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The Astrophysical Journal, 757:104 (13pp), 2012 September 20 doi:10.1088/0004-637X/757/1/104

C

!2012. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

AN ANALYTIC RADIATIVE–CONVECTIVE MODEL FOR PLANETARY ATMOSPHERES

Tyler D. Robinson1,3,4and David C. Catling2,3,4

1Astronomy Department, University of Washington, Box 351580, Seattle, WA 98195-1580, USA; robinson@astro.washington.edu

2Department of Earth and Space Sciences, University of Washington, Box 351310, Seattle, WA 98195-1310, USA

Received 2012 February 10; accepted 2012 July 30; published 2012 September 6

ABSTRACT

We present an analytic one-dimensional radiativ e – c o nve c t i ve model of t h e t h e r m a l s t r u c t u r e o f p l a n e t a r y

atmospheres. Our model assumes that thermal radiative transfer is gray and can be represented by the two-

stream approximation. Model atmospheres are assumed to be in hydrostatic equilibrium, with a power-law scaling

between the atmospheric pressure and the gray thermal optical depth. The convective portions of our models are

taken to follow adiabats that account for condensation of volatiles through a scaling parameter to the dry adiabat. By

combining these assumptions, we produce simple, analytic expressions that allow calculations of the atmospheric-

pressure–temperature proﬁle, as well as expressions for the proﬁles of thermal radiative ﬂux and convective ﬂux.

We ex p l o r e t h e g e n e r a l b e h a v i o r s o f o u r m o d e l . T h e s e i nve s t i g a t i o n s e n c o m p a s s ( 1 ) w o r l d s w h e r e a t m o s p h e r i c

attenuation of sunlight is weak, which we show tend to have relatively high radiative–convective boundaries; (2)

worlds with some attenuation of sunlight throughout the atmosphere, which we show can produce either shallow or

deep radiative–convective boundaries, depending on the strength of sunlight attenuation; and (3) strongly irradiated

giant planets (including hot Jupiters), where we explore the conditions under which these worlds acquire detached

convective regions in their mid-tropospheres. Finally, we validate our model and demonstrate its utility through

comparisons to the average observed thermal structure of Venus, Jupiter, and Titan, and by comparing computed

ﬂux proﬁles to more complex models.

Key words: convection – radiation mechanisms: general – planets and satellites: atmospheres – planets and

satellites: general

1. INTRODUCTION

Afundamentalaspectofplanetaryatmospheresistheverti-

cal thermal structure. A one-dimensional (vertical) model can

provide a reasonable estimate of a planet’s global-mean tem-

perature proﬁle. Simple models can provide insights into the

physics behind the thermal proﬁle of an atmosphere without be-

ing obscured by the details of numerical models that sometimes

can approach the complexities of real atmospheres. The best

simple models are those that incorporate the minimum amount

of complexity while still remaining general enough to provide

intuitive understanding.

The historical analytic approach to computing atmo-

spheric temperature proﬁles employs radiative equilibrium

only (Chandrasekhar 1960,p.293;Goody&Yung1989,

pp. 391–396; Thomas & Stamnes 1999,pp.440–450).Ther-

mal radiation is handled according to the two-stream equations

(Schwarzschild 1906), and the calculations typically use the

“gray” approximation, meaning that the thermal opacity of the

atmosphere is assumed to be independent of wavelength and is

represented with a single, broadband value. Such models have

been used to study the ancient (Hart 1978)andmodernEarth

(Pelkowski et al. 2008), and the stability of atmospheres to con-

vection (Sagan 1969;Weaver&Ramanathan1995). The temper-

ature proﬁles from purely radiative models resemble planetary

thermal proﬁles in two broad ways: the temperature falls with

height in the deep part of the atmosphere—the troposphere—and

above that a “stratosphere” forms, which may contain a temper-

ature inversion, and tends toward an isothermal proﬁle.

McKay et al. (1999)usedaradiativemodelwithtwoshort-

wave solar channels to study the antigreenhouse effect on Titan

3NASA Astrobi olo gy In stitute’s Virtual Planetary L abo rat ory.

4University of Washington Astrobiology Program.

and the early Earth. For Titan, the second solar channel improved

the ﬁt to the observed stratospheric structure by taking account

of stratospheric short-wave absorption. More recently, Hansen

(2008)andGuillot(2010)derivedsimilarmodelsforapplication

to hot Jupiters and studied variations in emerging radiation

and temperature proﬁles that result from the inhomogeneous

distribution of stellar irradiation across a planet.

In general, radiative equilibrium models tend to have regions

in their tropospheres where the temperature decrease implies

that convection should ensue, which is a process not incorpo-

rated into the models but is a part of the essential physics of

planetary atmospheres. Convection is common to all planetary

tropospheres known in the solar system (Sanchez-Lavega 2010)

and is predicted for exoplanet atmospheres (Seager 2010), so

radiative equilibrium models neglect the basic physics of ther-

mal structure. While Sagan (1969)andWeaverandRamanathan

(1995)investigatedtheconditionsunderwhichthetemperature

proﬁles generated by gray and windowed-gray radiative equi-

librium models will be convectively unstable, these authors did

not derive analytic radiative–convective equilibrium models.

Asimpleradiative–convectivemodel,employedinthe

limit that the atmosphere is optically thin at thermal wave-

lengths, joins a convective adiabat to an isothermal stratosphere

(Pierrehumbert 2010,pp.169–174).Sincethisopticallythin

limit seldom applies to realistic planetary atmospheres, it is

more common to join the convective adiabat to a gray radia-

tive equilibrium solution (Goody & Yung 1989,pp.404–407;

Nakajima et al. 1992)orawindowed-grayradiativeequilibrium

solution (Caballero et al. 2008). However, the aforementioned

models neglect short-wave attenuation of sunlight by the atmo-

sphere, leading to isothermal stratospheres that fail to represent

realistic planetary stratospheres for most planets of the solar

system with atmospheres, including all the giant planets.

1

The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

Realistic radiative–convective solutions to temperature pro-

ﬁles are commonly computed numerically, for example, using

convective adjustment. In this method, the statically unstable

proﬁle in the deep layer of the atmosphere that is calculated

from radiative equilibrium is ﬁxed to a dry or moist adiabatic

lapse rate, which accounts for convection. However, because set-

ting the vertical temperature gradient changes the energy ﬂuxes,

the tropopause altitude must be adjusted (and temperatures at

all altitudes in the troposphere shifted higher or lower) in nu-

merical iterations until the temperature and upwelling ﬂux are

continuous (Manabe & Strickler 1964;Manabe&Wetherald

1967).

Here, we present an analytic radiative–convective model that

uses two short-wave channels, thus allowing it to be realistically

applied to a wide range of planetary atmospheres. The generality

and novelty of our model is demonstrated by applying it to

adisparaterangeofworlds,includingJupiter,Venus,and

Titan. Given the wealth of new problems posed by exoplanets,

development of an analytic model with few parameters is likely

to be useful for future application to such worlds for which only

limited data are known.

In this paper, we ﬁrst derive our analytic model of atmospheric

structure for a planetary atmosphere in radiative–convective

equilibrium (Section 2). We assume that thermal radiative

transfer is gray, and we include two short-wave channels

for absorbed solar (or stellar) light so that the model can

compute stratospheric temperature inversions. A convective

proﬁle is placed at the base of the portion of the atmosphere

that is in radiative equilibrium, and the model ensures that

both the temperature proﬁle and the upwelling ﬂux proﬁle

are continuous across the radiation–convection boundary. In

Section 3we explore the general behaviors of variants of our

model, demonstrating its ability to provide clear insights, and

including an application to strongly irradiated giant planets,

including hot Jupiters. The utility, validity, and generality of our

model are demonstrated by comparing it to previous results, and

by applying it to Venus, Jupiter, and Titan (Section 4).

2. MODEL DERIVATION

In this section, we describe the steps to develop an analytic

radiative–convective model for a plane-parallel atmosphere, as

follows.

1. We derive a differential equation for the vertical thermal

radiative ﬂuxes in a gray atmosphere as a function of optical

depth and the temperature at each optical depth.

