TEMPERATURE STRUCTURE AND ATMOSPHERIC CIRCULATION...

TEMPERATURE STRUCTURE AND ATMOSPHERIC CIRCULATION OF DRY TIDALLY LOCKED ROCKYEXOPLANETS

Daniel D. B. Koll and Dorian S. AbbotDepartment of the Geophysical Sciences, University of Chicago, Chicago, IL 60637, USA; [email protected]

Received 2016 March 14; revised 2016 April 26; accepted 2016 April 29; published 2016 July 6

ABSTRACT

Next-generation space telescopes will observe the atmospheres of rocky planets orbiting nearby M-dwarfs.Understanding these observations will require well-developed theory in addition to numerical simulations. Here wepresent theoretical models for the temperature structure and atmospheric circulation of dry, tidally locked rockyexoplanets with gray radiative transfer and test them using a general circulation model (GCM). First, we develop aradiative-convective (RC) model that captures surface temperatures of slowly rotating and cool atmospheres.Second, we show that the atmospheric circulation acts as a global heat engine, which places strong constraints onlarge-scale wind speeds. Third, we develop an RC-subsiding model which extends our RC model to hot and thinatmospheres. We find that rocky planets develop large day–night temperature gradients at a ratio of wave-to-radiative timescales up to two orders of magnitude smaller than the value suggested by work on hot Jupiters. Thesmall ratio is due to the heat engine inefficiency and asymmetry between updrafts and subsidence in convectingatmospheres. Fourth, we show, using GCM simulations, that rotation only has a strong effect on temperaturestructure if the atmosphere is hot or thin. Our models let us map out atmospheric scenarios for planets such as GJ1132b, and show how thermal phase curves could constrain them. Measuring phase curves of short-period planetswill require similar amounts of time on the James Webb Space Telescope as detecting molecules via transitspectroscopy, so future observations should pursue both techniques.

Key words: astrobiology – methods: analytical – methods: numerical – planets and satellites: atmospheres – planetsand satellites: terrestrial planets – techniques: photometric

1. INTRODUCTION

1.1. Importance of Atmospheric Dynamics

Terrestrial exoplanets orbiting M-dwarfs are extremelycommon. Results from the Kepler Space Telescope show thatthere are at least ∼0.5 rocky planets per M-dwarf, half of whichcould even be habitable (Dressing & Charbonneau 2015). Justas important, near-future telescopes like the James Webb SpaceTelescope ( JWST) will be able to characterize the atmospheresof these planets (Deming et al. 2009; Beichman et al. 2014;Cowan et al. 2015), making them one of the most promisingobservational targets of the coming decade.

New theories are needed to understand the potentialatmospheres of these exoplanets, particularly their temperaturestructures and large-scale circulations. An atmosphere’stemperature structure and circulation critically influence aplanet’s surface and atmospheric evolution as well as itspotential habitability (Kasting 1988; Abe et al. 2011; Yanget al. 2013). An atmosphere’s temperature structure andcirculation are also important for interpreting observations.For example, a planet’s emission spectrum is determined by thevertical temperature distribution of its atmosphere, while theplanet’s thermal and optical phase curves are set by its day–night temperature gradient and cloud patterns (Seager &Deming 2009; Yang et al. 2013; Hu et al. 2015). Even transitmeasurements can be strongly influenced by chemical mixingand clouds, which in turn depend on the atmosphere’s large-scale circulation (Fortney 2005; Parmentier et al. 2013;Charnay et al. 2015; Line & Parmentier 2016).

Unfortunately there is a large gap between current theories ofterrestrial atmospheres and the wide range of potentialexoplanets. Planets accessible to follow-up observations willgenerally be in short-period orbits, experience strong tidal

forces, and thus tend to be either tidally locked or captured inhigher spin–orbit resonances (Kasting et al. 1993; Makarovet al. 2012). The solar system offers no direct analogs of suchatmospheres, and their dynamics are still poorly understood. Inthis work we focus on tidally locked (synchronously rotating)atmospheres because their dynamics would differ mostdrastically from rapidly rotating atmospheres, whereas planetsin higher spin–orbit resonances should resemble hybridsbetween tidally locked and rapidly rotating planets (seeSection 9).

1.2. Previous Work and Open Questions

Many groups have already used general circulation models(GCMs) to study the thermal structure and atmosphericcirculation of tidally locked terrestrial planets (e.g., Joshiet al. 1997; Merlis & Schneider 2010; Heng et al. 2011;Pierrehumbert 2011a; Selsis et al. 2011; Leconte et al. 2013;Yang et al. 2013; Zalucha et al. 2013; Wordsworth 2015;Kopparapu et al. 2016). These studies investigated a range ofprocesses that shape the atmospheres of tidally locked planets,including the large-scale day–night circulation, equatorialsuperrotation, heat transport by atmospheric waves, and thepotential for atmospheric collapse if the nightside becomes toocold. The development of theory to understand these processes,however, has not kept up with the rapid proliferation ofsimulations.Recent theories of rocky planets focused on planets for

which the horizontal heat redistribution is extremely efficient(Pierrehumbert 2011a; Mills & Abbot 2013; Yang &Abbot 2014; Wordsworth 2015). Among the latter, Pierrehum-bert (2011b) developed a scaling relation for the surfacetemperature of a planet that is horizontally completely uniform

The Astrophysical Journal, 825:99 (22pp), 2016 July 10 doi:10.3847/0004-637X/825/2/99© 2016. The American Astronomical Society. All rights reserved.

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and whose atmosphere is optically thick,

( )( )

tb

= ´G +

b

T T1 4

. 1s eqLW

1 4

Here Teq is the planet’s equilibrium temperature defined as[ ( ) ( )]* a sº -T L 1 4eq

1 4, tLW is the longwave opticalthickness, Γ is the Gamma function defined as ( )G ºa

( )ò -¥ -t t dtexpa

01 , and ( )b º R c np . L* is the stellar constant,

α is the planetary bond albedo, σ is the Stefan-Boltzmannconstant, R is the specific gas constant, cp is the specific heatcapacity, and n governs how optical thickness depends onpressure (Section 2). Similarly, Wordsworth (2015) developed atheory for the temperature structure of tidally locked atmo-spheres in the optically thin limit, in which atmospheres becomeparticularly vulnerable to atmospheric collapse. Wordsworth(2015) found a lower bound for the nightside temperature of atidally locked planet,1

( )t= ⎜ ⎟⎛

⎝⎞⎠T T

2. 2n eq

LW1 4

Common to both scalings is that they are not valid in thephysically important regime of optical depth unity, and indeedcontradict each other when extrapolated to this limit. Neither dothey explicitly account for horizontal atmospheric dynamics.2

Nevertheless, we expect that the dynamics of tidally lockedplanets should be sensitive to a range of additional processes,including the atmosphere’s radiative timescale, surface drag,and planetary rotation, all of which have not yet been addressedfor rocky exoplanets.

On a different front, recent work has begun to understand theatmospheric circulation of hot Jupiters (Perez-Becker &Showman 2013; Showman et al. 2015; Komacek & Show-man 2016). Perez-Becker & Showman (2013) developed aweak-temperature-gradient (WTG) theory that explains whythe hottest hot Jupiters also tend to have the highest day–nightbrightness temperature contrasts. WTG describes atmospheresthat are slowly rotating and are relatively cool, which allowsatmospheric waves to efficiently eliminate horizontal temper-ature gradients (Showman et al. 2013, pp. 277–326). Inequilibrium, the wave adjustment leads to subsidence, that is,sinking motions, in regions of radiative cooling (see Section 5).Perez-Becker & Showman (2013) showed that day–nighttemperature gradients become large once the radiative time-scale becomes shorter than the timescale for subsidence,

t trad sub. On hot Jupiters with sufficiently strong drag,temperature gradients are large when

( ) ´tt

tt , 3rad

wave

dragwave

where twave is the timescale for a gravity wave to horizontallypropagate across the planet and tdrag is a characteristic dragtimescale.

It would be tempting to assume Equation (3) applies equallywell to rocky exoplanets. That is not the case, and publishedGCM results of rocky planets are already at odds with it. We

show in Appendix A that, for most tidally locked terrestrialplanets, drag and wave timescales are comparable, »t tdrag wave.If Equation (3) applied to rocky planets, they should developlarge day–night temperature gradients when

( ) t

t1 . 4wave

rad

In contrast, the GCM simulations in Selsis et al. (2011) indicatethat tidally locked rocky planets can develop atmospherictemperature gradients at a surface pressure of about 1 bar (theirFigure 5), which translates to a much lower value of

~t t 0.05wave rad . Similarly, we found in Koll & Abbot(2015) that rocky exoplanets develop large day–night bright-ness temperature contrasts when -t t 10wave rad

2. Thedisagreement between the hot Jupiter theory and rocky planetshas not been explored yet. Here we will show that thequalitative threshold for a WTG atmosphere to develop largetemperature gradients, t trad sub, also applies to rocky planets.However, rocky planets end up behaving quite differently thanhot Jupiters because of the processes that determine the large-scale circulation and the subsidence timescale tsub.

1.3. Outline

In this paper we develop a series of models to understand theatmospheres of tidally locked rocky exoplanets. To show howour models complement previous theories we adopt ournondimensional analysis from Koll & Abbot (2015). Usingthe Buckingham-Pi theorem (Buckingham 1914), we showedthat the dynamics of a dry and tidally locked atmosphere withgray radiation are governed by only six nondimensionalparameters. This set of nondimensional parameters allows usto cleanly disentangle the atmospheric processes that need to beaddressed. One choice for the six parameters is given by

( )t t⎛⎝⎜

⎞⎠⎟

R

c

a

L

t

t

t

t, , , , , . 5

p

2

Ro2

wave

radSW LW

wave

drag

The convective lapse rate is controlled by R/cp. The nondimen-sional Rossby radius a L2

Ro2 governs the influence of planetary

rotation on equatorial waves. Here, a is the planetary radius,the equatorial Rossby deformation radius is defined as ºLRo

( )Wac 2wave , Ω is the planetary rotation rate, and cwave is thespeed of a gravity wave. Although cwave is a priori unknown,because it depends on an atmosphere’s vertical temperaturestructure, we can place a reasonable upper bound on it byassuming an isothermal atmosphere. This assumption leads to

= ´ = ´c R c gH R c RTp pwave eq , where g is theacceleration of gravity and ºH RT geq is the scale height. Thewave-to-radiative timescale ratio, t twave rad, compares the time ittakes for equatorial waves to redistribute energy across the planet,

ºt a cwave wave, to the atmosphere’s radiative cooling time,( )sºt c p g Tp srad eq

3 . The atmospheric shortwave and longwaveoptical thicknesses are tSW and tLW. The ratio of wave to dragtimescales =t t C a HDwave drag governs surface friction andturbulent heat fluxes (Appendix A).We only consider atmospheres that are transparent to

shortwave absorption (t = 0SW ), which ensures that the solidsurface substantially affects the atmospheric dynamics. Thismeans we exclude from our consideration potential “rocky”

1 We use Equation (2) instead of Equation (29) in Wordsworth (2015),because it does not assume a specific value for n.2 Wordsworth (2015) also developed a model that incorporates dynamics,which we revisit in Section 4.

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The Astrophysical Journal, 825:99 (22pp), 2016 July 10 Koll & Abbot

planet scenarios with a bulk silicate composition, but withgaseous envelopes several hundreds of bar thick (Owen &Mohanty 2016). We expect that the observable atmospheres ofsuch planets would resemble gas giants more than rockyplanets, with dynamics that are better captured by theoriesdeveloped for hot Jupiters (Perez-Becker & Showman 2013;Showman et al. 2015; Komacek & Showman 2016).

