Carone2014 Connecting the Dots - A Versatile Model for the Atmospheres of Tidally Locked Super-Earths

MNRAS 445, 930–945 (2014) doi:10.1093/mnras/stu1793

Connecting the dots: a versatile model for the atmospheres of tidallylocked Super-Earths

L. Carone,1‹ R. Keppens1 and L. Decin2

1Centre for mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium2Instituut voor Sterrenkunde, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium

Accepted 2014 September 1. Received 2014 August 27; in original form 2014 May 26

ABSTRACTRadiative equilibrium temperatures are calculated for the troposphere of a tidally lockedSuper-Earth based on a simple greenhouse model, using Solar system data as a guideline.These temperatures provide in combination with a Newtonian relaxation scheme thermalforcing for a 3D atmosphere model using the dynamical core of the Massachusetts Instituteof Technology global circulation model. Our model is of the same conceptional simplicitythan the model of Held & Suarez and is thus computationally fast. Furthermore, because ofthe coherent, general derivation of radiative equilibrium temperatures, our model is easilyadaptable for different planets and atmospheric scenarios. As a case study relevant for Super-Earths, we investigate a Gl581g-like planet with Earth-like atmosphere and irradiation andpresent results for two representative rotation periods of Prot = 10 d and Prot = 36.5 d. Ourresults provide proof of concept and highlight interesting dynamical features for the rotatingregime 3 < Prot < 100 d, which was shown by Edson et al. to be an intermediate regimebetween equatorial superrotation and divergence. We confirm that the Prot = 10 d case ismore dominated by equatorial superrotation dynamics than the Prot = 36.5 d case, whichshows diminishing influence of standing Rossby–Kelvin waves and increasing influence ofdivergence at the top of the atmosphere. We argue that this dynamical regime change relates tothe increase in Rossby deformation radius, in agreement with previous studies. However, wealso pay attention to other features that are not or only in partial agreement with other studies,like, e.g. the number of circulation cells and their strength, the role and extent of thermalinversion layers, and the details of heat transport.

Key words: methods: numerical – planets and satellites: atmospheres.

1 IN T RO D U C T I O N

There are many roads towards modelling and understanding of plan-etary atmospheres. Dry three-dimensional global circulation mod-els (3D GCMs) with idealized thermal forcing are one possiblemethod to gain insights into atmospheric large-scale dynamics andare even to this date relevant to the Earth climate community. Ina landmark paper, Held & Suarez (1994, hereafter HS94) reducedthe complexity of a 3D GCM to the very basics by replacing thedetailed radiative forcing, turbulence and moist convection descrip-tions with very simple forcing prescriptions in a full 3D hydrody-namical core. Dry 3D GCMs have the advantage that they can easilyadjust to varying planetary and atmosphere parameters and that theyare computationally efficient. This not only allows us to comparethe dynamical cores of different climate models with each other,

� E-mail: [email protected]

which was the initial intention for the HS94 Earth benchmark, butalso allows us to disentangle non-linear interdependences betweenforcing parameters and large-scale circulation, which is much moredifficult in more complex models (Held 2005). The HS94 bench-mark has been used to study, to name only a few examples, theeffect of temperature forcing on time-scales of variability in theextratropical atmosphere (Gerber & Vallis 2007), sensitivity of ex-tratropical circulation to tropopause height that might change due toglobal warming (Lorenz & Deweaver 2007), and winds in the upperequatorial troposphere (Kraucunas & Hartmann 2005). Despite thesimplifications, the main characteristics of the Earth atmospheredynamics already emerge from HS94 like Hadley, Ferrel, and polarcirculation cells, and jet streams (see e.g. Bordi et al. 2009). Identi-fication of uplifting circulation branches allows further identifyingregions of cloud formation that are particularly habitable.

Ultimately, complex models are needed that take into accountcomplex atmosphere chemistry (e.g. Hu, Seager & Bains 2012;Grenfell et al. 2013 among others), radiative transfer (e.g. Kataria

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Connecting the dots 931

et al. 2014) and hydrological cycles (Menou 2013). These mod-els aim to derive spectra (e.g. Bailey & Kedziora-Chudczer 2012;Hedelt et al. 2013) for comparison with observational data. Theirresults all critically depend on the dynamically established ther-modynamic equilibrium. Therefore, the following questions shouldbe addressed: Is the resolution sufficient? Are the numerical repre-sentation at the poles and the atmosphere boundaries (surface, top)adequate, and more importantly, are all relevant physical processesaccounted for? Keeping in mind that already for the Earth it is diffi-cult to parametrize subgrid processes, like eddies, clouds, planetaryboundary layer (PBL), wave drag, etc., it is widely understood thatthere are even less certainties for exoplanets, where even the com-position of their atmospheres and surface is unknown. The appeal ofdry idealized GCMs in the context of exoplanets is thus immediatelyclear: They are ideally suited to provide benchmarking.

We show in this paper that it is indeed possible to construct asimplified model suitable for all terrestrial planets with the samelevel of complexity as HS94, if one examines rigorously the under-lying assumptions of the temperature forcing given in HS94. Usinga simplified greenhouse model and the skin layer assumption, wedeveloped a new temperature forcing scheme that is suitable for atidally locked planet with negligible obliquity, and that can also beused for planets with different atmospheric composition. We pro-pose, therefore, a versatile and computationally efficient terrestrialplanet atmosphere model that retains full control over parametersand physical prescriptions – in particular subgrid processes – byincorporating empirical knowledge about Solar system planetaryatmospheres.

We aim to compute models of the atmospheres of habitable extra-solar planets that ultimately will be helpful for observations, makingit clear that tidally locked Super-Earths in the habitable zones of lateK and M-dwarf stars will be our first targets. This is because suchplanets will be easier to detect than truly Earth-sized planets due toa more favourable planet/star contrast ratio. Tidal interaction willthen synchronize the planet’s rotation with its revolution leading torotation periods between 8 and 100 d as pointed out by Edson et al.(2011).

Indeed, the rotation period Prot is one key parameter in shapingthe dynamics of exoplanet atmospheres. The atmospheres of fastrotating tidally locked Earth-like planets (Prot = 1 d) are accordingto Merlis & Schneider (2010) dominated by waves and eddies with awesterly jet at the equator, whereas for slow rotation (Prot = 360 d),it is dominated by divergent circulation. Edson et al. (2011) foundthe phase state transition between the fast and slow rotating regimebetween 4–5 d for dry planets and 3–4 d for aquaplanets. Anothertransition was found for a rotation period of 100 d. Interestingly,Edson et al. (2011) consistently report two circulation cells for eachhemisphere even for Prot = 100 d, whereas most other earlier studies– including Joshi, Haberle & Reynolds (1997) and Joshi (2003) –report just one.

In the following, we present our model in Section 2. We presenttwo representative simulations with Prot = 10 and Prot = 36.5 dfor the intermediate-rotation regime 3 ≤ Prot ≤ 100 d, identifiedby Edson et al. (2011), and discuss our results in Section 3. Weshow that our model yields results that are generally consistent withprevious studies using more complex GCMs. In particular, we dis-cuss superrotation and divergent flow, cyclonic vortices, circulation,and temperature distribution in context with possible habitabilityof tidally locked terrestrial planets. We also highlight differenceswith the published work and postulate possible reasons for theirdeviation. We conclude with a short summary of our results in Sec-tion 4 and give an outlook on future work. We emphasize that the

atmospheric dynamics is surprisingly complex in the intermediate-rotation regime (3 ≤ Prot ≤ 100 d) due to the mixture betweendivergent and superrotating circulation that warrants a dedicatedparameter study. We highlight the potential of our model to performsuch a parameter study and outline further possible investigationsto test basic model assumptions like surface friction and thermalforcing variation, in particular, at the nightside.

2 TH E MO D EL

In the following, we describe the specific set-up of the model forterrestrial exoplanets with a greenhouse atmosphere. We stress onceagain that our goal is to construct a conceptionally simple model thatyields the large-scale dynamics in the troposphere with maximumcomputational efficiency.