2. We relate the optical depth to pressure. Physics implies

apower-lawdependenceofopticaldepthonpressure,as

others have noted previously (Satoh 2004,p.372).

3. We deﬁne temperature in a (convective) troposphere as a

function of pressure (or optical depth using the relation from

Step 2). This temperature is used to derive a new expression

for the vertical thermal ﬂuxes from Step 1, using a boundary

condition of a reference temperature at a reference level,

such as the surface of a rocky planet or the 1 bar level in a

giant planet’s atmosphere.

4. We consider a balance of the net thermal radiative ﬂux with

absorbed stellar ﬂux and any internal energy ﬂux (which

is important for giant planets) to derive expressions for the

temperature and thermal ﬂux proﬁles in the radiative regime

above a troposphere.

5. We derive our radiative–convective model by requiring that

the analytic expressions for temperature and upwelling

thermal radiative ﬂux are continuous at the join between

the convective regime examined in Step 3 and the radiative

region evaluated in Step 4.

2.1. Gray Thermal Radiative Transfer

The ﬂux proﬁles of thermal radiation and the atmospheric

temperature proﬁle are key parameters in a radiative–convective

model, and these quantities are inter-related. The forms of these

proﬁles are different in the regions of the atmosphere that are

convection-dominated versus radiation-dominated. We start by

writing the general equations that describe the relationships

between these key quantities.

In a one-dimensional, plane-parallel atmosphere, the two-

stream Schwarzschild equations for the upwelling and down-

welling thermal radiative ﬂuxes (F+and F−,respectively)are

(Andrews 2010,p.84)

dF+

dτ=D(F+−πB)(1)

dF−

dτ=−D(F−−πB),(2)

where τis the gray infrared optical depth (0 at the top of the

atmosphere and increasing toward larger pressures), which is

used as a vertical coordinate. For brevity, we will simply refer

to these ﬂuxes as “thermal ﬂuxes” for the remainder of this

paper. The variable Bis the integrated Planck function, with

πB(τ)=σT4(τ),(3)

where σis the Stefan–Boltzmann constant (5.67 ×10−8Wm

−2

K−4)andT(τ)is the atmospheric temperature proﬁle. The

parameter Dis the so-called diffusivity factor, which accounts

for the integration of the radiances over a hemisphere. The value

of Dis often taken as 1.66, which compares well with numerical

results (Rodgers & Walshaw 1966; Armstrong 1968). Others

commonly set D=3/2(Weaver&Ramanathan1995), or

D=2, which is the hemi-isotropic approximation. For clarity,

we note that others (e.g., Andrews 2010 or Pierrehumbert 2010)

sometimes absorb the value of Dinto their deﬁnition of τ.

We choose to leav e i t i n o u r expressions s o t h a t t h e d i ffusiv i t y

approximation is not hidden. As we shall see in Section 2.3,

Equations (1)and(2)allowustosolvefortheupwelling

and downwelling thermal ﬂux proﬁles if T(τ)and a boundary

condition are speciﬁed.

The net thermal ﬂux, Fnet,isgivenby

Fnet =F+−F−(4)

so that we can combine Equations (1)and(2) to yield (see

Schuster 1905)

d2Fnet

dτ2−D2Fnet =−2πDdB

dτ,(5)

which is a differential equation that can be used to solve for

B(τ),andthustheatmospherictemperatureproﬁle,ifFnet (τ)

and a boundary condition is known.

2.2. Relating Optical Depth and Pressure

While the vertical coordinate of the equations governing the

transfer of thermal radiation is optical depth, we must relate this

to atmospheric pressure, which is the natural physical vertical

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The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

coordinate of planetary atmospheres. A vertical pressure coor-

dinate is particularly useful and unifying between atmospheres

when we consider lapse rates in tropospheres (Section 2.3).

We take t h e r e l a t i o n b e t w e e n g r a y t h e r m a l o p t i c a l d e p t h a n d

atmospheric pressure to be given by a power law in the form

τ=τ0!p

p0"n

,(6)

where pis atmospheric pressure, τ0is the gray infrared optical

depth integrated down from the top of the atmosphere to the

reference pressure p0,andnis a parameter that controls the

strength of the scaling. In the simplest case, when an absorbing

gas is well mixed and the opacity does not depend strongly on

pressure, we will have n=1. This can physically correspond

to Doppler broadening. A common scenario is to have a well-

mixed gas providing collision-induced opacity (e.g., H2in gas

giant atmospheres in the solar system) or pressure-broadened

opacity so that n=2. Typically, ntakes a value between 1

and 2, as has been parameterized into some radiative models

(Heng et al. 2012). In some cases, where the mixing ratio of the

absorbing gas depends strongly on pressure, larger values of n

have been proposed, such as for water vapor in Earth’s lower

troposphere (Satoh 2004,p.373;Friersonetal.2006).

2.3. The Convective Regime

In the convection-dominated region of a planetary atmo-

sphere, we take the temperature–pressure proﬁle to be similar

to a dry adiabat, although somewhat less steep because of, for

example, the latent heat released by condensation of volatiles

during convective uplift. By specifying T(p),wecanthenderive

the expressions for the upwelling and downwelling thermal ﬂux

proﬁles, which we can join to the proﬁles from the radiatively

dominated region of the atmosphere.

The dry adiabatic temperature variation in the lower, convec-

tive part of a troposphere is given by Poisson’s adiabatic state

equation (e.g., Wallace & Hobbs 2006,p.78):

T=T0!p

p0"(γ−1)/γ

.(7)

Here T0is a reference temperature at a reference pressure p0,

and γis the ratio of speciﬁc heats at constant pressure (cp)and

volume (cv):

γ=cp

cv

.(8)

Kinetic theory also allows us to relate the ratio of speciﬁc heats

to the degrees of freedom, N,fortheprimaryatmospheric

constituent(s), where

γ=1+ 2

N.(9)

Most appreciable atmospheres in the solar system are dominated

by linear diatomic gases, such as H2(Jupiter, Saturn, Uranus,

and Neptune), N2(Titan), or an N2–O2mixture (Earth). These

molecules have three translational and two rotational degrees

of freedom, so that N=5. Thus, γ=7/5=1.4 is the

same for these worlds. In the case of Venus and Mars, whose

atmospheres are dominated by CO2,thereare3translational,

2rotational,and∼2vibrationaldegreesoffreedom(weakly

depending on temperature; Bent 1965), so N=7andγ=1.3.

Thus, in Equation (7), the dry adiabatic temperature Tvaries

as p0.3and p0.2for diatomic- and CO2-dominated atmospheres,

respectively. These T–pscalings will also apply to exoplanet

tropospheres.

Planetary tropospheres do not follow a true dry adiabat,

though, because of the condensation of volatiles during the

convection process, such as the effects of water on Earth

or methane on Titan, or because the degrees of freedom for

the primary atmospheric constituents vary with altitude. Thus,

following Sagan (1962), we modify the temperature structure in

the convection-dominated region of a planetary atmosphere as

T=T0!p

p0"α(γ−1)/γ

.(10)

Here, αis a factor, typically around 0.6–0.9, which accounts for

deviations from the dry adiabatic lapse rate, primarily because

of latent heat release. Physically, αrepresents the average ratio

of the true lapse rate in the planet’s convective region to the dry

adiabatic lapse rate. While this parameterized representation

of the temperature–pressure proﬁle works well for all of the

worlds of the solar system with thick atmospheres, it breaks

down for very moist atmospheres (Pierrehumbert 2010,p.108),

and for one-component condensable atmospheres, which will

have temperature varying as 1/ln (p/p0)according to the

Clausius–Clapeyron relation (Satoh 2004,p.254).

By solving Equation (6) for p/p0and inserting this into

Equation (10), we can rewrite the temperature proﬁle with the

gray infrared optical depth as the vertical coordinate

T=T0!τ

τ0"α(γ−1)/nγ

=T0!τ

τ0"β/n

,(11)

where we deﬁne β=α(γ−1)/γ.Insertingthistemperature

proﬁle into Equation (1)andsolving(seetheAppendix), we

obtain an expression for the upwelling thermal ﬂux in the

convective portion of the atmosphere (see also Mitchell et al.