Of the six nondimensional parameters tSW and tLW governradiative transfer, R/cp sets the vertical temperature structure,and the remaining three parameters determine the horizontaldynamics. As an important starting point, we formulate ananalytical radiative-convective (RC) model for the temperaturestructure of tidally locked atmospheres that only depends on

( )b º R ncp and tLW and therefore addresses the first twoprocesses (Section 3). In the optically thick regime this modelreduces to the asymptotic limit found by Pierrehumbert(2011b). We then turn to horizontal dynamics. We show theday–night circulation acts as a heat engine, in which heatingand cooling balance frictional dissipation in the daysideboundary layer (Section 4). We use our heat engine theory todevelop an RC-subsiding (RCS) model that includes the effectsof t twave drag and t twave rad on temperature structure (Section 5).For cool/thick atmospheres the RCS model reduces to the RCmodel, whereas for optically thin and hot/thin atmospheres itreduces to the asymptotic limit found by Wordsworth (2015).Our RCS model explains why rocky planets develop large day–night temperature gradients at a significantly lower t twave radthreshold than hot Jupiters (Section 6). Next, we use GCMsimulations to address rapidly rotating planets and a L2

Ro2

(Section 7). We find that t twave rad has to be big for rotation tohave a strong effect on temperature structure, that is, causelarge eastward hot spot offsets or cold nightside vortices. Ourresults imply that planets like GJ 1132b or HD 219134b willlikely have significant day–night temperature contrasts, unlesstheir atmospheres are dominated by H2 (Section 8). Weestimate that detecting these potential contrasts via thermalphase curves will require about as much time with JWST asdetecting molecular signatures via transit spectroscopy. Finally,we discuss and summarize our results in Sections 9 and 10.Appendix A contains a derivation of the characteristic dragtimescale for tidally locked rocky planets, Appendix B explainshow we compute the wind speed scaling proposed byWordsworth (2015), Appendix C describes how we solve theRCS model, and Appendix D lists the atmospheric equations ofmotion and radiative transfer for reference.

2. METHODS

We compare our models with a large number of GCMsimulations. We use the FMS GCM with two-band gray gasradiative transfers and dry (non-condensing) thermodynamics.FMS has been used to simulate the atmospheres of Earth(Frierson et al. 2006), Jupiter (Liu & Schneider 2011), hotJupiters (Heng et al. 2011), tidally locked terrestrial planets(Merlis & Schneider 2010; Mills & Abbot 2013; Koll &Abbot 2015), and non-synchronously rotating terrestrial planets(Kaspi & Showman 2015). We use the same FMS configura-tion as Koll & Abbot (2015). The model version we usesimulates the full atmospheric dynamics and semi-gray (short-wave and longwave) radiation, and we include instantaneousdry convective adjustment. Drag is parameterized using astandard Monin–Obukhov scheme which self-consistentlycomputes the depth of the planetary boundary layer as well

as turbulent diffusion of heat and momentum. The surface isrepresented by an idealized “slab layer,” that is a single layerwith a uniform temperature and fixed depth. The “slab”temperature can be interpreted as a temperature average acrossthe surface’s thermal skin depth (Pierrehumbert 2011b). Oursimulations are all tidally locked, and orbits are assumed to becircular so that the stellar flux is constant in time.Because we only consider atmospheres that are transparent

to shortwave radiation (t = 0SW ), the incoming stellar flux andthe planetary albedo are degenerate in their effect on planetarytemperature. For simplicity we set the surface albedo to zero inall our simulations and vary the incoming stellar flux. Tospecify the relation between the longwave optical thickness andpressure, we use a standard power law of the form

( )tt

=⎛⎝⎜

⎞⎠⎟

p

p. 6

s

n

LW

The exponent n specifies how the optical thickness τ increaseswith pressure. For example, n = 1 if the opacity of a gasmixture is independent of pressure, and n = 2 if the opacityincreases due to pressure broadening (Pierrehumbert 2011b;Robinson & Catling 2012). Our GCM results assume n = 2 orn = 1. The longwave optical thickness tLW is set independentlyof the atmosphere’s bulk composition. To constrain tLW wenote that more complex radiative transfer calculations tend tofind values of tLW between ∼1 and ∼10 at ∼1 bar across a widerange of atmospheres, (Robinson & Catling 2014; Words-worth 2015). We extend these bounds by one order ofmagnitude in each direction and require that the opticalthickness at 1 bar satisfy t0.1 100LW,1 bar . The parameterbounds for our simulations are summarized in Table 1.Figure 1(a) shows the temperature structure of a representa-

tive, slowly rotating, and relatively cool, GCM simulation. Theplanet is Earth-sized ( = Åa a ), temperate ( =T 283eq K), has anorbital period and rotation rate of 50 days, has a moderatelythick N2-dominated atmosphere (ps = 1 bar), and a longwaveoptical depth of unity (t = 1LW ). The GCM does not explicitlymodel a host star, but the orbital period and equilibriumtemperature correspond to an early M-dwarf (M0 or M1;Kaltenegger & Traub 2009, Table 1). In terms of nondimen-sional parameters, ( )tR c a L t t t t, , , ,p

2Ro2

wave rad LW wave drag

( )= ´ -2 7, 0.12, 5 10 , 1, 1.43 . Because the temperaturestructure is approximately symmetric about the substellarpoint, we present this simulation in terms of a substellarlatitude, i.e., the angle between the substellar and antistellarpoint (also see Koll & Abbot 2015, Appendix B). Thetemperature structure in Figure 1(a) is comparable to thatfound by previous studies, and temperature contours arehorizontally flat outside the dayside boundary layer (dashedblack line). The flat temperature contours are characteristic ofthe WTG regime. However, WTG does not hold on large partsof the dayside where the absorbed stellar flux creates a regionof strong convection and turbulent drag, which damps atmo-spheric waves and allows the atmosphere to sustain horizontaltemperature gradients (Showman et al. 2013, pp. 277–326). Asnoted by Wordsworth (2015), this boundary layer will be ofcritical importance for understanding the atmospheric circula-tion of rocky planets.

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3. A TWO-COLUMN RC MODEL

In this section we present a two-column model for tidallylocked planets. We divide the planet into two (dayside andnightside) vertical columns, as shown in Figure 2. The daysideis heated by stellar radiation, which triggers convection andsets an adiabatic vertical temperature profile. We assume theconvective heat flux is large so that the temperature jumpbetween dayside surface and lowest atmospheric level is small.We also assume convection is deep and do not include astratosphere (i.e., a purely radiative layer in the upperatmosphere), so that the dayside column temperature profilein terms of optical thickness, τ, can be written as

( )tt

=b⎛

⎝⎜⎞⎠⎟T T , 7d

LW

where Td is the dayside surface temperature, tLW is the totaloptical thickness in the longwave, and ( )b º R c np is theadiabatic lapse rate in optical thickness coordinates. Next, weassume the WTG regime holds globally (i.e., also inside thedayside boundary layer). The atmosphere is therefore horizon-tally homogeneous, and the nightside temperature structure isalso described by Equation (7). Under these assumptions theentire atmosphere is in RC equilibrium, with convectiongoverned by the dayside surface temperature, Td. The nightsidesurface will generally be colder than the overlying air, whichleads to stable stratification and suppresses turbulent fluxesbetween the nightside surface and atmosphere. We idealize thissituation by assuming that the nightside surface is in radiativeequilibrium with the overlying atmosphere (see Figure 2).For a gray atmosphere on a dry adiabat, the top-of-

atmosphere (TOA) upward longwave and surface downward

Table 1Parameter Bounds for Our Simulations

Parameter Symbol Unit Minimum Value Maximum Value

Planetary radius a a⊕ 0.5 2Rotation rate Ω days−1 p2 100 p2 1Equilibrium temperature Teq K 250 600Surface gravity g 10 m s−2 ( )´ Åa a2

5( )´ Åa a5

2

Specific heat capacitya cp J kg−1 K−1 820 14518Specific gas constanta R J kg−1 K−1 189 4158Surface pressure ps bar 10−2 10Longwave optical thickness at 1 bar tLW ,1 bar — 0.1 100

Surface drag coefficient CD, via kvk — ´0.1 ´10

Notes. The shortwave optical thickness is set to zero, t = 0SW . CD is not a fixed parameter, so we vary the Von Karman constant, kvk, to increase and decrease CD byan order of magnitude (Appendix A). We vary R and cp, but require that R/cp stays within the range of diatomic and triatomic gases ( R c0.23 0.29p ).a Minimum values correspond to CO2, maximum values correspond to H2.

Figure 1. Temperature and circulation structure of a representative slowly rotating and weakly forced GCM simulation with ( ) ( )= ´ -a L t t, 0.12, 5 102Ro2

wave rad3 .

(a) Temperature as a function of substellar latitude (=0° at the terminator, =90° at the substellar point). (b) Vertical velocity in pressure coordinates as a function ofsubstellar latitude. (c) Area fraction of rising motion, where the dot shows the vertically averaged area fraction. The dashed black line in ((a), (b)) shows the top of theGCM’s boundary layer. Inside the boundary layer the temperature increases toward the substellar point, and air rises; outside the boundary layer the temperaturecontours are flat and air sinks. The region of rapidly rising motions, ( )w w< ´0.01 min , is narrowly focused on the substellar point while most of the atmosphereexperiences weak subsidence, w 0. We normalize temperature by the equilibrium temperature Teq, and the pressure velocity by the characteristic surface speed fromthe heat engine ´U p as s (Section 4). The planet’s physical parameters are ( )p t= = W = = =ÅT a a p283 K, , 2 50 days , 1 bar, 1,seq LW and ( ) ( )=R c R c, ,p p N2.

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longwave fluxes are (Pierrehumbert 2011b; Robinson &Catling 2012)

( ) ( )òt s stt

t= = +¢

¢tt b

t - - ¢⎛⎝⎜

⎞⎠⎟F T e T e d a0 , 8d d

4 4

0 LW

4LW

LW

( ) ( )( )òt t stt

t= =¢

¢t b

t t - ¢-⎛⎝⎜

⎞⎠⎟F T e d b. 8dLW

4

0 LW

4LW

LW

Using these expressions we write the dayside TOA, nightsideTOA, and nightside surface energy budgets3 as

( )

( )

* òa

s stt

t

-= +

¢

´ ¢ +

tt b

t

-

- ¢

⎛⎝⎜

⎞⎠⎟

LT e T

e d a

1

2

HT, 9

pd d4 4

0 LW

4LW

LW

( )òs stt

t= +¢

¢tt b

t- - ¢⎛⎝⎜

⎞⎠⎟T e T e d bHT , 9n d

4 4

0 LW

4LW

LW

( )( )òs stt

t= -¢

¢t b

t t- - ¢⎛⎝⎜

⎞⎠⎟T T e d c0 , 9n d

4 4

0 LW

4LW

LW

where Td is the dayside temperature, Tn is the nightsidetemperature, and HT is the day–night heat transport. Weexpress the stellar flux in terms of the equilibrium temperature,

( )* a s- =L T1 2 2p eq4 . Next, we combine the TOA

equations to eliminate HT and use the nightside surface budgetto write Tn in terms of Td. We find

The first term in the denominator is the atmosphere’scontribution to the TOA flux, and the second term is theTOA flux contribution from the dayside and nightside surfaces.In practice, we evaluate the definite integrals in theseexpressions numerically, but they can also be expressed interms of gamma functions (Robinson & Catling 2012).In the optically thick limit, these expressions reduce to the

result of Pierrehumbert (2011b). For t 1LW the exponentialterms t-e LW become negligibly small. The integrand in the up-ward flux decays exponentially at large t¢, which means wecan approximate the upper limit as infinity and replacethe integral with a gamma function, òt t t¢ ¢b t b t- - ¢e dLW

40

4LW

òt t t» ¢ ¢b b t- ¥ - ¢e dLW4

04 = ( )t bG +b- 1 4LW

4 . Similarly, in theoptically thick limit the downward flux at the surface has to

Figure 2. The two-column radiative-convective model. We assume convection on the dayside sets up an adiabatic temperature profile. Horizontal heat transport isassumed to be effective so that the atmosphere is horizontally uniform. The black dots indicate surface temperatures. The dayside surface and atmosphere are closelycoupled via convection, whereas the nightside surface is in radiative equilibrium with, and generally colder than, the overlying atmosphere.

( ) ( )( )

( )ò òs

s

t t=

¢ + + ¢t t

t

b t t t tt

b t t¢ - ¢ - ¢ - - ¢⎡⎣⎢

⎤⎦⎥

TT

e d e e da

2

2 1, 10d

4 eq4

0

4

0

4LW

LW

LWLW

LW

LW

( )( ) ( )

( )( )

( )

ò

ò òs

s t

t t=

´ ¢

¢ + + ¢

t tt

b t t

t tt

b t t t tt

b t t

¢ - - ¢

¢ - ¢ - ¢ - - ¢⎡⎣⎢

⎤⎦⎥

TT e d

e d e e db

2

2 1. 10n

4eq4

0

4

0

4

0

4

LW

LW

LW

LW

LW

LWLW

LW

LW

3 We implicitly use the dayside surface energy budget by assuming that thesurface-air temperature jump is negligible on the dayside.

