect to evaporation (e.g., the oceanic surface), the

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A. M. Makarieva et al.: Condensation-induced atmospheric dynamics 1051

spatial and temporal localization of condensation events is

likely to ﬂuctuate in a random fashion.

5 Discussion: condensation dynamics versus

differential heating in the generation of

atmospheric circulation

In Sect. 2 we argued that condensation cannot occur adia-

batically at constant volume but is always accompanied by

a pressure drop in the local air volume where it occurs. We

concluded that the statement that “the pressure drop by adi-

abatic condensation is overcompensated by latent heat in-

duced pressure rise of the air” (P¨

oschl, 2009, p. S12437) was

not correct. In Sect. 3 we quantiﬁed the pressure change pro-

duced by condensation as dependent on altitude in a column

in hydrostatic balance, to show that in such a column the

pressure drops upon condensation everywhere in the lower

atmosphere up to several kilometers altitude, Fig. 1c. The

estimated pressure drop at the surface increases exponen-

tially with growing temperature and amounts to over 20hPa

at 300K, Fig. 1b.

In Sect. 4 we discussed the implications of the

condensation-induced pressure drop for atmospheric dynam-

ics. We calculated the horizontal pressure gradients produced

by condensation and the efﬁciency of the atmosphere as a dy-

namic machine driven by condensation. Our aim throughout

has been to persuade the reader that these implications are

signiﬁcant in numerical terms and deserve a serious discus-

sion and further analysis. We will conclude by discussing

condensation in contrast to differential heating, the latter con-

ventionally considered the major driver dominating atmo-

spheric dynamics.

Atmospheric circulation is only maintained if, in agree-

ment with the energy conservation law, there is a pressure

gradient to accelerate the air masses and sustain the existing

kinetic energy of air motion against dissipative losses. For

centuries, starting from the works of Hadley and his prede-

cessors, the air pressure gradient has been qualitatively as-

sociated with the differential heating of the Earth’s surface

and the Archimedes force (buoyancy) which makes the warm

and light air rise, and the cold and heavy air sink (e.g., Gill,

1982, p. 24). This idea can be illustrated by Fig. 1c, where

the warmer atmospheric column appears to have higher air

pressure at some heights than the colder column. In the con-

ventional paradigm, this is expected to cause air divergence

aloft away from the warmer column, which, in its turn, will

cause a drop of air pressure at the surface and the resulting

surface ﬂow from the cold to the warm areas. Despite the

physics of this differential heating effect being straightfor-

ward in qualitative terms, the quantitative problem of pre-

dicting observed wind velocities from the fundamental phys-

ical parameters has posed enduring difﬁculties. Slightly more

than a decade before the ﬁrst signiﬁcant efforts in computer

climate modelling, Brunt (1944) as cited by Lewis (1998)

wrote:

“It has been pointed out by many writers that it is

impossible to derive a theory of the general circu-

lation based on the known value of the solar con-

stant, the constitution of the atmosphere, and the

distribution of land and sea .. . It is only possible

to begin by assuming the known temperature dis-

tribution, then deriving the corresponding pressure

distribution, and ﬁnally the corresponding wind

circulation”.

Brunt’s difﬁculty relates to the realization that pressure

differences associated with atmospheric temperature gradi-

ents cannot be fully transformed into kinetic energy. Some

energy is lost to thermal conductivity without generating me-

chanical work. This fraction could not be easily estimated by

theory in his era – and thus it has remained to the present.

The development of computers and appearance of rich satel-

lite observations have facilitated empirical parameterizations

to replicate circulation in numerical models. However, while

these models provide reasonable replication of the quantita-

tive features of the general circulation they do not constitute

a quantitative physical proof that the the observed circulation

is driven by pressure gradients associated with differential

heating. As Lorenz (1967, p. 48) emphasized, although “it

is sometimes possible to evaluate the long-term inﬂuence of

each process affecting some feature of the circulation by re-

course to the observational data”, such knowledge “will not

by itself constitute an explanation of the circulation, since it

will not reveal why each process assumes the value which it

does”.