2006;Mitchell2007;Caballeroetal.2008),

F+(τ)=σT4

0e−D(τ0−τ)+DσT4

0#τ0

τ!τ%

τ0"4β/n

e−D(τ%−τ)dτ%,

(12)

where we have used the boundary condition F+(τ0)=σT4

0,

which assumes that the reference location in the atmosphere is

either a solid surface or sufﬁciently deep so that the opacity is

large enough to drive the upwelling and downwelling thermal

ﬂuxes toward that of a blackbody radiating at the reference

temperature. Using the upper incomplete gamma function Γ

(deﬁned in the Appendix), we can write Equation (12)as

F+(τ)=σT4

0eDτ$e−Dτ0+1

(Dτ0)4β/n !Γ!1+ 4β

n,Dτ"

−Γ!1+ 4β

n,Dτ0""%,(13)

which provides an analytic expression for the upwelling thermal

ﬂux in the convective portion of the atmosphere. The incomplete

gamma function ought to be considered no different from a sine

or cosine, in the sense that it is evaluated with a single function

in modern programming languages, such as MATLAB, IDL,

or Python, so that the ﬂux can be computed readily, if all input

parameters are speciﬁed. Note that Equations (12)and(13)only

apply in the region τrc !τ!τ0,whereτrc is the optical depth at

3

The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

the boundary between the convection-dominated region and the

radiation-dominated region. Also, in deriving these expressions,

we have assumed that n,γ,andαare constant parameters

through the convection-dominated region. Physically, the ﬁrst

expanded term, σT4

0e−D(τ0−τ),inEquation(13)istheupwardly

attenuated contribution to the upwelling thermal ﬂux from the

reference level at temperature T0.Thesecondexpandedtermis

the ﬂux contribution from the atmosphere above the reference

level.

Asimilarlineofargument(seetheAppendix) allows us to

write an expression for the downwelling thermal ﬂux in the

convective region of the atmosphere

F−(τ)=F−(τrc)e−D(τ−τrc )

+DσT4

0#τ

τrc !τ%

τ0"4β/n

e−D(τ−τ%)dτ%,(14)

where the boundary condition is that the downwelling ﬂux

at the top of the convection-dominated region is equal to the

downwelling ﬂux coming from the radiation-dominated region

above. While this equation is not required in order to obtain the

solution to the radiative–convective thermal structure, it does

offer a means to compute the downwelling thermal ﬂux in the

convective region.

2.4. The Radiative Regime

Above the convection-dominated region of the planet’s atmo-

sphere, the temperature proﬁle tends toward radiative equilib-

rium. In this region, emitted thermal radiation, absorbed stellar

radiation, and any source of energy from the planet’s interior are

all in balance. Thus, by writing an expression for the proﬁle of

absorbed stellar ﬂux, we can solve for the thermal ﬂux proﬁles

and the temperature proﬁle using the results from Section 2.1.

The planetary atmospheres of the solar system exhibit ab-

sorption of solar radiation deep in the atmosphere (in the tropo-

sphere or at the surface) as well as in the stratosphere. The latter

can lead to an inversion in the atmospheric temperature proﬁle,

while the former can cause the upper part of the troposphere to

be radiatively dominated to considerable depth, as in the case of

Titan. Consequently, for generality it is necessary to distribute

the stellar ﬂux across two short-wave channels, and write the

net absorbed stellar ﬂux, F&

net,as

F&

net (τ)=F&

1e−k1τ+F&

2e−k2τ,(15)

where F&

1and F&

2are the top-of-atmosphere net absorbed stellar

ﬂuxes in the two channels, and k1and k2parameterize the

strength of attenuation in these two channels, and are a ratio

of the visible optical depth to the gray infrared optical depth.

(Note that k1and k2effectively incorporate a spatial and time

mean zenith angle.) This description of the absorbed stellar ﬂux

is similar to that of McKay et al. (1999)exceptthattheyonly

allowed for the attenuation of sunlight in one of their channels.

At the top of the atmosphere (i.e., where τ=0), the net absorbed

stellar ﬂux must obey

F&

net (0)=(1 −A)F&

4,(16)

where Ais the planet’s Bond albedo, F&is the stellar ﬂux

incident on the top of the planet’s atmosphere, and the factor of

four accounts for an averaging over both the day and night sides

as well as over the illuminated hemisphere.

Atemperatureproﬁleresultsfromabalanceofﬂuxessubject

to boundary conditions. Balancing the net thermal ﬂux with the

absorbed stellar ﬂux and any internal energy source requires that

Fnet (τ)=F&

net (τ)+Fi,(17)

where Fiis the energy ﬂux from the planet’s interior (e.g.,

∼5Wm

−2for Jupiter; Hanel et al. 1981), which is assumed to

be independent of pressure. By combining Equations (5), (15),

and (17), and using the boundary condition that there is no

downwelling thermal radiation at the top of the atmosphere

(i.e., where τ=0), we obtain the temperature proﬁle in the

radiation-dominated region (i.e., where 0 !τ!τrc)

σT4(τ)=F&

1

2$1+ D

k1

+!k1

D−D

k1"e−k1τ%

+F&

2

2$1+ D

k2

+!k2

D−D

k2"e−k2τ%

+Fi

2(1+Dτ).(18)

The upwelling and downwelling thermal ﬂux proﬁles in the

region where 0 !τ!τrc are then given by

F+(τ)=F&

1

2$1+ D

k1

+!1−D

k1"e−k1τ%

+F&

2

2$1+ D

k2

+!1−D

k2"e−k2τ%+Fi

2(2+Dτ)

(19)

F−(τ)=F&

1

2$1+ D

k1

−!1+ D

k1"e−k1τ%

+F&

2

2$1+ D

k2

−!1+ D

k2"e−k2τ%+Fi

2Dτ.

(20)

2.5. The Radiative–Convective Model

The formulae in the preceding sections describe our simple,

one-dimensional radiative–convective climate model. In apply-

ing the model, 10 parameters are often ﬁxed (p0,T0,n,γ,α,

F&

1,F&

2,Fi,k1,andk2), which leaves two variables,τ0and

τrc,thatthemodelprovidesinasolution.However,iftheto-

tal optical depth τ0is speciﬁed then one of the aforementioned

parameters (typically T0)canbecomeavariable.Thetemper-

ature, upwelling ﬂux, and downwelling ﬂux proﬁles follow

Equations (18), (19), and (20), respectively, in the radiative

region from the top of the atmosphere down to the optical depth

at the radiative–convective boundary, τrc.Foropticaldepths(or

pressures) below this level in the atmosphere, the temperature

proﬁle follows the adiabat described by Equation (11), and the

upwelling and downwelling ﬂux proﬁles follow Equations (13)

and (14), respectively.

The requirement that the temperature and upwelling ﬂux

proﬁles be continuous at the radiation–convection boundary will

implicitly solve for two variables in our model (the downwelling

ﬂux proﬁle is guaranteed to be continuous due to the boundary

condition applied in Equation (14)). Thus, at τrc,theupward

thermal ﬂuxes given by Equation (13)(fortheconvective

formulation) and Equation (19)(fortheradiativeformulation)

must be equal. We also require the convective temperature

4

The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

(Equation (11)) and radiative temperature (Equation (18)) to

be equal, so that

σT4

0!τrc

τ0"4β/n

=F&

1

2$1+ D

k1

+!k1

D−D

k1"e−k1τrc %

+F&

2

2$1+ D

k2

+!k2

D−D

k2"e−k2τrc %

+Fi

2(1+Dτrc).(21)

These two equalities result in analytic expressions. But these ex-

pressions contain transcendental functions and so must be solved

numerically, for example, using Newton’s scheme. Model pa-

rameters that remain after applying the requirements above must

be speciﬁed or ﬁt. Greater simplicity may be justiﬁed by the

transparency of an atmosphere, so that one of the stellar chan-

nels can be removed (eliminating F&

2and k2). Also, an internal

energy source is irrelevant for many worlds (e.g., Earth, Titan,

and Venus), setting Fi=0inmanycases.Asmentionedbefore,

the value of γis set by the degrees of freedom associated with

the primary atmospheric constituent, and αis around 0.6–0.9

for worlds in the solar system.