2.1 The dynamical core: MITgcm

The dynamical core of the Massachusetts Institute of Technologyglobal circulation model (MITgcm) developed at MIT1 (Adcroftet al. 2004) uses the finite-volume method to solve the primitivehydrostatical equations (HPE) that write as horizontal momentum

Dv

Dt+ f k × v = −∇p� + F v, (1)

vertical stratification

∂�

∂p= − 1

ρ, (2)

continuity

∇p · v + ∂ω

∂p= 0, (3)

equation of state for an ideal gas

p = ρ (R/μ) T , (4)

and thermal forcing equation

Dθ

Dt= θ

cpTFθ , (5)

where the total derivative is

D

Dt= ∂

∂t+ v · ∇p + ω

∂

∂p. (6)

In these equations, v is the horizontal velocity and ω = Dp/Dt isthe vertical velocity component in an isobaric coordinate system,where p is pressure and t is time. � = gz is the geopotential, with z

being the height of the pressure surface, g the surface gravity, andf = 2� sin ν is the Coriolis parameter, with � being the planet’sangular velocity and ν the latitude at a given location. k is the localvertical unit vector, T, ρ, and cp are the temperature, density, andspecific heat at constant pressure. θ = T(p/ps)κ is the potentialtemperature with κ being the ratio of the specific gas constant R/μ

to cp, where R = 8.314 J K−1mol−1 is the ideal gas constant andμ is the molecular mean mass of the atmosphere, and ps is thesurface pressure. Fθ,v are the horizontal and thermal forcing terms,respectively. Note that, in this case, F v is a vector and Fθ is a scalar.

1 http://mitgcm.org

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932 L. Carone, R. Keppens and L. Decin

The HPEs are solved in η vertical coordinates on a staggeredArakawa C grid using a curvilinear cubed-sphere horizontal coor-dinate system (Marshall et al. 1997; Showman et al. 2009). η is inthe absence of topography defined as

η = p − pt

ps − pt, (7)

where pt denotes the pressure at the top of the model atmosphere.The quasi-second-order Adams–Bashforth time-stepping scheme

with a stabilizing parameter εAB = 0.1 was used for all explicit termsin the momentum equations (see e.g. Durran 1991 for a discussionof time-stepping methods for the numerical simulation of advectionin the atmosphere). The model top at pt = 0 has vanishing verticalvelocity (ω = 0) as boundary condition. Pressure at the surface istreated in the implicit free surface form of the pressure equation(Marshall, Jones & Hill 1998), which allows us to fluctuate thesurface pressure around the constant ps.

For our study, we aim to investigate the dynamics in the tro-posphere, the ‘weather region’, that comprises the majority of theatmosphere’s mass. Therefore, we need to resolve this region withsufficient vertical spacing. Heng & Vogt (2011) used a linear verticalspacing with 20 levels separated at 500 mbar with a surface pressureof ps = 1 bar in accordance to HS94. Indeed, it was confirmed byWilliamson, Olson & Boville (1998) that this vertical resolution issufficient to resolve the general structure of the Earth’s troposphere.Therefore, we also use it for our study. We adopt the C32 cubed-sphere grid for the horizontal resolution that was tested successfullywith HS94 by Marshall et al. (2004) and is also used by Zalucha,Michaels & Madhusudhan (2013). The sphere is subdivided intosix tiles, each with 32 × 32 elements, thus there are 32 × 32 × 20volume elements per tile. This horizontal gridding corresponds toa global resolution in longitude–latitude of 128 × 64 or 2.◦8 × 2.◦8and translates to a mean horizontal resolution of �x, y ≈ 41 km fora planet with 1.45 Earth radii.

Heng & Vogt (2011) assumed that the spin-up of the atmosphericsystem from initial conditions at constant temperature T = 264 Kwould be concluded after tinit = 200 Earth days. Fig. 1 shows theglobally averaged kinetic energy per unit mass for our two runs withProt = 10 and 36.5 d, where we found that our simulations are justbarely at equilibrium after 200 d. Therefore, we have set tinit = 400Earth days to be on the safe side. Data generated before tinit werediscarded; the simulation was run subsequently for trun = 1000Earth days and averaged over this time period. Time averaging over

Figure 1. Kinetic energy per unit mass for our GCM runs after initializationat t = 0 d. The kinetic energy increases steeply and reaches, in any case,equilibrium after 400 d.

Table 1. Friction time-scales and planetary boundary layer extent (PBL)for some Solar system planets.

Planet τ s, fric pPBL Reference(d)

Earth 1 0.7 × ps Held & Suarez (1994)Venus 25 0.87 × ps Lee, Lewis & Read (2007)Mars 10–27.2, 0.2 (0.91–0.78) × ps Haberle et al. (1997),

Joshi et al. (1995)

1000 d should filter out small-scale variations and transient waves,allowing us to bring forth the basic stable large-scale dynamics.

2.2 Friction time-scale

We prescribe Rayleigh friction between the atmosphere and thesurface via

F v = − 1

τfricv, (8)

where τ fric is defined as

τfric = τs,fric max

(0,

p

ps− pPBL

ps

1 − pPBLps

), (9)

where pPBL is the pressure at the upper limit of the PBL and τ s, fric

is the maximum surface friction. Both values are somewhat uncon-strained even for terrestrial planets in the Solar system. HS94 useτ s, fric = 1 d and pPBL = 700 mbar for the Earth. For Mars, on theother hand, the possible values for τ s, fric range between 0.1and30 d(see Table 1). We assume for now τ s, fric = 1 d and pPBL = 700 mbarlike in HS94.

Like Zalucha et al. (2013), we prescribe an upper sponge-layerwith Rayleigh friction. This sponge layer prevents non-physicalwave reflection at the upper boundary but can also be physicallyjustified by assuming gravity wave braking at high pressure lev-els. We implemented the Rayleigh dampening profile suggested byPolvani & Kushner (2002) by adding to F v the term −kRv, wherekR is

kR = kmax

[psponge − p

psponge

]2

if p ≥ psponge

kR = 0 if p < psponge, (10)

with psponge = 0.1 bar and kmax = 1/80 d−1 appropriate for dampingin the middle stratosphere (Jablonowski & Williamson 2011).

2.3 Temperature forcing time-scale

The Newtonian relaxation scheme is applied through an externaltemperature forcing

FT = Teq(ν, φ, p) − T

τrad, (11)

where the temperature T in the model atmosphere is driven withthe relaxation time-scale τ rad towards an equilibrium temperatureTeq that will be defined in the following section. FT can easilybe transformed into Fθ , by using the definition of the potentialtemperature θ = T(p/ps)κ . The relaxation time-scale is estimated tofirst order by (e.g. Showman & Guillot 2002)

τrad = cpps

4 gσT 3s

, (12)

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Table 2. Temperature forcing time-scales for some planets.

Planet τ rad (radiative transfer) τ rad (first order) Reference(d) (d)

HD 209458b 2 1.1 Showman & Guillot (2002),Iro, Bezard & Guillot (2005)

Earth 40 20 Held & Suarez (1994)Venus 115–19 000 112 Pollack & Young (1975)Mars 2 0.6 Haberle et al. (1997)

where Ts is the mean surface temperature, ps is the surface pres-sure and σ is the Stefan–Boltzmann constant. This estimate yieldsrealistic values for low pressures p ≤ 1 bar (see Table 2). Forhigher pressures (p ≥ 10 bar), however, comparison with Venusbetween the computed radiative time-scales from more detailed ra-diative transfer calculations and the estimate given above (τ rad ≈100 yr instead of 112 d stemming from the first-order estimate) andcomparison between the first-order estimate and the time-scales cal-culated in Showman et al. (2008) show that equation (12) may bean underestimate by orders of magnitude.

For atmospheric models of dense atmospheres (ps � 1 bar), there-fore, either larger τ rad should be investigated or instead of assuminga single τ rad for the whole vertical atmosphere column, equation 6 ofShowman et al. (2008) should be used instead to calculate τ rad for agiven vertical layer with pressure p0 and temperature T0, followingthe steps outlined in that work. In this study, however, we aim tobuild an HS94-like model of a tidally locked terrestrial exoplanetwith an Earth-like troposphere with ps = 1 bar covering a limitedpressure range between 1000 to 100 mbar, for which the first-orderestimate is valid and sufficient.

In addition, we implement the dry convection scheme of Molteni(2002) to treat static instabilities at the substellar point. It diffusesvertically dry static energy s = cpT + �, if the static stability islocally violated. That is the case, if dθ/dp > 0, where we haveassumed a vertical diffusion time-scale of τ vds = 1 d.