In comparison to temperature-associated pressure differ-

ence, the pressure difference associated with water vapor

removal from the gas phase can develop over a surface of

uniform temperature. In addition, this pressure difference

is physically anchored to the lower atmosphere. Unlike the

temperature-related pressure difference, it does not demand

the existence of some downward transport of the pressure

gradient from the upper to the lower atmosphere (i.e., the

divergence aloft from the warmer to the colder column as

discussed above) to explain the appearance of low altitude

pressure gradients and the generation of surface winds.

Furthermore, as the condensation-related pressure differ-

ence δpsis not associated with a temperature difference,

the potential energy stored in the pressure gradient can be

nearly fully converted to the kinetic energy of air masses

in the lower atmosphere without losses to heat conductivity.

This fundamental difference between the two mechanisms of

pressure fall generation can be traced in hurricanes. Within

the hurricane there is a marked pressure gradient at the sur-

face. This difference is quantitatively accountable by the con-

densation process (Makarieva and Gorshkov, 2009b, 2011).

In the meantime, the possible temperature difference in the

upper atmosphere that might have been caused by the dif-

ference in moist versus dry lapse rates between the regions

of ascent and descent is cancelled by the strong horizontal

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1052 A. M. Makarieva et al.: Condensation-induced atmospheric dynamics

mixing (Montgomery et al., 2006). Above approximately

1.5km the atmosphere within and outside the hurricane is

approximately isothermal in the horizontal direction (Mont-

gomery et al., 2006, Fig. 4), see also Knaff et al. (2000).

Therefore, while the temperature-associated pressure differ-

ence above height zc, Fig. 1c, is not realized in the atmo-

sphere, the condensation-associated pressure difference be-

low height zcapparently is.

Some hints on the relative strengths of the circulation

driven by differential heating compared to condensation-

induced circulation can be gained from evaluating wind ve-

locities in those real processes that develop in the lower at-

mosphere without condensation. These are represented by

dry (precipitation-free) breezes (such as diurnal wind pat-

terns driven by the differential heating of land versus sea sur-

faces) and dust devils. While both demand very large temper-

ature gradients (vertical or horizontal) to arise as compared

to the global mean values, both circulation types are of com-

paratively low intensity and have negligible signiﬁcance to

the global circulation. For example, dust devils do not in-

volve precipitation and are typically characterized by wind

velocities of several meters per second (Sinclair, 1973). The

other type of similarly compact rotating vortexes – tornadoes

– that are always accompanied by phase transitions of water

– develop wind velocities that are at least an order of magni-

tude higher (Wurman et al., 1996). More reﬁned analyses of

Hadley circulation (Held and Hou, 1980) point towards the

same conclusion: theoretically described Hadley cells driven

by differential heating appear to be one order of magnitude

weaker than the observed circulation (Held and Hou, 1980;

Schneider, 2006), see also Caballero et al. (2008). While the

theoretical description of the general atmospheric circulation

remains unresolved, condensation-induced dynamics offers

a possible solution (as shown in Sect. 4.1).

Our approach and theory have other signiﬁcant impli-

cations. Some have been discussed in previous papers,

for example with regard to the development of hurricanes

(Makarieva and Gorshkov, 2009a,b) and the signiﬁcance of

vegetation and terrestrial evaporation ﬂuxes in determining

large scale continental weather patterns (Makarieva et al.,

2006, 2009; Makarieva and Gorshkov, 2007; Sheil and Mur-

diyarso, 2009). Recently accumulated evidence directly doc-

uments air ﬂows induced by the phase transitions of water va-

por (Chikoore and Jury, 2010). Other implications are likely

to be important in predicting the global and local nature of

climate change – a subject of considerable concern and de-

bate at the present time (Pielke et al., 2009; Schiermeier,

2010).

In summary, although the formation of air pressure gra-

dients via condensation has not received adequate theoreti-

cal attention in climatological and meteorological sciences,

here we have argued that this lack of attention has been un-

deserved. Condensation-induced dynamics emerges as a new

ﬁeld of investigations that can signiﬁcantly enrich our under-

standing of atmospheric processes and climate change. We

very much hope that our present account will provide a spur

for further investigations both theoretical and empirical into

these important, but as yet imperfectly characterized, phe-

nomena.