Using the thermal ﬂuxes from Equations (13)and(14), as

well as the parameterized net stellar ﬂux from Equation (15),

the convective ﬂux, Fconv ,canbeeasilycomputedaccordingto

Fconv (τ)=Fi+F&

net (τ)−(F+(τ)−F−(τ)) (22)

in the region τrc !τ!τ0.

2.6. The Simplest Radiative–Convective Model

It is worth considering the simplest form of our

radiative–convective model wherein there is no attenuation of

sunlight (k1=k2=0), so that

F&

net =F&

1+F&

2=(1 −A)F&

4(23)

at all locations in the atmosphere, which reduces our radiative

equilibrium expressions to those of a single short-wave channel.

In the radiative region, the temperature and thermal ﬂux proﬁles

then simplify to

σT4(τ)=F&

net +Fi

2(1+Dτ)(24)

F+(τ)=F&

net +Fi

2(2+Dτ)(25)

F−(τ)=F&

net +Fi

2Dτ.(26)

For dealing with the convective region, the equalities that ensure

temperature and upwelling ﬂux continuity now become

σT4

0!τrc

τ0"4β/n

=F&

net +Fi

2(1+Dτrc)(27)

and

σT4

0eDτrc $e−Dτ0+1

(Dτ0)4β/n !Γ!1+ 4β

n,Dτrc"

−Γ!1+ 4β

n,Dτ0""%=F&

net +Fi

2!2+Dτrc".(28)

The dependence on the Bond albedo, which is incorporated

into F&

net,canberemovedbyscalingthetemperaturestothe

equilibrium temperature, Teq,whichisdeﬁnedby

σT4

eq =(1 −A)F&

4+Fi.(29)

Note that the simple model described in this section is sim-

ilar to an exercise discussed by Pierrehumbert (2010;see

Problem 4.28, p. 311). However, we (1) generalized the

temperature–pressure relationship in the convection-dominated

region (Equation (10)), (2)generalizedtherelationshipbetween

optical depth and pressure (Equation (6)), and (3)usetheupper

incomplete gamma function to provide an analytic evaluation

of the upwelling thermal radiative ﬂuxes in the convection-

dominated region.

3. GENERAL PROPERTIES OF THE MODEL

The general behavior of the model described in the previous

section provides insights into phenomenon in planetary atmo-

spheres. In this section, we will explore the behavior of (1) our

simplest radiative–convective model, (2) a model with a single

attenuated stellar channel, and (3) a model with a single atten-

uated stellar channel and an internal energy source. Later (in

Section 5), we will see how the properties of atmospheres of

certain worlds within the solar system are explained by some of

the general properties discussed here.

3.1. Properties of the Simplest Radiative–Convective Model

Our simplest radiative–convective model (Section 2.6) allows

us to make straightforward deductions about the location of

radiative–convective boundary over a wide range of conditions.

We can combine Equations (27)and(28)toyieldanexpression

for τrc:

!τ0

τrc "4β/n

e−D(τ0−τrc)$1+ eDτ0

(Dτ0)4β/n !Γ!1+ 4β

n,Dτrc"

−Γ!1+ 4β

n,Dτ0""%=2+Dτrc

1+Dτrc

,(30)

which is independent of the net solar ﬂux or the internal energy

ﬂux, and only depends on τ0and the parameter 4β/n. Figure 1

shows contours of solutions to Equation (30)forDτrc over

arangeofvaluesforτ0and 4β/n.Sincetypicalvaluesof

4β/n are between 0.3 and 0.5 for worlds in the solar system

(taking n=2), the shaded region in this ﬁgure demonstrates

that Dτrc will typically be less than unity for realistic values

of 4β/n,placingthe“radiatinglevel”or“emissionlevel”

of Dτ=1intheconvectiveregionoftheatmosphere.As

was discussed in Sagan (1969), worlds with Dτrc <1 will

have “deep” tropospheres and “shallow” stratospheres, whereas

worlds with Dτrc >1willhave“deep”stratospheresand

“shallow” tropospheres.

Figure 1also shows that, in the limit that τ0'1andτ0'τrc,

the value of τrc becomes independent of the value of τ0.Inthis

limit, the upwelling ﬂux at the radiative–convective boundary

is no longer sensitive to the radiation coming from the deepest

atmospheric layers, and Equation (30)simpliﬁesto

Γ&1+ 4β

n,Dτrc'

(Dτrc)4β/n e−Dτrc

=2+Dτrc

1+Dτrc

,(31)

5

The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

Figure.Optical depth of the radiative–convective boundary, τrc, as a function

of 4β/n, computed from Equation (31). Note that these values agree with the

contours in Figurefor large values of τ0. Also shown are the values of τrc

from Sagan (1969), who took the radiative–convective boundary to be where

the radiative equilibrium proﬁle became unstable to convection. This approach

does not guarantee continuity in the upwelling ﬂux proﬁle and, as a result,

our radiative–convective boundaries are always at smaller optical depths. The

shaded region indicates values of 4β/n which are typical for the solar system.

Figure.Gray infrared optical depth of the radiative–convective boundary (τrc )

for a range of values for 4β/n and τ0for a model without solar attenuation.

For the solar system, the value of 4β/n is typically 0.3–0.5 (shown as a shaded

region), so that we expect deep tropospheres. Note that, for large values of τ0,

the value of τrc only depends on 4β/n.

which only depends on the value of 4β/n.Figure2shows the

solution to Equation (31), computed over a range of values for

4β/n,which,again,showsthat,forcommonvaluesof4β/n,

worlds will have deep tropospheres.

Several previous authors (e.g., Sagan 1969;Weaver&

Ramanathan 1995)haveshownthattherequirementforcon-

vective instability in a gray radiative equilibrium model without

solar attenuation is

dlog T

dlog p=Dnτ

4(1+Dτ)>γ−1

γ=!dlog T

dlog p"ad

,(32)

where the subscript “ad” indicates the (dry) adiabatic value. As

mentioned earlier, γtypically has a value of 1.3–1.4, so that the

right-hand side of this expression is typically between 0.23 and

0.29. Sagan (1969)tookthesolutiontoEquation(32)asdeﬁn-

ing the radiative–convective boundary, which is also shown in

Figure 3. Example temperature proﬁles from our model without short-wave

attenuation (labeled “k=0 (no atten.)”) and from our model with a single

attenuated short-wave channel for several different values of k.Pressurehas

been normalized to p0,andwetaken=2andτ0=2. Temperature, shown as

σT4, has been normalized to the sum of the net absorbed stellar ﬂux and the

internal energy ﬂux for the model without attenuation, and to the net absorbed

stellar ﬂux for the model with attenuation. Thick portions of the curves indicate

where the T–pproﬁle is unstable to convection for a dry adiabat with γ=1.4

(suitable for a world with an atmosphere dominated by a diatomic gas).

Figure 2(taking α=1). Sagan’s solution is not the same as

atrueradiative–convectivesolutionasheonlyensuresconti-

nuity in temperature across the radiative–convective boundary,

whereas a realistic radiative–convective model must also main-

tain upwelling ﬂux continuity across this boundary, which places

our radiative–convective boundary more correctly higher in the

atmosphere. At large values of Dτrc Sagan’s solution agrees

with ours because, in the optically thick limit, the upwelling

ﬂux approaches the local blackbody ﬂux, so that temperature

continuity and upwelling ﬂux continuity are equivalent. Note,

however, that high values of Dτrc are only achieved for values

of 4β/n that are far larger than values in the solar system.