2.4 Equilibrium temperature Teq

The equilibrium temperature Teq(ν, φ, p), which is used in thethermal forcing FT (equation 11), generally depends on the planet’slatitude ν, longitude φ and pressure level p and is prescribed asfollows:

Teq(ν, φ, p) = max (TNS(p), TDS(ν, φ, p)) , (13)

where TNS are nightside and TDS are dayside temperatures. The day-side thermal forcing is allowed to drop to very low temperatures atthe poles and the terminator, until the nightside temperature condi-tions are met. Furthermore, we assume that the temperature dropswith decreasing pressure (increasing height) following the dry adi-abat at the dayside and a nitrogen condensing moist adiabat on thenightside, until the atmosphere reaches the tropopause temperatureTtp. The temperature remains further on at Ttp and constitutes avertically isothermal layer on top of our troposphere.

The lower boundary of the radiative–convective temperature pro-file on the dayside is derived by calculating the surface temperatureusing a first-order greenhouse model adopting an optical depth atthe surface, τ s, appropriate for an atmosphere of Earth-like com-position. On the nightside, we set the lower boundary by assumingas surface temperature the nitrogen condensation temperature forps = 1 bar.

Figure 2. Visualization of the skin layer concept used to derive thetropopause temperature. An optically very thin atmosphere layer of verysmall absorptance and emittance εtp ≈ 0 is overlaid on top of a tropospherethat is heated by the blackbody flux of the atmosphere, σT 4

eff . The skinlayer’s contribution to the overall heat balance, however, is negligible.

To calculate the upper boundary, we adopt for both, day andnightside, the skin layer concept by Pierrehumbert (2010) to deter-mine the top of the convective layer, that is, the tropopause. First,we assume very small but non-zero values for optical depth andemissivity: τ tp ≈ 0 and εtp ≈ 0. It is further assumed that the skinlayer’s density is too small to transport heat efficiently into and fromthe layer by convection.

In this case, little of the upwelling infrared radiation is absorbedin the skinlayer due to its small absorptance εtp and it emits onlya small amount of infrared back radiation towards the lower atmo-sphere, εtpσT 4

tp. Therefore, the skin layer is only heated by infraredupwelling of the atmosphere with negligible contribution to theoverall heat budget and the energy balance for the tropopause un-der skin layer assumptions reads (Fig. 2 and see e.g. Pierrehumbert2010):

2εtpσT 4tp ≈ εtpσT 4

eff (ν, φ), (14)

and yields for the skin-layer temperature that defines thetropopause:

Ttp = 0.51/4Teff (ν, φ), (15)

where Teff(ν, φ) is the effective or blackbody temperature of theatmosphere that will be calculated in the following for the day andnightside, respectively.

2.4.1 Condensing nightside

In our nominal model, it is assumed that the temperatures at thenightside may plummet so low that they reach the condensationpoint for liquid nitrogen. For now, we prescribe a ‘Martian polescenario’ (Toon et al. 1980): it is assumed that the vertical tem-perature relaxes towards the saturated moist adiabat of the main

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ingredient of the atmosphere, N2, that is described in the limit of aone-component condensable atmosphere as

T∗(p) = Tref

1 − (R/μ)TrefL

ln p

pref

. (16)

Here, Tref and pref are the vaporization reference temperature andpressure, which is Tref = 77 K at pref = 1 bar for nitrogen, respec-tively, L = 1.98 × 105 J kg−1 is the vaporization latent heat release(Pierrehumbert 2010), R is the ideal gas constant, μ = 28 g mol−1

is the mean molecular mass of N2. Furthermore, we assume for nowthat surface pressure variation due to condensation is negligible. Itshould be noted that Zalucha et al. (2013) used a similar prescriptionfor the nightside of their tidally locked H2O atmosphere planet.

The vertically isothermal tropopause on top of the troposphereis calculated by assuming that a nitrogen condensing atmospheredoes not contain any greenhouse gases so that for the optical depthat the surface: τ s ≈ 0. In this case, the surface temperature T∗(ps)is equivalent to the effective temperature. Therefore, equation (15)can be applied to derive the tropopause temperature on the nightsidewith T∗(ps) = Teff(ν, φ) as

Ttp = 0.51/4T∗(ps). (17)

The total nightside equilibrium temperature description used inequation (13) is accordingly:

TNS(p) = max(Ttp, T∗(p)

). (18)

Pierrehumbert (2010) states that in a situation where τ s ≈ 0, theskin layer would extend down to the ground and the atmosphereassume Ttp, if radiative heating would be the only heat transfermechanism. However, because the surface temperature is warmerthan an atmosphere at Ttp, convection would kick in. Convectionwould then establish a vertical gradient along the dry adiabat fora non-condensing atmosphere. We assume that the same holdsequivalently for condensing atmospheres with the difference thatthe atmosphere relaxes along the above described saturated moistadiabat.

If the atmosphere is condensing nitrogen on the nightside in somevertical layers and not in others, then only the condensing layerswill follow the moist adiabat, where we neglect for now transportof latent heat to the neighbouring non-condensing cells. However,as will be shown later, this is not an issue with the two simulationsshown in this work, because the nightside temperatures are at allpressure levels well above the nitrogen condensation limit due tothe efficient dynamic heat transport from the dayside. We mayrevise this prescription in future work, in particular, when goingto lower surface pressures. Furthermore, we will investigate forspecific cases if it is possible to maintain the atmosphere in a partlyor fully condensing situation without total atmospheric collapse.

2.4.2 Illuminated dayside temperature with zero obliquity

For the dayside, it is assumed that the incident flux F is distributedwith the cosine of the zenith angle cos ζ across the sphere, whichis for zero obliquity and a tidally locked planet (see also Joshi et al.1997) geometrically related to the latitude and longitude by

cos ζ = cos φ cos ν. (19)

Using Stefan–Boltzmann’s law F = σT4, the horizontal temperaturedistribution has to be ∝ cos 1/4φ cos 1/4ν. This yields for the daysideeffective temperatures consequently:

Teff (ν, φ) = Teff,max cos1/4 ν cos1/4 φ, (20)

with the maximum effective planetary temperature:

Teff,max =[

I0(1 − α)

σ

]1/4

. (21)

The incoming flux is F = I0(1 − α), where α is the albedo of theplanet and the incident stellar flux I0 of a star with luminosity L∗,radius R∗, and effective temperature Teff, ∗ on a planet at distance ais

I0 = σT 4eff,∗

(R∗a

)2

. (22)

From the analytic solution of the two-stream approximation fora greenhouse atmosphere, the following relation for the daysidesurface temperature follows, where τ s is the optical depth at thesurface and the value of γ depends on which approximation is usedto treat scattering in the atmosphere:

TDS,s(ν, φ) = (γ τs + 1)1/4Teff (ν, φ). (23)

Zalucha et al. (2013) used in their equation 8 the Eddington approx-imation for which γ = 0.75. In this work, we use the hemi-isotropicor hemispheric mean approximation for which γ = 1 (see e.g. Toonet al. 1989; Pierrehumbert 2010; Heng, Mendonca & Lee 2014 fora discussion of different scattering treatments in the two-stream for-malism). We argue here that light from M dwarf stars, that are therelevant host stars for tidally locked planets with Earth-like ther-mal forcing, is less subject to Rayleigh scattering and that thereforeγ = 1 is here more appropriate (Toon et al. 1989). However, itis easy to change this assumption for other type of host stars andthe difference in temperature between the two approximations issmall.2

For an optically thin atmosphere (optical depth at surface τ s < 1),we may also formulate the surface dayside temperatures in terms ofthe leaky greenhouse model (see e.g. Marshall & Plumb 2008):

TDS,s(ν, φ) =(

2

2 − ε

)1/4

Teff (ν, φ), (24)

where the emissivity ε is connected with the surface optical depthτ s via

2

2 − ε− 1 = γ τs. (25)

In any case, the skin layer concept is applied again to definethe upper boundary of the convective troposphere as a verticallyisothermal tropopause for dayside conditions:

Ttp = 0.51/4Teff,max cos1/4 ν cos1/4 φ, (26)

combining equations (15) and (20).Assuming now that the vertical temperature gradient in the tro-

posphere follows the dry adiabatic index dT/dz = −�d = −g/cp

because the troposphere is subjected to convection, we have thefollowing prescription for the equilibrium temperature:

Teq(ν, φ, p) = max

{TNS(p),

max

[Ttp, TDS,s(ν, φ)

(p

ps

)R/cp] }

. (27)

2 It is ≈7 K for τ s = 0.62 and Teff, P = 255 K appropriate for Earth takenfrom Table 3.

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Table 3. Selected Solar system atmosphere data.