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approach unity, ( ) ( )ò t t t¢ ¢ »t b t t- - ¢e d 1

0 LW4LW LW . Combining

these approximations we find ( )t b» » G +b -T T T 1 4n d eq LW1 4,

which is the same as Pierrehumbert’s result (Equation (1)). Thedayside and nightside temperatures become equal in this limitbecause the atmosphere’s downward longwave emission becomeslarge enough to eliminate the temperature difference between thenightside surface and the air directly above it (which in turn isequal to the dayside surface temperature).

In the optically thin limit our model differs slightly from theresult of Wordsworth (2015). For t 1LW we can approximateall exponentials using the Taylor series, ( ) t= +t-e 1 LWLW .Retaining only the lowest order in tLW, we write the integrals inthe upward and downward fluxes both as òt t t¢ ¢b t b t- ¢e dLW

40

4LW

òt t t» ¢ ¢b t b- dLW4

04LW = ( )t b+1 4LW . The atmosphere’s TOA

upward and surface downward emission therefore become equal,which is a well-known property of gray radiation in the opticallythin limit (Pierrehumbert 2011b). Again discarding higher-orderterms in tLW, we find [ ( ( ))]t b» ´ - +T T2 1 3 4 1 4d

1 4eq LW

and ( )t b» ´ + -T T2 1 4n1 4

eq LW1 4 1 4. This nightside temper-

ature has the same asymptotic limit but is slightly warmer thanEquation (2) from Wordsworth (2015). That is because weassume the atmosphere remains fixed to an adiabat, whereasWordsworth (2015) assumes an atmosphere that is verticallyisothermal. We will use our RCS model (Sections 5–6) to showthat Wordsworth’s result is a limiting expression for atmospheresthat are very hot or thin, t t 1wave rad , whereas our results in thisSection apply for atmospheres that are cold or thick,

-t t 10wave rad4. Nevertheless, β is always of order unity, so

the difference between Equation (2) and our result is small in theoptically thin regime.

Next, we compare the previous scalings and our RC modelwith our GCM simulations. Figure 3 shows dayside (top) andnightside (bottom) average surface temperatures of manysimulations. To represent all GCM results in a single figure,we normalize surface temperatures using the equilibriumtemperature, Teq, of each simulation. We only show simulationswith b = 1 7, i.e., ( ) ( )=R c n R c, , , , 2p pN ,N2 2 . First, as weshowed above, the RC model tends toward the expressions ofPierrehumbert and Wordsworth in the optically thick and thinregimes (compare solid line with dashed and dotted lines).While the two approximate expressions diverge at t = 1LW , ourmodel provides a smooth fit in this region. Second, the RCmodel captures dayside surface temperatures very well, withdeviations between the RC model and the GCM simulationssmaller than ´ T0.1 eq. The RC model systematically over-predicts dayside temperatures because of its idealized geome-try, which represents the entire dayside as a single column. Forexample, the dayside-average temperature of an airless planetin pure radiative equilibrium is ´ »T T4 2 5 1.13eq eq,whereas the RC model predicts ´ »T T2 1.191 4

eq eq. Third,the RC model captures the general trend of nightside surfacetemperature with tLW. However, Figure 3 also shows thatnightside temperatures exhibit a much wider spread thandayside temperatures, which is not captured by the RC model.There are two reasons for the spread in nightside

temperatures: first, rapidly rotating atmospheres develophorizontally inhomogeneous nightsides, and second, tidallylocked atmospheres do not have an adiabatic temperaturestructure on the nightside. We address rotation in Section 7,here we consider the effect of temperature structure. Figure 4(a)shows the vertical temperature structure of a slowly rotating

Figure 3. Our radiative-convective (RC) model captures the basic dependency of surface temperature on tLW and joins previous asymptotic limits. Top: Averagedayside surface temperatures of many GCM simulations (N = 251). Bottom: Average nightside surface temperatures. Dashed and dotted curves show previouslyderived asymptotic scalings in the optically thick (t 1LW , Pierrehumbert 2011b) and optically thin limits (t 1LW , Wordsworth 2015). The solid curve shows theRC model (Section 3). While the RC model closely matches the GCM dayside temperatures, it does not account for the wide spread in nightside temperatures. Allshown simulations use n = 2 and ( ) ( )=R c R c, ,p p N2.

6


simulation. The gray lines show the vertical temperatureprofiles at each horizontal GCM grid point, which form a wideenvelope. The hottest temperatures at the right side of theenvelope correspond to the substellar point. These profiles areindeed adiabatic, which can be seen from the fact that they areparallel to the temperature profile of the RC model (dashed red-blue line). However, as the dayside and nightside averagedprofiles show, large parts of the atmosphere do not follow anadiabat (solid red and blue lines in Figure 4(a)). The deviationarises because WTG breaks down inside the dayside boundarylayer (Figure 1(a)). This allows the atmosphere outside theboundary layer to decouple from regions of convection, anddevelop a strongly non-adiabatic temperature profile. Inparticular, Figure 4 shows that the nightside average (blueline) forms a strong inversion below ~p p 0.6s , which meansthe nightside is stably stratified and far from RC equilibrium.Nightside inversions are robust features of tidally lockedatmospheres, and have been found in a range of simulations(e.g., Joshi et al. 1997; Merlis & Schneider 2010; Leconteet al. 2013), but are not captured by the RC model. As aconsequence, the RC model produces a warmer nightsideatmosphere and therefore a warmer nightside surface than theGCM (compare blue square and blue circle in Figure 4(a)). Wepresent a model that captures the nightside temperaturestructure in Section 5. However, to do so we have to accountfor atmospheric dynamics, which show up via the parameterst twave rad and t twave drag. To address the dynamics we first haveto develop a theory of large-scale wind speeds and theatmospheric circulation, which we turn to in the next section.

4. A HEAT ENGINE SCALING FOR WIND SPEEDS

Earth’s atmosphere acts as a heat engine — it absorbs heatnear the surface at a high temperature and emits heat to space ata low temperature, which allows the atmosphere to do workand balance frictional dissipation (Peixoto & Oort 1984). On

Earth the heat engine framework has been used to derive upperbounds on the strength of tropical moist convection (Renno &Ingersoll 1996; Emanuel & Bister 1996) and small-scalecirculations such as hurricanes (Emanuel 1986).In this section we idealize the atmospheric circulation of a

tidally locked planet as a single overturning cell between thesubstellar and antistellar point. We model the circulation as anideal heat engine to place an upper bound on its circulationstrength. The ideal heat engine is an upper bound becauseadditional physical processes, such as diffusion, lead to theirreversible production of entropy and decrease the efficiency of aheat engine below its ideal limit (Pauluis & Held 2002). As shownin Figure 5, the atmosphere absorbs heat near the dayside surfaceat a hot temperature and emits it to space at a cold temperature.Here we approximate the temperature at which the atmosphereabsorbs heat as equal to the dayside surface temperature, Td. Weapproximate the cold temperature as the planet’s effectiveemission temperature to space, i.e., its equilibrium temperature,Teq. The parcel does work against friction in the boundary layer,which is given by r=W C UD s s

3 (Bister & Emanuel 1998). HereW is the work, rs is the surface density, and Us is a surface windspeed, which we take to be the dayside-average surface wind(Figure 5). Using Carnot’s theorem,

( )h=W Q , 11in

where ( )h = -T T Td deq is the atmosphere’s thermodynamic

efficiency, and ( )s= ´ - t-Q T e2 1in eq4 LW is the amount of

energy that is available to drive atmospheric motion. We notethat the dayside-averaged incoming stellar flux is equal tosT2 eq

4 , but we additionally account for the fact that only afraction, - t-e1 LW, of stellar energy is available to theatmosphere, while the remainder is immediately re-radiatedfrom the surface to space.

Figure 4. Temperature structure of a slowly rotating GCM simulation ( )=a L 0.12Ro2 compared with the radiative-convective (RC, left) and the radiative-convective-

subsiding model (RCS, right). Solid curves correspond to dayside (red) and nightside-averaged (blue) GCM temperature profiles, and GCM temperature profiles ateach latitude and longitude (gray). Left: Although the RC model (mixed red-blue curve) qualitatively captures the temperature structure, it does not capture thenightside inversion and thus overpredicts the nightside surface temperature (compare blue square with blue circle). Right: the RCS model (dashed curves) accounts forthe imperfect day–night heat transport, and qualitatively captures the nightside inversion structure. This leads to a better fit of nightside surface temperature than for theRC model. The planet’s physical parameters are =T a400 K,eq = ( )pW =Åa p, 2 50 days , s = t0.5 bar, LW = 1, (R,cp) = ( )R c, p N2, and = -g 5 m s 2.

7


We find the following upper bound on the dayside averagesurface wind speed,

( )

( )

( ) ( ) ( )

s

r

s

s

=-

´ -

=-

´ -

= - ´ -

t

t

t

-

-

-

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

UT T

Te

T

C

T T

Te

RT T

C p

T T eR T

C p

12

12

12

, 12

sd

d D s

d

d

d

D s

dD s

eq eq4 1 3

eq eq4 1 3

eqeq4 1 3

LW

LW

LW

where we used the ideal gas law to substitute for rs in thesecond step. The only unknown in this equation is the daysidesurface temperature, Td. As we saw in Section 3, Td wasalready well constrained by the RC model (Figure 3). In thissection we therefore close the model using Td fromEquation (10a) (but note that we will self-consistently solvefor Td in Section 5).

Figure 6 compares dayside averaged surface wind speedsá ñUs with a numerical wind speed scaling from Wordsworth(2015, see Appendix B) and our analytical heat engine scaling.We note that Wordsworth considered the optically thin limit,whereas our results are valid for arbitrary tLW. Wordsworthderived a scaling by assuming that weak temperature gradientshold globally. In a WTG atmosphere, radiative cooling leads tosubsidence, which Wordsworth assumed in turn drives thelarge-scale circulation. Figure 6 shows that the GCM windspeeds span two orders of magnitude, from 3 m s−1 up to about300 m s−1. The Wordsworth (2015) scaling seems to matchthese wind speeds at ( ) 1 m s−1. However, it predicts a strongdecrease, down to less than 10−2 m s−1, which is several ordersof magnitude smaller than the GCM results (Figure 6(a)). Themismatch arises because Wordsworthʼs global WTG scalingassumes that winds are purely driven by radiative cooling,

tµUs LW (Appendix B), so Us should rapidly vanish in theoptically thin limit. Instead, Figure 1 shows that WTG balance

breaks down in regions that are strongly convecting. Theconvecting regions in turn govern the return flow from thenightside to the dayside, which means that the effect of frictionon the large-scale circulation cannot be neglected. Our heatengine scaling includes this effect and predicts very differentdynamics. For example, in the optically thin limit the daysidetemperature is approximately constant and t- »te1 LWLW , so

tµUs LW1 3 (Equation (12)). Figure 6(b) supports our theory. The

slope predicted by the heat engine provides an excellent fit tothe GCM results. Moreover, we expect the heat engine toprovide an upper bound on surface wind speeds. Ourexpectation is confirmed by the GCM simulations, which fallbelow the dashed black line in Figure 6(b). In addition, theoverestimate of á ñUs is small and generally amounts to less thana factor of four (gray dashed line in Figure 6(b)), with mostsimulations falling about a factor of two below the ideal limit.Next, we use the surface wind speed scaling to place an

upper bound on the strength of the day–night circulation. Ofparticular interest to us is the large-scale vertical motion on thenightside, which we will show governs the day–night heattransport and is critically important for the temperaturestructure on the nightside (Section 5). We express all verticalmotions using pressure coordinates, that is, using the pressurevelocity w º Dp Dt where w > 0 means sinking motions. Wetake the surface wind speed, Us, to be the characteristichorizontal velocity within the boundary layer. We relate thehorizontal velocity in the boundary layer to the pressurevelocity near the substellar point using mass conservation4

(Equation (51)),

( )w

~p

U

a. 13

s

sup

Figure 5. A diagram of the atmospheric heat engine. The heat engine is driven by dayside heating and cooling to space. Frictional dissipation in the dayside boundarylayer limits the strength of the resulting day–night atmospheric circulation.

4 A more accurate scaling than Equation (13) was pointed out to us at the timeof publication, w ~p U As sup up . This modification also affects Equation(15). However, as long as rising motions occupy less area than sinkingmotions, ( )~ -A A 10up down

1 , this modification leads to essentially the sameresults as Equation (15).