3.2 Properties of a Model with a Single

Attenuated Stellar Channel

The simplest model without a stellar channel is unstable to

convection, as shown in Figure 3.Here,theradiativeequilibrium

temperature proﬁle (labeled as “no atten.”), for an atmosphere

with n=2, has a portion that is unstable to convection (for

γ=1.4). We can increase the generality of our simplest

model by allowing for a single stellar channel with attenuation,

obtained from Equation (18) by eliminating terms in F"

2and

Fi,andbydroppingthesubscriptsontheremainingstellar

channel. Figure 3demonstrates example temperature proﬁles

(taking n=2) for such a model for different values of k,which

is the ratio of the stellar optical depth in the single channel

to the gray thermal optical depth. The logarithmic temperature

gradient, or lapse rate, in such a model is

dlog T

dlog p=nkτ

4#(D2−k2)e−kτ

kD +D2+(k2−D2)e−kτ$.(33)

Note that, for k>D, this expression is strictly negative, and,

thus, everywhere stable against convection, and the proﬁle has

atemperatureinversion,asshowninFigure3. This is similar to

an argument in Pierrehumbert (2010,p.212),albeitindifferent

notation. For k=D,thetemperatureproﬁleisisothermaland

is stable against convection for all values of n.However,fora

6

The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

Figure.Optical depth of the radiative–convective boundary, τrc, as a function

of 4β/n, computed from Equation (31). Note that these values agree with the

contours in Figurefor large values of τ0. Also shown are the values of τrc

from Sagan (1969), who took the radiative–convective boundary to be where

the radiative equilibrium proﬁle became unstable to convection. This approach

does not guarantee continuity in the upwelling ﬂux proﬁle and, as a result,

our radiative–convective boundaries are always at smaller optical depths. The

shaded region indicates values of 4β/n which are typical for the solar system.

Figure.Gray infrared optical depth of the radiative–convective boundary (τrc )

for a range of values for 4β/n and τ0for a model without solar attenuation.

For the solar system, the value of 4β/n is typically 0.3–0.5 (shown as a shaded

region), so that we expect deep tropospheres. Note that, for large values of τ0,

the value of τrc only depends on 4β/n.

which only depends on the value of 4β/n.Figure2shows the

solution to Equation (31), computed over a range of values for

4β/n,which,again,showsthat,forcommonvaluesof4β/n,

worlds will have deep tropospheres.

Several previous authors (e.g., Sagan 1969;Weaver&

Ramanathan 1995)haveshownthattherequirementforcon-

vective instability in a gray radiative equilibrium model without

solar attenuation is

dlog T

dlog p=Dnτ

4(1+Dτ)>γ−1

γ=!dlog T

dlog p"ad

,(32)

where the subscript “ad” indicates the (dry) adiabatic value. As

mentioned earlier, γtypically has a value of 1.3–1.4, so that the

right-hand side of this expression is typically between 0.23 and

0.29. Sagan (1969)tookthesolutiontoEquation(32)asdeﬁn-

ing the radiative–convective boundary, which is also shown in

Figure 3. Example temperature proﬁles from our model without short-wave

attenuation (labeled “k=0 (no atten.)”) and from our model with a single

attenuated short-wave channel for several different values of k.Pressurehas

been normalized to p0,andwetaken=2andτ0=2. Temperature, shown as

σT4, has been normalized to the sum of the net absorbed stellar ﬂux and the

internal energy ﬂux for the model without attenuation, and to the net absorbed

stellar ﬂux for the model with attenuation. Thick portions of the curves indicate

where the T–pproﬁle is unstable to convection for a dry adiabat with γ=1.4

(suitable for a world with an atmosphere dominated by a diatomic gas).

Figure 2(taking α=1). Sagan’s solution is not the same as

atrueradiative–convectivesolutionasheonlyensuresconti-

nuity in temperature across the radiative–convective boundary,

whereas a realistic radiative–convective model must also main-

tain upwelling ﬂux continuity across this boundary, which places

our radiative–convective boundary more correctly higher in the

atmosphere. At large values of Dτrc Sagan’s solution agrees

with ours because, in the optically thick limit, the upwelling

ﬂux approaches the local blackbody ﬂux, so that temperature

continuity and upwelling ﬂux continuity are equivalent. Note,

however, that high values of Dτrc are only achieved for values

of 4β/n that are far larger than values in the solar system.

3.2 Properties of a Model with a Single

Attenuated Stellar Channel

The simplest model without a stellar channel is unstable to

convection, as shown in Figure 3.Here,theradiativeequilibrium

temperature proﬁle (labeled as “no atten.”), for an atmosphere

with n=2, has a portion that is unstable to convection (for

γ=1.4). We can increase the generality of our simplest

model by allowing for a single stellar channel with attenuation,

obtained from Equation (18) by eliminating terms in F"

2and

Fi,andbydroppingthesubscriptsontheremainingstellar

channel. Figure 3demonstrates example temperature proﬁles

(taking n=2) for such a model for different values of k,which

is the ratio of the stellar optical depth in the single channel

to the gray thermal optical depth. The logarithmic temperature

gradient, or lapse rate, in such a model is

dlog T

dlog p=nkτ

4#(D2−k2)e−kτ

kD +D2+(k2−D2)e−kτ$.(33)

Note that, for k>D, this expression is strictly negative, and,

thus, everywhere stable against convection, and the proﬁle has

atemperatureinversion,asshowninFigure3. This is similar to

an argument in Pierrehumbert (2010,p.212),albeitindifferent

notation. For k=D,thetemperatureproﬁleisisothermaland

is stable against convection for all values of n.However,fora

6

The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

Figure.Optical depth of the radiative–convective boundary, τrc, as a function

of 4β/n, computed from Equation (31). Note that these values agree with the

contours in Figurefor large values of τ0. Also shown are the values of τrc

from Sagan (1969), who took the radiative–convective boundary to be where

the radiative equilibrium proﬁle became unstable to convection. This approach

does not guarantee continuity in the upwelling ﬂux proﬁle and, as a result,

our radiative–convective boundaries are always at smaller optical depths. The

shaded region indicates values of 4β/n which are typical for the solar system.

Figure.Gray infrared optical depth of the radiative–convective boundary (τrc )

for a range of values for 4β/n and τ0for a model without solar attenuation.

For the solar system, the value of 4β/n is typically 0.3–0.5 (shown as a shaded

region), so that we expect deep tropospheres. Note that, for large values of τ0,

the value of τrc only depends on 4β/n.

which only depends on the value of 4β/n.Figure2shows the

solution to Equation (31), computed over a range of values for

4β/n,which,again,showsthat,forcommonvaluesof4β/n,

worlds will have deep tropospheres.

Several previous authors (e.g., Sagan 1969;Weaver&

Ramanathan 1995)haveshownthattherequirementforcon-

vective instability in a gray radiative equilibrium model without

solar attenuation is

dlog T

dlog p=Dnτ

4(1+Dτ)>γ−1

γ=!dlog T

dlog p"ad

,(32)

where the subscript “ad” indicates the (dry) adiabatic value. As

mentioned earlier, γtypically has a value of 1.3–1.4, so that the

right-hand side of this expression is typically between 0.23 and

0.29. Sagan (1969)tookthesolutiontoEquation(32)asdeﬁn-

ing the radiative–convective boundary, which is also shown in

Figure 3. Example temperature proﬁles from our model without short-wave

attenuation (labeled “k=0 (no atten.)”) and from our model with a single

attenuated short-wave channel for several different values of k.Pressurehas

been normalized to p0,andwetaken=2andτ0=2. Temperature, shown as

σT4, has been normalized to the sum of the net absorbed stellar ﬂux and the

internal energy ﬂux for the model without attenuation, and to the net absorbed

stellar ﬂux for the model with attenuation. Thick portions of the curves indicate

where the T–pproﬁle is unstable to convection for a dry adiabat with γ=1.4

(suitable for a world with an atmosphere dominated by a diatomic gas).

Figure 2(taking α=1). Sagan’s solution is not the same as

atrueradiative–convectivesolutionasheonlyensuresconti-

nuity in temperature across the radiative–convective boundary,

whereas a realistic radiative–convective model must also main-

tain upwelling ﬂux continuity across this boundary, which places

our radiative–convective boundary more correctly higher in the

atmosphere. At large values of Dτrc Sagan’s solution agrees

with ours because, in the optically thick limit, the upwelling

ﬂux approaches the local blackbody ﬂux, so that temperature

continuity and upwelling ﬂux continuity are equivalent. Note,

however, that high values of Dτrc are only achieved for values

of 4β/n that are far larger than values in the solar system.