Earth Venus Mars Titan Jupiter

I0 (W m−2)1 13683 26143 5893 15 50.53

Albedo1 0.33 0.77 0.253 0.223 0.343

Teff, P (K)2 255 210 210 85 124.4b

Ttp (K)2 214 177 177 71.5 105b

γ 1 7/5 9/7 9/7 7/5 7/5Constituent 1, 3 N2 CO2 CO2 N2 H2

μ(g mol−1)d 28 44 44 28 2cp (J g K−1)c, 2 1.04 0.85 0.85 1.04 14.55H (km)2 8.7 15.6 11.1 20.7f 28g(m s−2)1, 3 9.8 8.9 3.7 1.35 24.8a

ps(bar)1, 3 1 92 0.006 1.47 1a

Ts(K)2 288 737 220 94 167a

ε e 0.8 – 0.33 0.66 –τ s

e 0.62 110 0.2 0.5 2.25

− dTdz dry

[ Kkm

]29.4 10.4 4.4 1.3 1.7

− dTdz real

[ Kkm

]13–9.4 7.7 1.6–4 0.56–1.24 1.7

Literature Marshall &Plumb (2008),Pierrehumbert(2010)

Irvine (1968),Kliore & Patel(1980), Lewis(1971),Pierrehumbert(2010)

Gierasch & Goody(1972),Pierrehumbert(2010)

McKay et al.(1997), McKay,Lorenz & Lunine(1999), Elachi et al.(2005), Brown,Lebreton & Waite(2010),Pierrehumbert(2010)

Lindal et al. (1981),Hanel et al. (1981),Pierrehumbert(2010)

Note: alower boundary of troposphere set at p=1 bar, bincluding 60 per cent contribution from internal heat flux, cκ = R/(μcp) = 1 − 1/γ , d

equal μ of main constituent, ecalculated using equations (24), (25), and (23), setting Ts = TDS, s. fDue to haze in the upper atmosphere, the realvalue is H = 40 km (Justus, Duvall & Kller 2004). 1literature, 2calculated using equation (15), chttp://nssdc.gsfc.nasa.gov/planetary/factsheet/.

Figure 3. Vertical equilibrium temperature profiles for Earth-like irradi-ation (I0 = 1368 W m−2 and α = 0.3) at the nightside/terminator region(coldest profile), i.e. cos ζ = 0 and for cos ζ = 0.025, 0.05, and cos ζ between0.1 and 1 in steps of 0.1.

In Table 3, we collect albedo, solar irradiance and calculate theeffective temperature and tropopause temperature for selected So-lar system bodies. From the mean global surface temperature Ts,assuming distribution of solar energy over the whole sphere, τ s canbe inferred via equations (23) and (24). The derived τ s-values serveus as ‘ground truths’. We also introduce here the scaleheight, whichis defined as H = RTs/μg.

Fig. 3 shows the vertical equilibrium temperature profile for anEarth-like atmosphere (ε = 0.8, α = 0.3, and ps = 1 bar) on atidally locked planet subjected to Earth-like stellar irradiation. Atboth hemispheres, the temperature drops continuously from the

surface temperature to colder temperatures with increasing height,following a dry adiabat at the dayside and the condensation profileat the nightside. Consequently, the effective temperature Teff andthe tropopause temperature Ttp are automatically reached at somepressure level. For our calculations, the tropopause is located at ap-proximately 450 mbar at the dayside and 200 mbar at the nightside.It has to be noted that an increase of optical depth at the dayside– and thus to higher surface temperatures than computed here –would lead to a shift of the effective temperature and tropopauselocation to higher altitudes, that is, to lower pressure levels. A de-crease of optical depth would lead correspondingly to a downwardshift in height. This connection between optical depth, temperatureand altitude explains why Teff is already reached at surface pressureat the nightside, where τ s ≈ 0 is assumed. The nightside tropopauseof the prescribed equilibrium temperature Teff is, however, locatedat higher altitude than the dayside tropopause because the verticaltemperature gradient due to condensation is much less steep thanthe vertical temperature gradient due to convection. Note, however,that dynamics will change this picture. It tends to decrease the ver-tical temperature gradient due to convection (Table 3) and, as willbe shown later, to increase the vertical gradient on the nightside asthe temperatures do not drop low enough to allow for condensation.Thus, the actual tropopause is shifted to higher altitudes at the day-side, whereas for the nightside the raising of surface temperatureand increase in vertical temperature gradient appear to compensatefor each other.

2.5 Optical depth and albedo

We assume that the optical depth is linearly dependent on pres-sure (e.g. Zalucha et al. 2013) and that therefore the optical depth

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decreases from its maximal value at the surface τ s to the tropopauseτ tp ≈ 0. Furthermore, non-condensing nitrogen-dominated atmo-spheres with surface pressures of the order of 1 bar are expected tobe optically thin (Table 3). It is thus justified to assume an opticallythin, leaky greenhouse model for the calculation of the equilibriumtemperature Teq at the dayside in this work. The effect of a variedcontent of greenhouse gases can then, in principle, be investigatedby varying the emissivity ε and thus the optical depth τ s betweenapproximately zero and one.3 τ s ≈ 0 corresponds then to an atmo-sphere with negligible greenhouse gas and τ s = 1 to an atmospherewith large greenhouse gas content, like the H2O atmosphere usedby Zalucha et al. (2013) that has indeed τ s = 1 for ps = 1 bar.

Another important factor that determines the thermal forcing ona tidally locked planet is the reflectivity of clouds, ice coverage, soilthat are encapsulated in the albedo coefficient α, which may also bevaried between 0 and 1. In this work, we want to demonstrate thatour model is indeed suitable for the investigation of atmosphericdynamics of terrestrial tidally locked planets. Thus, we focus fornow on Earth-like atmospheres to compare our results with previousstudies and assume thus ps = 1000 mbar, an albedo of α = 0.3, andε = 0.8 for the dayside as for the Earth (Table 3), and ε ≈ 0 andτ s ≈ 0 at the nightside because the temperature is allowed to dropso low that all greenhouses gases freeze out. We note, however, thatit is easy to change our model here to accommodate atmosphereswith different optical depths and albedo.

2.6 Investigated parameter space: tidally locked Super-Earthswith rotation periods in between 3 ≤ Prot ≤ 100 d

We have constructed a simplified versatile atmosphere model for atidally locked terrestrial planet with the same elegant simplicity asthe HS94 benchmark, without carrying over assumptions that areonly valid for the Earth. This is the drawback of the model of Heng& Vogt (2011) that retains some Earth-centric features.

Many previous studies focus on planets around M dwarf stars asthese planets are the most numerous main-sequence stars and alsothe ones for which terrestrial exoplanets are more readily detectedand observed. They broadly fall into two categories: complex Earthmodels that are used for tidally locked planets but are constrainedto the thermal forcing regime of Earth (e.g. Joshi et al. 1997; Merlis& Schneider 2010; Edson et al. 2011). On the other side, there areless complex models that allow us to get out of the comfort zonesof the Earth models in terms of forcing and composition, withmore or less sophistication in their radiative transfer schemes. Forexample, Zalucha et al. (2013) use a grey two-stream formalismwith Eddington approximation and Kataria et al. (2014) use themore complex SPARC radiative transfer code.

We use, like Zalucha et al. (2013), a grey two-stream formalismbut with the hemi-isotropic approximation for scattering for the sur-face temperature and calculate a radiative–convective equilibriumtemperature between the surface temperature and an upper bound-ary temperature. Furthermore, we focus on the investigation of thedynamics in the troposphere and cap our upper atmosphere witha vertically isothermal tropopause, using the skin layer concept toidentify the top of the convective troposphere. We will show thatour set-up is sufficient to resolve many dynamical features reported

3 To keep the skin layer concept, τ s can never be exactly zero and furthermoreτ tp < τ s has to hold. Indeed, this case was assumed for the derivationof the equilibrium temperature at the nightside in the limit of nitrogencondensation.