8


Figure 1(b) supports this scaling, and w wup near the substellarpoint is of order unity. However, Figure 1(b) also shows thatthere is a large asymmetry between rising and sinking motions.While air rises rapidly near the region of strongest convectionat the substellar point, it sinks slowly over a large area outsidethe boundary layer. Figure 1(c) quantifies the asymmetry usingA Aup down, the fraction of the atmosphere in which air risesversus sinks.5 As shown in the simulation, rising air nevercovers more than 20% of the atmosphere, while its verticallyaveraged value is about 10% (dot in Figure 1(c)). Theasymmetry in vertical motions arises from the geometricasymmetry of the incoming stellar flux, and is distinct from theasymmetry of rising and sinking motions in Earth’s tropicswhich is caused by the condensation of water duringconvection. Because upward and downward mass fluxes haveto balance across a horizontal slice of atmosphere,

( )r w r w=A A , 14up up down down

where ρ is the density of an air parcel, we can relate thepressure velocity on the nightside to the dayside surface wind,

( )w = ´A

A

p

aU . 15s

sdownup

down

Equation (15) explains how tidally locked planets sustain weakdownward motions despite very large horizontal wind speeds.The time for a parcel of air to be advected horizontally is

=t a Uadv s, whereas the time for a parcel to subside (that is, beadvected vertically) is w= = ´t p A A ts advsub down down up . For

~A A 10down up it takes a parcel of air ten times longer to sink

back to the surface on the nightside than to be advected fromthe nightside to the dayside. The same comparison alsoexplains why day–night temperature gradients of tidally lockedplanets are not set by the advective timescale and insteaddepend on the ratio of subsidence and radiative timescales(Section 6).Next, Figure 7 compares the pressure velocity, wdown, from

Equation (15) with the mass-weighted vertically and horizon-tally averaged pressure velocities, wá ñnight, from GCM simula-tions. In the comparison we use Equation (12) to predict Us butstill diagnose A Aup down directly from GCM output. First,because the heat engine provides an upper limit on Us it alsoprovides an upper limit on wdown. The GCM simulations indeedfall almost entirely below the dashed black line in Figure 6(b).We note that in deriving Equation (12) we neglected somefactors that we expect to be small (e.g., geometric factors),which explains why some GCM simulations slightly exceed thevalue predicted by the scaling. Second, we find that relativelyslowly rotating atmospheres (blue dots) closely follow the heatengine scaling, and most of them deviate less than a factor offour from it (gray dashed line). Third, rapidly rotatingatmospheres (red dots) still follow the scaling qualitativelybut wá ñnight is smaller than in slowly rotating atmospheres. Thelarger deviation arises because rapidly rotating atmospheresdevelop inhomogeneous nightsides (Section 7). In the extra-tropics the flow then becomes geostrophic, which in turnsuppresses vertical motions by ( ) Ro 1, where Ro is theRossby number (Showman et al. 2010, pp. 471–516).We conclude that atmospheres of dry tidally locked planets

are dominated by dayside boundary layer friction. The heatengine framework successfully constrains the amount ofdissipation and surface wind speeds within the boundary layer.Combined with the areal asymmetry between rising and sinkingmotions, we find an upper bound on the nightside vertical

Figure 6. Two different surface wind speed scalings compared with many GCM results (N = 271). Left: scaling for dayside surface wind speed from Wordsworth(2015), which was derived assuming optically thin atmospheres (t < 1LW ). Right: our scaling for average surface wind speed for an ideal heat engine (Equations (10a)and (12)). The GCM simulations are less efficient than ideal heat engines and therefore have smaller surface wind speeds. The gray dashed line corresponds to aninefficiency factor of 1/4.

5 Because the uppermost layers of the atmosphere show both weakly risingand sinking motions, we define Aup as the area with “significant” upwardmotion where ( )w w´0.01 min .

9


velocity. Our result is distinct from previous scalings that havebeen proposed for exoplanets. We will use our result in the nextsection to constrain the thermal structure of the nightside.

5. A TWO-COLUMN RCS MODEL

As we showed in Figures 3 and 4, to understand nightsidesurface temperatures of tidally locked planets we need toaccount for an imperfect day–night heat transport, and to betterconstrain the nightside atmospheric temperature structure. Inthis section we develop a two-column model that does so. Weagain divide the atmosphere into two dayside and nightsidecolumns, shown in Figure 8. As in Section 3, the daysidecolumn is strongly convecting, but we allow the nightsidetemperature profile to deviate from an adiabat. Both columnsare capped by a stratosphere, that is, a layer in pure radiativeequilibrium.

As in our RC model, convection sets an adiabatictemperature profile on the dayside. The dayside temperatureprofile is therefore

( )tt

=b⎛

⎝⎜⎞⎠⎟T T . 16d

LW

The nightside is in WTG balance. WTG balance followsfrom the thermodynamic equation (Equation (52)),

· ( )w

w¶¶

+ +¶¶

= +¶¶

+¶¶

uT

tT

T

p

RT

c p

g

c

F

p

g

c p, 17

p p p

where u is the horizontal velocity, ω is the pressure velocity(w > 0 for subsiding air), F is the net radiative flux (the sum ofupward and downward longwave fluxes, = - F F F ), and D

is the energy flux due to diffusion. The left side of thethermodynamic equation represents advection, the first term onthe right is heating/cooling due to compression/expansion asair parcels move vertically, the second term on the right isradiative heating/cooling, and the third term represents theeffect of small-scale convection inside the boundary layer. Inequilibrium ¶ ¶ =T t 0, and is negligible on the nightsidebecause the nightside is stably stratified. As long as horizontaltemperature gradients are small on the nightside the thermo-dynamic equation then reduces to the WTG approximation

( )w¶¶

- »¶¶

⎛⎝⎜

⎞⎠⎟

T

p

RT

c p

g

c

F

p. 18

p p

Equation (18) entails that radiative cooling is accompanied bysubsidence as follows. In a cooling layer the net radiative fluxdecreases toward the surface, ¶ ¶ <F p 0. The lapse rate has tobe smaller than or equal to the adiabatic lapse rate because theatmosphere would otherwise start convecting, ¶ ¶T p

( )RT c pp . It follows that w > 0.In Earth’s tropics the vertical temperature structure, ¶ ¶T p,

and radiative fluxes, ¶ ¶F p, are set by small regions of moistconvection, in which case WTG can be used to predict thelarge-scale ω (Sobel et al. 2001). In this section we pursue theopposite approach: because ω is set by the day–nightcirculation which, in turn, is limited by friction in the daysideboundary layer (Section 4), we will use WTG balance to solvefor T and F. For simplicity we replace ω with its verticalaverage w. Because we assume that horizontal variations aresmall, we also replace all partial derivatives with normalderivatives. We rewrite the WTG approximation in optical

Figure 7. The heat engine scaling provides a strong constraint on the day–night atmospheric circulation. Shown is the vertical velocity in pressure coordinatespredicted by the heat engine scaling (x-axis), compared with the average nightside pressure velocity in the GCM (y-axis). Rapidly rotating atmospheres, a L 12

Ro2 ,

develop inhomogeneous nightsides and can locally sustain smaller pressure velocities (Section 7).

10


depth coordinates and combine it with the Schwarzschildequation for the radiative flux, F (Equation (53)),

¯ ( )w

tbt t

- =⎜ ⎟⎛⎝

⎞⎠

c

g

dT

d

T dF

da, 19

p

( ) ( )t

st

- = -d F

dF

d T

db2 . 19

2

2

4

Given boundary conditions, these equations can be solved for Tand F. The left side of Equation (19a) represents the verticalenergy flux due to subsidence (in W m−2). In the WTG regimesubsidence is how the atmosphere transports heat betweendayside and nightside. Atmospheres with strong subsidence(large w) will tend to have nightsides that are close to anadiabat, while atmospheres with very weak subsidence willtend to approach pure radiative equilibrium on their nightsides(i.e., t »dF d 0).

To solve for T and F on the nightside we need to specify anupper boundary condition. A natural choice is the tropopause,t0, up to which convection rises on the dayside. Above t0 theatmosphere is in pure radiative equilibrium, t =dF d 0. Weassume that the stratosphere is horizontally uniform, whichmeans it has the same temperature structure as in Pierrehumbert(2011b),

( )t=

+⎜ ⎟⎛⎝

⎞⎠T T

1

2. 20strat eq

1 4

We can now specify the boundary conditions for the nightsideatmosphere and Equations (19). Because the WTG approx-imation is a first-order equation and the radiative equation is asecond-order equation, we require three conditions:

( ) ( ) ( )t t=T T a, 210 strat 0

( ) ( )t t =dF d b0, 210

( ) ( )t =F c0. 21LW

The first equation is temperature continuity at the tropopause.The second condition is the stratospheric energy budget, that is,pure radiative equilibrium. The third condition is the nightsidesurface energy budget. Because the nightside surface is inradiative equilibrium with the overlying atmosphere,

( ) ( )t t= F FLW LW , the net radiative flux = - F F F hasto vanish at the surface. The only unknown in these boundaryconditions is the tropopause height t0.The tropopause height, t0, is in turn governed by convection

on the dayside. On the dayside, the convective temperatureprofile (Equation (16)) has to match the stratospheric temper-ature profile (Equation (20)) at t0, so

( )

( )

tt

t

tt

t

=

=+

b

b⎜ ⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

T T

T T

,

1

2. 22

d

d

0

LWstrat 0

0

LWeq

01 4

Finally, we use the TOA energy budget to constrain Td. Theglobal TOA energy budget is

( ) ( ) ( )s = +T F F2 0 0 . 23eq4

day night

The left side is the incoming solar radiation and the right side isthe dayside and nightside outgoing longwave radiation (OLR).To specify these fluxes we note that the stratosphere is inradiative equilibrium, t =dF d 0, so the OLR has to match thenet flux at the tropopause, ( ) ( )t=F F0 0 . The net radiative fluxat the dayside tropopause is

( )

( )

( )

( )

òt s stt

t st

= +¢

´ ¢ -

t t

t

t b

t t

- -

- ¢-

⎛⎝⎜

⎞⎠⎟F T e T

e d T2

. 24

d d0 day4 4

LW

4

eq4 0

LW 0

0

LW

0

Figure 8. A diagram of the two-column radiative-convective-subsiding model. We assume convection sets an adiabatic temperature profile on the dayside, and abalance between radiative cooling and subsidence heating sets the temperature profile on the nightside. In addition, both columns are capped by a horizontally uniformand purely radiative stratosphere. The day–night circulation and the rate of subsidence are governed by the atmospheric heat engine.

11


The first two terms are the upwelling flux at the daysidetropopause (from the surface and atmosphere respectively), andthe third term is downward flux from the stratosphere(Robinson & Catling 2012). The global TOA energy budgettherefore is,

( ) ( )

( )

( )

òs s stt

t st

t

= +¢

´ ¢ - +

t t

t

t b

t t

- -

- ¢-

⎛⎝⎜

⎞⎠⎟T T e T

e d T F

2

2. 25

d deq4 4 4

LW

4

eq4 0

0

LW 0

0

LW

0

Equations (21)–(25) determine the tropopause height t0, thedayside surface temperature Td, and the nightside OLR ( )tF 0 .

Finally, we constrain the pressure velocity, w, on thenightside. We showed in Figure 7 that the heat engine scalingallows us to place an upper bound on w once we account for thefact that atmospheres are imperfect heat engines and once weknow the relative fraction of rising versus subsiding motions,A Aup down (we consider rotation in Section 7). In this sectionwe incorporate these effects via

( )w c= ´p U

a26s s

down

where χ captures the inefficiency of the heat engine as well asthe smallness of A Aup down. We again use Equation (12) tocompute Us, but now we self-consistently solve for the daysidetemperature, Td. To constrain χ we note that the asymmetrybetween rising and sinking motions is set by the tidally lockedgeometry and hence should not vary much between differentsimulations. We use »A A 0.1up down from Figure 1(c) as arepresentative value. Similarly, for slowly rotating atmosphereswe found that w falls between the value predicted by the heatengine and about a factor of four less (Figure 7), so we choosea representative inefficiency of 1/2. Combining these two, wefind c = 1 20. Because rapidly rotating atmospheres tend tohave weaker nightside subsidence (Figure 7), our choice of χ isan upper bound for wdown and will overestimate the day–nightheat transport on rapidly rotating planets. We emphasize that χis the only tunable parameter in our model and is fixed to asingle value. We do not change χ when we compare the RCSmodel with different GCM simulations.