3.2 Properties of a Model with a Single

Attenuated Stellar Channel

The simplest model without a stellar channel is unstable to

convection, as shown in Figure 3.Here,theradiativeequilibrium

temperature proﬁle (labeled as “no atten.”), for an atmosphere

with n=2, has a portion that is unstable to convection (for

γ=1.4). We can increase the generality of our simplest

model by allowing for a single stellar channel with attenuation,

obtained from Equation (18) by eliminating terms in F"

2and

Fi,andbydroppingthesubscriptsontheremainingstellar

channel. Figure 3demonstrates example temperature proﬁles

(taking n=2) for such a model for different values of k,which

is the ratio of the stellar optical depth in the single channel

to the gray thermal optical depth. The logarithmic temperature

gradient, or lapse rate, in such a model is

dlog T

dlog p=nkτ

4#(D2−k2)e−kτ

kD +D2+(k2−D2)e−kτ$.(33)

Note that, for k>D, this expression is strictly negative, and,

thus, everywhere stable against convection, and the proﬁle has

atemperatureinversion,asshowninFigure3. This is similar to

an argument in Pierrehumbert (2010,p.212),albeitindifferent

notation. For k=D,thetemperatureproﬁleisisothermaland

is stable against convection for all values of n.However,fora

6

The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

Figure.Optical depth of the radiative–convective boundary, τrc, as a function

of 4β/n, computed from Equation (31). Note that these values agree with the

contours in Figurefor large values of τ0. Also shown are the values of τrc

from Sagan (1969), who took the radiative–convective boundary to be where

the radiative equilibrium proﬁle became unstable to convection. This approach

does not guarantee continuity in the upwelling ﬂux proﬁle and, as a result,

our radiative–convective boundaries are always at smaller optical depths. The

shaded region indicates values of 4β/n which are typical for the solar system.

Figure.Gray infrared optical depth of the radiative–convective boundary (τrc )

for a range of values for 4β/n and τ0for a model without solar attenuation.

For the solar system, the value of 4β/n is typically 0.3–0.5 (shown as a shaded

region), so that we expect deep tropospheres. Note that, for large values of τ0,

the value of τrc only depends on 4β/n.

which only depends on the value of 4β/n.Figure2shows the

solution to Equation (31), computed over a range of values for

4β/n,which,again,showsthat,forcommonvaluesof4β/n,

worlds will have deep tropospheres.

Several previous authors (e.g., Sagan 1969;Weaver&

Ramanathan 1995)haveshownthattherequirementforcon-

vective instability in a gray radiative equilibrium model without

solar attenuation is

dlog T

dlog p=Dnτ

4(1+Dτ)>γ−1

γ=!dlog T

dlog p"ad

,(32)

where the subscript “ad” indicates the (dry) adiabatic value. As

mentioned earlier, γtypically has a value of 1.3–1.4, so that the

right-hand side of this expression is typically between 0.23 and

0.29. Sagan (1969)tookthesolutiontoEquation(32)asdeﬁn-

ing the radiative–convective boundary, which is also shown in

Figure 3. Example temperature proﬁles from our model without short-wave

attenuation (labeled “k=0 (no atten.)”) and from our model with a single

attenuated short-wave channel for several different values of k.Pressurehas

been normalized to p0,andwetaken=2andτ0=2. Temperature, shown as

σT4, has been normalized to the sum of the net absorbed stellar ﬂux and the

internal energy ﬂux for the model without attenuation, and to the net absorbed

stellar ﬂux for the model with attenuation. Thick portions of the curves indicate

where the T–pproﬁle is unstable to convection for a dry adiabat with γ=1.4

(suitable for a world with an atmosphere dominated by a diatomic gas).

Figure 2(taking α=1). Sagan’s solution is not the same as

atrueradiative–convectivesolutionasheonlyensuresconti-

nuity in temperature across the radiative–convective boundary,

whereas a realistic radiative–convective model must also main-

tain upwelling ﬂux continuity across this boundary, which places

our radiative–convective boundary more correctly higher in the

atmosphere. At large values of Dτrc Sagan’s solution agrees

with ours because, in the optically thick limit, the upwelling

ﬂux approaches the local blackbody ﬂux, so that temperature

continuity and upwelling ﬂux continuity are equivalent. Note,

however, that high values of Dτrc are only achieved for values

of 4β/n that are far larger than values in the solar system.

3.2 Properties of a Model with a Single

Attenuated Stellar Channel

The simplest model without a stellar channel is unstable to

convection, as shown in Figure 3.Here,theradiativeequilibrium

temperature proﬁle (labeled as “no atten.”), for an atmosphere

with n=2, has a portion that is unstable to convection (for

γ=1.4). We can increase the generality of our simplest

model by allowing for a single stellar channel with attenuation,

obtained from Equation (18) by eliminating terms in F"

2and

Fi,andbydroppingthesubscriptsontheremainingstellar

channel. Figure 3demonstrates example temperature proﬁles

(taking n=2) for such a model for different values of k,which

is the ratio of the stellar optical depth in the single channel

to the gray thermal optical depth. The logarithmic temperature

gradient, or lapse rate, in such a model is

dlog T

dlog p=nkτ

4#(D2−k2)e−kτ

kD +D2+(k2−D2)e−kτ$.(33)

Note that, for k>D, this expression is strictly negative, and,

thus, everywhere stable against convection, and the proﬁle has

atemperatureinversion,asshowninFigure3. This is similar to

an argument in Pierrehumbert (2010,p.212),albeitindifferent

notation. For k=D,thetemperatureproﬁleisisothermaland

is stable against convection for all values of n.However,fora

6

The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

Figure.Optical depth of the radiative–convective boundary, τrc, as a function

of 4β/n, computed from Equation (31). Note that these values agree with the

contours in Figurefor large values of τ0. Also shown are the values of τrc

from Sagan (1969), who took the radiative–convective boundary to be where

the radiative equilibrium proﬁle became unstable to convection. This approach

does not guarantee continuity in the upwelling ﬂux proﬁle and, as a result,

our radiative–convective boundaries are always at smaller optical depths. The

shaded region indicates values of 4β/n which are typical for the solar system.

Figure.Gray infrared optical depth of the radiative–convective boundary (τrc )

for a range of values for 4β/n and τ0for a model without solar attenuation.

For the solar system, the value of 4β/n is typically 0.3–0.5 (shown as a shaded

region), so that we expect deep tropospheres. Note that, for large values of τ0,

the value of τrc only depends on 4β/n.

which only depends on the value of 4β/n.Figure2shows the

solution to Equation (31), computed over a range of values for

4β/n,which,again,showsthat,forcommonvaluesof4β/n,

worlds will have deep tropospheres.

Several previous authors (e.g., Sagan 1969;Weaver&

Ramanathan 1995)haveshownthattherequirementforcon-

vective instability in a gray radiative equilibrium model without

solar attenuation is

dlog T

dlog p=Dnτ

4(1+Dτ)>γ−1

γ=!dlog T

dlog p"ad

,(32)

where the subscript “ad” indicates the (dry) adiabatic value. As

mentioned earlier, γtypically has a value of 1.3–1.4, so that the

right-hand side of this expression is typically between 0.23 and

0.29. Sagan (1969)tookthesolutiontoEquation(32)asdeﬁn-

ing the radiative–convective boundary, which is also shown in

Figure 3. Example temperature proﬁles from our model without short-wave

attenuation (labeled “k=0 (no atten.)”) and from our model with a single

attenuated short-wave channel for several different values of k.Pressurehas

been normalized to p0,andwetaken=2andτ0=2. Temperature, shown as

σT4, has been normalized to the sum of the net absorbed stellar ﬂux and the

internal energy ﬂux for the model without attenuation, and to the net absorbed

stellar ﬂux for the model with attenuation. Thick portions of the curves indicate

where the T–pproﬁle is unstable to convection for a dry adiabat with γ=1.4

(suitable for a world with an atmosphere dominated by a diatomic gas).