Table 4. Main Super-Earth planetary and atmosphericparameters for our two reference models.

Parameter

Planetary radius RP 1.45REarth

Planetary mass MP 3.1MEarth

Surface gravity g 14.3 m s−2

Obliquity 0◦cp 1.04 J g K−1

ps 1000 mbarMain constituent N2

Molecular mass μ 28 g mol−1

Atmospheric emissivity ε 0.8Surface optical depth τ s 0.62Adiabatic index γ 7/5Surface friction time-scale τ fric 1 dRadiative time-scale at substellar point τ rad 13 dRadiative time-scale at nightside τ rad 813 dRotation period Prot 10 d, 36.5 d

by other more complex and/or extended models but with the majorbenefit that we are very flexible with our parametrization.

In this work, we focus on Earth composition atmospheres, as mostdynamical studies have focused on this type of terrestrial planetsto address the prospect of habitability. We assume the parame-ters reported by Heng & Vogt (2011) as typical for a Super-Earth:RP = 1.45REarth and MP = 3.1MEarth, resulting in a surface gravityof g = 14.3 m s−2. We furthermore use Prot = 36.5 d, like in thenominal model of Heng & Vogt (2011), allowing us to intercompareour results directly. We further investigate Prot = 10 d to bridge thegap to the faster rotating regime.

The radiative time-scale τ rad is calculated using equation (12),and we use Earth-like composition and irradiation to set the equi-librium temperature profile Teq(p, ν, φ). It should be noted that wealso allow the nightside to relax towards the condensation temper-ature which is different from the procedure prescribed by Zaluchaet al. (2013) who allowed to cool the nightside without bound andinstantaneously reset the temperatures to the condensation verti-cal profile if the temperature dropped below the H2O condensationtemperature. Joshi et al. (1997) and Edson et al. (2011), on the otherhand, imposed no forcing on the nightside and allowed the nightsideto cool without bound.

Table 4 lists the planetary and atmospheric parameters used inour simulations.

3 R ESULTS AND DI SCUSSI ON

In the following, we will discuss flow patterns maintained as aconsequence of the imposed forcing averaged over 1000 d after themodel reached steady state, that is, after tinit = 400 d as is shown inFig. 1 and discussed in Section 2.1.

3.1 Horizontal flow

Fig. 4 shows the temperatures and horizontal flow for the lower,middle troposphere, and upper troposphere for Prot = 10 d andProt = 36.5 d. In both cases, the surface flow is directed towards thesubstellar point at zero longitude and latitude. This flow patternis consistent with the understanding that the substellar point isdynamically the Earth’s tropic equivalent with upwelling and thusconverging horizontal flow at the surface and divergence at the topof the atmosphere. The latter is indeed directly visible due to the

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divergence in horizontal flow for our slow rotating case at p = 225mbar embedded in a generally westerly flow that encompasses thesubstellar point.

For the fast rotating case (left-hand panels in Fig. 4), the equa-torial superrotation supersedes divergence. The local temperatureminimum at p = 225 mbar at the substellar point, however, indicatesthe top of the ascending circulation branch. This connection be-tween top horizontal flow and circulation cells is further confirmedby the meridional overturning stream function ψ that is defined inthe framework of the Eulerian mean, where a flow parameter A isdecomposed into a zonal average A and a longitudinally varying oreddy part A′, thus,

A = A + A′. (28)

ψ can then be calculated in pressure coordinates by (see e.g. Holton1992)

ψ(p, ν) = 2πRP cos ν

g

∫ p

0v dp (29)

and has units of kg s−1.

Fig. 5 plots this stream function ψ and shows two large circulationcells, one for each hemisphere, with upwelling at the equator anddownwelling at the poles. In the fast rotating case, we see embeddedin the large circulation another pair of circulation cells that we willfurther elaborate upon in the next section.

Our results for the horizontal flow are in agreement with Merlis& Schneider (2010) who found that slow rotating planets showdivergence at the top of the atmosphere and that fast-rotation periodslead predominantly to superrotation.

Due to conservation of angular momentum and because we haveimposed surface friction, the superrotation in the higher atmosphereis balanced by counterrotation, that is, easterly surface flow. Thiscounterrotation is visible as low-lying negative zonal wind valuesin the bottom layers of Fig. 6. The zonally averaged zonal windspeed shown there also reveals that the superrotation is not onlymore prominent in the fast rotating case, it is also stronger than inthe slower rotating model. This decrease in superrotation strengthwith increase in rotation period is in line with the findings by Edsonet al. (2011) that the equatorial wind speed continuously drops aftertransition from the fast regime for Prot ≥ 3 d. A survey with differentrotation periods will show if the anticyclonic vortices that appear in

Figure 4. Temperatures and streamlines of the flow, averaged over 1000 d, for pressure level p = 975 mbar (contour interval 10 K), p = 525 mbar, and p = 225mbar (contour interval 2 K), from top to bottom. The left-hand panel shows simulations with Prot = 10 d, the right-hand panel with Prot = 36.5 d.

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Figure 5. Meridional overturning cell in units of 109 kg s−1 for the fast (Prot = 10 d, left-hand panel, contour level: 10 × 109 kg s−1) and slow rotatingcase (Prot = 36.5 d, right-hand panel, contour level: 100 × 109 kg s−1). Note that positive values indicate clockwise and negative values counterclockwisecirculation.

Figure 6. Zonal mean of the zonal wind in m s−1 for the fast- (Prot = 10 d, left-hand panel), and slow-rotating case (Prot = 36.5 d, right-hand panel). Contourlevels are 1 m s−1.

the Prot = 36.5 d model east of the substellar point (right-hand panelof Fig. 4 for p = 525 mbar) are a by-product of reduced equatorialzonal wind speed or are actively decelerating the jet.

Furthermore, it is noticeable that the amplitude of the plane-tary large-scale wave is larger for Prot = 36.5 d compared with theProt = 10 d case (Fig. 4 in particular p = 525 mbar). This amplitudeincrease can be understood in terms of length-scale of the equatorialRossby wave of deformation (e.g. Showman & Polvani 2011):

LR =(√

gH

β

)1/2

, (30)

where g is the surface gravity, H is the scaleheight, β = 2�/RP isthe meridional variation of the Coriolis parameter f at the equator(cos ν = 1), where � = 2π/Prot is the planet’s rotation rate andRP the radius, respectively. Obviously, LR increases with increas-ing rotation period. Another useful property, in particular, whencomparing planets of different sizes is the wavenumber kER of anequatorial Rossby wave (e.g. Showman & Polvani 2011):

kER =( √

gH

2�RP

)1/2

. (31)

This dimensionless number expresses how much of the Rossbywave ‘fits’ on a planet, where we can assume that the whole planet

is filled if kER = 0.5 or LR = 0.5RP. Indeed, Edson et al. (2011)found that the transition from the fast- to slow-rotating regime –that is, their Prot = 3–4 d limit – coincides with the kER = 0.5-limit, confirming the role of the equatorial Rossby wave as animportant driving mechanism for superrotation. For slower rota-tion, the Rossby wave no longer ‘fits’ completely on the planet.This connection between Rossby wavenumber and superrotationalso explains why Edson et al. (2011) reported a different transi-tion rotation period for dry and wet planets: the temperatures andthus the scaleheights are different. However, Edson et al. (2011)also noted that a planet’s atmosphere can assume multiple equilib-rium states around the transition regime. Table 5 lists the equatorialRossby wavenumber for our two cases (kER = 1.5 for Prot = 10 dand kER = 2.8 for Prot = 36.5 d) and also some other useful char-acteristics like the maximum zonal wind speed umax, the maxi-mum strength of the meridional overturning stream function ψmax,and the buoyancy frequency N =

√−ρg2/θ × dθ/dp. The table

shows that both cases are in the kER � 0.5 regime. Nevertheless, thedynamics of the mid- and upper troposphere differs quite dramat-ically even though both show generally superrotating flow. Show-man & Polvani (2011), on the other hand, assume kER = 0.5–2as a valid range for a Hot-Jupiter-like dynamics regime dominatedby an equatorial superrotating jet, which is more in line with ourfindings that our faster rotating planet is superrotating whereas

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Table 5. Some atmospheric characteristics of the atmospheres of the modelled planets.