We numerically solve the model to find the nightsidetemperature, T, and radiative flux, F, the dayside surfacetemperature, Td, the nightside surface temperature, Tn, and thetropopause height, t0. The boundary conditions for T and F arespecified at the tropopause and at the surface, so we use ashooting method (Appendix C). We note that the Schwarzs-child equation (Equation (19b)) becomes difficult to solveaccurately in the optically thick limit because the radiativeboundary conditions at the tropopause and surface decouple atlarge tLW (Equations (21b) and (21c)). Nevertheless, theunderlying physics do not change qualitatively once t 1LW .We therefore avoid these issues by limiting our numericalsolver to atmospheres with t 15LW .

Figure 4(b) compares the RCS model with the same slowlyrotating GCM simulation as in Figure 4(a). The RCS modelproduces an adiabatic temperature profile on the dayside and aninversion on the nightside. Compared to the RC model (Figure4(a)), the RCS model produces a colder nightside and a warmerdayside because it does not assume that the day–night heattransport is necessarily highly effective. The predicted temperatures

match the GCM significantly better, particularly on the nightside.The RCS model also places the tropopause at ~p p 0.3s , whereasthe GCM tropopause is higher up at ~p p 0.1s . The hightropopause in the GCM arises because it is set by the deepestconvection and hottest temperatures near the substellar pointinstead of the average dayside temperature (Figure 4(b)), which theRCS model does not account for. Finally, the inversion structure inthe RCS model is somewhat skewed compared with the GCM, andthe inversion occurs higher up in the atmosphere (Figure 4(b)). Theraised inversion is likely due to our assumption of a verticallyconstant value of ω. Nevertheless, given the simplicity of the RCSmodel, we consider the fit between the RCS model and the GCMhighly encouraging. We emphasize that the RCS model is obtainedvia a simple numerical solution, and is conceptually much simpler(and computationally much cheaper) than the full GCM.Next, Figure 9 compares the RC and RCS models with many

GCM simulations. The top row compares the RC model withthe GCM, the bottom row does the same for the RCS model.We note that in rapidly rotating atmospheres ( a L 12

Ro2 ) the

WTG approximation does not hold at higher latitudes, and bothmodels should break down. However, the WTG approximationactually provides a good fit to the nightside structure even atrapid rotation, provided the atmosphere is not too hot or thin( ´ -t t 5 10wave rad

2). We explain this threshold in Sections6 and 7, here we simply mark simulations for which the RCSmodel could break down in red and all other simulations inblue. First, as we already explained for Figure 4(b), the RCSmodel generally predicts warmer daysides than the RC model,which already overestimates dayside temperatures slightly (seeright panels in Figure 9). To quantify the goodness of fitbetween the GCM and our models we compute r2 values for thesimulations marked in blue. For dayside temperatures we find

=r 0.822 with the RC model, and =r 0.232 with the RCSmodel. These values underline that the RC model alreadycaptures the basic structure of the dayside. Improving the fiteven further would require addressing the spatial inhomogene-ity on the dayside (Figure 1), whereas the reduced heattransport in the RCS model actually worsens its dayside fit.Second, as in Figure 3, Figure 9 shows that the RC modeloverpredicts nightside surface temperatures (top left panel). Incontrast, the RCS model fits the GCM values extremely well(bottom left panel). For nightside temperatures we find

=r 0.762 with the RC model, while the RCS model essentiallyreproduces the GCM results with a fit of =r 0.982 .To conclude, we have formulated an RCS model that utilizes

the WTG approximation combined with our heat enginescaling for the large-scale circulation to capture the day–nightheat transport and nightside temperature structure. Our modelcaptures the day–night temperature structure of many GCMsimulations extremely well. We provide an intuitive under-standing of the model results in the next section.

6. TRANSITION TO LARGE DAY–NIGHTTEMPERATURE GRADIENTS

In this section we explain the threshold at which atmospheresof tidally locked rocky planets develop large day–night temper-ature gradients. We point out again that hot Jupiter theoriessuggest this should occur when t t 1wave rad (Section 1). Incontrast, we show that on rocky planets the threshold is up to twoorders of magnitude smaller and temperature gradients becomelarge when ( ) -t t 10wave rad

2 . The small threshold isimportant because it means rocky exoplanets are relatively more

12


sensitive to the parameter t twave rad, so planets that are relativelycool or have thick atmospheres still exhibit large day–nighttemperature differences. Finally, we relate the RCS model back toprevious theories by showing that it reduces to our RC model for

-t t 10wave rad4 and to Wordsworth’s (2015) result for

t t 1wave rad and t 1LW .To start, we consider the thermodynamic equation under

the WTG approximation (Equation (19a)). The WTG approx-imation expresses a balance between subsidence heatingand radiative cooling, and we nondimensionalize it usingˆ =T T Teq and ˆ ( )s=F F Teq

4 . We find that the ratio ofsubsidence heating to radiative cooling is governed by twoparameters,

ˆ ˆ ˆ( )

tbt t

- =⎛⎝⎜

⎞⎠⎟

dT

d

T t

t

dF

d, 27sub

rad

where ( )b = R c np sets the adiabatic lapse rate, wºt pssub is acharacteristic subsidence timescale for a parcel of air, and =trad

( )sp c g Ts p eq3 is the radiative cooling timescale. Equation (27) is

the three-dimensional equivalent of the WTG scaling developed

by Perez-Becker & Showman (2013) using the shallow-waterequations. The lapse rate parameter, β, is always of order unitywhereas the subsidence timescale, tsub, is an emergent timescaleset by the large-scale dynamics. When t t 1sub rad radiativecooling is inefficient compared with subsidence heating, thenightside atmosphere is close to an adiabat, and the day–nighttemperature differences are small. When t t 1sub rad a parcel ofair cools significantly as it descends, the nightside developsinversions, and the day–night differences are large. Finally, for

t t 1sub rad the nightside is close to radiative equilibrium.The transition to large day–night temperature gradients occurs

at a wave-to-radiative timescale threshold of ~ -t t 10wave rad2.

Figure 10 shows temperatures from the RCS model as a functionof the timescale ratio t twave rad and optical thickness, tLW. Weassume a representative rocky planet scenario,6 and plot thedayside surface temperature,á ñTs day, nightside surface temperature,á ñTs night, and the atmospheric temperature just above the nightside

Figure 9. Comparison of surface temperatures predicted by the radiative-convective (RC) and radiative-convective-subsiding (RCS) models with many GCMsimulations (N = 241). The red dots represent simulations that are both rapidly rotating and have hot/thin atmospheres ( a L 12

Ro2 and t twave rad exceeds threshold

from Equation (34)), the blue dots show all other simulations. Top left: the average nightside temperature, RC model vs. GCM. Top right: the average daysidetemperature, RC model vs. GCM. Bottom left: the average nightside temperature, RCS model vs. GCM. Bottom right: the average dayside temperature, RCS modelvs. GCM. The RCS model captures nightside temperatures much better than the RC model. The RCS model breaks down only for atmosphere that are both rapidlyrotating and hot/thin (red dots; see Section 7).

6 We assume an Earth-sized planet, = Åa a , with an N2-dominatedatmosphere, ( ) ( )=R c R c, ,p p N2.

13


surface, á ñTatm night. Because the atmospheric temperature on thedayside is strongly coupled to the surface via convection,á ñ » á ñT Tsatm day day, the difference between á ñTs day and á ñTatm nightalso shows the day–night temperature gradient in the lowest partof the atmosphere. First, it is clear from Figure 10 that theatmospheric temperature gradient is small when twave

-t 10rad2. The transition to large temperature gradients spans

many orders of magnitude, but we take ~ -t t 10wave rad2 as a

representative value that ensures temperature gradients are largefor larger values oft twave rad. Second, once the day–nightatmospheric gradients are large their magnitude additionallydepends on tLW, with optically thicker atmospheres having largermaximum temperature gradients (compare maximum differencebetween á ñTs day and á ñTatm night). Third, because the nightsidesurface is in radiative equilibrium with the overlying atmosphere,the gradient in surface temperatures is at least as big as thegradient in atmospheric temperatures. However, it can be muchlarger in the optically thin limit because the nightside atmospherebecomes ineffective at radiatively heating the nightside surface. Atlow optical thickness the nightside surface is much colder than theoverlying air (Figure 10(a)), while at high optical thickness thenightside surface is closely tied to the overlying air temperature(Figure 10(c)).

Next, we explain why atmospheres develop large temper-ature gradients at ~ -t t 10wave rad

2. As we showed above, thenightside temperature structure is controlled by the ratio ofsubsidence to radiative timescales, t tsub rad. Here we analyzethe processes that control tsub. Using the heat engine and thearea ratio between upward and downward motions, we alreadyfound the pressure velocity on the nightside wdown.Equation (26) allows us to write

( )w c

= =tp a

U. 28s

ssub

down

Next, we scale the surface wind speed, Us, from the heat engine(Equation (12))

( ) ( )

( )

= - ´ - ´

» ´

t- ⎜ ⎟

⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥

⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

U T T ec

R

t

tc

Uc

R

t

tc

1 1 ,

,

29

s dp

sp

eq

2drag

rad

1 3

wave

2 3drag

rad

1 3

wave

LW

where in the second step we assumed an optically thickatmosphere, t 1LW , so that all incoming stellar flux goestoward driving atmospheric motion, - »t-e1 1LW . We alsodrop the dependence on the dayside temperature from

-T T 1d eq . We do so because in the optically thick limit Tdis approximately given by Equation (1), so ( )- »T T 1d eq

1 3

( [ ] )t bG + -b -1 4 1LW1 4 1 3 which is always of order unity.7

We combine Equations (28) and (29) and find

( )c

=⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟t

R

c

t

tt

1. 30

psub

2 3rad

drag

1 3

wave

Equation (30) gives us the subsidence time on the nightside.The day–night temperature gradients will be small if a parcel ofair cools slower than it sinks, >t trad sub. Conversely, the day–night temperature gradients will be large if a parcel cools fasterthan it sinks, <t trad sub. The threshold between these two

Figure 10. Day–night temperature gradients are large once the wave-to-radiative timescale ratio t twave rad exceeds the threshold from Equation (34) (vertical dashedlines). The panels show dayside surface temperature, á ñTs day, nightside surface temperature, á ñTs night, and the bottom-most atmospheric temperature on the nightside,á ñTatm night, from the radiative-convective-subsiding model (RCS, Section 5). In all cases, cool/thick atmospheres with -t t 10wave rad

4 have small temperaturegradients between dayside surface and nightside atmosphere. Surface temperature gradients additionally depend on optical thickness, and even cool/thick atmospherescan have large day–night surface temperature gradients if t 1LW (left panel). Black symbols show the nightside surface temperatures predicted by the radiative-convective model (RC), and the asymptotic scaling of Wordsworth (2015); the RCS model reduces to either in the limits -t t 10wave rad

4 and t t 1wave rad .

7 For example, assuming t = 2LW and a diatomic gas without pressurebroadening (b = 2 7), ( )- »T T 1 0.6d eq

1 3 . The gamma function[ ]bG + -1 4 1 4 does not vary significantly over the plausible range of

atmospheric gases. Similarly, the dependency on tbLW3 is negligible because

the exponent b 3 is always small.

14


regimes is

( ) ( )

c

c

ct

~

~

~

~

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

t t

tR

c

t

tt

tR

c tt

tR

c

t

tt

1

1 1

1for 1 . 31

p

p

p

rad sub

rad

2 3rad

drag

1 3

wave

rad2 3

2 3

drag

1 3

wave

rad

3 2wave

drag

1 2

wave LW

We can find a similar threshold for optically thin atmo-spheres, (t < 1LW ). We note that the standard radiativetimescale, ( )s=t c p g Tp srad eq

3 , is the cooling timescale of anoptically thick column of air. In contrast, an optically thincolumn of air only emits a radiative flux t s~ ´ TLW eq

4 so itsradiative cooling timescale is

( )t

=tt

. 32rad,thinrad

LW

WTG balance (Equation (27)) in the optically thin regime isstill governed by the ratio of subsidence to radiative timescales,but now trad has to be replaced by trad,thin.

