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An Analytic Radiative-Convective Model for Planetary Atmospheres

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We present an analytic one-dimensional radiative–convective model of the thermal structure of planetary 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 explore the general behaviors of our model. These investigations encompass (1) worlds where atmospheric 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.
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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.440450).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.
Asimpleradiativeconvectivemodel,employedinthe
limit that the atmosphere is optically thin at thermal wave-
lengths, joins a convective adiabat to an isothermal stratosphere
(Pierrehumbert 2010,pp.169174).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.404407;
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.
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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(FπB),(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 ×108Wm
2
K4)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
dτ2D2Fnet =2πDdB
dτ,(5)
which is a differential equation that can be used to solve for
B(τ),andthustheatmospherictemperatureprole,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"(γ1)/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,and2vibrationaldegreesoffreedom(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 Tpscalings 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(γ1)/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(γ1)/nγ
=T0!τ
τ0"β/n
,(11)
where we define β=α(γ1)/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
0eD(τ0τ)+DσT4
0#τ0
τ!τ%
τ0"4β/n
eD(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$eDτ0+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
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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
0eD(τ0τ),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)eD(ττrc )
+DσT4
0#τ
τrc !τ%
τ0"4β/n
eD(ττ%)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&
1ek1τ+F&
2ek2τ,(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.
Atemperatureproleresultsfromabalanceoffluxessubject
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
DD
k1"ek1τ%
+F&
2
2$1+ D
k2
+!k2
DD
k2"ek2τ%
+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
+!1D
k1"ek1τ%
+F&
2
2$1+ D
k2
+!1D
k2"ek2τ%+Fi
2(2+Dτ)
(19)
F(τ)=F&
1
2$1+ D
k1
!1+ D
k1"ek1τ%
+F&
2
2$1+ D
k2
!1+ D
k2"ek2τ%+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
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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
DD
k1"ek1τrc %
+F&
2
2$1+ D
k2
+!k2
DD
k2"ek2τrc %
+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 $eDτ0+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,whichisdenedby
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
eD(τ0τrc)$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)simpliesto
Γ&1+ 4β
n,Dτrc'
(Dτrc)4β/n eDτrc
=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)>γ1
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)asden-
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 Tpprofile 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
atrueradiativeconvectivesolutionasheonlyensuresconti-
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#(D2k2)ekτ
kD +D2+(k2D2)ekτ$.(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,thetemperatureproleisisothermaland
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)>γ1
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)asden-
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 Tpprofile 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
atrueradiativeconvectivesolutionasheonlyensuresconti-
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#(D2k2)ekτ
kD +D2+(k2D2)ekτ$.(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,thetemperatureproleisisothermaland
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)>γ1
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)asden-
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 Tpprofile 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
atrueradiativeconvectivesolutionasheonlyensuresconti-
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#(D2k2)ekτ
kD +D2+(k2D2)ekτ$.(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,thetemperatureproleisisothermaland
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)>γ1
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)asden-
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 Tpprofile 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
atrueradiativeconvectivesolutionasheonlyensuresconti-
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#(D2k2)ekτ
kD +D2+(k2D2)ekτ$.(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,thetemperatureproleisisothermaland
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)>γ1
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)asden-
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 Tpprofile 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
atrueradiativeconvectivesolutionasheonlyensuresconti-
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#(D2k2)ekτ
kD +D2+(k2D2)ekτ$.(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,thetemperatureproleisisothermaland
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)>γ1
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)asden-
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 Tpprofile 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
atrueradiativeconvectivesolutionasheonlyensuresconti-
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#(D2k2)ekτ
kD +D2+(k2D2)ekτ$.(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,thetemperatureproleisisothermaland
is stable against convection for all values of n.However,fora
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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
eD(τ0τrc)$1+ eDτ0
(Dτ0)4β/n !Γ!1+ 4β
n,Dτrc"
Γ!1+ 4β
n,Dτ0""%=1+D/k +(1D/k)ekτrc
1+D/k +(k/D D/k)ekτrc ,
(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
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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
Tpprofile 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
given by
dlog T
dlog p=nkτ
4$D2+(F&/Fi)(D