Prot β (m−1 s−1) N (s−1) H (km) kER umax (m s−1) ψmax (109 kg s−1)

10 d 1.57 × 10−12 0.0113 6.23 1.5 70 22136.5 d 4.31 × 10−13 0.0130 6.65 2.8 23 643

Figure 7. Eddy geopotential height in units of m and eddy wind in m s−1 at p = 225 mbar for the fast- (Prot = 10 d, left-hand panel), and slow-rotating case(Prot = 36.5 d, right-hand panel). Contour levels are 100 m. The longest wind vectors are 36.57 and 43.54 m s−1, respectively.

our Prot = 36.5 d-planet with kER > 2 exhibits a more divergentflow.

Indeed, the eddy geopotential height �′/g and eddy wind v′

(calculated via equation 28) at p = 200 mbar (Fig. 7, comparedwith the left-hand panels of figs 5 and 7 in Edson et al. 2011)reveal Rossby wave gyres at mid-latitudes and standing waves atthe equator similar to the Prot = 120 h case of Edson et al. (2011)and also comparable to the height fields calculated from the shallowwater model by Showman & Polvani (2011) for the τ rad = 100case. Showman & Polvani (2011) showed that standing Rossby–Kelvin waves are the underlying mechanism for generating theequatorial superrotation. The slow rotating Prot = 36.5 d case, onthe other hand, shows a mixture between divergent dynamics atthe substellar point and standing Rossby and Kelvin waves at otherlocations that are reduced in strength compared to the faster rotatingcase.

Thus, it appears that the kER = 0.5 transition denotes the onset ofdivergence on the top of the atmosphere countering superrotationthat decreases in strength until the flow becomes fully divergentfor Prot = 100 d according to Edson et al. (2011) and Merlis &Schneider (2010).

Shallow water models further suggest that not only changes inthe rotation period and thus in the Coriolis force, but also in thethermal forcing can lead to a transition between the divergent andsuperrotation regime for Hot Jupiters (Showman et al. 2013a) andterrestrial planets (Showman et al. 2013b).4 The results of Showmanet al. (2013b, their figs 17a and c) are indeed qualitatively verysimilar to the horizontal flow of the upper troposphere of our full3D-model, where strong forcing and thus a small radiative time-scale (τ rad = 0.016 d) correspond dynamically to our slow-rotationsimulation model with more divergent flow and weaker forcing(τ rad = 1.6 d) to our Prot = 10 d-simulation with superrotating flow.

4 Here, the two regimes are called eddy and jet dominated, respectively.

The model of Zalucha et al. (2013) has a higher thermal forcingthan our model, which suggests a tendency to a more divergentflow. The upper atmosphere levels show instead strong equatorialsuperrotation. Apparently, the divergence tendency of the strongforcing is overcome by the strong Coriolis force (because of thelarge planetary radius and fast rotation) that favours superrotatingflow. Indeed, Zalucha et al. (2013) found kER = 0.38, which placestheir model firmly in the superrotating regime. The authors note,furthermore, that the equatorial jet is substantially broader than theequatorial Rossby deformation radius, hinting at a deviation fromthe shallow water model by Showman & Polvani (2011).

3.2 Cyclonic vortices

A further interesting feature is the behaviour of the cyclonic vorticesin the mid- and upper troposphere. The slow rotating case showsthem at mid-latidude (60◦) between the antistellar point and dawnterminator (Fig. 4). Furthermore, they are associated with temper-ature anomalies: they are cold centres in the mid-troposphere andwarm centres at the upper troposphere for the slow-rotating case.Edson et al. (2011) also report mid-latitude vortices between anti-stellar point and dawn terminator with temperature anomalies andattribute them to the advection of cooler air towards the substellarpoint along the equator. This might explain why the centres of thevortices are located more polewards in our Prot = 10 d simulationcompared to the Prot = 36.5 d simulation: The stronger superrotatingjet at the equator pushes them towards the poles.

In contrast, Edson et al. (2011) find the mid-latitude vorticesfor Prot = 120 h, which is twice as fast as our fast rotating caseProt = 10 d, where the vortices are at the poles instead. We note,however, that the equatorial superrotation in the 120 h model ofEdson et al. (2011) is two times smaller compared with ourProt = 10 d case. Furthermore, we notice that the equatorial edges

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of the vortices coincide with additional spin-up of the zonal jet inour models (Fig. 6) that show two zonal wind maxima in the up-per troposphere (between p = 300 and 200 mbar) at latitudes 40◦

for Prot = 10 d and two maxima in the mid-troposphere (betweenp = 500 and 400 mbar) at latitudes 50◦ for Prot = 10 d. Thus, wespeculate that the vortices, the equatorial jet and the Rossby wavesare coupled with each other in our simulations.

The higher location of the maximum zonal winds in our fast-rotating case might indicate that the vortices are not only pushedpolewards, when compared to the slow-rotating case, but also up-wards. Furthermore, we see in the slow-rotating case at the top ofthe troposphere at p = 200 mbar (Fig. 6) the clear emergence ofa single maximum at the equator, indicating that the vortices aredecoupled from the zonal flow because the dynamics on top of theatmosphere is becoming divergent.

Indeed, we see in the latter Prot = 36.5 d-case that the vorticesbecome warm temperature anomalies at the top of the atmosphere,whereas the cyclones remain cold air anomalies in the fast-rotatingcase throughout the troposphere (Fig. 4, bottom panels). We spec-ulate that these warm temperature anomalies appear because ofadiabatic heating. When air that flows from the substellar to the an-tistellar point reaches the vortices, it falls ‘downwards’ because ofthe reduced vertical extent of the colder atmosphere in that region.A similar mechanism is responsible for winter polar warming in themiddle atmosphere at Mars (e.g. Forget et al. 1999).

In the fast-rotating case, the cold regions in the middle tropo-sphere are not confined to the vortices but comprise the polewardregions of the superrotating equatorial jet. Therefore, we speculatethat we see here warm temperature anomalies that are instead cor-related with strong horizontal temperature gradients at the flanks ofthe warm equatorial jet in the mid-troposphere, because the air at thetop of the troposphere ‘falls down the slopes’ of the superrotatingmid-troposphere jet and becomes heated (Fig. 4, left-hand panelsfor p = 525 and 225 mbar). We will later show that an adiabaticheating mechanism explains also the upper thermal inversions inthe vertical temperature profiles.

In contrast, upwelling of the flow at the substellar point shouldlead to adiabatic cooling and indeed the substellar point is in bothour simulations cooler at the top of the troposphere than the sur-rounding air (Fig. 4, bottom panels for p = 225 mbar). This doesnot invalidate, however, our argument that once the air has beenlifted up, it can stream away towards the nightside and in the pro-cess be adiabatically heated under the right conditions. We arguethat both processes may happen at the upper troposphere. Further-more, it should be noted that the involved temperature differencesare relatively small: �T = 12–16 K.

Edson et al. (2011) also describe cyclones at the same location asin our Prot = 36.5 d model with the difference that they extend downto the surface. Heng & Vogt (2011) also showed mid-latitude vor-tices between antistellar point and dawn terminator that can extenddown to the surface if the friction and radiative time-scales takenfrom the HS94 are increased by a factor of 30 for identical planetaryparameters than the ones we used here. Apparently, differences insurface friction prescriptions change the vertical extent of the cy-clones. It is interesting to note that Edson et al. (2011) donot seeour zonal wind maxima, but this difference in zonal wind strengthversus latitude might be attributed to a different circulation regime.They have two meridional circulation cells per hemisphere insteadof one. Heng & Vogt (2011), on the other hand, do exhibit maximaof zonal wind speeds at mid-latitude for their nominal model withProt = 36.5 d. These zonal wind maxima are, however, not addressedin their study.