Appendix A

On the physical meaning of Eq. (34) for condensation rate

Equation (34) expresses condensation rate as the difference

between (a) the total change of vapor density with height and

(b) the density change caused by adiabatic expansion. Here

we explore the physical meaning of this expression from a

different perspective. We shall show that Eq. (34) follows

directly from the condition that the vertical distribution of

moist air remains in equilibrium under the assumption that

condensation rate Sis linear over the amount of vapor (i.e.,

condensable gas) in the atmosphere.

A1 Linearity of condensation rate over the molar

density Nvof water vapor

The linearity assumption is justiﬁed by the particular physi-

cal nature and stoichiometry of condensation, with gas turn-

ing to liquid: condensation is a ﬁrst-order reaction over satu-

rated molar density Nvof the condensing gas. This can be

experimentally tested by considering condensation of wa-

ter with different isotopic composition (e.g., Fluckiger and

Rossi, 2003). (Note, for example, that the reverse process

(evaporation) is a zero-order reaction over Nv.)

The rate of ﬁrst-order reactions is directly proportional

to the molar density of the reagent, with the proportional-

ity constant having the dimension of inverse time: S=CNv,

where C(dimension s−1) is in the general case independent

of Nv. In chemical kinetics Cdepends on temperature and

the molecular properties of the reagent as follows from the

law of mass action. Since the saturated concentration Nvof

condensable gas depends on temperature as dictated by the

Clausius-Clapeyron law, we can ask what the proportionality

coefﬁcient Cphysically means in this case. Different sub-

stances have different partial pressures of saturated vapor at

any given temperature – this is controlled by the vaporization

constant Land the molecular properties of the substance.

Note too that for any given substance (like water) the satu-

rated concentration depends on various additional parameters

including the curvature of the the liquid surface and availabil-

ity of condensation nuclei. Therefore, a range of saturated

concentrations is possible at any given temperature. This al-

lows one to consider Cand Nvas independent variables in

the space of all possible combinations of Cand Nv.

A2 The equilibrium

The notions of equilibrium and deviation from it are key to

determining the rate of any reaction. For example, in the case

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A. M. Makarieva et al.: Condensation-induced atmospheric dynamics 1053

of evaporation the deviation from equilibrium is measured

by the water vapor deﬁcit: the deviation of relative humidity

from the (equilibrium) unity value. Atmospheric condensa-

tion is peculiar in being physically associated with air move-

ment in a particular direction – water vapor condenses as the

air moves vertically towards a lower temperature.

Here, in the context of this derivation, by invoking the con-

cept of equilibrium we mean the vertical distribution that the

water vapor would locally take in the absence of condensa-

tion, all other conditions being equal. Let us denote the in-

verse scale height of such an equilibrium distribution for kE.

Condensation rate Sis then proportional to the ﬁrst order de-

viation kvof the observed vertical distribution of water vapor

from the equilibrium:

kv= − 1

Nv

∂Nv

∂z −kE.(A1)

The physics of Eq. (A1) consists of the fact that the character

of the considered equilibrium distribution is not affected by

condensation. For example, for the case of hydrostatic equi-

librium any gas having molar mass M, temperature Tand

ﬁnding itself on a planet with acceleration of gravity gin the

presence of a temperature gradient ∂ T /∂z will have a distri-

bution of its molar density following −∂N /∂ z =kEN, where

kE=Mg/RT +(1/ T )∂ T /∂z. (But note that Eq. (A1) can

also be applied to describe physical equilibria of a different

nature. For example, in a vertically isothermal atmosphere in

the absence of gravity kE=0.)

Such a formulation (proportionality of condensation rate

to kv) presumes that the deviation kvof the vertical distribu-

tion of water vapor from equilibrium is due to condensation

alone. (This premise is empirically testable: where conden-

sation is absent, the vertical water vapor distribution should

have the same scale height as the non-condensable gases and

moist air as a whole.) This removes the need to consider Nv

as the saturated vapor concentration. When kv=0, the con-

densation rate is zero independent of whether water vapor is

saturated or not. When kv6= 0, Nvis saturated water vapor

by formulation.