To find the subsidence timescale, tsub, in the optically thinlimit we note that optically thin atmospheres are also lessefficient heat engines. The lower efficiency arises because,for t 1LW , the surface re-emits most of the incomingstellar flux directly back to space and only a fraction,

( )t t- = - - + »t-e1 1 1 ...LW LWLW , of the stellar flux isavailable to drive atmospheric motions. The dayside temp-erature, Td, is approximately constant in the optically thincase (Figure 3), so tsub is

( )c

=⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟t

R

c

t

tt

1. 33

psub

2 3rad,thin

drag

1 3

wave

Equation (33) only differs from Equation (30) through the useof trad,thin instead of trad. Our result for large temperaturegradients therefore also holds for optically thin atmospheres,once we replace trad with trad,thin.

To compare our result with the result for hot Jupiters weexpress the criterion for an atmosphere to develop largetemperature gradients in terms of the wave-to-radiative time-scale ratio t twave rad. The day–night atmospheric temperaturegradients are large once

( )c t

ct

t

´

´ <

⎧

⎨⎪⎪

⎩⎪⎪

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

t

t

c

R

t

t

c

R

t

t

if 1,

if 1.

34

p

p

wave

rad

3 2 drag

wave

1 2

LW

3 2

LW

drag

wave

1 2

LW

We emphasize that Equation (34) only ensures that atmospherictemperature gradients are large, but they remain significantuntil t twave rad becomes extremely small (Figure 10).

We draw three important conclusions from Equation (34).First, in the optically thick case the right hand side is dominatedby c » 1 20 (Section 5) while the other quantities do not varymuch in most cases of interest. The small value of χ causes thethreshold for large day–night temperature gradients to generallybe much smaller than one. As a representative high mean-

molecular-weight (MMW) scenario, we consider an N2

atmosphere with =T K300eq . In this case =c R 7 2p and=t t 1.4drag wave (Appendix A), so temperature gradients are

large when

( ) ´ -⎛⎝⎜

⎞⎠⎟

t

t5 10 . 35wave

rad high MMW

2

Our result explains why rocky planets develop large atmo-spheric temperature gradients at a threshold almost two ordersof magnitudes smaller than what one would expect based onthe results for hot Jupiters, t t 1wave rad .Second, hot H2-dominated atmospheres are a notable

exception to the first result and develop day–night temperaturegradients at larger values of t twave rad. The larger thresholdarises because the H2-dominated atmospheres have larger scaleheights than high-MMW atmospheres, which increases thedrag time, tdrag. For example, we consider an H2 atmospherewith =T K600eq . In this case =t t 40drag wave (Appendix A) sotemperature gradients are large when

( )⎛⎝⎜

⎞⎠⎟

t

t0.2. 36

H

wave

rad 2

The wave-to-radiative timescale threshold in this case is afactor of four larger than for high-MMW atmospheres, but it isstill almost an order of magnitude smaller than the result for hotJupiters, t t 1wave rad .Third, optically thin atmospheres are less prone to develop-

ing day–night temperature gradients than optically thickatmospheres because optically thin atmospheres cool lesseffectively. Although optically thin atmospheres are also lessefficient heat engines, the radiative effect dominates because

tµt 1rad,thin LW whereas tµ µt t 1sub rad,thin1 3

LW1 3. The WTG

approximation therefore holds even better in optically thinatmospheres than in optically thick ones. It also explains whythe stratospheres of our simulations, where the atmospherebecomes optically thin (Robinson & Catling 2012), are muchmore horizontally homogeneous than the lower atmosphere(Figure 1).We can now relate the RCS model to the results in Section 3.

First, when t t 1sub rad radiative cooling is inefficientcompared to subsidence heating. In this limit the sinkingparcels of air on the nightside remain close to an adiabat andthe RCS model reduces to the RC model.8 Figure 10 showsnightside surface temperatures in both models and demon-strates that the RCS model reduces to the RC model at arepresentative value of -t t 10wave rad

4 (compare blue lines toblack dots). Second, when t t 1sub rad radiative cooling ismuch stronger than subsidence heating. In this limit the WTGapproximation (Equation (27)) becomes t »dF d 0, so thenightside is in purely radiative equilibrium and F is verticallyconstant. To still satisfy the nightside surface budget(Equation (21c)), F has to be zero. The Schwarzschild equation(Equation (19b)) shows that, in this case, ( )s t »d T d 04 ,which means the nightside becomes vertically isothermal witha temperature that is set by the overlying tropopausetemperature, ( )tTstrat 0 . A lower bound for Tstrat (see

8 Because the RCS model also includes a stratosphere, it predicts slightlycolder nightsides in the limit < -t t 10wave rad

4 than the RC model, but theeffect is small (the black dot is slightly above the blue line in Figure 10, rightpanel).

15


Equation (20)) is given by the skin temperature º -T T2skin1 4

eq(Pierrehumbert 2011b). In the optically thin limit the nightsidesurface energy budget is then equal tos t s t s= ´ = ´T T T 2n skin

4LW

4LW eq

4 , and we recover Words-worth (2015)ʼs result ( )t=T T 2n eq LW

1 4. Figure 10 shows thatthe nightside temperature in the RCS model reduces to thislimit at a representative value of t t 1wave rad (compare blacksquare and blue line in the left panel).

Up to now we have focused on slowly rotating planets<a L 12

Ro2 . Next, we consider the effects of rapid rotation,

and how they interact with the threshold for large day–nighttemperature gradients.

7. EFFECTS OF RAPID ROTATION ON TEMPERATURESTRUCTURE

In this section we use GCM simulations to address how rapidrotation affects the circulation and temperature structure. Leconteet al. (2013) showed that tidally locked planets develop drasticallydifferent circulations when a L 12

Ro2 because equatorial waves

are not able to freely propagate into high latitudes once theplanetary radius, a, is larger than the equatorial Rossby radius,LRo. Rapidly rotating planets then develop standing Rossby andKelvin wave patterns. The standing wave patterns lead to strongequatorial superrotation, an eastward offset of the equatorial hotspot, and off-equatorial cold vortices on the nightside (Mat-suno 1966; Showman & Polvani 2011). Here we also find that thecirculation regime changes at ~a L 12

Ro2 . However, while rapid

rotation drastically alters the flow field, the effect on temperaturestructure is small unless the atmosphere is also prone todeveloping strong temperature gradients, ( ) -t t 10wave rad

2 .We perform a set of GCM simulations in which we vary

a L2Ro2 and t twave rad while keeping all other parameters fixed.

We vary a L2Ro2 by changing the rotation rate, Ω, and t twave rad

by changing the surface pressure, ps. All other parameters arefixed to the same values as the reference simulation in Figure 1,that is, = Åa a , =T K283eq , ( ) ( )=R c R c, ,p p N2, and t = 1LW .We explore ( )=a L 0.1, 0.5, 12

Ro2 and (= -t t 10 ,wave rad

3

)- -10 , 102 1 , which correspond to p2 60 days pW 2 6days and 5 bar p 0.05 bars .

We find that rapid rotation has a large effect on the circulation,but its effect on the temperature structure is small unless t twave radalso exceeds the threshold ´ -t t 5 10wave rad

2 from theprevious section. Figure 11 shows 2D maps of the circulationand temperatures in the upper atmosphere. The wind andtemperatures are mass-weighted averages taken over 0.3

p p 0.4s , and the substellar point is located at longitudel = 270 . Slowly rotating simulations are shown in the leftcolumn of Figure 11. As expected, the circulation consists of asubstellar-to-antistellar flow. At small values of t twave rad theday–night temperature differences are small, but the atmos-phere develops large day–night temperature gradients at twave

= -t 10rad1, consistent with our results in Section 6. The top row

of Figure 11 shows simulations with small t twave rad. As therotation rate increases, the atmospheric circulation changesdrastically. A strong equatorial jet develops and the nightsideatmosphere additionally develops standing Rossby waves in theform of off-equatorial vortices (Showman & Polvani 2011).Nevertheless, as long as = -t t 10wave rad

3, the maximumhorizontal temperature difference at =a L 12

Ro2 only reaches

T0.05 eq, or ∼15 K.

Figure 11 underlines that the day–night atmospheric temper-ature gradients are primarily controlled by t twave rad. However,the effect of rapid rotation can strongly enhance the temperaturegradients in the form of eastward hot spot offsets and coldnightside vortices. The slowly rotating simulation with a thinatmosphere ( =a L 0.12

Ro2 , = -t t 10 ;wave rad

1 bottom left) hasits hottest point located at the substellar point and a maximumhorizontal temperature difference of T0.2 eq, or ∼65K. Incontrast, the simulation with the same value of t twave rad but atrapid rotation shows an eastward hot spot offset and asignificantly larger maximal horizontal temperature differenceof T0.4 eq, or ∼110 K (bottom right). In the following section weconsider what our results imply for future observations.

8. IMPLICATIONS FOR OBSERVATIONS

Figure 12 summarizes some implications of our results forobservations of rocky exoplanets. We showed that atmosphericday–night temperature contrasts strongly depend on the parametert twave rad/ = ( )´a R c RTp eq/ ( )s´ g T c pp seq

3 µT pseq5 2 . The

planetary radius, a, and surface gravity, g, vary relatively little forplausible rocky planets, which means the day–night temperaturedifferences are to first order controlled by the equilibriumtemperature, Teq, the surface pressure, ps, and whether or notthe atmosphere is made of H2 (via R and cp). In Figure 12 weconsider these parameters for a GJ 1132b-sized planet9 withhypothetical CO2 and H2 atmospheres. The red region indicateswhen the atmosphere is hot/thin and develops large day–nightatmospheric temperature gradients (Equation (34) for t 1LW ).The blue region indicates when the atmosphere is cool/thick andthe day–night atmospheric temperature gradients become negli-gible ( -t t 10wave rad

4 from Section 6). Part of the CO2 phasespace is unstable to atmospheric collapse, which occurs when thenightside surface is cold enough for CO2 to condense. Wedelineate atmospheric collapse using two approaches. First, thesolid black line shows the empirical fit from Wordsworth (2015),who used a GCM with full radiative transfer to compute collapsethresholds up to =T K367eq . Second, we use the RCS model tocompute when nightside surface temperatures fall below thecondensation temperature of CO2. To specify the optical thicknesswe use Equation (6) and assume (t 1 bar ) = 1 and n = 1.Although we do not include non-gray effects, the RCS model(dashed black line) closely fits the GCM results (solid black line)over the range of parameters explored by Wordsworth (2015). Assuch, we consider the RCS model appropriate for predictingatmospheric collapse (also see Section 9). We repeat a similarcomputation for H2 atmospheres, and find that the entire phasespace in Figure 12(b) is stable against collapse.10 The symbols atthe bottom of Figure 12 show equilibrium temperatures11 of tworecently discovered rocky planets and of a hypothetical tidallylocked planet at Venus’ present-day orbit (Berta-Thompsonet al. 2015; Motalebi et al. 2015). Finally, we found thattemperature structure can be affected by rapid rotation. We markall rapidly rotating planet scenarios with a L 12

Ro using starsymbols (∗). For these cases we expect that strong rotationaleffects, such as large eastward hot spot offsets or cold nightsidevortices, occur inside the red region (see Section 7).

9 We assume = Åa g a, 1.16 , 11.7 m s−2 (Berta-Thompson et al. 2015).10 We use the Solar opacity value in Menou (2012a) and n = 1, and computenightside surface temperatures with the RCS model. We find that nightsidetemperatures always exceed the critical point of H2, 33.2 K.11 We assume a planetary albedo of zero.

16


Figure 11. Rapid rotation ( a L 12Ro2 ) does not have a strong effect on temperature structure unless the atmosphere is also hot or thin ( > -t t 10wave rad

2). Shown are2D temperature and wind fields, averaged over the upper troposphere ( p p0.3 0.4s ). Rotation increases from left to right, the wave-to-radiative timescale ratioincreases from top to bottom. Increased rotation changes the circulation drastically, from a day–night flow at slow rotation (left) to an equatorially superrotating jet andcold nightside vortices at rapid rotation (right). However, at low t twave rad, temperature gradients are small, even if rotation is rapid (top right). Large temperaturegradients, eastward hot spot offsets, and cold nightside vortices only emerge once an atmosphere is both hot/thin and rotates rapidly (bottom right). The substellarpoint is located at 270° longitude.