3.3 Circulation

In general, both our simulations show two giant circulation cells,one for each hemisphere. The presence of only two circulation cellsis in line with the understanding that our atmospheres are in arather slow rotation regime with Prot ≥ 10 d. Indeed, most authorsthat have investigated tidally locked slow-rotating planets find twocirculation cells: Joshi et al. (1997), Joshi (2003), and Merlis &Schneider (2010). Interestingly, Edson et al. (2011) report consis-tently four circulation cells even for Prot = 100 d which is actuallyquite puzzling as one would expect already from investigations ofplanets with non-tidally locked Earth-like forcing the emergence ofa single circulation cell per hemisphere that extends up to the poles.For example, Navarra & Boccaletti (2002) found the transition to asingle hemisphere circulation cell for Prot ≥ 144 h.

The difference in the number of circulation cells between oursimulations and those of Edson et al. (2011) also explains why theircirculation strength is reduced by one order of magnitude whencompared to our results. They got ψmax ∝ 10 × 109 kg s−1, whereasour circulation strength yields ψmax ∝ 200–600 × 109 kg s−1

(Table 5). Still, Joshi et al. (1995) have ψmax ∝ 80 × 109 kg s−1,which is a factor of 2.5 and 7.5 smaller, respectively. They let, how-ever, the nightside of their CO2/H2O atmosphere cool towards thecondensation temperature of CO2 which is about 150 K and thushigher than our 77 K nitrogen condensation temperature. There-fore, the difference in circulation strength may be attributable todifferences in nightside temperature forcing. To sum up, our sim-ulations agree in many features with the results from the study ofEdson et al. (2011), but differ greatly in something as fundamentalas the number of circulation cells. This alone shows that additionalinvestigations and intercomparisons are warranted.

Another interesting feature in the circulation of our modelledSuper-Earth atmospheres, for which we can find no counterpart inprevious studies for terrestrial exoplanets, is the emergence of asmaller counter rotating meridional cell (directed from high lati-tudes towards the equator) in mid-troposphere in the Prot = 10 dcase. Vertical cross-sections of the vertical winds are very usefulin this context and these are shown at substellar, antistellar, the ter-minators, and along the equator for slow rotation in Fig. 8 and fastrotation in Fig. 9. They confirm that the overall circulation is drivenby flow from the substellar point towards the poles and the night-side (Figs 8 and 9, upper panel). There is strong localized upwellingat the antistellar point and weaker general downwelling elsewherewith centres at the poles and the terminators, in agreement with theresults of Joshi et al. (1997). But there appears to be in addition aWalker-like circulation in the mid-troposphere parallel to the equa-tor with upwelling at high latitudes at the antistellar nightside andcorresponding downwelling at the terminator. This Walker circula-tion is more dominant in the faster rotating Prot = 10 d case (Fig. 9,lower panels) that shows upwelling at small latitudes at the antistel-lar nightside and corresponding downwelling at the terminator. Fur-thermore, a second Walker circulation cell in the other direction ispresent for the fast-rotating case: with upwelling at higher latitudesand corresponding downwelling at the nightside. We speculate thatthe interplay between these two cells results in a net-meridional cir-culation counterrotating to the large cells. The branches of the twoWalker circulations at the dawn terminator (LT = 6 h) apparentlyresult in a local meridional circulation cell (Fig. 10) with oppo-site direction to the large hemispheric cell (Fig. 5, left-hand panel).In the slow-rotating case, the connection between mid-tropospheredownwelling at dawn terminator and upwelling at the nightside isnot entirely clear (Fig. 8, lower panels). In addition, the origin of the

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Figure 8. Vertical slices of vertical velocity in units of [Pa s−1] for Prot = 36.5 d at substellar point (upper-left panel), along the equator (upper-right panel),at antistellar point (lower-left panel), and at dawn terminator (lower-right panel). Negative values denote upward, positive downward motion. Contour levelsare 0.02 Pa s−1 in the upper panels and 0.005 Pa s−1 in the lower panels.

Figure 9. Vertical slices of vertical velocity in units of [Pa s−1] for Prot = 10 d at substellar point (upper-left panel), along the equator (upper-right panel), atantistellar point (lower-left panel), and at dawn terminator (lower-right panel). Negative values denote upward, positive downward motion. Contour levels are0.02 Pa s−1 in the upper panels and 0.005 Pa s−1 in the lower panels.

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Figure 10. Schematic view for the two Walker-like circulations (dashedarrows, WaC) between antistellar point (LT = 0 h) and dawn terminator (LT= 6 h) for the Prot = 10 d simulation in the Nothern hemisphere. The blackarrows denote upwelling and downwelling, as seen in Fig. 9. Grey arrowsshow the resulting meridional circulation (MO) cells.

surface upwelling at dawn terminator is puzzling (Fig. 8, bottom-right panel) and has no counterpart in the fast-rotating case (Fig. 9,bottom-right panel).

3.4 Temperatures and habitability

When it comes to evaluating the possible habitability of a planet,surface temperatures are of the uttermost importance as they de-termine if and where liquid water is possible. When we inspectthe average nightside temperatures in our model, it is striking thatthey are relatively warm (Table 6 and Fig. 4), although we drive thenightside temperature to a nitrogen condensing equilibrium temper-ature. These moderate nightside temperatures despite tidal lockingconfirm once again the statement by Joshi et al. (1997) that a suffi-ciently dense atmosphere (p ≥ 100 mbar) distributes heat efficientlyfrom the illuminated dayside to the nightside. Indeed, given the hightime-scales for nightside cooling (τ rad = 813 d), it is not surprisingthat dynamical heating counteracts any strong cooling tendencies.

Still, our nightside temperatures barely drop below the freezingtemperature of water (273.15 K) for Prot = 10 d and are well abovethe freezing temperature for Prot = 36.5 d. Indeed, while our daysidetemperatures are comparable to the dry case of Edson et al. (2011),their nightside temperatures are lower by more than 100 K.

On the other hand, our nightside temperatures are comparableto those in the model of Joshi et al. (1997), but their dayside tem-peratures are of the order of 300 K, which are substantially lower(by approx. 50 K) than our temperatures, even though they pre-scribed the same flux surface distribution than in our model (F∝ I0 cos φcos ν) and a maximum surface equilibrium temperatureTs, max = 390 K versus Ts, max = 410 K used in our forcing, which isonly 20 K warmer (Fig. 3, calculated with equations 20, 21, and 24).The Earth-like aquaplanet with solar-like irradiation used by Joshi(2003), on the other side, yielded minimum nightside temperaturesof 200 K and maximum dayside temperatures of 320 K, in generalagreement with the wet models investigated by Edson et al. (2011),

which is not surprising since in both cases Earth-climate modelswere adapted to the tidally locked forcing scenario.

Therefore, at the current state it has to be concluded that there isa general agreement that tidally locked terrestrial planets allow forliquid water for solar-like insolation, in particular, at the substellarpoint because dynamics efficiently distributes heat from the dayto the nightside. The substellar point, in addition, appears to be adynamical Earth tropic analogue with upwelling and thus possiblecloud formation and precipitation, making it an ideal target forfuture searches for possible life.

However, there seems to be great disagreement in how this heattransport is achieved (with one or two circulation cells for terrestrialplanets and Prot ≥ 10 d) and how strong this transport is. Comparisonof different studies yields vast differences for nightside temperaturesthat can vary between 156 K (dry model of Edson et al. 2011 – whichwould actually allow for CO2 outfreezing – and 270–290 K (ourmodel and Joshi et al. 1997). The main difference in this contextappears to be how the thermal forcing at the nightside is treated.Thus, it is no surprise that dry models that prescribe condensationtemperatures at the nightside, like our model and Joshi et al. (1997),tend to agree, while others that impose in addition the presence ofan ocean, like Joshi (2003) and Edson et al. (2011), yield generalagreement with each other.

While in our model, the nightside temperatures are well abovethe prescribed nitrogen condensation temperature, there is anotherfeature that suggests that the nightside is indeed cooling down. Thevertical temperature profiles in Fig. 11 show the presence of sur-face thermal inversion reminiscent of Earth polar night temperatureprofiles at almost all locations, except the substellar point. Zaluchaet al. (2013) similarly report for their hot terrestrial GJ 1214b-likeplanet with Prot = 1.58 d global surface thermal inversion. However,although other studies also report surface thermal inversion, theyappear to be more localized phenomena: Joshi et al. (1997) findthem only at the antistellar point, and Edson et al. (2011) also findthem locally for some rotation periods. In fact, they do not explicitlyreport them but they can be inferred from the potential temperatureas shown in their fig. 4.