A3 Distribution of vapor, dry air and moist air as

a whole

We write the condition that moist air with molar density Nis

in equilibrium in the vertical dimension as:

−1

N

∂N

∂z ≡k=kE, N =Nv+Nd.(A2)

Condensation causes the distribution of vapor Nvto de-

viate from the equilibrium distribution. The condition that

moist air as a whole nevertheless remains in equilibrium

causes dry air Ndto also deviate from the equilibrium – but

in the opposite direction to the vapor:

−∂Nv

∂z =(k +kv)Nv,−∂Nd

∂z =(k +kd)Nd,(A3)

kvNv+kdNd=0,(A4)

kv= − 1

Nv

∂Nv

∂z −k, kd≡ − 1

Nd

∂Nd

∂z −k. (A5)

The value of kvdescribes the intensity of the mass sink.

In the case of water vapor kv>0 is caused by a steep ver-

tical temperature gradient that causes vapor to condense

(Makarieva and Gorshkov, 2007; Gorshkov et al., 2012,

Sect. 3). From consideration of the Clausius-Clapeyron law

and hydrostatic equilibrium one can see that

kv=L0

RT 2−Mg

RT ,(A6)

where Lis molar vaporization constant, 0≡ −∂T /∂ z is tem-

perature lapse rate, and Mis molar mass of air.

The value of kvis controlled by temperature lapse rate 0–

keeping all other variables constant, changing 0it is possible

for kvto take any value, −∞ < kv<∞. This validates our

assumption that kvcan be kept independent of Nvwhen in-

vestigating the limit behavior Nv→0 in Eq. (A10): for any

Nv(e.g., set by ambient temperature) any value of kvcan be

prescribed by changing 0.

A4 The limit behaviour ∂Nd/∂ x →0

Using Eqs. (32), (33) and ∂Nv/∂x =0 we obtain (see also

Gorshkov et al., 2012):

u∂Nd

∂x =(Sd−S) 1

γd,(A7)

where

Sd≡w∂Nv

∂z −γd∂Nd

∂z , γd≡Nv

Nd.(A8)

The magnitude of condensation rate Sin Eq. (A7) remains

unknown. Note that under terrestrial conditions 1/γd1.

Putting Eq. (A3) into Eq. (A7) using Eq. (A4) we obtain:

u∂Nd

∂x = −wkvNd1+Nv

Nd+S

wkvNv.(A9)

Now putting S=CNvinto Eq. (A9) we have

∂Nd

∂x = −wkvNd

u1+Nv

Nd+C

wkv.(A10)

We require that ∂Nd/∂x →0 at Nv→0 (no horizontal den-

sity gradient in the absence of condensable substance). This

condition follows from considering that, aside from conden-

sation, there are no processes in the atmospheric column that

would make the air distribution deviate from a static equilib-

rium. This limit is general and should apply to all conditions,

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1054 A. M. Makarieva et al.: Condensation-induced atmospheric dynamics

including cases where all other variables in Eq. (A10) are in-

dependent of Nv. From this condition we obtain C= −wkv

and

S= −wkvNv,(A11)

which is Eq. (34). An experiment to test this relationship

would be to consider a circulation with given vertical and

horizontal velocities wand u, set kvand Ndand change the

saturated molar density Nvby either changing the condens-

able gas or the amount of condensation nuclei in the atmo-

sphere or temperature (see below) or both. One will observe

that as the condensable gas disappears from the atmosphere,

the horizontal pressure gradients vanish. (It is interesting to

note the following. Given that the spatial distribution of Nv

is exponential, Nv(z) =N0exp(−z/hN), the local condition

Nv→0 corresponds to complete disappearance of the con-

densable component from the atmosphere and restoration of

equilibrium in the horizontal plane. In comparison, the local

condition kv→0 does not presume that condensation is ab-

sent everywhere else in the atmosphere (it is plausible that

kvchanges stepwise at the point where condensation com-

mences).)

In Eq. (A11) condensation rate Sis a linear function

of three independent variables: vertical velocity w, local

amount of vapor Nvand deviation kvof vapor from the equi-

librium distribution (kvcan be characterized as the “condens-

ability strength” of atmospheric vapor). Note an interesting

relationship: with Sgiven by Eq. (A11) and γ≡Nv/N we

have Sd−S≡Sγd≡Sdγ. When S=Sdwe have S≡Sd≡0:

condensation is absent.