Figure 12. CO2 atmospheres are more likely to develop large temperature gradients than H2 atmospheres. Atmospheric day–night temperature gradients are negligibleinside the blue region ( -t t 10wave rad

4) and are large inside the red region (Equation (34) for t 1LW ). CO2 atmospheres collapse inside the gray region (solid line:empirical fit to GCM results from Wordsworth (2015); dotted line: calculated using our RCS model). Bottom symbols show equilibrium temperatures of two nearbyrocky planets and of a hypothetical tidally locked Venus; (∗) marks scenarios for which rotational effects would additionally be important ( a L 12

Ro2 ). The shown

thresholds assume a GJ 1132b-sized planet ( = Åa g a, 1.16 , 11.7 m s−2).

17


Figure 12 allows us to make some tentative predictions.First, with a high MMW atmosphere like CO2, GJ 1132b andHD 219134b would have non-negligible day–night temperaturegradients. This conclusion holds even for surface pressures ashigh as that of Venus (ps = 92 bar). Second, GJ 1132b and HD219134b with CO2 atmospheres both satisfy the criterion forrapid rotation ( a L 12

Ro2 ). Should observations detect a large

eastward hot spot offset, it would favor surface pressures lessthan ( ) 1 bar (inside the red region). Third, Figure 12(a) showsthat a CO2 atmosphere with surface pressure comparable to thatof Mars ( = ´ -p 6 10s

3 bar) would be close to collapse on GJ1132b. We note that our collapse calculation does not accountfor rotational effects, and cold nightside vortices (Figure 11)would allow the atmosphere to collapse at even higherpressures. Fourth, if these planets managed to retain H2-dominated atmospheres against atmospheric escape, theywould be stable against collapse and exhibit much smallerday–night temperature differences than similar CO2 atmo-spheres. The increased stability and smaller temperaturegradient is due to a combination of H2ʼs large heat capacity,cp, which increases the radiative timescale, trad (Menou 2012a),its large gas constant, R, which increases the speed ofatmospheric waves, cwave, and thus decreases the wavetimescale, twave (Heng & Kopparla 2012), and its increasedscale height, which decreases the effect of friction(Appendix A). Fifth, H2-dominated atmospheres would besignificantly less affected by rotation than CO2 atmospheres.The smaller effect of rotation is also due to the reduced wavetimescale, twave, in H2 atmospheres. For example, assuming GJ1132b’(s) equilibrium temperature =T K579 ,eq the character-istic speed of gravity waves in a CO2 atmosphere is

= ´ =c R c RT 158pwave eq ms−1, whereas in a H2 atmos-phere = -c 838 m s .wave

1 It follows that the nondimensionalRossby radius, = Wa L a c22

Ro2

wave, is about five timessmaller in a H2 atmosphere. Our results also imply that rockyplanets with H2-dominated atmospheres are less likely toexhibit eastward hot spot offsets and cold nightside vortices.These predictions are qualitative because they do not considerthe optical thickness, tLW, which helps set the magnitude of theday–night temperature gradient (Figure 10). Quantitativelyinterpreting an observed day–night temperature gradient alsorequires constraining tLW, for example via transit spectroscopy(see Koll & Abbot 2015).

One way of distinguishing the scenarios in Figure 12 wouldbe through combined transit spectroscopy and thermal phasecurve observations with JWST. Previous feasibility studies havetended to emphasize the transit technique (e.g., Beichmanet al. 2014; Batalha et al. 2015). Here we compare the signal-to-noise ratio (S/N) that can be achieved by spending the sameamount of JWST time on low spectral-resolution transit andbroadband phase curve observations. We find that it would takeabout as much time to measure the broadband mid-IR phasecurve of a short-period rocky exoplanet as it would to detectmolecular signatures in its atmospheres through near-IR transitspectroscopy. The basic science goal for transit observationswould be to detect a molecular species from its spectralimprint; a flat spectrum could alternately be a cloudyatmosphere or no atmosphere. The basic science goal forphase curve observations would be to detect the day–night fluxdifference of a bare rock. An observed flux difference lowerthan that of a bare rock would imply the presence of anatmosphere thick enough to modify the day–night temperature

contrast (outside the red region in Figure 12). Similarly, hot/cold spot offsets would imply the presence of an atmospherethat is hot or thin enough that its thermal structure issignificantly affected by rotation (see above). We consider GJ1132b with a CO2-dominated atmosphere as a representativetarget. We assume GJ 1132b is tidally locked, which could beverified using optical phase curves with TESS (cf. Fujiiet al. 2014). We compute transit signals following Cowanet al. (2015), but assume that spectral features in the near-IRcause an absorption difference of three scale heights and have atypical width of 0.1 μm (see Table 2, Kaltenegger &Traub 2009). We compute the phase curve signal of a barerock following Koll & Abbot (2015). We estimate JWSTʼsprecision in the near-IR (1–4 μm in 0.1 μm bins, R ∼ 25 onNIRSpec) using the photon noise limit (see Koll &Abbot 2015). We similarly estimate the precision in the mid-IR (16.5–19.5 μm broadband, F1800W on MIRI) assumingphoton noise, but account for the imperfect instrumentthroughput of 1/3 (Glasse et al. 2010). We assume that bothtechniques bin photons over the length of one transit(45 minutes) and we multiply the noise by 2 to account forthe fact that both techniques compare two snapshots in time.We assume that a single transit measurement consists ofobserving the primary eclipse and an equal out-of-transitbaseline (cf. Kreidberg et al. 2014). We check our transitestimate by comparing our S/N with the detailed calculationsin Batalha et al. (2015), and find that we can reproduce theirresults up to a factor of two (not shown). We also note that ourestimate of GJ 1132b’(s) thermal emission is slightly higherthan the signal in Berta-Thompson et al. (2015), because theobserver-projected dayside temperature of a bare rock is higherthan its equilibrium temperature by ( ) »8 3 1.281 4 (Koll &Abbot 2015).Table 2 shows our results. Similar to previous estimates

(e.g., Batalha et al. 2015; Cowan et al. 2015), we find that asingle transit would not be sufficient to conclusively identifymolecular absorption features (S/N ∼ 1). The low S/N ariseslargely because of the high MMW atmosphere; for comparison,an H2 atmosphere on GJ 1132b should be detectable in a singletransit with S/N ∼ 22. For a CO2 atmosphere, 13 repeatedtransit observations would reduce the noise sufficiently to allowspectral features to be discerned with S/N ∼ 4. The time ittakes to measure 13 repeated transits of GJ 1132b is also equalto the time it takes to measure one half-orbit phase curve (fromtransit to secondary eclipse). We find that the thermal emissionof a bare rock would be detectable with a comparable S/N ∼ 4.We conclude that characterizing high MMW atmospheres ofrocky exoplanets will require relatively large investments ofJWST time. If such observations are pursued, however, thenthermal phase curves are a feasible technique that would yieldimportant complementary information about these planets (Koll& Abbot 2015).

9. DISCUSSION

The heat engine framework is well-established for Earth’satmosphere (e.g., Peixoto & Oort 1984). Similarly, Goodman(2009) pointed out that hot Jupiters can be viewed as heatengines, but did not develop his insight more quantitatively.Here we have demonstrated that the atmospheres of rockyexoplanets act as heat engines, which allowed us to develop anew constraint on their day–night circulations. We also foundthat surface wind speeds in most of our GCM simulations are

18


about a factor of two smaller than the value predicted by theheat engine (Figure 6). Because the work scales with the cubeof the surface wind speed, our simulations produce

( )~ =1 2 1 83 , as much work as an ideal heat engine.Interestingly, Earth’s atmospheric heat engine also producesabout an order of magnitude less work than its ideal limit, andtherefore has a similar inefficiency as our dry and tidally lockedsimulations (Peixoto & Oort 1984). Our result seems to be atodds with the usual understanding that the inefficiency ofEarth’s atmospheric heat engine is caused by its hydrologicalcycle (Pauluis et al. 2000; Pauluis 2010), and also raises thequestion of whether our scaling can be generalized to planetsthat are not tidally locked. We hope to address these issues infuture work. Our results also strongly suggest that hot Jupitersshould obey similar constraints as rocky planets. For example,using the heat engine framework, it might be possible toconstrain the day–night overturning circulation, which controlsthe vertical mixing and chemical equilibrium of hot Jupiteratmospheres. However, modeling these atmospheres as heatengines will require a better understanding of the mechanismsthrough which they dissipate kinetic energy, which couldinclude magneto-hydrodynamic drag, shocks, or shear instabil-ities (Li & Goodman 2010; Menou 2012b; Fromanget al. 2016).

Our results allow us to interpret previous GCM results thathave not been fully explained yet. First, Merlis & Schneider(2010) explored Earth-like atmospheres at different rotationrates. They found that although the strength of superrotation isstrongly dependent on rotation rate, the day–night surfacetemperature gradients are mostly insensitive to rotation rate(their Figure 15). Our results explain why: Merlis & Schneidervaried rotation rates while keeping the stellar flux fixed atEarth’s value. Their simulations were therefore in the rapidlyrotating ( >a L 12

Ro2 ) but relatively cool/thick regime

( < -t t 10wave rad2) in which temperature structure is not

strongly sensitive to rotation (Figure 11). Second, in Koll &Abbot (2015) we found that thermal phase curves are mainlysensitive to the nondimensional parameters t twave rad and tLW.Our result only broke down for hot/thin and rapidly rotating

atmospheres, -t t 10wave rad2 and a L 12

Ro2 . Our results

here explain both the wave-to-radiative timescale threshold of~ -t t 10wave rad

2 and why rotation is relatively unimportant(Figures 10 and 11).There are additional physical effects that might affect our

conclusions. We assume a broadband gray radiative transfer,but a wide range of plausible atmospheric compositions featuresignificant spectral window regions. As already noted byLeconte et al. (2013) and Wordsworth (2015), window regionsallow the nightside surface to cool even more effectively,which would increase the day–night surface temperaturegradients compared to that predicted by our gray models. Atthe same time, large spectral window regions would increasethe atmosphere’s radiative cooling timescale and thus reduceatmospheric temperature gradients, similar to the optically thincases we discussed in Section 6. We also did not considershortwave absorption. Shortwave absorption will shift heatingto lower pressures, which would decrease the heat intaketemperature of the atmospheric heat engine and reduce theatmospheric circulation strength. We therefore expect our heatengine theory to be an upper bound on wind speeds.Many planets might be able to retain a hydrologic cycle (e.g.,

H2O inside the habitable zone, or CH4 on Titan-like planets)against atmospheric escape and nightside collapse. Besideschanging the atmosphere’s radiative properties (e.g., H2Oeffectively absorbs both in the shortwave and longwave),condensation would also modify the atmospheric dynamics.Moist GCM simulations indicate that the temperature andcirculation structure sketched out in Figure 8 could still applyqualitatively (Merlis & Schneider 2010; Yang et al. 2013), butwith several modifications. First, latent heat transport wouldreduce the day–night temperature gradient compared to dryatmospheres (Leconte et al. 2013). Second, moist convectionwould lead to thick cloud cover on the dayside and coulddrastically change a planet’s appearance to remote observers(Fortney 2005; Yang et al. 2013). Third, dry atmospheresdevelop strongly turbulent daysides. The friction associatedwith this dry convection allows the nightside temperaturestructure to decouple from regions of convection (Figure 1,Voigt et al. 2012). In contrast, moist atmospheres such asEarth’s tropics maintain an adiabatic temperature profilethrough deep moist convection, while the dry turbulentboundary layer is relatively shallow. We therefore expect thatmoist atmospheres would be less dominated by friction, andwould be even better captured by WTG models similar to ourRC model (Section 3).It is an open question of how many rocky planets around

M-stars will actually be tidally locked. Leconte et al. (2015)found that thermal tides in relatively thick atmospheres ( p 1sbar) can prevent habitable-zone planets around early M-dwarfsfrom reaching a tidally locked state. Although thermal tidescould limit the application of our results to planets on longer-period orbits, they are less likely to apply to planets around lateM-dwarfs or hot exoplanets like GJ 1132b. Moreover, given thatrocky exoplanets are extremely common, we also expect thatfuture discoveries will find rocky exoplanets in a wide range ofrotational states. Optical phase curves could constrain therotation rates of these planets without relying on models (Fujiiet al. 2014), while future theoretical work should consider theconnection between the tidally locked limit we considered hereand planets in higher-order spin–orbit resonances.