Both, Zalucha et al. (2013) and our model, use Rayleigh fric-tion with τ fric = 1 d, inspired by HS94, whereas Joshi et al.(1997) and Edson et al. (2011) use Earth-like latent heat exchangeand report surface thermal inversion at the nightside. Therefore,it appears reasonable that the differences are at least in part at-tributable to difference in surface–atmosphere interaction descrip-tion. In principle, Heng & Vogt (2011) already showed that chang-ing the friction time-scales has a profound effect on the surfaceflow. However, first, they did not address thermal inversion, sec-ondly, they changed surface friction and radiative time-scales si-multaneously, thirdly, their temperature retained too many Earth-centric assumptions, and did not properly address conditions at thenightside.

Furthermore, we report additional thermal inversion at the top ofthe atmosphere for mid-latitudes in Prot = 10 d model as shown inFig. 11, left-hand panel. This thermal inversion pattern is also verysimilar to Zalucha et al. (2013), but, interestingly, it disagrees with

Table 6. Temperature of our models.

Prot Ts (K) Minimum nightside surface Maximum dayside surfacetemperature (K) (1000 d average) temperature (K) (1000 d average)

10 d 298.78 272.21 352.1836.5 d 292.61 286.72 357.62

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Figure 11. Temperature profiles for Prot = 10 d (left-hand panel) and for Prot = 36.5 d (right-hand panel). The data are averaged over 1000 d. Temperatureprofiles at two latitudes, equator (solid lines) and 60◦ (dashed lines), are shown. The longitude positions of the profiles are indicated in different shades of grey(see the legend) as local time LT, where LT = 0 h is the antistellar nightside and LT = 12 h is the substellar dayside. LT = 6 h and LT = 18 h are the dawn anddusk terminator west and east from the substellar point, respectively. The plots are comparable to those in Zalucha et al. (2013).

Figure 12. Vertical cross-section of the temperature at the antistellar side (LT = 0 h) for Prot = 10 d (left-hand panel) and for Prot = 36.5 d (right-hand panel).The data are averaged over 1000 d. Contour levels are 5 K.

the vertical temperature profiles for Prot = 36.5 d, where we havediminished thermal inversion on top.

The connection between dynamics and the temperature profilesis more readily apparent when consulting the vertical slices of tem-perature at the antistellar nightside (Fig. 12). Apparently, equatorialsuperrotating planets have a greater horizontal temperature gradientthan atmospheres with more divergent dynamics and thus reducedequatorial superrotation, because the dynamics in superrotating at-mospheres distributes heat predominantly along the equator. Con-sequently, the thermal inversions at the top of the atmosphere arestronger in atmospheres with faster rotation Prot period than com-pared to atmospheres with more divergent flow that distribute heatmore evenly. Indeed, Edson et al. (2011) report that the slope ofthe potential temperature becomes less and less steep for slowerrotation, which also corresponds to a decrease in horizontal tem-perature gradient, in agreement with our models. This confirms anadiabatic heating mechanism as the source of the upper atmospherethermal inversion as the air in the upper atmosphere ‘falls’ deeperat the cold nightside latitudes and is thus stronger heated than in theslow-rotating case. Furthermore, it has been shown in Section 3.1that the warm temperature anomalies at the top of the troposphereare correlated with strong horizontal temperature gradients in themiddle troposphere.

4 C O N C L U S I O N

We present here a conceptional simple and versatile model for atidally locked terrestrial planet by coupling a dry 3D-GCM witha Newtonian relaxation scheme using coherently derived radiativeequilibrium temperatures and friction prescriptions with the samelevel of complexity used in HS94 for the Earth.

We performed as proof of concept a small study for a tidallylocked Super-Earth planet with Earth-like atmosphere and Prot = 10and 36.5 d and found that already these two simulations show sur-prisingly different dynamics. The first case is still predominantlysuperrotating because of standing Kelvin and Rossby waves. Thelatter case shows on top of the atmosphere a divergent flow (Merlis &Schneider 2010) with cyclonic and anticyclonic vortices embeddedin a superrotating large-amplitude planetary wave. Furthermore, itshows a decrease in zonal wind speed compared to the faster rotationmodel in agreement with Edson et al. (2011).

We furthermore report for our two ‘proof-of-concept’ simula-tions, the emergence of zonal wind maxima that appear to be theresult of an interplay between cyclonic vortices and the superro-tating equatorial jet. It will be interesting to investigate how thezonal wind maxima change with increasing rotation period. Fur-thermore, it will be interesting to confirm that the vertical extent of

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such vortices can be changed when the surface friction time-scaleis varied.

We also report an upper atmosphere temperature inversion at thenightside mid-latitudes for faster rotation, also reported by Zaluchaet al. (2013). These diminish with increasing rotation period andthus decreasing horizontal temperature gradient suggesting adia-batic heating of the upwelling flow at the nightside-mid-latitudes.

Our study shows that dynamics is very efficient in transportingheat from the dayside towards the nightside, like Joshi et al. (1997)originally reported and has been reproduced in several studies sincethen (e.g. Joshi 2003; Edson et al. 2011). Indeed, the surface temper-atures in our model generally allow for liquid water at the surfaceand in the Prot = 36.5 d case even at the nightside. The planetsinvestigated here can thus be considered habitable.

Comparison with other studies shows, however, that there is gen-eral disagreement between models in the details of heat transport:We report one circulation cell per hemisphere (like, Joshi et al. 1997;Merlis & Schneider 2010), whereas Edson et al. (2011) report two.Our nightside temperatures agree with the model of Joshi et al.(1997), but are 100 K warmer than those of Edson et al. (2011) andJoshi (2003). Two obvious points that might explain these differ-ences are temperature forcing at the nightside and surface frictionassumptions.

Furthermore, we find near-global thermal inversion at the surface.But again, there seems to be disagreement about their prevalence.We find the best agreement with Zalucha et al. (2013), who alsouse Rayleigh friction with τ fric = 1 d, whereas Joshi et al. (1997)and Edson et al. (2011) use Earth-like latent heat exchange andreport surface thermal inversion at the nightside. Therefore, it ap-pears reasonable that the differences are at least in part attributableto difference in surface–atmosphere interaction description. Dueto the conceptional simplicity of our model, the connection be-tween surface friction and heat transport can be addressed easily bychanging the surface friction time-scales and extent of the PBL. In afuture study, we will explore variations between τ s, fric = 0.1–100 dand pPBL = (0.7–0.9) × ps, taking into account the uncertaintiesfrom Solar system planets (Table 1), to investigate the influenceof different surface friction scenarios on surface temperatures andcirculation.

Interestingly, we find dynamics that hasn’t been reported by otherclimate models. We identify Walker-like circulation along the equa-tor between the antistellar point and the dawn-terminator in themid-troposphere. We identify in the vertical velocities two suchcirculations in the meridional direction for Prot = 10 d and one forProt = 36.5 d. These warrant in itself further investigation.

We propose, therefore, that our dry 3D GCM with simplifiedforcing is particularly valuable in the regime between strictly su-perrotating and strictly divergent atmospheres, e.g. for 3 ≤ Prot ≤100 d. Incidentally, these are the rotation periods that are expectedfor tidally locked planets in the habitable zone of M dwarf stars. Wefurther conclude that our 3D model can extent the investigations ofShowman et al. (2013b) by exploring the transit between superrotat-ing and divergent flow regime not only for different thermal forcingsbut also under variation of Coriolis force. Last but not least, we caneasily and comprehensibly change our model to also investigatedifferent atmospheric compositions and surface pressures.

AC K N OW L E D G E M E N T S

We acknowledge support from the KU Leuven IDO projectIDO/10/2013. The simulations were calculated at the VSC (flemishsupercomputer centre) funded by the Hercules foundation and the

Flemish government. LC would further like to thank Adam Show-man for useful discussions about fundamental climate dynamicsand MITgcm properties during a research visit, Nikole A. Lewisfor advise on the adaptation of MITgcm, Andras Zsom for usefuldiscussions regarding the greenhouse model, and Olivia Venot foruseful comments on the paper. We would also like to thank theanonymous referee for helpful comments and remarks.

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Carone2014 Connecting the Dots - A Versatile Model for the Atmospheres of Tidally Locked Super-Earths