A5 Appendix summary

Equations (32) and (33), taken together, contain the infor-

mation that it is water vapor and not dry air that undergoes

condensation. Equation (34) contains information about the

magnitude of deviation from equilibrium that causes conden-

sation. Jointly considered, these facts are sufﬁcient to deter-

mine the horizontal pressure gradient produced by the vapor

sink.

Note that in Eq. (A7) any small difference of the order

of γdbetween Sand Sdis multiplied by a large magnitude

1/γd1 and thus has a profound inﬂuence on the magni-

tude of the horizontal gradient ∂Nd/∂z. We emphasize the

point we made in Sect. 4.2: if it were dry air to be in equi-

librium, i.e. kE= −(1/Nd)∂Nd/∂z, the same consideration

of the same equations would give ∂Nd/∂x =0 instead of

∂Nd/∂x =S/u as in the case when it is moist air that is in

equilibrium. The impact of this physical process on atmo-

spheric dynamics remains unexplored.

Acknowledgements. We thank D. R. Rosenfeld and H. H. G.

Savenije for disclosing their names as referees in the ACPD dis-

cussion of the work of Makarieva et al. (2008) and D. R. Rosenfeld

for providing clariﬁcations regarding the derivation of the estimate

of condensation-related pressure change as given by P¨

oschl (2009,

p. S12436). We acknowledge helpful comments of K. E. Trenberth

towards a greater clarity of the presentation of our approach.

The authors beneﬁted greatly from an exciting discussion of

condensation-related dynamics with J. I. Belanger, J. A. Curry,

G. M. Lackmann, M. Nicholls, R. A. Pielke, G. A. Schmidt, A. Sei-

mon and R. M. Yablonsky. We thank all people who discussed

our work, both in the ACPD discussion and in the blogosphere,

in particular, J. Condon, J. A. Curry, L. Liljegren, N. Stokes

and A. Watts for hosting the discussions on their blogs. We are

grateful to P. Restrepo for his interest and help in ﬁnding potential

reviewers for our work. AMM acknowledges the essential role of

S. McIntyre’s blog where she came in contact with J. A. Curry.

We sincerely thank our two referees, J. A. Curry and I. Held, for

their valuable input. We gratefully acknowledge the commitment

and insight of A. Nenes and the entire ACP Executive Committee

in handling our contribution. BLL thanks the US National Science

Foundation and UC Agricultural Experiment Station for their

partial support.

Edited by: A. Nenes

Editor Comment. The authors have presented an entirely new

view of what may be driving dynamics in the atmosphere. This

new theory has been subject to considerable criticism which any

reader can see in the public review and interactive discussion of the

manuscript in ACPD (http://www.atmos-chem-phys-discuss.net/

10/24015/2010/acpd-10-24015-2010-discussion.html). Normally,

the negative reviewer comments would not lead to ﬁnal acceptance

and publication of a manuscript in ACP. After extensive deliber-

ation however, the editor concluded that the revised manuscript

still should be published – despite the strong criticism from the

esteemed reviewers – to promote continuation of the scientiﬁc

dialogue on the controversial theory. This is not an endorsement or

conﬁrmation of the theory, but rather a call for further development

of the arguments presented in the paper that shall lead to conclusive

disproof or validation by the scientiﬁc community. In addition

to the above manuscript-speciﬁc comment from the handling

editor, the following lines from the ACP executive committee shall

provide a general explanation for the exceptional approach taken in

this case and the precedent set for potentially similar future cases:

(1) The paper is highly controversial, proposing a fundamentally

new view that seems to be in contradiction to common textbook

knowledge. (2) The majority of reviewers and experts in the ﬁeld

seem to disagree, whereas some colleagues provide support, and

the handling editor (and the executive committee) are not convinced

that the new view presented in the controversial paper is wrong.

(3) The handling editor (and the executive committee) concluded

to allow ﬁnal publication of the manuscript in ACP, in order to

facilitate further development of the presented arguments, which

may lead to disproof or validation by the scientiﬁc community.

Atmos. Chem. Phys., 13, 1039–1056, 2013 www.atmos-chem-phys.net/13/1039/2013/

A. M. Makarieva et al.: Condensation-induced atmospheric dynamics 1055

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