Figure 2(taking α=1). Sagan’s solution is not the same as

atrueradiative–convectivesolutionasheonlyensuresconti-

nuity in temperature across the radiative–convective boundary,

whereas a realistic radiative–convective model must also main-

tain upwelling ﬂux continuity across this boundary, which places

our radiative–convective boundary more correctly higher in the

atmosphere. At large values of Dτrc Sagan’s solution agrees

with ours because, in the optically thick limit, the upwelling

ﬂux approaches the local blackbody ﬂux, so that temperature

continuity and upwelling ﬂux continuity are equivalent. Note,

however, that high values of Dτrc are only achieved for values

of 4β/n that are far larger than values in the solar system.

3.2 Properties of a Model with a Single

Attenuated Stellar Channel

The simplest model without a stellar channel is unstable to

convection, as shown in Figure 3.Here,theradiativeequilibrium

temperature proﬁle (labeled as “no atten.”), for an atmosphere

with n=2, has a portion that is unstable to convection (for

γ=1.4). We can increase the generality of our simplest

model by allowing for a single stellar channel with attenuation,

obtained from Equation (18) by eliminating terms in F"

2and

Fi,andbydroppingthesubscriptsontheremainingstellar

channel. Figure 3demonstrates example temperature proﬁles

(taking n=2) for such a model for different values of k,which

is the ratio of the stellar optical depth in the single channel

to the gray thermal optical depth. The logarithmic temperature

gradient, or lapse rate, in such a model is

dlog T

dlog p=nkτ

4#(D2−k2)e−kτ

kD +D2+(k2−D2)e−kτ$.(33)

Note that, for k>D, this expression is strictly negative, and,

thus, everywhere stable against convection, and the proﬁle has

atemperatureinversion,asshowninFigure3. This is similar to

an argument in Pierrehumbert (2010,p.212),albeitindifferent

notation. For k=D,thetemperatureproﬁleisisothermaland

is stable against convection for all values of n.However,fora

6

The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

Figure.Optical depth of the radiative–convective boundary, τrc, as a function

of 4β/n, computed from Equation (31). Note that these values agree with the

contours in Figurefor large values of τ0. Also shown are the values of τrc

from Sagan (1969), who took the radiative–convective boundary to be where

the radiative equilibrium proﬁle became unstable to convection. This approach

does not guarantee continuity in the upwelling ﬂux proﬁle and, as a result,

our radiative–convective boundaries are always at smaller optical depths. The

shaded region indicates values of 4β/n which are typical for the solar system.

Figure.Gray infrared optical depth of the radiative–convective boundary (τrc )

for a range of values for 4β/n and τ0for a model without solar attenuation.

For the solar system, the value of 4β/n is typically 0.3–0.5 (shown as a shaded

region), so that we expect deep tropospheres. Note that, for large values of τ0,

the value of τrc only depends on 4β/n.

which only depends on the value of 4β/n.Figure2shows the

solution to Equation (31), computed over a range of values for

4β/n,which,again,showsthat,forcommonvaluesof4β/n,

worlds will have deep tropospheres.

Several previous authors (e.g., Sagan 1969;Weaver&

Ramanathan 1995)haveshownthattherequirementforcon-

vective instability in a gray radiative equilibrium model without

solar attenuation is

dlog T

dlog p=Dnτ

4(1+Dτ)>γ−1

γ=!dlog T

dlog p"ad

,(32)

where the subscript “ad” indicates the (dry) adiabatic value. As

mentioned earlier, γtypically has a value of 1.3–1.4, so that the

right-hand side of this expression is typically between 0.23 and

0.29. Sagan (1969)tookthesolutiontoEquation(32)asdeﬁn-

ing the radiative–convective boundary, which is also shown in

Figure 3. Example temperature proﬁles from our model without short-wave

attenuation (labeled “k=0 (no atten.)”) and from our model with a single

attenuated short-wave channel for several different values of k.Pressurehas

been normalized to p0,andwetaken=2andτ0=2. Temperature, shown as

σT4, has been normalized to the sum of the net absorbed stellar ﬂux and the

internal energy ﬂux for the model without attenuation, and to the net absorbed

stellar ﬂux for the model with attenuation. Thick portions of the curves indicate

where the T–pproﬁle is unstable to convection for a dry adiabat with γ=1.4

(suitable for a world with an atmosphere dominated by a diatomic gas).

Figure 2(taking α=1). Sagan’s solution is not the same as

atrueradiative–convectivesolutionasheonlyensuresconti-

nuity in temperature across the radiative–convective boundary,

whereas a realistic radiative–convective model must also main-

tain upwelling ﬂux continuity across this boundary, which places

our radiative–convective boundary more correctly higher in the

atmosphere. At large values of Dτrc Sagan’s solution agrees

with ours because, in the optically thick limit, the upwelling

ﬂux approaches the local blackbody ﬂux, so that temperature

continuity and upwelling ﬂux continuity are equivalent. Note,

however, that high values of Dτrc are only achieved for values

of 4β/n that are far larger than values in the solar system.

3.2 Properties of a Model with a Single

Attenuated Stellar Channel

The simplest model without a stellar channel is unstable to

convection, as shown in Figure 3.Here,theradiativeequilibrium

temperature proﬁle (labeled as “no atten.”), for an atmosphere

with n=2, has a portion that is unstable to convection (for

γ=1.4). We can increase the generality of our simplest

model by allowing for a single stellar channel with attenuation,

obtained from Equation (18) by eliminating terms in F"

2and

Fi,andbydroppingthesubscriptsontheremainingstellar

channel. Figure 3demonstrates example temperature proﬁles

(taking n=2) for such a model for different values of k,which

is the ratio of the stellar optical depth in the single channel

to the gray thermal optical depth. The logarithmic temperature

gradient, or lapse rate, in such a model is

dlog T

dlog p=nkτ

4#(D2−k2)e−kτ

kD +D2+(k2−D2)e−kτ$.(33)

Note that, for k>D, this expression is strictly negative, and,

thus, everywhere stable against convection, and the proﬁle has

atemperatureinversion,asshowninFigure3. This is similar to

an argument in Pierrehumbert (2010,p.212),albeitindifferent

notation. For k=D,thetemperatureproﬁleisisothermaland

is stable against convection for all values of n.However,fora

6

The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

Figure.Optical depth of the radiative–convective boundary, τrc, as a function

of 4β/n, computed from Equation (31). Note that these values agree with the

contours in Figurefor large values of τ0. Also shown are the values of τrc

from Sagan (1969), who took the radiative–convective boundary to be where

the radiative equilibrium proﬁle became unstable to convection. This approach

does not guarantee continuity in the upwelling ﬂux proﬁle and, as a result,

our radiative–convective boundaries are always at smaller optical depths. The

shaded region indicates values of 4β/n which are typical for the solar system.

Figure.Gray infrared optical depth of the radiative–convective boundary (τrc )

for a range of values for 4β/n and τ0for a model without solar attenuation.

For the solar system, the value of 4β/n is typically 0.3–0.5 (shown as a shaded

region), so that we expect deep tropospheres. Note that, for large values of τ0,

the value of τrc only depends on 4β/n.

which only depends on the value of 4β/n.Figure2shows the

solution to Equation (31), computed over a range of values for

4β/n,which,again,showsthat,forcommonvaluesof4β/n,

worlds will have deep tropospheres.