Table 2Transit vs. Phase Curve Observations of GJ 1132b with JWST

Method Observation TimeSignal(ppm)

Noise(ppm) S/N

Single transita one transitb = 90 minutes 19.9 19.7 1Stacked

transitsa13 transits = 19.5 hr 19.9 5.5 4

Thermal phasecurvec

one half-orbit = 19.5 hr 373 84 4

Notes. Transit spectroscopy and thermal phase curve measurements of a planetlike GJ 1132b will require similar amounts of JWST observation time. Theshown signal-to-noise ratios (S/N) are estimates for the most basicobservational goals: detecting molecular features in low-resolution near-IRtransit spectra, and detecting the day–night thermal emission contrast of a barerock in the mid-IR. We compute signals following Cowan et al. (2015) andKoll & Abbot (2015). We estimate noise assuming photon-limited precision,but include imperfect instrument throughput for MIRI (see Section 8).a 1–4 μm, NIRSpec, R ∼ 25, CO2-dominated atmosphere.b We assume a measurement lasts 45 minutes in-transit, plus 45 minutes out-of-transit baseline.c 16.5–19.5 μm, MIRI, broadband.

19


10. CONCLUSIONS

We have developed a series of theoretical models tounderstand the basic temperature structure and large-scalecirculations of tidally locked planets with dry atmospheres.These models are able to capture and predict many fundamentalaspects of much more complex GCM simulations, includingthe atmospheric temperature structure, dayside and nightsidesurface temperatures, as well as large-scale wind speeds. Wedraw the following conclusions from our work:

1. Our RC model describes tidally locked atmospheres withefficient day–night heat transport and applies in the limitof cool and thick atmospheres ( -t t 10wave rad

4). Itcaptures the basic temperature structure of tidally lockedplanets and extends the asymptotic theory for opticallythick atmospheres (t 1LW , Pierrehumbert 2011b) toarbitrary optical thickness.

2. The atmospheres of dry, tidally locked exoplanets act asglobal heat engines. Our heat engine scaling places strongconstraints on the day–night circulation strength of tidallylocked atmospheres.

3. Our RCS model describes tidally locked atmosphereswith limited day–night heat transport. It extends both ourRC model and the asymptotic theory for optically thinatmospheres (t 1LW , Wordsworth 2015), and capturesthe dynamics of a wide range of complex GCMsimulations. It breaks down in the limit of atmospheresthat are both rapidly rotating ( a L 12

Ro2 ) and hot/

thin ( ( )> -t t 10wave rad2 ).

4. Like hot Jupiters, the day–night atmospheric temperaturegradients of rocky exoplanets become larger once parcelsof air take longer to subside than to cool radiatively.Unlike hot Jupiters, the timescale for subsidence on rockyplanets is severely increased by the limited heat engineefficiency and the areal asymmetry between convectionand subsidence. Rocky planets develop large day–nightatmospheric temperature gradients when

( )c t

ct

t

´

´ <

⎧

⎨⎪⎪

⎩⎪⎪

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

t

t

c

R

t

t

c

R

t

t

if 1,

if 1.

37

p

p

wave

rad

3 2 drag

wave

1 2

LW

3 2

LW

drag

wave

1 2

LW

Optically thin atmospheres cool inefficiently, whichmakes them less likely to develop large temperaturegradients than optically thick atmospheres.

5. Rapid rotation ( a L 12Ro2 ) only has a strong influence

on temperature structure if the wave-to-radiative time-scale exceeds the above ratio, ( ) -t t 10wave rad

2 . Oncerotation is important its effects cannot be ignored for adetailed understanding of a planet’s atmosphere, includ-ing its thermal phase curve signature and the potential foratmospheric collapse.

6. Short-period rocky exoplanets with high MMW atmo-spheres and surface pressures of1 bar will likely exhibitsignificant day–night temperature gradients. Thermalphase curve observations of such planets will requiresimilar amounts of JWST time as transit observations.

Our work was completed with resources provided by theUniversity of Chicago Research Computing Center and supportfrom the NASA Astrobiology Institute's Virtual Planetary

Laboratory, which is supported by NASA under cooperativeagreement NNH05ZDA001C. We thank Malte Jansen andFeng Ding for insightful discussions, and an anonymousreviewer for their feedback. D.D.B. Koll was supported by aWilliam Rainey Harper dissertation fellowship.

APPENDIX ADRAG TIMESCALE

We start with the nondimensional parameters that we derivedin Koll & Abbot (2015):

( )t t⎛⎝⎜

⎞⎠⎟

R

c

a

L

t

t

C a

H, , , , , . 38

p

D2

Ro2

wave

radSW LW

Theoretical work on the atmospheric dynamics of hot Jupitersuses similar wave and radiative timescales (e.g., Perez-Becker& Showman 2013), but additionally introduces a dragtimescale. This is because the drag mechanisms on hot Jupitersare still not well-constrained, and friction is often parameter-ized as Rayleigh friction with an unknown damping timescale.To facilitate comparison of our work with the hot Jupiterliterature and to examine the importance of drag in theatmospheres of rocky planets, we rewrite the last of our sixnondimensional parameters as a ratio of wave-over-dragtimescales.Models of terrestrial atmospheres (including FMS) often

parameterize boundary layer friction as vertical momentumdiffusion with a source term that is quadratic in wind speed.The horizontal momentum equation takes the form

( )= +

¶¶

uD

Dtg

p... , 39m

where u is the horizontal wind speed, m is the diffusivemomentum flux due to surface drag, and the source term is

( ) ∣ ∣ r= u up Cm s D s s s. We take a vertical average across theboundary layer to find the average acceleration due to drag:

[ ( ) ( )]

( )

∣ ∣( )

ò ò

r

-= +

-¶¶

= +-

´ -

= +-

= +-

u

u

u

u u u

p p

D

Dtdp

g

p p pdp

D

Dt

g

p pp p

D

Dt

g p

p p

D

Dt

gC

p p

1...

...

...

... . 40

s p

p

s p

pm

sm s m

m s

s

D s s s

s

BL BL

BLBL

BL

BL

s s

BL BL

Here pBL denotes the top pressure level of the boundary layer,and we used the fact that drag has to disappear at the upperedge of the boundary layer, ( ) =p 0m BL . We then scale thisequation for fast atmospheric motions ~u cwave,

( )

r~

~

~

c

t

gC c

p

c

t

aC g

RT

c

ac

c

t

C a

H tc

1, 41

D s

s

D

s

D

wave

drag

wave2

wave

drag

wavewave

wave

drag wavewave

20


where we have assumed that the boundary layer is thick,p psBL (see Figure 1), used the ideal gas law in the second

step, r = - -p R Ts s s1 1, and used the wave timescale =twave

a cwave in the last step. This lets us derive a drag timescale

( )~tH

C at . 42

Ddrag wave

Note, in contrast to Rayleigh drag schemes where the dragtimescale is independent of u, here the drag timescale scaleswith u and thus with the dynamical timescale twave. Using thisdrag timescale we can rewrite the last nondimensionalparameter as ~C a H t tD wave drag and find an alternative setof six governing parameters:

( )t t⎛⎝⎜

⎞⎠⎟

R

c

a

L

t

t

t

t, , , , , . 43

p

2

Ro2

wave

radSW LW

wave

drag

In most cases,C a HD is of order unity so the drag timescale isgenerally comparable to the dynamical timescale. For example,assuming a planet of Earth’s size, ( ) ( )= Å Åa g a g, , , a highMMW atmosphere, =R RN2, a relatively cool temperature,

=T 300 Keq , and a standard value for the drag coefficient,= -C 10D

3, we find =t t1.4drag wave. Variations in the planetaryradius a or in the drag coefficient CD do not affect this resultmuch. For example, assume a neutrally buoyant boundary layer

[ ( )]=C k z zlogD vk 02 , where kkv is the von Karman constant, z

is the height above the surface, and z0 is the surface roughnesslength. Because CD only depends logarithmically on z0, the dragtimescale is not very sensitive to the surface properties. Thismeans we expect friction to generally be an important processinside the boundary layer of rocky planets.

The most important exception is a hot H2 atmosphere,through its effect on the scale height H. For example, repeatingthe above calculation with a hot H2-dominated atmosphere,

=R RH2 and =T K600 ,eq we find =t t40drag wave. This meanssurface friction is far less effective in H2 atmospheres than inhigh MMW atmospheres.

APPENDIX BWIND SPEED SCALING FROM WORDSWORTH (2015)

To compare the results of Wordsworth (2015) with our GCMsimulations we write Wordsworthʼs Equation (33) as

( )stz

=U Tp C

R

c4

2, 44

s D p0 eq

4 LW

where z º 1 3. The above equation reduces to WordsworthʼsEquation (33) by plugging in his Equation (12), i.e., byassuming a specific form for τ. We leave the equation in thisgeneral form. We identify Wordsworthʼs absorbed stellar flux( )- A F1 as sT4 eq

4 .The nondimensional equations in Wordsworth (2015) are not

affected. To find the surface wind speed we solve his Equations(44) and (45) numerically to find T and U . We then convert thenondimensional U into a dimensional quantity using the abovescale U0, i.e., ∣ ∣ ˜=u U U0 .

APPENDIX CNUMERICAL SOLUTION FOR THE RCS MODEL

The boundary conditions of the RCS model are specified attwo different points: the tropopause, and the surface. Instead of

matching both boundaries simultaneously, we first guess avalue of the nightside OLR, ( )tF 0 . Given a value of ( )tF 0 , wecan solve for all variables (Td, t0, ( )tT , and ( )tF ). Our guesswill, in general, not satisfy the nightside surface energy budget(Equation (21c)), so we iterate until ( )t =F 0LW is satisfied. Toiterate we use a bisection method, where the nightside OLR isbounded by ( ) t sF T0 0 eq

4 (the limits correspond to aplanet with zero and perfect day–night heat redistribution).We proceed as follows. Given a value of ( )tF 0 ,

Equations (22) and (25) can be rewritten as an implicitequation for t0,

[

( ) ( )

( )

( )ò

t t tt

tt

tt

s

+ =+

+¢

¢ +

bt t

t

t bt t

- -

- ¢-

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

e

e dF

T

22

1

2

. 45

0 0 LW

0

4

LW

40

eq4

LW 0

0

LW0

We solve for t0 using, again, a bisection method, where wenote that t t< <0 0 LW. We then use Equations (20) and (22)to find ( )tT 0 and Td:

( ) ( )tt

=+⎜ ⎟⎛

⎝⎞⎠T T

1

2460 eq

01 4

( )t tt

=+ b

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟T T

1

2. 47d eq

01 4

LW

0

Once we know Td we can find w (Equations (12) and (26)). Wethen have three boundary conditions that are specified at theupper boundary, the guessed ( )tF 0 , ( )t t =dF d 00 , and ( )tT 0 .We use SciPy’s VODE solver to integrate the WTG andSchwarzschild equations (Equation (19b)) down to the nightsidesurface, which gives us the net surface flux ( )tF LW . We iterateuntil we satisfy the nightside surface budget, ( )t =F 0LW . Afterhaving solved for ( )tT we find the nightside surface temperature,Tn, using the nightside surface energy budget (cf. Robinson &Catling 2012),

( )

( ) ( )( ) ( )òs t

st

s t t

=

= + ¢ ¢t t

t

tt t

-

- - - ¢-

T F

T e T e d2

. 48

n4

LW

eq4 0 4LW 0

0

LW0

APPENDIX DEQUATIONS OF MOTION

The FMS GCM integrates the primitive equations in pressurecoordinates, which are

( )f= - ´ - -

¶¶

uk u

D

Dtf g

p, 49m

( )f¶¶

= -p

RT

p, 50

· ( )w +

¶¶

=up

0, 51

( )w= +

¶¶

+¶¶

DT

Dt

RT

c p

g

c

F

p

g

c p. 52

p p p

21


Here ( )=u u v, is the horizontal wind velocity, = +¶¶

D

Dt t

· w + ¶¶

up

is the material derivative, q= Wf 2 sin is the

Coriolis parameter, k is the local vertical unit vector, f is thegeopotential, T is temperature, w º Dp Dt is the pressurevelocity,m and are the vertical diffusive fluxes of momentumand energy in the boundary layer, F is the net longwave flux, andan overview of the dimensional parameters can be found inTable 1. From the top, these equations express conservation ofmomentum, the hydrostatic approximation, conservation of mass,and conservation of energy. The net longwave flux F is governedby the two-stream Schwarzschild equations. For a gray gas thesecan be written in optical depth coordinates as

( ) ( )t

st

¶¶

- = -¶¶

FF

T2 . 53

2

2

4

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