Several previous authors (e.g., Sagan 1969;Weaver&

Ramanathan 1995)haveshownthattherequirementforcon-

vective instability in a gray radiative equilibrium model without

solar attenuation is

dlog T

dlog p=Dnτ

4(1+Dτ)>γ−1

γ=!dlog T

dlog p"ad

,(32)

where the subscript “ad” indicates the (dry) adiabatic value. As

mentioned earlier, γtypically has a value of 1.3–1.4, so that the

right-hand side of this expression is typically between 0.23 and

0.29. Sagan (1969)tookthesolutiontoEquation(32)asdeﬁn-

ing the radiative–convective boundary, which is also shown in

Figure 3. Example temperature proﬁles from our model without short-wave

attenuation (labeled “k=0 (no atten.)”) and from our model with a single

attenuated short-wave channel for several different values of k.Pressurehas

been normalized to p0,andwetaken=2andτ0=2. Temperature, shown as

σT4, has been normalized to the sum of the net absorbed stellar ﬂux and the

internal energy ﬂux for the model without attenuation, and to the net absorbed

stellar ﬂux for the model with attenuation. Thick portions of the curves indicate

where the T–pproﬁle is unstable to convection for a dry adiabat with γ=1.4

(suitable for a world with an atmosphere dominated by a diatomic gas).

Figure 2(taking α=1). Sagan’s solution is not the same as

atrueradiative–convectivesolutionasheonlyensuresconti-

nuity in temperature across the radiative–convective boundary,

whereas a realistic radiative–convective model must also main-

tain upwelling ﬂux continuity across this boundary, which places

our radiative–convective boundary more correctly higher in the

atmosphere. At large values of Dτrc Sagan’s solution agrees

with ours because, in the optically thick limit, the upwelling

ﬂux approaches the local blackbody ﬂux, so that temperature

continuity and upwelling ﬂux continuity are equivalent. Note,

however, that high values of Dτrc are only achieved for values

of 4β/n that are far larger than values in the solar system.

3.2 Properties of a Model with a Single

Attenuated Stellar Channel

The simplest model without a stellar channel is unstable to

convection, as shown in Figure 3.Here,theradiativeequilibrium

temperature proﬁle (labeled as “no atten.”), for an atmosphere

with n=2, has a portion that is unstable to convection (for

γ=1.4). We can increase the generality of our simplest

model by allowing for a single stellar channel with attenuation,

obtained from Equation (18) by eliminating terms in F"

2and

Fi,andbydroppingthesubscriptsontheremainingstellar

channel. Figure 3demonstrates example temperature proﬁles

(taking n=2) for such a model for different values of k,which

is the ratio of the stellar optical depth in the single channel

to the gray thermal optical depth. The logarithmic temperature

gradient, or lapse rate, in such a model is

dlog T

dlog p=nkτ

4#(D2−k2)e−kτ

kD +D2+(k2−D2)e−kτ$.(33)

Note that, for k>D, this expression is strictly negative, and,

thus, everywhere stable against convection, and the proﬁle has

atemperatureinversion,asshowninFigure3. This is similar to

an argument in Pierrehumbert (2010,p.212),albeitindifferent

notation. For k=D,thetemperatureproﬁleisisothermaland

is stable against convection for all values of n.However,fora

6

The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

Figure 4. Maximum value of the logarithmic temperature gradient from a

radiative equilibrium model with a single attenuated stellar channel. Solid lines

are for different values of n(Equation (6)), and the horizontal dashed lines

are the dry adiabatic lapse rates for γ=1.29 and γ=1.4, appropriate for a

CO2-dominated atmosphere and an atmosphere dominated by a diatomic gas

(e.g., Earth and the gas/ice giants), respectively. The thickened portions of the

curves indicate the values of kfor which models would possess a convectively

unstable region for the two aforementioned dry adiabatic lapse rates.

given nvalue, it is possible to have a value of k<Dbut still

be stable against convection, although in this case there is no

temperature inversion.

We investigated this other threshold by examining the maxi-

mum value of the lapse rate. Figure 4shows the maximum value

of the lapse rate, which is only a function of kand n.Thisﬁgure

demonstrates that, for a given value of nand γ(which deﬁnes

the dry adiabatic lapse rate), there is a threshold value of the

relative stellar absorption, represented by k,abovewhichthe

single stellar channel model is everywhere stable against con-

vection. The thick lines in this ﬁgure indicate the values of kfor

which proﬁles would be unstable to convection as compared to

dry adiabats in a CO2-dominated atmosphere and an atmosphere

dominated by a diatomic gas. For example, for the n=2case,

the threshold value of kin a CO2-dominated atmosphere is 0.2

and is 0.1 for an atmosphere dominated by a diatomic gas.

In general, the gray infrared optical depth of the

radiative–convective boundary in a model with only a single

attenuated stellar channel and no internal heat ﬂux depends on

τ0,k,and4β/n,andcanbecomputedfromtheexpression

!τ0

τrc "4β/n

e−D(τ0−τrc)$1+ eDτ0

(Dτ0)4β/n !Γ!1+ 4β

n,Dτrc"

−Γ!1+ 4β

n,Dτ0""%=1+D/k +(1−D/k)e−kτrc

1+D/k +(k/D −D/k)e−kτrc ,

(34)

which comes from combining the single-channel versions of

Equations (18)and(19)withEquations(13)and(11). Note that

this expression does not depend on the absorbed stellar ﬂux.

Contours of τrc as a function of τ0and kare shown in Figures 5(a)

and (b) for two different values of 4β/n (appropriate for a CO2-

dominated atmosphere and for an atmosphere dominated by a

diatomic gas, respectively).

Figures 5(a) and (b) demonstrate that, for small values of

k,wehavethatDτrc <1, which corresponds to a deep

troposphere. However, for values of klarger than about 0.1, we

have Dτrc >1, so that the troposphere is shallow because much

Figure 5. Gray infrared optical depth of the radiative–convective boundary, τrc,

for a model with a single attenuated short-wave channel as a function of kand τ0

for (a) 4β/n =0.46, which is appropriate for a dry adiabat in a CO2-dominated

atmosphere (assuming n=2) and (b) 4β/n =0.57, which is appropriate for

an atmosphere dominated by a diatomic gas (assuming n=2). In general,

increasing kstabilizes deeper portions of the atmosphere against convection,

pushing the radiative–convective boundary progressively lower.

of the lower atmosphere is stabilized against convection by the

absorption of stellar energy throughout the deep atmosphere,

as opposed to depositing this energy abruptly at p0,which

will happen in the limit of small values of k.Asweshallsee

later, cases corresponding to Figures 5(a) and (b) are applicable,

respectively, to Venus and Titan. For certain combinations of τ0

and k,thereexiststwovaluesofτrc which satisfy Equation (34),

but the larger of the two always represents an unphysical solution

where the lapse rate in the radiative regime (from Equation (33))

exceeds that for the adiabat in the convective regime at a range

of pressures above the radiative–convective boundary.

3.3 Properties of a Model with a Single Attenuated Stellar

Channel and an Internal Energy Source

Gray radiative equilibrium models with a single stellar chan-

nel and an internal energy source have been used to understand

certain properties of hot Jupiters (e.g., Hansen 2008; Guillot

2010). We can derive a similar model by dropping all terms

in F&

2in Equation (18)(andbydroppingthesubscriptsonthe

remaining stellar channel). The resulting radiative temperature

proﬁle only depends on n,k,andtheratiooftheabsorbedstellar

ﬂux to the internal energy ﬂux, F&/Fi.Figure6shows example

temperature proﬁles for this model for two different values of

7

The Astrophysical Journal, 757:104 (13pp), 2012 September 20 Robinson & Catling

Figure 6. Example temperature proﬁles for a strongly irradiated (F&/Fi=104)

gas giant for two different values of k, and taking n=2 and τ0=1. Pressure has

been normalized to p0, and temperature, shown as σT4,hasbeennormalized

to the net absorbed stellar ﬂux. Thick portions of the curves indicate where the

T–pproﬁle is unstable to convection for a dry adiabat with γ=1.4(suitable

for a world with an atmosphere dominated by H2).

kwhere thickened lines show regions unstable to convection.

Both models assume strongly irradiated (F&/Fi=104)deep

atmospheres with n=2. In Figure 6,thedeepconvectively

unstable zone at p/p0>100 is caused by the internal ﬂux of the

giant planet rather than the stellar ﬂux. In the case of the curve

with k/D=0.1, the stellar ﬂux is absorbed around the base of

adetached,convectivelyunstablezonewheretheatmosphereis

optically thick in the infrared, and this gives rise to the separate,

detached convective zone.

The lapse rate in the radiative portions of such a model is